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Thermal conductivity of ceramic green bodies during drying
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Thermal conductivity of ceramic green bodies during drying B. Nait-Ali a,∗ , S. Oummadi a , E. Portuguez b , A. Alzina b , D.S. Smith a a b
Ecole Nationale Supérieure de Céramique Industrielle, SPCTS, UMR 7315, 12 rue Atlantis, F-87068 Limoges, France Université de Limoges, SPCTS, UMR 7315, F-87000 Limoges, France
a r t i c l e
i n f o
Article history: Received 24 September 2016 Received in revised form 6 December 2016 Accepted 7 December 2016 Available online xxx Keywords: Thermal conductivity Drying Alumina Kaolin Analytical models
a b s t r a c t The thermal conductivity of alumina and kaolin green bodies has been studied as a function of the water loss during drying. Experimental measurements show strong variations with 3 distinct regimes. In the first regime, thermal conductivity increases during shrinkage. When shrinkage stops, a decrease in thermal conductivity with water loss is observed which becomes even stronger during the last phase of drying. This can be explained by the variations in the volume fractions of each phase and the effective thermal contacts between grains. Using analytical relations, the thermal resistance of an equivalent plane of small area grain-grain contacts is shown to increase strongly at the end of drying due to the removal of water. Finally, in certain drying conditions, if a portion of the heat required for drying, is supplied by conduction through the green body, then the rate of water evaporation increases with higher thermal conductivity. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Drying is an important step in many ceramic processes, including traditional sectors such as the tile and brick industry but also in more recent processing techniques such as tape casting or 3D printing [1–3]. It involves two types of transfer: heat transfer and mass transfer of water in liquid or vapour form. There are important issues for successful technology which are directly related to drying. First, this step requires a significant energy input, which in a context of reducing energy consumption and CO2 emissions has led to the development of new technology with a view to achieve more efficient drying [4]. A second issue is that mechanical stresses due to water removal may create macroscopic defects or cracks which are detrimental to the material. The mechanisms involved during drying of ceramic products, related to the kinetics of water evaporation, have been widely studied [5,6]. More recent contributions have described numerical modelling of heat and mass transfers in moist bodies [7–9]. However few studies concern the evolution of thermophysical properties during drying which are nevertheless required input data for modelling. If the heat capacity can be calculated simply as a function of the water content by the use of the rule of mixtures, the thermal conductivity on the other hand depends not only on the volume fraction of each phase but also on their spatial distribution within the material and other microstructural
∗ Corresponding author. E-mail address: [email protected] (B. Nait-Ali).
factors such as the nature of interfaces and the sizes of solid grains and pores. The role of water on the thermal conductivity has already been investigated, for example in the study of soils, or more recently, containers for radioactive waste and materials for buildings [10–13]. All these results show that the thermal conductivity increases with increasing water content. For example, in the case of samples of compacted bentonite, the thermal conductivity increases from 0.5 to 1 Wm−1 K−1 when the volume fraction of water in the pores is increased from 30 to 90% [12]. For zirconia samples, the thermal conductivity increases from 0.2 to 0.8 Wm−1 K−1 when the volume fraction of water in the pores is raised from 4 to 80% [14]. Changes in the thermal conductivity can be explained using an approach based on a mixture of phases by replacing a volume of air with a thermal conductivity of 0.026 of Wm−1 K−1 by a volume of water with a higher thermal conductivity: 0.61 Wm−1 K−1 . Many models have been proposed and tested in literature to describe the behaviour of the thermal conductivity with water content [10,15,14]. More recently, in the case of granular materials, it has been shown that variations in the thermal conductivity with water content cannot be explained solely by the contribution of water as an additional phase in the material [11]. The authors have reported an increase in the thermal conductivity of packed silica sand from 0.4 Wm−1 K−1 for the dry state to 1.8 Wm−1 K−1 with just 2vol% of water. They explain this increase by an effect on the interfaces between solid grains. The presence of a small amount of water reduces the thermal contact resistance between grains and the thermal conductivity is then strongly increased. In all of these studies the overall volume of the material
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Table 1 Water contents in the initial suspension and wet samples.
Alumina Kaolin
Water content in mass% in the initial suspension
Water content in mass% in the wet samples
50 65
22.3 28.8
which consists of solid, water and air did not vary when the water content was changed or this is not mentioned by the authors. In the case of drying, the volume of the sample may vary when the water is removed. The study of the thermal conductivity as a function of water content during drying must then consider the dimensional changes. For this reason, the measurement of shrinkage combined with weight loss should provide useful information to understand the relationship between the thermal conductivity and the water content in the sample. In this study of two technologically important materials, alumina and kaolin clay, we have measured the thermal conductivity as a function of the water content at different stages of drying in comparison to the linear shrinkage. Alumina was chosen as a model material because its thermal conductivity has been widely studied. The single crystal has a conductivity of 35 Wm−1 K−1 whereas for a polycrystalline sample, the value is decreased for small grains (<5 m), due to interfaces which provide additional thermal resistance [16]. Kaolin is a clay material used for the fabrication of many traditional ceramic products. Its thermal conductivity is much lower than that of alumina, less than 2 Wm−1 K−1 for fully dense material [17]. First a simple model based on the MaxwellEucken relation is used to explain the experimental results. Then values of the thermal resistance of an equivalent plane of small area grain–grain contacts were estimated from thermal conductivity data to investigate the mechanisms further. Observations of alumina during drying were also performed using environmental scanning electron microscopy (ESEM) in order to complement the analysis of the thermal conductivity results. As a final point, the influence of the thermal conductivity on the drying rate is also reported. 2. Materials and method 2.1. Samples preparation A fine ␣-alumina powder, P172SB (supplied by Alcan) was used, with a median grain size of 0.4 m and a specific surface area, measured using the BET method, of 7.5 m2 g−1 . The kaolin clay powder (BIP supplied by Imerys from Kaolins de Beauvoir – France) is constituted of 78 wt% of kaolinite, 17 wt% of muscovite and 4 wt% of quartz. The measured specific surface area is 10 m2 g−1 . The median grain size measured using laser light scattering (Mavern Mastersizer 2000) is equal to 9.5 m which indicates, taking into account the specific surface area, that the grains are constituted of smaller clay particles which have agglomerated. Powders were ball milled with water for 75 min according to the mass percent given in Table 1. A paste was formed after absorption of a part of the initial water by a plaster support. Samples with typical dimensions: 10 cm × 4 cm × 2 cm, were then cut into the body. The water content in mass percent in the samples, which is the starting point for the thermal conductivity study is given in Table 1. 2.2. Thermal conductivity measurement 2.2.1. Transient plane source method The thermal conductivity of the samples was measured using the transient plane source (TPS), method, operating at room temperature and supplied by Hot Disk AB (Sweden). This technique,
Fig. 1. Experimental setup for shrinkage measurement.
developed by Gustafsson, uses a thin probe, disk shaped and constituted of a nickel bifilar spiral sandwiched between two films of an insulating material [18]. The probe is placed between two blocks made of the material to be measured. The nickel wire is used, first, as a heat source using Joule’s effect and, second, as a sensor to monitor the temperature increase by measuring the electrical resistance of the wire. The two sample blocks enclosing the probe are considered to be an infinite medium during the experiment as well as homogeneous and isotropic. At t < 0, the material is in thermal equilibrium at the temperature T0 (uniform in the solid). The measurement consists of recording the increase in temperature after applying a constant heat flux from time t = 0. The thermal conductivity is determined from the time dependent temperature response. The probing depth of heat penetration is typically of the same order of magnitude as the probe diameter, which was 6.3 mm in our case. This probing depth, large compared to the mean pore size, allows measurements which average over the heterogeneous microstructure of our materials. Typical measuring time was 80 s. 2.2.2. Measurements at different stages of drying Drying was performed at ambient relative humidity and temperature (ca. 40% and 20 ◦ C). However during drying the temperature in the sample can vary strongly, depending on the water evaporation rate. Furthermore due to coupled heat and mass transfers, the temperature is not spatially uniform in the material. To ensure a constant and uniform temperature in the sample before each measurement, samples were encapsulated in a polyvinyl film, which stops the drying process, for a duration of 4 h. We have checked, using thermocouples that this delay is sufficient to reach a constant and uniform temperature in the sample. By cutting appropriate smaller specimens from the sample we have also checked, at different moments in the progression of drying, that water content is uniform after this delay. Furthermore, some samples were dried at 40 ◦ C until the mass had reached a constant value, and the thermal conductivity was then measured after cooling down to room temperature. 2.3. Linear shrinkage and pore volume fraction measurements Monitoring of shrinkage was performed using a linear variable differential transformer sensor which gives the change in length over time with an accuracy of 1 m. Simultaneously the sample is weighed with an accuracy of 1 mg. Fig. 1 presents the experimental setup. Measurements of sample height (H) and mass (m) are recorded using dedicated software. The linear shrinkage (s) and weight loss (w) are then calculated with the following relations: s=
H0 − H H0
w=
m0 − m m0
(1) (2)
where H0 and m0 are initial height and mass of the sample.
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shrinkage which then remains almost constant with further weight loss.
Fig. 2. Thermal conductivity and linear shrinkage as a function of weight loss for alumina samples.
After a complete drying at 40 ◦ C, the bulk density (bulk ) was determined from weight and dimensional measurements. The pore volume fraction was then calculated using the relation: vp = 1 −
bulk solid
(3)
where solid is the solid phase density. For alumina we used a value reported in literature from measurements of lattice parameters by X-ray diffraction: 3.986 g cm−3 [19]. For kaolin clay, a value of 2.64 g cm−3 for solid was measured with a helium pycnometer on powder after grinding in a mortar. 2.4. Microstructure observations The microstructure of an alumina sample was observed at various points in the drying process using an environmental scanning electron microscope (FEI Quanta 450 FEG). To avoid a rapid drying, the relative humidity, close to the sample surface, was varied step by step from 100%, by cooling down the sample using a Peltier stage fixed at 2 ◦ C and by adjusting slightly the pressure of water vapour in the chamber around 700 Pa. A fully dried sample covered with a platinum layer (approx. 5 nm) was also observed, in the same ESEM but without cooling down the sample and with a low pressure in the chamber of 2.4 × 10−3 Pa. These micrographs are used to illustrate the assumptions made on the distribution of water for the analytical models in the thermal conductivity study. However, only the surface of the sample can be examined and the technique cannot give information on what happens in the bulk of the sample during drying. 3. Results and discussion 3.1. Alumina samples 3.1.1. Thermal conductivity and linear shrinkage Thermal conductivity and linear shrinkage as a function of weight loss are plotted in Fig. 2. Thermal conductivity variations reveal 3 distinct regimes. At the beginning of drying, the value increases from 2.5 Wm−1 K−1 and reaches a maximum: 3.1 Wm−1 K−1 . Then the thermal conductivity decreases, weakly at first down to 1.5 Wm−1 K−1 and strongly at the end of drying to yield a value of 0.25 Wm−1 K−1 . These changes in the effective thermal conductivity as a function of the drying progression, corresponding to a factor of 12 between the maximum value and the end point, were examined in comparison to the linear shrinkage data. The thermal conductivity increases when the shrinkage occurs and the maximum conductivity value matches the end of strong linear
3.1.2. ESEM observations ESEM images of the sample at various stages in the drying process are presented in Fig. 3. In the first observation, in Fig. 3a, water occupies a large area of the drying surface and most alumina particles are surrounded by this continuous darker phase. In Fig. 3b, slightly later in the drying process, the area occupied by water is reduced but some of the alumina particles are still surrounded by water. It was also noticed, that from Fig. 3a and b, alumina grains move across the observation zone when water is removed. These observations are consistent with literature in the description of the first stage of drying, when shrinkage occurs. During this stage, water is evaporated directly from the external surface of the sample and solid grains are surrounded by water [6]. In Fig. 3c, water is not present in the form of a continuous film on the sample surface. Larger pores are empty but a part of water seems to be still present in the system, located at interfaces between small alumina particles. This situation should correspond to the regime of drying well after the shrinkage stage. A fully dried sample was also observed in Fig. 3d, which confirms that water located at interfaces between grains is now completely removed. 3.1.3. Simple model of the thermal conductivity as a function of the drying progress During the first stage of drying with strong shrinkage, up to ca. 6% of weight loss, the material is assumed to be constituted of uniquely the solid and water, meaning that the decrease in the apparent volume corresponds to the volume of water which has been evaporated [6]. As a consequence, the volume fraction of solid, which has a higher conductivity than water, increases. Furthermore solid particles, which at the beginning of drying are separated by stable films of water, evolve towards a closer packing increasing the overall grain-grain contact area. These two aspects, shown in the micrographs in Fig. 3 and schematically illustrated in Fig. 4, are responsible for the thermal conductivity increase when the shrinkage occurs. More insight into the role of each of these contributions is given by a simple model where the effective thermal conductivity has been calculated using the Maxwell-Eucken relation for a mixture of solid and water [20]. For this, the material is assumed to be constituted of isolated solid inclusions surrounded by water described by the following equation: eff = water
solid + 2water + 2vsolid (solid − water ) solid + 2water − vsolid (solid − water )
(4)
where water is the thermal conductivity of water, solid is thermal conductivity of the solid, vsolid is the volume fraction of solid. Eq. (4) is also equivalent to the Hashin-Shtrikman lower bound for an isotropic mixture of alumina particles and water [21]. Values calculated, with a solid phase thermal conductivity of 35 Wm−1 K−1 and a value of 0.61 Wm−1 K−1 for water, are 2.08 Wm−1 K−1 at the beginning of drying and 2.39 Wm−1 K−1 at the end of the shrinkage stage (Fig. 5). The analytical model assumes that there is no contact for the heat flow between particles, which is probably not strictly the case in the real material and can explain the slightly greater experimental values compared to the predictions. Based on the water and solid fraction volume changes, the calculated difference of 0.3 Wm−1 K−1 is less than the measured difference of 0.6 Wm−1 K−1 between the initial wet state and the end of shrinkage. This suggests that an additional factor, change in the particle-particle contact area, influences the thermal conductivity values during the first stage of drying. Once the first stage of drying is achieved, at 6% of weight loss, further drying gives a decrease in thermal conductivity. The effect has already been described in literature and is explained by the volume of water which is replaced by air with a lower thermal con-
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Fig. 3. ESEM observations of alumina at various stages in the drying. (a) In the first observation a large surface area is occupied by water which surrounds alumina grains. (b) The surface area of water has decreased but some alumina grains are still surrounded by water. (c) Water seems to be still present, located at interfaces between small grains. (d) The sample is fully dried, water is not present in the system.
Fig. 4. Schematic arrangement of solid grains and water at: (a) the start of drying and (b) the end of the shrinkage. The volume fraction of solid has increased as well as the overall contact area between grains.
ductivity: 0.026 Wm−1 K−1 or even less for small pores (<0.5 m) if the Knudsen effect is operating [22,23]. However, the final stronger decrease in the thermal conductivity seems to indicate that other mechanisms can be involved at the end of the drying process. After a complete drying at 40 ◦ C, the material is constituted of only the
solid and air. Using the Maxwell-Eucken model, i.e. assuming that the material is constituted of isolated solid inclusions surrounded by air, the predicted value from Eq. (4) on replacing water with air, is 0.1 Wm−1 K−1 , which is below the measured thermal conductivity (Fig. 5). Again the difference between experimental and calculated
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where water and air are the thermal conductivities of water and air, v air is the volume fraction of air in the second phase. This corresponds to a situation described successively in previous studies on porous zirconia ceramics in humid atmospheres [14,27]. The calculated values of solid matrix are given in Table 2 and are significantly less than 35 Wm−1 K−1 , the value characteristic of the intrinsic thermal conductivity of alumina. A first aspect of explanation can be attributed to the presence of interfaces in the equivalent solid matrix since the alumina phase is polycrystalline and originates from a compacted powder. From a pragmatic engineering point of view with respect to heat flow, we can treat the “solid matrix” as resistors in series, representing the particles (intrinsic thermal conductivity) and the thermal resistance at the interface regions. This can be written as:
Fig. 5. Schematic of thermal conductivity as a function of weight loss indicating 4 characteristic points (1, 2, 3, 4) in the drying behaviour. Calculated values (+) using the Maxwell-Eucken relation (Eq. (4)) with water (1, 2) or air (4) as the continuous phase.
values can be attributed to the contacts between particles which are not taken into account in the model. 3.1.4. Roles of particle-particle contacts and distribution of water To investigate in further detail, we have made a series of calculations at points 2, 3, and 4 of Fig. 5 illustrating the importance of information on the distribution of water in the solid-pore system during drying and in particular in relation to the grain-grain contacts. After loss of 6 wt.% water or more, the material exhibits an almost constant volume with a stable continuous phase of joined particles, termed in this paper as the solid matrix. This solid matrix is associated with a second continuous phase occupying the intervening space (or “open porosity”) which can be water, air or a mixture of both of them. The effective thermal conductivity (eff ) of this type of system can be estimated using effective medium percolation theory in the form of Landauer’s relation since the 2 phases are continuous with volumes close to 50% [24,25]. To calculate values corresponding to an equivalent solid matrix, Landauer’s relation can be solved to give: 2
solid matrix =
2eff − eff second phase (2 − 3vsolid ) second phase + eff (3vsolid − 1)
(5)
where second phase is the second phase thermal conductivity value. vsolid is the volume fraction of the solid. Alternatives such as the sigmoidal average or an exponential relation yield predictions of the effect of porosity on thermal conductivity which are close in value to Landauer’s relation but Eq. (5) is easy to apply [26]. We have assumed that: (i) at point 2, the material is only constituted by solid and water, (ii) at point 4, after a complete drying at 40 ◦ C, the material is only constituted of the solid and air, (iii) at point 3 when the slope in the thermal conductivity decrease changes, the material is constituted of the solid, water and air. A value for second phase was calculated by assuming that layers of water line the pore contours on the solid surface. Due to this specific distribution of the water inside the pores, with paths to vehicle heat across the pore, the Maxwell-Eucken model was chosen to estimate the thermal conductivity of the second phase.
second phase = water
air + 2water + 2vair (air − water )
air + 2water − vair (air − water )
(6)
1 1 = + nRcontact solid matrix grain
(7)
where solid matrix represents the thermal conductivity of the equivalent solid matrix (matrix = grains + interfaces), grain is the thermal conductivity of a grain and Rcontact is identified as a thermal resistance representing an interface plane with macroscopic dimensions (i.e. sample dimensions in the case of 100% dense material). For an idealized cubic grain assembly, it is assumed that interfaces parallel to the heat flow can be neglected and only interfaces perpendicular to the heat flow direction, presenting significant thermal resistance, need to be taken into account. The model of the equivalent solid matrix is thus simplified to one of flat slabs separated by interface planes at a distance of “average” linear grain size in the direction of heat flow, denoted by n. In the case of a perfectly dense ceramic, Rcontact is equivalent to an average grain boundary thermal resistance with a value of the order of 10−8 m2 KW−1 [16]. A value for n is obtained from the inverse median grain size of the powder (0.4 m). Using the values cited above for n, Rcontact and grain (35 Wm−1 K−1 ) in Eq. (7) a value of solid matrix = 18.7 Wm−1 K−1 is obtained, considerably greater than the values in Table 2. However in the case of a “green body”, the effective thermal conductivity is not only modulated by the pore volume fraction in the range 40–50%, conveniently handled here with a 2-phase model in the same manner as sintered porous ceramics, but also by reduced contact area between grains. In the present work, even if somewhat arbitrary, the effect of reduced contact area has been incorporated into the conductivity of the solid matrix (a similar approach could be adopted for microcracked ceramics yielding low values of solid matrix conductivity). Rcontact is then analogous to the thermal resistance of a mechanical contact between two separate bodies (typical values found in engineering heat transfer textbooks are 10−5 –10−3 m2 KW−1 ) [28]. Compared to the dense sintered body with a grain boundary network extending across the entire sample, the value of Rcontact for the green body is significantly amplified by the small solid-solid contact area between particles. The interface plane, described by Rcontact considered at the macroscopic scale of the sample cross section, represents both zones of solid-solid contact and narrow gaps of empty space. In fact the attribution of a negligibly small volume of pore fraction in the term Rcontact has very strong effect. The actual value of Rcontact , situated in the range 10−8 m2 KW−1 –10−4 m2 KW−1 , will also depend on the state of the microscopic contact interfaces in the green body. A representation of these different situations is given in Fig. 6. It should be noted that the schematic in Fig. 6b remains very idealized since the grain-grain contacts in a real ceramic green body will actually be at multiple levels. Hence from Eqs. (5)–(7) the thermal contact resistance (Rcontact ) has been estimated at 3 characteristic points in the drying behaviour shown in Fig. 5. These values are also given in Table 2 which can now be reviewed.
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Table 2 Data for analytical calculations, calculated values for equivalent solid matrix thermal conductivity and thermal resistance of planes representing the grain-grain contacts.
Volume fraction of solid Volume fraction of water Volume fraction of air Experimental effective conductivity (Wm−1 K−1 ) Calculated solid matrix thermal conductivity (Wm−1 K−1 ) Thermal contact resistance (m2 KW−1 )
Point 1
Point 2
Point 3
Point 4
0.47 0.53 0 2.5
0.52 0.48 0 3.1 7.8 4 × 10−8
0.52 0.07 0.41 1.6 5.1 6.6 × 10−8
0.52 0 0.48 0.25 0.7 5.3 × 10−7
Fig. 6. (a) Dense sintered body approximated by flat slabs separated by interface planes. (b) Green body represented as an idealised cubic arrangement of pores and particles with small solid-solid contact area. (c) Equivalent solid matrix for the green body where interface planes exhibit both solid-solid contacts and narrow gaps of empty space.
The calculated values of solid matrix thermal conductivity on applying volume fraction changes of air and water to Eqs. (5) and (6), decrease between points 2 and 4 from 7.8 Wm−1 K−1 to 0.7 Wm−1 K−1 . This suggests that the description of the behaviour with fixed values of intrinsic thermal conductivity and thermal contact resistance is not sufficient (in fact the value of solid matrix = 7.8 Wm−1 K−1 , corresponding to point 2, yields an upper limit for points 3 and 4 in the same spirit as Eq. (4) yields a lower limit in Fig. 5). It is necessary to vary the value of Rcontact with water content to find agreement between prediction and experiment. In particular, between point 3 and point 4, the strong decrease in the value of solid matrix thermal conductivity could be explained by the location of water in the system. Just before the end of the drying process, residual water is located in the neck regions between grains as shown in the micrograph in Fig. 3c and illustrated in Fig. 7a and acts like a “thermal grease” in a mechanical contact. The removal of water (with a higher thermal conductivity than air) in the neck region leads to the situation shown in Fig. 7b, explaining the strong increase in Rcontact , by a factor of 10 from point 3 to point 4. This analysis, though perhaps imperfect, points out the important roles of particle-particle contacts and the distribution of water in the effective thermal conductivity of the drying green body. 3.2. Kaolin samples Despite the much lower thermal conductivity of the solid phases, many aspects of the general behaviour during drying of kaolin samples are similar to alumina. Thermal conductivity and linear shrinkage as a function of weight loss are plotted in Fig. 8 for 3 sets of kaolin samples. As for alumina samples, 3 distinct regimes are observed. The situations for the two materials are compared. At the beginning of drying, the volume fractions of water in the samples are very close (53% for alumina and 51% for kaolin), then the higher initial value of thermal conductivity for alumina samples can be easily understood by the higher value of the intrinsic conductivity of alumina (35 Wm−1 K−1 ) compared to kaolin clay (<2 Wm−1 K−1 ). After a complete drying at 40 ◦ C, we can assume that the volume fraction of water is negligible and the material is only constituted by solid and air, with pore volume fractions of 48% for alumina and 44% for kaolin. At this stage, the conductivity value for alumina sample is lower than for kaolin (0.25 Wm−1 K−1
Fig. 7. Interface between grains with (a) water located in the neck region and (b) after a complete drying. Water with a higher thermal conductivity than air decreases the thermal resistance of the interface region (contact) between grains.
and 0.5 Wm−1 K−1 respectively). As discussed earlier, the thermal resistance for contacts between alumina grains can be responsible for a strong decrease in the equivalent solid matrix conductivity and consequently for the low effective value. Indeed, after a complete drying, alumina grains are very weakly bound, which results in a very brittle material in comparison with clay samples which can be easily handled without damage. The greater value of conductivity in kaolin samples can then be explained by more efficient contact between clay particles compared with alumina grains. 3.3. Influence of the thermal conductivity on the drying rate The drying rate of ceramic green bodies is well described in literature and exhibits two or three characteristic stages [5,6]. During the first stage, called the constant rate period (CRP), the drying rate expressed per unit area of drying surfaces is constant. The end of
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Fig. 8. Thermal conductivity and linear shrinkage as a function of weight loss for kaolin samples.
the CRP corresponds in most of cases to the end of the shrinkage and then also to the maximum of thermal conductivity according to Figs. 2 and 8. During the CRP, the main parameters which control the drying rate, for a convective drying, are the environmental characteristics i.e. the temperature, relative humidity and velocity of air. The rate of evaporation is given by the relation: 1 dm = k (pw (Ts ) − pa ) A dt
(8)
where A is the surface area exposed to drying, k is the mass transfer coefficient, pw is the saturation vapour pressure of water at the surface which depends on the surface temperature (Ts ) and pa is the vapour pressure of water far from the surface. If heat is only supplied by convection, then the surface temperature reaches the wet bulb temperature during the CRP. If heat is also partly supplied by conduction, then the surface temperature should lie above the wet bulb temperature during the CRP. As the saturation vapour pressure of water increases with temperature, the drying rate is increased, according to Eq. (8), and should depend on the thermal conductivity. To investigate the influence of the thermal conductivity, we have exposed to drying, alumina and kaolin samples with identical dimensions (15 mm × 15 mm × 40 mm). Only one face (15 mm × 15 mm) was exposed to drying with air at 20 ◦ C. The opposite face was put on a heating plate at 50 ◦ C, and the other faces were surrounded with polyvinyl film to prevent vapour loss along the bar, approximating a 1-dimensional mass transfer situation. The heat transfer situation involves axial conduction along the bar with a steady loss through the filmed surface; equivalent to a lossy transmission line. Close to the exposed surface, the lateral heat exchange should change direction when Ts at x = 0 is less than 20 ◦ C. The boundary condition for the volume element next to the exposed surface at x = 0 is of particular interest and can be written: =L
dT dm = h (A + px) (T∞ − Ts ) + A dt dx
(9) x=0
where is the energy per unit time consumed by the drying surface, L is the latent heat of evaporation, h is the convection heat transfer coefficient, p is the sample perimeter in cross section, x is the thickness of a volume element in the bar, T∝ is the temperature of the drying environment. Fig. 9 shows that during the CRP, the rate of evaporation of alumina samples is approximately 15% higher than for kaolin. This can be explained by the contribution of heat supplied by conduction along the bar, which should be greater in alumina than kaolin related to the higher value of the thermal conductivity during the CRP. The values vary between 2.5 and 3.1 Wm−1 K−1 compared to 1.4 and 1.6 Wm−1 K−1 for kaolin.
7
Fig. 9. Weight loss of alumina and kaolin samples, with identical dimensions and area exposed to drying, as a function of time. Samples of 40 mm height are put on a heating plate at 50 ◦ C to provide heat by conduction in addition to convection with air at 20 ◦ C.
We deduced that thermal conductivity, depending on the drying conditions, can then affect the drying rate during the CRP. 4. Conclusion A protocol to measure the thermal conductivity of ceramic green bodies as a function of the water loss during drying has been proposed and applied to alumina and kaolin samples. For these two materials, thermal conductivity evolves strongly with the water loss and Exhibits 3 distinct regimes. First, the thermal conductivity increases at the beginning of drying and reaches a maximum value at the end of the shrinkage. Second, a decrease is observed, when water which evaporates is replaced by air. Finally, at the end of drying, the thermal conductivity strongly decreases as a function of weight loss. To a first approximation, the behaviour can be described by a simple model based on the Maxwell-Eucken relation but this does not take into account the role of grain-grain contacts. The thermal resistance for an equivalent plane of grain-grain contacts at different stages of drying was estimated using analytical relations. A strong increase in the calculated values is observed at the end of drying that we can attribute to the removal of water located next to the solid grain-grain contacts. If thermal conductivity is a sensitive indicator of the drying stage, we have also shown that in certain drying conditions, this property can affect the drying rate of products during the CRP. Indeed, if a portion of the heat, which is required for drying, is supplied by conduction, then the rate of water evaporation increases with the thermal conductivity. In ambient conditions a higher drying rate, up to 15%, was observed for alumina compared with kaolin clay, for drying of samples placed on a heating plate at 50 ◦ C. This arrangement promotes heat supply by conduction. Future work involving numerical modelling of a material during drying should incorporate the variations of thermal conductivity during the process as well as information on the distribution of water in relation to the grain-grain contacts. Acknowledgments Siham Oummadi and Etienne Portuguez would like to thank the Limousin Region for financial support. We are also grateful to Yann Launay for his help with the SEM observations. References
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Thermal conductivity of ceramic green bodies during drying B. Nait-Ali a,∗ , S. Oummadi a , E. Portuguez b , A. Alzina b , D.S. Smith a a b
Ecole Nationale Supérieure de Céramique Industrielle, SPCTS, UMR 7315, 12 rue Atlantis, F-87068 Limoges, France Université de Limoges, SPCTS, UMR 7315, F-87000 Limoges, France
a r t i c l e
i n f o
Article history: Received 24 September 2016 Received in revised form 6 December 2016 Accepted 7 December 2016 Available online xxx Keywords: Thermal conductivity Drying Alumina Kaolin Analytical models
a b s t r a c t The thermal conductivity of alumina and kaolin green bodies has been studied as a function of the water loss during drying. Experimental measurements show strong variations with 3 distinct regimes. In the first regime, thermal conductivity increases during shrinkage. When shrinkage stops, a decrease in thermal conductivity with water loss is observed which becomes even stronger during the last phase of drying. This can be explained by the variations in the volume fractions of each phase and the effective thermal contacts between grains. Using analytical relations, the thermal resistance of an equivalent plane of small area grain-grain contacts is shown to increase strongly at the end of drying due to the removal of water. Finally, in certain drying conditions, if a portion of the heat required for drying, is supplied by conduction through the green body, then the rate of water evaporation increases with higher thermal conductivity. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Drying is an important step in many ceramic processes, including traditional sectors such as the tile and brick industry but also in more recent processing techniques such as tape casting or 3D printing [1–3]. It involves two types of transfer: heat transfer and mass transfer of water in liquid or vapour form. There are important issues for successful technology which are directly related to drying. First, this step requires a significant energy input, which in a context of reducing energy consumption and CO2 emissions has led to the development of new technology with a view to achieve more efficient drying [4]. A second issue is that mechanical stresses due to water removal may create macroscopic defects or cracks which are detrimental to the material. The mechanisms involved during drying of ceramic products, related to the kinetics of water evaporation, have been widely studied [5,6]. More recent contributions have described numerical modelling of heat and mass transfers in moist bodies [7–9]. However few studies concern the evolution of thermophysical properties during drying which are nevertheless required input data for modelling. If the heat capacity can be calculated simply as a function of the water content by the use of the rule of mixtures, the thermal conductivity on the other hand depends not only on the volume fraction of each phase but also on their spatial distribution within the material and other microstructural
∗ Corresponding author. E-mail address: [email protected] (B. Nait-Ali).
factors such as the nature of interfaces and the sizes of solid grains and pores. The role of water on the thermal conductivity has already been investigated, for example in the study of soils, or more recently, containers for radioactive waste and materials for buildings [10–13]. All these results show that the thermal conductivity increases with increasing water content. For example, in the case of samples of compacted bentonite, the thermal conductivity increases from 0.5 to 1 Wm−1 K−1 when the volume fraction of water in the pores is increased from 30 to 90% [12]. For zirconia samples, the thermal conductivity increases from 0.2 to 0.8 Wm−1 K−1 when the volume fraction of water in the pores is raised from 4 to 80% [14]. Changes in the thermal conductivity can be explained using an approach based on a mixture of phases by replacing a volume of air with a thermal conductivity of 0.026 of Wm−1 K−1 by a volume of water with a higher thermal conductivity: 0.61 Wm−1 K−1 . Many models have been proposed and tested in literature to describe the behaviour of the thermal conductivity with water content [10,15,14]. More recently, in the case of granular materials, it has been shown that variations in the thermal conductivity with water content cannot be explained solely by the contribution of water as an additional phase in the material [11]. The authors have reported an increase in the thermal conductivity of packed silica sand from 0.4 Wm−1 K−1 for the dry state to 1.8 Wm−1 K−1 with just 2vol% of water. They explain this increase by an effect on the interfaces between solid grains. The presence of a small amount of water reduces the thermal contact resistance between grains and the thermal conductivity is then strongly increased. In all of these studies the overall volume of the material
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Table 1 Water contents in the initial suspension and wet samples.
Alumina Kaolin
Water content in mass% in the initial suspension
Water content in mass% in the wet samples
50 65
22.3 28.8
which consists of solid, water and air did not vary when the water content was changed or this is not mentioned by the authors. In the case of drying, the volume of the sample may vary when the water is removed. The study of the thermal conductivity as a function of water content during drying must then consider the dimensional changes. For this reason, the measurement of shrinkage combined with weight loss should provide useful information to understand the relationship between the thermal conductivity and the water content in the sample. In this study of two technologically important materials, alumina and kaolin clay, we have measured the thermal conductivity as a function of the water content at different stages of drying in comparison to the linear shrinkage. Alumina was chosen as a model material because its thermal conductivity has been widely studied. The single crystal has a conductivity of 35 Wm−1 K−1 whereas for a polycrystalline sample, the value is decreased for small grains (<5 m), due to interfaces which provide additional thermal resistance [16]. Kaolin is a clay material used for the fabrication of many traditional ceramic products. Its thermal conductivity is much lower than that of alumina, less than 2 Wm−1 K−1 for fully dense material [17]. First a simple model based on the MaxwellEucken relation is used to explain the experimental results. Then values of the thermal resistance of an equivalent plane of small area grain–grain contacts were estimated from thermal conductivity data to investigate the mechanisms further. Observations of alumina during drying were also performed using environmental scanning electron microscopy (ESEM) in order to complement the analysis of the thermal conductivity results. As a final point, the influence of the thermal conductivity on the drying rate is also reported. 2. Materials and method 2.1. Samples preparation A fine ␣-alumina powder, P172SB (supplied by Alcan) was used, with a median grain size of 0.4 m and a specific surface area, measured using the BET method, of 7.5 m2 g−1 . The kaolin clay powder (BIP supplied by Imerys from Kaolins de Beauvoir – France) is constituted of 78 wt% of kaolinite, 17 wt% of muscovite and 4 wt% of quartz. The measured specific surface area is 10 m2 g−1 . The median grain size measured using laser light scattering (Mavern Mastersizer 2000) is equal to 9.5 m which indicates, taking into account the specific surface area, that the grains are constituted of smaller clay particles which have agglomerated. Powders were ball milled with water for 75 min according to the mass percent given in Table 1. A paste was formed after absorption of a part of the initial water by a plaster support. Samples with typical dimensions: 10 cm × 4 cm × 2 cm, were then cut into the body. The water content in mass percent in the samples, which is the starting point for the thermal conductivity study is given in Table 1. 2.2. Thermal conductivity measurement 2.2.1. Transient plane source method The thermal conductivity of the samples was measured using the transient plane source (TPS), method, operating at room temperature and supplied by Hot Disk AB (Sweden). This technique,
Fig. 1. Experimental setup for shrinkage measurement.
developed by Gustafsson, uses a thin probe, disk shaped and constituted of a nickel bifilar spiral sandwiched between two films of an insulating material [18]. The probe is placed between two blocks made of the material to be measured. The nickel wire is used, first, as a heat source using Joule’s effect and, second, as a sensor to monitor the temperature increase by measuring the electrical resistance of the wire. The two sample blocks enclosing the probe are considered to be an infinite medium during the experiment as well as homogeneous and isotropic. At t < 0, the material is in thermal equilibrium at the temperature T0 (uniform in the solid). The measurement consists of recording the increase in temperature after applying a constant heat flux from time t = 0. The thermal conductivity is determined from the time dependent temperature response. The probing depth of heat penetration is typically of the same order of magnitude as the probe diameter, which was 6.3 mm in our case. This probing depth, large compared to the mean pore size, allows measurements which average over the heterogeneous microstructure of our materials. Typical measuring time was 80 s. 2.2.2. Measurements at different stages of drying Drying was performed at ambient relative humidity and temperature (ca. 40% and 20 ◦ C). However during drying the temperature in the sample can vary strongly, depending on the water evaporation rate. Furthermore due to coupled heat and mass transfers, the temperature is not spatially uniform in the material. To ensure a constant and uniform temperature in the sample before each measurement, samples were encapsulated in a polyvinyl film, which stops the drying process, for a duration of 4 h. We have checked, using thermocouples that this delay is sufficient to reach a constant and uniform temperature in the sample. By cutting appropriate smaller specimens from the sample we have also checked, at different moments in the progression of drying, that water content is uniform after this delay. Furthermore, some samples were dried at 40 ◦ C until the mass had reached a constant value, and the thermal conductivity was then measured after cooling down to room temperature. 2.3. Linear shrinkage and pore volume fraction measurements Monitoring of shrinkage was performed using a linear variable differential transformer sensor which gives the change in length over time with an accuracy of 1 m. Simultaneously the sample is weighed with an accuracy of 1 mg. Fig. 1 presents the experimental setup. Measurements of sample height (H) and mass (m) are recorded using dedicated software. The linear shrinkage (s) and weight loss (w) are then calculated with the following relations: s=
H0 − H H0
w=
m0 − m m0
(1) (2)
where H0 and m0 are initial height and mass of the sample.
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shrinkage which then remains almost constant with further weight loss.
Fig. 2. Thermal conductivity and linear shrinkage as a function of weight loss for alumina samples.
After a complete drying at 40 ◦ C, the bulk density (bulk ) was determined from weight and dimensional measurements. The pore volume fraction was then calculated using the relation: vp = 1 −
bulk solid
(3)
where solid is the solid phase density. For alumina we used a value reported in literature from measurements of lattice parameters by X-ray diffraction: 3.986 g cm−3 [19]. For kaolin clay, a value of 2.64 g cm−3 for solid was measured with a helium pycnometer on powder after grinding in a mortar. 2.4. Microstructure observations The microstructure of an alumina sample was observed at various points in the drying process using an environmental scanning electron microscope (FEI Quanta 450 FEG). To avoid a rapid drying, the relative humidity, close to the sample surface, was varied step by step from 100%, by cooling down the sample using a Peltier stage fixed at 2 ◦ C and by adjusting slightly the pressure of water vapour in the chamber around 700 Pa. A fully dried sample covered with a platinum layer (approx. 5 nm) was also observed, in the same ESEM but without cooling down the sample and with a low pressure in the chamber of 2.4 × 10−3 Pa. These micrographs are used to illustrate the assumptions made on the distribution of water for the analytical models in the thermal conductivity study. However, only the surface of the sample can be examined and the technique cannot give information on what happens in the bulk of the sample during drying. 3. Results and discussion 3.1. Alumina samples 3.1.1. Thermal conductivity and linear shrinkage Thermal conductivity and linear shrinkage as a function of weight loss are plotted in Fig. 2. Thermal conductivity variations reveal 3 distinct regimes. At the beginning of drying, the value increases from 2.5 Wm−1 K−1 and reaches a maximum: 3.1 Wm−1 K−1 . Then the thermal conductivity decreases, weakly at first down to 1.5 Wm−1 K−1 and strongly at the end of drying to yield a value of 0.25 Wm−1 K−1 . These changes in the effective thermal conductivity as a function of the drying progression, corresponding to a factor of 12 between the maximum value and the end point, were examined in comparison to the linear shrinkage data. The thermal conductivity increases when the shrinkage occurs and the maximum conductivity value matches the end of strong linear
3.1.2. ESEM observations ESEM images of the sample at various stages in the drying process are presented in Fig. 3. In the first observation, in Fig. 3a, water occupies a large area of the drying surface and most alumina particles are surrounded by this continuous darker phase. In Fig. 3b, slightly later in the drying process, the area occupied by water is reduced but some of the alumina particles are still surrounded by water. It was also noticed, that from Fig. 3a and b, alumina grains move across the observation zone when water is removed. These observations are consistent with literature in the description of the first stage of drying, when shrinkage occurs. During this stage, water is evaporated directly from the external surface of the sample and solid grains are surrounded by water [6]. In Fig. 3c, water is not present in the form of a continuous film on the sample surface. Larger pores are empty but a part of water seems to be still present in the system, located at interfaces between small alumina particles. This situation should correspond to the regime of drying well after the shrinkage stage. A fully dried sample was also observed in Fig. 3d, which confirms that water located at interfaces between grains is now completely removed. 3.1.3. Simple model of the thermal conductivity as a function of the drying progress During the first stage of drying with strong shrinkage, up to ca. 6% of weight loss, the material is assumed to be constituted of uniquely the solid and water, meaning that the decrease in the apparent volume corresponds to the volume of water which has been evaporated [6]. As a consequence, the volume fraction of solid, which has a higher conductivity than water, increases. Furthermore solid particles, which at the beginning of drying are separated by stable films of water, evolve towards a closer packing increasing the overall grain-grain contact area. These two aspects, shown in the micrographs in Fig. 3 and schematically illustrated in Fig. 4, are responsible for the thermal conductivity increase when the shrinkage occurs. More insight into the role of each of these contributions is given by a simple model where the effective thermal conductivity has been calculated using the Maxwell-Eucken relation for a mixture of solid and water [20]. For this, the material is assumed to be constituted of isolated solid inclusions surrounded by water described by the following equation: eff = water
solid + 2water + 2vsolid (solid − water ) solid + 2water − vsolid (solid − water )
(4)
where water is the thermal conductivity of water, solid is thermal conductivity of the solid, vsolid is the volume fraction of solid. Eq. (4) is also equivalent to the Hashin-Shtrikman lower bound for an isotropic mixture of alumina particles and water [21]. Values calculated, with a solid phase thermal conductivity of 35 Wm−1 K−1 and a value of 0.61 Wm−1 K−1 for water, are 2.08 Wm−1 K−1 at the beginning of drying and 2.39 Wm−1 K−1 at the end of the shrinkage stage (Fig. 5). The analytical model assumes that there is no contact for the heat flow between particles, which is probably not strictly the case in the real material and can explain the slightly greater experimental values compared to the predictions. Based on the water and solid fraction volume changes, the calculated difference of 0.3 Wm−1 K−1 is less than the measured difference of 0.6 Wm−1 K−1 between the initial wet state and the end of shrinkage. This suggests that an additional factor, change in the particle-particle contact area, influences the thermal conductivity values during the first stage of drying. Once the first stage of drying is achieved, at 6% of weight loss, further drying gives a decrease in thermal conductivity. The effect has already been described in literature and is explained by the volume of water which is replaced by air with a lower thermal con-
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Fig. 3. ESEM observations of alumina at various stages in the drying. (a) In the first observation a large surface area is occupied by water which surrounds alumina grains. (b) The surface area of water has decreased but some alumina grains are still surrounded by water. (c) Water seems to be still present, located at interfaces between small grains. (d) The sample is fully dried, water is not present in the system.
Fig. 4. Schematic arrangement of solid grains and water at: (a) the start of drying and (b) the end of the shrinkage. The volume fraction of solid has increased as well as the overall contact area between grains.
ductivity: 0.026 Wm−1 K−1 or even less for small pores (<0.5 m) if the Knudsen effect is operating [22,23]. However, the final stronger decrease in the thermal conductivity seems to indicate that other mechanisms can be involved at the end of the drying process. After a complete drying at 40 ◦ C, the material is constituted of only the
solid and air. Using the Maxwell-Eucken model, i.e. assuming that the material is constituted of isolated solid inclusions surrounded by air, the predicted value from Eq. (4) on replacing water with air, is 0.1 Wm−1 K−1 , which is below the measured thermal conductivity (Fig. 5). Again the difference between experimental and calculated
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where water and air are the thermal conductivities of water and air, v air is the volume fraction of air in the second phase. This corresponds to a situation described successively in previous studies on porous zirconia ceramics in humid atmospheres [14,27]. The calculated values of solid matrix are given in Table 2 and are significantly less than 35 Wm−1 K−1 , the value characteristic of the intrinsic thermal conductivity of alumina. A first aspect of explanation can be attributed to the presence of interfaces in the equivalent solid matrix since the alumina phase is polycrystalline and originates from a compacted powder. From a pragmatic engineering point of view with respect to heat flow, we can treat the “solid matrix” as resistors in series, representing the particles (intrinsic thermal conductivity) and the thermal resistance at the interface regions. This can be written as:
Fig. 5. Schematic of thermal conductivity as a function of weight loss indicating 4 characteristic points (1, 2, 3, 4) in the drying behaviour. Calculated values (+) using the Maxwell-Eucken relation (Eq. (4)) with water (1, 2) or air (4) as the continuous phase.
values can be attributed to the contacts between particles which are not taken into account in the model. 3.1.4. Roles of particle-particle contacts and distribution of water To investigate in further detail, we have made a series of calculations at points 2, 3, and 4 of Fig. 5 illustrating the importance of information on the distribution of water in the solid-pore system during drying and in particular in relation to the grain-grain contacts. After loss of 6 wt.% water or more, the material exhibits an almost constant volume with a stable continuous phase of joined particles, termed in this paper as the solid matrix. This solid matrix is associated with a second continuous phase occupying the intervening space (or “open porosity”) which can be water, air or a mixture of both of them. The effective thermal conductivity (eff ) of this type of system can be estimated using effective medium percolation theory in the form of Landauer’s relation since the 2 phases are continuous with volumes close to 50% [24,25]. To calculate values corresponding to an equivalent solid matrix, Landauer’s relation can be solved to give: 2
solid matrix =
2eff − eff second phase (2 − 3vsolid ) second phase + eff (3vsolid − 1)
(5)
where second phase is the second phase thermal conductivity value. vsolid is the volume fraction of the solid. Alternatives such as the sigmoidal average or an exponential relation yield predictions of the effect of porosity on thermal conductivity which are close in value to Landauer’s relation but Eq. (5) is easy to apply [26]. We have assumed that: (i) at point 2, the material is only constituted by solid and water, (ii) at point 4, after a complete drying at 40 ◦ C, the material is only constituted of the solid and air, (iii) at point 3 when the slope in the thermal conductivity decrease changes, the material is constituted of the solid, water and air. A value for second phase was calculated by assuming that layers of water line the pore contours on the solid surface. Due to this specific distribution of the water inside the pores, with paths to vehicle heat across the pore, the Maxwell-Eucken model was chosen to estimate the thermal conductivity of the second phase.
second phase = water
air + 2water + 2vair (air − water )
air + 2water − vair (air − water )
(6)
1 1 = + nRcontact solid matrix grain
(7)
where solid matrix represents the thermal conductivity of the equivalent solid matrix (matrix = grains + interfaces), grain is the thermal conductivity of a grain and Rcontact is identified as a thermal resistance representing an interface plane with macroscopic dimensions (i.e. sample dimensions in the case of 100% dense material). For an idealized cubic grain assembly, it is assumed that interfaces parallel to the heat flow can be neglected and only interfaces perpendicular to the heat flow direction, presenting significant thermal resistance, need to be taken into account. The model of the equivalent solid matrix is thus simplified to one of flat slabs separated by interface planes at a distance of “average” linear grain size in the direction of heat flow, denoted by n. In the case of a perfectly dense ceramic, Rcontact is equivalent to an average grain boundary thermal resistance with a value of the order of 10−8 m2 KW−1 [16]. A value for n is obtained from the inverse median grain size of the powder (0.4 m). Using the values cited above for n, Rcontact and grain (35 Wm−1 K−1 ) in Eq. (7) a value of solid matrix = 18.7 Wm−1 K−1 is obtained, considerably greater than the values in Table 2. However in the case of a “green body”, the effective thermal conductivity is not only modulated by the pore volume fraction in the range 40–50%, conveniently handled here with a 2-phase model in the same manner as sintered porous ceramics, but also by reduced contact area between grains. In the present work, even if somewhat arbitrary, the effect of reduced contact area has been incorporated into the conductivity of the solid matrix (a similar approach could be adopted for microcracked ceramics yielding low values of solid matrix conductivity). Rcontact is then analogous to the thermal resistance of a mechanical contact between two separate bodies (typical values found in engineering heat transfer textbooks are 10−5 –10−3 m2 KW−1 ) [28]. Compared to the dense sintered body with a grain boundary network extending across the entire sample, the value of Rcontact for the green body is significantly amplified by the small solid-solid contact area between particles. The interface plane, described by Rcontact considered at the macroscopic scale of the sample cross section, represents both zones of solid-solid contact and narrow gaps of empty space. In fact the attribution of a negligibly small volume of pore fraction in the term Rcontact has very strong effect. The actual value of Rcontact , situated in the range 10−8 m2 KW−1 –10−4 m2 KW−1 , will also depend on the state of the microscopic contact interfaces in the green body. A representation of these different situations is given in Fig. 6. It should be noted that the schematic in Fig. 6b remains very idealized since the grain-grain contacts in a real ceramic green body will actually be at multiple levels. Hence from Eqs. (5)–(7) the thermal contact resistance (Rcontact ) has been estimated at 3 characteristic points in the drying behaviour shown in Fig. 5. These values are also given in Table 2 which can now be reviewed.
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Table 2 Data for analytical calculations, calculated values for equivalent solid matrix thermal conductivity and thermal resistance of planes representing the grain-grain contacts.
Volume fraction of solid Volume fraction of water Volume fraction of air Experimental effective conductivity (Wm−1 K−1 ) Calculated solid matrix thermal conductivity (Wm−1 K−1 ) Thermal contact resistance (m2 KW−1 )
Point 1
Point 2
Point 3
Point 4
0.47 0.53 0 2.5
0.52 0.48 0 3.1 7.8 4 × 10−8
0.52 0.07 0.41 1.6 5.1 6.6 × 10−8
0.52 0 0.48 0.25 0.7 5.3 × 10−7
Fig. 6. (a) Dense sintered body approximated by flat slabs separated by interface planes. (b) Green body represented as an idealised cubic arrangement of pores and particles with small solid-solid contact area. (c) Equivalent solid matrix for the green body where interface planes exhibit both solid-solid contacts and narrow gaps of empty space.
The calculated values of solid matrix thermal conductivity on applying volume fraction changes of air and water to Eqs. (5) and (6), decrease between points 2 and 4 from 7.8 Wm−1 K−1 to 0.7 Wm−1 K−1 . This suggests that the description of the behaviour with fixed values of intrinsic thermal conductivity and thermal contact resistance is not sufficient (in fact the value of solid matrix = 7.8 Wm−1 K−1 , corresponding to point 2, yields an upper limit for points 3 and 4 in the same spirit as Eq. (4) yields a lower limit in Fig. 5). It is necessary to vary the value of Rcontact with water content to find agreement between prediction and experiment. In particular, between point 3 and point 4, the strong decrease in the value of solid matrix thermal conductivity could be explained by the location of water in the system. Just before the end of the drying process, residual water is located in the neck regions between grains as shown in the micrograph in Fig. 3c and illustrated in Fig. 7a and acts like a “thermal grease” in a mechanical contact. The removal of water (with a higher thermal conductivity than air) in the neck region leads to the situation shown in Fig. 7b, explaining the strong increase in Rcontact , by a factor of 10 from point 3 to point 4. This analysis, though perhaps imperfect, points out the important roles of particle-particle contacts and the distribution of water in the effective thermal conductivity of the drying green body. 3.2. Kaolin samples Despite the much lower thermal conductivity of the solid phases, many aspects of the general behaviour during drying of kaolin samples are similar to alumina. Thermal conductivity and linear shrinkage as a function of weight loss are plotted in Fig. 8 for 3 sets of kaolin samples. As for alumina samples, 3 distinct regimes are observed. The situations for the two materials are compared. At the beginning of drying, the volume fractions of water in the samples are very close (53% for alumina and 51% for kaolin), then the higher initial value of thermal conductivity for alumina samples can be easily understood by the higher value of the intrinsic conductivity of alumina (35 Wm−1 K−1 ) compared to kaolin clay (<2 Wm−1 K−1 ). After a complete drying at 40 ◦ C, we can assume that the volume fraction of water is negligible and the material is only constituted by solid and air, with pore volume fractions of 48% for alumina and 44% for kaolin. At this stage, the conductivity value for alumina sample is lower than for kaolin (0.25 Wm−1 K−1
Fig. 7. Interface between grains with (a) water located in the neck region and (b) after a complete drying. Water with a higher thermal conductivity than air decreases the thermal resistance of the interface region (contact) between grains.
and 0.5 Wm−1 K−1 respectively). As discussed earlier, the thermal resistance for contacts between alumina grains can be responsible for a strong decrease in the equivalent solid matrix conductivity and consequently for the low effective value. Indeed, after a complete drying, alumina grains are very weakly bound, which results in a very brittle material in comparison with clay samples which can be easily handled without damage. The greater value of conductivity in kaolin samples can then be explained by more efficient contact between clay particles compared with alumina grains. 3.3. Influence of the thermal conductivity on the drying rate The drying rate of ceramic green bodies is well described in literature and exhibits two or three characteristic stages [5,6]. During the first stage, called the constant rate period (CRP), the drying rate expressed per unit area of drying surfaces is constant. The end of
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Fig. 8. Thermal conductivity and linear shrinkage as a function of weight loss for kaolin samples.
the CRP corresponds in most of cases to the end of the shrinkage and then also to the maximum of thermal conductivity according to Figs. 2 and 8. During the CRP, the main parameters which control the drying rate, for a convective drying, are the environmental characteristics i.e. the temperature, relative humidity and velocity of air. The rate of evaporation is given by the relation: 1 dm = k (pw (Ts ) − pa ) A dt
(8)
where A is the surface area exposed to drying, k is the mass transfer coefficient, pw is the saturation vapour pressure of water at the surface which depends on the surface temperature (Ts ) and pa is the vapour pressure of water far from the surface. If heat is only supplied by convection, then the surface temperature reaches the wet bulb temperature during the CRP. If heat is also partly supplied by conduction, then the surface temperature should lie above the wet bulb temperature during the CRP. As the saturation vapour pressure of water increases with temperature, the drying rate is increased, according to Eq. (8), and should depend on the thermal conductivity. To investigate the influence of the thermal conductivity, we have exposed to drying, alumina and kaolin samples with identical dimensions (15 mm × 15 mm × 40 mm). Only one face (15 mm × 15 mm) was exposed to drying with air at 20 ◦ C. The opposite face was put on a heating plate at 50 ◦ C, and the other faces were surrounded with polyvinyl film to prevent vapour loss along the bar, approximating a 1-dimensional mass transfer situation. The heat transfer situation involves axial conduction along the bar with a steady loss through the filmed surface; equivalent to a lossy transmission line. Close to the exposed surface, the lateral heat exchange should change direction when Ts at x = 0 is less than 20 ◦ C. The boundary condition for the volume element next to the exposed surface at x = 0 is of particular interest and can be written: =L
dT dm = h (A + px) (T∞ − Ts ) + A dt dx
(9) x=0
where is the energy per unit time consumed by the drying surface, L is the latent heat of evaporation, h is the convection heat transfer coefficient, p is the sample perimeter in cross section, x is the thickness of a volume element in the bar, T∝ is the temperature of the drying environment. Fig. 9 shows that during the CRP, the rate of evaporation of alumina samples is approximately 15% higher than for kaolin. This can be explained by the contribution of heat supplied by conduction along the bar, which should be greater in alumina than kaolin related to the higher value of the thermal conductivity during the CRP. The values vary between 2.5 and 3.1 Wm−1 K−1 compared to 1.4 and 1.6 Wm−1 K−1 for kaolin.
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Fig. 9. Weight loss of alumina and kaolin samples, with identical dimensions and area exposed to drying, as a function of time. Samples of 40 mm height are put on a heating plate at 50 ◦ C to provide heat by conduction in addition to convection with air at 20 ◦ C.
We deduced that thermal conductivity, depending on the drying conditions, can then affect the drying rate during the CRP. 4. Conclusion A protocol to measure the thermal conductivity of ceramic green bodies as a function of the water loss during drying has been proposed and applied to alumina and kaolin samples. For these two materials, thermal conductivity evolves strongly with the water loss and Exhibits 3 distinct regimes. First, the thermal conductivity increases at the beginning of drying and reaches a maximum value at the end of the shrinkage. Second, a decrease is observed, when water which evaporates is replaced by air. Finally, at the end of drying, the thermal conductivity strongly decreases as a function of weight loss. To a first approximation, the behaviour can be described by a simple model based on the Maxwell-Eucken relation but this does not take into account the role of grain-grain contacts. The thermal resistance for an equivalent plane of grain-grain contacts at different stages of drying was estimated using analytical relations. A strong increase in the calculated values is observed at the end of drying that we can attribute to the removal of water located next to the solid grain-grain contacts. If thermal conductivity is a sensitive indicator of the drying stage, we have also shown that in certain drying conditions, this property can affect the drying rate of products during the CRP. Indeed, if a portion of the heat, which is required for drying, is supplied by conduction, then the rate of water evaporation increases with the thermal conductivity. In ambient conditions a higher drying rate, up to 15%, was observed for alumina compared with kaolin clay, for drying of samples placed on a heating plate at 50 ◦ C. This arrangement promotes heat supply by conduction. Future work involving numerical modelling of a material during drying should incorporate the variations of thermal conductivity during the process as well as information on the distribution of water in relation to the grain-grain contacts. Acknowledgments Siham Oummadi and Etienne Portuguez would like to thank the Limousin Region for financial support. We are also grateful to Yann Launay for his help with the SEM observations. References
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