The Material Balance

Define the system and draw the system boundaries for which the material balance is to be made. 

Explain the difference between an open and a closed system. 

Write the general material balance in words including all terms. Be able to apply the balance to simple problems. 

Cite examples of processes in which no accumulation takes place no generation or consumption takes place no mass flow in and out takes place. 

Apply the material balance equation for the simplified case of input = output to the total mass of material and to an individual species. 

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When only the total system composition, pressure, and temperature (or enthalpy) are specified, the problem becomes a flash calculation. This type of problem requires simultaneous solution of the material balance as well as the phase-equilibrium relations. 

Proportions correspond to the material balance for catalytic cracking in Figure 10.3 showing streams (l)(2)(3)(4) and (5). 

The material balance equation relating produced volume of oil (Np stb) to the pressure drop in the reservoir (AP) is given by ... 

The prediction of the size and permeability of the aquifer is usually difficult, since there is typically little data collected in the water column exploration and appraisal wells are usually targeted at locating oil. Hence the prediction of aquifer response often remains a major uncertainty during reservoir development planning. In order to see the reaction of an aquifer, it is necessary to produce from the oil column, and measure the response in terms of reservoir pressure and fluid contact movement use is made of the material balance technique to determine the contribution to pressure support made by the aquifer. Typically 5% of the STOMP must be produced to measure the response this may take a number of years. 

Gas reservoirs are produced by expansion of the gas contained in the reservoir. The high compressibility of the gas relative to the water in the reservoir (either connate water or underlying aquifer) make the gas expansion the dominant drive mechanism. Relative to oil reservoirs, the material balance calculation for gas reservoirs is rather simple. A major challenge in gas field development is to ensure a long sustainable plateau (typically 10 years) to attain a good sales price for the gas the customer usually requires a reliable supply of gas at an agreed rate over many years. The recovery factor for gas reservoirs depends upon how low the abandonment pressure can be reduced, which is why compression facilities are often provided on surface. Typical recovery factors are In the range 50 to 80 percent. 

The primary drive mechanism for gas field production is the expansion of the gas contained in the reservoir. Relative to oil reservoirs, the material balance calculations for gas reservoirs is rather simple the recovery factor is linked to the drop in reservoir pressure in an almost linear manner. The non-linearity is due to the changing z-factor (introduced in Section 5.2.4) as the pressure drops. A plot of (P/ z) against the recovery factor is linear if aquifer influx and pore compaction are negligible. The material balance may therefore be represented by the following plot (often called the P over z plot). 

To reduce the material balance conditions (11,1) to differential equations for the composition and pressure, flux relations must be used to relate the vectors to the gradients of the composition and pressure... 

Combining these with the material balance conditions (11.1) we therefore obtain... 

Solution of the material balance equations gives the pressure and... 

Knowing the solution of the material balance equations It is easy to calcu>... 

When used in the material balance condition this again gives a single differ-ential equation for but it is not the same as the equation obtained... 

The material balance conditions (11.1) may be rewritten in terms of the total flux N and the diffusion fluxes J, when they take the form... 

For ease of exposition, let us limit attention to. two independent reactions--the generalization to more reactions is straightforward. Then the material balance equations take the form... 

Finally, it remains to find x and X2 as functions of position in Che pellet, and for this purpose we must return to the material balances... 

In order to Introduce thermal effects into the theory, the material balance equations developed in this chapter must be supplemented by a further equation representing the condition of enthalpy balance. This matches the extra dependent variable, namely temperature. Care must also be taken to account properly for the temperature dependence of certain parameters In... 

As In the case of the material balance equations, the enthalpy balance can be written in dimensionless form, and this introduces new dimensionless parameters in addition to those listed in Table 11.1. We shall defer consideration of these until Chapter 12, where we shall construct the unsteady state enthalpy and material balances, and reduce them to dimensionless form. 

Equation (12.29) then represents the material balance on species A, while equation (12.30) represents the overall material balance. The reaction rate... 

Simplified Design Procedure for Linear Equilibrium and Operating Lines. A straight operating line occurs when the concentrations are low such that and remain essentially constant. (The material balance is obtained from equation 35.) In cases where the... 

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 general criterion of chemical reaction equiUbria is the same as that for phase equiUbria, namely that the total Gibbs energy of a closed system be a minimum at constant, uniform T and P (eq. 212). If the T and P of a siagle-phase, chemically reactive system are constant, then the quantities capable of change are the mole numbers, n. The iadependentiy variable quantities are just the r reaction coordinates, and thus the equiUbrium state is characterized by the rnecessary derivative conditions (and subject to the material balance constraints of equation 235) where j = 1,11,.. ., r ... 

When these half-reactions are summed, there is no net reaction. Thus the material balance of the cell is not altered by overcharge. At open circuit, equation 19 at the negative electrode is the sum of a two-step process, represented by equation 15 and... 

Because a material balance on water must be satisfied during the drydown as well as afterward, the path from the initial concentration to equihbrium can be represented graphically by a material balance line and an equihbrium curve. The coordinates of the starting point on the material balance line are the initial water contents of the fluid to be dried and the desiccant. The slope of the line is the ratio of fluid mass to desiccant mass. The line terminates at its intersection with the equihbrium curve (Eig. 4). 

Fig. 4. Drydown path to equilibrium ia a closed system. A represents the equiUbrium curve and B, the material balance line.

Performance. The performance of an ED stack may be estimated by considering the material balance around the stack ... 

Formulate the constraining material-balance equations, based on conservation of the total number of atoms of each element in a system comprised of w elements. Let subscript k identify a particular atom, and define Ai as the total number of atomic masses of the /cth element in the feed. Further, let a be the number of atoms of the /cth element present in each molecule of chemical species i. The material balance for element k is then... 

Function F is identical with G because the summation term is zero. However, the partial derivatives of F and G with respect to are different, because function F incorporates the constraints of the material balances. 

During startup or discharge the material balance becomes... 

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