Balance equation for energy

Next, balance equations for energy and availability are written as... 

The thermodynamic constraints. The application of non-equilibrium thermodynamics to transformation processes is based on the formulation of two basic balance equations. The first one, a balance equation for energy, can, by virtue of the fact that the first law of thermodynamics assures energy to be a conserved quantity in any system, for a system in stationary state be expressed as ... 

Develop the balance equation for energy and entropy in open systems. 

On combining the balance equation for energy assuming the absence of velocity gradients, the first law of thermodynamics and assuming Gibbs equation for entropy production for the case of local equilibrium, a the entropy production per unit volume per unit time due to the occurrence of irreversible processes in the system is given by... 

Nonequilibrium thermodynamics is founded on the explicit expression for a in terms of the irreversible processes that we can identify and study experimentally. Before we begin deriving this expression, however, we shall write the explicit local forms of balance equations for energy and concentrations. 

The balanced equation for turbulent kinetic energy in a reacting turbulent flow contains the terms that represent production as a result of mean flow shear, which can be influenced by combustion, and the terms that represent mean flow dilations, which can remove turbulent energy as a result of combustion. Some of the discrepancies between turbulent flame propagation speeds might be explained in terms of the balance between these competing effects. 

BATCH ENERGY BALANCE EQUATIONS FOR FREE-RADICAL POLYMERIZATIONS AND COPOLYMERIZATIONS... 

C14-0083. Although the ATP-ADP reaction is the principal energy shuttle in metabolic pathways, many other examples of coupled reactions exist. For example, the glutamic acid-glutamine reaction discussed in the text can couple with the acetyl phosphate reaction shown in Example 14-10. Write the balanced equation for the coupled reaction operating in the direction of overall spontaneity and calculate A G ° for the overall process. 

The mass and energy balance equations for ideally mixed components where zero-order reaction proceeds are ... 

The energy and mass balance equations for reacting systems follow the same principles, as described previously in Secs. 1.2.3 to 1.2.5. 

The fluid model is a description of the RF discharge in terms of averaged quantities [268, 269]. Balance equations for particle, momentum, and/or energy density are solved consistently with the Poisson equation for the electric field. Fluxes described by drift and diffusion terms may replace the momentum balance. In most cases, for the electrons both the particle density and the energy are incorporated, whereas for the ions only the densities are calculated. If the balance equation for the averaged electron energy is incorporated, the electron transport coefficients and the ionization, attachment, and excitation rates can be handled as functions of the electron temperature instead of the local electric field. 

The energy balance equation for adiabatic operation becomes... 

This section treats the material and energy balance equations for a plug flow reactor. For steady-state operation the energy balance analysis leading to equation 10.1.4 is appropriate. 

The various energy transfer constraints enter into the analysis primarily as boundary conditions on the difference equations, and we now turn to the generation of the differential equations on which the difference equations are based. Since the equations for the one-dimensional model are readily obtained by omitting or modifying terms in the expressions for the two-dimensional model, we begin by deriving the material balance equations for the latter. For purposes of simplification, it is assumed that only one independent reaction occurs within the system of interest. In cases where multiple reactions are present, one merely adds an appropriate term for each additional independent reaction. 

Poly(3HB) syntheses coupled to consumption of energy (Eq. 4) resemble the general balance equation for the synthesis of biomass ... 

Eleat transfer occurs not only within the solid surface, droplet and vapor phases, but also at the liquid-solid and solid-vapor interface. Thus, the energy-balance equations for all phases and interfaces are solved to determine the heat-transfer rate and evaporation rate. 

The unit is divided in two pseudounits, each one corresponding to a different fluid. Pseudounits are interconnected by a pure energy flow that represents the heat transfer between them (see Fig. A.l). The balance equations for each pseudounit k can be stated as follows ... 

As discussed in Section 2.1, in high-Reynolds-number turbulent flows the scalar dissipation rate is equal to the rate of energy transfer through the inertial range of the turbulence energy spectrum. The usual modeling approach is thus to use a transport equation for the transfer rate instead of the detailed balance equation for the dissipation rate derived from (1.27). Nevertheless, in order to understand better the small-scale physical phenomena that determine e, we will derive its transport equation starting from (2.99). 

Equation (1.11) is now examined closely. If the s (products) total a number / , one needs (// + 1) equations to solve for the // n s and A. The energy equation is available as one equation. Furthermore, one has a mass balance equation for each atom in the system. If there are a atoms, then (/t - a) additional equations are required to solve the problem. These (// a) equations come from the equilibrium equations, which are basically nonlinear. For the C—H—O—N system one must simultaneously solve live linear equations and (/t - 4) nonlinear equations in which one of the unknowns, T2, is not even present explicitly. Rather, it is present in terms of the enthalpies of the products. This set of equations is a difficult one to solve and can be done only with modem computational codes. 

Figure 9.7 Graphical representation of energy balance equation for adiabatic operation. These are adiabatic operating lines.

We can also obtain these expressions from the energy-balance equation for the steady-state PFTR by simply transforming dzju dt with A,/ V replacing Pw/At. The solutions of these equations for the batch reactor are mathematically identical to those in the PFTR, although the physical interpretations are quite different. 

We solved the mass- and energy-balance equations for the CSTR in the previous chapter. The mass balance on species A is... 

Now we substitute this expression for X(T) into the energy-balance equation to yield a combined energy- and mass-balance equation for a first-order irreversible reaction in a CSTR,... 

Any fluid flow situation is described completely by momentum, mass, and energy balances. We have thus far looked at only simplified forms of the relevant balance equations for our simple models, as is done implicitly in aU engineering courses. It is interesting to go back to the basic equations and see how these simple approximations arise. We need to examine the full equations to determine the errors we are making in describing real reactors with... 

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