Unsteady state operation

There are special numerical analysis techniques for solving such differential equations. New issues related to the stabiUty and convergence of a set of differential equations must be addressed. The differential equation models of unsteady-state process dynamics and a number of computer programs model such unsteady-state operations. They are of paramount importance in the design and analysis of process control systems (see Process control). 

When an agitated bateh eontaining M of fluid with speeifie heat e and initial temperature t is heated using an isothermal eondensing heating medium Tj, the bateh temperature tj at any time 6 ean be derived by the differential heat balanee. For an unsteady state operation as shown in Figure 7-27, the total number of heat transferred is q, and per unit time 6 is ... 

The non-isothermal heating medium has a eonstant flowrate Wj, speeifie heat Cj, and inlet temperature Tj, but a variable outlet temperature. Eor an unsteady state operation ... 

The results quantitatively show the signifieanee of eonsidering both boundaries of the unsteady state operation. The load required at the end of the eyele is less than one-half the load required at the start of the eyele. Steve [13] inferred that reporting the value at only one boundary may be misleading. For example, the total heat load (provided by ehillers and heaters) from multiple reaetors with overlapping temperature adjustment eyeles may be skewed if the ehanges within those eyeles are not eonsidered. 

For unsteady-state operation the component balance equations, for each phase are now of the form... 

Batch processes are also examples of unsteady-state operation though the total material requirements can be calculated by taking one batch as the basis for the calculation. 

Equation based programs in which the entire process is described by a set of differential equations, and the equations solved simultaneously not stepwise, as in the sequential approach. Equation based programs can simulate the unsteady-state operation of processes and equipment. 

The input and output terms of equation 1.5-1 may each have more than one contribution. The input of a species may be by convective (bulk) flow, by diffusion of some kind across the entry point(s), and by formation by chemical reaction(s) within the control volume. The output of a species may include consumption by reaction(s) within the control volume. There are also corresponding terms in the energy balance (e.g., generation or consumption of enthalpy by reaction), and in addition there is heat transfer (2), which does not involve material flow. The accumulation term on the right side of equation 1.5-1 is the net result of the inputs and outputs for steady-state operation, it is zero, and for unsteady-state operation, it is nonzero. 

Unsteady-state operation means process control and obtaining uniformity of product more difficult (D) (but see England, 1982) Steady-state operation means process control and obtaining uniformity of product less difficult (A)... 

The design or performance analysis is complicated because of the unsteady-state operation. 

As in the case of a batch reactor for commercial operation, a CSTR is normally used for a liquid-phase reaction. In the laboratory, it may also be used for a gas-phase reaction for experimental measurements, particularly for a solid-catalyzed reaction, as in Figure 1.2. The operation is normally one of steady-state, except for startup, shutdown, and operational disturbances or upsets, in which cases unsteady-state operation has to be taken into account. 

The material-balance equation, in whatever form, is usually used to solve for V in steady-state operation, or to determine the changes of outlet properties with respect to time in unsteady-state operation for a particular V. Thus, for steady-state operation, with dnA/dr = 0, from equations 14.3-2 and -3,... 

For unsteady-state operation of a CSTR, the full form of the material balance, equation 14.3-2 or its equivalent, must be used. 

Although there is steady flow of feed, the reactor is in unsteady-state operation during the time t, since the outlet concentration cA (and hence fA) is continuously changing (cA decreasing from cAo, and fA increasing from 0). 

For unsteady-state operation, equation 20.1-1 constitutes a set of N ordinary differential equations that must be solved simultaneously (usually numerically) to obtain the time-dependent concentration within each tank. For a constant-density system, dnAl/dr is replaced by Vt dcAi/dr. We focus on steady-state operation in this chapter. 

In comparing the TIS and DPF reactor models, we note that the former is generally easier to use for analysis of reactor performance, particulariy for nonlinear kinetics and unsteady-state operation. 

Remark 1. Eq.(9) can be used on a CSTR in unsteady state operation or in a semibatch reactor when there is no phase change. 

The extruder would operate for several hours to days at steady state, and then for no apparent reason It would flow surge for several hours. After a period of time, the extruder would return to a steady-state operation and would remain there until the cycle repeated. Problem diagnosis was impossible without transient process data. Moreover, molten resin would frequently flow out the vent, especially during times of unsteady-state operation. 

In the batch reactor, or BR, of Fig. 5.1 the reactants are initially charged into a container, are well mixed, and are left to react for a certain period. The resultant mixture is then discharged. This is an unsteady-state operation where composition changes with time however, at any instant the composition throughout the reactor is uniform. 

Unsteady-state operations (semibatch) where the liquid composition changes with time... 

Batch-stirred tank reactor (BSTR) In this type of reactor, the reactants are fed into the container, they are well mixed by means of mechanical agitation, and left to react for a certain period of time. This is an unsteady-state operation, where composition changes with time. However, the composition at any instant is uniform throughout the reactor. 

Fig. 1.9. Continuous stirred-tank reactor showing steady state operation (a) and two modes of unsteady state operation (b) and (c)...

This phenomenon of increased conversion, yield and productivity through deliberate unsteady-state operation of a fermentor has been known for some time. Deliberate unsteady-state operation is associated with nonautonomous or externally forced systems. The unsteady-state operation of the system (periodic operation) is an intrinsic characteristic of this system in certain regions of the parameters. Moreover, this system shows not only periodic attractors but also chaotic attractors. This static and dynamic bifurcation and chaotic behavior is due to the nonlinear coupling of the system which causes all of these phenomena. And this in turn gives us the ability to achieve higher conversion, yield and productivity rates. 

Regenerative heat exchange in chemical reactors offers clear benefits, such as simplicity, robustness, low costs, and high efficiencies, against which must be set its inherently unsteady-state operation, the limited potential for an exact regulation of temperature profiles, and the restriction of its use to gaseous reaction media. 

Design equations for unsteady-state operation are needed for start-up of CSTRs or for semibatch operation. These equations must have the ability to predict accurately the concentration or conversion changes before steady-state flow is obtained. Starting with the general design equation, and assuming perfect mixing, we obtain... 

This chapter describes a forced unsteady-state operation technique as employed for continuous processes which represent the majority of heterogeneous catalysis applications. The catalyst life in these processes often lasts as long as several years. Traditional operation is in the steady-state, and automatic control systems are... 

However, the reactor performance obtained under optimal steady-state conditions does not determine a potential limit for a heterogeneous catalytic system. This performance can be improved further using forced unsteady-state operation. Such an operation is capable of substantial extending a gamut of the process features and allows for better use of nonlinear properties inherent in a catalytic reaction system. The positive effect can be generated by two major factors [1] ... 

Sufficient conditions for optimality of forced unsteady-state operation which provides J > Js, can be determined on the basis of analysis of two limiting types of periodic control [10]. The first limiting type is a so-called quasisteady operation which corresponds to a very long cycle duration compared to the process response time t. In this case the description of the process dynamics is reduced to the equations x(t) = /t(u(t)), where h is defined as a solution of the equation describing a steady-state system 0 = f(/t(u(t),u(t))). The second limiting type of operation, the so-called relaxed operation, corresponds to a very small cycle time compared to the process response time (tc t). The description of the system is changed to ... 

Several other examples for potential application of reverse-flow operated catalytic reactors are described in Ref. 9. Also, other potential techniques of forced unsteady-state operation which allow for combining chemical reaction and heat exchange in a fixed catalyst bed are discussed. One such technique is sequential switching between inlet and outlet ports of the reaction gas between two or more packed beds (Fig. 2(c)). In this case, the thermal wave travels continuously through a series of packed beds in one direction, as if along a closed ring. However, this operation is more complex and requires more catalyst than the reverse-flow operation. 

The mass balance in the reactor is derived under the following assumptions (i) unsteady state operation, (ii) convective laminar Newtonian flow in the axial direction z (the Re)molds number is below the transition regime), (iii) diffusion in the z direction is negligible with respect to convection, (iv) symmetry in the y direction (the lamp length is much larger than the reactor width), and (v) constant physical properties. The local mass balance for a species i in the reactor and the corresponding initial and boundary conditions are... 

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