In the strict sense of the PFR, elements of fluid enter and exit the reactor in exactly the same order. In truth, some degree of forward and backward mixing of fluid among elements occurs. This phenomenon is often referred to as dispersion. In order to create conditions that minimize dispersion, engineered reactors are generally modi- fied in one of two ways. For closed reactors with a “straight-through” flow path, packing is used to ensure a high degree of local mixing and lateral dispersion, which
0.1 1 10 100
0
x
1×103 2×103
t(x)
Figure e8.4.4 a plot of time required to accomplish the desired phenol reduction using the biomass concentration as the independent variable.
minimizes longitudinal dispersion. Many varieties of manufactured packing can be obtained and each has its advantages and disadvantages. For reactors that are open basins of overall rectangular shape, baffling is employed to route the flow in pathways such that elements of fluid follow a guided, longer path through the reactor than would be the case in the absence of baffling. For natural systems, we can identify flow regimes resembling plug flow by the presence of porous media in groundwater systems or by the long, narrow (when viewed from the large-scale perspective) reactor system presented by a river or stream.
Example 8.5 A reactor of design that may, with little error, be modeled as a PFR has a bed volume of 5 m3. The total volume of the reactor is 6.25 m3 and the total porosity of the packing material as packed in the reactor is 0.8. The bed volume is the volume of solution contained within the pores of the packing with which the reactor is filled and is the product of total volume and total porosity. Consider that this reactor is to be employed for a pseudo-first-order reaction having an overall rate constant k′ = 0.1/min. Consider that the design flow is to be 100 L/min and that the influent concentration of the substance targeted for removal from the flow stream is 0.1 mol/L. Let us investigate the relation between concentration of the target reactant and position within the reactor and then determine the concentration of the target substance in the effluent from the reactor.
Let us first compute the bed volume and residence time and assign some known values:
We may now invoke Equation 8.22b to compute the effluent concentration from the reactor:
Suppose that we observe this result and realize that the computed effluent concentration is lower than the design value, Ceff = 0.001 mol/L. We might wish to determine the flow rate that could be applied to the reactor to attain the target result.
We would employ Equation 8.21b for this computation. We could first compute the alternative residence time and then the alternative flow rate:
Here it would be of value to illustrate the concentration profile (value of concentration versus position in the reactor) across the reactor. We have stated nothing about the exact configuration of the reactor other than its strong resem- blance to an ideal PFR. Regardless of the exact configuration, we may suggest that a unique flow path exists, along which each element of fluid entering the reactor follows the element ahead of it and precedes the element behind it. We would also postulate that the fluid velocity through the reactor is constant. The fraction of the total residence time at position z relative to the total length of the flow path, LFP, is identical to the fraction of the total residence time associated with the travel time from the influent (z = 0) to position z.
We realize that the ratio
Tot
τz
τ is identical to
FP
z
L . We can visualize the concentration profile as the series of effluent concentrations associated with suc- cessively increasing the length of the flow path by an incremental Δz, correspond- ingly increasing the reactor residence time by Δtz. In this case, we will somewhat arbitrarily use Tot
z 50
t τ
∆ = . We write a function using the incremental value of t and plot the result in Figure E8.5.1:
As expected, the in figure E8.5.1 profile across the reactor is “log-linear,” conform- ing to the governing relation which is a decaying exponential.
A significant additional consideration with this context involves the sizing of the reactor itself when constrained by the influent and effluent concentrations of the target reactant and the influent flow rate. Let us suppose the influent and effluent concentrations
0 C(τZ)
Ceff
1×10–4 1×10–3 0.01 0.1
10 20 30
τZ
40 50
Figure e8.5.1 Plot of reactant abundance versus position in an ideal Pfr for arbitrary reactant with transformation governed by a pseudo-first-order rate law.
are 0.1 and 0.0001 mol/L, respectively, and the flow is 100 L/min. We can find the necessary residence time and, hence, the volume of the reactor using Equation 8.21b.
Since many of the parameters have been specified in our MathCAD worksheet, we need only specify the new target effluent value and invoke Equation 8.21b:
Our final consideration for this context involves examination of the pseudo-first- order rate coefficient. We have not considered a particular reaction here but merely suggested that it is pseudo-first-order, so we may have several factors potentially under our control through which we might manipulate the magnitude of the rate constant. Perhaps we can adjust the solution composition or the temperature, or employ an alternative set of supporting reactants. The necessary magnitude of the rate constant can be computed based on our final set of process constraints. Suppose the influent and effluent concentrations must again be 0.1 and 0.0001 mol/L, the flow would remain at 100 L/min, but the bed volume of the reactor would be limited to 2500 L. Let us determine the necessary value of the pseudo-first-order rate constant.
Since the MathCAD worksheet already contains many of the necessary param- eter values, we need only define new constraining reactor volume, and again, Equation 8.21b serves our purpose:
Then, knowing the necessary magnitude of the rate constant, we can examine the adjustments necessary to attain the desired performance.
The high-rate activated sludge process is accomplished in a reactor that is configured to attain a near-plug-flow configuration. The activated sludge process is employed to convert both soluble and insoluble biodegradable organic matter into biomass. High- rate systems are often designed specifically to address organic carbon substrates in a manner as efficient as possible with regard to reactor sizing and consumption of oxygen. Removal of nitrogen and phosphorus, if necessary, is left to additional processes. Our next example targets such a reactor.
Example 8.6 Consider a rectangular, baffled reactor (see Figure E8.6.1) with a total volume of 10,000 m3 (2.64 MG) that receives a flow of 0.694 m3/s (6 × 104 m3/ day, 15.85 MGD). The contaminant of concern is an oxygen-demanding organic material (both soluble and particulate) measured as chemical oxygen demand (COD).
The level of COD in the influent is 450 g/m3. The biomass employed in the reaction is provided via a recycle line through which a concentrated stream (4500 gVSS/m3, often called return activated sludge) is returned to the influent for introduction into the bioreactor. This biomass-containing recycle stream originates from the underflow
of the sedimentation basin immediately following the bioreactor. The quantity of VSS is the typical measure of the abundance of biomass in biological reactors. To determine VSS, solids are filtered from a known volume of sample, dried to obtain total suspended solids (TSS), and then fired in a muffle furnace at 550–600 °C to drive off the combustible portion. The difference in mass between the dried solids and the residual, after firing, is the VSS. The overall schematic of a reactor system with cell recycle, presented in Figure E8.1.1, is modified for inclusion here as Figure E8.6.2. Rather than nitrate, the reactant of interest here is merely biodegrad- able organic material, manifest as COD. The clarifier serves simply to separate the biomass from the effluent stream and produce a high-concentration stream for mixing with the influent to produce a seed of biomass to increase the rate of the reaction.
Examine the effect associated with the manipulation of the recycle flow rate (QR) upon the performance of the reactor.
The process under investigation is characterized as high-rate activated sludge.
Typical magnitudes of k ′ and Khalf suggested in the literature for the saturation-type reaction rate law are COD
VSS
10 g
g ⋅d and 40 gCOD/m3, respectively (Tchobanoglous
QR, recycled biomass from secondary clarifier underflow,
XR.min = ~3,000 gVSS/m3 XR.max = ~9,000 gVSS/m3 Q, influent flow –
split between two halves of reactor
effluent flow – to secondary clarifiers for
biomass recovery
Figure e8.6.1 schematic plan view of an activated sludge reactor arranged to approximate plug flow conditions.
et al., 2003). We construct a model of the bioreactor system, accounting for a variable recycle flow, using the master independent variable R = QR/Q. Given the baffling within the basin leading to a channeling of the flow, we consider the reactor to be ideal and plug-flow. We recognize that we have the flow split between two halves of the reactor and can consider half the flow introduced into half the volume, or can con- sider the two halves together, assuming that the total recycle flow is split exactly to each half of the reactor. Also, we recognize that as the flow passes through the reactor, biomass will grow, rendering the biomass concentration to be variable in position.
For extended aeration activated sludge systems, the level of biomass in the reactor is typically much larger than the rate of biomass growth, and we can with only small error neglect this growth of biomass. For high-rate systems, this assumption perhaps leads to measureable error. However, were we to consider the growth of biomass, our model would need a second mass balance (on biomass) and the level of sophistication of the effort would need to be upped significantly. We will address biomass growth along with reduction of COD later. Here, since the presence of additional biomass is to be neglected, we realize our results will likely be conservative; however, the analyses performed here will be useful for process understanding.
We assign values to pertinent parameters:
The return sludge line must be mixed with the influent line, using mixing prin- ciples from Chapter 7. Let us first try a recycle ratio (R = QR/Q) of 0.05, noting that the actual hydraulic residence time of the reactor and the biomass concentration will vary with the magnitude of R:
rCOD, VR, X Q, Cinf.COD
Clarifier
XR, QR
Biomass recycle Excess biomass to processing
Q, Ceff.COD Xinf ≅ 0
Reactor
Figure e8.6.2 schematic of the reactor and clarifier system used for an activated sludge recycle reactor.
Since we are considering a PFR and we have a saturation-type rate law, we may employ Equation 8.23b. The computation of the effluent concentration cannot be explicitly accomplished, but with the use of MathCAD’s given-find solve block, the effluent COD concentration is easily found:
Of note, MathCAD’s root() function might also serve for this computation.
Unfortunately, for implementation of the root() function, the initial guess tendered for Ceff must be quite close to the final value upon which the solution converges. We note that as R becomes infinitesimally greater than 0, due to the mixing of the influent stream (of concentration Cinf) with the recycle stream (of concentration Ceff), the actual concentration of the target reactant in the influent to the reactor is reduced from Cinf. If we ignore this for now, we must realize that our resultant com- putations are conservative with respect to the degree of reduction of the reactant concentration for any given set of constraining parameters. If the value of R is low, the error will be small. As R increases, the error of course increases. We will address mitigation of this error later, when we are ready to advance our capacity for mathematical modeling.
Now we have the form of the solution. Faced with determining the recycle ratio, R, necessary to attain a desired target effluent concentration, we realize we are either into a “guess and check” situation or we must harness some additional capa- bilities of MathCAD. Let us accomplish the latter. We can combine the use of the given-find capability with a MathCAD function. We first convert several of the computations to become functions of R:
We now may invoke the solve block, assigning the find statement into a function Ceff(R), which we may plot against R in Figure E8.6.3 to have a look at the performance of the reactor over a range of recycle ratios:
We may now define Ceff.target and make further use of the written function to iden- tify the value of R that would yield the target effluent concentration:
We double-check the effluent concentration for the determined target R value and of course it matches with our specification.
We may examine the effect of the dilution of the influent with the effluent by including the mass balance for mixing of the influent and recycle streams. This relation must accompany Equation 8.23b in the given-find block. With this addi- tional complication, we are no longer able to easily write a function of R that can be plotted against R. We must invoke this solve block for specific values of R and collect the Ceff results for comparison against our conservative approximations performed initially:
At this point, the most straightforward means of illustrating the comparison bet- ween this approximation and this more accurate solution is to select several values of
0 0.05 0.1
R 0.01
CPFR1(R)
0.1 1 10 100 1×103
0.15
Figure e8.6.3 Plot of effluent Cod concentration versus recycle ratio for a plug-flow with recycle reactor.
R, define a vector (Ceff1) using the approximation, and define a second vector (Ceff2) using the solve block to obtain effluent concentrations for each of the values of R:
The plot of the more accurate result (Ceff.2), considering the dilution of the influent concentration, and the approximation (Ceff.1), considering undiluted influent concentration, are plotted in Figure E8.6.4 against the recycle ratio to illustrate the magnitude of the error. We observe that our original approximate solution, ignoring the dilution of the influent concentration by the recycle stream, is indeed in error.