I
THOMAS J. HANRATTY, EDWARD M. ROSEN, and ROBERT L. KABEL' '
University of Illinois, Urbana, 111.
Effect of Heat Transfer on Flow Field at l o w Reynolds Numbers in Vertical Tubes A study of natural convection instabilities yields a better understanding of heat transfer at low Reynolds numbers description of heat transfer to fluids at low Reynolds numbers in tubes is complicated by the effect of heat transfer on the character of the flow field. Temperature variation over the tube cross section causes a variation of the fluid viscosity and density; if the variation is great enough, the velocity profile may be altered considerably from the parabolic streamline flow obtained for isothermal fields at low Reynolds numbers. Graetz ( 3 ) calculated the temperature distribution for the case in which the velocity profile is undisturbed by the heat transfer; he assumed that heat transfer occurs entirely by molecular conduction. The expression derived for the rate of heat transfer approximates data for liquids of large p / p and moderate AT (small Gr.) Yamagata ( 9 ) showed analytically that the disagreement for these liquids is largely due to viscosity variation. This has been corrected empirically by Sieder and Tate (7) by means of the ratio of viscosity at average bulk temperature to viscosity a t average wall temperature. The expression
ANALYTICAL
gave maximum deviations of approximately f 1 2 % from Re = 100 to Re = 2100 except for water ( 3 ) . Data for gases and for liquids having the consistency of water disagree with Equation 1, as density variations in the fluid distort the flow pattern because of natural convection effects. Martinelli and Boelter (4) and Pigford (5)'have derived expressions to correct for natural convection in vertical tubes. These expressions are based upon an assumed model for the flow field in which the field is not turbulent; they have been partially successful in describing heat transfer data. Cases of disagreement. probably arise from the inadequacy of the assumed flow model. An understanding of the heat transfer process for these cases might be attained through information on the effect of heat transfer upon the flow field. Present address, University of Washington, Seattle, Wash.
Experiments and calculations are reported here which illustrate the effect of heat transfer on water flow in a vertical tube a t low Reynolds numbers. This is a' case in which density variation plays a more important role than viscosity variation. A change in density causes a change in the force of gravity on a volume of fluid. At low Reynolds number gravity forces play an important role in determining the flow field. Temperature gradients may therefore retard portions of the flow field with respect to other portions. Two cases may be considered for flow in a vertical tube. Fluid near the wall is slowed down with respect to fluid in the center of the tube. This occurs in heating fluids in downflow or in cooling fluids in upflow. Fluid near the wall is accelerated with respect to fluid in the center. This occurs with upflow heating or with downflow cooling. The flow field for these two cases can be described by means of visual experiments. Exact analytical treatment of the general case is very difficult. However, a solution can be obtained for the special case of constant flux heat transfer far from the entrance of the tube. I n this region the velocity profile has reached a steady condition which does not change, and the temperature and pressure vary linearly with distance downstream. Calculated flow fields for this case are presented. The conditions represented by these calculations differ from those under which the experiments were conducted; however, they do show how natural convection effects can distort a flow field. Results of these calculations support the interpretation of visual experiments and give a rough approximation of the conditions under which observed instabilities occur.
glass tube. The tube was heated or cooled by circulating hot or cold water through a Lucite jacket fitted concentrically to the tube by two rubber stoppers. A 100-cm. length of tube preceding the jacket served as a calming section. Methylene blue dye was injected into the system through a No. 20 hypodermic needle. The tube was centered by a series of fin spacers along its length and was positioned in the tube such that the point of injection was about 50 cm. below the heat transfer section. Disturbances observed in the flow field then could not result from the presence of the injector tube. When no hot or cold water was circulated in the jacket, the injected dye stream appeared as a long, straight, thin filament in the center of the heat transfer section (Figure 1). The effect of heat transfer upon the field was reflected by distortion of the dye stream path. The flow field could be flooded with dye by injecting it at a very high rate for a short period of time. The dye flow was then turned off, and the dye was purged from the field by water flowing through the tube. 'The dye pattern in the purging operation reflected the nature of the flow field. Provisions were also made so that the
DRAIN) ROTAMETER
JACKET O U T L E T GLASS T U B E 219 C M I O .
200 C M LONG -PLASTIC JACKET 320CM.l.D 100 CM. LONG
- J A C K E T INLET
DYE INJECTOA
Visual Experiments S U P P O R T ARM
Equipment. The nature of the flow field was illustrated experimentally by injecting a thin speam of dye into flowing water or by flooding the flow field with dye. A schematic drawing of the equipment is' shown. Water was fed through a 2.19-cm. (inside diameter)
WATER SUPPLY TANK\
DYE SUPPLY T A N K A
Experimental equipment for determining flow patterns VOL. 50, NO. 5 e MAY 1958
815
dye and the water could enter a t the top of the tube. Experiments could thus be conducted for cooling and heating in downflow. Heating in Upflow or Cooling in Downflow. When water was heated in upflow, an over-all flow pattern (Figure 2) was observed. Three distinct flow regions were noted : I n the initial part of the heat transfer section the entering parabolic flow profile is distorted in such a way that the fluid in the center of the tube is slowed down and the fluid near the wall is accelerated. Eventually the fluid in the tube center is decelerated to such an extent that flow is reversed. This results in a region of inverted flow in which there is a very slow movement of fluid dowmvard in the central part of the tube and a rapidly moving upward flow in the annulus surrounding it. Far enough downstream this inverted flow becomes unstable and a region of turbulent flow is obtained. The existence of this velocity field is most vividly demonstrated by the experiments in which the tube is flooded with dye. Because of the existence of an inverted flow, the tube wall is not the only place at which the fluid velocity vanishes. The fluid possesses a velocity of zero a t points within the flow field. Upon purging from such a flobv field, the dye will remain in this region of low velocity for a relatively long time and will
appear as a paraboloid (Figure 3). A very slow movement of fluid downward uithin the paraboloid can be noted by watching the motion of dye striations. Decreasing the water temperature in the jacket or increasing the flow rate will cause the apex of the paraboloid to move to a position farther downstream. If dye is injected from the bottom of the tube, it will flow in the tube center until it reaches the apex of the paraboloid, where it disperses as if a solid were placed in its path. The existence of turbulence in the upper regions of the heat transfer section is illustrated by a haphazard motion of filaments of the dye. Experiments for cooling in downflow showed the same flow instabilities described above. indicating that distortion in the flow pattern resulted primarily from natural convection effects rather than variations of viscosity. Cooling in Upflow or Heating in Downflow. Cooling a fluid in upflow gives rise to an unstable flow field. Fluid near the wall is retarded, and the velocity gradient at the wall may become zero. Experiments on flow over solid surfaces have shown that such a condition leads to boundary layer separation and turbulence (6). -4Reynolds number aUAtP of 50
The distortion of the flow field evidenced by the visual experiments can be explained by considering the effect of natural convection upon the flow. This can be illustrated by obtaining a solution to the energy equation and the equations of motion for an incompressible fluid of constant viscosity in which density in the buoyancy term is allowed to vary. If the negative Z-axis is set in the direction of gravity, these equations may be expressed for flow upward in a cylindrical coordinate system possessing axial symmetry in the following manner:
a5 vk + - v+ - =all 0
and a temperature difference of 10' F. between the heat transfer medium and the incoming water produced complete turbulence throughout the flow field.
Detail of upstream flow pattern-apex
INDUSTRIAL AND ENGINEERING CHEMISTRY
R
Isothermal flow at Re, = 125
Dye flow pattern obtained heating water in upflow at Re,
Figure 3.
8 16
Quantitative Description
of F l o w Field
(7)
Figure 1.
Figure 2.
The same results were obtained for heating in downflow, indicating that instabilities arose from natural convection effects.
= 125
of paraboloid
dZ
(3)
H E A T TRANSFER A t L O W R E Y N O L D S NUMBERS
If the density variation with temperature is described as P
- PO/P
=
P(To - T )
(5)
and if a pressure, p , is defined as (6)
p=P+pLlgZ
The above equations may be expressed in terms of the following dimensionless variables: u =
u/u*v p’
v = V/UAV
= p/PuAV2
r = R/a
were conducted, is exceedingly difficult. However, if the case of heat transfer under conditions of constant heat flux is considered, the equations can be simplified by calculating the steady state velocity and temperature fields eventually attained at a distance far from the entry of the tube. Such a flow field would be represented by a velocity profile which is invariant in the direction of flow and by a pressure and temperature change which is linear with distance downstream. It represents the equilibrium condition eventually attained by the fluid in the tube when experiencing a given heat flux. Under these conditions
Constants a and c can be evaluated from the conditions that u = 0 at Y = 1 and
K1/4 -
2
a =
(19)
J1(K1l4)- Jo(K”4) Z1(K1/4)
If Equation 17 is substituted into Equa-tion 14, the following expression results for the temperature field after integration
z = Z/a
t = . T - To/Tc - T O
Re = a U . d p Pr = C,p/k Gr
=
I
-T -- To Tc - T O
and Equations 7 and 10 reduce to
a3p2gPIATl/pz
IATI = absolute value of ( T o- To)
Gr t 0 =Re2
- dp‘ -- +
-
Table 1.
Critical Gr/Re for Equilibrium Heat Transfer Gr/Re , Conditions Cooling in Upflow (Heating in Downflow) 49.2 (for tube) Velocity gradient zero 21.5 (for channel) at wall Heating in Upflow (Cooling in Downflow) 2 2 . 0 (for tube) For larger Gr/Revelocity profile develops dimple at center 121.6 (for tube) For larger Gr/Re reversal of flow occurs 298 (for tube) For larger &/Re reversed flow velocity profile develops dimple at center
dp’
Re2
dz A f(K) From the above equation AT = RePr.
for heating in upflow. The temperature may be eliminated from these to give the following expression for the velocity field v4u - Ku = 0
(15)
for cooling, and
v4u + Ku where the positive sign is taken for cooling in up flow or heating in downflow and the negative sign is taken for cooling in downflow or heating in upflow. v a- v+ u - a=v - - + 3P’ dr 3.z br
bT
,
(9)
bT
v - + u - = br a2
1 bT
Exact solution of the above equations for the condition of constaht wall temperature, under which the visual experiments
=
0
(16)
for heating.
K Gr Therefore, because - = -, the velocf ( K ) Re ity and temperature fields described by Equations 17 and 21 are unique functions of the parameter Gr/Re. Equilibrium Solution for Heating a Fluid Flowing up a Tube. Equations 16 and 14 describe the velocity and temperature for equilibrium heating in upflow. The solution to Equation 16 may be represented by the four Bessel functions
+
+
u = QJ~(K~/~&)b~~(~l/4.\/ir)
+
C Z O ( K ~ / ~ & ) d K 0 ( K ’ ’ ~ 4 r ) (23)
v2 =
av au z+;+-=o 3%
+
The constant A may be expressed in terms of IAT(,the difference between the center temperature and the wall temperature.
for cooling in upflow and
o = - - t Gr --+
+
[ Q J o ( K ~ / ~ dYo)( K1l4r)- do (K1/4) do( K”4) 1 [a - c - aJo(K1/4) cZo(Kl/4)]
dz
d2/dr2
+ l / r X d/dr
Equilibrium Solution for Cooling a Fluid Flowing Up a Tube. Equations 15 and 14 represent the velocity field and the temperature field, respectively. The solution to Equation 15 can be represented by the sum of four Bessel functions
Because the velocity is finite at r = 0 d = O
b=O
Both J o ( K 1 / 4 d ; y ) and I O ( K ~ ; Yare ) complex numbers and therefore = (a,
+
+
ia,)~0(~1/44ir)
(c,
+ ici) Z O ( K ~ / ~ & ) (24)
u = aJo(K114~)
+ bYo(K1’4r)+ cZo(K1/4r) + dKo(K114r) (17)
The real and the imaginary parts of the function Io(K1/4diy) have been tabulated (8) as
Because the velocity is finite a t r = 0
ber (K1/4r)= real part of Zo(K1’4t/i~)(25)
b = O
d=O
bei ( K % ) = imaginary part of VOL. 50, NO. 5
ZO(P4d&)
MAY 1958
817
The function ,J&W&) is the complex conjugate of1~(~1/42/;r) ber (K1l4r)= real part of J o ( K 1 / 4 4 & ) bei (K1I4r)= -imaginary part of Jo
( K 1 / 4 4 & ) (26) Substituting Equations 25 and 26 into Equation 24 yields u =
+
f 6er (K1j4r) g bei (K’/*r) (27)
f
(Q,
+
2 = (ai -
CT)
Ci)
Using the conditions u =
andl/z =
Oatr
=
1
h1
\AT1
=
the constants may be evaluated [fbei(K”*) + g -gber(K1’4)
(32)
Through Equation 32 the equations for the temperature field and the velocity field can be shown to be functions of Gr/Re. A solution for the more general case of fluid possessing sources and sinks of heat within it has been presented by Hallman (2) who used as a characteristic A parameter K = GrPr--. @TI Equilibrium Solution for Cooling a Fluid Flowing Upward between Two Parallel Planes. Equations similar to Equations 11 and 12 may be set up for a two dimensional field.
cos (Kl’d)
e = - d
cosh ( IY4)
Substituting Equation 37 into Equation 34 and integrating T
- TO= A Re Pr
C “
- F2cosh (K1l4y)
+
To - Ts = A Re Pr
cos
1) - d
urdr
(K1‘4) - l ) ]
(40)
Results
f= ber’ (K”*) ber (K”*)
+ bei‘ ( K 1 l 4bei)
(,TI’*)
(28)
Substituting Equation 27 in Equation 14 and integrating
Equation 34 may be substituted in Equation 33 to give
The solution to Equation 35 is
T - To=
“‘,Sr2 A
= aeK1’4y
[f bei (K114r)- g ber ( K 1 l 4 r )-
+
f bei (K1/4) p ber
(K1l4)]
(30)
-T= - To
To - To f bei ( K % ) - g ber ( K ’ b ) J bei (K’/*) g ber (W4) g ber (K114) - 2 - f bei (fW4) (31)
+ c sin
be-K1’* Y
(~l’*y)
dcos (K1!*y) (36)
Because the velocity profile is symmetrical a = b
+
u = e
c = O
+d
cosh (K1I4y)
COS
( K 9 ) (37)
Using the conditions u = o
Constant A may be related to IATI, the absolute value of the temperature difference between the tube wall and the fluid in the center.
1=
I
\
+
I
so1 aty 24
I
=
1
dY I
I
I
7
P
u
0:11 \\
0.2t
-04 02
04
08
08
0
10
I
I
I
I
J
02
04
OB
08
IO
\\ i
R/k
Ra /
Figure 4. Equilibrium velocity profiles for constant flux heating in upflowGr/Re range, 22.0 to 7 2 1.6
8 18
Equilibrium Field for Heating in Upffow. Calculated velocity and remperature profiles for heating in upflow are presented in Figures 4, 5, 6, and 7 , The shape of these profiles is a function of the parameter, Gr/Re. These equilibrium calculations show all of the characteristic flows displayed in the visual experiments. As the Gr/Re is increased the velocity profile is flattened until the value of Gr/Re = 22.0. At larger values of this parameter the calculations show that the profiles develop a dimple at r = 0, and the maximum velocity no longer occurs at the tube center. For Gr/Re > 121.6 a reversal of flow is obtained in the center of the tube and the maximum velocity is moved closer to the tube wall. Another marked change in the flow field is obtained for Gr/Re > 226. The velocity profiles assume a shape as indicated for Gr/Re = 298. U p to the condition where the velocity profile begins to develop a dimple (Gr/Re = 22.0) the temperature calculations depicted in Figures 6 and 7 do not show much change from that
Figure 5. Equilibrium velocity profiles for constant flux heating in upflowGr/Re range, 226 to 554
INDUSTRIAL AND ENGINEERING CHEMISTRY
Figure 6. Equilibrium temperature profiles for constant flux heating in upflow -Gr/Re range, 22.0 to 121.6
H E A T TRANSFER A T L O W R E Y N O L D S NUMBERS which would be obtained for a parabolic flow profile. Owing to reversal of flow occurring for Gr/Re > 121.6, there is a flow of hot fluid down the center region of the pipe. As a result it is possible for T - To the temperature parameter, ____ to T c - To' attain values greater than 1. The effects of changes in the flow field upon the rate of heat transfer are illustrated in Figure 8, where a modified Nusselt group is plotted against Gr/Re. The initial flattening out of the velocity profile has little effect upon the rate of heat transfer. However, a marked increase in the heat transfer rate occurs for Gr/Re > 22.0. Equilibrium Field fpr Cooling in Upflow. Calculated velocity and temperature profiles and rates of heat transfer for upflow cooling are shown in Figures 9, 10, and 11. As Gr/Re is increased, the velocity profile is continuously distorted as depicted in Figure 9. However, the temperature field and the heat transfer rate are little different from that calculated for a parabolic flow profile. When Gr/Re = 49.2 the velocity gradient a t the wall vanishes. This would probably be an unstable flow, and turbulence would exist (6). Significance of Calculations. Equilibrium heat transfer calculations for the constant flux case support the interpretation of the constant trmperature visual experiments. They show that observed distortion of the flow field can be explained by natural convection effects (Table I for summary). For heating in upflow (or cooling in downflow) a reversal of flow occurs a t Gr/Re = 121.6; for cooling in upflow (or heating in downflow) a velocity gradient of zero a t the wall occurs for Gr/Re = 49.2. These might be a rough approximation of the conditions under which the instabilities observed in this research would occur. Thus
I
00
0
,
I
I
:
0.4
I 02
I
I
I 06
I
08
Y IO
R/a
Figure 7. Equilibrium temperature profiles for constant flux heating in upflow -Gr/Re range, 226 to 554
two-dimensional flow for an equilibrium heat transfer in a similar manner as has been indicated for a tube. The shape of the velocity and temperature profiles are governed by the parameter, Gr/Re. When Gr/Re = 21.5, a vanishing velocity gradient a t the wall is obtained. This agrees a t least approximately with the value of 25 obtained above. ,
(Table 11) heating or cooling of a viscous oil is not likely to cause transition to turbulent flow as a result of natural convection effects. A transition will probably occur in operations involving fluids of approximately the same properties as water. Heat transfer to air represents an intermediate case. Few data are available in the literature with which to compare this approximation. Experiments reported in this research were performed a t Gr/Re
-
Discussion
= 400 (properties evaluated a t
120 entrance conditions). Experiments performed by Guerreri and Hanna (7) on the constant flux heating of air flowing downward in a channel present some interesting results on natural convection instability. The channel consisted of two plates, 12 X 8 inches, spaced 1.01 inches apart. The rate of heat transfer was measured ~
Table II. Values of Gr/Re for Flow in a Tube with 0.5-Inch Radius"
Gr/lATI
Fluid
p
10-8oc.-I
100Ocp. 0.80 Water Air
= p =
a
i
0.7 X
Gr/Re 0 . 7 X 10-8
5000 150
500 15
\AT\ = 50° F. and Re = 5bO.
along the channel length by measuring the temperature gradient at the wall. In some of the runs a transition to turbulence occurred, as evidenced by a sharp increase in heat transfer rate a t a point along the channel length. The results of some of these experiments are presented in Table 111. The average of the inlet and the outlet bulk gas temperatures was used in evaluating the temperature difference between the wall and the gas. The data indicate a critical Gr/Re of about 25. The equations of motion and the energy equation may be solved for a
:
100
Figure 8.
Some insight into causes for the partial failure of the models to describe the effect of natural convection upon heat transfer data has been obtained. The chief factor is the transition to turbulence brought about a t low Reynolds numbers. For cooling in upflow (or heating in downflow) natural convection should affect the rate of heat transfer only moderately as long as the flow is laminar. Transition to turbulent flow would give rise to a much more rapid heat transfer rate. For heating in upflow (or cooling in downflow) the acceleration of the fluid near the wall will cause an increased rate of heat transfer while the fluid is still in laminar flow. This is evidenced in the equilibrium heat transfer calculations by the increase in the transfer rate when the flow profile develops a dimple at its center. The transition to turbulence should be accomplished by an even greater transfer rate. A more exact knowledge of the effect of natural convection upon heat transfer will depend on the definition of the conditions under which the instabilities described will occur. Work is now under way at the University of Illinois to obtain this information. Acknowledgment The authors are grateful to Research Corp. for support of this work and to Michael H. Rogers for help in
%Re
I03
Heat transfer rates for constant flux heating in upflow VOL. 50,
NO. 5
MAY 1958
819
some of the mathematical aspects of the problem. Nomenclature
A a
dT/dz tube radius or half channel width ber ( x ) = real part of Z O ( ~ X ) bei ( x ) = imaginary part of lo(.\/&) ber’ ( x ) = d ber (x)/dx bei’ ( x ) = d bei (x)/dx C.. = heat capacitv I , s = acceleration of gravity = =
3 ~ 2 9,ApTI = ____ = Q
Gr ha
=
l o (x)
=
Il ( x )
=
Jo
=
(x)
Grashof num112 bkr based on radius heat transfer coefficient based on arithmetic average temperature difference modified Bessel function of the first kind of order zero modified Bessel function of the first kind of order one Bessel function of the first kind of order zero
Table 111. Experimental Data of Guerreri and Hanna ( I ) on Transition to Turbulence in Channel Caused by Natural Convection Position from Entrance of Run (Gr/Re)a Transition, Inches A1 39 5 A2 46 2 A3 45 1 A4 45 0.5 x4 37 3 Bla 21 No transition Blb 21 No transition B2a 32 7 B2b B3
B4 Cla Clb C2a C2 b
C3a C3b c4 Dla Dlb D2
29
10
34 34 16 14 24 19 25 24
6 4
No transition No transition No transition No transition No transition No transition
26
10
No transition
11
10 17 19
No transition No transition 03 No transition D4 20 No transition El 8 No transition E2 5 No transition E3 15 No transition E4 17 No transition a Based on average of bulk average inlet and outlet temperatures and an average wall temperature. Bessel function of the first kind of order one = Thermal conductivity - _ A Pr Gr
=
k
K 1
P
P P‘
Pr Re
R r
T R/h
Figure 9. Equilibrium velocity profile for constant flux cooling in upflow
t
U
IATI = length of = pressure = fJ pose = li/pLTav2 = C,p/k
tube
+
UAv
= average fluid velocity in the
U W
= U(UAV = weight rate of flow = fluid velocity in the
V
based on radius radial position R/a = temperature = wall temperature = center-line temperature = absolute value of ( T , - T O ) = T - To/T, - To = fluid velocity in the Z-direction
R-direc-
tion
Y
= =
YO( x )
=
2
=
P
=
P
=
P PO
=
V
=
V/U,, distance perpendicular to flow direction in a channel divided by a Bessel function of the second kind of order zero distance in direction of flow coefficient of cubical expansion coefficient of viscosity density density at To
literature Cited
(1) Guerreri, S. A., Hanna, R. J., “Local Heat Flux in a Vertical Duct with Free Convection in Opposition to Force Flow,” ONR Final Rept., Contract X-ONR-622(01) (Novernber 1952), (2) Hallman, T. M., “Combined Forced and Free Laminar Heat Transfer in Vertical Tubes with Uniform Heat Generation.” Am. SOC.Mech. Engrs., annual meeting, Chicago, Ill., November 1955. (3) Jakob, M., “Heat Transfer”, vol. I, p. 451, Wiley, New York. 1949. (4) Martinelli, R. C., Boelter, L. M. K., Unzv. Calzf. (Berkeley) Publs. Eng. 5, 23-58 (1942). Pigford, R. L., Chem. Eng. Progr., Symposzum Ser. 51, No, 17, 79 (1955). Schlichting, H., “Boundary Layer Theory,” p. 23, McGraw-Hill, New York, 1955. (7) Sieder, E. N., Tate, G. E., IND. ENG.CHEM.28.1429 11936). (8) Webster, A. G.,’ Brit. ‘Asso;. Advance Sci. Rept., 56 (1912). (9) Yamagata, K., Mem. Fuc. Eng., Kyushu Imp. Unzv., 8, 365 (1940). RECEIVED for review January 2, 1957 ACCEPTEDJuly 29, 1957
= a U A v p / p = Reynolds number = =
Z-direction
Division of Industrial and Engineering Chemistry, ACS, Symposium on Fluid Mechanics in Chemical Engineering, Lafayette, Ind., December 1956. Motion pictures of the flow field are available on request from Visual Aids Service, University of Illinois, 7131/2 South Wright St., Champaign, 111.
4 11
0.8
o.6[ 0.41
Figure 10. Equilibrium temperature profile for constant flux cooling in upflow
820
INDUSTRIAL AND ENGINEERING CHEMISTRY
Figure 1 1 .
I
Heat transfer rates for constant flux cooling in upflow
