Energy balances with variable properties

This is the same problem discussed in the foregoing section, except now we will allow the heat capacity, Cp, to be a function of temperature. 

To handle this problem properly, we must go back to the definition of the heat capacity in terms of the total energy. 

Since dTj./dt — 0 and p and Cp are not explicit functions of time, the accumulation term reduces to 

The term dV/dt is obtained from the material balance to be Fi — F2. Therefore 

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Tran and Mujtaba (1997), Mujtaba et al. (1997) and Mujtaba (1999) have used an extension of the Type IV- CMH model described in Chapter 4 and in Mujtaba and Macchietto (1998) in which few extra equations related to the solvent feed plate are added. The model accounts for detailed mass and energy balances with rigorous thermophysical properties calculations and results to a system of Differential and Algebraic Equations (DAEs). For the solution of the optimisation problem the method outlined in Chapter 5 is used which uses CVP techniques. Mujtaba (1999) used both reflux ratio and solvent feed rate (in semi-continuous feeding mode) as the optimisation variables. Piecewise constant values of these variables over the time intervals concerned are assumed. Both the values of these variables and the interval switching times (including the final time) are optimised in all the SDO problems mentioned in the previous section. 

A major limitation of the present work is that it deals only with well-defined (and mostly unidirectional) flow fields and simple homogeneous and catalytic reactor models. In addition, it ignores the coupling between the flow field and the species and energy balances which may be due to physical property variations or dependence of transport coefficients on state variables. Thus, a major and useful extension of the present work is to consider two- or three-dimensional flow fields (through simplified Navier-Stokes or Reynolds averaged equations), include physical property variations and derive lowdimensional models for various types of multi-phase reactors such as gas-liquid, fluid-solid (with diffusion and reaction in the solid phase) and gas-liquid-solid reactors. 

Particulate products, such as those from comminution, crystallization, precipitation etc., are distinguished by distributions of the state characteristics of the system, which are not only function of time and space but also some properties of states themselves known as internal variables. Internal variables could include size and shape if particles are formed or diameter for liquid droplets. The mathematical description encompassing internal co-ordinate inevitably results in an integro-partial differential equation called the population balance which has to be solved along with mass and energy balances to describe such processes. 

This chapter discusses the reaction engineering of chain-growth polymerization. In order to form polymers of specified properties, we observe that reactor temperature is a very important variable. To find this, the energy balance equation must be solved, along with mole balance relations of various species. In the study of copolymers, the quantities of practical interest are the relative distributions of the monomers on polymer chains and the overall rates of copolymerization. With these, it is possible to carry out the reactor design. 

The flow behavior of fluids is governed by the basic laws for conservation of mass, energy, and momentum coupled with appropriate expressions for the irreversible rate processes (e.g., friction loss) as a function of fluid properties, flow conditions, geometry, etc. These conservation laws can be expressed in terms of microscopic or point values of the variables, or in terms of macroscopic or integrated average values of these quantities. In principle, the macroscopic balances can be derived by integration of the microscopic balances. However, unless the local microscopic details of the flow field are required, it is often easier and more convenient to start with the macroscopic balance equations. 

The relationship between in the intensive and the extensive variable is the mass. Every mass specific property can be considered as an intensive variable. The conservation balances describe the accumulation rate based on mass, components, energy and momentum exchanges with the outside world in terms of intensive variables. In a PVT system these... 

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