ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
he
r
Effect of Expansion can Temperature Profiles H. L. TOOR Carnegie lnstifofe o f Technology, Pitfsburgh 13, Pa.
H E energy equation, Yhich determines the temperature distribution and heat transfer in the laminar flow of fluids, is usually obtained in usable form for the two special cases of an ideal gas and an incompressible liquid. The term incompressible is used here in the fiuid mechanics sense t o mean a fluid whose density is constant-independent of pressure and temperature. Deviations from the idealized fluids are of significance when the pressure gradient is large. For f l o ~through channels the effect of nonideality of fluids is important when frictional heat generation is large, as it map be in the flow of viscous liquids. All r e d liquids are somewhat compressible-i.e., density is dependent on temperature and pressure-and the effect of this compressibility is larger than might be expected. For example, if polystyrene at 400" F. enters an adiabatic channel a t a pressure of 1000 pounds per square inch and leaves a t atmospheric pressure, the mean temperature rise mill he approximately 4.4" F. (5). If the polymer density were constant a t its average value along the conduit, the equivalent temperature rise would he about 5.4" F., so that the expansion of the polymer has reduced the mean temperature rise by roughly 19y0. Since the rate of heat generation varies across a conduit, it might be expected t h a t a t certain points in the fiow the relative effect of compressibility would be larger than the above value. T o investigate this phenomenon and to clarify the equation for nonideal fluids when the effect of frictional heat generation is small, the energy equation is obtained in a form applicable t o real fluids and a simple example is solved. Energy equation in form applicable to real fluids
By means of an energy balance on a differential element fixed in a pure flowing fluid, Goldstein ( 1 ) derives the folloxing equation:
where
e =
-
Combining the first five equations gives one form of the general energy equation:
Since
(7) Equation 6 can be put in the alternate form PCP
dT T Edp - = - - $. kv2Ti + de J de
Equations 6 and 8 can also be derived from an equation given by Muller ( 4 ) or from one given by Hirechfelder, Curtiss, and Bird (3). It is convenient to separate the dissipation function into two terms-one that gives the rate of dissipation for a fluid of constant density and one that gives the additional contribution due to variations in density. For a Ten-tonian fluid ( 2 )
Combining Equation 9 n i t h the continuity equation (Equation 2) gives
Equation 10 can be put in another form, If p then by differentiating and using the definitions of
f (T, P ) , and 6
= E
The thermal conductivity is taken as constant in this equation although this is not necessary. The equation of continuity is
I n general, we can \>-rite for a change in internal energy of one component,
dU = c y d T where
+
[T
(g),-
1
=-. P
p] dg
(3) (4)
The relationship betxeen the coefficient of thermal expansion and the volume compressibility is
; (%)! =
922
(5)
Thus, a compressible A uid may have a zero volume compressibility. Substituting Equation 11 into 10 yields
Equation 6 or 8 with Equation 12 determines the temperature distribution in the flow of real fluids. If there is heat generated in the fluid in addition t o that generated by friction then an extra term must be added to Equation 12 to allow for this. Although Equations 6 and 8 are equivalent, their use can lead to some confusion. Thus, in the common heat transfer problem where dP is small, Equation 8 reduces to the Fourier-
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
voi. 48, N ~ 5.
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT on the temperature. If this term is neglected, Qj * ~ r and Equation 8 is identical with the equation for an incompressible fluid when E = 0, although p may be finite. It is also possible t o conceive of another type of hypothetical liquid in which j3 is zero although e is finite. I n this case (still neglecting the effect of compressibility on the dissipation function) Equation 8 is unchanged. Temperature profile in flow in a long tube with frictional heat generation
We will consider the problem of a compressible liquid flowing steadily through a tube long enough so that the temperature profile becomes independent of distance along the tube. At this point the temperature profile has adjusted itself so t h a t the net heat released by friction equals the heat lost through the tube walls by conduction. This type of steady state cannot be reached in an adiabatic channel where the temperature will continue increasing with length, but it will be reached in a channel with a constant wall temperature or with heat flux from the wall t o surroundings which are a t a fixed temperature. For steady flow in a channel of constant cross-sectional area and shape, where the flow is in the z direction only, Equation 8 becomes
Some time after the flow has started,
."k-----' / -0.60.0
Figure 1 .
0.4
0.2
r/ R
0.6
0.0
Temperature profile in a tube for a Newtonian fluid Equation
34,
R
= 2
Poisson equation with a generation term: p
c
dT
p
=~
kv2T
+
@
(13)
and this equation is correct for all fluids whether real or ideal. For an incompressible liquid c p = c!, but in all other cases the only correct heat capacity in Equation 13 is c,. When a = ai, Equation 13 is identical with the equation for an incompressible liquid (Equation 15). 1 1 For an ideal gas E = and p = - so t h a t Equations 8 and
dT
in Equation 16 ap-
proaches zero if the boundary conditions are independent of time. Neglecting cross-current components of the flow in Equation 16 in effect assumes t h a t the velocity profile is constant and is unaffected by temperature variations. This assumption-that the stress rate of strain relationship of the liquid is not temperature dependent-is incorrect for most liquids. Although the temperature dependence may be quite large, its inclusion makes the equations much more complex without adding a great deal to the information obtainable. The interaction between a temperature dependent stress rate of strain relationship and the temperature profile for the flow of an incompressible liquid under conditions similar t o those considered here was treated by Hausenblas (2). The assumption of negligible cross-current velocity components also implies t h a t natural convection is negligible. This restriction is not serious since frictional heat generation is only of significance for very viscous liquids or for very small tube diameters, and both of these conditions tend t o eliminate natural convection. I n order to include some types of non-Newtonian liquids as well as Newtonian ones, we will consider a class of liquids whose stress rate of strain relationship is given by
P
12 reduce to
When n = 2 this is the equation of a Newtonian liquid with 1
The usual form of the equation for ideal gases is obtained by neglecting the last term on the rightihand side. The definition of an incompressible liquid is t h a t dp/dO is zero. Equation 11 shows t h a t this assumes t h a t e and B are both zero. With this assumption Equations 8 and 10 give the energy equation for an incompressible liquid:
For a real liquid e and p are small, but finite. The last term on the right-hand side of Equations 10 and 12 under most conditions is small enough t o be neglected, since the actual change in the density of liquids is small. Small changes in density have little direct effect on the dissipation function or on the hydrodynamics, although i t is shown t h a t they may have a large effect
May 1956
p = 2
For a tube
For flow in a tube of radius R under these conditions, with no slip a t the wall, Equations 17 and 18 lead to
and
(- &)n-l
urn = F ARn2
The additional assumption made here is t h a t the pressure gradient is constant. This follows from the constancy of Equation 17 and from the fact t h a t liquid densities do not vary greatly.
INDUSTRIAL AND ENGINEERING CHEMISTRY
923
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Equation 16 and writing V z T for radial symmetry, we arrive a t
The last term on the right of Equation 26 can be considered as the net heat released due to the combined effects of heat generation and expansion. The dissipation function increases from zero a t the tube center to a maximum a t the wall, while the expansion term (which is always negative) decreases from zero at the wall t o a minimum a t the center. Consequently the sum of the two terms, vhich gives the net heat released, is negative in the region of the center of the tube and positive away from the center. Since e is multiplied by the absolute temperature, a fluid with a small volume compressibility cannot be treated as incompressible unless the term T Eis small. For flow in a long tube a steady temperature profile is reached with suitable boundary conditions and Equation 26 becomes 1 dT
[T + n (k)" - Te]
W 2Te
2+n
=0
(27)
n
- 1.41QO Figure 2.
I
0.1
1
a2
I
0.3
I
a4
I a5
r/
I 0.8
I
0.7
I
0.8
I
0.9
Conduction in the flow direction has been neglected, as is usual. The boundary conditions of interest are 1.0
R
T e m p e r a t u r e profile in a t u b e for a n o n - N e w t o n i a n fluid Equation
34, n = 5
The small changes in density also allow the compressibility contribution t o the dissipation function t o be neglected so t h a t Equation 9 reduces t o
We will assume that an average across the tube of the term T Emay be used and will let this average be E ' . The solution t o Equations 27 and 28 is - =T - -T,
R2W/k
n
+ 2 [ rn 4- 2s' (1 -[i]"") n 2(n + 2)2
This can be written as
and it is assumed that this applies t o non-Newtonian as well as t o Sevtonian liquids. Equations 17, 18, and 22 give the dissipation function in terms of the average rate of energy generation per unit volume:
The heat loss per unit area of pipe is q = h' ( T w - T,) and from Equation 30
WR
q = -(l
2
where
By a Bernoulli balance written on a time basis, neglecting kinetic and gravitational energy changes
-
E')
This is equal t o the net rate of heat release in the liquid contained by a length of tube of unit surface area. When h' = 03, T , = T,from Equation 30, and we have a constant wall temperature. If h' = 0, Equation 3Q becomes infinite since a steady profile cannot be reached in an adiabatic tube. For all other values of WR/h' the wall temperature decreases linearly as E' increases. When r = 0 in Equation 29 the center line temperature is obtained :
This is also obtained in another manner as shown in "Derivation of Equation 25." By substituting Equations 19, 23, and 25 into 924
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 5
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT 1.c
Discussion
.
Figures 1 and 2 show that as e ' increases the steady-state temperatures a t all values of T are 0.e reduced because of the expansion of the fluid. The reduction is most pronounced in the center of the tube, and for any value of e' the effect of 0.E the expansion increases as the velocity profile is flattened ( n increases). Even though heat is consumed in the region of 0.4 the center line, the temperature in this region may be either greater or less than the wall temperature because of conduction from other parts of the tube. Although the center line tem0.2 perature decreases rapidly with increasing t' and n, as shown in Figure 3, the maximum tempera4 ture (Figure 4) is relatively independent of n, 2 0.c and it decreases less rapidly with increasing e' 2 than the center line temperature. Figures 1 through 4 show t h a t real liquids cu-0.2 cannot be considered as incompressible in flon \ through a conduit if frictional heat generation is significant. Although the volume compressi3, -0.4 bility is small enough t o be neglected, the coefficient of thermal expansion of many liquids is P great enough to make e' too large to be neglected a t ordinary temperatures. - 0.6 T h e values of e' at 530' R. and 1 atm. (Figuie 3) for some liquids indicates the magnitude of the expansion effect. For a liquid such as ether the 0.8 temperature in the region of the center line actually falls below the wall temperature. Work on flowing polymers will usually be carried out a t considerably higher temperatures than 530' R., 1.0 so that the values of e' for these materials will be greater than those shown in Figure 3. When the temperature variation across the 1.2 tube is large enough to invalidate the assumption of a constant stress rate of strain curve, the calculated temperature profiles can be considered - 1.4 only as approximations to the true profiles. I I I 1 \ It is obvious that it would be very difficult t o reach the steady state in practice. For the low viscosity liquids it would be necessary t o use veiy 6' fine capillaries, and for the high viscosity ones the distance necessary to reach the steady stale Figure 3. Center line temperature in a tube as a function of e ' would be very great and the final temperature Equation 35 would be impossibly high. However, when the frictional heat generation is large the expansion term in Equation 8 is important whether the steady temperature profile has been reach6d or not. Equation and when e' = 0 26 indicates, for the general case of flow in a tube, t h a t a compressible fluid can be treated as an incompressible one with a (33) generation term containing the expansion effect. This net generation term is negative in the region of the tube center and positive away from the center. The effect of the expansion term Dividing Equation 29 by 33 gives is to redure the temperature below what it would be if the fluid 17-ere incompressible. T - T, n 2e'
-
-
-
-
-
___-
( T , - Tr)d=O
+
-
2(1
-[3)] (34)
and the dimensionless temperature ratio goes t o 1 when r = 0 and e' = 0. Equation 34 is shown graphically in Figure 1 for a Newtonian fluid ( n = 2 ) and in Figure 2 for a non-Xewtonian fluid where n = 5. The center line temperature in reduced form is obtained by combining Equations 32 and 33: -.
Tc-TT,
=I-- n + 4 ( ,
(T,- Tw)r'=O
2
(35)
This equation is shown in Figure 3. Values of e ' taken arbitrarily a t T = 530" R. and 1 atm. are shown on this figure for various liquids. The point of maximum temperature from Equation 34 is
Derivation of Equation 25
Consider an example of steady flow in the x direction through a channel in which kinetic and gravitational energy changes are small. A Bernoulli balance between two points a distance dx apart gives
Multiplying both sides by p U , yields
but =
pU, max
n
+ 2e
Equations 36 and 34 together give the maximum temperature which is shown in Figure 4. May 1956
WJ
so t h a t nT
=
INDUSTRIAL AND ENGINEERING CHEMISTRY
J dx
(4E)
925
ENGINEERING, DES! N, AND PROCESS DEVELOPMENT Equation 4E may be obtained for a.tube in another manner. By definition (5E)
If the stress rate of strain relationship for the fluid is dPL
5 = d.) then for no slip a t the mall
u
=
j-;
4(7)dr
and substituting Equation 7E into 5E yields
An integration by parts gives
It is interesting t o note that if g ( T ) is known for any fluid the mean velocity may be obtained immediately from Equation 9E. By definition
and if
\--. r--.
then 0.0
0.0
0.4
8.2
0.6
Q.8
1.0
E‘ Figure 4.
Maximum temperature in a tube as a function of E ’ Equations
2, q, z =
a
coordinate axes, ft.
= coefficient of volume compressibility. _. =
- 1 (-=)bb
dP
A
1
= dimensional constant defined by Equation 17, - X
hr.
8‘
(1g)l-n CP
CL
F
f,,9 h
J
=
k
=
12
=
P
= =
9 1’
R
T T*
Tw
Tc U
urn
c
V V
= = =
= = =
= = =
= =
2(!
=
TT’
=
926
e
= heat capacity a t constant pressure, B.t.u./lb.,,/” F. = heat capacity a t constant volume, B.t.u./lb.,/” F. = mechanical energy converted into heat, ft.-lb.p/lb.,,, = =
E
functional symbols heat transfer coefficient between inner tube wall and surroundings, B.t.u./hr./sq. ft./” F. mechanical equivalent of heat, ft.-lb.p/B.t.u. B .t .ti. coefficient of thermal conductivity, hr. sq. f t . / ” F./ft. dimensionless constant defined by Equation 17 pressure, lb.p/sq. ft. rate of heat transfer per unit area, B.t.u./hr./sq. ft. radial distance from tube center, ft. tube radius, ft. absolute temperature, O R. temperature of tube surroundings, O R. wall temperature, O R. center line temperature, O R. local velocity in 2: direction, ft./hr. mean velocity, ft./hr. internal energy, B.t.u./lb., local velocity in y direction, ft./hr. specific volume, cu.ft./lb., local velocity in x direction, ft./hr. average volume rate of energy dissipation across tube, B t.u./hr./cu. ft
p P T
@
0,
2
34 and 36
T ’
sq. ft./lb.p
coefficient of thermal expansion = 1 ( a ag )p, 1 / 3 R. 2: Tt across a tube = time, hr. = viscosity, 1b.F-hr./sq. f t . = density, lb.,/cu. ft. = shear stress, lb.F/sq. ft,. = rate of energy dissipation per unit volume, B.t.u./hr./ cu. ft. = rate of energy dissipation per unit volume a t a constant fluid density, B.t.u./hr./cu. ft. =
= average value of
= derivative following the motion of
a
= ~ + U ~ b+ v - + b U : -
v2
= Laplacian operator =
the fluid
b
az
82
az-
b2 + *-au3 2 -+ in rectangular a22
coordinates literature cited (1) Goldstein, S.,“Alodern Developments in Fluid Dynamics,” vol. 11, pp. 603-7, Oxford Univ. Press, S e w York, 1938. ( 2 ) Hausenblas, H., Ino.-Arch. 18, 151 (1950). (3) Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” p. 699, Wiley, New York, 1954 (4) Mliller, Wilhelm, Ostew. Ing.-Arch. 7, 77 (1953). ( 5 ) Toor, H. L., Eagleton, S. D., J . B p p l . Chem. 3,354 (1953). RECEIVED f o i r e r i e e liovember 3, 1955.
INDUSTRIAL AND ENGINEERING CHEMISTRY
ACCEPTED February 10, 1956.
Vol. 48, No. 5
