Radiative and Convective Heat Transfer Rates Pertaining to Heating Processes A Theoretical Comparison J A C K HUEBLER S U R F A C E C O M B U S T I O N C O R P O R A T I O N , T O L E D O 1 , OHIO
In recent years there has been a great focus of attention upon rapid heating methods employing radiation, convection, and induction. Concrete analyses of t h e effective value of convection versus radiation are difficult t o perform because of t h e inseparable character of t h e t w o phenomena. T h e present paper seeks t o compare, under theoretically ideal conditions, t h e effect of radiation and convection heating simultaneously applied t o a plate. For t h i s purpose a n equivalent radiation "heat transfer coefficient" i s utilized, and although t h e differential equations involved are insoluble, graphical methods of solution are applied. T h e results reveal t h e contribution made by convection alone t o t h e total heat transfer attainable under assumed conditions and its ratio t o heat transfer by radiation.
R
ECEKT denmistratioils oi iuiproved forging technique
and metal savings resuhing from est,reniely rapid heating have focused attention upon rapid heating methods. Extreniely rapid heating is practically scaleless; this gives lePs metal loss and provides the opport,unit,j for improved arid more accurate forgings. Although t,liere are a variety of possible methods fur accuniplishing rapid heating, such as induction heating. diffusion flames. radiation, and convection, practical considerations i n these e x t r e m temperature ranges place certain limitations upon the type of equipment t h a t will be useful and economical. As the range of t,cmperatures utilized in forging opt high. usuallj- above 2000" F., i t has bcen more or l e s ~common practice t o a s u m e that tlie contribution t80 the lientin:! rate of the work by convect,iveeffects is neg1igil)ie cnmpared t o r:rdiation. This rule of thumb has been coniniorily applicd as ail asioni i n furnace applicat,ion. Like all surh rules, after h11g use (.JiiC tends t o forget the liniit,aiions u1mi n-hich it v a s origiiiully liased and apply i t to furnaces radically diffewnt from tlit, corivcntional types, and conaequrntly obtain performarices far diffewiit than expected. Recent esperiences i\-ith a new tl-pe uf furiiace (descrited belon-), specifically clr~~ipned for high conv as high temperature, indicat,e that this rule of thudJ i i entirel!. inadequate. It is the purpose of the present paper to demonstrate theoretically that under ideal conditions it niay 111; reasonable to expect t'he nierhanisrii of convectivt: licit ti,niihfer to reduce heating tinies approsiniately 20' untltw pur't: radiation, in contradiction to the coninionly used design procedure. The calculation, although carried out for a tteoretically ideal case: can definitely be used as an argunierit to design couvt:etive as well as radiative heat transfer rnechanisnis into furnaces operating even in these extreme ranges of industrial teniperaturee. To clarify the sit'uation further, the calculation has been carried far enough SO that, for the ideal case, it is possible to t,ell at a glance a t what temperatures the convective and radiative heat transfer rnechanisnis contribute equally to the over-all heat transfer. This can be used as a rough guide in conPidr.ring any specific furnace design.
I n IJ&r t o carry out what might be an extremely involved i' riot impossible calculation, ccirtain simplifying assumptions must be made. These assumption.? might at first appear to be s(8 simplifying as t o make the problem impractical; however, although no thorough attempt has yet been made to verify thc computation quantitatively, the results are known from osperience to be of the correct order of magnitude. These obscrva[,ionswould appear t o lend some dcgrer of practirality to the niorlt of calrulat,ion. ASSUMPTIONS
Heat is ordinarily transferred froin the furnace to the i i u r h load by meclianisms: convection and radiation. Tht radiation occurs from both tlie hot gases and the furnace l\al! For the purposes of this calculation, the gas temperature i> assumed t o be exactly t h a t of the furnace wall, so t h a t the gas radiation factor does not enter the calculation. Further, it i b also assumed that' the work is at all times complctely surrounded hy the unifornilj- heated furnace wall, so t h a t only t h e surfwe area of the heat-receiving steel need entcr the calculations. Thc following rcitsonable numerical values have been aduptea tor t,lie variou.; physical constants t h a t are necessary to the solution of the problem. The convection cocfficicnt, is talicn a s 15 :he eriiis-ivity as 0.7, the conductiriiy of the steel as 290, tlic Ilensity as 490, and the specific hvar as 0 167. all in appropl'iatc unit. of the English systeni. T h e results of the calcu!aticiris that t'o!low are p1,iniariIy deptlnden~upon the mlues chosen for thix emissivity and the colit w t i o n hCat tranai'cr cocfficient. .is the author is not ptq)arrb' at p r w n t to s u b s t ' a n t k e the particular values clioscn, tlio wader i? justified in altering the results in accordance \Tit,h his onm coilwptitrn of tlie propcr values. HOX r, i t is possitile so trJ clesigr. u furnace that coiivection effects of unusually high order of magnitude can be obtained. Consequeiitly, one should not cxaniinc these figures in the light, of experience gaine(1 from convcnt,ionall\ liitsignrd fiirnarcs, CA LCU LATIONS
Considering, therefore, a piece l ~ f>tee1 a~ tcrnper:*iture 7 (J' surface area A , and emissivity E , completely surrounded liy n iurnace ~ v a l lof temperature T o ,the instantaneous lieat t,ran*frF rate in R.t,.u.per hour duP to convection Fvill be given by: I
'I arid the heat transfer rate due to racliatron will t c given by:
h, is the oonvect,ion heat tranufer coefficirnt in B.t.u. per hour per square foot per degree F . ; I is tlie radiation constant in I3.t.u per hour per square foot per degree F,;; and all ternperaturcs are i r . degrees Rankine.
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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
June 1948 I60
I60 140
\20 100
80 60
40
-6 10 .I-O
I
0 ,100
I
I
I
I
COLD S U R F A C L T i M P E R A T U R E
t-
700 900 1100 1300 1500 1700 1900 ill00
500
300
I OF:
Figure 1
Thc prublyni ilia>- be rnorc 3inipiy 1i:iiidkd i i an equivaieiit radiation licxt transier coefficient. /,it. is utilized and made t o e the sanit' meaning as the convwtion coefficient. To find his equivaltziit heat t,raneier coeffilcicnr , Equaiion 2 is placed in rht. roriii of Equation 1 by factoring 17' - 7'; from the r i g h t - h n d nirmli~~r of Eipation 2 . Thic gives f
'Iq = EII\To dtil.
- 1');'T-i- rz, .$ITo - 1')
13,
Puiely 13) analogy then, he iz t a l i ~ ~ to i i tic tht: coefficient o i i [ T n- T , in the right-hand meniher of Equation 3. Tbc~cfore. Ii R
=
E I ! T c - T I ! T,:+
:.2:
T'Z)
ilthough h, is actuallj, a con>tant,for a given gas velocity, h:, k a functioii of thc tc,iiiperature of h ~ lthe i furnace and thi. wrk. Thc t o i d hmt transfer ratc may tlleir lw rnlien as till. ~ 1 1 o1f t tie Iv)nwcti\-e anti radiative c o n i ~ i o n r ~ i t ~ .
heat-receiving surface teniperat,urc: for furnace temperatures from 3000' to 1400" I?. The results oi these calculations are plotted as a family of curves in Figure 1. Each cwrve corresponds to a constant furnace temperature. Because thew curves give the total heat t r a n s i ~ ~corfficient, r and the ronvtJotii)i. coefficient is constant at 15, it is possible to obtair, the value of the ryuivalent radiation coeficiem by auhtract,ing I 5 from any value found 011 t h e chart. I t is of some iIlTercSt to not(' 111 Figure 1 thai i i a constant furnaw teniprrature is: considered, t h t heat transfer coefficient increases as the work surface temperature increases. This means that for a Ziven furnace teniprrature more heat is transferrt4 to the xork per d r g r w of temperature cliflrrcntiai hct'n-een thc furnacrb and tht: work n.hc,n thar t r i l l pcrature differential i+ m a l l than wlren it is iarp 2300 2500 .Actually less heat i,> tmn>ferred a$ the w > r ktemger:lture rise., o f course, l,rwusc the ilrriiiuiit c~iheat tran-. Ferred is givrn tiy the protlucftloit he coeficicrit,anti rllt temperature differential, anti the differential d t ~ c r e a ~ t ~ nicire rapidly than the coefficient incrcases. This fact is shown i r Figure 2, nhere the heat, transfer in B.t.u. per liuur per q u a r e fool o i receiving surface is plortrd as a function of the n o r k tc,mpci,ature for a constant furnace temperature of 2200' F. Curves a r t > l i o n for heat transfer by convection alone, hy radiation alolit, a n d by both mechanisms together. The heat transfer l)! C O I I vtmvtim i. a niucli larger fraction of t h c wk~olra + Inn. SII.!:IIY tt'lripmttures than at high. Practical men are uauallj- more intcrc>ted I I L til(- Tinit: :Ldv:liitngcs t h a t can be gaiiicd by using convection in :tdtlition i o d i i r . tion than they arc in comparisons of heat trancfer rate5 a s tlw: may be visualized fruni Figures 1 and 2 . To r r w n l thc t,inic advant,agc in question it is necessary to solvc thc problem I): unsttmly state heating. Inasmuch as n o general solutioni o: availal)lr, the protilein t m y tip ~iilved l h i - rOlll[Jk!Y problem IO0
90
ao 70
60
50
40
1%
,ll4h
g
= ;I.;
= 1.31 I T
x IO--J ( 7',,
1.21 1
x
+ 1') (T;+
T?)=lita
10 - 9 (T,, t Tj[Pt
2771 A
-
t,
(t" -. i
1s; !\I
rr-heiciil the difference of the atisoluto teniperatures, I", i.5 replaced by its equivalent the differericrr of the temperaturrs, t, in degrees F. H , the total equivalent hcnt transfer coefficipiit, is p i w n by
H = 15
+ 1.211 X
10-
(l0j
H has hcen cnlculat,ed from F:qiuitiion 10 as a t'unctioii of flit=
1095
30
eo IO
0
Figure 2
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Vol. 40, No. 6
INDUSTRIAL AND ENGINEERING CHEMISTRY
unstcady st,at,ehcat fiow nit,hin the mebal. Hoiv obtained pvould not be considered sufficiently accurate even for comparison and the method is, therefore, not employed here. For the sake of simplification the heating rates of stwi plates are considered 'rather than the heating rates of routids 01' iithrr shapes. ,%lthough the heating of t,he latter shapes is or' much more practical interest, tlic problem i j much I n addition, the compariion being sought is as \vel heat,ing rates of the plates as it would be by more c~omplcsshapes. To demonstrate the time savings which may lie wtlizcd from convect,ion hcat trausfer over essentiallr pule radiation, t,he Schmidt iiiethod is applied to t x o different problems. Thc first problem is to find the heating time of various sized plat,es to any temperature up t,o 2000 F. as a funct,ion of the furnace tcniprrature. I n this probltxm the core temperature of c'ach plate is compukd as a function of the huating time untlt~rcnndirions t.liat, might well be encountered in the act,ual practice of rapid heating. These heat,ing rate curves are calculated for the condit,ions of heat transfer bl- convection alone, radiation alone, and by both mechanisms together for each of the various plates and teinperatures considered. The furnace temperatures assumed in the first problem are high, and tlie range of teniperatures covered is relatively narrow, making the results obtained correspondingly limited. To clarify further the importance of convect,ionin shortening heating times, the second problem deals with the time required tmnheat a s k e l plate of a given thickness to rvithin a definite number of degrees of the furnace temperature. This problem is solved for both high and low furnace temperatures. Figure 3
RESULTS
graphically by utilizing the accepted Schmidt method. The problem could bc solved generally if it mere assumed that the heated metal were always at a uniform temperature. This assumption n-odd greatly simplify the problem by ~cmovingthe
The core temperature of a steel plate has been calculated, in accordance with the first problem outlined above, as a function of the heating time for a 2-inch thick plate with furnace temperatures of 3000", 2800", 2600", and 2400" F., and for 0 3 - , 1-, 2-, and 4-inch thick plates with a furnace temperature of 2800" F. The heating rate curves under these conditions are shown in Figures 3, 4,and 5 , The particular conditions under ivhich each
2000
1800
I600
Lu
1400
0
Id
ff 1200 3
I-
d 1000 2p 800
rJ
ul
600 V
400 5 T € € L PLATES CONVECT ION
200
~
I
0
3
6
9
Figure 4
12
I
I
15
18
0
20
40
60
Figure 5
00
LOO
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
June 1948
1097
70 60
2I- 50 D 40
z 30 z J
zf-
=o 15
0
I
I
I
1
I
1
10 F 9
a 8
? 7 6
5
2400
Figure 6.
c
2500
2600
2700
2800
2900
Heating T i m e Temperature as Function of Furnace Temperature
curve has been calculated are appropriately marked on the curve. The curves of Figure 3 were calculated for heating by convection and radiation combined. Figures 4 and 5 were calculated for radiation alone and convection alone, respectively. The importance of convection heat transfer in shortening the heating time may be shown by comparing the curves of Figures 3 , 4 , and 5. Table I gives the time t o heat a plate t o a core temperature of the steel of 2000" F. under t,he various conditions. These values are taken from Figures 3, 4, and 5. From 15 t o 25% faster heating t o 2000" F. is obtained when the furnace is arranged t o obtain high convection over pure radiation. If the furnace is held a t 3000" F., a 2-inch plate may be heated t o a 2000" F. core temperature in 5.60 minutes by radiation and convection together, n-hereas i t nil1 take 6.46 minutes under the same conditions but by radiation alone. Thus, the convection heat transfer sares 0.86 minute or 15.3% of the actual heating time. In the same manner, if the furnace is held a t 2400" F., the convection heat transfer can reduce the heating time from 14.82 t o 11.98 minutes. This represents a t,ime saving of 2.84 minutes or 23.7%. Such time savings, although possibly not astounding, are industrially of such prime importance in scale reduction that it rrould appear most important t o employ convection, contrary to the generally practiced procedure in this temperature range. The results sho\rn in Table I, which are designed to exhibit the saving in heating time resulting from convection, could well be examined from another point of view. One important practical difficulty encountered with equipment, used for rapid heating is the lifetime of the refractory in the furnace. Whcn the furnace is operated above 2400' F., as i t must be, refract o r r life decreases rapidly . " with rise in temperature. It is apparent from the foregoing calcu80 lations that the same heat,ing rat,e should be In 6o Iobtained with both convection and radiation as 50 can be obtained with radiation alone when furnacc temperatures 100" lower are employed. If, f 40 then, a satisfactory heating rate can be obtained 30 with radiation alone at. some furnace temperaYJ t ture, the introduction of convection should make F: t o i t possible to decrease the furnace operating 0 temperature approximately 100" rvithout alterz ing t8he heating rate. Such a temperature re+ duction in this range probably would have a 4 profound influence upon the lifetime of the ? l o9 refractory. 8 1000 T o illustrate more clearly the implication of the calculations for the practical man, the
I
1 2 3 THICKNESS IN INCHES
3000
Figure 7.
4
Heating as Function of Thickness
data from Table I have been plotted in Figures 6 and 7. The time necessary t o heat a 2-inch plate t o a core temperature of 2000" F. by convection. by radiation, and by both mechanisms together is shown as a function of the furnace temperature in Figure 6. The expected relationship between the furnace temperature and the heating time may he seen in this figure. The heating time t o a core temperature of 2000" F. as a function of the thickness of the plate IS shown in Figure 7 for a constant furnace temperature of 2800" F. The heating time is shown relative t o the thickness of the plate-that is, the time t o heat per inch of thickness. The heating time per inch ie nearly constant with varying thicknesses when the heating mecha-
Table I. Furnace Tempera. ture 3000 2800 2600 2400
l i m e t o H e a t 2-Inch Plate t o 2000" F. Radiation and Convection 5.60 6.97 9.02 11.98
%
Radiation 6.46 8.27 10.83 14.82
45,9 62.1 62.5
Time Saved by Convection 15.3 18.6 20.0 23.7
8.8 20.6 45.9 119.2
22.4 19.2 18.6 14.8
Convection 42.1
With 2800O F. Furnace Plate Thickness, Inches 0.5 1 .o
2.0 4.0
1.61 3.33 6.97 15.34
1.97 3.97 8.27 17.62
: ~
z
-
1200
1400
I600
1800
2000
Figure 8
2200
2400
2600
2800
3000
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INDUSTRIAL A N D ENGINEERING CHEMISTRY
u s m IS radiation alurie 01 iadiatiun plus conveLtion l'lit heating times computed for plates niay be appro\imatcli cont r r t e d to heating times for bars bv dividing by 2. To help understand in x h a t region of tenipeiatuie- L U U ( ' i t x u l s over radiation, the time required to heat a 2-irich lhicA d a t e to nithin 400" of thc Euinace tempeiature has 1)e+i1( ( I ' I I puted for fuinacc temperatures iioiii 3000" to 1000" F. Figriit $1 At the highest tenipeiatuic (3000 F., the center Of tlLt plnrr ,s heated t o 2600" f , and at tht lone-t temperatuiL (1000 F I che center point is heated to 600" F. The heating t i r l , t * liaii 'een computed for convertion alone, iudiation alone, aid both nechanisins together. I t niay hr seen fioin Figure 8 that tlir onvection effect becomes mole and nioic important d" thr urnace tpmperaturr is reduced. At 3000" E'. the con\cctioIi w a t transfer reduces the heatlng time about I 1 minutes o x i t -adlation alone. K h e n the furnace temperature is 2000" L he time is reduced 5 5 minutes by convection and 42 miiiii[+ at 1000" F. It is also apparent that convective heat tinii-r8 I Decomes larger than radiative heat trnnqfer belon a t u i n ~ c t 'pmperature of 1350' F. Although the problem of heating a plate to a teriipeiatuit a00 O less than the furnace temperature has been arbitiaiilv >elected for illustrative purposes and further designed to make :he computations less laborious, the relations revealed are generally valid under most conditions pi actically encountered Alhough not calculated, i f the plates n ere heated to n ithin 50' or the furnace temperatwe, all the heating times nould be incieaaed bv roughly the same factor. Under the latter conditions the cemperature a t 13 hich convection approximates radiation R o d d h t slightly loizer.
Voi. 40. No. 6
,calls. The furnace \Talk are uniformly heated iri this nianner, a n d the combustion gases leaving the wall are a t the wall ternperature. The extremely high velocity rotary motion of the, gases gives an excellent convection cffect, for heat trnnsfer to tlic
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