Reactor Mass Balances

The second condition can be derived from the overall mass balance (reactor) for the reacting species. As we have assumed a gradientless recycle reactor (see Fig. 26), the fluid inside the reactor is supposed to be perfectly mixed. Thus, we have an ideal CSTR for which the mass balance of species i takes the form ... 

To illustrate the development of a physical model, a simplified treatment of the reactor, shown in Fig. 8-2 is used. It is assumed that the reac tor is operating isothermaUy and that the inlet and exit volumetric flows and densities are the same. There are two components, A and B, in the reactor, and a single first order reaction of A B takes place. The inlet concentration of A, which we shall call Cj, varies with time. A dynamic mass balance for the concentration of A (c ) can be written as follows ... 

As an alternative to deriving Eq. (8-2) from a dynamic mass balance, one could simply postulate a first-order differential equation to be valid (empirical modeling). Then it would be necessary to estimate values for T and K so that the postulated model described the reactor s dynamic response. The advantage of the physical model over the empirical model is that the physical model gives insight into how reactor parameters affec t the v ues of T, and which in turn affects the dynamic response of the reac tor. 

The equations that have been developed for design using these pseudo constants are based on steady-state mass balances of the biomass and the waste components around both the reactor of the system and the device used to separate and recycle microorganisms. Thus, the equations that can be derived will be dependent upon the characteristics of the reactor and the separator. It is impossible here to... 

From the mass balance equation for a batch reactor... 

These models are designed to define the complex entrance effects and convection phenomena that occur in a reactor and solve the complete equations of heat, mass balance, and momentum. They can be used to optimize the design parameters of a CVD reactor such as susceptor geometry, tilt angle, flow rates, and others. To obtain a complete and thorough analysis, these models should be complemented with experimental observations, such as the flow patterns mentioned above and in situ diagnostic, such as laser Raman spectroscopy. 

Secondly, these quotations emphasize the fact that the same river input that fuels longitudinal heterogeneity in reservoirs also forms a strong link between the reservoir and its watershed (e.g., [6]). This link has been conceptualized mostly in the form of load-response empirical models [7, 8], or mass-balance approaches [9]. Curiously, empirical modelers usually consider reservoirs as stirred reactors, ignoring the longitudinal spatial heterogeneity present in most situations and processes. 

Equations (1.1) to (1.3) are diflerent ways of expressing the overall mass balance for a flow system with variable inventory. In steady-state flow, the derivatives vanish, the total mass in the system is constant, and the overall mass balance simply states that input equals output. In batch systems, the flow terms are zero, the time derivative is zero, and the total mass in the system remains constant. We will return to the general form of Equation (1.3) when unsteady reactors are treated in Chapter 14. Until then, the overall mass balance merely serves as a consistency check on more detailed component balances that apply to individual substances. 

The difficulty disappears when the mixing and mass transfer steps are fast compared with the reaction steps. The contents of the reactor remain perfectly mixed even while new ingredients are being added. Compositions and reaction rates will be spatially uniform, and a flow term is simply added to the mass balance. Instead of Equation (2.30), we write... 

Most of this chapter assumes that the mass flow rate down the tube is constant i.e., the tube wall is impermeable. The reactor cross-sectional area Ac is allowed to vary as a function of axial position, Ac = Adz). Figure 3.1 shows the system and indicates the nomenclature. An overall mass balance gives... 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find = 0, which is the definition of a closed system. See... 

Overall and Phase Balances for Mass. The examples so far in Chapter 11 were designed to be simple yet show some essential features of gas-Uquid reactors. Only component balances for the phases, Equations (11.11) and (11.12), have been used. They are reasonably rigorous, but they do not provide guidance regarding how the various operating parameters can be determined. This is done in Section 11.1.2. Also, total mass balances must supplement the... 

In the above reactions, I signifies an initiator molecule, Rq the chain-initiating species, M a monomer molecule, R, a radical of chain length n, Pn a polymer molecule of chain length n, and f the initiator efficiency. The usual approximations for long chains and radical quasi-steady state (rate of initiation equals rate of termination) (2-6) are applied. Also applied is the assumption that the initiation step is much faster than initiator decomposition. ,1) With these assumptions, the monomer mass balance for a batch reactor is given by the following differential equation. 

A mass balance for an arbitrary liquid-phase component in the stirred tank reactor is thus written as follows dci... 

The utihty stream gets started at operating temperature and flow rate. In the following experiments, the utihty stream is heated so as to initiate the reaction. The main and secondary process tines are fed with water at room temperature and with the same flow rate as one of the experiments. Once steady state is reached, operating parameters are recorded. Process tines are then fed with the reactants, hydrogen peroxide and sodium thiosulfate. At steady state, operating parameters are recorded, and a sample of a known mass of reactor products is introduced in the Dewar vessel. Temperature in the Dewar vessel is recorded until equilibrium is reached, that is, until the reaction ends. This calorimetric method is aimed at calculating the conversion rate at the product outlet and thus the conversion rate in the reactor. The latter is also determined by thermal balances between process inlet and outlet of the reactor. Finally, the reactor is rinsed with water. This procedure is repeated for each experiment... 

Owing to the high computational load, it is tempting to assume rotational symmetry to reduce to 2D simulations. However, the symmetrical axis is a wall in the simulations that allows slip but no transport across it. The flow in bubble columns or bubbling fluidized beds is never steady, but instead oscillates everywhere, including across the center of the reactor. Consequently, a 2D rotational symmetry representation is never accurate for these reactors. A second problem with axis symmetry is that the bubbles formed in a bubbling fluidized bed are simulated as toroids and the mass balance for the bubble will be problematic when the bubble moves in a radial direction. It is also problematic to calculate the void fraction with these models. 

In the present study, the UASB reactor was modeled in terms of the dispersed plug flow and the Monod type of rate equations to constmct the differential mass balance equations fcs- the anaerobic biodegradation of single and multiple substrates components of the volatile fetty acids. 

Based on the kinetic mechanism and using the parameter values, one can analyze the continuous stirred tank reactor (CSTR) as well as the dispersed plug flow reactor (PFR) in which the reaction between ethylene and cyclopentadiene takes place. The steady state mass balance equations maybe expressed by using the usual notation as follows ... 

The vessels were indexed by the subscript "j" (j = 0 refers to the reactor and j from 1 to 4 to the UF cells) and oligomers were lumped in two categories "P" (Permeated) and "R" (Rejected). Let label "in" species entering a cell and "out" those leaving it. Instantaneous mass-balance in the stream leaving a cell and feeding the following one is ... 

At small space times, the C5 hydrocarbons could not account for all the PA converted, although no PA was found in the reactor outlet a similar observation was reported by La Vopa and Satterfield (17). But at higher space time the C5 hydrocarbons do account for more than 90% of the PA converted. Thus only the data at very high space time were used to calculate Kn (Fig. 5). The adsorption constants of DHQ were obtained in the same way. The mass balance of DHQ and CHE was always good. The resulting adsorption constants are given in Table 4. 

The catalyst prepared above was characterized by X-ray diffraction, X-ray photoelectron and Mdssbauer spectroscopic studies. The catalytic activities were evaluated under atmospheric pressure using a conventional gas-flow system with a fixed-bed quartz reactor. The details of the reaction procedure were described elsewhere [13]. The reaction products were analyzed by an on-line gas chromatography. The mass balances for oxygen and carbon beb een the reactants and the products were checked and both were better than 95%. 

Steady-state reactors with ideal flow pattern. In an ideal isothermal tubular pZi/g-yZovv reactor (PFR) there is no axial mixing and there are no radial concentration or velocity gradients (see also Section 5.4.3). The tubular PFR can be operated as an integral reactor or as a differential reactor. The terms integral and differential concern the observed conversions and yields. The differential mode of reactor operation can be achieved by using a shallow bed of catalyst particles. The mass-balance equation (see Table 5.4-3) can then be replaced with finite differences ... 

Cf, Cm, and c ui are reactant concentrations in the feed, and at the inlet and outlet of the catalyst bed, respectively and Vr is the reactor volume. The mass balance equation for mixing feed with recycle is as follows ... 

As will be shown later the equation above is identical to the mass balance equation for a continuous stirred-tank reactor. The recycle can be provided either by an external pump as shown in Fig. 5.4-18 or by an impeller installed within the reaction chamber. The latter design was proposed by Weychert and Trela (1968). A commercial and advantageously modified version of such a reactor has been developed by Berty (1974, 1979), see Fig. 5.4-19. In these reactors, the relative velocity between the catalyst particles and the fluid phases is incretised without increasing the overall feed and outlet flow rates. 

For a semibatch reaction between A already present in the reactor and B being fed into the reactor, each portion of B introduced is a source of vortices that grow by engulfement of the A-rich environment. The mass balance of component i in this growing zone is ... 

The mass balance for a continuous-flow, stirred-tank reactor with first-order reaction is... 

By simplifying the general component balance of Sec. 1.2.4, the mass balance for a batch reactor becomes... 

It becomes necessary to incorporate a total mass balance equation into the reactor model, whenever the total quantity of material in the reactor varies, as in the cases of semi-continuous or semi-batch operation or where volume changes occur, owing to density changes in flow systems. Otherwise the total mass balance equation can generally be neglected. 

For reactions involving heat effects, the total and component mass balance equations must be coupled with a reactor energy balance equation. Neglecting work done by the system on the surroundings, the energy balance is expressed by... 

The information flow diagram, for a non-isothermal, continuous-flow reactor, in Fig. 1.19, shown previously in Sec. 1.2.5, illustrates the close interlinking and highly interactive nature of the total mass balance, component mass balance, energy balance, rate equation, Arrhenius equation and flow effects F. This close interrelationship often brings about highly complex dynamic behaviour in chemical reactors. 

It is assumed that all the tank-type reactors, covered in this and the immediately following sections, are at all times perfectly mixed, such that concentration and temperature conditions are uniform throughout the tanks contents. Fig. 3.10 shows a batch reactor with a cooling jacket. Since there are no flows into the reactor or from the reactor, the total mass balance tells us that the total mass remains constant. 

The component mass balance, when coupled with the heat balance equation and temperature dependence of the kinetic rate coefficient, via the Arrhenius relation, provide the dynamic model for the system. Batch reactor simulation examples are provided by BATCHD, COMPREAC, BATCOM, CASTOR, HYDROL and RELUY. 

A total mass balance is necessary, owing to the feed input to the reactor, where... 

The component mass balance equation, combined with the reactor energy balance equation and the kinetic rate equation, provide the basic model for the ideal plug-flow tubular reactor. 

Figure 4.9. Mass balancing for a gas-phase, tubular reactor.

For a batch reactor, under constant volume conditions, the component mass balance equation can be represented by... 

The balances for the reactor liquid are as follows Total mass balance, assuming constant density... 

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