Isotherm irreversible

When the flow pattern is known, conversion in a known network and flow pattern is evaluated from appropriate material and energy balances. For first-order irreversible isothermal reactions, the conversion equation can be obtained from the R sfer function by replacing. s with the specific rate k. Thus, if G(.s) = C/Cq = 1/(1 -i- t.s), then C/Cq = 1/(1 -i-kt). Complete knowledge of a network enables incorporation of energy balances into the solution, whereas the RTD approach cannot do that. 

Conversion in a known network and flow pattern is evaluated from appropriate material and energy balances. For first order irreversible isothermal reactions, the conversion equation can be obtained from the transfer function if that is known by replacing the parameter s by the... 

For simple power law rate equations the effectiveness can be expressed in terms of the Thiele modulus, Eq 7.28. In those cases restriction is to irreversible, isothermal reactions without volume change. Other cases can be solved, but then the Thiele modulus alone is not sufficient for a correlation. 

As 5 is a thermodynamic property, ASsys is the same in an irreversible isothermal process from the same initial volume Vi to the same final volume V2. However, the change in entropy of the surroundings differs in the two types of processes. First let us consider an extreme case, a free expansion into a vacuum with no work being performed. As the process is isothermal, AU for the perfect gas must be zero consequently, the heat absorbed by the gas Q also is zero ... 

If the equilibrium is linear, exact analytical solutions for the column response can be obtained even when the rate expression is quite complex. In most of the published solutions, axial dispersion is also neglected, but this simplification is not essential and a number of solutions including both axial dispersion and more than one diffusional resistance to mass transfer have been obtained. Analytical solutions can also be obtained for an irreversible isotherm with negligible axial dispersion, but the case of an irreversible isotherm with significant axial dispersion has not yet been solved analytically. 

If Kd - Cm, this isotherm reduces to the rectangular or irreversible isotherm, Cs Cs max.134 For many natural and synthetic systems, the equilibrium dissociation constant is in the range of 10 6 and 10 n. 

A lack of significant intraphase diffusion effects (i.e., 17 > 0.95) on an irreversible, isothermal, first-order reaction in a spherical catalyst pellet can be assessed by the Weisz-Prater criterion [P. B. Weisz and C. D. Prater, Adv. Catal., 6 (1954) 143] ... 

If we fix our attention on two states 1 and 2, infinitesimally near together, through which the system passes during some particular irreversible, isothermal change, the work received by the system during this change will be, by (3.38)... 

The work performed by a system — w) as it undergoes an irreversible isothermal expansion is always less than when the expansion is conducted reversibly. To see this, return to the definition of work done on a system ... 

Calculate + A5su for the reversible and irreversible isothermal expansions... 

Comparison of a reversible and an irreversible isothermal expansion of an ideal gas for the same initial and final states (see text Example 7.12) ... 

Equation (4.31) gives the effectiveness factor for a first order, irreversible, isothermal reaction in a sphere. The actual reaction rate is the effectiveness factor times the ideal rate, the rate for reactant concentration throughout the pellet equal to the concentration at the external surface ... 

In the previous section we found that, in certain special cases, the directions of energy and mass transfer are limited by gradients in certain intensive properties. In this section we show that, during irreversible transfers of heat and work, not only are there constraints on the directions, but constraints also apply to the magnitudes. To develop the argument, we reconsider irreversible, isothermal, constant-mass work as discussed in 7.2.2. For such a process, we have already seen that the combined laws reduce to... 

We see from Figure 3.2-1 that the larger is the value of n, the more nonlinear is the adsorption isotherm, and as n is getting larger than about 10 the adsorption isotherm is approaching a so-called rectangular isotherm (or irreversible isotherm). The term "irreversible isotherm" is normally used because the pressure (or concentration) needs to go down to an extremely low value before adsorbate molecules would desorb from the surface. 

Figure 9.2-10.- Concentration profiles of free and adsorbed species in the case of irreversible isotherm...

Another feature of the irreversible isotherm case is the time taken for the particle to be completely saturated with adsorbate. This is obtained by setting F to unity in eqs. (9.2-52) and the results are tabulated in Table 9.2-5. This finite saturation time is only possible with the case of irreversible isotherm. For the cases of linear isotherm and nonlinear isotherm, the time it takes to equilibrate the particle... 

Table 9.2-6/ Fractional uptake for the case of irreversible isotherm with no film resistance Shape Fractional uptake Expression ...

We have discussed the behaviour of the bimodal solid with a linear isotherm. Now we discuss the other extreme of the isotherm, the irreversible isotherm. What we would expect in this case is that the concentration in the macropore behaves like a wave front, that is the adsorbed concentration in the region close to the pellet exterior is very close to the maximum concentration, while the region near the core is void of adsorbate in any form, either in free or adsorbed form. The position demarcating these two regions is the adsorption (wave) front position. How this wave front penetrates into the particle depends on the rate of macropore diffusion as well as the rate of diffusion into the micropore. 

Note the definition of the parameter y for this case of irreversible isotherm given in eq. (10.4-22b) compared to that for the case of linear isotherm in eq. (10.4-1). 

The dependence of y on temperature in the case of irreversible isotherm is opposite to what was observed for the case of linear isotherm. We have the following temperature dependence of the parameter y ... 

Thus the time scale for the case of irreversible isotherm is the same as that for the case of linear isotherm when micropore diffusion is the controlling mechanism and the intracrystalline diffusivity is a constant. 

We have dealt with the analysis of a zeolite pellet for the case of linear isotherm and the case of irreversible isotherm. These two isotherms represent the two extremes of the nonlinearity of the adsorption isotherm. In this section we will deal with the case of nonlinear isotherm and to make the formulation general we also add to it the heat balance equation to study the coupled effect of the nonlinear isotherm and the nonisothermality on the overall adsorption uptake in a zeolite pellet. Similar to the Section 10.3, we shall assume that the thermal conductivity of the zeolite pellet is high and the heat transfer resistance is due to that of the stagnant film surrounding the pellet. This means that the temperature of the pellet is uniform, and the model corresponding to this circumstance is called the lumped thermal model. 

The second term in the RHS of the above equation is generally larger than the first term, so the time lag equation for the case of irreversible isotherm is reduced to ... 

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