Power balance equation

Now taking the results for dU/dt from (2.4), Q from (2.6) and P from (2.7) and putting them into the power balance equation (2.2) of the first law, and combining all the volume integrals, gives... 

Another important point, closely connected with electron kinetics, concerns the self-consistent treatment of electron kinetics, of the particle and/or power balance equations for heavy particles (such as different ions and excited atoms or molecules), and of the Maxwell equations (or a reduced version such as the Poisson equation or appropriate electric circuit equations) to obtain a more complete description of all important plasma components as well as of the internal electric field. This self-consistent treatment is usually tricky and is based on an iterative approach to the solution of the various types of equations involved (Loffhagen and Winkler, 1994 Uhrlandt and Winkler, 1996 Yang and Wu, 1996). To integrate the electron kinetic equation in such an approach adequately, a very effective solution procedure for this equation is of particular importance, although remarkable progress with respect to the speed of computation has been achieved in recent years. 

The transient power balance equation of an individual generator prime-mover set may be written as -... 

In general, all of these heat transfer mechanisms can occur. The power balance equation is given by... 

If this is substituted into the power balance equation,... 

The numerical solution of the energy balance and momentum balance equations can be combined with flow equations to describe heat transfer and chemical reactions in flow situations. The simulation results can be in various forms numerical, graphical, or pictorial. CFD codes are structured around the numerical algorithms and, to provide easy assess to their solving power, CFD commercial packages incorporate user interfaces to input parameters and observe the results. CFD... 

Every chemical reaction can go in either forward or reverse direction. Reactants can go forward to products, and products can revert to reactants. As you may remember from your general chemistry course, the position of the resulting chemical equilibrium is expressed by an equation in which /Cec], the equilibrium constant, is equal to the product concentrations multiplied together, divided by the reactant concentrations multiplied together, with each concentration raised to the power of its coefficient in the balanced equation. Eor the generalized reaction... 

If the coefficients in a balanced equation are multiplied by a factor n, rite equilibrium constant is raised to the nth power ... 

SOLUTION We write the equilibrium constant with the partial pressure of the product, NH3, in the numerator, raised to a power equal to its coefficient in the balanced equation. We do the same for the reactants, but place their partial pressures in the denominator ... 

Each equilibrium expression described so far contains a ratio of concentrations of products and reactants. Moreover, each concentration is raised to a power equal to its stoichiometric coefficient in the balanced equation for the overall reaction. Concentration ratios always have products in the numerator and reactants in the... 

Here the square brackets indicate the concentration of the chemical species within the bracket. That is, [A] means the concentration of A, and so forth. [A]" means the concentration of A raised to the a power, where a is the value of the coefficient of A in the balanced equation for the chemical equilibrium. The value of the ratio of concentration terms is symbolized by the letter K, called the equilibrium constant. For example, for the reaction of nitrogen and hydrogen referred to in Sec. 19.3,... 

The term S represents the strength of the network. The power law exponent m was found to depend on the stochiometric ratio r of crosslinker to sites. When they were in balance, i.e. r = 1, then m - 1/2. From Equations (5.140) and (5.141) this is the only condition where G (co) = G (cd) over all frequencies where the power law equation applies. If the stochiometry was varied the gel point was frequency dependent. This was also found to be the case for poly(urethane) networks. A microstructural origin has been suggested by both Cates and Muthumkumar38 in terms of a fractal cluster with dimension D (Section 6.3.5). The complex viscosity was found to depend as ... 

As mentioned earlier, to solve explicitly for the temperature T2 and the product composition, one must consider a, mass balance equations, (/j, a) nonlinear equilibrium equations, and an energy equation in which one of the unknowns T2 is not even explicitly present. Since numerical procedures are used to solve the problem on computers, the thermodynamic functions are represented in terms of power series with respect to temperature. 

The expression for Kc is a fraction. The concentration of the product is in the numerator, and the concentrations of the reactants are in the denominator. Each concentration term must be raised to the power of the coefficient in the balanced equation. 

Balanced equations, diverse conversion factors, limiting re ents, and percent yield calculations all now tremble before you. Probably. Be sure your powers over them cire as breathtaking as they should be by checking your answers. 

An approach suggested by Williams et al. (117) uses an energy balance equation for initiation of flaw growth in a linearly viscoelastic material. A spherical flaw geometry was selected for simplicity since the expressions for the critical values of applied stress to cause fracture were similar for several flaw geometries. The critical conditions are based on a power (energy rate) balance... 

Equilibrium expression The expression (from the law of mass action) obtained by multiplying the product concentrations and dividing by the multiplied reactant concentrations, with each concentration raised to a power represented by the coefficient in the balanced equation. 

H.9 The powerful oxidizing strength of iFeO j2- is shown by its ability to liberate oxygen from water and to produce dmitrogen from ammonia Write balanced equations for these two reactions. 

Thus, Ksp always equals the product of the equilibrium concentrations of all the ions on the right side of the chemical equation, with the concentration of each ion raised to the power of its coefficient in the balanced equation. 

Flow between parallel-plate electrodes SFe etching silicon Plug flow Boltzmann equation and Monte Carlo for electron energy electron density power balance 77... 

Flow between parallel plate electrodes SFff-02 etching silicon Well mixed Boltzmann equation for electron energy power balance for electron density 78... 

Parts (a) and (b) illustrate the general result that AG for a reaction with halved coefficients is half the AG value for the standard coefficients. Further, K for the reaction with halved coefficients is the one-half power (i.e., square root) of the K for the standard reaction. Note that the value of the following ratio at equilibrium must be independent of the way we write the balanced equation ... 

You can see that a balanced chemical equation is a powerful tool. It allows chemists to predict the amount of products that will result from a reaction involving a known amount of reactants. As well, chemists can use a balanced equation to calculate the amount of reactants they will need to produce a desired amount of products. They can also use it to predict the amount of one reactant they will need to react completely with another reactant. 

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