Rate laws batch operations

For a constant-volume batch reactor operated at constant T and pH, an exact solution can be obtained numerically (but not analytically) from the two-step mechanism in Section 10.2.1 for the concentrations of the four species S, E, ES, and P as functions of time t, without the assumptions of fast and slow steps. An approximate analytical solution, in the form of a rate law, can be obtained, applicable to this and other reactor types, by use of the stationary-state hypothesis (SSH). We consider these in turn. 

The next several sections derive analytical solutions for some simple rate laws. Of course, the batch reactor is assumed to be operating at constant temperature in this discussion. 

One irreversible chemical reaction occurs in a constant-volume batch reactor. The reaction is exothermic and a digital controller removes thermal energy at an appropriate rate to maintain constant temperature throughout the course of the reaction. Sketch the time dependence of the rate of thermal energy removal, d 2/t< f)removai vs. time, for isothermal operation when the rate law is described by ... 

Finally a fourth boundary condition shall be valid to support the worst case character of the procedure. The reaction order necessary for the formal kinetic description of a process has a severe influence on the pressure/time and respectively the tempera-ture/time-profiles to be expected. Industrial experience has shown that approximately 90% of all processes conducted in either batch or semibatch reactors can be described with a second order formal kinetic rate law. But it remains uncertain whether this statement, which is related to isothermal or isoperibolic operation with a rather limited overheating, remains valid if the reaction proceeds adiabatically and if side reactions contribute to the gross reaction rate at a much higher degree. In consequence, it shall be assumed for a credible worst case evaluation that the disturbed process follows a first order kinetics. Any reactions occurring in reality will almost certainly proceed at a much lower rate. 

Viewed from the perspective of ethylene oxide, these reactions are competitive by contrast, from the perspective of the amines, they are consecutive. Consider a research scale batch reactor operating at 60°C and 20 bar to maintain all species in the liquid phase. Actual production of these commodity products on a large scale would be conducted in flow reactors, as described in Illustration 9.5. The rate laws are of the mixed second-order form (first-order in each reactant), with hypothetical rate constants ki, k2, and equal to 1,0.4, and 0.1 L-moCV min, respectively. MEA and DEA are both high-volume chemicals, while TEA is less in demand. The distribution of alkanolamine products obtained under the specified conditions can be influenced by controlling the initial mole ratio of EO to A and the time of reaction. 

The ability of Monod s empirical relation to fit kinetic data for biochemical reactions has its foundations in generalizations of two phenomena frequently observed for fermentation processes (1) nature places a cap on the quantity of microorganism that can be achieved during the exponential phase of growth in a bioreactor operating in a batch mode and (2) as the concentration of the limiting substrate approaches zero, the rate laws for biochemical reactions approach pseudo-first-order behavior with respect to that substrate. The cap indicated on the cell growth rate has been associated with the natural limit on the maximum rate at which replication of DNA can be achieved. 

This simple approach was adopted in order to circumvent the complications that are introduced by the fact that the volume of the liquid phase in the reactor varies with time. When the volume of the aqueous growth medium varies during the course of the reaction, an approach based on integration of a proposed rate law is problematic, although numerical integration would be possible. An additional reason for employing the differential approach below is that for rate laws that are other than those of the simple nth-order form (such as a Monod rate expression) a differential method of data analysis is often adequate for preliminary considerations involved in the design of a bioreactor that is intended to operate in a batch mode. 

Consider the operation of a chemostat with a working volume of 2 L. The results of preliminary batch cultivation studies of the biochemical reaction and growth medium of interest indicate that the rate law is of the Monod form with p = 0.1 h . However, the reported value of the half-saturation constant is suspect, although it is thought to be quite small compared to the concentration of substrate in the feed Sq Kg). Use this information to obtain a preliminary estimate of the maximum feed flow rate that can be accommodated by this apparatus when it operates at steady state. Employ two approaches for part A, obtain a crude estimate of the feed rate by assuming that Kg is zero then refine this estimate to obtain an improved value of Kg based on the information in part B. For Part B the effluent concentration of substrate may differ from that of Part A. 

The laws relating optimum operating temperature to medium composition are not well known. This relation was theoretically Investigated for processes whose rates saturate in substrate concentration. The Michaells-Menten reaction mechanism was modified to describe microbial biomass production and metabolite excretion in both batch and continuous reactors. 

For a rapid conversion of lab-scale results into an economically viable reaction-pervaporation system, an optimum value can be determined for each parameter. Based on experimental results as well as a model describing the kinetics of the system, it has been found that the temperature has the strongest influence on the performance of the system as it affects both the kinetics of esterification and of pervaporation. The rate of reaction increases with temperature according to an Arrhenius law, whereas the pervaporation is accelerated by an increased temperature also. Consequently, the water content fluctuates much faster at a higher temperature. The second important parameter is the initial molar ratio. It has to be noted, however, that a deviation in the initial molar ratio from the stoichiometric value requires a rather expensive separation step to recover the unreacted component afterwards. The third factor is the ratio of membrane area to reaction volume, at least in the case of a batch reactor. For continuous operation, the flow rate should be considered as the determining factor for the contact time of the mixture with the membrane and subsequently the permeation... 

This law can be applied to steady-state or unsteady-state (transient) processes and to batch or continuous reactor systems. A steady-state process is one in which there is no change in conditions (e.g., pressure, temperature, composition) or rates of flow with time at any given point in the system. The accumulation term in Equation (7.2) is then zero. (If there is no chemical or nuclear reaction, the generation term is also zero.) All other processes are unsteady-state. In a batch reactor process, a given quantity of reactants is placed in a container, and by chemical and/or physical means, a change is made to occur. At the end of the process, the container (or adjacent containers to which material may have been transferred) holds the product or products. In a continuous process, reactants are continuously removed from one or more points. A continuous process may or may not be steady-state. A coal-fired power plant, for example, operates continuously. However, because of the wide variation in power demand between peak and slack periods, there is an equally wide variation in the rate at which the coal is fired. For this reason, power plant problems may require the use of average data over long periods of time. However, most industrial operations are assumed to be steady-state and continuous. 

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