Moisture Isotherms of Hygroscopic Porous Solids M. El-Sabaawi and D. C. T. Pel’ Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G I
Two major mechanisms are believed to occur in the course of desorption. At high relative humidities, capillary evaporation dominates the desorption process. At low relative humidities, the equilibrium moisture content can be determined exclusively in terms of multilayer surface adsorption. A model has been developed which accounts for both mechanisms as well as their interactions.To evaluate the applicability of this desorption model, the moisture desorption isotherms of two porous samples were determined experimentally utilizing gravimetric vacuum sorption technique. Good agreement was obtained over a range of relative humidities.
Introduction A knowledge of the equilibrium moisture content of a hygroscopic material is essential for the efficient design and operation of systems for drying or storing the material. During the later stages of drying both the drying rate and the final moisture content depend upon the amount of moisture which is adsorbed on the material a t a given relative humidity. In order to measure the sorption isotherm of a material it is placed in an environment where both the temperature and the humidity are carefully controlled. The equilibrium moisture content is measured a t each value of the ambient relative humidity. Unfortunately, it usually takes long times to determine these equilibrium points. It would be a significant help, therefore, to be able to describe the entire isotherm with as few data points as possible. In achieving this, a knowledge of the mechanisms involved in a sorption process a t various ranges of relative humidities will be required in order to make such an extrapolation possible. It is believed that, over the whole range of relative humidities, surface adsorption of water molecules due to physical adsorption forces occurs to an extent dependent on the value of the ambient humidity. However, in the case of hygroscopic porous solids, the moisture condenses in the capillaries at relative humidities less than unity. The capillary condensed moisture also contributes significantly to the equilibrium moisture content at high relative humidities. Several theoretical models were developed and reported in the literature in predicting the sorption isotherm (Brunauer et al., 1938; Langmuir, 1918; Rounsley, 1961). These equations are very successful over a major portion of the humidity range. However, for capillary porous solids, these equations usually deviate significantly from the experimental data collected at high relative humidities where capillary condensation predominates. Because of the particular importance of the knowledge of the desorption isotherm in the drying of hygroscopic porous materials, it would be of considerable value to develop a model to predict the entire range of the desorption isotherm with only one mathematical equation. This proposed model will take into account both the mechanisms involved as well as their interactions. L i t e r a t u r e Review The majority of the isotherms which result from physical adsorption may for convenience be grouped into five classes. These types are shown in Figure 1. It must be pointed out that types IV and V usually possess a “hysteresis loop”. The lower branch of the loop represents measurements obtained by progressive addition of vapor to the system; the upper branch represents progressive withdrawal of the vapor. There was a time when the hysteresis loop was believed to
be spurious and was attributed to volatile impurities desorbed from the adsorbent surface (Coolidge, 1924). However, by the careful and extensive work of Allmand and collaborators (1929), Lambert and Foster (1932), Foster (1934), and Burrage (1934),the phenomenon has been established as real. That hysteresis results from the porous structure was confirmed by the experimental results of Kislev (1958) and Carman and Raal (1951). They found that when a powder material, normally showing a type I1 isotherm with no hysteresis loop, is compacted under pressure it will show a type IV isotherm with a clearly defined hysteresis loop. Compaction is believed to reduce the spaces between the particles to values low enough to initiate capillary condensation. While there has been a diversity of opinion (Zsigmondy, 1911; Rao, 1941; Kraemer, 1931; McBain, 1935; Cohan, 1938; Katz, 1949; Pierce and Smith, 1950; Carman and Raal, 1951; Foster, 1952; Pierce and Smith, 1953; Kiselev, 1958) as to the course of events along the two sides of the loop whether a multilayer is gradually built up on the walls and eventually merge to fill up the pores and/or capillary condensation fills up the pores at pressures below the saturation value, all the theories that have been advanced were based on capillary evaporation or condensation which is governed by the Kelvin equation (Thomson, 1871). In - =--2uy
(,”,>
rRT
‘Os’
(5)
For most solid-liquid systems the value of the contact angle, $, is much less than 90” and is often assumed to be zero. Thus
the vapor pressure over a liquid contained in a capillary will be lower than that over a plane surface of the liquid a t the same temperature. If a solid containing cylindrical pores of radius r is exposed to a vapor, the pressure of which is gradually increasing, the vapor would condense as liquid in the pores as soon as its pressure reaches the value p given by the Kelvin equation. Conversely, if the pores already contain liquid, evaporation would not commence until the vapor pressure in the system falls t o p , a value given by the Kelvin equation. In developing a mathematical expression for the adsorption isotherm, Langmuir (1918) approached the adsorption problem kinetically. He regarded the surface of the solid as an array of adsorption sites, each site being capable of adsorbing one molecule and the following equation was developed
X --_
BP X, 1+Bp Thus, the equation predicts an isotherm which approaches a maximum value asymptotically as the vapor pressure is increased. It has been shown that eq 1 satisfactorily describes Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977 321
x c 0 300
FBP
2ook/ 0 EXPERIMENTAL ADSORPTION
FOlKTS
a ISOTHERM PREDICTED BY EWATION ( 3 ) b ISOTHERM PREDICTED BY EQUATION ( 2 ) c ISOTHERM PREDICTED BY EPUATION ( 4 1
Figure 2. Adsorption of Nz on iron-aluminum oxide catalyst.
Figure 1. Five classes of adsorption isotherms. the adsorption isotherm of water vapor on wood a t 20 OC up to a relative humidity of 0.215 (Gorling, 1972). In 1938, Brunauer et al. (1938) extended the Langmuir mechanism to second and multiple molecular layers. For n layers adsorbed on the surface, Brunauer et al. (1938) developed the following equation
For an infinite number of layers, the equation becomes
-X- -
C$ [I + (C - 1)$1(1- $1
(3) Xm which is frequently referred to as the BET equation. It should be noted that eq 2 could be regarded as a general equation which reduces to Langmuir equation with n = 1. The BET equation has been proven successful up to a relative pressure of about 0.5 (Rounsley, 1961).Gorling (1972)has used the BET multilayer concept to follow the moisture isotherms of vegetable materials. About five layers describe the behavior of wood and potatoes a t room temperature, four layers for potatoes at 80-100 OC. Rounsley (1961) simplified and extended the original work of Brunauer et al. (1938) in order to develop a multilayer adsorption equation that could be applied beyond the BET normal range. By means of a statistical thermodynamic derivation, Rounsley developed the following equation
X _ -
x,
c+
l+(C-l)$
[-I
1-$
(4)
which represents the isotherm for the most probable distribution of molecules between layers. In this equation n represents the maximum number of layers which can be adsorbed, and the constant C has the same definition as that in the BET equation. I t should be noted that in this model the amount of adsor322
Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
bate in each layer is complete only a t saturation. Therefore, as $ approaches unity, the maximum equilibrium sorbate content is simply the product of the number of layer times the amount of sorbate in each layer. Rounsley (1961) has applied eq 4 to a large number of sets of sorption data, mainly for water vapor adsorbed on hygroscopic materials. He concluded that most hygroscopic materials will have pronounced capillary condensation if the humidity is sufficiently high. He referred to this as the reason for the deviation of the data from his predictions a t high relative humidities. T o illustrate this point and to compare the BET equation with the Rounsley equation, the data of Brunauer et al. (1938) are plotted in Figure 2 with three curves representing eq 2,3, and 4. It is noted that the experimental point a t $ = 0.7 deviates from these multilayer adsorption models. Moreover, it appears that the Rounsley's equation gives better predictions than the BET equation at higher values of relative humidities.
Theoretical Development From the previous review it is clear that the mechanism of multilayer surface adsorption adequately describes the moisture isotherm of hygroscopic porous solids over a moderate range of relative humidities. However, at higher humidities, the multilayer adsorption models fail to predict the equilibrium moisture content of porous materials. This is due to the fact that moisture condenses in the capillaries of the material at ambient vapor pressures less than the saturation value. This moisture contributes significantly to the total equilibrium moisture content at high values of relative humidities. Therefore, in order to develop a model that can describe the whole range of the moisture desorption isotherm of capillary porous materials, both the multilayer adsorption mechanism and the capillary evaporation mechanism must be accounted for. Physical Description of the Desorption Model. Consider a capillary porous solid with all the pores filled with water a t saturation pressure PO.Assume that the pore entries (or necks) are cylindrical although the whole pore has an irregular shape. Also assume that the angle of contact between water and the solid surface is zero which means that a hemispherical meniscus will be formed a t the neck of each pore. The radius of the meniscus will, therefore, be equal to the pore radius at the cross section where the meniscus is held, as shown in Figure 3. When the ambient water vapor pressure is reduced, evap-
Figure 3. Sketch of the assumed pore shape.
oration will not occur from the pore until the pressure falls to a value p 1 corresponding to the radius r l as given by the Kelvin equation ( 5 ) for the case of hemispherical meniscus (@ = 00). Once evaporation starts a t radius r l it proceeds with the meniscus stretching across the pore since any meniscus with radius larger than rl will be unstable at vapor pressure p1. Obviously, evaporation will cease only when the meniscus assumes a smaller radius than r l , e.g., at r2 which requires further lowering of the vapor pressure to a value of p2 in order to affect evaporation. Thus when the ambient vapor pressure is reduced from p o to p 1 , the liquid contained in pores with entry radii Lrl will evaporate. When it is further reduced to p2, the liquid contained in pores with entry radii >rz will evaporate, and so on. This will proceed progressively until the last portion of the pores empties at a pressure corresponding to the smallest entry pore radius. Therefore, in order to describe the desorption isotherm in terms of the capillary evaporation mechanism one has to know the pore size distribution of the sample. The most suitable and logical pore-size distribution determination technique to be employed for that purpose is mercury porosimetry. In this technique mercury is forced, under high pressure, to penetrate through the pores of a preevacuated porous sample. The pressure P required to force the mercury into a pore of radius r is given by (Moelwyn-Hughes, 1950)
P=
-2y cos @
r where y is the surface tension of mercury and C$ is the angle of contact between mercury and the solid surface. According to eq 6, a higher pressure has to be applied to force the mercury into a narrower pore. In order that the mercury may penetrate the irregularly shaped pore shown in Figure 3, a pressure of P1 corresponding to the entry radius r l , has to be applied. Once the mercury penetrates through rl it fills the pore up to the narrower cross section which has a yet smaller radius of r2. To penetrate through r z , a higher pressure of Pz has to be applied. Obviously, the mercury will fill up all the void space a t a pressure corresponding to the smallest entry radius. In practice, however, this is limited by the mercury porosimeter (e.g., 15000 psc porosimetry will go down to about 100-A pores).
Using the mercury porosimeter, the pressure is progres, penesively increased and the volume of mercury, V H ~that trates through the sample a t each value of P is measured. Plotting the values of the capillary pressure P (or alternatively , cuthe values of the entry radii, r ) vs. the values of V H ~the mulative size distribution curve can be constructed. Development of Desorption Model. The similarity between the mechanism of water evaporation from a pore filled with water and that of mercury intrusion through a preevacuated pore is obvious. Both mechanisms are controlled by the pore entry radius rather than the bulge radius. Evaporation from a water-filled pore with a meniscus of radius r l would start a t a relative pressure of p1IpO. Similarly, mercury would penetrate this pore through that entry radius r l under pressure P I . Evaporation of water would proceed until the meniscus assumes a smaller radius r2 where further lowering of relative pressure to p2Ipo is required to affect evaporation. Similarly, mercury would penetrate and fill the pore up to that cross section of radius r2 where a higher pressure of P2 has to be applied for penetration to proceed. Thus the volume of water evaporated upon lowering the relative pressure to p1Ipo will be approximately equal in volume to that of the mercury which penetrated through the pores with entry radius L r l . Let the volume of mercury which would fill up all the void space of the porous sample be VH,O, and the volume which ,). let the penetrates a t any pressure P, be V H ~ ( ~ Moreover, volume of water left in the porous sample in equilibrium with then the relative humidity q1 be X+,, (7) The above treatment assumes that all of the liquid water contained in a pore will evaporate if the relative humidity is reduced to a value corresponding to the radius of the pore neck. However, there are layers of water molecules adsorbed on the pore surface as a result of the physical adsorption forces. The effect of the presence of this adsorbed multilayer can be accounted for in two ways: (1)it reduces the radius of the meniscus; (2) the amount of water in the multilayer left on the walls of the pore will contribute to the total moisture content of the sample after the evaporation process has taken place. Effect of the Multilayer in Reducing the Meniscus Radius. When evaporation proceeds through the pore neck, the radius of the meniscus receding through the neck will assume a value of (ri - 2 ) rather than the neck radius r,, where 2 is the thickness of the multilayer left on the neck wall. Thus, the value of the relative humidity $i required for evaporation will be given by In $i =
-2yu (ri - 2 ) R T
If the thickness of the multilayer is not the same a t every point along the pore neck, then the maximum value of the thickness will determine the minimum radius the meniscus would assume along the neck. Consequently, it will control the value of )I needed to affect evaporation. T o evaluate the value of 2, the Rounsley multilayer adsorption equation will be used. According to Rounsley’s model, the multilayer builds up on the solid surface in an orderly efficient manner. All of the layers build up simultaneously upon increasing the relative humidity from zero to unity, but a t different rates. The lower layers build up more rapidly at lower humidities while the higher layers grow rapidly at higher humidities. If n is the maximum number of layers that could be adsorbed, then at any value of relative humidity there will be n layers adsorbed on the surface but with varying amounts of water molecules in each layer. Each layer will be completed only a t saturation. Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
323
It may be assumed, therefore, that at higher humidities there will be a point along the pore neck where the multilayer assumes its maximum thickness given by 2 = n-D, where D is the molecular diameter of the water molecule and n may be obtained from Rounsley's model. It is this maximum value which controls the relative humidity required for evaporation. The Amount of Moisture in the Multilayers Left on the Surface after Capillary Evaporation. When the relative humidity is reduced from saturation to any value gL,the liquid in pores with the corresponding neck radii will evaporate. However, after the vaporation process has taken place at this particular $ I ,there will still be a multilayer of moisture left on the surface. This multilayer is in equilibrium with the ambient vapor pressure. The amount of water in this multilayer, X ' G ~should , be added to the amount X + ,given by eq 7 in order to get the total moisture content of the material at $,. The value of XI+, could be estimated from the relation X'+, = f,.X,$,, where Xs$, is the amount of moisture adsorbed on the surface at $, as predicted by the Rounsley's multilayer equation. f , is the fraction of surface area which is available for multilayer adsorption at $,. It is equal to the surface area of the pores from which evaporation had taken place at divided by the total surface area of the sample. In order to have a reasonable estimate of f I , the following procedure is suggested. (1) The cumulative mercury porosimetry curve is divided into N divisions. Each division represents the volume of pores . ~ V H ~whose ( ~ ) entry radii lie between rr and r,+l. (2) Each division is treated as equivalent to one long cylindrical pore of volume L . I V H ~ (and ~ ) of radius (r, r,+J/2. (3) The length, Lequlr,of this long cylindrical pore is calculated from
+
(9)
(4) The surface area, Si,of this equivalent pore is obtained by
(5) The fraction fi of the surface area of the pores which have entry radii Lri will be given by i-1 P
O
J=1
Obviously, the value of ZN-l,,l SJ does not represent the actual total surface area of the sample. However, dividing EL-l1=1S, by ZN-',=l SI gives a reasonable estimate of f L Analytical Expression for the Desorption Model. Considering both the mechanisms of capillary evaporation and multilayer adsorption and their interaction, the amount of moisture XfiLleft in the porous material upon lowering the relative humidity from saturation to any value qL,will be given by (12) = V H 2 - VHg(r,) + fr-l,xsi
x$,
X,,is the value predicted by Rounsley's equation and VH,O, V H ~ ( ~P,, , ) , and fL-l are obtained from the mercury porosimetry data. At high relative humidities near saturation, eq 12 indicates the predominance of the capillary evaporation mechanism since the fraction fL-l, and consequently the surface adsorption term (fI-l.Xs,),will be small. As the relative humidity $j is reduced, f i - l becomes larger and the contribution of surface adsorption to the total mixture content increases. As $iis 324
Ind. Eng. Chem., Fundam.. Vol. 16, No. 3, 1977
I
1
1
L-a 1 I
1
Figure 4. Sketch of the setup used for measuring the sorption isotherm: B, electrobalance; S, sample container; W1, water bath; Wa, water bath; C, control unit; R, recorder; f, water reservoir; MI, thermometer; Ml, thermometer; H, mercury manometer; L, cold trap; D, drying filter.
further reduced, the value of V H ~ ( approaches ~,) that of VH,O while the fraction f i - l approaches unity and, eventually, eq 12 reduces to Rounsley's equation. To conclude, eq 12 predicts the moisture content of a capillary porous solid over the whole range of relative humidities during desorption. It takes into account the interaction of the two major mechanisms encountered in a desorption process, namely, the capillary evaporation and the multilayer surface adsorption. Experimental Investigation In order to test the applicability of the desorption model, experimental investigation of the moisture sorption isotherm of some capillary porous solids has been conducted utilizing the gravimetric vacuum sorption technique. Selection of the Samples. The samples required for this study must be capillary porous solids with strong hygroscopic properties to have capillary condensation at high relative humidities. Therefore, the samples must have pore sizes in the range of tens to hundreds of angstrom units. Two catalyst materials were found to meet this condition: (1)Girdler G-3 iron oxide catalyst, l/4 X lh-in. tablets; (2) Girdler G-3A chromium-promoted iron oxide catalyst, 3/8 X 3/16-in.tablets. Chemically, both samples are mainly magnetic iron oxide, Fe:j04. The first sample was provided by Chemetron Corporation, Chicago, Ill., while the second sample was provided by Girdler Chemical Inc., Louisville, Ky. Experimental Apparatus. The experimental setup as shown in Figure 4 is composed of the following: (a) the Cahn electrobalance (B) to follow the weight changes of the sample upon changing the ambient relative humidity; (b) the sample container (S) which is immersed in a water bath controlled to f0.05 "C; (c) the manometer (H) where the readings were taken by means of a cathetometer; (d) the water reservoir (f) which supplies the system with water vapor at controlled temperature; and finally, (e) the cold-trap (L) to prevent the water vapor from reaching the vacuum pump. The connecting tubes were all wrapped with heating tapes and insulated with 1 in. of aluminum-backed asbestos. The tubes were heated to a temperature slightly above the isotherm temperature (the temperature of the sample) in order to prevent condensation in the system during the run. The temperature at points designated by "T" were measured by copper-constantan thermocouples. The mercury porosimetry measurements were made on a mercury porosimeter (American Instrumentation Co.) with a maximum pressure of 15 000 psi. The procedure recommended in the operational manual provided by the company was followed.
EXPERIMENTAL ADSORPTICU POINTS EXPERIMENTAL DESORPTW WlNTS ISOTHERM PREDICTED BY EOUATION 141 ISOTHERM PREDICTED BY EOUATION ( 7 I ISOTHERM PREDICTED BY EOUATDN 112)
0
-J
le-
0
b C
515-
1
i * 512-
s
6L I>: , t , ,
~,
, : ,
II
0 0
01
02
03
04 += PIP0
05
06
07
08
10
09
F i g u r e 5. Experimentally determined and predicted moisture sorption isotherms of sample no. 1 a t 40.1 "C.
0
a b
"t
I
A 2
C
EXPERIMLNTAL ADSORPTIW POINTS EXPERIMENTAL DESORPTION POINTS ISOTHERM PREMCTED B Y EOUATION ( 4 1 ISOTHERM PREDICTED BY EOUAllON 1 7 1 ISOTHERM PREDICTED BY EOUATIW 1121
C
i
F i g u r e 7. Mercury porosimetry curves for sample no. 1 and no. 2.
I
O
'
3-
F i g u r e 6. Experimentally determined and predicted moisture sorption isotherms of sample no. 2 a t 40.1 "C.
Experimental Results. The experimentally determined moisture adsorption and desorption isotherms of samples I and I1 are shown in Figures 5 and 6. The isotherms were determined at 40.1 "C. Both isotherms show hysteresis loops in the high humidity range. The mercury porosimetry curves of both samples I and I1 are shown in Figure 7. Each curve approaches the saturation value a t very high pressures. At 15 000 psi, the value of VH,O for sample I is 46.1 $L and that for sample I1 is 21.8 $L. Discussion Since the desorption isotherm represents measurements obtained by successive reduction of the ambient water vapor pressure from the saturation value, it is applicable to the drying process of the material considered. Therefore, the discussion will be limited to the desorption part of the isotherm. In Figures 5 and 6 the isotherms predicted by the Rounsley model, eq 4, are plotted along with the experimental data. Since Rounsley's equation describes only the multilayer surface adsorption mechanism, it shows good agreement with the data obtained a t low values of relative humidities. Over this low range of humidities, the mechanism of multilayer surface adsorption provides an adequate description of both the adsorption and desorption processes. Rounsley's equation follows the experimental data up to a relative humidity of about 0.70 for sample I and 0.74 for sample 11. For sample I, the value
of the constant C is 26.05, the value of X, is 0.9033, and that of n is 4.2614. For sample 11, the values of C, X,, and n are 18.7, 0.9428, and 4.1819, respectively. The C values for both samples are higher than 2, which is characteristic for type IV isotherms, Le., isotherms showing a knee at the lower end of the relative pressure range. The constant C is a function of the heat of adsorption and the temperature. Both sample I and I1 are mainly iron oxide and the isotherms were determined at the same temperature. Therefore, the values of N and C for sample I are very close to those for sample 11. The calculated values of n are within the range of the values obtained by Rounsley (1961). A t relative humidities higher than 0.70 for sample I, and 0.74 for sample I1 the experimental data of both adsorption and desorption deviate appreciably from the curve predicted by Rounsley's multilayer theory. This deviation is due to the capillary condensation of moisture which fills up some of the pores and contributes significantly to the total equilibrium moisture content of the material at a given relative humidity. Thus, the capillary condensation of moisture has to be accounted for in order to predict the high relative humidity data as well. However, it should be pointed out that the miltilayer adsorption mechanism affects the total equilibrium moisture content a t all relative humidities. In both Figures 5 and 6 desorption isotherms predicted by eq 7 , which is based exclusively on the capillary evaporation mechanism, are shown. Since it does not make an account of the effect of multilayer surface adsorption at high values of relative humidities, eq 7 utilizes only the mercury porosimetry data to predict the amount of moisture left in the material after evaporation takes place at a given relative humidity. Both curves show poor agreement with the experimental data a t high relative humidities and completely fail t o account for the data obtained at lower humidities. It is also obvious that both curves show a sharp discontinuity with Rounsley's; both mechanisms of multilayer adsorption and capillary evaporation have to be simultaneously considered in order that the whole range of the desorption isotherm may be predicted. On the other hand, the desorption curve predicted by eq 12 shows good agreement Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977 325
with the desorption experimental data over the whole range of humidities. Moreover, eq 12 provides a smooth transition in the desorption isotherm from a region where the capillary condensed liquid predominates the equilibrium moisture content to a region where the multilayer surface adsorption exclusively determines the equilibrium moisture content. However, the agreement between the desorption curve predicted by the model and the experimentally determined data over the transition region is not as good as it is over the other regions of the isotherm as shown in Figures 5 and 6. Equation 1 2 predicts a higher value of $ to evaporate the moisture from the narrowest portion of the pores. This may indicate that the actual radius of the water meniscus in a pore neck is smaller than the assumed hemispherical value. This effect becomes more significant for the narrower pores. In order to apply eq 12, the constants C, X,, and n have to be determined as dictated by the Rounsley model. This requires the knowledge of four or five experimental points in the low humidity range. Nevertheless, it should be pointed out that during these experiments, the time required to reach equilibrium at low relative humidities was relatively short. While some equilibrium points were reached after two days in the high humidity range, it took only about 1-2 h to determine an equilibrium point at low relative humidities. Although the applicability of the new model has been experimentally verified using a mercury porosimeter of 15 000 psi maximum pressure and porous samples which were saturated with mercury a t that pressure, it is believed that its applicability may be extended to samples with finer pores if porosimeters with higher maximum pressure were used. Therefore, the new model may be used to predict the desorption isotherm for any hygroscopic porous material provided that a few experimental points in the low humidity range are known and mercury porosimetry data are available. Finally, this model may be used to predict the desorption isotherms of materials having the same chemical composition and/or surface properties but different pore-size distributions. The knowledge of a few experimental sorption points obtained in the low humidity range for one material only is required to determine the parameters C, n, and X,. The other material can be assumed to have the same values of C and n. However, the value of X, for each material can be estimated by multiplying the determined value of X , by the ratio of the specific surfaces of the material for which X , is to be estimated and that of the original material. Knowing the values of C, n, and X, along with the mercury porosimetry data for each material, the desorption isotherm of any material can then be predicted by eq 12.
Conclusions A model which accounts for both mechanisms of capillary evaporation and multilayer surface adsorption is developed. It combines both mechanisms in one mathematical equation which can be used to predict the moisture desorption isotherms of hygroscopic porous materials over the whole range of relative humidities. The model shows the predominant
326 Ind. Eng. Chem., Fundam., Vol.
16, No. 3, 1977
contribution of capillary evaporation to the desorption process a t high relative humidities and at low relative humidities, it reduces to Rounsley's multilayer surface adsorption equation.
Nomenclature B = parameter in eq 1varying with the gas and the solid C = constant given by, C = e ( E l - L ) / R T ) D = molecular diameter E1 = heat of adsorption in the first layer f i = fraction of surface area of the pores which have entry radii 2 ri L = heat of condensation Lequiv = length of the equivalent pore n = maximum number of layes which can form on the surface p = ambient vapor pressure po = saturation vapor pressure r = radius of the capillary R = gas constant S = surface area of the equivalent pore T = absolute temperature u = molar volume of the liquid V H ~= volume of mercury which penetrates the porous sample under pressure P V H =~ volume ~ of mercury which fills up all the void space of the sample X = equilibrium moisture content, wt % X, = amount of moisture in a complete monolayer, wt % 2 = maximum thickness of the multilayer y = surface tension of the liquid 4 = angle of contact between the liquid and the solid surface $ = relative humidity, equal to P/Po Literature Cited Allmand. A. J., Hand, P. G. T., Manning, J. E., J. Phys. Chem., 33, 1694 (1929). Brunauer, P. H., Emmett, P. H.. Teller, E., J. Am. Chem. Soc., 60, 309 (1938). Burrage, L. J., Trans. faraday SOC.,30, 317 (1934). Carman. F. C.. Raal. F. A., Proc. Roy. SOC.London, Ser. A, 209, 59 (1951). Cohan, L. H., J. Am. Chem. SOC.,60, 433 (1938). Cook, M. A,, J. Am. Chem. Soc., 70,2925 (1948). Coolidge, A. S.,J. Am. Chem. SOC.,46, 596 (1924). Foster, A. G., J. Am. Chem. SOC.,74, 1806 (1952). Foster, A . G., Roc. Roy. SOC.London, Ser. A, 146, 129 (1934). Gregg,S.J.. Sing, K. S.W., "Adsorption, surface area and Porosity" pp 45, 135, Academic Press, London and New York, N.Y., 1967. Gorling. P.. "Drying Principles and Practice", R. B. Keey, Ed., p 25, Pergamon Press, New York, N.Y., 1972. Halsey, G. D., J. Chem. Phys., 16, 931 (1948). Katz, S. M., J. Phys. Chem., 5 3 , 1166 (1949). Kiselev, A. V., "The Structure and Properties of Porous Materials", p 95, Butterworth. London, 1958. Kraemer, E. 0.."A Treatise on Physical Chemistry", H. S. Taylor, Ed., p 1661, Macmillan, New York, N.Y., 1931. Lambert, B., Foster, A. G., Proc. Roy. SOC.London, Ser. A, 136, 363 (1932). Langmuir, I., J. Am. Chem. SOC.,40, 1361 (1918). McBain. J. W., J. Am. Chem. SOC.,57, 699 (1935). Moelwyn-Hughes, E. A . , "Kinetics of Reactions in Solutions", p 7, Oxford, 1950. Pierce, C., Smith, R. N., J. Phys. Chem., 54, 784 (1950); 57, 61 (1953) Rao, K. S.,J. Phys. Chem., 45, 500 (1941). Rounsley, R. R., A./.Ch.E.J., 7 (2), 308 (1961). Thomson, W.T., Phil. Mag., 42, 448 (1871). Zsigmondy, R., 2.Anorg. Chem., 71, 356 (191 1).
Received for review February 13,1976 Accepted March 16, 1977
