ENGINEERING AND PROCESS DEVELOPMENT Ud
= velocity deficiency, defined by Equation
7,dimensionless
urn. = maximum velocity, ft./sec.
y f = distance parameter, defined by Equation 1, dimensionless Yd = distance from nearest wall, ft. yo = distance between walls, ft. LJ = kinematic viscosity, sq. ft./sec. u = specific weight, Ib./ou. ft. TO = shear a t wall, Ib./sq. ft.
6.0 j *
kid
U >
;
44
u
literature Cited
w 0 I
(1) Bakhmeteff,
>
c
0 i w 5
20
0.1
0.5
0.2 RELATIVE POSITION
Figure 5.
IN CHANNEL d*2
Smoothed Values of Velocity Deficiency
p = radius parameter 27 o, dimensionless r: R e = Reynoldsnumber tanh = hyperbolic tangent u = point velocity, f t . / s e s
u,
u+
= friction velocity
4%
ft./sec. = velocity parameter, defined by Equation 2, dimensionless 01
B. A ,
“The Mechanics of Turbulent Flow,” Princeton, N. J., Princeton University Press, 1941. ( 2 ) Deissler, R. G., Natl. Advisory Comm. Aeronaut., Tech. Note 2138 (1950). (3) Dunn, L. G., Powell, W. B., and Seifert, H. S., “Heat-Transfer Studies Relating to Power Plant Development,” Third Anglo-American Aeronautical Conference, 1951, published 10 by the Royal Aeronautical Society (England). (4) K&rm&n,Th., von., Nachr. Ges. Wiss. Gottzngen, Math.-physik. Klasse, 1930,p. 558. (5) Laufer, J., Natl. Adbisory Comm. Aeronaut, Tech. Note 2123 (1950). (6) Sikuiadse, J., Forsch. Gebiete Ingenzeurw., 3, Suppl., Forschungsheft, No 356, 1-36 (1932). (7) Page, F., Jr., Schlinger, W. G., Breaux, D. K., and Sage, B. H., Washington, D. C.,Am. Doc. Znst., Doc. 3294 (1951). (8) Page, F., Jr., Schlinger, W. G., Breaux, D. K., and Sage, B. H., IND.ENG.CHEM., 44,424 (1952). (9) Skinner, G., thesis, California Inst of Technology, 1950. (10) Wattendorf, F. L.,and Kuethe, A. M., Physics. 5, 153 (1934). RECEIVED for review June 22, 1953.
.4CCEPTED
August ?, 1953.
Heat Transfer in Forced
Convection Film Boiling LEROYA. BROMLEY, NORMAN R. LEROY,AND JAMES A. ROBBERS (r
*
Radiufion Laborafory and Division o f Chemical Engineering, Universify o f Culifornia, Berkeley, Culif.
T
HERE are two distinct types of boiling phenomena for the
boiling of liquids from heated surfaces, depending upon the heat input to the system. The first type, and that most generally preferred because of the fairly large heat transfer coefficients encountered, is nucleate boiling, where the vapor originates from individual points on the hot surface. The second type of boiling phenomena is film boiling, where there is a continuous vapor film between the boiling liquid and the heated surface. Film boiling is generally distinguished by low values of the heat transfer coefficient and high values of the temperature of the heated surface compared t o nucleate boiling. Bromley has discussed the occurrence of film boiling ( 8 ) . Natural convection film boiling, studied b y Bromley ( 8 ) , is khat form of film boiling in which the liquid motion past the heated surface is caused by viscous drag forces of the rising vapor acting on the liquid. Forced convection film boiling is that type of film boiling where the liquid is forced past the heated surface. As
December 1953
would be expected, the heat transfer coefficients increase with increasing liquid flow rates past the heated surface, because of the decrease in the vapor film thickness. It is the purpose of this study to develop a theory for predicting heat transfer coefficients to be expected in forced convection film boiling from a horizontal tube, and t o verify by experiment the resulting expressions. Coefficients of Heat Transfer Can Be Calculated
Film Boiling Theory. Mathematical relationships have been developed which make i t possible to calculate the coefficients of heat transfer t o be expected in upward flow forced convection film boiling from the outside of a horizontal tube. The vapor film is in dynamic equilibrium, for as i t rises under the action of buoyant and drag forces, vapor is added to it from the boiling
I N D u s T R I A L A N D E N G I N E E R I N G’ c H E M I s T R Y
2639
ENGINEERING AND PROCESS DEVELOPMENT the heat does not pass clear through the vapor film, as some of the heat is required to heat the film itself. Therefore, the above expression can not be exact but should give a fairly good approsimation, as long as the latent heat is t’he major item in A’. The correct A’ to be used has been derived ( 1 i and is given in 14;qu;ttion 25. Let us apply Bernoulli’s theorem t,o t,he element de, neglecting the kinetic energy of the vapor in t h e film ( 2 ) . Consider a unit mass of vapor entering this differeiltial clement in viscous flow.
liquid. The heat required to vaporize the liquid and heat the vapor is supplied by radiation and conduction through the film. This mechanism appears to be the situation on about the lower half of the tube, where from visual observations there appears to be a smooth continuous vapor film. The situation is very complicated on the upper half of the tube, for in this region the bubbles form before rising. The point on the tube a t which the
From the application of
R
material lsalance we obtain
w = pVaL
(5)
From the geometry of the sj-iteni we obtain dx = I2 sin @dB
(6)
rWATES CCnET
t
f
t
t
LIQUID VELOCITY
Figure 1.
u
Theory of Film Boiling
thickness of the vapor film approaches infinity (compared to the normal thickness) may be called the separation point. Because most of the heat is transferred on the bottom section of the tube up to the separation point, it would seem most important t o have the theory fit the mechanism on this part of the tube. The concept is represented graphically in Figure 1. The tube is immersed in a body of fluid which is moving upward a t a uniform velocity equal to U . If P represents the pressure a t a point in the film, it has been shown (6) that
P = Po
+ E Rpi
COS
6
sc
TO MANWETER
+ STRAIGHTENING
-.--TO
A heat balance may be written for the element of vapor in the film enclosed , b y de for a tube of lengt,h L. dq = h,,AtdA = hcoRLdOAt = X’dw
(2)
where A‘ represents the effective heat content of the vapor film in excess of the heat content of the liquid. Consider that the vapor film is in laminar flow such that heat travels through the vapor film by conduction only. If a represents the average thickness of the vapor film a t any angle e, then
h,, =
k
-
a
(3)
In this case h,, reuresenta the heat transfer coefficient due t o convection with h,, the coefficient due t o radiation, equals zero. All
2640
MANOMETER
STRAIGHTENING VWES
ORIFICE SECTION
~ u xSTEAM
wL TO
FLOW
V
MANOMETER
/ C-.,.J/
Figure 2.
Forced Convection Film Boiler
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 12
ENGINEERING AND PROCESS DEVELOPMENT It is necessary t o restrict ourselves to a tube of such diameter that the film thickness is a negligible quantity compared to the radius of curvature of the film. The vapor film can be considered as in viscous flow between two parallel plates, one of which is stationary while the other is moving at a velocity of 2U sin 0. The frictional drag force has been shown (9)to be represented by
ye1t 100
-P
1
E
(7)
\
a
The use of the numerical value of 12 implies that the velocity distribution in the liquid a t the liquid-vapor interface is unaffected b y the vapor drag. Combining the values of dF from Equation 4,d p from the differentiation of Equation 1 , and dx from Equation 6, and substituting the value of V from Equation 5, the value of a from Equations 2 and 3, and sin 20 for sin 0 cos 8, we obtain the following equation
4plU2 sin 28 -__
2
pgc
12pX'W sin e
dw
gR(p1 -
3
(z)
=
p)
P.90
0 3 8 7 inch
A
U
Figure 3.
0
0496
0
0637
' "
t
1
ft/sec
Film Boiling of n-Hexane
2
-k pg,k2RL2At2 (22)
The terms in Equation 8 may be designated as the static drag effect, the gravity pressure effect, the velocity pressure effect, and the velocity drag effect, respectively. As we have not been able t o derive an exact solution to Equation 8, let us derive an approximate solution from which we may predict the coefficients of heat transfer due to convection. The term 0' is defined as the separation point for the flow upward across the tube. Let us restrict ourselves to the consideration of the heat transfer from the region of 0 t o e', as visual observation indicates that i t is from this lower section of the tube that the major portion of the heat transfer takes place. The value of 8' will depend upon the velocity of the liquid impinging upon the tube. For high flows 8' approaches 90 degrees, while for low values e' tends toward a value of 180 degrees and equals 180 degrees at sufficiently low flow rates. An analysis of the no flow case ( 2 ) indicates that dw/de does not vary greatly over a large section of the tube and hence it is possible t o substitute W/e' for dw/dO in the least significant term, t h e velocity drag term, in Equation 8. Thus
12pht3w dw p2gck3R2L4At3
TUBE DIAMETER
G 5
sin
e
TUBE DIAMETER
2
A
10-
0 3 8 7 inch
0496 a 0637 0
" ''
I
-
-
,
+
TUBE DIAMETER
e
2plU2 sin 28
12pXT.J sin
PBC
+ %ok2RL2At2
(3) 2
(')
Multiply the above equation by pgc/2p~U2,take the cube root, and separate thevariables; alsointegrate from 0 to eland rearrange.
U
Figure 5.
tt/sec
Film Boiling of Benzene
tion 10 and neglect for the present the heat transferred above e'; hence
sin e + sin 261 3'1
(10) The weight of liquid vaporized per unit time on one half the tube, W , can also be evaluated from the integrated form of Equation 2.
I n this equation hoa(o-8 ' ) is the value of the heat transfer coefficient for convection averaged over the portion of the tube from zero to e'. This equation, as stated previously, neglects the heat transferred by radiation from the specified portion of the tube. Let us substitute the value of W from Equation 11 in Equa-
December 1953
The following equationresults:
v) e+
3D p +h
co2
sin
a 14
sin 2 8 / ' " d e ]
(13)
This equation has been evaluated numerically in Table I. A more complete derivation is given by Robbers ( 7 ) .
INDUSTRIAL AND ENGINEERING CHEMISTRY
2641
ENGINEERING AND PROCESS DEVELOPMENT
With Equation 15 or 16 the coefficient of heat transfer may be calculated for almost the entire range of velocity. If Equations 15 and 16 are adopted as two dimensionless groups, a third group must be added to correlate all the variables. Such groups as ( Lr/ (Atkpi 'PA'&), or combinations with the above might he used
a),
TUBE DIAMETER
c OCP
i
I
I :' A I
I
Figure 6.
b ' d ' 4 ' d ' L ' l'o'
111
6
0 3 8 7 inch
0
0.637
'
I IL'1,
a
I 1 2 ' 1:'
ft/sec
U
Film Boiling of Ethyl Alcohol
Radiation Affects Convection Film Boiling
For the case of natural convection film boiling the following equation has been used (2): h =
Lo
+3/4b
(17)
This equation is accurate within 5% for cases where the value of h, is small compared to the value of hm. -4 similar development assuming that all heat transferred above 0' is by radiation results ( 7 ) in the expression
5
a 0
2= 0
b
e
0
0
A
BENZENE 0
ETHYL ALCOHOL
m
n-HEXANE
at high velocities 0'-
r / 2 and 7
h = h,, ' 0
I
2
3
4
5
7
6
8
1011
9
1 2 1 3 1 4 1 5
U / m
Figure 7.
Table 1.
B =
3T
(19)
The term h, may be calculated by the following equation for parallel plates:
Effect of Velocity and Diameter on Heat Transfer Coefficients
Numerical Evaluation of Integral i o r Equation 13
[-'( F sin e + sin 29)1/' P 0 0.01 0.10 0.50 1.00 2.00 10.00 m
For F For F
> 2; < 2:
0' = n
F = - 2 cos 8 '
db'Ia/'
B 0.429 0.438 0,451 0.521 0.608 0.814 1.293 0.721
where T L is the temperature of the boiling liquid in degrees R T Tis the temperature of the hot tube in degrees R e is the emissivity of the hot tube O( is the absorptivity of the liquid u is the Stefan-Boltzman constant, 0.1713 X B.t.u./ (hour) (eq. foot)(' R4)
8
8'1/4
900 90' 17: 920 52 104: 29' 120 1800 l8O0~ 180
Thb is plotted as the solid curve in Figure 8.
I n Figure 8 the experimental data lie approximately in a straight line somewhat above the predicted curve. Or
At very low flows this equation reduces to
which checks the natural convection work ( 2 ) . At very high flows 9 ' - + ~ / 2 and Equation 14 becomes 2642
+ 81. h,
01
For the graphite tube E lyas taken = 0.8 and for the liquids used was taken = 1.0.
Dimensionless Groups Are Evaluated Knowing Physical Properties of Vapor Film
It is necessary to know the correct awrage values of the physical properties of the vapor film in order to evaluate the dimensionless groups utilized in the theory. A general procedure for evaluating the film properties is to determine the values of the arithmetic average temperature. This approach, although usually adequate, is not very exact. Therefore, expi essions are developed for evaluating the best average value of each physical property individually. In order to simplify the derivation, other properties were assumed constant when considering any single property. Thermal Oonductivity. The thermal conductivity of most vapors varies with temperature in such a manner that a plot of the logarithm of the thermal conductivity, k , versus the logarithm of the temperature, T , is approximately a straight line vith a slope equal to n. The equation of the straight line can be written as IC = CTn where C is a constant.
(21)
With this it may be derived ( 7 ) that
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
Vol. 45, No. 12
ENGINEERING AND PROCESS DEVELOPMENT
*
Y
0
*
I!3 .o
I
We now have an expression for the average value of the thermal conductivity that involves a knowledge of the values of the thermal conductivity at the temperature a t the tube surface and at the temperature of the boiling liquid. Equation 22 can be reduced t o a simpler form, as for any given liquid a t its boiling point the value of k ~ T t / n + l is a constant. Density of Vapor. Assuming ideal gas, it may be derived ( 7 ) that
I
0.8
SIMPLE
THEORY
0.6 ETHYL ALCOHOL V 0
0.3 O ' F
I
I
I
a2
0.3
I 0.4
I
I
I
I
2.0
3.0
I
0.6 0.8 1.0
TUBE 0.387 0.496 0.658
DIAMETER INCH INCH INCH
I 4.0
I
I
I
6.0
60
10.0
~PLTL
PS"
= __ At
-
Figure 8.
where p~ = density of the vapor a t the liquid boiling temperature, Viscosity. The average viscosity of the vapor film was determined from an analogy to an expression given by McAdams ( 6 ) . McAdams evaluates the average viscosity of the liquid film produced by film condensation a t a temperature that is lower than the average film temperature. The viscosity for the vapor film also should be evaluated a t a temperature closer to the tube temperature than to the liquid temperature. The temperature a t which the viscosity should be evaluated is denoted by T p . If the fluidity is assumed linear with temperature, then
Tp
=
TL
+ 0.75At
Forced Convection Film Boiling of Ethyl Alcohol
Average Temperature Method. The only physical property for which the error in the calculated coefficient is serious (>5%) is that for A'. Accordingly, it is recommended that all physical properties except A' be evaluated a t the average temperature of the vapor film for simplicity. Forced Convection Film Boiler
Is Used to Secure Data (24)
Figure 2 is a diagram of the forced convection film boiler. The film boiling apparatus was used for continuously circulating a liquid a t its boiling temperature past a graphite heating element where the liquid underwent filmboiling. Acondenser was provided to condense the resulting vapors for recirculation. The heat input to the heating e!ement was controlled by means of avoltageregulator and current transformer. Instruments were provided for measuring the amperage, v o l t a g e , t e m p e r a t u r e , and liquid velocity. Auxiliary equipment was included for pumping the liquids t o storage and for filling the film boiler. Safety measures have been devised for protection against the toxic vapors and the fire hazards resulting from the liquids to be studied in the apparatus. A more complete description has been given by LeRoy ( 4 ) and Robbers (7). E xp e r i m e n t a 1 Procedure. TUBE DIAMETER The graphite heating tube to be A 0.387 INCH.':, 4496 INCHES used in a series of runs was 0.638 INCH:" " . carefully polished with crocus cloth before being installed in I I t I I I I I I I I the test section. The outside 0.2 Q3 0.4 0.6 0.8 1.0 2.0 3.0 4.0 61) 8.0 10.0 diameter of the graphite tube was determined with a micrometer and recorded. The c o p per sleeves which fitted the Figure 9. Forced Convection Film Boiling of Benzene
Heat Content, A'. The "effective" latent heat of vaporization in the case of condensation has been derived ( 1 ) . A consideration of the equations indicates that for natural convection film boiling the same type of equation is involved; although it may be slightly in error for forced convection film boiling, the error should not be serious.
2 3.o ,0
