Mass balance nonisothermal system

It should be understood that this rate expression may in fact represent a set of diffusion and mass transfer equations with their associated boundary conditions, rather than a simple explicit expression. In addition one may write a differential heat balance for a column element, which has the same general form as Eq. (17), and a heat balance for heat transfer between particle and fluid. In a nonisothermal system the heat and mass balance equations are therefore coupled through the temperature dependence of the rate of adsorption and the adsorption equilibrium, as expressed in Eq. (18). 

For a system with n components (including nonad-sorbable inert species) there are n — 1 differential mass balance equations of type (17) and n — 1 rate equations [Eq. (18)]. The solution to this set of equations is a set of n — 1 concentration fronts or mass transfer zones separated by plateau regions and with each mass transfer zone propagating through the column at its characteristic velocity as determined by the equilibrium relationship. In addition, if the system is nonisothermal, there will be the differential column heat balance and the particle heat balance equations, which are coupled to the adsorption rate equation through the temperature dependence of the rate and equilibrium constants. The solution for a nonisothermal system will therefore contain an additional mass transfer zone traveling with the characteristic velocity of the temperature front, which is determined by the heat capacities of adsorbent and fluid and the heat of adsorption. A nonisothermal or adiabatic system with n components will therefore have n transitions or mass transfer zones and as such can be considered formally similar to an (n + 1)-component isothermal system. 

The specific models will be further subdivided into isothermal and non-isothermal models. This distinction is justified because mathematical modeling of a nonisothermal system involves a heat balance in addition to coverage equations (or reactor mass balances), and therefore introduces strong Arrhenius-type nonlinearities into the coverage equations. Nonisothermal processes are much more dependent on the reactor type and the form of the catalyst (supported, wire, foil, or single crystal). Thus these heat balance equations that describe them must take into account the type of catalyst and... 

In isothermal systems the general mass conservation-reaction rate expression of equation (1-15) is sufficient to describe the state of the system at any time. In nonisothermal systems this is not so, and expressions for both the conservation of mass and the conservation of energy are required. In reacting systems the energy balance most conveniently is written in terms of enthalpies of all the species entering and leaving a reference volume such as that of Figure 1.3a. Chemical reaction affects this balance by the heat that is evolved or consumed in the reaction. The balance that is required in addition to equation (1-15) is... 

We first discuss the various simple models, and start with linear models, favoured for the possibility of analytical solution which allows us to study the system behaviour in a more explicit way. Next we will discuss nonlinear models, and under special conditions such as the case of rectangular isotherm with pore diffusion analytical solution is also possible. Nonisothermal conditions are also dealt with by simply adding an energy balance equation to mass balance equations. We then discuss adsorption behaviour of multicomponent systems. 

As an example of the application of equilibrium theory to nonisothermal systems we consider here a plug flow system, with one adsorbable component, in which the concentration of adsorbable species and the temperature changes are both small enough to validate the constant velocity approximation. For such a system the differential mass and heat balance equations are... 

Except for the limiting case of the irreversible isotherm discussed above the prediction of the temperature and concentration profiles requires the simultaneous solution of the coupled differential heat and mass balance equations which describe the system. The earliest general numerical solutions for a nonisothermal adsorption column appear to have been given almost simultaneously by Carter and by Meyer and Weber. These studies all deal with binary adiabatic or near adiabatic systems with a small concentration of an adsorbable species in an inert carrier. Except for a difference in the form of the equilibrium relationship and the inclusion of intraparticle heat conduction and finite heat loss from the column wall in the work of Meyer and Weber, the mathematical models are similar. In both studies the predictive value of the mathematical model was confirmed by comparing experimental nonisothermal temperature and concentration breakthrough curves with the theoretical curves calculated from the model using the experimental equilibrium... 

For a nonisothermal catalytic packed bed, the energy balance Equation 5.158 is coupled to the mass balances and the system therefore consists ofN + I (number of components -F 1) of ordinary differential equations (ODEs), which are solved applying the same numerical methods that were used in the solution of the homogeneous plug flow model (Chapter 2). If the key components are utilized in the calculations, the system can be reduced to S -F 1 (number of reactions + 1) differential equations—provided that the number of reactions (S) is smaller than the number of components (N). 

In order to analyze the nonisothermal systems, an energy balance must be added to the mass balance within the solid particle. Although this is not essential in the development that follows, we shall assume for the sake of simplicity that both and the concentration profiles are unaffected by temperature. These assumptions should be reasonable when the temperature difference across the product layer is small. 

In systems that fall in regime II, it is possible that temperature gradient due to the heat effect of the reaction may play a role in determining the overall rate. The analysis of the general nonisothermal systems involves the heat balance equation in addition to the mass balance, and usually requires complicated and time-consuming numerical solution. However, under certain circumstances, simple approximate solutions are possible. We shall limit our discussion to these cases. 

Write the steady-state mass and heat balance equations for this system, assuming constant physical properties and constant heat of reaction. (Note Concentrate your modeling effort on the adiabatic nonisothermal reactor, and for the rest of the units, carry through a simple mass and heat balance in order to define the feed conditions for the reactor.)... 

To turn these heat-balance equations into nonisothermal heat balance design equations, we define the rate of reaction per unit volume (or per unit mass of catalyst, depending on the system) and the heat transfer per unit volume of the process unit (or per unit length), whichever is more convenient. 

A more quantitative analysis of the batch reactor is obtained by means of mathematical modeling. The mathematical model of the ideal batch reactor consists of mass and energy balances, which provide a set of ordinary differential equations that, in most cases, have to be solved numerically. Analytical integration is, however, still possible in isothermal systems and with reference to simple reaction schemes and rate expressions, so that some general assessments of the reactor behavior can be formulated when basic kinetic schemes are considered. This is the case of the discussion in the coming Sect. 2.3.1, whereas nonisothermal operations and energy balances are addressed in Sect. 2.3.2. 

Assuming a steady state, for first-order reaction-diffusion system A -> B under nonisothermal catalyst pellet conditions, the mass and energy balances are... 

Example 9.11 Modeling of a nonisothermal plug flow reactor Tubular reactors are not homogeneous, and may involve multiphase flows. These systems are called diffusion convection reaction systems. Consider the chemical reaction A -> bB described by a first-order kinetics with respect to the reactant A. For a nonisothermal plug flow reactor, modeling equations are derived from mass and energy balances... 

For nonisothermal cases, higher-order reactions (>1), and for systems with coupled reactions, the mass and energy balances for gas-liquid reactors are solved numerically. An example can be seen in Figure 7.22, in which p-cresol is chlorinated to mono- and dichloro-p-cresol following the reaction scheme below [ 11-13]. 

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