Damkohler numbers single

The parameter term (k x) which is called Damkohler Number I, is dimensionless and is now the single governing parameter in the model. This results in a model simplification because originally the three parameters, x, k and Cao. all appeared in the model equation. 

Fig. 10. Fractional conversion versus Damkohler number for half,-, first- and second-order reactions taking place in a single ideal CSTR. Shaded areas represent possible conversion ranges lying between perfectly micromixed flow (M) and completely segregated flow (S). Data taken from reference 32. A = R —...

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... 

C.J. van Duijn, Andro Mikelic, I.S. Pop, and Carole Rosier, Effective Dispersion Equations for Reactive Flows with Dominant Peclet and Damkohler Numbers Mark Z. Lazman and Gregory S. Yablonsky, Overall Reaction Rate Equation of Single-Route Complex Catalytic Reaction in Terms of Hypergeometric Series A.N. Gorban and O. Radulescu, Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited... 

The governing equations - that is, mainly the component and the total mass balances in the anode channels - are provided here in dimensionless form. The five ordinary differential equations (ODE) with respect to the spatial coordinate describe the development of the five unknowns in one single anode channel, namely the mole fractions, with i = CH4, H2O, H2, CO2, as well as the molar flow density inside the anode channel, y. Here, the Damkohler numbers, Da/, are the dimensionless reaction rate constant of the reforming and the oxidation reaction, respectively, the rj are the corresponding dimensionless reaction rates, and the v, j are the stoichiometric coefficients ... 

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). 

The PFR is efficient for screening solid catalyst in a single fluid phase. It can also be used in later research stages to assess commercial criteria. Consider the evaluation of the ultimate commercial performance of a newly developed fixed-bed catalyst. The theory of similarity teaches that for the laboratory and the industrial reactor, the Damkohler number (NDa), the Sherwood number (Nsh), and the Thiele modulus (<)>) need to be kept constant (Figure 2). As a result, the laboratory reactor must have the same length as the envisioned commercial reactor (7). In this case, scale up is done by increasing the diameter of the reactor. This example further illustrates that laboratory reactors are not necessarily small in size. 

Damkohler number, primary bubble size (m) secondary bubble size (m) catalyst decay constant (1/s) particle diameter (m) tube diameter (m) activation energy (kcal/kmol) energy input rate (W) function (-) feed rate (mol/s) feed rate of i (kmol/s) solids circulation rate (kg/s) flow rate in a single tube (kmol/s) total feed rate (kmol/s)... 

At relatively low pressures, what dimensionless differential equations must be solved to generate basic information for the effectiveness factor vs. the intrapellet Damkohler number when an isothermal irreversible chemical reaction occurs within the internal pores of flat slab catalysts. Single-site adsorption is reasonable for each component, and dual-site reaction on the catalytic surface is the rate-limiting step for A -h B C -h D. Use the molar density of reactant A near the external surface of the catalytic particles as a characteristic quantity to make all of the molar densities dimensionless. Be sure to define the intrapellet Damkohler number. Include all the boundary conditions required to obtain a unique solution to these ordinary differential equations. 

Based on their abihty to convert reactants to products via first-order irreversible chemical kinetics, in rectangular channels with various aspect ratios at large Damkohler numbers (i.e., p = 1000) in the diffusion-limited regime. Reactant molar density vs. channel length follows a single exponential decay for those deposition profiles that are not underlined. 

The boundary conditions at the external surface of the catalyst are T = Tsurface and Ca = Ca surface, and A effeciive is the effective thermal conductivity of the composite catalyst structure (i.e., 1.6 x 10 J/cm s K for alumina). Initially, the surface temperature and concentration of reactant A in Uie vicinity of a single isolated catalytic peUet are chosen to match the inlet values to the packed reactor. If external mass and heat transfer resistances are minimal, then bulk gas-phase temperature and reactant concentration at each axial position in the reactor represent the characteristic quantities that should be used to calculate the intrapellet Damkohler number for nth-order chemical kinetics ... 

Biofilters are chemically enhanced absorbers, and therefore mass transfer limited (see Absorption with Chemical Reaction in Sec. 14). The magnitudes for the Hatta [= Damkohler II = (Thiele modulus)2] numbers are quite low, perhaps below 5. Nevertheless, for design simplicity, mass-transfer limitation is generally assumed to be in the liquid phase (the biofilm). For a single-component biofilter, the simplified biofilter model and design equation is... 

The summation included for completeness is over all mineral species k. It reduces to the last term for the case of a single fluid and mineral species containing oxygen. Figure 13 shows the solutions of the equation for different Peclet and Damkohler I numbers calculated for a fluid composed of water and a one-dimensional rock column. Bowman et al. (1994) presented many more examples. The curves shown in Figure 13a are solutions for an infinite Damkohler I number at a Peclet number of 100. In this case of rapid... 

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