July, 1930
I N D U S T R I A L A N D ENGI,VEERISG CHEMISTRY Acknowledgment
72 1
Literature Cited
This investigat,ion was carried out under the Henry Marison Byllesby ~ ~Research~ Fellowship ~ in ~ ~ ~ i~ Grateful acknowledgment8 is also expressed for the assistance rendered by R. F.Smith.
(1) Cantelo, Chem. M e f . Eng., 33, 680 (1926). (2) Cantelo, 19, i ~ ~Simmons, l ~ Giles,~and Brill, ~ IND.iENG.CHEM., ~ ~ 989 (1927) . (3) Donnan and Masson, J . SOL.Chem. I n d . , 39, 236T (1920). (4) Lewis, J , ENG, CHEM,, s, s25 (1916), ( 5 ) Whitman and Keats, Ibid.. 14, 186 (1922).
Heat Transfer from a Gas Stream to a Bed of Broken Solids-II’i’ C. C. Furnas NORTH CESTRAL
EXPERIMEKT STATION
OF THE
u. s. BEREAUO F
hfIhES, h\IINSEAPOLIS,I f I S S .
This paper represents a continuation of a study pubR-ISSFERRISG heat Schumann’s Method lished under the same title in a previous issue of Into or from a fluid dustrial and Engineering Chemistry (6). The first It is possible that the exs t r e a m t o b e d s of article reported work on beds of iron balls, while t h e perimental difficulties would broken material is one of the data for t h e present article are for materials of more have proved insurmountable most common industrial procindustrial interest-iron ores, coke, limestone, coal, had it not been for a very esses. yet quantitative inforand a typical blast-furnace charge. important theoretical paper mation on the subject is alCurves are given for the temperature history of a cold recently published by Schumost entirely lacking. As far bed of broken solids being heated by a hot fluid. A mann (21). S c h u m a n n ’ s as the author know, there method of determining the coefficient of heat transmethod of attack was somehave been only two previous fer between a gas stream and a bed of particles is given. w h a t a s follows: Assume experimental studies bearing The temperature history curves of a bed are shown to that a uniform fluid stream is directly on the problem. One be applicable when there is a heat of chemical reaction. allowed t o flow through a bed was conducted by the ComEquations have been developed for the steady state in of broken solids at some unitiustion L-tilities Corporation countercurrent heat flow equipment. form temperature initially (28) and the other by the presAn illustrative problem involving a foundry cupola is lower than the temperature ent author (6). Inbothcases given, using the equations and experimental data preof the stream. If there is no the beds were made up of sented. uniformly sized iron balls and heat loss through the walls of the apparatus, the entire bed although the data and the relations betjveen variables are of theoretical value, they are of material will eventually arrive a t the initial temperature of the fluid. Given the thermal properties of the fluid and solid, not directly applicable to industrial apparatus. The theoretical considerations of the fundamental physics it should be possible to develop the mathematical relations of heat transfer are c.xceptionally complicated and no satis- for the temperature history of any point in the bed. factory general theory has ever been developed. However, The problem is difficult, but Schumann presents a very a very uqeful method of attack has been found along almost clever and skilful solution. The derivation is rigorously purely mat heinatical lines-mainly by methods of dimensional exact only for systems where the thermal properties are conanalysis. stant and for a noncompressible fluid and where the solid parwas probably the first to make an adequate ticles are so small that there is no temperature gradient formulation of the variables controlling heat transfer. His within the piece at any time. However, as will be shown pioneer work was later supplemented by Boussinesq ( l ) , later, it a-as found experimentally that this solution applied Susselt ( 2 1 ) . llciidams (20),Rice (24),Russell (RB), Stanton quite accurately to systems where the fluid is a gas and where (ZQ), and others. This theoretical work, coupled with a there are considerable variations in thermal properties and the mass of experiments of recent investigators, has resulted in solid pieces are large. Schumann’s mathematical procedure will not be presented a very satisfactory quantitative theory for liquids flowing here, but his computed curves are important and are given through conduits (50). However. none of these studies are directly applicable to in Figures 1 and 2. The symbols used have the following heat transfer to beds of broken solids, so in 0rdt.r to obtain definitions : quantitative information regarding the operation of blast T o = initial uniform temperature of the fluid furnaces the Bureau of Mines has found it necessary to take T , = temperature of fluid a t any point at any time T , = temperature of solid a t any point at any time up this experimental study. It is part of a general program of blast-furnace research which this bureau is conducting in the laboratory and in the field (3, 11 to I ? , 25). The method used in the previous investigation (6) was y = -k- x that of alternately measuring the temperature of the gas h, 21 stream and of the solid. Although this method Tvorked satis- where x = distance from bottom of column in centimeters factorily when the bed was made of iron balls, it mas inadet = time in seconds quate when applied to beds of iron ore and other irregular w = fluid velocity in cubic centimeters per second per square centimeter cross-sectionalarea of bed pieces of low thermal conductivity. f = fractional voids in bed (no units) h, = heat capacity of solid in calories per cubic centimeter 1 Recened May 10, 1930. Presented before the American Institute of per degree Chemical Engineers, Detroit, Mich , June 4 t o 6, 1930. h,= heat capacity of fluid in calories per standard cubic 2 Published b y permission of the Director, U. S. Bureau of Mines. centimeter per degree (Not subject to copyright )
T
INDUSTRIAL AND ENGINEERING CHEMISTRY
722
Figure 1-Schumann's
k
Curves for Temperature History of Solid for Values of Y from 0 to 10
= coefficient of heat transfer in calories per degree per
second per cubic centimeter of volume of bed
Since the groups of variables of Equations 1 and 2 are dimensionless, any system of units may be used instead of the metric, provided they are mutually consistent. This paper is concerned primarily with gas streams, and for such a system the quantity x / v is negligible compared with the time, t, so the last term of Equation 1 may be dropped and 2 may be defined as kt hs(1 - f)
z=-
(3)
Thus, in Figures 1 and 2 the abscissas are proportional to time; the ordinate is the temperature; and the different values of Y on the different curves are proportional to the height of column being considered. L i m i t s of Application
A l t e r n a t i v e M e t h o d of S o l u t i o n
T., T,. T o ,Y , and 2 have already been defined. J, ( 2 i d k ) is a so-called modified Bessel function of the first kind and zeroth order. Schumann's nomenclature differed slightly from that ordinarily accepted, and the function he used is This ordinarily represented (9) by I , (2&%). symbol will be used in the following discussion. From the above equations it can be shown by simple algebra that
7." T"
2-8
=
2b'e-Y-z
=
I,(2&qe-Y-Z
(9)
From Equations 4 and 9, if Y is constant T"
= Io(2dYZ)e-Y-ZGZ
(10)
The curve of T,, then, for each value of Y can be obtained by plotting values of the product of the two functions I , (2dZ) and e-'-' against 2 and integrating graphically from 2 = can be computed exactly, 0 to Z = m . The function e-'-' but the I , function, as mentioned above, is known only for small values of 2dE. However, Equation 10 holds the key to the determination of this function. Indicating the formal integration of (lo),
Jo
Obviously, if the thermal properties of the system are known, the solution of a problem of temperature history becomes one of extreme simplicity, after the curves have been computed, for the temperatures for any given time or position may be read directly from the curves. However, the curves of Figures 1 and 2 are of limited application, for the solu," tion of a problem involving a column more than 50 cm. high, over 100 seconds of time, and a coefficient of 0.01 is beyond the limit of the ud curves. For small coefficients of heat transfer the size of column considered could be somewhat larger but still very limited. 06
5'01. 22, No. 7
J O
From the physical conditions of the system it can be seen that the value of this integral from zero to infinity must be equal to unity, for the total possible change in solid temperature, T , is equal to To. Therefore
ZIP
The formal integration of the differential equations which Schumann ( d ? ) developed results in an infinite series, each term of which is itself an infinite series-a complicated arrangement. These infinite series are related to the so-called modified Bessel functions of the first kind' the r e a s o n t h a t Figure 2-SLhumanA's Curves for Temperatke History of Gas for Values of Y Schumann did not compute the curves for from 1 to 10 higher values of Y and 2 is that only the small values of these modified Bessel functions have been computed and the direct computation for higher values involves enorI , ( ~ ~ / Y Z ) ~ - Y - Z G Z= 1 (12) mous labor. However, it is possible to solve the equations by graphical integration and for any desired values of Y and 2. The K i t h this criterion, a solution may be assumed for the Bessel method is a fine illustration of the unique power of graphical function and then tested for its validity. The details of these methods. Schumann showed that: computations are reported elsewhere (8). ~~
Lrn
I.VDUS'TRIA L L V D E-L'GI-VEERIiYG CHEMISTRY
July, 1930
When tlic values of the Bessel function for the higher values of have thus been established, they may be used in turn to perform the graphical intrgration of Equation 10 and thus to obtain the curve of T,/T, against 2, as described in a previous paragraph. Graphical integration to the limit a , of course, is impossible, but the curves of the function of Equation 10 start at zero, rise to a iiiaximum approximately a t the point where Y = %, and then fall and approach zero so rapidly that the area under the curve for higher values of iiiay be neglected without loss of accuracy. This very fortunate circumstance makes the graphical iiiethod possible. The method just described furnishes a means of computing the curve of the temperature history of the solid. From Equation 4 it is seen that for anv value of 2 the temDerature of the gas IS
723
dj??
di??
.
111
Figure 4-Computed
Temperature History of Gas for Values of Y from 9 to 25
and changing process, and that to have any significance it is necessary to obtain data which are the statistical averages Ifelice, to obtain the temperature of the gas it I S only neces- of the temperature of an entire plane. The only practical zary to add the nunierical slope of the curve. 'The tempera- way to do this is to force all the gas passing a given plane in ture-history curves for both gas and solid have been computed the bed through a relatively small orifice and measure the for values of Y up to 500 and are shown in Figures 3 to 8. temperature of this gas stream. This value of Y is large enough for very long columns of maLbTARATuS-The apparatus used for doing this is shown terial. in Figures 9 and 10. Hot flue gases were pulled through the column of material and the temperatures measured continuExperimental Determination of Coefficients of Heat ously a t the top and bottom of the column. The hot gases Transfer were supplied by a pot furnace, F , which was heated by a gas Interesting as this theoretical development may be, it is burner, B. This furnace was mounted on a four-wheeled useless without knowledge of the values of the coefficients of truck on a track so that it could be rolled out from under the heat transfer. A method mas found which worked satis- apparatus when desired. The gas and air connections t o the factorily for systems of fairly large iron balls ( 6 ) . When the burner were flexible hose. A screw-lifting device was atbailie method was applied to beds of iron ore, the results were tached to the truck so that the furnace could be lifted quickly erratic and most unsatisfactory. It had been suggested by 10 to 15 cm. A lip of burned Carbofrax cement was made Burke and Schumann (28) that the theoretical time-tem- around the hole in the top of the lid and this lip fitted into perature curves should furnish a means of determination of the base of the shell containing the material to be tested. This gave a fairlv close fit between the furnace and --_ --1-7 the cchmn. The rotary blower, R, would pull the hot gases from the pot up through the column of broken solids, C, through the dust catcher, D,through the orifice of nichrome plate, 08,though the water cooler, W , consisting of a pipe 10 cni. in diameter and 3 meters long in a bath of running water, and thus to the pump and exhaust to the open air. The gases could be further cooled by a fine water spray, SIinside the pipe. The column of material was held in a sheet metal cylinder made of 16-gage black iron and was surrounded by Sil-0-Cel insulation packed in an outer metal shell made of 20-gage sheet. Two sizes of cylinders were used, the smaller approximately 15 cm. in diameter and 50 cni. high, in an insulating shell 45 cm. in diameter; the largcr cylinder was approxiniately 23 cin. in diameter I Figure 3-Computed Temperature History of Solid for Values of Y from 9 t o 25 xiid 1 meter high and was packed in an insulating shell 61 cni. in diameter. the coefficient if the temperature of the solid F\ere measured Three refractory rings, made of one-half fire clay and oneat all times at some point in the bed. However, when this half Carbofrax, were placed a t the bottom of the column as method was tried it failed to give curves which coincided with sho~vnin Figure 9 to minimize end heat loss. The middlc the computed curves. Other methods were tried and all ring constricted the passage to 5 cm. in diameter in order to facilitate the mixing of the hot combustion gases and whatwere failures. From the erratic nature of the observations it appears that Cvpr air was being pulled in with them. The charge mas held for coluinns of irregular solids of low tlieririal conductivity in place a t the 1)ottoin by means of heavy iron screen placed the passage of gas and transfer of heat is a very uncertain on iron bars. A similar screen was placed on the top of the
724
INDUSTRIAL AND ENGINEERING CHEMISTRY
os
"8
"1
"6
c'lca5
"I
us "2
Figure 5-Computed
Temperature History of Solid for Values of Y from 25 to 100
Yol. 22, KO.7
means of pump R. After thermocouple TZ had come to a steady state, the furnace was removed and the column cooled down with air a t room temperature being pulled through the bed. This left the surrounding insulation warm, approximating a mean temperature between the beginning and end of a heating-up period, thus tending to eliminate errors due to heat loss through the walls. After the preliminary heating, the process was repeated for the taking of data. Thermocouple T I ,which was a t the bottom of the column, was kept approximately constant. The mass rate of flow of gas as measured by orifice 03 was kept constant. The linear rate of flow of gas over the thermocouple beads was maintained a t a t least 10 meters per second, in order to insure a reading of the true gas temperature (IO). The data were recorded on a Leeds and Northrup curve-drawing recording potentiometer. Similar data were taken for the cooling of the bed, when the furnace was removed and cold air pulled through the column. TEMPERATURE RANGEAND GASCOMPOSITIONGas temperatures up to 1100" C. were used on iron ores. For work on coke and coal the gas temperatures were kept below 300" C. to prevent ignition. The hot gases pulled through the column normally contained about 5 per cent carbon dioxide, 10 per cent water vapor, 10 per cent oxygen, and 75 per cent nitrogen. The density, v i s c o s i t y , specific heat, and thermal conductivity of such a gas are much the same as for air, so the data obtained may be considered as being those for air. Wide Application of Theory
The theorv of the comDuted temperature-history curves presumably iS Strictly applicable Ody to a noncompressible fluid and for solid particles of infinitely great thermal conductivity. Fortunately, as has been found experimentally, it also applies to almost a n y sort of gas-and-solid system. A typical example of the excellent agreement between the curve computed for liquids and the observed data for gases is shown in Figure 11. This highly satisfactory degree of correspondence held throughout the work on various kinds of materials. I n systems of large solid particles wherein the thermal conductivity of the particles is small it becomes necessary to explain the term "temperature of the solid," for there is a gradient within the solid piece itself a t all times during the heating. I n such a case the temperature of the solid must be considered the mass average temperature as determined by the heat capacity. If the observed data are plotted on a logarithmic scale, as has been done for the time-coordinate in Figure 11, and compared with a group of computed curves for different values of Y , the plot of observed data may be displaced along the abscissa until it coincides throughout its length with one of the computed curves. The value of Y on the coinciding computed curve is then the correct value of Y for the data and k can be computed by means of Equation 2.
Figure 6-Computed Temperature History of Gas for Values of Y from 25 to 100
column. On top of this was placed a fire-clay plug, P , which was sealed tightly around the edge with h e clay, forcing all gases through the central 5-cm. hole. Two thermocouples in specially constructed shields were placed in the bottom of the bed (at T I ) and in the center of the passage of the fireclay plug (at Tz). The details Sf the construction of the thermocouple shields are also shown in Figure 9. Gases were pulled through these shields by means of a vacuum line and measured by means of the orifices, 01 and 02. The temperature of the gases passing through the orifice 0 3 in the main line was measured by two small base metal thermocouples, T3 and TA. The thermocouples were all made of calibrated chromel-alumel wire. Orifices O1 and O2 were 0.793 cm. in diameter in a pipe 5.24 cm. in diameter. Orifice 03 was 5.08 cm. in diameter in a pipe 10.3 cm. in diameter. The usual orifice formulas were used in estimating the rate of gas flows. The orifice 03 was checked against a standard orifice which had been calibrated by means of a wet-gas meter. PROCEDURE-The principal data of the experiments consisted of a continuous record of the gas temperature a t the top of the column as obtained by thermocouple T2. I n order to minimize errors due to heat losses through the walls of the tube, a preliminary heating was made by pulling the furnace, F , under the column, lifting it by the screw device, and pulling the hot gases through the column by
Heat Capacity of Charge
Tlie method under discussion not only furnishes a means of determining the coefficient of heat transfer, but also gives a value of the heat capacity of the charge. The heat-transfer
I S D C S T R I A L ,450 E S G I S E E R I S G CHEMISTRY
July, 1930
coefficient 1s deterniiried by the value of Y which fits the ciiive regardless of the amount which the abscissa has been ihifted to accommodate the observed data. I n other words, the coefficient is dependent only on the shape of the curve. On tlie other hand, the length of time the apparatus takes to acquire a gir en temperature is determined by the heat capacity ~f the solid. Thus the heat capacity detwmines the position ui the curl e on the absciqsa. Thic may he easily seen froiii Equation 3. A rearrangement . t i thii equation gives - \ -
where h . i k
Z
= = = =
=
1,
heat capacity per unit of volume of solid time in seconds heat-transfer coefficient per unit of volume of bed value of Z taken from computed curves fractional value of voids in bed
Example
The folloning is tlie computation of tliernial properties from the t h t a presented in Figure 11. The iiiaterial is Danube iron ore from the Moaahi range >creenetl to pass 0.746-inch (18.9-mi.) and stay on 0.325inch (13.3-nim.) screen. The mean particle size is 1.6 ciii. The mluiiin Tvas 44.5 cni. high in a circular iron container of 153 >q. cni. cross section and made of 18-gage sheet. The average f l r m u-as 0.111 standard liter per second per sqnare centinirtrr. The gas passing through the columii contained approxitiiately 5 per cent carbon dioxide, 10 per cent w t e r . 10 per cent oxygen. and 75 per cent nitrogen. The thermal properties of this gas arc' \-cry nearly those of air. The heat capacity 01' the gab at the mean temperature of 530" C. was taken a. 0.000324 calorie per degree per standard cubic centiiiieter. From Figure 11 it is seen that the clata points fit tlie shape of the curve for Y = 6.0. Transposing Equation 2 and substituting these clata.
725
agreement throughout' the mass of experimental data which hare been gathered for conduits. It was also found t o \)e true for a bed of iron balls. On the other hand, it would seem reasonahle to assume that in the case of beds of iron ore or coke, Jvherein part of the resistance to transfer of heat resides within the particles themselves, the effect of changing gas velocity would be less than for other systems. A series of runs was made at different gas velocities on Corsica iron ore through a 0.523-inch (13.3-1nm.) on 0.371-inch 19.-1-nini.) screen. The apparatus was not designed to handle great variations iii rate of flow, but a span of 3 to 1 iii mass velocity of the gas Tvas obtainable. The data of this series of experiments are plotted in Figure 12 on logarithmic coordinate paper. The straight line drawn through the points has a slope of 0.7. This nieans that the coefficient for niateriak of low thermal conductivity yaries as the 0.7 poll-er of tlie niass velocity of the gas, or where k
=
k ~ ~ 0 . 7 (15) rate of flow in standard liters per second or any other appropriate units
Tliis exponent varies but little froiii tlie d u e for conduits and iron balls. It s e e m runarkable that the high resistance to heat flow on the interior of refractory particles should hare such a small effect. The rates of flow studied have all been relatirely hi&; and particularly in the study of ordinary combustion processes
ti 0 X 0 000324 X 111 = , o o ~ Fcalorie ) pet. 44.5 second per degree cubic: centimeter of bed
k =
The voids in the l x d are 62.5 per cent. In Figure 1 1 the computed curve, which is a function of Z. n-as shifted along the ahscissa until it coincided with the data points. I t will be seen that when Z = 2.0, t = 172. (The same factor, of course, holds for the rest of the curve. For instance when Z = 9.0, t = 77*5.) Substituting these values in Equation 14. ii. =
0 O(l-49 X 1i2
2(1 - 0 625) centimeter ~~
Figure 7-Computed
Temperature History of Solid for Values of Y from 100 to 500
Figure 8-Computed
Temperature History of Gas for Values of Y from 100 to 500
= 1.12calories per degree per cubic
This figure includes the heat capacity of the metal shell. which amounts to approximately 0.06 calorie per degree per cubic centimeter of bed. Henre, for this ore, h,
=
1.06
The specific gravity of this ore was determined as 3.99. Therefore. the specific heat of the ore is 1.0ti/:3.99 = 0.266. Tliis figure is in good agreement with figures obtained froin direct observation of heat capacity. Effect of Gas Velocity 111practically all caws of heat transfer to or froiii a fluid the coefficient has been found to vary approximately ae the 0.8 power of the fluid velocity. This one figure has heen almost the sole point of
INDUSTRIilL .4;L'D EA%INEERIllrG CHEMISTRY
726
the interest might be centered on lower rates, but the figures given later in this paper may be extrapolated by means of Equation 15 with considerable confidence, for all laws of heat flow seem to apply over a wide range. I t must be
ir
two temperatures the mean heat capacity of the gas (air) increased by about 2.5 per cent (IQ), and the heat capacity of the iron oxide was increased approximately 50 per cent ( 2 ) . Then the increase of the average heat capacity - of the pas and solid system was about 26 per cent when the mean absolute temperature was approximately doubled. If the variation is assumed to be in an exponential manner, then, from these figures the heat capacity, and hence the coefficient of heat transfer, varies approximately as the 0.3 power of the absolute temperature. Therefore, it will be considered that k = c~o.3 (16) where G
Figure 9-Diagram
of Heat-Transfer Equipment
admitted, however, that a study of the lorn rates of flow would be preferable to the very best extrapolation. Effect of T e m p e r a t u r e
A study of Equation 2 shows that for Y to be constant, if
Vol. 22, No. 7
=
some constant
I t is recognized that this is an unsatisfactory method of obtaining the variation with temperature, but the very fact that the observed data fit the curves computed for constant transfer coefficient and heat capacity precludes the possibility of direct measurement. This is an unfortunate circumstance which cannot be remedied unless some other satisfactory experimental method can be found. I t is quite surprising that the temperature should have such a small effect, considering its importance in some other heat-transfer systems, but the data cannot be interpreted in any other manner. Effect of Voids
The degree of packing in a bed has a very large influence the rate of flow is constant, the ratio k / h j must be constant. Since the computed curves have been found to fit the experi- on the resistance to fluid flow (4). Most things which inmental data in all cases, it follows that this condition must crease resistance to flow may be expected to increase the cohold within the range of experimental error. To test this efficient of heat transfer. To study the quantitative effect item further, two runs \-ere made on a bed of Corsica ore of this variable, a series of tests was made on a Corsica iron through 0.525-inch (13.3-mm.) on 0.371-inch (9.4-mm.) ore. The ore was sized to pass a 0.371-inch (9.4-mm.) and screen, one run with the gas temperature equal to 300" C. stay on a 3-mesh screen. The average particle size was 0.80 and the other, 1100" C. The gas flow in both cases Tvas 0.11 cm. The desired degree of packing was produced by a standard liter per second per square centimeter. The column foundry vibrator, such as is used in making sand molds. The determination of the amount of voids in a bed of ore of material was 42.5 cm. high. The value of Y for both runs was 14.0. This meant that the ratio k/h, was constant for is a somewhat arbitrary matter because the particles possess these temperatures, within the range of experimental error, a certain amount of porosity. The effective specific gravity The coefficient of heat transfer is 0.0114 for the temperature of a particle, then, depends upon whether or not the pore range 25" to 300" C. and 0.0017 for the temperature range volume is considered as part of the piece. It is common usage to speak of the specific gravity when the pore volume 25" to 1100" c. Investigation of Equation 1 shows that if the values of 2 is included as the "apparent" specific gravity, whereas the for observed data are to be consistent with the computed value for the particle exclusive of the pore volume is called curves, k / h , must be constant a t all temperatures. This the "true" specific gravity. It was found that the apparent means that the coefficient of transfer must be proportional to specific gravity gave the best correlation of data so this was. the heat capacity of the solid. This is not consistent with the the quantity used. The results are shown in Figure 13. The condition required by Equation 2, for the heat capacity of absolute value of k as shown by these data has no significance a gas and a solid usually do not vary to the same degree with when applied to other sizes or other materials, but the change change of temperature. The only reason, then, that the of IC with voids is important and the data may be expressed observed data are consistent with the theoretical curves is by the equation that the inconsistencies which occur lie within the range of Alogi k = 1 68(fz - f i ) - 3.56(fi2 - fi') (17) experimental error. The general relation between the cofractional voids in original bed efficient of heat transfer and temperature, then, is that the where ffzi = = fractional voids in bed under consideration coefficient varies in the same proportion as the heat capacity Effect of P a r t i c l e Size of the system as a whole. This furnishes the means of obtaining a rough approximation of the variation of the coefExperiments have been made upon various sizes of Corsica ficient of heat transfer with temperature. I n considering the variation of the heat capacity of the iron ore, Danube iron ore, and coke, upon one size each of system with temperature, half of the capacity must be as- Vermilion ore, bituminous coal, anthracite, and limestone; signed to the gas and half to the solid, for all the heat acquired and upon a typical blast-furnace charge. The blast-furnace from the solid must have come from the gas. I n the case charge consisted of 10 kg. of coke 6.3 to 7.6 cm. in diameter] under consideration, the mean temperature for the low-tem- 20 kg. of run-of-mine Evergreen ore, and 3 kg. of limestone perature run was about 425" C. absolute and for the high- 3 cni. in diameter. The average particle mas about the same temperature run, about 825' C. absolute. Between these as for a normal blast-furnace charge, about 4 cm. In com-
July, 1930
I N D U S T R I A L A N D ENGINEERING CHEMISTRY
mercial furnaces much of the coke is larger, hut there are also some smaller pieces. Some of the data are shown in Figure 14. The points are plotted on coiirdinates graduat.ed on a logarit,hmic scale on both axes. The points vary considerably, as might be expccted in a systeni so complicated, brit it seems iust,ifiable to draw straight lines through each set of points. Each of the lines is parallel and lias a slope of about -0.9. This ineans that for the differential materials
where c = some cons t a n t characteristic of system
T h e v a l u e s of k s h o w n are t h o s e computed for a flow of 0.1 st,andnrd liter per second per square centimeter. This rat,e of flow was chosen for eoinparison as it, is in the range used in blastfurnace work. Equation 15 was used for intermlation to t,his rate Of flow. It is evident that t,here is a considerable variation between the constants of the different materials.
127
dead air spaces in the coke afford good heat insulation. As niight be expected, the value of k, for the blast-foriiace charge msuines a value intermediate between that of coke and ore. Anomalous Behavior of Cooling Coke The data presented thus far have hem for the passage of hot gas-that is, for the beating of the bed. The same mimher of data were nbtained for the cooling of the beds with a stream of air. For iron ores t,he values of the coefficients for cooling u-cru practically the sarne as for heating and were subject to ahout the same degree of variabion as is shown in Figure 14. There was an apparent tendency for the cooling valoe3 to be somevhat higher, but. tire difference was not great enoiigli to have any partieiilar meaning, unless a siifficicntly large niirnber of data were t,aken to have siabistical significance. On the other hand, cooling coke wit11 air revealed a much higher coefficient of transfer than for the heating of the same system. Further, the rate of ehmge of the c.ooeffieientwibh particle size is inuc11greater for cooling than for heating. The data are presented in Figure 15. It is seen that the coeffieient atza rate of flow of 0.1 liter per seoond p ~ square r ecntimet,er varies approximatcly as the inverse 1.30 power of the particle diameter. No adequate explanation has been found for this unespeeted behavior of coke. A similar though smaller effect was found in syst,ems where iron halls were cooled with water but this portion of the heat-tramfer rtndy is not discussed in t,he present paper. It is quite evident that there are some phases of the heat-transfer mechanism .yhieli are not fully understood a t the present time.
Standard of Comparison of Materials
I n order to cornpare different materials there must be some standard hasis of comparison. For this article this standard of comparison will arbitrarily be set up as follows: The standard coefficient. of heat transfer shall be at a rate of flow of 0.1 standnrd liter per second per square centimeter, at an average particle dianieter of 1 CNI. at 50 per cent voids, and at a mean tempcraturc of 500' C. The synibol k, will be used to designate the coefficient of heat transfer under these conditions. The units are calories per second per degree difference Centigrade between gas and solid, per cubic centimeter i f volume of the bed. Interpolations to this standard may he made with the aid of Equations 15, 16, 17, and 18. The resultsof such a standardization are shown in Table I. Table I 4 o e S c i e n r of Heat Transfer for Various Materials under Standard Conditions
The table shows that there is no significant difference between the different iron ores studied if they are put upon a coinmon void basis. The coke bed has only onehalf the transferring povver of iron ore. There may he two reasons for this: (1) The honeycomb structure of coke affords relatively&few points of contact with the fluid stream; (2) the
Fiyure 11-Typical
Temperature History of Gae Sfrasrn at TOP of Column
Heats of Transition
Ferric oxide has at, least two transforniatinn points below 1OOO" C. (Sg). It nlight be expected that such points of isotherrnal absorption of enerRy would cause distortions in the time-temperature curves. Contrary t,o such anticipat.ion, all the heating and cooling curves were found to be very snioot,h and to fit the t.lreoretical temperature-history curves wiiieli were developed on a theory that did not take account of any transformation pints. The reason for t,his probably lies in the fact that the absorption of the transformation heat is distributed more or less evenly over t,he entire period of temperature rise as the temperature change advances up the column. IXence the only effect of heats of transforination is to increase the over-all heat capacity of the system. Such a circumstance is very fortrrnate for the utilization of this method of determining coefficients of heat t,ransfer.
I S D U S T R I A L A.YD E S G I S E E R I S G CHEMISTRY
728
The total heat involved in the transformations mentioned is about 7500 calories per mol, enough to raise the tetnperature of untransfornicd material about 200" C. If transition 0 02,
1-01. 2%. s o . 7
heat capacity of calciuin oxide is between 0.20 and 0.25 calorie per gram per degree, v-hereas the figure obtained for the fourth test was 0.33. Within the range of experimental error there was no change of the coefficient of heat transfer as the surface of the particlewas converted to lime. Lifting the Solid Charge
I
I
0 002
I
I I l l 1
0 03 0W (1 06 0 OB 0 I ?LOW STANUAHU LITERS PLH SECOND PER JQUAHF CEbTIMETFR
0'02
Figure 12-Effect
of Gas Velocity o n Coefficient of Heat Transfer
Material-Corsica
-1series of experiments was made on systems in which the solid particles of ore used mere so small that they n-ere lifted by the gas stream A sinall screen a t the top kept them within the cylinder, but hufficient room was allowed for the particles to niove in a free volume from one to three time5 a t great as the original bed. Because of the turbulent state of the solid phase. the teniperature-history curves of these systems did not coincide, eren approximately, with the computed curves. To date no nieans of correlating these data have been found. The matter is mentioned to indicate the limits of applicability of this inethod of experimentation. Heat Transfer per Unit of Area
iron ore 1 13 cm in diameter
points tend to cause serious discrepancies in the results obtained by this method, it would be expected that they would be observed when the energies involved are of this magnitude. The fact that they are not adds considerable confidence to the method. I t is problematical, of course, whether or not the temperature-history curves would be valid where very large heats of transforination are involved.
A11 the coefficients of heat transfer have been reported on the basis of a unit volume of the bed of material. I n most heat-transfer studies it is customary to report coefficients in terms of surface area. It is not practicable to do so in the present case. for two reasons: (1) I t is not possible to get a n exact measurement of the surface area of irregular particles; and ( 2 ) there is no way of determining what portion of the surface is actually in contact with the gas and is effective in transferring heat.
Calcination of Limestone
Table I1 gives the coefficient of heat transfer from a gas stream to limestone. It is worth while discussing this matter more in detail, for the material was heated to the calcining point and the heat of reaction of the decomposition of the calcium carbonate was a factor in the experiment. I t was desired t o determine whether or not the theoretical temperature-history curves were still valid in a system where a considerable heat of reaction was involved, and it was most gratifying to see that the observed data points fitted one of the theoretical curves almost perfectly in every case. The heat of calcination of limestone is in the neighborhood of 400 calories per gram, which is enough heat to raise the temperature of the undecoinposed material 1500" to 2000" C. The details of the experiments are given in Table 11. Table 11-Data MAX.
TEMP.
ya
c. A B C D
860 860 840 840
10 12 12 10
on Heat Transfer to Limestone RATEOF
FLOW
Liter pev sec. p e r sq. cm. 0 10 0 10 0 08 0 10
R
APPARENT
HEAT
CAPACITY
Cal. p e r sec. degree per cc. 0 0034 0 0040 0 0033 0 0034
Cal. per degree per gram CaO 0 0 0 0
65 53 37 33
a Values of Y obtained by comparison of plotted d a t a \$ith theoretical temperature-history c u n e s
Figure 13-Effect
voins. PER CENT of Voids o n Coefficient of Heat Transfer
Material-Corsica iron ore 0.8 cm. in diameter R a t e of gas flow-0.1 liter per second per square centimeter
It has been found (.5) that only 10 to 20 per cent of the total cross-sectional area of a bed is effective in sllon-ing the passage of a gas. Thus it would seem that the bed is filled with dead pockets or flow shadows which keep a large portion of the surface from being in direct contact with the stream. Summary of Equations
The preceding discussion may be summarized in a simple equation. ~v0.iT0.31()l.68/
The apparent heat capacity of the qolid includes not only the actual sensible heat but also the heat of reaction of calcination; thus the figure obtained depended upon the rate of calcination which became slower with each test as the amount of unchanged calcium carbonate became less. The same sample was used for all tests. The maximum temperatures attained mere only in the lower part of the zone of rapid decomposition, so the reaction was prolonged from test to test and was not eren finished in the fourth test, for the actual
k =
d0.l
- 3.66/'
-
(19)
where A = some constant characteristic of substance
This equation applies for all systems except that where coke is cooled with air. For this case
For the metric system,
I S D CS T RI A L A SD E S G I S E E RI S G CHEMISTRY
July, 1930 k
=
u =
T
=
d
=
j'
=
calories per second per degree Centigrade per cubic centimeter standard liters per second per square centimeter of crosssectional area of bed degrees Centimeter absolute particle diameter in centimeters fractional value of voids in bed (no units)
with tinie. A diagrani of such a set-up along with the type of teinperature curves to be expected are shown in Figure 16. The equations developed are applicable to any countercurrent flow system and are not limited to this particular arrangement of materials.
SOXIESCLATURE
For tlie English system, k v
= =
T
=
d = =
72'3
B. t. u. per second per degree Fahrenheit per cubic foot standard cubic feet per second per square foot of crosssectional area of bed degrees Fahrenheit absolute particle diameter in feet the fractional value of voids in bed (no units)
t: = temperature of gas a t bottom of furnace (entrance) i, = temperature of solid a t bottom of furnace (exit) tl = temperature of combustion with cold reactants T,, = temperature of gas a t any position, x, in furnace T , = temperature of solid a t any position, .Y, in furnace f3
