Component Mass Balances

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. 

The nonlinear dynamic model of this fed-batch reactor consists of a total mass balance, component balances for three components, an energy balance for the liquid in the reactor, and an energy balance for the cooling water in the jacket ... 

Total mass balance Component mass balance Equilibrium relationship ... 

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 ... 

The mass-balance restrictions are the C balances written for the C components present in the system. (Since we will only deal with non-reactive mixtures, each chemical compound present is a phase-rule component.) An alternative is to write (C — 1) component balances and one overall mass balance. 

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... 

Energy balances differ from mass balances in that the total mass is known but the total energy of a component is difficult to express. Consequently, the heat energy of a material is usually expressed relative to its standard state at a given temperature. For example, the heat content, or enthalpy, of steam is expressed relative to liquid water at 273 K (0°C) at a pressure equal to its own vapor pressure. 

For the mass balance of component A, diffusion velocity and the corresponding diffusion factor are defined with regard to the mean molar velocity V, defined by the equation... 

The competitive adsorption isotherms were determined experimentally for the separation of chiral epoxide enantiomers at 25 °C by the adsorption-desorption method [37]. A mass balance allows the knowledge of the concentration of each component retained in the particle, q, in equilibrium with the feed concentration, < In fact includes both the adsorbed phase concentration and the concentration in the fluid inside pores. This overall retained concentration is used to be consistent with the models presented for the SMB simulations based on homogeneous particles. The bed porosity was taken as = 0.4 since the total porosity was measured as Ej = 0.67 and the particle porosity of microcrystalline cellulose triacetate is p = 0.45 [38]. This procedure provides one point of the adsorption isotherm for each component (Cp q. The determination of the complete isotherm will require a set of experiments using different feed concentrations. To support the measured isotherms, a dynamic method of frontal chromatography is implemented based on the analysis of the response curves to a step change in feed concentration (adsorption) followed by the desorption of the column with pure eluent. It is well known that often the selectivity factor decreases with the increase of the concentration of chiral species and therefore the linear -i- Langmuir competitive isotherm was used ... 

Li and Hsiao [143] provide a useful approach to the environmental problem of removing (by stripping) volatile organics from solution in a contaminated water stream by using fresh air as the stripping medium. It should be noted that a number of industrial firms perform this stripping with steam. The mass balance on the VOC component around the column (trayed or packed) as shown in Figure 8-55 uses the symbols of Reference 143. 

General Material Balances. According to the law of conservation of mass, the total mass of an isolated system is invariant, even in the presence of chemical reactions. Thus, an overall material balance refers to a mass balance performed on the entire material (or contents) of the system. Instead, if a mass balance is made on any component (chemical compound or atomic species) involved in the process, it is termed a component (or species) material balance. The general mass balance equation has the following form, and it can be applied on any material in any process. 

Here, m(tp) is the mass of the system contents at final time t and m(t ) is the mass at initial time t . As before, both component and overall mass balances may be written in integral form. 

If a concentration gradient exists within a fluid flowing over a surface, mass transfer will take place, and the whole of the resistance to transfer can be regarded as lying within a diffusion boundary layer in the vicinity of the surface. If the concentration gradients, and hence the mass transfer rates, are small, variations in physical properties may be neglected and it can be shown that the velocity and thermal boundary layers are unaffected 55. For low concentrations of the diffusing component, the effects of bulk flow will be small and the mass balance equation for component A is ... 

Mass Balance Estimates. Based on National Pollutant Discharge Elimination System monitoring reports, the total daily discharge of trace elements into the main stem Willamette River is of the order of 100 pounds per day. Seventy-five percent of the total is zinc with the bulk of the remainder due to chromium and copper. Table 6 identifies industrial and natural sources of trace elements into the Willamette basin. The table indicates that an average of 97 percent of all trace element loading to the basin is natural in origin. The natural component is due to weathering of soil and rocks in the basin and this... 

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. 

Closure normally begins by satisfying the overall mass balance i.e., by equating the input and outlet mass flow rates for a steady-state system. For the present case, the outlet flow was measured. The inlet flow was unmeasured so it must be assumed to be equal to the outlet flow. We suppose that A and B are the only reactive components. Then, for a constant-density system, it must be that... 

Chapter 11 treats reactors where mass and component balances are needed for at least two phases and where there is interphase mass transfer. Most examples have two fluid phases, typically gas-liquid. Reaction is usually confined to one phase, although the general formulation allows reaction in any phase. A third phase, when present, is usually solid and usually catalytic. The solid phase may be either mobile or stationary. Some example systems are shown in Table 11.1. 

Operating Modes. The component and mass balances are quite general and apply to any operating mode e.g., batch, semibatch, or steady state. Table 11.2 gives examples for the various modes. 

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... 

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

Based on this configuration, the reformer and combustor are modeled with partial differential equations. Since the thickness of the plates is relatively small, only the flow direction is considered. Using the equation of continuity, the component mass balances are constructed and the energy balance considering with heat loss and momentum balance are established as follows. 

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. 

The UASB tractor was modeled by the dispensed plug flow model, considering decomposition reactions for VFA componaits, axial dispersion of liquid and hydrodynamics. The difierential mass balance equations based on the dispersed plug flow model are described for multiple VFA substrate components considaed... 

A dynamic model should be consistent with the steady-state model. Thus, Eqs (1) and (4) should be extended to dynamic form. For the better convergence and computational efficiency, some assumption can be introduced the total amounts of mass and enthalpy at each plate are maintained constant. Then, the internal flow can be determined by total mass balance and total energy balance and the number of differential equations is reduced. Therefore, the dynamic model can be established by replacing component material balance in Eq. (1) with the following equation. 

Allowing for Eq. (1.35), we can write the condition for mass balance of the reacting component j as... 

Uncharged reaction components are transported by diffusion and convection, even though their migration fluxes are zero. The total flux density Jj of species j is the algebraic (vector) sum of densities of all flux types, and the overall equation for mass balance must be written not as Eq. (4.1) but as... 

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 ... 

Fluid density and component brownian diffusivity D are also assumed constant. A steady-state component mass balance can be written for component concentration c ... 

Steady-state component and solvent mass balances can be written for single-pass operation by considering an incremental area element dA in the axial or flow direction for a feed channel, module, or membrane... 

An unsteady-state component mass balance, Eq. (20-68), can be written for batch operation by assuming a uniform average retentate concentration c, within the system. Assuming a constant solvent concentration and a 100 percent passage, the solvent balance becomes Eq. (20-69). 

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