Damkohler-Peclet number

This ratio is equivalent to the inverse of the Damkohler-Peclet number usually proposed for the preliminary design and optimization study of membrane reactors. °... 

Cob. molar concentration (mol L" ) of alkaline hydroxide DaPe Damkohler-Peclet number... 

In any circumstances, it can be expected that and (5x are algebraic functions of turbulence length scale and kinetic energy, as well as chemical and molecular quantities of the mixture. Of course, it is expedient to determine these in terms of relevant dimensionless quantities. The simplest possible formula, in the case of very fast chemistry, i.e., large Damkohler number Da = (Sl li)/ SiU ) and large Reynolds Re = ( Ij)/ (<5l Sl) and Peclet numbers, i.e., small Karlovitz number Ka = sjRej/Da will be Sj/Sl =f(u / Sl), but other ratios are also quite likely to play a role in the general case. 

Fluid flow and reaction engineering problems represent a rich spectrum of examples of multiple and disparate scales. In chemical kinetics such problems involve high values of Thiele modulus (diffusion-reaction problems), Damkohler and Peclet numbers (diffusion-convection-reaction problems). For fluid flow problems a large value of the Mach number, which represents the ratio of flow velocity to the speed of sound, indicates the possibility of shock waves a large value of the Reynolds number causes boundary layers to be formed near solid walls and a large value of the Prandtl number gives rise to thermal boundary layers. Evidently, the inherently disparate scales for fluid flow, heat transfer and chemical reaction are responsible for the presence of thin regions or "fronts in the solution. 

The only two parameters appearing in eqn. (65) are the dispersion number, DjiiL, or inverse Peclet number, and the Damkohler number, or dimensionless rate group, t/jCa," - Solutions to eqn. (65) are therefore functions only of these two groups. If term (4) in eqn. (65) is absent, then... 

The Peclet number, uLjD, when written in the form (L /D)l(Llu) is seen to be a ratio of characteristic dispersion time to characteristic residence time and the Damkohler number can, in similar manner, be considered as a ratio of characteristic residence time, L/u, to characteristic reaction time, l/feCA " [59]. 

In this section, we will obtain the non-dimensional effective or upscaled equations using a two-scale expansion with respect to the transversal Peclet number Note that the transversal P let number is equal to the ratio between the characteristic transversal timescale and longitudinal timescale. Then we use Fredholm s alternative to obtain the effective equations. However, they do not follow immediately. Direct application of Fredholm s alternative gives hyperbolic equations which are not satisfactory for our model. To obtain a better approximation, we use the strategy from Rubinstein and Mauri (1986) and embed the hyperbolic equation to the next order equations. This approach leads to the effective equations containing Taylor s dispersion type terms. Since we are in the presence of chemical reactions, dispersion is not caused only by the important Peclet number, but also by the effects of the chemical reactions, entering through Damkohler number. 

Figure 10.4 [1] shows the results for theoretical calculations [4] for the ratio n, the number of viable cells leaving the holding section of a continuous sterilizer, to Hq, the number of viable cells entering the section, as a function of the Peclet number (Pe), as defined by Equation 10.7, and the dimensionless Damkohler number (Da), as defined by Equation 10.8 ... 

Figure 22.3 One-dimensional concentration profiles at steady-state calculated from the diffusion/advec-tion/reaction equation (Eq. 22-7) for different parameter values D (diffii-sivity), x (advection velocity), and kr (first-order reaction rate constant). Boundary conditions at x = 0 and x - L are C0 and CL, respectively. Pe = 7. vx ID is the Peclet Number, Da = Dk/v] is the Damkohler Number. See text for further explanations.

Explain qualitatively the significance of the Damkohler Number and of the Peclet Number. 

Note that for the special case kTW= kr s, k is krw/fw. The extra factor (f yl compensates for the fact that in Eq. 25-46 k is multiplied only with the dissolved BC concentration, whereas here we assume that the hydrolysis affects the total (dissolved and sorbed) BC concentration. The Peclet Number (Eq. 25-13, with x0 = 30 m) is affected neither by sorption nor by reaction Pe = 10. The Damkohler Number is (Eq. 25-48) ... 

Again, the full set of equations can be solved, but appropriate simplifications are more instructive. For common LPCVD reactions and conditions the Damkohler and Peclet numbers are small (<<1 and 1, respectively). Moreover, the annulus is narrow (10-20 mm) compared with the deposition zone (500 mm). Therefore, radial variation may be neglected, and the following dimensionless equation results... 

Note that our program runaxialdispDa is specialized to run for just two sets of values for Pen and PeM as 4 and 8 or as 8 and 4, respectively. We have not attempted to make runaxialdispDa truly global by internal parameterizations for other Peclet data. It is obvious, however, how our readers can modify the program with a bit of trial and error to work for other sets of Peclet numbers PeH and PeM- The main task for this consists of figuring out - by trial and error - how to set up the various integration subintervals differently and to experiment with the direction of integration and the step size used for different Damkohler numbers. 

It is often useful to write a model equation such as Equation 8-121 in terms of dimensionless variables. This introduces the Peclet number NPe = uL/De>1, which represents the ratio of characteristic dispersion time to characteristic convection time (residence time), and the Damkohler number,... 

Here we have denoted y conversion, 0 Frank-Kameneckii dimensionless temperature, Da Damkohler number, Pe Peclet number for axial mass dispersion, Pe Peclet number for Sxial heat dispersion, Y dimensionless activation energy, B dimensionless adiabatic temperature rise, 3 dimensionless cooling parameter, 6 temperature of the cooling medium, A mass capacity, AT heat capacity. 

The external electric field is in the direction of the pore axis. The particle is driven to move by the imposed electric field, the electroosmotic flow, and the Brownian force due to thermal fluctuation of the solvent molecules. Unlike the usual electroosmotic flow in an open slit, the fluid velocity profile is no longer uniform because a pressure gradient is built up due to the presence of the closed end. The probability of the particle position is obtained by solving the Fokker-Planck equation. The penetration depth is found to be dependent upon the Peclet number, which is a measure of significance of the convective electroosmotic flow relative to the Brownian diffusion, and the Damkohler number, which is a ratio of the characteristic diffusion-to-deposition times. 

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

Here, C( , z, t) is the scaled solute concentration in the fluid phase, Cw the solute concentration at the wall, 6 the normalized adsorbed concentration (O<0< 1), K the adsorption equilibrium constant, p the transverse Peclet number, T represents the adsorption capacity (ratio of adsorption sites per unit tube volume to the reference solute concentration), and Da is the local Damkohler number (ratio of transverse diffusion time to the characteristic adsorption time). We shall assume that p 4Cl while T and Da are order-one parameters. (In physical terms, this implies that transverse molecular diffusion and adsorption processes are much faster compared to the convection.)... 

After suitable non-dimensional variables are substituted into the equations, following the same procedure as outlined in sect 1.2.5, the important dimensionless groups are obtained for the problem in question. These are the Reynolds number, the Schmidt number, the Peclet number, Pe = Re Sc = ul/D, and the Damkohler number, Daj = Ir/u. The u and I are the characteristic velocity and length scales, respectively, for the velocity field, and r denotes a characteristic chemical reaction rate. 

Finally, we studied the effect of liquid dispersion on catalyst performance by comparing the performance of the powder catalyst in a bubble column reactor with the HyperCat-FT system. As shown in Figure 6, the CO conversion is much lower for a low Peclet number (bubble colunm reactor with a back-mixed liquid phase) as opposed to a higher Peclet number for the HyperCat system. The tests were conducted under the same process conditions and Damkohler number. The change in Peclet number did not change the liquid product... 

A small spherical particle of radius a is immersed in a uniform flow with velocity U far from the fixed particle. On the surface of the particle a chemical reaction takes place in which a solute c in the fluid is consumed according to a first-order reaction, r = —kc. Estimate the net rate of consumption, R, of c, in dimensionless form, when the fluid far from the particle has solute concentration co. The molecular difiusivity of the solute is a constant D. R depends on two parameters, a Peclet number and a Damkohler number R(Pe, Da). 

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)... 

Figure 3.4. Contours of log(req), which is equal to log(Xeq), for a range of Damkohler and Peclet numbers. The dashed line separates a region where req and Xeq depend only on Da (reaction dominated) from a region where they depend only on Pe (transport or advection dominated). The shaded areas are discussed in the text. The arrows point to the region where the modeling grid size can be less than 100 meters.

To evaluate the exit concentration given by Equation (14-26) or the conversion given by (14-27), we need to know the Damkdhler and Peclet numbers. The Damkohler number for a first-order reaction. Da = ti. can be found using the techniques in Chapter 5. In the next section, we discuss methods to determine the Peclet number. 

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