Heat Transfer to Granular Materials Settled Beds Moving Downward through Vertical Tubes M. S . BRINN. S. J. F R I E D M A N , F . A . G L U C K E R T , AND E. 1 . DU
R.
L.PIGFORD'
P O N T DE NEMOURS A N D
Practically n o design information is available i n t h e published literature relating t o equipment for transferring heat t o or f r o m granular materials. One method of performing this operation consists of passing t h e granular material, as a settled bed, downward through vertical jacketed tubes. Effective t h e r m a l conductivities, as calculated f r o m heat transfer data for beds of granular materials moving downward through vertical steam-jacketed tubes, assuming rodlike flow, agreed closely w i t h t h e conductivities measured i n separate experiments w i t h stationary beds. T h i s close agreement indicated t h a t t h e beds moved substantially as rods. Rodlike flow was visually observed for O t t a w a sand by t h e use of a split-tube technique. Theoretical equations have been developed and design charts prepared for t h e general case of countercurrent heat transfer t o a material i n rodlike flow i n a tube. Equations have also been developed for t h e case of parallel flow.
C O M P A N Y , W I L M I N G T O N , DEL.
niight be obtained when this method is used tu hcat or cooi granular solids. This investigation examined the factors which influence the heat t,ransfcr rates in this type of solids heatexchange equipment, adapted existing heat transfer theory and equations for rodlike flow of fluid t o this casc, estended tht, theoretical equations to cover case8 which previously had not tirteu solved analytically, verified the simpler equations esperinientally, and developed design charts for usc in predicting heat transfer rates t o solids flowing in a settled condition inside vertical p i p s A N A L Y S I S OF PROBLEM
If it is assumed that. the solids flow downward in the pipe ar a solid rod of material, analytical espressions can be developed which represent the heat transfer that occurs. These expreasions vary with the conditions under which tht. heat is being transferred. Preliminary considerations indicated the casrs listed in Table I might be encountered. Case I would occur when the solids are heated by condensing vapors or cooled by boiling liquid? outside t.he tube. It is also generally applicable when only a small temperature drop occurs OL hearing iir cooli11y gra~iular 01' pulve1,ized in the heat t,rarisfer fluid. The majority of the applications will solida is receiving increased anent ion in the clieniirxl indusfall in subclass 1 in this group, as condensing and boiling coeffitry. The "fluidized solids" and hypersorptiori prrrcessc's are cients are generally high. I t is conceivable that large quantities particularly co~iceriirdwith t,his problem. T h r triBrid toward of air might be used for cooling or heat,ing, when subclass 2 would higher operating temperatures and higher fuel prices n i a k c ~ ~ he encountered. waste heat recovery from solids and efficient hc*atirtg of solids Case 11, which covers countercurrent f l o ~ ,is normally enincreasingly important. Savings can oftc,n hr rc,alizcd I coolcountered when solids are coolcd and is also met when solids arr ing chemicals 1)rfore packaging. heated with liquids or hot, flue gasefi. Subclnss A of I1 usually I n many rasps of cwoling or heating graniilar -.oli(i-. i t iz ncccsoccurs when primary consideration is being given to the heating sary or advantagrous to k e t y thct solids phyGally 3cyarated or cooling of the solid, while subclass B will be encountered when from the cooling or heating medium. This is particularly true the solids are being used to heat or cool a limited qunntit,y of fluid if the solids react wit,h or are contaminated by flue gasc~sor damp air, as in the lir,ating of readily osidiaable materials brfort: reacrion and t h p cooling of hygroscopic solids liefore packaging. When ext,reniely fine solids are handled, the solids milst be kept Table I. P r e l i m i n a r y Considerations separated from the cooling or heating fluid if count ercurrent I. Heat transfer fluid at constant temperature operation is to be obtained. Espensive recovery syett~msare 1. Negligible resistance to heat flow offered by heat transoften required when the solid3 are directly contbcted ivith the fer fluid licating 0 1 ' cooling fluid. If crystal shape and particle size are 2. Appreciable resistance t o heat flow offered by heat transfer fluid vritical, t,he solid cannot be subject to the grinding action and the attrition t h a t might normally occur if the solid is suspt.nded in [I. Heat transfer fluid flowing countercurrently to flow of solids A . Temperature increase or decrease of the solids is greater the heating or cooling medium and sul~sequentlyseparat,t~Ib. a than that of heat transfer fluid cyclonc: or other type of collector. 1. Xegligible resistance to heat flow offered by heat transWhere it is advantageous to keep I,lie solid physically sepai.atcd fer fluid f r o m the heating or cooling medium, one method of accomplishing 2. Appreciable resistance to heat flow offered by hear transfer fluid the, desirrti heat exchange is t o a l h v the xolid t,o flow in a settled 13. Temperature increase or decrease of solids i i less than thal, iwndition through a single vertical pipe or a bundle of vert,ical of heat transfer fluid pipes which are jacketed, either individually (douhle-pipe esI . Xegligible resistance to heat flow offered hy heat trans(.hanger) 01' collectively (shell-and-tube eschanger) wii h the heatfer fluid 2. Appreciable resistance to heat flow offered by heat ing or cooling fluid. Equipment, of this type is readily available, transfer fluid wlatively inexpensive, and simple and cheap to operat?. 111. Heat transfer fluid flowing parallel to flow of solids I-nfortunately there arc> no data available in the literature 1. Negligihle resistance to heat flow offered by heat tranawhich can t i t , used to predict t h e h w t transfer c ' o c ~ f f i c ~ i c ~that nt~ fer fluid 2. Appreciable resistance t o heat flow offered by heat transfer fluid
THVY
)>.
1050
1051
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1948
. When tvater or sotlie other fluid is used as thr hrat transfer fluid, the heat transfer will generally fall into siildass 1 under TI-=\ and 11-B. IVlieti gitsrs are the heat transfer fluid, subclass 2 under 1I-.l and 11-B \vi11 apply. ('she 111, which covers parallel flow, is seldom encountered, arid it is listcd primarily t o include all possibilities. Only special oprrating rircumstancxes or conditions justify the use of this arrarigrnierir , From an exprrinii>ntal standpoint Case 1-1 !vas considered the c.asic>st to evaluatc. By using condensing steam outside a vertical tube through which a settled granular solid was flowing, teniperat urw could be measured and controlled accurately and r1ii.r~~as no necessity for evaluating the jacket heat transfer ciwfficsicnt precisely. Furthermore, it \vas desirable to heat the *()lid from the ambient tctnperature IT-hich it could attain uniftrrnily by prolonged standing rather than at,tempt to cool a solid \vhich had been heated to greater than ambient temperature. inasniuch as i t is difficult, to heat, a solid uniformly.
LARED GLASS
SAND DISCHARGE DETAIL ABOVE)
(:SEE THEORY
OF
CASE 1-1, U N I F O R M T U B E - W A L L T E M P E R A T U R E
Tht. theoretical equation for heat transfer to a fluid in rodlike How in a tuhfl n i t h a uniform wall tpniperature has long been knov-n ' 1 ) :
AND VENT STEAM OUT Figure 1.
Apparatus for Heating G r a n u l a r Solid
abscissa of Figure 10 and from the cIefinitii?n of h,,,n tiy tho following relationship: n-htsrr !(!/I
and
;/
= 0.B!)2r.-5.7s!' A 0 . 1 3 1 ~ " ~ = akl, w n . This c n r r ( ~ ~ p o it iodEquation ~ .t23h.
E X P E R I M E N T A L E Q U I P M E N T AND PROCEDURE
For V X ~ U Wof ( t 2 - t i ) i Z , - f i ~ 1 Rrc'ater th:lIi ahout 0.55, olll>the first tcirm of thr infinite ,series i3 significant, so that (t?
- tl\.'(T,-
f l )
- O.B02e-5.:6@
(3)
= 0.692~-'.'*~
(3a)
= 1
Heat-Transfer Experiments with Moving Beds of Granular Solid. Tiit, npp:~ratusl i d for t l i i w , t~spi,i'iinc~nt> is illust~~:itt~il iii Figiij,tss 1 : i r i i I 2 .
or
(T6-
- til
f2)/(Th
I'lottiiiy Fkpation 3a on .riiiiltrgtrrithiilic. papc.r, as haz been
In Figurrh 10, ~ v i l lrt-sult in a straight line for ralues of : T 8 - t:) 'iT,* - i,'l lesl: than ahout 0.45. For d u e s of (t? - tl) ( T ~- i,! less than 0.55, it is desirahle T o plot the theoretical equation on logarithmic paper \\-it11 i t , - t l ) / ( T , - t , ) as thr ordinate and u ~ , . ' k Las t,he abscissa. Thr curve falls off to the right, approsimat,ing a straight, line,. Thew are several interesting points to lie noted about heat transfpr to fluids in rodlike flo~i-:
t h i r
I . The transfer of heat t o a fluid in rodlike flow in a tube is niatht-matically identical to unsteady-state heating of a long cylinder with negligible surface resistance. The Graetz number, w r p / k L , used i n the rodlike-flow equation is equivalent t o
Fourier's modulus,
used in the unsteady-state heating of FcPR'
a caj.linder. The time of passage through the tube is H =
r;
whei,t. Tit = aR2L, cubic feet
and ii =
w/p,
cubic feet, per hour
'ep
(4)
(5) (61
1. For a given weight rate of flow through a tube, the teniprrature change is independent of the diameter. Any value for the heat transfer coefficient may be obtained merely by changing the tube diameter. 3. If there are other resistances in series with that of the rodlike fluid, it is not possible to obtain the over-all resistance by adding the individual resistances. The solution of this case is given by Fquations -4-22, -4-23, and '4-25. 4. T h r heat transfer corfficient for materials in rodlike f l o ~ is useful only for obtaining a comparison of resistances. I - a l u r ~ nf b?,. r a n be calculated frniii i l l ( . value> ( ~ the f ortlinatiz : r i i i i tiit.
Figure 2.
Apparatus for Heating Granular Solid
1052
INDUSTRIAL AND ENGINEERING CHEMISTRY
Three vertical steam-jacketed aluminum tubes were employed: a 20-foot, 0.870-inch inside diameter tube; a 10-foot, 0.870-inch inside diameter tube; and a 10-foot, 0.622-inch inside diameter tube. Outside diameters were 1.00, 1.00, and 0.840 inch, respectively. Granular solid a t approximately 20" C. was fed in a t the tops of the tubes from hoppers. The flow rate of the solid was regulated by inserting corks, which held glass tubes from 7 to 15 mm. inside diameter in the bottoms of the aluminum tubes.
.%L
I
1
I
, ,
I
,
32 ' 60 '100 200 60d800 2000
40 00 400 1000 SIZE OF APERTURE, MICRONS Figure 3. Particle Size Data
Because this bottom discharge device gave the largest resistance to flow, the granular solid flowed in a consolidated or settled condition through the tubes. Because of the inherent characteristics of flowing granular solids, variations of level in the hopper did not materially affect the rate of discharge. To be sure that thermal equilibrium was obtained, a t least twice the holdup of the tubes was allowed to flow through the tubes before data were taken. The particle flow rate was determined several times during each run by weighing the discharge from the tubes for timed intervals. The inlet temperature of the solid was measured with a 0.1" C. thermometer placed at a n expansion in the tube about 8 inches above the steam jacket. An attempt was made to determine the exit temperature of the solid with a thermocouple, but because of t,hermal striations in the
'
Vol. 40, No. 6
material leaving the tube, it was necessary to measure this temperature calorimetrically. Thermos bottles of 1-quart capacity each containing 700 cc. of water at a known temperature, were used for calorimeters. The n-ater equivalent of each Thermos bottle was determined by cooling 700 cc. of water with a weighed amount of ice. I n the determination of both the solid temperature and the n-at,er equivalent of the bott'les, the original water temperature was adjusted so that room temperature was approximately half-way between the initial and the final temperatures of the water. Three determinations of t,he exit, solid temperature were made for each run. These gave an average deviation of less t,han 1%. Steam temperature readings taken a t the inlet and the exit ends of the jacket agreed within 0.5' C. ThP inlet steam temperature, measured 11-ith a 0.1 C. thermometer. was believed to be the more accurate of the tlvo readings and was used as the over-all steam temperature. Two sands n-ere studied in this invest,igation: a silica sand, designated as Ottan-a sand; and a n ilmenite ore sand. Particle size data for both sands iwre determined in a standard Ro-Tap analyzer. The results are shown in Figure 3. Because the specific heat, bulk density, and thermal conductivity of the sands were required for the heat transfer calculations, these quantitiec! were determined experimentally. It was part,icularly necessary to know the specific heat accurately, because this quant,ity was used to calculate the exit temperature of the sand. Changes in specific heat over the range studied (20 to 150' C.) were found sufficiently great t o make possible significant errors if this variation was neglected. The specific heats were measured by allowing the granular solid to remain in the steam-jacketed tubes for several hours, until its temperature reached that of the steam, and then collecting the solid in water-filled Thermos bottles. The initial and the final water temperatures were measured. From these values, the weight of the sand collected, its initial temperature, and the previously determined water equivalent,s of the Thermos bottles, the mean specific heats were calculated. Check determinations of t,he specific heats agreed t o better than 1%.
RUBBER STOPPER
- JACKET GLASS STEAM-COPPER
TUBE
- H O T JUNCTION OF TH ERMO COUPLE
RUBBER STOPPER WITH CAPILLARY TUBE PLACED ATCENTER TO GUIDE THERMOCOUPLE WIRE CONDENSATE AND VENT STEAM OUT
\
Figure 4. Apparatus for Determining Effective T h e r m a l Conductivity of Stationary Bed of G r a n u l a r Solid
Figure 5. Velocity Distribution O t t a w a Sand i n Vertical Tube
of
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1948
1053
less than that of a moving bed of the sanie material. This difference in bulk densities (168 and 153 pounds per cubic
0.24
-
p 0.22 a
t0.20
3J
*0.18
uno.I6
TEMPERATURE. Figure 6.
OC.
Specific H e a t Data
Values of c p are average f r o m
203 C. to temperature indicated
Bulk densities of stationary beds were determined in graduates and in the vertical steam-jacketed tubes by Tveighing a measured volume of the sand; bulk densities were also det,ermined for the granular materials flowing t,hrough tubes. This was done by eollecting and weighing the sand remaining in the t,ube when the bed level receded from the hopper and reached t,he top of the tube. Thermal conductivit,ies of the sands when stationary were determined by a method similar t o that described by Waddams (9). Fhich involves measuring the unsteady-state heat transfer t o a cylinder of sand. The apparatus is shown in Figure 4. The iron-constantan thermocouple was st,retched tautly b e b e e n the capillaries. The junction was located half-viay along the ~ from t'ube, which was long enough for the heat to f l o radially the surface to the granular mat,erial at this point (9). .1 t,hermometer Kas inserted in 6he jacket to measure t,he steam temperature. The procedure comisted of pouring the granular material into the tube and then admitting atmospheric steam into the jacket which was cont,inuously vented. Time-teniperat,ure readings were obtained every 30 seconds. Because of the low heat transfer resistance of the copper tube and the high condensing coefficient, the steam temperature \vas used as the surface t>emperature of the cylinder of granular material. Visual Experiment to Demonstrate Rodlike Flow. In the experiment carried out to demonstrate visually the existence of rodliicc, flow, a split copper tube 1 inch in diameter mas employed. When the split halves were joined, diameters measured a t 90" differed by only 1/64 inch. This tube was mounted vertically and filled partially with white Ottan-a sand. A thin layer of colored sand \vas added and the tube was then filled nit,h white sand and the sand column was allowed t o flow don-nivard until the thin colored layer had moved approsiniately 3 feet. A thin piece of sheet metal was slid between the two halves of the tube in the direction of the sand flow (to accentuate differences in velocity if the sheet disturbed the sand) and one half of the tube was removed. K h e n the halves were separated, it \vas foundby examining the cross section of the colored sand layer (see Figure 5)-that the local velocity ;vas uniform, except very close t o the tube wall, where the velocity was 15 to 2 0 5 lower.
foot for stationary and moving beds, respect,ively) is larger than the probable experimental error in their detcrmination, and indicates a closer packing of the small particles while in motion. The thermal conductivities of the two sands in the stationary state were ohtailled from the data on unsteadystate heat transfer to cylindiical beds of the materials. The time-temperature history of the center line of the cylinder was plotted as shown in Figure 7, wlierc the logaritshin of the unaccomplished temperature change, (1'* - f m ) j ( T 8 t o ) , is plott,ed against time. By comparing the slopes of the straight-line portions of these curves n-ith the slope of the theoretical center-line temperature curve for unsteady-state heat transfer to cJ-linders with no surface resistance ( S ) , the thermal conduetivitics of the two materials were calculated,
EXAMPLE.The negative slope of one of the curves for Ottawa sand was found (from Figure 7 ) to be 0.128 per minute, or 7.68 per hour. From the theoretical relationship (3, 9 ) , the theoretical slope is 5.77 (kO/pc,R2). Therefore, 5.77 k ' p c , R 2 = 7.68. Since p = 100.5 pounds per cubic foot, average cp = 0.182 pound centigrade unit per pound X C., and R = 0.0833 foot, then
In this manner, the average value of thermal conductivity for Ottawa sand and ilmenite sand in stationary beds was found to be 0.172 and 0.132 P.c.u. per hour (sq. foot)(' C. per foot), respectively. These values represent average conductivities over the range from 20" to 100" C. Mchdams (4) reports a value of 0.19 for the thermal conductivity of dry sand. An ewellent re-
Table I I . B u l k Densities of O t t a w a and I l m e n i t e Sands i n Various Containers, under Flow and under Static Conditions F!ow-
Material Ottawa sand
So
Container 500-cc. graduate 500-cc. graduate
K O 0
1000-cc. graduate 20-ioot, 0.870-inch i.d. tube PO-foot. 0.870-inch i.d. tube
Ing
KO
s
Yes Crushed ilmenite sand
50
h-0 N0
Yes Yes
B,ilk Density Lb./Cu. Ft.
pi
100 10G (graduate shaken to settle sand) 102 106 103 152 153 134 167 169
I .o
0.8
0.6 0.4
EXPERl M ENTAL RESULTS
Physical Properties of Sands. The exp(~rimcntal1ydetermined 6pecific heats of Ottan-:i nnd ilnionitr wnds arc shon-n in Figure 6 as a funcstion of s:md tcmpcrature. V:ilues on this graph represent the average specific heat between 20' C. and the temperature in question. For purposes of comparison, values of the average specific heat of quartz and fused silica obtained from the international Critical Tables ( 2 ) , and of pure ilmenite as determined by Shomate et ai. ( 7 ) , are also shon-n. The esperiniental vdues agree reasonably well with the published data. Bulk densities of the sands under various conditions are given in Table 11. The bulk densit,! of the Ottawa sand n-as approximately the same in both moving and stationarj- beds. With ilmenite sand, however, the stationary bulk density n-as slight'ly
0.2
0.I 0.08 0.06
0
4
8 12 16 20 HEATING T I M E , MIN.
24
I B
Figure 7. Variation in Center Temperature w i t h T i m e for Stationary G r a n u l a r Beds See Figure 4 for apparatus used
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
1054
T a b l e I I I. K81n S o ,j' , b
Tub exit trmgeraturc n a - rririi.urrd ~~l~iii:ii~trir~i!l?. ( ' 1 1 1 1 tiirriiiocouple used to measure exit sand teniperatiirr in this T U I I . .I. i l l yi5coiii f l o v of liquids, heat-transfer coefficient ha,,h i- of little iiti!iry. as it n-111 vary T r i t l i dianietw of tube for a fiue(1 >ani1 flm rate. I t i j i n w r r w . 1 1,l reciprocal of h a m to-othrr resi.srancei in series in order to o11t~i11~ \ f , r - a lrcqiitance l
Table IV.
D a t a on Heating of Crushed I l m e n i t e Ore t?
1 I l l - f o i J t . 0.870 inch
i.ri
I O - f o u t , 0.622 incii i.d.
I O - f o o t , 0.870 inch i.d.
i40i 333 19.5 1129 112.5 383 247 91 3 11 14li 74
wti
I O - f o o t , 0.870 inch i . d
20-foot, 0.870 i n c h i.d. 10-foot, 0.622 inch i.d,
131 84 80.5 131
I,
1 i t j , .i 1 i l i .5
176.3 101 .i 177 0 176.0 173.0
Iii.0 176, ,j 175.7 lii.5 101 8 Iii.? 177 .I 17.5 3 17:3 4
7 s - I
~L
T , - 11
I,
i t i ti
51.6
'38.1
75.1
0.341 0.489 0.622
0,639 0 211 0 378 0.499 0.311
0.174 0.173 0.177 0 173 0.175 0.177 0.178 0,1785 0.17.53 0 179 0.180 0.174 0.179 0,180 0.177 0.177
96.7 129.3 141.6 130.4 103.4 1.59 ;,
172. I 81,i 134,ii 1.75.1 1118.7 128. I
vitaw of thti 1itci.atui.e on tlir t,her~nalwiiduetivity of graiiulai solids is pwseiited by \l-addams (9). Heat Transfer to Moving Granular Beds. .I hu~iiiiiai~y of tliv Iic~itt-tr:~ii~fi*~~ (lata t o iiioving granular beds is prcswntid i i i Tah1c~111 m d I Y . l'h(wj data arc' rrprtl.vnttd grap!iic~:'lly iiy p l i i t s (if unaccomplished temperature change, ( T 3- t 2 ) (Z'3 - t i l , against a modified Graetz modulus, .L/wc,. The Ottawa sand data are presented in Figure 8, and the data for ilmenite sand iri Figurc 9. In calculating the quantity, L/wcp, the average specific, htbat tictween t z and tl was usrd. The good correlation ohtitilied in these plots iiidicatoti that tht. effects of diauwter, length, and temperature driviiig forcat, on the heat t,raiisfer rate ~ v e r eideritical with thosc predicted l)y c'quations for rodlike flow of fluids through pipes. As effect of dianieter was substantially t,hat expected theoretically, it caii be conrluded that there was negligible surface resistance to ht>at flow. For values of the ordinates less than about 0.5, it is apparc'nt that the data can bc represented as a straight line on the seinilogarithrnic plots, as would he expected from rodlike-flow thiviry. By comparing the slopes of the straight-line portion of t h i w graphs with the theoretical sloptt for rodlike flow a theriiial coiiduc>tivity for thc moving beti could be calculated.
From Equation 3a, the negative slope of the theoretical curve is 5.78. The ratio of the two slopes is (L/u.c,)!(~kl,!'u.c,,~,equal to r k . Thus 5.78 k = 2.80, and k = 0.155 P.c.11. per (hour) ($11,foot)(' C. pw foot).
11
1 - - ti
110.7 til , 6
EXAMPLI:. The iiegativt: slope uf the experimental curve foi Ottawa sand (see Figure 8) is 2.30 P.c.u.per (hour)(f'oot) ( " C.).
-
- t,
f
/g
93. I 40.3 76.7 107,3
0,501 0.489
0,698 0,773 0,823 0.516 0.896 0 969 0.754 0.731 0.838 0.7.53 0.692
115.6
123.6 77.9 137.9 148. 61. I 115.1 136.6 112.7 102,l
0,302 0.225 0.177 0 484 0.1043 0.0312 0 . 24j.i 0.269 0.141 0.246 0 307
I' LLLP 0,0623 0.168 0.283 0.172 0,100 0.289 0.449 0,602 0 180 0 .i R 2 1.475 0,369 0.419 0.653
0.36; 0,423
*Ly
I
ltj.0.5 4 95 3.31 5.82 10,oo 3.46 12.23 1 .66 5.56 I .33 0.68 2.71 2.3s 1.53 2.74 2,Ri
sumption of rodlike flow is 11ume out, not only by the agreemriii of the heat transfer equations with the experimental data. hut also by the visual observations made with the split tubc. Comparison of the experimental data ryith the rodlike theory ih furt,her demonstrated by Figure 10, in which the unaccomplished temperature change is plot,ted against the reciprocal of t,hr Graetz modulus, using values of thermal conduct,ivity obtained from the stationary-bed esperiment,s. The theoretical equation reprwents the experiiiiental data Tvithiii =t157.
1.0 0. 8 0.6 0.4
0.2
0. I 0.0 8 0.0 6
0.04
0.02
0.0I
WCP
Figure 8.
H e a t Transfer D a t a for O t t a w a Sand
,
1055
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1948
1.0
0.8 0.6
ILMENITE DATAL=20ft., D.0870 in - - v L = I O ft, D.0870 in 0 L = I O ft, D80.622 in $ O p , O Ts= 1 7 4 O C b,v,* T,-IOO'C 0
L = loft., D-0.870in. L = 10 ft., D=0.622ln. 0,n.o T -, 174-C. ---e,A,* Ts= I00 'C. 3
0
0.4
0.4
-6
0.2
ni 0.68
I
I
0.06 0.04
ILMENITE DATA L = 20ft., D.0.870in v L = loft.. D.0.870in
I
I
0.04
+
\,
I 1
I
I
I
0.02
14
I
L -
THEORETICAL ROD EO.
wcP
Figure 9.
0.0I
Heat Transfer Data for Crushed Ilmenite Ore
0
0. I
02
A% wcP
The points plutted near the low(~rt~ridof the line 0 1 1 Figurt: 10 rrprosent data taken pvhen t,he exit sand temperatuw was nearly equal to thc' steam temperature. On Figure 11, however, these same points fall near the flat upper asymptote of the tht3oretical curve, and the data taken under conditions of only partial approach to the steam temperature fall along the descending branch of the curve. The close agreement of the data 11-ith theory on both plots, both when the difference betn-een the exit sand temperature and the steam temperature was small and when it \vas large, lends further support, to the assumption of rodlike flon-. GENERALIZED EXPANSION
.\lthough data for heating or cooling granular solids flowing in a settled state inside vertical tubes have been shown to be representable by esisting theoretical equations for heat transfer to a fluid in rodlike flow \Then t.he thermal conductivity of a stationary bed is used to evaluate the Graetz modulus, the theoretical equations available previously are useful only when the shell-fluid temperature is constant. Since niany practical problems encountered involve a temperature change in the jacket fluid (countercurrent or parallel f l o ~ )theoretical , equations have been developed to cover the more general cases. The solution for the case of countercurrent flow when negligible resistance t o heat flow is offered by the heat transfer fluid is given by Equation A-23a and Figure 12. These plots are similar to Figures 10 and 11, except, that solutions are shown for various values of the parameter, Z, which represents the ratio of the total heat capacity of the granular material stream to that of the jacket, stream. The lines for Z = 0 are identical to those shoIvn on Figures 10 and 11. -1special solution was required for Z = 1, which is the case for equal temperature change in the tn-o streams. One method of approximating the heat transfer occurring when the tube-will temperature varies because of changing jacket fluid 6eniperature is to assume an average tubc-wall temperature equal to the arithnietic average of the inlet, and exit jacket fluid temperature and to use the solution for the rod-flory equation for constant tube-wall teniperature ( Z = 0 ) . This approximate method will givv const.i~atireresult.? vhen ~t..- tl) ''(1'. - t,) > 0.5 and unsafe rc~sulrswlitBii .? > 2, as can h which gives ~ h ratio r of the t u l x length actually r c q u i r d to that obtainrd by the approsimatt~nwthod. Thr ratio is plotted as H funrtion ( ~ f%. \ \ i t t 1 I i ~ l ,i i i r ~ c ~ c o m ~ I i ~ ~tt.nip(1r:itur.i. ic.ti change
Figure 10. Comparison of Experimental Data w i t h Theoretical Rodlike-Flow Equation Close temperature approach
k = experimental value for stationary bed
as the parameter. -1s shonn by Figure 13, the eiror involved 111 the approviniate calculation becomr5 verj gieat as the produrt (1
+ ;)(&I+)
approaches unit,y. The approximate calculation is impossible lor larger values of the product, even though heat transfer can actually be obtained because of the countercurrent action. Although the relationships given in Figures 12 and 13 will solve the majority of problems, an occasional case may be encountered where the resistance of the jacket fluid to heat transfer may be significant. This case, which is completely general, has been developed and solved (see Equations 6 2 2 and h-24). The solution is represented graphically by Figures 14, 15, and 16, where the usual funcrions arc plot,ted with 6 (the ratio of the resistance of the jacket fluid to heat flow to the resistance in the granular material) as a parameter. Figure 14 is applicable in cases where Z = 0 and could have been obtained from the GurneyLurie charts (3) for unsteady-state heat transfer to cylinders. 1.0 0.8
(26 12- 11 T-, 1, 0.4
7
03 0.2 2 3 4 6 SAND DATA 0 L=20f t., D= 0.8 7 0 in. A L=IOft..D=0.870in. 0 L = IOft.; D=O.6 2 2in. O,A,o T, 0 I 7 4 O C. *.A,. T= , IOO°C.
20
810
E k
L
30 40 60 80100 ILMENITE DATA 0 L=20ft.,D=0870in. v L- IOft..D- 0.870 In. 0 L = IOft.: D=0.622 in. 6 0 0 Ts=I74OC 6V T,=IOO°C.
+
Figure 11. Comparison of Experimental Data w i t h Theoretical Rodlike-Flow Equation k
Remote temperature approach = experimental value for stationary bed
1056
Vol. 40, No. 6
INDUSTRIAL AND ENGINEERING CHEMISTRY 10
0.1
T w w
b
0.01
o'oo10
0.2 0.4
0.6
I
OS
1.0
1.2
1.4
&
NG= Figure 12.
WCp
Theoretical Curves for Heat Transfer t o Fluids i n Rodlike Flow in Cylindrical Pipes Curves for 0 = 0
Figures 15 and 16 are for the cases where Z = 0.5 and 1.5, renpectively. Interpolation can be used to obtain values of the temperature functions a t intermediate values of the parameters. The general solution for the case of parallel flon- has h e n developed and is given by Equations A-25 and -4-26. Because viscous materials flow in an almost rodlilie condition while being heated, it is expected that the preceding relationships will give a good approyimation of the temperature changes occurring when these fluids are heated. A P P L I C A T I O N TO D E S I G N
The folloning examples are given to illustrate the use of the design charts that have been presented. Example 1. Sand having a specific heat of 0.2 P.c.u. per (pound)(" C.) and a thermal conductivity of 0.15 P.c.u. per (hour)(sq. foot)( O C. per foot) is to be cooled from 140' to 40" C. a t the rate of 1000 pounds per hour by water entering at 30 O C. and leaving a t 40" C. I t is desired t o use a bundle of vertical tubes no greater than 10 feet in length, and with a diameter of 2 inches. SOLUTIOS'. The water film coefficient can be assumed to be = 500 P.c.u.per (hour)(sq. foot)(' C.).
W
(o'15)(10) = 62.5 pounds per hour (0.2)(0.12)
This represents the rate through one tube. of tubes is then: i v
=
1000 62.5
--
The total number
16
rllthough the diameter of the tubes was specified, actually tubes of any diameter could be employed, provided that the weight flow through each tube equals 62.5 pounds per hour and the diameter is not small enough to make p significant.' Example 2. Assume, for purposes of comparison, that the same cooling required in Example 1 is to be performed with air, increasing from 30" to 40 ' C. in temperature. SOLUTION.An air film coefficient of 9 P.c.u. per (hour) (sq. foot)(" C.) probably can be obtained.
The air rate is (1000)(0'2)(140-40), - ~(40-30) ~__ (0.25) -___ or 8000 pounds per hour, assuming the specific heat of air to be 0.25 P.c.u. per (pound) ( " C.)
k = 0L 15(12) p = =0.0036 U'R (500)(1) For practical purposes, p = 0. The cooling water rate is (1000)(0.2) ~ _(14040) _ 40-30
_
or 2000 pounds per hour.
Tz Tz
- ts - ti
=
30-40 = 0.091 30-140
Interpolating b e k e e n Figures 14 and 15 for p = 0.2, 2 = 0.1, and ( T z - t 2 ) / ( T 2- t l ) = 0.091
FromFigure 12 ( p = 0), for Z = 0.1 and (T2 - t * ) / ( T 2- ti) = 0.091
IZL
- = 0.12 WCP
The use of air instead of water as the cooling medium increases the size of the cooler by 62%-
105'1
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1948
and Table V.
T h e r m a l Conductivities
[Determined from heat transfer d a t a for granular solids flowing in vertical tubes (assuming rodlike flow) and those obtained in stationarj-bed thermal conductivity apparatus ] Thermal Conductivity. k, P.c.u./ Bulk (Hr.j(Sq. Ft.) Den-ity, hIaterial Condition i°C./Ft.j p , Lb./Cu.Ft. Ottawa sand Ottawa sand Crujhed ilmenite Crushed ilmenite
Stationary bed Flowing settled bed in veTtical tube Stationary bed Flowing settled bed in vertical tube
0.172
101
0.155 0.132
103 149
0.141
168
d-gidr" = p
La
(bJtlbr71)dz
e-ps
(A-8b)
Cse is made of these properties to reduce the original partial differential equation to a subsidiary ordinary differential equation involving only a single independent variable, T . The quantity p is a constant, yet to be determined. The operation of transformation is carried out on both Equations A-1 and A-6 by multiplying each term by e-Pzd: and integratinq, as indicated by Equation A-7. The result is
k d
;d;
(r
2) -
pc,pzyg
=
- pc,pzjt,
(A-la)
D E R I V A T I O N OF EQUATIONS
Derivation of Basic Differential Equation. A hctat balance written around a ring-shaped diffcwntial volume of lmgth dt, thirkness dr. and radius r lead? to t h e following partial diffcrential eqwit ion : 1.
(A-1) The left side of this equation evpresses the net rate at Khich heat accumulates in the differential element oning to flow in and out along a direction parallel t o the axis. The right side represents the net rate at which heat is lost by conduction in a radial direction. Conduction in the i: direction is neglected, as is customary in problems of this type. 2. Derivation of Boundary Conditions. 'a' -'IT EYTRISCE TO TUBE. The core-fluid temperature i. a\sumed uniform. aq expressed by writing t =
tl
= constant for z = 0, 0
0.
pdp,r)
,-
There
{A-ga)
and the transform of the average temperature is
where p = k / l ; ' R and Z
=
wc,/TVCP.
4. Evaluation of Inverse Transform. I n ordrr rn dctt,rniine the solution for the tempcmturc., an expre.qbion must be found which will give the right >idr of Ilquation AI-t5when transformed according to Equation A-7. In other ~ o r t l s ,Equation .1-15 must be substituted on the left of Equation A-7, FThich then must
Vol. 40, No. 6
INDUSTRIAL AND ENGINEERING CHEMISTRY
1058 D
I
8
I
i
6
4
2 -I
3" (3
I
0.8
z
0.6
2
0.4
\
0.2
0.1
3
Figure 13.
Comparison of Exact Method of Evaluating No, w i t h T h a t Based on Use of
Average Jacket-Fluid Temperature =
T a b l e VI.
" ~~
3
"; for Case where 8 = 1
Characteristic T i m e Constants for H e a t Conduction i n a Cylinder
5.4831 30.47 13 74.8865 13R.O3!l.j 222.9318
326.561 3.3685 30. 00fiir 14.4888 138,6412 218 53.512 321;. 127 4 , fiY 16 29.4556 7 3 8809 138.0366 213.0?24 325.520
11
I1
0.2
1
3.3893 28,4288 72,8667 137,0304 204,7475 324,576 1,4957 27.1838 71.6.562 135.816 197,712 323,352 - 4 i1976 24.4748 B8.8867 133.0401 189 3431 320.553 3.9593 ?2.2124 .38.0294 112,8332 187.1014 280.8305 0.8851 13.6768 60.2183 104.4965 I7..i21i
272.133
0.5
0 2
2 2 1 2300 3i 4.564 112,483 186.871li 280 0963
(A- 18j \\
he1 P k " ( p i , / ) = (n' dp)F(p,r),n h r r e p = pL.
( i-l%)
F(Pk,T.) = 0
(A-1911)
ant1
0 .3
2 11
0.0494 1 4 . 781i 4'9 , 3 183 103.B36.5 177.8223 271.590
1.5
0.2
- 2 3323 !9 G9d8 .ID 4737 111 8 4 5 4 180 43eG 280, X Q
1.5
2
4453 1; 58(r9 30.18i3 104,4852 178. 3110 272.283
1.5
20
-0.0494 14.7809 49.3183 103.6365 177.6223 271.550
-0
The .values of pk are the "time constaiit." f o r 1 he problem, arid are determined, according to Equlitionq A-16 and A-lRh, by solving for the rootq of the equation X 2J1(z) J " ( r ) = B 222 f Z
(.4-20)
This equation may be solved by succes-ive Iiuinerical approximation, or graphically by the use of a plot such its that s h u ~ v nby Figure 17. A few values of the roots, calculated numerirally for several values of p and Z , are listcd in Table VI. For large values of .i-z anti large values of il o r %, t h p roots ari' givtxri ai)proxiniattxly 117 .i'
=
a 5a Ba
~,1, y, . I
4k + 1 . (--i--)
K,
(A-21)
By suhstitut,ing in Equatioii .\-Is, t l i t , fiiiril c i i l u t i o i t fi)r thr average esit temperaturr of ttic, c:)rr t l i i i t l i:: fouri(1 t o t w
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1948
L
0.001 1.0
0.0
I
i
0.2
0.4
0.6
L
NGZ
S
0.8
1.0
12
1.4
A
wcp
Figures 14 (upper), 15 (center), 16 (lower). Theoretical Curves for H e a t Transfer t o Fluids i n Rodlike Flow in Cylindrical Pipes. Various Values of Z and B
1059
INDUSTRIAL AND ENGINEERING CHEMISTRY
1060
Vol. 40, No. 6
Figures 13, 14, 15, and 16 are based on Equations A-22 and 6-23. 5. Special Case of Equal Flow Rates, ( iwP = TT-C',). Ileaviside's formula cannot bcl used directly for this case, but bpeeial methods inay be employed (6). The result, is I
I
\
t? - i.
TI
- II
=
1
-E -
48
(?A) R2
4
exp [ - + ? i a l . K ~ J ;
(w Ib./hr) .[?(XI
I I
\ \
I
I
1
I
I
Tz
Illustration of H e a t Balance Used for Setting U p Differentia I Eq uations
The followliig 6pecial Ciibt'b are d intereat: (a) No surface resistance = h G'R = 0 , :
c A-14)
6. Parallel Flow of Core and Shell Fluids. Equations may be developed for this case, using t h r same operational method as that described piw,iounly. The result F are
I
J
i2
Figure 17.
= -- ( 8 : 2 ) x Z , 5 # O
- tl -ti-
__ 1
+ z- 4
with thr valuea of x deterininrtl from i.1-26 j
ih
Constant surfacr teiiiptBiitture (X
= ICC,
li'(',,
= 0):
as dhowii hy Figure 18.
\Then, in additiori, t1ir.w is nu estwnal tkwnial resistance, this case reduces t o the faniiliar equation for unsteady-state heating of a cylinder wit,h the "t.ime constants" det,ermined by che roots of J&c) = 0.
.Ipprosimate iorniulas, involving the error integral, could be derived for cases where iaL R 2 ) is small. These are useful within a narrox- range of conditions where the slow convergence of the above infinite series requires that several terms of the series be evaluated for accuracy. NOMENCLATURE
bpeeific heat uf granulai wlid 01 fluid in rodlihe motion, P.c.u./(hour)(" C.) C, = specific heat of jacket fluid, P.c.u. ' (hour)(" C.) D = inside diameter of tube. feet e = base of natural logarithms = 2.718 S G=~Graetz number, wc,lkL - V G ~ , ~=%Graetz . number based on use of a n average jacketL~
=
Ti
+ Tz
fluid temperature equal to __ 2 h a , = film heat-transfer coefficient, based on arithmetic mean temperature difference, P.c.u./(hour)(sq. foot) ( ' C.! I: = apparent. thermal conductivity of granular material o r fluid in rodlike motion, P.c.u./(hour)(sq. foot) ( C., foot) L = jacketed length of tube, feet L,, = jacketed length of tube basedpn use of a n average jacketfluid temperature equal to
-__ '1
T2,
feet
>V = total number of tubes r = radial coordinate inside tube, feet 12 = inside radius of tube, fret tl = entering temperature oi material in rodlike motion, ' C'. tl = average exit, temperature of material in rodlike motion, O C :
Y
t,
Figure 18.
Graphical Solution for T i m e Constants
-
= temperature at longitudinal axis of stationary cylindrical bed of granular solid after a time e, O C.
t , = uniform initial temperature of stationary cylindrina! bed of granular C. T , = tube-wall temperature, C. TI = exit temperature of jacket fluid, ' C. I', = entering temperature of jacket fluid, C. c" = coefficient of heat transfer from jacket fluid to iiiside tube surface, based on inside surface area, P.P.U..' (hour)(sq. foot)(' C.j 1-l = volume of tube, cu. feet
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 CITED
(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 January 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 o f e q u i p m e n t e m p l o y i n g h e a t t r a n s f e r f r o m a f l o w i n g f l u i d t o a bed o f loose solids. I n t h i s investigation, design d a t a in t h e f o r m o f u n s t e a d y - s t a t e h e a t t r a n s f e r coefficients f r o m a i r t o a bed o f g r a n i t i c gravel are presented. T h e size o f gravel e m p l o y e d 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 s t a n d a r d c u b i c feet per m i n u t e per square f o o t o f cross-sectional area were used; a n d t h e e n t r a n c e a i r t e m p e r a t u r e s m a i n t a i n e d were over t h e r a n g e 100” t o 250” F. T h e gravel was packed i n t o t h e bed in s u c h a m a n n e r t h a t n o r m a l voids were o b t a i n e d . 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 e q u a t i o n :
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 I N V E S T I G A T I O N S
w h e r e i n h i s t h e h e a t t r a n s f e r 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 c u b i c f o o t o f bed v o l u m e per degree F a h r e n h e i t 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, p o u n d s per h o u r per square f o o t o f bed cross section, a n d d is t h e e q u i v a l e n t spherical d i a m e t e r o f t h e particles in feet. C h a n g e in t e m p e r a t u r e o f t h e e n t e r i n g a i r h a d no a p p r e ciable 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 ilamos, S . \I.
