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INDUSTRIAL AND ENGINEERING CHEMISTRY
denses on the surfaces and causes costly rejections. The lowering of the dew point to a t least 20" below the prevailing dry-bulb temperature, by means of an activated alumina dehumidifier, completely eliminated this difficulty. I n another plant 5 per cent relative humidity a t 76" F. is being maintained for the purpose of testing electrical apparatus under specific conditions. Another activated alumina installation is used in Conjunction with the drying of celluloid products where the dry air prevents blemishes which form on the product when the moisture concentration becomes too great. These several applications are merely indicative of the variety of operations which may be improved by controlled humidity.
obtain comfort a t the lowest possible cost. (c) Those who have been concerned with adsorbent drying have been engaged in the standardization of equipment to be used in existing and less competitive lines. It is believed that adsorbent dehumidification, in conjunction with refrigeration or other methods of cooling, has a place in comfort conditioning. This place will be fixed by the prevailing costs of power, gas, cooling water, and by any future progress which may be made in adsorption equipment specially adapted to this application. Adsorbents have been known for a long time but their commercial use is new. The success which has been attained so far is the basis for predicting a promising future for the greater use of complete and partially dried gases.
Literature Cited
Comfort Air Conditioning Much less progress has been made in the comfort air conditioning of houses and public buildings than in the industrial field. The major reasons are as follows: (a) I n addition to dehumidification, some sensible heat must be eliminated. (b) Each large installation must be separately engineered to
VOL. 30. NO. 4
(1) Bower, Bur. Standards J . Research, 12, 241 (1934). (2) Dover and Marden, J . Am. Chem. SOC., 39,1609 (1917). (3) Johnson, Ibid., 34, 911 (1912). (4) Munro and Johnson, IND.ENQ.CHEM.,17,88 (1925). (5) Yoe, Chem. News, 130,340 (1926). RXICEIVED February 7, 1938.
Drying Materials in Trays Evaporation of Surface Moisture
T
HE removal of moisture from a material by passage of heated air over the surface of the wet material represents the most common form of industrial drying today. The mechanism involved in this form of drying has been studied experimentally and theoretically by a number of investigators during the past ten years, but very little advance has been made in developing the results obtained to the point of practical application. This is probably due largely to the difficulty of developing theoretical formulas which will consistently cover the entire drying period of different materials. I n general, the drying of most solids may be divided into three distinct periods: a heating period, a constant rate of evaporation period, and a falling rate of evaporation period. The heating period is usually short in comparison with the total drying time and hence is of minor practical importance. The constant rate period varies in industrial drying processes from a negligible fraction to a major portion of the drying time. This is the period most susceptible to the application of theoretical considerations, and the influence of most of the variables affecting the mechanism of drying in this period has been fairly well established (9). The primary purpose of the present paper is to contribute new experimental data to this field and a t the same time to present a rational method for the practical application of these data. The falling rate period also may vary from a small proportion of the drying time to practically the entire drying period. Owing to the fact that the rate of drying in this period is continuously decreasing, the falling rate period may constitute the greater portion of the total drying time even when a large proportion of the total moisture is removed during the constant rate period. A considerable amount of experimental work has been performed in the study of the falling rate period, but the results have been so diverse that no great progress has been made in the development of a general theory.
C. B. SHEPHERD, C. HADLOCK, AND R. C. BREWER E. I. du Pont de Nemours & Company, Wilmington, Del.
The tray drying of surface moisture from nonhygroscopic materials and the effects of various drying conditions o n the constant drying rate have been studied. Two samples of Ottawa sand (20-30 and 50-70 mesh) were employed. The drying variables studied with the ranges covered were : air temperature, 115-300 O F.; relative humidity of air, 10-60 per cent; air velocity, 150-1375 feet per minute; material depth, 0.5-2 inches; and insulation of tray, none and 1-inch cork. The evaporation of water in trays under similar air conditions was also studied for comparison. I n the runs with sand, nearly all the water was removed during the constant rate period. The results obtained chiefly concern this period. The constant drying rate was found nearly identical for
APRIL, 1938
INDUSTRIAL AND ENGINEERING CHEMISTRY
389
FIGURE 1. EXPERIMENTAL DRYER
Further experimental data are indicated t o be necessary before a complete picture of the drying mechanism in this period can be set forth.
Equipment and Procedure The present study has been confined t o the drying of nonhygroscopic materials in trays. The materials consisted of Ottawa sand of two different sizes, 20-30 mesh and 50-70 mesh. The individual particles of this sand appeared spherical. The physical characteristics of the two sizes are shown in Table I. T h e evaporation of water i n trays was also measured for comparison. The drying tests were performed in a wind-tunnel type of dryer designed to give close control of air temperature, humidity, and velocity, and a t the same time to permit these variables to be changed over a wide range. Figure 1 gives the essential features
sand and water under the same drying conditions. This rate has been expressed in terms of either the heat transfer coefficient or the mass transfer coefficient between the wetted material surface and air. The results indicate that these coefficients vary with approximately the 0.8 power of the air velocity over the range covered. However, the heat transfer coefficient is preferable from the standpoint of reliability and convenience of use. The coefficients for perfectly insulated trays, based on experimental data, may be employed for the prediction of the constant drying rate of any similar material under any given drying conditions in an uninsulated tray. If the critical moisture content of the material is low, the rate so found will enable an approximate calculation of the total drying time.
TABLE I. PHYSICAL CHARACTERISTICS OF OTTAWA SAND 20-30 Mesh Cumulative wt. % retained on: 20 mesh 28 mesh 35 mesh 48 mesh 6 5 mesh 100 mesh Specific gravity Bulk density, lb./cu. ft. Voids, % Av. particle diam., in.
0.4 99.6 100.0
...
50-70 Mesh
... ...
0.06 1.6 99.5
100.0
2.66
100 40 0.0314
2.66 92
44 0.0118
of the dryer. The tests were conducted in the horizontal section of the dryer with air assing over the trays parallel to the tray surface. The inside Slimensions of this section were 24 inches high and 18 inches wide. The horizontal length of the section upstream from the tray was 10 feet. Two alternative methods were used for determining the ~ O S Sin weight of the wet material during the drying: (a) The tray was supported from a calibrated spring which was so connected to the recording mechanism of an ordinary pressure recorder that a continuous plot of the weight was obtained; and (b) the tray was supported on a balance pan, and readings of the weight were taken a t frequent intervals. Both of these methods gave reliable results. The temperature of the heated air used for drying was measured by two copper-constantan thermocouples located about % inch upstream and downstream with respect to the tray, and about 1 inch above the level of the tray surface. The dry-bulb temperature ahead of the tray was also checked by a mercury thermometer. The wet-bulb temperature was measured at the same point by a mercury thermometer whose bulb was covered with several folds of light gauze. The gauze was wetted with distilled water a t approximately the wet-bulb temperature each time a reading was to be made. Temperatures of the material during drying were read by nine copper-constantan thermocouples located in three groups of three couples each. One of these groups was placed 1 inch from the upstream edge of the tray, one in the center of the tray, and one 1 inch from the downstream edge. In each group the three thermocouples were placed one over the other, one located inch below the top surface, one l / ~inch above the bottom surface, and one in the center of the material bed. These groups were located in a line equidistant from the sides of the tray and parallel to the direction of air flow. The procedure followed in the experimental tests was essentially as follows: A tared tray, 12 inches square (inside measurements), was filled level with weighed amounts of sand and water which had been intimately mixed; sufficient water had been added to the dry sand to fill the voids completely but not so much that any large pools of water were in evidence. The thermocouples measuring the material temperatures were next inserted in the wet sand. The tray containing the sand was then weighed and placed in the dryer. In all tests the dryer was operated at the desired conditions of temperature, humidity, and
.
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VOL. 30, NO. 4
velocity for at least 1 hour previous t o the start of a drying test. The loss in moisture was determined either continuously by means of the automatic recorder or at frequent time intervals by means of the balance until no further change in weight was observed. (In all cases the final moisture content was less than 0.05 per cent, dry basis, and usually less than 0.01 per cent.) The tray was then removed and weighed, and a representative sample of the dried m a t e r i a l was placed in an oven at 220" F. t o d e t e r m i n e the bone-dry weight.
Experimental Results
z Z Z Z Z%
x
000 000
24
?2?4$%$%262zz%z%
0.0 o c o o o o o o ~ ~ o c , ~ o
00
00o0ccoococcoo
A
Tests were run on sand to determine the effect on drying rate of the following variables: air velocity (1251370 feet per minute), temperature (114-303' F.), and relative humidity (10-60 per cent); m a t e r i a l depth ( l / ~ 2inches) and particle size (20-30 and 50-70 mesh); and tray insulation. The standard drying conditions w e r e c h o s e n as 150" F. dry bulb, 30 per cent relative humidity, and 300 feet per minute velocity, since these values represent an approximate average of the conditions commonly employed in i n d u s t r i a 1 practice. The experimental r e s u 1 t s obtained are summarized in Table IIA. The d a t a a r e a l s o s h o w n graphically in the form of plots of drying rate us. moisture content of the material to illustrate the effects of each of the above variables (Figure 2 ) . Several tests were a l s o m a d e on the rate of evaporation of water in trays in order to compare these rates with the drying rates obtained with sand. The results are summarized in Table IIB and shown graphically on Figure 3. I n order to analyze and correlate the test results reported, the individual data on each test were corrected and averaged as follows: The dry-bulb measurements of the air as recorded by the upstream thermocouple were corrected for r a d i a t i o n from the dryer walls by the method of McAdams (6). The wet-bulb temperatures as recorded by a wick thermometer were similarly corrected, although in this case it was necessary to set up a balance between radiation, convection, and evaporation. The wet-bulb correction was negligible in most of the t e s t s b e c a u s e the temperatures of the dryer walls were very close to the measured wet-bulb temperature. The dry- and wet-bulb temperatures immediately preceding the drying tray were used as a basis for all the test data rather t8hanaverage values .of the wet- and dry-bulb temperatures across the tray, since the former are the values generally known in practice.
APRIL, 1938
INDUSTRIAL AND ENGINEERING CHEMISTRY
391
PERCENT MOISTURE CONTENT -DRY BASIS
FIGURE 2. DRYING OF SAND Experimental conditions were as follows, with variations as noted for each set of curves: air temperature ljOo F relative humidity, 3 0 , p e r cent: air velocity, 300 feet/minute: katerial a e p t h , 1 inch; particle size, 20-30 mesh; insulated tray. 9
PERCENT MOISTURE CONTENT
-
D R Y BASIS
Tray Insulation I n practically all commercial types of dryers in which the drying takes place from only one surface of the material, an appreciable amount of heat is generally introduced through the remaining unlagged surfaces, and this heat serves to increase the rate of moisture removal. When drying takes place from two surfaces, especially when the distance between the two surfaces is small, the addition of heat through the unwetted surfaces is generally negligible. I n the present investigation two runs were made with sand in uninsulated trays, and the remainder in trays insulated with a 1-inch thickness of cork covered with tin foil. A comparison of drying rates in insulated and uninsulated trays is shown in Figure 2A. The increased rate of evaporation obtained in the unlagged trays is in agreement with that predicted by heat transfer calculations, as will be shown later. A similar increase in rate of evaporation was also noted for water in an uninsulated tray as compared with that in an insulated tray (tests W-3 and W-2).
Air Temperature and Humidity The effect of air temperature on the rate of drying of sand is shown in Figure 2B. I n the constant rate period the rate
A. B. C.
Effect of tray insulation Effect of air temperature Effect of relative humidity
D.
Effect of air velocity
8'.
Effect of material depth
E. Effect of particle size
of moisture removal was found to be directly proportional to the temperature difference between the drying air and the surface of the material (for the same degree of insulation). These results are in accord with theoretical predictions. The influence of relative humidity a t a fixed dry-bulb temperature is shown graphically in Figure 2C. I n the constant rate period the drying rate was directly proportional to the pressure difference between the partial pressure of water vapor in the air stream and the vapor pressure of water a t the material surface, except in tests 6 and 6-2 which were made a t a very high humidity. I n these cases accurate values of the pressure difference were difficuIt to obtain because of the large variations in pressure differences due to very small variations i n surface temperature. However, the drying rate for different relative humidities varied directly with the temperature difference between the drying air and the material surface, even in the high humidity tests. The latter also held true in the evaporation of water (tests W-2 and W-5). Actually the effects of relative humidity and dry-bulb temperature are always linked together, inasmuch as these two variables combine to determine the temperature difference or the partial pressure difference between the material surface and the drying air. The question of whether to use temperature difference or partial pressure difference as a basis for
392
INDIJSTIIIAL AliD ENGINEERING CHEMISTRY
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showing the effect of air temperature and relative humidity isdiscussed later.
the top of the tray until the level reached tlie bottom of tile tray.
Air Velocity
Heating Period
The effect of air velocity on drying rate is sliown in Figure 20. In the constant rate period bhe rate varicil as the 0.76
In all the tests reported for sand, tlie heating period was relatively short (approximately 5 per cent of the total drying time). This is generally the case when driing mnterial c o n t a i n i n g an appreciable amount of moisture. Far materials cont a i n i n g 1 to 2 per cent moist.ure or less, however, the heating period may extend over a large portion of the Ootal drying time. By g r a p h i c a l l y determining the mean t e m p e r a t u r e difference between air and the material surface during the h e a t i n g period, heat transfer coefficients were calculated and found to be approximately the same as those developed below for the c o n s t a n t rate period. The quantity of tho moisture removed during the heating period was only a small fraction of the moisture removed during the constant rate period. For p 11 r p o s e s of calculation, it may be asmmed that the material is first heated to the wetbiilb temDeraturc b e f o r e any evaporation takes place and that theevaporation then proceeds along the constant rate line.
power of the air velocity witliin the range of velooities investigated (125-1370 feet per minute). The effect of velocity on the rate of evaporation of water in trays checked the results obtained with sand (tests W-I, W-2, XT--4.and W-6). This is in accordance with the established results on heat transfer and mass transfer in the turbulent region. At lonm velocities it would be expected that natural convection would control and that the air velocity would have less effect on the drying rate.
Particle Size The effect of the particle size of the material being dried is shown graphically in Figure 2E. Very little difference was observed in the rate of evaporation during the constant rate period between the two sizes of sand and water alone. These results do not confirm those reported by Ceaglske and Hougen (1) which indicated a definite difference in drying rate between water, coarse sand, and fine sand, the rate decreasing in this order.
Depth of Bed The influenceof the depth of the material tied 011 thc drying rate is shown graphically in Fiigurc 2 F . The rate of drying in the co&ant rato period v;as essentially the same for I-, and 2-inch depths of sand, altlioiigli theoretically the increased insulating effect of the greater depth should slightly reduce the amount of heat transferred from the bottom of the tray and hence the over-all drying rate. I n the evaporation of water in trays, tlie rate remained practically constant in two test.s from the time when the wmter surface was flush with
Falling Rate Period The scope of this paper does not warrant a discussion of the falling rate period of drying in any detail. In the present tests this period was relatively short, as shown by representative drying curves of moist.ure content vs. drying time in Figure 4. Owing to the rapid change in rate, the rate data obtained in this period cannot be considered so reliable as those obtained in the constant rate period, and further data are desirable before an accurate analysis of the drying of sand in this period is attempted. However, the falling rate curves for sand all followed approximately the ssnie shape, as shown in Figure 2.
Critical Moisture Content In all the tests employing a 1-inch depth of 20-30 mesh sand in an uninsulated tray, tlie critical moisture'content (i. e., the moisture content. at which the constant rate period ends and the falling rate period commences) occurred within a range of 2 to 4 per cent water. Ceaglske and Horigen reported critical iooistures of the order of 10 per cent and Sherwood and Comings (IO) reported 5.5 per cent for sand within the same particle size range. A range of air temperature, humidity, and velocity was covesed so as to result in a variation of 700 pcr cent in tlie absolute drying rate, and yet no definite effect on the critical moisture content mas observed. Within the limit of experimental error this substantiates Ceaglske and IIougen's rcsults, which showed a variation in absolute drying
INDUSTRIAL AND ENGINEERING CHEMISTRY
APRIL, 1938
rate of 50 per cent, and indicates that the movement of moisture from the interior to the surface of the sand bed may well be due to capillary forces. The particle size of sand over the range studied in this investigation apparently does not have a considerable effect on the critical moisture content, but the value appears to increase slightly for the finer sand'(Figure 2E). Ceaglske and
393
rw = latent heat of vaporization of water a t surface
temperature
d W / d s = instantaneous rate of evaporation of water
The equilibrium temperature (tu) reached under these conditions is the wet-bulb temperature. The value of h/kr, depends on the temperature and humidity of the air and, according to psychrometric charts, it should vary from 0.29 to 0.24 for the conditions employed with water in air in the present investigation (when the temperature is expressed in degrees Fahrenheit and the partial pressure in millimeters of mercury). If heat is supplied to the wet surface by radiation as well as convection from the surroundings, h in Equation 1is replaced h,.), and a slightly higher surface temperature is by (h, attained with a resultant increase in the rate of evaporation. If heat is supplied to the material by convection and conduction through the unwetted surfaces of the tray, the wetted surface temperature and the rate of evaporation are correspondingly higher. This effect was noted by Sherwood in his work on the tray drying of clays (8). I n order to correlate the present test data, the dry- and wetbulb temperatures of the air were corrected as previously described and, together with the measured surface temperature, were used as a basis for computing the partial pressure difference of water in the air and a t the drying surface and also the temperature difference between the air and drying surface. Mass transfer and over-all heat transfer coefficients were then computed b y the application of Equation 1. These coefficients are given in Table I1 as pounds of water/(hour) (square foot) (mm. mercury) and B. t. u./ (hour) (square foot) ( O F.), respectively.
+
EVAPORATION
TIME
-
HOURS
OF WATERFROM FREESURFACE FIGURE3. E~APORATIOX
(Standard experimental conditions with exceptions a s noted)
Hougen's data also indicated a negligible increase in critical moisture with increase in fineness of the sand. Over the same range of particle size, the data of Sherwood and Comings check this conclusion. However, with particle sizes less than 200 mesh these authors observed a substantial increase in critical moisture c o n t e n t n a m e l y , 10 per cent for the 200325 mesh sand and 21 per cent for sand finer than 325 mesh. (Subsequent tests by the present authors on three sizes of glass beads-3-, 6-, and 12.7-mm. diameter-showed a decided increase in critical moisture content with increase in particle size of the beads; the values obtained were 14, 15, and 21 per cent, respectively.) The depth of sand in the tray had a definite increasing effect on the critical moisture content, as shown in Figure 2F. Ceaglske and Hougen also observed this increase with depth and explained it satisfactorily on the basis of the capillary movement of moisture within the material. A method for predicting the critical moisture content of granular materials was advanced by Ceaglske and Hougen, but no attempt was made to predict the critical moisture content of the sands in the present investigation by this method.
"26
g I2 5 6 2 4
E 2 '0
I
2
3 4
5
6
7 8 9 10 II DRYING TIME
12 13 14 15 16 17 18 19
- HOURS
x)
21 22
FIGURE 4. MOISTURE CONTENT vs. DRYING TIMEAT VARIOUS AIR VELOCITIES (Air temperature 150' F . ; relative humidity, 30 per cent. material depth, 1 ihch; particle size, 20-30 mesh; insulated trky)
Correlation of Results I n a drying process where heated air or gas supplies all the heat for the evaporation of water directly to the wetted surface, a dynamic equilibrium is set up between the rate of heat input to the material and the rate of moisture removal from the wet surface. This equilibrium may for practical purposes be expressed as follows : r,dW/d@
- t,) = kAr,(p, - pa) - Pa - h t, - t, k rw
hA(t,
(1)
Pw
or where
=
heat transfer and mass transfer Coefficients, respectively A = area of wetted surface t,, t , = temperatures of air and surface (in this case equivalent to wet-bulb), respectively pa, p , = partial pressures of water in'air and at wetted surface, respectively h, k
=
The mass transfer coefficients represent the mass transfer a t the wetted surface and need not be corrected for heat input through unwetted surfaces, since this had already been taken into account by the use of the actual surface temperatures. The heat transfer coefficient, however, is an over-all coefficient which includes the heat transferred through all surfaces by both convection and radiation. In order to obtain a heat transfer coefficient representing heat transfer to the wetted surface due to convection only, the over-all coefficients were corrected for heat transfer by radiation and conduction through the unwetted surfaces and radiation to the wetted surface. The resulting coefficients, h,, are given in Table I1 and represent the coefficients which would be obtained if the trays were perfectly insulated and the radiation to the wetted surface was negligible. A brief discussion of the heat transfer considerations involved in the drying of material in trays is desirable a t this
INDUSTRIAL AND ENGINEERING CHEMISTRY
394
point, since in many plant installations the heat input t o the unwetted surfaces of the trays is often as large as that to the wetted surfaces, and this directly affects the rate of drying from the wetted surfaces. This is shown by the high drying rates obtained in tests 1 and 11 using unlagged trays. Such a discussion will serve to indicate the method used for correcting the over-all heat transfer coefficients obtained in the present detailed investigation and a t the same time to point out a logical approach to the solution of practical problems. When material is dried in a tray in a stream of air, heat may be transferred to the wetted surface in three ways: (a) by direct convection from the air stream, ( b ) by radiation from the dryer walls or the surrounding trays, and ( c ) by heat transfer through the material bed (which in turn is comprised of convection and radiation to the unwetted tray surfaces and conduction through the trays and the material being dried). This may be expressed as follows:
+ +
where ht h, h, h,
hi = ho hr ha (2) =e over-all heat transfer coefficient for total heat to wetted surface = heat transfer coefficient for convection to wetted surface = heat transfer coefficient for radiation to wetted surface [equals actual h', as predicted by radiation formulas corrected to same basis as h, (multiplied by ta - t a / t w - L)l = over-all heat transfer coefficient for heat to wetted surface through unwetted surfaces
VOL. 30, NO. 4
the thermal conductivity of the material appeared to be of the order of 2 B. t. u./(hour) (square foot) ( O F./foot), whereas that of dry sand is only 0.2 and that of water about 0.4. This surprisingly high value appears to be real, since it is well outside the range of any experimental error that might occur in the temperature measurements. One possible explanation is that an evaporation-condensation phenomenon occurs within the bed of the material which materially assists in the transfer of heat from a lower to a higher stratum. This matter warrants further investigation. With the value of ICl known or estimated, the only unknown factor in the above expressions becomes hl. Both Equations 3 and 4 may then be expressed as follows : (5)
where C is a constant depending on the thermal conductivities of both material and tray; then in the general case,
As an approximation, it may be assumed that the over-all heat transfer coefficient t o the wetted surface from the sides of the trays is the same as that from the bottom of the trays. (The average convection coefficients should be about the same, and, although the heat entering the sides has only half as far to go on an average, it may be considered as only half as effective owing to interference with heat flow from the bottom.) The over-all heat transfer through the unwetted surfaces may then be broken down as follows: ha! =
1
LI
La
(3)
hX+I;,a,+k,s, where h,
=
kl = k* =
L1
=
Lp
=
Az
=
A, = Ai =
heat transfer coefficient by convection and radiation to outer unwetted surfaces thermal conductivity of material being dried thermal conductivity of tray and insulation (if any) thickness of material bed, feet thickness of tray and insulation, feet ratio of outside unwetted surface t o wetted surface ratio of unwetted inner tray surface to wetted surface ratio of average tray surface to wetted surface ( A a = !2L&h)
For an unlagged tray such as is generally used in plant practice, A , = A1 = AB. Moreover, the resistance to conduction of heat through the tray itself when made of metal is negligible, and the above expression simplifies to : (4)
/
u I n the above expressions for h, the thicknesses (L1 and Lz) and the area ratios (Au,A I ,and A*) are known for any given case. The thermal conductivity of the tray itself and of any insulation can be readily obtained from available data. The thermal conductivity of the wet material is less readily determined. On the basis of the temperatures recorded by the thermocouples located in the wet sand in the present tests,
I B
I 10
500
1000
ZOO0
3000
MASS VELOCITY (G)-LBs/(SQ.
5000
FT.)(HR3
FIGURE 5. EVAPORATION OF WATER A. B.
Effect of air velocity on mass transfer Coefficients Effect of air velocity on heat transfer coeffioients
Substituting for h, in Equation 2, the following equation is obtained:
The coefficient h, includes both convection and radiation to the unwetted tray surfaces. It is reasonable to assume that the convection coefficient will be the same to both wetted and unwetted surfaces. Moreover, in most industrial dryers the net radiation to and from a tray of material surrounded by other trays is practically zero. I n this case, the above equation simplifies to:
This equation shows that, in order to obtain the total heat transfer coefficient, it is necessary to multiply the coefficient for c o n v e c ti o n to the wetted surface by a factor of (1 A,
.+ m).
Courtesy, Proctor & Schwartr, Inc.
A COMMERCIAL TRAYDRYER
Values of A , and C are known, as previously pointed out. Similarly, this same factor may be used to determine the increase in rate to be obtained with an uninsulated tray as compared with a perfectly insulated tray. If radiation to the tray is an appreciable factor, an approximation of this effect may be obtained by assuming the same value of h, for both wetted and unwetted surfaces; in this case Equation 7 becomes :
determine the value of h, by means of Equation 8. Sufficient experimental data were available from each run to compute all the values except h,; hence h, could be obtained directly.
Correlation of Present and Previous Experimental Data I n Table 11,the mass transfer coefficients and the corrected heat transfer coefficients are reported for each of the experimental runs of this investigation. These apply only to the constant rate period of drying. (Each of these coefficients as reported is the average of the coefficients calculated from temperature data taken periodically during the runs, Hence these average values should not necessarily agree with those calculated from the average temperature and pressure data, also reported in Table 11.) A close examination of the data shows that the mass transfer coefficients are considerably more erratic than the heat transfer coefficients. I n fact, in two cases (tests 6 and 7) some of the individual data showed negative mass transfer coefficients. This wider variation in the mass transfer coefficients is due both to the inaccuracy in the determination of the partial pressures of water in air by means of psychrometric charts and also to the large errors in the calculations of pressure differences resulting from small errors in surface temperature measurements. The error involved in computing temperature differences and from them, heat transfer coefficients, is considerably smaller. Hence in all the tests the heat transfer coefficients were reasonably uniform, both for individual readings during the same test and for two check tests under the same conditions. The mass transfer coefficients computed from the present data are plotted in Figure 5A against the mass velocity of the drying air, together with coefficients calculated from other available data in the literature on the evaporation of water or on the constant rate period of drying solids. The best line through the present data is a straight line on log-log paper with a slope of 0.75. I n other words, the mass transfer coefficient varies with the 0.75power of the mass velocity between the values of 500 and 6000 pounds/(hour) (square foot), equivalent to 125-1370 feet/minute a t 150' F. and normal barometer.
As an example of the use of Equation 8, consider the corrections t o be applied to ht for run 1 .in order to obtain h,: ht = 6.77B. t.u./(hr.)(sq.ft.)('F.) h, = 0 (since temperatures of dryer walls and of material
surface are practically the same) A , = 1.33 A1 = 1.33 L1 = 1 in. = 0.0834 ft. kt = 2 B. t. u./(hr.) (sq. ft.) ( O F./ft.)
Substituting these values in Equation 8:
('
+
6'77 = hc +1 0.042 1.33 X h, h, = 3.12 B. t. u./(hr.)(sq. ft.)(OF.)
)
Actually, h, may be slightly different for the unwetted surfaces and for the wetted surfaces, since the temperature a t the former surfaces will be the higher, depending on the thermal conductivity of the material, tray, and lagging (if any). For unlagged metal trays, the difference in radiation coefficient will not be sufficient to affect the value of ht appreciably, and Equation 8 should be satisfactory. A knowledge of the mechanism of heat transfer to a material drying in an air stream is very important in the application of theory to industrial practice or in the application of small-scale drying tests to a large-scale dryer. Equations 7 and 8 show that a prediction of the over-all drying rate of the material in trays, during the constant rate period, may be made provided the coefficient of heat transfer by convection to the wetted surface is known. In the present tests on sand and water, the over-all values of ht obtarined from the experimental data were corrected to 395
IKDUSTRIAL AND ENGINEERING CHEMISTRY
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The data of the other investigators indicate lower coefficients than the present data, although the variation with velocity is closely similar. The results of Powell and Griffiths (6) and Hinchley and Himus (8) are low in both cases. I n these investigations the evaporation of water was being studied, the temperature of the water being higher than that of the air. This fact may explain the low values of the mass transfer coefficients obtained, since the liquid surface film resistance may cause the surface temperature, and hence the vapor pressure difference, to be lower than that actually measured and used in the caIculation. I n the tests of Lurie and Michailoff (4) and of Thiesenhusen ( I l ) , the evaporation of water was carried out adiabatically as in drying; the water assumed a temperature close to that of the wet bulb. The water evaporation data reported by Rohwer (7) were considered to be too widely divergent among themselves to warrant inclusion in the plot. The data of Kamei, Mizuno, and Shiomi (3) were obtained during the constant rate period of the drying of clay, and those of Ceaglske and Hougen ( I ) during the constant rate period of the drying of sand. All of these data agree reasonably well with the present data. When the same data, obtained either under conditions of adiabatic drying or evaporation of water, are plotted as heat transfer coefficients (Figure 5B), the agreement with each other and with the present data is much more pronounced. This fact demonstrates one of the reasons for preferring the use of heat transfer coefficients to that of mass transfer coefficients in interpreting the constant drying rate of water from wet material. As with mass transfer coefficients, the heat transfer coefficients based on the present data vary with the 0.76 power of the mass velocity.
Practical Application of Results No reliable method has yet been advanced for the prediction of commercial drying times and dryer performance which Ceaglske and Hougen made a notable start in their recent work on granular materials, but their method of attack requires verification for materials other than sand, and it will probably not prove applicable to a wide range of materials. Sherwood offered the following relation between moisture content and drying time as a possible approach to the estimation of drying times :
is generally applicable to a variety of materials.
where
.
ec,
total drying time, drying time in constant rate period, and drying time in falling rate period, respectively To,Tc,T E = initial, critical, and equilibrium moisture content, respectively T = moisture content at time e, dry basis a = aconstant OF
=
The application of this equation requires a knowledge of the critical moisture content and the constant a, neither of which is usually obtainable except by means of actual drying tests. Moreover, the logarithmic relation applying to the falling rate period holds only for certain types of falling rate curves and only approximately in these cases. Another disadvantage of Equation 9 is that it does not readily show the relative influence of the variables affecting the over-all drying time. I n the present paper it is proposed primarily to develop the available knowledge on the constant rate period of drying to the point of practical application to dryer design and performance. This is considered as merely the first step in the problem of developing a rational basis for the estimation of commercial drying times and optimum drying conditions for a given material. It appears evident, however, that a considerable amount of fundamental information must still be
VOL. 30, NO. 4
developed on the falling rate period of drying and the critical moisture content before any successful and practical solution can be obtained for the complete drying mechanism. The data reported above on the constant rate period have served to establish absolute values for the rate of moisture removal from sand during this period (expressed in the form of either mass transfer or heat transfer coefficients). I n addition, the similarity between the rates of evaporation from sand and from water surfaces lends considerable weight to the assumption that the evaporation rate will not vary greatly in the constant rate period for most materials. Sherwood and Comings (10) also reported that the rates of evaporation of water alone and from various materials are closely similar. The results reported from Kamei’s work on clay (Figure 5B) confirm this assumption. TABLE 111. COMPARISON OF HEATTRANSFER AND MASSTRANSFER COEFFICIENTS, CALCELATED ON VARIOES BASES Run No.
1 2 3 3-2 4 4-2 5 5-2 6
6-2 7 7-2 8 9 10 11 12 13 14 14-2
Heat Transfer Coefficients Based on: Readine: from (to - t r ) (to - t w ) Fig. 5B F. F. 3.12 2.58 3.4 3.4 3.56 3.31 3.35 3.22 3.21 3.4 3.42 3.44 3.4 3.47 3.77 3.4 3.39 3.46 1.7 1.53 1.46 2.0 2.21 2.13 3.3 3.73 3.87 3.2 3.80 4.16 7.74 6.38 6.0 6.21 6.49 6.0 2.65 2.88 3.2 3.79 3.55 3.8 3.40 3.27 3.35 2.33 3.4 2.99 3.46 3.55 3.4 3.65 3.63 3.4 11.33 10.94 11.0 10.68 10.59 10.5
Mass Transfer Coefficients Based on: Readine from( p a - PO) (PW - p4) Fig. 5A Mm. Hg M m . Hg 0.0131 0.0207 0.0125 0.0135 0.0109 0.0125 0.0122 0.0130 0,0120 0,0144 0.0140 0,0125 0,0180 0.0347 0,0125 0.0143 0.0159 0.0125 0.0057 0,0070 0.0064 0.0074 0,0089 0.0072 0.0192 0,0840 0,0120 0.0256 0,0120 .... 0.0302 0.0225 0.0241 0.0225 o’oib 0.0130 0.0120 0.0089 0.0157 0.0134 0.0140 0,0109 0.0122 0.0120 0,0208 0.0120 0.0125 0.0118 0.0130 0.0125 0,0147 0.0153 0.0125 0.0490 0.0615 0.037 0.0396 0.037 0.0376
The computation of drying problems on the basis of a heat transfer mechanism rather than a mass transfer mechanism is definitely to be preferred in the constant rate period. Heat transfer coefficients calculated from the present experimental data have been shown to be much more consistent than mass transfer coefficients determined from the same data. Consequently, the appIication of heat transfer coefficients in determining drying rates would be more reliable than the use of mass transfer coefficients. Furthermore, unless the temperature of the drying surface is measured, it must be calculated by means of heat transfer considerations before it is possible to apply mass transfer coefficients for drying rate predictions. I n practice, surface temperature measurements are rarely made, and i t is usually assumed that the surface is essentially a t the wet-bulb temperature of the air during the constant rate period. I n Table I11 both heat transfer and mass transfer coefficients have been calculated for the present tests based on (a) actual material surface temperatures (as in Table II), (b) measured wet-bulb temperature of air corrected for radiation, and (c) average curves of the coefficients shown in Figure 5 . As shown in Table 111, the assumption that the surface is a t the wet-bulb temperature of the air introduces a more serious error in the computation of mass transfer than of heat transfer coefficients. For an unlagged tray, however, a significant error is introduced in both coefficients. I n the application of the present results to practical drying problems, it is recommended that heat transfer coefficients be used as the basis for calculations of drying rate. Figure 6 permits a ready estimate to be made of the relative rates of drying for various air temperatures and relative humidities.
APRIL, 1938
INDUSTRIAL AND ENGINEERING CHEMISTRY
The chart is based on the difference between the dry-bulb and wet-bulb temperatures of the entering stream of air. The absolute drying rates shown are based on a heat transfer coefficient of 3.1 B. t. u./(hour) (square foot) (” F.) for air a t 300 feet/minute velocity, 150” F., and 30 per cent relative humidity. They also take into account the change in air density with temperature. This value was obtained from Figure 5B and was chosen slightly (i. e., 10 per cent) below that given by the smooth curve representing the present data in order to give some weight to the other data studied. Until further data on other materials are available, this chart may be assumed approximately correct for any material in the constant rate period of drying.
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When the drying takes place in unlagged solid bottom trays, the surface temperature of the material may be calculated reasonably accurately by the application of the same heat transfer methods and the chart readings corrected by the factor where t, = dry-bulb temperature, t, = surface t o - t, temperature, and t, = wet-bulb temperature. The fact that the proposed method for computing drying rates is applicable only to the constant rate period of drying definitely limits its scope. There are a number of possible uses for the method, however, which were not investigated in the present study. The over-all drying time of some materials exhibiting a distinct falling rate period has been found to be influenced by velocity, humidity, and temperature in approximately the same degree as is the case for the constant rate period. For these materials a knowledge of the drying time under any given set of conditions can be used to predict the drying time under other conditions by analogy to the constant rate period. Again, for processes where air is blown up or down through a bed of a granular material, the drying is often nearly 100 per cent within the constant rate period. I n this case the equivalent area can be determined by a single test, or possibly by particle size distribution data, and the drying can then be predicted from constant rate data. The same possibility applies to the drying of materials in a rotary dryer. Finally, the above correlation of data for the constant rate drying period makes it possible to use an over-all drying expression of the type proposed by Sherwood without the necessity of a number of drying tests, provided Tc is known. For instance, the constant in Sherwood’s Equation 9 could be determined from constant rate data as follows:
(-)
I n this expression only Tc is of uncertain value for any given material, since the equilibrium moisture content ( T E ) , the wetted surface area ( A w ) the , dry weight, and the latent heat (T,) are likely to be known, and the temperature difference (t, - tJ and over-all heat transfer coefficient (h,) are calculable from Figure 6 and Equation 8. FIGURE 6. RATEOF EVAPORATION CHART
Acknowledgment
A correction curve t o correct for the air velocity in any given problem is incorporated in Figure 6. This curve is based on the variation of drying rate with the 0.8 power of the velocity. There is some question as t o the shape of this curve below a velocity of about 150 feet/minute, where natural convection begins to be significant, but the dotted curve as drawn should be conservative. The use of this chart in practical drying problems is as follows: Since the drying conditions of temperature and relative humidity are fixed, the corresponding absolute drying rate is read from Figure 6. This value is then multiplied by the correction factor corresponding to the air velocity employed (also given on Figure 6). The rate so obtained, however, does not include any effects of radiation or of conduction through unwetted surfaces. I n the case of a material drying from two sides, the latter effect is generally negligible, and the radiation can be readily computed. The rate determined by Figure 6 is then corrected by the factor (h, h,)/h,. If drying takes place from only one surface, as in most cases of industrial tray drying, a reasonably approximate correction can be made for heat transfer through the unwetted surfaces, as indicated earlier in the paper in the discussion of heat transfer to material drying in trays.
+
The authors wish t o acknowledge the cooperation of A, P. Colburn in analyzing and calculating the heat transfer through the Unwetted surfaces of the trays. They also are indebted to W. H. McAdams, T. H. Chilton, and T. K. Sherwood for their constructive reviews of the manuscript.
Literature Cited Ceaglske, N. H., and Hougen, 0. A., IND.ENQ.CHEM., 29,80513 (1937): Trans. Am. Inst. Chem. Engrs., 33,283-312 (1937). Hinchley, J. W., and Himus, G. W., Trans. Inst. Chem. Engrs. (London), 2,57-64 (1924). Kamei, S., Mizuno, S., and Shiomi, S., J . SOC.Chem. Ind. Japan, 38,460-3B (1935). Lurie, M., and Miohailoff, N., IND. ENG.CHEM.,28, 345-9 (1936). McAdams, W. H.,“Heat Transmission,” 1st ed., pp. 222-4, New York, McGraw-Hill Book Co., 1933. Powell, R. W., and Griffiths, E., Trans. Inst. Chem. Engrs. (London), 13,175-92 (1935). Rohwer, C., U . S. Dept. Agr., Tech. Bull. 271 (1931). Sherwood, T. K., IND ENQ.CHEM., 21,976-80 (1929). Sherwood, T. K.,Trans. Am. Inst. Chem. Engrs., 32, 150-68 (1936). Sherwood, T.K.,and Comings, E. W., Ibid., 27,118-33 (1932). Thiesenhusen, H., Gesundh.-Ing., 53, 113-19 (1930). RECEIVED February 7, 1938.
Additional papers presented a t this symposium will appear in subsequent issues.
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