June 1948
INDUSTRIAL AND ENGINEERING CHEMISTRY
= volume flow rate of the material in rodlike motion, cu.
1;
feet per hour I = weight rate of Bow per tube of the material inrodlike motion, pounds per hour i f - = weight flow rate per tube of jacket fluid, pounds per hour I = dimensionless time constant relating to heat conduction in the core fluid, calculated by solving Equation A4-20 y = lrkL/UY, = wc,/m%, 0 = thermal diffusivity of material in rodlike motion, k l c p p j = L/C’R 9 = heating time, hours x = 3.1416 (2 = bulk density of granular material, pounds per cubic foot
z
1061
LITERATURE C I T E D
(1) Drev. T. B.. Trans. Am. Inst. Chem. Enors.. 26. 26-80 (19311 (2) International Critical Tables, 1’01. 5 , p. 165, Xew York, McGraw-
Hill Book Co., 1929. (3) hIc..ldams, W. H.. “Heat Transmission,” Fig. 11. 2nd ed.. New York, McGrav-Hill Book Co., 1942. (4) Ibid., p. 384. ( 5 ) Marshall, 11’. R., and Pigford. R. L., “;lpplication of Differen-
tial Equations to Chemical Engineering Problems,” University of Delaware, 1947. (6) Pipes, L. A , “.ipplied 3Iathematics for Engineers and Physicists,” Sew l o r k . 1IcGraw-Hill Book Co. 1946. (7) Shomate, C. H., Kaylor, B. F., and Boericke, F. S.,Bur. Mines, Rept. Invest. 3864 (1946). ( 8 ) Waddams, L. h.,Chemistry & Industry, 1944,206. (9) Waddams, L. A., J . SOC.Chem. Ind., 63, 337-40 (1944). RECEIVED J a n u a r y 12, 1948.
Unsteady-State Heat Transfer Air and loose Solids G.0.G . L O F ’A N D R. W. H A W L E Y ‘ U N I V E R S I T Y O F COLORADO. B O U L D E R . COLO,
O n l y meager d a t a are available f o r t h e design of equipm e n t e m p l o y i n g heat transfer f r o m a flowing f l u i d t o a bed of loose solids. I n t h i s investigation, design data in t h e f o r m o f unsteady-state heat transfer coefficients f r o m a i r t o a bed of g r a n i t i c gravel are presented. T h e size o f gravel employed ranged f r o m 4-mesh t o 1.5 inches; a i r r a t e s f r o m 12.05 t o 66.3 standard cubic feet per m i n u t e per square f o o t of cross-sectional area were used; a n d t h e entrance a i r temperatures m a i n t a i n e d were over t h e range 100” t o 250” F. T h e gravel was packed i n t o t h e bed in such a m a n n e r t h a t n o r m a l voids were obtained. It was f o u n d t h a t t h e results c o u l d be correlated b y t h e equation:
change. Knowledge of the relationship between these variables was desired in the form of the simplest) possible equat>ion,in order that the relation could be easily 11wd in the design of heat t,ransfer or heat st,orage beds. PREVIOUS INVESTIGATIONS
wherein h is t h e h e a t transfer coefficient a t a given p o i n t a n d t i m e , B.t.u. per h o u r per cubic f o o t of bed v o l u m e per degree Fahrenheit difference between a i r a n d solid a t t h e given p o i n t a n d t i m e , G i s t h e a i r flow rate, pounds per h o u r per square f o o t o f bed cross section, a n d d is t h e equivalent spherical diameter of t h e particles in feet. Change in t e m p e r a t u r e of t h e e n t e r i n g a i r had no appreciable effect o n t h e coefficient.
Studies uf the problem made by Schumann (16), Furnas ( 7 , 8), and Saunders and Ford (14) were directed tonyard the establishment of basic methods and design data for numerous applications. Schumann formulated and solved the complicated theoretical heat transfer rat,e equations for the simple case of an incompressible fluid passing uniformly t.hrough a bed of solid particles with perfect conductivity. I n this analysis it was assumed that (a) the part.icles were so small or had such a high thermal diffusivity that any given lump could be considered as being a t a uniform temperature at any given instant; (b) the rcsist.ance to transfer of heat by conduction in the fluid itself or in the solid it,self v a s negligible; (c) the rate of heat transfer from fluid t o solid at any point n-as proportional to the average difference in temperature betn-een fluid and solid at that point; (d) change in volume of the fluid and solid due to change in temperature n-as nv~ligible; and ( e ) the therms1 constants were independent of temperature.
0
Based on t.hese assumptions, Schuniann derived two equations relating the solid and fluid temperatures to the heat transfer coefficient,, the physical properties of the solid material and the bed, the time, and the positionin the bed. These equations are:
h = 0.79:@’) : (
F THE several types of industrial equipment which require a knowledge of rates of heat transfer between fluids and
broken solids, a few of the more important are iron blast furnaces, noke dry quenchers, limekilns, furnace regenerators, packed still columns, petroleum catalytic converters, and pebble bed heat ehchangers. Published basic data on unsteady-state heat transfer rates for use in the design of such equipment are ratht,r limited, honever, and it was for this reason that the present study Kas made. The particular inforniation desired was the value of the coefficient of heat tranqfer betneen heated air and beds of loose gravel, under conditions of lo^ air flow rates, comparatively lon temperatures (200’ F.), and xhere the primary purpose of the bed was for heat storage rather than heat ex1 3
Present address, 1719 Mariposa .Ive.. Boulder, Colo. Present address. Los Alamor Scientific Labnratory, Los i l a m o s , S . \I.
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
1062
wherein t, is the solid temperature, f,,O is the initial solid tt>iiipc.rature f is the fluid temperature, t,,o is the entering fluid teniperature: is a function of position in the bed, and Z is a function of time since start of operation. Further nomenclature is tahulatcd at, the rnd of the paDer. t - t, (1 t, - t,,o fo1 value. Srhuiiiann calculatcd value.: of and f,,o - t,,a t,,,i - t- 0 I J t I- and Z from 0 to 10 and Dresented the results i n crranhical form. The curves presented 6y Schumann were too 1 h i h in range for practical use in most heat transfer calculations, anti were therefore extended by Furnas ( 7 , 8 ) to values of Y anti Z of 500. Figure 1 is a typical set of Schuniann curves as extmtlcd by Furnas and plotted on semilogarithmic scale. By use of thwe curves and from a knowledge of the bed properties and the esprrimental time-trmperature relations in t.he solid or fluid, a nit>an hoar transfer coefficient in an existing brd can be calculate~i.
fT
_ _
2 -hek(#-')
Figure 1.
Computed Temperature History of Gas Schurnann curves
The paper by Furnas, published in 1832, describes t w o bets of experiments directed toward the establishnwnt of caquations permitting prediction of heat transfer coefficients. In thc first, si%riixs,heated air \vas paswd through a bed of iron halls, and attempts to measure the temperature of thr air and balls simultaneously by high velocity thermocouples were made. SI)many difficulties were encountered in securing accurate tcmpcrature iiieasurenients that this method was abandoned.
In the sccond series, hot combustion gases w e r ~p a ~ upd wards through cylinders filled with iron ores, coke, coal, anthracite, limestone, iron halls, crushed hrick, slag, and a typical blast, Furnace charge. Thcx temperature of the gascs leaving thc brti was recorded throughout an espc~rinient,and thc exit gas tcmpei,ature-time curve thus obtained was compared with Pchumann'h theoretical curves for different assumed hrat, transfer coefficients. The theoretical curve Jl-hich most nearly coincided i n shape lvith the experimental one mas then found, and a value, of t h e heat transfer coefficient was calculated. From these idculatc,tl v:iluw of the corfficirnt. Furnas postulated the following c,qiiatioii:
\\herein h is the volunictric heat transfer coefficitxnt, .1 i p a cmihtant dependent, on the bed material. G is the mass velociry of the fluid, 7' is an average of the entering air temperature and tlir initial bed temperature, .f is the fraction voids in ttic hctl, a n d d i:. the avc'rage particle diametcxr. This iridirrct riiethod usi~dliy E'uriias in t h e cic,tcrliiillatioli ~f the tixiinfer coefficient has onr outstanding advantage tlirwt methods. In the diwct nicthods i t is vtxry tlifficuli t o nieasurc accurately the continually chaiiyirig tenipc,i'atnre [liifcrenccx between the fluid and solid. This difficult>-iiiay 1 1 ~ t1o large accidental errors ivhich can remain undctectcd. \T.hell ttit, Schumann curves are employed, i t is necessary to iilt'asurc' only the exit air tr~iripc~rature over a period of time. Thvir applicabilit'y depends of course on the rr~liabilityof thc assumptions made for t.heii. dt~rivatioii. uv18r
Vol. 40, No. 6
Though the work of Furnas \vas estensive, many niateriab have not yct been studied, and extension and verification of the data for the equation have heen necidtd for its applicatioii i o somewhat different, conditions, particularly thoso involving r d u c w l gas velloc.itit,i. Sauncltw and Ford ( I . $ ) appI'lJ2Whd thr problciii iiy diriiriiarid found that for h d s of spherical particles, thc, ransicr should lxs govc~rntdby t,he dimensionltJss
L
-I n t h t w groups, i: is the fluid viLliir*d' ity, e is thib tinic', c ' and c are volumetric specific hcia and hcd matrrial, rcsspectively, d is thts particle diamet thermal conductivity of the particles, and L is the len hrd. T h r thcory was t w t e d by performing experiments in which hot air v a s passtd through beds of various depths of stcel, l(,ad,01 gla.;s s p h ~ r c s . It \vas found that uxidri the particular coritlitioiir of this ttssts, the heat transfer rate could tw correlated with v8c'~dc aloni~at a singlc, value of the ratio of bcd length to particlt. tlianieter. 13ei.ausc. t h c conductivity of thc. particles is not involvrtl. this siniplts relationship indicated that the effect of ttaniperaturt diffcrcnces xithin the particles is nrgligible in the Pxperinic~nts, The variations of gas temperature t,hrough a given twd at R I I ~ given niomcrit~,and the amount of hrat transferred can hr. calrulatrd by rrfcrring to curves presented in the papcr. So hmt, tranafcir coefficients were calculated, but it can be shon.n t)y ailother comparison that the effectivpncss of heat transfer is conridcrahly grcater than that' obtained by Furnas. Only sphixres ivert: used by Saundcrs and Ford, hoivever, and the results are thc:ri:irectly applicable to crushed or broken solids. studies of the rate of heat transfer between ga solids have b w n made under steady-state ran( Killianis (20) has correlated the data of a number of invwtigators \Tho stu(1ic.d the heat transfer between air and uniform sphcws. Under otherwise uniform conditions, the coefficient of hwt traiiufc>r,per unit area, is proport,ional to Gamson, Thodos, and Hougen (9) studied the drying of spht'rical and cylindrical pellets of various catalyst, carriers in the constant, rate period with through circulation of air. On thr assumption that, adiabatic conditions prevailcd, heat, transfer coefficients w r e computed under these steady-state conditions. I n the turbulent flow region, ahove a modified Reynolds number of 350. the coefficient, of heat transfer, per unit area, was reported t o bt. proportional to G./0.55dd".41. I n the streaniline region, below a modified Reynolds number of 40, the coefficient is stated to he independent, of G, and proportional t,o 1 ~ ' d . I n tests on a pebble heater, operating tlitions? with downflow of heated refractor cold gas, S o r t o n (12) measured over-all h Initial temperatures of prllrts and cxsit gas temperaturcs \YIW iii t,he vicinity of 2000" F. T w o values of the coefficiisnt t w i n w i l air a i d pebbles are rcyorted at s p ~ c i f i i dconditions, tiut n o I,I)I'rtxlittion of data ih presentrd.
g r o u p r8c' dc, i'dr' k , and
Thc~rcst~-nis t o be no hasic theorclt iral rt'itson Lvhy hvat trariaft:r cor~fficientsdetcxrmiried under steady-statc conditions should 1101 tic applicable to thc t#ypc' of unsteady-state operation involvvtl w h ~ na hot gas is passed t,hrough a ,stationary bed ( ~ t '(:old partirlw, provided that the coefficient is propc,rly used in ari intcyiyition \vit,h rrspcct to time and position. In an actual conipai,isoii, thvrc, appcws t o hc, considrrahle possibility that suc.11 thc. iiirlthods of' averaging t timprrature diff~~i~enc~c+ wiili ~ n i vin length of zone in \vhivh heat transfvr. i c t aliirig plac~xmay likcly cause a large error i n applying data ol)tairic,tl irl on(' typv of study to t.he other use. In as coniplicat(d R p i ~ o m w a. this, a drtc~rmination of pc,rfornianc.t, charactcti i9tic.s uni1c.i. condit i o r i h approaching as clowly as possiblc the actual o f USI' is generally dcsirable. I n thib case as the, r w v w or thcir c,quivalcnt must bc utilizcd if t,he cotJfficitirit i > i o lx' ciiiploycd i n the design of a heat csehangc bcd, it is f(1li i l i a 1 tht>ir use is of advaiitag(3 in determination of the cw~flicic~iit~. Thus, wrtairi inherent iriaccuraeiw in method can tit, lai,gt\ly coiiipcuaattd. .ilthough the techniqutis of determining hwi t ranit'er cwffirient s under unst eady-st at e operat ions invol ving contiiiual variatiou i n conditions with respect to both timimid position are niuch niore difficult t,han those ncxtd~idi r i sttxdy-state operat,ion, it is t'c.lt that thcl additional clifficul1y i. iu.1 i f i c d .
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1948
M E T H O D FOR COEFFICIENT D E T E R M I N A T I O N
Ii rhc, rontlltions of a particular h i a a t trarisfrr tr,st are Surh that Prhuniann'P axruniptions are applicable, all of tht=data ner for calculatinn of the tempcaturc, history of the fluid or solid art' kiioivn or niay be easily found, t w r p t thtj coefficirnt of heat transfer. Thr coefficicnt can thrrcxforrj he drtcrmined by the previousl>- outlined mrthod of Furnas. I n thc particular cas? wherr the fluid is a gas, the ratio of chtaiic-c from entering end of 110d to fluid velocity through t ) d , which is the term 1: '1' in th(3 . . anti may t)e iirplectrti. Time is therefor(: proportional t,o %. Thy t - f; i,talation t i c a r \ w ~ nL*-i. and thca time 0, may tie d(1trrniined i,,, - fa,,, i ~ \ ; p ~ ~ ~ i r i i e nfor t ~ iail!. l l ~ .givrn tr.*t c~iriditionand loi,;ition jn the t)cd and the, rvhulting curve niay l x ~U J I I I I ~ U ( ~with thc, Schuniann i w v w ~vhirh plot.tcd with Z as the abscissa. The value ot I. fol, tht, wrv(2 n-hich fits most calosrly the cspei,imr-.ntal i,csult? is thC.11 u 5 t ~ i it o c~:ilrulat~~ th(3 heat transfcxr c w f f i c ~ i c v i t ,h , 1)y uuirig tht. rt'latioii h = I;,'/, .r or h = I-r'G .cp. ~ o n i p a r i w t iof t h o th(~oi,c~tiral arid vspt~rimontalCUI'YC+ ill t,tic ahovc nivt hoti of ralculation involvt..G u,t' of the fact that c u r v c 1)reparcd o i i c~~iiiilngaritliniic paprr niay he plotted in any unit.* or tranqioscd any dietanc(A along the logarithmic asis without vhanging the shape of thix c u r w in aiiy way. The theoretical I'UI'VI'S niay therefort, be plotted in this mariner and tmheexpeririiental data plotted as tcmperaturc, ratio on a rectangular ordinate against time on a logarithmic abscissa. The experimental and theoretiral curves may then he supwiniposed and the correct thtwret,ical curve found by transposing the experiniwtal one along thr horizontal axis until the two ('urves which match niost i*lowlyare found. Furnas found that his data agreed \ well with the hheoof Srhuniann deapitcl the. fart that the conditions of iiis teats \vert$ not in strict accordanre with the assumptions usid hy the 1attc.r in the derivation of the rquations. He concludwl. thcreforo, that variations in thti trst conditions had little i x f l ' w t on tht? validity of thr Schuniann cwrves and that the system was a(.ting as it' tlir caondition;. ivrre h h u w hpcifird hy Schumann. r
APPARATUS
'fhcs c.quipuimt ustd i n the determination of thc heat transfer
iwLfficirnts ir: shown diagrammatically in Figure 2. Compressed iiir \vas rnrnjshrd t o two storage tanks through a prrssuri~control valvr. Thc storagci tanks served to dampen t h e pulw of the i-ompresww and supply air at a coiictant pressure. After being 1 in a standard orifice, the air \vas hrated to the dwired I tmperaturc' i n a pas-fired heatrr. Closr temperature control \vas obtaincd by t h c use of needle valve. in the fuel gas and primary air lincs to the heater, and by regulation of a nerdle valvr in a linr. which prrmittrd a portion of the coniprrssed air t o hy-pass thts tic.at,cr. Bwause thc orifice nirter was not suitablt. fur nieasuri'nitwt nf low air f l o w , a rotsnietcBr wis u s i 4 in a riunihc~rof 1'Uli..
VENT
AIR
-~
I1
41R
*A-ER
1 Q
--
f
8
,
L . J /
- -~_.
Figure 2.
T C DENOTES -PE'E9'JocOL.PI
Diagram of Apparatus
1063
After the air was metered and heated, i t n a s passed through a n insulatcd, horizontal, rectangular box, filled x i t h gravel of nearly uniform size. B vent in the line entering the bed was provided for use at the start of a run during the period in which ail" temperature and flow rate Jvere adjusted. All of the air lines from the heater were insulated with 2 inche: of magnesia pipe insulation, and fittings were covered with SIlpeJ 66 insulation cement.. The calihatcd orifice meter (f 1, located i n a l.5-inch standard pipe, was provided with t w o sharp-edged orifice plates for use in obtaining the greatest accuracy in the diffc.rent velocity ranges. One platv had an opening of 0.625-inch diameter and the other, 1.000-inch. F l a n p taps lid to a U-tube niariometer filled with water. Thcs rotamr'tt'r was of conventional typc, calibrated by thc factor!- from 0 to 12 standard cubic f't1c.t of air prr minute. TO 5 P b F T GAGE
cn
PINE
worn
THERMOPlLt
I NSUILA T ' O N
Figure 3.
Heat Transfer Bed
Th(1gravel bed container is shown in detail in Figure 3, and the appearance and arrangement of the equipment are seen in Figures 4 and 5 . It consisted of a n.ood box with over-all dimensions of ti feet by 12.9 inches by 11.25 inches. The bos was lined with abbestos paper and insulated esternallg on all sides by 4 inches of fiber glass mineral wool held in place by thin plywood. Three perforated, sheet steel baffles in the entrance section of thr box served to eliminate cross-sectional variations in air f l o ~ . The gravel was contained b e t m e n vertical, 0.25-inch mesh wire screms placcd 36 inc-heP apart. The inside cross-sectional artla of the bed, normal to air flon-, \vas 0.77 square foot. A piezomcter arrangement comisting of three taps with connecting pipe leading to an inclined manometer \vas installed at thr, inlet and outlet of t h r bed. ,iny hrat loss from the I n order to eliminatr the normally o( bed, six separate electric heating coils I wound on the exterior surface of t h r container tjrneath thr in tion. Variable resistances \ w r r provided for adjustment of the hmt, gcnerated in each coil. Thc temperatures on the inside and outside of the boxat the midpoint of each coil were measured at intervals with chromelconstantan foil thermocouples, and thc adjusted so that these two temperatures as possihle. To minimize the diffrrtJnce and th(x sides and bottom, and to prevent an!' possibility of air channt~lingover the top of the brd, a sheet of corrugated alumin u m foil \vas placed between the top of the grawl and the cover. miprrature of the air at the orifice \vas measui,cd with a . thcrnionieter: the temperatures at the rotanietc>r and rt' determined hy use of standard iron-constantan therinoT o avoid nonunifurinity in tied tenipcraturr, therniosed for measurement of ternperat,uws in the brfore each run. Thc voltage produced by n i r ~ a x m d with a portahle potentiometer. Tynipri,aturrs of thc air a t brd inlet and outlct during the fours? of a run wrre nirasurc~tlI)>- five-junction iron-constantan therinopiles connr,cted to a multiple point rwording potentiomeuic pyronivter. .inice hath n'as u ~ o das thr cold rt~frrrncejunction for. nll thcriiic,c,ouplcs.
1064
Figure 4.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Gravel Bed w i t h Cover Removed t o Show Interior
The gravel, as received, was separated into vxriou5 ~'i,a(.tiiJti,. in standard gravel sieves. The fractions employed in these t ( wnsistcd of particles 0.19 t,o 0.38 inch. 0.38 to 0.5 inc I , 0.75 to 1 inch, and 1 to 1.5 inches in diameter. Closer sizing could have been carried out, but it \vas deRired to have the data :tpplicable t o commercial gravels, v-hicli arch not generally rivailal)li. in narrower size ranges. Brfore a test, was made on a gravel of 1):lrticular size se\ its physical properties were det,ermined. Thr average sp iliameter of the particles, the t,rue and apparent densi :lit material, and the percentage of voids present in the bed \vert: till mcasiired in one comparatively simple determination. .I bucket, of known volume was filled with the gravel by counting each particle as it n'as placed. The bucket, of gravel was weighcd, and the void spaces then filled with water. The volume of voids vas calculated from the weight of water added, and the tliffercnce betwecn the void3 and the original knoivn total volumc equalcd the volunie of gravel in the bucket. From t h c i e figurrs, the density of tjhe gravel, the apparent density of the bed, the average equivalent spherical diameter of the particlc.~, and the per cent voids were calculated. When granulated or broken material is put into a container, the size and number of void spaces is dependent on tlie manncr in which the particles are placed. Hon-ever, if each particlo is dropped into the cont,ainer and allowed t o come to rest, before the nest particle is dropped, the fraction of void space in t h e resulting bed will be reproducible within very close limits. The free space obtained in the bed by this technique is called norm;ll voids, and each different particle size has a somen-hat different percentage of free space corresponding t o normal voids. In t,h(J tests described here, the condition of normal voids was a l w a p employed, and the results are therefore a.pplieable to beds p w pared in the usual manner by dumping in tlie material at H moderate rate. Before the start of a run, uniformity of bed temperatures w a k first determined by thermocouple readings. If variation was d, air a t room temperature Tvas passed through the bed until a uniform temperature was established. Heated air w:i> then tiirected from the heater and meter through the vent whili, thermal equilibrium in the piping system was being establishrd and the t'eniperature and f l o rate ~ were adjusted to the desircd values. After the proper conditions had been obt,ained, the heated air was passed into the bed. Readings of orifice differential, orifice temperature, orifice pressure, bed temperatures, insidc a,nd outside wall temperatures, pressure drop through bed, and barometer were made a t frequent intervals; temperatures of inlet and outlet air were aut,omatically recorded. In runs including use of low air rates, orifice readings were replaced by rotameter measurements. During all runs t,he current, in the heating ciiils on the hos was atijiistptl t o nraintnin, A S closelv 3,s
Vol. 40, No. 6
possik)le, the same temperature 011 tlir inside and outside of the container. K h e n the teniperature of the exit air had risen to a value within a few degrees of the inlet temperature, the heating run was discontinued and the hot air routed through the vent again until the heater could be by-passed completely. *lir a t room temperat'ure was then passed t,hrough the system until the piping had cooled to a constant temperature. The cooling r u n was finally started and carried out in the same manner as the heating run, but with use of cool air a t room temperature rather than hot air. The run was discontinued when t'he difference b e t w e n the teniperatures of air entering and leaving the bed was negligible. 11-it.hthe entering air temperature a t 200 F., sets of runs \yere made n-ith gravel of the four different particle sizes and with air rates of 12.05, 18.1, 30.1, 22.2, and 66.3 standard cubic feet per minute per square foot of t'otal cross-sectional area. Corresponding superficial velocities are roughly 0.2 to 1.0 foot per second, but because a normal barometer of about 630 mm. prevailrct, the actual superficial velocities a t 200" F. ranged from 0.3 to 1.6 fect per second.
Figure 5. Gravel Bed Complete Assembly
I n ~ I aI t t,t>iiip,tI(J dett~rnimrthe eft'rct o f inlet tcniperature t,he coefficient,, single runs )yere made a t temperatures of loo", l!iO", 200", and 2 3 " F. with a constant air rate and one particle size, and single run? n'ew made at, 150 O, 200 O, and 250' F. rvit,h the same air rate and u smaller sized gravel. Finally, the iirecision of the results W B P approsiniated by making four run3 with the same gravel under ident,ical condit,ions of temperature and air rate. #.in
CALCULATION OF COEFFICIENTS
~ ' o l l o ~ v i ithe ~ g measurement, of the time-air t,emperarurr relationship at, thc esit end of the bed, the heat transfer coefficient, was det,erniinrd by use of the Schumann curves. The exact, metshodcan best be described by presenting a samplc ealculat,ion which is shojvn for run 10.
Conditions. Air ra.tc = 35 cuhic feet per minute measured at 760 mni. and 70" F. This corresponds to 42.2 standard cubic feet per minute per square foot cross section or 204.5 pounds per hour per square foot total cross section. . t,quiv,zlrnt t o caritri~irigair temperature) t,,o = 1Y4.3' F. tp,iJ
- Lr = 1 ~ I ~ O < y of rwults. In the runs uiiilr~t~ varying writlition.. l i i ) \ v t , -he constant in tlic drrclopcd (,quation is B calculwtc~cliiit'iiti fiir tirenty runs; arr'rage deviation of thceo conqtiints Ironi I Ires i i i i x i i i.i 6.2%.
0.4
0.6
Figure 8.
1.0
2.0 a;d
4.0
ti.u 10.0
Correlation of D a t a
(Units in thousands)
'r
June 1948
INDUSTRIAL AND ENGINEERING CHEMISTRY
ininat ion of heat transfer coefficients, it is necessary that the timc required for an individual particle to become heated to practically uniform temperature be negligibly m a l l in comparison with the timc for heating thr entire bed. This is equivalent t o a requirement that nearly thc entire resistance to heat transfer be in the gas phase. That this condition esistcd in the, present studj- was confirmed by two calculations, one in which thr Gurney-Lurie method ( 2 1 ) for unsteady-state heating of a sph(,rr, \vas employcd, and the other. in n-hich t,he value of
*
UdCl --
k was compared with that stated by daundcrs and Ford to reprew i t no dependence of heat transicr rate on particle properties. Thti maximum value of this ratio encountered was slightly over 5 , which falls well below the range of 8 t o 12 stated to tie a safe 11-orking limit without, appreciable influence of temperature gradients within the particles. The apparatus was drsigned in auch a manner that the heat loss was eliminated by heating thr external surface of t,he box. Conscquently, during the hcat'ing period a large amount of heat was actually stored in the insulation. As it \vas impractical t,o install cooling coils to maintain the outside wall temperature at the level oi the inside wall during the cooling portion of the runs, a portion of the heat stored in the insulation was transferred back through the \valls during the cooling cycle and caused an erroneously high value for the heat transfer coefficient. Furnas )vas confronted Tvith the same effect but claimcd that thrre was no adequate explanation for the fact that the co(lfficicnt in the cooling runs\vas always higher than in the heating runs. The equipment used by Furnas n-as of greater heat capacit>-than the apparatus described in this paper, and no attempt. \vas made to balance inside and outside vall tcmperatures. Consequctntly it Tvould be espected that the diffrrence behveen his heating and cooling cocfficirnts nould be greater than those found in this investigation. Comparison of both scts of results s h o w that cooling coefficients of Furnas average 365: greater than his heating coefficients wlwreas the present data shoiv an approsiniate difference of 2 3 5 . Because the results of these cooling runs are knoirn to be in error, they have not been employctl in establishing the relatioilships for the coefficient. It is hlieved that the heating and cooling coefficients are identical for all practical purposes, provided that the particle size is not so large that appreciable temperature gradients prevail in an individual particle for a considerable time. I n other words, if the conditions satisfy the requirrments for application of Schumann's mcthod, the cooling and heating coefficients are equal. The same conclusion was reached by Saunders and Ford, who obtained temperature gradients in beds of spherical particlcs during cooling which m r e identical with those during heating. It may be observed that in the tquation developed by Furnas, trrms involving the temperature of entering air and the fraction of voids are includrd. Examination of the original data recorded by Furnas shoivs that the dat'a on coefficient variation with temperature and with fraction voids in the bed are meager and cover a limited range. The degree to which these quantities affect the coefficient is therefore somen-hat approximate. I n the present, work these two factors have been lumped into the constant because of the minor, but unmeasured, rffect, they have and bwause thrs range of variation in the temperature and fraction voids was riot prcat enough to show conclwive effects on the coefficient. The fraction voids ran tie intentionall>- varied over a r able range b y rapid dumping of large volumcs of particles in one case and by mechanical vibration of a bed of particles in another case. The difference can be made even larger by employing perfect spheres in t,hc vibrated bed and irregular shapes i n the dumped bed. It is recognized that large differences in heat transfer coeffiricnt vi11 result when such variation in voids is int,roduced. According to the Furnas equation, changing the voids from 50yoto 25Vc, b>- the above artificial packing means, will produce more than a fourfold variation in h. However, variation in fraction of voids is usually unimportant because the variation is nearly ahvays small and normal voids are grnerally employed in practical applications. The, constant, .4, in the equation of Furnas, wap explained as
1067
being characteristic of the bed material. Heat transfer data in tho literature rvould not indicate the need for such a parametei, and Saunders and Ford ( 2 1 ) found that the coefficient was indrpendent of the composition of the bed material. It is therefort! rcasonable to suspect that this constant. is not dependent on tht: iwniposition of the material, but rather on some physical property, such as the type of surface, the general shape of the particle, or the degree of size uniformity. The value of the cxponcnt of 0 in the present work agrees cxact1.v n-ith that obtaincld by Furnas but the exponent of d i.5 different, 0.7 rather than 0.9. There is a difference in the definition of d , honever. The Furnas value was obtained by computing the reciprocal of the mean of the reciprocals of the particle diameters as determined by screen analyses. The particle diameter in the present work was calculated tiy dividing a mea>ured net volume of particles by the. number present, and using the computed mean particlr volume to obtain the m ~ a nparticlt? diameter of a corresponding sphere. The solids in both studi:,S w r e of irregular shape and differencea in equivalent dianietcsis ~ o u l dbr expected by use of the trvo methods. It is felt that t.he method used here for obtaining d is simpler than screen analysis, and can he easily uied by others in determining the proper valuc of d for use in the equation. As pointed out. by Furnas, corihiderable variation in d can be obtained by different. methods of calculating averages. h second point of difference between the t,wo st,udiea is in range of gas velocities and temperatures employed. The air rates used in the prescnt study \yere 58 to 322 pounds per hour per squai~efoot, and those used by Furnas were 457 to 2000. Temperatures employed by Furnas tvere generally about 1800 ' F., whereas 200" F. was t,he usual temperature in the present, work. Because considerable heat transfer can take place by radiatioii from flue gas and from one particle t,o another a t temperature!, as high as 1800" F.,the coefficients determined by Furnas arc' no doubt higher than if only convection and conduction had beeii involvcd, a;? at 200" F. I t is possible t,hat, over these widely different. ranges of air ratcs and temperatures, coefficicnt,s may vary with particle diameters in somewhat different manners. In order t,hat a comparison !vit,h t,he coefficient,s of Furiias could be made, the values which he computed for several materials at a particle diameter of 1.0 em., an air rate of 0.1 st'andard liter per second per square em., a temperature of 500" C., and normal voids were recalculated in engineering units by use of his equation for extrapolation into the range of 1 emperatures and flow rate': employed in the present work. The above flovc. rate, corresponding to 197 cubic feet per minute per square foot,, is considerably above the maximum of 66.3 used, but as the variation of h with G is identical in both studies, it, is felt that, the ext,rapolationof Furnas' results t o the l o w r velocity is reasonably reliable. The validity of the temperature extrapolation it questionable inasmuch as the temperature relat,ion recornmend~l by Furnas is based more on theoretical considerations than 011 actual data. -In additional approximation in the ternperaturc factor is that T in the Furnas equation is merely an average of brd and gas temperatures, taken as the mean of the temperaturcl*. of rnt,ering air and the bed of particles at start of heating. An t h t b
Table
II.
Comparison
of
H e a t Transfer Coefficients
fCornputed for 0.394-inr,li particle diameter, 50 std. cu. ft. per min., s'q. ft. air rate, 900' F . enterini. temperature, and normal voids (8, 1 4 ) l Heat Transfrr Coefficient, B.t u. per hour, F., cubic foot Material Iron ore4 Limestone Coke Blast furnace charge Coal Fire brick Iron halls Lead, gla-s, steel spheres 114) Grax-el in pre-ent s t u d y
406 t o 1200 460 a47 %GO 400 584 668 1121 3Y 'j
1068
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 40, No. 6
wit,h the rough irregular shapes used in the present study. The close packing possible with spherical particles causes a decreased voids fraction and results in higher values of linear gas velocity, surface area per unit volume of bed, and heat transfer coefficient. The fraction voids employed by Saunders and Ford covered the range 37.5 t,o 38.0Yc, whereas the present work involved voids of 42.6 to 4 5 . 4 5 . Their considembly higher coefficients should therefore be expected. The reason for difference in the exponent on t,he G / d ratio is not so apparent. According to Saunders and Ford's data on spheres, hd. . IS independent of d or v ; hence h is directly proporthe ratio 2 3
Figure 9.
Comparison w i t h Saunders and Ford ( 1 4 )
latrr, holyever, use of the teniperature re1:ttion U 1 J p ' : i h tiring th(. t i y o sets of results into fairly satisfact,ory agreenient.
~ l l , , \ 11 j
10
ln Table 11, the heat, transfer coefficients for a variety of ma,rerials used by Furnas, for the spheres used by Saunders and Ford, and for the gravel used in the present study are compared. 'rhc values have been computed s t a particle diameter of 0.394 inch, an air rate of 50 st'andard cubic feet per minute per square foot, a mean temperature of 135" F., and normal voids. -411 of rhe results except those for gravel and spheres were calculated by the Furnas equation as described above; the gravel coefticieniwas found from the equation presented here. The coefficient for the spheres was obtained by use of the equation derived from the Saunders and Ford result's. The value for the gravellies rr-ithin the rangr of results for several other materials. COMPARISON W I T H RESULTS OF SAUNDERS AND FORD 111 the work of Saunders and Ford (,I.$: a m l l designc~dapparatus was used for a heat transfer study of perfect spheres of lead. glass, and stecl. The problem was approached from the tlirwry of tiimensions, but the complicated equations mere not solvrd for I he heat transfer coefficient'. The rrsults were presented rather a family of curves plotted with gas teniperat,urc fraction
/; 'y,o
-L o
- f m o as a function of the term vBc'/Lc(l
- j ) , in the prt:reni
terminology. Each curve on the chart represents a different value of the ratio of the lengt'h of t.he bed to the diamet'er of the particles ( L j d ) . It is claimed that, within certain limits, this set of curves should apply to any spherical material and a,ny size bed. I t is conclusively shown, in the present tests, however, chat these curves are not, valid for broken solids. In Figure 9, tor a value of L/d of 27.4, the 8aunders and Ford values differ considerably from those calculat,ed for t8hepresent work on 1.5inch gravel, but t,he results determined with t,he 1.5-inch gravel are reasonably consistent among themselves. A com'parison of results a t an L/d ratio of 43, where wall effects are less pronounced, shows a difference of about t,he same magnitude. I t is believed that the difference in the two sets of data is caused primarily by the shape of particle employed, perfect spheres in one case, and irregular broken pieces in the other.
A comparison between the heat t,ransfer coefficients calculated from the Saunders and Ford results and those obtained in the present study indicates further the considerable difference in performance of beds with spheres and beds wit,h irregular particles. By use of their measured temperature history curves, and the Schumann curves, Saunders and Ford computed a mean value ofothe group (hdlvl;" c.) to be 1.85 X calories per c c . per C. Recalculation in new terms and engineering unit,s G results in the relation, h = 0.152 - , (present nomenclature). In d this equation the coefficient for spheres is proportional to the first power of G j d rather than the 0.7 power as determined in the present work. Use of this relation yields a coefficient of 1121 for sphrres at the conditions employed for the results in Table 11. The sharper temperature rises and the higher coefficients obtained by Saunders and Ford are due largely to the fact that the smooth, uniform spheres used in their work permitted much more effectivc and complete gas-solid cont,act than \\-as possible
G
tional to or -, rather than t,o the ratio to the 0.7 power. I t is d d possible that again the major difference lies in the shapes of the particles used in the two investigations. Another factor is t,he difference in range of the t'wo experiments. Unfortunately, Saunders and Ford have not reported their range of air rates, except, by referring t o runs with superficial velocities of 2 feet per second and 4 feet per second and by stating that 1.5 feet per second vias about average for the runs. A graph of a friction correlation indicates runs at lolver velocities, but there is no information as to whet'her heat transfer data were also procured in those runs. In the absence of information, it is assumed that the minimum velocities employed by Saunders and Ford were about 0.5 to 1.0 foot' per second. Their values for the G/d rat,io would then range from about 7500 or 15,000 up to 240,000, as compared to a range of 530 to 7820 covered in the present study. Actual particle diameters differed also: 0.0625 inch to 0.25 inch for spheres and 0.314 inch to 1.312 inches for gravel. Superficial air velocities covered the range 30 or.60 standard feet per minute to 240 feet per minute in Saunders and Ford's work, and 18 to 66 feet per minute in the present work; values of ($2). the modifietl Reynolds number (5),varied from 50 in the present study and about' 20 in the Saunders and Ford rrork, up to about 500 in both investigations; and the temperature of air entering the cool bed was usually about 200" F. in each case. Because the zone of transition from streamline t o turbulent flow is a t values of the modified Reynolds number from 40 to 350, most of the data in both studies are in the transitional region. Slight differencesin condit,ions of testing could therefore have marked effects on the resulting coefficients, and considerable difference in performance of spherical particles and irregular particles could be expected. This situation, along with the fact' that the present work covers considerably larger part,icle sizes and lower air rates, makes direct comparison of the heat, transfer relations of questionable accuracy and value. Still another possible source of discrepancy in the relations is the difference in uniformity of particle size in the two studies; Saunders and Ford used perfectly uniform sized spheres, and the present workers used screened fractions having variation in particle diameters of 30 to 6 5 7 , in a given fraction. GENERAL CORRELATION OF COEFFICIENTS
For a general correlation of existing data, the results of Furnas, Saunders, and Ford, and the present authors, were recalculated to the same basis and plotted in Figure 10. The coefficient, h, in B.t.u. per hour, O F . , per cubic foot, is shown as a power function of G/d, where G is the superficial mass velocity, pounds per hour per square foot empty bed, and d is the equivalent spherical diameter, in feet. Definition of d varies somewhat between the investigators, however. Saunders and Ford have used the actual diameter of the uniform spheres, Furnas has used a weighted mean based on screen analysis, and the present authors have used a value based on the weighted mean volume of the particles. The equation of the Saunders and Ford relation is: h = 0.152 (Gld) The general equation for the Furnas relations is: h. = (a constant)
(;)""
X
1
where the constant is dependent 011 the nature of The material, the fraction voids, and the average temperature; and the equation of the present correlation Jyith a granitic type of gravel is: h = 0.79(G,'d)','
~
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
The Furrias r e k i o n s were computed, by nieans of his aquatioil, to an entering air temperature of 200" F.; the fraction voids arid applicable constant for each material were obtained from his data summary and incorporated in the equation constant. The resulting constants for several materials are: Limestone 0,452 Corsica iron ore 0,627 Coke 0.344 Iron balls 0.661 Because the relations of Saunders and Ford and the present authors involve only the variable (G;'d),single lines are obtained. The Furnas equation involves d to a power other bhan that on G; hence, different lines are obtained for each diameter. It is seen in Figure 10 that a large total range in values of G / d and the coefficient is covered, even though t h e ranges of individual groups of materials in the Furnas study are small. The solid portion of each line represents the actual range of conditions employed, whereas the dotted portion is extrapolated. The general significance of Figure 10 is that: (a)the Saunders and Ford coefficients for spheres are much above all other values; ( b ) the results of the present investigation extend to much lower values of G / d than any reported hitherto; and (c) t.he coefficients obtained in the present investigat.ion have values intermediate between the limits of those obtained by Furnas, when his results are reduced to the same temperature basis by using h proportional to To.3. Use of the temperature relation recommended by Furnas for computing a coefficient at temperatures widely different from those at which the coefficient was measured appears to be a reasonably good approximation, and in the absence of furt,her data it is recommended that the heat transfer coefficient be considered proportional to 2 3 . 3 .
1069
tion. ('al~~iilation I J ~ 'thv actual rittr oi heat tranaier or the biz(, (Jf hra t transfer unit rcquired for a part,icular duty necessitate,? use IJT the t heoretiid Scluiniann cquat,ions or curves in conjunct,ioii nit h the computed or nieaiured heal, transfer coefficient. The follo~vingillustrative problem indiwtes the method of rising these coefficients:
A bed of granitic t,ype gravel coiisisting of 1.C-inch particles has 45% voids and is initiallv at a uniform temperature of 50" F.
If 60.0 pounds of air per hour per square foot. of cross sectional area a t 200 F. are flowing into one end of the bed for 6 hours, how long must the bed be if the exit air temperature is not to exceed 90" F.? A d d i t F l data: Heat, capacity of gravel, 0.25 H.t.u. pvr pound, F. = 41.3 I3.t.u. per cubic foot,, F.; actual density of gravel, 165 pounds per cubic foot; heat capacity of air, 0.237 B.t.u. per pound, ' F. = 0.0191 B.t'.u. per standard cubic foot, "
F.
Solution: Using t,he equation to determine t h c heat transfer coefficient :
t - tso E'ro~nthe curve for >-against Z for various valut>>of E' i, 0 - t, 0 (Figure l ) , the \ d u e of Y corresponding to a valuts of 7 ot 21.0 t - t,o and a value oi e - of 0.267 iz found: E' = 24.5.
t,
USE O F T H E COEFFICIENT
-Lo
Lon from the equation of I*, thta length of bed niap 1~ valcu-
The coefficient determined in t,his study is applicable t u the entire heat transfer bed but it cannot be used directly in design because the system is in unsteady state and the temperature difference is continually varj-ing with respect to time arid posi-
latPll:
By use of siniilar methods, the temperaturc proliile in the bcxd at any given time can be computed, or ilir t,enipcsrature history of a given point in tht. txvl n i : a \ I I I ,
i
-I
0.2
d
ployPtl. 0.1
ACKNOWLEDGMENT
0.06 0.04
0.4
0.6
1.0
Figure 10.
2.0
10.0 '20.0 G/d
1.0 6.0
50.0
100.0
General Comparison of Coefficients (Units i n thousands)
300.0
.\ppreriation is exprcssed for the full support, given thir study by the Anicrican Rlndow Glass Company of Pittsburgh, Pa., ttnd to E. R. Irish and J. W. Corn of the Engineering Experiment Station, University o f Colorado, for valuable assistance i n the construction of t'he apparatus.
,
1070
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 40, No. 6
NOMENCLATURE
'1 = constant in I ~ ' u r ~ i('quat ar iori ,- = heat, capacity of unit volume of nixterial of t l i t s pauticlrh, l3.t.u. p r cubic foot. F. = heat capacity of unit volunic of ga- at C C I ~ S ~ N I Ipriwilre. I I.' 13.t.u. ncr cuhic. foot, F.
J
=
fraction void9 in the b d
I , = Ir,ngth of bcd, feet ,! = a n y poaitivr nurnht~r T = temperature' in Furnah tquaticin, avc'rage ( i t cnteriiig tiuid tempcraturt' and initial hrd tenqx~raturr,A' 13. F. t C . ( , = ronstant rsntcring trmpcraturc of thc fluid, I, = the fluid ttlmperature at any point and timt., O F'. I,,,,, = initial conatant solid tempcratuw, ' F. = the solid teniperaturc at any point anti t,inir. F'. 4 = time after start of heating, hours = average volumetric fluid rate through tied, crlliii, t!,i$t pt'r hour, squaw foot total cross scction L = riistancc from cntr,ring c.nd of hPd. f w t V = h x ' c ' c = hx ~ , c ' G f,
1,
i)
o
.
= densit,. of fluid, pounds per cubic foot = viscosity of fluid. pounds per hour. foot
Heat Transfer into Cylindrical Columns of Bone Char v. R .
DElTZ
AND
H. E. R O B I N S O N . N A T I O N A L B U R E A U O F S T A N D A R D S ,
M a t h e m a t i c a l expressions have been derived f r o m t h e o r y f o r t h e temperature of g r a n u l a r materials such as bone c h a r heated in c y l i n d r i c a l r e t o r t s o f k n o w n w a l l temperatures. T h e c h a r is considered t o move t h r o u g h t h e cylinder w i t h constant velocity a n d w i t h o u t mixing. T h e t e m p e r a t u r e of t h e char a t t h e axis of t h e cylinder and t h e temperature averaged over t h e cross section o f t h e c y l i n d e r were f u n c t i o n s of t h e t h r o u g h p u t , i n i t i a l a n d w a l l temperatures, a n d l e n g t h of t h e retort. A simpleapproxim a t i o n o f a n exact s o l u t i o n of t h e problem has been f o u n d w h i c h i s v a l i d f o r t h e t e m p e r a t u r e rangeof interest in bone
B
ONE char is a graiiular adsorbent urtid chiefly i n large scale operations for the purificat,ion of sugar liquors. Efficient revivification is the key to the economic SUC('CSP of the bone char process of purification. In the revivification kiln with stationary retorts the char descends by gravit,y through the vertical retorts about 8 fert long which arr surrounded by t,he hrating medium. The developmcnt of the hone char kiln with stationary retorts reached a ronvcmtional design in the 1880's and 1890's. Little modification has bwn made sinw Thii time, though sonic changes have been proposed for the structiirc. of thP retorts uf these kiln. ( 4 ) . The proposed changes related to thP nuitcrialb 01' (*onbtrurtiiili and the geometrical shape as shown in Figure 1. Thv modt. of the transmission of heat into hone char has an important in-
WASHINGTON, D. C
c h a r revivification. T h e r e s u l t s a r e presented in t w o useful dimensionless curves of general applicability. For given i n i t i a l , axial, a n d w a l l temperatures, t h r o u g h p u t is independent of t h e diameter of t h e retort, a n d t h e average t e m p e r a t u r e o f t h e c h a r i s independent of t h e diameter of t h e retort. Experimental results observed w i t h b o n e c h a r k i l n s having 240 3 - i n c h c h r o m e steel r e t o r t s are in good agreement w i t h thecalculations. Residual m o i s t u r e in t h e entering char is shown t o decrease t h e capacity of t h e k i l n significantly. V a l u e s f o r t h e t h e r m a l diffusivity of some bone c h a r samples are reported.
Huence on the design of the kiln for heating hone rhar t,o revivifiration temperatures of 750" to 950" F. This paper is coilrerned only with the problem of the transfer of heat into the slowly moving column of char under 'certain idealized conditions. -1 theoret,ical basis is presented upon which to calculate thr transfer of heat into cylindrical retorts of circular cro. DERIVATION OF HEAT TRANSMISSION FORMULAS
It is proposed to calculate the axial and the average tcmpcra.turPs of bone rhar as it slowly drscends by gravity in cylindrical caolumns through thin-walled tuhes. The walls of the tubes art' assumed to br at constant temperature by contact with rapidly circulating hot gases or some other heating medium. The kmperature of the wall, T a ,is assumed to be equal to the temperature
t,
*
