Convective Heat Transfer. Mathematical and Computational Modelling of Viscous Fluids and Porous Media

646 Pages·2001·9.967 MB·English

by  Ioan Pop and Derek B. Ingham (Auth.)

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
radius of cylinder or sphere, or Ki permeabilities of layered porous
major axis of elliptical cylinder, or media
body curvature, or :K micropolar parameter
amplitude of surface wave l length scale, or
ac radius of core region length of plate
A reactant L convective length scale, or
AT transversal heat dispersion constant length of vertically moving cylinder
A amplitude of surface temperature ~L Lewis number
b thickness of plate, or m exponent in power-law temperature,
minor axis of elliptical cylinder, or or power-law heat flux, or
thickness of sheet, or power-law potential velocity
width of jet slit, or distributions
body curvature n stratification parameter, or
B product species power-law index
C body shape parameter, or n unit vector
aspect ratio N buoyancy parameter
pc specific heat at constant pressure Nu Nusselt number
C concentration p pressure
,sC Cs skin friction coefficients Pc characteristic pressure
D chemical diffusion Pe P~clet number
Dm mass diffusivity of porous medium Pr Prandtl number
DT transversal component of thermal sq energy released from line heat
dispersion tensor source
e~ stress tensor q~ wall heat flux
E activation energy q" heat flux per unit area
transpiration parameter Q strength of radial source/sink, or
g magnitude of acceleration due to total line heat flux, or
gravity volumetric flow rate in film
Vr Grashof number r radial coordinate
Gr* modified Grashof number ~(~) axial distance
h(x) film thickness R buoyancy parameter, or
h constant solid/fluid heat transfer gas constant
coefficient ~T temperature or heat flux parameter
2I second invariant of strain rate tensor Ra Rayleigh number for viscous fluid,
J microinertia density or modified Rayleigh number for
k conjugate parameter porous medium
kf thermal conductivity of fluid Rah , Rat modified Rayleigh numbers
km thermal conductivity of porous Ra; local non-Darcy-Rayleigh number
medium Re Reynolds number
ks thermal conductivity of solid Re* modified Reynolds number
1nik thermal conductivity of near-wall Reb Reynolds number for jet
layer ReD Reynolds number based on the
K permeablility of porous medium diameter of cylinder
K* 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 e small quantity
Sc Schmidt number transformed x-coordinate, or
Sh Sherwood number elliptical coordinate
t time 0~ quantity related to local Reynolds
T fluid temperature number
T* reference temperature, or ( similarity, or
reference heat flux pseudo-similarity variable in
% boundary-layer temperature y-direction
% core region temperature, or /7 similarity, or
plume centreline temperature pseudo-similarity variable, or
~T temperature at exit elliptical coordinate
ST temperature in fluid ~/(~) viscosity function
oT temperature of outside surface 8 non-dimensional temperature, or
of plate or cylinder angular coordinate
~T temperature of solid plate, or bO conjugate non-dimensional
of sheet boundary-layer temperature
~T wall temperature 0~ non-dimensional wall temperature
T~(x) stratified temperature 0 characteristic temperature
U fluid velocity along x-axis, or ~t vortex viscosity
in transverse direction A mixed convection parameter
Uc plume centreline fluid velocity ~A Richardson number
velocity outside boundary-layer A inclination parameter
velocity of moving sheet, or H configuration function
of moving cylinder It dynamic viscosity
u(~) velocity of potential flow in *tI consistency index
x-direction it0 consistency index for non-
~u characteristic velocity Newtonian viscosity
u~ velocity of moving plate u kinematic viscosity
V fluid velocity along y-axis, or p density
in radial direction a heat capacity ratio
V fluid velocity vector a(x) wavy surface profile
W fluid velocity along z-axis T non-dimensional time
w(z) velocity of potential flow in )~'(T shear stress
z-direction jiT strain rate tensor
Wc characteristic velocity ~V wall skin friction
x, y, z Cartesian coordinates o~ inclination angle, or
,cY cZ characteristic coordinates porosity of porous medium
r non-dimensional concentration, or
Greek Letters angular distance
energy activation parameter r stream function
f~c thermal diffusivity of fluid w vorticity
~.~c effective thermal diffusivity of
porous medium Subscripts
fl thermal expansion coefficient, or f fluid
Falkner-Skan parameter ref reference
*lf concentration expansion coefficient s solid
7 eigenvalue, or w wall
gradient of viscosity x local
~" shear rate tensor oc ambient fluid
F conjugate parameter
boundary-layer thickness, or Superscripts
plume diameter - dimensional variables, or
(~T, O~t thermal boundary-layer thicknesses average quantities
~f( momentum boundary-layer ' differential with respect to
thickness independent variable
A C concentration difference, Cw- ooC "~ - non-dimensional, or
AT temperature difference, T~ - ~oT boundary-layer variables
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
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 pip-
ing 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, are other examples of natural convection flows.
In the ensuing chapters of this book, a uniform format is adopted to present
theoretical (analytical and numerical) results for the most important situations of
the buoyancy convective flows obtained over the last few years. Most of these results
refer to cases which have never, or only partially, been presented in review articles or
handbooks. The most important fluid flow and heat transfer results are presented
in terms of mathematical expressions as well as in tabular and graphical form to
display the general trends. We believe that such tables are very important since
they can serve as reference tests against which other exact or approximate solutions
can be compared in the future. Due to the vastness of the results presented in this
book, computer codes are not presented. However, frequent references are made
to papers and/or books which contain extensive numerical methods collected from
worldwide sources.
We begin by considering a heated (or cooled) body which has, in general, a
variable surface temperature or variable surface heat flux immersed in a fluid which
has a uniform or variable (stratified) temperature. Apart from any motion that
is generated by density gradients, we suppose that the fluid is motionless. The
complete dimensional form of the continuity, momentum, thermal energy and mass
diffusion (concentration) equations for a viscous and incompressible fluid, simplified
only to the extent that we assume that all the fluid properties, except the density, are
constant and neglect viscous dissipation, diffusion-thermal (Dufour) and thermal-
diffusion (Soret) effects, are given by, see Gebhart et .la (1988) or eejan (1995),
v. v - 0 (i.a)
OV I l_ P~ ooP
T-o + (V. V) V - + + ~ g (I.2)
ooP ~cP
OT
-- + (v. - (I.3)
CO
-- + (V. V) C- - DV2-C (I.4)
~o
where V is the velocity vector, T is the fluid temperature, C is the concentration,
is the pressure, t is the time, g is the gravitation acceleration vector, u is the
kinematic viscosity, p is the fluid density, p~ is the constant local density, cff is the
thermal diffusivity, D is the chemical diffusivity and ~2 is the Laplacian operator.
For many actual fluids and flow conditions a simple and convenient way to express
the density difference (p-poo) in the buoyancy term of the momentum Equation (I.2)
6 CONVECTIVE FLOWS
is given by, see Gebhart et al. (1988),
(I.5)
when the thermal gradient dominates over the concentration (mass diffusion) gradi-
ent and
p - - (T- - (V- (I.6)
when both the thermal and concentration (mass diffusion) gradients are important.
Here fl and *lf are the thermal and concentration expansion coefficients and ooT 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 ~o~ ~ and if the
density varies linearly with both T and C then p and *~ in Equation (I.6) are given
by r176176 U,~ and r ~~-a(l0 ~,'"b~ the expansion coefficients ~and
~* may be evaluated anywhere in the ranges oT( - Too) and (Co - Coo), where oT
and oC are the other bounding conditions on the flow.
Equations (I.5) and (I.6) are good approximations for the variation of the density,
especially when (To-Too) and (Co-Coo) are small, and they are known as Boussinesq
(1903) approximations. The interested reader should also consult Oberbeck (1879).
Other recent considerations of these approximations can be found in the book by
Gebhart et al. (1988). Itowever, if the density variation is substantially nonlinear
in T or both in T and C over the ranges of their values in the buoyancy region,
then the expressions for r and ~* must in general be much more complicated to
yield an accurate representation in .Equations (I.5) and (I.6). This occurs for large
temperature differences in any fluid and it also may arise, for example, in thermally
driven motion in cold water, see Gebhart et al. (1988).