Reactor Damkohler number

For a specific comparison of the two different reactor types, channels of 300 pm diameter were considered. The equivalent pellet size for that case is 675 pm. As a characteristic quantity, the conversion at the reactor exits was computed for different flow velocities and a range of Damkohler numbers spanning three orders of magnitude. The results for the two different reactor types obtained in such a way were practically indistinguishable. This suggests that the different reactors considered in this study are equivalent as far as chemical conversion is concerned. 

In order to implement the PDF equations into a LES context, a filtered version of the PDF equation is required, usually denoted as filtered density function (FDF). Although the LES filtering operation implies that SGS modeling has to be taken into account in order to capture micromixing effects, the reaction term remains closed in the FDF formulation. Van Vliet et al. (2001) showed that the sensitivity to the Damkohler number of the yield of competitive parallel reactions in isotropic homogeneous turbulence is qualitatively well predicted by FDF/LES. They applied the method for calculating the selectivity for a set of competing reactions in a tubular reactor at Re = 4,000. 

Data on residual reactant concentration as a function of Damkohler number for the PER and laminar flow reactors. 

Table 7 presents Hilder s data for the residual concentration in a PFR and in a laminar flow reactor, as predicted by both eqns. (56) and (57) over a wide range of Damkohler numbers. 

By a reactor model, we mean a system of equations (algebraic, ordinary, or partial differential, functional or integral) which purports to represent a chemical reactor in whole or in part. (The adequacy of such a representation is not at issue here.) It will be called linear if all its equations are linear and simple if its input and output can be characterized by single, concentration-like variables, Uo and u. The relation of input and output will also depend on a set of parameters (such as Damkohler number. Thiele modulus, etc.) which may be denoted by p. Let A(p) be the value of u when w0 = 1. Then, if the input is a continuous mixture with distribution g(x) over an index variable x on which some or all of the parameters may depend, the output is distributed as y(x) = g(x)A(p(jc)) and the lumped output is... 

The Damkohler numbers are useful measures of the characteristic transport time relative to the reaction time. If the surface Damkohler number (sometimes referred to as the CVD number see reference 7) is large, mass transfer to the surface controls the growth. For small Damkohler numbers, surface kinetics governs the deposition. Similarly, if the gas-phase Damkohler number is large, the reactor residence time is an important factor, whereas if it is small, gas-phase reactions control the deposition. 

If the reaction is first-order, what is the value of the Damkohler number Da in order to achieve a conversion of 0.75 at the exit of the reactor ... 

We can validate our formulas and code against the earlier computed conversion rate of 72.64% at the end of the tubular reactor for the Damkohler number Da = 1.4 and Pe = 15.0 by running conversionDa(15,0.7264) with the inputs Pe = 15 and the conversion rate percent = 0.7264 (= xa) in MATLAB. This call computes the Damkohler number Da = 1.4007 correctly to within 0.05%. 

Calculations were carried out for different values of the Damkohler Number (i.e., for various reactor lengths) and various concentrations of the activator [A], which determines the number of active centers and therefore the final degree of polymerization. The ratio Da/Da was used, as a... 

The ratio of reactor and reaction timescale is the (dimensionless, of course) Damkohler number of the first kind Da] = k-t (for a first-order reaction). In a... 

Other factors limiting the overall rate can be external or internal mass transfer, or axial dispersion in a fixed-bed reactor. Pertinent dimensionless numbers are the Biot number Bi, the Damkohler number of the second kind Dan, or the Bodenstein number Bo (Eqs. (5.46)—(5.48)]. 

Effect of Compositional Nonuniformities on the Unifying Ability of Characteristic Time Ratios to Analyze the Dynamic State of Reactions Figure 11.10, plotting the dimensionless initial reactant concentration as a function of the Damkohler number, Da = ties/tr for both batch and continuous reactors. This analysis assumes a well-mixed reacting system, (a) What will the effects of poor mixing be and how will they influence this analysis (b) What is the maximum allowable striation thickness between the reacting species for the system to be considered well mixed ... 

Analysis and Experimental Investigation of Catalytic Membrane Reactors In this equation, Da is the Damkohler number [14]... 

For all three reactor configurations the conversion could be expressed as a function of the Damkohler number (defined relative to the forward reaction) ... 

Fig. 12.12. Calculated dependence of cyclohexane conversion (Eq. (37)) as a function of the Damkohler number (Eq. (41)) for a) the conventional fixed-bed reactor b) the diluted fixed-bed reactor and c) the membrane reactor with an optimized thickness ofVycor glass membrane (the dashed line corresponds to a hypothetically higher membrane selectivity, Sm). The range in which the membrane reactor experiments were performed is also indicated. Parameters T = 473 K, x r H =...

Note that the overall mass balance requires the equality of the feed and product flow rates, F0 = F4. Consequently, the Damkohler number accounts for the production rate (F ), reactor design (V) and reaction kinetics (k). 

The first inequality characterizes recycle systems with reactant inventory control based on self-regulation. It occurs because the separation section does not allow the reactant to leave the process. Consequently, for given reactant feed flow rate F0, large reactor volume V or fast kinetics k are necessary to consume the whole amount of reactant fed into the process, thus avoiding reactant accumulation. The above variables are grouped in the Damkohler number, which must exceed a critical value. Note that the factor z3 accounts for the degradation of the reactor s performance due to impure reactant recycle, while the factor (zo — z4) accounts for the reactant leaving the plant with the product stream. 

The dimensionless variables and parameters are axial coordinate 0 < < 1, conversion temperature 0(1 ), recycle flow rale f and reactor-inlet concentration z3 plant Damkohler number Da, activation energy y, adiabatic temperature rise B, heat-transfer capacity p, coolant temperature 0 concentration of recycle and product streams z3, z4. For convenience, X = x(l) will stand for conversion at reactor outlet. 

Little is known about the kinetics of the bioprocesses. A reasonable assumption is that the reaction rates are proportional to the amount of micro-organisms catalyzing the reactions. The influence of the other reactants is more complex, for example a nutrient in high concentration often has an inhibiting effect. Moreover, factors such as pH, salt concentrations, temperature, can have effects that are difficult to quantify. For this reason, we assume first-order kinetics and include all the other factors influencing the process rate in two Damkohler numbers. The following dimensionless reactor model is obtained ... 

Sizing of the absorption column started from a base case that assumed complete recovery of FeEDTA2- in the bioreactor. Then, sensitivity studies provided the values of the G/L interfacial area and of the absorber volume giving maximum performance. The values of the two Damkohler numbers characterizing reactor performance were found after relaxing the assumption of complete FeEDTA2-recovery. Finally, the specification of NO concentration in the purified gases was checked, for different feed conditions. 

When the process pN- n-> pv in the mobile phase or stationary phase can be represented by first-order or pseudo-first-order interconversion kinetics and as a reversible binding event, the resolution of the interconverting species can be evaluated319 by treating the column as a chemical reactor with properties specified by the corresponding Damkohler number Da and the corresponding interconversion rate constants derived. Thus,... 

Da is the Damkohler number, Pe the Peclet number based on the length of the reactor and is given by... 

In the above equations, cjtS is the surface/wall concentration of species j, rw (intrinsic rate of surface reaction, Das is the reactor scale Damkohler number, which are given by... 

Comparing this with the slow reaction case, we note that the effective velocity has increased (by a factor 1.83), the dispersion coefficient is reduced by a factor 3 while the apparent reactor scale Damkohler number changed from Das to 8jp. 

Fig. 18. Variation of conversion (X) with the Damkohler number, Daa, for a bimolecular second-order wall-catalyzed reaction occurring in a tubular reactor.

The accuracy of low-dimensional models derived using the L S method has been tested for isothermal tubular reactors for specific kinetics by comparing the solution of the full CDR equation [Eq. (117)] with that of the averaged models (Chakraborty and Balakotaiah, 2002a). For example, for the case of a single second order reaction, the two-mode model predicts the exit conversion to three decimal accuracy when for (j>2(— pDa) 1, and the maximum error is below 6% for 4>2 20, where 2(= pDd) is the local Damkohler number of the reaction. Such accuracy tests have also been performed for competitive-consecutive reaction schemes and the truncated two-mode models have been found to be very accurate within their region of convergence (discussed below). 

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