9.8 MOdELing rEactiOns WitH tHE PLUg-fLOW WitH
9.10.2 Modeling Pseudo-first-Order reactions
Let us continue with the reactor system of Figure 9.10 and apply the TiS, PFD, and SF models in the context of a pseudo-first-order reaction carried out in the hypothetical reactor under the flow conditions of the RTD analysis. We will con- sider a biological reaction for the reduction of organic matter manifest as biode- gradable chemical oxygen demand (bCOD). Tchobanoglous et al. (2003) provide values of typical coefficients from which we may approximate a pseudo-first- order rate law. These are coefficients used in the Michaelis–Menton relation (Equation 8.11), which is most definitely of the saturation type. If we decide to neglect the substrate concentration term in the denominator, the overall relation reverts to pseudo-first-order form. Although some inaccuracies will accompany the modeling work accomplished, the results will be sufficient to illustrate appli- cations of the nonideal models and to make comparisons both among them and with the ideal PFR model.
Example 9.3 Employ the ideal PFR and the nonideal TiS, PFD, and SF models to analyze a pseudo-first-order reaction carried out in the reactor system of Figure 9.10.
Employ the models used with and results from Examples 9.1 and 9.2 in applying the TiS and PFD models.
We define the kinetic parameters using a growth rate coefficient that is in the low range of the stated typical values. We need to choose a biomass concentration also, and for a PFR-like reactor, we do not require a large abundance of biomass in the reactor to effect large reduction in the targeted reactant:
We gather flow and volume information from Examples 9.1 and 9.2, taking care to employ consistent time units:
We compute the predicted bCOD concentration from the ideal PFR:
We employ the result for the CMFRs in series model that arose from application of the RTD density function and compute the predicted effluent bCOD concentration were the reactor a string of CMFRs in series. We would be tempted to simply employ N as a noninteger value with the following result:
Conversely, we can visualize a string of 12 CMFRs, each with residence time
SM, t
N followed by a thirteenth reactor of residence time (0.15) tSM.
N This approach requires that we perform some programming to step through the first 12 reactors in the series employing a loop and then using the effluent from the twelfth reactor as the influent to the much smaller, thirteenth reactor. We illustrate two MathCAD functions that are quite useful for obtaining integer values from decimal values. We arrive at an alternate result, which is more in line with the theoretical visualization of the nonideal reactor as a string of CMFRs in series:
We computed (not shown) the effluent concentration with the fractional reactor as the first and sixth in the series with results identical to that shown. Of great significance
is that via the programming we have accomplished, a vector containing the thirteen effluent values has been created. We will find this capacity immensely useful in later analyses.
We might visualize the TiS model using a third and a fourth configuration as a string of Ncount CMFRs each of residence time SM
count
N t
N N , or as a string of NTot CMFRs each of residence time SM
Tot
t N
N N :
We observe the 12-reactor and 13-reactor strings to return effluent values that are, respectively, greater than and less than that from stepping through the reactors.
Adjusting N to an integer value and proportionately increasing or decreasing the residence times of the individual CMFRs in the series might seem reasonable, but we are cautious that these modifications, as well as simply using N as a noninteger value, result in significantly different outcomes. We really cannot definitively judge which of the three methodologies employed with the TiS model in this example would be most correct. However, the implementation accomplished by stepping through the reactors and including a reactor whose volume is a fraction of that of the others in the series most closely follows the derivation of the CMFRs in series rela- tion for the pseudo-first-order reaction case. With this approach, in order to quantita- tively “step into the reactor” and understand the process within the reactor, we must decide the relative location of the partial reactor. Its location certainly affects the profile of the target concentration through the reactor. Given this difficulty, perhaps the most convenient and certainly the conservative approach would be to truncate N and accordingly adjust the volume of the N identical reactors. We can certainly use this approach to generate a profile of target reactant concentration through the reactor, allowing us to quantitatively “step through” the reactor with small steps rather than integer values from 1 to N. We will investigate this idea in later examples.
For the PFD model, we gather the appropriate data from Examples 9.1 and 9.2 and convert to a time unit in days:
Rather than employing the particular solution for the PFD model as one massive relation, we will compute the R1, R2, C1, and C2 coefficients and use Equation 9.19 in its simplest form. R1 and R2 are functions only of the superficial velocity, disper- sion coefficient, and pseudo-first-order rate coefficient so we may set their values as constants:
We then set the initializing value for the concentration derivative at the reactor outlet from the PFR solution:
The remainder of the solution is most conveniently executed as a programmed loop. We use the initialized K to compute Cout and employ Cout to compute a new K from Cin and
0
,
z
dC
dz = both of which are constant. We compute the relative change in K from one iteration to the next and when that relative change remains greater than a threshold value (here we chose 10−6), we perform an additional iteration after updating K from Knew. When the relative change becomes less than the criterion for convergence, the computation exits the loop and we have our value of Cout. A capture of the MathCAD program is shown in Figure E9.3.1 and the output from that program is shown as the Cout.PFD vector:
We have programmed the loop to create a vector of Cout values, corresponding with the results from each of the iterations. Herein nine iterations are required to arrive at a suitable approximation for Cout. For our rather simple numeric solution here, each new K is computed using the old K, resident in the relations for C1 and C2.
We might reduce the number of iterations were we to invoke an implicit solution for each new K value, involving simultaneous solution of the relations for Knew, C1 and C2. A given-find block would be assembled and written into a function, which can be employed within the programmed loop. Here, the savings of a few iterations of the loop at the expense of the added complexity of including the given-find block within the programmed loop are likely not war- ranted. A profile of concentration through the reactor can be computed using Equation 9.19 employing c1 and c2 as defined in section 9.8.1 based on the known influent and effluent concentrations.
For the SF model prediction, we employ the t, Et, and SR vectors defined for Example 9.1. We find it to be most convenient to convert the pseudo-first-order rate constant for use of t in minutes. We create a vector of effluent concentra- tions for the seventeen parallel reactors visualized for the SF model using the relation yielding the effluent from an ideal PFR. We then employ Simpson’s 1/3 rule, using h (the interval) and n (the number of intervals of the original C(t) data set) from Example 9.1, to obtain the predicted effluent bCOD value:
Figure e9.3.1 Capture of mathCad code for the iterative solution of the Pfd model for a pseudo-first-order reaction rate law.
We have computed the percent error associated with the assumption of an ideal PFR relative to a nonideal reactor based on the TiS, PFD, and SF models for the system and conditions of Example 9.3, as the ratio of the difference between the PFR prediction and the real reactor prediction to the real reactor prediction. Relative errors are in the range of 70%.
The percent error associated with the prediction from the PFR model relative to the predictions from the non-ideal reactor (NIR) models was computed from the relation:
= . − .
.
% 100 eff PFR eff NIR
eff NIR
C C
R C C
E
The percent relative errors are in the range of 70%.
We easily observe from Example 9.3 that the TiS, PFD, and SF models all yield similar results but are certainly not in exact agreement. After all, they are three distinct methods employed to model nonideal reactors. Further, with the TiS model, we can choose among numerous specific means of application. One important consideration is that we would like to use these models to quantita- tively understand processes that occur within the reactor so that we may perform enlightened design of the systems in support of the reactor.