Two other methods might occur to one faced with a problem of the type treated here. One is to linearize and decouple the equations by taking the coefficients of the derivatives to be given by a trial solution-for example, approximate yldyz/dx by y1Odyz/dx. Then the decoupled equations are solved one after another in a cyclic process producing new functions to be used as a trial solution. I n general, the convergence behavior is poorer than for the present method, although there are special problems where the coupling is not strong and the method works. A second method would be to treat the problem as an initial value problem and to fabricate the needed initial conditions. This method requires little storage space, but the adjustment of the added initial conditions so as to satisfy the boundary conditions a t x = L can be tricky or impossible. This method is similar to plate-to-plate distillation calculations, which are very sensitive to, say: the value used for the mole fraction of a light component in the bottom product. The errors in the present method arise from three sources: convergence errors for the nonlinear problem (which can be made negligibly small here), errors in the difference approximation to the differential equations (which decrease with the mesh distance, h ) , and round-off errors in the computer (which increase as the mesh distance is decreased). Convergence may not be possible if there are sharp variations of the unknowns in some region of x; in such a case a singular-perturbation method may be appropriate. A remark is in order for first- and third-order differential equations. For the purpose of computation, a third-order equation involving, say, d 3 ~ t / d x 3could be replaced by two second-order equations (with w = dyJdx and y2 = dre/dx) or a first- and second-order equation (with w = y l and j z = dai/dx). I n this way the finite difference forms still involve only the 1. For a first-order equation it is points at j - 1, j , and j
+
probably better to use a backward difference rather than a central difference. The order of the approximation will still be h2 if the coefficient takes on its average value:
K d j i l d x = '/z
[K(j)
+ K ( j - l)][Yi(j) Y l ( j - l)l/h +O(h2)
(25)
Availability of Computer Programs
A general program for solving coupled, linear, difference equations as outlined here has been developed and is available from the author. This can be used as a subroutine for programs which set up the coefficients A i , k , Bi,k, etc., and control the iteration procedure. This report (UCRL-17739) also contains two distillation programs, the earlier one (Newman, 1963) with improvements in the calculation of compositions and the adjustment of flow rates to satisfy the enthalpy balances and one which uses the more general linearization procedure. literature Cited
Grens, E. A , , 11, Tobias, C. W., Ber. Bunsenges. Physik. Chem. 68, 236-49 (1964). Newman, John, Hydrocarbon Process. Petrol. Refiner 42, No. 4, 141-4 (196'3). \----,.
Newman, John, IND. ENG. CHEM.FUNDAMENTALS 5 . 525-9 (1966). Newman, John, Hsueh, Limin, Electrochim. Acta 12, 417-27 (1967). Newman, John, Tobias, C. W., J . Electrochem. Sod. 109, 1183-91 (1962).
JOHN XEWMAN Uniuersitj of California Berkeley, Calif. 94720 RECEIVED for review August 21, 1967 ACCEPTEDMarch 4, 1968 Work supported by the U. S. Atomic Energy Commission,
FILM BOILING FROM A SPHERE An analysis of film boiling from a sphere to a saturated liquid in forced convection is presented. The heat transfer coefficient from a sphere is similar to that obtained by Bromley for forced convection film boiling from cylindrical tubes.
c
onsider a sphere in a floxving saturated liquid with film boiling occurring a t the sphere surface. Kobayashi (1965) performed an analysis of film boiling for this case in a manner similar to that of Bromley et al. (1953) for a cylinder. Kobayashi's analysis was refined by Hesson and Witte (1966). The analysis of Kobayashi resulted in a set of equations which were amenable to a numerical solution only. This paper presents an analysis which results in a simple expression for the heat transfer coefficient from a sphere to a flowing, saturated liquid.
-c
Figure 1. Model for film boiling around a sphere
Figure 1 shows the model representing the film boiling phenomenon. A solution for the heat transfer coefficient for this case can be achieved under the following assumptions. The
vapor and liquid are both pure and incompressible. The sphere surface temperature is uniform. Thermophysical properties are constant with temperature. The flow is laminar. T h e vapor layer is thin compared with the sphere radius and the liquid-vapor interface is smooth. The velocity and temperature profiles in the film are linear. Radiative heat transfer may be neglected. The liquid velocity a t the interface can be calculated from potential flow theory. An energy-mass balance a t the liquid-vapor interface for a small differential element of vapor adjacent to the sphere surface gives
k~ATdA 6 - hf,'dw
(11
where k is the thermal conductivity, AT is the sphere wall temperature minus the saturation temperature, dw is the rate of vapor produced in the element, and hfo' is the latent heat of vaporization as derived by Rohsenow (1956) hiv' = h,, VOL.
+ 0.68 C,AT
7 NO. 3
AUGUST 1 9 6 8
(2) 517
T h e average vapor velocity can be written as _ii
=
(3/4) U, sin
e
(3)
where Urnis the liquid free stream velocity. rate change, da’, becomes
The mass flow
dw = pd [ (3//2) U,RP sin2 e ]
(4)
and several organic liquids. The author‘s analysis of cylindrical film boiling (like the analysis described above) yields
h = 0.636
[&]Phfg’Um “’
However, Bromley found empirically that
Phf,’Um T h e energy balance, Equation 1, can be written as
4
kAT
3 phf,’U,R
-
R
6 d [sin20 (6/R)] sin 0 d8
h = 2.70 [
(5)
By making the substitution
one may write Equation 5 as
dz dB
+ 4 (cot e) z = 2 csc e
(7)
A solution to this equation is sin4 0
2 sin3 0 dB
k
l/Z
~
]
provided a good correlation for experimental cylindrical film boiling data. An explanation for the large difference in the constants of Equations 16 and 17 is not readily apparent. T h e constant in Equation 17 is based on experimental data and must be regarded as correct within the limits of experimental error. The simplifying assumptions involved in the derivation of Equation 16 undoubtedly contribute to the fact that the constant is so small; however, it is difficult to believe the effect of these assumptions is so large. Therefore, the difference in the constants in Equations 16 and 17 must remain unexplained and will require further investigation. By analogy to the cylindrical case, the heat transfer coefficient for film boiling from a sphere to a flowing, saturated liquid becomes
(8)
h = 2.98
[&I”
PhfO’Urn
or The heat transfer coefficient from a sphere to a flowing, saturated liquid in film boiling is similar to the coefficient for the cylindrical case, differing only by a constant. The difference is attributable to geometry.
T h e expression for local heat transfer on the sphere is
Nomenclature
A
J
LJo
C,
D
T h e total heat transferred from the sphere is given by the integral of Equation 10 over the sphere surface
so
98
q t = 2a[(3/4)kph’fQ~r,R3AT]1’2 p(0)de
(11)
where 6, represents the point on the sphere at which the vapor film grows very thick. High speed motion pictures of a heated sphere falling through nearly saturated water show that this “separation” occurs at approximately 90 degrees on the sphere. T h e integrand in Equation 11, p(O), is defined as
h = hf, = hfo’ = k =
q R T AT G U, ze!
z
= = = = = = = =
area of sphere, sq. cm. specific heat of vapor, cal./g. O C. diameter of sphere, cm. heat transfer coefficient, cal./sq. cm. sec. a C. latent heat of vaporization, cal./g. “effective” latent heat of vaporization, cal./g. thermal conductivity of vapor, cal./cm. sec. a C. heat flux, cal./sq. cm. sec. radius of sphere, cm. temperature, a C. wall temperature minus saturation temperature, a C. average vapor velocity, cm./sec. velocity (free-stream), cm./sec. mass flow rate, g./sec. transformation variable, dimensionless
GREEKSYMBOLS p
6
e
= = =
density of vapor, g./cc. thickness of vapor film, cm. angle measured relative to free stream, degrees
literature Cited
By defining the heat transfer coefficient as
h = q t / ( A ) ( A T ) = qt/(47rR2)(AT)
= = =
(13)
one obtains
Bromley, L. A., LeRoy, N. R., Robbers, J. A., Znd. E n g ; Chem. 45 (12), 2639-46 (1953). Hesson, J. C., \Vitte, L. C., J . h’ucl. Sci. Technol. 3 (lo), 448 (1966). Kobayashi, Kiyoshi, J . iVucl. Sci. Technol. 2 (2), 62-7 (1965). Rohsenow, W.M., Trans. ASME 7 8 , 1645-48 (1956).
L. C. WITTE1 Argonne National Laboratory Argonne, Ill. 60439
T h e evaluation of the integral gives
PhfQlUrn [km] 112
h = 0.698
The above heat transfer coefficient is similar to that obtained by Bromley et a l . (1953) for film boiling over cylindrical tubes. Bromley’s relation correlates experimental data well for water 518
l & E C FUNDAMENTALS
RECEIVED for review August 31, 1967 ACCEPTED January 29, 1968 Present address, Department of Mechanical Engineering, University of Houston, Houston, Tex. 77004. Work performed under the auspices of the U. S. Atomic Energy Commission, Argonne National Laboratory, operated by the University of Chicago, under contract No. W-31-109-eng-38.
