Her focus lies on fluid dynamics in fuel cells, especially on multiphase flow in the porous layers of fuel cells and on the impact of flow field design on fuel cell performance and degradation. In addition to his active research in gas adsorption and the thermodynamics of confined fluids, his other active research interests include molecular simulation techniques, including reference calculations, uncertainty quantification, and simulation best practices.
Microbial processes that allow pollutant biodegradation in toxic and inaccessible micro-environments within porous media. Werth received a B. His research focuses on the fate and transport of pollutants in the environment, the development of innovative catalytic technologies for drinking water treatment, and the mitigation of environmental impacts associated with energy production and generation.
Werth has published peer-reviewed journal articles, and his research is supported by grants from both government agencies and private companies. White is a staff scientist at Lawrence Livermore National Laboratory. He received a B. He currently leads the Subsurface Flow and Transport Group at the laboratory.
His research focuses on integrating field monitoring techniques with large-scale computing to improve our understanding of complex geologic systems. Current applications of interest include carbon sequestration, unconventional energy production, and induced seismicity. Progress and challenges in the textural characterization of nanoporous materials.
In addition he leads a number of authoritative bodies in his field including: Network models for physical and biological processes in porous media. With regards to porous media, he is developing network models to describe a range of processes such as biological invasions, capillary condensation or fluid flow. Update your browser to view this website correctly. Update my browser now. Detailed timetable calendar file. In order to enable an iCal export link, your account needs to have an API key created. This key enables other applications to access data from within Indico even when you are neither using nor logged into the Indico system yourself with the link provided.
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Permanent link for all public and protected information:. In this theory, the total moisture flux is assumed to be made up of three components: By using the above conservation equations, three interdependent partial differential equations involving variables , , and can be obtained as where the kinetic coefficients depend not only on temperature and moisture content but also on material properties and drying conditions. For example, where is the coefficient of moisture conductivity, is the moisture capacity, is the density of the dry solid, and is the thermogradient coefficient related to the moisture content difference.
For more details on the computation of these kinetic coefficients, refer Luikov [ 45 ], Irudayaraj and Wu [ 47 ], and Lewis and Ferguson [ 48 ].
In practice, it is often not possible to obtain these parameters to solve the full system of equations. These equations are still commonly employed today and quite often solved by the finite element method. Krischer is also among the first researchers who had investigated the role of heat and mass transfer during drying of porous media. The research work of Krischer was and is still used today as a basis for much of the development in drying theory. In his work [ 49 ], which was first published in , Krischer proposed a set of equations to describe the moisture transport for several geometries plate, cylinder, and sphere.
Krischer assumed that moisture transfer is controlled by the combined influence of capillary flow of liquid and diffusion of vapor. By using the theory of Krischer, Berger and Pei [ 50 ] included the sorption isotherm empirically obtained into the model as a coupling equation among liquid, vapor, and heat transfer. In this model all phenomenological coefficients e. Heat transfer is assumed to take place only by conduction through the solid skeleton.
The overall mass and heat balance equations proposed by Berger and Pei [ 50 ] are expressed in terms of moisture content and vapor pressure as follows: In addition, as for the previous models, experimental tests are needed to ensure its validity. In the late s and early s, Whitaker [ 51 , 52 ] presented a set of equations to describe the simultaneous heat, mass, and momentum transfer in porous media.
Based on the traditional conservation laws, the model proposed by Whitaker, an important milestone in the development of drying theory, incorporated all mechanisms for heat and mass transfer: By using the volume averaging method, the macroscopic differential equations were defined in terms of average field quantities.
Based on the model of Whitaker, a system of governing equations can be built to represent the drying process, in which the most important equations are the conservation equation for water in the liquid and the gas phases, the conservation equation for air in the gas phase, and the conservation equation of energy of the whole porous system under consideration. These equations can be solved numerically and form the framework for the numerical simulation of the drying process, in which simultaneous mass and heat transfer together with phase changes vaporization can be taken into account.
We will briefly discuss these equations here. The first equation governing the drying process is the conservation equation for water in both liquid and gas phases. This equation can be written as where , , and are the mass density of the liquid, vapor, and gas phases; and are the volume fractions of the liquid and gas phases; and are the velocities of the liquid and gas phases, and is the effective diffusivity tensor. The conservation equation for water states that the mass of water at each point inside the porous body is conserved. The second equation governing the drying process is the conservation equation for air in the gas phase: The third equation is the conservation equation of energy, which can be formulated as.
Note that in the above system of equations, the velocity of water can be computed using the equation of motion for the liquid phase: In addition to the above governing equations, the conditions for mass and heat transfer at the external drying surfaces of the porous medium must be specified. For example, it can be assumed that, at the external drying surfaces, the fluxes of mass and heat are described for convective drying by the boundary layer theory with Stefan correction: Additionally, we can assume that the gas pressure at the external drying surfaces is fixed at the pressure of the bulk drying air: These parameters are either functions of moisture content or temperature or both moisture content and temperature.
In addition, solving the coupled equations of heat and mass transfer, which are strongly nonlinear, requires very complicated numerical methods. The theory of Whitaker was further developed and applied in drying analysis of various porous media, for example, in the drying analysis of sand by Whitaker and Chou [ 53 ], Hadley [ 54 ], Oliveira and Fernandes [ 55 ], and Puiggali et al.
In these works, the model was usually quite successfully matched against experimental data. Whitaker and Chou [ 53 ] simplified the theory to obtain two nonlinear equations for the distribution of saturation and temperature. In this work, the gas pressure was assumed as constant, the gas momentum equation was ignored, and a quasisteady state was applied. It is interesting to note that there is a resemblance of these two equations to the equations proposed by Luikov and by Philip and De Vries [ 46 ]. In this simplified case, the comparison between theory and experiment was made by Hougen et al.
The important conclusion is that the gas phase momentum equation must be included in solving the comprehensive set of equations. To validate the model, a comparison of one-dimensional drying to the experimental drying of apple and potato was presented and a good agreement was found. The obtained partial differential equations in one dimension were solved by a three-point, two-level implicit finite difference method.
The calculated results were compared with experimental results and showed a quite good agreement. Ferguson [ 67 ] focused on a two-dimensional problem of the high-temperature drying of spruce.
The numerical results highlighted the advantage of the discretization technique control volume finite element method in solving the problem with structured and unstructured meshes. A numerical investigation was conducted by Boukadida et al. The work analyzed the influence of the properties of the surrounding drying agent temperature, gas pressure, and vapor concentration as well as the initial medium conditions temperature and moisture content on the drying process by considering several configurations.
However, the full investigation of the effect of the boundary layer on the coupled heat and mass transfer still requires further work, as concluded by the authors. By using the volume averaging method, a set of equations for multiphase systems was applied to porous media. Numerical results showed a good agreement with the experimental data of kaolin drying.
In their work, the drying of two quite different porous media—clay-brick and softwood—was investigated. By considering bound water, the driving potential for bound water migration was assumed to be proportional to the gradient in the bound moisture content. The control volume method was applied to solve the nonlinear partial differential equations. The mathematical schemes for equidistant and nonequidistant meshes were discussed [ 61 ]. The authors also investigated the sensitivity upon model parameters by numerically varying the effective diffusivity, effective thermal conductivity, intrinsic and relative permeabilities, and external drying conditions heat and mass transfer coefficients.
With the rapid development of the computer technology, modern computers allow the simulation of drying not only in one dimension but also in two and three dimensions. Besides, numerical methods are also more efficient in obtaining accurate results and reducing the computational time. In this work, a homogeneous model, which employed the full set of conservation equations, was considered. A cube of light concrete isotropic medium and a board of wood anisotropic medium were chosen to investigate the influence of the number of exchange faces. Several simulation results for low- and high-temperature drying of softwood were presented and discussed.
By comparing the different simulation results, the study showed that a three-dimensional model is required to describe correctly the drying behavior of porous media. The variation of the material property information such as capillary pressure and absolute permeability was taken into account with the help of experiments [ 77 , 78 ].
The material information of wood obtained from this work was later applied to a two-dimensional heterogeneous drying model [ 79 ]. In this work, the effects of material heterogeneity and local material direction on the heat and mass transport during drying were investigated. Two cases of low- and high-temperature drying were considered. Following this direction, more recently, Truscott [ 80 ] and Truscott and Turner [ 81 ] developed a three-dimensional heterogeneous drying model for wood.
The work considered the heterogeneity of the material properties, which vary within the transverse plane with respect to the position that defines the radial and tangential directions. Two nonlinear partial equations for moisture content and temperature pressure was assumed as constant were solved. By solving the system 23 — 30 resulting from the model of Whitaker, the drying process in porous media can be simulated numerically taking into account complex mass and heat transport phenomena. Different numerical methods can be used to solve the above system of equations, for example, the finite element method, the finite difference method, or the control volume method.
In simple cases, the numerical solution can be obtained with relatively small computational effort. One of such examples can be found in [ 52 ], where numerical simulation was compared with experimental measurement [ 14 ] of a sand plate under isothermal drying conditions. In the work, the above system 23 — 30 was applied. The plate of sand has a thickness of 5. The drying took place by air. One surface of the plate was considered impermeable.
The other surface was in contact with air. Since the width and length of the plate are much larger than its thickness, the problem can be reduced to be 1-dimensional. The numerical solution was obtained with the help of the finite difference method. Note that during the experiment [ 14 ], the averaged saturation was determined at different time instants.
Corresponding to these time instants, the saturation profiles were measured. The numerical solution was then compared with the experimental result as shown in Figure 3. In Figure 3 , the saturation profile is plotted as a function of the normalized distance from the impermeable surface of the plate. The comparison shows that the model of Whitaker delivers a reasonable result. In this work, the control volume method is chosen to solve the model of Whitaker. The reason for using the control volume method lies in the fact that this method satisfies the conservation requirement of the basic physical quantities such as mass and energy at any discrete level.
This means without the need to enforce this requirement by using additional constraints, the heat and mass flows across a boundary of a control volume element or over the boundary of the whole porous medium are automatically conserved. In this work, we will not discuss the details of the application of the control volume method in solving the system 23 — 30 and limit ourselves in presenting a numerical example in order to demonstrate the capability of the approach using the model of Whitaker.
For more details on the approach and numerical implementation, refer [ 12 ]. In order to dry the sample, the sample is put in an oven with dried air. At the boundary of the sample, heat and mass transfer takes place during the drying process. Since the sample is symmetric and the boundary conditions applied to the sample can also be considered as symmetric, the drying problem of our sample can be solved by the control volume method in one-dimensional space.
For other details concerning transport properties, refer [ 12 ]. As numerical results, the temporal evolution of moisture, temperature, and pressure for approximately every 0. In Figure 4 , the dashed curve presents the average moisture content of the whole sample. From the numerical results presented here, some important drying characteristics can be observed. This is called the first drying period or constant rate drying period. During the first drying period approximately The moisture gradient increases relative permeability k w decreases , and our analysis shows that the moisture profiles as the function of radius appear fairly flat.
Within this period, the pressure stays constant at atmospheric pressure Figure 6. In the second drying period, the dominating forces are viscous forces. Our analysis shows that a front separating the regions of adsorbed water and free water recedes from the surface into the sample. This process is finished when the moisture content everywhere in the sample is below the irreducible value, that is, when all free water of the sample has been removed. During this period, heat transfer is almost unchanged resistance in the sample is slightly increased , but mass transfer experiences an important additional resistance.
Heat is used not only to evaporate water but also to increase the temperature of the sample. Therefore, the temperature of the sample starts to rise from the wet bulb temperature. The center of the sample stays cooler than the outside Figure 5. This is due to the fact that the evaporation of water takes place not at the surface but at a place inside the sample front. The free water region heats up until a new equilibrium is attained if we assume a stationary front. At the front, heat is consumed for evaporation.
As we can see from Figure 6 , in the second drying period, an overpressure appears due to the Stefan effect. When the receding front has passed through the whole sample, the sample becomes dry and the entire porous medium is in the hygroscopic zone. The moisture content goes down to the equilibrium value.
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By considering the numerical example presented here, it is easy to see that the implemented simulation framework based on the model of Whitaker can be used effectively in studying the complex drying process of porous media. Especially important is the incorporation of the simultaneous heat and mass transfer processes together with the evaporation process in the simulation. In this work, a review of the development of some drying models, their application, and their restrictions is presented. Among the most complex and modern models is the model developed by Whitaker.
The result is a system of equations governing the drying process of porous media, which can be solved numerically using modern numerical methods, in particular the control volume method.