Shorter communications

Chemical Engineering Science, 1965, Vol. 20, p. 711. Pergamon Press Ltd., Oxford. Printed in Great Britain

Heat transfer from a fin to a boiling liquid

(Received 28 November 1964; revised 1 December 1964)

FOR many years fins have been used in contact with fluids having poor heat transfer characteristics, usually gases or slow-moving liquids. Under such conditions the heat transfer coefficient, h, between the fin and the fluid is nearly constant along the length of the fin. The common textbooks on heat transfer, including a recent one devoted entirely to finned surfaces [1] consider h as a constant. The formulae first derived by HAaPrat and BROWN [2] and those of nearly all subsequent researchers assume a constant h.

An interesting situation in which h definitely is not constant occurs when heat is transferred from a fin to a boiling liquid. Such an arrangement is attractive for nuclear reactors which are cooled with terphenyls, other organic liquids and possibly water [3]. Rather short, stubby fins have been used in this manner. A second example is provided by the Compagnie Francaise Thomson-Houston of France [4] which uses fins to cool large radio power tubes. In this case, the fins extend into boiling water.

The proper technique for design of a fin in a boiling liquid is unknown, and the design problem promises to be a lively subject for some years. An analytical procedure can be used only with simple functionalities of h vs. length. For example, HAN and LEEKOWaTZ [5] solved for the temperature distribu- t ion and fin efficiency for the selected case in which h = h~(y + 1)(x/L)v, for y >__ 0. COMO et al. [6] have presented a numerical solution for the case where the heat flux is proportional to the third power of the wall to liquid tempera- ture difference. The true functionality when all types of boil- ing are present is far more complex.

Photographs illustrate the complexities of boiling on a fin. The accompanying figures show isopropyl alcohol boiling at atmospheric pressure on a horizontal cylindrical copper fin, 1.207 in. long and of 0"25 in. diameter. The fin passes through a Teflon packing gland in the wall of the boiler and connects to a 350-W electric heater outside the boiler. Three thermocouples on the geometric axis of the fin permit calculations of the temperature and axial heat flux at the base of the fin (the packing gland).

Figure 1 shows nucleate boiling along a 0"48 in. portion of the fin near its base (right end) plus nucleate boiling at a single nucleation site at the free tip. Free convection occurs everywhere else on the fin. The value of h is about 1670 B.Th.U/hr ft 2 °F at the hot end and about 168 at the cool end.

Figure 2 shows that if the base temperature is increased sufficiently, film boiling occurs at the hot end. The wave structure and large bubbles associated with film boiling are evident. The center of the fin shows vigorous nucleate boiling. Between the film boiling region and the nucleate boiling region is a narrow zone in which transition boiling exists. The left portion of the fin has a few nucleation sites and also shows considerable free convection. Thus, four distinct modes of heat transfer are occurring simultaneously, each following its own physical laws. The variation in h is from about 45 B.Th.U/hr ft 2 °F at the hot end, to 2300 at the centre, to 325 at the cool end. The variation in h is by a factor of 50 to 1, and it is obvious that use of the traditional fin formulae is out of the question.

Figure 3 shows the cold end of the rod covered by nucleate boiling at the maximum h of 2300 B.Th.U./hr ft z °F. Most of the rod is covered by film boiling, and h at the base is about 42.

Figure 4 shows that film boiling can take place on the entire fin. The h is fairly uniform at about 42. This case is the only one of the four illustrated which can be reasonably approximated by use of the published fin formulae.

An organized study using a variety of liquids and a selec- tion of geometries is now underway, sponsored by the National Science Foundation. A fascinating question which is under study is, what is the optimum shape for a fin in contact with a boiling liquid?

K. W. HALEY J. W. WESTWATER

Department of Chemistry and Chemical Engineering, University of Illinois, Urbana, Illinois.

NOTATION

h Local heat transfer coefficient B.Th.U/hr ft 2 °F ha Average heat transfer coefficient B.Th.U/hr ft 2 °F L Length of fin ft

AT Temperature difference between fin and liquid °F x Distance from base to fin ft y Arbitrary constant dimensionless

REFERENCES

[1] I ~ u s A. D., Extended Surfaces. Spartan Books, Baltimore 1964. [2] ~ P E R D. R. and BROWN W. B., Nat, Advis. Comm. Aeronaut. Rept. 158 1922. [3] DEMAN~ Ph., DOUGUET E. and L E F ~ N c J. D., Rev. Tech. Compagnie Francaise Thomson-Houston, Paris, No. 3

May, 1963. [4] BEUR~l~T C., Rev. Teeh. Compagnie Francaise Thomson-Houston, Paris, No. 24, Dec., 1956. [5] HAN L. S. and LE~OWITZ S. G., ASME Paper 60oWA-41 1960. [6] CUMO M., LOPEZ S. and PINCm~RA G. C., 7th Nat. Heat Transfer Conf. Cleveland, Ohio. Paper 31. American Institute

Chemical Engineers 1964.

711

Shorter communications

Chemical Engineering Science, 1965, Vol. 20, pp. 712-713. Pergamon Press Ltd., Oxford. Printed in Great Britain

Comments on the pseudo-steady state approximation for moving boundary problems

(Received 30 July 1964; revised 1 December 1964)

IN A RBcE~rr paper BmQaorF [1] proposed that heterogeneous solid-fluid chemical reactions, which occur when the solid reactant is supported on an inert, spherically-shaped sub- strate, may be described by the following partial differential equation:

1 8 [ z Oc~ Oc r "] e-~ [r ~r} = e -~ (1)

subject to the fo l low ing boundary condi t ions:

c(1, t) = 1, c(re, t) = O, (2)

Ocl = - ~ e d r e ~ r rc co -d-7 ' r c = l a t t = 0 and f o r n > _ 1

As no analytic solutions are known for this equation (subject to these boundary conditions), a perturbation solution involving the parameter e-----(R~/tcD) < 1 was given. Although no consideration was given to the exact physical interpretation of the characteristic time, tc, the following

-discussion will demonstrate the necessity for a precise specification of te.

In the foregoing equations there appears only one char- acteristic time, that is, the characteristic time for diffusion: rD = R~/D. In the strictest sense arbitrary characteristic times cannot be freely introduced into a model. These times must appear either in the model or be derivable f rompara - meters which appear in the equations. A second character- istic time may be found if the boundary condition: c(re, t) = 0 is modified to include cases for which the surface reaction rates are finite. In this case the boundary condition, for a first order chemical reaction, would be:

_O~r _ R R re coD m = - - -~ kc (3)

The characteristic time for the heterogeneous chemical reaction would be ~e = R / k and the most natural definition of the perturbation quantity e would be

~'n Rk re D

If e ~ 0, diffusion times are very short when compared to reaction times and, hence, the process is governed by the rate of chemical reaction, whereas e ~ oo the process is governed by the rate of diffusion.

Now, if both c and re are expanded in a perturbation series and if these expansions are substituted into the partial differential equation (1) and the boundary conditions (2), as modified herein, the following equations for the perturba- tion functions are found, by collecting terms which have the same power of e:

for n = O (o)

1 O [rgOC ~ r ~Or~, ~r /=O (4)

(o) cO , t) = 1,

(o) oc (o) = O, Or rc

(o) dre _ _ I , o ,

xp-~s/ I r c '

(0) r e = 1 a t t = 0

(n) ( n - I ) 1 8 [ 2 8c

(n) c(l, t) = 0,

(n) 8c ( O ) = ( n - t ) er re C,

(n)

drc (col(;) (o) ~ = - - \p-~s/ r~

(n) r c = O a t t = O

A solution for the zeroth and first order terms yield:

(o) (o) co c = 1 r c = l - - - - t

P,

c = - - ( r c ) 2 -- 1 ;

O) 1 (o) 1 r = + ~ [I - (re)~] - ~ [1 - (,o)(°) 81,

l [cot~ 2[1 2 c o l = + ~ - / - g ~ - t j

and solutions for c and re are:

c = l - - e [ 1 - - c o t ] ' 1)

j_q{cot?[1 2 co "I rc = 1 -- p~ t + \ 2 ] \ p , ] L - 3 p" tJ + 0(~2) (7)

Higher order corrections, in principal, may be obtained by solving equation (5).

The discrepancy between the result given in equation (7) and that given by BtscHOrr for re should be noted. The differences arise because in the previous work [1] an incom- plete boundary condition was imposed on the concentration at the free surface. It can be easily shown that this boundary

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