Drying of acidic macroporous styrene-divinylbenzene resins

Drying of acidic macroporous styrene-divinylbenzene resins

65 Reactive Polymers, 21 (1993) 65-76 Elsevier Science Publishers B.V., Amsterdam Drying of acidic macroporous styrene-divinylbenzene resins M. Ibo...

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65

Reactive Polymers, 21 (1993) 65-76

Elsevier Science Publishers B.V., Amsterdam

Drying of acidic macroporous styrene-divinylbenzene resins M. Iborra, C. Fit6, J. Tejero *, F. Cunill and J.F. lzquierdo Chemical Engineering Department, Faculty of Chemistry, UniL'ersity of Barcelona, Mart[ i Franqubs 1, 08028 Barcelona, Spain

(Received February 12, 1993; accepted in revised form June 25, 1993)

Abstract

Drying of the acidic macroporous resin Lewatit K 2631 has been studied. Oven drying at 84-112°C and 0.015 atm water partial pressure shows that the resin behaves as a fibrous solid. Drying by methanol percolation at 20°C leads to moisture contents in the resin slighty lower than in the previous method. Finally, oven drying of the resin previously washed with methanol shows that the residual water content increases with increasing amounts of the methanol. Estimates of the water-effective diffusivity suggest that the last method gives rise to a lower polymer shrinkage. The changes in the polymer morphology probably result from the formation of a large number of small pores. Keywords: resin drying; water-effective diffusivity; Lewatit K 2631; methanol

Introduction

Macroporous resins consist of aggregates of nearly spherical microparticles of polymer gel interspersed by macropores. Commercial polymers are styrene-divinylbenzene (DVB), homogeneously sulfonated and containing 10-30% DVB. In the dry resins, gel structure is collapsed and polymeric chains are as close as atomic forces will allow, so that the matrix is completely impervious to molecules unable to swell the resin. Polar molecules permeate through the network of hydrogen bonds between sulfonic groups, insert into it

*

Corresponding author.

and, as a result, swell the polymer. Moreover, in the case of organic molecules, the longer the aliphatic chain of the permeating molecule, the greater its affinity to the polymeric matrix [1,2]. Both interaction types work together in the resin swelling and, generally, the higher the polarity of the permeating molecules, the greater the swelling. Specifically, water, which is able to form up to four hydrogen bonds with the sulfonic groups, permeates easily, so that ion-exchange resins can retain relatively large amounts of water. In this way, cationic resins have been proposed as desiccants of organic solvents [3-6]. Sorption capacity for water depends mainly on the chemical nature of the cation. For monopositive cations, a higher

0923-1137/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

M. Iborra et al./ React. Polym. 21 (1993) 65-76

66

charge density of the cations results in a larger amount of adsorbed water. Consequently, as hydrogen is the smallest monovalent ion, acidic resins sorb more water than the other cationic resins. However, the presence of water is unfavourable in most industrial applications of resins in non-aqueous systems. For instance, catalytic activity of macroporous resins, which are widely used as catalysts because they offer the possibility of operating in low-polar media, generally decreases in the presence of water. Moreover, water can promote undesired side-reactions. Consequently, resins should be dried to remove as much sorbed water as possible before using them as catalysts. It should be emphasized that the pore structure can collapse in the drying process owing to the cohesional forces between the polymer chains enhanced by the loss of water. In macroporous resins this collapse effect is stronger in smaller than in larger pores, and it decreases on increasing degree of crosslinking [7]. The collapse can be avoided by washing the resin, before drying, with solvents similar to water but with smaller

TABLE 1 Physical properties of Lewatit K 2631 Skeleton Structure Active group Nominal degree of crosslinking Moisture a, maximum Bead diameter (minimum 99%) Mean bead diameter b Uniformity coefficient b Porosity Mean porous diameter Surface area c Exchange capacity d Temperature stability range a Determined b Determined c Determined Determined

Styrene-DVB Macroporous Sulfonic 18% 58-60% 0.25-1.6 mm 0.73 mm 1.5

50%o(v/v) 650 A 36 m 2 / g 4.83 m e q / g dry resin - 2 0 to 120°C

by Karl-Fischer titration. in the laboratory by sieving. by the method of Fisher and Kunin [9]. by the BET method.

solvating power, such as methanol, to remove the water sorbed on the resin [8]. To reduce the water content, among other methods, resins can be dried in an oven at atmospheric pressure or by percolation of methanol. Both methods are simple and can be applied in the industry. Consequently, this paper deals with the effect of both drying methods on the final water content and the structure changes of Lewatit K 2631. This resin was selected because it is used as a catalyst in numerous industrial reactions, in particular in the etherification processes to obtain methyl tert-butyl ether (MTBE), ethyl tert-butyl ether (ETBE) and tert-amyl methyl ether (TAME).

Experimental Lewatit K 2631 (previously SPC 118 BG; Bayer, Leverkusen, Germany), with a mean water content of 59.3%, is a macroporous sulfonic styrene-DVB resin whose characteristics are given in Table 1. Three series of experiments were performed in our laboratory which was held at 20°C and 65% relative humidity (0.015 atm water partial pressure). In the first series, the resin was dried in an oven (800 W) at 84112°C and atmospheric pressure. A 150-g amount of wet resin was placed in a shallow capsule and dried by blowing air across the surface. At given times, 0.5-2 g of resin were taken out, introduced into a sealed vial and cooled at 20°C. These samples were distributed in four parts, and their water content was determined by the Karl Fischer titration method. The average of these measures was taken as the moisture content of the sample. Karl Fischer titrations were carried out in a Karl Fischer titrator (Baird and Tatlock AF-5) using methanol and Karl Fischer solution (Merck, Darmstadt, Germany) consisting of pyridine, 802, 12 and 2-methoxyetha-

M. Iborra et al./React. Polym. 2I (1993) 65-76

67

nol. To avoid problems of water diffusion within the polymer, a 5-min extraction with methanol before titration was carried out. In the second series, drying by methanol percolation (HPLC Petrochem, 99.8% pure, < 0.05% water) was carried out at 20°C. Resin beds (20-80 cm 3, 10-40 cm length) were placed in a glass column (3 cm I.D.), and the methanol was added up to 1 cm over the bed. Then, the methanol was allowed to flow down through the bed at a flow-rate of 3 cm3/min. The water content in the eluted methanol was measured by the Karl Fischer titration method. Finally, in the third series, 150 g of wet resin, previously washed with methanol in the glass column by flowing through a given volume of the alcohol, were dried in the oven as mentioned above.

Re sul ts and d i s c u s s i o n

Application method

of the Karl Fischer titration

The Karl Fischer titration method is the most convenient method (accuracy + 2%) for fast measurements of water contents higher than 1 wt% in ion-exchange resins [10,11]. In % water

,~

ss

!

l i 60

i ! 55

T

2

g 0 0

J 2

T

+

• [

/ 4

6

0 t (rain)

10

12

14

16

Fig. 1. Karl Fischer titrations of Lewatit K 2631. e, Commercial (wet) resin; A, resin dried in the oven at 112°C for 48 h. The intervals for a 95% confidence level are shown.

order to get reliable values of the moisture contents it is essential that previous extraction of water by methanol will be carried out long enough [12]. As can be seen in Fig. 1, for a completely swollen (wet) resin, extraction with methanol is not necessary, whereas for low water contents (air-dried resins), extraction should take at least 5 rain. So, to be sure that the moisture contents measured are reliable, an extraction time prior to titration of 5 rain was established.

Ot~en drying of the wet resin Fig. 2 shows the drying curves obtained at 84, 90, 104 and 112°C. Their shapes suggest that Lewatit K 2631 behaves as a fibrous solid [13], holding moisture as an integral part of its structure. Drying curves show a short constant-rate period, which ends at high values of moisture content (the critical moisture content), and an only falling-rate period. The bulk of the drying occurs in the fallingrate period which is controlled by liquid diffusion through the solid. The equilibrium moisture contents are high, indicating that a significant amount of water is closely held in the resin structure. The slope of the constant-rate period (drying rate) increases with temperature and, as a result, the constant-rate period decreases. Exact measurements of the critical moisture are difficult because it is essential to determine accurately the end of the constant-rate period: a mean value of 0.6 _+ 0.1 kg water per kg dry resin (7 mol water per mol sulfonic group) is acceptable. In the constant-rate period, drying proceeds with the solid not directly influencing the drying rate which is independent of the moisture content. At steady state, the temperature of the wet solid is approximately the wet bulb temperature of the air, and the water evaporated from the surface is continuously replaced by the movement of liquid from the interior of the solid. During this

68

M. Iborra et a t / R e a c t . Polym. 21 (1993) 65-76

period the moisture flow to the surface is so fast that a continuous film of water always covers the solid surface, and the drying-rate is controlled by the heat transfer rate to the solid surface. As heat is tranferred by natural convection, the temperature of the solid surface, T/, can be computed by [13]

dW dt

. . . .

hfA ( T - T,.) msh

amount of liquid at the surface is insufficient to maintain a continuous film covering the entire drying area, and the solid temperature increases up to the oven temperature. This temperature change is accompanied by a sharp increase in the drying rate. From this moment, diffusion of water within the solid controls the drying rate, which is lower and lower. The water content at the surface is at, or very near, the equilibrium value. The velocity of the air has little or no effect on this value, and its humidity influences the process mainly through its effect on the equilibrium moisture content. For a bed of nearly spherical beads, moisture-drying time data are ade-

(1)

The solid temperature (20 + 8°C) is much lower than the oven temperature, since the water partial pressure is very low. When the water content is under the critical value, the

W (kg water/kg dry resin)

C)

A)

1.5

0.6

i I

0

t

I

I

l

]

D)

B)

1.5

0.5

O

'

0

5

10

16

i

20

*

i

x

26

30

0

5

10

15

i

J

i

20

25

30

35

t (h) Fig. 2. Drying curves (total moisture on a bone-dry basis versus drying time) of Lewatit K 2631 in the oven. (A) 84°C; (B) 90°C; (C) 104°C; (D) 112°C.

69

M. lborra et al./ React. Polym. 21 (1993) 65-76 TABLE 2 Residual water content in the resin dried in the oven and effective diffusivity values for water within the polymer T

(°C) 84 90

104 112

VMeOH/ Vresin

We × 102 (kg w a t e r / kg dry resin)

DXl012 (m2/s)

0 0.5 0 (/.5 1 2.5 0 0 0.5

4.42 4.97 3.36 3.86 3.87 4.14 2.04 1.90 2.02

1.3 12.5 2.2 5.6 8.4 35.5 3.8 3.6 11.2

quately described, if ( W < 0.6, by [14]

6 I

Wcrit - W e = 77-2 e x p

- D

We)//(Wcrit- IV) (t

/'crit)

1 (2)

Control by water diffusion starts at a mean water content in the resin of 0.4 kg water per kg dry resin (between 4 and 5 mol of water per mol of sulfonic group). These 4 mol of water can be regarded as water of solvation of the sulfonic groups while the remainder is the free water which cause the swelling of the polymer network until the contractile tension balances the osmotic swelling [15-17]. Water-effective diffusivity can be estimated by fitting eqn. 2 to the moisture content of the solid. Table 2 shows the values obtained, which increase with temperature as could be expected since diffusion is a rate process. Testing their reliability is difficult because water diffusivity values within nonswollen polymers are scarce in the literature. Even so, it is worth mentioning that values of Table 2 practically agree with effective diffusivities found for water sorption from alcoh o l - w a t e r mixtures by alcohol-saturated resins ( - 1 × 10 -12 m 2 / s ) [18]. From the temperature dependence of the water-effec-

tive diffusivity, an activation energy of 35 k J / m o l was found, which is in good agreement with those of diffusion processes in resins with more than 8% DVB (25-42 k J / m o l ) [19-21]. The equilibrium moisture content decreases when the temperature increases (see Table 2), as well as the drying times at which these values are reached. It is noteworthy that more than 14 h are needed at 112°C. Equilibrium contents are equal to 0.5-0.2 mol water per mol sulfonic group. Each molecule of water is hydrogen-bridged to three sulfonic groups on the average [22]. As surface area of the water molecule is 12.5 .~2 [23], the total amount of water at the surface would be 9.6 × 10 -3 kg water per kg dry resin assuming a monolayer coverage. This value is less than a half of the measured values. As a consequence, we can infer that most of the equilibrium moisture remains within the polymer, in agreement with the fact that sulfonic group concentration within the polymer is twice that of surface [24]. Adsorption equilibrium data for water at about 100°C are scarce in the open literature. Table 3 compares our data with those of Kabel and Johanson [25], who determined water adsorption isotherms at 79-116°C and 0-0.1 atm water pressure on the microTABLE 3 Comparison of moisture equilibrium data with those of Kabel and Johanson [25] T (o)

We × 102 (kg w a t e r / kg dry resin)

n~ (mol water/ mol group sulfonic)

This work 84 90 104 112

4.42 3.36 2.04 1.90

0.51 0.39 0.24 0.22

Kabel and Johanson 84 90 104 112

5.4 4.4 2.7 2.1

0.56 0.45 0.28 0.21

M. Iborra et aL / React. Polym. 21 (1993) 65-76

70 -In We

V(MeOH)/V(resin)=-O ~l-- V(MeOH)/V(resin)=0.5

3

"-.... 2

2.5

i

J

2.6

2.7

i

i

2.8 1/T 10 3 (K" 1)

2.9

3

Fig. 3. Moisture equilibrium versus the reciprocal of temperature. Solid line, oven drying; dotted line, oven drying after washing with methanol.

porous resin Dowex 50X-8. Values of Table 3 were obtained by interpolating their data through a Langmuir model. The data of Kabel and Johanson [25] are slightly higher since water adsorption increases as the DVB content decreases because of the higher number of acidic groups. When comparison is done on a mol w a t e r / m o l sulfonic group

basis, both series agree fairly well within the limits of the experimental error. Fig. 3 shows the dependence of the equilibrium moisture content on temperature. As water partial pressure was nearly the same at each temperature, Fig. 3 is, in fact, an isobaric plot. The heat of adsorption can be computed from the slope, assuming that at

TABLE 4 Comparison of heat of adsorption of water at 0.015 atm partial pressure of water with literature data and key resin properties Ref.

A H~os (KJ/mol)

Resin

This work: drying in the oven This work: drying in the oven after percolation with methanol Cunill et al. [26] Sundheim etal. [15]

-31

Lewatit K 2631

Kabel and Johanson [25] Gottifredi et al. [27] R e d i n h a and Kitchener [16]

+4

-36 + 2 - 34 + 3 -37 +5 - 34.0 + 0.5 - 37.2 + 0.7 - 39.6 - 39.6 - 45.4

Lewatit K 2631 Lewatit SPC 112

D O W E X 50X-8 Amberlite IR200 Zeo-Karb 225 Dowex 50

DVB (%)

Resin type

18

Macro

18 12 2 13 23 8 20 4.5 8 16

Macro Macro Micro

Micro Micro Micro

71

M. Iborra et al./React. Polym. 21 (1993) 65-76

low water pressure adsorption follows a Langmuir model. As shown in Table 4, the obtained value, 31 +_ 4 k J / m o l , agrees with those found in the literature within the limits of the experimental error. It is worth noting that these values were found through different experimental methods, i.e. a d s o r p t i o n desorption experiments [15,16,25], kinetic study of a non-aqueous reacting system in which the decreasing effect on the reaction rate of little amounts of water was measured [26], and kinetic study of a system in which water was a reaction product [27]. So, we can conclude that true equilibrium values have been found. Moreover, as the mean standard liquefaction enthalpy of water at 1 atm and 25-100°C is - 4 1 k J / m o l , we can conclude that water adsorbs physically.

% water

A)

20 mi

80



40 mt (I)



40 ml (2)



40 ml (31



80

rnl

60

40

20

W

(kg water/kg dry

resin)

1.5

:a/

Drying by methanol percolation

t

0.5

Beds of 20, 40 and 80 cm 3 of wet resin were dried by methanol percolation. Fig. 4A shows a plot of the water content in the eluted m e t h a n o l versus the volume of methanol flowing through the bed given as VMeoH/Vresi n. AS can be seen, the experimental curves are practically the same irrespective of the bed size. As a result, drying by methanol percolation is not controlled by external mass transfer. Moreover, as the resin is in a swollen state, we can expect that percolation is not controlled by diffusion processes either. At first, the water content in the eluted methanol lessens swiftly, but for VMeoH/Vresin > 3 it lessens very slowly to reach the water content present in the commercial methanol. The moisture content of the resin can be estimated from Fig. 4A, since the removed water is given by the area u n d e r the curve. As can be seen from Fig. 4B, for VM~oH/Vr~si . > 3 the removal of supplementary moisture is a methanol-consuming process with an insignificant reduction of moisture. At VM~oH/Vresi n = 5 , the water content in the resin, almost in equilibrium

methanol

in

100

0

~--

0

J

=

J

'

0.5

1

1.5

2

J

2.5

i

3

'

',

3.5

'

~

4

'

I

4.5

'

5

V(MeOH)/V(resin)

Fig. 4. Percolation experiments with methanol. (A) Water content in the eluted methanol versus the volume of methanol percolated at different bed volumes; (B) resin moisture cersus the volume of methanol percolated.

with methanol, is 0.017 kg water per kg dry resin. This value is slightly lower than the equilibrium values found in the oven drying method• Oven drying of the resin previously washed with methanol As shown in the above section, methanol percolation is an effective m e t h o d of water removal. The question still remains to what is the nature of the removed water. Therefore, resin samples previously washed with methanol, using VMeOH/l/resin ratios of 0.5, 1

72

M. Iborra et al. /React. Polym. 21 (1993) 65-76

and 2.5, were dried in the oven at 84, 90 and 112°C. These V M e o H / V r e s i n ratios were selected to assure the removal of a significant amount of water before oven drying. As shown in Figs. 5 and 6, only the falling-rate period is completely characterized. The presence of methanol greatly increases the experimental error because water solved in the methanol was titrated jointly with water still retained in the resin. Fig. 5 shows that the falling-rate period starts when the solid moisture is about 0.35 kg water per kg dry resin (4 mol water per mol sulfonic group),

W (kg water/kg dry resin) 0.6

A) 0.5 0.4

"44"

+

4-

++

0.3

0,2 0.1

t

0

i

i

i

i

B)

0.5 0.4 0.3 .v+

+

0.2

W (kg waterlkg dry resin)

0.6

0.1

A)

q-

0

0.5

*

J

I

i

J

c) 0.4

0.6

0.3

0.4

0.2

0.3

0.1

0.2 0.1

i

0

B) 0

0.5

I

I

t

l

I

I

5

10

15

20

25

30

0.4

35

t (h)

0.3

Fig. 6. Drying curves of resin samples percolated with methanol at 90°C. (A) VMeoH/V~esin = 0.5; (B)

0.2

VMeOH / Vresin

= 1;

(C) VMeOH / Vresin = 2.5.

0.1 0

c)

0.5 0.4 0.3 0.2 0.1

x...

0 0

5

10

15

20

25

30

35

t (h)

Fig. 5. Drying curves of resin samples percolated with methanol using a VmeoH/ ~resin ratio of 0.5. (A) 84°C; (B) 90°C; (C) 112°C.

this being the time at which this period starts independent of temperature within the limits of the experimental error. However, as can be seen from Fig. 6, the falling-rate period starts at lower drying times when the VMeoH/Vresin ratio increases, as the total amount of water in the resin is lower when the used V M e o H / V r e s i n ratio increases. As Table 2 shows, the equilibrium moisture contents are higher than those obtained without methanol percolation, and increase with the volume of washing methanol. A similar fact was reported for the vacuum drying of Lewatit SPC 118 at 95°C, even

M. Iborra et al./ React. Polym. 21 (1993) 65-76

though it was explained by the experimental error [12]. Nevertheless, as shown in Fig. 3, the dependence of equilibrium moisture with temperature is similar with that of drying without methanol pretreatment. From the slope we can deduce a heat of adsorption of -36_+ 2 kJ/mol, which agrees fairly well with the value obtained without methanol percolation and with literature data, as Table 4 shows. The solid behaviour in the falling-rate period agrees with a diffusion mechanism, but a clear variation of the effective diffusivity with temperature was not found because of higher experimental errors. These effective diffusivity values are higher than those obtained without methanol washing (see Table 2) and increase with the volume of methanol used. These facts suggest that the action of methanol gives rise to a more open structure with more of small pores. As a result, water diffusion is easier and more sulfonic groups, on which water can adsorb, are likely available. To check the possible changes in the resin structure caused by the drying processes, the surface area (BET method) of some samples was measured, and their morphology studied by scanning electron microscopy. BET surface area is not influenced by the drying method except for the resin washed with a VMeoH/Vresi . ratio of 5 and then oven-dried (Fig. 7). The higher suface area of this sample responds to the formation of a large number of small pores [8]. Moreover, as can be seen in Fig. 8, samples washed with methanol and then dried in the oven show a less collapsed structure than those oven-dried without pretreatment with methanol and also than those percolated with methanol. This fact could be due to the formation of small nodules from bigger aggregates, and agrees with the increase of water-effective diffusivities observed. However, as only the sample washed with a VMeOH/Vresi n ratio of 5 clearly shows a higher surface area, we can assume

73

4O

Surface area (m2/g)

38

34

32

A"

Oven drying



. . * h . n o , * . . h I . Q • o v , . d.y*nQ



Methanol percolation

30

0

1

2

3

4

V(MeOH)/V(reein)

Fig. 7. Surface area of Lewatit K 263l from the different drying methods. The intervals for a 95% confidence level are shown.

that withdrawing significant amounts of water is essential in order to observe surface area increments. In the oven drying, removal of free water, whose sorption is responsible of the resin swelling [17], occurs in the constant-rate pe-

Fig. 8. (A) Microphotography of a sample dried by methanol percolation. (B) Microphotography of a sample washed with methanol and subsequently dried in the oven. VMeOH/ Vresi. = 5.

74 riod and, at first, of the falling-rate one until water diffusion starts. The second step is accompanied by a sharp rise of the temperature of the solid. Thereby, the resin structure collapses because of free water withdrawal. However, as the DVB content of Lewatit K 2631 is high, the collapse only affects to the phase gel, and the surface area of the dried polymer is relatively high (about 36 m2/g). After the collapse, water diffusion is slow, and the resin adapts to the most favorable configuration from an energetic standpoint. So, in the dried resin, residual water is bridged to three sulfonic groups [22,28]. It is surprising that samples percolated with methanol do not show higher surface areas than those oven-dried. As the methanol solubility parameter (29.7 x 10 3 jl/2 m3/2) is lower than that of water (48.6 x 103 j1/2 m 3/2) [8], we can expect a slight increase of surface area for a macroporous polymer containing 18% DVB when the eluted alcohol is free of water [29]. In our case, percolation was stopped when eluted methanol still contained some water, so that methanol-solvated regions and water-solvated regions coexisted together in the resin. This suggests that the residual water keeps the most stable distribution, solvating mainly the sulfonic groups, whereas the methanol interacts with the polymer and also with the sulfonic groups. As a result, BET areas of ca. 36 mZ/g were also found. When samples washed with methanol are oven-dried, methanol and free water are withdrawn jointly giving rise to a lower shrinking of the resin beads and, as a consequence, to a less collapsed structure. The evaporation of methanol, which in the presence of water mainly interacts with the polymer chains, results in a more rigid polymer than when drying from water, and the pores available in the swollen state do not disappear completely [29]. Thus, higher water-effective diffusivities and, at very high VMeOH / V~esin ratios, higher surface areas than those

M. Iborra et al. /React. Polym. 21 (1993) 65-76

found for the resin oven-dried from water can be obtained. As a consequence of the enlargement of the pores, distances between chains are long enough to hinder the bridging of residual water molecules to three sulfonic groups. Thus, in a significant part of the beads, water molecules are two hydrogen-bonded to the same sulfonic group, or at most shared by only two groups. What results is an increase in the moisture equilibrium contents.

Conclusions Lewatit K 2631 behaves as a fibrous solid. Consequently, the moisture equilibrium contents are rather high. Drying curves clearly show constant-rate and falling-rate periods. In the constant-rate period the resin temperature is ca. 20°C. This period ends when 6 - 7 mol water per mol sulfonic group remain in the polymer. At first of the falling-rate period the resin temperature increases to the air within the oven. This temperature change, which is relatively fast, ends at a moisture content of about 4 mol water per mol sulfonic group. At this moment, the free water is already removed. The water of solvation is subsequently removed by a diffusion mechanism until the moisture equilibrium content is reached. It is worth noting that the remaining moisture content of the resin after percolation with methanol is lower than in the other methods. Oven drying of samples previously washed with methanol shows that the process also fits a diffusion mechanism. It is worth mentioning that moisture equilibrium contents are higher than in the drying without pretreatment with methanol. This fact, the higher values for water-effective diffusivity computed and the higher BET areas measured, can be explained by the formation of small pores.

M. lborra et al./React. Polym. 21 (1993) 65-76

List of symbols A D h;

Heat transfer area (m 2) Water-effective diffusivity (m2/s) Heat transfer coefficient [W/(K" m2)]

ms r

T

l

/crit W

~/crit

A Haa s

A

Mass of solid (kg) Mean particle radius (m) Temperature (K) Solid temperature during the constant-rate period (K) Drying time (h) Time of constant-rate drying (h) Total moisture on a bone-dry basis (kg/kg) Critical moisture on a bone-dry basis (kg/kg) Equilibrium moisture on a bone-dry basis (kg/kg) Heat of adsorption of water (kJ/mol) Standard liquefaction enthalpy of water (kJ/mol)

Acknowledgement The authors thank the oil company Repsol Petroleo for the financial support of this work.

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