We analyse the steady-state operation of two types of reactor cascade without recycle. The first is a standard reactor cascade in which the feed stream enters into the first reactor. The second is a step-feed reactor cascade in which an equal proportion of the feed stream enters each reactor in the cascade. The reaction is assumed to be a biological process governed by Monod growth kinetics with a decay coefficient for the microorganisms. The steady-states of both models are found for an arbitrary number of reactors and their stability determined as a function of the residence time. We show that in a step-feed reactor cascade the substrate and biomass concentrations leaving the reactor of the cascade are identical to those leaving the first reactor of the cascade. We further show that this result is true for a general specific growth rate of the form μ (S,X). Thus for such processes the non-standard cascade offers no advantage over that of a single reactor. This is surprising because the use of a non-standard cascade has been proposed as a mechanism to improve the biological treatment of wastewater.
Funding statement: Research funding: Sultan Qaboos University (Grant/Award Number: “IG/SCI/DOMS/08/04”, “IG/SCI/DOMS/14/04”).
During part of this work MIN was a Visiting Fellow in the School of Physical Environmental & Mathematical Sciences (PEMS), UNSW@ADFA. He thanks the members of PEMS for their collegiality.
A.1 Steady-state analysis
Here we show that the solution of eq. (16) in the second reactor of a cascade () is only physically meaningful ( and ) when the negative square root sign is taken. In doing so the only property of and that we use is that they are positive.
The steady-state equations for the concentrations inside the third reactor of a cascade () are identical to those of the second reactor in the cascade except that the all indices have increased by one. It immediately follows that the physically meaningful steady-state solution in the third reactor, and hence any reactor in the cascade, is given by the negative square root in eq. (16).
In Section A.1.1 we establish the desired result for the non-degenerate case with . In Section A.1.2 we show that the steady-state solution is physically meaningful in the degenerate case that .
A.1.1 Positivity of the steady-state solution ()
Consider the function
The coefficient is strictly positive because , and .
The roots of eq. (25) are given by
By calculation we have
as , and .
In the case when the calculations (27) and (28) lead to the conclusion that eq. (25) always has two solutions: one in the region and one in the region . The solution with is not physically meaningful: eq. (14) shows that the corresponding concentration of microorganisms is negative. It follows that the solution of interest corresponds to the negative square root sign in eq. (26).
In the case when the calculations (27) and (28) lead to the conclusion that eq. (25) always has two solutions: a positive solution in the region and a negative solution . The latter solution is not physically meaningful. It follows that the solution of interest corresponds to the negative square root sign in eq. (26).
A.1.2 Positivity of the steady-state solution ()
In this section we show that the solution of eq. (16) is positive in the degenerate case when . The case happens when . Now a straightforward calculation shows that
B.1 Symbols used
A subject j refers to a property of the jth reactor in a reactor cascade containing n reactors.
|F||Flowrate through the bioreactor.|
|Dimensionless substrate concentration.||(—)|
|Dimensionless substrate concentration along the no-washout solution branch.||(–)|
|Substrate concentration in the feed.|
|Dimensionless substrate concentration in the feed.||(—)|
|SFRC||Step-feed reactor cascade.|
|SRC||Standard reactor cascade.|
|Volume of a bioreactor.|
|Concentration of microorganisms.|
|Dimensionless microorganism concentration.||(—)|
|Concentration of microorganisms in the feed.|
|Dimensionless microorganism concentration in the feed.||(—)|
|Dimensionless death coefficient.||(—)|
|n||The number of reactors in a SRC or a SFRC.|
|Specific growth rate model.|
|Maximum specific growth rate.|
|Average residence time in a SFRC.|
|Nominal residence time through a SFRC with reactors of equal size.|
|Dimensionless residence time.||(—)|
|The value of the dimensionless residence time at the transcritical bifurcation.||(—)|
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