hottel 1927 heat transmission by radiation from non luminous gases

7/29/2019 Hottel 1927 Heat Transmission by Radiation From Non Luminous Gases

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88 8 I N D U S T R I A L A N D E N G I NE E R IN G C H E M I S T R Y Vol. 19, No . 8

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Heat Transmission b y Radiation fromNon-Luminous Gases

absorption coefficients, thethickness of th e gas layer,and the tem perature of t hegas.Of t he gases encounteredin heat transmission equip-me nt, carbon monoxide, thehydrocarbons, water vapor,an d carbon dioxide are theo n l y o n e s w i t h e miss io n

By H. C . HottelFUEL N D GAS ENGINEERING,ASSACHCSETTSNSTITUTEF TECHNOLOGY,AMBRIDGS,ASS.

HE imp ortanc e of ra diatio n from hot gases as one ofthe major factors involved in man y problems of hea tT ransfer has, until recently, been overlooked. Co-efficients of he at transm ission based o n the assum ption ofconvection as the mechanism controlling heat transfer havebeen reported for ma ny types of industrial heat-transmissionequipment in which radiation from the hot non-luminousgases was actually the controlling factor. Such coefficients,while satisfactory as long as their application is limited toapparatus s imilar to that which furnished the data, lead tofalse conclusions when conditions are different. Heattransfer by convection varies widely with gas velocity andsize of ga s passage, some what with tem pera ture of gas, andalmost none a t all with gas composition. He at transfer byradiation is independent of ga8 velocity. varies with the sizeof ap pa fat us in- a manner -entirely different from con-vection heat transfer, andi s h i g h l y s en s i t i v e t o achange in temperature. Itis obvious, then, that safeextra pola tion of da ta neces-si tates the assumption ofthe proper mechanism ofh e a t t r a n s f e r . T he roleplayed by radiation fromhot gases mag be indicated

with very gre at thicknesses of gas layers (Figu re 1). A stillgreater sp ectra l dispersion of t he radi ation will resolve th eban ds them selves into a series of sm aller bands. As th ethickness of t he ga s layer is increased, the in tensitie s of t hedifferent bands increase and approach as a limit the intensityof black-body radiation at t he wave length of th e band inquestion, as given by the Planck radiation equation forenergy distribution in the spectrum of a black body. Th erat e a t which black-body intensity is approache d as thicknessof gas laye r is increased is dete rmin ed by th e absorbing ch ar-acteristics of the gas for radiation of the wave length inquestion, and varies not only from one band to the next,but even within a band. It is apparent , then, that the areaunde r th e wave length-intensity curve, which is proportiona lto the energv em itted from the gas. is a function of th e width

The amoun t of heat transmitted from a gas to itsbounding surface may be calculated when we knowthe gas and surface temperatures, the gas composition,and the shape of the apparatus. Figures 3, 5, and 6 ,together with equations (6 ) and (9), are sufficient tosolve mos t problems involving this type of heat transfer.Three examples are given illustrating the method ofusing the plots.

by the fact that in cracking coils roughly 40 per cent of t h etotal heat transferred is by radiation from the products ofcombustion; in open hearth furnaces, 90 per cent.Although the general nature of therm al radiation fromgases has been known ever since th e early work of Julius,Paschen, and others, and ha s been studied by ma ny invest i-gators since, these experimenters have been interested morein th e resolution of characteristic gas radiation into ba nds,groups, an d families for the purpose of studying molecularstructure, than in the qu anti tat ive determination of the tota lenergy emitted by a gas. Not unti l the importa nt invest iga-tio n of Schack1128*was there any sat isfactory at tempt todetermine heat transmission by radiation from non-luminousgases. It is the purpose of this p aper t o outline the methodof using dat a obtained from investigations on the infra-redspectra of gases, in order to calculate the qu an tity of h eattransmitted f rom those gases; to present char ts for use insuch calculations; an d to indicate the m ethod of using th ech art s for solution of proble ms in design of h ea t transfe requipment.

General Picture of Gas RadiationIf the radiant energy f rom a gas layer is passed througha prism to a receiving instrument capable of m easuring energyintensity, and if t he intensity is plotted against wave leng th,the resulting figure will not be a continuous curve similar tothat obtained with radiat ion f rom a black body, but willconsist of peaks or bands separated by wave-length regionsfrom which there is app aren tly no radiation wh atever, even

1 Presented before the meeting of the American Institute of Chemical* Numbers refer to bibliography at end of paper.Engineers, Cleveland, Ohio, May 31 to June 3, 1927.

content to merit consideration. Moreover, carbon mon-oxide and the hydrocarbons are present in combustion prod-ucts in such small amounts as to be negligible comparedwith water vapor and carbon dioxide. Th e last two, the n,are the only ones we need consider. Their emission bandsmay be grouped into three spectral regions for each gas,which will hereafter be spoken of as the f irs t , second, andthird band of carbon dioxide or of water vapor, the wavelength of t he band increasing with band numbe r.Outline of Derivation

Let us consider a hollow evacuated space with walls inthermal equilibrium, the surface element dA radiat ing thequan t i ty U of energy of wave length X to the surroundingsurface. As there is to be no change in temperature, thesurrounding surface must radiate back the same amountof energy U t o dA. If now a gas a t the tempera tu re of th espace is introduced, there will be no change in temperatureand t he walls wil l continue to rad iate the amount U towardsd A . ThendA receives U - U 1 from the surrounding walls. Since itstemperature has not changed, it must still be radiating theamount U , and must consequently receive a total of U . Itfollows that the gas itself radiates an amount UI to th esurface dA, in order that the t otal radiat ion f rom walls andgas may equal U .Thi s conclusion-that the amount of energy radiated bya gas to a surface is equal to the a mo unt of energy from thesuiface which is absorbed in passing through the gas-isa special form of Kirchoffs law. It enables one, on the as-sumption that absorption coefficient is independent of tem-pera ture, to use measurem ents of absorption of radiatio n from

Let t he gas absorb the por t ion U1 of this energy.


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August , 1927 I N D U S T R I A L A N D E N G I NE E R I- \- G C H E J f I S T R Y 889a surface at temperature T b y a cold gas, in determining th ea m o u n t of radiation which the gas itself would emit were itat temperature T .Since the rat io U , / U i s f ixed by the absorbing character-istics of t he gas an d is indep ende nt of tem peratu re provide dabsorption coefficients do not change wi th t empera ture , U1w-ill follow the sam e law of tem pera ture change t ha t U , the

I

WAVE - LENGTHFigure 1-Comparison of Radiation from a Black BodySharpnes s of Boundariesith That from a Thick Gas Layer.Exaggerated

radiatio n from a black surface, follows-i. e., Plancks lawof the distribu tion of energy in the spectrum of a black body.T h a t i s , c, 1 - 5 (1)E x =ecz / xT - 1in which Ex is monochromatic radiat ion intensi ty (energyradiated from a uni t surface throughou t a solid hemisphericalangle-Ex is frequ ently defined in terms of unit solid angle,in which case the factor l / s enters before the (?I of equation(1)-above th e surface, in un it t ime, in unit, wave-lengthrange) of wave length A, T is absolute temperature, e isthe n atural base of logari thms, and C1 a n d C2 re constants ,Th e radiat ion from an infini te ly thick gas layer which absorbsin the wave-length range a t o b , and consequent ly acts l ikea black body throughout that range of wave length, is givenby the in tegra l

R =fixh (2)in which R is rad iant energ y per unit t ime per un it of boun d-ing surface, and Ex is defined by equation (1).T o consider the effect of thic kne ss of gas layer we mustintroduce the absorpt ion lawin which J is the intensity of beam of initial intensity J o .after passage through a thic kn ess of gas 2; i s the absorpt ioncoefficient of the gas for the wa re length in question. Sincek varies widely throughout the spectral range a o b, i t i snot p ermit ted t o use an average value of 72 . The assumptionof lin ear vari atio n of k from a value of 0 t o k,,,, t h roughoutthe ranee of the band l eads to the eaua t io n

J =J o e--k= (3)

- 1 - - k ~1--o -- =Jo k x (4)in which the left side is the frac tion of radiatio n enterin g agas, which is absorbed b y passage through thickness 5 , whenth e maxim um coefficient of th e band is k . The va l id i tyof the assum ption as to variat ion of k , and the d erivat ion of(4) ro m (3) are discussed more fully in Appendix I of an-other paper.3The effect of concentration of radiating constituent ofthe gas is the same as that of thickness of gas layer; i . e.,a layer 1 foot thick, containing 10 per cent carbon dioxidea n d 90 per cent of a const i tuent not radiat ing at the wavelength in quest ion, wil l radiate the same amount as a layer

2 feet thick, containing 5 per cent carbon dioxide. Thisinterc han gea bility of pa rtial pressure an d thicknes s of gaslayer is valid so long as the total pressure of th e gas is main-tained constant ,4$5 nd i t i s immaterial what inert const i tuentis used to maintain this constancy of total pressure. Wemay then subs t i tu te for z i n equa t ion (3) or (4) t he t e rm PL,in which P s partial pressure of rad iating c onstitu ent inatmospheres , and L is thickness of gas layer in feet. givingk the dimensions, atmos.-l x feet-.A com binatio n of equatio ns (2 ) a n d (4) gives

There remains to be considered the effect of gas shape on R.It will be remembered th at Ex of equation (5) refers to rad ia-t ion throughout the total sol id angle above the surfaceelement d A , and s ince the L of equat ion ( 5 ) 13 a constantouts ide the integral s ign, tha t equat ion represents the radia-t ion from (or absorpt ion of) a solid hemisphere of gas ofradius L. located above the surface element d d , when thebeam of radiant energy proceeds in all directions to (or from)dd. Actually, the length of path of the beam varies withth e angle of incidence of t he beam s t rik ing d A ; and th e rela-tion between angle of incidence of be am a nd leng th of p at hthrou gh gas is a function of t he partic ular gas shape beingconsidered. Figure 2a s h o w a c ros s s ec tion through a bankof tubes, as encountered in cracking coils or water-tubeboilers. Th e gas seen by a surface element on one of t hetubes is shaded. This may be compared (Figure 2b) withthe shape of gas seen by the surface element di l , as assumedin equat ion ( 5 ) . Th e rat io of th e energy radiated by th eactual shape to that radiated by the hemisphere of radiusL equal to a characteristic dimension (to be explained later)of the actual shape, will be called the shape factor, and bedesignated by + A more coniplete mathematical con-sideration of + and its evaluation for various gas shapes,

Figure 2-Path of Radiant Beam for Different Gas Shapesi s given in the paper al ready ci ted.3 For our purpose i tis sufficient to k now t ha t +is a func tion of th e sha pe of g as,an d of the pro duct term, kPL. Talues for it are given inFigure 4. By multiplying the right side of equation (5)by + we obtain the t rue radiat ion from a gas mass of tem-perature T, characteristic dimension (or effective thickness)L, composition P , in the wave-length range a o b. It hasalready been ment ioned that there are three such wai-e-length regions or bands for each of the two const i tuents ,carbon dioxide and water vapor. We are now in a posit ionto construct charts for calculations of heat transfer.


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890

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I ND U ST R I A L A N D E NG I NE E RI NG CH E M I STR Y Vel. 19 , No. 8LF: ? he

C O R R E C T I O N FOR SUPER-IMPOSED RADIATIONS FROMFIRST INFRARED BANDS OF4

12-IO-0 9-08-0 7-06- -P,+LP, O R A N D ,WHIFHCOS AND H1O.CA S CaNTAIHI BOT"., .

1003 ...,,04- """4

'....- 0 2 0

00 5020-018-016- 0014-012-0 I*

' 008-006-

004-002-

24%300

0 -Figure 3

The calculation of the value of R for each of the bands ofeach radiating constituent necessitates a knowledge of themaximu m absorption coefficient of each band and its effectivewave-length width. &hack 2 has made an exhaustive surveyof available da ta on infra-red absor ption spectra of carbondioxide and water vapor, and the values recommended byhim will be used, with b ut slight modifications. Table Igives these values. Those familiar with the infra-red spec traof carbo n dioxide and wa ter vapor will perhaps wonder a tthe choice of band boundaries. It must be rememberedth at th e calculations are based on the assump tion of linearvariat ion of absorption coefficient within a band a nd th a tthis is not a true representation of the course of absorption.It is therefore necessary so to adjus t the band boundariesused as to obtain a resulting band energy agreeing mostnearly with the true value throughout the greatest range ofthickness of gas layer. Ev en with such adjustmen t equation( 5 ) , with constants from Table I, predicts values of R forthe second and third bands of COz,which are too great n-henP L is small. Th e measuremen ts of tota l abso rptio n of un-reso lv ed ra di at io n b y An gs tr om , G er la ch , a n d C o b l e n t ~ ~ ~ s ~ ~ J ~provide a means of determining a correction factor for thetwo bands ment i0ned. l~~Using the da ta of Ta ble I to determine the R value foreach band in accordance with equation (5), we could con-struc t families of curves representing the radiation from theindividual bands of COZ and H20. There would be onefamily of curves for each of t he th ree ba nd s of each con-stituent. Eac h curve in a family would represent the varia-tion of R with temperature, for a constant value of P L , inwhich P is the partial pressure of the radiating con stituenta n d L is the effective thickness (the meaning of this term

will be considered late r) of gas layer over th e surface 're-ceiving the radiation. Th e choice of scales for R vs . Twould be determined by the form of the Planck equation.The older R ie n energy distribution equ ation, differing fromth at of Planck b y the absence of th e subtrac ted term "unity"in the denominator, is known to agree well with that ofPlanck when AT is small, and is of such form as to give astraight line when the Iogarithm of Ex is plotted againstreciprocal of absolute temperature. Since the range ofvariation of X for each band is small, we may expect anapproximately straigh t line when R is plotted on a logarithmicscale against reciprocal of absolute temperature.

Table:&-Constants Used in CalculationsWAVE-LEXCTHANGE k,a, BIBLIOGRAPHYBAND -0. P Ft.-l atmos.-l REFERENCES1st cot 2 .64 to 2 .84 4 . 9 6, 72nd COz 4 . 1 3 t o 4 .49 550.0 6, 73rd COZ 1 3 . 0 t o 1 7 . 0 24.0 8 , 9, 10, 6

1 s t Hz0 2 . 5 5 t o 2 . 8 4 6 . 5 1 1 , 1 22nd Hz0 5 . 6 to 1 . 6 " 13.5 7, 113rd HzO 1 2 . 0 t o 2 5 . 0 0.3b 7, 11. This band actual1y:extends from 4 . 8 ~o 8.2p, but contains regionsof such low absorption as:to be inef fective. It s effective width is bu t I#.One-half area under curve from 5 . 6 ~o 7.6,~ s used.6 This is an average, rather th an a maximum value, as a resul t of whichthe term, 1- 1 e - k P L ) / k P L , of eauation (51 is replaced by t he expression1-e-kPL for this band only.To obtain th e total radiation to a sq uare foot of boundingsurfa ce from a gas containing carbon dioxide, we would addthe three R values for the three ba nds of the gas, one fromeach family of curves. If the gas contains both carbondioxide and w ater, we a re justified in ad ding all six band effect4only under certa in conditions. An inspection of Table Ishows that the first band of COz and first band of HzO li ein the same wave-length range. Either constituent willconsequently be somewhat opaque to the radiation fromthe other, and the total radiation to a surface, owing to thecomb ined effect of th e first ba nd s of the two constituents,will be somewhat less than tha t obtained by adding the value sas calculated independently by equation (5 ) . The errorintroduced by the la tter m ethod will be negligible when th egas layer is thin or the percentage of radiating constituentlow, but will be quite large fo r thi ck layers of gas or highpercentage carbon dioxide and water vapor. The correctionterm to allow for this superimposed radia tion is a complicatedfunction of temp eratu re, gas composition an d .thickness, andabsorption coefficients of the bands in question. Figure 3is an alignment ch ar tt for the determination of this cor-rection term, to be subtracted from the added R 's as calcu-lated by equat ion (j), r as read from Figures 5 and 6 .It will be found that when

the correction term is negligible. Although Table I wouldseem to indicate th at th ere is some overlapping of t he thi rdbands of carbon dioxide and water vapor, the water vaporban d is actually composed of a great numb er of sm all ban dsseparated by non-absorbing regions, and the interference isnegligible.The determination of total radiation from a gas involves aconsideration of its shape. Before addin g the R's for thedifferent bands, we must multiply each value by its corre-sponding shape factor, + which we have found is a functionof th e produ ct term k P L , and which therefore varies fromone band to the next as k varies. Th e sha pe factor of a gasmass is found to vary about th e value, unity, and t o approachuni ty at high values of k P L . The radiation from a tall ,narrow cylinder (Figure 2c) of gas to un it are a of i ts basewill be less tha n th at f rom a hemisphere (Figure 2b) t o uni tarea a t the center of its base, when the height of the cylinderequals the radius of the hemisphere. Consequently the;shape factor will be less than 1.

L(4.9 Pcoz +6. 5 Palo)


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August , 1927 I N D U S T R I A L A N D E N G I N E ER I N G CHEiMISTR Y 891Looking at the subject from another angle , the effect ivethickness of g as layer above th e base of th e narrow cyl inderis less tha n t ha t of t he hemisphere of gas above the centerof its base. W hen , however, the ratio of diameter to heightof cylinder is large, 4 for the cyl inder will be greater th an 1.By prope r choice of effective thickness or ch aracteristicdimension, L, of the gas shap e in quest ion; tha t is , by le t t ingL = 011 x height of cylinder, or a2 x clearlznce betweentubes in a cracking coil , e tc ., th e shap e factor can be m a d e t oosci l la te about uni ty as kPL varies, so that the correct ion

equa t ion , except th a t th e t empera ture sca le has been l abe ledin O F. instead of O R , for engineering convenience. Aftermaking the assumption of 4 = 1 in calculating Figures 5a n d 6, the only furth er use we mak e of Figure 4 is to deter-mine t ,he proper value to use fo r L, th e effective thick ness ofthe gas layer, and to f ind by inspect ion how gr eat an error isintrod uced by th e use of the to tal radiat ion ch arts ins tead ofthose giving individual band radiat ions .The to ta l hea t exchange by radia t ion f rom a gas t o i t sbounding surface is given by the equat ionterm for each R value as calculated from equat ion (5) willbe smal l. Shap e factors fo r different gas shapes of in dus trialimportance are given as a func t ion of ~ P Ln Figure 4, to -Q / ~ ,A (c, - + - 1~:) - (K, K ~ ) (6)

in which the subscripts g a n d s refer to the value of C, W ,gethe r with the I-alue of k t o use for each band, a nd th e char -acteris t ic dimension L f the gas shape. We are now able

or K a t t h e m e a n g a s t e m p e r at u r e , TU> r the lnean surfacetempera ture , T ~ JespectiTrely* P is the black-bodyto calculate rildiation from a gas shape, to a square footofbounding surface. c s i n g t h e v a lu e of L specific,d in ~i~~~ 4for the shape in quest ion, we would turn to the s ix famil iest h e R for each band, at t h e temperature of the gas a n d o fthe surface , subt rac t ing th e low f rom th e h igh va lue for eachband. Ea ch resul tant R would then be mult ipl ied by i tscorresponding shape factor as read from ~i~~~~ 4, and thesefinal values of R added. T he las t s tep would consis t inmult iplying the resul t by the black-body coeffic ient of thereceiving surface, which will usually lie between 0.6 a n d 0.9.It is apparenth a t t h e m e t h o d just outlined is exceedinglytedious and , in t he of a gas containing both c;arbondioxidevapor, w-ould involve th e reading of six pairs ofvalues from the famil ies of curves represent ing individualband radiat ion, their subtract ion by pairs , the mult iplyingof the s ix resul tant term s by s ix corresponding term s as readf rom F igure 4, and th e su btract ion , in Some cases , of a cor-rection term for superimposed radiation from ~i~~~ 3 .A simplification is almost indispensable fromstandp oint . Fortun ately, this is possible witho ut appre- temperature from to T-dT* Thenciable sacrifice of accu racy . If there were noshape factor considerat ion, nothing would pre-vent the three R values for the three bands ofone const i tuent from being added before beingplo t t ed as a func t ion of T and of P L . This

cient of the receiving surface. K is the correction some-times necessary for superimposed rad iation of carbon dioxideand water vapor, determined from Figure 3 .ent rance t o exit end of passage, the question arises as t o t h eproper average gas temperature to use fo r in terms Ofentrance and exi t gas temperatures . A l i t t le considerationleads to the conclus ion that the temperature-space relat ionthrough the gas passage wil l di ffer , depending on whetherradiation Or Convection is controlling, an d th at & /e onse-quent ly wil l va ry w i th th e typ e of hea t t rans fe r contro ll ing ,fo r f ixed terminal condi t ions . Le t us f i rs t consider thecase in which r adiat ion is so large compared with convectionthat i t pract ical ly determines the temperature-space relat ionin the apparatus . Le t th e mass of gas f lowing throu gh peruni t t ime v, pecific heat = S, inlet gas temperature =T iJ exi t t empera ture = T2, and average sur face t empera -ture = Ts. L et CT+W T K T , he to ta l rad ia t ion at tem-pera ture T , be des ignated by RT. Consider a differentialengineering length of Passage of surfa ce area dA , with t he gas fal l ing in

of curves constructed by means of equat ion (j), nd read When the re i s a large drop in temperature of gas from

I GA S INSIDE S3 GAS BETWEENPARAL L EL PL4 G A S OUTSIDE

U 2 GA S IN INFINI L = DIAMETER, FT

would be equivalent to assuming 4 t o e q u a l u n i t ythroughout . JJ-e hare a l r ea d y f o un d t h a t b y a AS 4 . EXCEPT TUBEe O S C L E A R A N C E L = 3 8 . U E A R A N C EPARALLELOPIPEO OF G A 5 ,roper choice of L for a gas shape i ts 4 m a ybe made to osc i l l a t e about uni ty wi th a maxi-m um deviat ion of ab out 10 per cent . For ex - g loam ple , if we consider the radiation from a gassphere to a unit element of its surface, 4 willva ry f rom /a to 1 as kPL varies from zero toinfinity, if th e diameter of the sp here is used forL. If , however, we use diam eter as t h eeffective thickness, L, of the gas layer, 4 passesf rom 1 t h rough a minimum value of 0.955 a n dback to 1. Likewise, in the case of a gas be-tween parallel planes a distance D a p a r t , d varies

E2$

09

0f rom 2 t o 1 when L quals D; whereas, if we let,L equal 1.8 imes D , 6 varies from 1.111 hroughK. P. L-igure 4-Shape Factors and Equivalent Thick ness of Gas Layers-a minimum of 0.94 a n d b a c k t o 1. I n a s imilar manner, by d Q / d e = L7s.dT = p dA . (RT - R.)proper choice of L , t he shape fac tor for any shape m a y b emade t o s t a y c lose to 1. The assumption of unrt shape fac-tor for a l l three bands of a gas then leads t o small errorswhich part ly counterbalance, and n-hich are wel l wi thin theaccuracy of t he cons tant s of Table I. Figures 5 a n d 6, basedon this assumption, present th e total R due to al l three bands .(Th e cons tants of th e Planck radiat ion equat ion, used in cal-culating Figures 5 an d 6, are those given in Internat ionalCritical Tables.) ZR for carbon dioxide will be designatedb y C; ZR f o r w a t e r v a p o r b y W. The scales used are thosealready ment ioned as jus t i fied b y the form of the Planck

J A P . d AL 2 T ldTvs = RT - R,

F o r L7s 1I-e may subs t i tu te Q/Ae / (T ,- T2). obtaining(7)TI T ,_ - p . A .8 . d T


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892 I N D U S T R I A L A N D EhTGINEERIiVG CH EM IST RY Vol. 19, No . 83oooo

TOTAL RADIAT ION 2OOOODU E TO CARBON DIOXIDEP=PARTIAL PRESSURE O F

CARBON DIOXIDE, ATM. IO000L=CHARACTERISTIC DIMENSION 8ooo

OF GAS SHAPE (EFFECTIVETHlCKNESS OF GAS LAYER),FT 6000SEE SHAPE FACTOR d O T

4000THIS PLOT IS BASED ON ASSUMP-TlON OF UNI T SHAPE FACTOR. 3000a20005tL0;1000 v)3600 G

am300Loo

800>m

1008060

, - I 1 . 1 I I l T l I I l l l I I I I I I 1 4 0r0 0 0 8 0 0 8 8 8 8 8 8 82 0 8 !3 o L - R 3 8 % W $ % $ %0- 0 0 0 00TEMPERATURE DEGRESS FAHRENHEIT0@

Figure %Total Radiation from Carbon Dioxidewhich is the des ired relation. To integrate the denominatorwe must express R T as a function of T . Over a sh ort rangeof v ari atio n of T this may be done sat is factori ly by a powerfunct ion, bu t th e result i s unwieldy, and t he power funct ionis different for different ranges of tem per atur e. Th e in-tegration can be performed graphically by plot t ing thet e r m , 1/(Rt - R,) vs. T, and t ak ing the a rea under theresulting curve, between the limits TI n d T z . A simplermethod will be presented shortly.Equa t ion ( 7 ) gives Q /e when radiation controls the tem-perature-space relat ion through the apparatus . For thecase in which convection fixes that relation, a method ofderivation s imilar to tha t used to obtain equat ion ( 7 ) eads to

As with equation ( T ) , a graphical solution may be obtainedby plot t ing R T / ( T - T ,) vs. T , and t ak ing the a rea underthe curve between T 1 a n d Tz.The determinat ion of Q/e by use of equation ( 7 ) or (8)i s tedious , and t he quest ion arises as to w hether i t is possibleto use a me an tem pera ture of gas, expressed as a functionof T 1 ,Tz, n d T ,. T w o methods , with a semi-mathematicalbas is , both lead to resul ts which are poorer th an t he em piricalmethod of using for gas temperature in equat ion ( 6 ) , t h etemperature of the surface T, , plus the logari thmic meantem per atur e difference of gas an d surface. I n other words,

Table I1 presents a comparison of t he tru e value of Q/Ae,calculated graphically by equation ( 7 ) or (S), with the value

obtained by us ing gas temperature as defined by equat ion(9), and by us ing the ari thm etic mean tem perature of th egas. It will be not iced t ha t Q/ile ased on a gas temp eratureas calculated from equat ion (9) l ies, in every case, betweenthe two tru e values. Since neither radiation nor con-vection ever controls to th e exclusion of th e other, t he useof a mean gas tempe ratu re as defined by equation (9) isas fully justified as the use of equation (7 ) or (8).

Table 11-Comparison of Values of Q / A 9

TIF.3000

TzF.2000

T. .0

100015001800

@ / A 9 A SCALCD.BY En. (7)

OR ( 8 )56605842495550623617369823682445

Q / A e Q / A eU S I N G U S I N GTg AS ARITHMETIDEFINED MEANBY Eo. (9) Te5840 59505020 52903670 40902385 3085

1550 1695 186017851296 1369 1669600 { g 1469 1 O i O 1480025800 { 1168

2000 1000 0 K!Q Factor controlling temperature-space relation: R = radratlon, C =convection.

Before illustrating the use of the plots, let us considerbriefly their probable accuracy. Th e assump tions involvedin their derivat ion are (1) that the course of absorptionthrough a band may be expressed as a l inear function ofwave length, provided a compensat ing adjus tment of b a n dboundaries is made, (2) that the absorbing characteris t icsof th e gas do not change with tempe rature, (3 ) t h a t t h e s h a pefactor is 1. The fi rs t assumption undoubtedly leads toerrors of considerable magn itude.2 Th e second assump tionis necessary to ma ke use of absorption coefficients determined


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August , 1927 INDUSTRIAL ,4YD ENGIiVEERING C H E M I S T R Y 89 3

I---'-+-,/ ~ _ _ _ ~ _ _ ~ _c 4 4-/ I !8 g a c u ru Num rn00- 00Q 2 (0 01 12 N 3TEMPERATURE , DECREES FAHRENHEITFigure 6-Total Radiatio n from Wa t e r Vapor

a t room tempera tu re . Th e work of E va 1-011 Bahr16 indi-cates th at there is an increase in absorptio n with rise intempera tu re , a t l eas t up to 900" C. By neglecting thisincrease and using values obtained at room temperature,we obtain minimum values of &/-4e, pro vid ed, of course,that the absorption coefficient does not again decrease athigh temperatures. Th e third assumption has already beenconsidered, and leads to errors of m uch smaller mag nitudetha n the other two. Th e data on carbon dioxide are muchmore rel iable th an those on water vapor . I t is probableth at the maximum error in determining &/+4erom theplots is not over 30 per cent , and probably less . This isgreate r th an the error involved in using many of th e bestequations for heat t ransfer by convection, but i t at leastleads to results in the r ight neighborhood. Work is underway a t the M assachusetts Inst i tute of Technology to de-termine the total radiat ion f rom carbon dioxide and watervapor at high temperatures, to provide a more accuratebasis for Figures 5 an d 6.Hea t Transfe r Calcula t ions I l lus t ra t ing Use of Plots

Since the principles involved in heat transfer by radiationfrom gases are different from the more familiar ones used inconvection heat t ransfer , a few representatil-e examples willbe giren to indicate th e use of th e charts .A s a first examp le let us consider a gas flowing with a m assvelocity of 0.4 pound per second per square foot. carbondioxide content 20 per cent , no water rapor , specif ic heat0.3, through a 6 b y 6 inch square duct , the gas enter ing at2000" F. Suppose an average surface temperature of 800" F.Assuming al l the heat to be transferred by radiat ion f rom

the gas, how long would the flue hai-e to be to cool the gasto 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, an dCSW = 380. Th en , assu min g a black-body coefficient of0.9 for the surface of the duct ,Q/Ae = 0.9 (1450-380) = 963 B. t . u./sq. ft./hour.The total heat t ransferred per unit t ime isQ/.O =mass velocity X cross sectional area X specific heat Xrise in temperature = 0.4 X 3600 X 0.25 X 0.3 X (2000-1000) = 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 equivalentcoefficient h,, as used in convection he at tran sfer calculations,is 963,laverage At = 963/558 = 1.73, which is of the sameorder of m agn itud e as we would expect h, to be under suchconditions, indicating that roughly half the total heat trans-ferred mould be due to radiation. It is t o be rememberedth at the expression of he at transfe r by radiation in the formof a coefficient is a p urely artificial p roce dure, used simp lyfor comparison with h,,.Sup pose it is desired to find the effect of do ublin g the m assvelocity of th e gas in the above duc t, maintainin g the sameentrance gas temperature and average surface temperatureand f lue length. We have the equat ' ion

0.8 X 3600 X 0.2-5 X 0.3 X (2000- Tz)112.9 ( C , - Cm) =


7/7

894 I ,V D CSTR I A L A X D E S G I N E E R I X G C H E M I S T R Y . Vol. 19, K O . 8which reduces to C, =4284 - .142 Tz ,which must be solvedby t r ia l an d error . Assuming T z =1200, equation (9) givesa gas temp erat ure of 1528, for which C, = 1935. 4284 -2.142 X 1200 = 2094, as opposed to 1935; therefore, theassum ption of 1200 " F. was slightly low. A second trialindicates an exit gas tem per atu re of 1230" F. Thus we seeth at the re is an increased amo unt of h eat t ransferred, thoughat lower efficiency, when the velocity is increased.( Q / A 0 ) c - 0.8 - (2000- 1230)0.8 - 1,532(Q/A8) ,- 0. 4 - -(2000 - 1000)0.4This is analogous to the effect of velocity on heat transferby convection, in w hich th e coefficient increases as the0.5: 0.8 power of t he velocity, althoug h th e cause is entirelydifferent, being du e in the case of radiation to t he use ofa greater portion of th e flue are a for heat transmission a t thehigh temperature at which i t i s most effect ive.As a final example, let us consider a continuous billet-reheating furnace such as is used in rolling mills. A repre-sentat iv e furnace of th is typ e has an effective hea rth areaof 40 by 11.5 feet , and a de pth of gas above th e billets of3 .5 fee t a t the b urner end and 1.5 feet a t the cold end. Sup-pose the gas enters a t a tem per atu re of 3450" F. an d is cooledto 1300" F., and that the s teel , f lowing countercurrent tothe gas , is heated from 80" t o 2275" F. Let the furnaceburn 100 cubic feet of coke-oven gas per second, producinga flue gas of 7 .5 per cen t COZ, 20.0 per cen t HzO, a n d t h e r e s tCO, Oz, a n d N2. W h a t is the rate of hea t t ransmission atthe two ends of t he furnace? Turning to the shape fac torplot (Figure 4), we f ind tha t the gas shapes neares t tha tunde r consideration a re (3) the space between infinite parallelplanes , and (6) a rectangular parallelepiped 1 X 2 X 6.At the burner end where gas l ayer is t h ick , and the gas shapemore n early approximates (6) , we shal l use 1.4 X 3 .5 feet,or 4.9 feet as the effective L . At the cold end , where theflat gas mass approx imates case (3) of Figure 4, we use 1.6 X1.5 feet, or 2.4 feet as effective thickness, L. Consideringfi rs t th e burner end, P , L = 0.075 X 4. 9 =0.37, and P , L =0.20 X 4. 9 = 0.98. Using Figures 5 a n d 6 for gas and s teeltemp erature s of 3450" an d 2275" F., respectively, and p =0.8, we have, on subst i tut in g into equat ion (6 ) ,Q / A 0 =0.8( (17500-7000) +(25000-9600) - 6900-2300)} =17,050 B. t. u./sq. ft./hourIt wil l be not iced that for this case the correct ion due t osuperimposed radiat ion is appreciable , owing to the highvalues of P L . As a m att er of inte rest, the value of th ecoefficient of heat t ransfer , as used in convect ion, may beobtained for comparison:If we use the Weber equ ation" for convection coefficientof gases flowing inside conduits, th e result ob tained is h, =1.58. It i s appa rent , then , th a t at the hot end of t he furnaceabo ut 90 per cent of t he tot al heat t ransferred is by radiat ionfrom the hot gases . At th e cold end of the furnace, however.condi tions are reversed. At this end, P , L =0.18, P , L =0.48, To= 1300" F., T , = 80" F. On subs t i tu t ion in toequat ion (6) , we f ind Q/A8 = (0.8)(1500-0 +1800-0 -150 +0) =3150. Th e equivalent h, =2.46, while the v alueof h, from the Weber equat ion is 2.44. Consequent ly, a tthis end of t he furnace heat t ransferred due to convect ionand th a t due to rad ia tion a re about equa l . The over -a llH due to convec t ion and gas radiat ion combined (obtainedby adding the indiv idua l h, an d h,) var ies f rom about 16 a tthe hot end to 5 at the cold end of th e furnace. Actualexper imenta l da ta on a furnace such as that described hereindicate averag e over-all coefficients ranging from 5 t o 12 ,six ou t of seven of th em lyin g above 10. Such high valueswould not have been expected on the assumption that a l lth e heat was being t ransferred by convect ion.

h, = 17050/(3450 -2275) =14.5

A c k n o w l e d g m e n tSpecial acknowledgment is d u e R. T. Haslam, of theMassachusetts Institute of Technology, for critically readingthis paper and for valuable suggest ions .Ex =x =Q , ' 8 =A =P =L =T =c, =It', =C =?.I/ =K =P =v =+ =R =s =

-1 -2 -Q -s =e -H

----

N o m e n c l a t u r eenergy intensity at wave length Xwave length, pB. t . u. transmitted per hourarea of surface, squar e feetpartial pressure of radiating constituent, atmosphereseffective thickness of gas layer, fee ttemperature, O F., in all equations after ( 5 )heat radiated due to th e nth band of COZheat radiated due to th e nth b and of Hz Ototal heat radiated due to CO1, B . t. u./sq. ft./hourtotal heat radiated due to H20, B . t. u./sq. ft./hourcorrection due to superimposed radiation, same unitsblack-body coefficient of surfacemass velocity of gas, Ibs./sec./sq. f t . cross-section areaspecific hea t of gasshape factor of gas massC +W - K n equations after ( 5 )gas enteringgas leavinga t the temperature of t he gasa t the temperature of the surfacedue to carbon dioxide (or convection)due to water vapor

as C an d W

Subscripts

Bibl iographyt For derivation of the correction term an d alignment cha rt th erefor, see1-Schack, Miil. Warmeste l le Dlis se idor f , 56 (1924).2-Schack, Z. ech. Physik, 5, 266 (1924).3-Hottel, Presente d at the meeting of the Ins t i tute of Chemical Engi-

neers, Ma y 31 to June 3 , 1927.4-Von Bahr, Ann. Physik , 29 , 780 (1909).5-Hertz, Ve rhand l . deut . physik. Ges . , 13, 617 (1911).6-Rubens and Laden burg, I b i d . , 7, 170 (1905).i--Von Bahr, I b i d . , 16, 721 (1913).8-Hertz, Dissertation , Berlin, p. 26 (1911).9-Burmeister, Verhandl. deut . p h y s i k . Ges., 15, 610 (1913).

Appendix IV of Ref. 3.

10-Rubens an d Aschkinass, Ann. Physik, 64, 5 8 4 (1898).11-Rubens an d Hett ner , Verhandl. deul . physik. Ges., 18, 154 (1916).lZ-Sleator, A s t r o p h y s . J., 48, 124 (1918).13--Angstrom, Ann . Phrs ik , 39, 267 (1890); 6, 163 (1901).14--Gerlach, I b i d . , 50, 233 (1916).15-Coblentz, Bu r . Standards, Sci. Paper 357 (1919).16-Von Bahr, Ann . Physzk, 38, 206 (1912).17--Walker, Lewis, an d McAdam s, "Principl es of Chemical Engineering,"

p. 148, McGraw-Hill Book Co., Inc., 1923.

Information StorageWe are indebted to Dan Gut leben , an engineer of thePennsylvania Sugar Company, for the chart (page 895),selected from a drawing man y feet in length, which i l lus t ratesthe app l ication of the engineer 's language to the s torageof information . Th e original outline represented a beet

sugar fac tory f rom the bee t s torage sheds to the sugars torage house. Th e purpose is to f ix sys tematical ly thes tat is t ical information for use jn the draft ing room, in theerect ion work, and as a permanent record of the operators .The out l ine is in fact a f low sheet for the ent i re factory,carrying descriptions of app arat us and o ther equipment .The descript ions indicate the sal ient features with shopnumbers and manufacturers ' names for ident i f icat ion.As all the equip me nt is arranged in order, the informationis automatical ly indexed. Some plants s tore their s ta t is t icalinformation in volumes wri t ten in the language of S o ahWebster , but in this out l ine, which presents the facts in thelangua ge of th e engineer, m ay be found pra ctical suggestionswhich can be appl ied in many industr ies and to variousbranches of the chemical industry.

hottel 1927 heat transmission by radiation from non luminous gases

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