Equations of conservation

In addition to the Burke and Schumann model (34) and the Displacement Distance theory, a comprehensive laminar diffusion flame theory can be written using the equations of conservation of species, energy, and momentum, including diffusion, heat transfer, and chemical reaction. 

Bather than carrying out the calculation for the general case, which yields rather unwieldy expressions, only equations sufficient to obtain certain approximations will be developed. If we multiply the Boltzmann equation, Eq. (1-39), by 1 = i%( 2)3r )) (0.9>)> the resulting equation is simply the equation of conservation of mass, since integrating unity over the collision integral gives zero ... 

All other coefficient equations are identically zero. The second of Eq. (1-101), combined with the equation of conservation of mass, Eq. (1-77), shows that the motion is isentropic ... 

For pipe flow, HEM requires solution of the equations of conservation of mass, energy, and momentum. The momentum equation is in differential form, which requires partitioning the pipe into segments and carrying out numerical integration. For constant-diameter pipe, these conservation equations are as follows ... 

To control, optimize, or evaluate the behavior of a chemical plant, it is important to know its current status. This is determined by the values of the process variables contained in the model chosen to represent the operation of the plant. This model is constituted, in general, by the equations of conservation of mass and energy. 

Although this equation is derived in the classic works of Webster [2] and Sommerfeld [5], only Webster notes that the equation of conservation of energy... 

These three equations of conservation may be looked upon as defining any three of the four variables p, p, U, u in terms of the 4th, if it is assumed that the equation of the medium, f(p,p,T)=0, as well as the dependence of internal energy of any pair of these variables of state is known. Therefore, the properties of a stationary shock wave follow from the knowledge of the velocity of the piston maintaining the wave, which is also the material (particle) velocity, u... 

A more sophisticated approach is to avoid the postulate of a shock and instead to state the differential equations of conservation of mass, momentum, and energy to include more properties of a real fluid. Including the effects of viscosity, heat conditions, and diffusion along with chem reaction gives eqs with a unique solution for given boundary conditions and so solves the determinacy problem. The boundary conditions are restricted by the assumption that the reaction begins and is completed with the region considered. 

The restriction that no chem reaction occurs in the flow field is removed but consideration is limited to exothermic reactions. It is assumed that the chem reaction occurs instantaneously, so that the reaction zone is of zero width. Under this assumption the jump forms of the equations of conservation of mass, momentum, and energy are again justified... 

Ref 66, p 145). Solutions for the Chapman-Jouguet steady detonation wave are obtd from the equations of conservation of mass, the conservation of momentum, the conservation of energy, an equation of state and the C- J condition. Explicit solutions are reported by Eyring et al (Refs 9 22a) and by Taylor (Ref 26, pp 87-89)... 

There has always been an interest in having correlations for heat- and mass transfer for different geometries in view of the difficulties in describing turbulence in a meaningful way, and in solving the differential equations of conservation of momentum, energy, and mass transfer for real bodies. 

Combining the first two basic equations of a steadily propagating regime, specifically the equations of conservation of mass and momentum, so as to obtain the equation... 

The course of change of the specific volume and pressure of the material in a detonation wave corresponding to these conceptions are studied exclusion of the states indicated above ( 1) and selection of a specific value of the velocity are consequences of the mechanism of the beginning of the chemical reaction described in 2 and of the equations of conservation which lead (Todes, Izmailov) to a linear relation between the pressure and volume in the absence of losses. 

The equations of conservation of mass for gas and particle phases are given, respectively, by... 

The mathematical model of the batch reactor consists of the equations of conservation for mass and energy. An independent mass balance can be written for each chemical component of the reacting mixture, whereas, when the potential energy stored in chemical bonds is transformed into sensible heat, very large thermal effects may be produced. 

The obvious reason is that in this system a relation apart from the equations expressing the conservation of the elements exists, viz., that the number of atoms of two of the elements are the same in all compounds occurring in the mixture. This causes two of the equations of conservation to merge into one, so that the number of independent reactions must be increased by one. If generally there are r such relations, of which the... 

The most important category of dimensionless groups is that of the numerics connected with transport (of mass, energy and angular momentum). Scheme 3.1 shows the three fundamental equations of conservation, written in their simplest form (i.e., one-dimensional). A complete system of numerics can be derived by forming "ratios" of the different terms of these three equations, as was suggested by Klinkenberg and Mooy (1943). This system is reproduced in Scheme 3.2. 

Equation of conservation of Local change + Change by + convection Change by diffusion + Change by production = 0 Boundary condition... 

Equations (1) are to be solved together with the equations of conservation of momentum and mass ... 

The third level of complexity in airshed modeling involves the solution of the partial differential equations of conservation of mass. While the computational requirements for this class of models are much greater than for the box model or the plume and puff models, this approach permits the inclusion of chemical reactions, time-varying meteorological conditions, and complex source emissions patterns. However, since this model consists only of the conservation equations, variables associated with the momentum and energy equations—e.g., wind fields and the vertical temperature structure—must be treated as inputs to the model. The solution of this class of models will be examined here. 

If we divide the airshed into L cells and consider N species, LN ordinary differential equations of the form (15) constitute the airshed model. As might be expected, this model bears a direct relation to the partial differential equations of conservation (7). If we allow the cell size to become small, it can be shown that (15) is the same as the first-order spatial finite difference representation of (7) in which turbulent diffusive transport is neglected—i.e,. 

This model is similar to several of the those cited above in that the reactor is represented as a series of plug-flow zones. The equation of conservation for first order reactions in each zone is ... 

When we substitute (2.32) in (2.1). the equation of conservation of species in the presence of an external force Held becomes... 

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