Steady-state performance

Equations (4.1) or (4.2) are a set of N simultaneous equations in iV+1 unknowns, the unknowns being the N outlet concentrations aout,bout, , and the one volumetric flow rate Qout- Note that Qom is evaluated at the conditions within the reactor. If the mass density of the fluid is constant, as is approximately true for liquid systems, then Qout=Qm- This allows Equations (4.1) to be solved for the outlet compositions. If Qout is unknown, then the component balances must be supplemented by an equation of state for the system. Perhaps surprisingly, the algebraic equations governing the steady-state performance of a CSTR are usually more difficult to solve than the sets of simultaneous, first-order ODEs encountered in Chapters 2 and 3. We start with an example that is easy but important. 

The method of false transients converts a steady-state problem into a time-dependent problem. Equations (4.1) govern the steady-state performance of a CSTR. How does a reactor reach the steady state There must be a startup transient that eventually evolves into the steady state, and a simulation of... 

There is a general trend toward structured packings and monoliths, particularly in demanding applications such as automotive catalytic converters. In principle, the steady-state performance of such reactors can be modeled using Equations (9.1) and (9.3). However, the parameter estimates in Figures 9.1 and 9.2 and Equations (9.6)-(9.7) were developed for random packings, and even the boundary condition of Equation (9.4) may be inappropriate for monoliths or structured packings. Also, at least for automotive catalytic converters. 

Solution The reactions are the same as in Example 12.5. The steady-state performance of a CSTR is governed by algebraic equations, but time derivatives can be useful for finding the steady-state solution by the method of false transients. The governing equations are... 

Example 12.8 The batch reactor in Example 12.7 has been converted to a CSTR. Determine its steady-state performance at a mean residence time of 4 h. Ignore product inhibition. 

Unlike stirred tanks, piston flow reactors are distributed systems with one-dimensional gradients in composition and physical properties. Steady-state performance is governed by ordinary differential equations, and dynamic performance is governed by partial differential equations, albeit simple, first-order PDEs. Figure 14.6 illustrates a component balance for a differential volume element. 

However, the simulation of the steady-state performance for a heat exchanger with a known heat transfer surface area will demand an iterative split boundary solution approach, based on a guessed value of the temperature of one of the exit streams, as a starting point for the integration. 

In this paper, we summarize results from a small scale methane direct oxidation reactor for residence times between lO and lO seconds. For this work, methane oxidation (using air or oxygen) was studied over Pt-10% Rh gauze catalysts and Pt- and Rh-coated foam and extruded monoliths at atmospheric pressure, and the reactor was operated autothermally rather than at thermostatically controlled catalyst temperatures. By comparing the steady-state performance of these different catalysts at such short contact times, tiie direct oxidation of methane to synthesis gas can be examined independent of the slower reforming reactions. 

The fluid bed operation allows the continuous removal of a portion of deactivated catalyst and continuous replacement of catalyst during operation. This results in a steady state performance. In addition, a fluid bed reactor is nearly isothermal by design, which minimizes catalyst deactivation due to exposure to excessive heat. 

The drug product s steady-state performance is equivalent to a currently marketed, noncontrolled release or controlled-release drug product that contains the same active drug ingredient or therapeutic moiety and that is subject to an approved full NDA. 

Arkun, Stephanopoulos and Morari (1978) have added a new twist to control system synthesis. They developed the theory and then demonstrated on two example problems how to move from one control point to another for a chemical process. They note that the desirable control point is likely at the intersection of a number of inequality constraints, the particular set being those that give optimal steady-state performance for the plant. Due to process upsets or slow changes with time, the point may move at which one wishes to operate. Also, the inequality constraints themselves may shift relative to each other. Arkun, Stephanopoulos and Morari provide the theory to decide when to move, and then develop alternative paths along which to move to the new... 

An important feature of a reactor operating with reversing flows is a gradual decrease of temperature of the packed bed outlet that allows for higher conversion in an adiabatic catalyst bed than for steady-state performance of an exothermic reversible reaction such as SO2 oxidation or ammonia synthesis. Conventional operation can provide only the temperature rise along the adiabatic catalyst bed. 

Table 6 shows an example of the cyclic steady state performance of the SERP concept using an admixture of a SMR catalyst (noble metal on alumina) and a CO2 chemisorbent (K2CO3 promoted hydrotalcite) in the reactor [20]. The reactor temperature was 490°C. The feed H20 CH4 ratio was 6 1. The concept can directly produce 95% H2 product (dry basis) with a CH4 to H2 conversion of 73%. The trace impurities in the product gas contained less than 40 ppm COx- The corresponding product gas composition (thermodynamic limit) of a SMR reactor operated without the CO2 chemisorbent will be -67.2% H2, 15.7% CH4, 15.9% CO2, and 1-2% CO (dry basis), and the CH4 to H2 conversion will be only 52%. Thus, the SERP concept may be attractive for direct production of a CO free H2 stream for fuel cell applications. 

There are a number of general techniques suggested by the problem formulation. At the most detailed level of design, the design parameters need to be optimized in relation to performance criteria based on a nonlinear dynamic model. This points to a need for effective tools for dynamic optimization. At a more preliminary level in a hierarchy of techniques, it might be useful to evaluate steady-state performance or to carry out tests on achievable dynamic performance to eliminate infeasible options. Appropriate screening techniques are therefore needed. All these methods can use nominal models for initial analysis, but a full analysis should be based on design with uncertainty. 

This last class of methods provides a way of avoiding the repeated optimization of a process model by transforming it into a feedback control problem that directly manipulates the input variables. This is motivated by the fact that practitioners like to use feedback control of selected variables as a way to cormteract plant-model mismatch and plant disturbances, due to its simphcity and reliability compared to on-line optimization. The challenge is to find functions of the measured variables which, when held constant by adjusting the input variables, enforce optimal plant performance [19,21]. Said differently, the goal of the control structure is to achieve a similar steady-state performance as would be realized by an (fictitious) on-line optimizing controller. 

Fig. 6 shows performance predictions obtained with the equilibrium-dispersive model for such single-column recycling with and without ideal solvent removal (TSR). The same requirements were used as in section 3. The process is basically infeasible without ISR. Also shown is the steady state performance of an SMB-based process (6 columns, ISR cf Fig. 3a). As is often found, the SMB process achieves a lower productivity, but at the same time allows for significantly lower solvent consumption.

Table 10.3 provides an example of the cyclic steady-state performance of the SERP concept using a 6 1 H20 + CH4 feed gas at a pressure of 11.4psig and a temperature of 490°C.S1 The process can directly produce an essentially COx-free H2 product (-94.4% H2 + 5.6% CH4), which is suitable for H2 fuel-cell use. The conversion of CH4 to H2 was -73.0%. The table also shows the equilibrium compositions of the H2 product from a conventional plug-flow reforming reactor operating under identical conditions. Both the H2 conversion and product purity were rather poor in the latter case, which demonstrates the advantage of the SERP concept. Theoretical models of the above-described SERP concept and its variations for H2 production by SMR have been developed, and theoretical parametric studies of the process have been conducted by various authors.62,63... 

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