Accumulation term material balance

Unsteady material and energy balances are formulated with the conservation law, Eq. (7-68). The sink term of a material balance is and the accumulation term is the time derivative of the content of reactant in the vessel, or 3(V C )/3t, where both and depend on the time. An unsteady condition in the sense used in this section always has an accumulation term. This sense of unsteadiness excludes the batch reactor where conditions do change with time but are taken account of in the sink term. Startup and shutdown periods of batch reactors, however, are classified as unsteady their equations are developed in the Batch Reactors subsection. For a semibatch operation in which some of the reactants are preloaded and the others are fed in gradually, equations are developed in Example 11, following. 

For all mechanisms except axial dispersion, the transition can be centered just as well using cf because of Eq. (16-138). For axial dispersion, the transition ould be centered using /if provided the fluid-phase accumulation term in the material balance, Eq. (16-124),... 

A special case of the above equation applies to a continuous steady-state flow process when all of the rate terms are independent of time and the accumulation term is zero. Thus, the differential material balance for any component i in such a process is given by... 

Note that ai will gradually increase during the course of the reaction and will reach its saturation value, agjKu, when B is depleted. Dropping the accumulation term for ai i) represents a form of the pseudo-steady hypothesis. Since component B is not transferred between phases, its material balance has the usual form for a batch reactor ... 

The general material balance of Section 1.1 contains an accumulation term that enables its use for unsteady-state reactors. This term is used to solve steady-state design problems by the method of false transients. We turn now to solving real transients. The great majority of chemical reactors are designed for steady-state operation. However, even steady-state reactors must occasionally start up and shut down. Also, an understanding of process dynamics is necessary to design the control systems needed to handle upsets and to enable operation at steady states that would otherwise be unstable. 

All the previous material balance examples have been steady-state balances. The accumulation term was taken as zero, and the stream flow-rates and compositions did not vary with time. If these conditions are not met the calculations are more complex. Steady-state calculations are usually sufficient for the calculations of the process flow-sheet (Chapter 4). The unsteady-state behaviour of a process is important when considering the process start-up and shut-down, and the response to process upsets. 

There are a variety of limiting forms of equation 8.0.3 that are appropriate for use with different types of reactors and different modes of operation. For stirred tanks the reactor contents are uniform in temperature and composition throughout, and it is possible to write the energy balance over the entire reactor. In the case of a batch reactor, only the first two terms need be retained. For continuous flow systems operating at steady state, the accumulation term disappears. For adiabatic operation in the absence of shaft work effects the energy transfer term is omitted. For the case of semibatch operation it may be necessary to retain all four terms. For tubular flow reactors neither the composition nor the temperature need be independent of position, and the energy balance must be written on a differential element of reactor volume. The resultant differential equation must then be solved in conjunction with the differential equation describing the material balance on the differential element. 

Consider the segment of tubular reactor shown in Figure 8.3. Since the fluid composition varies with longitudinal position, we must write our material balance for a reactant species over a different element of reactor (dVR). Moreover, since plug flow reactors are operated at steady state except during start-up and shut-down procedures, the relations of major interest are those in which the accumulation term is missing from equation 8.0.1. Thus... 

Consider the schematic representation of a continuous flow stirred tank reactor shown in Figure 8.5. The starting point for the development of the fundamental design equation is again a generalized material balance on a reactant species. For the steady-state case the accumulation term in equation 8.0.1 is zero. Furthermore, since conditions are uniform throughout the reactor volume, the material balance may be... 

The input and output terms of equation 1.5-1 may each have more than one contribution. The input of a species may be by convective (bulk) flow, by diffusion of some kind across the entry point(s), and by formation by chemical reaction(s) within the control volume. The output of a species may include consumption by reaction(s) within the control volume. There are also corresponding terms in the energy balance (e.g., generation or consumption of enthalpy by reaction), and in addition there is heat transfer (2), which does not involve material flow. The accumulation term on the right side of equation 1.5-1 is the net result of the inputs and outputs for steady-state operation, it is zero, and for unsteady-state operation, it is nonzero. 

Third, for a CSTR, the accumulation term in the material-balance equation 14.3-2 or -3 becomes... 

A material balance analysis taking into account inputs and outputs by flow and reaction, and accumulation, as appropriate. This results in a proper number of continuity equations expressing, fa- example, molar flow rates of species in terms of process parameters (volumetric flow rate, rate constants, volume, initial concentrations, etc.). These are differential equations or algebraic equations. 

Similarly the unsteady material balance of a CSTR has an accumulation term added to it, for example,... 

Here the rate of accumulation term represents the rate of change in the total mass of the system, with respect to time, and at steady state, this is equal to zero. Thus, the steady-state material balance is seen to be a simplification of the more general dynamic balance. 

For reactions involving heat effects, the total and component material balance equations must be coupled with a reactor energy balance equation. Neglecting work done by the system on the surroundings, the energy balance is expressed by where each term has units of kj/s. For steady-state operation the accumulation... 

In the ideal CSTR, the fluid concentration is uniform and the fluid flows in and out of the reactor. Under the steady state condition, the accumulation term in the general material balance, eq. (3.70), is zero. Furthermore, the exit concentration is equal to the concentration in the reactor. For a volume element of fluid (F,), the mass balance for the limiting reactant becomes (Levenspiel, 1972)... 

The liquid composition in the CSTR is uniform and equal to that of the exit stream, and the accumulation term is zero at steady state. Thus, the material balance for a reactant A is given as... 

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 ... 

The basic equation for a tubular reactor is obtained by applying the general material balance, equation 1.12, with the plug flow assumptions. In steady state operation, which is usually the aim, the Rate of accumulation term (4) is zero. The material balance is taken with respect to a reactant A over a differential element of volume 8V, (Fig. 1.14). The fractional conversion of A in the mixture entering the element is aA and leaving it is (aA + SaA). If FA is the feed rate of A into the reactor (moles per unit time) the material balance over 8V, gives ... 

When a series of stirred-tanks is used as a chemical reactor, and the reactants are fed at a constant rate, eventually the system reaches a steady state such that the concentrations in the individual tanks, although different, do not vary with time. When the general material balance of equation 1.19 is applied, the accumulation term is therefore zero. Considering first of all the most general case in which the mass density of the mixture is not necessarily constant, the material balance on the reactant A is made on the basis of FA moles of A per unit time fed to the first tank. Then a material balance for the rth tank of volume V (Fig. 1.17) is, in the steady state ... 

For a CSTR operating at steady state, the accumulation term, dA) jdt, is equal to zero, and the material balance equation has the following form ... 

Since the process is at steady stale there can be no buildup of anything in the system, so the accumulation term equals zero in all material balances. In addition, since no chemical reactions occur, there can be no nonzero generation or consumption terms. For all balances, Equation 4.2-2 therefore takes the simple form input = output. 

For a differential balance on a continuous process (material flows in and out throughout the process) at steady-state (no process variables change with time), the accumulation term in the balance (the rate of buildup or depletion of the balanced species) equals zero. For an integral balance on a batch process (no material flows in or out during the process), the input and output terms equal zero and accumulation = initial input — final output. In both cases, the balance simplifies to... 

The procedures for deriving balances on transient systems are essentially those developed in Chapters 4 (material balances) and 7 (energy balances). The main difference is that transient balances have nonzero accumulation terms that are derivatives, so that instead of algebraic equations the balances are differential equations. 

In Eq. (2.1) the accumulation term refers to a change in mass or moles (plus or minus) within the system with respect to time, whereas the transfers through the system boundaries refer to inputs to and outputs of the system. If Eq. (2.1) is written in symbols so that the variables are functions of time, the equation so formulated would be a differential equation. As an example, the differential equation for the O2 material balance for the system illustrated in Fig. 2.1 might be written as... 

Most, but not all, of the problems discussed in this chapter are steady-state problems treated as integral balances for fixed time periods. If no accumulation occurs in a problem, and the generation and consumption terms can be omitted from consideration, the material balances reduce to the very simple relation... 

Recall from Eq. (2.1) that for a specific chemical compound the steady-state material balance for a reactor is (the accumulation term in zero)... 

Although A/j is a negative value, our equation takes account of this automatically. The accumulation is positive if the right-hand side is positive, and the accumulation is negative if the right-hand side is negative. Here the accumulation is really a depletion. You can see that the term p, the density of water, cancels out, and we could just as well have made our material balance on a volume of water. 

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