Energy balances over reactors

There are a variety of limiting forms of equation 8.0.3 that are appropriate for use with different types of reactors and different modes of operation. For stirred tanks the reactor contents are uniform in temperature and composition throughout, and it is possible to write the energy balance over the entire reactor. In the case of a batch reactor, only the first two terms need be retained. For continuous flow systems operating at steady state, the accumulation term disappears. For adiabatic operation in the absence of shaft work effects the energy transfer term is omitted. For the case of semibatch operation it may be necessary to retain all four terms. For tubular flow reactors neither the composition nor the temperature need be independent of position, and the energy balance must be written on a differential element of reactor volume. The resultant differential equation must then be solved in conjunction with the differential equation describing the material balance on the differential element. 

An energy balance over the differential length of reactor for steady-state operating conditions... 

The steady-state energy balance over an element of reactor volume, AV, may be written... 

The performance of semibatch reactors under isothermal conditions was studied in Sec. 4-8. When the temperature is not constant, an energy balance must be solved simultaneously with the mass-balance equation. In general, the energy balance for a semibatch reactor (Fig. 3-1 c) will include all four items of Eq. (3-2). Following the nomenclature of Sec. 4-8, let Fq and iq be the total mass-flow rates of feed and product streams and Hq and the corresponding enthalpies above a reference state. Then, following Eq. (3-2) term by term, the energy balance over an element of time At is... 

The only difference here is that the temperature at the reactor inlet depends on the outlet temperature and the recycle. Hence, we have to solve Eq. 9.4.22 subject to the initial condition that at t = 0, 9(0) = 9i (the dimensionless temperature at point 1). To determine 9i, we write an energy balance over the mixing point. 

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. 

In an energy balance over a volume element of a chemical reactor, kinetic, potential, and work terms may usually be neglected relative to the heat of reaction and other heat transfer terms so that the balance reduces to ... 

Again we write an energy balance over a differential reactor volume to arrive at. 

This paper presents the physical mechanism and the structure of a comprehensive dynamic Emulsion Polymerization Model (EPM). EPM combines the theory of coagulative nucleation of homogeneously nucleated precursors with detailed species material and energy balances to calculate the time evolution of the concentration, size, and colloidal characteristics of latex particles, the monomer conversions, the copolymer composition, and molecular weight in an emulsion system. The capabilities of EPM are demonstrated by comparisons of its predictions with experimental data from the literature covering styrene and styrene/methyl methacrylate polymerizations. EPM can successfully simulate continuous and batch reactors over a wide range of initiator and added surfactant concentrations. 

The ideal continuous stirred tank reactor is the easiest type of continuous flow reactor to analyze in design calculations because the temperature and composition of the reactor contents are homogeneous throughout the reactor volume. Consequently, material and energy balances can be written over the entire reactor and the outlet composition and temperature can be taken as representative of the reactor contents. In general the temperatures of the feed and effluent streams will not be equal, and it will be necessary to use both material and energy balances and the temperature-dependent form of the reaction rate expression to determine the conditions at which the reactor operates. 

In principle, if the temperatures, velocities, flow patterns, and local rates of mixing of every element of fluid in a reactor were known, and if the differential material and energy balances could be integrated over the reactor volume, one could obtain an exact solution for the composition of the effluent stream and thus the degree of conversion that takes place in the reactor. However, most of this information is lacking for the reactors used in laboratory or commercial practice. Consequently, it has been necessary to develop approximate methods for treating... 

In addition to flow, thermal, and bed arrangements, an important design consideration is the amount of catalyst required (W), and its possible distribution over two or more stages. This is a measure of the size of the reactor. The depth (L) and diameter (D) of each stage must also be determined. In addition to the usual tools provided by kinetics, and material and energy balances, we must take into account matters peculiar to individual particles, collections of particles, and fluid-particle interactions, as well as any matters peculiar to the nature of the reaction, such as reversibility. Process design aspects of catalytic reactors are described by Lywood (1996). 

Numerous reactions are performed by feeding the reactants continuously to cylindrical tubes, either empty or packed with catalyst, with a length which is 10 to 1000 times larger than the diameter. The mixture of unconverted reactants and reaction products is continuously withdrawn at the reactor exit. Hence, constant concentration profiles of reactants and products, as well as a temperature profile are established between the inlet and the outlet of the tubular reactor, see Fig. 7.1. This requires, in contrast to the batch reactor, the application of the law of conservation of mass over an infinitesimal volume element, dV, of the reactor. In contrast to a batch reactor the existence of a temperature profile does not allow us to consider the mass balances for the reacting components and the energy balance separately. Such a separation can only be performed for isothermal tubular reactors. 

For a steady-state perfectly mixed flow reactor the energy balance can be made over the complete reactor ... 

Consider a PFR operating at nonisothermal conditions (refer to Figure 9.4.1). To describe the reactor performance, the material balance. Equation (9.1.1), must be solved simultaneously with the energy balance. Equation (9.2.7). Assuming that the PFR is a tubular reactor of constant cross-sectional area and that T and C, do not vary over the radial direction of the tube, the heat transfer rate Q can be written for a differential section of reactor volume as (see Figure 9.4.1) ... 

The formal similarity allows us to carry over the equations for mass and energy balances in the tubular reactor, Eqs. (3.4.11)-(3.4.14). The momentum equation has no meaning. Care must be taken however to distinguish between a batch reactor working at constant volume and one that works at constant pressure. The latter has the Eqs. (3.4.12) or (3.4.14) which were derived from an enthalpy balance. In the former case the heat added would be equated to the internal energy change. Thus in this case c should replace Cp and the internal energy of reaction replace the heat of reaction. These... 

To evaluate x it is necessary to know the temperature of the product stream leaving the reactor. This requires an energy balance, which again may be written over the total volume element V. The fourth term in Eq. 

The differential energy balance equation for steady-flow reactors widi no mechanical work is obtained by differentiating Eq. 5.2.48 over a reactor volume element (IVr ... 

One-dimensional models basically assume that species concentrations and fluid temperature vary only in the axial direction. The only transport mechanism operating in this direction is the overall convective flow. The conservation equations may be obtained from mass and energy balance on a reference component A, over an elementary cross section of the tubular reactor. For a single reaction, the steady state conservation equations can be written for the pseudo-homogeneous model as follows ... 

---