Unsteady-State Systems, Accumulation

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. 

The general material balance of Section 1.1 contains an accumulation term that enables its use for unsteady-state reactors. This term is used to solve steady-state design problems by the method of false transients. We turn now to solving real transients. The great majority of chemical reactors are designed for steady-state operation. However, even steady-state reactors must occasionally start up and shut down. Also, an understanding of process dynamics is necessary to design the control systems needed to handle upsets and to enable operation at steady states that would otherwise be unstable. 

In a steady state continuous distillation with the assumption of a well mixed liquid and vapour on the plates, the holdup has no effect on the analysis (modelling of such columns does not usually include column holdup) since any quantity of liquid holdup in the system has no effect on the mass flows in the system (Rose, 1985). Batch distillation however is inherently an unsteady state process and the liquid holdup in the system become sinks (accumulators) of material which affect the rate of change of flows and hence the whole dynamic response of the system. 

Because the system either gains or loses mass, drop either of the rate terms for depletion or accumulation. To apply Equation 3.2 to a specific situation, the first decision requires determining whether the process operation is steady or unsteady state. The unsteady-state operations are ... 

For unsteady-state diffusion into a quiescent medium with no chemical reaction, the mass transfer Peclet number does not appear in the dimensionless mass transfer equation for species i because it is not appropriate to make variable time t dimensionless via division by L/ v) if there is no bulk fluid flow (i.e., (d) = 0). In this case, the first term on each side of equation (10-11) survives, which corresponds to the unsteady-state diffusion equation. However, the characteristic time for diffusion of species i over a length scale L, given by L /50i,mix. replaces L/ v) to make variable time t dimensionless. Now, the accumulation and diffusional rate processes scale as CAo i.mix/A, with dimensions of moles per volume per time. Since both surviving mass transfer rate processes exhibit the same dimensional scaling factor, there are no dimensionless numbers in the mass transfer equation which describes unsteady-state diffusion for species i in nonreactive systems. 

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. 

With respect to open systems, we make the additional distinction between steady-state, and unsteady-state operation. At steady state the accumulation terms, dU /dt, dS /dt and dM /dt, are zero. This reduces the energy balance from a differential equation into an algebraic equation. For this reason, steady-state processes are much simpler to calculate compared to unsteady-state processes. 

Given that the right-hand-side term in (7.7) is equal to 20 (different from 0), we can state that the system (tank) is in an unsteady-state condition and, in addition, because the term is positive, the system is accumulating water at a ratio of 20 kg/h. As seen in Fig. 7.8, M(0) = 600, indicating that the mass in the tank at time 0 is 600 but, as was shown, is increasing at a ratio of 20 kg/h. 

Derive the differential equation for unsteady state for diffusion and reaction for this system. [ Hint First make a mass balance for A for a Az length of tube as follows rate of input (diffusion) + rate of generation (heterogeneous) = rate of output (diffusion) + rate of accumulation.]... 

Chapter 11 deals with dynamic (unsteady-state) mass balancing of a technological system involving inventories. A straightforward method of reconciliation is presented avoiding accumulation of small systematic errors in the time series of measurements. The method has proved successful in practice and can also be supported by theoretical arguments. 

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 Chapter 1 we developed a steady-state model for a stirred-tank blending system based on mass and component balances. Now we develop an unsteady-state model that will allow us to analyze the more general situation where process variables vary with time. Dynamic models differ from steady-state models because they contain additional accumulation terms. 

Equations 2-2 and 2-3 provide an unsteady-state model for the blending system. The corresponding steady-state model was derived in Chapter 1 (cf. Eqs. 1-1 and 1-2). It also can be obtained by setting the accumulation terms in Eqs. 2-2 and 2-3 equal to zero,... 

These accumulation terms are added to the appropriate steady-state balances to convert them to unsteady balances. The circumflexes indicate averages over the volume of the system, e.g.. 

By its nature process control is concerned with the dynamic behaviour of systems. It is no longer sufficient to make the steady-state assumption. Material and energy balances for unsteady systems must include the accumulation terms so far omitted. Because of the extra mathematical complexity involved in a quantitatve treatment of control this section will, instead, concentrate on general concepts rather than detailed analysis of... 

The penetration theory can be viewed as the original surface-renewal model. This model was formulated by Higbie [51]. This model is based on the assumption that the liquid surface contains small fluid elements that contact the gas phase for a time that is equal for all elements. After this contact time they penetrate into the bulk liquid and each element is then replaced by another element from the bulk liquid phase. The basic mechanism captured in this concept is that at short contact times, the diffusion process will be unsteady. Considering that the fluid elements may diffuse to an infinite extend into the liquid phase, the model formulation developed earlier for diffusion into a semi-infinite slab can be applied describing this system. After some time the diffusion process will reach a steady state, thus the penetration theory predictions will then correspond to the limiting case described by the basic film theory. However, when the transient flux development is determining a notable amount of the total flux accumulated, the two models will give rise to different mass transfer coefficients. 

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