Coupled mass and energy balances

We consider the general balance equations of mass and energy in the absence of chemical reactions, and electrical, magnetic and viscous effects. The partial differential equations of these general balance equations represent the mathematically and thermodynamically coupled phenomena, which may describe some complex behavior due to interactions among various forces and flows within a system. 

For a binary mixture under mechanical equilibrium and without chemical reaction, the general balance equations are 

Without the external mass and heat transfer resistances, the initial and boundary conditions with the y-coordinate oriented from the centerhne (y = 0) to the surface (y L) arc 

the cross coefficients are eliminated by using the heats of transport. These equations may be solved by using appropriate initial and boundary conditions in Eq. (7.105). 

If we use Eq. (7.101) instead ofEq. (7.105) and a thermal diffusion coefficient for one-dimensional heat and diffusion flows for a binary mixture, we have the following coupled balance equations  

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At steady state and considering molecular transport, coupled mass and energy balances in Eqs. (7.109) and (7.110) become... 

Up to this point we have been avoiding the complications of the coupled mass and energy balances by treating cs and Ts as known. The surface temperature is in general unknown and c, depends on it. To determine Ts we need to write an energy balance on the particle... 

Consideration of the coupled mass and energy balances for the CSTR have led to possible behaviors that may seem surprisingly complex for even the simplest kinetic mechanism, an irreversible first-order reaction. Just because these behaviors are possible does not mean that they are normally observed in reactor operation for something as simple as A goes to B. 

Finally, we often assume that the diffusivity, thermal conductivity and partial molar enthalpies are independent of temperature and composition to produce the following coupled mass and energy balances for the steady-state problem... 

The coupled mass and energy balance in the dense phase that consists of particles and interstitial gas phases is given in the following equation ... 

By substituting Eqns (7.26) and (7.27) into Eqns (7.52) and (7.53), we find the mathematically and thermodynamically coupled mass and energy balances... 

If the reaction rate is a function of pressure, then the momentum balance is considered along with the mass and energy balance equations. Both Equations 6-105 and 6-106 are coupled and highly nonlinear because of the effect of temperature on the reaction rate. Numerical methods of solution involving the use of finite difference are generally adopted. A review of the partial differential equation employing the finite difference method is illustrated in Appendix D. Figures 6-16 and 6-17, respectively, show typical profiles of an exothermic catalytic reaction. 

Rigorous and stiff batch distillation models considering mass and energy balances, column holdup and physical properties result in a coupled system of DAEs. Solution of such model equations without any reformulation was developed by Gear (1971) and Hindmarsh (1980) based on Backward Differentiation Formula (BDF). BDF methods are basically predictor-corrector methods. At each step a prediction is made of the differential variable at the next point in time. A correction procedure corrects the prediction. If the difference between the predicted and corrected states is less than the required local error, the step is accepted. Otherwise the step length is reduced and another attempt is made. The step length may also be increased if possible and the order of prediction is changed when this seems useful. 

Notice how the catalyst particle balances are coupled to the reactor mass and energy balances. Thus, the catalyst particle balances [Equation (10.2.7)] must be solved at each position along the axial reactor dimension when computing the bulk mass and energy balances [Equation (10.2.6)]. Obviously, these solutions are lengthy. 

Finally, we showed several ways to couple the mass and energy balances over the fluid flowing through a fixed-bed reactor to the balances within the pellet. For simple reaction mechanisms, we were still able to use the effectiveness factor approach to solve the fixed-bed reactor problem. For complex mechanisms, we solved numerically the full problem given in Equations 7.84-7.97, We solved the reaction-diffusion problem in the pellet coupled to the mass and energy balances for the fluid, and we used the Ergun equation to calculate the pressure in the fluid. 

This problem requires an analysis of coupled thermal energy and mass transport in a differential tubular reactor. In other words, the mass and energy balances should be expressed as coupled ordinary differential equations (ODEs). Since 3 mol of reactants produces 1 mol of product, the total number of moles is not conserved. Hence, this problem corresponds to a variable-volume gas-phase flow reactor and it is important to use reactor volume as the independent variable. Don t introduce average residence time because the gas-phase volumetric flow rate is not constant. If heat transfer across the wall of the reactor is neglected in the thermal energy balance for adiabatic operation, it... 

As discussed in Section 22-2.2, it is necessary to include the appropriate effectiveness factor E( A A, intrapellet) from Section 27-5 in the reaction rate expression which appears in the coupled plug-flow mass and energy balances for... 

One further way of making the numerical computations more viable and stable is to define an averaged and fixed temperature inside the different reactors. The fixed temperature inside these reactors reduces the nonlinearity of the system, which is mainly related to reaction rates and to the coupling of mass and energy balances. 

We have considered the case of multicomponent adsorption under isothermal conditions in the last section. Such an isothermal condition occurs when the particle is very small or when the environment is well stirred or when the heat of adsorption is low. If these criteria are not met, the particle temperature will vary. Heat is released during adsorption while it is absorbed by the particle when desorption occurs, leading to particle temperature rise in adsorption and temperature drop in desorption. The particle temperature variation depends on the rate of heat released and the dissipation rate of energy to the surrounding. In the displacement situation, that is one or more adsorbates are displacing the others, the particle temperature variation depends also on the relative heats of adsorption of displacing adsorbates and displaced adsorbates. Details of this can only be seen from the solution of coupled mass and heat balance equations. 

The basis of a population balance is that the number of crystals in a system is a balanceable quantity. Such balances are coupled to the usual mass and energy balances describing any system, and the crystals are... 

The dynamic model is based on mass and energy balances for the solid phases and mass and energy balances for the gas phases which are coupled with moisture evaporation, black liquor devolatilization, char combustion and gas-phase combustion. 

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]. 

The endeavors of this textbook are to define the qualitative aspects that affect the selection of an industrial chemical reactor and coupling the reactor models to the case-specific kinetic expressions for various chemical processes. Special attention is paid to the exact formulations and derivations of mass and energy balances as well as their numerical solutions. 

The equations presented so far for the multigrain model are mass- and energy-balance equations in a spherical catalyst particle used for conventional heterogeneously catalyzed reactions subjected to a moving boundary due to polymer formation. To predict polymer properties such as chain length and chemical composition, these monomer and temperature profiles must be coupled with an additional set of equations that describes polymerization and termination mechanisms... 

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