Fundamentals of Drying E. R. GILLILAND Massachusetts Institute of Technology, Cambridge, Mass.
The basic equations for the rate of heat and mass transfer are reviewed. The mass transfer coefficients are considered in the light of the film and turbulent-core concepts as well as on the basis of empirical dimensional equations. When data on the rate of evaporation of liquids into a turbulent air stream in a wetted-wall tower are correlated on the basis of the film and turbulent-core concept, they indicate that the resistance of the core and film are of the same order of magnitude, the ratio of the two depending on the diffusivity of the system in question. Data on the simultaneous transfer of heat
I
N DRYING and air conditioning, as with most of the unit operations of chemical engineering, the fundamentals are concerned with the basic principles of equilibrium and reaction rate. The reaction rate furnishes a measure of the time required to approach within a given distance of equilibrium. The operations of drying and air conditioning involve mainly the transportation of energy (heat) and fluids. The size of the required equipment depends largely on the rate a t which the required heat and fluid can be added or removed. I n the case of heat transfer alone, a large amount of experimental work has been carried out, and the data have been correlated empirically. While these empirical correlations serve as a basis for the prediction of transfer coefficients, basically they leave much to be desired as to the mechanism of the transfer. In the case of mass transfer the experimental data are meager, and ,for most cases empirical correlations are not available. I n addition, the problem in itself has the inherent difficulty that diffusion implies a mixture of two unlike species of molecules, while in heat transfer the experiments were often on pure substances. The physical properties necessary for a study of the transfer are fairly well known or easily obtained for pure substances, whereas the corresponding properties for systems containing two or more components are generally not known and a great deal of effort is often necessary to obtain them. Thus the viscosity of pure gases is, in general, predictable from equations based on kinetic theory with an accuracy sufficient for most uses. However, for mixtures of only two gaseous components the viscosity varies widely from the predictions of kinetic theory, often going through a maximum or minimum as a function of composition. I n spite of these difficulties, considerable progress has been made in special important cases. The present concepts of the transfer of heat and mass are based largely on two fundamentally different kinds of motion. The first is molecular motion due to the kinetic energy of the molecules, and the second is transfer due to bulk motion of the fluid. The following discussion will be largely concerned with mass transfer, although an analogous picture applies to the transfer of heat.
However, the proportionality factor often becomes so complicated, unless it is broken down into its fundamental factors, that the difficulties of correlation are very great. In analyzing the fundamental factors of the proportionality constant, the film and turbulent core concept have proved invaluable. In the film where turbulence is a t a minimum, the transfer is mainly by molecular diffusion; in the main body, or turbulent core, the transfer is largely by convection as a result of the rapid eddying. Thus a substance being transferred must pass through the two consecutively, being carried through the main body by turbulence, and then by true diffusion through the film from the interface. The principles of the transfer through the film by molecular diffusion were given by Maxwell in his kinetic theory of gases and have been adapted to various conditions by later investigators, particularly Stefan. For the transfer of a vapor or gas through a stagnant film of another gas, Maxwell’s derivation indicates that the resistance to diffusion will be proportional to the relative velocity of the two types of molecules, to the diffusion distance, and to the concentration of the nondiffusing molecules which block the path. The equations specifying these conditions were integrated by Stefan to give
Mass Transfer In drying and air conditioning one of the most important problems is often the transfer of one fluid from a solid or liquid interface into another fluid; in general, the latter is in relative motion to the first. The equilibriuln Conditions for such a condition are simple and obvious-namely, that the concentration of the first fluid is the same a t all points in the second fluid. For the rate of reactions under such conditions it is possible to write the simple equation
where d N a / d A = molal rate of diffusion per unit area D , = molal diffusivity Bj = film thickness (length of diffusion path) p B = pressure of nondiffusing gas i = indicates the interface, and prime (’) the outside of the diffusion path
where the rate of mass transfer per unit area is equated to a proportionality factor times the concentration gradient.
where k
This equation may be rearranged to give
’ -- B/(P5)>rm
506
D,
and Air Conditioning’ solved so satisfactorily as the problem of transfer by molecular diffusion. This is largely due to the lack of adequate knowledge of the fundamentals of convection. The mechanism in this case is pictured as the transfer of one fluid by another fluid due to the bulk motion of the latter fluid. Thus in this turbulent core the carrier gas is moved around and mixed by the eddies existing in it, and any of the other fluid present will be moved and mixed by the same motion. Experimental data for the evaluation of the rate of transfer by eddy diffusion are meager, but consideration of the mechanism would indicate that for given flow conditions the current density of transfer should be independent of the physical properties of the component being exchanged, as long as the latter does not appreciably alter the flow conditions or turbulence. Thus a turbulent body of fluid would have the same resistances for all substances as long as its own properties were not altered significantly. J I n engineering calculations the net transfer coefficient is desired rather than the individual coefficient for the film and main body. The mass transfer from the interface to the main body of the gas can be visualized as meeting two resistances in series, that of the turbulent main body of gas and that of the gas film. The gross gas-phase resistance, r, is then the sum of the two individual resistances, r, r,. Algebraically expressed,
and mass are considered on the basis of the same concept ,and lead to an analogy between heat and mass transfer, which does not involve the inclusion of the analogy to fluid friction. I n the case of drying, the problem is complicated by the lack of knowledge of the mechanism of fluid motion through solids. The use of the diffusion equation in this last case is questionable, especially where the driving force causing diffusion is based on the weight of liquid per unit weight of solid, because the movement of the fluid may be in the opposite direction to that indicated by the driving force in these units. 4
+
I n other words, for the case of molecular diffusion through a stagnant gas film, the proportionality constant, kf, is proportional to the diffusivity and inversely proportional to the film thickness and the logarithmic mean of the pressures of the nondiffusing gas. At atmospheric pressure or under conditions where the perfect gas laws apply within a few per cent, the above equation has been shown to represent the experimental data of a large number of investigators. In agreement with Maxwell’s theory, the diffusivity, D,, for a given binary mixture is found to be independent of relative proportions of the two gases and the total pressure, and to be directly proportional to the square root of the absolute temperature. For different mixtures, D , is proportional to the square root of the sum of the reciprocals of the molecular weights of the two gases, and is also a function of the effective volume of the molecules. While the above equation was developed for diffusion through stagnant films, a little consideration shows that it should apply to a film flowing in laminar or streamline motion under conditions such that the transfer is across the stream-i. e., under conditions where the amount of diffusing substance carried by the film in the direction of flow is negligible. Such a condition is often approximated in actual operations, and the equation is then applicable to the film. I n the case of liquids the equations for molecular diffusion are, in general, much less satisfactory than in the case of gases. This is largely due to the fact that the basic kinetics of the liquid are unknown. Experimentally the problem for liquids is complicated by the effects of ionization, association, and dissociation, and to the difficulty of eliminating natural convection. The problem of mass transfer through the turbulent core of the fluid phase by convection or eddy diffusion has not been
where the resistance of gas film, r/ = 1is
L
’*,D7n
giving
Values of kQ can be measured for two substances with different diffusivities when these substances are allowed to diffuse separately or simultaneously through a third component, under conditions such that the diffusing substances do not alter the flow conditions of the third component. These two values of ko, together with known values of D, and the fact that the flow conditions were such that r, and Bf were the same for both substances, allow the calculation of both re and B,. This equation has been applied to the evaporation of pure liquids into an air stream in a wetted-wall tower, and for these conditions the data may be expressed as
where 903/Re0.8represents the resistance of the turbulent core and 135/DRe0.8 represents the resistance of the gas film. The agreement of the experimental data with this equation is shown in Figure 1. The fraction of the total resistance to mass transfer due to the turbulent core is ’03 135/D 903 (135/D) and the fraction due to the film is 903 + (135,D). Thus, for water diffusing through air, where D = 0.267 sq. cm. per second, the resistance of the film is only 36 per cent of the total resistance. This fraction of the resistance would be expected to vary widely; it is only 18per cent for hydrogen through air and as great as 70 per cent for n-octane through air. These figures indicate that a change in flow conditions
+
1 This paper was presented as part of the Symposium on Drying and Air Conditioning a t the Fourth Chemical Engineering Symposium held under the auspioes of the Division of Industrial and Englneering Chemistry of the American Chemical Society at the University of Pennsylvania, April, 1938. Two papers of the symposium were published in April, 1938; others will be printed in subsequent issues.
507
INDUSTRIAL AND ENGINEERING CHEMlSTRY
508 IO 8
ferences of the two methods of plotting. Although the method of correlation used in Figure 2 is convenient, it is incompatible with films and core concept; in the film the rate is proportional to the first power of the diffusivity, while in the turbulent core the transfer should be independent of D,, and the actual effect of diffusivity should be a sum of such terms and not a power function. However, Murphree (9) gave a picture of flow involving no film, with which this type of function could be compatible.
6
i
s
4
a‘ I:
A
2 rn
VOL. 30, NO. 5
2
\
rn J
g
1.0
y 2
04
Simultaneous Heat and Mass Transfer
W
0.6 0.4
0.2
0*I 0.I
0.2
0.4
0.6 0.8 1.0 F I G U R1~
2
4
in a type of apparatus which decreased the thickness of the gas film but did not affect flow conditions of the main body would have only a small effect on the transfer of hydrogen through air but would make a large difference in the transfer of n-octane through air. I n general, for cases of gas film controlling, the relative suitability of different equipment would be expected to vary with the system under consideration, depending on whether the main resistance was in the film or in the main body. Prandtl’s study of the relation between heat transfer and friction would indicate that r, decreased with increasing velocity faster than r,, and that a t very high velocities the main resistance to transfer would be eddy diffusion. The experimental data on mass transfer are not sufficiently precise to evaluate the effect of velocity on the relative value of the two resistances, but they indicate that any such effect is relatively minor in comparison with the large change in magnitude of both re and r,. However, special experiments indicate agreement with Prandtl’s conclusions. Mass transfer coefficients have also been correlated by empirical methods similar to those used for heat transfer. Thus k,d ( p ~Em) / D , is found to be a function of Reynolds number and of a group p/M,D,, which is analogous to the Prandtl group in heat transfer. Another method of correlating mass transfer data has been to visualize a fictive film of thickness B such that its resistance to molecular diffusion would equal the observed transfer resistance:
6
8
IO
A problem often met in drying and air conditioning operations is the simultaneous transfer of heat and mass. In a number of cases this involves the evaporation of a liquid from a fluid or solid through a gas film and the simultaneous transfer of the heat for the evaporation through the same gas film. Consider the adiabatic interaction of a gas with water, enough water being added a t temperature T to replace that evaporated. The gas is cooled from tl to t2, and the humidity increases from HI to Hz. An over-all heat balance given is (Hz - H ~ ) ( t z- T TZ) = 81(tl - tz)
+
If the surface is sufficiently large so that the gas is cooled to saturation a t the definite saturation temperature, t,, the heat balance becomes (HI
- H1)(tI -
T
+ r,)
=
- t3
When T i s equal to saturation temperature t,, the equation is
When the relation of H vs. t, given by the above equation, is plotted on the humidity chart, adiabatic saturation curves
The values of B so calculated are then correlated by dimensionless functions (4) as shown in Figure 2. Thus the same data as are given in Figure 1may be correlated by
B=
0.023
(
&,)0’44
The experimental results correlate with about the same percentage variation in Figures 1 and 2. This is due to the fact that the variations in D are not large enough to test the dif-
(1000)~~~~
FIQURE 2
MAY, 1938
INDUSTRIAL AND ENGINEERING CHEMISTRY
are obtained. The curves are concave upward and end a t H I and t, (Figure 3). I n actual operation the air leaves unsaturated, but the adiabatic curves may be used as “path curves” to determine the final conditions. vrovided the make-up water enters a t the adiabatic saturation temperature. The w e t - b u l b t h e r mometer is one of the important cases of the simulH taneous transfer of mass and heat. C o n s i d e r a thermometer with a wetted surface area of A ta t temperature T , giving FIGURE 3 a vapor pressure of water P a t the interface, and let t and p be the corresponding values in the surrounding gas. The rate of evaporation is then:
I
dW/A& = k(P
-
p)
Heat will flow from the air to the wetted surface in accordance with Newton’s law: p = hA(t
-
2’)
where h = coefficient of heat transfer through gas film on surface of wetted bulb p = instantaneous rate of heat transfer When a condition of dynamic equilibrium is reached, the heat transferred from the surroundings will exactly supply the heat of vaporization, T,, of the water evaporating at temperature t,, and the corresponding heat balance is p = r,W
= kAr,(P,
- p)
= hA(t
- t,)
This equation is most often used to determine the unknown partial pressure of water in the air. Since t and t, may be observed, and since the vapor pressure, P,, can be taken a t the wet-bulb temperature, t,, a knowledge of the term h/kr, enables one to determine the amount of water in the air. The evaluation of h/kr, requires knowledge of both h and k. The heat transferred to the wet bulb flows both by radiation and by conduction and convection. Since for given temperature conditions the radiation from the surrounding is constant, independent of the air velocity, i t can be made negligible in comparison to the heat transfer by conduction and convection by increasing the air velocity. Coefficient h in the above equation is the sum of h, for radiation and he for conduction and convection. Designating h/h,, or (h, h,)/h, by the symbol a, the equation becomes,
509
( H , - H)/(M,/M,) is approximately equal to ( P w- P)/p,, and the wet-bulb equation can be written:
PBMB where k’ = ~ MA
If ah/k’ is equal to s, then Equation 2 becomes identical in form with Equation 1, and the adiabatic saturation temperature and the wet-bulb temperature are identical. I n the case of water and air the value of h,/k’s is approximately 1, and there is little difference between the wet-bulb temperature and the adiabatic saturation temperature; for other mixtures, such as air and toluene vapors, where h,/k’s is approximately 1.8, the wet-bulb temperature of unsaturated air is much higher than the adiabatic saturation temperature, even when the radiation correction has been made. The slope of the chord connecting points H,, t, and H1,11 is much greater than the slope of the chord connecting points H., 1, and HI, tl. As H and t proceed up the adiabatic saturation curve, H , and t, decrease and approach H , and t, for saturated air (Figure 3). This effect is due to the fact that the diffusivity of toluene through air is much lower than the diffusivity of water vapor through air; therefore the wet-bulb relation (Equation 2) necessitates that the diffusion driving force increase relative to the heat transfer driving force. This result is accomplished by H , and t , becoming greater than H , and t,. The values of ( H , - H ) / ( t - t,) or ah,/sk’ are best determined by calibration; however, since these ratios are characteristic of the gas and liquid employed, it is possible to estimate their value by the use of coefficients of heat transfer and mass transfer. The relation of these coefficientshas been considered by Arnold (1) and by Chilton and Colburn (8) on the basis of analogies between heat transfer and mass transfer and fluid friction, and by the ratio of empirical equations. On the basis of the film and turbulent core concept, it ia not necessary to include friction in an analogy between heat and mass transfer. Thus, the gas close to the surface is flowing in streamline motion, whereas beyond this film (of thickness B,)the gas is mixed by the eddies in the turbulent stream. Excluding the effect of radiation, which is allowed for separately, the over-all thermal resistance l/h, by conduction and convection is the sum of the conduction resistance B,/X of the film and the convection or eddy resistance re1’. Applying the resistance concept to the mass transfer of vapor from the wet bulb to the gas stream gives
+
(P,
- p)
=O r h C (t kr,
-t3
Since the surroundings are usually a t the dry-bulb tempersture, radiation raises the temperature of the wet bulb. As the motion of air over the wet bulb is increased, a decreases and ah/kr, also decreases, asymptotically approaching a constant value. As would be expected, the value of ah/kr, obtained when calibrating hygrometers of the stationary type is always higher than when high gas velocities are used. Since it is diEcult to control the air velocity when low, and small changes make large changes in the coefficient,it is advisable to use a high air velocity. I n order to be able to plot this equation on the humidity charts, the driving force in the pressure units (Pw- P)will be converted to humidity units ( H , H ) . In those cases where the fractional variation in the inert pressure is small,
-
where ref = resistance to mass transfer through turbulent portion of gas stream D , = molal diffusivity of vapor through true gas film M B = molecular weight of vapor-free gas The eddy resistances, T,” and r l , should be independent of the vapor diffusivity or thermal conductivity, and depend only on the turbulent flow conditions. By definition To” and T,‘ equal A(2 - t’)/(dQ/dO) and A(H’ - H ) / ( d W / d B ) , respectively. Consider the transfer of a differential mass of gas, do, by the turbulent eddies in time de from the region where conditions t’ and H’ exist, to the main body of the gas identified by t and H ; for such a convection transfer dQ/dO is equal to d w ( d ) ( t - t’), dW/dB equals dw(H’ - H ) , and the substitution of these quantities in the definition of T,” and T,’ gives:
INDUSTRIAL AND ENGINEERING CHEMISTRY
5 10
where s' is the humid heat a t the outer boundary of the gas film, and the term r,'/r,''s' should be independent of velocity and other factors affecting turbulence. By taking the ratio of h, to k' and rearranging, one obtains: 1 --" knc
TABLEI. EXPERIMENTAL VALUESOF PSYCHROMETRIC RATIO h,/k's' = rw(Hw - H)/cYs' (t - 1,) FOR VARIOUS LIQUIDS AND
h,
- =&MB Ls,
VOL. 30, NO. 5
Liquid
k'x
-
, XTr'
4.0
2.0
.*x \
V
c
1.0
(8)
(1)
1:6l
1:45 1.69
.. ., ..
2:OO
1:79 1.76 1.76
1177 1.83
..
Toluene Chlorobenzene Carbon tetrachloride n-Propyl acetate Ethyl propionate
2.0
4.0
FIGURE4
Figure 4 is a plot of the experimentally determined value of the psychrometric ratio h,/k's' vs. the dimensionless ratio X/D,MBs', The dotted line is based on Equation 3, using the empirically determined vaIue of ro'/B, based on the toluene run. The predicted curve agrees satisfahtorily with the experimental values. The same wet-bulb data that were used for the calculations in Table I have been correlated (IO) on the basis of the relations given by Arnold and by Chilton and CoIburn. The agreement is approximately of the same degree as that obtained in Table I. If the value of X/D,MBs' is approximately 1, as is the case for water and air, h,/k's' will be 1. It is interesting to note that h/lc's for eddy diffusion is equal to I, independent of the system under consideration, but that h/k's for laminar flow is equal to X/D,MBs, which may be greater or less than 1. For systems that give values of X/D,Mj+ less than 1, a greater wet-bulb depression should be obtainable with streamline flow than with turbulent flow.
Drying Most of the experimental data on drying have been obtained under constant drying conditions, in an attempt to determine the various factors involved.
2.78 2.88 2.92 2.97
2:OO 2 07
..
Xv1en.-
..
..
1.60 1.68 1.72 1.75 1:8l 1.83 1.85 1.85 1.90 1.92
Consider the case of the drying of a slab in which the initial concentration is uniform and sufficiently high so that the surface of the slab is completely covered by a water film. The following discussion assumes that there is no vapor pressure lowering due to soluble salts in the water. If the sheet is brought into contact with a warm air stream, the water on the surface begins to evaporate and the vapor diffuses through the surrounding air film; then it is carried away by turbulence in the air stream. I n general, the slab soon approaches the wetbulb temperature since the latent heat for evaporating the water is usually high, relative to the sensible heat of the slab. The sheet therefore acts as a wet-bulb thermometer, with the heat transferred from the warm air and surroundings to the surface being quantitatively consumed in evaporating water from the surface. As the water on the surface is consumed, additional water may flow to the surface from the interior of the slab. This period under constant drystancy acterized ing conditions of evaporation by the is charcon-
0.8 0.6 0.8 1.0
(14)
Calcn. from Equstion 3 1.22
Values of h,/k's calculated from Equation 3 are given in the last column of Table I (I6),together with the experimental values. The experimental value of h,/k's equal to-1.79 for toluene was taken for the calculation of r,'/Bf.
0.6
-Citation-
1
This equation makes possible the calculation of r,'/B, from experimentally determined wet- and dry-bulb temperatures. This ratio should be independent of the nature of the liquid on the wet bulb and be only slightly affected by gas velocity as long as the flow is turbulent. One experimental determination, therefore, makes it possible to predict the wet-bulb conditions for other liquids. By rearranging, this last equation becomes
x
D,MB8'
AIR
I
#
pq: __________ D
r a t e a n d i s often 5 P I called the "constantf 3 r a t e period." It FALL1 wn corresponds to curve OE BD of Figure 5. However, as t h e FIGURE 5 drying is continued, the water on the surface becomes so depleted that the surface no longer behaves as it does when thoroughly wet, and the rate of evaporation decreases and the so-called falling-rate period has begun. Since the rate of evaporation is now less than before, the rate of heat consumption likewise decreases; this requires a decrease in the temperature difference since the heat transfer coefficient and heat transfer area are the same. With constant air temperatures this necessitates a rise in the temperature of the slab. Although this increase in temperature increases the partial pressure difference of water,vapor across the air film, the wetted surface is decreasing, and the rate of drying continues t o decrease. The moisture content at which the rate of drying starts to decrease is called the "critical water content" r,, and is usually expressed as weight of total water per unit weight of bonedry stock. Water may evaporate only at or near the surface during the falling-rate period, or may evaporate in the interior of a porous material, Depending upon the thickness of the slab and upon other factors such as the rate of evaporation during the constant-rate period and the ease with which liquid water flows through the interior, the rate curve in the falling-rate period may have various shapes. The instantaneous rate of evaporation per unit surface d W/AdO, when plotted against
MAY, 1938
INDUSTRIAL AND ENGINEERING CHEMISTRY
r , the weight ratio of total water to dry stock, may give a curve that is convex upward, linear, or concave upward. However, as the drying proceeds, the rate curve eventually approaches a zero ordinate a t point E , which corresponds to the equilibrium water content (Figure 5). If in the falling-rate period the “rate” -&/do is linear in (T - E ) , as indicated in Figure 5-4. e.:
.
-dr/dE
= K(r
- t.)
period, the rate of evaporation from various materials under comparable conditions should be the same. The following table shows the rate of evaporation during the constant-rate period when drying various materials in a pan over which air a t 130 F. and of low humidity was blown a t a velocity of approximately 11 feet per second (13): O
Material
- E)
This equation shows that the time in the falling-rate period in such a case is proportional to the fractional reduction in free water; i. e., a given time is required to halve the existing value of (T - E ) . CONSTIANT-RATE PERIOD.The rate of evaporation in this period is fixed by the drying conditions chosen. The air velocity and its angle of incidence upon the surface determine both the rate coefficient, IC’, for the transfer of water vapor from the surface of the sheet to the air and the coefficient of heat transfer, h,, by conduction and convection from the air to the surface of the sheet. The humidity of the air is also fixed arbitrarily. These factors are related by the following equation ; A[h,(t
511
+ h,(t” - t.)] + b = k‘A,(H,
- H)(r,)
(4)
where b is a factor to correct for the initial sensible heat of the slab, and, if the constant-rate period is appreciable in extent, b is in general small enough to be negligible. If A is greater than A,--i. e., the heat transfer surface is greater than the wetted area-then t, will not be constant and will vary according to its position relative to the wetted surface. In this last case Equation 4 should be modified for the transfer of heat through the dry surface. Assuming equilibrium a t the interface, H a is the humidity of saturated air a t temperature t, of the surface of the sheet. Under conditions where the wetted area, A,, and the heat transfer area, A , are equal, and where the air velocity and temperature conditions are such as to make h,(t”- 1,) small relative tto h,(t t,), the equation becomes identical with the wet-bulb equation, and surface temperature t, becomes the Even when these conditions are wet-bulb temperature t,. not satisfied, the heat balance requires that t, be constant during the constant-rate period. In order to reduce edge effects, laboratory experiments on slabs are often carried out with the edges waterproofed. When the total heat transfer area becomes greater than the area for e v a p o r a t i o n , (t - t,)/(H, - H ) be1 , i 0004comes lese than (t - t,)/ 3 ( H , - H ) , which neces0.003 sitates that t, and H , become greater than t, and i 0002 H,; as shown in Figure e 6 ( I I ) , a higher rate of 0.001 evaporation will be obZ 0 0.5 1.0 1.5 2.0 tained than when drying < RATIO DRY TO WETTED SURFACE b slabs for which A equals FIGURE6 A,. I n Figure 6 the-rate of drying per unit of wetted surface, y, which is equal to Ic’(H, - H ) , divided by the wet-bulb depression is plotted against the value of dry to wetted surface. When material supported in trays is dried, heat conduction through the bottom and sides of the tray will increase the heat supply and raise the temperature of the wetted surface, thus increasing the rate of drying. If the surface acts completely wet during the constant-rate
Water Whiting pigment Brass filings Brass turnings Sand (five sizes) Clays
Evapn. Rate G,/(hr.) (SQ. cm. wet face) 0.27 0.21 0.24 0.24 0.20-0.24
0.23-0.27
The ratio of the dry to wetted surfaces was 1.65, and therefore the rates of evaporation were higher than would have been obtained with A , equal to A . The rates are nearly all the same, and as expected the rate for water is slightly higher, since heat can be more easily transferred from 0.06 the dry surface to the interface in this case be0.04 cause of the convection in thewater. 0.02 Often in a laboratory dryer only one sample is 0 present, and it,is sub0 4 8 12 16 jected to radiation from WET BULB DEPRESSION, DES. C. all sides. The effect of FIGURE 7 radiation will be much greater than in commercial dryers where only a few of the outside samples are exposed to radiation. The rate of drying in the constant-rate period may be increased in several ways: (a) by increasing the temperature difference by the use of warmer and/or less humid air, (a) by using radiation and metallic conduction in addition to heat
-
.---
-z. v
h
t:
AIR VELOCITY, METERS PER SECOND
FIGURE8
transfer from the air itself, (c) by increasing h, and k’ by increasing the air velocity, and (d) by choosing a gas other than air through which water will diffuse a t a faster rate. The effect of increasing the wet-bulb depression is shown in Figure 7, where the data on the rate of drying obtained by Kamei, Mizuno, and Shiomi (6) ,are plotted as a function of the wet-bulb depression. The correlation is in excellent agreement with that predicted by Equation 4. Figure 8 (6) illustrates the effect of increasing air velocity and indicates that for the case in question-e. g . , slab of clay 3 cm. thick with A , / A equal to 0.6 and with air a t 40” C. and a relative humidity of 60 per cent-the exponent of velocity is about the 0.62 power. It is expected that this exponent will be less than the customary 0.8 power, owing to the fact that the heat transferred through the dry areas has an additional heat resistance (due to the solid) that is independent of air velocity.
VOL. 30, NO. 5
INDUSTRIAL AND ENGINEERING CHEMISTRY
512
By using radiation, the net heat transfer to the slab can be increased and thereby give a higher net rate of evaporation; and by utilizing metal screens for supports, the effect of heat transfer fins can be obtained. The use of gases other than air for increasing the rate of drying is limited to those systems which give higher heat and mass transfer coefficients. Thus hydrogen a t comparable Reynolds number and partial pressure conditions should give rates of evaporation about three times those obtained with air. In the latter case the surface temperature would increase because h/ks for water vaporhydrogen is appreciably greater than 1. FALLINQ-RATE PERIOD.If the drying is carried far enough, a falling rate period is generally encountered. During this period the evaporation may be at or near the surface or may occur below the surface. The relative resistance to the movement of water and heat through the gas film and through the solid determine the locus of evaporation. Thus if liquid water moves easily through the solid and the rate of evaporation is low, the water will be able to equalize throughout with only a small concentration gradient in the solid, and evaporation will take place mainly near the surface. However, as the rate of evaporation is increased, larger and larger concentration gradients are required within the solid to transport the liquid to the surface. Lower surface concentrations result, and eventually a condition will be reached where the evaporation plane will recede and thereby increase the resistande to mass and heat transfer relative to the internal resistance to liquid water. However, under such conditions the evaporation plane will retreat farther and farther into the solid as the run proceeds. The liquid from the central part of the slab flows to the plane of vaporization, and the resulting vapor diffuses out through the pores in the solid and through the gas film into the surrounding air.
I t 0.04
1
I
I/
I
IRATE
I
1
I
r,
I
/ I
I
1
,
,I u ! I I
I 4.0
-
-2 I
9
BRICK CLAY
0
2
5 IO I5 20 25 PER CENT TOTAL MOISTURE, DRY MSlS
0
0.06
0
3.0
=+-
5 Y
L 1L
0.04
2.0
0.02
1.0
5
6 LL
2 B I-
0
0 5 20 40 60 60 100 120 PERCENT FREE MOISTURE, DRY BASIS
I
FIGURE9
By measuring the over-all heat transfer coefficient from the midplane of the slab to the main body of the air, it is possible to determine whether or not the plane of evaporation is retreating within the slab. If the evaporation continues to take place a t the surface, the resistance to heat transfer will be largely in the gas around the slab and will remain essentially constant; however, if the plane of evaporation retreats, the heat must pass through the solid outside the evaporation eone, and the over-all coefficient decreases. Data for the drying of a brick clay are shown in Figure 9 (IS). The coefficient of heat transfer showed no substantial decrease in the fallingrate period; this indicates surface evaporation. Data for a sulfite-pulp block ( l a ) are also shown in Figure 9, and the decreasing value of U indicates subsurface evaporation.
Figure 10 shows data of McCready and McCabe (7) for drying slabs of paper pulp with air of the same temperature, humidity, and velocity. I n the falling-rate period subsurface evaporation occurred in the more porous sample, which was similar to blotting paper, but evaporation occurred a t the sur2,0 0.20 face of the less porous pulp, 0.1 6 1.6 2 which was similar to writing 0.12 1.2 S paper. 0.08 0.8 The p r o b l e m of the move004 0.4 2 ment of water through solids is complicated by0 the- fact that 0 ~ 0 40 80 120 160 200 0the quantitative mechanism of 20 32020 t the movement of liquid water 1.6 5 0.16 Y through solids has not been 5 0.12 1.2 ; satisfactorily solved. Diffusion s .I; 0.0 8 0.6 e q u a t i o n s have been widely : : 4 0.04 0.4 p used to describe this movement.
-
~
2
proximation of the driving force causing the movement of water in certain cases such as wood. But in porous solids, such as sand, and in other nonhygroscopic materials, the movement of water is undoubtedly by capillarity, and the movement of water may be in the opposite direction to that indicated by a driving force in the above units ($6). If during the falling-rate period the resistance is mainly due to the surrounding air film, the drying time would be reduced by increasing the air velocity or by increasing the wet-bulb depression-i. e., by increasing the air temperature or by lowering the absolute humidity. For a given percentage decrease in moisture, the drying time will be shorter, the thinner the slab. On the other hand, if the resistance of the surface film is negligible compared to the internal resistance and evaporation proceeds a t the surface, then air velocity and humidity difference and other factors affecting the surface resistance should have no effect on the drying coefficient, K, except in so far as they affect the temperature of the stock and the corresponding diffusivity, D. This latter case is encountered in drying very thick materials or materials of low diffusivities. During the falling-rate period, the rate curves may not necessarily be linear in (7 - E ) , and K varies with the free water. The problem can be handled by fitting straight lines to approximate the several sections of the rate curve. CRITICALMOISTURE7,. When the surface concentration, C, falls below a critical value, C,,, the falling-rate period begins. It has been suggested that C,, may be estimated by extrapolating the plot of equilibrium water, E,to 100 per cent relative humidity and multiplying the value of E , so obtained by p, the weight of dry stock per unit total volume: Cm =
EP
At high humidity the equilibrium moisture varies widely with the material, and this procedure would predict considerable variation in C,, from one material to another. The measured values of C,c are meager; the indications are that they are greater for the more hygroscopic materials. The critical moisture content is not merely a property of the material being dried. Consider what happens during the constant-rate period of the drying of a slab of homogeneous material having initially a uniform moisture concentration CO. If the rate of evaporation is low and if the liquid will flow to the surface owing to a small concentration gradient, the critical moisture content will be but little greater than the equilibrium moisture content. However, if a very high rate
INDUSTRIAL AND ENGINEERING CHEMISTRY
MAY, 1938
of evaporation is employed, such a steep concentration gradient a t the surface is required that virtually no water will have been removed before Ca,is reached; hence the equilibrium moisture content will be but little less than the original water q,and the constant-rate period will be over in a short time. Although neither of these extremes is usually encountered, both can be approached-the first, by slowly drying a thin sheet, and the second, by rapidly drying a thick sheet. I n order to utilize the experimental data under varying drying conditions, the above rate of drying equations must be integrated, and a t the same time the varying drying oonditions must be taken into account. I n continuous dryers the humidity of the air is generally not constant but is a function of the water evaporated. I n such a case no constant-rate period of drying will be encountered, although the sheet may be thoroughly wet and at a constant temperature, since the humidity and temperature of air are actually changing. I n addition, the critical moisture content is likely to be widely different from that obtained in laboratory experiments because of different temperature and rate-of-drying conditions. The constant-temperature period may be expressed by -d(r
- E)/&
= J(H,
513
(5-4)
and Equation 6 gives
-
The fractional humidification (actual humidification H Z H I , divided by that obtainable with an infinite apparatus, H , - H I ) takes the form,
which is analogous to the Murphree rectification efficiency for a bubble plate, where gas-phase resistance is controlling. A similar expression may be obtained from Equation 5A. The fractional cooling or humidification given by a bubble plate is of the order of 0.8. ADIABATIC DEHUMIDIFICATION. The problem in this case is complicated by the fact that the interface is not a t a constant temperature. Consider a given cross section in a direct-
- H)
and during the falling rate, -d(7
- E)/&
f
(T
-
E)
A t the critical point both equations apply, and J(Ha
- H)
j
(7,
H
- E)
These equations are combined with the water balance, -d(T
-
E ) = mdH 1, DEG. F
and integrated.
FIGURD 11
Air Conditioning Air conditioning is the controlling of the temperature and humidity of the air. The rate of transfer of sensible heat between the main body of the air a t t and the interface a t ti is: -wsdt
= ha(adV)(t
-
ti)
(5)
The rate of transfer of water vapor between the main body of the gas and the interface is given by the equation: --wd H = k’a d V ( H
- Hi)
(6)
The ratio of Equation 6 to Equation 5 gives the significant relation: (7)
For air and water vapor h/k’s is substantially 1, and Equation 7ma8ybe written
dH - -- (H - Hi) dt
(t
- ti)
Equation 8 applies to either the heating or cooling of air, provided that the transfer is through a thoroughly wet surface. The value of ti and H c depends on the manner in which heat is supplied to the interface from other sources than the air itself. Thus in the case of an adiabatic humidifier, the heat transferred from the air to water tends to heat the water, but the evaporation tends to cool it. Where the unevaporated water is recirculated, it attains the wet-bulb temperature and the water temperature remains unchanged. Since ti is substantirlly constant a t t,, integration of Equation 5 gives
contact dehumidifier where the main body of the air is unsaturated, as represented by the point having coordinates H I and tl in Figure 11. The conditions at the air-water interface will be represented by HI-ts coordinates on the saturation curve. The air will be dehumidifed by water vapor flowing from the main body of the gas through the gas f l m and liquefying a t the interface. Since ti is less than t , the air will be cooled, and the sensible heat so transferred will flow through the same gas film. As a result of these conditions, the temperature of the water will rise. As the water flows through the apparatus, the temperature ti of the interface will rise. At each cross section Equation 8 applies, which means that the instantaneous slope of the path is dependent on the prevailing ratio of the humidity difference to the temperature difference. The situation is shown in Figure 11, and it is seen that the path curve will be convex upward. If data are available for the rate coefficients, ha for the gas phase and hL for the liquid phase, the necessary size of apparatus for given operating conditions can be calculated by graphical integration of the equations. The equation can be approximated by substituting finite increments for the differential terms and making stepwise calculations, with intermediate calculations of t,, assuming that d H / d t is constant over short finite intervals (Figure 11).
Nomenclature A B B
= area = fictive film thickness
thickness of true film of gas, ft. d, Cnal ,: CO = concentration of liquid, weight per unit total origivolume; C, corresponds to the surface, C,, t o the surface
D
at the critical point, and C, to the original concehtration = volumetric diffusivity, sq. cm./sec.
INDUSTRIAL AND ENGINEERING CHEMISTRY
5 14
molal diffusivity, gram moles/(cm.)(sec.) e uilibrium water, weight per unit weight of bone-dry stock absolute humidity, weight per unit weight of %nk$ii; H corresponds to the main body of the air, H, to saturation at the wet bulb temperature, H, to saturation at the surface temperature in the constant rate period, and H‘ to the boundary between the air film and the main body of the air J = proportionality constant R = drying coefficient, in units of the reciprocal of time M A , Mg = molecular weight of diffusing gas and nondiffusing gas, respectively N A = moles transferred per unit time P = ressure of liquid CJ = [eat transferred, B. t. u. Re = Reynolds number T = temperature of liquid U = heat transfer coefficient V = volume W = weight a = surface area per unit volume 6 = sensible heat factor d = diameter f = function relating to falling-rate period h,, h,, h ~h ,~ = , heat transfer coefficient, by conduction and convection, by radiation, for the gas phase, and for the liquid phase, respectively I%, k’, k~ = transfer coefficient as Ib./sq. ft./atm., IbJsq. f t . / unit humidity difference, and lb. moles/sq. ft./atm., respectively Im = logarithmic mean 7n = weight ratio of bone-dry air to bone-dry stock p g = partial pressure of nondiffusing gas p a , p’, pi = partial pressure of diffusing gas, in the main body of the as, at the boundary between the main body and the gas Elm, at the liquid gas interface, respectively P, = saturation pressure at wet-bulb temperature q = rate of heat transfer, B. t. u./hr. re, rot, r/ = resistance to mass transfer, for the turbulent core using pressure units, for the turbulent core using humidity units, and for the laminar film, respectively T = over-all resistance r2!,rs,rW = latent heat of evaporation r,, = resistance to heat transfer in turbulent core s , s’ = humid heat
D,
E
=
VOL. 30, NO. 5
t,, ta, t’,
=
w
z a ,6 y
X
t i , t” = temperature of wet bulb, of saturation, at the boundary between turbulent core and laminar film, at the liquid-gas interface, and of the surroundings, respectively = rate of flow of bone-dry air through dryer, weight per unit time = distance
+
= h, h,!h, = proportionality constant = rate of drying, grams/sq. = thermal conductivity
cm.
time weight of bone-dry stock per unit original volume 7 , TO, T~ = total moisture content, weight per unit weight of bone-dry stock; T corresponds to e, 70 to the original time, and T~ to. the critical point, respectively p = viscosity w = mass of gas-vapor mixture transferred by convection 0 p
= =
Literature Cited (1) Arnold, J. H., Phusics, 4, 255, 334 (1933). (2) Chilton, T. H., and Colburn, A. P., Iivn Eao. CHEM.,26, 1183 (1934). (3) Comings, E. W., and Sherwood, T. K., Ibid., 26, 1096 (1934). (4) Gilliland, E. R., and Sherwood, T. K., Ibid., 26, 516 (1934). (5) Hougen, 0. A., and Ceaglske, N. H., Ibid., 29, 805 (1937). (6) Kamei, S., Minuno, S., and Shiomi, S., J. SOC. Chem, I n d . (Japan), 38, Suppl. Binding 456-73B (1935). (7) McCready, D. W., and McCabe, W. L., Trans. Am. Inst. Chem. Engrs., 29, 131 (1933). (8) Mark, J. G., Ibid., 28, 107 (1932). (9) Murphree, E. V., IND. ENQ.CHEM., 24, 726 (1932). (10) Sherwood, “Absorption and Extraction,” p. 5 5 , New York. McGraw-Hill Book Co., 1937. (11) Sherwood, T. K., IND. ENG.CHEM.,21, 976 (1929). (12) Ibid.. 22. 132 (1930).
5 ) Walker, Lewis, McAdams, and Gilliland, “Principles of Chemical Engineering,” 3rd ed., p. 593, New York, McCraw-Hill
Book Co.. 1937. REPXZIVBD January 24, 1938.
Dermatitic Properties of Tung Oil M. W. SWANEY Ellis Laboratories, Inc., Montclair, N. J.
T
HIS paper is the result of a severe case of dermatitis contracted by a worker in the varnish laboratories;
this irritation was attributed to tung oil. I n view of the widespread use of tung oil in paint and varnish manufacture, a few explanatory remarks should be made. It is not to be construed from this discussion that tung oil is a notoriously insidious material, nor are any aspersions to be cast upon its efficacy as a paint and varnish base. On the other hand, only a small minority of persons who possess the necessary allergic sensitivity are likely to be affected, and then probably only by the heated oil. By the vast majority, this oil can be handled with impunity. There would appear to be no danger a t all in handling or using the finished tung oil varnishes or varnish products. In the present case the affected person, although not actively engaged in handling tung oil, was present in the laboratory in which tung oil was being utilized in varnish making. The irritation appeared soon after this person was admitted to the laboratory and continued for many months before it was eventually terminated. During this time, however, tkpe irritation was most evident in the region of the groin and about the face and eyes, although on several oc-
casions it covered the entire body. It was most apparent as a reddening of the skin, accompanied by violent itching and burning. On certain parts minute water blisters were often present. After a long period of ailment, it was suspected from daily observations that the source of the trouble lay in tung oil. By careful correlation it was found that the greatest activity followed the days on which closest contact was had with the vapors of heated tung oil. When contact with the oil’s vapors was avoided after a severe attack, the irritation invariably lessened in intensity but did not show immediate signs of complete disappearance; it even continued in a semimild manner for several months after the subject was removed from the tung laboratory. The variety of tung oil used a t that time was dark in color and was probably of Chinese origin. After all conceivable topical applications were found to be of no avail, relief was obtained in the following manner.
Treatment Since the irritation was evidently being caused by materials present in the vapors issuing from tung oil, it was thought possible that injection of such material into the body right prove
