Unsteady-state calculations

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. 

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. 

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. 

The general approach to the solution of unsteady-state problems is illustrated in Example 2.15. Batch distillation is a further example of an unsteady-state material balance (see Volume 2, Chapter 11). 

The behaviour of processes under non-steady-state conditions is a complex and specialised subject and beyond the scope of this book. It can be important in process design when assessing the behaviour of a process from the point of view of safety and control. 

The general approach to the solution of unsteady-state problems is illustrated in Example 2.15. 

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In the theoretical treatment, the heat- and mass-transfer processes shown in Fig. 6 were considered. Simultaneous solution of the equations describing the behavior of the unsteady-state reaction system permits the temperature history of the propellant surface to be calculated from the instant of oxidizer propellant contact to the runaway reaction stage. 

Figures 9.17-9.19 clearly show that, as the Biot number approaches zero, the temperature becomes uniform within the solid, and the lumped capacity method may be used for calculating the unsteady-state heating of the particles, as discussed in section (2). The charts are applicable for Fourier numbers greater than about 0.2.

The heat transfer problem which must be solved in order to calculate the temperature profiles has been posed by Lee and Macosko(lO) as a coupled unsteady state heat conduction problem in the adjoining domains of the reaction mixture and of the nonadiabatic, nonisothermal mold wall. Figure 5 shows the geometry of interest. The following assumptions were made 1) no flow in the reaction mixture (typical molds fill in <2 sec.) ... 

The calculations for the experimental reaction rates are based on an unsteady state heat transfer analysis. We computed the overall heat transfer coefficient of the system and estimated the experimental rates as follows dT... 

Chapter 14 and Section 15.2 used a unsteady-state model of a system to calculate the output response to an inlet disturbance. Equations (15.45) and (15.46) show that a dynamic model is unnecessary if the entering compound is inert or disappears according to first-order kinetics. The only needed information is the residence time distribution, and it can be determined experimentally. 

A fugacity level I calculation may be 6 compartment equilibrium, no reaction, no advection, steady state a level II may be equilibrium, with reaction and advection, steady state level III may be non equilibrium, with reaction and advection, steady state,and level IV and EXAMS are non equilibium, with reaction and advection, unsteady state. 

Confined or unconfined Unsteady state Hantush Calculation Aquifer homogeneous, anisotropic, and... 

Semi confined Unsteady state Hantush Calculation ... 

As a result of self-focusing, power propagating within the core f iz) inereases behind the region of unsteady-state regime (Fig. 10). Meanwhile the total power of the light beam calculated within the computational window decreases due to emitting of radiation field outside the window. 

Unsteady State Diffusion. The apparatus, experimental procedures, and the computational procedures used to calculate the diffusion parameter D /r (where D is the diffusion coefficient and r is the diffusion path length) have been described in detail previously (6, 8). A differential experimental system was used to avoid errors caused by small temperature fluctuations. In principle, the procedure consisted of charging the sample under consideration with argon to an absolute pressure of 1204 12 torr (an equilibrium time of about 24 hours was allowed) and then measuring the unsteady state release of the gas after suddenly reducing the pressure outside the particles back to atmospheric. 

Mathematical models derived from mass-conservation equations under unsteady-state conditions allow the calculation of the extracted mass at different bed locations, as a function of time. Semi-batch operation for the high-pressure gas is usually employed, so a fixed bed of solids is bathed with a flow of fluid. Mass-transfer models allow one to predict the effects of the following variables fluid velocity, pressure, temperature, gravity, particle size, degree of crushing, and bed-length. Therefore, they are extremely useful in simulation and design. 

Fig. 7.9 The effect of dilution rate (D) on the fraction of plasmid-carrying cells during steady-state operation of CSTF. The initial value of f was calculated by assuming that the number of generations required for the step inoculation and initial batch and unsteady-state continuous fermentation was 20.

First, it is of common interest to unsteady processes and their models. Chemical unsteadiness must be taken into account in many cases. For example, studies with variations in catalyst activity, calculations of fluidized catalyst bed processes (when the catalyst grain "is shaking in a flow of the reaction mixture and has no time to attain its steady state), analyses of relaxational non-stationary processes and problems of control. Unsteady state technology is currently under development [14,15], i.e. the technology involving programmed variation of the process parameters (temperature, flow rate, concentration). The development of this technology is impossible without distinct interpretation of the unsteady reaction behaviour. 

A direct kinetic problem consists of calculating multi-component reaction mixture compositions and reaction rates on the basis of a given kinetic model (both steady-state and unsteady-state) with the known parameters. Reliable solution for the direct problem is completely dependent on whether these parameters, obtained either on theoretical grounds or from special experiments, have reliable values. Modern computers can solve high-dimensional problems. Both American and Soviet specialists have calculated kinetics for the mechanisms with more than a hundred steps (e.g. the reac-... 

In the intriguingly entitled publication "The steady-state approximation, fact or fiction by Farrow and Edelson [41] presents calculated data on the unsteady-state behaviour of a complex chemical reaction including 81 steps. The reaction mixture consists of 50 substances. Numerical calculation shows a great variety of unsteady-state characteristics of a complex reaction. This variety cannot be interpreted in the narrow framework of the quasi-steady-state hypothesis. Nevertheless, the authors discriminate between the ranges of parameters and time intervals within which this hypothesis is confirmed by numerical experiments. 

Inaba, A. and Kashiwagi, T. A calculation of thermal degradation initiated by random scission. II. Unsteady state radical concentration. European Polymer Journal 1987 23 871. 

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