Model Development and Experimental Verification of Liquid Desiccant

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Model Development and Experimental Verification of Liquid Desiccant Drying of Gelcast r-Alumina Ceramic Objects Aboulfazl Barati,*,† Hamid Reza Norouzi,‡ Sedighe Khaleghi Rostamkolaei,† and Saba Sharafoddinzadeh† † ‡

Department of Chemical Engineering, Faculty of Engineering, Arak University, Arak 38156-8-8349, Iran School of Chemical Engineering, College of Engineering, University of Tehran, PO Box 11155-4563, Tehran, Iran ABSTRACT: In this work, the effect of molecular weight of poly(ethylene glycol) (PEG) on the drying rate of gelcast R-alumina objects was investigated. Moreover, the effect of the sample shape (cylindrical, spherical, and slab-like shapes) and various effective thicknesses of these shapes were studied. Thermogravimetric analysis and differential scanning calorimetry were used in order to study the diffusion of PEG in the porous network structure. A general model was proposed for drying of green objects in liquid desiccant solutions. The results show that using higher average molecular weight of PEG increases the drying rate. PEG chains diffuse into the network during the drying process and the amount of diffused PEG decreases as the molecular weight of polymer increases and the concentration of solution decreases. The proposed model successfully predicts the drying rate and the dynamic volume change of ceramic objects in various desiccant solutions.

1. INTRODUCTION Gelcasting is a powerful method that is based on in situ polymerization of monomers (acrylamide, for example) to form green ceramic objects. This method is used in the manufacturing of advanced structural ceramic objects in various industries.13 Gelcasting enjoys considerable advantages compared to other methods based on polymer binders such as extrusion, slip casting, and injection molding. The advantages are so significant that gelcasting is considered to be an alternative for such methods due to the intriguing properties of near-net-shape forming, high green strength, and low binder concentration.4 However, there is a disadvantage that limits the industrialization of this method. The disadvantage is the drying of molded objects.5,6 Conventional methods for the drying of gel-cast ceramic objects are usually based on the use of dry air or gas with particular conditions of humidity and temperature. Nevertheless, such methods usually confront many difficulties. Nonuniform and differential drying in various regions are caused by the solvent gradient. The solvent gradient induces structural and residual stresses that cause defects and malformation.7,8 In addition, the formation of heat stresses caused by temperature differences within the object with low conductivity cause a greater problem during the drying. These malformations can be minimized or eliminated by using the appropriate low temperature polymer solution (i.e., environment temperature) as the liquid desiccant, which releases residual stresses. In the previous works of this group (based on the results of Janney and Kiggans9), liquid desiccant drying of gelcast objects was developed.1012 The effect of factors such as solid loading, concentration of desiccant, and type of solvent (aqueous or nonaqueous) on the drying rate and shrinkage of the object were studied. The effect of desiccant concentration on the sintering characteristics of BaTiO3-based ceramic objects has been recently reported.13 However, there have not been many studies carried out in this field, and there are various aspects of this method which are left unstudied. Therefore, in this work we carried out a set of experiments with regard to the drying of ceramic objects by r 2011 American Chemical Society

using a liquid desiccant in order to complete our previous studies, and to present new characteristics of this drying method. In addition to these experiments, a general and comprehensive model was developed to describe the drying characteristics of these objects. This model is able to predict the flow pattern of the solvent in the part as well as the amount of shrinkage and the drying rate. This model also gives some information about gradient of solvent’s concentration in the object. This information can be used for the analysis of residual stress as well as for predicting the probability of the formation of cracks in the dried objects. In this work, the gelcasting is performed by using 35 vol. % of slurry R-alumina and acrylic acid monomers.14 Three different molds in spherical, cylindrical, and slab-like shapes were prepared in order to study the different geometries of the objects. The drying process took place in PEG solutions with different concentrations and molecular weights and in a thermostatic condition. In order to determine the amount of diffused PEG polymer chains into the porous structure of green ceramic, and to determine the drying mechanism, TGA-DSC analyses were carried out on the dried objects. A model was developed to predict the drying kinetics and the amount of shrinkage of the objects at the same time. The attempt was to develop a simple model, based on the physics of the drying process. This model is a general one, and it is not limited to a special geometry. Therefore, it can be used for all multidimensional geometries. The volume of the part is directly related to the shrinkage factor in the equation of volume change. The diffusion coefficient of solvent in this model does depend on the network structure. In the end, outputs of this model were compared to the experimental results in order to study the capability of the model in predicting the drying characteristics of these parts. Received: November 28, 2010 Accepted: May 10, 2011 Revised: May 7, 2011 Published: May 10, 2011 7504

dx.doi.org/10.1021/ie102391d | Ind. Eng. Chem. Res. 2011, 50, 7504–7512

Industrial & Engineering Chemistry Research

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2. MODELING The drying kinetics of gelcast objects is based on the isothermal and counter current diffusion of polymeric and solvent phases. The following relation is valid in each finite element of this object. wp þ ws ¼ 1

ð1Þ

The equation of mass conservation for both polymeric and solvent phases is as follows: Dwi þ r:ð BJ i Þ ¼ 0 Dt

uw ¼ up wp þ us ws

ð3Þ

BJ i ¼ Jiuw þ wi uw

ð4Þ

The combination of eqs 14 yields the following equation, which is the continuity equation: ð5Þ

2.1. Diffusion in Porous Media. Equation 4 requires a relation that relates the diffusion flux to mass fraction gradient. The diffusion mechanism of solvent and polymer in the porous network of gelcast parts are Fickian, which are not depend on the amount of loaded ceramic or the concentration of desiccant solution.12 In this type of diffusion, the molecular relaxation caused by Brownian-like motion of polymer chains is ignored compared to the molecular diffusion. A large portion (more than 95 wt %) of the object, on the dry basis, is the porous ceramic and the polymeric network, which does not play dominant role in the diffusion. Therefore, the ceramic object behaves like a porous network in which solvent and polymer molecules diffuse among these pores. According to the Fick’s law, the diffusion flux for phase i is written as follows:

Jiuw ¼ Def f rðwi Þ

ð6Þ

where Deff is the effective diffusion coefficient of the solvent into the ceramic network. The diffusion of solvent molecules in porous media is affected by hydrodynamic hindrances and the tortuosity of the network. The hydrodynamic hindrance is a friction against movement of molecules throughout the porous media and increases with the increasing ratio of the diameter of the diffusing molecule to the diameter of the pores. The tortuosity relates to the bent or twisted diffusion path available for the movement of molecules. An increase in tortuosity leads to high friction and a reduced effective gradient in the network. These effects are included in the effective diffusion coefficient of the solvent in the porous network that is defined as follows: Def f ¼ f ðεÞ D0

Def f ε2 ¼ D0 1ε

ð2Þ

Where BJ i stands for the mass flux of phase i, and it is equal to the mass fraction of this phase that passes through the area of each element in one second. The mass flux is divided into two parts, convection flux and diffusion flux. If uw is defined as the mean mass velocity, then it can be used as the base velocity to calculate the mass flux in the following way:

r:ðuw Þ ¼ 0

of f(ε), the model used by Bisschops is used.15 This model is based on the analogy between diffusion of solvents through a porous network on the microscopic scale and viscous flow through a bed of particles on the macroscopic scale. In this model, Darcy’s law is used for describing the viscous flow in the porous network and then, for completion, the well-known KozenyCarman relation is employed. Eventually, this relation is utilized to describe the effective diffusion coefficient in the porous network of the ceramic object.

2.2. Equation of Volume Change. Since the drying process is concurrent with the change in volume, this factor must somehow be included in the modeling process. The volume change of the ceramic object is assumed to be proportional to the amount of the solvent within it. Achanta assumed that the shrinkage occurred only at the radial direction and the amount of the volume change in the polymer was equal to the volume of the removed solvent from the polymer.16 By drawing volume versus the solvent content, he showed that the volume change of the polymer has a linear relation to the amount of the solvent within it. The relation between the volume of the ceramic object and the solvent content is shown in the following equation:

VðtÞ ¼ V0 ðβ1 þ β2 ws Þ

ð10Þ

Taking into account the V(t) = V0 and ws = w0, s as initial conditions, the above equation can be converted to a nondimensional one: VðtÞ ¼ V0 ð1 βFÞ

ð11Þ

β1 ¼ 1 β2 w0

ð12Þ

β ¼ β2 ðw0 we Þ

ð13Þ

F ¼

w0 wt w0 we

ð14Þ

where F stands for the fractional approach to equilibrium and β is the total shrinkage factor. β is directly related to the shrinkage of the object. When the β value is high, it indicates a high amount of shrinkage in the object and vice versa. It was further assumed that the ratio of dimensions of the object is constant during the drying process, i.e., for a cylindrical part, the ratio of length over diameter of the cylinder was considered constant. The experimental results prove this assumption. The results will be discussed in the following sections. 2.3. Solution of Equations. Taking into account the homogeneity of the solvent content in the ceramic part at the beginning, and constant concentration of solvent in the bath during the drying process, the initial and boundary conditions are as follows:

ð7Þ

The void volume in the ceramic parts is well estimated by the volume occupied by the solvent. To determine the functionality

ð8Þ

wsðt ¼ 0Þ ¼ w0, S

ð15Þ

wsðt > 0Þ ¼ w¥, s

ð16Þ

DwS ¼0 D nB

ð17Þ

Equation 17 is the symmetrical boundary condition, and B n is the coordinate vector where the symmetrical boundary condition 7505

dx.doi.org/10.1021/ie102391d |Ind. Eng. Chem. Res. 2011, 50, 7504–7512


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Table 1. Specification of Materials and Instruments Used in This Work materials material

function

chemical formula

characteristics

supplier

R-alumina

ceramic powder

A12O3

Figure 1(ac)*

Good-Fellow

acrylic acid

monoflmctional monomer

CH2CHCOOH

Mw = 72.06 colorless liquid

Merck

N,N0 -methylene bisacrylamide

cross-linker

(C2H3CONH2)2CH2

Mw= 154.20 solid white powder

Sigma

ammonium persulphate

initiator

(NH4)2S2O8

Mw = 228.20 solid white powder

Merck

TEMED

accelerator

C6H16N2

Mw = 116.2 liquid, yellow

Merck

poly(ethylene glycol)

liquid desiccant

HO(C2H4O)nH

ave. Mw = 600, 1000, 2000 and 10000

Merck

ammonium sail of Poly(acrylic acid)

dispcrsant

(C3H5O2NH4þ)n

ave Mw = 15000 40 wt.% aqueous solution

R.T. Vanderbilt

instruments

*

instrument

model

supplier

X-ray diffraction

X’Pert PRO Alpha-1

Philips

scanning electron microscopy

XL30

Philips

simultaneous DSC-TGA

STA 6000

Perkin-Elmer

pH meter

CH-9101

Metrohm, AG

see reference.12

is held, i.e., for a sphere, this condition is utilized in the r direction at r = 0. As there are no analytical methods for solving the equations, numerical methods were employed in this work. In order to convert the partial differential equations to a set of algebraic equations, the finite volume method of Patankar was used.17 The geometry being studied was at first divided into discrete finite volumes (10 15 20 cells for slab, 12 20 for cylinder, and 15 cells for sphere). The discretization of the equations was carried out according to the fully implicit algorithm. Finally, Taylor Series was used in order to determine the gradients on the control volume surfaces. Applying the boundary conditions, the differential equations were converted to a set of linear algebraic equations. To perform these calculations, a program code was developed. This program reads the experimental data from a text file and then compares them with the results of the model. Afterward, it saves the squares of errors, the experimental and modeling results in terms of the fractional approach to equilibrium, and the volume of the object in a separate text file.

3. EXPERIMENTAL SECTION Table 1 includes the characteristics of the materials used in the experiments. It also includes the characteristics of the instruments used in experiments. A gelcasting process was carried out according to Young and his co-workers’ studies.18 Commercial alumina powder with an average particle size of four micrometers was used in order to produce raw objects. The characteristics of alumina powder used in this work were similar to those mentioned elsewhere.12 Epoxy molds were prepared in sphere, cylinder, and slab shapes, in two different sizes. The size of the produced objects is shown in Table 2. The samples were taken out of the molds after the molding process and were immersed in desiccant solution after being rinsed with distilled water. The amount of desiccant solution in comparison with the object volumes was great enough for the solution concentration to be considered constant during the drying process. Four different molecular weights of PEG (600, 1000, 2000, and 10000) were used in the experiments, and for each polymer, 20, 40, 60, and

Table 2. Specification of Gelcast Molded Parts with Various Geometries sample identification

geometry

small cylinder

cylinder

dimensions diameter = 7 mm, L/D = 4 diameter = 11 mm, L/D = 4

large cylinder small sphere large sphere

sphere

diameter = 7 mm diameter = 11 mm

slab

slab

7 14 21 (mm)

80 wt % solutions were prepared. However, 60 and 80 wt % solutions of PEG 10000 could not be prepared due to their low water solubility, and 80 wt % solution of PEG 2000 was used in the saturated form. Objects were taken out of the solution at proper time intervals, rinsed with distilled water, and their surfaces were dried by tissues. Then they were weighed and their dimensions were precisely measured by a micrometer. The process was repeated until the weights of the objects did not vary notably in two successive weighing processes. When the objects were dried in the desiccant solution, they were kept at room temperature and in a dry environment for a week. Therefore, the remaining water inside the objects was vaporized. In the end, TGA-DSC analysis was done on the dried objects.

4. DRYING MECHANISM It is necessary to present a theoretical mechanism for the drying of gelcast objects in polymeric solutions before discussing the results. Primary theoretical studies on this triple system, consisting of hydrogel, solvent (water), and linear polymer, show that the linear macromolecule does not diffuse into the gel network. Brochard ignored the interaction between polymer and network in his calculations.19 Rejecting the pervious assumptions, Poh,20 Adachi,21 Hamurcu, and Baysal22 believe that polymer diffusion inside the network depends on the thermodynamic parameters of the system which leads to gel shrinkage inside the polymeric solution. According to Gehrke, Freitas, and Cussler,23 7506



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Figure 1. TGA-DTG analysis results of samples dried in desiccant liquid solution containing 20 wt % of various molecular weights of PEG.

Figure 2. TGA-DTG analysis results of samples dried in PEG 2000 solution with various concentrations.

the concentration of diffused linear polymer in the network structure depends on the size and the molecular weight of the macromolecule. Studies carried out by Kayaman express that linear PEG with a molecular weight higher than 2000 is not able to diffuse into a neutral network structure such as acrylamide.24 It is also shown that the polymer diffusion into the gel structure takes place only due to the osmotic pressure difference caused by chemical potential difference between the solvent and the polymer inside the network and solution. Taking into account the above-mentioned studies and due to the non-neutral presence of ceramic particles (alumina) in the network structure,12 the system consists of four parts, and the ceramic particles influence the drying mechanism as active components. A few facts explain why water leaves the wet gelcast object when it is immersed in PEG solution. The most important reason is the osmotic pressure caused by the concentration difference of PEG between the surrounding solution and the water inside the object. This osmotic pressure increases when polymer concentration in desiccant solution increases. The second reason is the formation of a hydrophobic complex through hydrogen bonds between PEG chains diffused into the network and carboxylic groups of the network. This formation increases the hydrophobicity of the network.25 In lower pH environments and as the length of PEG chains increases, the complexes are more probable to form.26 It must be noted that as polymeric chains diffuse into the gelcast porous network, polymer concentration in the outer layers of the object increases. This increase leads to a positive concentration gradient within the object. As a result, water molecules move toward the outer surface of the object and the concentration gradient inside the object is reduced. Therefore, the stresses caused by the concentration difference are reduced, especially during the first stage of drying. The third reason is the octahedral formation of the aluminum ions, which leads to the formation of Br€onsted acid sites on particles’ surface and strong interactions between PEG chains and OH groups. This theory is applicable to any solid particle that is able to form Br€onsted acid sites on its surface.27

desiccant liquid solution containing 20 wt % of various molecular weights of PEG. Figure 2 shows these curves for samples dried in different concentration of desiccant liquid solution of PEG 2000. As it is observed in these figures, each curve shows a few stages. The first stage, up to about 90 C, shows the weight reduction of the sample due to the loss of the physically adsorbed water. The weight reduction for all samples in this experiment was lower than 2 wt %. The second stage starts at the temperature of about 160 C and continues up to 250 C. In this stage, hydroxyl groups of PEG chains were dehydrated and PEG chains were degraded, and aldehydes, ketones, ethers, and water were produced.27 The last stage in the thermo-gravimetric diagram occurs at temperatures higher than 350 C showing the degradation of acrylic acid gel network.28 The temperature in the thermo-gravimetric diagrams at which the rate of weight loss is maximum, introduced the degradation temperature of the sample. The sharp peak of DTG curves around 200 C illustrates this temperature. This peak shows the point in which the rate of mass loss in the sample is maximal. Degradation temperature of pure PEG samples depends on the molecular weight, and increases as the molecular weight increases. It varies between 300 and 360 C for various molecular weights.29 PEG degradation temperature decreases due to the interaction between hydroxyl groups and aluminum ions (here around 200 C). This interaction is extremely stronger than the one that takes place between hydroxyl groups and carboxylic acid groups formed by acrylic acid hydrolysis on network chains. The higher the concentration of the hydroxyl group is, i.e., the higher the concentration of the PEG inside the network matrix is, the higher the interaction between the aluminum ions and the hydroxyl groups will be. Therefore, a stronger complex between these groups is formed, and the degradation of polymeric chains takes place at a lower temperature. It has been reported that solid acids cause various polymers to degrade faster.3032 The point in which the DTG curve reaches zero is known as the initial degradation temperature of the sample. As it is shown in Figure 2, the initial degradation temperature of PEG in the presence of aluminum ions is between 155 and 160 C. By comparison between Figures 1 and 2, it is clear that at a given concentration but different molecular weights, polymer degradation starts at a higher temperature. However, at a given molecular weight but different concentrations, polymer degradation starts at lower temperature.

5. RESULTS AND DISCUSSION 5.1. Results of TGA-DSC Analysis. Figure 1 shows the results obtained from the thermo-gravimetric and differential thermal gravimetric (DTG) analyses of the alumina samples dried in

7507



Figure 3. DSC analysis results of samples dried in desiccant liquid solution containing 20 wt % of various molecular weights of PEG.

As illustrated in Figure 1, the mass loss of samples increases when PEG molecular weight increases at a constant concentration. When the molecular weight of PEG increases, the lower amount of PEG chains diffuse into the network structure due to the growth in the size of polymer molecules. Thus, fewer PEG molecules diffuse into the network and less mass loss of samples occurs. The most significant difference is between PEG 600 and PEG 100, and between PEG 1000 and PEG2000, while this difference vanishes between PEG200 and 10000. It seems that further increases in the molecular weight of PEG do not further advance the polymer diffusion into the network. When the PEG concentration within the liquid desiccant solution increases, the polymer chemical potential in the solution increases as well. Therefore, the polymer diffusion flux grows, and a higher amount of PEG diffuses to the network during the drying process (Figure 2). The large difference between mass (%) curves in Figure 2 shows that the effect of concentration is stronger than that of molecular weight (molecules’ size). Therefore, the PEG concentration overcomes the influence of the network prevention on the diffusion of PEG into the network matrix. DTG curves in Figure 2 also confirm the above-mentioned conclusions. Increasing the PEG concentration does not change the maximal points of DTG curves, but it increases the maximal value of DTG curves indicating the higher rate of degradation of the sample. DSC data of gelcast objects in Figure 3 show an endothermic peak at temperatures below 100 C when the objects lose the physically adsorbed water. It also shows two exothermic peaks at about 180 and 380 C, indicative of the PEG and gel network degradation, respectively. The extent of the area below the curves (enthalpy) in PEG 600 and its reduction with the increase in the molecular weight indicates that the increase in the interactions between hydroxyl groups in PEG chains and Br€onsted acid formed by aluminum ions existing in alumina particles, has caused a decrease in the required energy for the thermal decomposition of PEG chains. The results of these diagrams are in good accordance with the results obtained by TGA analysis. The mentioned results show that PEG chains with different molecular sizes ranging from 600 to 10 000 diffuse into the row ceramic network. 5.2. Drying Experiments. Figure 4 shows the drying curves of small cylindrical objects in PEG 2000 solutions with various concentrations. As seen, the drying process can be divided into

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Figure 4. Drying curves of small cylindrical objects in PEG 2000 solution with various concentrations.

Figure 5. Drying curves of small cylindrical objects in 20 wt % solution of PEG with various molecular weights.

three stages. In the first stage, referring to the first minutes of the drying process, the drying rate is constant therefore, its curve is linear. In this stage, most of the free water inside the pores of the porous structure of the ceramic exits to the surface of the object and the rate of water elimination is very high. The 20 wt % solution soaks up 0.5% water per minute while this value increases to 1% per minute by further increase in the solution concentration to 80 wt %. The second zone named “high rate of drying” is located between the linear and the low rate of drying zones. Most shrinkage takes place in this stage. The mechanism of water movement inside the object is mostly diffusion. As there is still a great osmotic pressure difference between the surrounding solution and inside the object, water leaves the object at a fast rate. As this process continues, higher amounts of PEG diffuse into the network. In addition, the pores become smaller due to the shrinkage. Both of these factors slow down the water diffusion outward. The last stage is called “low rate of drying”. In this stage, the drying process by liquid desiccant solution has become more than 90% complete, and the object does not lose any more water. The drying process can be considered complete at this stage. The dotted line separating the two zones of high and low rates of drying shows the time of the end of the drying process. As the concentration of desiccant solution increases 7508



Figure 6. Drying curves of small cylindrical objects in 40 wt % solution of PEG with various molecular weights.

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Figure 8. Drying curves of small cylindrical ceramic objects in PEG 2000 solution with different mass concentration, Lines show the results of the model.

Figure 7. Axial shrinkage of cylindrical objects versus their radial shrinkage, all dried in 20 wt % solutions.

Figure 9. Drying curves of large spherical objects in the 20 wt % of PEG with different molecular weights, lines show the results of the model.

from 20 to 80 wt %, the drying time lessens from 400 to 180 min. This reduction of time is achieved through more water absorption by the desiccant solution. Figure 5 shows the drying curves of small cylindrical objects in 20 wt % solution of PEG with molecular weights of 600, 1000, 2000, and 10 000. The figure shows that by increasing the polymer molecular weight in a constant concentration, the amount of the water that the object loses gets increased. The reason is that the increase in the molecular weight results in an increase in the chains length. Consequently, a lower amount of PEG diffuses into the object. The osmotic pressure difference between two environments lasts longer, and finally the object loses more water. For instance, a 40 wt % solution of PEG 2000 and a 20 wt % solution of PEG 10 000 eliminated similar amounts of water from wet cylindrical objects. In addition, the time of drying process increases from 300 to 450 min as the polymer molecular weight increased in 20% concentration. Figure 6 shows the drying curves of small raw cylindrical objects in 40 wt % solution of PEG with various molecular weights. This figure has been included in order to be compared with the previous figure. As seen in this figure, in a 40% concentration, increasing

molecular weight does not have a notable effect on the final amount of the water absorbed from the object. Moreover, the objects are dried in approximately similar durations. The curves of PEG 600 and PEG 1000 almost match each other and the distance between other curves is significantly reduced. But this does not happen for the 20 wt % solution of these polymers. Therefore, the concentration of desiccant solution seems to be a much more effective factor than the molecular weight of the polymers in drying the objects. This effect shows itself better in higher concentration while the molecular weight decreases. The amount of shrinkage of objects during and after the drying process and the retaining of their initial shape after the drying process are very important in drying technology. For example, the cylindrical objects shrink in both radial and axial directions. Figure 7 shows the axial versus radial shrinkage of cylindrical objects. The figure depicts cylinders dried in solutions with various molecular weights. The time parameter has been left out in this chart. The proximity of the points to the diagonal line indicates that the ceramic objects have shrunk equally in two radial and axial dimensions. In fact, the ratio of the length of the object to its diameter remains constant during the drying process 7509



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Table 3. Free Solvent Diffusion Coefficient and Total Shrinkage Factor Utilized in the Model 20 wt % D0 1010 PEG 600

8.34

R2* PEG 1000

6.77

R2 a

D0 1010

0.205

9.49

0.258

6.32

D0 1010

80 wt % β

D0 1010

0.311

26.6

β

0.215 0.274 0.978

0.252

9.88

0.283 0.971

0.293 0.970

β

9.76

0.976 5.13

60 wt %

0.972

0.968

R2 PEG 10000

β

0.969

R2 PEG 2000

40 wt %

10. 2

21.6 0.978

0.324 0.958

0.295 0.968

R2 is the coefficient of determination.

Figure 10. Drying curves of ceramics objects with different geometries.

and the ratio is equal to its initial value. This is true for the slab objects too, and the ratio between their dimensions remained almost constant during the process. 5.3. Modeling Results. Figure 8 shows the drying curves of small cylindrical ceramic objects in the solution of PEG 2000 with different mass concentrations. The vertical axis shows the fractional approach to the equilibrium defined by eq 14, and the horizontal axis shows the period of drying. As can be seen, the model can satisfactorily predict the experimental drying curve. The drying process can be considered complete when the fractional approach to equilibrium is 0.9. Thus, the time required for drying of objects reduced from 400 to 180 min as the concentration of PEG increases from 20% to 80%. Figure 9 illustrates the drying curve of large spherical objects in the 20 wt % of PEG with different molecular weights. The decrease in molecular weight of PEG leads to an increase in the required time for the drying of the objects. The free diffusion coefficient of solvent increases when the molecular weight of PEG increases. This trend can be seen in Table 3. Thus, the time of drying completion increases with molecular weight of PEG. In addition, when the solution of low molecular weight PEG is used, objects lose only a little amount of water and shrink less. As mentioned, this model can be used for different shapes and sizes of ceramic objects and its use is not limited to a special geometry. Figure 10 represents the comparison of model results with experimental drying curves of ceramic objects with different geometries and sizes. All of these objects were dried in 40 wt %

Figure 11. Measured volume vs predicted volume of cylindrical ceramic objects.

solution of PEG. In all of these cases, the free diffusion coefficient and total shrinkage factor are identical in the model. As seen in this figure, by knowing the free diffusion coefficient and the total shrinkage factor, one can predict the drying curve of the various ceramic objects with different sizes and geometries. This figure also shows that the small spherical object has the shortest, and the big cylinder object has the longest drying time. This is attributed to the characteristic diffusion length (the ratio of volume to the diffusion area) in these objects. It is expected that the drying rates of samples increase while the drying period decreases. The characteristic diffusion lengths for small sphere, small cylinder, big sphere, slab-like and big cylinder objects are 1.2, 1.75, 1.85, 1.91, and 2.75 mm. Thus, the fastest drying rate belongs to the small sphere, and the slowest one belongs to the big cylinder. 5.3.1. the Reliability of the Model for Predicting the Volume of Ceramic Objects. In Figure 11, the measured volume of objects in the experiment during the drying is plotted vs the predicted volume of the object in the model. This comparison is done for drying of the large cylindrical object. The closer the points are to the diagonal line, the better prediction of the volume is done by the model. As can be seen in this figure, the points are all around diagonal line that indicates that the model 7510



Figure 12. Concentration profile within the spherical ceramic objects during the drying process, (a) small sphere and (b) large sphere.

could satisfactorily predict the volume change of objects. As mentioned before, β, the total shrinkage factor is directly related to the volume change of the objects. The values of β are listed in Table 3. When the concentration of PEG increases, or when molecular weight of PEG increases, the total shrinkage factor increases too. It is worth mentioning that the effect of molecular weight at 40 wt % or higher concentrations is less than the effect of molecular weight at 20 wt % solution. One of the most valuable elements that is not available via the experiment is the concentration profile within the object during drying process. Figure 12 is our prediction model of the concentration profile of solvent along the radius of large and small spheres, which are plotted at different drying times. The linear shrinkage of spheres in radius direction is clear. In both graphs, the concentration gradient near the surface is more than that of inner parts. This persistent concentration gradient helps the drying process that is; it increases the ability of the object to remove water. In addition, the concentration profile within the small sphere is more uniform than that of the large sphere. High concentration profile of solvent leads in bending, structural, and residual stresses after drying of objects.7,8 It is recommended to use low concentration of PEG solutions for safe water removal of objects without bending and other malformations.

6. CONCLUSIONS Ceramic objects were synthesized by the gelcasting method in various sizes and cylindrical, spherical, and slab-like shapes. The drying rate of these objects in PEG solutions with various molecular weights and concentrations was measured. Subsequently, in order to study the diffusion of PEG chains into the network and to study

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the interactions between these chains and the network structure, TGA-DSC analyses were carried out on dried samples, and the following results were gained: • PEG chains diffuse into the network during the drying process. The diffusion decreases as the molecular weight of polymer increases. Moreover, increasing the concentration of the desiccant solution increases the chains’ diffusion inward and the effect of changing molecular weight becomes so insignificant that in higher concentrations, it is negligible. • According to the drying mechanism presented in this work, the existing alumina in the network acts as an active component. Due to the formation of strong complexes between polymeric chains and Br€onsted acid sites formed by alumina molecules, the hydrophobicity of the network increases when PEG diffuses into it. These interactions also decrease the degradation temperature of PEG from about 300 360 C to about 160250 C. • The drying rate of the objects and the amount of water absorption from them increase with the increasing of the molecular weight of the polymer and with the increasing of the concentration of polymeric solution. Concentration is an effective factor. In high concentration (about 60%), no significant difference is observable in the ability of solutions with various molecular in absorbing water; however, in low concentration the difference in the absorbing ability is considerable. • The proposed model could satisfactorily predict the drying rate and the volume change of all samples. Various desiccant solutions of PEG with different molecular weights and concentration can be studied through this model.

’ AUTHOR INFORMATION Corresponding Author

*Tel: þ98 (861) 4173446; Fax: þ98 (861) 4173450; E-mail [email protected].

’ ACKNOWLEDGMENT The authors thank Arak University for the financial support of this work. They thank technical staff in the department of chemical engineering of Arak University for their help during various stages of the work. ’ NOMENCLATURE D0 free diffusion coefficient, m2s1 Deff effective diffusion coefficient, m2s1 F fractional approach to equilibrium J mass flux, (kg/m2s) T time, sec u velocity, ms1 V Volume, m3 mean mass velocity uw w weight fraction w mean weight fraction β1, β2 shrinkage factors β total shrinkage factor ε void fraction ’ SUBSCRIPTS 0 initial condition P polymer 7511


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solvent bath condition

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Model Development and Experimental Verification of Liquid Desiccant

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