Accumulation term

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. 

Enthalpy balances also will have accumulation terms. 

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),... 

Accumulation The rise of pressure above the MAWP of the pro-tec ted system, usually expressed as a percentage of the gauge MAWP. Note The MAWP and accumulation terms are not included in the ANSI definitions since they relate to the protec ted system instead of the relief device. 

A steady-state process is one in wliich there is no change in conditions (temperature, pressure, etc.) or rates of flow with time at any given point in die system. The accumulation term in Eq. (4.5.1) is dien zero. If diere is no cheniieid reaetion, the generation tenn is also zero. All other processes are unsteady state. 

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... 

The ODEs governing the unsteady CSTR are obtained by adding accumulation terms to Equations (4.1). The simulation holds the volume constant, and... 

These equations are obtained by setting the accumulation terms to zero. 

The accumulation term is zero for steady-state processes. The accumulation term is needed for batch reactors and to solve steady-state problems by the method of false transients. 

The global design equations for packed beds—e.g.. Equations (10.1), (10.9), (10.39), and (10.40)—all have a similar limitation to that of the axial dispersion model treated in Chapter 9. They all assume steady-state operation. Adding an accumulation term, da/dt accounts for the change in the gas-phase inventory of component A but not for the surface inventory of A in the adsorbed form. The adsorbed inventory can be a large multiple of the gas-phase inventory. 

Solution Example 11.5 treats a system that could operate indefinitely since the liquid phase serves only as a catalyst. The present example is more realistic since the liquid phase is depleted and the reaction eventually ends. The assumption that the gas-side resistance is negligible is equivalent to assuming that a = ag throughout the course of the reaction. Equilibrium at the interface then fixes a = ag/Ku at all times. Dropping the flow and accumulation terms in the balance for the liquid phase, i.e.. Equation (11.11), gives 0 = kiAiV(ag/KH - ai) - Vikafi... 

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 ... 

Example 11.6 ignored the accumulation term for a/f) in Equation (11.11). How does the result for a change if this term is retained Consider only the asymptotic result as f oo. 

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. 

These accumulation terms are added to the appropriate steady-state balances to convert them to unsteady balances. The circumflexes indicate averages over the volume of the system, e.g.. 

The three accumulation terms represent the change in the total mass inventory, the molar inventory of component A, and the heat content of the system. The circumflexes can be dropped for a stirred tank, and this is the most useful application of the theory. 

A well-mixed stirred tank (which we will continue to call a CSTR despite possibly discontinuous flow) has p = Pout- The unsteady-state balance for total mass is obtained just by including the accumulation term ... 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified 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 mass balance is seen to be a simplification of the more general dynamic balance. 

The accumulation term for the gas phase can be therefore written in terms of number of moles as... 

The accumulation term in the energy balance equation can be rewritten as... 

Thus the accumulation term has the units of (energy)/(time), for example J/s. 

For steady-state operation of a continuous stirred-tank reactor or continuous stirred-tank reactor cascade, there is no change in conditions with respect to time, and therefore the accumulation term is zero. Under transient conditions, the full form of the equation, involving all four terms, must be employed. 

The main advantage of the above estimation procedure is that there is no need to assume steady-state operation. Since practice has shown that steady state operation is not easily established for prolonged periods of time, this approach enables the determination of average specific rates taking into account accumulation terms. 

Bunimovich et al. (1995) lumped the melt and solid phases of the catalyst but still distinguished between this lumped solid phase and the gas. Accumulation of mass and heat in the gas were neglected as were dispersion and conduction in the catalyst bed. This results in the model given in Table V with the radial heat transfer, conduction, and gas phase heat accumulation terms removed. The boundary conditions are different and become identical to those given in Table IX, expanded to provide for inversion of the melt concentrations when the flow direction switches. A dimensionless form of the model is given in Table XI. Parameters used in the model will be found in Bunimovich s paper. 

For a steady-state process the accumulation term will be zero. Except in nuclear processes, mass is neither generated nor consumed but if a chemical reaction takes place a particular chemical species may be formed or consumed in the process. If there is no chemical reaction the steady-state balance reduces to... 

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. 

All the examples of energy balances considered previously have been for steady-state processes where the rate of energy generation or consumption did not vary with time and the accumulation term in the general energy balance equation was taken as zero. 

While we laud the virtue of dynamic modeling, we will not duphcate the introduction of basic conservation equations. It is important to recognize that all of the processes that we want to control, e.g. bioieactor, distillation column, flow rate in a pipe, a drag delivery system, etc., are what we have learned in other engineering classes. The so-called model equations are conservation equations in heat, mass, and momentum. We need force balance in mechanical devices, and in electrical engineering, we consider circuits analysis. The difference between what we now use in control and what we are more accustomed to is that control problems are transient in nature. Accordingly, we include the time derivative (also called accumulation) term in our balance (model) equations. 

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. 

The accumulation term is just the time derivative of the number of moles of reactant A contained within the reactor (dNJdt). This term also may be written in terms of either the extent of reaction ( ) or the fraction conversion of the limiting reagent (fA). (A is presumed to be the... 

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... 

An initially clean activated carbon bed at 320 K is fed a vapor of benzene in nitrogen at a total pressure of 1 MPa. The concentration of benzene in the feed is 6 mol/m3. After the bed is uniformly saturated with feed, it is regenerated using benzene-free nitrogen at 400 K and 1 MPa. Solve for both steps. For simplicity, neglect fluid-phase accumulation terms and assume constant mean heat capacities for stationary and fluid phases and a constant velocity. The system is described by... 

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