Dynamic component balances

The dynamic component balance equations for each of the two phases in turn. 

The dynamic component balances on the gas phase in each lump assume a constant average molar density of pav = 1.124 kmol/m3 (at 535 K and 50 bar). 

The rate-based model often employed is based on the two-film theory and comprises the material and energy balances of a differential element of the vapor and of the liquid phase. The dynamic component balances for the liquid and the vapor are given by ... 

The generation of residue curves is described mathematically by a dynamic molar balance of the liquid in the vessel Muq and two dynamic component balances for components A and B. The rate of vapor withdrawal is V (moles per time). 

Dynamic component balance equation for retentate compositions in each cell... 

Since there is no energy balance, nor an overall mass balance, the only dynamic balance that remains is a component balance for component A and/or B. To describe the outlet concentration of component A, the dynamic component balance of A is of interest It can be written as ... 

The dynamic component balances for the column are reflux drum ... 

The dynamic component balances for the column are the conventional ordinary differential equations, except for the reactor effluent remrn trays ... 

To illustrate the development of a physical model, a simplified treatment of the reactor, shown in Fig. 8-2 is used. It is assumed that the reac tor is operating isothermaUy and that the inlet and exit volumetric flows and densities are the same. There are two components, A and B, in the reactor, and a single first order reaction of A B takes place. The inlet concentration of A, which we shall call Cj, varies with time. A dynamic mass balance for the concentration of A (c ) can be written as follows ... 

Unlike stirred tanks, piston flow reactors are distributed systems with one-dimensional gradients in composition and physical properties. Steady-state performance is governed by ordinary differential equations, and dynamic performance is governed by partial differential equations, albeit simple, first-order PDEs. Figure 14.6 illustrates a component balance for a differential volume element. 

Adding the above two component balance equations gives the dynamic equation for the complete stage as... 

For gas absorption, this problem can often be circumvented by the assumption of a quasi-steady-state condition for the gas phase. In this, the dynamics of the gas phase are effectively neglected and the steady state, rather than the dynamic form of component balance is used to describe the variation in gas phase concentration. 

Dynamic Difference Equation for the Component Balance Dispersion Model... 

In this section we shall use the standard notation employed by biochemical engineers and industrial microbiologists in presenting the material. Thus if we denote by Xv the viable cell (cells/L) or biomass (mg/L) concentration, S the limiting substrate concentration (mmol/L) and P the product concentration (immol/L) in the bioreactor, the dynamic component mass balances yield the following ODEs for each mode of operation ... 

The dynamic process model involves a component balance, energy balance, kinetics and Arrhenius relationship. Hence... 

The dynamic component mass balances in terms of concentrations are as follows ... 

Note that the entries of f feed have already been incorporated into the equations (A) to (E) on p. 226 to 227. The eight component functions Fjy for k = 1,. ..,4 and j = 1,2 are the dynamic mole balance equations in normalized form. They are given in equations (4.103) to (4.106) as follows. 

The dynamic mathematical model describing the system consists of a total mass balance, two component balances, an energy balance on the reactor liquid, and a jacket energy balance ... 

The nonlinear dynamic model of this fed-batch reactor consists of a total mass balance, component balances for three components, an energy balance for the liquid in the reactor, and an energy balance for the cooling water in the jacket ... 

The ordinary differential equations describing a steady-state adiabatic PFR can be written with axial length z as the independent variable. Alternatively the weight of catalyst w can be used as the independent variable. There are three equations a component balance on the product C, an energy balance, and a pressure drop equation based on the Ergun equation. These equations describe how the molar flowrate of component C, temperature T, and the pressure P change down the length of the reactor. Under steady-state conditions, the temperature of the gas and the solid catalyst are equal. This may or may not be true dynamically ... 

The reactor is modeled by three partial differential equations component balances on A and B [Eqs. (6.1) and (6.2)] and an energy balance [Eq. (6.3) for an adiabatic reactor or Eq. (6.4) for a cooled reactor]. The overall heat transfer coefficient U in the cooled reactor in Eq. (6.4) is calculated by Eq. (6.5) and is a function of Reynolds number Re, Eq. (6.6). Equation (6.7) is used for pressure drop in the reactor using the friction factor /given in Eq. (6.8). The dynamics of the momentum balance in the reactor are neglected because they are much faster than the composition and temperature dynamics. A constant... 

Residue curve (RCM) and distillation curve (DCM) maps are today standard tools for designing distillation systems dealing with nonideal mixtures involving azeotropes. A residue curve characterizes the evolution of the liquid composition in a vessel during a batchwise distillation experiment. The whole compositional space may be spanned by residue curves considering different initial mixture compositions. For nonreactive mixtures the RCM is obtained by solving the component dynamic material balance expressed by the following differential equation ... 

Throughout this book, we have seen that when more than one species is involved in a process or when energy balances are required, several balance equations must be derived and solved simultaneously. For steady-state systems the equations are algebraic, but when the systems are transient, simultaneous differential equations must be solved. For the simplest systems, analytical solutions may be obtained by hand, but more commonly numerical solutions are required. Software packages that solve general systems of ordinary differential equations— such as Mathematica , Maple , Matlab , TK-Solver , Polymath , and EZ-Solve —are readily obtained for most computers. Other software packages have been designed specifically to simulate transient chemical processes. Some of these dynamic process simulators run in conjunction with the steady-state flowsheet simulators mentioned in Chapter 10 (e.g.. SPEEDUP, which runs with Aspen Plus, and a dynamic component of HYSYS ) and so have access to physical property databases and thermodynamic correlations. 

The dynamic mass balance for component i and section j in the liquid phase is... 

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