General property balance

Therefore, the general property balance equation (II-9) becomes for total particle volume ... 

Figure 3.1 Introductory scheme for the equation of a general property balance.

A General Molecular Transport Equation and General Property Balance... 

Figure 2.3-1. Molecular transport of a property (a) plot of concentration versus distance for steady state, (b) unsteady-state general property balance.

In Section 2.3 we derived an equation for a general property balance of momentum, thermal energy, or mass at unsteady state by writing Eq. (2.3-7). Writing a similar equation but specifically for heat transfer. 

In Section 2.3 a general property balance for unsteady-state molecular diffusion was made for the properties momentum, heat, and mass. For no generation present, this was... 

Particle Size Development. Now that a general total property balance equation has been developed (equation (II-9)), one can use it to obtain ordinary differential equations (ode s) which will describe particle size development. What is needed with equation (II-9) is an expression for dp(t,t)/dt, where p denotes a specific property of the system (e.g. particle size). Such an expression can be written for the rate of change of polymer volume in a particle of a certain class. The analysis, which is general and described in Appendix III, will finally result in a set of ode s for Np(t), Dp(t), Ap(t) and Vp(t). 

Kiparissides, et al. (8) developed mathematical models of two levels of sophistication for the vinyl acetate system a comprehensive model that solved for the age distribution function of polymer particles and a simplified model which solved a series of differential equations assuming discrete periods of particle nucleation. In practice, the simplified model adequately describes the physical process in that particle generation generally occurs in discrete intervals of time and these generation periods are short in duration when compared with operation time of the system. The simplified model is expanded here for a series of m reactors. The total property balances for number of particles, polymer volume, conversion, and area of particles, are written as ... 

Suppose sys(f) is the total energy (internal + kinetic + potential) of a system, and ihm and /hout are the mass flow rates of the system input and output streams. (If the system is closed, these quantities each equal zero.) Proceeding as in the development of the transient mass balance equation, we apply the general energy balance equation (11.3-1) to the system in a small time interval from t to t + 1st, during which time the properties of the input and output streams remain approximately constant. The terms of the equation are as follows (see Section 7,4) ... 

Flow processes for wliich the accumulationtermof Eq. (2.28), d mU v/dt, is zero are said to occur at steady state. As discussed with respect to tire mass balance, tliis means tliat tire mass of the system within the control volume is constant it also means that no changes occur with time in tire properties of tire fluid witliin the control volume nor at its entrances and exits. No expansion of the control volume is possible under these circumstances. The only work of the process is sliaft work, and the general energy balance, Eq. (2.28), becomes ... 

The simplest three-periodic hyperbolic surfaces are "Infinite Periodic Minimal Surfaces" (EPMS, named by Alan Schoen [6]). For these surfaces, the mean curvature is constant on the surface, and everywhere identically zero. This is a defining characteristic of minimal surfaces. For these structures, the sub-volumes can be geometrically identical. This occurs if the IPMS contains straight lines. Such surfaces have been called "balance surfaces" by Koch and Fischer [7]. We focus primarily on IPMS in this book. Some further discussion of general properties of minimal surfaces is in order here, since a number of their geometrical and topological properties will be required for later chapters. 

The Mass Balance Equation of Chromatography and Its General Properties... 

The Mass Balance Equa Hon of Chroma tography audits General Properties... 

Jacob et al. used the method of characteristics to discuss the general properties of the system of mass balance equations in multicomponent preparative gas chromatography (GC) [34-36], assuming either a linear or a nonlinear isotherm. The GC problem is more complicated than the HPLC one because the gas mobile phase is much more compressible than a solution and the mobile phase velocity is very different inside and outside a high concentration band because the partial molar volumes of compounds are much larger in the gas mobile phase than in the condensed stationary phase (the sorption effect). They showed that the method of characteristics appHes to multicomponent systems as well as to single component... 

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