Conservation equation energy

The heat coming in through AA per unit time at Xj is represented by pucTAA 

The heat produced by the chemical reaction in the volume AAAx per unit time is given by 

The energy in AAAx is conserved according to pucTAA + (-XdT/dx)AA + coQAAAx = pucTAA-t- d/dx(pwcT)AAAx  

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Flow and Performance Calculations. Electro dynamic equations are usehil when local gas conditions (, a, B) are known. In order to describe the behavior of the dow as a whole, however, it is necessary to combine these equations with the appropriate dow conservation and state equations. These last are the mass, momentum, and energy conservation equations, an equation of state for the working duid, an expression for the electrical conductivity, and the generalized Ohm s law. 

The energy conservation equation is not normally solved as given in (9.4). Instead, an evolution equation for internal energy is used [9]. First an evolution equation for the kinetic energy is derived by taking the dot product of the momentum balance equation with the velocity and integrating the resulting differential equation. The differential equation is... 

Thermal plumes above point (Fig. 7.60) and line (Fig. 7.61) sources have been studied for many years. Among the earliest publications are those from Zeldovich and Schmidt. Analytical equations to calculate velocities, temperatures, and airflow rates in thermal plumes over point and line heat sources with given heat loads were derived based on the momentum and energy conservation equations, assuming Gaussian velocity and excessive temperature distribution in... 

For each layer the energy conservation equation is solved. The individual terms are the absorbed radiation in the layer and the radiative and convective heat exchange to the adjacent panes, to the room, or to the exterior. 

Considered are mass conservation of air and species (contaminants and humidity). Momentum equations are not considered on a global scale but have been used in some cases for the definition of the airflow-pressure relation of the individual links. Heat fluxes and thus energy conservation equations are not considered. 

A detailed study of the influence of viscous heating on the temperature field in micro-channels of different geometries (rectangular, trapezoidal, double-trapezoidal) has been performed by Morini (2005). The momentum and energy conservation equations for flow of an incompressible Newtonian fluid were used to estimate... 

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

However, a number of theories on flame quenching were also developed. Some of these were based on arbitrary assumptions concerning the conditions for quenching [7,11]. Another group of theories were based on the energy conservation equation for the flame, including heat loss [19-21]. 

The internal energy of the O2 fragment is found from the total kinetic energy distribution using the energy conservation equation... 

Applying the continuity and energy conservation equations, the mass flux can be found by... 

Using the same assumptions that were made in the vapor-layer model, the energy-conservation equation for the incompressible 2-D vapor phase can be simplified to a 1-D equation in boundary layer coordinates ... 

Since combustion processes generate significant sensible energy during reaction, the species conservation equations of Eq. (2.67) become coupled to the energy conservation equation through the first law of thermodynamics. 

To determine the laminar flame speed and flame structure, it is now possible to solve by computational techniques the steady-state comprehensive mass, species, and energy conservation equations with a complete reaction mechanism for the fuel-oxidizer system which specifies the heat release. The numerical... 

For a system where there is simultaneous heat generation and heat loss, the overall energy conservation equation takes the form... 

As mentioned, to include nonisothermal effects, an overall thermal energy balance needs to be added to the set of governing equations. The energy conservation equation can be written for phase k in the... 

The energy conservation equation in the gas phase for steady-state burning is given by Eq. (3.41). If one assumes that the physical parameters Xg and Cg are constant in the gas phase, Eq. (3.14) can be represented... 

Replacing ds/dt by the entropy balance equation, du/dt by the energy conservation equation, and dpkidt by the mass balance equation, we have... 

For steady, one-dimensional flow without body forces, with local mean velocity u(x) in a channel of constant cross-sectional area A, the energy conservation equation becomes, approximately (13) ... 

Taking into account the axial conduction of heat in the solid phase, the energy conservation equation for the gas is... 

Nondimensionalization of the species- and energy-conservation equations follows a procedure that is analogous to that for the Navier-Stokes equations. For two-dimensional steady axisymmetric flow of a perfect gas, the full equations are given as... 

For homogeneous gas-phase kinetics one may incorporate arbitrarily complex reaction mechanisms into the mass and energy conservation equations. Aside from questions of units, there is almost no disagreement in the formulation of the elementary rate law the rate of progress of each reaction proceeds according to the law of mass action. The CHEMKIN software [217] is widely used in the kinetics community to aid in the formulation and solution of gas-phase kinetics and transport problems. 

Calculations taking into account the strong coupling between the radiant energy conservation equation and the momentum, mass, and heat balances call for a model describing this interaction as a function of space and time. 

The present equations will be useful when the acceleration occurs deep interior or when the gravitational energy release is important in the energy conservation equation. In such cases we will have different structures from that of the usual solutions. Further investigation will be desired in many cases such as red giants or very massive stars. 

Temperature has a strong influence on fcasignificant effect on the kinetics of adsorption, then energy conservation equations may have to be included in the analysis to establish the temperature to every point in the bed. In these circumstances the additional space coefficients, kx, ht, and ke must be added to the previous list. A point coefficient analogous to k d8 is not included, because it has been assumed that the temperature behavior of the particle can be faithfully represented by assuming a homogenous material. With this simplification it is not... 

Here k denotes reaction rate constant, superscripts / and b stand for forward and backward reactions respectively, subscripts SMR and WGS stand for steam methane reforming and water gas shift reactions respectively, and the square brackets represent the mole fraction of the specie. The heat sources due to these reactions, to be added to the energy conservation equation are given by ... 

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