Continuous mass balance

Aging and redispersion phenomena of electrocatalysts are currently little understood, and no models exist for their quantitative description. In analogy to supported catalysts for gas phase reactions (266), we can write a continuity (mass balance) equation for the change of surface concentration nj of a particle consisting of j atoms, by capture and emission of i atom particles (7>2).-... 

The Geilikman et al. model relates yield front velocity to the volumetric rate of solids production by mass balance. A front between yielded and intact zones moves radially away from the wellbore as solids production continues. Mass balance analysis allows determination of the position of the moving front. Fluid production enhancement is dependent on the instantaneous solids flux, as well as the radius of the yielded zone around the wellbore (i.e., the solids production history). Viscous resistance to oil flow is reduced when the granular matrix is also flowing. The volumetric fluid production, Qf(t), is given as... 

The continuity (mass balance) means that the mass of fluid flowing into the tank is equal to the mass of the filled tank, so this... 

Design and operation of recirculation systems can be compHcated. Problems are avoided by using a sludge-blanket clarifier, in which feed enters below a blanket of accumulated and flocculated soflds which become fluidized in the zone-settling regime by the upflowing feed. Feed soflds are trapped in the blanket. The soflds content of the blanket continuously increases and part must be bled off in order to maintain the mass balance. 

Mass Balance Constraints. Erom the schematic diagram of a continuous crystallizer shown ia Eigure 11, the foUowiag mass balance on solute can be constmcted ... 

Material Balances Whenever mass-transfer applications involve equipment of specific dimensions, flux equations alone are inadequate to assess results. A material balance or continuity equation must also be used. When the geometiy is simple, macroscopic balances suffice. The following equation is an overall mass balance for such a unit having bulk-flow ports and ports or interfaces through which diffusive flux can occur ... 

Mass Balance, Continuity Equation The continuity equation, expressing consei vation of mass, is written in cartesian coordinates as... 

Mass balances for one vessel in a series of continuous fermenters... 

This is an old, familiar analysis that applies to any continuous culture with a single growth-limiting nutrient that meets the assumptions of perfect mixing and constant volume. The fundamental mass balance equations are used with the Monod equation, which has no time dependency and should be apphed with caution to transient states where there may be a time lag as [L responds to changing S. At steady state, the rates of change become zero, and [L = D. Substituting ... 

Figure 24-23 is a sketch of continuous culture with recycle. The symbols for flow rates and organism concentrations are F and X, respec tively Assuming perfect mixing and steady state so that the derivatives can be set to zero, mass balances lead to ... 

The population balance accounts for the number of particles at each size in a continuous distribution and may be thought of as an extension of the more familiar overall mass balance to that of accounting for individual particles. 

It has been shown that an increase in crystallizer residence time, or decrease in feed concentration, reduces the working level of supersaturation. This decrease in supersaturation results in a decrease in both nucleation and crystal growth. This in turn leads to a decrease in crystal surface area. By mass balance, this then causes an increase in the working solute concentration and hence an increase in the working level of supersaturation and so on. There is thus a complex feedback loop within a continuous crystallizer, illustrated in Figure 7.11. 

Differential and Integral Balances. Two types of material balances, differential and integral, are applied in analyzing chemical processes. The differential mass balance is valid at any instant in time, with each term representing a rate (i.e., mass per unit time). A general differential material balance may be written on any material involved in any transient process, including semibatch and unsteady-state continuous flow processes ... 

It may be noted that the energy and mass balance equations assume that the fluid is continuous. This is so in the case of a liquid, provided that the pressure does not fall to such a low value that boiling, or the evolution of dissolved gases, takes place. For water... 

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. 

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 ... 

At the end of 24 hours of continuous process the system was shut down. The knowledge of flowed buffer volumes and of the optical densities inside and downstream each ultrafiltration stage allowed to estimate product distribution (see appendix for mass-balance equations and the calculation procedure). The content of each cell was recovered and ffeeze-dried in order to be stored and used for subsequent kinetic experiments. A schematic flow-sheet of the whole procedure is illustrated in figure 1. 

In the steady state, the total flux is constant along the entire path. This condition (i.e., that of flux continuity) is a reflection of mass balance nowhere in a steady flux will the ions accumulate or vanish (i.e., their local concentrations are time invariant). The condition of continuity of the steady flux is disturbed in those places where ions are consumed (sinks) or produced (sources) by chemical reactions. It is necessary to preserve the balance that any excess of ions supplied correspond to the amount of ions reacting, and that any excess of ions eliminated correspond to the amount of ions formed in the reaction. 

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. 

In the previous discussion of the one- and two-compartment models we have loaded the system with a single-dose D at time zero, and subsequently we observed its transient response until a steady state was reached. It has been shown that an analysis of the response in the central plasma compartment allows to estimate the transfer constants of the system. Once the transfer constants have been established, it is possible to study the behaviour of the model with different types of input functions. The case when the input is delivered at a constant rate during a certain time interval is of special importance. It applies when a drug is delivered by continuous intravenous infusion. We assume that an amount Z) of a drug is delivered during the time of infusion x at a constant rate (Fig. 39.10). The first part of the mass balance differential equation for this one-compartment open system, for times t between 0 and x, is given by ... 

Liquid flows continuously into an initially empty tank, containing a full-depth heating coil. As the tank fills, an increasing proportion of the coil is covered by liquid. Once the tank is full, the liquid starts to overflow, but heating is maintained. A total mass balance is required to model the changing liquid volume and this is combined with a dynamic heat balance equation. 

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

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. 

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 mass balance relationships for the feed plate, the plates in the stripping section, of the column and for the reboiler must, however, be modified, owing to the continuous feed to the column and the continuous withdrawal of bottom product from the reboiler. The feed is defined by its mass flow rate, F, its composition xp and the thermal quality or q-factor, q. The column bottom product is defined by its mass flow rate, W, and composition, xw and is controlled to maintain constant liquid level in the reboiler. 

The balances for the continuous reactor are as follows The total mass balance, assuming constant density is... 

This section describes the continuous flux melting model used in Bourdon et al. (2003) and has many similarities with the model of Thomas et al. (2002). A significant difference is that the model described here keeps track of the composition of the slab as it dehydrates. This model is based on mass balance equations for both the mantle wedge and the slab. We assume secular equilibrium in the U-series decay chain initially ... 

The reactor system, where the kinetic experiments were carried out can be described as a semi-batch reactor. Only the synthesis gas (H2 and CO) was fed into the reactor continuously during the experiments, while 1-butene and the solvent were in the batch mode. All reactions took place in the liquid phase. The mass balance for an arbitrary component in the gas is given by... 

The dynamic mass balance for nonviable cells in a fed-batch or continuous culture is... 

As mentioned at the beginning of Section 3.2.3 the separation process can be modeled by mass balances. Two mass balances have to be made as the liquid appears in two phases dispersed in the gas phase (the droplets) and continuous in the accumulated state. Figure 3.2.12 describes the process and the conditions in addition to what was explained at the very beginning in Section 3.2.1. 

The mass balance of soil air may be described by the classic continuity equation for compressible fluids ... 

At constant density, the mass balance of a continuous flow mixed tank is simply... 

The mass balance of a continuous flow stirred-tank reactor (CSTR) with a first-order chemical reaction is very similar to the problem in Section 2.8.1 (p. 2-20). We just need to add the chemical reaction term. The balance written for the reactant A will appear as ... 

Example 10.7 A solution of sucrose in water is to be separated b. A mass balance on the solvent gives by crystallization in a continuous operation. The solubility of... 

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