INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1948
Substituting these values into Equation 16 and dropping negligible terms, Z(396 - 200)(55.9) 02 = (A5 5 - 23.3)e4'83"l 4.ns>>er.
ZI =
1039
F. or ' C., provided same scale is used t hrounhout At = tempe&ture$ifference betlyeen gas anti film at element-gas interface. I?. r = linear distance measured from point of attachment of temperature-measuring unit along the path it follows, ft. Subscripts. a, b = either of t x o thermocouplr wires e = element j = film betneen solid and gas g = gas whose temperature is being measured i = insulation p = protecting tube s = gas surrounding element c = convection r = radiation T = total of convection and radiation u = liquid well fluid 1 = free end of the unit 2 = position at end of insulation or protecting tuhe, if the element protrudes considerably heyond = a unit of infinite length = the unit a t its point of attachmrnt t o the wall of the equipment t = temperature,
1.7 feet
This less conservative calculation s h o m t h a t the error due to cwnduction can be reduced to less than 0.2' F. by installing a unit 1.7 feet long, and t h a t the simpler calculation was overconservative to the extent of 0.2 foot. Normally, this extra length can be tolerated, and t h e leas complicatrd calculation is sufficient t o set tlic design. NOMENCLATURE
.I.. dehncd by Equation 2a = ft.-2 lrdefined by Equation 6a = i t . ? A T defined by Equation 7a = f t . ? H , defined b y Equation 13c = f t . 3 BT defined hv Equation 17c = ft.-2 C, defined by Equation 13d = ft.-2 C T defined bv Equation 17d = ft.-* c', c" = arbitrary constants L), = outside diameter of temperatuic,-nira4uiing unit, i t . D, defined b y Equation 13a = ft.-' DT defined by Equation 17a = f t . - 1 E, defined by Equation 13b = Et. 1 ET defined by Equation 17b = ft.-1 F , defined by Equation 21 = it.-? F T defined b y Equation 23 = ft.? G = mass velocity of gas, pounds pel hour-.q. ft. h = coefficient of heat transfer, B.t.u. per hour-sq. f t . - " F. k = thermal conductiritr, B.t.u. per hour-ft.-" F. P = outside pcrimrtrr, perpendicular to the axis, ft. D = total prw-ure ot gas. a t n i o ~ p h f ~ i r s Z2 = outside iadius, ft. r = h i d e radius, ft. Y = cross-vctional aira p~ipeiidirulait o axi.. bq ft.
(19
Subscripts are frequently used together. of the proterting tuhe a t its free end. Q
Thus,
tpl
is the temperature
LITERATURE CITED
W.H., " H e a t Transmission," 2nd ed., pp. 63, 221, 223, 3SO-1, New York. llrGraw-Hill Book Co., 1942. (2) Rohsenow, W. &I.. T r a n s . Bm. Soc. J I f 2 c h .Engrs., 68, 195-8
(1) McAdarns, (19461.
(3) If-alker. W ,€1.. Lcwis. W.K., Slc*Adains,W ,H., and Gilliland, E. R . . "Principle. of Chemical Ligiiieering." 3rd ed., p. 156, S e w Yolk, SIcGraw-Hill Book c'o., 1937. K L C C I V E D Januarv 14, 1948. Contribution from Gulf Research & Development Company'? \Iultiple Felloxship. 3relion I n - t i t u t e , Pittshiireh. P a .
Heat Transfer in a Recirculating Furnace M . J . SINNOTT
AND
C.A. S l E B E R T
U N I V E R S I T Y OF M I C H I G A N . A N N A R B O R . M I C H .
T h e heating of steel in a recirculating a i r furnace f r o m r o o m temperature t o 600", 800°,lOOO", and 1200" F. has been studied and t h e effect of a i r temperature and a i r velocity on t h e r a t e of heat transfer has been investigated. T h e value of h,,t h e instantaneous convective heat transfer coefficient, has been found t o be independent of t h e t e m perature difference between t h e m e t a l and t h e air for a given constant a i r temperature. T h e values of hc for various air temperatures and a i r velocities have been comp u t e d ; these values when used w i t h h,, t h e instantaneous radiation coefficient of heat transfer, m a k e possible t h e calculation of t h e length of t i m e required t o heat a m e t a l i n a recirculating a i r furnace t o a n y given temperature.
T
HEHE arr nunit~rouhoccasioiis in wurkiiig n i t h nitstal+ n - h t ~ 1 1 a relatively low heat-treatment temperature is uxrd; thf. solution treatmpnt of aluminum alloyh, process annt,aliiig, tc'nipering aftcr qucnching, P t r . , are' typical examples. Although practically any type of furnace may be used for these operations, the rrcirculating furnace, in which the hot products of combustion are continually circulated over the work rather than drawn off through the stack, has several advantages. The forced circulation ensures a greater rate of convective heat t.ranafer: a more uniform tc.mperature distribution within the furnace can he obtaincd; a greater economy of operation can he shown: and
ucually this typr of iurnacr is adapt ahlr. t n very (~lnst~ temperature c~ontrol. These factors have been knonn in a qualitat,ive way, hut no quant,itative data are availahlr to show how the ratt' of heat transfer varies with either temperature or rate of recirculation; this investigation was undertak l i ~ t clear. The values of h , are small wvhen compared to thv va1uc.k of h;, especially at. the highrr temperatures. This is t h r reasoii x h y so many investigators ncy&ct convective heat transfer at rcmperatures above 1000" F. If, hoivever, a inrtal is t)ririg 1it.ated to t,hese teniperatures and h low emissivity, the valutl of h , may actually exceed the value of h,, arid in thokc caws convtxction beconies thc c w ~ i t ~ ~ ~ lfactor l i n g iri dett'riiiining thc) rat(, i)t' heat transfw.
S u m m a r y of Instantaneous Coefficients
(Furnace teiiipt'rature, 800' F. Instantaneous Teillperaturr Over-all Difference, Coeficienr T j - 7'6 (Figure 7 )
450
Vol. 40, No. 6
10.20
9.85
Air velocity, 2.1A f r e t per s c c o l ~ r l , Kailiarioii Coefficient !Emiss. 0.81 10 39 9.76 c3.22 8 61 S 1.i 7.68 7.23 6.80 6.41 6.03 i 68 .5,RK .i04
Ctinvection Coefficient (by Difff,renrrj 3 81 R 64 3.62 3 49 3.40 3.37 3 37 3.41) 3 44 3 45 3.47 3.44 :i 31
Figure 5.
Time-Heat Absorption Curves
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1948
Table V.
S u m m a r y of Sample Calculation
(Furnace trinperature, 1000" F. hc = 3.0 B.t.u. per square f o o t X hours X ' F.) T,, Surface TI - T s , B.t.u.;footl q . ~ . t . u . q , a . ~ . t . u . ~ ~ o ~H . r BCUI., ?iIin. F. F. >Iin. Temperature, HO~II0 . F, I - = ttc ' h r i r t c AT 1.1 T, - T,) 0 , 39 100 900 7 "I 10.21 ,39 122 18,400 I ,53 150 850 7.jY 10.39 340 18,000 1.14 200 800 7.96 10 99 343 17.350 1.18 2.71 2 30 i50 8.33 11.39 339 17,050 300 700 8.80 11.80 372 16,550 3 30 650 9.26 12.26 385 15,950 I 4.j 6.77 400 600 9 74 12.74 398 15,300 1.36 8.33 450 550 I O . 38 13 38 41: 14,600 :00 10.82 13.82 430 13.850 .,a0 460 11.40 14.40 442 13,990 2.11 14 00 600 400 12 00 1.5 00 452 12,000 2.28 16.28 650 350 12.68 13.68 4i0 11,000 700 300 13 31 18.31 484 9,800 14 03 li.03 500 8.530 3.52 2.5.12 750 250 800 200 14.80 li.80 ,510 7,130 4.30 29. 42 830 150 13.52 18.52 323 ,5,560 900 100 10.40 19.40 ?35 3,880 950 50 17.29 20 29 Ai 2,030 ~ 6 . 2 0 59. j3 9i5 23 18.00 21 . O O 2i5 1,030 1,j.iO 7 5 . 2 3
1043 be used is the Stefan-Boltzman equation: y/e = 0.172eA(0.01T,4 -O.OlT,4)
,
where e is the emissivity of the surface receiving heat; A the area receiving heat; 2 j t h e furnace temperature; and T , the temperature of the surface receiving heat. The emissivity of the steel surface as taken from McAdamp (4)is 0.85. The t,erm A can bP any area of the plate we wish to choose, in the present case 1 square foot per side, but because the plate is being heated from both sides, A then equals 2 square feet. h, = q/O,/A(Tj - T,) hy definition; therefore, h, = 0.172 eA4(O.0lTj4- 0.01T,4);.4(T1 T,). By assuming various values of T,, Tj being fixed at 1000" F., 4 d u e s of h, may be computed. ;is the over-all coefficient, L', is equal t o h, h,, t o obtain values of IT the value of h, is added t o the computed value of h,. As the "&lues of C , d,and (Tj- T,) heen fixed, the term p,?o he computed, because = CACl) - T J . I n order to solve for 8, the time, the value of p must be determined.,, This can be done by the use Of the relatioxiship ?! = W C , A I , where TY is the weight of the piece, in the present case 64.1 pounds for the 1 square foot chosen, c, is the specific heat, and AT is the temperature increment through which the piece is being heated. After y has been determined, 0 can be found, thus giving t'he time required to raise t'he temperature over the chosen range ( T j - T,). T o find the total time t o heat the metal t o a particular temperature, the tirne increments must be
:::; ::: ;::::
:;:
i:;: iy:;:
a,!;
-
+
COMPARISON WITH OTHER DATA
30 \Yolk Of the tYIJc' Studit'd ill this iliV~3StigLitiOlihas k'l'rl reported in the literature, There have been numcrous puhliratioils on tile heating of metals in various types of furnaces but 110 attc,nipt,v at llieajuririg the actual coefficients or heat trarlsfpr were reported. The only studies lvhich xere carried out undt3r rest conditioris similar t o those used in the present research arc> those of Kistrier ( 2 1 arid Koflcr (3),\Tho studied ttle unstpadv state heating coolirlg regerlerators;. Botll ir igators reported that /I,, the convection coefficient, i i indepe Of the temperature drop from air or gas to the hrickn.orli of the regenerators. This is confirnied h>- the present invejtigation, whirh shows that hc is independent of thc temperature drop from air t o metal. Kistner showed that h , varied as the square root, of thra air or gas velocity and t,his conclueion is confirmed in the present instancp. .\ plot, of / I c agaitist air velocity on log-log paper shows that h , varied as the 0.45 power of the air velocity. he data that are available for the value of h , under te transfer conditioris are compared with the values 11 the present, investigation, it appears that all these coefficients arc ( i f the same order of magnitude-i.e., 1.5 to 8.0 re' foot, X hour X F. for velocities up to 10
q , j O
O
SAMPLE CALCULATION
In order to SIIIJ\Y how the information (JtJtaiIld iii this wsearcki may he used to computt. the heating curvp in a particular illstancr, thv fnllo~virrgI,\-ampli. is givcri: steel plate 1.5 inch thick has bern sura queiiched and is *to tempr:reti in a rirculating draw furnace at 1000" F. ( T j ) . Circulation in the furnace i> at the rate of 2 feet per second. From Figures 7 and 8 the value of h , IS found to be 3.0 B.t.u. per square foot X hour X ' F., and this value is independent of (1; Tal. The radiarion coefficient, h,, i i iiot independent of (7; ; 7'J; so a sunimation of terms must he used in IJrder to diAtermine t,he anlourit, of heat carried t o thc piccca hv thiy iiierhnnisin. T h e vquation tn
Figure 6.
Instantaneous Over-All Transfer Coefficient vs. (Ti - T , )
-
am
IOM
TEMPERATURE OF FURNACE -'F
Figure 7.
Convection Coefficient vs, Furnace T e m peratu re
Figure 8.
Convection Coefficient Ve Ioc ity
VI.
Air
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 CITED (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
