Unsteady-State Balances

The unsteady state balance of acetic anhydride in a CFSTR. 

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 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. 

Perform the unsteady state balance and solve to obtain the startup transient in the product stream concentrations of anhydride and acid. 

In Section 14.3.1 we used the system shown in Figure 14-12, with bypass flow rate and dead volume to model our real reactor system. We shall inject our tracer, T, as a positive-step input. The unsteady-state balance on the nonreacting tracer T in the reactor volume, Vs is... 

Batch processes are also examples of unsteady-state operation although the total material requirements can be calculated by taking one batch as the basis for the calculation, unsteady-state balances are needed for determination of reaction and separation times. 

Step 4 Develop the unsteady-state balance(s) for the process. In an elemental time At, the height of the water in the tank drops A/i. The mass of water leaving the tank is in the form of a cylinder 5 cm in area, and we can calculate the quantity as... 

Next we make an unsteady-state balance for the compound in the solution (the system... 

Clements, W. C., Unsteady State Balances, AIChE Chemi Series No. F5.6, American Institute of Chemical Engineers, New York (1986). 

Dynamic flowsheeting is based on the unsteady state balance of mass and energy, and may be formulated in general terms by the following equation ... 

The unsteady state balance equations must be completed with constitutive equations, which are relations between some state variables, usually expressing natural laws or the kinetics of transport phenomena. Examples are PVT relations, as equations of state, or kinetics expressions, phase-equilibrium factors, etc. Specifying the initial and boundary conditions completes the problem formulation. 

With dimensions of motes per time, each term in this unsteady-state balance is evaluated separately ... 

The most important thing to remember about accumulation is that it deals with the actual amount of stuff in the system, not any sort offlow rate. If you remember if you re dealing with actual system properties rather than flow rates, it will help keep the terms straight in unsteady-state balances. 

Obtain the steady-state equation as a special case of the general unsteady state balance. 

Consider the mixing vessel shown in Figure 20.19. The feed stream flow rate, and composition, Cj, are considered to be disturbance variables. The feed is mixed with a control stream of flow rate, F2, and constant known composition C2. To ensure a product of constant composition, it is also possible to manipulate the flow rate, Fj, of the product stream. Perform a degrees-of-freedom analysis and suggest alternative control system configurations. Note that unsteady-state balances are required. 

The unsteady-state balance of the nth module is given by the following equation, written as a general expression ... 

The unsteady-state balance equation for this system (air within the engine housing) is the first law of thermodynamics for open systems with changes in kinetic and potential energies neglected. Also, the temperature and composition of the system contents are assumed independent of position and no phase changes occur. This gives the equation... 

Unsteady-state balances arise not only in batch processes but also when conditions in continuous processes are changed. The simple case of a two-component material balance will be considered. Consider the application of equations (2.1) and (2.2) to the vessel depicted in figure 2.4. The feed flow rate is Wp kg s, the discharge rate wd kg s, the density of the liquid p kg m and the volume of liquid in the. vessel V m Thus from equation (2.1) ... 

In the previous chapter, we have dealt with single-component balances, thus with the case when only one quantity is balanced around each node. A multicomponent balance is a set of several component balances. In chemical process networks, the components are certain chemical species present in the streams as the components of a mixture. Generally, the species can be transformed into one another by chemical reactions so their quantities may not be conserved individually. The individual balances are then not independent, as they must obey stoichiometric laws. This chapter deals mainly with steady-state chemical species balancing where chemical reactions are admitted the wording steady state is explained below. Formally precise unsteady-state balancing brings some problems, mainly because the holdup (accumulation) of a component in a unit is difficult to identify, due to spatial variability of the concentrations. See Section 4.7. 

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. 

A simple unsteady-state balance equation for the fth zone is given by... 

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