Periodic operation equations

Rate information is displayed on a meter in decades per minute, and since it is used by the operator to monitor the rate of change of power during startup, it is termed startup rate. Startup rate (SUR) equates to reactor period using Equation 6-10. 

A general approach to the analysis of low amplitude periodic operation based on the so-called Il-criterion is described in Refs. 11. The shape of the optimal control function can be found numerically using an algorithm by Horn and Lin [12]. In Refs. 9 and 13, this technique was extended to the simultaneous optimization of a forcing function shape and cycle period. The technique is based on periodic solution of the original system for state variables coupled with the solution of equations for adjoin variables [Aj, A2,..., A ], These adjoin equations are... 

The fundamental relationships dealing with continuous interest compounding can be divided into two general categories (1) those that involve instantaneous or lump-sum payments, such as a required initial investment or a future payment that must be made at a given time, and (2) those that involve continuous payments or continuous cash flow, such as construction costs distributed evenly over a construction period or regular income that flows constantly into an overall operation. Equation (12) is a typical example of a lump-sum formula, while Eqs. (23) and (25) are typical of continuous-cash-flow formulas. 

By comparison with equation (12), er,(+) is simply Iy, as expected. During the acquisition period the magnetization undergoes CSE and relaxation. Applying the CSE operator, equation (26), gives... 

In the case of an electric potential applied to the metal, as in electrocatalysis, U(r) = U(r) + eV. However, the modified energy is the total energy, so the kinetic operator has to be also modified in an unknown form. We define the Bloch electrons as those obeying the periodic Schrodinger equation and the free electrons as those obeying a zero periodic potential. 

Consider a CSTR under periodic operation governed by the state equations dyi... 

Simulation results show that response similar to Figure 2 with a shift towards left indicating that under period operation higher yield of B can be obtained for a given 7. in the case of periodically varying simultaneously A and B in the feed. In what follows we will assume the simultaneous variation of A and B in the feed as per Equation 6. 

Aris et al. have primarily analyzed whether the steady-state multiplicity features in a CSTR arising from a cubic rate law also can arise for a series of successive bimolecular reactions [26]. Aris et al. have showed that the steady-state equations for a CSTR with bimolecular reactions scheme reduces to that with a cubic reaction scheme when two parameters e(=k,Cg/k j) and K(=kjC /k j) arising in system equations for the bimolecular reactions tend to zero. Aris et al. have shown that the general multiplicity feature of the CSTR for bimolecular reactions is similar to that of the molecular reactions only at smaller value of e and K. The behavior is considerably different at larger values of e and K. Chidambaram has evaluated the effect of these two parameters (e and K) on the periodic operation of an isothermal plug flow reactor [18]. 

Now let us consider the periodic operation of an isothermal tubular reactor with autocatalytic reactions given by Equation 1. It is assume here [16] that the component A, has an activating influence on the rate constant K, represented as... 

Whereas, for ordinary reactions (i.e., when P = 0) the reactor model equations are linear and, hence, the yield at any y is the same as at y = 1. The effect of P on the average yield of B under periodic operation is shown in Figure 9. For a given value of y, a higher value of average yield is obtained for a larger values of p. The value of y, at which a steep increase in yield is obtained, decreases with decrease in p. As y decreases, during the first reaction of a period, the concentration... 

The problem whether the periodical operation of a technical reactor with input concentrations which change periodically provides higher selectivities and yields than stationary operating was examined with the example of benzene oxidation into malein anhydride [103], A rather complex example was the oxide-hydrogenation of isobutyric aldehyde to methacrolein [100], Based on dynamic experiments a reaction scheme is proposed and estimation of kinetic parameters of the main reaction using an Eley-Rideal type rate equation was carried out. The examples revealed that the wave-front analysis provides valuable qualitative and quantitative kinetic information of heterogeneous catalytic reactions. 

Fig. 3. Operational equation of radioactive deoxyglucose method in comparison to the general equation for measurement of the reaction rates with tracers. T represents the time at the termination of the experimental period X equals the ratio of the distribution space of deoxyglucose in the tissue to that of glucose equals the fraction of glucose which, once phosphorylated, continues down the glycolytic pathway Km, Vm and Km, Vm represent the familiar Michaelis-Menten kinetic constants of hexokinase for deoxyglucose and glucose, respectively. These six constants collectively constitute the lumped constant (equivalent to the isotope-effect correction factor of the general equation). The other symbols are the same as those defined in Fig. 2. (Reproduced with permission from Sokoloff, 1978.)...

Two other constraints (the constancy of plasma glucose and of the rate of glucose utilization during the period of measurement) impose minor restrictions on experimental design. The recent derivation of a modified operational equation permits the deoxyglucose technique to be applied quantitatively to conditions in which the moderate, dynamic alterations in plasma glucose levels occur during the measurement of... 

The following example reveals the type of results which can be obtained with the help of Equation (5-27). The pipe section is again the same one which has been used for the explanation of static behavior. However, now it is assumed that the pipe section is subjected to a dynamic stress with a characteristic/CC = 187.5. A crack characteristic A"/ = 4.7 has already been determined for the pipe section. During the operation time given, the pipe section is scheduled to sustain N = 2500 stress reversals of the stress magnitude KG. The question is What is the failure probability of the pipe section at the end of the stress period (operation period) ... 

From the mechanism it can be seen that material is added to or depleted from the gas phase by adsorption/desorption with the exception of hydrogen which is assumed to be consumed directly from the gas phase. In formulating a theoretical model for the system it was assumed that the adsorption/desorption kinetics played an important role in the dynamics of the periodic operation and these kinetics were incorporated into the dynamic equations. Furthermore, it was assumed that there was neither bulk nor pore diffusional heat and mass transfer resistances, that the reactor was isothermal (both in the bulk gas phase and locally) and that the flow pattern in the reactor could be approximated by plug flow. Most of the above assumptions (i.e. plug flow, bulk isothermal conditions, no pore diffusion limitations) could be... 

These process components are related in series, thus if any one of the components fails, the entire system fails. The failure rates for the various components are given in Table 3. The rehabiflty and failure probabiflty are computed for each individual component using equations 1 and 2 and assuming a one-year period of operation. The results are shown in Table 4. 

The generalized transport equation, equation 17, can be dissected into terms describing bulk flow (term 2), turbulent diffusion (term 3) and other processes, eg, sources or chemical reactions (term 4), each having an impact on the time evolution of the transported property. In many systems, such as urban smog, the processes have very different time scales and can be viewed as being relatively independent over a short time period, allowing the equation to be "spht" into separate operators. This greatly shortens solution times (74). The solution sequence is... 

Time-Dependent Cascade Behavior. The period of time during which a cascade must be operated from start-up until the desired product material can be withdrawn is called the equiUbrium time of the cascade. The equiUbrium time of cascades utilizing processes having small values of a — 1 is a very important quantity. Often a cascade may prove to be quite impractical because of an excessively long equiUbrium time. An estimate of the equihbrium time of a cascade can be obtained from the ratio of the enriched inventory of desired component at steady state, JT, to the average net upward transport of desired component over the entire transient period from start-up to steady state, T . In equation form this definition can be written as... 

Unsteady material and energy balances are formulated with the conservation law, Eq. (7-68). The sink term of a material balance is and the accumulation term is the time derivative of the content of reactant in the vessel, or 3(V C )/3t, where both and depend on the 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 conditions do change with time but are taken account of in the sink term. Startup and shutdown periods of batch reactors, however, are classified as unsteady their equations are developed in the Batch Reactors subsection. For a semibatch operation in which some of the reactants are preloaded and the others are fed in gradually, equations are developed in Example 11, following. 

Equation (9-245) shows that in this particular case the fixed-capital cost per unit of input energy CpJW) must not exceed 160,000 (GJh" )" or 576 per kilowatt, to have a 1-year payback period if the heat pump is operational for 8000 h/year. For this case the corresponding value of y is about 0.12 for a heat pump with an operating life of 10 years purchased with money borrowed at a 10 percent rate of interest. 

Equivalent-Area Concept The preceding equations for batch operations, particularly Eq. 11-35 can be appliedforthe calculation of heat loss from tanks which are allowed to cool over an extended period of time. However, different surfaces of a tank, snch as the top (which would not be in contact with the tank contents) and the bottom, may have coefficients of heat transfer which are different from those of the vertical tank walls. The simplest way to resolve this difficulty is to nse an equivalent area A in the appropriate equations where... 

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