1044
INDUSTRIAL AND ENGINEERING CHEMISTRY
added cumulatively. Table V is the summary of the results of these computations for the particular example chosen. SUMMARY
The convection coefficient of heat transfer for air to steel in a recirculating furnace operating a t temperatures of 600 ’, 800 ’, lOOO”, and 1200” F. has been investigated, and found to be independent of the temperature difference between the metal being heated and the furnace heating the metal. Increased velocity of recirculation results in a n increased convection transfer coefficient, the coefficient increasing approximately as the square root of the velocity. At temperatures above 1000° F. the convection coefficient decreases in value in comparison to the results obtained at lower teinperatures of operation.
Vol. 40, No. 6
The transfer coefficients obtained, when used in conjunction with the radiation transfer coefficient, make possible the determination of the time-temperature curve for any material placed in a recirculating air furnace operating a t a given temperature. LITERATURE C I T E D
(1) Kelley, K. K., U. S. Bur. Mines, BuU. 371 (1934).
(2) Kistner, H., Arch. Eisenhuttenw., 3,751 (1930). (3) Kofler, F., Ibid., 3, 41 (1929). (4) Mcildams, W. H., “Heat Transmission,” 2nd ed., p. 45, New York, McGraw-Hill Book Co., 1942., RECEIVEDJanuary 14, 1948. P a r t of a dissertation submitted to the Horace H. Rackham School of Graduate Studies in partial fulfillment of t h e requirements for the degree of doctor of science in t h e University of blichigan.
Fin Heat Transfer by Geometrical Electrical Analogy CARL
F. K A Y A N
COLUMBIA UNIVERSITY, NEW YORK. N. Y .
W
ITH extended surface of
F o r a fin integral w i t h i t s base material, or attached t o i t of the “surface conductance” through substantially negligible t h e r m a l resistance, anthe integral fin-base strips then permit the estabtype or equivalent-i.e., with alysis of steady-state heat transfer and of resultant t e m lishment of electrical ponegligible thermal resistance peratures throughout t h e structure is rendered difficult tential difference across the between fin and base-two b y t h e distorted temperature conditions w i t h i n t h e fin working surface of the simuassembly, particularly a t t h e base. By t h e method of problems dealing with the lating model. This potenelectrical analogy, t h e performance t o be expected under predicted steady-state periial difference becomes the formance under different condifferent conditions of surface conductance and material counterpart of the temperaditions loom u p as worthy conductivity has been investigated for two fin structures ture difference between the of attention. These concern of given proportions. Temperature distributions w i t h i n working fluids of the fin themselves with distribution t h e structure and along t h e surfaces, as well as over-all struct,ure, inasmuch as the of temperatures throughout heat transfer values, m a y be established through t h e electrical resistances of thc the fin element itself, and m e d i u m of dimensionless temperature-distribution and different parts of t,he simulatdistribution of temperatures relative thermal-resistance charts included in t h e paper. ing circuit are in scale cs. the throughout the primary Calculated fin surface values predicated on fin-base test thermal resistances. Potenb%se t o which the fin ie values are included for comparison w i t h those determined tial measurements representaattached. Additional probthrough t h e analogy method. tive of the temperature conl e m are encountered when ditions throughout are posthe fluid surface conductsible through the use of a ances on the fin side vary, particularlv when the conditions o r i movable probe in the electrical circuit. Further measurement of the two sides of the fin are themselves different. the electrical characteristics of the model permits translation into thermal performance predictions. EXPERIMENTAL APPROACH The possibility of this mode of attack in fin analysis was indiI n view of the difficulty of determining temperature distrilucated by the author in a discussion on the excellent treat’ment af tions and attendant heat flom- under the above diverse conditions extended surface by Gardner (1) in 1945. Attention was drawn by means of orthodox mathematical, graphical, and other techto some of the customary assumptions in fin analysis of uniform niques-including direct experiment-the “fin problem” has been temperahre across the fin section, and of uniform boundary studied by means of the “geometrical” electrical analog for twoconductance over the surface studied. The problem of temperadimensional steady-state heat flow. This involves construction of ture distribution throughout t,he fin base, under assumed condia simulating electrical model t o scale, using an electrically conductions of boundary conductance, has hitherto received scant treattive flat sheet on which t o represent the cross section of the heat ment in the literature, although this problem may assume special transfer member studied. This has been described elsewhere importance in particular circumstances. (3, 4, 6). Herein the resistance effect of the boundary fluid may Study of heat transfer conditions by means of the electrical be represented by equivalent resistance of a length of conductive analogy Analogger is subject to the same restrictions as similar sheet beyond the profile of the exposed surface of the structural st.udies by other analyt,ical methods. Thus, thermophysical member. properties t,hroughout, such as conductivity and boundary conThus, in application t o the fin problem, the boundary effects ductances, are assumed, and are normally regarded as indepcnda t the fin-side surfaces and at the opposite lower surface of base ent of temperature. The laws of steady-stat,e heat flow are may be simulated in a n Analogger model. Electrodes a t the end considered the same as for electrical flox expressed essent,ially by
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1948
Ohm’s law-i.e, thermal circuits are the counterpart of electrical circuits. Thus, thermal resistance effects at fluid surfacesdefined through boundary conductances-are presented in scale to the thermal resistance of the solid structure. Thereby the electrical model is comprised of a n equivalent electrical resistance structure of flat sheet, with the electrical potential differences proportionately equivalent t o the temperature differences of the thermal arrangement. It has been possible, therefore, t o present the final results of performance predictions in terms of dimensionless ratios, t o cover a variety of operating circumstances.
1045
Figures 3, 4, and 5 . For application of these results to temperature predictions, use is made of the potential difference ratio, c, defined: c = Aez/Ae (1) From the Analogger values of c, equivalent temperatures throughout the fin-base assembly are determined: tz =
tb
+ cAt
(2)
Following the procedure established in a previous paper (9), use has been made of a general correlating ratio, m,for the electrical results. Whereas different forms of m may be used, the one herein employed establishes the thickness of the base as the reference section:
(3)
vi = r h / r w
Throughout the studies in P a r t .4, the boundary conductance effect on the base side was maintained constant. Thus, the auxiliary operating ratio, m’,was maintained constant: nit TYPE SECTION STUDIED
Figure 1.
Section of F i n
The program herein presented covers two cases of rectangular fins at wide heat transfer extremes. Case -1deals with a type of fin structure normally of thin metal which could be extruded, cast, or otherwise formed Lvith negligible thermal ,resistance between the fin and the base, with the fin relatively long as compared to its thickness. Case B deals Kith an equivalent fin structure, primarily though not. necessarily nonmetallic, with a relatively short and thick fin section. I n dealing n-it,h Case A, the basic correlating system will be developed.
=
(4)
rhl/ru
This constant value for base side m’ was 28.7 ( = JI’ as hereafter noted). Figure 3 shon-s the .halogger results for fin-side conditions, V L = 28.7, thus = m’. Values of c by test are shown for the fin surface and also for the upper and lower surfaces of the base. Within the fin-assembly profiles are shown lines of constant c,
I I
Ill
1
CASE A
A thin fin structure, normally of metal, is considered, with section as shown in Figure 1. The length of the fin extending beyond the primary surface is 20 t,imes its own thickness; the base thickness is assumed the same a s the fin thickness; t,he fin spacing is taken equal t o the fin length. The particular section studied has been identified in Figure 1, symmetrical conditions being assumed. The fluid conductance on the underside of t8hcflat base has been taken as constant, with results obtained for the varied conditions of t’he fin-side conductance. On the basis of the section chosen, an Analogger model t o enlarged scale was constructed, and is represented in Figure 2. The length of thestripson the fin side, representative of the boundary conductance effect, was varied for the different values of conductance; auxiliary resistances here made possible adjustment for a wide variety of values; and recognition was taken of the model configuration in establishing the electrical properties of the circuit. With potential difference applied to the over-all electrode system, by means of the probe it was possible t o establish the potential distribution within the structure profile. Electrical measurements of voltage and current permitted the determination of the over-all resistance value of the fin-base assembly. Similar st,udies could, of course, have been carried out for the fin or for t,he base separately, but, in the present circunist’ance i t xvas considered desirable to confine t,he study to the composite assembly. Figures 3, 4, and 5 show the results for three diflerent values of fin-side conductance, with a constant surface conductance on the lon-er side of the primary base. The effect of boundary conductance may be represented in terms of proportionate resistance corresponding to this effect. Although this may be expressed in a variety of ways, for convenience it. has been referred t o the onedimensional resistance of the primary base section. Isopotential patterns for the different conditions are shown in
/
W 0 [ 0 r
8
I
I
W
Figure’2.
Analogger-Model C i r c u i t for Fin-Base Assern bly
1046
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 40, No. 6
Table I . Value of h (70of T o t a l Value) for Different Positions Outward f r o m Corner ( 0 of T o t a l Distance), for Fin and z a s e tin
Position
ii
t
c
0
58 3 97 0
Hd>e h
i
2 3
34.0 49 5 67.0 83.0 91 , 5 100.0
22.3 27.5
ion o
1’O.LtlClll
c-7
10.0 20.0 30.0 Il0.0 n o .0
i~i~prvwiits the traiisitiori bridge between the xiiiwurid r1c~c.trical How of tlie model assembly, and t,he predicted th(mna1 flow for thv i t r t u a l fin aswmbly of given structural proportions. Thni
.\I
! t i :=E
=
Rh R,, allti
f
=P
VI’
= Kt
= -If’ =
Kh’
!71
K,
H,,
(81
Froin Flquation 8, i t fo1lon.s thai
Rt 1.1
=
R,,F 1
=
/,‘T,
Kc
,!lJ
For tile determination oi F , from Figure 6 anti t m c d on t h e fixed value of J I ’ , the value of .\I = R h / R , = (l;L,)(k;h) must firat, be set up. -1s indicated, this folloivs from the data on tlii. base thickness L,, k , and h on the fin side. The value of I; thus determined holds for the present fin-assembly proportions, and, as noted, for thp specified value of JI‘. -4s shown in Equation 9, 1‘(, necessary for the calculation nf the owr-all htvit, iritnsfcr, may then b13 specified.
Figure 3.
Variation of c for m = 28.7
their shape in the base section being particularly intrrestirig. The lines are relatively flat, wit,hin the fin itself, suggesting thr. acceptability in this case of the usual analytical assumption originally noted. Also shown for comparison is the c value fcr the fin surface computed according to the conventional HarperBrown relationship, based on the acceptance of the fin-potential value c’ at t’he fin base as given by the Analogger results, and with the over-all fin length augmented by half the fin thickness. The equation may be shown in the following form for a rectangular fin: c
*
1.00
-
(1.00 -
c’)
+ en
