Component balance differential

One way of dealing with this is to replace those component balance differential equations, having low time constants (i.e., high K values) and fast rates of response, by quasi-steady-state algebraic equations, obtained by setting... 

A differential balance written for a vanishingly small control volume, within which t A is approximately constant, is needed to analyze a piston flow reactor. See Figure 1.4. The differential volume element has volume AV, cross-sectional area A and length Az. The general component balance now gives... 

The most important characteristic of an ideal batch reactor is that the contents are perfectly mixed. Corresponding to this assumption, the component balances are ordinary differential equations. The reactor operates at constant mass between filling and discharge steps that are assumed to be fast compared with reaction half-lives and the batch reaction times. Chapter 1 made the further assumption of constant mass density, so that the working volume of the reactor was constant, but Chapter 2 relaxes this assumption. 

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. 

For a first-order process the time constant can be found from the defining differential equation as shown in Sec. 2.1.1.1. For the case of the aeration of a liquid, using a stirred tank, the following component balance equation applies... 

Writing unsteady-state component balances for each liquid phase results in the following pair of partial differential equations which are linked by the mass transfer rate and equilibrium relationships... 

Generally for modelling chromatograph systems, component mass balances are required for each component in each phase. The differential liquid phase component balances for a chromatographic column with non-porous packing take the partial differential equation form... 

State variables appear very naturally in the differential equations describing chemical engineering systems because our mathematical models are based on a number of first-order differential equations component balances, energy equations, etc. If there are N such equations, they can be linearized (if necessary) and written in matrix form... 

The component fluxes N entering into Eqs. (A1)-(A3) are determined based on the mass transport in the film region. Because the key assumptions of the film model result in the one-dimensional mass transport normal to the interface, the differential component balance equations including simultaneous mass transfer and reaction in the film are as follows ... 

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... 

It is often convenient to assume a closed chemical system, which imposes one mass balance constraint for each chemical component. In differential form, each equation is... 

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 fed-batch reactor model is commonly built using classical thermal and mass balance differential equations [11], Under isothermal conditions, the material balance for each measured component in the H KR of epichlorohydrin with continuous water addition is expressed by one of the following equations (Eqs. 12-17). These equations can be solved using an appropriate solver package (Lsoda, Ddassl [12]) with a connected optimization module for parameter estimation. 

The concentrations in kmol/m may be found also as soon as we know the masses, M,. The concentrations may now be used along with the temperature to calculate the reaction rate densities, ry, y = 1 to M. Now we know the inflow and outflow of each component, and the rate at which it is being produced or used up by the chemical reactions. Hence we may integrate the N mass-balance differential equations forward by one timestep. 

The constant-density and ideal-mixture cases for the three reactor operations are summarized in Table 4.1. We have ris + 2 unknowns and the component balances at the top of Table 4.1 provide Us equations. The remaining two equations for the constant-mass, constant-volume and specified-outflow operations are provided lower in the table. In these cases, these two extra equations are either differential equations or merely specify the values of constants. 

The general equation of state case for the three reactor operations is summarized in Table 4.2. The Us component balances are the same as in Table 4.1. But for the remaining two equations, we have a choice. We can either specify algebraic equations to obtain DAEs for the model, or differentiate the algebraic relations to obtain ODEs. DAEs may be preferred because the DDEs require the partial derivatives of the equation of state, fj df IdCj, as shown in Equation 4.57. Depending on the form of the equation of state, these derivatives may be complex expressions. In the following example, we set up and solve a nonconstant-density case, in which the products are.denser than, the reactants,... 

Notice that in the total material balance the argument of the derivative involves this mixture density. This as we have seen is the sum of the two concentrations of the two components, not the density of the solid alone, nor that of the liquid alone. On the right-hand side of the same equation, we note that the two input terms do involve the densities of the solid and the liquid in their pure states. This is because they are being delivered to the system as pure "feeds." The outflow term, however, includes the mixture density, the same density that appears in the argument of the differential. This is critical to understand. It says that everywhere in the control volume the density is the same at any time and that the material exiting the control volume also has the same density as the material in the tank. This is the consequence of assuming the system is well-mixed. The same analysis can be made for the two component balances. They show the well-mixed assumption because they include the corresponding mixture concentrations in the differential and the out-flow term. 

The component balances can be integrated in the same way. The initial condition is that the volume in the tank is Vo of pure solvent and the concentration of the solid is zero. To find the analytical solutions to these equations, we specify V[t] and then we use DSolve to simultaneously solve for the concentrations, calling the set of two solutions "a." Two functions are named and then extracted from the solution set and assigned to these names. Finally, the two new functions are placed back into the original differential equations and tested for validity. 

In the following the diffusion of component i in a sphere is discussed. The component balance for a sphere element leads to the following partial differential equation ... 

The quantity k is the so-called gradient-energy coefficient. The set of terms in the summation in Eq. (A.6) is the added term to Tick s first law of diffusion (see Section 2.2), which provides the effect of concentration gradient to the diffusional flux of material. For a purely diffusive process, Eq. (A.6) is combined with the differential component balance expression to give... 

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