Enthalpy conservation equation

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,... 

In reality however, situations also exist where a more complex form of the rate expression has to be applied. Among the numerous possible types of kinetic expressions two important cases will be discussed here in more detail, namely rate laws for reversible reactions and rate laws of the Langmuir-Hinshelwood type. Basically, the purpose of this is to point out additional effects concerning the dependence of the effectiveness factor upon the operating conditions which result from a more complex form of the rate expression. Moreover, without going too much into the details, it is intended at least to demonstrate to what extent the mathematical effort required for an analytical solution of the governing mass and enthalpy conservation equations is increased, and how much a clear presentation of the results is hindered whenever complex kinetic expressions are necessary. 

The main advantage of using a Lagrangian framework for dispersed phase particles is that particle-level phenomena can be modeled rigorously. Species and enthalpy conservation equations for individual particles can be written ... 

Other conservation equations (enthalpy and species) for multiphase flows can be written following a similar general format. For example, the enthalpy conservation equation is written ... 

These considerations have been accomplished in an example to model the SPS behavior of graphitic elements that are inserted between the two stainless steel rams used in the model 515S system (Sumitomo), which is schematically shown in Fig. 6.30 [42]. The geometry of the graphitic elements has been designed in such a way that it not necessary to consider the effect of vertical interfaces between them. As a result, the horizontal contact resistances can be excluded in the mathematical models. A 2D model in cylindrical coordinates based on the usual enthalpy conservation equation that takes into account the Joule heat generation is developed, which is coupled to density current balances expressed in terms of the RMS portion of the electric potential and the mechanic equilibrium equations due to elastic behavior and thermal expansion of the materials. Thermophysical properties of... 

The physics and modeling of turbulent flows are affected by combustion through the production of density variations, buoyancy effects, dilation due to heat release, molecular transport, and instabiUty (1,2,3,5,8). Consequently, the conservation equations need to be modified to take these effects into account. This modification is achieved by the use of statistical quantities in the conservation equations. For example, because of the variations and fluctuations in the density that occur in turbulent combustion flows, density weighted mean values, or Favre mean values, are used for velocity components, mass fractions, enthalpy, and temperature. The turbulent diffusion flame can also be treated in terms of a probabiUty distribution function (pdf), the shape of which is assumed to be known a priori (1). 

Most physical phenomena in fires can be described as the transfer of heat and mass and chemical reactions in either gas or condensed phases. The physics of the fluid flow are governed by the conservation equations of mass, momentum, and enthalpy ... 

The region over which this balance is invoked is the heterogeneous porous catalyst pellet which, for the sake of simplicity, is described as a pscudohomoge-ncous substitute system with regular pore structure. This virtual replacement of the heterogeneous catalyst pellet by a fictitious continuous phase allows a convenient representation of the mass and enthalpy conservation laws in the form of differential equations. Moreover, the three-dimensional shape of the catalyst pellet is replaced by assuming a one-dimensional model... 

The conservation equations for mass and enthalpy for this special situation have already been given with eqs 76 and 62. As there is no diffusional mass transport inside the pellet, the overall catalyst effectiveness factor is identical to the film effectiveness factor i/cxl which is defined as the ratio of the effective reaction rate under surface conditions divided by the intrinsic chemical rate under bulk fluid phase conditions (see eq 61). For an nth order, irreversible reaction we have the following expression ... 

In view of equation (39), the similarity in the forms of equation (40) (for the thermal enthalpy Jto p dT) and equation (41) (for the mass fractions 1 ) is striking. Equations (40) and (41) are the energy- and species-conservation equations of Shvab and Zel dovich. The derivation given for these equations required neither that any transport coefficient or the specific heat of the mixture is constant nor that the specific heats of all species are equal. Coupling functions may now be identified from equations (40) and (41). 

A thermodynamic quantity of considerable importance in many combustion problems is the adiabatic flame temperature. If a given combustible mixture (a closed system) at a specified initial T and p is allowed to approach chemical equilibrium by means of an isobaric, adiabatic process, then the final temperature attained by the system is the adiabatic flame temperature T. Clearly depends on the pressure, the initial temperature and the initial composition of the system. The equations governing the process are p = constant (isobaric), H = constant (adiabatic, isobaric) and the atom-conservation equations combining these with the chemical-equilibrium equations (at p, T ) determines all final conditions (and therefore, in particular, Tj). Detailed procedures for solving the governing equations to obtain Tj> are described in [17], [19], [27], and [30], for example. Essentially, a value of Tf is assumed, the atom-conservation equations and equilibrium equations are solved as indicated at the end of Section A.3, the final enthalpy is computed and compared with the initial enthalpy, and the entire process is repeated for other values of until the initial and final enthalpies agree. 

The stationary-state heat release rate may also be interpreted from the measured temperature excess in well-stirred flow systems. The energy conservation equation for a well-stirred flow system is similar to equation (6.13) but an additional term is required to represent heat transport via the outflowing gases (a-Cp(T- Tafltres) as shown in equation (4.4). The inflowing gases are assumed to be pre-heated to the vessel temperature, Ta- Under constant pressure conditions, normally applicable to flow reactors, Cp replaces C, and A.H replaces AU in equation (6.13). The heat release is obtained from a summation of the product of individual reaction rates and their enthalpy change (-AH)jRj) in equation (5.4)). 

Peclet Number, Pe dimensionless number appearing in enthalpy or species mass conservation equations (defined for heat transfer and mass transfer, respectively). It is interpreted again as the ratio of the convective transport to the molecular transport and is defined 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 ... 

The energy conservation equation for laminar flows can be written in terms of enthalpy as... 

A wide range of physical models is available in most commercial CFD software. At a minimum, the flow field will be calculated by solving the conservation equations for mass and momentum. In addition to flow, many of the problems encountered in the process industry involve heat transfer also. For such applications, the temperature field can also be calculated, which is commonly done by solving a conservation equation for enthalpy. For problems involving chemical reaction, the transport equations for the chemical species involved in the reaction(s) will be solved. The creation and destruction of the species due to the reaction are modeled by means of source terms in these equations. The reaction rates determining these source terms are calculated locally, based on the values of species concentrations and temperature at each... 

Often it is more convenient to work with enthalpy rather than internal energy. Using the definition of enthalpy, i = u + Pip, and the mass conservation equation, Eq. 1.41, Eq. 1.52 can be rearranged to give... 

The energy conservation equation can be written using the sensible enthalpy as dependent variable ... 

The equations that govern the evaporation process of a drop are the conservation equations for mass, species and energy/enthalpy for the gas and liquid phases, together with boundary conditions and compatibility conditions at the liquid-gas interface. The momentum conservation is neglected since drop drag is not considered in this chapter. The conservation equations for both the gas and liquid phases, are given by (cf. Byron Bird et al. [8])... 

Densification behaviors of electrical conducting Cu and insulator AI2O3 with SPS have been compared and modeled, with the portion of the system schematically shown Fig. 6.24 [39]. In this modeling study, stainless steel electrodes (rams) are included in the model. The boundary conditions of axial cooling and the consequent temperature distribution change dramatically. The mathematical equations used in the modeling, i.e., the enthalpy and current density conservation equations in cylindrical coordinates applied to the elements, are solved in full 3D version through CFD-Ace (FEM). Temperature dependences of thermophysical properties for aU materials have been considered. 

In CFAST, a set of equations that predict state variables (pressure, temperamre, etc.) are solved based on the enthalpy and mass flux over small increments of time. These equations are derived from the conservation equations for mass, momentum, energy and the ideal gas law together with plume models, vent flow equations, radiation and combustion models. Forney and Moss reviewed that there are 11 variables to be solved the mass, internal energy, density, temperature and volume for the upper and lower layers (Mu, Eu, qu, Tu, Vu and ML, EL, qL, TL, VL), and the pressure R Because there are seven constraints, any four of those variables have to be chosen as solution variables. The four variables solved are the pressure... 

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