Znd. Eng. Chem. Res. 1987, 26, 1597-1603
NSERC for partial financial support of the work. Nomenclature am,,= group interaction parameter C = constant in eq 12 Dlk = binodal residual of data point k in data set 1 Elk = tie-line residual of data point k in data set 1
F = objective function p = constant in eq 12
group surface area qi = effective molar volume, van der Waals surface area R = gas constant Rk = group volume ri = van der Waals volume T = temperature Uij = binary interaction parameter Xi = group concentration xi = mole fraction 2 = coordination number Qi =
Greek Symbols
ri= individual group contribution to the activity coefficient yi = activity coefficient
vi, = number of group species j in molecule i
1597
p = parameter vector in eq 13 4i = area fraction Bi = volume fraction qm,, = binary parameter defined by eq 11 Registry No. Butyric acid, 107-92-6; formic acid, 64-18-6;
toluene, 108-88-3. Literature Cited Badakhshan, A.; Chowdhury, A. I.; Leung, R. J . Chem. Eng. Data 1985, 30, 416. Derr, E. L.; Deal, C. H. Inst. Chem. Eng. Symp. Ser. 1969,32(3), 40. Fredenslund, Aa.; Gnehling, J.; Michelsen, M. L.; Rasmussen, P.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Dev. 1977,16,450. Leung, R. W. K. M.Sc. Thesis, University of Calgary, Calgary, Alberta, Canada, 1986. Magnussen, T.; Rasmussen, R.; Fredenslund, Aa. Ind. Eng. Chem. Process Des. Deu. 1981, 20, 331. Nedler, J. A.; Mead, R. “A Simpler Method for Function Minimization”, Comput. J. 1965, 7, 308-312. Othmer, D. F.; Tobias, P. E. Ind. Enp. Chem. 1942. 34, 690. Skjold-Jergensen,S.; Kolbe, B.; Gmehling, J.; Rasmussen, P. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 714.
Received f o r review August 27, 1986 Accepted April 13, 1987
Water Vapor Uptake by Ceramic Microspheres Enio K u m p i n s k y Polymer Products Department, D u Pont Experimental Station, Wilmington, Delaware 19898
Experimental studies on the water vapor uptake by ceramic microspheres have been carried out a t atmospheric conditions. Hysteresis has been found for the adsorption/desorption isotherm with respect to the relative humidity. Similarly, it is shown that the transient paths for the uptake exhibit a quick weight increase in the early stages of the process, followed by a n asymptotic period of very small weight gain. T h e process was modeled by simultaneous mass-transfer equations, which show that the rate-limiting step is molecular diffusion and not condensation of water vapor on the surface of the powder. Fillers are defined as relatively inert small particles which are added to plastics compounds for specific purposes (Schwartz and Goodman, 1982). Fillers are blended with polymers to provide satisfactory tensile strength and to improve stiffness, abrasion resistence, tear resistence (Billmeyer, 1984),gloss control, and thermal conductivity. Likewise, rheological properties can be improved by selecting the right filler and choosing the proper loading. In addition to modifying the composition’s properties, fillers can reduce its costs. Structurally, filler particles range from highly irregular masses to precise geometrical forms and can constitute either a major or a minor part of a composition (Grayson and Eckroth, 1980). The most commonly used fillers are kaolins and other clays, silicas, aluminas, calcium carbonates, talcs, barium sulfate, powdered metals, metal oxides, nylons, and poly(tetrafluoroethylene) (PTFE). Very fine fillers, such as carbon black and titanium dioxide, can act as pigments. Fillers are of paramount technological significance as shown by the thousands of patents and publications concerning filled polymers. It is not our intent to provide detailed information on fillers since other publications on the subject are available, such as Grayson and Eckroth (1980) and Katz and Milewski (1978). It is our intention to explore a problem whose final outcome is the degradation of matrix polymers during processing, i.e., the presence of hygroscopic moisture in fillers and pigments. Even small amounts of water can cause major damage to the matrix. For example, moisture in excess of 0.1% by 0888-5885/87/2626-1597$01.50/0
weight polymer can result in significant degradation of Hytrel (trade name), a polyester elastomer ( H y t r e l Handbook, 1981). Some publications on the adsorption of vapors by powders are found in the literature. The dynamical aspect of adsorption is covered in detail by de Boer (1953),while the adsorption in micropores of porous powders is explored by Dubinin (1966) and Pesaran (1983). Adams et al. (1970) studied the uptake of organic vapors by Saran (trade name)-carbon fibers and powders, finding adsorption hysteresis for some compounds. Likewise, adsorption hysteresis was reported by Tabibi (1982) during the uptake of water vapor by sugar powders. Transient curves of moisture adsorption by carbamazepine powders were obtained by Kaneniwa et al. (1984), who showed that different crystalline forms of the same compound can adsorb quite diverse amounts of water vapor. El-Dib and Aly (1977) present the effect of time on the adsorption of phenylamides on powdered carbon. Their results are in qualitative agreement with ours, although the systems are quite different. Gammage and Brey (1972) obtained stepped adsorption isotherms of water vapor on ground thorium oxide. This system also exhibits adsorption/desorption hysteresis that is qualitatively identical with that obtained here. Only a few experiments are found in the literature describing the transient moisture uptake by commercial powders. Additionally, there is no detailed description on the rate-determining step for the adsorption of hygroscopic 0 1987 American Chemical Society
1598 Ind. Eng. Chem. Res., Vol. 26,
No. 8, 1987 Table I. Dimensionless GMUPS
Mean diameter is besed on the particle of average volume, which is determined by light scattering of a laser beam on particles flowing through a sample cell. Signals emitted by the scattered light provide statistical parameters, such as particle size distribution and external surface area. The Model Figure 2 shows the terms involved in the mass transfer
of water vapor in the bed of tiller. For the solid phase we have (accumulation) dt = d W,,from which
a
[(l - t)S dz dC,] = (k,a)(C, - C,*)S dz at
Figure 1. Zwospherec n / R , magnifwd 1.800 times
water. We present here such a study, in which ceramic microspheres are allowed to adsorb water vapor under different relative humidities a t 20 OC. Some results on other powders are also reported. It is our intention to help the user of industrial tillers to handle these materials more effectively prior to processing. The hollow ceramic microspheres studied are Zeeospheres (trade name), which can be chemically described as a silica-alumina alloy. A micrograph of Zeeospheres Type 0/8 enlarged 1800 times is shown in Figure 1. Experimental Section Thin aluminum trays with dimensions 1.5 cm (height) by 6 cm (diameter) held the h p h e r e s (bed height 1cm) to be submitted to hygroscopic water uptake. The powder was maintained overnight in a Hotpack 633-4 oven under an absolute pressure of 17 kPa at 200 " C in nitrogen a t mosphere. This was enough t o eliminate any condensed water from the particles' surface. The impervious and smooth tray walls were slightly tapped once loaded with the powder to form a uniform bed with no short circuiting of the diffusion path, in the bed itself and between the bed and side wall. To obtain the adsorption isotherms, the aluminum trays containing the powder were then transferred immediately into a desiccator having a Bacharach Precision Model 22-7059 relative humidity indicator. Either a sulfuric acid solution with known specific gravity or a salt in contact with a saturated aqueous solution of the given solid was introduced into the desiccator to provide a desired relative humidity. The closed desiccator was then placed into a room with constant temperature (20 "C), and after 48 h, the samples were weighed in a Sartorius 1712 MP8 analytical balance. A Bendix Psychrometer Model 566 was utilized to measure relative humidities (RH) during the transient uptake experiments,which were carried out at 20 O C and constant RH. Most experiments with Zeeospheres were performed with Type 0/8. However, we were interested in the effect of particle size distribution on the water vapor uptake, so that the original material was separated into three individual cuts. A Donaldson Classifier A-12-3s was used for the separation. Size distributions were obtained by means of a Microtrac (trade name) 7991-01 particle size analyzer.
(1)
Assuming pseudoequilibrium between solid and gas phases, we can write Cs*= mC,, where m repreaenta the interface equilibrium constant and is determined through the adsorption isotherm. The equation for the solid phase is then (1 - 4
ac. at = (k.a)(C, - mCJ
(2)
For the gas phase the balance is (accumulation) dt =
-[(W,), - (W,),,,]- dW., which can be rewritten as
or
The initial conditions are C,=C.=O
at t = O and the boundary conditions can be written as
dC,
-De-=O dz
(5)
at r = H
With the dimensional groups of Table I, we obtain the above equations in dimensionless form as gas phase:
solid phase:
am
- -8(1 + y - w)
a?
initial conditions: y = - 1 and w = O boundary conditions:
=0
at r = O
(9)
Mathematical Solution The system is self-adjoint (Amundson, 1980) and eq 7-10 were solved analytically by means of a finite Fourier
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1599 exposed top
Following inversion, the solutions are expressed in terms of infinite series
Y(t,T) = and
(22)
U(E9.)
sealed bottom
x (YPn)Un($)
"=I
=
"=l
(W7tJn)Un&)
(23)
Figure 2. Schematic representation of a bed of filler,showing the differential element in which the mas8 balances are applied,
The average of y and w with respect to the bed height can be written as
transform (Ramkrishna and Amundson, 1980), with the associated eigenvalue problem
Y(T) = x B . ( y , u , )
-
"=I
(24)
and m
m(T)
and the inner product
(f(t,T),
=
u&))
J1f(E,d %(E) d t
(12)
After the system is transformed, the obtained ordinary differential equations are set in the matrix form
and the solution is found in terms of the eigenvalues of E,using Sylvester's formula (Amundson, 1966). The inner products of the desired functions are (YJ") =
and
=
(25)
x&(UJ,U,) *=1
In the neighborhood of the bed top, and especially when the steady states were approached, n m l y loo0 serial terms were required to provide good accuracy to the numerical solutions. As eq 17 is written, we found it very difficult to achieve conversion for n values lower than five, especially for high k,, in which case eigenvalues corresponding to larger serial numbers were approached. However, numerical convergence ceased to he a problem when we performed the inversion A,'/'
= (n - 1)r
+ arctan
(+) -
(26)
After this transformation, the characteristic equation converged very fast, independent of the initial guess. As n approaches infinity, we have A,'/* (n - 1)r. For convenience, the following variables are defined
g = 1 - y = c*/c.
(27)
u = (lOOC,Mww)/(mP,)
(28)
and The variable, u,is actually the water vapor uptake of the powder in weight percent.
where
Y
=-
tan
n = 1,2, 3, etc.
A,'/2
I
+ y2 + sin
-cos (An'/')]
X
+ sin
(17)
Results and Discussion Most experiments were performed with the conditions displayed in Table I1 for Zeeospheres 0/8.This table also shows the constants and expressions used here in most cases. C. is based on psychrometric charts (Perry and Green, 1984). DABis found in the same publication at 0 "C and is corrected to 20 "C, using an exponent of 1.75 for the temperature. De is based on some preliminary computations showing good agreement between model and experiments and on the work of Evans and Kenney (1966). We found the mass-transfer coefficient for stagnant conditions by taking the Sherwood number for a sphere equal to 2 (e.q., Satterfield and Sherwood, 1963). This was done by reducing the circular top of the plate containing the powder to a sphere with equivalent external surface area, as described by Gamson et al. (19431, who made measurements for nonspherical particles. Values for k,a are not very important, as will he discussed later, hut an estimate was made based on the onset of adsorption by means of eq 2, for which C. is essentially equal to zero and the experimental adsorption rate is experimentallyknown. Figure 3 shows adsorption and desorption isotherms for Zeeospheres 0/8 a t 20 "C. The steady-state amount of water vapor adsorbed on the surface of the powder in-
1600 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 Table 11. Physical Data for the Adsorption of Hygroscopic Humidity by Zeeospheres 0/8 k,a = 271s C, = 9.87 X 104RH
DAB= 2.49
X
= lOoC,(l - E)Mw
m2/s
UqPb
De = 0.67DAB
u
.c m
M , = 18 kg/kg-mol
H = 0.01 m
e = -
k , = 1.7 X m/s (for stagnant air)
pb
Pp
- Pb
r-
0
PP
oL.
= 900 kg/m3
0 1
&, = 2400
kg/m3
=
0
or
20
60
40
80
40
:
80 ~
T
6.2686 X 10-'RH(l + 0.29285) 0 5 RH C 62 1 + 0.29285RH ueq= 0.2769 + 6.6364 X 10-3(RH- 62)(1 - 0.01561) 62 5 RH 5 100 1 - 0.01561(RH - 62) Ueq
2*
3
c
100
RH (%)
Figure 3. Adsorption and desorption isotherms for hygroscopic water by Zeeospheres 018 at 20 "C.
creased very quickly at lower values of relative humidity, reaching a stable value at RH approximately equal to 20% and staying virtually unchanged up to RH = 62%. For low RH, there were only singly adsorbed molecules, apart from each other. As RH increased the condensed molecules became closer together, creating a new surface that allowed an additional water vapor molecule to be adsorbed when RH reached 62%. This is the case of multilayer adsorption, as depicted in Figure 3. Table I1 displays equations for the equilibrium uptake for adsorption covering the entire range of relative humidities. The monolayer adsorption is well described by a Langmuir isotherm, while the multilayer condensation is well represented by an empirical formula with a negative sign in the denominator (see Perry and Green, 1984). The desorption isotherm, as displayed in Figure 3, is described by a polynomial of degree 3. The presence of hysteresis is not unique to this system, as it has been observed by Adams et al. (1970),Gammage and Brey (1972), and Tabibi (1982) for other gas-solid pairs. There are reasons to believe we are dealing with monolayers surface condensation followed by multilayer adsorption. Firstly, the shapes of the adsorption curves in Figure 3 are in very good agreement with those of the literature (e.g., de Boer, 1953). Secondly, Figure 4 provides enough information on the existence of chemical and physical adsorptions, which are usually associated with monolayer and multilayer adsorptions, respectively. The powder, allowed to adsorb water vapor under the conditions given in Figure 4, is then heated in a vacuum oven over a period of 8 h. The filler increasingly loses weight until the temperature reaches 70 "C, where a plateau is reached. The water vapor that had been physically adsorbed on the second and subsequent layers is now com-
~
120 ~
,
160 ~
:
200 i
("C)
Figure 4. Thermal desorption a t various temperatures in nitrogen atmosphere with P = 17 kPa for Zeeospheres 0/8 previously allowed to adsorb water vapor at 20 OC with P = 101 kPa and RH = 81%. Table 111. Materials Used To Obtain Figures 4 and 5 specific mean surface diameter area, material symbol (vol), um m2/cm3 Zeeospheres A 0 4.54 2.25 Zeeospheres B 0 1.93 4.35 Zeeospheres C 4 2.62 3.30 Zeeospheres D A 4.78 1.97
pletely removed. No additional water vapor is lost until the temperature is 93 "C. At this stage the decoupling of the chemisorbed monolayer starts and the required high temperature stems from the fact that the heat of chemisorption is 3-4 greater than the heat of physical adsorption (de Boer, 1953). As the temperature is further increased, a new plateau is attained a t 125 "C, corresponding to complete desorption. A subject to be discussed is whether we are dealing with surface adsorption or internal solid-phase diffusion. Let us examine the monolayer case. The maximum monolayer weight gain, according to Figure 3, is 0.27 g of water/100 g of dry powder. For a particle density of 2.4 g/cm3 and specific surface area of 2.25 m3/cm3,one obtains 2.9 X g of water/m2 of powder surface. The collision diameter of water molecules from the Lennard-Jones potential, as determined from viscosity data (Reid et al., 1977), is 2.64 A. If these molecules are placed in rows, side by side, one arrives at 5.9 X lov4g of water/m2 of surface. Thus, the water uptake by the powder is about 5 times as large as that predicted by this simple calculation. If we were dealing with intraparticle diffusion, that difference could be several orders of magnitude. The Lennard-Jones potential assumes that the molecules are spherical and nonpolar, which is not the case. Zeeospheres are made from molten fly ashes and are essentially dense and nonporous. However, micrographs of this powder, enlarged 30000 times, show that the surface is quite rough, so that the actual surface area is considerably larger than that shown in Table 111. Therefore, we are probably underestimating the amount of water that can be spread on a surface in a monolayer fashion and underestimating the actual surface area of Zeeospheres, which leads us to conclude that indeed we have surface adsorption of water. The original Zeeospheres 018 were separated into three cuts with different size distributions. Table I11 shows properties and symbols for the original material (label A) and the three cuts (labels B-D), while Figure 5 displays the respective cumulative particle size distributions. On a weight basis, cuts B, C, and D represent 9.1%, 35.4%, and 55.5% of Zeeospheres A. Figure 6 exhibits the transient uptake of water vapor by the four types of Zeeospheres. A rapid weight gain is seen in the early stages
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1601
cn
0 0
0.1
10
1
Diameter
40
0.2
0.4
0.6
5
0.8
1.0
Figure 7. Calculated dimensionless gas-phase profiles in the bed as functions of time for Zeeospheres 0/8 at 20 OC and R H = 15%.
luml
Figure 5. Cumulative particle size distribution for Zeeospheres A-D. Symbols as in Table 111.
t
0 41
I
3
13
0.0
0 2
0 4
0.6
0.8
1.0
P
Figure 8. Calculated dimensionless solid-phase profiles in the bed as functions of time for Zeeospheres 0/8 at 20 OC and R H = 15%. 0
60
120
180
240
300
360
420
t hin)
1
Figure 6. Transient water vapor uptake for Zeeospheres A-D at 20 "C with R H = 15%. Symbols as in Table 111. Solid lines are the result of exponential regression. For Zeeospheres A, it also represents the curve generated by the model with k, = 0.017 m/s.
of the process, followed by a period of slow uptake, corresponding to the asymptotic approach to the steady state. As expected, the larger cut, D, has the lowest uptake, while the smaller cut, B, has the largest weight percent gain. We also observe that cuts A and C have intermediate values of uptake. The fact that Zeeospheres A show uptake higher than that of Zeeospheres C is nonetheless surprising since the former has a lower specific surface area than the latter, according to Table 111. The possibility that both materials might have different compositionswas explored. However, X-ray fluorescence and X-ray diffraction revealed no compositional differences between these two powders. Moreover, since the results are reproducible, the fact that experimental error might have occurred was also discarded. An explanation can be devised as follows. The bulk densities of both Zeeospheres A and C are identical, even though A has a broader size distribution than C. In these circumstances C must be more closely packed than A, meaning that the former has less surface area available for adsorption than the latter. This justifies the higher water vapor uptake by A. Figure 6 also shows how well model and experiments agree for Zeeospheres A (i.e., 018). Air velocities away from the tray containing the powder were measured a t approximately 0.6 m/s. This will give values for k, very close to that used in the calculations, according to the expression of Ranz and Marshall (1952) for convective heat transfer around a sphere and adapted to mass transfer by way of the Chilton-Colburn analogy (Bird et al., 1960). Figures 7 and 8 display the calculated time-dependent dimensionless concentration profiles for water vapor in the gas and solid phases, respectively. They represent purely mathematical results but are probably reliable because the
t (minl Figure 9. Effect of the adsorption rate k,a on the calculated water vapor uptake by Zeeospheres 0/8 at 20 "C, R H = 15% and k, = 0.017 m/s.
1
0 0
I
c 120
iao
240
300
t (min) Figure 10. Effect of the dispersion coefficient De on the calculated water vapor uptake by Zeeospheres 0/8 at 20 O C and R H = 15% with k, = 0.017 m/s.
solid-phase, spaced-averaged concentrations agree very well with the experiments. We can see that the shapes of both g and u as functions of the dimensionless distance for the same exposure time are very similar. This may be an indication that molecular diffusion controls the process, since adsorption occurs whenever water vapor is available
1602 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
t iminl
Figure 11. Influence of the mass-transfer coefficient k , on the calculated water vapor uptake by Zeeospheres 0/8 at 20 "C and RH = 15%. r
_
1
.
0 3*
100
200
300
1
400
t (min) Figure 12. Transient water vapor uptake by Zeeospheres 0/8 at 20 O C , RH = 67%, and very slow air circulation. Computations made with three different values of k,.
and not after the bed is virtually saturated with hygroscopic water. The concept outlined above is corroborated by the results displayed in Figures 9 and 10. Figure 9 shows how the transient path for the solid phase is affected by the choice of the adsorption rate constant, k,a. Unless k,a is very low (2 orders of magnitude smaller than the estimated value), no major changes in transient uptake occur. On the other hand, slight changes in De within the same order of magnitude can impose large variations in the uptake, as shown in Figure 10, thereby confirming the fact that molecular diffusion of water through the gas-filled interstices of the bed is the rate-limiting step for the process. Figure 11 shows how variations in the mass-transfer coefficient at the bed's upper surface affect the transient water vapor adsorption. The lowest k, corresponds to conditions of stagnant air and does not provide good agreement between calculations and experiments. The intermediate and largest k,'s assume that the air velocity away from the powder circulates at 0.6 and 80 m/s, respectively. These results show that for higher velocities the air circulation is not critical. However, this becomes a problem at low air velocities. Figure 12 shows the uptake performed with minimal air circulation in the room. The agreement is not good if k , = 0.0017 m/s is chosen, Le., zero air velocity. However, some very small and interrupted circulation did occur, and if we choose k , = 0.0025 m/s, we obtain good agreement. This k corresponds to an air velocity of less than 0.003 m/s, waich was not actually measured. The curve for k , = 0.017 m/s would require an air velocity of 0.6 m/s. Figure 12 was obtained under conditions of multilayer adsorption, and we might expect to see some discontinuity in the experimental data when the chemisorption was completed. However, this did not occur, meaning that the multilayer adsorption is also quick and controlled by molecular diffusion.
Table IV. Weight Gain Experienced by Different Fillers When Exposed for 8 h to an Atmosphere with a Relative Humidity of 15% at 20 "C filler type wt gain, % Al-Sil-Ate" NC-calcined kaolin 0.337 Al-Sil-Ate" S-calcined kaolin 0.140 Al-Sil-Ate" W-calcined kaolin 0.088 Atomite" calcium carbonate 0.016 Blanc Fixe" HD-80 barium sulfate 0.038 Micral" 932 alumina trihydrate 0.912 Novacite" L-337 silica 0.023 nylon 66 fine powder 0.258 Ti-Pure" R-101 titanium dioxide 0.150 Translink" 445 surface-treated kaolin 0.039 Zeeospheres" 0/8 ceramic microspheres 0.226 'Trade name.
Finally, Table IV displays the experimental steady-state results of water vapor uptake by different commercial fillers. The materials were dried overnight in a vacuum oven at 150 "Cand absolute pressure of 17 kPa in nitrogen atmosphere. Note that alumina trihydrate will start losing its hydrate water a t 230 "C,so the weight gain reported in Table IV is in fact due to hygroscopic water. Table IV shows that these fillers adsorb water to different degrees, depending on chemical composition, specific surface area, and compactation. If polymer degradation and bubble formation during compounding are critical matters, it is important to measure the amount of water vapor adsorbed by the filler. This will have to be dried prior to blending it with the polymer, and a quick estimate with a mathematical model can be useful to determine the wait period the powder can tolerate before being processed.
Summary and Conclusions Experiments have been conducted to measure the isothermal water vapor uptake by ceramic microspheres and other commercial fillers. A mathematical model was developed and then solved, giving excellent agreement between experimental work and calculations. It was determined that molecular diffusion of water through the gasfilled interstices of the powder beds is the rate-limiting step for the uptake of hygroscopic water by the powders. Typically, the process is described by a quick weight gain in the early stages of exposure, followed by an asymptotic approach to the steady state. We found, a t apparent equilibrium, that ceramic microspheres adsorb atmospheric water vapor in a monolayer fashion for lower values of relative humidity. At higher relative humidity, multiple layer adsorption occurs. Hysteresis was also observed, as the desorption isotherm does not match the adsorption curves. Among the commercial fillers examined here, calcium carbonate, barium sulfate, silica, and calcined kaolin (surface treated with aminosilanes) are the ones which adsorb less hygroscopic water. Before compounding a filler with polymer, it is advisable to obtain information on how much hygroscopic water a filler adsorbs under the prevailing ambient conditions. The present model may then be used to estimate the time over which the filler can be exposed to the atmosphere prior to blending powder and resin. This will lead to a product with the desired properties, without degradation and free from gas bubbles in the matrix polymer unless, of course, chemical reactions and gas evolution take place between the two phases. Acknowledgment
I am thankful to William L. Holt, who participated in the experimental portion of this project, and to the Du
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1603 Pont Company for the use of facilities and equipment.
Nomenclature a = surface area per unit volume, m2/m3 An = serial constant defined by eq 18 B, = serial constant defined by ea 19 C, = water vapor concentration in the ambience, kg-mol/m3 C, = molar concentration of water vapor in the moist air, kg-mol/m3 C,* = interfacial molar concentration of water vapor in the gas phase, assuming it is in equilibrium with C,, kg-mol/m3 C, = number of kg-mol of water vapor adsorbed per unit volume of solid phase, kg-mol/m3 D m = binary diffusivity of water vapor in air, m2/s De = dispersion coefficient of water vapor in air inside the bed of filler, defined in Table 11, mz/s E = matrix of coefficients for the transformed system, eq 13 f = function used for the definition of the inner product, eq 12
J’ = array of constants for the transformed system, eq 13
g = dimensionless gas-phase concentration, defined by eq 27
H = bed height, m k, = mass-transfer coefficient of water vapor from the atmosphere to the bed of filler, m / s k, = mass-transfer coefficientfor water vapor at the gas-solid interface, m/s m = equilibrium constant at the gas-solid interface, dimensionless M, = molecular weight of water vapor, kg/kg-mol P = absolute pressure, kPa S = cross-sectional area of the bed, m2 t = time, s or min u = uptake of water vapor by the powder, defined by eq 28, wt%
u, = normalized eigenfunctions, given by eq 16 w = dimensionless water vapor concentration in the solid phase, defined in Table I W = rate of mass transfer in the gas phase, kg-mol/s = rate of mass transfer in the solid phase, kg-mol/s y = dimensionless water vapor concentration in the gas phase, defined in Table I P = array of dependent variables defined by eq 13 z = space coordinate, m Greek Symbols CY,@, y = dimensionless groups defined in Table I 6 , i = defined by eq 21, i = 1, 2 e = void fraction of filler in_the bed, defined in Table I1 vn,i = eigenvalues of matrix E,i = 1, 2 A, = eigenvalues of the adjoint problem F = dimensionless space coordinate, defined in Table I Pb = bulk density of the powder, kg/m3 of bed pp = density of particles, kg/m3 of solids T = dimensionless time, defined in Table I Subscripts eq = equilibrium, steady-state condition n = serial number
Superscript - = space-averaged value
Registry No. Nylon 66,32131-17-2; HzO, 7732-18-5; CaCO,, 471-34-1; BaS04,7727-43-7; Al(OH),, 21645-51-2; TiOz,13463-67-7. Literature Cited Adams, L. B.; Boucher, E. A.; Everett, D. H. Carbon 1970,8, 761. Amundson, N. R. Mathematical Methods i n Chemical Engineering-Matrices and Their Applications; Prentice-Hall: Englewood Cliffs, NJ, 1966; pp 194, 196. Amundson, N. R. T h e Mathematical Understanding of Chemical Engineering Systems-Selected Papers of Neal R. Amundson; Aris, R., Varma, A., Eds.; Pergamon: New York, 1980; pp 740-742. Billmeyer, F. W. Textbook of Polymer Science, 3rd ed.; Wiley: New York, 1984; p 471. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. de Boer, J. H. Dynamical Character of Adsorption; Oxford University Press: Glasgow, 1953. Dubinin, M. M. J . Colloid Interface Sci. 1966, 23, 487. El-Dib, M. A.; Aly, 0. A. Water Res. 1977, 11, 617. Evans, E. V.; Kenny, C. N. Trans. Znst. Chem. Eng. 1966,44, T189. Gammage, R. B.; Brey, W. S., Jr. J. Appl. Chem. Biotechnol. 1972, 22, 31. Gamson, B. W.; Thodos, G.; Hougen, 0. A. Trans. AZChE 1943,39, 1. Grayson, M.; Eckroth, D. Kirk-Othmer Encyclopedia of Chemical Technology, 3rd ed; Wiley: New York, 1980; Vol. 10,p 198. Hytrel Handbook; E. I. du Pont de Nemours & Co.: 1981; Bulletin HYT-IOIA, Chapter on Rheology and Handling, p 3. Kaneniwa, N.; Yamaguchi, T.; Watari, N.; Otsuka, M. Yakugaku Zasshi 1984, 104, 184 (translated from Japanese). Katz, H. S.; Milewski, J. V. Handbook of Fillers and Reinforcements for Plastics; Van Nostrand-Reinhold: New York, 1978. Perry, R. H.; Green, D. Perry’s Chemical Engineering Handbook, 6th ed; McGraw-Hill: New York, 1984; pp 12-3 to 12-13,16-10 to 16-19. Pesaran, A. A. Energy Report UCLA-ENG-8310, 1983; US Department of Energy, Washington, DC. Ramkrishna, D.; Amundson, N. R, T h e Mathematical Understanding of Chemical Engineering Systems-Selected Papers of Neal R. Amundson; Aris, R., Varma, A., Ed.; Pergamon: New York, 1980; pp 744-766. Ranz, W. E.; Marshall, W. R., Jr. Chem. Eng. Prog. 1952, 48, 141-146, 173-180. Reid, R. C.; Prausnitiz, J. M.; Sherwood, T. K. T h e Properties of Gases and Liquids, 3rd ed.; McGraw-Hill, New York, 1977; p 679. Satterfield, C. N.; Sherwood, T. K. T h e Role of Diffusion i n Catalysis; Addison-Wesley: Reading, MA, 1963; p 47. Schwartz, S. S.; Goodman, S. H. Plastics Materials and Processes; Van Nostrand-Reinhold: New York, 1982; p 467. Tabibi, S. E. PbD. Dissertation, University of Maryland, Baltimore, 1982.
Received for review June 25, 1985 Revised manuscript received December 19, 1986 Accepted March 10, 1987
