Investigation of the Effects of Baffle Orientation, Baffle Cut and Fluid Viscosity on Shell Side Pressure Drop and Heat Transfer Coefficient in an E-Type Shell and Tube Heat Exchanger Koorosh Mohammadi Investigation of the Effects of Baffle Orientation, Baffle Cut and Fluid Viscosity on Shell Side Pressure Drop and Heat Transfer Coefficient in an E-Type Shell and Tube Heat Exchanger A Dissertation accepted by the Faculty of Energy Technology, Process Engineering and Biological Engineering of the University of Stuttgart in Partial Fulfillment of the Requirements for the Degree of Doctor of Engineering Sciences (Dr.-Ing.) by Koorosh Mohammadi, B. Sc., M. Sc. born in Abadan, Iran Institute for Thermodynamics and Thermal Engineering University of Stuttgart, Germany February 2011 Supervisor : Prof. Dr. Dr.-Ing. habil. H. Müller-Steinhagen Co-Referee : Prof. Dr.-Ing. habil. Holger Martin Date of Oral Examination: 28. February, 2011 Dedication This work is dedicated to my wife, Parvaneh Jafari, without whose caring supports it would not have been possible, and to the memory of my father, Fathollah Mohammadi, who passed on a love of reading and respect for education. v Contents Contents page v Acknowledgment vii Nomenclature xi Zusammenfassung xix Abstract xxi 1. Introduction 1 1.1 Industrial Heat Exchangers 1 1.2 Shell and Tube Heat Exchangers: Applications and Main Components 2 1.2.1 Tubes 2 1.2.2 Tube Sheets 3 1.2.3 Shell and Shell-Side Nozzles 3 1.2.4 Tube-Side Nozzles 3 1.2.5 Pass Divider 4 1.2.6 Baffles 4 1.3 Tubular Exchangers Manufacturers Association (TEMA) Design Code 6 1.4 Shell and Tube Heat Exchangers with E-Shell 7 2. Shell and Tube Heat Exchanger Design Methods 9 2.1 Performance of Heat Exchanger 9 2.2 Basic Design Equations and Methods 9 2.3 Calculation of Shell-Side Heat Transfer Coefficient and Pressure Drop 10 2.3.1 Heat Transfer and Pressure Drop for Unbaffled Tube Bank 11 2.3.2 Shell-Side Heat Transfer and Pressure Drop 12 3. Limitation of Common Calculation Methods with Respect to the Effect of Baffle Orientation 17 3.1 Geometrical Difference in Baffle Orientation 17 3.2 Definition of Baffle Orientation 20 3.3 Minimum Shortcut Distance 21 3.4 The Necessity of the Investigation of the Effect of Baffle Orientation 24 4. Application of CFD for the Present Heat Exchanger Investigations 25 4.1 Model Characteristics 25 4.1.1 Mesh Qualification 25 4.1.2 Determination of Mesh Size and Structure 26 4.2 Numerical Model 28 4.2.1 Governing Equations 28 4.2.2 Turbulence 29 4.2.3 Near Wall Treatment of the Flow 29 5. Effect of Baffle Orientation, Baffle Cut and Fluid Viscosity on Pressure Drop and Heat Transfer Coefficient in the Inlet Zone of Shell and Tube Heat Exchangers without Leakages 31 5.1 Geometry and Mesh Structure 31 5.2 CFD Model for the Study of Inlet Zone Effects 31 5.2.1 Boundary Conditions 34 5.2.2 Thermophysical Properties of Working Fluids 35 5.2.3 Settings 36 5.2.4 Mesh Validations 36 5.2.4.1 Mesh Dependency 36 5.2.4.2 Reliability of Mesh Structure for Wall Function Treatment 37 5.3 Performance of the Inlet Zone in the Domain of Laminar and Turbulent Flow 38 vi 5.4 Validation and Sensibility Analysis 52 5.4.1 Validation with Experimental Data for Ideal Tube Banks 52 5.4.2 Error Analysis 52 5.4.3 Sensibility Analysis 55 5.4.4 Validation with VDI Method 56 5.5 Performance of the Inlet Zone Subject to Different Shell-Side Fluid Viscosities 57 5.5.1 Pressure Drop 57 5.5.2 Heat Transfer Coefficient 59 5.6 Semi-Analytical Model for the Performance of the Inlet Zone of Shell and Tube Heat Exchangers without Leakages 65 5.6.1 Performance Factor of Water at Baffle Cut 24% 65 5.6.2 Consideration of Different Working Fluids 67 5.6.3 Effect of Different Baffle Cuts 70 6. Effect of Baffle Orientation and Fluid Viscosity on Shell-Side Pressure Drop and Heat Transfer Coefficient in a Complete Shell and Tube Heat Exchanger without Leakages 75 6.1 Geometry 75 6.2 Meshing and Grid Configuration 75 6.3 Boundary Conditions 76 6.4 Modelling Options and Numerical Setups 78 6.5 Validity of Wall Function Treatment 79 6.6 Shell-Side Fluids 79 6.7 Stability and Iterative Error of Calculation 80 6.8 Final Results and Discussion 80 7. Effect of Baffle Orientation and Fluid Viscosity on Shell-Side Pressure Drop and Heat Transfer Coefficient in a Complete Shell and Tube Heat Exchanger with Leakages 85 7.1 Geometry: Complete Shell and Tube Heat Exchanger with Leakages 85 7.2 Mesh Structure 85 7.3 Boundary Conditions and Physical Properties of Shell-Side Fluids 86 7.4 Comparison 87 7.5 Effect of Leakages on Performance 88 7.5.1 Stream Analysis 88 7.5.2 Discussion of the Numerical Results 92 8. Conclusions and Outlook 111 References R1 Appendix A: TEMA Designation System A1 Appendix B: HTRI Bundle Specification A3 Appendix C: Derivation of Sensibility Analysis Equation A5 Appendix D: Tube Layout for Shell and Tube Heat Exchanger with 660 Tubes A9 vii Acknowledgments First and foremost, “In the name of the Lord of both wisdom and mind, To nothing sublimer can thought be applied” (Ferdowsi, 940 – 1020). This thesis would not have been completed without the help, support, aid, guidance and fruitful collaboration of numerous people. The unforgettable people whose names, memories and teachings will be always kept in me as a precious treasure. Like all other lucky Iranians of my generation, I could pass salubriously through very harsh and difficult years, what a great serendipity! Throughout those hard and remorseless years, it was the tenacious resistance, strenuous efforts and unbounded encouragement of my parents which gave me a boost to go to university and then to do my postgraduate education. During my M.Sc., my reliance was on, beside my parents, my siblings especially Lotfollah, Farshid and Soheyla. I am deeply appreciative of all my parents and siblings have done for me. Next to my family I feel indebted to my supereminent professor and great teacher Prof. Dr. Mohammad Jamialahmadi. During my Bachelor and Master educations in Petroleum University of Technology (PUT, Ahvaz, Iran), I've learned a lot from him. He was also the very first person who introduced me the opportunity to do the PhD when I was in PUT. I would like also to express my appreciation to Dr. Mohammad-Reza Izadpanah and Dr. Amir Sarrafi not only for their helps and teachings during my Master education and military service, but also for their advices which convinced me to do my PhD in overseas. Before I travel to Germany, I had to do my military service in Tehran for about 2 years. Military service in Iran was compulsory at that time and without the certificate of military service I could not apply even for passport. During my residence in Tehran, Saeid Kalantari and Taheri family have supported me. I am deeply thankful for their moral supports. Applying for student visa was itself a kind of “Seven Labours of Rostam”. Without the aid of my brother Lotfollah Mohammadi and the cooperation of Dr.-Ing. Jochen Sohns (the deputy head of ITW at that time), it was impossible to proceed by all of those bureaucracies. I herewith would like to express my gratefulness for their kindness and assists. Render thanks each moment from thy heart, for gratitude is not the work of the tongue alone. Saadi (1184 – 1283/1291) The essence of all beautiful art, all great art, is gratitude. Friedrich Wilhelm Nietzsche (1844 – 1900) Each of us has cause to think with deep gratitude of those who have lighted the flame within us. Albert Schweitzer (1875 – 1965) As we express our gratitude, we must never forget that the highest appreciation is not to utter words, but to live by them. John F. Kennedy (1917 – 1963) viii I am extremely indebted to my supervisor, Prof. Dr. Dr.-Ing. habil. Hans Müller-Steinhagen. I am sincerely grateful for his encouragement and acclaim, and providing enthusiasm and direction when it was lacking. I am heartily thankful for his wise supportive ideas and also affording assistance when it was needed. Without his suggestions, comments and corrections this thesis would not read the way it does. A part of this thesis has been accomplished in the frame of a project, financed by Heat Transfer Research, Inc. (HTRI). I gratefully acknowledge this support. Many thanks are also due to Prof. Joseph W. Palen, the deceased Dr. Richard Stanley Kistler and Dr. Kevin J. Farrell for their precise comments which winnowed out all the inaccuracies in the early stage of my CFD simulations. I am also tremendously obliged to Prof. Dr.-Ing. habil. Holger Martin (Institute of Chemical Engineering, Karlsruhe Institute of Technology) for providing me with experimental data on heat transfer and pressure drop in ideal tube banks, and also for introducing me the generalized Lévêque equation. Additionally, his meticulous reading of my thesis has helped me to finalize my thesis more precise than before. I am very thankful for his demanding effort. My special thanks go to Dr.-Ing. Wolfgang Heidemann. His guidance, patience, unwavering support and encouragement through many stimulating discussions helped me a lot to complete this thesis. He is more than a co-supervisor, a supportive friend and a trustworthy gentleman. Beyond his high character, gentle trait and even disposition and habitude, he is in faith like a real brother to me. I am also beholden to Dipl.-Ing. Gerrit Barthau. He was a great secular scientist and a real positivist. His method of epistemology was very attractive: sometimes he was a pure empiricist and sometimes a purposeful rationalist. I am certainly very lucky that I could meet him during the last three years of his life when he was working in ITW. Our conceptual discussion about thermodynamics, nature of fluid flow and heat transfer, physics of instabilities, meaning of physical properties, the inmost ingenuity in the definition of dimensionless numbers, the perception of transportation phenomena and the possibility of existence of a sophisticated mathematical model which can explain and predict any behavior observed and/or occurred in nature, and also our disputation in the field of evolution of life, philosophy, politic, sociology, art and even language sometimes became protracted until midnight or even dawn. Herewith, by the reminiscence of his name and his individual character I would like to express my thanks to the gone scientist Gerrit Barthau. Thanks also go to Prof. em. Dr.-Ing. Erich Hahne who always freely gave of his time to improve the quality of my scientific knowledge. It would not be a proper acknowledgement if I did not make a special mention to my friends Reza Zehtaban and Dr. Mohammad-Reza Malayeri. Whenever I needed help, they gave me a hand and they always supported and inspired me to finalize my PhD thesis. Many thanks are due to Dr. Mohamed Salam Abd-Elhady (Department of Mechanical Engineering, Beni-Suief University, Egypt) for his help while I was writing my dissertation. A big round of thanks goes to my colleagues from ITW who have made my time as a PhD student and then as an academic employee, cheerful, happy and memorable. I am furthermore indebted to the administration staff of University of Stuttgart and ITW especially Mses. Doris Walz, Viktoria Heuser and Thi My Dung Ta and also Dipl.-Ing. Thomas Brendel for their help and assistance in administrative affairs. A particular thank goes to Apl. Prof. Dr.-Ing. Klaus Spindler, provisional director of ITW, for his scrupulous care about my dissertation defense presentation. His thoughtfulness and valuable suggestions winnowed out the factual errors and mistakes in my presentation. ix I would like to thank Prof. Dr.-Ing. Eberhard Göde (Institute of Fluid Mechanics and Hydraulic Machinery, IHS), Prof. Dr.-Ing. Joachim Groß (Institute of Thermodynamics and Thermal Process Engineering, ITT), Prof. Dr.-Ing. Eckart Laurien (Institute of Nuclear Technology and Energy Systems, IKE) and Prof. Dr. techn. Günter Scheffknecht (Institute of Combustion and Power Plant Technology, IFK) for reading thoroughly my thesis and for their valuable comments. Particularly, a very special appreciation is due to my wife, Parvaneh, not only for her persistent efforts, constant encouragement and unreserved support in every circumstance but also for her love and sacrifice, without which completing of this thesis would not have been possible. Moreover, I would like to express my thanks to my adorable baby Mani for his love and endearing smiles, which fill my life with joy and happiness. And once more, at last and most, my utmost praises be to the Almighty, the Lord of the worlds. Verily, “Laudation is due the most High, the most Glorious, Whose worship bridges the Gap and Whose recognition breeds beneficence. Each breath inhaled sustains life, exhaled imparts rejuvenation. Two blessings in every breath, each due a separate salutation...The shower of His merciful bounty gratifies all, and His banquet of limitless generosity recognizes no fall.” Saadi (1184 – 1283/1291) Koorosh Mohammadi Institute for Thermodynamics and Thermal Engineering (ITW) Stuttgart, Germany August 2011 xi Nomenclature All dimensional parameters and variables are in SI units.  Latin symbols Symbol Definition 4rh Hydraulic diameter (=4Amin nrcdoXl/AH) A Area / Magnitude of AሬሬԦ AH Heat transfer area Amin The minimum free flow area in the tube bank BC Segmental baffle cut percentage C Heat capacity rate (=Mሶ cp) C0, C1, C2, C3 Constant coefficients in the semi-analytical model for the performance factor of water at baffle cut 24% / Constant coefficients in polynomial functions of physical properties of shell-side fluid cp Fluid constant-pressure specific heat cs Control surface cv Control Volume Cε1, Cε2 Coefficients in approximated turbulent transport equations in RNG k-ε model C Coefficient in k- eddy viscosity formulation (for RNG k- model this coefficient is equal to 0.0845) dHsh Hydraulic shell diameter for heat transfer calculation dHsp Hydraulic shell diameter for pressure drop calculation Dn Inside diameter of inlet/outlet nozzle do Tube outside diameter Dotl Diameter of the circle circumscribed to the outermost tubes of the tube bank Ds Shell inside diameter E Empirical constant in the law of the wall (=9.81 for RNG k- model) es Sensible enthalpy F Function describing the influence of main cross-flow stream on other streams f Friction factor G Geometrical function of shell and tube heat exchanger h Convective heat transfer coefficient i Direction i of coordinate xi / Direction i of axis x in Cartesian coordinate j Direction j of coordinate xj / Direction j of axis y in Cartesian coordinate Jb Correction factor for bypass flow in Delaware method for calculation the shell-side heat transfer coefficient Jc Correction factor for baffle cut and spacing in the Delaware method for calculating the shell-side heat transfer coefficient xii Jl Correction factor for baffle leakage effect including both shell to baffle and tube to baffle leakages in the Delaware method for calculating the shell-side heat transfer coefficient Jr Correction factor for adverse temperature gradient build-up in the Delaware method for calculating the shell-side heat transfer coefficient Js Correction factor for variable baffle spacing in the Delaware method k Kinetic energy of turbulence fluctuations per unit mass kf Fluid thermal conductivity L Tube or heat exchanger length / Length / Effective flow length Lbc Central baffle spacing Lbch Baffle cut height Lbi Inlet baffle spacing Lbo Outlet baffle spacing Ln Nozzle minimum length or nozzle neck length Lsb Inside shell-to-baffle clearance (diametral) Ltb Diametral clearance between tube outside diameter and baffle hole ltp Tube pitch m Power factor in polynomial functions of shell-side physical properties Max Maximum of a function MSD Minimum shortcut distance n Refers to measurable variables in a system / Refers to the particular number of an geometrical entity like tubes, tube rows, tube columns, segments, baffle windows or baffles and etc. / Iteration NMSD Normalized minimum shortcut distance NMSDR Normalized minimum shortcut distance ratio nt Tube number O Orientation function explaining the effect of baffle orientation and baffle cut on stream s p Static pressure Pn Tube pitch normal to flow direction Pp Tube pitch parallel to flow direction Pt Tube pitch Q Heat flow rate ARQ Aspect ratio quality EASQ EquiAngle skewness R Flow resistance of stream rs Shell inside radius T Static temperature t Time/ Time average in Reynolds-averaged Navier-Stokes equations Tr Function explaining the transfer rate of heat and momentum tubesl Number of tubes in the longitudinal direction, i.e. in the direction of Xl tubest Number of tubes in the transverse direction, i.e. in the direction of Xt U Conductance or total heat transfer coefficient of heat exchanger u Friction velocity xiii u Characteristic velocity fluctuation of turbulence (= 2k 2 j 2 i uuu  ) / Characteristic velocity of fluid motion u׳. Velocity fluctuation umax Maximum fluid velocity between the tubes at the central row V Instantaneous velocity magnitude of fluid w Velocity of the fluid in the empty cross section of the channel x x axis, coordinate or component in Cartesian coordinates / Axis of a desired coordinate system / Distance / Displacement in two-point velocity correlation tensor Xl Longitudinal pitch ratio (=Pp/do) Xt Transverse pitch ratio (=2Pn/do) y y axis, coordinate or component in Cartesian coordinates / Distance to the wall y* Dimensionless, sublayer-scaled, distance y+ Dimensionless, sublayer-scaled, distance z z axis, coordinate or component in Cartesian coordinates  Greek symbols Symbol Definition  Thermal diffusivity  Gain factor ≡ h/p ∆ Difference  Dissipation rate of kinetic energy of turbulence fluctuations per unit mass Nu Relative error in Nusselt number calculation or evaluation T Relative error in static temperature calculation or evaluation Δp Relative error in pressure drop calculation or evaluation ζ Kolmogorov microscale of length / Face of the computational cell ph Physical property of the shell-side fluid ϴ Angle between MSD and the axis which passes through the shell center and is parallel to the inlet nozzle neck ϴca Centri-angle of the baffle cut intersection with the inside shell wall  Ratio of tube wall temperature to inlet temperature ≡ Twall/Tin o Ratio of outlet temperature to inlet temperature ≡ Tout/Tin Ԃmax Maximum angle in radian between the edges of elements Ԃmin Minimum angle in radian between the edges of elements ιi Average length of the edges in a coordinate direction (i) local to the element  Fluid dynamic (absolute) viscosity  Fluid kinematic (molecular) viscosity ξ Displacement between two points in turbulent flow  Fluid density σε Effective turbulent Prandtl number for ε τ Kolmogorov microscale of time W Surface (wall) shear stress xiv υ Kolmogorov microscale of velocity φ Void fraction of tube bank ϕ Hypothetical function ω Specific dissipation rate ω in k-ω model for turbulence modeling  Special and mathematical symbols Symbol Definition Average value / intermediate value | Mathematical symbol which means “restriction of one value or function to a parameter” or “as” • Dot product in vector algebra : Mathematical symbols which means “such that” | | Absolute value || || Absolute deviation / Absolute error / Norm { } Shows a set of property, variables, parameters or entities ( , ) Open interval ۃ ۄ. Average value → In limit of a function or parameter means “approaches to” ן Proportionality: is proportional to, varies as ≈ Mathematical symbol which means “approximately equal to” ≡ Mathematical symbol which means “it is equivalent or congruent to” ب Strict inequality: is much greater than ٿ Logical conjunction: and ׊ Universal quantification: for all, for any, for each ׌ Existential quantification: there exists, there is, there are ׌! Uniqueness quantification: there exists exactly one א Set membership: is an element of AሬሬԦ. Area of control surface in vector notation bfሬሬሬԦ. Baffle-vector ƒ Hypothetical function ƒA Nondimensional geometrical correction factor presented in correlation of heat transfer coefficient of tube bank ƒB Bypass-stream factor in VDI method ƒG Geometrical factor in VDI method ƒL Leakage-stream factor in VDI method ƒW Correction factor in calculation of shell-side heat transfer coefficient by use of VDI method (=ƒGƒLƒB) iԦ. Unit vector of x axis in Cartesian coordinates iԦi or iԦj Cartesian unit vector in the direction of the coordinate xi or xj jԦ. Unit vector of y axis in Cartesian coordinates kሬԦ. Unit vector of z axis in Cartesian coordinates ℓ Turbulence length scale / Integral length scale ℓH Hydraulic length of heat transfer for heat exchanger Mሶ . Mass flow rate xv max{ } The largest value of a set of parameters min{ } The smallest value of a set of parameters nzሬሬሬԦ. Face-vector of inlet-plane imagined in nozzle Oന. Order of magnitude / Magnitude of a parameter or value Oനሼƒ, Xሽ Order of magnitude of function ƒ with respect to parameter X Uഥ. Mean velocity component in x, y, z direction uሬԦ. Instantaneous velocity of fluid in vector notation VሬሬԦ. Instantaneous velocity of fluid in vector notation relative to control volume of fluid ׊. Volume of fluid / Volume of system xሬԦ. Displacement vector in two-point velocity correlation tensor {X} Set of geometrical parameters (={tube layout, tube outside diameter, tube length}) ξԦ. Displacement vector between two points in the flow ׏ሬԦ. Divergence (= kzjyix   ) ห׏ሬԦηห n, Magnitude of   normal to face  )b,a(  Counter-clockwise angle of rotation of vector a towards vector b  Subscripts Symbol Definition 0 Related to the value at x=0 24% | Water,ISO Refers to a value at baffle cut 24% for water as the shell-side fluid with constant physical properties at 20 °C and 1 atm. 95%∞ Related to 95% of ultimate value of  ס൫bfሬሬሬԦ,nzሬሬሬԦ൯ Shell and tube heat exchanger with baffle orientation equal to ס൫bfሬሬሬԦ,nzሬሬሬԦ൯ A Related to tube-baffle leakage stream (flow stream A) Air Refers to the air at desired operating conditions B Related to main effective cross-flow stream (flow stream B) baffle Related to the baffle BC Refers to baffle cut BC bp Related to bypass streams C Related to tube bundle bypass stream (flow stream C) CFD Refers to the CFD calculation cl Related to the central line which presents the middle of the inlet cold Related to the cold fluid or cold fluid side E Related to baffle-shell leakage stream (flow stream E) F Related to bypass stream in tube pass partition (flow stream F) fluid 1 or 2 Related to the shell-side fluid 1 or 2 H Refers to the heat transfer h Values, variables, parameters or dimensionless numbers base on hydraulic diameter 4rh hor./horizontal Shell and tube heat exchanger with horizontal baffle orientation xvi hot Related to the hot fluid or hot fluid side i Refers to direction i of coordinate xi in a defined coordinate in/inlet Inlet nozzle / Property calculated at inlet temperature ISO Related to the reference temperature and pressure according to the International Organization for Standardization (p=1 atm, T=20 °C)j Refers to direction j of coordinate xj in a defined coordinate max Refers to the maximum value min Refers to the minimum value o Refers to the outside diameter of tube out/outlet Outlet nozzle / Property calculated at outlet temperature rc Refers to the total number of tube rows crossed the flow s Refers to stream flow Semi-Analytical Refers to the calculation based on the semi-analytical model shell Related to shell-side / Property calculated at average shell-side temperature (Tin+Tout)/2 turb Refers to turbulent flow ver./vertical Shell and tube heat exchanger with vertical baffle orientation wall Related to tube wall / Property calculated at tube wall temperature Water Refers to the water at desired operating conditions WF Refers to working fluid window Related to baffle window / Refers both upper and lower baffle x, y, z Refers to the x, y and z axis direction in Cartesian coordinates φ Refers to modified values base on void fraction of tube bank  Superscripts Symbol Definition * Sublayer-scaled value + Sublayer-scaled value csf Cross sectional flow area ms Main stream / Related to the stream in heat exchanger exclusive n Refers to iteration n SATP Standard Ambient Pressure and Temperature (p=100 kPa, T=25 °C)  Dimensionless numbers Symbol Definition f Modified Fanning friction factor ≡ 2rh∆p ሺρu2Lሻ⁄ Nk Shell-side Kârmân number ≡ ρdHsp3 ൫Δpshell Lbc⁄ ൯ μ2ൗ Nu Nusselt number ≡ (h characteristic length)/kf Nu0 Nusselt number at zero Reynolds number Nu0, bank Nusselt number of ideal tube bank defined by Gnielinski Nushell Shell-side Nusselt number ≡ hshelldHsd/kf Pe Peclet number ≡ ρVL/Ω Pr Prandtl number ≡ (Cp/kf) xvii Re Reynolds number ≡ u(characteristics length)/ Reinlet Reynolds number based on the inlet nozzle diameter, the fluid velocity in the inlet nozzle and the physical properties at inlet conditions ≡ uinletDn/ Reo Reynolds number based on the maximum fluid velocity between the tubes at the central row and the tube diameter as the characteristic length ≡ inletumaxdo/inlet Retrans,ll Lower limit transitional Reynolds number Retrans,ul Upper limit transitional Reynolds number Reφ Modified Reynolds number based on the stream length πdo/2, the velocity of the fluid in the empty cross section of the channel and the void fraction of the tube bank ≡ wπdo/(2φ) St Stanton number ≡ Nu/(Re.Pr) γ. Dimensionless number used in the equation of sensibility analysis ≡ ൫μin μ⁄ ൯ሺℓH Dn⁄ ሻሺAin AH⁄ ሻ Γ Gain factor ≡ Nu/Nk ΘBC Baffle cut preference ≡ BC/24% ΘWF Working fluid preference ≡ (WF/Water)|BC  Performance factor ≡ (Γshell)hor./ (Γshell)ver. xix Zusammenfassung Zur Bestimmung des Einflusses der Umlenkblechanordnung und des Umlenkblechausschnitts sowie der Viskosität des Arbeitsfluids auf den mantelseitigen Wärmeübergang und Druckverlust eines Rohrbündelwärmeübertragers im laminaren und turbulenten Bereich wird das kommerzielle CFD-Programm FLUENT eingesetzt. Luft, Wasser und Motoröl werden als mantelseitige Arbeitsmedien betrachtet. Die betrachteten Rohrbündelwärmeübertrager erfüllen die TEMA-Standards. Die Untersuchung wurde in drei Schritten durchgeführt: 1. Der Rohrbündelwärmeübertrager besteht aus 660 glatten Rohren mit festem Außendurchmesser, die in Dreieckteilung versetzt angeordnet sind. Es wird eine horizontale und vertikale Anordnung der Umlenkbleche sowie drei Öffnungsweiten, 20%, 24% und 30% des Mantelinnendurchmessers betrachtet. Die Leckageströme in den Bohrungsspielräumen und im Spalt zwischen Umlenkblech und Mantel werden nicht berücksichtigt. Die Untersuchung wurde auf die Einlasszone angewendet, um den Effekt der Umlenkblechanordnung, des Umlenkblechausschnitts und der Viskosität des mantelseitigen Fluids auf die mantelseitige Leistung in der Einlasszone zu bestimmen. Für die jeweiligen Umlenkblechanordnungen, Umlenkblechausschnitte und Arbeitsfluide werden verschiedene Strömungsgeschwindigkeiten am Einlass untersucht. Diese Geschwindigkeiten werden durch die Reynoldszahl am Einlass Reinlet charakterisiert, welche sich auf die Geschwindigkeit in dem Einlassstutzen, den Innendurchmesser des Einlassstutzens und die physikalischen Eigenschaften des mantelseitigen Fluids bei Einlassdruck und –temperatur bezieht. Wärmeübergang und Druckverlust werden als allgemeine Nusselt-Zahl (Nu oder Nushell) beziehungsweise Kârmân-Zahl (Nk) angegeben. Die Definition von Nushell erfolgt entsprechend dem VDI Wärmeatlas [VDI-2006]. Ergebnisse für alle geometrischen Variationen zeigen, dass sich Nk proportional zu Re2 und Nu zu Rem verhält, wobei 0,6 ≤ m ≤ 0,8. Ein für die Bewertung von Rohrbündelwärmeübertragern geeigneter mantelseitiger Gewinnfaktor wird als Verhältnis von mantelseitigem Wärmeübergangskoeffizienten zu mantelseitigem Druckverlust eingeführt. Um die Unterscheidung zwischen horizontaler und vertikaler Orientierung der Umlenkbleche zu vereinfachen, wird ein Leistungsfaktor Φ, als das Verhältnis des Gewinnfaktors von horizontal angeordneten Umlenkblechen zum Gewinnfaktor bei vertikaler Umlenkblechanordnung, verwendet. Die Simulationsergebnisse zeigen den Vorteil der horizontalen Umlenkblech- orientierung im Vergleich zur vertikalen Orientierung, insbesondere für Luft (d.h. Gas) als mantelseitigem Fluid. Bei einem Umlenkblechausschnitt von 30% erreicht der Leistungsfaktor seinen Maximalwert für alle mantelseitigen Fluide, während der minimale Wert des Leistungsfaktors bei einer Umlenkblechöffnung von 24% beobachtet wird. Simulationsergebnisse für den Einlassbereich zeigen, dass die horizontale Umlenkblechorientierung im Vergleich zur vertikalen Orientierung einen bis zu 20% höheren Druckverlust zur Folge hat. Weiterhin zeigen die Ergebnisse, dass die Nusselt-Zahl für die horizontale Umlenkblechanordnung etwa 15% bis 52% höher ist, als die Nusselt-Zahl für die vertikale Anordnung. Für Wasser und Motoröl ist der Gewinnfaktor Γ für die horizontale Umlenkblechorientierung bis zu 20% größer als der Gewinnfaktor für die vertikale Orientierung. Für Luft als mantelseitigem Fluid ist der Wert für Γ für die horizontale Orientierung bis zu 40% größer als bei der vertikalen Orientierung. 2. Um einen vollständigen Rohrbündelwärmeübertrager zu simulieren, wird ein Rohrbündelwärmeübertrager mit denselben geometrischen Abmessungen wie im xx vorhergehenden Schritt betrachtet. Wieder werden keine Leckageströme berücksichtigt. Für die numerischen Berechnungen wird der Wärmeübertrager in acht verschiedene Strömungsbereiche, wie Ein- und Auslasszone und sechs mittlere Strömungsabschnitte, die sich zwischen benachbarten Umlenkblechen befinden, geteilt. Um den Einfluss der Viskosität auf den Wärmeübergang und den Druckverlust zu bestimmen, werden Simulationen für die beiden Arbeitsfluide Luft und Wasser durchgeführt. Für alle Umlenkblechorientierungen und –ausschnitte, sowie für alle Arbeitsfluide werden Simulationen für fünf Einlassreynolds-Zahlen, 3,9×104 ≤ Reinlet ≤ 1,16×106, durchgeführt. Die Simulationsergebnisse zeigen den Vorteil der horizontalen gegenüber der vertikalen Umlenkblechanordnung, insbesondere in der Ein- und Auslasszone, für alle untersuchten mantelseitigen Arbeitsfluide. Der Leistungsfaktor für die horizontale Anordnung der Umlenkbleche ist in den mittleren Umlenkblechbereichen etwa gleich dem Leistungsfaktor für die vertikale Anordnung, wenn flüssiges Wasser als mantelseitiges Fluid verwendet wird. Für Luft ist ein Vorteil der vertikalen gegenüber der horizontalen Umlenkblechanordnung erkennbar. 3. Um einen realen vollständigen Rohrbündelwärmeübertrager zu simulieren, wird ein Rohrbündelwärmeübertrager bestehend aus 76 Rohren mit festem Außendurchmesser betrachtet. Die Rohre sind in Dreieckteilung versetzt angeordnet. Die Leckageströme in den Bohrungsspielräumen und den Spalten zwischen Umlenkblechen und Mantel werden mit einbezogen. Wie bei den vorhergehenden Schritten wird die horizontale und vertikale Umlenkblechanordnung berücksichtigt, der Umlenkblechausschnitt jedoch auf 20% des Mantelinnendurchmessers festgelegt. Dadurch wird das Verhältnis der Wärmeübergangsfläche der Rohre im Umlenkblechfenster zur Wärmeübergangsfläche der Rohre in einem Umlenkblechzwischenraum annähernd so groß wie der zugehörige Wert des Rohrbündelwärmeübertragers mit 660 Rohren. Um den Einfluss der Viskosität auf den Wärmeübergang und den Druckverlust zu bestimmen, werden Simulationen für die drei Arbeitsfluide Luft, Wasser und Motoröl bei Prandtl-Zahlen zwischen 0,7 und 1798,8 bezogen auf SATP-Bedingungen durchgeführt. Für alle Umlenkblechorientierungen und -ausschnitte, sowie für alle Arbeitsfluide werden Simulationen für fünf Einlassreynolds-Zahlen im Bereich von 2,0×104 < Reinlet < 105 durchgeführt. Die Simulationsergebnisse zeigen den Vorteil der horizontalen gegenüber der vertikalen Umlenkblechanordnung, insbesondere in der Ein- und Auslasszone, für alle untersuchten mantelseitigen Arbeitsfluide. Die Simulationsergebnisse zeigen einen signifikanten Einfluss der Umlenkblechorientierung auf den mantelseitigen Druckverlust und Wärmeübergang von Rohrbündelwärmeübertragern. Im Gegensatz zu den Ergebnissen der vorausgegangenen Simulationen zeigen die Ergebnisse, dass die vertikale Umlenkblechorientierung in Rohrbündelwärmeübertragern mit Verlustströmen vorteilhafter als die horizontale Orientierung ist. Dieser Vorteil (einer vertikalen gegenüber einer horizontalen Anordnung der Umlenkbleche) ist bei Gasen deutlicher. Als Fazit ergibt sich, dass ein Vergleich der Rechenergebnisse mit und ohne Berücksichtigung der Leckageströme unterschiedliche Verhalten aufzeigt. Dies macht deutlich, dass die Berücksichtigung von Leckageströmen in den Bohrungsspielräumen und im Spalt zwischen Umlenkblech und Mantel sowie von Bypassströmen von Bedeutung ist. xxi Abstract The commercial CFD code FLUENT is used to determine the effect of baffle orientation and baffle cut as well as viscosity of the working fluid on the shell-side heat transfer and pressure drop of a shell and tube heat exchanger in the domain of laminar and turbulent flow. Air, water and engine oil are considered as shell-side fluids. The shell and tube heat exchangers considered follow the TEMA standards. The investigation has been completed in three stages: 1. The shell and tube heat exchanger consists of 660 plain tubes with fixed outside diameter which are arranged in a triangular layout. Horizontal and vertical baffle orientations as well as three baffle cuts, 20%, 24% and 30% of shell inside diameter, are considered. No leakage flow in tube-to-baffle gaps and baffle-to-shell gaps is considered. The investigation has been applied for the inlet zone, in order to find the effect of baffle orientation, baffle cut and viscosity of shell-side fluid on the shell-side performance of the inlet zone. For each baffle orientation, baffle cut and working fluid, different flow velocities at inlet are investigated. These velocities are introduced as inlet Reynolds number Reinlet which is defined based on the velocity at the inlet nozzle, inside diameter of the inlet nozzle and the physical properties of the shell-side fluid at inlet pressure and temperature. Heat transfer and pressure drop are reported as overall Nusselt number (Nu or Nushell) and Kârmân number (Nk), respectively. Nushell is defined according to VDI Wärmeatlas [VDI-2006]. Results for all geometrical variations show that Nk is proportional to Re2 and Nu is proportional to Rem, where 0.6 ≤ m ≤ 0.8. A shell-side gain factor suitable for the assessment of shell and tube heat exchangers is introduced as ratio of the shell-side heat transfer coefficient to the shell-side pressure drop. To facilitate the decision between horizontal and vertical baffle orientation, a performance factor Φ is used as ratio of the gain factor for horizontally orientated baffles to the gain factor for vertical baffle orientation. The simulation results show the advantage of the horizontal baffle orientation over the vertical orientation, especially for air (i.e. gas) as shell-side fluid. At baffle cut 30%, the performance factor reaches its maximum value for all shell-side fluids, while the minimum value of performance factor is observed at baffle cut 24%. Simulation results for the inlet region show that the horizontal baffle orientation produces up to 20% higher pressure drop than the pressure drop in vertical baffle orientation. The results also show that the Nusselt number for horizontal baffle orientation is approximately 15% to 52% higher than the Nusselt number for vertical orientation. For water and engine oil, the gain factor Γ for horizontal baffle orientation is up to 20% more than the gain factor for vertical baffle orientation. For air as shell-side fluid, the value of Γ for horizontal baffle orientation is up to 40% more than the value of Γ for vertical baffle orientation. 2. In order to simulate the complete shell and tube heat exchanger, a shell and tube heat exchanger with the same geometrical aspects used in the previous stage is considered. Again, no leakage flows are taken into account. For the numerical investigations the heat exchanger is subdivided into eight different flow sections such as the inlet zone, six intermediate flow sections located between adjacent baffles and the outlet zone. In order to determine the effect of viscosity on heat transfer and pressure drop, simulations are performed for the two working fluids; air and water. For each baffle orientation, baffle cut and working fluid, simulations are performed for five inlet Reynolds numbers; 3.9×104 ≤ Reinlet ≤ 1.16×106. xxii The simulation results show the advantage of the horizontal baffle orientation over the vertical orientation, particularly in the inlet and outlet zone for all investigated shell-side fluids. The performance factor for horizontal baffle orientation is approximately equal to the performance factor for vertical baffle orientation at intermediate baffle spacing zones when liquid water is used as shell-side fluid. For air, the benefit of vertical baffle orientation on horizontal baffle orientation is noticeable. 3. In order to simulate a real complete heat exchanger, a shell and tube heat exchanger consisting of 76 tubes with fixed outside diameter is considered. The tubes are arranged in a triangular layout. The tube-to-baffle and baffle-to-shell leakages are also taken into account. Similar to the previous stages, horizontal and vertical baffle orientations are considered, but the baffle cut is fixed to 20% of shell inside diameter. This will make the ratio of the heat transfer area of the tubes in the baffle window to the heat transfer area of the tubes in one baffle spacing zone similar to the corresponding ratio for the heat exchanger with 660 tubes. In order to determine the effect of viscosity on heat transfer and pressure drop, simulations are performed for three working fluids air, water and engine oil with Prandtl numbers in the range of 0.7 to 1798.8 based on the standard ambient pressure and temperature. For each baffle orientation, baffle cut and working fluid, simulations are performed for five inlet Reynolds numbers in the range 2.0×104 < Reinlet < 105. The simulation results show the advantage of the horizontal baffle orientation over the vertical orientation, particularly in the inlet and outlet zone for all investigated shell-side fluids. The simulation results show a significant influence of the baffle orientation on the shell- side pressure drop and heat transfer of shell and tube heat exchangers. Contrary to the outcomes of the previous simulations, the results show that in shell and tube heat exchanger with leakage flows the vertical baffle orientation seems to be more advantageous than the horizontal orientation. The benefit of vertical baffle orientation over horizontal baffle orientation is more noticeable for gases. As a final conclusion, the comparison of calculation results with and without leakage flows identifies different behaviour and underlines the importance of a consideration of tube-to- baffle and baffle-to-shell leakage and bypass streams for the prediction of the performance factor of technical heat exchangers. 1 1. Introduction 1.1 Industrial Heat Exchangers Most operations that are carried out by engineers involve the production or absorption of thermal energy. The laws governing the transfer of heat and the types of apparatus that have for their main object the control of heat flow are therefore of great importance [McCabe, 2005]. A heat exchanger is a piece of equipment in which heat is transferred from a hot fluid to a colder fluid. In most applications the fluids do not mix but transfer heat through a separating wall which takes on a wide variety of geometries. In industries, heat exchangers are widely used in refrigeration, air conditioning, space heating, electricity generation, and chemical processing. Heat exchanger design aims in most cases at minimum cost, balancing the cost of pumping the fluids and initial cost of the exchanger against the savings resulting from heat transfer. In some cases, such as missile, aircraft or shipboard applications, design may be governed by necessity of minimizing either volume or weight [Palen, 1986]. Shell and tube heat exchangers are probably the most widespread and commonly used basic heat exchanger configuration in oil refineries and other large chemical processes and are suited for higher-pressure applications. Reasons for this general acceptance are several: The shell and tube heat exchanger provides a comparatively large ratio of heat transfer area to volume and weight (up to 1000 m²/m³) [Hesselgreaves, 2001]. It provides this surface in a form which is relatively easy to construct in a wide range of sizes (Figure 1.1 and 1.2) and which is mechanically rugged enough to withstand shop fabrication stress, shipping and field erection stress, and normal operating conditions [Palen, 1986; Driedger, 1996; Wolverine, 2001]. There are many modifications of the basic configuration which can be used to solve special problems. [Saundres, 1988; Wolverine, 2001]. It is essential to mention that a heat exchanger is not only an apparatus for transferring heat from one medium to another, but is at the same time a pressure and/or containment vessel. In addition to heating up or cooling down fluids in just a single phase, shell and tube heat exchangers can be used either to heat a liquid to evaporate (or boil) it or used as condensers to condense a vapor back to a liquid. Figure 1.1: Left: shell and tube heat exchanger made by Addison Fabricators Inc., Addison, Alabama, USA (8000 tubes/90 tonnes). Right: titanium shell and tube heat exchanger made by TITAN Metal Fabricators, Camarillo, California, USA (1.07 m shell outside diameter, 7.3 m maximum length, 25 tonnes, 1282 tubes). 2 1.2.1 Tubes The tubes are the basic component of the shell and tube exchanger, providing the heat transfer between one fluid flowing inside the tubes and the other fluid flowing across the outside of tubes. From the point of view of thermal design features, the most substantial characteristics of tubes are tube diameter, tube wall thickness, tube length and tube pitch. •Tube diameter Compact, economical units are obtained by using small diameter, closely spaced tubes but the surface may foul up quickly and may be difficult to clean by mechanical means. The problems of fouling and cleaning may be overcome by using large-diameter, widely spaced tubes, but then the unit will be less compact and more costly. The selection of the tube diameter is therefore a compromise taking into account the fouling nature of the fluids, the space available and the cost. Tube of 19.05 and 25.4 mm (¾ -1 in) outside diameter are most widely used, but small units with clean fluids may use tubes as small as 6.35 mm (¼ in) outside diameter, and units handling heavy tars may use tubes up to 50.8 mm (2 in) outside diameter. •Tube wall thickness The tube wall thickness must be checked against the internal and external pressure separately. However, in many cases the pressure is not the governing factor determining the wall thickness. For example, a steel tube with 19.05 mm (¾ in) outside diameter, 2.11 mm thick (0.083 in), at 350 °C, is suitable for design pressure up to 200 bar, which is adequate for many applications. Except when pressure governs, tube thickness is selected to provide an adequate margin against corrosion, resistance to flow-induced vibration, axial strength, standardization for the stocking of spare parts and cost. Figure 1.2: Shell and tube heat exchanger for temperature control systems made by Exergy LLC (inside shell diameter = 22.5 mm, maximum length = 248 mm and tube number = 24) 1.2 Shell and Tube Heat Exchangers: Applications and Main Components In shell-and-tube exchangers, many tubes are mounted inside a shell; one fluid flows through the tubes (the tube side) and the other flows outside the tubes but inside the shell (the shell-side). Heat is transferred from one fluid to the other through the tube walls, either from tube side to shell-side or vice versa. The fluids can be either liquids or gases on either the shell or the tube side. In order to transfer heat efficiently, a large heat transfer area should be provided. Shell and tube heat exchangers with only one phase (liquid or gas) on each side are called single-phase heat exchangers. Two-phase heat exchangers are usually condensers or boilers. From the design point of view, while there is an enormous variety of specific design features that can be used in shell and tube heat exchangers, the number of basic components are relatively small. The construction of a shell and tube heat exchanger is illustrated by Figure 1.3, in which the main components, discussed below, are numbered [Saundres, 1988]. 3 •Tube length For a given surface area the cheapest exchanger is one which has a small shell diameter and a long tube length, consistent with the space and handling facilities at site and fabricator’s shop. Therefore, the incentive is to make exchangers as long as possible, limited only by the tube length available from tube suppliers. Tube lengths of 2438, 3658, 4877, 6096 and 7315 mm (8, 12, 16, 20 and 24 ft) are often regarded as standard tube length. 1.2.3 Shell and Shell-Side Nozzles The shell is simply the container for the shell-side fluid, and the nozzles are the inlet and exit ports. 1.2.4 Tube-Side Nozzles The tube-side nozzles (and channels) control the flow of the tube-side fluid into and out of the tubes of the exchanger. Figure 1.3: Diagram of typical shell and tube heat exchanger. A: Tubes, B: Tube Sheets, C: Shell, D: Tube-Side Inlet (Outlet) Nozzle, E: Tube-Side Outlet (Inlet) Nozzle, F: Pass Divider, G: Baffles, H: Shell-Side Inlet (Outlet) Nozzle, I: Shell-Side Outlet (Inlet) Nozzle. A B B C D E F G H I A B C D E F G H I •Tube pitch It is customary practice to arrange the tube pitch (center-center distance) such that it is not less than 1.25 times the outside diameter of the tubes. In certain applications involving clean fluids and small tubes, e.g. 12.7 mm (½ in) outside diameter and less, the pitch/diameter ratio is sometimes reduced to 1.20. Different tube pitch arrangements are shown in Figure 1.4. In Table 1.1 typical pitch angles for various flow regimes and nature of shell-side fluids are listed. For a given pitch/diameter ratio and shell inside diameter, about 15% more tubes can be accommodated for 30° and 60° pitch angle compared with 45° and 90°. To achieve compactness the incentive is to use 30° and 60° pitch angles, which is satisfactory for clean services. However, these patterns are not suitable if external mechanical cleaning is required. 1.2.2 Tube Sheets Tube sheet is usually a single round plate of metal that has been suitably drilled and grooved to take the tubes in the desired tube layout, the gaskets, and the bolt circle where it is fastened to the shell. 4 Pitch pattern Pitch angle Nature of shell-side fluid Flow regime triangular 30° clean turbulent and laminar rotated triangular 60° clean rarely used (30° is better to use) square 90° fouling turbulent rotated square 45° fouling laminar 1.2.5 Pass Divider A pass divider is required in one channel or bonnet for an exchanger having two tube-side passes, as the one illustrated in Figure 1.3. They are needed in both channels and bonnets for an exchanger having more than two passes. 1.2.6 Baffles One of the most important parts in shell and tube heat exchangers are the baffles. Baffles serve mainly two functions:  Fixing of the tubes in the proper position during assembly and prevention of tube vibration caused by flow-induced eddies.  Guidance of the shell-side flow across the tube field, increasing the velocity and the heat transfer coefficient. The most common baffle shape is single segmental as shown in Figure 1.5. The segmental baffle cut must be less than half of the shell inside diameter in order to ensure that adjacent baffles overlap at least one full baffle tube row. For liquid flows on the shell- Flow direction Pt Pn Pp Pitch 30° Pitch 60° Triangular (30°) Rotated triangular (60°) Pitch 45° Pitch 90° Square (90°) Rotated square (45°) Figure 1.4: Tube layout and arrangement. The definition of tube pitch and tube pitches parallel and normal to flow (Pt, Pp and Pn) is typically shown for equilateral triangular arrangement. Table 1.1: Selection of pitch angle based on the nature of shell-side fluid and the flow regime. 5 Edge of baffle 1 Edge of baffle 2 1 2 Baffle cut / Baffle window Overlap with at least one full length tube to insure support Shell 1 2 1 Baffle spacing side, a baffle cut of 20 to 25 percent of the shell inside diameter is common. For low pressure gas flows, 40 to 45 percent is more common, in order to minimize the pressure drop. The baffle spacing should be correspondingly chosen to make the free flow areas through the baffle window and across the tube bank roughly equal [Ball, 2000]. For many high velocity gas flows, the single segmental baffle configuration results in an undesirably high shell-side pressure drop [Palen and Taborek, 1969; Perry and Chilton, 1999]. One way to retain the structural advantages of the segmental baffle and reduce the pressure drop is to use double segmental baffles as shown in Figure 1.6 [Poddar and Polley, 2000]. For sufficiently large units, it is possible to go to triple segmental arrangement. 1/32 inches (0.8 mm) for tubes larger in outside diameter than 1¼ inches (31.8 mm). If the maximum unsupported tube length exceeds 36 inches (914 mm) or if the tube outside diameter is less than 1¼ inches (31.8 mm), the baffle-tube clearance is 1/64 inches (0.4 mm) [TEMA, 1999]. In some air-conditioning applications, the tubes are expanded after assembly in order to eliminate the clearance altogether. This significantly complicates tube replacement, but might solve some potential tube vibration problems. The outer diameter of the baffle must be less than the shell inside diameter to allow assembly. But the clearance which is called shell-baffle clearance, should be as little as possible to minimize the shell-to-baffle leakage flow rate. The shell-to-baffle leakage typically is the greatest penalty against the shell-side heat transfer coefficient. If the shell-to-baffle leakage reaches a substantial magnitude, e.g. more than 15% of the total shell-side flow, the design effectiveness is poor and double segmental baffles or other modifications should be considered [Taborek, 1979; Kuppan, 2000]. Figure 1.5: Schematic configuration of single segmental baffles. Other baffle patterns and configurations such as disc-and-donut and orifice baffles have been used in the past but are seldom seen now. A small gap (clearance) between tube outside diameter and baffle-hole diameter, which is called tube-baffle clearance, is required in order to allow assembly of the tube bundle and tube replacement if required. Excessive clearance provides too little tube support and possible vibration, as well as excessive leakage of fluid across the baffle. Too little clearance makes assembly and tube replacement difficult. According to the standards of the Tubular Exchanger Manufacturers Association (TEMA), the tube-baffle clearance is. Figure 1.6: Schematic configuration of double segmental baffles. Edge of baffle 2 Edge of baffle 2 3 1 Baffle window Each overlap with at least one full length tube to insure support Shell 1 2 1 Baffle spacing 33 2 6 A shell and tube heat exchanger may be divided into different zones or spacing due to the presence of baffles. The first zone is the sector between the inlet nozzle and the first baffle and therefore may be called inlet zone. Moreover, the inlet baffle spacing may also refer to the longitudinal distance of the inlet zone and will be represented by Lbi. Similar to the inlet zone, the sector between the last baffle and the outlet nozzle is the outlet zone and will also be named as outlet baffle spacing. The outlet baffle spacing may also refer to the length of the outlet zone and will have the symbol Lbo. The second baffle zone or the second baffle spacing is the region between the first baffle and the second baffle. Using this definition, a shell and tube heat exchanger could have different baffle zones or baffle spacing zones. In most heat exchanger designs, the longitudinal distances between first, second, third and etc. baffle are equal. Therefore, these equal lengths are termed as central baffle spacing or Lbc. In most handbooks, the region consisting of inlet baffle spacing and the first baffle zone is called the inlet region. Similar to the inlet region, the outlet region is defined as a region of shell and tube heat exchanger which includes the outlet zone and its neighbouring baffle spacing. This definition is again used because of the effect of outlet zone on the pressure drop and heat transfer of shell and tube heat exchanger. 1.3 Tubular Exchangers Manufacturers Association (TEMA) Design Code A wide variety of configurations are available in shell and tube heat exchanger designs. The pressure parts of a shell and tube heat exchanger are designed in accordance with pressure vessel design codes such as ASME (American Society of Mechanical Engineers), BSS (British Standards Specifications), AD Merkblätter, and so on, but a pressure vessel code alone cannot be expected to deal with all the special features of shell and tube heat exchangers. To give guidance and protection to designers, fabricators, and purchasers alike, a supplementary code is desirable that provides minimum standards for design, materials, thicknesses, corrosion allowances, fabrication, tolerances, testing, inspection, installation, operation, maintenance, and guarantees for shell and tube heat exchangers [Hewitt, 1992]. A widely accepted standard is published by the Tubular Exchanger Manufacturers Association TEMA, which is intended to supplement the ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, although most of the information may be used to complement other pressure vessel codes if required [Saundres, 1988; ASME, 2004]. TEMA was founded in 1939 and is a group of leading manufacturers who have pioneered the research and development of heat exchangers for more than seventy years. The TEMA standard was prepared by a committee comprising representatives of 27 USA manufacturing companies and their combined expertise and experiences provide exchangers of high integrity at reasonable costs. 3 4 Inlet region 1 5 6 8 7 2 Outlet region Intermediate region 7×Lbc Lbi Lbo Outlet nozzle Inlet nozzle 9 Inlet zone Outlet zone Baffle zone Figure 1.7: Schematic representation of different zones and regions in a shell and tube heat exchanger. In order to minimize the pressure drop in inlet and outlet zones, the inlet baffle spacing and the outlet baffle spacing are longer than the central baffle spacing. The region consisting of all the other baffle zones is termed the intermediate region. Figure 1.7 represents schematically the different baffle zones and regions in a shell and tube heat exchanger. 7 TEMA also provided and developed a standard type designation and notation system for the major types of shell and tube heat exchangers. This standard system simplifies the specifications and identifies by three letters the basic configuration of shell and tube heat exchangers. The first letter identifies the front head, the second letter describes the shell and the third letter explains the rear head. Figure A.1 in “Appendix A” shows the TEMA designation system for shell and tube heat exchangers [TEMA, 1999]. Of the various shell geometries available, the simplest is the so called E-Shell. With a single tube pass unit and good baffling, the flow can be considered as pure counter current flow. This flow arrangement makes the best use of available temperature driving force and results in the smallest exchanger size [Poddar and Polley, 2000], see Figure 1.8. Even though the E- Shell is the most common configuration, variety of other designs, namely shell types F, G, H, J, K, and X, are used (Appendix A) [Palen, 1986]. Some typical TEMA shell and tube heat exchangers with E-shell design are shown in Figure 1.9. BEM shell and tube heat exchangers with fixed tube sheet are a standard choice with one or two tube side passes, as it is shown in Figures 1.9 (a) and (b). In order to clean the inside of the tubes, the front head piping must be unbolted and the heads must be removed. However, it is not possible to clean the outside surface of the tubes as these are inside the fixed part. AEM design is very similar to BEM, but the removable cover allows the inside of the tubes to be inspected and cleaned. An AEM shell and tube heat exchanger is schematically illustrated in Figure 1.9 (c). Figure 1.8: Schematic configuration of E-Shell based on TEMA notation system. 1.4 Shell and Tube Heat Exchangers with E-Shell The most common design of the shell-side of shell and tube heat exchangers is the E-shell due to its simplicity and its acceptable temperature driving force. For instance, most shell and tube heat exchangers in nuclear power plants are one; two, or four pass designs on the tube side and have an E-shell design on the shell-side. Another example of E-shell design are steam turbine condensers. Figure 1.9: Schematic representation of typical TEMA shell and tube heat exchangers with E-shell. (a) BEM shell and tube heat exchanger with one tube side pass (b) BEM shell and tube heat exchanger with two tube side passes (c) AEM shell and tube heat exchanger with two tube side passes (d) AES shell and tube heat exchanger with split ring (floating head with backing device) and two tube side passes 8 The AES design is excellent for applications where the difference in temperature between the hot and cold fluid causes unacceptable stresses in the axial direction of the shell and tubes. The floating head can move, i.e. provides the possibility to expand in the axial direction. For maintenance both the front and rear end head including the backing device, must be disassembled. An example of an AES shell and tube heat exchanger is shown in Figure 1.9 (d). Due to the importance of shell and tube heat exchangers with E-shell design, the present work will focus on the thermo-hydraulic behavior of this type, even though some results may be valid for other shell and tube heat exchanger as well. 9 2. Shell and Tube Heat Exchanger Design Methods 2.1 Performance of Heat Exchanger The design of a heat exchanger involves a consideration of both, heat transfer rates between the fluids and mechanical pumping power expended to overcome fluid friction and move the fluids through the heat exchanger. For a heat exchanger operating with high density fluid, the friction power expenditure is generally small relative to the heat transfer rate. However, for low density fluids, such as gases, it is very easy to expend as much mechanical energy in overcoming friction power as is transferred as heat. It can be readily shown that for most flow passages the heat transfer rate per unit of surface area can be increased by increasing fluid flow velocity. This rate varies as something less than the first power of the velocity. The friction power expenditure increases also with increasing flow velocity, but in this case the power varies by as much as the third power of the velocity and never less than the square [Kays and London, 1984]. If the friction power expenditure in a particular application tends to be high, the designer may reduce the velocities by increasing the number of flow passages in the heat exchanger. This will also decrease the heat transfer rate per unit of surface area, but according to the above relations the reduction in heat transfer rate will be considerably less than the friction power reduction. The lost heat transfer rate is then compensated by increasing the surface area like lengthening the tubes, which in turn also increases the friction power expenditure, but only in the same proportion as the heat transfer surface area is increased. In gas flow heat exchangers the friction power limitations generally force the engineers to arrange the design for moderately low mass velocities. Low mass velocities, together with the low thermal conductivity of gases (compared to most liquids), results in a low heat transfer rate per unit of surface area. Thus a large amount of surface area becomes a typical characteristic of gas flow heat exchangers. Gas-to-gas heat exchangers may require up to 10 times of the surface area of liquid-to-liquid heat exchangers. Summarizing the archetypical problem in heat exchanger design is to evaluate the thermal and pressure drop behaviour. 2.2 Basic Design Equations and Methods The steady state overall adiabatic heat exchanger behavior can be presented in terms of dependent fluid outlet temperatures or as functions of four operating variables and three design controlled parameters: with C and U as heat capacity rate and overall heat transfer coefficient, respectively. Equation (2.1) contains six independent and one or more dependent variables for a given heat exchanger flow arrangement. Any independent variable and/or parameter in Equation (2.1) can be made dependent if unknown. In that case, one of the three dependent variables in Equation (2.1) becomes an independent variable and/or parameter. Thus the most general heat (2.1) Thot, outlet, Tcold, outlet, Q = ϕ ( Thot, inlet, Tcold, inlet, Chot, Ccold, U, A, flow arrangement) operating variables parameters under control of designer dependent variables independent variables and parameters 10 exchanger design problem is to determine any two unknown variables from this set when the rest of them are known. For heat exchanger analysis, it is difficult to understand and work with such a large number of variables and parameters as outlined in Equation (2.1). From dimensional analysis, three dimensionless groups are formulated from six independent and one or more dependent variables of Equation (2.1). The reduced number of nondimensional variables and parameters simplifies much of the analysis, provides a clear understanding of the performance behavior, and the results can be presented in more compact graphical and tabular forms. The specific form of these groups is to some extent optional. Four such options have been used, depending on which method of heat transfer analysis has been selected: the effectiveness–number of heat transfer units method, the mean temperature difference (MTD) method, the nondimensional mean temperature difference–temperature effectiveness method and the generalised mean temperature difference method (GMTD) [Gardner, 1945; Kakaç, 1981; Bačlič, 1990; Hewitt, 1992; Bott, 1995; Oosthuizen, 1999; Sekulić, 1999; Hesselgreaves, 2001; Kraus and Aziz, 2001; Lienhard, 2002; Naterer, 2003; Nellis, 2003; Shah, 2003; VDI, 2006; Luben Cabezas-Gómez, 2007; Utamura, 2008]. 2.3 Calculation of Shell-Side Heat Transfer Coefficient and Pressure Drop The capital investment in heat exchangers throughout the world is exceedingly large and their maintenance and renewal are often costly hence proper initial design is an important economical consideration. Good design requires an accurate prediction of the pressure drop and heat transfer both inside and outside the tubes, in order that safety factors need not be made excessively large and that economic balances can be made between pumping and exchanger costs. Although satisfactory correlations are available for flow inside tubes, the status of information for flow across tube banks leaves much to be desired and therefore offers an attractive field for research. Research on pressure drop and heat transfer coefficient for flow outside of tubes is restricted either to results for flow across single banks of tubes, or to selected results on the industrial type of baffled, cylindrical shell and tube heat exchanger. In subsection 2.3.1, the correlations which evaluate the heat transfer coefficient and pressure drop of an ideal tube bank will be explained. An ideal tube bank can be defined as an unbaffled tube bank in which the tubes are arranged in-lined or staggered (in-lined arrangement refers to the tube layout Square 90°, while the staggered arrangements refer to the tube layouts Triangular 30°, Rotated triangular 60° and Square 45°, as it is illustrated in Figure 1.4) and the fluid flows across the tubes and normal to the tube lengths. Most of the correlations and methods which evaluate the heat transfer coefficient and pressure drop of a shell and tube heat exchanger are based on data and correlations valid for the ideal tube bank. In subsection 2.3.2, the correlations and methods for prediction of shell-side heat transfer coefficient and pressure drop of shell and tube heat exchangers will be discussed. Analytical and semi-analytical approaches have shown that these correlations can be presented in a general form as shown in Equation (2.2) [Achenbach, 1971; Eckert, 1972; Görtler, 1975; Shames, 1982; Hesselgreaves, 2001; Zlokarnik, 2002; Zwillinger, 2003]: In Equation (2.2), ሼXሽ denotes a set of dimensionless geometrical parameters and f is the friction factor. Δp= ρumax 2 2 ƒሺRe, f, ሼXሽሻ and Nu=ƒቆRe, Pr, f, ሼXሽ, μμwall ቇ (2.2) 11 2.3.1 Heat Transfer and Pressure Drop for Unbaffled Tube Bank Early studies of heat transfer and pressure drop done by U.S. and German sources go back to the 1910’s and are usually based on ideal tube banks [Palen, 1986]. The heat transfer equations assumed the basic tube side form. The pressure drop was correlated as a function of the maximum mass flux based on the minimum cross-sectional area in flow direction, total number of tube rows crossed by the flow and friction factor. For studying the pressure drop in tube banks, substantial experimental contributions have been made by Huge [1937], Pierson [1937] and Ter Linden [1939]. Based on the data then available, general correlations for pressure drop and heat transfer coefficient of tube banks were presented [Chilton and Genereaux, 1933; Grimison, 1937; Jakob, 1938; Colburn, 1942; McAdams, 1942; Gunter and Shaw, 1945; Boucher and Lapple, 1948]. As part of a comprehensive research program on pressure drop and heat transfer on the shell- side of tubular heat exchangers, Bergelin et al. conducted experiments in the region of viscous flow on seven different ideal tube banks. The results were presented graphically and the correlations for pressure drop and heat transfer coefficient of ideal tube banks were recommended [Bergelin, 1950]. Kays, London and Lo studied heat transfer and flow friction for flow normal to ideal tube bank for six staggered tube layouts and one in-lined tube arrangement in laminar and turbulent domain. Data were also provided so that the influence of the longitudinal and transverse pitch and the number of tube rows on the mean coefficient may be estimated accurately. They presented graphically correlations for the heat transfer coefficient and the pressure drop as a function of Reynolds number [Kays, 1954]. Žukauskas analysed the pressure drop in a tube bank using the pressure drop around a single tube by comparing the separation angle of a single tube and the frontal tube in a bank with in- line arrangement [Žukauskas, 1972]. He recommended seven correlations for in-line arrangement and nine correlations for staggered tube layouts. Each correlation was defined for different Reynolds numbers based on the free stream velocity and the tube outside diameter as well as different longitudinal pitches. Gnielinski presented a method for calculating the average heat transfer coefficient for a bank of tubes [Gnielinski, 1978]. The Reynolds number used in this method was modified to include the velocity of the fluid in the empty cross-section of the channel, the void fraction and the stream length. The method introduced by Gnielinski covers a wide range of experimental data obtained by different authors like Colburn, Grimison, Huge, Pierson, Bergelin, Kays et al., Bressler [1958] and Žukauskas. This method plays a significant role in the calculation of the shell-side heat transfer coefficient of shell and tube heat exchangers [VDI, 2006]. Martin introduced a new method based on the generalized Lévêque equation and discovered a new type of analogy between pressure drop and heat transfer that may be used in the corrugated channels of plate heat exchangers, in packed beds, in tube bundles and in many other space-wise periodic arrangements [Martin, 2002]. The author believes that this method can be proposed as the best available method to calculate the heat transfer coefficient of an ideal tube bank due to its simplicity and superb validity. The abovementioned correlations and methods are the basic concepts to calculate the shell- side heat transfer and pressure drop of a shell and tube heat exchanger, since the core of the tube bundle in a shell and tube heat exchanger may be considered as an ideal tube bank. However, due to some other geometrical factors like the cylindrical shape of the shell, the gap between the tube bundle and the shell wall, the configuration of baffles and the effect of baffle windows, the presence of tube-baffle and baffle-shell leakages, various modifications have to be considered. In the following subsection, the correlations, methods and approaches which 12 may predict the shell-side heat transfer coefficient and pressure drop of practical shell and tube heat exchangers will be assessed. 2.3.2 Shell-Side Heat Transfer and Pressure Drop The available methods for the prediction of shell-side heat transfer coefficient and pressure drop of shell and tube heat exchangers can be divided into five groups:  The early developments based on flow over ideal tube banks or even single tubes.  The analytical approach based on Tinker’s multi stream model and his simplified method [Tinker, 1947].  The stream analysis method, which utilizes a rigorous reiterative approach based on Tinker’s model [Tinker, 1947; Serth, 2007].  The Delaware method which uses the principles of the Tinker model but applies them on an overall basis without iterations [Bergelin, 1958].  The integral approach, which recognizes baffled cross-flow modified by the presence of windows. Initially, treatment of the problem was on an overall basis without consideration of the modifying effects of leakages and bypass flows. In the following, firstly the early model developments will be discussed in a few words. Secondly, the stream analysis of Tinker will be explained succinctly. This analysis will be explained in more detail in the subsequent chapters of the present work, where the effect of baffle cut and baffle orientation will be discussed. Then the Delaware method [Bergelin, 1958] will be outlined. Finally, the integral approach [VDI, 2006] will be explained. •Early developments: It was recognized in the early 1930's that baffled shell-side flow will behave similarly to flow across ideal tube banks. The first heat transfer correlation suggested is due to Colburn [1933]. The validity of this correlation was restricted to turbulent flow and staggered tube layout. In 1937; based on the equation for ideal tube banks and Colburn’s correlation, Grimison suggested a correlation which was modified to include the nonisothermal effects [Grimison, 1937]. For very quick estimations, Grimison’s correlation may still be used due to its simplicity. • Stream analysis method: In the late 1940’s it became obvious that baffled shell-side flow is so complex that it cannot be adequately expressed on a general basis by simple correlations and approaches [Emerson, 1963]. Only parts of the fluids take the desirable path through the tube nest, whereas a potentially substantial portion flows through the leakage and the bypass areas between tube bundle and the shell wall. However, these clearances are inherent to the manufacturing and assembly process of shell and tube exchangers, and the flow distribution within the exchanger must be taken into account. The early analysis of the shell-side flow is based on the schematic flow pattern depicted in Figure 2.1. Tinker applied an analytical approach for the shell-side method and suggested a schematic flow pattern, where the shell-side flow is divided into a number of individual streams. The individual streams of the shell-side flow may be defined as follows:  Stream A is the leakage stream in the orifice formed by the clearance between the baffle hole and tube wall.  Stream B is the main effective cross-flow stream, which can be related to flow across ideal tube banks.  Stream C is the tube bundle bypass stream in the gap between the bundle and the shell wall.  Stream E is the leakage stream between the baffle edge and the shell wall. 13  Stream F is the bypass stream in flow channel due to omission of tubes in tube pass partition. This stream has been appended to the original Tinker model [Palen and Taborek, 1969]. This stream behaves similarly to stream C, but will be present only in some tube layouts. In principle the stream analysis states that the pressure drop of the cross-flow stream B will act as a driving force for the other streams, forcing part of the flow through the leakage and bypass clearances [Tinker, 1951 and 1958]. • Delaware method: From 1947 to 1963 the Department of Chemical Engineering at the University of Delaware carried out a comprehensive research program on shell-side fluid flow and heat transfer in shell and tube heat exchangers, beginning with measurements of heat transfer and pressure drop during flow across ideal tube banks. These efforts were successively extended to introduce the various design features characteristic of shell and tube heat exchangers in commercial use. Sequentially, various baffle cuts and spacing configurations were investigated inside a cylindrical shell with no baffle leakage and minimal bypass clearance. Baffle leakages between baffles and shell, and between the tubes and baffles were added in later studies. Finally, bypass flow around the bundle between the outer tube limit and the shell inside diameter was studied together with the effect of sealing devices. The first and second report were published in 1950 [Bergelin, 1950] and1958 [Bergelin, 1958], respectively, and in 1960 a preliminary design method for E shell exchangers was published [Bell, 1960]. The final report was published in 1963 [Bell, 1963]. The shell-side heat transfer coefficient is given by the following equation: In Equation (2.3) the effect of baffle cut and spacing is shown by Jc. Jl represents the correction factor for baffle leakage, including both shell to baffle and tube to baffle leakages. Jb is the correction factor for the bypass flow and Js is the correction for variable baffle spacing. Jr is the laminar heat transfer correction factor for adverse temperature gradient. This gradient lowers the local and the average heat transfer coefficient with increasing distance. The correction factor Jr has been worked out mathematically for flow in well defined geometries like inside the round tubes, but it is also found experimentally to exist during flow ቀ shell-side heat transfer coefficient ቁ=ሺJcJlJbJsJrሻ ቀ heat transfer coefficient  of an ideal tube bank ቁ (2.3) Figure 2.1: Flow paths of various streams through the shell of a cross-baffled shell and tube heat exchanger. E A A B C E B A C F B B B B C 14 across tube banks. For large heat exchangers in deep laminar flow, it can result in a decrease in the average heat transfer coefficient by a factor of two or more compare to what would have been predicted based on flow across a 10-row tube bank. This correction factor; Jr, applies only if the shell-side Reynolds number is less than 100 and is fully effective only in deep laminar flow characterized by shell-side Reynolds numbers less than 20. The shell-side pressure drop is correlated as a linear function of the pressure drop in one cross flow section and the pressure drop in one baffle window section without leakage or bypass flow. However, in this correlation, three correction factors are considered for the effect of bypass flow, leakages, and also inlet and outlet zone on the pressure drop. •Integral approach: Donohue [1949] and Kern [1965] published shell-side methods based on overall data from baffled exchangers which assumed that the baffles are used to direct the shell fluid perpendicularly to the tubes. Due to the limited number of available data only an insufficient variation of basic geometrical parameters like baffle spacing, baffle cut and tube layout were presented. To overcome this deficiency, safety factors were introduced which lead to poor accuracies for the prediction of the shell-side heat transfer coefficient and pressure drop. The Donohue method became quite popular for its simplicity while presenting a more systematic treatment than anything known before. The heat transfer correlation was based on a flow area that is the geometric mean between the minimum cross-flow area at the inside of the shell and the baffle window longitudinal flow area. The Nusselt equation has a form similar to Grimison’s correlation, except for the above mentioned interpretation of the mass velocity. For pressure drop, a set of friction factor curves based on Grimison’s work with a large safety factor was used. Even though Donohue’s method represented a step in the right direction, being based on a non-systematic set of data, it provided up to several hundred percent over-prediction on pressure drop and heat transfer coefficient. Kern’s book was used as a virtual industrial standard for many years. The merits of Kern’s method are not so much that better correlations are used, but rather in the way the overall design problem is approached as an entity, including numerous practical hints and calculated examples. Both heat transfer and pressure drop are presented for 25% baffle cut only, which is reasonably close to the best design. The length term in the Nusselt and Reynolds numbers is an equivalent diameter based on longitudinal flow projection, in order to account for the tube layout variations. No account is taken for variations in bypass or leakage areas. Pressure drop prediction are almost invariably on the safe side and usually more than 100%, whereas heat transfer may vary from slightly unsafe to very safe, because of the poor treatment of the bypass and leakage effects. The prediction accuracy decreases further in laminar flow, because very few data were available at that time and the simple method is not equipped to handle the complex problem. Although Kern’s method cannot be recommended any more, many of the practical comments on design remain qualitatively valid. Gnielinski and Gaddis [1977, 1978 and 1983] developed a method based on the integral approaches which is used for ideal tube banks, to evaluate the shell-side heat transfer coefficient and pressure drop. This method predicted a large number of experimental data with a good validity. Due to its systematic treatment and the acceptable validity, the VDI Wärmeatlas recommends this method [VDI, 2006]. Therefore, in the present work, this method will be termed as VDI method. In the VDI method, the average shell-side Nusselt number Nuതതതതshell is calculated as follows: Nuതതതതshell=ƒWNu0, bank (2.4) 15 The Nusselt number Nu0, bank and the average shell-side Nusselt number Nuതതതതshell are based on the stream length πdo/2 as characteristic length. All physical properties are calculated for average bulk temperature. The correction factor ƒW is a geometrical factor and has to be calculated as follows: In Equation (2.5) the correction factor ƒW describes the effect of geometry on the heat transfer. The influence of baffle window or baffle cut is considered by the geometrical factor ƒG. The effect of leakages is taken into account by considering the leakage-stream factor ƒL and the bypass effect by considering the bypass-stream factor ƒB. ƒG depends on the total number of tubes located in both upper and lower baffle windows, and the total number of tubes in the shell. ƒL is a function of the total area of tube-baffle leakages and baffle-shell leakages, and the minimum free flow area in tube bundle. ƒB depends on the number of pairs of sealing strips, the number of the tube rows located between the baffles, and the minimum free flow area in tube bundle. Figure 2.2 presents the minimum free flow area in tube bundle for three tube layouts. The method recommended by VDI Wärmeatlas to calculate the pressure drop is based on the integration of different pressure drops in different domains consisting of inlet and outlet zones, inlet and outlet nozzles, baffle windows and the tube banks which are located between the baffle. ƒW=ƒGƒLƒB (2.5) Figure 2.2: Definition of minimum free flow area for different tube layout in a shell and tube heat exchanger. The dotted lines represent this area. Flow direction Flow direction Flow direction Minimum free flow area 17 3. Limitation of Common Calculation Methods with Respect to the Effect of Baffle Orientation 3.1 Geometrical Difference in Baffle Orientation It is well known that the inlet and outlet zones have a significant influence on the performance of a shell and tube heat exchanger due to the presence of nozzles. Therefore, it is valuable to study the effects of the end-zones on the shell-side heat transfer and pressure drop. In this study, only the inlet zone where the shell-side fluid enters the shell will be considered. The horizontal and vertical baffle orientation is presented by case (a) and (b) in Figure 3.1. Even though the effect of the baffle orientation is not taken into account in the VDI and Delaware methods, due to the definition of flow direction and minimum cross-flow area, it is possible to calculate the shell-side heat transfer coefficient and pressure drop for both cases shown in Figures 3.1 and 3.2. In this case it is assumed that there is no leakage; which means that the leakage-stream factor ƒL (see Equation (2.5) in subsection 2.3.2) in the VDI method is equal to 1. Depending on the orientation of baffles, the hypothetical flow direction defined for the VDI and Delaware methods is different. Because of this flow direction, the minimum cross-flow area will be different in case (a) and case (b) as it is shown by the dotted lines in Figure 3.2. Figure 3.1: Inlet zone for two shell and tube heat exchangers. Case (a) and (b) represents the horizontal and vertical baffle orientation, respectively. (a) (b) Figure 3.2: Hypothetical flow direction due to the presence of baffles according to the definition of VDI and Delaware methods for case (a) and case (b). The minimum cross-flow areas are shown by dotted lines. (a) Flow direction (b) Flow direction 18 Reφ= wπdo 2φ ∆p [p a] 1000 3000 5000 7000 90000 2000 4000 6000 8000 10000 250 750 1250 1750 0 500 1000 1500 2000 Horizontal baffle orientation Vertical baffle orientation 1000 3000 5000 7000 90000 2000 4000 6000 8000 10000 25 75 125 175 0 50 100 150 200 Horizontal baffle orientation Vertical baffle orientation Reφ= wπdo 2φ N uതതതത sh el l The shell-side pressure drop and average Nusselt number are calculated at different Reynolds numbers for a typical shell and tube heat exchanger. The shell and tube heat exchanger considered is an ideal heat exchanger without leakages and consists of 660 tubes. The shell inside diameter is 23.3” and the tube outside diameter is ⅝”. The tube layout is triangular 30° and the tube pitch is 13/16”. The baffle cut is 24% of the shell inside diameter. The shell-side fluid is water and the heat transfer process is heating. Shell-side pressure drop and average Nusselt number versus modified Reynolds number calculated according to the VDI method are presented in Figures 3.3 and 3.4, respectively. Figure 3.3: Shell-side pressure drop versus modified Reynolds number for horizontal and vertical baffle orientation and 24% baffle cut according to the VDI method. The shell-side fluid is water and the heat transfer process is heating. Figure 3.4: Average shell-side Nusselt number versus modified Reynolds number for horizontal and vertical baffle orientation and 24% baffle cut according to the VDI method. The shell-side fluid is water and the heat transfer process is heating. 19 Figure 3.5: ∆phorizontal ∆pvertical⁄ versus Reinlet for 24% baffle cut according to the VDI method. The shell-side fluid is water and the heat transfer process is heating. Reinlet= uinletDn  ∆p ho riz on ta l ∆p ve rti ca l ⁄ 101 102 103 104 105 106 107 0.75 0.80 0.85 0.90 0.95 1.00 1.05 Both shell-side pressure drop and average Nusselt number are calculated for horizontal and vertical baffle orientation. The shell-side pressure drop and heat transfer coefficient for the vertical baffle orientation is greater than the shell-side pressure drop and heat transfer coefficient for the horizontal baffle orientation, as it is shown in Figures 3.3 and 3.4. The effect of baffle orientation on the pressure drop and heat transfer coefficient of the shell- side are analyzed by representing the values of the pressure drop ratio ∆phorizontal ∆pvertical⁄ and the Nusselt number ratio ሺNuതതതതshellሻhorizontal ሺNuതതതതshellሻvertical⁄ at different Reynolds numbers. The Reynolds number is based on the conditions at the inlet nozzle. In Equation (3.1) uinlet and Dn is the fluid velocity at the inlet nozzle and the inside diameter of the inlet nozzle, respectively. The shell-side pressure drop ratio ∆phorizontal ∆pvertical⁄ versus Reinlet is plotted in Figure 3.5. Figure 3.5 indicates that the shell and tube heat exchanger with vertical baffle orientation presents higher pressure drop than that with horizontal baffle orientation. A comparable behavior for the heat transfer coefficient should be expected due to the analogy between the pressure drop and the heat transfer coefficient. The local maximums of graph in Figure 3.5 are due to the mathematical formulation presented in VDI heat atlas [VDI, 2006]. Figure 3.6 represents the shell-side Nusselt number ratio ሺNuതതതതshellሻhorizontal ሺNuതതതതshellሻvertical⁄ as a function of Reinlet. The shell-side pressure drop behaviour according to the Delaware method corresponds with the VDI method. However, the value of ሺNuതതതതshellሻhorizontal ሺNuതതതതshellሻvertical⁄ is equal to 1 for all Reynolds numbers according to the Delaware method. Reinlet= uinletDn  (3.1) 20 3.2 Definition of Baffle Orientation The baffle orientation is not a flow property, but a geometrical characteristic which declares the orientation of the baffles with respect to the nozzles. Therefore, it is important to state clearly and mathematically the definition of the baffle orientation. In order to define the baffle orientation of a heat exchanger, reference planes and vectors are introduced as given in Figure 3.7.  The “baffle-orientation-plane” is parallel to the tube-bundle axis and touches the baffle edge.  The “inlet (outlet)-plane” contains the inlet (outlet) area of the inlet (outlet) nozzle.  The “tube-sheet-plane” contains the tube-sheet at the inlet zone.  “Face-vectors” are normal to the planes considered and directed to the centre of the shell.  The “baffle-vector” is normal to the baffle-orientation-plane and directed towards the outside of the shells. The description of the abovementioned system will be in Cartesian coordinates with y-axis being in opposite direction of the face-vector of the inlet-plane and z-axis in direction of the face-vector of the tube-sheet-plane. If jԦ and kሬԦ denote the unit vectors of y and z axes, respectively, the positive direction of x-axis can be found by its unit vector iԦ according to the following equation: In the x-y plane of this Cartesian coordinates, the counter-clockwise angle between the “baffle-vector” and the “face-vector” of the “inlet-plane” characterizes the baffle orientation in each baffle zone. In the typical sketch shown in Figure 3.7, (“baffle-vector”, “face-vector” of “inlet-plane”) represents the counter-clockwise angle between the “baffle-vector” and the “face-vector” of the “inlet-plane”. iԦ= jԦൈ kሬԦ (3.2) 101 102 103 104 105 106 107 0.90 0.95 1.00 1.05 1.10 1.15 Reinlet= uinletDn  ሺ N uതതതത sh el lሻ h or iz on ta l ሺ N uതതതത sh el lሻ v er tic al ⁄ Figure 3.6: ሺNuതതതതshellሻhorizontal ሺNuതതതതshellሻvertical⁄ versus Reinlet for 24% baffle cut according to the VDI method. The shell-side fluid is water and the heat transfer process is heating. 21 In the present work two different baffle orientations will be considered: horizontal and vertical. In the heat exchanger with horizontal baffle orientation, (“baffle-vector”, “face- vector” of “inlet-plane”) at the inlet and outlet zone is equal to 0º, and at the central baffle spacing zones is equal to 180º or 0º. In the heat exchanger with vertical baffle orientation, (“baffle-vector”, “face-vector” of “inlet-plane”) at the inlet and outlet zone is equal to 270º, and at the central baffle spacing zones is equal to 90º or 270º. 3.3 Minimum Shortcut Distance The residence time of the working fluid in the inlet zone of the shell and tube heat exchanger depends on baffle orientation, baffle cut, and number of tubes. The comparison of residence times of the inlet zone of two shell and tube heat exchangers with identical number of tubes and baffle cut but with different baffle orientation is done by contrasting the rectilinear distance between the inlet nozzle and the baffle window of these two inlet zones. In the present work, the rectilinear distance between the inlet nozzle and the baffle window will be termed as minimum shortcut distance (MSD). Since there are infinite numbers of MSDs, it is meaningful to present an average or normalized distance. Figure 3.9 illustrates the variables used to calculate the normalized minimum shortcut distance (NMSD). Equation (3.3) presents the integral form of NMSD based on the variables shown in Figure 3.9. NMSD= ׬ MSDሺϴሻdϴϴmaxϴmin ׬ dϴϴmaxϴmin (3.3) Figure 3.7: System definition of Cartesian coordinates, baffle orientation and outlet nozzle arrangement Now it is possible to define the horizontal and vertical baffle orientation by use of a set of angles. Thereby the first angle refers to the inlet zone, the second angle refers to the first central baffle spacing zone and ditto. The last angle refers to the outlet zone. Figure 3.8 represents schematically two E type shell and tube heat exchangers with three central baffle spacing zones. Using the aforementioned definition, the shell and tube heat exchanger with horizontal baffle orientation (see Figure 3.8) has a set of angles ={0°,180°,0,180°,0°}. In the same manner, the shell and tube heat exchanger with vertical baffle orientation presented in Figure 3.8, has a set of angles ={270°,90°,270°,90°,270°}. The abovementioned mathematical definition could be considered as an appropriate method to define the other baffle orientations for a segmentally baffled shell and tube heat exchanger. k×j=i  j  k  (“baffle-vector”, “face-vector” of “inlet-plane”) = 0° 22 Figure 3.9: The variables used to calculate the normalized minimum shortcut distance or NMSD for the inlet zone. Case (a) and (b) represents the horizontal and vertical baffle orientation, respectively. (a) ϴca ϴmax ϴmin ϴ rs MSD(ϴ) Lbch (b) rs Lbch ϴca MSD(ϴ) ϴmax ϴmin ϴ Figure 3.8: Presentation of E type shell and tube heat exchanger with three central baffle spacing zones and two different baffle orientations: (a) horizontal baffle orientation and (b) vertical baffle orientation. The value presented in  refers to the counter-clockwise angle between the presented vectors and the “face-vector” of the “inlet-plane”. (a) E type shell and tube heat exchanger with horizontal baffle orientation: ={0°,180°,0°,180°,0°}. (b) E type shell and tube heat exchanger with vertical baffle orientation: ={270°,90°,270°,90°,270°}. =0° =180° =0° =0° =180° “baffle vector” of zone 1 “face vector” of “outlet plane” “face vector” of “inlet plane” “baffle vector” of zone 2 “baffle vector” of zone 3 “baffle vector” of inlet zone “baffle vector” of outlet zone =270° =270°=270°=90° =90° “baffle vector” of zone 1 “baffle vector” of zone 2 “baffle vector” of zone 3 “face vector” of “outlet plane” “face vector” of “inlet plane” “baffle vector” of inlet zone “baffle vector” of outlet zone 23 The integral form above is solved and expressed as a function of ϴca. ϴca is the centri-angle of the baffle cut intersection with the inside shell wall, and depends on the baffle cut height Lbch and the shell inside radius rs. In Equation (3.4), BC is the segmental baffle cut percentage. The mathematical function of NMSD for horizontal and vertical baffle orientation is given by the following equations: The comparison between (NMSD)horizontal and (NMSD)vertical is done by introducing the normalized minimum shortcut distance ratio or NMSDR. Figure 3.10 represents the plot of NMSDR versus BC for segmental baffle cut percentage less than 45. Figure 3.10 confirms that the value of NMSD for horizontal baffle orientation is greater than the value of NMSD for vertical baffle orientation. Therefore, the residence time in a shell and tube heat exchanger with horizontal baffle orientation is more than the residence time in a shell and tube heat exchanger with vertical baffle orientation. This means that a shift in baffle orientation from vertical to horizontal will increase the mixing level and consequently the rate of heat transfer and the value of pressure drop. Hence, the ratio of ሺNuതതതതshellሻhorizontal ሺNuതതതതshellሻvertical⁄ shown in Figure 3.6 seems to be rational, at least for Reinletذ6×103. However, the behaviour of ∆phorizontal ∆pvertical⁄ represented in Figure 3.5 is not logical. It is important to restate the assumption of no leakages in the aforementioned analysis. However, the concept of NMSDR is useful to explain the effect of baffle orientation on performance of a real shell and tube heat exchanger with leakage streams. ϴca= arccosቆ rs-Lbch rs ቇ= arccos ൬1- BC 50 ൰ (3.4) BC=50 Lbch rs (3.5) ሺNMSDሻhorizontal= 2 sinϴca ϴca tan൭ϴca 2ൗ ൱ ln ൥sec൭ϴca 2ൗ ൱+ tan൭ϴca 2ൗ ൱൩ rs (3.6) ሺNMSDሻvertical= cosϴcaϴca ln ൥ 1+ sinϴca +ඥ2ሺ1+ sinϴcaሻ 1- sinϴca +ඥ2ሺ1- sinϴcaሻ ൩ rs (3.7) NMSDR= ሺNMSDሻhorizontalሺNMSDሻvertical = 4 1- tan2 ൭ϴca 2ൗ ൱ ln ൥sec൭ϴca 2ൗ ൱+ tan൭ϴca 2ൗ ൱൩ ln ቈ1+ sinϴca +ඥ2ሺ1+ sinϴcaሻ 1- sinϴca +ඥ2ሺ1- sinϴcaሻ ቉ (3.8) 24 3.4 The Necessity of the Investigation of the Effect of Baffle Orientation The discussions in Sections 3.1 and 3.3 confirm that the effect of baffle orientation is not taken into account yet. In fact, all the available investigations and methods are only based on one baffle orientation, namely horizontal. Moreover, NMSDR shows that both heat transfer and pressure drop will increase when the baffle orientation changed from horizontal to vertical. Therefore, it is important to investigate the effect of baffle orientation on the performance of shell and tube heat exchangers. BC 5 15 25 35 450 10 20 30 40 N M SD R 1.5 2.5 3.5 4.5 1.0 2.0 3.0 4.0 5.0 √2 Figure 3.10: NMSDR versus Bc for baffle cuts less than 45% of shell inside diameter. This graph shows that the value of NMSD for horizontal baffle orientation is greater than the value of NMSD for vertical baffle orientation. 25 4. Application of CFD for the Present Heat Exchanger Investigations The technological value of computational fluid dynamics (CFD) has become undisputed in the last decade. CFD allows to compute flows that can be investigated experimentally only at reduced Reynolds numbers, or at greater cost, or not at all. Another significant feature of CFD is to evaluate and quantify the data in a high level of details without changing the geometry. This is similar to an ideal experimental method in which measuring all the required data without inserting or installing a set of measurement devices is possible. Measuring the necessary data in a high level of detail without changing the real geometry is an ultimate tool in research and development [Ferziger, 2002]. A distinguishing feature of the present state of computational fluid dynamics is that large commercial CFD codes have arisen, and have found widespread use in industry. One of the most powerful commercial CFD tools is the state-of-the-art computer program Fluent which offers different and suitable solutions and models and provides valuable geometrical tools for modelling fluid flow and heat transfer in complex geometries [Fluent, 2008]. In the present work, Fluent is applied as CFD tool for the heat exchanger investigations. 4.1 Model Characteristics In order to build up a CFD model, it is necessary to find a suitable spatial discretization for the geometrical calculation domain which is in the present work the shell-side geometry of the considered shell and tube heat exchanger. Based on the requirement to reduce numerical errors (numerical diffusion, mesh independence and the alignment of the mesh elements with the main flow direction), the applied mesh scheme is a conformal, non-hybrid, structured mesh, that is, a grid scheme with quadrilateral-faced hexahedral elements [Thompson, 1985; Ruppertt, 1995; Wesseling, 2001; Russell, 2002; Shewchuk, 2005]. A brief description of some important features of this grid scheme is given in the following subsections 4.1.1 and 4.1.2. 4.1.1 Mesh Qualification The mesh quality has a considerable impact on the computational analysis in terms of the quality of the solution and the required computational time. The evaluation of the quality of the mesh is very useful because it provides some indication of how suitable a particular discretization is for the analysis type under consideration [Babuska and Aziz, 1976; Křížek, 1992]. For the present work two quality ratios have been applied namely the aspect ratio which defines the dilation of a mesh element and the equiangular skewness which measures the skewness of a mesh element. For hexahedral elements, the aspect ratio QAR is defined as: where ιi is the average length of the edges in a coordinate direction i local to the element. QAR=1 describes an equilateral element. However, the mesh structure for the shell-side of a medium size heat exchanger could have mesh elements with aspect ratio up to 15. The equiangular skewness, or EquiAngle, is a normalized measure of the distortion of a mesh element. For hexahedral elements, the EquiAngle skewness QEAS is expressed as follow: QAR= maxሼιiሽ minሼιiሽ (4.1) 26 where Ԃmax and Ԃmin are the maximum and minimum angle in radian between the edges of a element, respectively. A hexahedral element with QEAS=0 is a cuboid, while QEAS=1 describes a completely degenerate element. In general, high quality three dimensional meshes contain elements which show an average QEAS value of 0.4 [Gambit, 2007]. 4.1.2 Determination of Mesh Size and Structure The required computational memory for solving the governing equations (see section 4.2) at each mesh element is about 1.5 Kbyte [Fluent, 2008]. Hence, the maximum number of elements which can be solved is about 1,400,000, since the available computational memory for the present work is limited to 2GB. A midsize heat exchanger with one tube pass typically consists of about 500 tubes. Hence the question has to be answered how it is possible to mesh the shell-side inlet zone of a medium size shell and tube heat exchanger with 1,400,000 quadrilateral-faced hexahedral elements. For this reason, preliminary grid investigations for different ideal (without leakages) shell and tube heat exchangers (baffle cut 15%, horizontally orientated baffles, triangular tube layout 30°) are carried out. For a shell and tube heat exchanger consisting of 140 tubes (tube outside diameter 19.05 mm, shell inside diameter 304.8 mm), source faces are introduced in order to mesh the inlet zone. Figure 4.1 (a) represents schematically the inlet zone of the heat exchanger with 140 tubes. The source faces are two-dimensional surfaces between the tubes which construct the tube sheet. Figure 4.2 (a) shows a typical source face. The source faces are meshed using a quad-pave meshing scheme in a manner that the 2D aspect ratio of cells will be around 1, and at least 10 elements will be located between two adjacent tubes. This ensures the compatibility of the mesh structure with the turbulent flow. Figure 4.2 (b) represents a meshed source face. Figure 4.1 (b) shows a part of the meshed tube sheet obtained by juxtaposing the meshed source faces. Finally, the cooper mesh scheme is applied in order to generate the three dimensional mesh for the inlet zone. In the cooper scheme, the meshed source faces are swept along the tube length with a certain extrusion number or size. The quad-faced hexahedral elements of the 3D mesh structure have an aspect ratio less than 2 and the maximum EquiAngle skewness equal to 0.53. The application of the mesh procedure described results in 60,000,000 mesh elements for the inlet zone of the shell and tube heat exchanger with 140 tubes. The same meshing procedure as described for the heat exchanger with 140 tubes is applied for the grid generation for the inlet zone of two smaller ideal heat exchangers with 10 and 24 tubes, respectively (see Figures 4.3 and 4.4). The baffle cut, the baffle orientation and the tube layout are identical with heat exchanger with 140 tubes. The tubes outside diameter and the shell inside diameter are 16 mm and 90 mm for the heat exchanger with 10 tubes and 19.05 mm and 205 mm for the heat exchanger with 24 tubes. The resulting mesh structures contain 1,200,000 and 1,500,000 elements for the heat exchangers with 10 and 24 tubes, respectively. About 99.6% of the mesh elements of both grids have an EquiAngle skewness less than 0.5. The number of 3D elements required for meshing the inlet zone can be expressed as a function of the tube number, as it can be concluded from the mesh structure generated for the inlet zone of the heat exchangers with 140, 10 and 24 tubes. The function that will estimate the number of mesh cells is named as the mesh number estimation function or MNEF. Besides the tube number, MNEF depends on the shell inside diameter, the number of 2D elements between two adjacent tubes, the 2D and the 3D aspect ratios. In Figure 4.5 the curves of MNEF as a function of the tube number with the shell inside diameter as a QEAS=max ൜ 2Ԃmax-π π , π-2Ԃmin π ൠ (4.2) 27 parameter are depicted. In Figure 4.5, the 2D and 3D aspect ratios are 2 and 4, respectively, and the number of mesh elements between two adjacent tubes is 10. The analysis of MNEF for the TEMA shell and tube heat exchangers shows that the value of MNEF is proportional to the square of the tube number, the cube of the number of 2D elements between two adjacent tubes, and the inverse of the 2D and 3D aspect ratios. Figure 4.1: TEMA shell and tube heat exchanger with 140 tubes: (a) inlet zone (b) two dimensional mesh structure. 10 elements are located between two adjacent tubes. The 2D aspect ratio is about 1. Approximately, 99.7% of elements have the EquiAngle skewness less than 0.3, and the rest of elements have the skewness less than 0.53. The elements with blue colour have the best skewness, i.e. 0, and the red elements have the maximum skewness, 0.53. (a) (b) Figure 4.2: Basic two dimensional surface: (a) source face (b) meshed source face using the pave scheme. (a) (b) 28 Considering Figure 4.5, the only way to mesh the inlet zone of a medium size shell and tube heat exchanger with 1,400,000 cells hence is to reduce the number of elements between tubes and to increase the 2D and 3D aspect ratios. A simple analysis on MNEF shows that the inlet zone of an ideal shell and tube heat exchanger consisting of around 600 tubes can be meshed with 1,200,000 cells by inserting 8 elements between two adjacent tubes and by applying the 2D and 3D aspect ratios equal to 10 and 15, respectively. This will be presented and discussed in chapter 5. 4.2 Numerical Model 4.2.1 Governing Equations In a steady state study, as the present work, CFD uses basically the Eulerian formulation for analysis and computation. That is, the conservation equations will be solved at every fixed Tube number 0 50 100 150 200 250 300 N um be r o f t hr ee d im en si on al m es h el em en ts × 1 06 1 3 5 7 9 11 13 15 shell inside diameter = 205.0 mm shell inside diameter = 254.5 mm shell inside diameter = 304.8 mm shell inside diameter = 336.6 mm shell inside diameter = 387.4 mm shell inside diameter = 443.2 mm shell inside diameter = 494.0 mm Figure 4.5: Estimated number of three dimensional elements required for meshing the inlet zone versus tube number. The tube layout is triangular 30°. The number of cells between two adjacent tubes is 10 and the 2D and 3D aspect ratios are about 2 and 4, respectively. Figure 4.4: Inlet zone of a shell and tube heat exchanger with 24 tubes. Figure 4.3: Inlet zone of a shell and tube heat exchanger with 10 tubes. 29 point of the domain. The steady state governing equations in the Cartesian coordinate are as follows [Ferziger, 2002; Fluent, 2008]: Equation (4.3) is the continuity or mass conservation equation. Equation (4.4) is the momentum conservation equation in the direction i of the non-accelerating Cartesian frame for Newtonian fluids and without body forces. In Equation (4.4), iԦj represents the Cartesian unit vector in the direction of the coordinate xj. Equation (4.5) describes the energy equation for a one-component fluid in the direction i of the Cartesian coordinate. In Equation (4.5) es is the sensible enthalpy, kf, turb is the turbulent thermal conductivity and μturb is the turbulent or eddy viscosity. 4.2.2 Turbulence The Reynolds stress model (RSM), the standard k-ω model, the renormalization group k-ε (RNG k-ε), the realizable k-ε and the standard k-ε models are the most suitable turbulence models for the present work [Wilcox, 1998; Ferziger, 2002; Fluent, 2008]. Compared with the k-ε models, the RSM requires additional memory and CPU time due to the increased number of transport equations for Reynolds stresses. On average, the RSM requires 50-60% more CPU time per iteration and 15-20% more memory compared to the k-ε models [Fluent, 2008]. In order to find the difference between the RSM, the k-ω and the k-ε models, the inlet zone of the shell and tube heat exchanger shown in Figure 4.4 is meshed and then solved by applying different turbulence models. The comparison between the final results of the RSM, the k-ω and the k-ε models does not show any significant qualitative and quantitative difference. However, the k-ε models are more satisfactory since the k-ω model are fairly new and have not been examined as well as the k-ε models. Moreover, the RSM needs more memory than k-ε models. Both the realizable and RNG k-ε models have shown substantial improvements over the standard k-ε model where the flow features include strong streamline curvature, vortices, and rotation. However, the realizable k-ε model is still relatively new and it is not clear in exactly which instances the realizable k-ε model consistently outperforms the RNG model. On the other hand, the RNG theory provides an analytically-derived differential formula for effective viscosity that accounts for low-Reynolds-number effects [Fluent, 2008]. Hence, the RNG k-ε model is the preferred turbulence model implemented in the present work. 4.2.3 Near Wall Treatment of the Flow Turbulent flows are significantly affected by the presence of walls. Obviously, the mean velocity field is affected through the no-slip condition that has to be satisfied at the wall. However, the turbulence is also changed by the presence of the wall in non-trivial ways. Very close to the wall, viscous damping reduces the tangential velocity fluctuations, while kinematic blocking reduces the normal fluctuations. Toward the outer part of the near-wall region, however, the turbulence is rapidly augmented by the production of turbulence kinetic energy due to the large gradients in mean velocity [Bradshaw, 1971; Tennekes, 1972]. ׏ሬԦ·ሺρuሬԦሻ=0 (4.3) ׏ሬԦ·ሺρuiuሬԦሻ=- ∂∂xi ൬p+ 2 3 μ׏ሬԦ·uሬԦ൰+׏ሬԦ· ቈμቆ∂ui∂xj + ∂uj ∂xiቇ i Ԧj቉ (4.4) ∂ ∂xi ቈui ቆρes+ρ ui2 2 ቇ቉= ∂∂xi ቈ൫kf+kf, turb൯ ∂T ∂xi +uj൫μ+μturb൯ ቆ ∂ui ∂xj + ∂uj ∂xi - 2 3 ∂uj ∂xjቇ቉  (4.5) 30 Numerous experiments have shown that the near-wall region can be largely subdivided into three layers. In the innermost layer, called the viscous sublayer, the flow is almost laminar, and the molecular viscosity plays a dominant role in momentum and heat or mass transfer. In the outer layer, called the fully-turbulent layer, turbulence plays a major role. Finally, there is an interim region between the viscous sublayer and the fully turbulent layer where the effects of molecular viscosity and turbulence are equally important. This interim region is called blending region or buffer layer. The important parameter to distinguish the different viscous layers in a flow is the dimensionless sub-layered distance y+. The value of y+ depends on the friction velocity uτ. The friction velocity uτ is defined as ඥτW ρ⁄ where τW is the surface or wall shear stress. The friction velocity can be interpreted as the disturbance velocity induced by shear stress of the solid walls. The subdivisions of the near-wall region can be presented as the plot of u/uτ versus y+ in a semi-log coordinates [Kutateladze, 1964]. One common approach to model the near-wall region is the usage of so-called “wall functions”. In doing so, the viscosity-affected inner region, i.e. viscous sublayer and buffer layer, is not resolved. Instead, semi-empirical formulas called “wall functions” are used to bridge the viscosity-affected region between the wall and the fully-turbulent region. The use of wall functions obviates the need to modify the turbulence models to account for the presence of the wall. Depending on the turbulence model, three choices of wall function approaches are available: standard wall functions which are based on the proposal of Launder and Spalding, non- equilibrium wall functions, and enhanced wall treatment [Kutateladze, 1964; Launder and Spalding, 1974; Kader, 1981; Fluent, 2008]. Since both non-equilibrium wall functions and enhanced wall treatment require sufficiently fine mesh structure near the walls, the standard wall functions is implemented in the present work. 31 Figure 5.1: Main dimensions of the inlet zone of the shell and tube heat exchanger consisting of 660 tubes. The baffle thickness (6.35 mm), the tube pitch (3.20 mm) and the tube partition width (1.97 mm) are not indicated in this figure. Dn=154.18 mm Ln=192.73 mm Lbc=262.41 mm do=15.88 mm Ds=590.93 mm Lbch=118.19, 139.78 and 177.28 mm 5. Effect of Baffle Orientation, Baffle Cut and Fluid Viscosity on Pressure Drop and Heat Transfer Coefficient in the Inlet Zone of Shell and Tube Heat Exchangers without Leakages As shown in the previous chapter, it is possible to mesh the inlet zone of a medium size ideal shell and tube heat exchanger with around 1,200,000 cells. The meshed domain then will be solved numerically by applying the RNG k-ε turbulence model and standard wall functions. This procedure makes it feasible to study the influence of baffle cut and baffle orientation on the shell-side performance of a shell and tube heat exchanger. In the present chapter, a shell and tube heat exchanger consisting of 660 tubes is considered. The geometrical data according to the HTRI data sheet is presented in Appendix B. Additional information regarding the tube layout may be found in Appendix D. 5.1 Geometry and Mesh Structure Three baffle cuts and two baffle orientations are considered for the inlet zone of the shell and tube heat exchanger with 660 tubes. The baffle cuts are 20%, 24% and 30% of the shell inside diameter, and the baffle orientations are horizontal and vertical. The heat exchanger has equal baffle spacing. No leakage flows are taken into account. Figure 5.1 represents the inlet zone of the shell and tube heat exchanger with horizontal baffle orientation. Figure 5.2 shows schematically the tube partition width for the heat exchanger and the configuration of tube pitch. Table 5.1 provides the geometrical layout of the shell and tube heat exchanger in more detail. For shell and tube heat exchangers with single- segmental baffles, the most frequently used baffle cut is 25% [Kara and Güraras, 2004]. Since a baffle cut of 25% introduces mesh elements with skewness more than 0.7 around the baffle tips, a baffle cut of 24% is taken into account instead of 25%. In addition to the geometry for the inlet zone with horizontal baffle orientation, a corresponding geometry is also prepared for the shell and tube heat exchanger with vertical baffle orientation. The geometry with horizontal baffle orientation consists of 1,198,478 three dimensional 5.2 CFD Model for the Study of Inlet Zone Effects 32 control elements, while the vertical baffle orientation includes 1,187,508 elements. The mesh structure is generated by use of the cooper mesh scheme (see subsection 4.1.2) and includes only quadrilateral-faced hexahedral elements. Figure 5.3 represents the source faces near the baffle and baffle window. The combination of source faces looks like a honeycomb, as can be seen in Figure 5.3. Figure 5.4 shows the surface meshes of the shell wall and the baffle window. The cooper mesh scheme aligns the mesh elements with the tube length and consequently reduces the numerical errors. Item Symbol Size Tube number nt 660 Tube outside diameter do 15.875 mm (5/8 in) Nozzle inside diameter Dn 154.178 mm (6.07 in) Nozzle minimum length Ln 192.786 m (7.59 in) Tube partition width -- 9.525 mm (0.375 in) Baffle spacing (for the inlet, outlet and intermediate regions) Lbc 262.407 mm (10.331 in) Baffle thickness -- 6.350 mm (¼ in) Shell inside diameter Ds 590.931 mm (23.265 in) Baffle cut height Lbch 118.19, 139.78 and 177.28 mm Baffle cut percentage BC 20%, 24% and 30% Tube pitch ltp 20.638 mm (13/16 in) Reduction of mesh number under 1,200,000 elements is achieved by inserting 8 elements between tubes and by adjusting the two dimensional and the three dimensional aspect ratios Figure 5.2: Configuration of the tube partition width for the shell and tube heat exchanger with (a) vertical baffle orientation and (b) horizontal baffle orientation, and also the tube pitch. Table 5.1: Geometrical measurements of the inlet zone of the shell and tube heat exchanger with 660 tubes. (a) Vertical baffle orientation Tube partition width baffle window (b) Horizontal baffle orientation Tube partition width baffle window flow from inlet nozzle 60° tube pitch (c) Configuration of tube pitch 33 about 10 and 15, respectively. Figure 5.5 shows the meshed geometries with horizontal and vertical baffle orientations. In both geometries, approximately 98% of the total mesh elements have EquiAngle skewness less than 0.4. Figure 5.6 shows the distribution of EquiAngle skewness for these two geometries.   vertical baffle orientation Baffle windows horizontal baffle orientation Figure 5.5: Meshed inlet zone of shell and tube heat exchanger with 660 tubes for both horizontal and vertical baffle orientation. The shell side fluid is meshed by use of the cooper mesh scheme. The total number of quadrilateral-faced hexahedral elements is 1,198,478 and 1,187,508 for horizontal and vertical baffle orientation, respectively. Figure 5.3: Meshed source faces around the tubes. The red region shows the baffle window. Figure 5.4: Surface meshes in baffle window and shell wall. 34 5.2.1 Boundary Conditions The conjugate heat transfer boundary condition for tube walls is more realistic than the constant temperature boundary condition. However, the conjugate boundary condition requires excessive mesh elements on the tube side. Therefore, a constant temperature boundary condition is considered for the tube walls. All other solid walls, i.e. the baffle, the tube sheet, the nozzle and the shell wall, are defined as adiabatic walls. The boundary condition for the inlet, i.e. the inlet nozzle, is the velocity inlet boundary condition since the velocity vectors and the temperature at inlet are known. Moreover, the shell-side flow is incompressible and the first solid obstruction, i.e. the first tube row, is not too close to the inlet boundary. This makes the velocity inlet boundary condition for the inlet more applicable [Fluent, 2008]. Appropriate boundary conditions for the outlet, i.e. the baffle window, are the outlet pressure boundary condition and the outflow boundary condition [Fluent, 2008]. The outlet pressure boundary condition requires the specification of a static gauge pressure at the outlet boundary. This boundary condition is also used if the flow reverses direction at the outlet. The reverse flow or backflow at outlet will cause difficulties in the numerical solution. The outlet pressure boundary condition will minimize these numerical difficulties. The simulation results obtained by applying the outlet pressure boundary condition have shown that the backflow behaviour at outlet is negligible (less than 3% of the total flow reverses direction at outlet). On the other hand, the static gauge pressure at the outlet boundary is not faithfully known. Therefore, the outlet boundary condition is not defined for the outlet. The outflow boundary condition is used to model flow exits where the details of the flow velocity and pressure are not known prior to solution of the flow problem. From the numerical point of view, all required information for outflow boundary condition will be extrapolated from the interior. The outflow boundary conditions consist of zero normal derivatives at the boundary for all quantities. The zero-derivative condition is intended to represent a smooth continuation of the flow through the boundary. Hence, the outflow boundary conditions can Figure 5.6: Distribution of EquiAngle skewness of hexahedral mesh structure generated for the inlet zone. 96.89% 2.18% 0.92% 0.01% horizontal baffle orientation 97.99% 0.25% 1.75% 0.01% vertical baffle orientation 0.0 ≤ QEAS ≤ 0.4 0.4 < QEAS ≤ 0.6 0.6 < QEAS ≤ 0.8 0.8 < QEAS < 1.0 35 be considered as a Dirichlet boundary condition and are derived following an approach analogous to the Dirichlet-to-Neumann method [Ol’shanskii, 2000]. Importantly, the outflow boundaries cannot be used if the flow is compressible or if the modelling encounters unsteady flows with varying density. However, when the Mach number is less than 0.1, compressibility effects are negligible and the variation of fluid density with pressure can safely be ignored in flow modelling [Jin and Barza, 1993]. In the present work, the fluid velocity is less than 33 m/s. Moreover, the flow velocity and pressure at the outlet are not known, and the outlet backflow is negligible. Hence, the outflow boundary condition is the appropriate boundary condition for the outlet. The boundary conditions used for the present stage of study are summarized in Table 5.2. Boundary Boundary condition Tube outside walls Constant temperature equal to 370K Tube sheet wall Adiabatic Baffle wall Adiabatic Nozzle wall Adiabatic Shell wall Adiabatic Inlet at inlet nozzle Velocity inlet: defined velocity, all velocity vectors are normal to the boundary. The inlet temperature is 30 K more than the tube wall temperature, i.e. 400 K. Outlet at baffle window Outflow: zero normal derivatives for all quantities, Mach number < 0.1 5.2.2 Thermophysical Properties of Working Fluids Three fluids with constant physical properties are considered as shell-side fluids: air, liquid water and engine oil. The physical properties are obtained at 8 bar and 385 K [Touloukian, 1972; Rohsenow, 1998; Incropera, 2006; VDI, 2006]. Operating pressure 8 bar is an arbitrary high pressure that will facilitate minimization of the difficulties in numerical solution [Fluent, 2008]. As it is shown in Table 5.2, the tube wall temperature and the inlet temperature are equal to 370 K and 400 K, respectively. The temperature of the tube walls and the inlet temperature are selected based on typical operating conditions of shell and tube heat exchangers in oil refineries [Gary and Handwerk, 2001]. Therefore, the average bulk temperature, which is the operating temperature, is equal to 385 K. Table 5.3 shows the physical properties of the shell-side fluids. Physical property Symbol Unit Shell side fluid Variation Gaseous air Liquid water Liquid engine oil Density ρ kg/m³ 7.25 998.20 828.96 Constant Dynamic viscosity μ kg/(m.s) 2.250×10-5 1.003×10-3 1.028×10-2 Constant Thermal Conductivity kf W/(m.K) 0.032 0.600 0.135 Constant Heat capacity surface cpcp J/(kg.K) 1018.63 4182.00 2307.00 Constant Prandtl number Pr -- 0.716 6.991 175.674 Constant Table 5.2: Boundary conditions for the inlet zone of the heat exchanger with 660 tubes and without leakages. Table 5.3: Physical properties of three shell-side fluids. All physical properties are assumed to be constant and are obtained at 8 bar 385 K. 36 5.2.3 Settings The overall numerical setup for the present investigation is summarized in Table 5.4 [Torrance, 1986; Shyy, 1994; Anderson, 1995; Wesseling, 2001; Ferziger, 2002; Fluent, 2008]. Model/Treatment Algorithm/Theorem Formulation/Method/Discretization Turbulence model k-ε Model RNG with following constants: Cμ=0.0845, Cε1=1.42, Cε2=1.68, σε =0.85 Near wall treatment Wall function Standard wall function base on Launder and Spalding Flow solver Pressure-base Segregated / Implicit Velocity coupling method Pressure-base SIMPLE Other sources of heat Radiation Viscous heating Neglected For engine oil with inlet velocity more than 20 m/s (Br ≥ 1) Gradients and Derivatives Green-Gauss Green-Gauss cell-based gradient evaluation Discretization Taylor First-order 5.2.4 Mesh Validations 5.2.4.1 Mesh Dependency Ignorance of mesh dependency can sometimes be an embarrassment in numerical calculations. The numerical results are not trustworthy when the numerical simulation depends on the mesh size. Mesh structures need to be developed to eradicate the mesh dependency. The inlet zone of a shell and tube heat exchanger with 140 tubes, presented in Figure 4.1 (a), is meshed with different mesh sizes. Herein, the mesh size is the total number of quadrilateral-faced hexahedral elements. The main parameters that define the mesh size are: the number of elements between two tubes, the number of elements on the tube perimeter, and the number of elements on the tube length. The mesh structure characterised by 8×12×30 indicates that 8 elements are between two tubes, 12 elements are on the tube perimeter, and 30 elements are on the tube length. The mesh 8×12×30 is similar to the mesh presented in Figure 5.3 and Figure 5.4. The total number of mesh elements, i.e. the mesh size, for the inlet zone of the heat exchanger with 140 tubes is 250,000 when the mesh is 8×12×30. A refined and a coarse mesh are also generated for the inlet zone of the heat exchanger with 140 tubes. The refined mesh is characterized by 10×24×45 with the mesh size of around 1,000,000. The coarse mesh is a 6×12×25 mesh and includes about 160,000 elements. All three meshes (8×12×30, 10×24×45 and 6×12×25) are used to perform calculations applying the CFD setup presented in Table 5.4 and the boundary conditions shown in Table 5.2. The shell-side fluid was liquid water with the physical properties as presented in Table 5.3. The qualitative results of these three simulations were very similar. Moreover, the deviation of the quantitative results of these simulations, i.e. the outlet temperatures and pressures, were less than 0.10%. This means that the numerical simulation is mesh- independent. The mesh structure generated for the inlet zone of the shell and tube heat exchanger with 660 tubes (described in section 5.2) is 8×12×30 mesh. A refined and a coarse mesh are also generated for this mesh. The refined mesh is 8×12×33 with about 1,300,000 elements. This is the maximum possible level of refinement because refining the mesh structure with 10×24×45 Table 5.4: CFD setup 37 mesh will produce around 4,500,000 elements. The coarse mesh is 6×12×25 mesh and included about 750,000. The numerical results obtained from the simulations of the refined mesh, i.e. 8×12×33 mesh, and the coarse mesh, i.e. 6×12×25 mesh, are compared with the numerical results obtained from the simulation of the original mesh, i.e. 8×12×30 mesh. The relative absolute difference between the outlet temperature of the original mesh and the refined mesh is less than 0.002%, while the same difference for the coarse mesh is less than 0.117%. For the outlet pressure, the maximum relative absolute difference between the original mesh and the refined mesh is 0.054%. The same difference for the coarse mesh amounts to 0.129%. The comparison of the outlet temperature and pressure obtained from the simulations of the refined, coarse and original meshes at seven different Reynolds numbers are presented in Figure 5.7. The identical numerical results obtained from the simulations of the refined, coarse and original meshes show that the mesh structure generated for the inlet zone (described in section 5.2) guarantees the mesh independency of the simulation. 5.2.4.2 Reliability of Mesh Structure for Wall Function Treatment As it is described in subsection 4.2.5 and shown in Table 5.1, a semi-empirical function based on the proposal of Launder and Spalding, bridges the viscosity-affected region between the wall and the fully turbulent region [Launder and Spalding, 1974]. This semi-empirical function comprises laws for mean velocity and temperature, which yields: Maximum temperature deviation% (8×12×33 mesh) 0.0010 0.0012 0.0014 0.0016 0.0018 0.0020 M ax im um p re ss ur e de vi at io n% (8 ×1 2× 33 m es h) 0.002 0.013 0.024 0.035 0.046 0.057 Maximum temperature deviation% (6×12×25 mesh) 0.086 0.092 0.099 0.105 0.112 0.118 M ax im um p re ss ur e de vi at io n% (6 ×1 2× 25 m es h) 0.110 0.114 0.118 0.122 0.126 0.130 Reinlet×10-5:1.5|1.8|2.1|2.5|2.8|3.1|3.4 8×12×33: 6×12×25: Figure 5.7: Comparison of the outlet temperature and pressure obtained from the simulations of the refined, coarse and original meshes at seven different Reynolds numbers. The refined mesh is characterised by 8×12×33 and the coarse mesh is described by 6×12×25. 38 with In Equation (5.1) and (5.2),  is the von Kârmân constant equal to 0.4187, E is the empirical constant in the law of the wall (equal to 9.81 for RNG k- model), y* is the dimensionless sublayer-scaled wall distance, k is the turbulent kinetic energy, τW is the surface or wall shear stress, Uഥ is the mean velocity of the fluid at the distance y from the wall, and Cμ is the constant coefficient in the k- eddy viscosity formulation. The distance from the wall at the wall-adjacent cells is usually measured in the wall unit y* or y+, where y+ is another type of the dimensionless sublayer-scaled wall distance. The logarithmic law for mean velocity is known to be valid for y*>30 ~ 60 [Hinze, 1959; Bradshaw, 1971; Wilcox, 1998; Pope, 2000; Ferziger, 2002]. For the RNG k-ε model, logarithmic law is employed when y* is greater than 11.225 [Kutateladze, 1964; Launder and Spalding, 1974; Ferziger, 2002]. Moreover, the comparable values of y* and y+ show that the first cell is placed in the fully turbulent region [Ferziger, 2002]. Figure 5.8 shows the values of y+ and y* as a function of the inlet Reynolds number obtained from 25 numerical simulations. In these simulations, which will be discussed in section 5.8, the shell-side fluid is liquid water, the baffle orientation is horizontal, and the baffle cut is 24%. It can be seen from Figure 5.8 that the values of y+ are similar to the values of y*. The turbulent flow occurs at Reinlet ≈ 105. At Reinlet > 105 the value of y+ exceeds 11.225. y+ versus the inlet Reynolds number in the domain of turbulent flow for three baffle cuts and three shell-side fluids are presented in Figure 5.9. Although the shell-side fluids are air, liquid water and engine oil, the values of y+ are independent of the shell-side viscosity. Since the values of y+ are more than 14 for all Reynolds numbers, the mesh structure is satisfactory for implementing the standard wall function. 5.3 Performance of the Inlet Zone in the Domain of Laminar and Turbulent Flow 25 simulations are accomplished for the inlet zone of the shell and tube heat exchanger with 660 tubes. The baffle cut is equal to 24% and the shell side fluid is liquid water. The inlet fluid velocities are between 0.1 and 2.2 m/s, which cover an inlet Reynolds number range in the laminar and turbulent flow domains (4.6×103 ≤ Reinlet ≤ 3.4×105). The results of the shell-side pressure drop and the shell-side heat transfer coefficients are presented by the shell-side Kârmân number and the shell-side Nusselt number, respectively. ρUഥ τW ටCμk 24 = 1  ln൫Ey *൯ (5.1) y*= ρy μ ටCμk 24 (5.2) y+= yඥρτW μ (5.3) Nk= ρdHsp3 ൫Δpshell Lbc⁄ ൯ μ2 (5.4) Nu= hshelldHsh kf (5.5) 39 Reinlet 103 104 105 106 y+ an d y* 0 5 10 15 20 25 30 y+ y* y+=11.225 Reinlet×10 -5 1.5 2.0 2.5 3.0 3.5 y+ 14 18 22 26 30 34 Bc=20% Bc=24% Bc=30% Figure 5.8: Values of y+ and y* versus the inlet Reynolds number for horizontally orientated baffles (baffle cut 24%). Figure 5.9: Values of y+ versus the inlet Reynolds number for horizontal baffle orientation and for baffle cuts 20%, 24% and 30%. 40 Reinlet×10 -5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 N k× 10 -8 0 1 2 3 4 N u 0 35 70 105 140 Nkhorizontal Nkvertical Nuhorizontal Nuvertical In Equations (5.4) and (5.5), dHsp is the hydraulic diameter for shell-side pressure drop and dHsh is the hydraulic diameter for shell-side heat transfer. Equations (5.4) and (5.5) yield dHsh=1.099do and dHsp=1.041do for the present investigation. The shell-side Kârmân number, Nk, and the shell-side Nusselt number, Nu, are presented in Figure 5.10 as a function of the Reynolds number of the inlet. As it is shown in Figure 5.10, the shell-side pressure drop and the shell-side heat transfer coefficient for horizontal baffle orientation are greater than for vertical baffle orientation. This can be explained by considering the effect of baffle orientation on the residence time and the mixing level of the shell-side fluid. As it is explained in section 3.3, at baffle cut 24% the value of NMSD for horizontal baffle orientation is approximately 84% greater than the value of NMSD for vertical baffle orientation (see Equation (3.8) and Figure 3.10). Therefore, the residence time and mixing level of the shell-side fluid for horizontal baffle orientation are greater than for vertical baffle orientation. Consequently, the horizontal baffle orientation will result greater values of pressure drop and heat transfer coefficient than the vertical baffle orientation. dHsp= Ds2-ntdo 2 Ds+ntdo (5.6) dHsh= Ds2-ntdo 2 ntdo (5.7) Figure 5.10: The effect of baffle orientation on the shell-side pressure drop and heat transfer coefficient for the inlet zone of the shell and tube heat exchanger with 660 tubes. The baffle cut is 24% and the shell-side fluid is water. 41 The static pressure distribution in the inlet zone for horizontal and vertical baffle orientation is presented in Figure 5.11. In order to have a better view of pressure gradients, the static pressure distribution on the tube walls is also presented in Figure 5.12. In Figures 5.11 and 5.12, the heat transfer processes is heating. p (bar) 7.91 7.83 7.76 7.68 7.61 7.53 7.46 7.38 7.31 Figure 5.11: Pressure field in the inlet zone for horizontally (left) and vertically (right) orientated baffles (water, baffle cut 0.24%, Reinlet=3.4×105). The heat transfer processes is heating. p (bar) 7.91 7.83 7.76 7.68 7.61 7.53 7.46 7.38 7.31 Figure 5.12: Pressure field on the tube walls of the inlet zone for horizontally (left) and vertically (right) orientated baffles (water, baffle cut 0.24%, Reinlet=3.4×105). The heat transfer processes is heating. Horizontal baffle orientation Vertical baffle orientation Horizontal baffle orientation Vertical baffle orientation 42 T (°C) 126 125 123 121 120 118 116 114 113 Horizontal baffle orientation Vertical baffle orientation Figure 5.13: Temperature field in the inlet zone for horizontally (left) and vertically (right) orientated baffles (water, baffle cut 0.24%, Reinlet=3.4×105). The heat transfer processes is heating. T (°C) 126 125 123 121 120 118 116 114 113 Horizontal baffle orientation Vertical baffle orientation Figure 5.14: Temperature field on the tube walls of the inlet zone for horizontally (left) and vertically (right) orientated baffles (water, baffle cut 0.24%, Reinlet=3.4×105). The heat transfer processes is heating. The horizontal baffle orientation and vertical baffle orientation have closely comparable values of pressure drop near the inlet nozzle, as it is shown in Figures 5.11 and 5.12. However, the overall pressure drop, especially near the baffle window, for horizontal baffle orientation is significantly greater than the overall pressure drop for vertical baffle orientation. The static temperature distribution in the inlet zone for horizontal and vertical baffle orientation is presented in Figure 5.13. The static temperature distribution on the tube walls is also shown in Figure 5.14. In Figures 5.13 and 5.14, the heat transfer processes is heating. 43 From Figures 5.13 and 5.14 it can be seen that the heat transfer is greater for the arrangement with horizontally orientated baffle than for the arrangement with vertical baffle orientation. Moreover, the baffle window for horizontal baffle orientation is significantly more effective in heat transfer than the baffle window for vertical baffle orientation. In the heat exchanger with horizontal baffle orientation, about 89% of the baffle window area is up to 8 °C colder than the inlet temperaure. However, only 33% of the baffle window area is 6 °C colder than the inlet temperature for the heat exchanger with vertical baffle orientation. In order to have a better understanding of the effect of baffle orientation on the shell-side heat transfer and pressure drop, the profiles of temperature, pressure and velocity are determined at the middle of the inlet zone. The middle of the inlet zone is presented by a plane shown in Figure 5.15. The temperature, pressure and velocity profiles of the heat exchangers with horizontal and vertical baffle orientation are obtained on this plane. Figure 5.16 shows the temperature profile at the middle of the inlet zone of the shell and tube heat exchanger with horizontal and vertical baffle orientaion. For the shell and tube heat exchanger with horizontal baffle orientation, the baffle window is located isobilateral with respect to the inlet nozzle. Therefore, the temperature profile at the middle of the inlet zone with horizontal baffle orientation is symmetrical. However, the asymmetrical profile of temperature in the shell and tube heat exchanger with vertical baffle orientation is due to the lopsided positioning of the baffle window with respect to the inlet nozzle. The effect of the baffle orientation on the pressure drop, presented in Figure 5.17, is very perspicuous. For the inlet zone with horizontal baffle orientation, the highest pressure drop is obsereved near the shell wall due to the the bypass flow. However, the highest pressure drop is obsereved in the baffle window for the inlet zone with vertical baffle orientation. The velocity magnitude profile for the inlet zone with horizontal and vertical baffle orientation is shown in Figure 5.18. The velocity magnitude |u| is an scalar value and is equal to ට|uሬԦx|2+หuሬԦyห2+|uሬԦz|2 where uሬԦx, uሬԦy and uሬԦz are the x, y and z components of velocity vector in Cartesian coordinates. Figure 5.15: The plane which presents the middle of the inlet zone of the shell and tube heat exchanger with horizontal baffle orientation (left) and vertical baffle orientation (right).   vertical baffle orientation horizontal baffle orientation central line central line 44 Figure 5.16: Temperature profile at the centre of the inlet zone for both horizontally (top) and vertically (bottom) orientated baffles of a shell and tube heat exchanger with 660 tubes (Reinlet=3.4×105). 122 °C 123 °C 124 °C 125 °C 126 °C Horizontal baffle orientation 0.2 0.1 0.0 -0.2 -0.1 0.05 0.10 0.15 0.20 0.25 126 125 124 123 122 122 °C 123 °C 124 °C 125 °C 126 °C 127 °C Vertical baffle orientation 0.2 0.1 0.0 -0.2 -0.1 0.05 0.10 0.15 0.20 0.25 126 125 124 123 121 122 45 Figure 5.17: Pressure profile at the centre of the inlet zone for both horizontally (top) and vertically (bottom) orientated baffles of a shell and tube heat exchanger with 660 tubes (Reinlet=3.4×105). 0.2 0.1 0.0 -0.2 -0.1 0.05 0.10 0.15 0.20 0.25 -15140 -15160 -15180 -15200 -15240 -15220 -15120 Horizontal baffle orientation -15260 pa -15240 pa -15220 pa -15200 pa -15180 pa -15160 pa -15140 pa -15120 pa -15100 pa -15080 pa -15060 pa -15100 0.2 0.1 0.0 -0.2 -0.1 0.05 0.10 0.15 0.20 0.25 -14000 -15000 -16000 -17000 -13000 Vertical baffle orientation -17900 pa -16840 pa -15780 pa -14720 pa -13660 pa -12600 pa -11540 pa -10480 pa -9420 pa -8360 pa -7300 pa -12000 46 Figure 5.18: Velocity magnitude profile at the centre of the inlet zone for both horizontally (top) and vertically (bottom) orientated baffles of a shell and tube heat exchanger with 660 tubes (Reinlet=3.4×105). -0.2 -0.1 0.0 0.2 0.1 0.25 0.20 0.15 0.10 0.05 1.0 0.8 0.6 0.4 1.2 Horizontal baffle orientation 1.4 0.2 m/s 0.4 m/s 0.6 m/s 0.8 m/s 1.0 m/s 1.2 m/s 1.4 m/s 1.6 m/s 1.8 m/s 1.6 1.8 0.2 -0.2 -0.1 0.0 0.2 0.1 0.25 0.20 0.15 0.10 0.05 1.0 0.8 0.6 0.4 1.2 Vertical baffle orientation 1.4 0.2 m/s 0.4 m/s 0.6 m/s 0.8 m/s 1.0 m/s 1.2 m/s 1.4 m/s 1.6 m/s 1.6 0.2 47 For the inlet zone with horizontal baffle orientation, the highest velocity magnitude is observed in the bypass region. However, the highest velocity magnitude is observed in the baffle window for the inlet zone with vertical baffle orientation. In order to have a more comprehensible analysis of the velocity profile, the profiles of uሬԦx, uሬԦy and uሬԦz are presented in Figures 5.19, 5.20 and 5.21, respectively. The x, y and z axes of the Cartesian coordinate system are presented in Figure 5.15 and defined in subsection 3.2. At the middle of the inlet zone with horizontal baffle orientation, the x-velocity profile, i.e. uሬԦx profile, is asymmetrical, however, the profile of the magnitude of the x-velocity, i.e. |uሬԦ୶| profile, is symmetrical. Mathematically, the x-velocity profile at the middle of the inlet zone with horizontal baffle orientation can be expressed as uሬԦxሺx, zሻ ~ - uሬԦx൫2xcl-x, z൯, where xcl defines the location of the central line presented in Figure 5.15 on the x-axis. This can be explained only by considering the existence of vortices. Therefore, the x-velocity profile shows intensive vortices in the inlet zone of the shell and tube heat exchanger with horizontal baffle orientation. In contrast to this, the x-velocity profile at the middle of the inlet zone with vertical baffle orientation shows redirection of the flow to the baffle window without effective vortices. The y-velocity profile at the middle of the inlet zone is presented in Figure 5.20. The profile of uሬԦy for horizontal baffle orientation shows an intensive bypass flow and an effective flow in the tube bank region toward the baffle window. The profile of uሬԦy for vertical baffle orientation indicates that the y-velocity flow in the tube bank region is not as effective as the y-velocity flow for horizontal baffle orientation. Therefore, the tubes located far from the inlet nozzle for horizontal baffle orientation are more effective in transferring heat and generating pressure drop than for vertical baffle orientation. The z-velocity profile at the middle of the inlet zone is presented in Figure 5.21. The magnitude of uሬԦz is negligible compared to the magnitude of uሬԦx and uሬԦy for horizontal baffle orientation. However, the profile of uሬԦz represents a high level of mixing in the inlet zone when using the horizontal baffle orientation. The profile of uሬԦz at the middle of the inlet zone with vertical baffle orientation confirms the redirection of the flow to the baffle window without effective vortices. The profiles presented in Figures 5.16 to 5.20 explain the behaviour shown in Figure 5.10, that is the heat transfer rate and the pressure drop of the inlet zone with horizontal baffle orientation is greater than the heat transfer rate and the pressure drop of the inlet zone with vertical baffle orientation. Since the heat transfer coefficient relates to the energy recovered by the heat exchanger and the pressure drop refers to the work which is necessary to maintain the shell-side fluid flow, a shell-side gain factor suitable for the assessment of shell and tube heat exchangers may be introduced as ratio of the shell-side heat transfer coefficient to the shell-side pressure drop: To facilitate the judgment between the horizontal and vertical baffle orientation, a performance factor is defined as: Γshell= NushellNkshell ן hshell Δpshell (5.8) Φ= ሺΓshellሻhor.ሺΓshellሻver. (5.9) 48 Figure 5.19: x-velocity profile at the centre of the inlet zone for both horizontally (top) and vertically (bottom) orientated baffles of a shell and tube heat exchanger with 660 tubes (Reinlet=3.4×105). Horizontal baffle orientation -0.10 m/s -0.05 m/s 0.00 m/s 0.05 m/s 0.10 m/s -0.2 -0.1 0.0 0.2 0.1 0.20 0.15 0.10 0.05 0.25 -0.10 0.00 -0.05 0.05 0.10 Vertical baffle orientation 0.0 m/s 0.2 m/s 0.4 m/s 0.6 m/s 0.8 m/s 1.0 m/s 1.2 m/s -0.2 -0.1 0.0 0.2 0.1 0.20 0.15 0.10 0.05 0.25 0.8 0.6 1.0 1.2 0.2 0.0 0.4 49 Horizontal baffle orientation -1.8 m/s -1.6 m/s -1.4 m/s -1.2 m/s -1.0 m/s -0.2 -0.1 0.0 0.2 0.1 0.20 0.15 0.10 0.05 0.25 -1.4 -0.6 -0.8 -0.4 -0.2 -1.0 -1.2 -1.6 -1.8 -0.8 m/s -0.6 m/s -0.4 m/s -0.2 m/s Vertical baffle orientation -1.4 m/s -1.2 m/s -1.0 m/s -0.8 m/s -0.6 m/s -0.2 -0.1 0.0 0.2 0.1 0.20 0.15 0.10 0.05 0.25 -1.0 -0.4 -0.2 0.0 -0.6 -0.8 -1.2 -1.4 -0.4 m/s -0.2 m/s 0.0 m/s Figure 5.20: y-velocity profile at the centre of the inlet zone for both horizontally (top) and vertically (bottom) orientated baffles of a shell and tube heat exchanger with 660 tubes (Reinlet=3.4×105). 50 Horizontal baffle orientation -0.02 m/s -0.01 m/s 0.00 m/s 0.01 m/s 0.02 m/s -0.2 -0.1 0.0 0.2 0.1 0.20 0.15 0.10 0.05 0.25 0.00 0.02 0.03 0.01 -0.01 -0.02 0.03 m/s Vertical baffle orientation 0.0 m/s 0.2 m/s 0.4 m/s 0.6 m/s 0.8 m/s -0.2 -0.1 0.0 0.2 0.1 0.20 0.15 0.10 0.05 0.25 0.4 0.8 1.0 0.6 0.2 0.0 1.0 m/s Figure 5.21: z-velocity profile at the centre of the inlet zone for both horizontally (top) and vertically (bottom) orientated baffles of a shell and tube heat exchanger with 660 tubes (Reinlet=3.4×105). 51 Φ=1.00 Φ=0.98 Figure 5.22: Inlet zone performance factor Φ at baffle cut 24% for liquid water as a function of Reynolds number. The heat transfer processes is heating. Reinlet×10 -5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5  0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 Applying the Equation (5.8) in Equation (5.9), the performance factor is simply as: A performance factor  greater than one indicates that a heat exchanger with horizontally orientated baffles is more desirable than one with vertical baffle orientation. Rationally, no advantage exists between different baffle orientations when the performance factor is equal or near to one. Figure 5.22 shows the performance factor Φ at baffle cut 24% for liquid water. As it is shown in Figure 5.22, at Reynolds number 4.6×103 the value of Φ is about 1.22 and then at Reynolds number 7700 it reaches a maximum value about 1.24. Next to this local maximum, the performance Φ decreases as the Reynolds number increases. The minimum value for Φ is approximately 0.98 at Reynolds number 92000. In this range of Reynolds number (4.6×103 ≤ Reinlet ≤ 9.2×104), the value of y+ is near 10. Next to this Reynolds number range, the value of Φ increases with increasing the Reynolds number. At Reinlet≈1.7×105 the value of Φ is equal to 1. In this range, y+ is between 11.225 and 16. Next to it, the performance Φ ascends continuously with increasing the Reynolds number. At Reinlet≈3.4×105 the value of Φ reaches 1.1. It seems that the trend of Φ at high Reynolds number approaches to a constant value around 1.1. Figure 5.22 shows the benefit of using horizontal baffle orientation compared to the vertical baffle orientation. Even though the results confirm the predicted behaviour discussed in Φ= ൫hshell Δpshell ⁄ ൯ hor. ൫hshell Δpshell⁄ ൯ver. = ሺhhor. hver.⁄ ሻshell ൫Δphor. Δpver.⁄ ൯shell = ሺNushell Nkshell⁄ ሻhor. ሺNushell Nkshell⁄ ሻver. = ሺNuhor. Nuver.⁄ ሻshell ሺNkhor. Nkver.⁄ ሻshell  (5.10) 52 subsection 3.3, the CFD resuls have to be compared with experimental data in order to ensure the validation of the numerical results. 5.4 Validation and Sensibility Analysis The numerical results are based on a set of convergence criteria. The nature of discretization makes it impossible to have an exact conformity between the numerical results and the hypothetical exact analytical solution. Therefore, an error analysis of the numerical results is required. Moreover, it is necessary to compare the numerical results with experimental data. On the other hand, any design parameter is obtained from the measurement of fundamental quantities. For example, the heat transfer coefficient is obtained from the heat capacity rate, the heat transfer area and the inlet and outlet temperatures. The evaluation of the heat transfer coefficient depends on the measurement of the outlet temperature if the capacity rate, the heat transfer area and the inlet temperature are considered as known variables. Any deviation on the basic quantities will cause deviations on the pertinent parameters. Therefore a sensibility analysis is performed which is suitable to explain the exactness of the results achieved. 5.4.1 Validation with Experimental Data for Ideal Tube Banks A CFD model is implemented for different ideal tube banks. The mesh structure is based on the mesh scheme explained in section 5.2 and the numerical setup is according to Table 5.4. The ideal tube banks are based on the studies presented by Kays and London [1954]. The numerical results obtained from the simulation of these different ideal tube banks are compared with the experimental data published by Kays and London [1954]. These experimental data are also used by Martin [2002]. The comparison between the experimental data of pressure drop and the pressure drops obtained from the CFD simulations is shown in Figure 5.23. The pressure drop is presented as modified Fanning friction factor f. The CFD simulation can predict the pressure drop of 52 experimental data with a relative absolute error less than 10%, as it is shown in Figure 5.23. Additionally the experimental data of the heat transfer coefficient, presented as Colburn j- factor for heat transfer jH, are compared with the results obtained from CFD simulations (see Figure 5.24). An absolute relative error less than 10% for heat transfer coefficient confirms the validation of the simulation. Even though the flow on the shell-side of an ideal shell and tube heat exchanger is not simillar to the flow on an ideal tube bank, the presented validation proves the reliability and trustworthiness of the CFD setup and simulation procedure. 5.4.2 Error Analysis The convergence of the numerical calculation is presented as the residuals of continuity, velocities, k, ε and energy. A typical convergence progress for the present investigation is depicted in Figure 5.25. The horizontal branch of the convergence progress shows that the iterative error is very low and in fact very near to zero. The maximum order of the iterative error is 10-5%. Hence, the numerical error base on the convergence criteria is adequately satisfactory. The other two important parameters for error analysis are the unbalanced values of mass and energy. The unbalanced value of mass is always equal to zero, since the outflow boundary condition is in some manner a mirror of the velocity inlet boundary condition due to the mass rate. The maximum unbalanced value for energy conservation, however, is about 0.2%. Figure 5.26 presents the unbalanced value of energy as a function of the inlet Reynolds numbers. 53 Figure 5.23: Comparison between the experimental data of pressure drop and the pressure drops obtained from the CFD simulations. 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 628 1 (BC > 50%) Figure 7.7: Schematic presentation of the effect of Lbch/Lbc and baffle cut BC on the shell-side flow pattern. Baffle Shell B B E E 90 As it is denoted in Figure 7.7, the main flow stream B crosses the tube bank and generates intensive eddies around the baffle walls for BC < 30%. The intensive eddies around the baffle walls stimulate the turbulent momentum diffusion and cause rapid mixing in the region near the baffle walls. Hence, the rates of momentum, heat, and mass transfer will increase in the baffle region [Tennekes, 1972]. The main stream B will produce blowing or suction for tube-baffle leakages, which consequently will develop a jet-kind stream flow. The flow of stream B near the baffles can be compared with a pure shear flow over a porous wall. Thus, for BC < 30%, the effect of tube-baffle leakages on the performance factor is indisputable due to the substantial rates of momentum and heat transfer around the baffle walls and the noticeable pressure drop in the tube-baffle leakages. Figure 7.8 and 7.9 show the path lines coloured by velocity magnitude for the shell and tube heat exchanger with 76 tubes. The baffle orientation is horizontal and the baffle cut equals 20% of the shell inside diameter. These figures are comparable with the conceptual flow for baffle cut less than 30% of the shell inside diameter (see Figure 7.7). The main stream B crossing the tube bundle as well as the tube bundle bypass stream C is presented in Figure 7.8. The flow in tube-baffle leakages as well as the bypass stream C is shown in Figure 7.9. In Figures 7.8 and 7.9, the heat transfer process is heating, the shell-side fluid is water and the inlet Reynolds number is about 105. The bypass streams C and F cause a non-uniform flow distribution over the heat transfer surface. In fact, only a portion of the total stream will flow through the active tube surface including the tube-baffle leakages and another portion of the total stream will flow through inactive flow regions, i.e. bypass regions and baffle-shell leakages. Hence, the velocity and the heat transfer coefficient will decrease. This is the negative effect of bypass streams C and F. Another adverse effect of bypasses is when the bypass stream reaches the outlet without a significant change in temperature or, in the worst plausible case, with the same temperature as at the inlet. Considering the negligible change in physical properties with temperature and pressure, the effect of bypasses on the outlet temperature can then be simply represented as: In Equation (7.1), the subscript “bp” refers to the bypass streams and the superscript “ms” refers to a portion of the stream flow without bypasses, i.e. main stream flow. Taking into the account the cooling and heating processes yields: The bypasses may reduce the effective shell-side heat transfer coefficient, depending on extent of the bypass fraction and the inlet temperature, as can be concluded from Equations (7.1) and (7.2). Therefore, the bypasses, especially between tube bundle and shell wall, have to be minimized. However another important cause of degradation of the shell-side heat transfer coefficient and consequently the shell-side gain factor Γshell is the existence of stationary eddies. These eddies energize the main stream and are not swept away. Such zones appear where the normal fluid escape routes through the annular orifices between baffles and tubes are blocked. Under such circumstances, not only the heat transfer coefficient and effective temperature difference suffer, but the pressure drop as well. Nevertheless, the reduction on the gain factor can be drastic due to the tube-baffle leakages. Toutlet=Toutletms + Mሶ bp Mሶ ሺTinlet െ Toutlet ms ሻ (7.1) ቐ for cooling : Tinlet>Toutletms ֜Toutlet>Toutletms for heating : Tinlet 0 (C.12) ƒ൫Reinlet, Pr൯หReinlet=0=ƒ൫0, Pr൯=0 (C.13) A7 If the order of magnitude is presented by Oന, it results: The correlations reported in the Delaware method and in the VDI handbook fulfill condition (C.14). According to the VDI method, the value of Nu0 is: Based on the specific Reynolds number Reφ, ƒ൫Reinlet, Pr൯ is defined as: Therefore, the order of magnitude of ƒ൫Reinlet, Pr൯with respect to the Reynolds number according to the VDI method is between 0.5 and 0.8. An analysis based on the Delaware method shows that the order of magnitude of ƒ൫Reinlet, Pr൯ with respect to the Reynolds number lies between 0.333 and 0.612. Therefore, it is sense full to consider the utmost order of magnitude of ƒ൫Reinlet, Pr൯ with respect to the Reynolds number equal to 0.8. Knowing the properties of the function that describes the shell-side Nusselt number, it is straightforward to evaluate the sensibility of the Nusselt number (and also the pressure drop) with the outlet temperature of the shell-side fluid. Differentiating Equation (C.6) yields: On the other hand: Applying Equation (C.19) into Equation (C.18) and rearranging the result yields: At very high Reynolds number where Tout approaches Tin, the average values of cp and kf tend to their local values at Tout, too. Moreover, a comprehensive study for different shell-side 0 < Oന൛ƒ൫Reinlet, Pr൯, Reinletൟ < 1 (C.14) Nu0=0.3ƒWƒA (C.15) ƒ൫Reinlet, Pr൯=ƒWƒA ൞0.4409ReφPr 2 3⁄ + 0.00137Reφ1.8Pr 2 ቂ2.443 ቀPrഥ 2 3⁄ -1ቁ+Reφ0.1ቃ 2ൢ 1 2⁄   (C.16) Max Oന൛ƒ൫Reinlet, Pr൯, Reinletൟ = 0.8 (C.17) 1=ሺθ-1ሻ dNu dθo γReinletPr-Nu d൫γReinletPr൯ dθo ൫γReinletPr൯2 expቆ- NuγReinletPr ቇ  (C.18) d൫γReinletPr൯ dθoൗ γReinletPr = d൫cp kf⁄ ൯ dθoൗ cp kf⁄ = dcp dθo⁄ cp - dkf dθoൗ kf = 1 θo-1ቆ cp cp - kf kf ቇ  (C.19) dNu Nu = ቊ 1θ-1 γReinletPr Nu expቆ NuγReinletPr ቇ+ 1θo-1ቆ cp cp - kf kf ቇቋ dθo  (C.20) A8 fluids over a wide range of temperatures, pressures, as well as inlet Reynolds numbers shows that the following assumption is legitimated specially when the inlet velocity is high or the heat transfer area is small. By considering the negligibility presented in Equation (C.21), a small change in θo, will lead a change in the Nusselt number of Since θo/θo is equal to Tout/Tout, the relative error of the Nusselt number, ||Nu||, can be represented as a function of the relative error of the outlet temperature, ||T||. Equation (C.23) can be presented as a function of the inlet Stanton number Stinlet, since Nu=Nu0+ƒ൫Reinlet, Pr൯: The shell-side pressure drop is approximately proportional to the square of the Reynolds number. Considering Equation (C.17), the shell-side pressure drop is proportional to the shell-side Nusselt number as follows: Therefore, Applying Equation (C.24) into Equation (C.27), the relative error of the shell-side pressure drop, ||p||, can be represented as a function of the relative error of the outlet temperature, ||T||. 1 θ-1 γReinletPr Nu expቆ NuγReinletPr ቇب 1θo-1ቆ cp cp - kf kf ቇ (C.21) ฬ∆Nu Nu ฬ≈ θo|θ-1| γReinletPr Nu0+ƒ൫Reinlet, Pr൯ expቆNu0+ƒ൫Reinlet, Pr൯γReinletPr ቇ ฬ∆θoθo ฬ  (C.22) ԡεNuԡ=± θo|θ-1| γReinletPr Nu0+ƒ൫Reinlet, Pr൯ expቆNu0+ƒ൫Reinlet, Pr൯γReinletPr ቇԡεTԡ  (C.23) ԡεNuԡ=± θo|θ-1| γ Stinlet exp ൬Stinletγ ൰ ԡεTԡ (C.24) Oനሼ∆p, Reinletሽ ؆ 2 (C.25) Min Oനሼ∆p, Nuሽ = 2.5 (C.26) ฬd∆p∆p ฬ ≈2.5 ฬ dNu Nu ฬ (C.27) ฮε∆pฮ=±2.5 θo|θ-1| γ Stinlet exp ൬Stinletγ ൰ ԡεTԡ (C.28) A9 Appendix D: Tube Layout for Shell and Tube Heat Exchanger with 660 Tubes 5 4 3 2 1 6 9.524 mm do=15.875 mm ltp=20.638 mm Ds=590.931 mm 24.611 mm 18.967 mm Dotl=578.250 mm Horizontal baffle orientation: BC=20%, BC=24% and BC=30% Vertical baffle orientation: BC=20%, BC=24% and BC=30% 1 32 5 4 6