Material Balance Methods

Material balance method (pressure decline method)... 

The tray temperature gradient method requires only one steady-state simulation at design conditions, while the shift in material balance method requires two simulations with different product stream compositions. The maximum tray temperature gradient of 7.6% of 17.7°C or 1.35 C/tray occurs at stages 15 and 16 as shown in Figure 4.1. These observations show... 

If the sphere in Fig. 6.2-3a is evaporating, the radius r of the sphere decreases slowly with time. The equation for the time for the sphere to evaporate completely can be derived by assuming pseudo-steady state and by equating the diffusion flux equation (6.2-32), where r is now a variable, to the moles of solid A evaporated per dt time and per unit area as calculated from a material balance. (See Problem 6.2-9 for this case.) The material-balance method is similar to Example 6.2-3. The final equation is... 

Solving, L = 72.5 kg and V = 27.5 kg, which is a reasonably close check on the material-balance method. 

This material balance method would appear to have wide application for studying process rates in natural streams regardless of their location since less than maximum permissible concentrations of were used while radionuclide levels in periphyton and water were sufficient to monitor for at least six weeks following the release.. . . Since P has a short physical half-life (14.3 days), there is no danger of high residual concentrations in a stream ecosystem following the release because of radioactive decay. Thus, repeated releases may be made to the same stream to determine changes in biomass, primary production rate, and primary consumption rate on a seasonal basis. 

Elwood, J. W., and D. J. Nelson. 1972. Periphyton production and grazing rates in a stream measured with a P material balance method. Oikos 23(3) 295-303. 

The amount of nonfreezable water (defined relative to a very low temperature, e.g., - 100°C) may be determined by plotting the total melting enthalpy of water per unit weight of sample versus the sample composition and extrapolating to zero enthalpy [8], i.e., the nonfreezable water content is equated with the maximum amount of water for which no enthalpic peak has been detected [147]. We used a material balance method for this determination. During the investigation... 

As the drive water is injected, it displaces the polymer slug. Next, the location of the rear of the polymer slug as it is displaced through the porous rock is computed with the material-balance method. Table 5.26 contains values of S, and from the polymer... 

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 oscillating jet method is not suitable for the study of liquid-air interfaces whose ages are in the range of tenths of a second, and an alternative method is based on the dependence of the shape of a falling column of liquid on its surface tension. Since the hydrostatic head, and hence the linear velocity, increases with h, the distance away from the nozzle, the cross-sectional area of the column must correspondingly decrease as a material balance requirement. The effect of surface tension is to oppose this shrinkage in cross section. The method is discussed in Refs. 110 and 111. A related method makes use of a falling sheet of liquid [112]. 

Current Efficiency. Current efficiency for caustic production in diaphragm and membrane cells can be estimated from collection of a known amount of caustic over a period of time and from a knowledge of the number of coulombs of electricity passed during that time period. An alternative method involves analysis of the gases evolved during electrolysis and determining the anolyte composition. Material balance considerations (7) show the expression for the caustic efficiency for membrane cells to be... 

A total material balance assay is a Fischer assay in which the retort gases are collected. A complete material balance closure and yields in excess of those expected from Fischer assay results are achieved. More complete descriptions of both the Fischer assay and the Tosco material balance assay methods have been reported (9). 

Multiple-Effect Evaporators A number of approximate methods have been published for estimating performance and heating-surface requirements of a multiple-effect evaporator [Coates and Pressburg, Chem. Eng., 67(6), 157 (1960) Coates, Chem. Eng. Prog., 45, 25 (1949) and Ray and Carnahan, Trans. Am. Inst. Chem. Eng., 41, 253 (1945)]. However, because of the wide variety of methods of feeding and the added complication of feed heaters and condensate flash systems, the only certain way of determining performance is by detailed heat and material balances. Algebraic soluflons may be used, but if more than a few effects are involved, trial-and-error methods are usually quicker. These frequently involve trial-and-error within trial-and-error solutions. Usually, if condensate flash systems or feed heaters are involved, it is best to start at the first effect. The basic steps in the calculation are then as follows ... 

Operating Lines The McCabe-Thiele method is based upon representation of the material-balance equations as operating lines on the y-x diagram. The lines are made straight (and the need for the energy balance obviated) by the assumption of constant molar overflow. The liqmd-phase flow rate is assumed to be constant from tray to tray in each sec tiou of the column between addition (feed) and withdrawal (produc t) points. If the liquid rate is constant, the vapor rate must also be constant. 

The constant-molar-overflow assumption represents several prior assumptions. The most important one is equal molar heats of vaporization for the two components. The other assumptions are adiabatic operation (no heat leaks) and no heat of mixing or sensible heat effects. These assumptions are most closely approximated for close-boiling isomers. The result of these assumptions on the calculation method can be illustrated with Fig. 13-28, vdiich shows two material-balance envelopes cutting through the top section (above the top feed stream or sidestream) of the column. If L + i is assumed to be identical to L 1 in rate, then 9 and the component material balance... 

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. 

Evaluation of the integral in Eq. (14-86) requires a knowledge of the liquid-phase bulk concentration of B as a function of y. This relationship is obtained by means of a material balance around the tower, as shown in Eq. (14-73). Numerical integration by a quadrature method such as Simpson s nrle normally will be required for this calculation. 

Example 4 Shortcut Calculation Case B Let iis solve the problem in Example 2 hy assuming case B. The solute (acetic acid) concentration is low enough in the extract so that we may assume that the mutual solubilities of the solvents remain nearly constant. The material balance can be calculated by an iterative method. 

For the methods, we consider Eq. (16-52), the material balance for a fixed bed, written in the form... 

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. 

By convention, Cj is measured in the bulk feed, not at the membrane. Clearly, the concentration at the membrane is the important one, but the convention is well estabhshed and it simphfies calculations on yield and material balance. Concentration at the membrane, may be calculated by the method shown in Eq. (22-91). 

The use of process flow diagrams and material balances are worthwhile methods to quantify losses or emissions and provide essential data to estimate the size and cost of additional equipment, other data to evaluate economic performance, and a basefine for tracking the progress of minimization efforts (Ref. 3). Material balances should be apphed to individual waste streams or processes and then utilized to construct an overall balance for the facility. Details on these calculations are available in the literature (Ref. 8). In addition, an introduction to this subject is provided in the next section. 

The conservation of mass law finds a major application during the performance of pollution-prevention assessments. As described earlier, a pollution-prevention assessment is a systematic, planned procedure with the objective of identifying methods to reduce or ehminate waste. The assessment process should characterize the selected waste streams and processes (Ref. 11)—a necessaiy ingredient if a material balance is to be performed. Some of the data required for the material balance calciilation may be collected during the first review of site-specific data however, in some instances, the information may not be collected until an actual site walk-through is performed. 

References A variety of mathematical methods are proposed to cope with hnear (e.g., material balances based on flows) and nonhnear (e.g., energy balances and equilibrium relations) constraints. Methods have been developed to cope with unknown measurement uncertainties and missing measurements. The reference list provides ample insight into these methods. See, in particular, the works by Mah, Crowe, and Madron. However, the methods all require more information than is tvpicaUy known in a plant setting. Therefore, even when automated methods are available, plant-performance analysts are well advised to perform initial adjustments by hand. 

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