Semi-empirical equations continuity

This simplified form of (1) is the so-called semi-empirical equations of continuity, or, as we have referred to them, the working equations. It is... 

Several approaches to airshed modeling based on the numerical solution of the semi-empirical equations of continuity (7) are now discussed. We stress that the solution of these equations yields the mean concentration of species i and not the actual concentration, which is a random variable. We emphasize the models capable ot describing concentration changes in an urban airshed over time intervals of the order of a day although the basic approaches also apply to long time simulations on a regional or continental scale. 

Ham, J.H. and Platzer, B. (2004) Semi-empirical equations for the residence time distributions in disperse systems - part 1 continuous phase. Chem. Eng. Technol, 27 (11), 1172-1178. 

Discrete drops continue to be formed with increasing velocity of the dispersed phase up to a critical velocity u.. The dispersion formed at each velocity is mono-disperse and the drop diameter, d, can be obtained from a semi-empirical equation developed by Haywo th and Treybal (2) and based on the balance of forces. 

These equations predict a continuously diminishing rate of creep. Many empirical and semi-empirical models of creep-strain have been made and are described by Ward [24], One of these has been used successfully to describe the later stages of creep in polymers such as oriented polyethylene. The Arrhenius equation was modified by Eyring to apply to the rate of creep (deJdt) in the following way ... 

Flooding-point. Because the flooding-point is no longer synonymous with that for spray towers, equations 13.34 and 13.35 predict only the upper transition point. Dell and Pratt 30 1 adopted a semi-empirical approach for the flooding-point by consideration of the forces acting on the separate dispersed and continuous phase channels which form when coalescence sets in just below the flooding-point. The following expression correlates data to within 20 per cent ... 

We see, as for the species continuity equations, that when we perform the averaging procedure we obtain new dependent variables, in this case the terms pu iu j. These new terms are usually called the turbulent momentum fluxes or the Reynolds stresses, but, as before, we have more dependent variables than equations. Thus, some means to evaluate the turbulent momentum fluxes must be developed. Although no rigorous method of obtaining a closed set of equations is known, a number of semi-empirical approaches have been proposed which yield qualitative or semi-quantitative results for appropriately chosen classes of prr blems. 

To calculate micelle size and diffusion coefficient, the viscosity and refractive index of the continuous phase must be known (equations 2 to 4). It was assumed that the fluid viscosity and refractive index were equal to those of the pure fluid (xenon or alkane) at the same temperature and pressure. We believe this approximation is valid since most of the dissolved AOT is associated with the micelles, thus the monomeric AOT concentration in the continuous phase is very small. The density of supercritical ethane at various pressures was obtained from interpolated values (2B.). Refractive indices were calculated from density values for ethane, propane and pentane using a semi-empirical Lorentz-Lorenz type relationship (25.) Viscosities of propane and ethane were calculated from the fluid density via an empirical relationship (30). Supercritical xenon densities were interpolated from tabulated values (21.) The Lorentz-Lorenz function (22) was used to calculate the xenon refractive indices. Viscosities of supercritical xenon (22)r liquid pentane, heptane, decane (21) r hexane and octane (22.) were obtained from previously determined values. 

In chemical reaction engineering single phase reactors are often modeled by a set of simplified ID heat and species mass balances. In these cases the axial velocity profile can be prescribed or calculate from the continuity equation. The reactor pressure is frequently assumed constant or calculated from simple relations deduced from the area averaged momentum equation. For gases the density is normally calculated from the ideal gas law. Moreover, in situations where the velocity profile is neither flat nor ideal the effects of radial convective mixing have been lumped into the dispersion coefficient. With these model simplifications the semi-empirical correlations for the dispersion coefficients will be system- and scale specific and far from general. 

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