Heat Transfer near the Critical Point
. .
W B HALL
Nuclear Engineering Department. University of Manchester. Manchester. England
I . Introduction
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . .
I1. Physical Properties near the Critical Point . .
A. Thermodynamic Properties . . . . . . .
B. Molecular Structure near the CriticalPoint
.......
. . . . . . .
.................
C . Transport Properties
D . The Implications of Physical Property Variation on Heat
Transfer . . . . . . . . . . . . . . . . . . . . . .
I11. The Equations of Motion and Energy . . . . . . . . . . .
A . Boundary Layer Flow . . . . . . . . . . . . . . . .
B. ChannelFlow . . . . . . . . . . . . . . . . . . . .
C . The Turbulent Shear Stress and Heat Flux . . . . . . .
IV. Forced Convection . . . . . . . . . . . . . . . . . . .
A . Methods of Presentation of Data . . . . . . . . . . . .
B. Experimental Data . . . . . . . . . . . . . . . . .
C . Correlation of Experimental Data . . . . . . . . . . .
D . Semiempirical Theories . . . . . . . . . . . . . . .
V . Free Convection . . . . . . . . . . . . . . . . . . . .
A . Experimental Results . . . . . . . . . . . . . . . . .
B . Theoretical Methods and Correlations . . . . . . . . . .
VI. Combined Forced and Free Convection . . . . . . . . . .
A . Experimental Results . . . . . . . . . . . . . . . . .
B. A Proposed Mechanism for the Heat Transfer Deteriorations
VII . Boiling . . . . . . . . . . . . . . . . . . . . . . . .
A . Nucleate Boiling . . . . . . . . . . . . . . . . . . .
B. Film Boiling . . . . . . . . . . . . . . . . . . . .
C. PseudoBoiling . . . . . . . . . . . . . . . . . . .
Nomenclature . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . .
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2
W. B. HALL
I. Introduction
The rapid growth of research activity in supercritical heat transfer
over the past ten or fifteen years is a consequence of several trends in
engineering. There has been a steady development of steam plant towards
supercritical conditions, and supercritical water has been considered as
a coolant for several types of nuclear reactors. Helium is used at nearcritical conditions as a coolant for the conductors of electrical machines,
and rocket motors are frequently cooled by pumping fuel through
cooling pipes at supercritical pressure.
From a fundamental standpoint, the problem has been regarded as
one in which the variation of physical properties with temperature
becomes extremely important. Effects which, with most fluids, may be
treated as small perturbations of the “constant property” idealization,
sometimes become dominant, rendering existing theoretical models and
empirical correlations useless. In some cases phenomena appear which
have no counterpart with constant property fluids. At the same time
experimental difficulties have hampered the investigation of these effects.
These are not merely the difficulties of operating equipment at high
pressures, but also the problems of compressibility (which becomes very
high near the critical point and makes the density sensitive to relatively
small pressure variations) and of specific heat (which also becomes
large and hinders the accomplishment of thermal equilibrium).
It might be thought that heat transfer experiments of such complexity
would have little to contribute to the understanding of basic mechanisms.
It is true that in constructing models of the process one is forced to
introduce additional assumptions which are difficult to test; nevertheless,
there are some cases where extreme property variations afford a much
more stringent test of some aspects of current theories than could be
obtained in other ways. An example of this is the interaction between
forced and free turbule‘nt convection; with a supercritical fluid the trend
of the results is in the opposite sense to that which one would expect.
This may well lead to a reexamination of the same problem for fluids
with small property variations.
The near-critical region may be thought of as that region in which
boiling and convection merge. When the pressure is sufficiently subcritical or supercritical, the problem tends towards either a boiling
problem or a constant property convection problem; under such conditions existing theoretical and empirical methods are generally adequate.
We shall concentrate on the region rather close to the critical point where
the property variations are severe and where there are very significant
heat transfer effects. Such effects are usually found in a range of pressures
HEATTRANSFER
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CRITICALPOINT
3
from the critical up to about 1.2 times the critical; they are generally
largest when the temperatures of the hotter surface and the fluid span
the critical temperature.
We begin with a brief description of the behavior of thermodynamic
and transport properties near the critical point. T h e equations of continuity, momentum, and energy are then examined with a view to
revealing the effect of variable properties and deciding whether the same
simplifications can be made as are common with a constant property
fluid. A discussion of the various modes of heat transfer then follows,
particular attention being given to the interaction between forced and
free convection.
11. Physical Properties near the Critical Point
A. THERMODYNAMIC
PROPERTIES
T h e properties of a fluid near its critical point have interested thermodynamicists for the past hundred years. This is hardly surprising in view
of the singular behavior in this region: the classical description indicates]
for example, that the compressibility and the specific heat at constant
pressure both become infinite at the critical point. These factors make
experimentation difficult; it is evident that as (avjap),becomes large,
the hydrostatic pressure variation in the fluid will lead to significant
density variations even for small changes of height and also that the
approach to thermal equilibrium will be slow as cp becomes large.
T h e present state of knowledge of thermodynamic behavior is not
entirely satisfactory, either from a theoretical or from an experimental
standpoint; nevertheless, it is probably true to say that an understanding
of heat transfer in the critical region is limited more by lack of knowledge
of the heat transfer processes (e.g., turbulent diffusion, effect of buoyancy
forces) than by uncertainties in the thermodynamic properties. In these
circumstances, the classical description of the critical point may still
be adequate.
1 . The van der Waals Model
I n 1873, van der Waals proposed an explanation of thermodynamic
behavior near the critical point. His model, in which an allowance is
made for the attractive and repulsive forces between molecules, leads to
an equation of state of the following form:
W. B. HALL
4
The physical arguments underlying the equation are well known and
need not be repeated here; it is sufficiently to note that the constant b
accounts for the strong, short range repulsive forces (imposing a limit
to the reduction of volume as pressure is increased), and the term a i r 2
represents the long range attractive forces between molecules. Figure 1
illustrates the shape of isotherms on a p , V diagram, according to van der
Waals equation.
Consider a particular isotherm, marked abcdef in Fig. 1. The fluid
I
!?'
3
Lo
a
!?'
Volume, V
-
FIG. 1. The van der Waals isotherms.
can exist in a homogeneous state along the section of the isotherm marked
abc and def; the section cd represents conditions in which the thermodynamic inequality
( W W T
<0
is not satisfied, and the fluid would separate into two distinct phases.
The regions bc and de represent, respectively, superheated liquid and
subcooled vapor; the extent of these metastable regions is indicated by
broken lines in Fig. 1. Equilibrium between the liquid and vapor phases
(with a plane interface between them) is achieved with states marked b
and e. (Note that unstable equilibrium between liquid and vapor can be
achieved with a curved interface along bc and de. In these cases, surface
tension forces at the bubble or droplet surface lead to a difference
between the liquid and vapor pressures.)
The isotherm marked o in Fig. 1 is known as the critical isotherm and
HEATTRANSFER
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CRITICAL
POINT
THE
5
passes through the critical point. It represents the isotherm for which
the points bcd and e all coincide, thus giving a point of inflection at the
critical point (CP on Fig. I), so that
(ap/aV)c,= 0
(aZpplaP2);
=0
2. The Law of Corresponding States
The behavior of the critical isotherm, as embodied in Eqs. (2) and (3),
can be used to eliminate the constants a and b in the van der Waals
equation as follows: Equation (1) may be written as
p
= R T / ( P - b)
-
alpz
and, using Eqs. (2) and (3),
-RTc
2a
(*);=
0 = (VC b y t($)+
2RTc -- 6a
-
(P)S
(Be
(P)4
1
3
a/b2;
T e = 8a/27bR
- b)3
from which we find that
VC
= 3b;
=
Introducing the “reduced” quantities,
the van der Waals equation becomes
( p * + 3/(V*)’)(3V* - 1)
= 8T*
(4)
An interesting aspect of this equation is the fact that it involves only
p*, V*, and T* and not any quantities that are characteristic of a
particular substance. I n the above form it applies only to substances for
which the van der Waals equation is true; however, the same principle
may be stated in more general terms by asserting that there is a unique
relationship between p*, V*, and T* for all substances. This is known as
the princzple of corresponding states and is frequently stated in the form
2 = Z ( p * , T*)
(5)
6
where
W. B. HALL
z =~ P ~ R T
(For those substances for which van der Waal’s equation is true
Zc = pcpe/RTc = 318,
and
Z
=
$P*V*/T*)
T h e “reduced” isotherms, p* = p*(V*), as determined by the
principle of corresponding states, have the same shape for all substances;
we may therefore conclude that for the same value of T*, all substances
conforming to this principle must have the same reduced saturated
vapor pressures and the same reduced specific volumes of saturated
vapor and liquid. Further, the reduced enthalpy of evaporation,
h,,/RTc must be the same function of T* for all substances.
Thus
h,,/RTC =f(T * )
(6)
(This function tends to a constant value of approximately 10 at temperatures appreciably below the critical temperature.)
T h e importance of the principle of corresponding states in the present
context is that it provides a qualitatively accurate description of thermodynamic behavior near the critical point.
3 . Heat Capacities near the Critical Point
Rowlinson (1) has reviewed the state of knowledge concerning
singularities in the thermodynamic and transport properties near the
critical point. While the specific heat at constant volume is always finite
for a van der Waals gas, it has been shown experimentally that this model
is inadequate at the critical point; it appears that cv does in fact become
infinite but that the infinity is much weaker than that in cp , the specific
heat at constant pressure. In most heat transfer problems we are more
concerned with the value of cp ,and, in this case, the van der Waals model
does predict a value of infinity at the critical point; this may be demonstrated as follows: T h e difference in the specific heats is given by the
thermodynamic identity
c, - c y =
w%J/w
v12/(aP/av)T
and it may be shown (2) that the slope of the critical isochor ( 8 ~ j a T ) , ~
is equal to the slope of the vapor pressure curve at the critical point,
which is finite. On the other hand, (@/8V),c is zero, so that cp - cv
becomes infinite.
T h e question of the precise nature of the singularities at the critical
HEATTRANSFER
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CRITICAL
POINT
7
point is somewhat academic in most practical situations because of the
extreme difficulty of achieving critical conditions precisely. In many heat
transfer systems the pressure will be maintained somewhat above the
critical value; under these circumstances the singularities will be avoided
although the property variations may still be severe. For example, the
peak in the specific heat is large even at pressures considerably greater
than the critical, as may be seen from the data for CO, shown in Fig. 2.
!I
18
-
16
U
.$.'4
- 12
7
P
-P
U
c
10
u
c
a
v)
6
4
2
20
30
40
50
Temperature ("C,
FIG. 2. Specific heat (at constant pressure) of carbon dioxide near the critical point.
4. Compressibility and the Velocity of Sound
T h e compressibility of the fluid may be defined for an isothermal or
for a reversible adiabatic (isentropic) process as follows:
Isothermal coefficient of bulk compressibility,
K~
=
-(aV/ap),/V
Isentropic coefficient of bulk compressibility,
K~
=
-(aV/ap),/ V
I t will be evident from what has been said in section 11, A, 1 that the
isothermal coefficient, K ~ is, infinite at the critical point. It may be
shown (2) that the two coefficients are related in the following manner:
l/KS
= l/KT
+ Tv(ap/aT);/CV
(7)
W. B. HALL
It may also be shown that the vapor pressure curve is continuous with
the critical isochor beyond the critical point. At the critical point,
therefore, (i?p/i?T),is equal to the limiting value of the slope of the vapor
pressure curve, which is finite. Provided that the value of cy at the critical
point is not zero, the isentropic compressibility will be finite.
T h e velocity of sound, c, is given by
From Eqs. (7) and (8) it will be clear that a maximum in c y will lead
to a maximum in K, and a minimum in c near the critical point. Measurements in carbon dioxide at the critical temperature have indicated a
minimum velocity of sound of 140 mjsec at a pressure about 0.5 atm
higher than the critical pressure; at the critical pressure the measured
value was 172 m/sec, compared with the calculated value of 155 mjsec (3).
B. MOLECULAR
STRUCTURE
NEAR
THE
CRITICALPOINT
T h e transition from a subcritical to a supercritical temperature at a
slightly supercritical pressure does not, of course, involve a change of
phase. While from a macroscopic standpoint the change of density is
continuous, there is conclusive evidence that on a molecular scale the
fluid is far from homogeneous. T h e phenomenon of “critical opalescence’’
indicates the presence of a structure large enough to produce scattering
of light; moreover, X-ray diffraction patterns characteristic of the liquid
have been detected at supercritical temperatures when the macroscopic
density is much less than that of the liquid.
An interesting survey of the information on structure has been made
by Smith (4). It appears that as the temperature is increased (at a slightly
supercritical pressure), the liquid structure gives way to liquidlike
clusters in a matrix of gas; these reduce in size until the situation is
virtually one of a gas with a high degree of association. I n most Auid flow
and heat transfer problems it is probably reasonable to regard the fluid
as a continuum because the dimensions of the system will usually be much
greater than the scale of the molecular structure.
C. TRANSPORT
PROPERTIES
T h e pattern of variation of viscosity and thermal conductivity with
temperature and pressure is illustrated in Figs. 3 and 4, which refer to
carbon dioxide in the near-critical region. T h e theoretical basis for
describing the variation of transport properties is less well-developed
than that for the thermodynamic properties, and the problems of
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20
THE CRITICAL
POINT
30
40
Tern p e r a t u re ( "C I
9
50
FIG.3. Viscosity of carbon dioxide near the critical point.
I
20
I
I
30
40
Temperature
I
50
('C)
FIG.4. Thermal conductivity of carbon dioxide near the critical point: (a) data of
N. V. Tzederberg and N. A. Morosova, Teplmergetika No. 1, 75 (1960); (b) data of
J. V. Sengers and A. Michels, Progr. Int. Res. Thermodyn. Tramp. Prop., Pap. Symp.
Thermophys. Prop. 2nd 1962 (1963).
W. B. HALL
measurement are even more severe. I t appears that the thermal conductivity certainly becomes infinite near the critical point ( I ) , but there
is less certainty about the viscosity.
T h e effect of viscosity variations on fluid flow and heat transfer to
low pressure gases is generally dealt with by using approximate expressions of the form
in which T , is some reference temperature, and ps is the corresponding
viscosity. It is quite clear that this technique will fail completely near
the critical point, and one must seek a more general relationship between
the transport properties and the temperature and pressure. One possible
approach is to attempt to describe the transport in terms of “reduced”
quantities in an analogous manner to the description of thermodynamic
properties by the principle of corresponding states. Such an approach has been proposed by Borishansky et al. (5) and has been
applied by them to the generalization of heat transfer processes at
subcritical pressures. Hirschfelder, Curtiss, and Bird (6) have reviewed
various methods by which the transport properties may be correlated.
T h e most suitable, for our present purpose, relates the viscosity and
thermal conductivity to the reduced temperature and reduced pressure
in the following manner
(Pc )2131 - P*(P*, T*)
P* = P ( R p ) l / 6 / [ ~ 1 / 2
k*
= KM1’2(RTc)1’6/[R(pc)2’9]
= A*@*,
T*)
While the extension of thermodynamic similarity to heat transfer is
certainly interesting, it seems likely that when it is taken together with
the conditions for dynamical similarity, the resulting requirements for
complete similarity will be extremely restrictive in all but the simplest
cases.
T h e matter is considered in more detail in Sections IV, C and V, B,
in which the correlation of experimental data for forced and free convection is considered.
T h e lack of accurate data on transport properties for most fluids near
the critical point makes it essential in reporting experimental work to
quote the raw data so that new theories or proposals for generalization
may be tested against them.
D. THEIMPLICATIONS
OF PHYSICAL
PROPERTY
VARIATION
ON
HEATTRANSFER
T h e problem of physical property variation in an extreme form is a
central feature of all near-critical heat transfer processes. I n many cases
HEATTRANSFER
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CRITICALPOINT
11
it produces a quantitative difference in behavior, and in some cases a
phenomenon appears which at first sight has no parallel in constantproperty heat transfer. These effects will be considered in detail later;
in this section we consider some of the more general implications of
physical property variation.
1. The Effect of Temperature Diflerence
Heat transfer processes may be divided into two categories depending
upon whether or not physical property variation forms an essential part
of the process. Conduction and forced convection both may take place
in the absence of variation in any property but the heat content of the
substance involved. Boiling and condensation, on the other hand,
involve phase changes with distinct properties for the separate phases;
free convection involves a fluid flow pattern which is a direct result
of density variation caused by heating or cooling.
T h e existence of a meaningful constant property situation is frequently
useful in analyzing forced convection data; the concept of a limiting
value of the heat tranfer coefficient as the temperature difference tends to
zero is often used to separate the effects of property variation from the
inherent heat transfer and fluid flow processes. Provided that they are
relatively small, property variations may be represented by the inclusion
of an empirical function of temperature difference. From a mathematical
standpoint, the constant property situation is important because the fluid
flow and heat transfer problems are separable, and the energy equation is
linear in temperature; complicated boundary conditions may therefore
be built up by the superposition of solutions with simpler boundary
conditions.
In contrast to the case of forced convection, boiling is a process which
necessarily involves variations in physical properties throughout the
fluid. It is true that property variations within the separate liquid and
vapor phases may be small, but the relative proportions in which these
phases are present will, in general, depend upon the rate at which heat is
added to the system, and therefore on the temperature difference between
the fluid and the heating surface. There is, therefore, no meaningful
limit of the heat transfer coefficient for boiling as the temperature
difference tends to zero.
T h e necessity for a simultaneous solution of the equations of motion
and energy renders free convection a more difficult problem, at least
from a mathematical standpoint, than forced convection. T h u s even
when all properties (except density) are constant, superposition is
prohibited by the nonlinearity of the equations; neither does the heat
12
W. B. HALL
transfer coefficient tend to a constant value as the temperature difference
decreases. Nevertheless, it is possible to make a simplification when the
property variations are small. This arises because the term in the
momentum equation that links it (through temperature variation) with
the energy equation involves the difference between the fluid density
and the density that would be obtained if there were no heat transfer;
this term is still important even when changes in density are entirely
negligible in those other terms of the equation where it occurs as a factor.
This simplification, which is implicit in most free convection theory, is
of doubtful validity near the critical point at all but the smallest temperature differences.
2. The Limit as Temperature Di#erence Tends to Zero
While this limit may be of little practical significance, it is often useful,
as mentioned, as a device for separating the effects of property variation
from heat transfer and fluid flow phenomena. It has been suggested in the
preceding section that this limit is relevant only to the cases of conduction, forced convection, and, in a more restricted sense, to free
convection; these cases are considered in more detail in the following.
a. Conduction. Consider the problem of steady conduction through
a slab of fluid at a slightly supercritical pressure, the two surfaces of the
slab having temperatures which span the critical temperature. If the
temperature difference is large, then there will be a thin layer of fluid
in the interior of the slab in which the thermal conductivity exhibits a
peak. If the thickness of the layer is small in relation to the thickness of the
slab, the effect of the peak in conductivity will be negligible. If, however,
the temperature difference is small, the whole of the fluid may have a
conductivity equal to the peak value. Is it then possible that a reduction
in temperature difference could, by increasing the average conductivity
of the fluid, increase the heat flux through the slab ? T h e answer may be
obtained as follows: From the definition of conductivity, k,
=
--~(~)a~/ax
where q is the heat flux through the slab and x is the distance in the
direction of heat flow. Thus
where b is the thickness of the slab, T I is the temperature of the cold
surface, and Tz is the temperature of the hot surface. Equation (10)
HEATTRANSFER
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CRITICAL
POINT
13
is illustrated in Fig. 5 , which is drawn for a fluid which has an infinite
value of K at the critical temperature, but for which JK(T)dT through
the critical temperature is finite.
'I
'2
Temperature, T
+
FIG.5. Illustration of conduction through a supercritical pressure fluid, Eq. (10).
(Note: if, following Rowlinson (I),we express K( T) near the critical
point of the form
k = C I T - TC (-0.2
then
1K
dT
=
C I T - TC1O.*/0.8
which remains finite as we pass through the critical point.)
Referring to Fig. 5 and Eq. (lo), we see that for a slab of fixed thickness,
the heat flux is proportional J k( T) dT. If TI is held constant, then q can
only decrease as T, is decreased. On the other hand, if the temperature
14
W. B. HALL
difference is maintained at a constant small value, the heat flux will,
of course, change as the temperatures traverse the range. T h e matter
may be summarized by saying that it is not the fact that the conductivity
may become infinite which is important, but that its integral through
the critical temperature remains finite.
Unsteady conduction depends upon the density and specific heat of
the fluid as well as on the thermal conductivity. With constant properties
it is possible to group these three quantities into a single parameter,
klpc,, the thermal diffusivity, which governs the rate of transmission of
temperature changes through the medium. With variable properties the
parameters are k/pc, at some reference condition plus parameters which
express the property variations with temperature. T h e question then
arises as to whether the fact that k/pc, becomes zero at the critical point
(because c, has a stronger infinity than that in k at the critical point)
has any heat transfer significance. T h e matter may be resolved by referring to the unsteady conduction equation which, for a constant pressure
system, may be written in the two identical forms
and
where qz ,qsl ,qz are the heat fluxes in the x-, y-, x-direction.
T h e first equation contains quantities (c, and k) which become infinite
at the critical point. However, it has been shown above that $ k dT
remains finite through the critical temperature; the same is true of J’cP dT.
T h e second equation therefore does not contain any singularities, and the
only heat transfer effect will be that %/at will momentarily become zero
as the temperature of the fluid passes through the critical temperature.
A zero value of the thermal difiusivity does not imply in this case that
the fluid is impervious to heat!
b. Forced Convection. Steady state forced convection is governed by
equations of motion together with an energy equation which is formally
similar to that for unsteady conduction, (see Section 111). As with
conduction, therefore, the singularities in k and cp do not have the implications that might at first be attributed to them; i.e., the zero value of
klpc, does not prevent the diffusion of heat into the flow.
As the temperature difference tends to zero, convective heat transfer
HEATTRANSFER
NEAR
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CRITICALPOINT
15
tends to a constant property process, and one might expect the usual
correlations to apply, i.e., for a pipe flow
or@
0.023 Reo.8(c,p/k)u.4
which, for a constant mass flow in the pipe gives a proportional to
kO .6,-Cp”. 4,
.4.
In other words, the heat transfer coefficient becomes infinite at the
critical point as the temperature difference tends to zero. This fact has
little practical significance again because the integrals of k and cp with
respect to temperature remain finite through the critical temperature.
Thus the heat flux will remain finite as the temperature difference tends
to zero.
c.
Free Convection. T h e remarks made about the singularities in k and
cp in connection with conduction and forced convection apply equally to
free convection. There is, however, an additional aspect which deserves
mention. I t will be shown in Section I11 that one of the dimensionless
parameters governing free convection, the Grashof Number, arises
naturally from the basic equations in the form
Gr
= gd3 dp/v2p
where d p is a characteristic density difference, usually that between the
fluid at the heated surface and that outside the thermal layer. Because
most normal fluids have a fairly constant expansion coefficient, /3, it is
convenient to write the Grashof Number in the form
Gr
= gd3,6A T / v 2
If this form is used at the critical point, then difficulties will arise
because /3 becomes infinite. Reverting to the original expression for Gr,
however, we see that the true Grashof number remains finite because dp
(corresponding to a given value of A T ) remains finite as the system
temperature traverses the critical temperature.
111. The Equations of Motion and Energy
Well established techniques exist for simplifying the basic equations
when they are to be applied to constant property fluids in particular
circumstances. Thus, for example we frequently use “boundary layer’’
forms of the equations and sometimes neglect the effects of viscous
dissipation, buoyancy forces, and pressure gradients. We now consider
16
W. B. HALL
whether the same techniques may be applied to variable property flows.
We begin with the equation of continuity, momentum, and energy for a
two-dimensional, nonturbulent boundary layer flow (with the x-coordinate in the upward direction)
Continuity:
= u - - - + Tay
ax + per ax
ay
ay
ah
pu -
Energy:
ap
ah
a4
au
(13)
I t is an advantage to use the above form of the shear stress and heat
flux terms when the physical properties p and k vary, as they do, in an
eccentric manner near the critical point. Thus, even though k may
become infinite at the critical point, aT/ay will simultaneously
become zero, the product q = --K 8Tjay remaining finite at some value
between zero and the maximum value q,, at the wall. It is much easier to
make reasonable estimates of q and T than of aT/ay and aujay.
Similar equations may be written down for a turbulent flow. Putting
u =B
u', etc., for the mean and fluctuating components, we find that
the equations become
+
-x
-a@
dp
pu-+pv-=--+--pg
ax
ay
ax
a7
-
ay
where T and q are now given by
Our first concern will be to establish the conditions under which
we may neglect the first and third terms on the right-hand sides of
Eqs. (15) and (16) with respect to the second term in each equation.
I n establishing criteria for neglecting these terms we shall employ
constant property empirical relationships for such quantities as boundary
layer thickness, 6, friction factor,f, and Stanton number, St. T h e criteria
HEATTRANSFER
NEAR
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CRITICAL
POINT
17
will therefore be very approximate and will certainly give no indication
of the effect that the terms may have if they are large. We then investigate
the relative importance of the turbulence terms in Eqs. (17) and (18), and
attempt to express them in terms of time mean flow quantities.
We shall consider a boundary layer and a “fully developed” channel
flow. T h e latter type of flow frequently arises with constant property
fluids but is less likely when properties change significantly in the direction of flow; the analysis must therefore be treated with reserve until the
hypothesis can be tested adequately.
A. BOUNDARYLAYERFLOW
T h e turbulent form of the equations will be used, since turbulent
conditions are of the greatest practical interest. If Eq. (15) is applied
to the (nonturbulent) free stream outside the boundary layer, we find
-dWx
= P P , au,lax
+ p,g
(19)
which may be inserted in Eqs. (15 ) and (16), giving
-a@
pu-
ax
+ -ai
pvay
au,
= Ps%%
aT
++ (Ps -P)g
aY
T h e first term on the right-hand side of Eqs. (20) and (21) represents
the effect of the acceleration of the free stream; the last term in Eq. (20)
represents the effect of bouyancy forces; the last term in Eq. (21)
represents the effect of viscous dissipation. We now consider the
magnitudes of these terms in relation to & / ~ J J and aqjay.
1. Buoyancy Eflects
T h e magnitude of 1
I is of the order ~ ~ (where
/ 6 T~ is the wall
shear stress and 6 the boundary layer thickness), and the buoyancy term
therefore may be neglected if
(Ps
-P)gS/70
<1
This will have a maximum value at the wall where
criterion may therefore be written
2 Gr 6
<I
fW,
P
= po
, and the
W. B. HALL
18
where
Gr
Re = u,x/vs ;
= ( p S - p0)gx3/pSv,2;
if
= TO/pSus2
and x is the distance from the start of the boundary layer.
Empirical data for a turbulent boundary layer with a uniform velocity
and Qf = 0.0295 Re-0.2
free stream show that S/x = 0.037
(interpreting 8 as the momentum thickness).
Thus the above criterion becomes, approximately,
Gr/Re2
<< 1
2. Dissipation
T h e dissipation term in Eq. (21) may be neglected if
This will have a maximum value at the wall where
so that the criterion may be written
where
St
= qO/pSuS
Ah;
7
&/ay = r o 2 / p 0 ,
E = u,'/Ah
and Ah is the enthalpy difference from surface to free stream. T h e group
E is the Eckert number, ususally quoted for a constant property situation
(at constant pressure) as u,"/c, d T .
Using again empirical data for a turbulent boundary layer, the
preceding equation becomes, approximately,
E
< 1000
(25)
(The viscosity ratio has been omitted since it will always be between
about 0.5 and 1.0.)
3 . The Effect of Free Stream Acceleration
An acceleration of the free stream produces a pressure gradient which,
acting uniformly on the boundary layer, causes it also to accelerate.
T h e difference, psus &,/ax - P. &/ax, (Eq. (20)), is available for
modifying the shear stress gradient, a ~ / a y and
,
is greatest at the wall
where pU Z j a x = 0. Thus the change in the shear stress gradient at
the wall can be neglected only if psu, au,/ax is much less than ~ ~ / 6 .
HEATTRANSFER
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19
Thus acceleration of the free stream is negligible if
i.e., if
2
s
K-Re-<I
where
f
r
Using empirical data for a turbulent boundary layer, this reduces to
the form
KRe<1
(27)
In a similar manner, the criterion for neglecting the acceleration term
in Eq. (21) is found to be
KKeE<I
(28)
[Note: I n Sections 111, A, 1 , 2, and 3, it has been assumed that the order
' qo:8,
8
respecof magnitude of I &/dy I and I dq dy I is given by ~ ~and
tively. While this is true for the greater part of the thickness of the
boundary layer, it is not true very close to the wall where, for a boundary
layer with a uniform velocity free stream, both &lay and aq,lay become
zero. This means that, for example, viscous dissipation could be significant in a thin layer close to the wall even if the above criterion (Eq. (25))
is satisfied. However, when one reflects that it is the integrated effect
across the boundary layer that matters, the discrepancy is of little
importance. I n view of the fact that the equations are to be integrated,
it would perhaps be better in the first place to compare for exampIe qo
with Ji T a&,laydy.]
T h e virtue of using the shear stress and heat flux in the preceding
equations is seen to be that one can more easily establish criteria for
neglecting certain terms in the eqautions. This is difficult when the
transport properties are introduced, in view of their very drastic variation.
If, however, one wishes to predict the effect of buoyancy forces
(rather than establishing when they may be neglected), then the problem
is much more difficult: the matter is considered in Section VI.
B. CHANNELFLOW
T h e terms containing the pressure gradient can also be eliminated
when Eqs. (15) and (16) are applied to channel flow. Suppose we have a
W. B. HALL
20
channel formed by parallel planes a distance 2b apart, and that fully
developed conditions are established (i.e., p T aii/ax independent of y ) .
Integrating Eq. (15) from the wall to the center plane of the channel
gives
or
wheremis the mass flow through the channel per unit width. [Note that the
assumption of uniform aii/ax across the channel is inconsistent with the
acceleration resulting from a pressure gradient which is uniform across
the channel; this would give uniform jiiaiijax. However we shall get
a good estimate of the pressure gradient if we take the gradient of the
mean velocity, diim/dx, in Eq. (29).]
Substituting Eq. (29) into Eqs. (15) and (16) we get, noting that
p. = 0,
As in the case of the boundary layer, we now consider the conditions
under which dissipation, buoyancy forces, and acceleration may be
neglected.
1. Buoyancy Effects
If we consider the case where dii,/dx
=
0 then Eq. (30) reduces to
+ ( a ~ / a+~ G) m -
0 = (~o/b)
And if we use the approximation -&/ay M T ~ / Z J (which is certainly
true if (pm - p ) g is small), then T~ is indeterminate. The point is that
when buoyancy effects are present, they are balanced by the change in
&/ay from its initial value of ~ , / b .Thus, if the buoyancy force assists
the pressure flow near the wall, then I a ~ / a Iy will be greater near the
wall. Buoyancy forces will be significant, therefore, if they are able to
y
the value 7o/b. Thus once
produce a significant change in a ~ / a from
again the criterion becomes
&Pm
-P)g/To
<< 1
HEATTRANSFER
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21
As in the boundary layer case, this may be expressed in terms of the
Grashof and Reynolds numbers
Gr/(Re)l.s
< 0.1
(32)
where in this case
Buoyancy effects are frequently important in forced convection with
supercritical fluids, and have rather large and unexpected results; these
are discussed in Section VI.
2. Dissipation
Following the same argument as that for a boundary layer we find
which, using empirical data for channel flow, becomes
Reo.* E
< 100
where, in this case, Re = u,(4b)/vm and E
(33)
=
urn2/&
3 . Acceleration Efects
Whereas with the boundary layer the acceleration was imposed by
applying a pressure gradient to the free stream, in the case of a channel
of uniform cross section it will occur because of the expansion of the
fluid as it is heated. T h e effect will be greatest at the wall where
P. &/ax = 0 in Eq. (30), and the acceleration term will be negligible
with respect to rO/bif
m
2b dx
b
T~
-
li2 du,
2r0 dx
T h e value of du,/dx may be assessed by calculating the rate of expansion of the fluid as it is heated by the specified heat flux q,, through the
channel walls. As an approximation, this calculation is based on mean
values of the variables taken across the channel (denoted by subscript m).
We find that
dii,,,ldx w e,,$ dhm/dx = 2 q o ~ m ~ m 1 ~ ~ , ,
22
W. B. HALL
T h e criterion then becomes
If we again assume that St m f / 2 this becomes
(Note that for a perfect gas this would become ATIT. T h e group
3/, dh,C,, may in fact have a value of order unity near the critical point,
so that it will not normally be safe to neglect acceleration effects. However, we shall usually find that, unless the pipe diameter is very small,
buoyancy effects are even more important.)
Turning to the energy equation, Eq. (31), we find that the term
k , / b (and the acceleration term (iifi/2b)(diiWJdx),which will usually be
less than U T , / ~ except very close to the critical point where it may be
of the same order) can be neglected if
or
(provided again that St w f / 2 )
C. THETURBULENT
SHEAR
STRESS
AND HEAT FLUX
Equations (17) and (18) define the turbulent shear stress and heat
flux for a variable property fluid. If we are to draw on the large volume of
data on turbulent diffusion in constant property fluids, we must make
( 18) and theircountersome estimate of the differences between (1 7) and
parts for constant property fluids (i.e,, T~ = -pu'v' and q1 = ph'v').
1. The Effect of Variable Properties on T~ and qt
T h e relative magnitude of the terms in Eqs. (17) and (18) may best
be displayed by expressing the equations in dimensionless form. Thus,
we define the dimensionless quantities
HEATTRANSFER
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THE
23
and the subscript s refers to a reference condition, e.g., the free stream.
Equations (17) and (18) then become
7t =
-
-
-@,2 U'V."- pau,U R'U'
qt =
-
Ah H'V'
-
p , ~ , 2R'U'V'
(36)
+ pI Ah 6 R H ' + p p , Ah R H ' V '
__
(37)
T h e fluctuations, R' and H', may be expressed in terms of the dimensionless temperature fluctuations, 8', provided that we may assume that
the turbulent pressure fluctuations are small compared with the mean
pressure.
Thus, if 6' = T'jAT, where AT = To - T , ,
AT
R'=8'-Ps
( aapT )
P
-
-8'f-AT/3,
where
Ps
1
a=-(+
v
av
a2
We now make the assumption that 8' m U', and that the correlations
between 0' and other quantities are the same as between U' and those
quantities. This is the crucial assumption and can be justified only by
appealing to an analogy between the temperature and x-direction velocity
fields. Thus the velocity fluctuations u', normalized by the overall velocity
difference, us are compared with the temperature fluctuations T ' ,
normalized by the overall temperature difference; both are seen as the
result of turbulent transport by the fluctuating velocity 71' in the transverse direction. While this argument is of doubtful validity with extreme
property variations, it probably gives the correct order of magnitude of
the terms.
With these assumption, Eqs. (36) and (37) become
-@82[U'V' - (ti/uS)/3 AT U'U' - /3 AT U'U'V']
___
=~u~c~AT[UV'-(U/U~)B~T
-/3ATU'U'V']
U'U'
~
T~
~
=
~
qt
(38)
(39)
T h e following comments can be made about the quantities in these
equations:
I U'U' I because the correlation between u' and v'
(i) I U'V' I
is usually about 0.4
-
(ii)
u"v'<u
<" because
I U' I
<1
<
us
(iii) for a boundary layer or channel flow 3
(iv) the product /3 AT, which for a perfect gas is ATiT, probably
does not exceed unity except very close to the critical point.
W. B. HALL
24
On this basis, therefore, the last two terms in Eqs. (38) and (39) are
probably negligible, and the expressions revert to the constant property
form. There is one very important point to note, however: while the
expressions for T~ and qt may remain the same, the magnitudes of the
correlations &’ and h’v’, when expressed in terms of the mean flow
quantities, will almost certainly be different. Thus, for example, Hall
el a1 (7) have suggested that the effect of the relatively large expansion
of the fluid undergoing turbulent diffusion may significantly affect the
turbulence level. I t seems likely that such effects will outweigh any errors
resulting from the omission of the lower order terms in Eqs. (38) and (39),
and we therefore proceed using the constant property forms for T~ and q1 .
2.
r t and ql
in Terms of Time Mean Flow Quantities
There is, at present, insufficient data to assess accurately the effect of
T h e approach
variable properties on the correlations UTand
adopted in Sections IV, V, and VI is to use a “mixing length” model
of turbulent diffusion in which
m.
where I is the “mixing length,’) and E is the turbulent diffusixity.
(Townsend (8) has shown that such a model may be derived from the
equation governing the production, dissipation, convection, and diffusion
of the kinetic energy of turbulence, under circumstances when the
production and dissipation are in local equilibrium.)
An assumption concerning the variation of I throughout the flow thus
enables the turbulent shear stress and heat flux to be related to gradients
of the time mean quantities ?i and h. A common assumption is that
I = 0.4y, at any rate in the region close to the wall. From what has been
said, this relationship might well be affected by variable property effects
in supercritical fluids.
An alternative method of specifying the level of turbulent diffusion
is to establish a relationship between the turbulent diffusivity, E , and
position in the flow. This is quite permissible, but one must remember
HEATTRANSFER
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25
that the expressions in common use are based on measurements in
which the shear stress distribution in the flow has its normal constant
property value. From the definition of E we see that it may be expressed
as
and
where d is a characteristic dimension, and s refers to a reference condition
such as the free stream.
Thus, provided that the shear stress distribution, T / T ~is always the
same, that ps/p M 1, and that 1 can be expressed in terms of y, then
E / V , = f (Re, y/d). I n
many supercritical heat transfer problems,
however, buoyancy forces are large enough to change significantly
the shear stress distribution; it is then important to avoid the use of an
expression for E/V, which has in it an implicit assumption concerning
T / T ~
. T h e matter is discussed in more detail in Section VI.
IV. Forced Convection
Almost the whole of the data on forced convection near the critical
point has been obtained using pipes or channels of uniform cross section,
in most cases with a uniform heat flux boundary condition. This apparently simple situation has nevertheless yielded a diversity of experimental
results that is matched only by the range of correlations produced to
describe them! One is faced not only with discrepancies between correlations, but also between sets of data which have been obtained under
apparently similar conditions. This interesting situation cannot be
attributed solely to inadequate experimental techniques, and one must
question whether there are some important factors which were not controlled in the experiments.
We begin by discussing the manner in which forced convection heat
transfer data are commonly presented and the difficulties that arise when
physical property variations are severe. T h e apparent discrepancies
between typical sets of experimental data and the deficiencies of current
methods of correlation are then demonstrated. Finally, the prediction
of heat transfer by numerical solution of the semiempirical equations
of motion and energy is discussed.
26
W. B. HALL
A. METHODS
OF PRESENTATION
OF DATA
Forced convection heat transfer data are frequently presented in a
form in which neither the temperature of the heat transfer surface nor
that of the fluid is given explicitly. T h e implication is that the heat
flux is proportional to the temperature difference between surface and
fluid, and that any effect due to the general level of temperature can be
adequately expressed by evaluating the physical properties of the fluid
at some characteristic temperature, e.g., the bulk mean fluid temperature.
Proportionality of heat flux and temperature difference occurs with
constant property fluids and is a consequence of the facts that the energy
equation is linear in temperature, and the heat transfer process does not
affect the flow process. Presentation of data in this form is inappropriate
for fluids near their critical point, and confusion often results when data
are forced into such a pattern.
T h e matter is best illustrated by presenting the same data in a variety
of forms; this is shown as follows for the case of carbon dioxide at a
pressure of 75.8 bars flowing in a downward direction in a heated
vertical pipe of diameter 1.90 cm (9). (The pressure of 75.8 bars is
somewhat above the critical pressure (73.8 bars). It is usual to relate the
behavior of the fluid to the "transposed critical temperature," Tpc,
the temperature at which the specific heat reaches its maximum value.
At the pressure of 75.8 bars this temperature is 32"CJrather than the
critical temperature of 31.04"C.) Data were obtained for upward flow,
but these gave anomalous results which will be discussed in Section VI.
1. The Experimental Measurements
T h e quantities measured in the experiments were the mass flow of
CO, the fluid inlet temperature, the heat input (which was arranged to
give a very nearly uniform wall heat flux into the fluid), and the temperature of the pipe wall. T h e latter temperature was measured at
intervals of one pipe diameter along the length of the test section. Typical
results are given by the full lines in Fig. 6 , which shows the variation
in wall temperature, T o , along the pipe for three different heat fluxes
but with the same mass flow and inlet fluid temperature. T h e complete
set of results for this mass flow involves three different fluid inlet temperatures each with five different heat flux levels, i.e., fifteen curves in all.
Results in this form embody the whole of the information and should
always be made available in tables if not also in graphs. Subsequent
operations all involve either the input of physical property data (sometimes of doubtful accuracy) or the use of some simplifying hypothesis such
as the elimination of an experimental variable or its combination with
HEATTRANSFER
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CRITICAL
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27
another. Difficulties are caused not by the well-intentioned attempt to
find simpler means of expressing the results but by the rejection of
information; it may be impossible to recover this from partly processed
results. It is unfortunate that a great deal of the heat transfer data on
supercritical fluids are of considerably less value than they should be
because of such omissions.
;
W a l l temperature
_----_----
-__--,------_--,---__----
Bulk ternperoture
\-/--___----*_ _ - &
*_--
-#
--
&&*-k==
0
20
40
60
a0
100
Distance from start of heatmg (diameters)
120
I
140
FIG. 6. Temperature distribution along a I .9-cm diameter vertical pipe for downward
flow. Carbon dioxide at a pressure of 75.8 bars and a mass flow of I60 gmlsec. (a) wall
heat flux, 3.09 W/cmZ;(b) wall heat flux, 4.05 W/cmz; (c) wall heat flux, 5.19 W/cm2.
2. Description in Terms of Local Conditions Only
One of the most useful concepts in forced convective heat transfer is
that of “fully developed” fluid flow and heat transfer conditions in a
pipe. It is asserted that both the velocity and the temperature distributions across a pipe will become invariant aftes a certain distance from the
pipe inlet. (Note that in the case of temperature it is only the shape of the
distribution that remains unchanged since the bulk temperature must, of
course, increase as heat is added.) There is ample evidence to show that
such conditions are established in about 10 to 20 pipe diameters with a
turbulent constant property fluid and a uniform cross section pipe; and
if we therefore exclude the “entry region,” the heat transfer coefficient
will be independent of position along the pipe. I n the case of a fluid
whose properties change with temperature, and therefore with distance
along the pipe, the hypothesis of a “fully developed” condition is less
plausible and must be tested experimentally.
28
W. B. HALL
Figure 7 shows the results, of which Fig. 6 is a sample, presented in the
form of heat flux against wall temperature, with the bulk fluid temperature
T , as parameter. For this purpose it is necessary to calculate the bulk
fluid temperature along the pipe (shown by dotted lines in Fig. 6);
this may be done by applying a heat balance from the pipe inlet to the
point in question and requires an accurate knowledge of enthalpy as a
function of temperature. The hypothesis is then made that the heat
transfer rate (expressed as heat flux for particular values of wall temperature and bulk temperature) is independent of the point along the
pipe at which these particular conditions occur; the result is Fig. 7.While
w a i l temperature
("c)
FIG. 7. Heat flux versus wall temperature for various bulk fluid temperatures. Same
data as for Fig. 6. Bulk fluid temperature: ( O ) , 19°C; (+), 22°C; (v), 25°C; ( x ) , 28°C;
(O), 31°C.
there is a good deal more scatter of the experimental results than in
the curves of Fig. 6 (in which the experimental points fall within 0.1"C
of the smooth line drawn through them), the result does suggest that
the heat transfer process is very largely governed by local conditions.
The lower parts of the curves are drawn in broken lines because no
results were obtained in this region; it is possible, however, to fix the
point at which they intersect the To axis because at this point q = 0
and To = T , . Incidentally, the slopes of the curves as they cross the To
axis give the limiting value of the heat transfer coefficient as the temperature difference tends to zero.
HEATTRANSFER
NEAR THE CRITICAL
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29
3 . Presentation in Terms of a Heat Transfer Coejicient
This form of presentation is shown in Fig. 8 in which the points are
the same as those of Fig. 7. The heat transfer coefficient is by no means
independent of either the wall temperature or the bulk temperature.
This being so, one must question whether the concept of a heat transfer
coefficient has any useful purpose to serve since the results could equally
well be presented as in Fig. 7. It may perhaps be useful to see whether the
results show a tendency towards a constant heat transfer coefficient in
certain limiting conditions; for example, it appears that this may be so as
the wall temperature increases, the bulk temperature still being below
the transposed critical temperature. However, one might easily be
misled by Fig. 8 into thinking that high heat fluxes could be achieved
with small temperature differences whereas Fig. 7 shows that this is not
the case.
0.5
"
0
N
5
-3\
0.4
C
m
.r
c
"
0)
0
L
2
0
--
03
0
0
0
I
+ A x
0
0.2
I
I
20
I
I
40
Wall
temperature
I
I
60
I
J
80
("c)
FIG. 8. Heat transfer coefficient versus wall temperature for various bulk fluid
temperatures. Same data as for Figs. 6 and 7. Bulk fluid temperature: (a),19°C; (+),22"C;
(A), 25°C; ( x), 28°C; ( o ) ,31°C.
30
W. B. HALL
T h e usefulness of the heat transfer coefficient when applied to supercritical fluids has been questioned by Goldman (10); he has suggested
that rather than expressing heat transfer results as a relationship between
Nusselt number, Reynolds number, and Prandtl number, it is more
appropriate to collect together all the terms in the dimensionless groups
that are temperature dependent. Thus starting from the assumption that
Nu
=c
Ren Prs
(42)
where c, n, and s are constants, he obtains
qod'-"/(P)"
= f(T0 I
Tm)
(43)
This presentation is rather like that of Fig. 7 except that it postulates a
specific form of variation with pipe diameter, d, and mass velocity, pu,
whereas the data of Fig. 7 are for one pipe diameter and one mass velocity
only. Equation (43) is, however, no more valid than Eq. (42) from which
it was derived; there is no a priori reason to suppose that with a supercritical fluid Reynolds number and Prandtl number effects are adequately
represented by an equation of the form of Eq. (43).
4. Presentation in Terms of Dimensionless Groups
T h e data of Fig. 8 are shown in dimensionless form in Fig. 9. T h e
form of correlation that has been used is due to Miropolsky and Shitsman
(11) and is of the form
Nu,
= c(Re,)"
(Prmjn)8
(44)
T h e physical properties in the Nusselt number and the Reynolds
number are evaluated at bulk temperature, but the Prandtl number is
the lower of the two values obtained by evaluating properties at the wall
temperature and at the bulk temperature. T h e exponent n has been
chosen as 1.4 to give the best fit to the results, and thus N u , , / ( R ~ , ) ~ . ~
has been plotted against Prmin.
While the correlation appears at first sight to be reasonably good,
it should be noted that in fact the scatter is a good deal more than in the
original data or in the form of presentation shown in Fig. 7. Bearing in
mind the restriction which must be placed on its use in conditions
outside the range of the data on which it is based, the use of such a
correlation is of doubtful value. An even more serious criticism is the
fact that it is impossible to recover the original data from such a presentation of results.
HEATTRANSFER
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31
t
FIG. 9. Correlation of the data shown in Fig. 8. (Physical properties based on bulk
temperature for N u and Re; Prmj,,is the lesser of the values at wall and bulk temperatures.)
B. EXPERIMENTAL
DATA
Existing data for forced convection have been reviewed by Hall,
Jackson, and Watson (12). Most experimenters have used circular cross
section pipes with a uniform heat flux boundary condition. I n spite of the
very considerable amount of data that exists, the situation is still somewhat confused; it is not simply that one is unable to correlate the results
in terms of the usual parameters, but rather that one suspects that there
may be some important parameters that have not been controlled. T h e
situation is made worse by the fact that in some cases experimental
results have been presented in “correlated” form, and the physical
property data used in the correlation has not been quoted. Little purpose
would be served in presenting a detailed review of experimental data
here; it is perhaps more useful to identify the important discrepancies and
to consider what physical factors may have been responsible for them.
Figures 10 and 1 1, which are based on uniform heat flux measurements
in a circular pipe using water (2.3-16), illustrate some of the apparent
discrepancies between experiments. In all cases the measurements
were made in pipes of circular cross section with a uniform wall heat
flux, 4, into the water. Provided that the entry conditions are similar,
one would expect the wall temperature to be a function of the bulk
enthalpy, the mass velocity of the water, the pipe diameter, and the wall
heat flux. T h e latter three parameters are quoted on the figures, and the
W. B. HALL
32
580
560
540
5 20
500
4 80
u 460
.e
P
4 40
420
400
380
360
340
1600
1800
2000
2200
2400
2600
2800
Bulk e n l h o l p y (J/grn)
FIG. 10. Experimental wall temperature distributions as a function of local bulk
enthalpy along a pipe. p* = 1.05 (W. B. Hall, J. D. Jackson, and A. Watson, “Symp,
Heat Transfer and Fluid Dynamics of Near Critical Fluids.” Proc. Znst. Mech. Eng. 182.
Part 31 (1968)). (a) Shitsman (13): q = 34 W/cm*, ni/A = 43 gm/sec cm2, d = 0.8 cm
vertical. (b) Shitsman (13):q = 28.5 W/cm*, m/A = 43 gm/sec cmz, d = 0.8 cm vertical.
(c) Shitsman (13):q = 28.0 W/cm*,& / A = 43 gm/sec cmB,d = 0.8 cm vertical. (d) Domin
(IS):q = 72.5 W/cmz, &/A = 68.6 gm/sec cm*, d = 0.2 cm horizontal. ( e ) Domin (15):
q = 72.5 W/cm2, m/A = 72.4 gm/sec cm2, d = 0.2 cm horizontal.
first is used as abscissa; Fig. 10 is for a pressure closer to the critical than
Fig. 11. While it is not possible to make direct comparisons between the
various experiments, it is difficult to believe that the sharp peaks of wall
temperature in Shitsman’s data, the broader peaks of Domin, of Vikrov
and Lokshin, and of Schmidt, and the sharp depression of wall temperature observed by Domin (Fig. l l ) , all form part of a single consistent
pattern. Nevertheless one can identify certain trends, as follows:
(i) I n all cases where the wall temperature behaves in an anomalous
manner, it does so just before the bulk temperature reaches its critical
value.
(ii) There is a strong heat flux effect evident in all four sets of data;
that is to say, the heat transfer coefficient is strongly dependent on heat
flux. This is strikingly illustrated by Shitsman’s data (curves a, b, and c
on Fig. 10).
(iii) There is evidence both for a local improvement and also for a
local deterioration in heat transfer when the critical temperature lies
between the wall temperature and the bulk fluid temperature.
HEATTRANSFER
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33
Recent data have shown that one of the factors which is important in
forced convection is the orientation of the pipe. Of the above sets of data,
only Shitsman’s was obtained for a vertical pipe, the remainder being
horizontal. This matter will be dealt with in more detail in Section VI.
T h e experimental evidence for local increases and decreases in heat
transfer coefficient are summarized in the following.
“
/
Bulk fluid
temperalure
/
1750
2000
2250
2500
Bulk enthalpy (J/gm)
FIG. 11. Experimental wall temperature distributions as a function of local bulk
enthalpy along a pipe. p * = 1.15 (W. B. Hall, J. D. Jackson, and A. Watson, “Symp.
Heat Transfer and Fluid Dynamics of Near Critical Fluids.” Proc. Inst. Mech. Eng, 182,
Part 3 I (1968)). (a) Vikrev and Lokshin (16): q = 69.9 W/cm2, rit/A = 100 gm/sec cma,
d = 0.8 cm horizontal. (b) Vikrev and Lokshin (16): q = 69.9 W/cm2,1h/A= 40gm/seccm2,
d = 0.8 cm horizontal. (c) Schmidt ( 1 4 ) : q = 58 W/cmz,rit/A = 61 gm/sec cm2,d = 0.5 cm
horizontal.(d) Schmidt (14): q = 82 W/cm2,rit/A = 61 gm/sec cm2,d= 0.5 cm horizontal.
(e)Domin (15):q = 91 W/cma,rh/A = 101 gm/sec cm2,d = 0.2 cm horizontal.(f) Shitsman
(13):q = 39.6 W/cm2, rh/A = 44.9 gm/sec cm2, d = 0.8 cm vertical.
1. Local Increases in Heat Transfer Coefficient
One example of this has already been mentioned (Fig. 11, curve e ) .
A similar effect has also been found in experiments with CO,, three
examples of which follow:
(a) T h e data of Figs. 7 and 8 clearly show an enhancement in the
heat transfer coefficient for conditions in which the heat flux is small and
the critical temperature lies between the wall and bulk fluid temperatures.
W. B. HALL
34
As the heat flux is increased the enhancement becomes less marked.
It is important to note that these data were obtained with a downward
flow in a 1.905-cm diameter vertical tube; results for upward flow can be
quite different, as will be shown later.
2 .o
TPC
0
0
I
O I
0 O1
0
28
30
Bulk
32
34
temperature
36
38
("C)
FIG. 12. Variation of the heat transfer coefficient with bulk temperature for forced
convection in a heated pipe. Data of H. Tanaka, N. Nishiwaki, and M. Hirata, Turbulent
heat transfer to super-critical carbon dioxide. Nippon Kikai Gakkai Rombunshu ( 1 967).
(Carbon dioxide at a pressure of 78.5 bars flowing upwards in a 1.0-cm diameter vertical
pipe.) 0 Theory; ( A ) exp.: G = 140 f 4.4kg/hr, q = 1.44 W/cma. @ theory;
( X ) exp.: G = 140 f 3.1 kg/hr, q = 2.73 W/crne. @ theory; ( 0 )exp.: G = 280 f 5.6
kg/hr, q = 3.32 W/cm2. @ theory; ( 0 ) exp.: G = 280 If 7.8 kg/hr, q = 5.20 W/cm2.
(b) T h e data of Tanaka, Nishiwaki, and Hirata (17) are shown in
Fig. 12. Peaks in heat transfer coefficient occur when the bulk temperature is slightly below the transposed critical temperature, the peaks
being more marked when the heat flux is low, i.e., when the wall temperature is also close to (but slightly above) the transposed critical tem-
HEATTRANSFER
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35
Hot wall ternDeroture ("C)
FIG.13. Variation of heat flux through a 1 .O cm high channel (formed by horizontal
planes) with the temperature of the upper (heated) wall. (The lower (cooled) wall temperature is kept constant at 28.25"C.) (Carbon dioxide at a pressure of 75.8 bars; mass
velocity 37 gm/scc cm2). Data ofS. A. Khan, Ph.D. Thesis, University of Manchester, 1965.
perature. These results were obtained in a 1-cm diamater vertical tube
with upward flow.
(c) Hall, Jackson, and Khan (7) measured the overall heat transfer
coefficient for a flat duct 1 x 18 cm in cross section with one of the
longer sides heated and the other cooled. This arrangement produces a
situation which is basically simpler than that in a tube because it is
possible to arrange that the fluid temperature does not change in the
direction of flow. There is thus no convection, and the experiment is a
direct measure of the diffusive power of the turbulent stream. A sample of
the results is shown in Fig. 13 in which the heat flux is plotted against the
temperature of the heated wall, with the temperature of the cooled wall
as parameter. Again, there is a sharp increase in heat flux as the heated
wall passes through the transposed critical temperature (32°C in this
case), and the increase becomes larger as the cooled wall approaches
the transposed critical temperature. T h e flow in these experiments was
horizontal with the upper surface of the duct heated.
2. Local Decreases in Heat Transfer Coeflcient
There appear to be two distinct types of situations in which a local
reduction in the heat transfer coefficient occurs, both of which are
t
p = 2 4 5 bars
p - 2 4 5 bars
515
-
35 25
0
\
*
45-
8b
55-
7b
65 -
6b
75
5b
85 -
95 100
0
I
100
\ \a
200
4 4,h \
\
300
1
400
tw , t b ( O C )
FIG. 14. Wall temperature, t , , and bulk
temperature, te , as a function of distance ( x / d )
along a vertical heated pipe (1.6-cm diameter).
Upward flow of water at a pressure of 245 bars.
Data of M. E. Shitsman, “Symp. Heat Transfer
and Fluid Dynamics of Near Critical Fluids.”
Proc. Inst. Mech. Eng. 182, Part 31 (1968).
(1) m/A = 382 gm/sec cm2, q = 27 W/cm2.
(2) m/A = 382 gm/sec cm2, q = 37 W/cm2.
(3) &/A = 400 gm/sec cmz, q = 45 W/cmP.
(4) m/A = 375 gm/sec cm2, q = 52 W/cm2.
FIG. 15. Wall temperature, t , , and bulk
temperature, t B , as a function of distance ( x / d )
along a vertical heated pipe (1.6-cm diameter).
Downward flow of water at a pressure of 245 bars.
Data of M. E. Shitsman, “Symp. Heat Transfer
and Fluid Dynamics of Near Critical Fluids.”
Proc. Inst. Mech. Eng. 182, Part 31 (1968).
(5) m/A = 400 gm/sec cma, q = 27 W/cmz.
(6) A / A = 400 gm/sec cm2, q = 36 W/cmz.
(7) m/A = 393 gm/sec cm2, q = 43 W/cme.
(8) m/A = 381 gm/sec cm2, q = 50 W/cm2
HEATTRANSFER
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37
illustrated in Figs. 10 and 11. T h e data of Domin, Schmidt, and Vikrov
and Lokshin, all of which are for horizontal pipes, show rather broad
peaks of temperature at higher heat fluxes; similar peaks have been
reported by Griffiths and Shiralkar (18). Shitsman’s data, on the other
hand, show sharp peaks of temperature when the flow is upwards in a
vertical pipe.
(a) Figs. 14 and 15 show some of Shitsman’s data (19) in more detail.
Wall and bulk temperature are plotted against the dimensionless distance
x / d from the pipe inlet, for several (uniform) heat fluxes. Fig. 14 is for
upward flow and Fig. 15 for downward flow. I t is seen that while there
is no anomalous behavior for downward flow, the wall temperature for
upward flow rises to a sharp peak once a particular value of heat flux is
exceeded.
(b) A very similar behavior has been found by the author’s colleagues,
J. D. Jackson and K. Evans-Lutterodt (20) using carbon dioxide
(Fig. 16). These results form part of the same series as those presented
in Fig. 6 but in this case the flow is in the upward direction. T h e pipe,
of diameter 1.905 cm, had some 200 thermocouples distributed along
its length, so that the shape of the temperature peaks could be accurately
100 110
90 -
-
-
nu
80
-
70
-
2
2 60 ?
;
c
3
50
-
40 30
-
20
-
10
‘
I
0
I
20
I
40
Dlstance from
I
60
I
80
I
I00
I
I20
start of heating ( d i a m e t e r s )
FIG. 16. Temperature distribution along a 1.9-cm diameter vertical pipe for upward
flow. Carbon dioxide at a pressure of 75.8 bars and a mass flow of 160gm/sec. Wall heat
flux (a) 3.09 W/cma, (b) 4.05 W/cm2,(c) 5.19 W/cm2,(d) 5.67 W/cm*.
38
W. B. HALL
determined. Again, there is a sharp deterioration in heat transfer once
a particular heat flux is exceeded with the flow in the upward direction.
There is one interesting difference between these results and those of
Shitsman; while it seems that the deteriorations occur in the CO, results
only after the wall temperature has passed through the transposed critical
temperature, they occur in Shitsman’s experiments even when it is
substantially below. They do however also occur in some of Shitsman’s
experiments when the wall temperature has passed the transposed critical
temperature.
(c) T h e data of Tanaka et al. (Fig. 12) was obtained under conditions
rather similar to those described in (b) above, and yet no localized
deteriorations in heat transfer were found. T h e heat flux, mass velocity,
and fluid temperatures are all similar to those used by Jackson and
Evans-Lutterodt, but the tube diameter was 1 cm rather than 1.905 cm;
it will be shown later in Section VI that this difference can account for
the differences in heat transfer behavior.
T h e deteriorations that occur in horizontal tubes are generally less
localized than those in vertical tubes with upflow. Only in a few cases
has the temperature distribution around the tube been measured;
Fig. 17 shows the temperature distribution along the upper and lower
surfaces of a heated horizontal pipe carrying a flow of supercritical
pressure water (21);there is a very considerable difference in temperature,
corresponding to a reduction in the heat transfer coefficient for the upper
surface, when compared with the lower, by a factor of about four.
While such temperature variations may have been suppressed in other
experiments by conduction around the pipe wall, there could then have
been large variations in heat flux around the circumference of the pipe.
I t is probably not worth attempting detailed comparisons between sets
of data for horizontal pipes until the question of circumferential variations
has received much more attention.
3 . Gaps in the Experimental Data
I t has been clear for some time that many of the apparent discrepancies
between different sets of heat transfer data are the result of differences
in the experimental arrangement (sometimes wrongly assumed to be of
no importance and therefore inadequately described). As mentioned,
there is strong evidence to suggest that sharp local reductions in heat
transfer occur with upward flow only and that even then they can be
suppressed by a reduction in heat flux or tube diameter. Also confusion
has been caused by the presentation of results in dimensionless form
without giving the raw measurements or the property data used.
HEATTRANSFER
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39
FIG. 17. Temperature distributions as a function of local bulk enthalpy along heated
vertical and horizontal pipes (1.6-cm diameter). Data of Z. L. Miropolsky, V. J. Picus,
and M. E. Shitsman, Proc. Imt. Heat Tvansj'er Conf., 3rd, Chicago, 1966 Vol. 11, Paper
No. 50 (1967). Water at a pressure of 245 bars, I ~ / =
A 60 gm/sec cm2; q = 52 W/cm'.
(1) Horizontal pipe, upper surface. (2) Horizontal pipe, lower surface. (3) Vertical pipe,
upward flow. (4) Fluid temperature.
In spite of this rather confused situation, the writer believes that the
basic forced convection problem of steady flow in vertical tubes with
uniform heating could be resolved by a relatively small amount of
experimental work. Once it is recognized that buoyancy forces are
responsible for large differences between upward and downward flow
(see Section VI), the data begin to make sense. Because of difficulties
in generalizing physical property data it is still necessary to carry out
experiments with a wide range of fluids; there is already good data for
CO, , less detailed but reasonably satisfactory data for water, and rather
inadequate data for most other fluids. Experiments must cover upward
and downward flow, and should preferably involve a range of pipe sizes.
It would be helpful if the experiments could be planned so that it is
possible to determine the limiting situation as the temperature difference
tends to zero at a range of fluid temperatures spanning the critical
temperature. Very detailed pipe wall temperature measurements are
required both in the axial and the circumferential directions; Jackson
and Evans-Lutterodt (20),for example, use some 200 thermocouples on
40
W. B. HALL
a 1.9-cm diameter pipe of a length of 3 meters. Such experiments are not
to be undertaken lightly since they make large demands on experimental
skill if the results are to be reliable. It is to be hoped, therefore, that those
engaged in the work will maintain closer liaison with each other than
has been the case so far.
More detailed work will certainly be required on horizontal pipes and
also on the effect of heat transfer boundary conditions with both
horizontal and vertical pipes.
Note added in proof: A recent publication by B. Shiralkar and P. Griffith
[The effect of swirl, inlet conditions, flow direction and tube diameter on
the heat transfer to fluids at supercritical pressure, ASME Paper
No. 69-WA/HTl (1969)] provides an opportunity to compare three sets
of data on CO, with both upward and downward flow in a vertical pipe;
the other two sets of data are those of references (20) and (51).
T h e three sets of data differ only in the diameter of the test section
on which they were obtained; moreover it is possible to choose conditions
giving approximately the same Reynolds number in each case. In the
absence of buoyancy and acceleration effects (see Sections 111, 3 and
IV, C, 3, c) one would expect identical temperature distributions under
these conditions, provided that the data are compared at similar values
of qwd, where qw is the wall heat flux, and d is the pipe diameter. A study
of the basic equations (Section 111, B) shows that the acceleration effect
is the same for each pipe under the conditions imposed above; on the
other hand, the buoyancy effect, characterized by the parameter
Gr/(Re)1.8of equation (32), is certainly very different for the three cases.
A comparison of the relationships between the wall temperature and the
bulk temperature for the three sets of data might therefore be expected
to yield differences that are attributable to buoyancy effects.
T h e conditions for three approximately similar experiments, one
taken from each of the three sets of data, are shown in Table I ; in each
case results were obtained for upward and downward flow. Figure 17a
shows the wall temperature as a function of the bulk enthalpy for the
three cases shown in Table I. Shiralkar and Griffith’s data are represented
by a single curve since they found that there was no significant difference
between upward and downward flow in their tests. Very different results
were obtained for upward and downward flow in the larger pipes.
In both cases the sharp peaks mentioned in section IV, B, 2 were found
with upward flow. T h e results for the two larger pipes are very similar
in form; the differences in level may be accounted for by the fact that
it was not possible to choose identical values of pressure, Reynolds
number, and qwd for the comparison.
HEATTRANSFER
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CRITICALPOINT
41
TABLE I
COMPARISON
OF DATA
FOR CO,
Reference
Shiralkar and Griffith
Jackson and
Evans-Lutterodt (20)
Bourke et al. (54)
IN
VERTICAL
PIPES
Ra
d
(cm)
(W/cm2)
0.635
I .905
15.8
5.67
1.24
2.285
5.1
0.82
QW
1 .o
G“
$ q,d
(W/cm)
Pressure
(bars)
1
21
10.0
10.8
75.8
75.8
46.5
11.6
74.5
‘R = Reynolds number/Reynolds number (Shiralkar and Griffith); G = Grashof
number/Grashof number (Shiralkar and Griffith).
Three extremely interesting points emerge from the comparison:
(i) T h e sharp wall temperature peaks observed with the two larger
pipes are present only in upward flow whereas the rather broad
peak obtained in the small pipe is insensitive to flow direction.
Except
for the sharp peaks themselves, the wall temperatures
(ii)
for upward flow are everywhere less than those measured on the
small pipe.
(iii) T h e wall temperatures for downward flow in the two larger
pipes show no sign of peaks and are everywhere considerably
lower than those for the small pipe.
T h e results illustrate, in a rather striking manner, the remarks made
in Section IV, B, 2 concerning the two types of temperature peak that
have been observed with supercritical fluids; they are undoubtedly of
different origin (see also Fig. 10). Griffith and Shiralkar propose a
mechanism for the deterioration in heat transfer which depends essentially on physical property variations across the pipe. Thus when the
pipe wall passes through the critical temperature there appears at the
wall a low conductivity “gaslike” layer, the core remaining in a “liquidlike” state moving with a relatively low velocity; the heat transfer coefficient is therefore reduced. As a greater proportion of the fluid is heated
through the critical temperature, the flow velocity increases and the
heat transfer coefficient is thereby restored to something like its initial
value.
While the above mechanism may be valid in the absence of buoyancy
effects (i.e., at low values of Gr/Relas), it is radically modified when these
effects are large. A mechanism for the effect of buoyancy is proposed in
Section VI, where it is suggested that the shear stress distribution across
the pipe, and hence the turbulence production, is drastically modified.
42
W. B. HALL
1
0
FIG. 17a. Comparison of the data of Shiralkar and Griffith, and J. D. Jackson and
K. Evans-Lutterodt, Rept. N.E.Z., Simon Engineering Labs., University of Manchester,
1968, and P. J. Bourke, D. J. Pulling, L. E. Gill, and W. H. Denton, Atomic Energy
Research Establishment, Harwell, Rept. No. AERE. R5952, for forced convection of
carbon dioxide flowing upwards and downwards in vertical heated pipes. Key: (--.--.-)
Shiralkar and Griffith; (-)
Jackson and Evans-Lutterodt; (- - -) Bourke et al.
With upward flow the shear stress is rapidly reduced to zero in the core
as the wall passes through the critical temperature and is then reversed,
thus reestablishing turbulence production; the heat transfer coefficient
thus passes through a minimum and then increases. For downward flow
the effect of the buoyancy forces is always to increase the shear stress
in the core of the flow and thus to improve heat transfer. The effect
described by Shiralkar and Griffith may also be present, but at the higher
values of Cr/Re1.8, it appears to be completely dominated by the
buoyancy effect.]
HEATTRANSFER
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43
C . CORRELATION
OF EXPERIMENTAL
DATA
T h e great virtue of dimensionless correlations is that, by grouping
the variables, one is able to describe a particular situation by a smaller
number of parameters. With constant property fluids there is, in this
respect, no distinction between the physical properties and the other
parameters governing the flow; thus we may achieve Reynolds number
similarity between two systems by adjusting any of the component
parameters, p, d, u , or p. T h e situation is very different when the property
variations are large. Formally, one might introduce further dimensionless
ratios to describe the property variations; for example the variation of
viscosity with temperature could be expressed by a series of dimensionless
ratios
P(Tl)IP( Ts), P V 2 ) / P V J ,
P.(T J I P V s ) ,
etc.
where T , is some reference temperature. Unless the variation is particularly simple one might require a large number of such groups and thus
reach the rather ridiculous situation where the result of dimensional
analysis was a series of groups far greater in number than the original
quantities needed to specify the problem. I n this situation it is simpler
to specify the fluid, the boundary conditions (flow and heat transfer) and
accept the fact that there is little to be gained by comparisons between
different fluids.
This is perhaps an unduly pessimistic view of the situation as far as
supercritical fluids are concerned, for it is possible that one might be able
to make use of reduced coordinates (of temperature and pressure) to
describe property variations in similar classes of fluids in the critical
region. On the other hand it cannot reasonably be claimed that the use
of dimensionless parameters has done much, so far, to clarify the
situation.
T h e problems of correlation can best be illustrated by referring to the
momentum and energy equations developed in Section 111, (Eqs. (12),
(13), (15), and (16)). I t will be recalled that this particular form of the
equations, in which the shear stress and heat flux appear explicitly, was
chosen in order to facilitate comparison of the terms on the right hand
side of the equations. If we are to integrate the equations it will be necessary to express T in terms of the velocity gradient, and q in terms of the
temperature (or enthalpy) gradient. Suitable relationships were developed
in Section 111, C.
We confine our attention to turbulent flow in channels since this is
the case which has received the most attention experimentally. T h e
appropriate momentum and energy equations are obtained by combining
Eqs. (30), (31), (40), and (41) of Section 111. If all the terms in these
W. B. HALL
44
equations are retained, we find the trivial result that, for strict similarity,
two systems must be identical in every respect! (That is, we cannot
compare different sizes of systems using different fluids under different
flow conditions.) We therefore restrict our arguments to cases where the
dissipation, acceleration, and buoyancy terms are negligible although
we shall examine separately the very important buoyancy effect in
Section VI.
1. The Efects of Dissipation, Acceleration, and Buoyancy
In order to illustrate the nature of the above restrictions we shall
estimate now the magnitude of the dissipation, acceleration, and buoyancy
effects in the case of the results presented in Fig. 7. We take wall and
bulk temperatures of 35°C and 29"C, respectively, and the wall heat flux
will therefore be about 2.5 W/cm2. T h e other relevant quantities will
be approximately as follows:
po = 0.3
lit
=
gm/cm; pm
160 gm/sec; u,
= 0.7
M
gm/cm3;
70 cm/sec;
vm =
8.10-* cm2/sec; Re m lo5
cDm= 6
J/gm°C;
Prn = O.O38/OC.
(i) Referring to Section 111, B, 2, we see that dissipation may be
neglected if ReO.8 E
100 or, in this case
<
E
< 106)/Re0-8
=
T h e enthalpy difference between the wall and bulk conditions is about
100 J/gm or loDcm2/sec2. Thus E = u,2/dh M 5 x
T h e effects
of dissipation on the energy equation are thus seen to be negligible.
(ii) Referring to Section 111, B, 3, the criterion given by Eq. (34) is in
this case
PmAh/c,, = 0.64
and thus acceleration effects cannot safely be neglected. However, we
shall find that bouyancy effects (which have a rather similar effect on
&/ay near the wall when the flow is upwards) are even more important.
(iii) Referring to Section 111, B, 1, the criterion for the neglect of
buoyancy effects is
Gr/(Re)1.8 0.1
I n the present case
<
HEATTRANSFER
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Thus
THE
CRITICAL
POINT
45
Gr/(Re)1.8 = 5 x 10n/lOB= 5
This is greatly in excess of the value 0.1, and buoyancy effects cannot
therefore be neglected. This is confirmed by the very large difference
between the results for downward flow (Figs. 6 and 7) and upward flow
(Fig. 16). There may also be an acceleration effect present, and, in the
absence of buoyancy, this effect should be the same for upward as for
downward flow. I t is conceivable that the rather broad temperature peaks
mentioned in Section IV, B, 2 could be due to this rather than to
buoyancy. They do, in fact, seem to arise with horizontal or with small
diameter pipes and in both of these cases buoyancy effects would tend
to be less important.
Summarizing these very approximate calculations, we see that the
effect of dissipation is certainly negligible, acceleration effects may
be important, and buoyancy effects are certainly important, at any rate
when the flow is vertically upwards. None of the correlations so far
proposed have taken account of acceleration and buoyancy effects;
indeed the problem of correlation quickly becomes intractable if they
are included. We shall nevertheless proceed to deal with those cases
where both effects are small, recognizing, however, that the range of
applicability will be severely restricted.
2. Correlations in Which Buoyancy, Acceleration and Dissipation Are
Neglected
Under these conditions Eqs. (30) and (31), together with Eqs. (40) and
(41) reduce to:
(45)
These may be put into dimensionless form by transforming the
variables as follows:
Y
= y/b,
X
= x/b,
H
= h/Ah,
U
= tl/U,,,
Equations (45) and (46) then become
f
4
a
0 =-f--[-(1
2
Re EJY
-
_--
P
pm
1 12
au
+-Y”-Re/-i)-]
4
b2
ay
aiJ
ay
(47)
46
W. B. HALL
(Note: we have assumed that ah/aT = cp in Eq. (46), which implies
constant pressure. This is correct since the pressure is constant in the
y-direction.)
I n most forced convection systems the pressure variations in the
x-direction will also be small compared with the absolute pressure, and
we may therefore write
ahlax = c, a q a X
(Noting that for a supercritical fluid cp is a strong function of temperature.) Thus Eq. (48) may be rewritten in terms of B = ( T - Tv,)/
( T o- T,) = ( T - T,)/dT
T h e solution of the above equation will yield the temperature gradient
at the wall and, thus, the heat flux. T h e Nusselt number can then be
obtained
Nu = a4b/k, = q,,4b/KO AT = 4(a6/aY),
Thus we may write the solution formally as
where it is implied that pip,, etc., are expressed as functions of temperature.
Once again it is necessary to stress that the preceding arguments apply
only when it is permissible to neglect buoyancy forces and accelerations.
This is a severe restriction, and taken together with the necessity to
specify the fluid (because of the property ratios in Eqs. (50) and (51))
it is questionable whether the attempt to find a general correlation in
terms of dimensionless groups is worthwhile.
3 . Limiting Forms of Correlations
a . Small Temperuture Dajferences. T h e limiting situation as the
temperature difference AT tends to zero is one of constant physical
properties, and one might therefore expect that in this limit any correlation should reduce the same form as those for constant property fluids.
Measurements under such conditions tend to be somewhat inaccurate,
and it is a good plan to determine the Nusselt number by measuring
HEATTRANSFER
NEAR
47
THE CRITICAL P O I N T
the limiting slope of a curvc of heat flux against wall temperature as AT
tends to zero (see, for example, Fig. 7).
It should be noted that, as AT
0 and the fluid temperature everywhere becomes T I ) &it, is unnecessary to retain the property ratios,
p , ’ ~ , , ~etc.,
,
and the correlation then becomes
---f
Nu
=
Nu(Ke, Pr,n , X )
(52)
While it is certainly of interest to test whether any praposed correlation
is consistent with constant property data, the result does not throw much
light on the manner in which physical properties must be introduced
into the correlation when d T is finite.
b. Large Temperature DqfJerence. T h e form of property variation in
the region of the critical temperature is such that the greatest changes are
restricted to a rather small temperature range spanning the critical
temperature. I n particular, the rate of change with temperature soon
becomes small as the temperature increases above the critical value.
T h u s with turbulent flows in which the wall is heated and in which the
wall and fluid temperatures span the critical temperature, it may be
reasonable to treat the important region close to the wall as one having
constant properties equal to the values at the wall. There will still, of
course, be a layer of fluid in which the property variation is severe;
however the thickness of this layer will decrease as the temperature
difference increases. This suggests the use of a two-region model in
which the property variation is characterized by two discrete vaIues
(corresponding to the wall temperature and the fluid temperature, i.e.,
free stream temperature in a boundary layer or bulk temperature in a
pipe) rather than by the relationships given in Eqs. (50) and (51).
Introducing the ratios of properties at the wall temperature (subscript 0)
to those at the bulk temperature (subscript m),Eq. (51) becomes
It will be seen later that this forms the basis of many correlations; while
it may be made consistent with the constant property form as d T
0,
the arguments that may be adduced in support of it are, as shown, very
tenuous indeed. In particular, it should be noted that it is possible to
achieve the same two values of specific heat cpoand cpmin different waysone in which the two temperatures span the critical temperature and one
in which they do not. Such a form of correlation is unlikely to have any
general validity, although, as previously shown, it may be a reasonable
approximation when the temperature difference is large.
--f
W. B. HALL
48
c. Correlation Based on a Similar “Reduced” Temperature Distribution.
If we postulate that the physical properties are not significantly affected
by the pressure variations in the fluid, then it may be possible (Section 11)
to describe the properties in the following manner: Taking the case of
viscosity as an example,
(54)
P =Pmf(T*)/f(Tm*)
where pnLis the viscosity at the reduced temperature T,*
and T , is the bulk mean temperature. Thus
dpm
=.f((To* - Tm *) 0
=
T,/Tc,
+ Tm*)lf(Tm*)
where B = ( T - T,)/(To- T,,), the variable employed in Eq. (49).
T h e property variations can then be characterized by the parameters
To* and Tm*, the reduced wall and bulk temperatures. Equation (50)
would then become
Nu
= Nu(Re, Pr,
, X , To*,Tm*)
(5 5 )
This has the great advantage over Eq. (51) in that it may be used to
generalize data for different fluids, albeit with the restriction that To*
and T,* must have the same values for all cases.
T h e correlation is still extremely restrictive, particularly when one
recalls that, in addition to containing the parameters To* and Tm*,
it also involves implicitly the assumption that dissipation, buoyancy, and
acceleration effects are negligible. Nevertheless, it would be interesting
to attempt a correlation between different fluids within the limits imposed by these restrictions. Suitable data for such a comparison is at
present very limited.
4. Existing Correlations
Table I1 shows some of the correlations that have been used to
describe supercritical forced convection heat transfer. I t will be seen that
most are of the form used for constant property fluids, expressing the
Nusselt number as a simple function of the Reynolds number and
the Prandtl number, with extra terms involving property ratios. T h e
expressions by Petukhov et al. (22) and by Kutateladze and Leontiev (23)
reduce to the constant property form in the limiting case of small
temperature difference. I n all cases it is implicitly assumed that the
Reynolds number, Prandtl number, and property ratio effects are
separable, i.e., it is assumed that a change in Prandtl number or in the
property ratios does not affect the functional relationship between
HEATTRANSFER
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49
TABLE I1
CORRELATIONS
FOR VARIABLE
PROPERTY
FORCED
CONVECTION
Author/Fluid used
(symbol from Fig. 18)
Correlation
Mirapolsky and
Shitsman ( 0)/H2O
Nu, = 0.023 Re': Pr":
where Prmtn is the lesser of Pr, and Pro
Petukhov,
Krasnoschekov, and
Protopopov (m)/CO,
NU, = N ~ o ( p ~ / p O )(km/kO)-0.a3
~."
(c^,/~,~)'.'s
where Nu, = 0.1251 Re,Pr,/[12.7(5/8)1'2(Pr~'3
1 = 1/(1.82 log,, Re, - 1.64)2
and
t, = (h, - h,,,)/(T,- Tm)
Kutateladze and
Leontiev (A)
Nu,
Bishop, Sandberg, and
Nu, = 0.0069 Re': ~ r ~ 6 6 ( p 0 / p m )(1
0 ~ 4 s2.4/(L/D))
A
where Pr, = t p p m / k m
Tong ( v ) / H 2 0
=
0.023 Re:'
Pr~4[2/((p,/po)"2
-
1)
+ 1.071
+ l)]*
+
Swenson, Carver, and
Kakarala (+)/H,O
Nu, = 0.00459
~'813(p0/p,)0.231
A
where Pro = i?9po/k,
Touba and
McFadden ( x )/H,O
Nu,,
=
0.0068 Re:'
A
Pr, exp[2.19(hm/h,, - 0.801)]
Nusselt number and Reynolds number. There is no a priori justification
for this, and it has not yet been tested adequately by experiment.
T h e accuracy with which the correlations are claimed to fit their
respective data is of the order of 1 1 5 % . Some authors have made comparisons with data other than their own: e.g., Petukhov, Krasnoschekhov,
and Protopopov (22) report that data from the water experiments of
Miropolsky and Shitsman (11) and Dickenson and Welch (24, and the
carbon dioxide experiments of Bringer and Smith (25),lie within f20%
of their correlation. Figure 18 shows a comparison of the correlations
when applied to water at 254 bars ( p / p c = 1.15); the heat transfer
coefficient is shown as a function of the wall temperature for a constant
bulk temperature of 360°C. Agreement, in terms of the level of the heat
transfer coefficient, is seen to be poor; the trend, as the wall temperature
approaches the transposed critical value (387°C in this instance), is
fairly consistent, however.
W. B. HALL
50
Distributions of the heat transfer coefficient, a , along a pipe reveal
striking differences when compared with those for the constant property
case. Koppel (26),using CO, , made detailed measurements of the axial
variation of wall temperature; he observed that unusual variations of the
4.0r
I
Y
U
in
El 3.0 -
N
--.
7
L
C
‘?
.u
2.0 e
0
W
L
al
+
g
e
1.0-
*
0
I
I
I
0
360 370 300
I
I
I
I
390 400 410 420
W a l l temperature (“CI
I
I
430
440
I
I
450 460
FIG. 18. Heat transfer coefficient for water in a 0.8-cm diameter pipe as predicted
by various correlations (W. B. Hall, J. D. Jackson, and A. Watson, “Symp. Heat Transfer
and Fluid Dynamics of Near Critical Fluids.” Proc. Inst. Mech. Eng. 182, Part 31 (1968))
( p * = 1.15; tiz/A = 200 gm/sec cm2; T, = 360°C). Key: (m) Z. L. Miropolsky and
M. E. Shitsman, Zh. Tekhn.Fig. 27, No. 10 (1957); ( H) B. S.Petukhor,E.A. Krasnoschekhov, and V. S.Protopopov, Int. Develop. Heat Transfer, Proc. Heat Transfer Conf., 1961
(1963); ( A ) S. S. Kutateladze and A. I. Leontiev, “Turbulent Boundary Layers in Compressible Gases.” Arnold, London, 1964; (v) A. A. Bishop, R. 0. Sandberg, and L. S.
Tong, Forced convection heat transfer to water at near-critical and super-critical pressures.
A.I. Ch. E.-I. Chem. E. Symp. Ser. No. 2, 1965; (+) H. S. Swensen, J. R. Carver, and
C. R. Kakarala, J . Heat Transfer, November (1965);( x ) R. F. Touba and P. W. McFadden,
Combined Turbulent Convection Heat Transfer to Near Critical Water. Tech. Rept.
No. 18 COO-1177-18, Purdue Res. Foundation, Indiana, 1966.
heat transfer coefficient occurred along the tube depending upon the
proximity of the inlet bulk temperature to Tc and that local maxima
and minima in 01 could be produced. H e pointed out that these axial
variations could have contributed to the scatter among data based on
values of 01 at some arbitrary point along a tube. For low and moderate
heat fluxes, however, Koppel’s results were in agreement with the trend
indicated by the correlations, i.e., towards improved heat transfer under
near-critical temperatures.
HEATTRANSFER
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I t must be concluded that existing correlations are inadequate as a
means of predicting heat transfer in the critical region. This is hardly
surprising in view of the additional restrictions that are placed on
similarity by large physical property variations. Some of the experimental
data have been affected by parameters that are not even included in the
correlation; the orientation of the tube in conjunction with buoyancy
forces is noteworthy in this respect.
One of the alternatives to the conventional dimensionless correlation
is presented in the following section; this consists of solving the equations
of motion and energy numerically using empirical data on turbulent
diffusion. Unfortunately the technique of correlation focuses attention
on the problem of finding suitable mathematical functions to fit an
empirical result sometimes to the detriment of the physical understanding
of the phenomenon. A numerical method does not suffer from this
disadvantage.
D. SEMIEMPIRICAL
THEORIES
T h e difficulties encountered in any attempt to correlate empirical
heat transfer data by means of the usual dimensionless parameters stem
not so much from a lack of understanding of heat transfer mechanisms as
from the interaction between these and the variation of physical properties. This being so, it seems profitable to approach the problem from a
rather different angle and to attempt a numerical solution of the equations
of motion and energy into which physical property data may be fed in
tabular form. In the case of turbulent flow it is necessary also to feed
in empirical information concerning the effect of turbulence on the
diffusion of heat and momentum; sometimes this information is backed
up by an hypothesis concerning the mechanism by which turbulence
operates. Such calculations can be made for a range of conditions, and
the results compared directly with experimental data. This technique
has already proved useful with constant property fluids, and preliminary
results for supercritical fluids are quite promising.
T h e crucial assumption to be made in calculating heat transfer in a
turbulent flow is that concerning turbulent diffusion, and, unfortunately,
this has not yet been adequately tested in situations where the property
variations are large. Some of the ways in which the description of
turbulent diffusion may be affected have already been discussed in
Section 111, C. T h e mixing length model forms the basis of most theories,
and various methods have been used to allow for the effect of variable
properties on the magnitude of the mixing length.
52
W. B. HALL
1. Specijication of the Turbulent Shear Stress and Heat Flux'
As shown in Eqs. (40) and (41), the turbulent shear stress and heat
flux may be written in terms of the turbulent diffusivity 6 as follows:
rt = PE
aulay;
qt = p E ahlay
4t
aTPY
or, at constant pressure,
PCP"
This result followed from the mixing length theory. A more general
specification would allow the diffusivities in the two equations to take
different values, eM and e H , and in fact there is some evidence which
suggests that e H / e M= f (Re, Pr, y/a), and that eH is slightly greater
than eM for the values of Pr with which we are concerned (1 < Pr < lo),
(27)-(29).However, the dependence of eH/eM on Re, Pr, and y / a is not
well established even for constant properties in this range of Pr, and its
value under supercritical pressure conditions is likely to remain a matter
of speculation for some time, T h e differences between the semiempirical
theories are therefore centered around the manner in which they specify
E
~ Deissler
.
(30) attempted to predict the heat transferred to supercritical water at a reduced pressure, pip" = 1.8; he then proposed an
improved analysis (31) which was used for the case of common gases
at normal pressure but with large temperature differences. It is the second
of these analyses with which we are concerned. A two-region model was
employed in which the following expressions for e M were developed from
a dimensional analysis of the problem:
0 < y+ < y1';
y+
= y(T~/po)~/~/Vo
;
U+ = u / ( T o / p ~ ) ~ / ~
= +u+y+[ 1 - exp( -u&u+y+/v)],
yl+ < y+
< a+;
EM
_
-- K2(d~+/dy+)s
vg
(56)
(57)
(d2u'/dy+"*
Values of n, K , and yl+ were selected so as to give the best possible
agreement between predicted and measured distributions of u+ versus y+
for the case of unheated, fully developed turbulent pipe flow. Thus,
n = 0.124, K = 0.36, and yl+ was chosen as the value of y+ at which
E ~ / V= 1.92 ;these values were then assumed to apply also to the variable
This section is based on reference (12), and its content is very largely the work of
the author's colleague, J. D. Jackson.
HEATTRANSFER
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53
property case. Deissler put forward tentative arguments, based on the
effect of viscosity on turbulent diffusivity near the wall, in support of the
exponential damping terms in Eqs. (56) and (57). T h e equations give
results which are in good agreement with velocity distributions and
pressure drop-flow relationships for unheated pipe flow ; this is perhaps
not surprising in view of the fact that the three constants n, K , and y,+ are
available for adjustment. I t is well known that several different approaches
(e.g., Van Driest (32))give equally good results when applied to unheated
flows; this, of course, is no reason to expect the same methods to give
good predictions for variable property conditions.
We next consider a rather different approach which has been used by
Wiederecht and Sonnemann (33) for liquids with large property variations, and later by Hess and Kunz (34)for supercritical pressure hydrogen
( pipc = 1.5). I t is an application of the unheated flow model of Van
Driest (32), and is based on a modification of the Prandtl mixing length
model to include wall damping. Thus Van Driest obtained the following
expression for e M :
cM/vo= K2(y+)2[1
- exp(-y+/A+)I2 I du+/dy+ I
(58)
Good agreement with experiment for unheated pipe flow is obtained
with K = 0.4 and A+ = 28. Wiederecht and Sonnemann assumed that
Eq. (58) could be used under variable property conditions, but Hess and
Kunz suggested that the damped layer thickness (defined in terms of the
relaxation distance A+)should be allowed to vary; the property determining the value of A+ was taken to be kinematic viscosity, which appears
also in the damping terms of the Deissler expressions (Eqs. (56) and (57)).
By applying Eq. (58) to the supercritical hydrogen data of Hendricks et al.
(35),Hess and Kunz decided that A+ should vary according to the relationship A+ = 30.2 exp(--0.0285 vo/vnr).T h e Van Driest formulation is
convenient because it can be put into the form of a single expression for
E~ as a function of y+:
However, it will be apparent that the problem of deciding on the manner
in which kinematic viscosity variations might cause a change in the damping of turbulence in the wall layer is a difficult one, and the application
of the model under conditions of extreme property variation must be no
less tentative than Deissler’s model.
We turn finally to the model proposed by Goldmann (36) which
involves a rather different method of taking account of the effect of
W. B. HALL
54
property variations on eM . Goldman defines “variable property”
universal parameters for velocity and distance from the wall as follows:
and
Hence
and
du+/dy+ = (po/p)(dU++/dy++)
Goldman then assumes that u++ and y++ will be related under variable
property conditions in exactly the same manner that uf and yf are
related for the case of the unheated pipe flow; it follows that similar
relationships exist also between du++/dy++ and y+ and between du+/dy+
and y+. The effect of Goldmann’s hypothesis can be illustrated by using
Van Driest’s model; this gives
-EM
= - - [ 11 v- 1 1 + 4 7 [ K y
vo
2 vo
70
+
( 1 - exp ( - ~ ) ) ] e / l ’ z ]
VO
(62)
which may be compared with Eq. (59). Unfortunately there are insufficient accurate data on velocity profiles under variable property conditions
to check directly the basic assumption in Goldmann’s model.
2 . Application of the Theories to Supercritical Heat Transfer
Almost all the calculations that have so far been performed using the
discussed models of turbulent diffusion have assumed that the shear
stress distribution across the channel is affected to only a minor extent by
physical property variations. As we have seen, this constitutes a serious
limitation to their usefulness; in Section VI we shall attempt to make an
allowance for the effect of buoyancy forces in modifying the shear stress
distribution.
Comparisons between the various theories has been made difficult by
the fact that they have been applied to widely differing situations. Hall
et al. (12) have therefore obtained numerical solutions using three rather
different theories and employing in each case the same sets of physical
property data; the results of these calculations were then compared with
experimental data. While it cannot be claimed that the degree of agree-
HEATTRANSFER
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55
ment is satisfactory, certain trends were reproduced which were reflected
in the experimental data. Certainly, the agreement was no worse than
that between the empirical correlations and the experimental data.
I n spite of the rather limited success of the above methods at present,
the general line of attack seems to be worth following. I t will certainly
be necessary in future work to allow for buoyancy forces and possibly
also for acceleration effects.
V. Free Convection
Free convection systems were among the first in which the unusual
heat transfer properties of a supercritical fluid were demonstrated. In
1939, Schmidt et al. (37) made measurements on a loop filled with
ammonia at critical conditions; later measurements by Schmidt (38,39)
using ammonia and CO, in a closed vertical pipe, the bottom end of
which was heated and the upper end cooled, gave heat transfer rates
several thousand times greater than would be achieved by conduction
in a copper bar of the same dimensions. These effects were attributed to
the abnormally large values of the expansion coefficient and the specific
heat, and the relatively low viscosity of fluids near the critical point;
thus a large Grashof number could be achieved in a relatively small
system with a small temperature difference. By the same token, turbulent
conditions could readily be achieved.
In contrast to the case of forced convection, free convection data show
a fairly unambiguous trend towards higher transfer coefficients as conditions approach the critical. It appears fairly certain, in fact, that the
basic theoretical models established for constant property fluids are still
adequate although their application to supercritical fluids poses a
formidable problem which has not yet been satisfactorily solved for a
wide range of conditions.
Many of the difficulties encountered in predicting forced convection
are present with free convection also; the same reservations must be made
about the use of a heat transfer coefficient, and the problem of predicting
the effect of property variations on turbulent diffusion is at least as great.
I n the remainder of this section we shall refer only briefly to these common problems and shall concentrate on the points of difference.
A. EXPERIMENTAL
RESULTS
Most of the measurements of free convection with supercritical fluids
have been made using horizontal wires or vertical plane surfaces. I n
56
W. B. HALL
some cases the temperature difference has been kept small so that conditions approach those of constant properties while in others the temperature difference is large and frequently spans the critical temperature.
T h e two ranges of temperature difference are illustrated by the samples
of data discussed in the following; as in the case of forced convection
the data quoted are by no means exhaustive and are intended to illustrate
the more important trends.
1. Small Temperature Dtyeerences
Simon and Eckert (do), using an interferometer to measure density
variations, experimented with a heated vertical plate in CO,; the sensitivity of their technique allowed them to employ extremely small
temperature differences (0.001"C to 0.01"C). In addition to measuring
the overall density difference from fluid to heated plate, they were able
to determine the density gradient, and hence temperature gradient, at the
plate surface; from these measurements, together with the heat flux,
they were able to compute not only the heat transfer coefficient but also
the thermal conductivity at the wall. They found that the heat transfer
coefficient and also the thermal conductivity increased wih the heat flux
when the fluid density was close to its critical value. T h e smooth curves
which they fitted to their results have been replotted in Fig. 19 so that
a comparison can be made between the relative increases in heat transfer
coefficient and thermal conductivity. T h e full lines represent the heat
transfer coefficient, a, normalized by dividing by its value at a fluid
density of 0.4 gm/cm3, and a heat flux of 5.75 x 10-6 W/cm2; similar
curves of the normalized thermal conductivity are shown by broken lines.
I t will be seen that the increase in the heat transfer coefficient is generally
somewhat greater than the increase in thermal conductivity; both are
very significant, and show peaks near the critical density (0.468 gm/cm3).
The striking aspect of these results (Fig. 19) is the fact that in all
cases the temperature of the fluid is everywhere within 0.01"C of the
value measured at some distance from the plate. This temperature range
would normally be expected to result in quite negligible property variations (ens., a temperature difference of 0.01"C would give a density
variation of about 0.2% under these conditions and the variations in
conductivity and viscosity would be of the same order). If the measurements of thermal conductivity are to be believed, therefore, they must
represent an effect of heat flux rather than temperature. Simon and
Eckert suggest that the effect may be connected with the existence of
clusters of molecules in the fluid at near-critical conditions. Thus they
write, "one might agree that the formation and break up of such clusters
HEATTRANSFER
NEAR THE CRITICAL
POINT
57
3.0-
2.01
\
-r
\
U
*
1.0-
0.4
0.45
Density (gm /cm3 )
0.5
0.55
FIG. 19. Replot of data of H. A. Simon and E. R. G. Eckert [Laminar free convection
in carbon dioxide near its critical point. Intern. /. Heat Muss Trunsfer 6,681-690 (1963)]
for free convection from a vertical plate to CO, . Heat transfer coefficient, a, and thermal
conductivity, k, normalized with respect to the values at a fluid density of 0.4 gm/cmS
W/cma (ar,k,). (a) heat flux = 2.11 x
W/cm2;
and a heat flux of 5.75 x
(b) heat flux = 3.72 x
W/cm2;(c) heat flux = 5.75 X 10-6W/cm2.Key:(-)a/a,;
(-
- -) k/k, .
near the heated plate surface through the shear within the boundary
layer is the cause of the dependence of the thermal conductivity on heat
rate." No further light appears to have been thrown on the subject by
subsequent work. While there does not appear to be any obvious reason
to suspect the use of an optical measurement of density under these
conditions, it would perhaps be useful to attempt to repeat the measurements of the heat transfer coefficient using direct measurements of the
wall and fluid temperatures.
Dubrovina and Skripov (41) have measured the heat transfer for a
horizontal wire (29 x IO-O-cm diameter) to CO, in the region of the
critical point. Most of their measurements were made at a temperature
difference of 0.5"C; this is considerably larger than that used by Simon
and Eckert, and the property variations throughout the fluid will certainly
be significant (a temperature variation of 0.5"C around a temperature
of 32°C and a pressure of 75.8 bars gives a density variation of about 20
to 30%). On the other hand, the small size of the wire reduces the
Grashof number, for given fluid conditions, by a factor of about lo9
compared with that for Simon and Eckert's experiments. Their deter-
W. B. HALL
58
mination of the heat transfer coefficient as a function of pressure, for
a number of fluid temperatures, is shown in Fig. 20. Unfortunately,
it is not possible to make comparisons with Simon and Eckert’s data
because of the difference in shape of the heat transfer surface. It is worth
noting, however, that Dubrovina and Skripov find that for conditions
close to the critical, the heat transfer coefficient increases as the temperature difference is decreased.
I
12-
3
2
10 -
.*
N
E
:0 8 -3
-
-3
c
?
2
06-
0
W
L
W
4
t 04-
FIG. 20. Free convection heat transfer coefficient for a 2.9 x 10-%rn diameter wire
in carbon dioxide as a function of pressure. Data of E. N. Dubrovina and V. P. Skripov, in
“Heat and Mass Transfer” (A. V. Lykov and B. M. Smol’skii,eds.), Vol. I. Israel Program
for Scientific Translations, 1967. (1) Fluid temperature 31.S”C; (2) Fluid temperature
32.0”C; (3) Fluid temperature 34.0”C; (4) Fluid temperature 37.0”C.
2. Large Temperature oisferences
“Large” means, typically, greater than 1°C. I n such cases there may
be variations in density by as much as a factor two from point to point in
the fluid, and it is not relevant to make comparisons with constant
property data.
Doughty and Drake (42) made measurements of the free convection
HEATTRANSFER
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59
from a 0.025-cm diameter horizontal wire in Freon 12. For fluid close
to the critical temperature and pressure, they found that the maximum
heat transfer coefficient occurred with a temperature difference of about
5°C (An error of a degree or two in the temperature of fluid or wire would
make a very great difference to the computed value of the heat transfer
FIG. 21. Heat transfer from a horizontal 0.025-cm diameter wire to carbon dioxide
at a pressure of 89.6 bars. (Fluid temperature 9.5"C; wire temperature 154°C; heat flux,
46.5 W/cm2.) Data of K. K. Knapp and R. H. Sabersky, I n t . J. Heat Muss Transfer 9,
41-51 (1966). Magnification x 10 approx.
W. B. HALL
60
coefficient as the temperature difference decreases; it is possible that
such errors may account for the apparent discrepancy between these
results and those of Dubrovina and Skripov (41) which indicate a
maximum heat transfer coefficient as A T -+ 0).
Knapp and Sabersky (43) reported the appearance of a “bubble-like”
flow in experiments with a heated wire in CO,; Fig. 21 illustrates this
phenomenon at a pressure well above the critical, and with fluid and wire
temperatures spanning the critical. It is very difficult to accept the idea
of a sharp “phase boundary” under these conditions, and yet the
“bubbles” beneath the wire do give such an impression. Short of a gross
error in determining the fluid conditions, which is highly unlikely, the
only other possibility seems to be contamination by a second component.
I n this context, Draper (44) has reported the appearance of a separate
phase in the form of droplets on a heated wire in sulphur hexaflouride;
in this case it was identified as a separate component, but beyond the
fact that it contained water, its composition was not established.
Boiling and free convection from a heated horizontal wire immersed
in sulphur hexaflouride have been studied by Draper, Figs. 22-24 show
his results for supercritical conditions (results for subcritical pressures
shown in Figs. 30-32). Each figure refers to one fluid temperature
(the critical temperature is 456°C) and on each are plotted several
- 30
x
=
I
‘I00
Temperature difference, To -Ts (“C)
FIG. 22. Free convection heat transfer from a 0.01 I -cm diameter wire immersed in
supercritical pressure sulphur hexaflouride.Fluid temperature 23.3”C.
(Data of R. Draper,
M.Sc. Thesis, University of Manchester, 1968.)
HEATTRANSFER
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61
FIG.23. Free convection heat transfer from a 0.01 I-cm diameter wire immersed in
supercritical pressure sulphur hexaflouride. Fluid temperature 39.6"C.(Data of R. Draper,
M.Sc. Thesis, University of Manchester, 1968.)
FIG.24. Free convection heat transfer from a 0.011-cm diameter wire immersed in
supercritical pressure sulphur hexaflouride. Fluid temperature 43.1% (Data of R. Draper,
M.Sc. Thesis, University of Manchester, 1968.)
W. B. HALL
62
curves of heat flux against temperature difference for a range of pressures
(the critical pressure is 37.7 bars). It is of interest to note that the whole
range of results (for different fluid temperatures and pressures) fall
within & l o % of a single curve. I n all cases the fluid temperature is
below the critical, and the measurements were not sufficiently sensitive
to determine accurately any anomalous results as the wire temperature
passed through the critical temperature. For the greater part of the range
the temperature difference is sufficiently large to establish something like
the "two-region" pattern described in Section IV. Reference to Figs. 3032, which are for subcritical pressures, shows that the results for film
boiling are also very close to those for supercritical pressures.
U
0
10
Temperature
20
30
d i f f e r e n c e ("Cl
FIG. 25. Free convection from a 6-in high, 0.5-in wide vertical ribbon to water
at a pressure of 223 bars. Data of J. R. Larson and R. J. Schoenhals, J. Heat Transfer,
November, 407 (1966). Bulk temperature: (*) 374"C,( A ) 368.5"C,( x ) 376.8"C,( 0 )379"C,
( 0 ) 383"C, (+) 385.6"C. T,,.= 374.9"C.
Free convection data for a 15-cm high vertical flat plate in supercritical
pressure water have been obtained by Larson and Schoenhals (45)' and
are shown in Figs. 25 and 26. When comparing these data with Draper's
data (Figs. 22-24), it should be noted that in Fig. 25 only two and on
Fig. 26 only three of the curves refer to conditions in which the fluid
temperature is below the transposed critical temperature (whereas all
Draper's data were for this situation). For these curves there appears to
be a stronger effect of fluid temperature than was obtained by Draper;
HEATTRANSFER
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63
however the fluid temperatures are relatively closer to the critical, and
the range of temperature difference is much smaller. T h e conditions are
also different in that Draper used a horizontal wire.
n
"0
I
10
Temperature
I
20
30
difference ("C)
FIG. 26. Free convection from a 6-in high, 0.5-in wide vertical ribbon to water at
a pressure of 228 bars. Data of J. R. Larson and R. J. Schoenhals, J. Heat Transfer,
November, 407 (1966). Bulk temperature: (+) 375.8"C, (0)376.4"C, (0)376.7"C,
( x ) 365.8"C, ( A ) 377.9"C, (*) 378.3"C, (0)
382.9"C. T,, = 376.5OC.
B. THEORETICAL
METHODS
AND CORRELATIONS
Most of the free convection problems that arise in engineering are
likely to give rise to turbulent boundary layers. This may be illustrated
by considering the case of a vertical plane surface at a temperature, T o ,
of 400°C, immersed in water at 250 bars (critical pressure, 221.2 bars) at
a temperature, T , , of 370°C (critical temperature, 374°C). T h e relevant
physical properties are as follows:
ps = 0.540 gm/cm3;
p,$= 6.35
x
po = 0.166
gm/cm sec;
gm/cm3,
Y, =
:.
( p s - po)/ps = 0.692
1.17 x 10-9
cm2/sec
W. B. HALL
64
Thus the Grashof number,
Gr -"
8
gx3 - 5.0
x 108x9
Ps
v2
If we assume that the criterion for transition to a turbulent boundary
layer is still approximately the same as for fluids with small property
variations, (i.e., G r M 2 x log), then this would occur in the above
example with a plate 1.5 cm high.
1. The Basis of Correlations
T h e dimensionless parameters governing turbulent free convection
may be obtained from Eqs. (14), (20), and (21), together with Eqs. (40)
and (41). We proceed in a similar manner to that employed in Section
IV, C, 2, noting, however, that in this case there is no characteristic
velocity which may be extracted as a parameter. Thus we employ the
following change of variables:
X
= x/d;
Y =y/d;
6' = ( T - T,)](To - T,)
=
H
=
(h - h,)/(ho - h,)
( T - Ts)/AT;
Ul
=
(h - hs)/Ah;
= ud/vs ;
Vl
= vd/vS
We also assume, as in Section IV, C, 2, that dissipation effects are
negligible, and also that u, = 0. T h e following equations, analogous
to Eqs. (47) and (48) for free convection, are then obtained:
pU,aR
-ps
,iiaH
ax+p,=
where
If we again assume that pressure variations throughout the system are
small compared with the absolute pressure, we obtain the equation
analogous to Eq. (49):
HEATTRANSFER
NEAR
THE
CRITICAL
POINT
65
Thus, we see that the solution is formally similar to that for forced
convection, except that the Grashof number replaces the Reynolds
number. Thus
Similar arguments to those used in Sections IV, C, 3, a , b, and c can
also be proposed. With reference to the limit as LIT --f 0, it has already
been pointed out in Section 11, D, 2, c that the Grashof number does
not become infinite at the critical point as has sometimes been suggested.
Nevertheless, the problem remains more difficult than that of forced
convection because of the interconnection between the momentum and
energy equations (the term ( p , - p ) / ( p , - po), which is a function of
temperature, links Eqs. (63) and (64)); thus the energy equation remains
nonlinear in temperature even if the physical properties are constant.
It would be interesting to follow up the possibility of using reduced
coordinates to make comparisons between different fluids. As with
forced convection, this would involve the comparison of data at the same
values of reduced pressure, p * and the same values of reduced surface
temperature To* and reduced fluid temperature, T,*. It is doubtful,
however, whether sufficient data exist at the present time to make such
a comparison.
2. Theoretical Calculations
It seems likely that numerical integration of the basic equations will
eventually prove the most profitable line of attack although the difficulties
to be overcome are greater than with forced convection. As has been
demonstrated previously, most practical problems are likely to involve
turbulent conditions, and far less is known about turbulent diffusion in
free convection than in forced convection boundary layers. For want of
a better model, one might adopt similar descriptions of, for example, the
mixing length, 1, in Eqs. (63)-(65) as those which are used in forced
convection; this hypothesis has not yet been tested for constant property
fluids, however.
A number of attempts have been made to solve the free convection
equations for nonturbulent conditions, usually by means of numerical
methods. Some of these involve the use of the same similarity variables
as are used in the constant property case, which seems questionable.
Unfortunately it has not been possible adequately to test the calculations;
this is not surprising when one considers the ease with which turbulence
is produced, and the small temperature differences that are required if
66
W. B. HALL
it is to be avoided. Some of the calculations are described in references
(46)-(49).
VI. Combined Forced and Free Convection
It has long been recognized that the very large density differences and
small kinematic viscosities that occur with supercritical fluids produce
ideal conditions for free convection. It is therefore surprising that more
attention has not been directed towards free convection effects in forced
flows. Most of the forced convection experiments have not, in fact, been
designed to detect such effects; a change of flow direction in a vertical
pipe, or measurements around the circumference of a horizontal pipe,
are necessary for this purpose. One of the earliest investigations in which
such effects were clearly present in a horizontal pipe was presented by
Mirapolsky et al. (50). Results for upward and downward flow in vertical
pipes have been reported more recently by Shitsman (19) using water,
by Jackson and Evans-Lutterodt (20) and by Bourke et al. (51) using
CO, . Shitsman clearly identified the sharp deteriorations in heat transfer
coefficient which he observed for upward flow with a free convection
effect; Hall et al. (12) proposed a mechanism for the effect in terms of the
redistribution of the turbulent shear stress across the pipe.
There is now little doubt that localized reductions in heat transfer in
vertical pipes can be caused by buoyancy effects. I t was very difficult
to understand, until a detailed mechanism was proposed, how free convection could combine with an upward forced convection to give a lower
heat transfer coefficient than that measured with forced convection alone.
This was particularly the case since the opposite effect had been predicted
by a number of theoretical investigations of combined forced and free
convection under nonturbulent conditions. (The model we shall describe
refers to turbulent flows.) One interesting sidelight on the problem is
the close connection between it and the phenomenon of “laminarization”
of a turbulent boundary layer by means of an acceleration of the flow (52).
Indeed, attempts were made to describe the supercritical heat transfer
deteriorations in terms of an acceleration of the mean flow (following the
rapid reduction in density as the fluid is heated through the critical
temperature), but it soon became clear that buoyancy forces were more
important than acceleration forces in this case. It seems likely that in
both phenomena a reduction of turbulent diffusion follows from a
modification of the shear stress distribution; this modification is produced
in the one case by inertia effects in an accelerated flow and in the other by
buoyancy forces in a flow with large density differences.
HEATTRANSFER
NEAR
THE
CRITICAL
POINT
67
Apart from the work of Miropolsky et nl. (50) there is very little
detailed information on free convection effects in horizontal pipes. This
is a very important area in which experimental data will be required
before many practical applications can be adequately assessed. For the
remainder of the present section, however, we shall restrict the discussion
to vertical pipes.
A. EXPERIMENTAL
RESULTS
T h e most revealing comparison in the present context which can be
made is that between upward and downward flow in a heated vertical
pipe; this is illustrated in Figs. 6 and 16, both of which are based on the
work of Jackson and Evans-Lutterodt (20). T h e curves for two particular heat fluxes are superimposed in Fig. 27 to ease comparison. It will
be seen that at a low heat flux there is only a small difference between the
results for the two flow directions, but as the heat flux is increased,
a large peak occurs in the wall temperature for upward flow; as the heat
flux is increased further, the peak sharpens and moves towards the inlet
of the pipe. A very similar behavior has been obtained by Shitsman (19)
and by Bourke et al. (51). T h e peak occurs, typically, when the surface
temperature is above and the fluid temperature below the critical
temperature, although Shitsman has observed similar effects when both
temperatures, just before the peak, are below the critical value. T h e magnitude of the circumferential variation of temperature around the pipe,
shown by vertical lines in Fig. 27, is seen to increase in the region of the
peaks in upward Aow.
T h e results presented in Figs. 6 , 16, and 27 form part of an extensive
range of data in which the fluid inlet temperature, the mass flow, the
pressure, and the heat flux were varied (9). T h e temperature peaks are
present for upward flow at all pressures investigated (from the critical
pressure to about 1.1 times the critical pressure) and for all Reynolds
numbers (covering the range 2.5 x lo4 to lo5). T h e peaks generally
disappear as the fluid temperature at the pipe inlet approaches the critical
temperature and are replaced by a uniformly lower heat transfer coefficient (when compared with the corresponding downward flow results).
There is also a tendency for an increase in pressure or mass flow to raise
the heat flux at which the peaks begin to occur.
Were it not for the fact that the temperature peaks occur only with
upward flow, one would be tempted to explain the phenomenon in
terms of a reduction in thermal conductivity in the viscous wall layer as
the fluid in this region passes from a subcritical to a supercritical temperature. (This effect has, in fact, been shown not to result in a reduced
W. B. HALL
68
heat transfer coefficient (7) presumably because of the corresponding
thinning of the viscous layer as the viscosity decreases). The same
difficulty arises with any explanation based on the acceleration of the
flow as it is heated; were this the case, it should presumably occur with
downward as well as upward flow.
110
A
-
100 -
90 -
-:
80 -
0
70-
608
n
5
50-
:
c
4030 20
I
20
Distance
I
40
I
60
I
80
I
100
I
I20
I
from start of heating (diameters)
FIG.27. Comparison of data of Figs. 6 and 16 for upward and downward flow.
(a) Wall heat flux = 3.09 W/cma, (b) Wall heat flux = 5.67 W/cm2.(-)
upward flow;
(- - -) downward flow.
B. A PROPOSED
MECHANISM
FOR
THE
HEATTRANSFER
DETERIORATIONS
The following analysis is substantially the same as that described by
Hall and Jackson (52). I t may assist the reader to follow the argument if
it is preceded by a brief statement of the more important steps.
We begin by evaluating the approximate magnitude of the buoyancy
forces in a typical case. It is found that even a very thin low density layer
near the pipe wall is sufficient to generate forces which are of the same
order as the turbulent shear force on the pipe wall. The shear stress
distribution across the pipe is therefore drastically changed, and with it
the amount of energy being fed into the turbulence (c.f. Section 111, C, 2.)
Consequently the level of turbulence, and with it the turbulent diffusivity,
decreases. At a later stage of development of the low density layer adjacent
HEAT TRANSFER
NEAR
THE
CRITICALPOINT
69
to the wall, the shear stress is reversed, the energy input to the turbulence
is restored, and with it the heat transfer coefficient.
1. The Effect of Buoyancy Forces in Modifying the Shear Stress Distribution
T h e sharpness of the density change in passing from a subcritical
to a supercritical temperature allows one to use, with a fair degree of
accuracy, a two-region pattern of density when calculating the shear
stress distribution across a pipe. We assume that there is a layer of fluid
at a uniform low density pw adjacent to the wall, and that the core is at
a uniform high density pe; we also neglect inertia effects in the wall layer.
T h e result is a modification to the shear stress distribution as shown
diagrammatically in Fig. 28 for three different thicknesses of wall layer.
T h e particular thickness, A, for which the shear stress at the edge of
the layer (and throughout the core) is zero, is calculated below.
Typical values of pw and pe corresponding to carbon dioxide are
Pipe
‘
E
l
i
i
1:
c ,
Y)
P ’
Y ) ’
o /
r,
I
v)
I
,
/
0/
/
/
,
/
/
I
I
FIG. 28. Diagram showing the shear stress distribution across a channel for three
thicknesses of the low density Iayer adjacent to the wall (upward flow).
W. B. HALL
70
0.3 gm/cms and 0.7 gm/cm3, respectively. (These values are appropriate
to wall and bulk temperatures of about 35°C and 29°C.) If the shear stress
is to be reduced to zero at y = A, then the buoyancy force per unit length
of pipe, B , must balance the wall shear force per unit length of pipe, S.
T h e latter force increases slightly as the wall layer thickness increases (53),
but the value for a uniform density equal to that in the core is still
a good approximation when the shear stress in the core is reduced to zero;
at a Reynolds number of lo5 in a 1.9-cm pipe this gives a value for S of
about 15 dyn/cm.
Thus
and
B
h
= ndh(p,
- pw)g
=
S
= S / d ( p , - pw)g 11
= 6.4
5 / ~x 1.9 x 0.4 x 981.
x lo-* cm
Thus a very thin layer of low density fluid at the wall is capable of reducing the shear stress in the core to zero. This will have a profound effect
on the production of turbulence and, therefore, on the turbulent diffusivity, as shown in the following section.
2. The Effect of the Shear Stress Distribution on Turbulence
T h e turbulence in the flow is maintained by an energy input which
arises from the shearing of the turbulent fluid by the mean velocity
gradient; this input is equal to r 1 afijay, and has its greatest value close
to the wall. (Not at the wall, because the flow there is nonturbulent, and
71 = 0.) One might suspect, therefore, that the modifications to the
shear stress shown in Fig. 28 reduce the level of turbulence and thus the
turbulent diffusivity, e; this will certainly be the case in the core where
both r d and ai?/ay become zero. I n the wall layer the net effect will
depend upon the relative changes in T~ and &jay, and can be estimated
by using a mixing length model of turbulent diffusion.
T h e mixing length model asserts that
rt =
-
p uw
= pi2
I aqay I aqay
T h e turbulent diffusivity is then given by
= Tti,j aqay = i ( T t l , j ) l / z
Thus, provided that the mixing length 1 is not changed significantly
by the density variation, a reduction in total shear stress will involve a
reduction in T~ and an associated reduction in the turbulent diffusivtiy.
HEATTRANSFER
NEAR
THE
CRITICALPOINT
71
As shown diagrammatically in Fig. 28, the shear stress very close to
the wall is increased by the presence of the low density layer. However,
this increase will usually be in a region in which turbulence is damped
out by the wall and in which the turbulent shear stress is small. T h e
increase will not, therefore, give rise to an increase in turbulence production.
T h e above analysis is open to question on at least two grounds; firstly,
it is by no means certain that the mixing length will be unaffected by the
presence of the low density layer; secondly, and perhaps more fundamentally, the use of a mixing length model may not be justified in these
circumstances. I t was mentioned in Section 111, C, 2, that a description of
the distribution of turbulence based on the turbulent kinetic energy
equation (which balances the production, dissipation, convection, and
diffusion of the kinetic energy of turbulence) reduces to the mixing length
model for an equilibrium boundary layer when the convection and
diffusion terms in the equation are omitted; this implies local equilibrium
between the production and dissipation of turbulence. I n the present
application it is likely that the turbulence level will change quite rapidly
in the direction of flow and that convection of turbulent kinetic energy
will become important. Nevertheless, it is felt that the mechanism
proposed above is qualitatively correct.
3 . A Mechanism for the Local Deteriorations in Heat Transfer Coejicient
T h e consequence of the events described previously is that the turbulent diffusivity is reduced in upward flow when the low density wall
layer becomes thick enough to reduce materially the shear stress in the
region where energy is normally fed into the turbulence. This will,
of course, reduce the diffusivity for heat, and, therefore, the heat transfer
coefficient. As the process develops along the tube the wall temperature
rises, the density difference becomes greater, and the buoyant layer
thickens; both these effects accentuate the laminarization. It is possible
that the low density layer eventually becomes sufficiently thick for the
sign of the shear stress to be changed in the central region, as shown in
Fig. 28. T h e wall layer will then exert an upward force on the core:
the production of turbulence will be restored, and the turbulent diffusivity will increase. T h e model, therefore, accounts not only for the
deterioration of the heat transfer coefficient but also for its subsequent
improvement.
This progression of events is illustrated in Fig. 29 which is based on
an approximate theoretical model of the flow of supercritical pressure
CO, between parallel planes 1.5 cm apart at a Reynolds number of
W. B. HALL
72
lo6 (53).It was assumed in the calculations (53) that the flow in the low
density wall layer was nonturbulent, and the velocity distribution in the
core was described by a “velocity defect” law in which the characteristic
shear stress was taken as that at the interface between core and wall
layer rather than that at the wall. Values of y+ at the interface are marked
on the curves in Fig. 29 and show that the assumption of nonturbulent
flow in the wall layer is a reasonable approximation. A comparison is made
in (53)between this simple model and one that uses a van Driest formulation of the velocity distribution, which avoids the necessity for separate
treatment of the wall layer and the core.
- 20
Shear stress
0
1
2
3
4
5
Distance
6
7
from wall
8
9
1
0
1
1
1
2
(10-3cm)
FIG.29. Calculated velocity and shear stress distribution near the wall of a channel
with upward flow and three thicknesses of low density layer adjacent to the wall. (W. B.
Hall and J. D. Jackson, Laminarisation of a turbulent pipe flow by buoyancy forces.
1l t h National Heat Transfer Conf., Special Session on Laminarization of Turbulent
Flows, Minneapolis, Paper No. 69-HT-55, 1969. W. B. Hall, The Effect of Buoyancy
Forces on Forced Convection Heat Transfer in a Vertical Pipe. Rept. N.E.1, Simon
Engineering Labs., University of Manchester, 1968.)
It is interesting to note that when the shear stress falls to zero at the
interface, (curve B of Fig. 29), the core is completely decoupled from the
wall, and is moving upwards as a true “plug flow.” The wall layer, on
the other hand, is entirely motivated by buoyancy forces, which are
acting in precisely the region where they are required to overcome
viscous shear (rather than being transmitted by a shear process from the
core-a process which normally maintains the production of turbulence).
HEATTRANSFER
NEAR
THE
CRITICAL
POINT
73
If the wall layer is so thin that turbulence in it is supressed by the wall,
then the whole flow could, if the situation persisted long enough, become
nonturbulent even though the Reynolds number was well above the
transition value.
An approximate criterion for the conditions under which heat transfer
may be impaired by buoyancy forces can be obtained by the methods
that were developed in Section 111, €3, 1. In this case we apply the condition that the modification to the shear stress distribution is not merely
significant (with respect to the initial distribution given by &lay =
- ~ , / b ) , but large enough to reduce the shear stress to zero where it
really matters, i.e., in the region y + w 30, where turbulence production
is normally a maximum. If this is to happen (Fig. 28),
&/ay = - ~ ~ / h , where
and
(pc - p w ) g must
h = ~OV,,/(T,/~,)~/*
be comparable in magnitude to ~ , / h i.e.,
,
-
This may be expressed in the form
Gr/Re2.7
where
1.2 x 10-4
(In obtaining Eq. (67) it has been assumed that the usual relationships
between T~ and Re apply, i.e., T~ =f ip&u,Z and Q f= 0.023 Re-Oe2, and
that vc/vzo 1).
If we insert the data appropriate to Figs. 6 and 16 (i-e., that quoted in
Section IV, C, I), we find that,
-
Gr/Re2.7m 0.7 x lo4
Thus it appears that the criterion established in Eq. (67)is approximately satisfied in the case of these results, (which certainly show
buoyancy effects). The main difficulty in applying the criterion lies in
the selection of suitable values of pw and pc . The above analysis is based
on the assumption that a “two-region” model is adequate, as far as the
density variation across the channel is concerned; the validity of this
assumption depends very much on the wall heat flux level. This matter is
discussed in the following.
74
W. B. HALL
4. The Injuence of Heat Flux
The experimental results show that at low heat fluxes there is no
sudden deterioration in heat transfer and that the wall temperature
distribution for upward flow is not greatly different from that for downward flow. As the heat flux tends to zero, the temperature variation (and
hence the density variation) across the flow becomes so small that a
thick wall layer would be required in order to affect the shear stress
significantly; the shear stress reduction would then take place too far
from the wall to modify the turbulence production greatly. It seems,
therefore, that in addition to the criterion expressed in Eq. (67), we must
also stipulate a heat flux which is sufficiently high to produce a fairly
sharp density change between wall and core regions.
There will be a range of conditions where the “two-region” model of
density is inadequate, e.g., a small density difference may be compensated
by a low Reynolds number, and deteriorations may still occur. T h e
analysis of such conditions is considerably more difficult; the most
promising line appears to be the numerical solution of the flow and energy
equations for a range of boundary conditions. Such work is in progress
using a mixing length model of turbulent diffusion, and it is proposed to
follow this by a more sophisticated model based on the turbulent kinetic
energy equation. The most that can be said at the present time is that
the criterion based on Eq. (67) must be regarded with caution when the
heat flux is so low as to produce a temperature difference which does not
span the region of rapid property variation.
VII. Boiling
Boiling can occur only at subcritical pressures and is, thus, strictly
outside the scope of this article. Nevertheless, it is of interest to touch
briefly upon the characteristics of the boiling process at pressures
approaching the critical pressure. In this respect the critical point may
be regarded as the condition where the boiling and convection merge.
T h e particular characteristics of the boiling process which set it apart
from convection are, of course,
(i) the coexistence of two separate phases each with its own density,
enthalpy, and transport properties
the
phenomenon of surface tension at the interface between
(ii)
the phases; unless the interface is plane, this implies a pressure
difference between the phases.
HEATTRANSFER
NEAR
THE
CRITICAL
POINT
75
At the critical point the distinction between the two phases disappears,
as does the surface tension. As the critical point is approached, therefore,
boiling is characterized by diminishing property differences between
phases and diminishing surface tension.
T h e ease with which a vapor bubble may be nucleated in a liquid
depends upon the enthalpy change associated with the change of phase
and upon the surface tension. The effect of surface tension is to increase
the pressure in a vapor bubble (by an amount that increases as the bubble
size decreases) and thus to make it necessary for the liquid to be superheated before the bubble can grow. For a given liquid and superheat
there is a particular size of vapor bubble that can exist in equilibrium
with the liquid. T h e equilibrium is unstable and a slight increase in
bubble size will cause it to grow indefinitely, whereas a slight decrease
will cause it to shrink and eventually disappear. For nucleation to occur
it is necessary to create a bubble that exceeds this equilibrium size;
associated with this is a critical value of enthalpy input for the nucleation
process. It is found that nucleation takes place preferentially at a solid
surface rather than in the body of the liquid; also that the superheat
required is frequently much less than the theoretical value. This suggests
that in practice the process must depend upon the stabilization of a
relatively large nucleus (i.e., one containing many thousands of molecules), possibly at some cavity in the solid surface. T h e main effect of
near-critical conditions on nucleation is the reduction in the superheat
required to initiate bubble growth at a given size of cavity. I t will be
shown later, however, that the range of heat fluxes over which nucleate
boiling occurs is reduced as the critical point is approached, and film
boiling occurs more readily.
Surface tension has a large effect on the initial rate of growth of the
vapor bubble; high surface tensions cause the equilibrium pressure in
the bubble to be greater, and it is the excess of this pressure over that
of the liquid that causes the rapid initial growth. This is one of the factors
which causes the high heat transfer coefficient in nucleate boiling; the
effect will be less marked at near-critical conditions because of the
reduction in surface tension. At the same time the free convection that
precedes nucleate boiling will generally be more effective as one approaches the critical point. These factors operate so as to produce a less
marked change in the heat transfer coefficient when nucleate boiling
begins; indeed it is sometimes difficult to be sure, from measurements
of the heat transfer coefficient, that boiling has commenced.
One of the most striking differences observed at near-critical conditions occurs in the case of film boiling. At low pressures the transition
from nucleate to film boiling presents a rather confused visual appearance.
76
W. B. HALL
Large surface tension and buoyancy forces combine to cause the bubble
departure from the surface to be a somewhat unstable and violent process.
Near the critical point, however, both these factors are smaller, and
film boiling presents a surprisingly orderly appearance.
These effects are briefly illustrated, using the data of Draper (sulphur
hexaflouride and a horizontal 0.1-mm diameter tungsten wire) and
Grigull and Abadzic (CO, and Freon 13 and a horizontal 0.1-mm
platinum wire).
BOILING
A. NUCLEATE
Figures 30-32 show Draper's results for SF, (critical pressure 37.7
bars and critical temperature 45.6"C) at three different fluid temperatures
(44). On each figure the heat flux is plotted against the temperature
difference between wire and fluid. It will be seen, from the saturated
vapor pressure corresponding to each fluid temperature, that the liquid
is, in all cases, subcooled. T h e surface heat flux was controlled, and
therefore departure from nucleate boiling led directly to film boiling,
and no points in the transition region could be obtained. It will be seen
that the effectiveness of nucleate boiling in improving heat transfer
rapidly diminishes as the pressure approaches its critical value. Figure 33
shows the wire under nucleate boiling conditions; the fluid is subcooled
(at a temperature of 21.3"C compared with a saturation temperature of
Temperature difference, T,-T,
PC)
FIG. 30. Pool boiling from a 0.01I-cm diameter wire in sulphur hexaflouride. Fluid
temperature 233°C. (Data of R. Draper, MSc. Thesis, University of Manchester, 1968.)
*
P
01
0
I
10
I
5
Temperature
I
15
I
1
25
20
difference ,To-Ts
('C)
FIG. 31. Pool boiling from a 0.011-cm diameter wire in sulphur hexaflouride. Fluid
temperature 39.6"C. (Data of R. Draper, MSc. Thesis, University of Manchester, 1968.)
i
10
I
-
U
/
*
I
/
Temperature difference ,To-Ts
("C)
FIG. 32. Pool boiling from a 0.01 1-cm diameter wire in sulphur hexaflouride. Fluid
temperature 43.1"C. (Data of R. Draper, M.Sc. Thesis, University of Manchester, 1968.)
78
W. B. HALL
31.5"C). There appears to be little coalescence of bubbles, and their
condensation as they leave the wire can be seen clearly.
Figure 34 shows the results of Grigull and Abadzic for CO, boiling
under saturated conditions (critical pressure 73.8 bars and critical temperature 31.1"C). I n this case hysteresis in the transition between
nucleate and film boiling was observed by reducing the heat input under
film boiling conditions until nucleate boiling was again established.
FIG. 33. Nucleate boiling from a 0.011-cm diameter wire in sulphur hexaflouride
at a pressure of 27.6 bars. Data of R. Draper, MSc. Thesis, University of Manchester,
1968. Fluid temperature 21.3"C; wire temperature 34.4"C; heat flux, 23.8 W/crn*.
Magnification x 25 approx.
T h e full lines at the lower left hand corner of the figure represent free
convection conditions. Since the fluid was at its saturation temperature,
and since the superheat required to initiate nucleate boiling was only
2.4"C at the pressure of 55.7 bars and 0.1"C at 71.2 bars, the free convection condition was difficult to achieve; the authors found that once
nucleate boiling was established the superheat of the wire surface could
be reduced below the value at which boiling first occurred, and that
natural convection could only be reestablished by disconnecting the
power supply for a few minutes. T h e results obtained for Freon 13 were
generally similar to those for CO, .
HEATTRANSFER
NEAR
THE
CRITICAL
POINT
79
30
"c
FIG. 34. Heat flux as a function of temperature difference between a 0.01-cm diameter
horizontal wire and carbon dioxide. Data of U. Grigull and E. Abadzic, "Symp. Heat
Transfer and Fluid Dynamics of Near Critical Fluids." Proc. Inst. Mech. Eng. 182,
Part 31 (1968). Key: ( A ) T = 18.7"C; ( A ) T = 18.9"C; ( 0 ) T = 20.1"C; ( V ) T =
20.7"C; ('I) T = 22.3"C; (11) T = 23.2"C; ( 0 ) T = 24.2"C;
T = 25.4"C;
( 0 ) T = 26.5"C; (u) T = 27.6"C;(+) T = 29.5"C;(0)
T = 30.7"C;( X ) T = 30.9"C.
(a)
B. FILMBOILING
T h e film boiling process near the critical point exhibits some interesting
characteristics. Movies of the process have been produced by Grigull(54)
and by Draper (44);Figs. 35 and 36 are reproduced from that by Draper.
T h e behavior of the vapor film is much more orderly than its is at low
pressures, and exhibits instabilities which are very reminiscent of those
proposed by Zuber in connection with departure from nucleate boiling
on a horizontal plate. As conditions approach more closely to the critical
point, the vapor layer tends firstly to form into tubes which subsequently
break up into bubbles (Fig. 35) and then into thin sheets of vapor in the
form of festoons rising from the wire (Fig. 36). Grigull and Abadzic (54)
have produced some very beautiful pictures of these phenomena using
co, .
Draper has compared his data for departure from nucleate boiling
80
W. B. HALL
FIG. 35. Film boiling from a 0.011-cm diameter wire in sulphur hexaflouride at
a pressure of 34.8 bars. Data of R. Draper, M.Sc. Thesis, University of Manchester, 1968.
Fluid temperature 25.2"C; wire temperature 1658°C; heat flux 34.3 W/cme. Magnification
x 25 approx.
FIG.36. Film boiling from a 0.011-cm diameter wire in sulphur hexaflouride at
a pressure of 36.2 bars. Data of R. Draper, M.Sc. Thesis, University of Manchester, 1968.
Fluid temperature 41°C; wire temperature 221.3"C; heat flux 42.9 W/cmZ.Magnification
x 25 approx.
HEATTRANSFER
NEAR
THE
CRITICALPOINT
81
under subcooled conditions with the method of prediction proposed
by Zuber, Tribus and Westwater (57). Figure 37 show the comparison;
the degree of agreement is encouraging. Grigull and Abadzic (54,
however, found that their data for saturated conditions did not compare
well with this type of prediction.
A
e-
I
Fluid temp
\
pi
I
FIG. 37. Comparison between the results of R. Draper, M.Sc. Thesis, University of
Manchester, 1968 and the theory of N. Zuber, M. Tribus, and J. W. Westwater, The
hydrodynamic crisis in pool boiling of saturated and subcooled liquids. Int. Deoelop.
Heat Transfer, Proc. Heat Transfer Conf., 1961 Pt 11, Paper No. 27 (1963) (shown by
curves) for departure from nucleate boiling in a subcooled liquid. (Sulphur hexaflouride;
critical pressure 37.7 bars, critical temperature 456°C.)It should be noted that while
the experimental data are for a wire, the theory refers to a horizontal plane surface.
C. PSEUDOBOILING
There have been many attempts to explain unusual heat transfer
behavior of supercritical pressure fluids in terms of a “pseudo-boiling”
W. B. HALL
phenomenon. Thus, an increase in heat transfer coefficient has been
attributed to the occurrence of something like nucleate boiling, and a
decrease to the onset of film boiling. It appears to the writer to be both
irrational and unnecessary to introduce this concept at pressures which
prohibit the existence of two distinct phases. It is true that the grouping
of molecules into clusters is in some respects similar to a phase change;
it seems likely, however, that the scale of this phenomenon will usually
be small compared with the scale of the system, and that the fluid may
therefore be treated as a continuum. The high level of turbulence that
can be produced at a heated surface in a supercritical fluid has also been
attributed to “pseudo boiling,” but it is not necessary to look further
than the very large density changes, and consequently violent free
convection, for an explanation.
The one piece of evidence that is still difficult to explain in terms other
than “pseudo boiling,” has already been mentioned in Section V, and
is illustrated in Fig. 21. I t has not been possible to repeat this behavior
using SF, under similar conditions (44,and until further evidence is
produced the writer remains unconvinced that it is necessary to invoke
the concept of boiling to explain heat transfer at supercritical pressures.
ACKNOWLEDGMENT
I t is a pleasure to acknowledge the help of my colleagues and research students at the
Simon Engineering Laboratories, University of Manchester. I wish particularly to thank
J. D. Jackson whose pertinacity in our many arguments has forced me to think harder
about the subject than I would otherwise have done.
NOMENCLATURE
U
A
b
B
C
CII
Ce
d
E
f
g
Gr
pipe radius; constant in Eq. ( I )
constant in Eq. (58)
half width of channel; constant in
Eq. (1)
buoyancy force per unit length of
Pipe
velocity of sound
specific heat at constant pressure
specific heat at constant volume
pipe diameter
Eckert number (defined in Section 111, A, 2)
friction factor (defined in Section 111, A, 2 )
gravitational acceleration
Grashof number (defined in Section 111, A, 1)
h
H
k
K
1
rh
M
n
Nu
P
9
R
Re
S
St
enthalpy
( h - ha)/(& - he) or ( h - M / ( h o- hm)
thermal conductivity
acceleration parameter (defined in
Section 111, A, 3)
mixing length (defined by
Eqs. (40) and (41))
mass flow rate
molecular weight
constant in Eq. (56)
Nusselt number
pressure
heat flux
p / p , ; molar gas constant
Reynolds number
shear force per unit length of pipe
Stanton number
HEATTRANSFER
NEAR
t
T
Tllc
U
U
UI
V
V
Vl
X>Y, 2
X
Y
Z
a
B
8
€
e
KT
K8
h
time
temperature
transposed critical temperature
velocity in the x-direction
THE
CRITICAL
POINT
p
v
p
T
4%
83
viscosity
p / p , kinematic viscosity
density
shear stress (turbulent component
7,)
ud/v
velocity in the y-direction
v / u , ; specific volume (P = molar
volume)
SUBSCRIPTS
c
vdlv
m
0
s
Cartesian coordinates
xjd or xjb
Y/_d or Y / b
P V W
heat transfer coefficient
coeflicient of expansion,
(1 /v)(av/aT)”
boundary layer thickness
eddy diffusivity (c,, for heat, eM
for momentum)
( T - T.)I(To- TJ or
( T - T,fl)/(To - T m )
- ( l j V ) ( a V / a p ) , ,isothermal compressibility
-(I/V)(aV/ap), , isentropic compressibility
thickness of wall layer for which
T = 0 in core of pipe (Section
VI, B, 1)
w
in the turbulent core
bulk mean
at a wall
in the free stream, or, when stated,
at some other reference condition
in a wall layer
SUPERSCRIPTS
c
*
-
+
++
value of parameter at the critical
point
reduced coordinate: e.g.,
P* = PIPC
fluctuating component
time mean value of a fluctuating
quantity
“universal” parameter (see Eq. 56)
variable property “universal”
parameter (see Eq. 60)
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