General integral material balance

For any transient process that begins at time t and is terminated at a later time tp, the general integral material balance equation has the form... 

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 ... 

The material balance relationship (i.e., Equation 1.5) holds for any reactant. If the liquid in a reactor is completely stirred and its concentration is uniform, we can apply this equation to the whole reactor. In general, it is applicable to a differential volume element and must be integrated over the whole reactor. 

To study different operating conditions in the pilot plant, a steady-state process simulator was used. Process simulators solve material- and energy-balance, but they do not generally integrate the equations of motion. The commercially-available program, Aspen Plus Tm, was used in this example. Other steady-state process simulators could be used as well. To describe the C02-solvent system, the predictive PSRK model [11,12], which was found suitable to treat this mixture, was applied. To obtain more reliable information, a model with parameters regressed from experimental data is required. 

If the compositions vary with position in the reactor, which is the case with a tubular reactor, a differential element of volume SV, must be used, and the equation integrated at a later stage. Otherwise, if the compositions are uniform, e.g. a well-mixed batch reactor or a continuous stirred-tank reactor, then the size of the volume element is immaterial it may conveniently be unit volume (1 m3) or it may be the whole reactor. Similarly, if the compositions are changing with time as in a batch reactor, the material balance must be made over a differential element of time. Otherwise for a tubular or a continuous stirred-tank reactor operating in a steady state, where compositions do not vary with time, the time interval used is immaterial and may conveniently be unit time (1 s). Bearing in mind these considerations the general material balance may be written ... 

This is the integrated form of equation 1.17 obtained previously it may be derived formally by applying the general material balance to unit volume under conditions of constant density, when the Rate of reaction term (3) is simply 9lA and the Accumulation term (4) is ... 

In these equations Vcr/v- rr, the residence time in tank r. Equation 1.46 may be compared with equation 1.37 for a tubular reactor. The difference between them is that, whereas 1.37 is an integral equation, 1.46 is a simple algebraic equation. If the reactor system consists of only one or two tanks the equations are fairly simple to solve. If a large number of tanks is employed, the equations whose general form is given by 1.46 constitute a set of finite-difference equations and must be solved accordingly. If there is more than one reactant involved, in general a set of material balance equations must be written for each reactant. 

Consider any quantity V m associated with a mixture of phases. For an arbitrary material macroscopic control volume Vm t) bounded by the surface area a generalized integral balance can be postulated stating that the... 

Material balances (mass or quantity balances) can be general or total material balances over the complete system and must be distinguished from material balances for individual mixture components (Fig. 1-7). Using the terms in Fig. 1-7, for an open system the general integral balance equation is... 

First, for now, these nine points are simply a list of recommendations, but we will analyze these recommendations later when we focus on developing an integral and general procedure for approaching, formulating, and solving material balance problems. Second, we will exemplify the relevance of these tips on warm-up examples and when solving problems in Sect. 7.9 (solved... 

The approach to be followed in the determination of rates or detailed kinetics of the reaction in a liquid phase between a component of dissolved gas and a component of the liquid is, in principle, the same as that outlined in Chapter 2 for gas-phase reactions on a solid catalyst. In general, the experiments are carried out in flow reactors of the integral type. The data may be analyzed by the integral or the differential method of kinetic analysis. However, for a single reaction, two continuity equations, in general, are required one for the absorbing component A in the gas phase and one for A in the liquid phase. In addition, a material balance is required, linking the consumption of B, the reactant of the liquid phase, to that of A. The continuity equations for A, which contain the rate equations derived in... 

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