Damkohler numbers first order irreversible reaction

Figure 2.5 Product yields and unreacted fraction of the key reactant f, as function of the first Damkohler number. First order irreversible reaction, v = 0.5.

The Damkohler number for a first-order irreversible reaction is... 

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 reactors. The Damkbhler 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 irreversible reactions the Damkbhler numbers are... 

It is not necessary to introduce a critical spatial coordinate because the rate of disappearance of reactant A is extremely small when its molar density approaches zero in the central core of the catalyst at large values of the intrapellet Damkohler number. One-dimensional diffusion and first-order irreversible chemical reaction in rectangular coordinates is described mathematically by a frequently occurring... 

What is the analytical expression for the effectiveness factor vs. the intrapellet Damkohler number that corresponds to one-dimensional diffusion and first-order irreversible chemical reaction in catalytic pellets with cylindrical symmetry The radius of the cylinder is used as the characteristic length in the definition of the intrapellet Damkohler number. 

If L = R for spherical pellets, as defined in Section 18-1, then the intrapellet Damkohler number (i.e., A ) is nine-fold larger than Ay g, and the analytical and numerical solutions for radial diffusion and first-order irreversible chemical reaction proceed as follows. Now, the dimensionless independent spatial variable r) ranges from 10 near the center of the catalyst to 1 at the external surface, and the geometric factor a is 3 for spheres. 

What is the defining expression for the isothermal effectiveness factor in spherical catalysts Reactant A is consumed by three independent first-order irreversible chemical reactions on the interior catalytic surface. Your final expression should be based on mass transfer via diffusion and include the reactant concentration gradient at the external surface of the catalyst, where t) = 1. Define the intrapellet Damkohler number in your final answer. 

Effect of the Damkohler Number on Conversion in Square Ducts. More conversion is predicted at higher Damkohler numbers because the rate of surface-catalyzed chemical reaction is larger. At a given axial position z, reactant conversion reaches an asymptotic limit in the diffusion-controlled regime, where oo. Actual simulations of I Abuik vs. f at /i = 20 are almost indistinguishable from those when p = 1000. The effect of p on bulk reactant molar density is illustrated in Table 23-5 for viscous flow in a square duct at = 0.20, first-order irreversible chemical reaction, and uniform catalyst deposition. These results in Table 23-5 for the parameter A, as a function of the Damkohler number p can be predicted via equations (23-80) and (23-81) when C = A and... 

Rectangular duct simulations performed over the following range of Damkohler numbers for first-order irreversible chemical reaction ... 

One first-order irreversible chemical reaction occurs within a porous catalyst that exhibits rectangular symmetry. The center of the catalyst corresponds to rj = 0, and the external surface is at = 1. The intrapellet Damkohler number for reactant A is 1, and the Arrhenius number is 8.6. 

The influence of different process and geometrical parameters on conversion was estimated in case of an irreversible first-order catalytic reaction. The influence of temperature nonuniformity (when temperature varies between the channels) had the largest impact on conversion in a nonideal reactor as compared to non-uniform flow distribution and nonequal catalyst amount in the channels. Obtained correlations were used to estimate the influence of a variable channel diameter on the conversion in a microreactor for a heterogeneous first-order reaction. It was found that the conversion in 95% of the microchannels varied between 59 and 99% at = 0.1 and Damkohler number of 2. Figure 9.1a shows conversion as a function of Damkohler number for an ideal microreactor and a microreactor with variations in the channel diameter (aj = 0.1). It can be seen that although the conversion in individual channels can vary considerably, the effect of nonuniformity in channel diameter on the overall reactor conversion is smaller. The lower conversion in channels with a higher flow rate is partly compensated by a higher conversion in channels with a lower flow rate. Due to the nonlinear relation between the channel diameter and the flow rate, the effects do not cancel completely and a decrease in reactor conversion is observed. 

The mass balance with homogeneous one-dimensional diffusion and irreversible nth-order chemical reaction provides basic information for the spatial dependence of reactant molar density within a catalytic pellet. Since this problem is based on one isolated pellet, the molar density profile can be obtained for any type of chemical kinetics. Of course, analytical solutions are available only when the rate law conforms to simple zeroth- or first-order kinetics. Numerical techniques are required to solve the mass balance when the kinetics are more complex. The rationale for developing a correlation between the effectiveness factor and intrapellet Damkohler number is based on the fact that the reactor design engineer does not want to consider details of the interplay between diffusion and chemical reaction in each catalytic pellet when these pellets are packed in a large-scale reactor. The strategy is formulated as follows ... 

Two expressions are given below to calculate the effectiveness factor E. The first one is exact for nth-order irreversible chemical reaction in catalytic pellets, where a is a geometric factor that accounts for shape via the surface-to-volume ratio. The second expression is an approximation at large values of the intrapellet Damkohler number A in the diffusion-limited regime. 

At high-mass-transfer Peclet numbers, sketch the relation between average residence time divided by the chemical reaction time constant (i.e., r/co) for a packed catalytic tubular reactor versus the intrapeUet Damkohler number Aa, intrapeiiet for zeroth-, first-, and second-order irreversible chemical kinetics within spherical catalytic pellets. The characteristic length L in the definition of Aa, intrapeiiet is the sphere radius R. The overall objective is to achieve the same conversion in the exit stream for all three kinetic rate laws. Put all three curves on the same set of axes and identify quantitative values for the intrapeiiet Damkohler number on the horizontal axis. 

Simulations are presented below in tabular and graphical forms when the temperature at the external surface of the pellet is constant at 350 K. The effective thermal conductivity of alumina catalysts is 1.6 x 10 J/cm s K. The chemical reaction is first-order and irreversible and the catalysts exhibit rectangular symmetry. Most important in Tables 27-5 to 27-8 and Figures 27-1 to 27-3, the diffusivity ratio a(0) varies with temperature in the mass transfer equation. This effect was neglected in Tables 27-1 to 27-4. Notice that in all of these tables (i.e., 27-1 to 27-8), numerical simulations reveal that the actual max exceeds I + fi, except when the intrapellet Damkohler number is small enough and 4 a( = 0) > 0 because the center of the catalyst is not reactant starved in the chemical-reaction-rate-controlled regime. 

Use the following data to analyze the performance of a packed catalytic tubular reactor that contains porous spherical pellets. The heterogeneous kinetic rate law is pseudo-first-order and irreversible such that / surface, with units of moles per area per time, is expressed in terms of the partial pressure of reactant A, only (i.e., surface = i.siufacePA), and ki, surface has dimensions of moles per area per time per atmosphere, ki, surface is not a pseudo-volumetric kinetic rate constant. Remember that the kinetic rate constant in both the intrapellet and interpellet Damkohler numbers must correspond to a pseudo-volumetric rate of reaction, where the rate law is expressed in terms of molar densities, not partial pressures. 

If the intrinsic reaction rate is fast compared to the internal and/or external mass transfer processes, the reactant concentration within the porous catalyst and on its outer surface is smaller compared to the bulk concentration, whereas the concentration of the intermediate will be higher. Consequently, the consecutive reaction is promoted and the yield diminishes. The degree of yield losses depends on the ratio between transfer time and the intrinsic rate of the consecutive reaction, which is characterized by the corresponding Thiele moduli and Damkohler numbers referred to the consecutive reaction. For irreversible first-order reactions, the equations are as follows ... 

---