Damkohler second

Da Second Damkohler number K l2 ID K = first-order reaction rate constant l = characteristic length D = diffusion coefficient... 

The relative importance of reaction with respect to diffusion can be described in terms of the nondimensional (second) Damkohler number [30-36] (also called Thiele modulus), in terms of the reaction layer thickness [37,38] or in terms of lability criteria [39,40]. 

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 —...

Perry and Lee [28,29] offer an enhancement of QPA, based upon use of dual heat flux sensors and additional thermocouples in autoclave curing. This enhancement entails determining heat transfer properties during the cure, then using these properties in conjunction with PID regulatory control of autoclave temperature. Using the additional sensors, Perry and Lee employ an on-line Damkohler number in lieu of the second time-derivative of temperature to avoid exothermic thermal runaway within the prepreg stack thermoset resin. The Damkohler number is defined as ... 

The second approach consists in deliberately abandoning certain similarity criteria and checking the effect on the entire process. This technique was used by Gerhard Damkohler (1908-1944) in his trials to treat a chemical reaction in a catalytic fixed bed reactor by means of dimensional analysis. 

First, recall that the nondimensional Damkohler number, Da (Eq. 22-11 b), allows us to decide whether advection is relevant relative to the influence of diffusion and reaction. As summarized in Fig. 22.3, if Da 1, advection can be neglected (in vertical models this is often the case). Second, if advection is not relevant, we can decide whether mixing by diffusion is fast enough to eliminate all spatial concentration differences that may result from various reaction processes in the system (see the case of photolysis of phenanthrene in a lake sketched in Fig. 21.2). To this end, the relevant expression is L (kr / Ez)1 2, where L is the vertical extension of the system, Ez the vertical turbulent diffusivity, and A, the first-order reaction rate constant (Eq. 22-13). If this number is much smaller than 1, that is, if... 

Develop and discuss a set of boundary conditions to solve the Graetz problem. Take particular care with the effects of surface reaction, balancing heterogeneous reaction with mass diffusion from the fluid. A second Damkohler number should emerge in the surface boundary condition,... 

If the reaction is second-order and the numerical value of the Damkohler number Da is the same as in part 1, find the exit conversion using the solution of the nonlinear two point boundary value differential equation. 

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)]. 

The second requirement, for reactions that are not diffusion controlled to reach completion, is that the Damkohler number be larger than 10. The previous discussion and Fig. 11.12 strongly indicate that for SSEs, where at t 0.757 there is an almost vertical ascent of F(t), we use Da = 0.75 t/tr > 10 as the requirement for completion. 

Figure 11. Equivalence between the droplet diffusion model (81) and the IEM model for a zero-order reaction and a second-order reaction in a CSTR. The Damkohler numbers are such that f = 0.5 for perfect micromixing. The agreement is excellent for the second-order reaction, more approximate for the zero-order one.

Fig. 12.1. Dependence of conversion on Damkohler number for a) a first-order reaction (Eq. (13)) b) a second-order reaction (Eq. (14)) and c) a second-order reaction with two reactants and a nonstoichiometric feed composition (Eqs. (15) and (16), here for X = 0.5).

The first Damkohler number quantifies the ratio between the heat transferred from the flame (conduction time) and the energy required to heat the reactants to the ignition temperature (residence time). Extinction will occur when heat cannot be transferred fast enough. Equation 3.53 is written in terms of the ratio between conduction and convection (residence time), but in a more general form, could include all forms of heat transfer like gas-phase radiation. The second Damkohler number indicates if the reaction has sufficient time to proceed. In general, extinction is attained when either of the Damkohler groups is reduced below unity. 

A different way to look at extinction is by reducing the oxygen concentration and thus, increasing the characteristic chemical time. This will result in a decrease of the second Damkohler number. This mechanism of extinction is analyzed in a standardized manner by the limiting oxygen index (LOI) [32],... 

Notice that here Assumption 4.2 is expressed as a ratio of the characteristic time for the chemical reaction and the characteristic time for convection, being thus equivalent to considering that the second reaction has a low Damkohler number. 

The maximum effective reaction rate is obtained for the limiting value of cs = 0. This means that the product of the effectiveness factor and the second Damkohler number can never exceed unity. A comparison of the definition of the Weisz modulus (eq 56) with the definition of Dan (eq 78) gives the equivalence... 

Figure 13 shows how the steady-state exit conversion X[— 1 — Cnm(z — 1)/C 5jii m] varies with the Damkohler number Da for different values of the dimensionless mixing time t](= tmix/r). The figure shows how non-uniform feeding could significantly reduce the conversion as compared to premixed feed for the case of a bimolecular second-order reaction (e.g. by a factor of 2 for the case of f] — 0.1), when mixing limitations are present in the system. 

Fig. 15. Variation of conversion with Damkohler number for a bimolecular second-order reaction for uniform and distributed feeding in a CSTR.

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). 

The Damkohler is a dimensionless number that can give us a quick estimate of the degree of conversion that can be achieved in continuous-flow reactions. The Damkdhler number is the ratio of the rate of reaction of A to the rate of convective transport of A at the entrance to the reactor. For first- and second-order ineversible reactions the Damkohler numbers are... 

Later Mayle, 1970 [400] continued their research by performing measurements of velocity and pressure within the fire whirl. He found that the behavior of the plume was governed by dimensionless plume Froude, Rossby, second Damkohler Mixing Coefficient and Reaction Rate numbers. For plumes with a Rossby number less than one the plume is found to have a rapid rate of plume expansion with height. This phenomenon is sometimes called vortex breakdown , and it is a hydraulic jump like phenomena caused by the movement of surface waves up the surface of the fire plume that are greater than the speed of the fluid velocity. Unfortunately, even improved entrainment rate type models do not predict these phenomena very well. 

Bo = - k- ax Dal = x/tT Dali = tm/tT pP — ud 1 e x Re = c AtJEj ° RT% Qr — -L. J Dm Sh = krt CX dVk<7 DnV T Dm Bodenstein number Capillary number first Damkohler number second Damkohler number axial Peclet number Reynolds number (channel) particle Reynolds number heat production potential Schmidt number Sherwood number (channel) Sherwood number (particle)... 

The importance of external mass transfer between the bulk fluid and the outer surface of a catalyst is characterized by the ratio of the mass transfer time, fm, to the characteristic reaction time, the second Damkohler number. 

Figure 7.1 Snapshots of the spatial distribution for the autocatalytic reaction (7.1) in the alternating sine-flow (2.66) for a subcritical Damkohler number, at times t =0, 3, 6 (first row), 9, 12, 15 (second row), 18, 21, 24 (third row) with the time unit based on the period of the flow. Dark corresponds to the unreacted state C = 0 and light gray is the fully reacted state C = 1.

This second-order ordinary differential equation given by (16-4), which represents the mass balance for one-dimensional diffusion and chemical reaction, is very simple to integrate. The reactant molar density is a quadratic function of the spatial coordinate rj. Conceptual difficulty arises for zeroth-order kinetics because it is necessary to introduce a critical dimensionless spatial coordinate, ilcriticai. which has the following physically realistic definition. When jcriticai which is a function of the intrapellet Damkohler number, takes on values between 0 and 1, regions within the central core of the catalyst are inaccessible to reactants because the rate of chemical reaction is much faster than the rate of intrapellet diffusion. The thickness of the dimensionless mass transfer boundary layer for reactant A, measured inward from the external surface of the catalyst,... 

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