Accumulation term mass balance

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 ... 

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. 

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... 

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. 

A steady-state condition is assumed that is, the accumulation term in the mass balance is zero. 

Consider the thermal wave given in Fig. 4.4. If a differential control volume is taken within this one-dimensional wave and the variations as given in the figure are in the x direction, then the thermal and mass balances are as shown in Fig. 4.5. In Fig. 4.5, a is the mass of reactant per cubic centimeter, Cj is the rate of reaction, Q is the heat of reaction per unit mass, and p is the total density. Note that alp is the mass fraction of reactant a. Since the problem is a steady one, there is no accumulation of species or heat with respect to time, and the balance of the energy terms and the species terms must each be equal to zero. 

There are not many models that do transients, mainly because of the computational cost and complexity. The models that do have mainly been discussed above. In terms of modeling, the equations use the time derivatives in the conservation equations (eqs 23 and 68) and there is still no accumulation of current or charging of the double layer that is, eq 27 still holds. The mass balance for liquid water requires that the saturation enter into the time derivative because it is the change in the water loading per unit time. However, this treatment is not necessarily rigorous because a water capacitance term should also be included,although it can be neglected as a first approximation. 

A mass balance on one compound in our box is based on the principle that whatever comes in must do one of three things (1) be accumulated in the box, (2) flux out of another side, or (3) react in the source/sink terms. If it seems simple, it is. 

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 gas energy and mass balance equations, unlike the corresponding solid balances, do not have a term for accumulation. This is because the high convective flow of gas through the channels of the monolith makes accumulation of heat or reactants in the gas phase negligible. In practice, the accumulation term in the solid mass balance could also be removed as, in general, it also tends to be small. However, it is included in our models as it enables the equations to be solved numerically more easily. 

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 ... 

From the mass balance equation (4.49) we can obtain the unsteady-state equations for a dynamic model by using an accumulation term as follows. 

If we write the mass balance for a reaction of the type A—-—> P, at steady state, there is no accumulation term, thus the mass balance becomes [6]... 

This form of mass balance is also called the continuity equation. At steady state, the accumulation term (dmldt) becomes zero, and we have... 

In deriving the integral mass balance for a closed system in Section 4.2c we eliminated the input and output terms, since by definition no mass crosses the boundaries of a closed system. It is possible, however, for energy to be transferred across the boundaries as heat or work, so that the right side of Equation 7.3-1 may not be eliminated automatically. As with mass balances, however, the accumulation term equals the final value of the balanced quantity (in this case, the system energy) minus the initial value of this quantity. Equation 7,3-1 may therefore be written... 

A total mass balance necessarily has the form [accumulation = input - output], since mass can neither be generated nor consumed. The accumulation term is always dM < dt. where M t) is the mass of the system contents. Once you have determined M(t) by solving the differential balance equation, you may have to verify that the mathematical solution remains within the bounds of physical reality—that it does not become negative, for example, or that it does not exceed the total capacity of the system. 

Write an expression for the total amount of the balanced species in the system [K(m )p(kg/m ) for total mass, V(m )CA(mol A/m ) or n,otai(mol)xA(mol A/mol) for species A], Differentiate the expression with respect to time to obtain the accumulation term in the balance equation. 

Eq. (8) states that the accumulation of mass in both free and adsorbed form (LHS) is balanced by the diffusion rate into the particle of gas (first term in RHS) and the adsorbed species (second term in RHS). The diffusion rate of adsorbed species in particle is related to the observed adsorbed concentration (Eq. 6) so that the effect of size exclusion (Eq. 5) has... 

The first 1 species (fly with 7 = 1,...,/) are assumed to be the intermediates, whereas the remaining T = A -1 species (fly, 7 = / -t- 1,..., / -I- T) are terminal species. If intermediates are present in very low concentrations, and their high rate of production is balanced by a high rate of consumption, the quasi-steady-state assumption allows dropping the accumulation term from the mass balance of an intermediate. 

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... 

During batch operation, the concentrations of reactants and formed products change as a function of time. In the case of perfect mixing, i.e., in an ideal batch reactor, both temperature and composition (and if present, also catalyst concentration) are uniform throughout the reactor. Since in a batch reactor there is no inflow or outflow of reactants and products during reaction, a mass balance for species A orUy contains an accumulation (left) and a production term ... 

Where t is time, z are the axial position in the column, qt is the concentration of solute i in the stationary phase in equilibrium with Cu the mobile phase concentration of solute /, u is the mobile phase velocity, Da is the apparent dispersion coefficient, and F is the phase ratio (Vs/Vm). The equation describes that the difference between the amounts of component / that enters a slice of the column and the amount of the same component that leaves it is equal to the amount accumulated in the slice. The fist two terms on the left-hand side of Eq. 10 are the accumulation terms in the mobile and stationary phase, respectively [109], The third term is the convective term and the term on the right-hand side of Eq. 10 is the diffusion term. For a multi component system there are as many mass balance equation, as there are active components in the system [13],... 

After having defined the accumulation terms of the general mass balances (Eqs. 6.1-6.6), the transport and source terms are evaluated as follows. Mass transport in... 

There are many ways to combine the various finite differences that may be used for each of the terms of the mass balance equation, and there are as many ways to approximate a partial differential equation by a finite-difference scheme. The choice is limited in practice, however, for two reasons. First, we need the numerical calculation to be stable, and there is a condition to satisfy to achieve numerical stability. Second, we need to control the numerical errors that are made during the calculations. Replacing a partial difference term with any of the possible finite difference terms gives a tnmcation error. These tnmcation errors accumulate during the calculation of a numerical solution. The error contribution... 

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