Damkohler number transport control

Where dissolution or precipitation is sufficiently rapid, the species concentration quickly approaches the equilibrium value as water migrates along the aquifer the system is said to be reaction controlled. Alternatively, given rapid enough flow, water passes along the aquifer too quickly for the species concentration to be affected significantly by chemical reaction. The system in this case is transport controlled. The relative importance of reaction and transport is described formally by the nondimensional Damkohler number, written Da. 

A notable aspect of this equation is that L appears within it as prominently as the rate constant k+ or the groundwater velocity vx, indicating the balance between the effects of reaction and transport depends on the scale at which it is observed. Transport might control fluid composition where unreacted water enters the aquifer, in the immediate vicinity of the inlet. The small scale of observation L would lead to a small Damkohler number, reflecting the lack of contact time there between fluid and aquifer. Observed in its entirety, on the other hand, the aquifer might be reaction controlled, if the fluid within it has sufficient time to react toward equilibrium. In this case, L and hence Da take on larger values than they do near the inlet. 

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. 

The effective diffusivity Dn decreases rapidly as carbon number increases. The readsorption rate constant kr n depends on the intrinsic chemistry of the catalytic site and on experimental conditions but not on chain size. The rest of the equation contains only structural catalyst properties pellet size (L), porosity (e), active site density (0), and pore radius (Rp). High values of the Damkohler number lead to transport-enhanced a-olefin readsorption and chain initiation. The structural parameters in the Damkohler number account for two phenomena that control the extent of an intrapellet secondary reaction the intrapellet residence time of a-olefins and the number of readsorption sites (0) that they encounter as they diffuse through a catalyst particle. For example, high site densities can compensate for low catalyst surface areas, small pellets, and large pores by increasing the probability of readsorption even at short residence times. This is the case, for example, for unsupported Ru, Co, and Fe powders. 

Figure 6. Channel length at the same point in time versus Damkohler number. In the transport-controlled regime, the channel length is independent of the reaction rate. As the Damkohler number decreases from 0.1 to 0.01, the length of channels increase (see text for discussion). At Damkohler numbers below about 0.01, dissolution becomes pervasive and the channel does not propagate.

For small Damkohler numbers (Da <3C 1) the mass transport is much faster than the surface reaction itself and therefore the mass transport effect may be ignored. On the other hand, if the Damkohler munber is high (Da 1) the sensorgram profile is completely controlled by the diffusion mass transfer and is it not possible to determine rate constants of the surface reaction. 

As pointed out, the influence of mass transfer on the observed reaction rate depends on the ratio between the characteristic reaction time and the characteristic time for mass transfer. By increasing the temperature, the intrinsic reaction rate increases more strongly (exponentially) than the rates of external and internal mass transfer. Consequently, the Thiele modulus and the second Damkohler number augment with increasing temperature, and transport phenomena become more and more important and will finally control the transformation process. In addition, the temperature dependence of the observed reaction rate will change as indicated. 

At Earth surface conditions, gypsum (CaS04 2H20) dissolution appears to be controlled partly by the rate of Ca and release at the mineral surface and partly by the rate of transport of these species away from the dissolving surface. The second Damkohler number, Eq. (7.18), can be used to determine the relative importance of reaction rate versus diffusion rate for gypsum dissolving into a static solution in a fracture with a width, L, of 1.0 x 10 m... 

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