Simultaneous mass and energy balances

Figure 1.19. Information flow diagram for modelling a non-isothermal, chemical reactor, with simultaneous mass and energy balances.

These solution methods are restricted to very dilute solutions. In more concentrated solutions, flow rates are not constant, solutes may not have independent equilibria, and tenperature effects become inportant. When this is true, more complicated computer solution methods involving simultaneous mass and energy balances plus equilibrium are required. These methods are discussed in the next section and the conputer simulation is explored in the chapter appendix. 

The mass and energy balances usually must be solved simultaneously by using numerical methods. Equation 6-47 becomes... 

This chapter has introduced the concepts of mass and energy balances. These are essential steps in the analysis of any process. Simple examples have been used to illustrate the different steps, including not only formulating mass and energy balances but also simultaneously solving mass and energy balances together (the analysis of the condenser in the distillation unit). 

Solution of Equation (10.2.1) provides the pressure, temperature, and concentration profiles along the axial dimension of the reactor. The solution of Equation (10.2.1) requires the use of numerical techniques. If the linear velocity is not a function of z [as illustrated in Equation (10.2.1)], then the momentum balance can be solved independently of the mass and energy balances. If such is not the case (e.g., large mole change with reaction), then all three balances must be solved simultaneously. 

Example 10.2.1 illustrates the simultaneous solution of the mass and energy balances for an adiabatic, fixed-bed reactor with no fluid density changes and no transport limitations of the rate, that is, rj = 1. Next, situations where these simplifications do not arise are described. 

Homemade models are often mass and energy balance spreadsheets, simplified kinetic models, or the simultaneous solution of the convection diffusion and heat equations together with nonlinear isotherms. All levels of models have their place. 

The structure of the chapter should be clear from Fig. 7.1. The design must be based on the proper mass and energy balances for the reactor. When there is only one reaction, conversion and yield are equivalent concepts, but with simultaneous reactions, the primary concern of the design... 

In Example 5-3 the temperature and conversion leaving the reactor were obtained by simultaneous solution of the mass and energy balances. The results for each temperature in Table 5-7 represented such a solution and corresponded to a diiferent reactor, i.e., a different reactor volume. However, the numerical trial-and-error solution required for this multiple-reaction system hid important features of reactor behavior. Let us therefore reconsider the performance of a stirred-tank reactor for a simple single-reaction system. 

The structure and interrelationship of the batch conservation equations (population, mass, and energy balances) and the nucleation and growth kinetic equations are illustrated in an information flow diagram shown in Figure 10.8. To determine the CSD in a batch crystallizer, all of the above equations must be solved simultaneously. The batch conservation equations are difficult to solve even numerically. The population balance, Eq. (10.3), is a nonlinear first-order partial differential equation, and the nucleation and growth kinetic expressions are included in Eq. (10.3) as well as in the boundary conditions. One solution method involves the introduction of moments of the CSD as defined by... 

As a consequence of the complete mixing, a continuous flow stirred tank reactor also operates isothermally. Therefore, in the steady state it is not necessary to consider the mass and energy balances simultaneously. Optimum conditions may be computed on the basis of the material balance alone, and then afterwards the energy balance is used, in principle (see Sec. 10.4), to determine the external conditions required to maintain the desired temperature. 

If the temperature of the feed to the drum, Tp, is the specified variable, the mass and energy balances and the equilibrium equations must be solved simultaneously. You can see from the energy balance, Eq. (2-7 why this is true. The feed enthalpy, hp, can be calculated, but the vapor and liquid enthalpies, and depend upon x, which are unknown. Thus a sequential solution is not possible. 

Although the mass and energy balances, equilibrium relations, and stoichiometric relations could all be solved simultaneously, it is again easier to use a trial-and-error procedure. This problem is now a double trial and error. 

A6. What type of specifications will lead to simultaneous solution of the mass and energy balances A7. Specifications for a distillation column cannot include all three flow rates F, D, and B. Why not ... 

For non-isothermal conditions the energy balance equation must be solved simultaneously, but in either case the principles behind sizing calculations are simply the application of mass and energy balances. The straightforward example, given in exercise 7.1 illustrates how equation (7.4) can often be simplified. 

In Equation 5.220, P and Pq denote the total pressure at the reactor outlet and inlet, respectively. For an exact calculation, the pressure drop expression, Equation 5.216, needs to be solved simultaneously with the mass and energy balances of one- and two-dimensional models. 

Derive steady-state and nonsteady-state mass and energy balances for a catalyst monolith channel in which several chemical reactions take place simultaneously. External and internal mass transfer limitations are assumed to prevail. The flow in the chaimel is laminar, but radial diffusion might play a role. Axial heat conduction in the solid material must be accounted for. For the sake of simplicity, use cylindrical geometry. Which numerical methods do you recommend for the solution of the model ... 

As in all packed-column operations, the fxmdamental model equations consist of differential balances taken over each phase the principal novelty here is tiie simultaneous use of mass and energy balances. 

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