Unsteady Integral Balances

Here the balance is taken over a well-stirred tank or compartment such as the water basin we considered in Illustration 2.1. To describe the process, we use the scheme Rate in - Rate out = d/dt contents, which was given by Equahon 2.1. It follows from tiiis expression that all unsteady integral balances lead to first-order ODEs. When the time derivative in Equation 2.1 is zero, the system reverts to a steady state and the result is an algebraic equahon (AE). 

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As is usual in systems of some complexity, a number of different balances can be made depending on the choice of balance space. We may choose, for example, to make an unsteady integral balance about the still, or to make a similar balance about the receiver. They can be instantaneous or cumulative in time and can involve total or component mass balances. 

Cumulative balances such as the two simple expressions (Equations 2.15a and Equation 2.15b) are often overlooked in modeling, or else written out without much thought to their origin. It is important to note that they are quite independent of the unsteady integral balances mentioned previously and consequently serve as additional tools that can be used to supplement the model equations. Typically, they are used as adjuncts to instantaneous balances in batch distillation. We can, for example, solve for either still or distillate composition and obtain... 

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 ... 

We are concerned in this text primarily with differential balances applied to continuous steady-state systems and integral balances applied to batch systems between their initial and final states. In Chapter 11, we consider general balances on unsteady-state systems and show how integral and differential balances are related—in fact, how each can be derived from the other. 

Rigorously, an unsteady-state balance should take into account variable concentration profiles in units such as columns, or even in pipelines. Such a balance then, in fact, takes the form of a differential balance cf. Appendix C. In some cases, the balance can be simplified for example in a stirred reactor, we can approximate the -th species content (accumulation) as Y where V is (fixed) volume, is (generally time-dependent) averaged (integral mean) volume concentration of species. Then the unsteady-state balance is again an integral (volume-integrated) balance, extended by accumulation terms. 

We start the derivation of the model by composing the unsteady integral mass balances for the system ... 

Equation (20-70) is the unsteady-state component mass balance for fed-batch concentration at constant retentate volume. Integration yields the equations for concentration and yield in Table 20-19. 

It is not possible to predict a priori which of the possible stationary-states is actually attained, based on steady-state operating considerations. It may be done by integrating the material-balance equation in unsteady-state form, equation 14.3-2 or equivalent, with the given rate law incorporated. For this, the initial concentration of A in the reactor cA(f) at t= 0 must be known this is not necessarily the same as cAg. 

After substituting Equations 3.1.2 and 3.1.3 into Equation 3.1.1, the oxygen mole balance reduces to Equation 3.1.4 in Table 3.1.1. Because Equation 3.1.4 is an unsteady-state, first-order differential equation, we need an initial condition to calculate the constant of integration. Initially, the tank contains air, which has an oxygen concentration of approximately 21 % by volume. We could also write the mole balance for nitrogen, but in this case it is more convenient to write the total mole balance, which results in Equation 3.1.5. Once we write Equations 3.1.4 to 3.1.6, the nitrogen mole balance is not an independent equation. Equation 3.1.7 states that the molar flow rate is equal to the product of the molar density and the volmnetric flow rate. 

By inserting all terms derived into the general integral heat balance for cooled ideal reactors, including the thermal inertia O, the unsteady state heat balance of the cooled SBR is obtained in a partially dimensionless form ... 

In Chapter 4 on steady-state heat transfer and Chapter 5 on unsteady-state heat transfer new overall energy balances were made on a finite control volume for each new situation. To advance further in our study of heat or energy transfer in flow and nonflow systems we must use a differential volume to investigate in greater detail what goes on inside this volume. The balance will be made on a single phase and the boundary conditions at the phase boundary will be used for integration. 

As we saw in Chapter 2, batch operation is essentially unsteady-state. The energy balance is drawn up at some instant t the average cooling rate over the entire batch Q is given by integration with respect to time ... 

It is not immediately clear at the outset how this information is to be obtained. Because the permeation process is unsteady, a good way to start is to set up an integral imsteady moisture balance, in which the rate of... 

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