Batch reactor dimensionless

As in the case of batch reactors, dimensionless energy balance Eq. 5.2.53 is not conveniently used because the heat capacity of the reacting fluid, (FjCpj), is a function of the temperature and reaction extents and, consequently, varies along the reactor. To simplify the equation and obtain dimensionless quantities for heat transfer, we define the heat capacity of the reference stream and relate the heat capacity at any point in the reactor to it by... 

BATCHD - Dimensionless Kinetics in a Batch Reactor System... 

Dynamics of an Equalisation Basin 560 Dimensionless Kinetics in a Batch Reactor 235 Batch Reactor with Complex Reaction Sequence 240 Single Solute Batch Extraction 442 Mixing and Segregation 394... 

The introductory example may be reworked using the Gamma distribution, since the special case given there is n = 1. Let c(x, 0) = Cogn( ) where C0 is the total initial concentration. Let the first order rate constant be k(x) = kx and make time dimensionless as kt. This reaction time or intensity of reaction—severity of reaction as the oil people have it—is really the Dam-kohler number, Da, for the reactor, with t the time of reaction if it is a batch reactor or the residence time if a PFTR. Thus... 

In a more recent study [15], Westerterp and Molga introduced a set of dimensionless numbers (cooling, reactivity, and exothermicity numbers) characterizing the stability of heterogeneous slow liquid-liquid reactions in the semi-batch reactor. They demonstrated that the key parameter is the cooling number Co ... 

Thus, the equations describing the thermal stability of batch reactors are written, and the relevant dimensionless groups are singled out. These equations have been used in different forms to discuss different stability criteria proposed in the literature for adiabatic and isoperibolic reactors. The Semenov criterion is valid for zero-order kinetics, i.e., under the simplifying assumption that the explosion occurs with a negligible consumption of reactants. Other classical approaches remove this simplifying assumption and are based on some geometric features of the temperature-time or temperature-concentration curves, such as the existence of points of inflection and/or of maximum, or on the parametric sensitivity of these curves. 

For a safe operation, the runaway boundaries of the phenol-formaldehyde reaction must be determined. This is done here with reference to an isoperibolic batch reactor (while the temperature-controlled case is addressed in Sect. 5.8). As shown in Sect. 2.4, the complex kinetics of this system is described by 89 reactions involving 13 different chemical species. The model of the system consists of the already introduced mass (2.27) and energy (2.30) balances in the reactor. Given the system complexity, dimensionless variables are not introduced. 

The effect of heat transfer area is illustrated in Figure 4.3. Three different areas are used. The temperature controller is proportional with a gain of 0.1 (dimensionless using a 50-K temperature transmitter span and split-range hows shown in Figure 2.1). The set-point is ramped to 340 K in 60 min. Clearly in the numerical example, a jacket-cooled batch reactor of the size selected (2 m diameter) and with the given heat of reaction would produce runaway reactions. An external heat exchanger with 4 times the jacket area would be required to catch the reaction. 

By application of this transformation under conditions of constant t, the dimensionless solution to the characteristic population balance for a batch reactor can be found to be... 

We now move to the general case of a continuous description the pragmatical usefulness of such a type of description in real-life kinetic problems has been discussed by Krambeck (1991a,b). Let c(x,0) be the initial distribution of reactant concentrations in the batch reactor, and let c(x,t) be the concentration distribution at any subsequent (dimensional) time t. We assume that both the label x and the concentration c have already been normalized so that = = 1. Furthermore, we assume that a (dimensional) frequency factor k(x) can be identified, and that x has been normalized so that k(x) = k x, where k is the average value of k(x) at r = 0. One then normalizes the time scale as well by defining the dimensionless time t as k t. The overall concentration C(t) is defined with a weighting function that is identical to unity, C(t) = , 0(0) = 1. 

Suppose one has performed experiments with the mixture under consideration in a batch reactor, and one has obtained experimentally the overall kinetics—the R ) function such that dCldt = —R(C). For instance, one could obtain R C) = if the intrinsic kinetics are in fact first order and the initial concentration distribution is (l,x) = exp(—x). If one were to regard R(C) as a true (rather than an apparent) kinetic law, one would eonclude that in a CSTR with dimensionless residence time T the exit overall concentration is delivered by the (positive) solution of TC +C = 1. The correct value is in fact C = , and the difference is not a minor one. (To see that easily, consider the long time asymp-... 

To characterize the generic behavior of chemical reactors, it is preferred to describe their operations in terms of intensive dimensionless quantities. To convert the reaction extents to intensive quantities, dimensionless extents are defined. For batch reactors, the dimensionless extent, Z , of the mth independent reaction is defined by... 

To reduce the design equation of an ideal batch reactor, Eq. 4.3.8, to dimensionless form, we first select a reference state of the reactor (usually, the initial state) and use the dimensionless extent, Z , of the mth-independent reaction, defined by Eq. 2.7.1 ... 

Equation 4.4.4 is die dimensionless, reaction-based design equation of an ideal batch reactor, written for die mth-independent reaction. The factor ( / Co) is a scaling factor that converts die design equation to dimensionless form. Its physical significance is discussed below (Eqs. 4.4.13-4.4.15). 

Adams et al. (/. Catalysis 3, 379, 1964) investigated these reactions and expressed the rate of each as second order (first order with respect to each reactant). Formulate the dimensionless, reaction-based design equations for an ideal batch reactor, plug-flow reactor, and a CSTR. 

Formulate the dimensionless, reaction-based design equations for ideal batch reactor, plug-flow reactor, and CSTR using the heuristic rule. ... 

Equation 5.2.18 is the dimensionless, differential energy balance equation of ideal batch reactors, relating the reactor dimensionless temperature, 0(t), to the dimensionless extents of the independent reactions, Z (t), at dimensionless operating time T. Note that individual dZ /dfr s are expressed by the reaction-based design equations derived in Chapter 4. 

Equation 5.2.20 is a dimensionless differential energy balance equation of batch reactors and can be simplified further by defining two dimensionless groups ... 

The design equations and the species concentration relations contain another dependent variable, 6, the dimensionless temperature, whose variation during the reactor operation is expressed by the energy balance equation. For ideal batch reactors with negligible mechanical shaft work, the energy balance equation, derived in Section 5.2, is... 

The main difficulty in determining the reaction rate r is that the extent is not a measurable quantity. Therefore, we have to derive a relationship between the reaction rate and the appropriate measurable quantity. We do so by using the design equation and stoichiometric relations. Also, since the characteristic reaction time is not known a priori, we write the design equation in terms of operating time rather than dimensionless time. Assume that we measure the concentration of species j, Cj(t), as a function of time in an isothermal, constant-volume batch reactor. To derive a relation between the reaction rate, r, and Cj(t), we divide both sides of Eq. 6.2.4, by obtain... 

Below, we describe tbe design formulation of isothermal batch reactors with multiple reactions for various types of chemical reactions (reversible, series, parallel, etc.). In most cases, we solve the equations numerically by applying a numerical technique such as the Runge-Kutta method, but, in some simple cases, analytical solutions are obtained. Note that, for isothermal operations, we do not have to consider the effect of temperature variation, and we use the energy balance equation to determine tbe dimensionless heat-transfer number, HTN, required to maintain the reactor isothermal. 

To reduce the design equation to dimensionless form, we have to select a reference state and define dimensionless extents and dimensionless time. The reference state should apply to all operations, including those with an initially empty reactor, and should enable us to compare the operation of a semibatch reactor to that of a batch reactor. Therefore, we select the molar content of the reference state, (A tot)o. as the total moles of species added to the reactor. The dimensionless extent is defined by... 

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