Damkohler IV number

Three different dimensionless groups are of importance for estimating this temperature rise in reactive extrusion the Damkohler IV (Daiv) number, the Brinkmann (Br) number, and the Graez (Gz) number. The Damkohler IV number gives the ratio between the heat released by the... 

For reactive extrusion the Damkohler IV number, the Brinkmann number, and the Graez number can conveniently be rewritten as... 

If the Brinkmann number is much smaller than the Damkohler IV number (or Daiv/Br 1), the heat released by viscous dissipation can be neglected with respect to the heat of reaction. If the Graez number is small, the heat removed by the cooled wall will be negligible, and the process is nearly adiabatic. This will generally be the case in situations where large extruders are used. 

The three terms account for the convective heat, the heat of reaction, and the conduction to the barrel wall, respectively. In combination with this balance, the Damkohler IV number plays an important role in the thermal scale-up. 

For small Damkohler IV numbers, the exponent equation obtained from Eq. (12.41) is... 

Although Damkohler traced numbers I to IV back to the above combinations of named dimensionless numbers known at that time, numbers I to IV have come to be known as the four Damkohler numbers Da to DaIV in chemical literature. We will not identify them in this way, instead we will only refer to the new, genuine reaction kinetic pi-number... 

We illustrate this reaction behavior further using our concentration and temperature profile plotting code plugflowxAy. m for a Damkohler number such as 107-456 just below ignition in Figure 5.16 and then for Da = 107 463 which indicates ignition and y uj) = 2 near the end of the tube for iv < 1 in Figure 5.17. 

This Damkohler criterion is the Damkohler number of type I (Dof). Other Damkohler numbers were defined [12] type II used to characterize the material transport at the surface of a solid catalyst, type III used to characterize the convective heat transport at the catalyst surface, and type IV used to characterize the temperature profile in a solid catalyst. For batch reactions, the reaction time xr is defined at a reference temperature, the cooling medium temperature, as... 

Damkohler s original analysis [2] resulted in four dimensionless numbers which today are referred to as Damkohler numbers I through to IV and are given by... 

Fluid-fluid reactions are reactions that occur between two reactants where each of them is in a different phase. The two phases can be either gas and liquid or two immiscible liquids. In either case, one reactant is transferred to the interface between the phases and absorbed in the other phase, where the chemical reaction takes place. The reaction and the transport of the reactant are usually described by the two-film model, shown schematically in Figure 1.6. Consider reactant A is in phase I, reactant B is in phase II, and the reaction occurs in phase II. The overall rate of the reaction depends on the following factors (i) the rate at which reactant A is transferred to the interface, (ii) the solubihty of reactant A in phase II, (iii) the diffusion rate of the reactant A in phase II, (iv) the reaction rate, and (v) the diffusion rate of reactant B in phase II. Different situations may develop, depending on the relative magnitude of these factors, and on the form of the rate expression of the chemical reaction. To discern the effect of reactant transport and the reaction rate, a reaction modulus is usually used. Commonly, the transport flux of reactant A in phase II is described in two ways (i) by a diffusion equation (Pick s law) and/or (ii) a mass-transfer coefficient (transport through a film resistance) [7,9]. The dimensionless modulus is called the Hatta number (sometimes it is also referred to as the Damkohler number), and it is defined by... 

Now, all of the tools required to calculate the molar density of reactant A on the external surface of the catalyst are available to the reactor design engineer. It is important to realize that Ca, surface is the characteristic molar density, or normalization factor, for all molar densities within the catalyst. Hence, Ca, surface only appears in the expression for the intrapellet Damkohler number (i.e., excluding first-order kinetics) when isolated pellets are analyzed. Furthermore, intrapellet Damkohler numbers are chosen systematically to calculate effectiveness factors via numerical analysis of coupled sets of dimensionless differential equations. Needless to say, it was never necessary to obtain numerical values for Ca, sur ce in Part IV of this textbook. Under realistic conditions in a packed catalytic reactor, it is necessary to (1) predict Ca, surface and Tsurface, (2) calculate the intrapellet Damkohler number, (3) estimate the effectiveness factor via correlation, (4) predict the average rate of reactant consumption throughout the catalyst, and (5) solve coupled ODEs to predict changes in temperature and reactant molar density within the bulk gas phase. The complete methodology is as follows ... 

In this chapter, we applied dimensional analysis to chemical processes. We used homogeneous batch reactions, plug flow reactions, and porous solid—catalyzed reactions as examples. We demonstrated how to derive the dimensionless parameters for these examples, then we showed how to combine them to form the Group I, II, III, and IV Damkohler numbers, as well as the Reynolds number. We demonstrated how these dimensionless parameters and numbers are used during upscaling and downscaling. 

Here x is the extent of the reaction (or scaled concentration of the reagent B), X2 is the normalized temperature of the complex liquid-solid medium, Pei and Pc2 are the Peclet numbers for mass and heat transport, Le is the Lewis number, is the longitudinal spatial coordinate. Da is the Damkohler number, 7 is the normalized activation energy of the reaction, /) is the transverse residence time of fluid in the reactor determined by the rate of cross-flow, b is the adiabatic temperature rise for the empty reactor (without packing), / iv is the surface heat transfer coefficient, and X2w the temperature of the reactor walls [22],... 

---