Stage material balance

Consider the single-stage phase split shown in Fig. 3.1a. Overall material balances and component material balances can be written as... 

Analytical models using classical reservoir engineering techniques such as material balance, aquifer modelling and displacement calculations can be used in combination with field and laboratory data to estimate recovery factors for specific situations. These methods are most applicable when there is limited data, time and resources, and would be sufficient for most exploration and early appraisal decisions. However, when the development planning stage is reached, it is becoming common practice to build a reservoir simulation model, which allows more sensitivities to be considered in a shorter time frame. The typical sorts of questions addressed by reservoir simulations are listed in Section 8.5. 

The design of countercurrent contactors is considerably simplified when the solvents A and B are not significantly miscible. The mass flows of A and B then remain constant from one stage to the next, and the material balance at any stage can be written... 

The value functions appearing in equation 3 may be expanded in Taylor series about x and, because the concentration changes effected by a single stage are relatively small, only the first nonvanishing term is retained. When the value of is replaced by its material balance equivalent, ie, equation 4 ... 

FIG. 13-30 Material-balance envelope around the bottom end of the column. The partial rehoiler is equilibrium stage 1,... 

For each stage J, the following 2C -1- 3 component material-balance (M), phase-equilibrium (E), mole-fraction-summation (S), and energy-balance (H) equations apply, where C is the number of chemical species ... 

Next, Eq. (13-68) is summed over the C components and over stages 1 throiign j and combined with Eqs. (13-7o (13-71), and X Z-ij — 1.0 = 0 to give a total material balance over stages 1 through J ... 

Replace the holdup derivatives in Eqs. (13-149) to (13-151) by total-stage material-balance equations (e.g., dMj/dt = Vj + i + Ej- — Vj — Lj) and solve the resulting equations one at a time by the predictor step of an explicit integration method for a time increment that is determined by stability and truncation considerations. If the mole fraclions for a particular stage do not sum to 1, normalize them. 

In case A the solvents are immiscible, so the rate of feed solvent alone in the feed stream F is the same as the rate of feed solvent alone in the raffinate stream R. In like manner, the rate of extraction solvent alone is the same in the stream entering S as in the extract stream leaving E (Fig. 15-12). The ratio of extraction-solvent to feed-solvent flow rates is therefore S /F = E /R. A material balance can be written around the feed end of the extrac tor down to any stage n (see Fig. 15-12) and then rearranged to a McCabe-Thiele type of operating line with a slope of F /S [Eq. (15-11)]. 

Similarly, the same operating line can be derived from a material balance around the raffinate end of the extractor up to stage n [Eq. (15-12)]. 

Algebraic Comptttation This method starts with calculation of the quantities and compositions of all the terminal streams, using a convenient quantity of one of the streams as the basis of calculation. Material balance and stream compositions are then computed for a terminal ideal stage at either end of an extraction battery (i.e., at Point A or Point B in Fig. 18-81), using equilibrium and solution-retention data. Calculations are repeated for each successive ideal stage from one end of the system to the other until an ideal stage which corresponds to the desired conditions is obtained. Any solid-hquid extraction problem can be solved by this method. 

For more precise values, computer programs can be used to calculate soluble recoveiy as weh as solution compositions for conditions that are typical of a CCD circuit, with varying underflow concentrations, stage efficiencies, and solution densities in each of the stages. The calculation sequence is easily performed by utihzing material-balance equations around each thickener. 

Material and energy balances of common types of reactors are summarized in several tables of Sec. 7. For review purposes some material balances are restated here. For the /ith stage of a CSTR batteiy,... 

During this preparation stage, analysts will frequently find that there is insufficient quantity or quality of measurements to close the material balance. Analysts should make eveiy effort to measure aU stream flows and compositions for the actual test. They should not rely upon closing material balances by back-calculating missing streams. Tne material balance closure will provide a check on the vahdity of the measurements. This preparatoiy material balance will help to identify additional measurements and schedule the installation of the additional instruments. 

A similar material balance for the stripping stages which are labelled 1 to m + r yields... 

The operating company must underwrite the emissions associated with the plant through environmental impact reporting. Such accounting has become an important part of the design. Rather than pass over the need for a closed heat and material balance at the study stage of a project, it is better to get this job done as early as possible. Persistence is sometimes required. 

Equilibrium data must be obtained for material balance showing raffinate and extracted phases. A simple separation funnel for single-stage extraction using amyl acetate as organic solvent is shown in Figure 7.13. 

For any given stage, n, the component material balance equations for each phase are thus defined by... 

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