Optimal Wall Temperatures

The method of lines formulation for solving Equation (8.52) does not require that T aii be constant, but allows T aiiiz) to be an arbitrary function of axial position. A new value of T aii may be used at each step in the calculations, just as a new may be assigned at each step (subject to the stability criterion). The design engineer is thus free to pick a T au z) that optimizes reactor performance. 

Optimization requires that a-rtjl have some reasonably high value so that the wall temperature has a significant influence on reactor performance. There is no requirement that 3 AtlR be large. Thus, the method can be used for polymer systems that have thermal diffusivities typical of organic liquids but low molecular diffusivities. The calculations needed to solve the optimization are much longer than those needed to solve the ODEs of Chapter 6, but they are still feasible on small computers. 

Optimization requires that ct/t/R2 have some reasonably high value so that the wall temperature has a significant influence on reactor performance. There is no requirement that be large. Thus, the method can be used for poly- 

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For the case of continuous wall temperature profiles, a function fitting approach was used. Although there are no end conditions on Tw(z) to be satisfied, for all the cases considered in this paper Tw(z) is expected to be high near the entrance to the reactor and lower near the exit. Thus, with a decreasing optimal wall temperature sequence in mind, a Laguerre expansion (with exponential weighting) of the wall temperature was chosen. The expansion can be represented as... 

Figure 8. Example of fourth order Laguerre approximation of the continuous optimal wall temperature.

Initial results on the two zone wall temperature optimization of bulk polymerizers in tubular reactors shows that the molecular weight distribution and product quality can be controlled for conversion levels aroung 20%. Further investigations into the use of optimal wall temperatures in tubular polymerizers are underway. 

H.-S. Huang, L.T. Fan, and C.L. Hwang. Optimal wall temperature control of a heat exchanger. Technical Report 15, Institute for Systems Design and Optimization, Kansas State University, Manhattan, 1969. 

This simple criterion based on the least irreversibility is technically interesting for choosing the most adequate fluid in the optimal design of a certain spray-cooling application. It should be stressed that the optimal wall temperature corresponds to the working temperature of the heat-dissipating surface, which the spray-cooling system is required to maintain at a constant value. 

The theoretical model may be used to optimize the cure cycle for a given formulation. Wall temperature and the required shell thickness are the most important parameters that determine the process time. 

Constrained optimization in a two-zone reactor where the wall temperature is prescribed as two constant temperature regions, the zone lengths being equal. 

Constrained optimization with continuously varying wall temperatures. 

Figure 3. Effect of desired conversion on the optimal first zone wall temperature (normalized by T n) ...

Figure 5. Variation of the optimal first and second zone wall temperatures along lines of constant desired conversion.

Figure 10. Comparison of exiting polymer weight fraction profiles as the optimal two-zone wall temperatures are approached.

Simple methods have been presented to study constrained and free optimization in two-zone reactor systems and in systems with a continuous wall temperature profile. 

Both FFF and SEC require careful control of the temperature for universal calibration. For SEC and Fl-FFF, this means controlling the temperature of the room or of the channel/column. For Th-FFF, it is important to maintain the specified cold-wall temperature, Tc. Fortunately, the temperature at the center of gravity of a component is independent of the field strength in Th-FFF, so that universal calibration constants do not change when AT is tuned to optimize the analysis of a particular range in M, provided Tc is held constant. 

The effect of the wall temperature discontinuity may also be mitigated by an alternative implementation of the wall boundary condition to that described in the previous section. The two equations at each interior boundary node (l,tj), j = 2,3,...,m, are dropped, together with one equation at each of (1,-1) and (1,1). These 2m equations are replaced by application of the boundary conditions at two points within each subinterval on the line y 1. The Gaussian points are not necessarily the optimal choice, but were used in the absence of any other guideline. 

The preliminary investigation showed that lower pressures result in significantly higher heat fluxes. It s possible to approaeh 50 W/em while maintaining the wall temperature below 85 °C with little optimization. A more eomprehensive parameterization study will be addressed such as pore size, geometry, and other effeets as the limits of graphite foam evaporator performance [34-35]. 

The rebum test runs have been performed under constant conditions in the combustion reactor. Besides the wall temperature (1300 °C) the hiel, the fUel feeding rate, and the probe positions stayed unchanged. The hard coal G ttelbom was burnt in the main combustion zone at an air ratio of 1,15. The fuel feeding rate amounts to 1 kg/h. The position of the burnout air probe remained unchanged. It was adjusted to a calculated residence time of 1 n /h pyrolysis gas of 2 s in the reduction zone. Those settings (residence time, wall temperature) have been proved to be optimal for rebuming in earlier projects [7, 8]. Only the burnout air was varied in order to get a constant O2 concentration of 3 % in the flue gas. 

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