Distributed parameter reactor

The sum of squares as defined by Equation 7.8 is the general form for the objective function in nonlinear regression. Measurements are made. Models are postulated. Optimization techniques are used to adjust the model parameters so that the sum-of-squares is minimized. There is no requirement that the model represent a simple reactor such as a CSTR or isothermal PER. If necessary, the model could represent a nonisothermal PFR with variable physical properties. It could be one of the distributed parameter models in Chapters 8 or 9. The model... 

Level (3) global e.g., reactor model some key parameters reactor volume, mixing/flow, residence time distribution, temperature profile, reactor type... 

Optimization of a distributed parameter system can be posed in various ways. An example is a packed, tubular reactor with radial diffusion. Assume a single reversible reaction takes place. To set up the problem as a nonlinear programming problem, write the appropriate balances (constraints) including initial and boundary conditions using the following notation ... 

Flow patterns in a stirred tank (lumped parameter system) and a tubular reactor (distributed parameter system). 

The effect of HDM reaction selectivity variations on the metal distribution parameter at the reactor entrance for Ni-T3MPP follows. 

In addition to catalyst pore structure, catalytic metals content can also influence the distribution of deposited metals. Vanadium radial profile comparisons of aged catalysts demonstrated that a high concentration of Co + Mo increases the reaction rate relative to diffusion, lowering the effectiveness factor and the distribution parameter (Pazos et al., 1983). While minimizing the content of Co and Mo on the catalyst is effective for increasing the effectiveness factor for HDM, it may also reduce the reaction rate for the HDS reactions. Lower space velocity or larger reactors would then be needed to attain the same desulfurization severity. 

Most commonly, distributed parameter models are applied to describe the performance of diesel particulate traps, which are a part of the diesel engine exhaust system. Those models are one- or two-dimensional, non-isothermal plug-flow reactor models with constant convection terms, but without diffusion/dispersion terms. 

For analysis of distributed-parameter systems, such as a tubular fixed bed reactor, numerical simulation of periodic operation at various values of control parameters is typically applied. Asymptotic models for quasisteady and relaxed steady states are valuable instruments for a substantial simplification of the original distributed-parameter system. A method allowing for... 

The study of the Vanadium deposition profiles in spent catalyst particles from resid hydro-processing confirms that HDM is a sequential reaction. Furthermore, it is shown that the distribution parameter for the deposited Vanadium Qv is constant through the reactor for each catalyst type and that Qv is proportional to the efficiency of the Vanadium removal reaction. 

Vanadium molecular size distributions in residual oils are measured by size exclusion chromatography with an inductively coupled plasma detector (SEC-ICP). These distributions are then used as input for a reactor model which incorporates reaction and diffusion in cylindrical particles to calculate catalyst activity, product vanadium size distributions, and catalyst deactivation. Both catalytic and non-catalytic reactions are needed to explain the product size distribution of the vanadium-containing molecules. Metal distribution parameters calculated from the model compare well with experimental values determined by electron microprobe analysis, Modelling with feed molecular size distributions instead of an average molecular size results in predictions of shorter catalyst life at high conversion and longer catalyst life at low conversions. 

P. Dufoijr P, F. Couertne, Y. Toure 2003 Model predictive control of a catalytic reverse flow reactor. Control of industrial spatially distributed parameter processes. Special issue of IEEE Transactions on Control System Technology, 11(5) 705-714. 

K), nj is the effectiveness factor for reaction j (to be computed from the diffusion reaction equation at each point in the reactor), (—AHj) is the heat of reaction for reaction j (KJ/Kg mole), Rt is the catalyst tube radius, m, U is the overall heat transfer coefficient in (KJ/mP.K) and To> is the wall temperature (K) which is determined through the coupling between the above model equations for the catalyst tube and the model equations for the combustion chamber (the furnace). A distributed parameter model for the combustion chamber is being developed by (CREG) for both top fired and side fired furnaces. 

This example found the reactor throughput that would give the required annual production of product B. For prescribed values of the design variables T and V there may not be a solution. If there is a solution, it is unique. The program uses a binary search to And the answer, but another root finder could be used instead. For the same accuracy, Newton s method (see Appendix 4.1) requires about 3 times fewer function evaluations (i.e., calls to the Reactor subroutine). The saving in computation time is trivial in the current example but could be important if the reactor model is a complicated, distributed parameter model such as those in Chapters 8 and 9. 

Optimization requires that at/R have some reasonably high value so that the wall temperature has a significant influence on reactor performance. There is no requirement that 33 jR be large. Thus the method can be used for polymer systems that have thermal diffusivities typical of organic liquids but low molecular diffusivi-ties. The calculations needed to optimize distributed-parameter systems (i.e., sets of PDEs) are much longer than needed to optimize the lumped-parameter systems (i.e., sets of ODEs) studied in Chapter 6, but the numerical approach is the same and is still feasible using small computers. 

Analogous to the experimental approaches discussed in the previous section, mathematical models have been developed to describe mass transfer at all three levels—cellular, multi-cellular (spheroid), and tissue levels. For each level two approaches have been used—the lumped parameter and distributed parameter models. In the former approach, the region of interest is considered to be a perfectly mixed reactor or compartment. As a result, the concentration of each region has no spatial dependence. In the latter approach, a more detailed analysis of the mass transfer process leads to information on the spatial and/or temporal changes in concentrations. Models for single cells and spheroids were reviewed in Section III,A and are part of the tissue-level models (Jain, 1984) hence, we will focus here only on tissue-level models. 

By and large we can describe the results of the analysis of distributed parameter systems (i.e., flow reactors other than CSTRs) in terms of the gradients or profiles of concentration and temperature they generate. To a large extent, the analysis we shall pursue for the rest of this chapter is based on the one-dimensional axial dispersion model as used to describe both concentration and temperature fields within the nonideal reactor. The mass and energy conservation equations are coupled to each other through their mutual concern about the rate of reaction and, in fact, we can use this to simplify the mathematical formulation somewhat. Consider the adiabatic axial dispersion model in the steady state. 

In case of packed columns, a qualitatively different behavior can be found for finite and infinite intra-partide mass transfer resistance. For vanishing mass transfer resistance inside the catalyst a small number of solutions, typically three, can be observed. Note, that this is consistent with the TAME case discussed above. Instead, for finite transport inside the catalyst a very large number of solutions can be observed. An example is shown in Fig. 10.17, right. It was conjectured by Mohl et al. [74], that this behavior is caused by isothermal multiplidty of the single catalyst pellet and is therefore similar to the well-known fixed-bed reactor [38, 77]. However, further research is required to verify this hypothesis. Further, it was shown by Mohl et al. [74] that in both cases the number of solutions may crucially depend on the discretization of the underlying continuously distributed parameter system. A detailed discussion is given by Mohl et al. [74]. 

The state of mixing in the reactor—uniform distribution ( lumped parameters ) or uneven distribution ( distributed parameters ). Associated with this is the question of whether the reactor is maximally mixed or totally segregated, that is, whether the reactor is to be considered an ideal stirred vessel or an ideal tube reactor. 

The analysis of a hybrid bioartificial membrane pancreas (HBMP) in which porcine islets of Langerhans were segregated in the shell side of a hollow fibre module, showed the importance of convective fluxes in determining reactor performance. A distributed parameter model, taking into account... 

Distributed systems These are systems in which the state variables are varying in one or more direction of the space coordinates. The simplest chemical engineering example is the plug flow reactor. These systems are described at steady state by either an ordinary differential equation [when the variation of the state variable is only in one direction of the space coordinates, (i.e., one-dimensional models) and the independent variable is the space direction] or partial differential equations [when the variation of the state variables is in more than one direction of the space coordinates (i.e., two-dimensional models or higher), and the independent variables are these space directions]. The ordinary differential equations of the steady state of the one-dimensional distributed parameter models can be either initial-value differential equations (e.g., plug flow models) or two-point boundary-value differential equations (e.g., models with super-... 

Perhaps the first published analysis of an RFBR was by Raskin, et al. ( ), who developed a quasicontinuum distributed parameter model for a radial ammonia synthesis reactor. General conclusions were limited, however, since the model was specifically concerned with ammonia synthesis and later carbon monoxide conversion ( ), where both processes are second order and reversible. However, these authors did note that "radial reactors are anisotropic", i.e., they observed higher ammonia yields for centripetal radial flow (CPRF " periphery to the center) than for centrifugal radial flow (CFRF — center to periphery) ( ). ... 

A few distributions of VCM suspensions in water viewed by light microscopy into specially designed pressure cells appear in the literature (23,24), but no analyses of droplet size distribution under different conditions of reactor agitation or polymeric additive addition have been reported. A technique for fixing VCM emulsions by osmium tetroxide (25) may prove useful to study the VCM/water system in greater detail. Mersmann and Grossmann (26) have studied the dispersion of liquids in non-miscible two-phase systems, which include chlorinated liquids such as carbon tetrachloride in water. The influence of stirrer type and speed on the development of an equilibrium droplet size distribution is discussed. Different empirical relationships to calculate the Sauter mean diameter of droplet distributions from reactor operating parameters are also reviewed. 

In some eases, the independent and dependent process variables can vary along a spatial coordinate. A well-known example is a tnbnlar reactor, where temperatures, concentrations and other process variables vary with the axial coordinate of the reactor. In the absence of micro-mixing, the system variables eonld even vary with the radial coordinate of the reactor. Spatial variations of process variables lead to eomplex models and consequently the solution of the model is difficult. Models that aeeonnt for spatial variations are called distributed parameter models. A widely used method that approximates the behavior of distributed parameter systems is the division of the spatial coordinate into small sections, within each section the system properties are assumed to be constant Each section can then be considered as an ideally mixed section and the entire process is approximated by a series of ideally mixed sub-systems. Such a system approximation is cdled a lumped parameter system. In the case of a chain process, such as a distillation tray colmnn, usually every tray is considered to be lumped, i.e. process conditions on each tray are averaged and assumed constant. 

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