Oxygen consumption in Pfr-Like reactors

9.8 MOdELing rEactiOns WitH tHE PLUg-fLOW WitH

9.11.2 Oxygen consumption in Pfr-Like reactors

Here, we need some development regarding the prediction of the rate of oxygen con- sumption. In biological waste treatment, we often express the equivalent concentration of biodegradable organics as the ultimate biochemical oxygen demand (BODult) or biodegradable chemical oxygen demand (bCOD). In theory, in the absence of inor- ganic oxygen demanding substances, these two parameters should be equal. Further, were we able to compute the theoretical oxygen demand (ThOD), we would find the three parameters to be essentially the same. Typically, we are not able to determine the theoretical oxygen demand of reactants in wastewaters to be treated. Most often degradable matter includes a suite of specific organic compounds, some of which have yet undefined formulae and structures. ThOD is, however, very useful in cali- brating tests for chemical and biochemical oxygen demand.

00 100 200 300

20 40

z (m, D/S from influent)

c (gbCOD/m^3)

60 PFRN CMFRs SF

80

Figure e9.6.4 a plot of the predicted concentration profiles across a Pf-like reactor as predicted by the Pfr, n-Cmfrs, and sf models.

Fortunately, investigators in the wastewater treatment area have examined biomass from wastewater treatment systems and determined the typical elemental content of the volatile portion of the biomass (VS). The most often published empirical formula for VS is C5H7O2N. Biodegradable organic matter, manifest as BOD or COD is in part converted from its original state to biological cell matter. This portion of the original organic carbon, which is the energy source and carbon source for heterotrophic bacteria, when tied up in cellular mass, acts as an offset for the consumption of oxygen by the process. Thus, if we can compute the ThOD of VSS, we can determine the oxygen consumption offset associated with the net production of biomass.

For VS, we may write a half reaction for the oxidation of the organic carbon to carbon dioxide and couple that with a half reaction for the reduction of molecular oxygen to water. We will address redox half reactions in more detail in Chapter 12:

5 7 2 2 2

C H O N 8H O+ →5CO +20e−+23H+

2 2

5O +20e−+20H+→10H O

The sum of these reactions, after balancing the donated electrons with the accepted electrons, yields the stoichiometric relation we seek for the oxygen consumption offset:

5 7 2 2 2 2

C H O N 5O+ →5CO +3H++2H O

In this analysis, the most important aspect of the result is the ratio of five moles of oxygen offset per unit empirical VS formula. If we convert to mass, we find that the five moles of oxygen comprise 180 g and that the unit empirical formula for the VS is 113 g. The ratio (mass stoichiometry) is that 1.42 g theoretical oxygen demand (or bCOD or BODult) is offset per gram VS produced.

We may return to Chapter 8 and use a combination of Equations 8.11 and 8.13 to produce a hybrid relation from which we may express the rate of oxygen consump- tion in terms of the rates at which biodegradable organic matter is utilized and VS is produced:

2 2

O S O /VSS X.obs

r = r +F r (9.23a)

O2

r is the volume specific rate of oxygen consumption O2

R

M ,

V t

 

 ⋅ 

  FO /VSS2 is

O2 VSS

1.42 g / g , and rS and rX. obs retain the exact definitions of Chapter 8. Oxygen is a reactant and will be consumed, so we expect the rate will have a negative sign. In most cases, the net production of VS will be positive, as a product. However, in some cases, when biodegradable organic matter becomes scarce, the death/decay of bio- mass can occur at a rate higher than the true growth. Equation 9.23a is most useful if we employ units of biodegradable chemical oxygen demand (bCOD ≈ BODult ≈ ThOD) as the unit of measure describing degradable organic substrate abundance.

We can populate the relation for the oxygen consumption rate using the RHSs of Equations 8.11 and 8.13 and perform some algebraic rearrangement to obtain conve- nient relations. We employ the pseudo-first-order approximations used in Example 9.3 to obtain one relation for the specific oxygen-utilization rate:

2 2 2

O .pfo O /VSS max O /VSS D

half

1 XS

r F F k X

Y K

à

 

 

=  −     − (9.23b)

We then employ the saturation rate law of Example 9.4 to produce another relation for rO2:

2 2 2

O .sat O /VSS max O /VSS D

half

1 XS

r F F k X

Y K S

à

 

 

=  −   + − (9.23c)

We can now address the point-wise distribution of oxygen consumption rates using profiles through PFR-like reactors of target reactant generated by our ideal and real reactor models.

Seemingly, we may employ Equations 9.23b and 9.23c directly with concentration profiles generated using the PFR, N CMFRs, and SF models to predict oxygen consumption along the reactor flow path. Such is truly the case for the PFR and TiS models. However, we must look deeper into the assumptions we have made with the SF model. We have subdivided the real reactor into n parallel reactors of equal length so we may correlate reactant concentrations and derived parameters with position across all n parallel reactors. In the limit as n→∞, the flow received by an arbitrary reactor i would be defined as follows:

= Tot⋅ ( )

i i

Q Q E t dt

The volume of the overall reactor is defined as the product of flow and statistical mean residence time:

τ

= ⋅

R Tot SM

V Q

Then, the volume of the ith of the n + 1 parallel reactors is similarly defined:

R.i i i Tot ( )i i

V =Q⋅ =τ Q ⋅E t dt⋅τ

The fraction of the total volume comprising the ith reactor is then further defined:

τ τ

τ τ

⋅ ⋅

= =

Tot ⋅

R Tot

R Tot SM SM

( ) ( )

i i i i

i

V Q E t dt

E t dt

V Q

Lastly, we require that all n reactors have flow path length of L and we choose to subdivide each of the reactors into an equal number of spatial steps. When we sum reaction rates, specific to reactor volumes, across the n reactors we must weight them by the volumes of the respective reactors. Then at any position z along the flow path of the overall reactor, we can integrate any volume-specific reaction rate across the n reactors at position z using the ratio

SM

τi

τ to weight the respective reactor volumes:

τ

= τ

 

= ∫  SM 

0

( )

i

n

z i z i

i

E t dt

r r (9.24)

where rz is the weighted-average specific reaction rate at position z, and

zi

r is the computed specific reaction rate at position z in arbitrary reactor i. We can apply Equation 9.24 to any specific reaction rate operative along the flow path length across all of the n parallel reactors.

We are now ready to compute some oxygen consumption rates along the flow paths of our ideal and nonideal reactors.

Example 9.7 Continue with Example 9.6 and compute the profiles of oxygen con- sumption rates for the reactor performance predictions developed using the PFR, N CMFRs, and SF models.

We have the profiles of Cpfr(z), CNCMFRs(z) and CSF.z(z) and the matrix of CSF values from Example 9.6 that we can use with Equation 9.23b. Then, in order that we can compute the overall oxygen consumption rate by two distinct methods, we will also use these concentration predictions to compute associated predictions of biomass growth and use these in developing a check.

We compute the point-wise specific oxygen consumption rates for the ideal PFR and integrate over LFP to obtain RO .pfr2 :

We then compute the profile of VS production and integrate that along LFP to obtain overall VS production were we to consider biomass growth:

Now we can use the difference between the effluent and influent concentrations and flow with the VS production for another prediction of the overall oxygen con- sumption rate:

We observe reasonably good agreement between the two methods, validating them.

We now examine the stepped CMFRs data for predictions of

O2.

R Here, we compute the rate for each of the Ncount CMFRs in the series. The overall rate is the sum of the products of each rO2 and the volume of each of the reactors. We also use the flow and difference between Cout and Cin and the overall production of VS for a second prediction:

We observe fairly close agreement between the two estimates. We also note that the overall consumption predicted by the N CMFRs approach yields a lower overall oxygen consumption rate than that for the PFR, consistent with the higher value of the effluent target reactant concentration. The N CMFRs in series approach using the stepped profile apparently yields accurate estimates of the oxygen consump- tion. However, the stepped profile is of lesser utility in point-wise predictions than would be a smooth prediction.

We turn our attention to the SF model predictions. We first generate the matrix of rO j,k2 values using Equations 9.23b and 9.24:

We now integrate across the n parallel reactors to obtain the

O2

r versus z profile.

We integrate this profile to obtain the overall oxygen consumption rate.

Then to check with our SF estimate, we first generate the matrix of rGX.SF and integrate that across the n parallel reactors to obtain the rgXz.SF versus z profile and integrate along LFP to obtain RgX.SF:

We now combine the overall production of VS with the change in the target reactant concentration across the reactor to obtain a second estimate or RO2.SF:

We have not attained perfect agreement for the SF model, but certainly the two estimates are quite close. We also note that the overall oxygen consumption rate is lower than that predicted from the N CMFRs in series model, commensurate with the higher predicted value of the target reactant concentration by the SF model.

We are fully aware that the analyses performed in the previous two examples are sim- plistic and likely in error owing to the assumption that the biomass concentration along the reactor flow path is constant. Had we not embraced this assumption, the insightful analyses could not have been completed using closed-form analytic solutions for our PFR or SF reactor systems. Consideration of spatial variability due to biomass growth adds significant complexity to the effort. The CMFRs in series model, as a strictly algebraic solver, can be implemented for multiple reactions—we would simply employ a given-find solve block within the program for each reactor in the series, including rela- tions arising from the mass balance on biomass.

The stepped N CMFRs in series approach is accurate, but of limited use owing to the stepped nature of the C versus z profile. Then by the process of elimination, we under- stand that our best option to accurately model processes within PFR-like reactors is through application of the segregated flow model. Examples hereafter, employing coupled substrate conversion and biomass growth will bear out that postulation.