Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media by Ioan I. Pop, Derek B. Ingham • ISBN: 0080438784 • Pub. Date: February 2001 • Publisher: Elsevier Science & Technology Books Preface Interest in studying the phenomena of convective heat and mass transfer between an ambient fluid and a body which is immersed in it stems both from fundamental considerations, such as the development of better insights into the nature of the underlying physical processes which take place, and from practical considerations, such as the fact that these idealised configurations serve as a launching pad for modelling the analogous transfer processes in more realistic physical systems. Such idealised geometries also provide a test ground for checking the validity of theoretical analyses. Consequently, an immense research effort has been expended in exploring and understanding the convective heat and mass transfer processes between a fluid and submerged objects of various shapes. Among several geometries which have received considerable attention are flat plates, circular and elliptical cylinders and spheres, although much information is also available for some other bodies, such as corrugated surfaces or bodies of relatively complicated shapes. It is readily recognised that a wealth of information is now available on con- vective heat and mass transfer operations for viscous (Newtonian) fluids and for fluid-saturated porous media under most general boundary conditions of practi- cal interest. The number of excellent review articles, books and monographs, which summarise the state-of-the-art of convective heat and mass transfer, which are avail- able in in the literature testify to the considerable importance of this field to many practical applications in modern industries. Given the great practical importance and physical complexity of buoyancy flows, they have been very actively investigated as part of the effort to fully understand, calculate and use them in many engineering problems. No doubt, these flows have been invaluable tools for the designers in a variety of engineering situations. How- ever, it is well recognised that this has been possible only via appropriate heuristic assumptions, see for example the Boussinesq (1903) and Prandtl (1904) boundary- layer approximations. Today it is widely accepted that viscous effects, although very often confined in small regions, control and regulate the basic features of the flow and heat transfer characteristics, as for example, boundary-layer separation and flow circulation. As a result, these characteristics depend on the development of the vis- cous layer and its downstream fate, which may or may not experience transition to turbulence and separation to a wake. Numerous numerical schemes have been devel- xii CONVECTIVE FLOWS oped and these have proved to be fairly reliable when compared with experimental results. However, applications to real situations sometimes brings difficulties. As mentioned before, it is only in the last two decades that various authors have prepared excellent review articles, books and monographs on the topic of convective heat and mass transfer. However, to the best of our knowledge, the last monograph on this topic is that published by Gebhart et al. (1988). Therefore, it is pertinent now to emphasise some of the important contributions which have been published since then, and, indeed, these are very numerous. On studying the published books and monographs on convective heat and mass transfer, we have noticed that much emphasis is given to the traditional analytical and numerical techniques commonly employed in the classical boundary-layer theory, most of which have been known for several decades. In contrast, rather little attention has been directed towards the mathematical description of the asymptotic behaviours, such as singularities. With the rapid development of computers then these asymptotic solutions have been widely recognised. In fact, in the last few years a large number of such contributions have appeared in the literature, especially those concerning the mixed convection flows and conjugate heat transfer problems. Therefore, we decided to include in the present monograph more on the asymptotic and numerical techniques than what has been published in the previous books on convective heat and mass transfer. This book is certainly concerned with very efficient numerical techniques, but the methods per se are not the focus of the discussion. Rather, we concentrate on the physical conclusions which can be drawn from the analytical and mlmerical solutions. The selection of the papers reviewed is, of course, inevitably biased. Yet we feel that we may have over-emphasised some contributions in favour of others and that we have not been as objective as we should. However, the perspective outlined in the book comes out of the external flow situations with which we are most personally familiar. In fact, we have knowingly excluded certain areas, such as, convective compressible flows and stability either because we felt there was not sufficient new material to report on, or because we did not feel sufficiently competent to review them. However, we have made it clear that the boundary-layer technique may still be a very powerful tool and can be successfully used in the future to solve problems that involve singularities, such as separation, partially reversed flow and reattachment. It should be mentioned again, to this end, that the main objective of the present book is to examine those problems and solution methods which heat transfer researchers need to follow in order to solve their problems. The book is a unified progress report which captures the spirit of the work in progress in boundary-layer heat transfer research and also identifies the potential difficulties and future needs. In addition, this work provides new material on con- vective heat and mass transfer, as well as a fresh look at basic methods in heat transfer. We have complemented the book with extensive references in order to stimulate further studies of the problems considered. We have presented a picture of the state-of-the-art of boundary-layer heat transfer today by listing and com- PREFACE xiii menting also upon the most recent successful efforts and identifying the needs for further research. The tremendous amount of information and number of publica- tions now makes it necessary for us to resort to such monographs. It is evident, from the number of citations in previous review articles, books and monographs on the topic of heat transfer that these publications have played a significant role in the development of convective heat flows. The book will be of interest to postgraduate students and researchers in the field of applied mathematics, fluid mechanics, heat transfer, physics, geophysics, chemical and mechanical engineering, etc. and the book can also be recommended as an advanced graduate-level supplementary textbook. Also the wide range of methods described to solve practical problems makes this volume a valuable asset to practising engineers. Acknowledgements A number of people have been very helpful in the completion of this work and we would like to acknowledge their contributions. First, we were impressed with the warm interest and meaningful suggestions of Professor T Y. Na and Dr. D. A. S. Rees, the reviewers of this work. Secondly, the formatting of this book and the preparation of the figures were performed by Dr. Julie M. Harris, and we are very appreciative of her patience and expertise. Thirdly, we are indebted to Mr. Keith Lambert, Senior Publishing Editor of Pergamon, for his enthusiatic handling of this project. Cluj/Leeds Ioan Pop/Derek B. Ingham October, 2000 Nomenclature ac A AT A b B C cp C Cs, Cs D Dm DT e~ E g Vr Gr* h(x) h I2 J k kf km ks kin1 K K* radius of cylinder or sphere, or Ki major axis of elliptical cylinder, or body curvature, or K: amplitude of surface wave l radius of core region reactant L transversal heat dispersion constant amplitude of surface temperature L~ thickness of plate, or m minor axis of elliptical cylinder, or thickness of sheet, or width of jet slit, or body curvature n product species body shape parameter, or n aspect ratio N specific heat at constant pressure Nu concentration p skin friction coefficients Pc chemical diffusion Pe mass diffusivity of porous medium Pr transversal component of thermal qs dispersion tensor stress tensor q~ activation energy q" transpiration parameter Q magnitude of acceleration due to gravity Grashof number r modified Grashof number ~(~) film thickness R constant solid/fluid heat transfer coefficient T~ second invariant of strain rate tensor Ra microinertia density conjugate parameter thermal conductivity of fluid thermal conductivity of porous medium thermal conductivity of solid thermal conductivity of near-wall layer permeablility of porous medium Rah , Rat Ra; Re Re* Reb ReD permeabilities of layered porous media micropolar parameter length scale, or length of plate convective length scale, or length of vertically moving cylinder Lewis number exponent in power-law temperature, or power-law heat flux, or power-law potential velocity distributions stratification parameter, or power-law index unit vector buoyancy parameter Nusselt number pressure characteristic pressure P~clet number Prandtl number energy released from line heat source wall heat flux heat flux per unit area strength of radial source/sink, or total line heat flux, or volumetric flow rate in film radial coordinate axial distance buoyancy parameter, or gas constant temperature or heat flux parameter Rayleigh number for viscous fluid, or modified Rayleigh number for porous medium modified Rayleigh numbers local non-Darcy-Rayleigh number Reynolds number modified Reynolds number Reynolds number for jet Reynolds number based on the diameter of cylinder inertial (or Forchheimer) coefficient, Re~,, Reo~ Reynolds numbers for moving or or modified permeability for fixed plate power-law fluid s heat transfer power-law index xviii CONVECTIVE FLOWS S(x), S(r body functions Sc Sh t T T* % % T~ TS To T~ T~ T~(x) U Uc u(~) u~ u~ V V W w(z) Wc x, y, z Yc, Zc Schmidt number Sherwood number time fluid temperature reference temperature, or reference heat flux boundary-layer temperature core region temperature, or plume centreline temperature temperature at exit temperature in fluid temperature of outside surface of plate or cylinder temperature of solid plate, or of sheet wall temperature stratified temperature fluid velocity along x-axis, or in transverse direction plume centreline fluid velocity velocity outside boundary-layer velocity of moving sheet, or of moving cylinder velocity of potential flow in x-direction characteristic velocity velocity of moving plate fluid velocity along y-axis, or in radial direction fluid velocity vector fluid velocity along z-axis velocity of potential flow in z-direction characteristic velocity Cartesian coordinates characteristic coordinates Greek Letters energy activation parameter c~f thermal diffusivity of fluid c~.~ effective thermal diffusivity of porous medium fl thermal expansion coefficient, or Falkner-Skan parameter fl* concentration expansion coefficient 7 eigenvalue, or gradient of viscosity "~ shear rate tensor F conjugate parameter boundary-layer thickness, or plume diameter (~T, t~O thermal boundary-layer thicknesses (f~ momentum boundary-layer thickness A C concentration difference, Cw- Coo AT temperature difference, T~ - To~ e small quantity transformed x-coordinate, or elliptical coordinate ~0 quantity related to local Reynolds number ( similarity, or pseudo-similarity variable in y-direction 7/ similarity, or pseudo-similarity variable, or elliptical coordinate ~/(~) viscosity function 8 non-dimensional temperature, or angular coordinate Ob conjugate non-dimensional boundary-layer temperature 0~ non-dimensional wall temperature 0 characteristic temperature t~ vortex viscosity A mixed convection parameter A~ Richardson number A inclination parameter H configuration function It dynamic viscosity It* consistency index it0 consistency index for non- Newtonian viscosity u kinematic viscosity p density a heat capacity ratio a(x) wavy surface profile T non-dimensional time T('~) shear stress Tij strain rate tensor V~ wall skin friction ~o inclination angle, or porosity of porous medium r non-dimensional concentration, or angular distance r stream function w vorticity Subscripts f fluid ref reference s solid w wall x local oc ambient fluid Superscripts - dimensional variables, or average quantities ' differential with respect to independent variable ~" - non-dimensional, or boundary-layer variables Table of Contents Convective Flows: Viscous Fluids. 1. Free convection boundary-layer flow over a vertical flat plate. 2. Mixed convection boundary-layer flow along a vertical flat plate. 3. Free and mixed convection boundary-layer flow past inclined and horizontal plates. 4. Double-diffusive convection. 5. Convective flow in buoyant plumes and jets. 6. Conjugate heat transfer over vertical and horizontal flat plates. 7. Free and mixed convection from cylinders. 8. Free and mixed convection boundary-layer flow over moving surfaces. 9. Unsteady free and mixed convection. 10. Free and mixed convection boundary-layer flow of non-Newtonian fluids. Convective Flows: Porous Media 11. Free and mixed convection boundary-layer flow over vertical surfaces in porous media. 12. Free and mixed convection past horizontal and inclined surfaces in porous media. 13. Conjugate free and mixed convection over vertical surfaces in porous media. 14. Free and mixed convection from cylinders and spheres in porous media. 15. Unsteady free and mixed convection in porous media. 16. Non-Darcy free and mixed convection boundary-layer flow in porous media. CONVECTIVE FLOWS: VISCOUS FLUIDS 3 A body which is introduced into a fluid which is at a different temperature forms a source of equilibrium disturbance due to the thermal interaction between the body and the fluid. The reason for this process is that there are thermal interactions between the body and the medium. The fluid elements near the body surface assume the temperature of the body and then begins the propagation of heat into the fluid by heat conduction. This variation of the fluid temperature is accompanied by a density variation which brings about a distortion in its distribution corresponding to the theory of hydrostatic equilibrium. This leads to the process of the redistribution of the density which takes on the character of a continuous mutual substitution of fluid elements. The particular case when the density variation is caused by the non- uniformity of the temperatures is called thermal gravitational convection. When the motion and heat transfer occur in an enclosed or infinite space then this process is called buoyancy convective flow. Ever since the publication of the first text book on heat transfer by GrSber (1921), the discussion of buoyancy-induced heat transfer follows directly that of forced convection flow. This emphasises that a common feature for these flows is the heat transfer of a fluid moving at different velocities. For example, buoyancy convective flow is considered as a forced flow in the case of very small fluid velocities or small Mach numbers. In many circumstances when the fluid arises due to only buoyancy then the governing momentum equation contains a term which is propor- tional to the temperature difference. This is a direct reflection of the fact that the main driving force for thermal convection is the difference in the temperature be- tween the body and the fluid. The motion originates due to the interaction between the thermal and hydrodynamic fields in a region with a variable temperature. How- ever, in nature and in many industrial and chemical engineering situations there are many transport processes which are governed by the joint action of the buoyancy forces from both thermal and mass diffusion that develop due to the coexistence of temperature gradients and concentration differences of dissimilar chemical species. When heat and mass transfer occur simultaneously in a moving fluid, the relation between the fluxes and the driving potentials is of a more intricate nature. It has been found that an energy flux can be generated not only by temperature gradi- ents but also by a composition gradient. The energy flux caused by a composition gradient is called the Dufour or diffusion-thermal effect. On the other hand, mass fluxes can also be created by temperature gradients and this is the Soret or thermal- diffusion effect. In general, the thermal-diffusion and the diffusion-thermal effects are of a smaller order of magnitude than are the effects described by the Fourier or Fick laws and are often neglected in heat and mass transfer processes. The convective mode of heat transfer is generally divided into two basic pro- cesses. If the motion of the fluid arises from an external agent then the process is termed forced convection. If, on the other hand, no such externally induced flow is provided and the flow arises from the effect of a density difference, resulting from a temperature or concentration difference, in a body force field such as the grav- 4 CONVECTIVE FLOWS itational field, then the process is termed natural or free convection. The density difference gives rise to buoyancy forces which drive the flow and the main difference between free and forced convection lies in the nature of the fluid flow generation. In forced convection, the externally imposed flow is generally known, whereas in free convection it results from an interaction between the density difference and the grav- Rational field (or some other body force) and is therefore invariably linked with, and is dependent on, the temperature field. Thus, the motion that arises is not known at the onset and has to be determined from a consideration of the heat (or mass) transfer process coupled with a fluid flow mechanism. If, however, the effect of the buoyancy force in forced convection, or the effect of forced flow in free convection, becomes significant then the process is called mixed convection flows, or combined forced and free convection flows. The effect is especially pronounced in situations where the forced fluid flow velocity is low and/or the temperature difference is large. In mixed convection flows, the forced convection effects and the free convection ef- fects are of comparable magnitude. Both the free and mixed convection processes may be divided into external flows over immersed bodies (such as flat plates, cylin- ders and wires, spheres or other bodies), free boundary flow (such as plumes, jets and wakes), and internal flow in ducts (such as pipes, channels and enclosures). The basically nonlinear character of the problems in buoyancy convective flows does not allow the use of the superposition principle for solving more complex prob- lems on the basis of solutions obtained for simple idealised cases. For example, the problems of free and mixed convection flows can be divided into categories depend- ing on the direction of the temperature gradient relative to that of the gravitational effect. It is only over the last three decades that buoyancy convective flows have been isolated as a self-sustained area of research and there has been a continuous need to develop new mathematical methods and advanced equipment for solving modern practical problems. For a detailed presentation of the subject of buoyancy con- vective flows over heated or cooled bodies several books and review articles may be consulted, such as ~k~rner (1973), Gebhart (1973), Jaluria (1980, 1987), Marty- nenko and Sokovishin (1982, 1989), Aziz and Na (1984), Shih (1984), Bejan (1984, 1995), Afzal (1986), Kaka(~ (1987), Chen and Armaly (1987), Gebhart et al. (1988), Joshi (1990), Gersten and Herwig (1992), Leal (1992), Nakayama (1995), Schneider (1995), Goldstein and Volino (1995) and Pop et al. (1998a). Buoyancy induced convective flow is of great importance in many heat removal processes in engineering technology and has attracted the attention of many re- searchers in the last few decades due to the fact that both science and technology are being interested in passive energy storage systems, such as the cooling of spent fuel rods in nuclear power applications and the design of solar collectors. In particu- lar, for low power level devices it may be a significant cooling mechanism and in such cases the heat transfer surface area may be increased for the augmentation of heat transfer rates. It also arises in the design of thermal insulation, material processing [...]... friction and wall heat flux f"(O) Iml-4 F"(O), 0'(0) - lml 1 G'(0) (1.90) ~s I-~l ~ ~ The two sets of Equations (1.83) - (1.85) and (1.87) - (1.89) have been solved numerically by Henkes and Hoogendoorn (1989) for Pr = 0.72 and different combinations of sgn (M) and n The variation of f"(0) and 0'(0) with m for n - 1 and sgn (M) - 1 is shown in Figure 1.5 Also, some fluid velocity and temperature profiles.. .CONVECTIVE FLOWS" VISCOUS FLUIDS 5 and geothermal systems In particular, it has been ascertained that free convection can induce the thermal stresses which lead to critical structural damage in the piping systems of nuclear reactors The buoyant flow arising from heat rejection to the atmosphere, heating of rooms, fires, and many other such heat transfer processes, both natural and artificial,... receives heat and becomes hot and therefore rises Fluid from the neighbouring areas rushes in to take the place of this rising fluid On the other hand, if T < Too, or ~ < 0, the plate is cooled and the fluid flows downward It is the analysis and study of this steady state flow that yields the desired information on heat transfer rates, flow rates, temperature fields, etc In practice the temperature of the... important Here fl and fl* are the thermal and concentration expansion coefficients and Too and C ~ are the temperature and concentration of the ambient medium If the density varies linearly with T over the range of values of the physical quantities encountered in the transport process, ~ in Equation (I.5) is simply ~ p ~ ~ and if the o~ density varies linearly with both T and C then p and ~* in Equation... = 1 and for a limited m-range by Sparrow and Gregg (1958); (iii) n = - 1 , sgn (M) = 1 and a limited m-range by Cheesewright (1967) and Yang et al (1972); (iv) n = 0, sgn (M) = 1 and the complete m-range by Merkin (1985a); and 24 C O N V E C T I V E FLOWS (v) n - 1 , sgn (M) - - 1 and the complete m-range by Henkes and Hoogendoorn (1989) We next present some of the results reported by Henkes and. .. 1.2: Variation of (a) f " (0), and (b) 0' (0), with m as obtained from numerical integration (solid lines) and asymptotic solutions for P r = 1 The symbol 9 shows the position of the exact solution 0 ~(0) = 0 for m = 5" 3 14 1.3.1 CONVECTIVE FLOWS m ,-,~0 An approximate solution of Equations (1.32) - (1.34) near m - 0 can be obtained by expanding f(~/) and 0(7/) in a power series in m of the form f(rl)... However, we shall review in this chapter some of the most recent and novel results which have been recently published on the problem of steady boundary-layer free and mixed convection over a vertical flat plate We consider a heated vertical flat plate of temperature Tw, or which has a heat flux ~ , oriented parallel to the direction of the gravitational acceleration and placed in an extensive quiescent medium... fluid and it also may arise, for example, in thermally driven motion in cold water, see Gebhart et al (1988) Chapter 1 Free convection boundary-layer flow o v e r a v e r t i c a l flat p l a t e 1.1 Introduction The problem of free convection due to a heated or cooled vertical flat plate provides one of the most basic scenarios for heat transfer theory and thus is of considerable theoretical and practical... y ) , and (b) the temperature, 0(~), profiles for n 1 and sgn ( M ) - 1 when P r - 0.72 26 CONVECTIVE FLOWS (~) (5) 1.5 i 0.0 ,, -\ , , ,,,,, ,,, 1 f"(O) -0.0104 lm[Z o'(0) / -0.5 1.0 : 0.5913m88 1.3900 Im1-88 0 5 -5 ~ 0 9026m ~ 0 rn -1.0 5 -5 0 m 5 Figure 1.7: Variation of (a) f"(O), and (b) O'(O), with m f o r n - - 1 and s g n ( M ) - 1 when P r - 0.72 flow along the heated plate: f'(~7) and 0'(r/)... when using a relatively crude grid The results were reported for three values of the Prandtl number, namely P r - 0.01 (liquid metals), 0.7 (air) and 7 (water) The variation with x of the reduced skin friction, f " ( x , 0), and the reduced heat transfer, O'(x, 0), for some values of the parameter ,4 are shown in Figures 1.12 and . Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media by Ioan I. Pop, Derek B. Ingham • ISBN:. cylinders and spheres in porous media. 15. Unsteady free and mixed convection in porous media. 16. Non-Darcy free and mixed convection boundary-layer flow in porous media. CONVECTIVE FLOWS: VISCOUS. smaller order of magnitude than are the effects described by the Fourier or Fick laws and are often neglected in heat and mass transfer processes. The convective mode of heat transfer is generally