Batch mass balance

Mass balances for common, unsynchronized batch culture give ... 

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

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. 

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. 

Figure 4.42. Trend analysis over 46 batches of a bulk chemical produced according to the same manufacturing procedure Small and scaled-up batch size [kg], HPLC and Titration assays [%], resp. individual HPLC impurity levels [%], versus batch number. The lack of full correlation between assays indicates that the titration is insensitive to some impurities detected by HPLC. The mass balance, where available, suggests that all relevant impurities are quantified. Impurities B and C, for instance, are highly correlated (r = 0.884, p = 0.0002).

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. 

Thus, the initial value of the initiator concentrations, [Il]° and [I2]°, are calculated with Equation 15, for given values of the initial loading, feed rates, temperature, and time for the main semi-batch step, and [M]° is fixed according to experimental data from the base case semi-batch step. The nonlinear differential equation for [M] in terms of [II] and [I2] is given by Equation 16. Equation 10, with a redefinition of terms, is the differential equation mass balance for [II] and [12]. In the finishing step, only one of the initiators would be added for residual monomer reduction. Thus, Qm = 0,... 

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). 

Equation (20-70) is the unsteady-state component mass balance for fed-batch concentration at constant retentate volume. Integration yields the equations for concentration and yield in Table 20-19. 

N, and solute passage Si needed to produce desired retentate product with impurity concentrations Cj and retentate product yield Mj/Mjo. Permeate product characteristics for batch operation can be determined by mass balances using a permeate volume of Vp = Vo(l 1/X + N/X), a mass of solute i in the permeate as Mj perme e = Mjo(l — and the permeate concentration as the ratio of the... 

For a batch system, with no inflow and no outflow, the total mass of the system remains constant. The solution to this problem, thus involves a liquid-phase, component mass balance for the soluble material, combined with an expression for the rate of mass transfer of the solid into the liquid. 

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. 

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. 

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

For the semi-batch operation the total mass balance is... 

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 batch or perfusion culture yields. 

Constraints (5.1) states that the inlet stream into any operation j is made up of recycle/reuse stream, fresh water stream and a stream from reusable water storage. On the other hand, the outlet stream from operation j can be removed as effluent, reused in other processes, recycled to the same operation and/or sent to reusable water storage as shown in constraints (5.2). Constraints (5.3) is the mass balance around unit j. It states that the contaminant mass-load difference between outlet and inlet streams for the same unit j is the contaminant mass-load picked up in unit j. The inlet concentration into operation j is the ratio of the contaminant amount in the inlet stream and the quantity of the inlet stream as stated in constraints (5.4). The amount of contaminant in the inlet stream to operation j consists of the contaminant in the recycle/reuse stream and the contaminant in the reusable water storage stream. Constraints (5.5) states that the outlet concentration from any unit j is fixed at a maximum predefined concentration corresponding to the same unit. It should be noted that streams are expressed in quantities instead of flowrates, which is indicative of any batch operation. The total quantity of water used at any point in time must be within bounds of the equipment unit involved as stated in constraints (5.6). Following are the storage-specific constraints. 

The mass balance constraints given above would suffice if the process were continuous. However, due to the fact that the processes dealt with are batch processes, additional constraints are required to capture the discontinuous nature of the process. This implies that the time related constraints are necessary. 

The model is based on fixed batch size, as mentioned previously. The batch size is fixed to the appropriate level in constraint (8.2). A product mass balance over a unit is given in constraint (8.3). This constraint states that the mass product produced at a time point is the amount of raw material used at the previous time point less the residue left in the unit. 

If the assumption that the contaminant mass in the wastewater is relaxed, then the additional raw material in the form of the contaminant mass has to be accounted for. The wastewater in this case not only supplements the water in the raw material, but also any other raw materials used in product formulation. The raw material balance given in constraint (8.1) is reformulated to account for the additional raw material source. Constraint (8.1) is split into a water balance and a raw material balance for the other components required in product formulation. The water balance is given in constraint (8.52). The balance for the other components used in the product formulation is given in constraint (8.53). Due to the fixed ratio of water and other components in product formulation and the fixed batch size, the amount of water and the amount of other components are fixed. Therefore, in constraints (8.52) and (8.53) the amount of water and amount of other raw material is fixed. The water balance, in constraint (8.52), states that the amount of water used in product is comprised of freshwater, water from storage and directly recycle/reused water. Constraint (8.53), the mass balance for the other components, states that the mass of other components used for product is the mass from bulk storage, the mass in directly recycled/reused water and the mass in water from storage. 

The first minor change to the mass balance constraints from the scheduling formulation is found in constraint (8.2), which defines the size of a batch. In the synthesis formulation, the batch size is determined by the optimal size of the processing unit. Due to this being a variable, constraint (8.2) is reformulated to reflect this and is given in constraint (8.59). The nonlinearity present in constraint (8.59) is linearised exactly using Glover transformation (1975) as presented in Chapter 4. 

The most relevant contribution for global discrete time models is the State Task Network representation proposed by Kondili et al. [7] and Shah et al. [8] (see also [9]). The model involves 0-1 variables for allocating tasks to processing units at the beginning of the postulated time intervals. Most important equations comprise mass balances over the states, constraints on batch sizes and resource constraints. The STN model covers all the features that are included at the column on discrete time in Table 8.1. 

Batch and continuous processes may also be compared by examining their governing mass-balance relations. As an elaboration of equation 1.5-1, a general mass balance may be written with respect to a control volume as ... 

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