INDUSTRIAL AND ENGINEERING CHEMISTRY
888
Vol. 19, No. 8
Heat Transmission by Radiation from Non-Luminous Gases’ By H. C. Hottel FUELA N D GAS ENGINEERING, MASSACHCSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGS, MASS.
HE importance of radiation from hot gases as one of
with very great thicknesses of gas layers (Figure 1). A still the major factors involved in many problems of heat greater spectral dispersion of the radiation will resolve the transfer has, until recently, been overlooked. Co- bands themselves into a series of smaller bands. As the efficients of heat transmission based on the assumption of thickness of the gas layer is increased, the intensities of the convection as the mechanism controlling heat transfer have different bands increase and approach as a limit the intensity been reported for many types of industrial heat-transmission of black-body radiation at the wave length of the band in equipment in which radiation from the hot non-luminous question, as given by the Planck radiation equation for gases was actually the controlling factor. Such coefficients, energy distribution in the spectrum of a black body. The while satisfactory as long as their application is limited to rate at which black-body intensity is approached as thickness apparatus similar to that which furnished the data, lead to of gas layer is increased is determined by the absorbing charfalse conclusions when conditions are different. Heat acteristics of the gas for radiation of the wave length in transfer by convection varies widely with gas velocity and question, and varies not only from one band to the next, size of gas passage, somewhat with temperature of gas, and but even within a band. It is apparent, then, that the area almost none a t all with gas composition. Heat transfer by under the wave length-intensity curve, which is proportional radiation is independent of ga8 velocity. varies with the size to the energv emitted from the gas. is a function of the width of appafatus in- a manner e entirely different from conabsorption coefficients, the The amount of heat transmitted from a gas to its vection heat transfer, and thickness of the gas layer, bounding surface may be calculated when we know i s h i g h l y sensitive to a and the temperature of the the gas and surface temperatures, the gas composition, change in temperature. It gas. and the shape of the apparatus. Figures 3, 5, and 6, is obvious, then, that safe Of the gases encountered together with equations (6) and (9), are sufficient to extrapolation of data necesin heat transmission equipsolve most problems involving this type of heat transfer. sitates the assumption of ment, carbon monoxide, the Three examples are given illustrating the method of the proper mechanism of hydrocarbons, water vapor, using the plots. h e a t t r a n s f e r . The role and carbon dioxide are the o n l y o n e s w i t h emission played by radiation from hot gases mag be indicated by the fact that in cracking coils roughly 40 per cent of the content to merit consideration. Moreover, carbon montotal heat transferred is by radiation from the products of oxide and the hydrocarbons are present in combustion prodcombustion; in open hearth furnaces, 90 per cent. ucts in such small amounts as to be negligible compared Although the general nature of thermal radiation from with water vapor and carbon dioxide. The last two, then, gases has been known ever since the early work of Julius, are the only ones we need consider. Their emission bands Paschen, and others, and has been studied by many investi- may be grouped into three spectral regions for each gas, gators since, these experimenters have been interested more which will hereafter be spoken of as the first, second, and in the resolution of characteristic gas radiation into bands, third band of carbon dioxide or of water vapor, the wave groups, and families for the purpose of studying molecular length of the band increasing with band number. structure, than in the quantitative determination of the total Outline of Derivation energy emitted by a gas. Not until the important investigaLet us consider a hollow evacuated space with walls in tion of Schack1128*was there any satisfactory attempt to determine heat transmission by radiation from non-luminous thermal equilibrium, the surface element dA radiating the gases. It is the purpose of this paper to outline the method quantity U of energy of wave length X to the surrounding of using data obtained from investigations on the infra-red surface. As there is to be no change in temperature, the spectra of gases, in order to calculate the quantity of heat surrounding surface must radiate back the same amount transmitted from those gases; to present charts for use in of energy U to dA. If now a gas a t the temperature of the such calculations; and to indicate the method of using the space is introduced, there will be no change in temperature charts for solution of problems in design of heat transfer and the walls will continue to radiate the amount U towards dA. Let the gas absorb the portion U1 of this energy. Then equipment. dA receives U-U1 from the surrounding walls. Since its General Picture of Gas Radiation temperature has not changed, it must still be radiating the If the radiant energy from a gas layer is passed through amount U , and must consequently receive a total of U. It a prism to a receiving instrument capable of measuring energy follows that the gas itself radiates an amount UI to the intensity, and if the intensity is plotted against wave length, surface dA, in order that the total radiation from walls and the resulting figure will not be a continuous curve similar to gas may equal U . This conclusion-that the amount of energy radiated by that obtained with radiation from a black body, but will consist of peaks or bands separated by wave-length regions a gas to a surface is equal to the amount of energy from the from which there is apparently no radiation whatever, even suiface which is absorbed in passing through the gas-is a special form of Kirchoff’s law. It enables one, on the as1 Presented before the meeting of the American Institute of Chemical sumption that absorption coefficient is independent of temEngineers, Cleveland, Ohio, May 31 to June 3, 1927. perature, to use measurements of absorption of radiation from * Numbers refer to bibliography at end of paper.
T
INDUSTRIAL A N D ENGINEERI-\-G CHEJfISTRY
August, 1927
a surface a t temperature T by a cold gas, in determining the amount of radiation which the gas itself would emit were i t at temperature T . Since the ratio U,/U is fixed by the absorbing characteristics of the gas and is independent of temperature provided absorption coefficients do not change with temperature, U1 w-ill follow the same law of temperature change that U , the
I
-
WAVE LENGTH
Figure 1-Comparison of Radiation f r o m a Black Body w i t h T h a t f r o m a Thick Gas Layer. Sharpness of Boundaries Exaggerated
radiation from a black surface, follows-i. e., Planck’s law of the distribution of energy in the spectrum of a black body. That is,
c,
1-5
(1) E x = ecz/xT - 1 in which Ex is monochromatic radiation intensity (energy radiated from a unit surface throughout a solid hemispherical angle-Ex is frequently defined in terms of unit solid angle, in which case the factor l / s enters before the (?I of equation (1)-above the surface, in unit time, in unit, wave-length range) of wave length A, T is absolute temperature, e is the natural base of logarithms, and C1 and C2are constants, The radiation from an infinitely thick gas layer which absorbs in the wave-length range a to b, and consequently acts like a black body throughout that range of wave length, is given by the integral
R
=
fix dh
889
2 feet thick, containing 5 per cent carbon dioxide. This interchangeability of partial pressure and thickness of gas layer is valid so long as the total pressure of the gas is maintained constant,4$5and it is immaterial what inert constituent is used to maintain this constancy of total pressure. We may then substitute for z in equation (3) or (4) the term PL, in which P is partial pressure of radiating constituent in atmospheres, and L is thickness of gas layer in feet. giving k the dimensions, atmos.-l x feet-‘. A combination of equations (2) and (4) gives
There remains to be considered the effect of gas shape on R. It will be remembered that Ex of equation (5) refers to radiation throughout the total solid angle above the surface element d A , and since the L of equation ( 5 ) 13 a constant outside the integral sign, that equation represents the radiation from (or absorption of) a solid hemisphere of gas of radius L. located above the surface element d d , when the beam of radiant energy proceeds in all directions to (or from) dd. Actually, the length of path of the beam varies with the angle of incidence of the beam striking d A ; and the relation between angle of incidence of beam and length of path through gas is a function of the particular gas shape being considered. Figure 2a s h o w a cross section through a bank of tubes, as encountered in cracking coils or water-tube boilers. The gas “seen” by a surface element on one of the tubes is shaded. This may be compared (Figure 2b) with the shape of gas seen by the surface element dil, as assumed in equation ( 5 ) . The ratio of the energy radiated by the actual shape to that radiated by the hemisphere of radius L equal to a characteristic dimension (to be explained later) of the actual shape, will be called the shape factor, and be designated by A more coniplete mathematical consideration of +, and its evaluation for various gas shapes,
+,
(2)
in which R is radiant energy per unit time per unit of bounding surface, and Ex is defined by equation (1). To consider the effect of thickness of gas layer we must introduce the absorption law J
=
J o e--k=
(3)
in which J is the intensity of beam of initial intensity Jo. after passage through a thickness of gas 2; k is the absorption coefficient of the gas for the ware length in question. Since k varies widely throughout the spectral range a to b, it is not permitted t o use an average value of 72. The assumption of linear variation of k from a value of 0 to k,,,, throughout the ranee of the band leads to the eauation
-
Jo -J - = Jo
1--
1 - e-k~ kx
(4)
in which the left side is the fraction of radiation entering a gas, which is absorbed by passage through thickness 5 , when the maximum coefficient of the band is k . The validity of the assumption as to variation of k , and the derivation of (4)from (3) are discussed more fully in Appendix I of another paper.3 The effect of concentration of radiating constituent of the gas is the same as that of thickness of gas layer; i. e., a layer 1 foot thick, containing 10 per cent carbon dioxide and 90 per cent of a constituent not radiating a t the wave length in question, will radiate the same amount as a layer
Figure 2-Path
of Radiant Beam for Different Gas Shapes
is given in the paper already cited.3 For our purpose it is sufficient to know that is a function of the shape of gas, and of the product term, kPL. Talues for it are given in Figure 4. By multiplying the right side of equation (5) by +, we obtain the true radiation from a gas mass of temperature T, characteristic dimension (or effective thickness) L , composition P , in the wave-length range a t o b. It has already been mentioned that there are three such wai-elength regions or bands for each of the two constituents, carbon dioxide and water vapor. We are now in a position to construct charts for calculations of heat transfer.
+
890
INDUSTRIAL AND ENGINEERING CHEMISTRY
Le
?he
LF:
CORRECTION FOR SUPERIMPOSED RADIATIONS FROM
I;
I .4
FIRST INFRARED BANDS OF
12-
09
IO-
08
0 9-
080 7-
06-
LP,+LP,
:.., . 0 4-
1003
04-
"""4
...
,
OR AND WHIFH CAS CaNTAIHI BOT"
COS AND
H1O.
.'
-
A
.... 020
020-
300
24%
0
014-
0120 I* '
1 s t Hz0 2.55to 2.84 6.5 11,12 2nd Hz0 5 . 6 to 1 . 6 " 13.5 7, 11 3rd HzO 12.0 to25.0 0.3b 7, 11 . This band actual1y:extends from 4 . 8 ~to 8.2p, b u t contains regions of such low absorption as:to be ineffective. Its effective width is but I#. One-half area under curve from 5 . 6 to ~ 7.6,~is used. 6 This is an average, rather than a maximum value, as a result of which the term, 1 (1 - e - k P L ) / k P L , of eauation (51 is replaced by the expression 1-e-kPL for this band only.
-
005
018016-
will be considered later) of gas layer over the surface 'receiving the radiation. The choice of scales for R vs. T would be determined by the form of the Planck equation. The older Rien energy distribution equation, differing from that of Planck by the absence of the subtracted term "unity" in the denominator, is known to agree well with that of Planck when AT is small, and is of such form as to give a straight line when the Iogarithm of Ex is plotted against reciprocal of absolute temperature. Since the range of variation of X for each band is small, we may expect an approximately straight line when R is plotted on a logarithmic scale against reciprocal of absolute temperature. Table:&-Constants Used in Calculations WAVE-LEXCTH RANGE k,a, BIBLIOGRAPHY P BANDh-0. Ft.-l atmos.-l REFERENCES 1st cot 2 . 6 4 t o 2.84 4.9 6, 7 2nd COz 4.13to 4.49 550.0 6, 7 3rd COZ 13.0 t o 1 7 . 0 24.0 8, 9, 10, 6
,,
03
Vel. 19, No. 8
008006-
004-
0020-
Figure 3
To obtain the total radiation to a square foot of bounding surface from a gas containing carbon dioxide, we would add the three R values for the three bands of the gas, one from each family of curves. If the gas contains both carbon dioxide and water, we are justified in adding all six band effect4 only under certain conditions. An inspection of Table I shows that the first band of COz and first band of HzO lie in the same wave-length range. Either constituent will consequently be somewhat opaque to the radiation from the other, and the total radiation to a surface, owing to the combined effect of the first bands of the two constituents, will be somewhat less than that obtained by adding the values as calculated independently by equation (5). The error introduced by the latter method will be negligible when the gas layer is thin or the percentage of radiating constituent low, but will be quite large for thick layers of gas or high percentage carbon dioxide and water vapor. The correction term to allow for this superimposed radiation is a complicated function of temperature, gas composition and .thickness, and absorption coefficients of the bands in question. Figure 3 is an alignment chartt for the determination of this correction term, to be subtracted from the added R's as calculated by equation (j),or as read from Figures 5 and 6 . It will be found that when
The calculation of the value of R for each of the bands of each radiating constituent necessitates a knowledge of the maximum absorption coefficient of each band and its effective wave-length width. &hack2 has made an exhaustive survey of available data on infra-red absorption spectra of carbon dioxide and water vapor, and the values recommended by him will be used, with but slight modifications. Table I gives these values. Those familiar with the infra-red spectra of carbon dioxide and water vapor will perhaps wonder a t the choice of band boundaries. It must be remembered that the calculations are based on the assumption of linear variation of absorption coefficient within a band and that L(4.9 Pcoz 6.5 Palo) < 0.3 this is not a true representation of the course of absorption. the correction term is negligible. Although Table I would It is therefore necessary so to adjust the band boundaries used as to obtain a resulting band energy agreeing most seem to indicate that there is some overlapping of the third nearly with the true value throughout the greatest range of bands of carbon dioxide and water vapor, the water vapor thickness of gas layer. Even with such adjustment equation band is actually composed of a great number of small bands ( 5 ) , with constants from Table I, predicts values of R for separated by non-absorbing regions, and the interference is the second and third bands of COz, which are too great n-hen negligible. The determination of total radiation from a gas involves a P L is small. The measurements of total absorption of unof its shape. Before adding the R's for the resolved radiation by Angstrom, Gerlach, and C o b l e n t ~ ~ ~ sconsideration ~~J~ provide a means of determining a correction factor for the different bands, we must multiply each value by its corresponding shape factor, +, which we have found is a function two bands menti0ned.l~~ Using the data of Table I to determine the R value for of the product term kPL, and which therefore varies from each band in accordance with equation (5), we could con- one band to the next as k varies. The shape factor of a gas struct families of curves representing the radiation from the mass is found to vary about the value, unity, and t o approach individual bands of COZ and H20. There would be one unity a t high values of kPL. The radiation from a tall, family of curves for each of the three bands of each con- narrow cylinder (Figure 2c) of gas to unit area of its base stituent. Each curve in a family would represent the varia- will be less than that from a hemisphere (Figure 2b) to unit tion of R with temperature, for a constant value of PL, in area a t the center of its base, when the height of the cylinder which P is the partial pressure of the radiating constituent equals the radius of the hemisphere. Consequently the; and L is the effective thickness (the meaning of this term shape factor will be less than 1.
+
August, 1927
INDUSTRIAL AND ENGINEERING CHEiMISTR Y
891
Looking a t the subject from another angle, the “effective equation, except that the temperature scale has been labeled thickness” of gas layer above the base of the narrow cylinder in O F. instead of O R , for engineering convenience. After is less than that of the hemisphere of gas above the center making the assumption of 4 = 1 in calculating Figures 5 of its base. When, however, the ratio of diameter to height and 6, the only further use we make of Figure 4 is to deterof cylinder is large, 4 for the cylinder will be greater than 1. mine t,he proper value to use for L, the effective thickness of By proper choice of effective thickness or characteristic the gas layer, and to find by inspection how great an error is dimension, L, of the gas shape in question; that is, by letting introduced by the use of the total radiation charts instead of L = 011 x height of cylinder, or a2 x clearlznce between those giving individual band radiations. The total heat exchange by radiation from a gas to its tubes in a cracking coil, etc., the shape factor can be made to oscillate about unity as kPL varies, so that the correction bounding surface is given by the equation term for each R value as calculated from equation (5) will Q / ~ p , A (c, + - 1~:) - (K,- K ~1 ) (6) be small. Shape factors for different gas shapes of industrial importance are given as a function of ~ P in L Figure 4, to- in which the subscripts g and s refer to the value of C, W , gether with the I-alue of k to use for each band, and the char- or K a t the mean gas temperature, TU>or the lnean surface P is the black-body J acteristic dimension L of the gas shape. We are now able temperature, T ~ respectiTrely* cient of the receiving surface. K is the correction someto calculate rildiation from a gas shape, to a square foot of bounding surface. c s i n g the value of L specific,d in ~i~~~ 4 times necessary for superimposed radiation of carbon dioxide for the shape in question, we would turn to the six families and water vapor, determined from Figure 3 . When there is a large drop in temperature of gas from of curves constructed by means of equation (j),and read entrance t o exit end of passage, the question arises as to the the R for each band, at the temperature of the gas and of in terms Of the surface, subtracting t h e low from the high value for each proper average gas temperature to use for entrance and exit gas temperatures. A little consideration band. Each resultant R would then be multiplied by its corresponding shape factor as read from ~i~~~~ 4, and these leads to the conclusion that the temperature-space relation final values of R added. The last step would consist in through the gas passage will differ, depending on whether consemultiplying the result by the black-body coefficient of the radiation Or Convection is controlling, and that quently will vary with the type of heat transfer controlling, receiving surface, which will usually lie between 0.6 and 0.9. It is apparent that the method just outlined is exceedingly for fixed terminal conditions. Let us first consider the tedious and, in the of a gas containing both c;arbondioxide case in which radiation is so large compared with convection vapor, w-ould involve the reading of six pairs of that it practically determines the temperature-space relation values from the families of curves representing individual in the apparatus. Let the mass of gas flowing through per specific heat = S, inlet gas temperature = band radiation, their subtraction by pairs, the multiplying unit time of the six resultant terms by six corresponding terms as read TiJ exit temperature = T2, and average surface temperafrom Figure 4, and the subtraction, in Some cases, of a cor- ture = Ts. Let CT W T - K T , the total radiation at temrection term for superimposed radiation from ~i~~~ 3 . perature T , be designated by RT. Consider a differential engineering length of Passage of surface area dA, with the gas falling in A simplification is almost indispensable from to T - d T * Then standpoint. Fortunately, this is possible without appre- temperature from ciable sacrifice of accuracy. If there were no shape factor consideration, nothing would prevent the three R values for the three bands of I GAS INSIDE S U 2 GAS IN INFINI L = DIAMETER, F T one constituent from being added before being 3 GAS BETWEEN PARALLEL PL plotted as a function of T and of PL. This 4 G A S OUTSIDE would be equivalent to assuming 4 to equal unity e throughout. JJ-e hare already found that by a AS 4 . EXCEPT TUBE OS‘CLEARANCE L=38.UEARANCE proper choice of L for a gas shape its 4 may PARALLELOPIPEO OF G A 5 , be made to oscillate about unity with a maxi- 2 mum deviation of about 10 per cent. For ex- g l o ample, if we consider the radiation from a gas $ sphere to a unit element of its surface, 4 will vary from ‘/a to 1 as kPL varies from zero to infinity, if the diameter of the sphere is used for L. If, however, we use diameter as the 09 effective thickness, L, of the gas layer, 4 passes from 1 through a minimum value of 0.955 and back to 1. Likewise, in the case of a gas be0 tween parallel planes a distance D apart, d varies K. P. L from 2 to 1 when L equals D; whereas, if we let, Figure 4-Shape Factors a n d Equivalent Thickness of Gas Layers L equal 1.8 times D , 6 varies from 1.111through a minimum of 0.94 and back to 1. I n a similar manner, by d Q / d e = L7s.dT = p d A . ( R T - R.) proper choice of L , the shape factor for any shape may be made to stay close to 1. The assumption of unrt shape facJAP.dA tor for all three bands of a gas then leads to small errors vs = which partly counterbalance, and n-hich are well within the L 2 T l dT accuracy of the constants of Table I. Figures 5 and 6, based RT - R, on this assumption, present the total R due to all three bands. (The constants of the Planck radiation equation, used in cal- For L7s 1I-e may substitute Q / A e / ( T ,- T2). obtaining culating Figures 5 and 6, are those given in International TI- T , _ -- p . A . (7) Critical Tables.) ZR for carbon dioxide will be designated 8 by C; Z R for water vapor by W . The scales used are those .dT already mentioned as justified by the form of the Planck
&/e
v,
+
E
-
-
’
Vol. 19, No. 8
INDUSTRIAL A N D EhTGINEERIiVG CHEMISTRY
892
3oooo
TOTAL RADIATION DUE TO CARBON DIOXIDE
2OOOO
P=PARTIAL PRESSURE O F CARBON DIOXIDE, ATM.
IO000
8ooo
L=CHARACTERISTIC DIMENSION OF GAS SHAPE (EFFECTIVE THlCKNESS OF GAS LAYER),FT SEE SHAPE FACTOR d O T
6000 4000
THIS PLOT IS BASED ON ASSUMPTlON OF UNIT SHAPE FACTOR.
3000
a 20005 LL t
0;
1000 v)
800
> 3
600 G m
am 300
Loo
100 80
60
, -
0 0 @
,-
1
r
1 . 1
I
0
0
0
2
0TEMPERATURE 8
0
o
3 ! DEGRESS
0
8
L
-
I
I
0 0
0 0
l
T
l
I
FAHRENHEIT
I
l
l
l
I
I
I
I
I
I
8 8 88888
0
0
R
3 8 %W
%$%$140
Figure %Total Radiation from Carbon Dioxide
which is the desired relation. To integrate the denominator we must express R T as a function of T . Over a short range of variation of T this may be done satisfactorily by a power function, but the result is unwieldy, and the power function is different for different ranges of temperature. The integration can be performed graphically by plotting the term, 1/(Rt - R,) vs. T, and taking the area under the resulting curve, between the limits TIand T z . A simpler method will be presented shortly. Equation (7) gives Q/e when radiation controls the temperature-space relation through the apparatus. For the case in which convection fixes that relation, a method of derivation similar to that used to obtain equation ( 7 ) leads to
obtained by using gas temperature as defined by equation (9), and by using the arithmetic mean temperature of the gas. It will be noticed that Q/ilebased on a gas temperature as calculated from equation (9) lies, in every case, between the two “true” values. Since neither radiation nor convection ever controls to the exclusion of the other, the use of a mean gas temperature as defined by equation (9) is as fully justified as the use of equation (7) or (8). Table 11-Comparison
@/A9
T.
TI
Tz
F.
’ F.
3000
2000
0
F.
1500 1800 2000
1000
0
AS
USING
Q/Ae USING
CALCD. Tg AS ARITHMETIC B Y En. ( 7 ) DEFINED MEAN OR ( 8 ) B Y Eo. (9) Te
1000
As with equation ( T ) , a graphical solution may be obtained by plotting R T / ( T - T,) vs. T , and taking the area under the curve between T 1 and Tz. The determination of Q/e by use of equation ( 7 ) or (8) is tedious, and the question arises as to whether it is possible to use a mean temperature of gas, expressed as a function of T1, Tz,and T,. Two methods, with a semi-mathematical basis, both lead to results which are poorer than the empirical method of using for gas temperature in equation ( 6 ) , the temperature of the surface T,, plus the logarithmic mean temperature difference of gas and surface. I n other words,
of Values of Q / A 9 Q/Ae
K!
600
{g
800
{:
5660 5842 4955 5062 3617 3698 2368 2445 1550 1785 1296 1469 1025 1168
5840
5950
5020
5290
3670
4090
2385
3085
1695
1860
1369
1669
1OiO
1480
Factor controlling temperature-space relation: R = radratlon, C = convection. Q
Before illustrating the use of the plots, let us consider briefly their probable accuracy. The assumptions involved in their derivation are (1) that the course of absorption through a band may be expressed as a linear function of wave length, provided a compensating adjustment of band boundaries is made, (2) that the absorbing characteristics of the gas do not change with temperature, (3) that the shape factor is 1. The first assumption undoubtedly leads to Table I1 presents a comparison of the true value of Q/Ae, errors of considerable magnitude.2 The second assumption calculated graphically by equation (7) or (S), with the value is necessary to make use of absorption coefficients determined
INDUSTRIAL ,4YD ENGIiVEERING CHEMISTRY
August, 1927
A
---'-+-,/
~
___
~~~~
/
I 0 0
Q
- 0
2
0
0
(0
01
0 0
12
TEMPERATURE
0
,
Figure 6-Total
N
893
_ _ ~
I
_c
4
I
8
3
DECREES FAHRENHEIT
g
a
c
u
ru
4-
!
Numrn
Radiation from Water Vapor
a t room temperature. The work of Eva 1-011 Bahr16 indicates that there is an increase in absorption with rise in temperature, a t least up to 900" C. By neglecting this increase and using values obtained a t room temperature, we obtain minimum values of &/-4e, provided, of course, that the absorption coefficient does not again decrease a t high temperatures. The third assumption has already been considered, and leads to errors of much smaller magnitude than the other two. The data on carbon dioxide are much more reliable than those on water vapor. I t is probable that the maximum error in determining &/+4efrom the plots is not over 30 per cent, and probably less. This is greater than the error involved in using many of the best equations for heat transfer by convection, but it a t least leads to results in the right neighborhood. Work is under way a t the Massachusetts Institute of Technology to determine the total radiation from carbon dioxide and water vapor a t high temperatures, to provide a more accurate basis for Figures 5 and 6. H e a t T r a n s f e r Calculations I l l u s t r a t i n g Use of Plots
Since the principles involved in heat transfer by radiation from gases are different from the more familiar ones used in convection heat transfer, a few representatil-e examples will be giren to indicate the use of the charts. A s a first example let us consider a gas flowing with a mass velocity of 0.4 pound per second per square foot. carbon dioxide content 20 per cent, no water rapor, specific heat 0.3, through a 6 by 6 inch square duct, the gas entering a t 2000" F. Suppose an average surface temperature of 800" F. Assuming all the heat to be transferred by radiation from
the gas, how long would the flue hai-e to be to cool the gas to 1000" F.? To determine average gas temperature, we use equation (9). To = 800 4- (2000-1000),'2.3 log (2000-800)/(1000-800) = 1358" F.
PL
= (6/12) X 0.20 = 0.1.
Using Figure 5 lye read, for P L = 0.1, C I ~ = ~ S1460, and CSW = 380. Then, assuming a black-body coefficient of 0.9 for the surface of the duct, Q/Ae
=
0.9 (1450-380)
=
963 B. t. u./sq. ft./hour.
The total heat transferred per unit time is Q/.O = mass velocity X cross sectional area X specific heat X rise in temperature = 0.4 X 3600 X 0.25 X 0.3 X (20001000) = 108,000 B. t. u./hour given up by the gas. Then the necessary area is 108,000/963 = 112 square feet,
corresponding to a flue length of 56 feet. The equivalent coefficient h,, as used in convection heat transfer calculations, is 963,laverage At = 963/558 = 1.73, which is of the same order of magnitude as we would expect h, to be under such conditions, indicating that roughly half the total heat transferred mould be due to radiation. It is t o be remembered that the expression of heat transfer by radiation in the form of a coefficient is a purely artificial procedure, used simply for comparison with h,,. Suppose it is desired to find the effect of doubling the mass velocity of the gas in the above duct, maintaining the same entrance gas temperature and average surface temperature and flue length. We have the equat'ion 0.9 (C, - Cm)
=
0.8 X 3600 X 0.2-5 X 0.3 X (2000- Tz) 112
I,VDCSTRIAL A X D ESGINEERIXG CHEMISTRY
894
which reduces to C, = 4284 - 2.142 T z ,which must be solved by trial and error. Assuming T z = 1200, equation (9) gives a gas temperature of 1528, for which C, = 1935. 4284 2.142 X 1200 = 2094, as opposed to 1935; therefore, the assumption of 1200" F. was slightly low. A second trial indicates an exit gas temperature of 1230" F. Thus we see that there is an increased amount of heat transferred, though at lower efficiency, when the velocity is increased. (Q/A0)c (Q/A8),
-
-
0.8
-
0.4
(2000 - 1230)0.8 - 1,532 (2000 - 1000)0.4
This is analogous to the effect of velocity on heat transfer by convection, in which the coefficient increases as the 0.5: 0.8 power of the velocity, although the cause is entirely different, being due in the case of radiation to the use of a greater portion of the flue area for heat transmission a t the high temperature a t which it is most effective. As a final example, let us consider a continuous billetreheating furnace such as is used in rolling mills. A representative furnace of this type has an effective hearth area of 40 by 11.5 feet, and a depth of gas above the billets of 3.5 feet a t the burner end and 1.5 feet at the cold end. Suppose the gas enters a t a temperature of 3450" F. and is cooled to 1300" F., and that the steel, flowing countercurrent to the gas, is heated from 80" to 2275" F. Let the furnace burn 100 cubic feet of coke-oven gas per second, producing a flue gas of 7.5 per cent COZ, 20.0 per cent HzO, and the rest CO, Oz, and N2. What is the rate of heat transmission a t the two ends of the furnace? Turning to the shape factor plot (Figure 4), we find that the gas shapes nearest that under consideration are (3) the space between infinite parallel planes, and (6) a rectangular parallelepiped 1 X 2 X 6. At the burner end where gas layer is thick, and the gas shape more nearly approximates (6), we shall use 1.4 X 3.5 feet, or 4.9 feet as the effective L. At the cold end, where the flat gas mass approximates case (3) of Figure 4, we use 1.6 X 1.5 feet, or 2.4 feet as effective thickness, L. Considering first the burner end, P , L = 0.075 X 4.9 = 0.37, and P , L = 0.20 X 4.9 = 0.98. Using Figures 5 and 6 for gas and steel temperatures of 3450" and 2275" F., respectively, and p = 0.8, we have, on substituting into equation (6), Q / A 0 = 0.8( (17500-7000)
+ (25000-9600) - (6900-2300)} = 17,050 B. t. u./sq. ft./hour
It will be noticed that for this case the correction due t o superimposed radiation is appreciable, owing to the high values of PL. As a matter of interest, the value of the coefficient of heat transfer, as used in convection, may be obtained for comparison: h, = 17050/(3450 - 2275)
=
14.5
If we use the Weber equation" for convection coefficient of gases flowing inside conduits, the result obtained is h, = 1.58. It is apparent, then, that at the hot end of the furnace about 90 per cent of the total heat transferred is by radiation from the hot gases. At the cold end of the furnace, however. conditions are reversed. At this end, P , L = 0.18, P , L = 0.48, To = 1300" F., T , = 80" F. On substitution into equation (6), we find Q/A8 = (0.8)(1500-0 1800-0 150 0) = 3150. The equivalent h, = 2.46, while the value of h , from the Weber equation is 2.44. Consequently, at this end of the furnace heat transferred due to convection and that due to radiation are about equal. The over-all H due to convection and gas radiation combined (obtained by adding the individual h, and h,) varies from about 16 at the hot end to 5 a t the cold end of the furnace. Actual experimental data on a furnace such as that described here indicate average over-all coefficients ranging from 5 to 12, six out of seven of them lying above 10. Such high values would not have been expected on the assumption that all t h e heat was being transferred by convection.
+
+
.
Vol. 19, K O . 8
Acknowledgment Special acknowledgment is due R. T. Haslam, of the Massachusetts Institute of Technology, for critically reading this paper and for valuable suggestions. Nomenclature Ex = energy intensity at wave length X x = wave length, p Q,'8 = B. t. u. transmitted per hour A = area of surface, square feet P = partial pressure of radiating constituent, atmospheres L = effective thickness of gas layer, feet T = temperature, O F., in all equations after ( 5 ) c, = heat radiated due t o the nth band of COZ It', = heat radiated due to the nth band of HzO C = total heat radiated due to CO1, B. t. u./sq. ft./hour ?.I/ = total heat radiated due t o H20, B. t. u./sq. ft./hour K = correction due to superimposed radiation, same units as C and W P = black-body coefficient of surface mass velocity of gas, Ibs./sec./sq. f t . cross-section area s = specific heat of gas shape factor of gas mass R = C W - K in equations after ( 5 )
v =
+ = + 1 2
Q
s e H
Subscripts - gas entering gas leaving - a t the temperature of the gas = a t the temperature of the surface - due to carbon dioxide (or convection) - due to water vapor -
-
Bibliography
t For derivation
of the correction term and alignment chart therefor, see Appendix IV of Ref. 3. 1-Schack, Miil. Warmestelle Dlisseidorf, 56 (1924). 2-Schack, Z. tech. Physik, 5, 266 (1924). 3-Hottel, Presented a t the meeting of the Institute of Chemical Engineers, M a y 31 t o June 3, 1927. 4-Von Bahr, A n n . Physik, 29, 780 (1909). Verhandl. deut. physik. Ges., 13, 617 (1911). 5-Hertz, 6-Rubens and Ladenburg, Ibid., 7, 170 (1905). i--Von Bahr, Ibid., 16, 721 (1913). 8-Hertz, Dissertation, Berlin, p. 26 (1911). Verhandl. deut. p h y s i k . Ges., 15, 610 (1913). 9-Burmeister, 10-Rubens and Aschkinass, Ann. Physik, 64, 584 (1898). 11-Rubens and Hettner, Verhandl. deul. physik. Ges., 18, 154 (1916). lZ-Sleator, A s t r o p h y s . J., 48, 124 (1918). 13--Angstrom, A n n . P h r s i k , 39, 267 (1890); 6, 163 (1901). 14--Gerlach, Ibid., 50, 233 (1916). Bur. Standards, Sci. Paper 357 (1919). 15-Coblentz, 16-Von Bahr, A n n . Physzk, 38, 206 (1912). 17--Walker, Lewis, and McAdams, "Principles of Chemical Engineering," p. 148, McGraw-Hill Book Co., Inc., 1923.
Information Storage We are indebted to Dan Gutleben, an engineer of the Pennsylvania Sugar Company, for the chart (page 895), selected from a drawing many feet in length, which illustrates the application of the engineer's language to the storage of information. The original outline represented a beet sugar factory from the beet storage sheds to the sugar storage house. The purpose is to fix systematically the statistical information for use jn the drafting room, in the erection work, and as a permanent record of the operators. The outline is in fact a flow sheet for the entire factory, carrying descriptions of apparatus and other equipment. The descriptions indicate the salient features with shop numbers and manufacturers' names for identification. As all the equipment is arranged in order, the information is automatically indexed. Some plants store their statistical information in volumes written in the language of S o a h Webster, but in this outline, which presents the facts in the language of the engineer, may be found practical suggestions which can be applied in many industries and to various branches of the chemical industry.
