Let us return to our ODE describing the overall mass balance in terms of the general specific reaction rate. We may insert the pseudo-first-order rate law into the relation, divide through by VR, and rearrange the relation to yield a succinct ODE:
Cin C k C dC
dt
τ τ
− − ⋅ ′ ⋅
=
This result is separated and integrated (easily done using MathCAD’s calculus palette) between the limits of C0 and Ct and t0 and t:
0 in 0 in
0
ln ln
1 1
1
t t
C C C k C C C k
k k t t
k
τ τ
τ τ
τ ′ τ
− + ⋅ ⋅ ′ − + ⋅ ⋅ ′
−
+ ⋅ ′ + ⋅ ′ = −
+ ⋅
This intermediate result may be rearranged to yield either an explicit relation for time as a function of Ct or for Ct as a function of time:
0 in 0
0 in
ln 1
t t
C C C k
C C C k
t t
k τ τ
τ τ
− + ⋅ ⋅ ′
− + ⋅ ⋅ ′
− =
+ ⋅ ′
(8.27a)
in ( 0 in 0 ) exp 0(1 )
t 1
C C C C k t t k
C k
τ τ τ
τ
−
+ − + ⋅ ⋅ ′ ⋅ − + ⋅ ′
= + ⋅ ′
(8.27b)
Equations 8.27a and 8.27b are useful if we are interested in analyzing the progres- sion of a reactor between startup and the steady-state condition or the progression from one steady-state condition to another, where influent conditions need to be changed. We can certainly observe significant similarities between Equations 8.27 and 7.5. The inclusion of reaction with mixing simply requires an enhancement of the basic structure of the mathematical result.
Use of the saturation rate law for unsteady-state analyses in CMFRs leads to a differential equation for which a closed-form analytic solution would be extremely difficult to obtain at best:
in
half
C C
dC k X C
dt τ K C
− ′ ⋅ ⋅
= −
+
Attempts to develop that closed-form solution are beyond the scope of the current discussion. However, numerous methods are available for obtaining approximate, numeric solutions for both single and systems of ODEs.
8.3.5.2 The Fed-Batch reactor
For the fed-batch reactor, we simply include a flow and associated concentration as influent to the batch reactor in the mass balance. A significant complication arises in that the volume of the reactor is no longer a constant. We write the overall mass balance as an ODE, employing first the pseudo-first-order rate law.
R in R
( )
d V C
Q C V k C
dt = ⋅ − ⋅ ′ ⋅
We expand the LHS using the chain rule, use the definition that dVR
dt =Q, and employ the relation that VR = V0 + Qãt:
0 in 0
( )dC ( )
Q C V Q t Q C V Q t k C
⋅ + + ⋅ dt = ⋅ − + ⋅ ⋅ ′ ⋅
We rearrange the result to collect coefficients of dC/dt and C, and rearrange further such that the coefficient of dC/dt = 1:
in
0 0
Q C
dC Q
dt V Q t k C V Q t
⋅
+ + ⋅ + ′ = + ⋅
This result is a linear, first-order ODE. We use the integrating factor prescribed by Wylie (1966) to obtain the general solution and employ the initial condition that at t = 0, C = C0 to evaluate the constant of integration. Some further rearranging yields the final relation for C as a function of time:
in in 0
0 0
e e
k t V k t
C C Q
C C
k k Q V Q t
′⋅ − ′⋅
⋅
= ′ − ′ + ⋅ ⋅ + ⋅ ⋅ (8.28)
This result may then be applied for analyses of reactions in fed-batch reactors with a pseudo-first-order rate law, when k′ is constant.
Consideration of the saturation rate law for a fed-batch reactor leads to a sim- ilar, but more mathematically complex, ODE than that for the pseudo-first-order rate law:
in
0 half
( )
( )
Q C C
dC k X C
dt V Q t K C
− ′ ⋅ ⋅
= −
+ ⋅ +
We quickly observe that the resultant ODE is nonlinear, with noninteger powers of C on the RHS. Attempts to develop a closed-form solution for this case would be beyond the scope intended herein. We realize that since we have isolated dC/dt on the LHS, we would be able to readily develop numeric approximations using Euler’s or one of the Runge–Kutta methods.
8.3.5.3 Time-Variant Analyses of the pFr
We might ponder development of relations for the unsteady case for the PFR. These would undoubtedly result in partial differential equations (PDEs) in time and posi- tion. Development of closed-form analytic solutions or examinations of numerical approximations of these PDEs are well beyond the scope of the discussions intended herein.
8.4 aPPLicatiOns Of rEactiOns in idEaL rEactOrs
In order that we may apply reaction/reactor principles in analyses of environmental systems of interest we must first answer a few questions:
1. What is the specific system in question – are the boundaries real and physical or must we envision a representative reactor volume (RRV)?
2. What is the reactant of interest and what is the overall reaction?
3. What is the law expressing the rate of reaction in terms of rate coefficients and abundances of products and reactants?
4. What ideal reactor system can best describe the system in which the process takes place?
Often the system is well defined by real boundaries – the walls of an open biological reactor basin, the sediment–water and vapor–liquid interfaces of a water body, or the tankage associated with a fermentative process for ethanol production. In natural sys- tems, the boundaries are not always well defined – the mixing and reaction zone beneath a solid-waste landfill or the porous medium through which contaminated groundwater flows. In these cases, we need to examine whether we need real bound- aries or whether we must visualize the boundaries of the system to be investigated.
For this latter case, we must apply the concept of the RRV. A RRV would comprise a unit cross-sectional area and known reactor length to which we may apply the appropriate reactor model. The most significant property of a RRV is that the RRV is identical in all ways to all other such volumes comprising the overall system repre- sented by the RRV.
Research into the specific reactants of interest, oftentimes toxic or hazardous components or oxygen-demanding substances, is necessary to identify the specific reaction. Particularly with many pseudo-first-order reactions, the abundances of all reactants contribute to the forward rate of the reaction while abundances of prod- ucts tend to reduce the forward rate of the reaction. Herein, we consider reactions
with rates that are predominantly dependent upon reactants while having little or no inverse dependence upon the abundances of products. Specific reactions have been very well defined by chemists, biochemists, and microbiologists, and herein we do not attempt to compile a database of chemical and biochemical reactions. Rather, our focus will be upon the process for application of the chemistries once they are iden- tified and understood. Most often, environmental process analysis requires specific research by the engineer to develop necessary understandings of the process or chemistry.
The science associated with chemical reaction kinetics is highly empirical in practice. The magnitudes of rate coefficients and dependences of those rate coeffi- cients upon the abundances of reactants are derived almost wholly from experiment.
Certain reactions have well-known (as a consequence of significant experimental examination) kinetics while the kinetics of others are poorly quantitatively under- stood. Thus, the rate laws employed in environmental process analysis often need to be defined from the scientific and engineering literature on a case-by-case basis as necessary to analyses of targeted environmental systems. Herein, we do not attempt to develop a database of kinetic information, but rather use information that we are able to identify in the examination of targeted systems. Our focus is upon the use of quantitative kinetic information, once known, in the analyses of environ- mental processes.
The choice of a reactor system is dependent upon the character of the environ- mental system in question. No real reactor is perfectly ideal in the context of either plug flow or complete mixing. In some cases, analyses of real reactors using ideal reactor principles would lead to significant error. In a subsequent chapter, we examine some strategies useful in analyses of nonideal reactors. Herein, however, we strive to identify the aspects of examined systems that would permit us to select one of the three ideal reactor systems (or one of the modifications discussed earlier) with which we may apply the known chemistry and quantitative understandings of reaction rates.