The Equation of Continuity for Species Mass Transport Phenomena

Mass and Species Transport

Mass conservation can be expressed in terms of the density, p,

Or in terms of the individual gaseous species, Y_a, where:

Y_(t = Mass fraction of species a,

D_a = Diffusion coefficient of species a, lit" = Mass production rate of species a per unit volume, m"'_a = Mass production rate of species a by evaporating droplets or particles.

Here mb' = Z_a m"'_a is the production rate of species by evaporating droplets or particles. Summing these equations over all species yields the original mass conservation equation because ZY_a = 1 and Zm"'= 0 and Zm"'_a = mJT, by definition and because it is assumed that Z pD_HVY_a = 0.

This last assertion is not true in general. However, transport equations are solved for total mass and all but one of the species, implying that the diffusion coefficient of the implicit species is chosen so that the sum of all the diffusive fluxes is zero.

Momentum Transport

The momentum equation in conservative form is written as follows:

The term UU is a dyadic tensor. In matrix notation, the dyadic is given by the tensor product of the vectors u and u^T. The term V-pUU is, thus, a vector formed by applying the vector operator to the tensor. The force term f_b in the momentum equation represents external forces such as the drag exerted by liquid droplets. The stress tensor x^ is defined as follows:

The term Sjj is the symmetric rate-of-strain tensor, written using conventional tensor notation.pis the dynamic viscosity of the fluid. The overall computation can either be treated as a DNS, in which the dissipative terms are computed directly, or as an LES, in which the large-scale eddies are computed directly, and the subgrid-scale dissipative processes are modeled. The numerical algorithm is designed so that LES becomes DNS as the grid is refined. Most applications of FDS are LES. For example, in simulating the flow of smoke through a large, multiroom enclosure, it is not possible to resolve the combustion and transport processes directly. However, for small- scale combustion experiments, it is possible to compute the transport and combustion processes directly.

Energy Transport

The energy conservation equation is written in terms of the sensible enthalpy, h_s:

The sensible enthalpy is a function of the temperature:

q = the heat release rate per unit volume from a chemical reaction, q_b = the energy transferred to the evaporating droplets, q = the conductive and radiative heat fluxes, к = the thermal conductivity, q_r = the radiative heat flux, e = Dissipation rate.

Equation of State

The equation of state can be expressed in the form 3.6 below:

W = molecular weight of the gas mixture. An approximate form of the Navier-Stokes equations appropriate for low Mach number applications is used in the model. The approximation involves the filtering out of acoustic waves while allowing for large variations in temperature and density. This gives the equations an elliptic character, consistent with low speed, thermal convective processes. In practice, this means that the spatially resolved pressure, p(x, y, z), is replaced by an "average" or "background" pressure, p_m(z, t), which is a function of time and height above the ground.

where:

Y„ = Molecular weight of the gas mixture,

W_a = Molecular weight of the gas species a.

Taking the material derivative of the background pressure and substituting the result into the energy conservation equation yields an expression for the velocity divergence, VU, which is an important term in the numerical algorithm because it effectively eliminates the need to solve a transport equation for specific enthalpy. The source terms from the energy conservation equation are incorporated into the divergence, which appears in the mass transport equations. The temperature is determined from the density and background pressure via the equation of state.

LES Approach

A small term in the energy equation is known as the dissipation rate, e, the rate at which kinetic energy is converted to thermal energy by viscosity:

This term is usually neglected in the energy conservation equation because it is very small relative to the heat release rate of the fire. To understand where this term originates, develop an evolution equation for the kinetic energy of the fluid by taking the dot product of the momentum equation with the velocity vector.

As mentioned above, e is a negligible quantity in the energy equation. However, its functional form is useful in representing the dissipation of kinetic energy from the resolved flow field. Following the analysis of Smagorinsky, the viscosity |r is modeled.

where C_s is an empirical constant and Д is a length on the order of the size of a grid cell. The bar above the various quantities denotes that these are the resolved values, meaning that they are computed from the numerical solution sampled on a coarse grid (relative to DNS). The other diffusive parameters, the thermal conductivity and material diffusivity, are related to the turbulent viscosity as follows:

The turbulent Prandtl number Pr_t and the turbulent Schmidt number Sc_t are assumed to be constant for a given scenario. The model for the viscosity, p_LES, serves two roles: first, it provides a stabilizing effect in the numerical algorithm, damping out numerical instabilities as they arise in the flow field, especially where vorticity is generated. Second, it has the appropriate mathematical form to describe the dissipation of kinetic energy from the flow. In the parlance of the turbulence community, the dissipation rate is related to the turbulent kinetic energy (often denoted by k) as e=k^3/2/L, where L is a length scale. Based on simulations of smoke plumes, C_s is 0.20, and P_rt and S_ct are 0.5. There are no rigorous justifications for these choices other than by comparison with experimental data [25].

The most distinguishing feature of any CFD model is its treatment of turbulence. Of the three main techniques of simulating turbulence, FDS contains only LES and DNS. There is no Reynolds-averaged Navier-Stokes (RANS) capability in FDS. Pyrosim uses a computational simplification called LES to enhance the speed at which complex flows can be solved numerically.

While LES employs a computational shortcut where only large eddies are directly solved, and the dissipative energy generation of small eddies is modeled as a byproduct of large eddies, the CFD programs utilize DNS where all the equations are solved for all sizes of turbulent eddies. LES does not adopt the conventional time- or ensemble-averaging RANS approach with additional modeled transport equations being solved to obtain the so-called Reynolds stresses resulting from the averaging process. In LES, the large- scale motions (large eddies) of turbulent flow are computed directly and only small-scale (SGS) motions are modeled, resulting in a significant reduction in computational cost compared to DNS. LES is more accurate than the RANS approach since large eddies contain most of the turbulent energy and are responsible for most of the momentum transfer and turbulent mixing; moreover, LES captures these eddies in full detail directly whereas they are modeled in the RANS approach. Furthermore, the small scales tend to be more isotropic and homogeneous than the large ones, and thus modeling the SGS motions tend to be easier than modeling all scales within a single model, as in the RANS approach. Therefore, currently, LES is the most viable and promising numerical tool for simulating realistic turbulent and transitional flows and produces fully validated results for many fire problems. LES method models the dissipative processes (viscosity, thermal conductivity, material diffusivity) that occur at length scales smaller than those that are explicitly resolved on the numerical grid. This means that the parameters ц, k, and D in the equations in the appendix cannot be used directly in most practical simulations. They must be replaced by surrogate expressions that "model" their impact on the approximate form of the governing equations. Appendix A contains a simple explanation of how these terms are modeled in FDS.

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Source: https://ebrary.net/195681/engineering/mass_species_transport

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