Unsteady conditions with accumulation terms

Unsteady material and energy balances of reacting systems are formulated with the conservation law, 

Inputs = Outputs + Sinks + Accumulations The sink term of a material balance is Vrra and the accumulation term is the time derivative of the content of reactant in the vessel, or d(VrCa)/dt, where both Vr and Ca depend on 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 the conditions do change with time but are taken account of by the sink term. Startup and shut down periods of batch reactors, however, are classified as unsteady. 

When a batch tank is being filled with reacting liquid, the material balance is 

For a semibatch operation in which some of the ingredients are preloaded and the others are fed in gradually, the equations are developed in problem P4.09.09. 

Similarly the unsteady material balance of a CSTR has an accumulation term added to it, for example, 

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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. 

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. 

This law can be applied to steady-state or unsteady-state (transient) processes and to batch or continuous reactor systems. A steady-state process is one in which there is no change in conditions (e.g., pressure, temperature, composition) or rates of flow with time at any given point in the system. The accumulation term in Equation (7.2) is then zero. (If there is no chemical or nuclear reaction, the generation term is also zero.) All other processes are unsteady-state. In a batch reactor process, a given quantity of reactants is placed in a container, and by chemical and/or physical means, a change is made to occur. At the end of the process, the container (or adjacent containers to which material may have been transferred) holds the product or products. In a continuous process, reactants are continuously removed from one or more points. A continuous process may or may not be steady-state. A coal-fired power plant, for example, operates continuously. However, because of the wide variation in power demand between peak and slack periods, there is an equally wide variation in the rate at which the coal is fired. For this reason, power plant problems may require the use of average data over long periods of time. However, most industrial operations are assumed to be steady-state and continuous. 

In Equation 6.72, the first term represents the unsteady accumulation term S is the volumetric source or sink term given by the three-phase electrochemical reaction region in the active layer of the electrodes. The volumetric reaction term is neglected from the mass species transport equation for the electrode if the active layer is assumed as the electrode-membrane interface with surface reaction and this is taken into account in the assigned boundary condition at the interface. 

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