Stokes equation, mass transport

The governing equations for mass flow, energy flow, and contaminant flow in a room will be the continuity equation, Navier-Stokes equations (one in each coordinate direction), the energy equation, and the mass transport equation, respectively. 

With the increased computational power of today s computers, more detailed simulations are possible. Thus, complex equations such as the Navier—Stokes equation can be solved in multiple dimensions, yielding accurate descriptions of such phenomena as heat and mass transfer and fluid and two-phase flow throughout the fuel cell. The type of models that do this analysis are based on a finite-element framework and are termed CFD models. CFD models are widely available through commercial packages, some of which include an electrochemistry module. As mentioned above, almost all of the CFD models are based on the Bernardi and Verbrugge model. That is to say that the incorporated electrochemical effects stem from their equations, such as their kinetic source terms in the catalyst layers and the use of Schlogl s equation for water transport in the membrane. 

Fluid dynamics in a tubular fuel cell has significant effects on different other phenomena such as the electrochemical reactions, which need species to be transported to the reaction site, and mass and heat transfer. In the porous structures, an equation that accounts for the fluid flowing in the pores - such as Brinkman s or Darcy s equations (Equation (3.4)) - must be used usually the velocities are low. Fluid-dynamic is modelled through Navier-Stokes equation in the channels (Equation (3.3)). 

For the simulation of gas-evolving electrochemical processes, a new approach is proposed that combines numerical models for electrolyte flow, ion transport and gas evolution. The mass and momentum conservation of the electrolyte flow is modelled by the incompressible Navier-Stokes equations, solving for the fluid velocity u and the pressure p ... 

The velocity field in these flows is governed by the incompressible Navier-Stokes equation (1.385) and the corresponding continuity relation (1.382). For incompressible fluids a generic transport equation for scalar species mass concentration fields can be deduced from (1.454) and expressed as ... 

The momentum equation, as represented by the Navier-Stokes equation, is not restricted to a single-component fluid but is valid for a multicomponent solution or mixture so long as the external body force is such that each species is acted upon by the same external force (per unit mass), as in the case with gravity. In the following section we consider external forces associated with an applied external field, which differ for different species. The reason for there being no distinction between the various contributions to the stress tensor associated with diffusive transport is that the phenomenological relation for the stress is unaltered by the presence of concentration gradients. This is seen from the fact that the stress tensor must be related to the spatial variations in fluid... 

As a solution to momentum transport (Navier-Stokes equation) is usually not available for bioreactor velocity field, we are left with mass and heat balances. A local balance describing mixing, flow and interface transfer, including reaction is known as the dispersed model (3) ... 

Brenner [6] proposes a different description of fluid dynamics. He proposes that Vv in Newton s rheological law should not be based on the mass-based velocity of the fluid, but on its volume velocity. This is a controversial idea. The adapted Navier-Stokes equations provide a hydrodynamic description of thermophoresis and thermal creep. Bedeaux et al. [7] describe how this alternative approach of the transport equations can be validated experimentally by means of thermophoresis. 

Physical modeling of CVD processes means solving the Navier-Stokes equations, partial differential equations for mass and heat transport in fluids given the constant boundary conditions of the reactor. These processes affect the uniformity of the deposit in all parts of the reactor. The proper name for CVD reactor physics is chemical vapor technology (CVT) and it is a subject of some significance for industrial reactor design. (This branch of continuum physics is outside the scope of this book, which is concerned with materials rather than machinery.)... 

Several stochastic models have been developed to reconstract the GDL microstructure. With a reconstracted GDL, Wang etal [34] further presented a detailed DNS study to investigate the transport phenomena of mass, reactant, electron, and heat inside a GDL. In this approach, the Navier-Stokes equation was apphed directly to the void space with no slip boundary condition at the sohd matrix surface. The oxygen and water transport can be expressed in a similar way to that in the DNS of CLs. Figure 30.8 shows the mass flow through the microstructure of the GDL. The results were used to evaluate the GDL properties, such as the permeabihty and tortuosity. They found that the tortuosity of the considered pore structure is 1.2 whereas the permeability fC is 3.1 x 10 m [34]. 

The dynamics is obtained hy numerical solving a set of the coupled Boltzmann-BGK transport equations (d. eqn [35]) on a spatial lattice in discrete time steps with a discrete set of microscopic vdodties. At each time step, the prohahility density evolved hy each LB equation is adverted to nearest neighhoting lattice sites and modified by molecular collisions, which are local and conserve mass and momentum. As a result, a LB fluid is shown to obey the Navier-Stokes equation (in the limit of a small lattice spacing and small time step). For dilute polymer solutions, the method typically involves phenomenological coupling between the polymer chain and the flowing fluid in the form of a linear friction term based on an effective viscosity. 

The flow in the gas channels and in the porous gas diffusion electrodes is described by the equations for the conservation of momentum and conservation of mass in the gas phase. The solution of these equations results in the velocity and pressure fields in the cell. The Navier-Stokes equations are mostly used for the gas channels while Darcy s law may be used for the gas flow in the GDL, the microporous layer (MPL), and the catalyst layer [147]. Darcy s law describes the flow where the pressure gradient is the major driving force and where it is mostly influenced by the frictional resistance within the pores [145]. Alternatively, the Brinkman equations can be used to compute the fluid velocity and pressure field in porous media. It extends the Darcy law to describe the momentum transport by viscous shear, similar to the Navier-Stokes equations. The velocity and pressure fields are continuous across the interface of the channels and the porous domains. In the presence of a liquid phase in the pore electrolyte, two-phase flow models may be used to account for the interaction between the gas phase and the liquid phase in the pores. When calculating the fluid flow through the inlet and outlet feeders of a large fuel cell stack, the Reynolds-averaged Navier-Stokes (RANS), k-o), or k-e turbulence model equations should be used due to the presence of turbulence. 

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