Unsteady Differential Balances

It was previously noted that when the 1-D pipe model is applied to an unsteady process, the mass balance will lead to a PDE. We can generalize this into the following statements Whenever a variable is distributed in both time and distance — that is, varies with t and x (y, z) — the resulting mass balance will be a PDE. 

We can summarize these results as follows A mass balance can be time-independent or time-dependent, and it can be applied over a finite entity or a differential increment of either time or distance. It can further depend on a single space variable, several such variables, or none. Depending on which combination of factors applies to a particular problem, this will result in either an AE, an ODE, or a PDE. These features are summarized in Table 2.1. 

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The unsteady material balances of tracer tests are represented by linear differential equations with constant coefficients that relate an input function Cj t) to a response function of the form... 

Such a differential equation together with a rate equation for the main reactant constitutes a pair that must be solved simultaneously. Take the example of a CSTR for which the unsteady material balance is... 

The procedure for the solution of unsteady-state balances is to set up balances over a small increment of time, which will give a series of differential equations describing the process. For simple problems these equations can be solved analytically. For more complex problems computer methods would be used. 

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. 

However, as long as any term varies with time, the derivative on the left side of Equation 11.1-2 remains part of the equation. We thus conclude that the balance equation for an unsteady-state system at an instant of time is a differential equation (hence the term differential balance). 

Porter, R. L., Unsteady State Balances—Solution Techniques for Ordinary Differential Equations, AIChE Chemi Series No. F5.5, American Institute of Chemical Engineers, New York (1986). 

The rate of reactant now entering at time t through side entrances in the volume element is Qc0(t)f( )d. An unsteady material balance on the element gives the partial differential equation describing the concentration profile c(, t). 

Presented below is a brief recap of the definition of linear equations in the context of differential equations. Following the recap are examples of unsteady mass balances, which lead to linear first-order problems. Also presented are examples involving chemical reactions that can be treated as linear first-order problems. 

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. 

In unsteady states the situation is less satisfactory, since stoichiometric constraints need no longer be satisfied by the flux vectors. Consequently differential equations representing material balances can be constructed only for binary mixtures, where the flux relations can be solved explicitly for the flux vectors. This severely limits the scope of work on the dynamical equations and their principal field of applicacion--Che theory of stability of steady states. The formulation of unsteady material and enthalpy balances is discussed in Chapter 12, which also includes a brief digression on stability problems. 

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 ... 

Note that since there are two independent variables of both length and time, the defining equation is written in terms of the partial differentials, 3C/dt and 3C/dZ, whereas at steady state only one independent variable, length, is involved and the ordinary derivative function is used. In reality the above diffusion equation results from a combination of an unsteady-state mass balance, based on a small differential element of solid length dZ, combined with Pick s Law of diffusion. 

Writing unsteady-state component balances for each liquid phase results in the following pair of partial differential equations which are linked by the mass transfer rate and equilibrium relationships... 

The TDE moisture module (of the model) is formulated from three equations (1) the water mass balance equation, (2) the water momentum, (3) the Darcy equation, and (4) other equations such as the surface tension of potential energy equation. The resulting differential equation system describes moisture movement in the soil and is written in a one dimensional, vertical, unsteady, isotropic formulation as ... 

An unsteady state mass balance over the typical reactor element lying between x and x + 5x gives rise to the partial differential equation... 

In this section we develop a dynamic model from the same basis and assumptions as the steady-state model developed earlier. The model will include the necessarily unsteady-state dynamic terms, giving a set of initial value differential equations that describe the dynamic behavior of the system. Both the heat and coke capacitances are taken into consideration, while the vapor phase capacitances in both the dense and bubble phase are assumed negligible and therefore the corresponding mass-balance equations are assumed to be at pseudosteady state. This last assumption will be relaxed in the next subsection where the chemisorption capacities of gas oil and gasoline on the surface of the catalyst will be accounted for, albeit in a simple manner. In addition, the heat and mass capacities of the bubble phases are assumed to be negligible and thus the bubble phases of both the reactor and regenerator are assumed to be in a pseudosteady state. Based on these assumptions, the dynamics of the system are controlled by the thermal and coke dynamics in the dense phases of the reactor and of the regenerator. 

An unsteady-state mass balance for solids in a differential element with a small perturbation in a system otherwise in equilibrium yields... 

When an agitated batch containing M of fluid with specific heat c and initial temperature t is heated using an isothermal condensing heating medium Tt, the batch temperature t2 at any time 0 can be derived by the differential heat balance. For an unsteady state operation as shown in Figure 7-27, the total number of heat transferred is q, and per unit time 0 is ... 

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. 

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