The Steady State Model

The mass- and heat-balance equations for the steady-state model are the equations (7.25) to (7.47). In the following, we describe a simple procedure to compute the model parameters. 

The two phase parameters are computed as follows. The bubble velocity Ub (in cm/sec) is given by 

Therefore, the area occupied by the bubble and cloud phase is 

Here A represents gas oil, A2 gasoline, and A3 represents coke and dry gases. The rate of disappearance of gas oil, the rate of appearance of gasoline, and the rate of appearance of coke and light gases are given by the equations (7.25), (7.26), and (7.27), respectively. The rate constants can be written in Arrhenius2 form as follows  

These starting values are used as initial guesses for fitting the model to industrial data and the preexponential factors are changed to obtain the best fit. This is done because the kinetic parameters depend upon the specific characteristics of the catalyst and of the gas oil feedstock. This complexity is caused by the inherent difficulties with accurate modeling of petroleum refining processes in contradistinction to petrochemical processes. These difficulties will be discussed in more details later. They are clearly related to our use of pseudocomponents. But this is the only realistic approach available to-date for such complex mixtures. 

---

Material and energy balances of the steady-state model. 

A dynamic model should be consistent with the steady-state model. Thus, Eqs (1) and (4) should be extended to dynamic form. For the better convergence and computational efficiency, some assumption can be introduced the total amounts of mass and enthalpy at each plate are maintained constant. Then, the internal flow can be determined by total mass balance and total energy balance and the number of differential equations is reduced. Therefore, the dynamic model can be established by replacing component material balance in Eq. (1) with the following equation. 

Step 2. The qualitative value of the desired change is propagated through the steady-state model equations of the plant equipment, following the constraint propagation procedure of Steele (1980). Manipulations that cause the desired change and that are feasible are identified as White Knights and are constrained to lie before the situation of interest s, in accordance with the truth criterion. 

Under steady-state conditions, variations with respect to time are eliminated and the steady-state model can now be formulated in terms of the one remaining independent variable, length or distance. In many cases, the model equations now result as simultaneous first-order differential equations, for which solution is straightforward. Simulation examples of this type are the steady-state tubular reactor models TUBE and TUBED, TUBTANK, ANHYD, BENZHYD and NITRO. 

Model interpretation takes a different bent when minimum values for the respective diffusion coefficients are incorporated in the steady-state model, i.e., 1013 cm2/s for the stratum corneum and 10 9 cm2/s for the follicular shunt route. Inserting these values, everything else held constant, suggests there should be a substantial upgrading of the importance of the transfollicular contribution. Data with steroids seem to indicate, however, that the transepidermal route retains a dominant position in the steady state even in this case. 

A plot of the initial reaction rate, v, as a function of the substrate concentration [S], shows a hyperbolic relationship (Figure 4). As the [S] becomes very large and the enzyme is saturated with the substrate, the reaction rate will not increase indefinitely but, for a fixed amount of [E], it reaches a plateau at a limiting value named the maximal velocity (vmax). This behavior can be explained using the equilibrium model of Michaelis-Menten (1913) or the steady-state model of Briggs and Haldane (1926). The first one is based on the assumption that the rate of breakdown of the ES complex to yield the product is much slower that the dissociation of ES. This means that k2 tj. 

When operating conditions were changed, transient phenomena were sometimes observed that first move in one direction and in the reverse direction on going to the final steady state. To study these transients and to design an improved control strategy for the unit, a dynamic response model was needed. With the inclusion of the fast coke in the model, it became possible to extend the steady-state model to obtain useful dynamic response results by the addition of time-dependent accumulation terms (Weekman et al., 1967). 

Primary outputs are produced essentially by sedimentation and (to a much lower extent) by emissions in the atmosphere. The steady state models proposed for seawater are essentially of two types box models and tube models. In box models, oceans are visualized as neighboring interconnected boxes. Mass transfer between these boxes depends on the mean residence time in each box. The difference between mean residence times in two neighboring boxes determines the rate of flux of matter from one to the other. The box model is particularly efficient when the time of residence is derived through the chronological properties of first-order decay reactions in radiogenic isotopes. For instance, figure 8.39 shows the box model of Broecker et al. (1961), based on The ratio, normal-... 

Pinczewski and Sideman [63] have introduced additional transient terms in the steady-state model of Ruckenstein. Eqs. (209) to (211) were therefore replaced by the unsteady equations... 

The next problem of the Langmuir-Hinshelwood kinetics, the validity of the rate-determining step approximation, has not been rigourously examined. However, as has been shown (e.g. refs. 57 and 63), the mathematical forms of the rate equations for the Langmuir-Hinshelwood model and for the steady-state models are very similar and sometimes indistinguishable. This makes the meaning of the constants in the denominators of the rate equations somewhat doubtful in the Langmuir—Hinshelwood model, they stand for adsorption equilibrium constants and in the steady-state models, for rate coefficients or products and quotients of several rate coefficients. 

Overall, the steady-state model for physiochemical weathering provided a good description of observed variations in congener distributions for sur-ficial sediments in the Twelve Mile Creek-Lake Hartwell system. In general,... 

Further studies examining time-variable behavior of PCBs in the Twelve Mile Creek-Lake Hartwell system and sensitivity of model calculations to various system parameters are presently being performed. The steady-state modeling results presented in this chapter, however, provide a reasonable base for an initial assessment of the fate of PCBs in the Twelve Mile Creek-Lake Hartwell system. The cumulative removals of PCBs from the system by volatilization and burial are shown as percents of the total PCB... 

If the feed to the tank increases from 2.3 m3/hour to 3.4 to3/hour and the valve opening remains the same, i.e., the valve coefficient remains the same, calculate and plot the change of height with time. Find the final height using the dynamic model and the steady-state model and make sure that they both give the same result. 

Industrial Verification of the Steady State Model and Static Bifurcation of Industrial Units... 

In this section we develop a dynamic model from the same basis and assumptions as the steady-state model developed earlier. The model will include the necessarily unsteady-state dynamic terms, giving a set of initial value differential equations that describe the dynamic behavior of the system. Both the heat and coke capacitances are taken into consideration, while the vapor phase capacitances in both the dense and bubble phase are assumed negligible and therefore the corresponding mass-balance equations are assumed to be at pseudosteady state. This last assumption will be relaxed in the next subsection where the chemisorption capacities of gas oil and gasoline on the surface of the catalyst will be accounted for, albeit in a simple manner. In addition, the heat and mass capacities of the bubble phases are assumed to be negligible and thus the bubble phases of both the reactor and regenerator are assumed to be in a pseudosteady state. Based on these assumptions, the dynamics of the system are controlled by the thermal and coke dynamics in the dense phases of the reactor and of the regenerator. 

The above dynamic model equations are defined in terms of the same state variables as the steady-state model was. The parameters used are also the same as those of the steady-state model except for the additional four dynamic parameters chr, chg, mr, and cmg-... 

The soil and air concentrations of p,p -DDT were used in a soil-air exchange model developed by Harner et al. [103] to estimate p,p -DDT emissions to air from soils under steady-state conditions. The volatilization flux of p,p -DDT from soil to air was larger than fluxes in runoff over the same time period. For example 15.7 kg of DDT was volatilized during the period March-June, if it was assumed that an area of 300 km2 of land was formerly treated with DDT the steady-state model results were used and calculated monthly, with adjustment for mean monthly temperature. Re-deposition in rainfall was about 15 g ofp,p -DDT to the watershed, based on concentrations in precipitation measured at the field station during May-August 2000. The extent of re-deposition of gas-phase DDT compounds on plants and soils was not known. 

Equations (5.73) and (5.74) are based on Equation (5.68). In the middle of the cell, the temperature is the cell temperature (Tc) calculated from the steady state model. The 2-D temperature distribution thus obtained is as shown in Figure 5.30. Though only three temperatures are calculated along x -direction for each layer the interpolation done by the plotting software gave detailed contours that show the distinction... 

Steady-state modeling is not sufficient if one faces various disturbances in RA operations (e.g., feed variation) or tries to optimize the startup and shutdown phases of the process. In this case, a knowledge of dynamic process behavior is necessary. Further areas where the dynamic information is crucial are the process control as well as safety issues and training. Dynamic modeling can also be considered as the next step toward the deep process analysis that follows the steady-state modeling and is based on its results. 

Equations (36) and (37) represent the steady-state model. After achieving... 

The reaction rate is, of course, the same as that used in the steady-state model... 

For noninteracting control loops with zero dead time, the integral setting (minutes per repeat) is about 50% and the derivative, about 18% of the period of oscillation (P). As dead time rises, these percentages drop. If the dead time reaches 50% of the time constant, I = 40%, D = 16%, and if dead time equals the time constant, I = 33% and D = 13%. When tuning the feedforward control loops, one has to separately consider the steady-state portion of the heat transfer process (flow times temperature difference) and its dynamic compensation. The dynamic compensation of the steady-state model by a lead/lag element is necessary, because the response is not instantaneous but affected by both the dead time and the time constant of the process. 

The results of the steady-state model for the reactor under the same operating conditions are displayed as the solid lines in Figure 2. The predicted catalyst and gas temperatures are shown at each of the axial collocation points. As discussed earlier, a priori values of kinetic parameters were used ( 1, 2) similarly, heat and mass transfer parameters (which are listed in Table II) were taken from standard correlations (15, 16, 17) or from experimental temperature measurements in the reactor under non-reactive conditions. The agreement with experimental data is encouraging, considering the uncertainty which exists in the catalyst activity and in the heat transfer parameters for beds with such large particles. 

Gunn and coworkers (1,2) were the first to propose a steady-state model, and their predictions agreed very well with the Hanna UCG test results. In an analysis of the different versions of permeation models that have appeared in the literature, Haynes (3) judged the steady-state model superior for most applications since reaction kinetics are taken into account and only a modest computational effort is required. Despite these desirable features, applications of the steady-state model have not been as widespread as one might anticipate. 

One reason for the apparent reluctance to utilize the steady-state model may be the numerical problems that must be circumvented in order to obtain a solution to the system equations. These numerical difficulties are discussed for the first time in this report. Also, the present formulation differs from the original... 

Orthogonal collocation on two finite elements is used in the radial direction, as in the steady-state model (1), with Jacobi and shifted Legendre polynomials as the approximating functions on the inner and outer elements, respectively. Exponential collocation is used in the infinite time domain (4, 5). The approximating functions in time have the form... 

---