Medium conservation equations

Adoption of this approach to microbial process development cannot occur until methods exist for determining the influence of reactor design and operating parameters on single-cell metabolic control actions and reaction rates. If this Information is available, population balance equations and associated medium conservation equations provide the required bases for reactor analysis (1, ). For example, for a well-mixed, continuous-flow Isothermal mTcroblal reactor at steady-state, the population balance equation may be written ... 

Equation 5.1.13 shows how heat release acts a volume source. Assuming that the combustion takes place in a uniform medium at rest (Mach < 0), and writing for small perturbations, a = a + a a = p, p, v), the linearized conservation equations for mass and momentum can be used to eliminate the density in 5.1.13 to obtain a wave equation for the pressure in the presence of local heat release ... 

The formulation of combustion dynamics can be constructed using the same approach as that employed in the previous work for state-feedback control with distributed actuators [1, 4]. In brief, the medium in the chamber is treated as a two-phase mixture. The gas phase contains inert species, reactants, and combustion products. The liquid phase is comprised of fuel and/or oxidizer droplets, and its unsteady behavior can be correctly modeled as a distribution of time-varying mass, momentum, and energy perturbations to the gas-phase flowfield. If the droplets are taken to be dispersed, the conservation equations for a two-phase mixture can be written in the following form, involving the mass-averaged properties of the flow ... 

The mass conservation equation only relates concentration variation with flux, and hence cannot be used to solve for the concentration. To describe how the concentrations evolve with time in a nonuniform system, in addition to the mass balance equations, another equation describing how the flux is related to concentration is necessary. This equation is called the constitutive equation. In a binary system, if the phase (diffusion medium) is stable and isotropic, the diffusion equation is based on the constitutive equation of Pick s law ... 

It must be noted that R-H. equations, being derived from the Conservation equations, are valid regardless of the equation of state of the medium. Nevertheless, they cannot be solved explicitly, nor even plotted without specifying a suitable equation of state. This can be done as explained in Ref 4, p 951 Ref 7, pp 66-72 and Ref 10, pp 45, 127 165-66... 

The enthalpy conservation equation of the multiphase medium, obtained from the sum of the appropriate balance equations of the constituents includes the heat effects due to phase changes and hydration (dehydration) process, as well as the convectional and latent heat transfer,... 

This equation shows that the conservation of a property depends on the fortuitous or natural displacement of the property produced by vector w, when that is generated through a volume (Jy/-) or/and by a surface process (vector The mentioned displacement is supplemented by a diffusion movement (Dj- in the right part of the conservation equation). This movement is characterized by steps of small dimension occurring with a significant frequency in all directions. When the diffusion movement takes place against the vector w it is often called counterdiffusion. In the case of a medium, which does not generate the property, the relation can be written as follows ... 

In a continuous medium, the classical conservation equations apply. Restricting our analysis to incompressible flows, these equations are... 

Example 2.4 Diffusion in a Finite Domain The steady-state conservation equation for diffusion of species i in an finite medium can be expressed as... 

The additional sink is added to the usual conservation equations corrected for the volume fraction of the porous media. The governing equations look similar to those for Eulerian multiphase flow processes (Section 4.2.2) except that the volume fraction of the porous medium is not a variable. In the enthalpy equation, it is possible to include influence of porous media by considering an effective thermal conductivity, fceff, of the form ... 

Note that there are as many droplet media (1.15), (1.16) as many are typical radii rvr2,...,rN in the droplet ensemble. It has also been assumed here that (i) the flow situation is steady (if) no droplets change their size as well as there are no breaks up or coalescence for them what is true to some extent for spraying cooling (Hi) there are no internal pressure and shear stresses within each continuous droplet medium [468], All the droplets have their own temperatures t(x, z r) and the air vapour concentration close to their surfaces e(x, z r) that are considered as scalar fields along with the fields of the corresponding quantities for the air flow, T(x, z) and E(x, z). The following heat conservation equation for any individual droplet is valid ... 

A conservation equation can be written for a unit volume of porous medium ... 

Conservation Equations. Mass balance equations for the gaseous and aqueous phases are written in standard reservoir simulator form 10, 93). For the nonwetting foam or gas phase in a one-dimensional medium,... 

The effluent concentration can be predicted by solving the mass conservation equation. The conservation equations of particulate matter consider the change in concentration of particulate and change of porosity with time. The amount of fines retained in the porous medium is represented by a, while u signifies the superficial velocity of the incompressible transport fluid. For constant volumetric, incompressible flow, neglecting dispersion and gravitational effects, the one dimensional conservation equation follows. 

The principles of conservation of momentum, energy, mass, and charge are used to define the state of a real-fluid system quantitatively. The conservation laws are applied, with the assumption that the fluid is a continuum. The conservation equations expressing these laws are, by themselves, insufficient to uniquely define the system, and statements on the material behavior are also required. Such statements are termed constitutive relations, examples of which are Newton s law that the stress in a fluid is proportional to the rate of strain, Fourier s law that the heat transfer rate is proportional to the temperature gradient. Pick s law that mass transfer is proportional to the concentration gradient, and Ohm s law that the current in a conducting medium is proportional to the applied electric field. 

When bulk fluid flow is present (v 0), concentration profiles can be predicted from Equation 10-17, subject to the same boundary and initial conditions. This set of equations has been used to describe concentration profiles during micro-infusion of drugs into the brain [33]. In addition to Equation 10-17, conservation equations for water are needed to determine the variation of fluid velocity in the radial direction. Relative concentrations are predicted by assuming that the brain behaves as a porous medium (i.e., velocity is related to pressure gradient by Darcy s law, see Equation 6-9). Water introduced into the brain can expand the interstitial space this effect is balanced by the flow of water in the radial direction away from the infusion source and, to a lesser extent, by the movement of water across the capillary wall. 

Mass conservation equations apply to water and air. When the porous medium is deformable, the momentum balance equation (mechanical equilibrium) is also taken into account. In non-isothermal problems, the internal energy balance for the total porous medium must be considered. The basic equations solved by the finite element code CODE BRIGHT are ... 

According to this conservation equation, the viscosity of a suspension of N spheres dispersed in pure solvent, treated as a continuous fluid, is equal to the viscosity of a suspension of N — K spheres dispersed in an effective medium consisting of the solvent and the K remaining spheres, to each of which is assigned an appropriately modified diameter. 

Conservation equations together with constitutive equations describe a phenomenological model of the continuous medium (continuum). 

Drying is a macroscopical phenomenon involving simultaneous heat and mass transfer. Suppose that heat conduction and moisture diffusion are dominant during the overall transfer process in a homogeneous medium. The multidimensional conservation equations for an anisotropic medium can then be expressed as... 

Depending on the resolution of the mathematical model, different forms of the species conservation equations may be considered in the porous electrodes. For instance, in the multi-scale modehng of Khaleel et al. [18], a mesoscale lattice Boltzmarm model of the electrodes resolves the species transport in the gas, on the surface of the electrode, and through the bulk solid of the electrode. In this model, Eq. (26.1) is solved in three separate domains with corresponding transport properties and source terms. In contrast, in the macroscale distributed electrochemistry model of Ryan et al. [19], the porous medium of the SOFC electrodes is not explicitly resolved but is included in the model via effective properties. In the effective properties model, the diffusion coefficient of Eq. (26.1) is replaced with an effective diffusion coefficient, which is discussed in Section 26.3.3. 

In fuU fuel-cell modeling and simulation, it is difficult to incorporate the DNS-based CL sub-model owing to the computational burden, that is, the large number of unknowns involved. A macroscopic approach is more suitable in this regard. The CL is usually treated as a homogeneous medium with macroscopic properties such as effective permeability and diffusion coefficients. Darcy s law usually applies to describe the flow. In contrast to the common flow equation, mass exchange exists at the solid-fluid interface in CLs, and a mass term must therefore be added to the conservation equation as follows [29] ... 

Note 5.4 (On the permeability andflow in a porous medium). The seepage equation can be obtained by substituting Darcy s law into the mass conservation equations of fluid and solid phases, as described above. The effects of the micro-structure and microscale material property are put into the hydraulic conductivity k, which is fundamentally specified through experiments. It is not possible to specify the true velocity field by this theory, whereas by applying a homogenization technique, we can determine the velocity field that will be affected by the microscale characteristics. In Chap. 8 we will outline the homogenization theory, which is applied to the problem of water flow in a porous medium, where the microscale flow field is specified. 

Let us consider the problem of the diffusion of a multi-component fluid (see e.g.. Drew and Passman 1998) in a non-deformable porous medium which is saturated with the solution. Note that in ensuring adsorption, desorption is also taken into account unless otherwise specified. Referring to (3.246), the mass conservation equation for the multi-component fluid is... 

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