Correspondence Comment on “Wet Oxidation Lumped Kinetic Model for Wastewater Organic Burden Biodegradability Prediction”
and the solution to eq 6 is then written as
The authors (1) are to be commended for the development of a lumped kinetic model that establishes the link between the chemical oxygen demand (COD) and the biochemical oxygen demand (BOD) of wastewater during wet oxidation. We will show that the lumped kinetic model can be solved analytically by a straightforward procedure. The simple analytical solution may be used conveniently by people with or without the numerical experience to understand the lumped kinetic model and to predict the biodegradability of wastewater after wet oxidation. The lumped kinetic model developed by the authors (1) is in a system of four first-order ordinary differential equations (ODEs):
[A](k3/(k1+k2))
[BL] )
1 p1(A)
(∫
[A]
-
A[0]
(∫
k2
[A]
(-k2)
A[0]
(k1 + k2)
d[A] ) (k1 + k2)[A] dt
(1)
-
d[BL] ) k3[BL] - k2[A] dt
(2)
-
d[BS] ) k4[BS] - k3[BL] dt
(3)
)
where ξ is a dumpy variable. Evaluating the integral in eq 8 expresses lump BL explicitly as a function of lump A:
[BL] ) -R1[A] + R2[A](k3)/(k1+k2)
(9)
where
k2
(10)
(k1 + k2 - k3)
and
R2 )
k2
d[C] ) -k4[BS] - k1[A] dt
BL[0]A[0]((-k3)/(k1+k2)) (11) Dividing eq 3 by eq 1 gives
d[BS] (4)
The initial conditions are [A] ) A[0], [BL] ) BL[0], [BS] ) BS[0], and C ) C[0] ) 0 at t ) 0. By separating variables, eq 1 is integrated directly to give
[A] ) A[0]e-(k1+k2)t
A[0]((k1+k2-k3)/(k1+k2)) +
(k1 + k2 - k3)
d[A] -
ξ((-k3)/(k1+k2)) dξ + BL[0]A[0]((-k3)/(k1+k2)) (8)
R1 ) -
)
p1(ξ) dξ + BL[0]p1(A[0]) )
(k1 + k2)
-
(-k3) [BL] 1 [BS] ) (k1 + k2) [A] (k1 + k2) [A] k4
(12)
Subsequently, substituting eq 9 into eq 12 gives
d[BS] d[A]
-
k4 1 [B ] ) (k1 + k2) [A] S (-k3) [-R1 + R2[A]((k3-(k1+k2))/(k1+k2))] (13) (k1 + k2)
(5)
which shows that the concentration of lump A decays exponentially with the reaction or wet oxidation time, t. Since the four first-order ODEs given by eqs 1-4 are linear and autonomous, the concentrations of lumps BL, BS, and C can be expressed exclusively as functions of the concentration of lump A, respectively. Dividing eq 2 by eq 1 gives
Equation 13 is also a first-order linear ODE with variable coefficients. Following the procedure outlined by Hindebrad (2) again, an integration factor for eq 13 is defined as
p2(A) ) e∫((-k4)/(k1+k2))(1/[A]) d[A] ) e((-k4)/(k1+k2)) ln[A] ) [A]((-k4)/(k1+k2)) (14) and the solution to eq 13 is then written as
d[BL] d[A]
-
k2 1 [BL] ) (k1 + k2) [A] (k1 + k2) k3
(6)
Equation 6 is a first-order linear ODE with a variable coefficient. Following the procedure outlined by Hindebrad (2), an integration factor for eq 6 is defined as
p1(A) ) e∫((-k3)/(k1+k2))(1/[A])d[A] ) e((-k3)/(k1+k2))ln[A] ) [A]((-k3)/(k1+k2)) (7) 10.1021/es020910b CCC: $25.00 Published on Web 02/05/2003
2003 American Chemical Society
[BS] )
1 p1(A)
(∫
[A]
(-k3)
A[0]
(k1 + k2)
[-R1 +
)
R2ξ((k3-(k1+k2)/(k1+k2)))]p2(ξ) dξ + BS[0]p2(A[0]) )
(∫
[A](k4/(k1+k2))
[A]
k3
A[0]
(k1 + k2)
[R1 -
R2ξ((k3-(k1+k2)/(k1+k2)))]ξ((-k4)/(k1+k2)) dξ +
)
BS[0]A[0]((-k4)/(k1+k2)) (15) VOL. 37, NO. 6, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
9
1225
Evaluating the integral in eq 15 expresses lump BS explicitly as a function of lump A:
[BS] ) R3[A] - R4[A](k3/(k1+k2)) + R5[A](k4/(k1+k2)) (16)
BOD ) [BL] + [BS] ) (R3 - R1)[A] +
where
R3 )
(R2 - R4)[A]((k3)/(k1+k2)) + R5[A]((k4)/(k1+k2)) (23)
k3
R1
(17)
(k1 + k2 - k4)
R4 )
k3
R2
(18)
(k3 - k4)
and
R5 ) BS[0]A[0]
((-k4)/(k1+k2))
- R3A[0]
((k1+k2-k4)/(k1+k2))
+
Adding eqs 1-4 together gives a simple relation among the reaction rates:
d[A] d[BL] d[BS] d[C] + + + )0 dt dt dt dt
(20)
(21)
where the concentrations of lumps BL and BS are given by eqs 9 and 16, and CT[0] is the total initial concentration defined as
CT[0] ) A[0] + BL[0] + BS[0] + C[0]
(22)
Equation 21 with eq 22 is the overall mass balance for the reaction system (batch reactor or plug flow reactor at the steady state). Given that lumps A, BL, BS, and C are expressed in equivalent concentrations such as in oxygen demands, the total concentration in the reaction system remains constant or invariable with the reaction or wet oxidation time. The concentrations of lumps BL, BS, and C are expressed explicitly as elementary functions of the concentration of lump A in eqs 9, 16, and 21, respectively. By substituting eq 5 into eqs 9, 16, and 21, the concentrations of lumps BL, BS, and C can also be expressed explicitly as elementary functions of wet oxidation time t, respectively. The lumped kinetic model (1) relates the concentrations of the four lumps to the BOD and the COD of wastewater by [A] ) COD - BOD, [BL] + [BS] ) BOD, and [C] ) COD0 COD, where COD0 is the initial COD at t ) 0. With those
9
COD ) [A] + BOD ) (1 + R3 - R1)[A] + (R2 - R4)[A]((k3)/(k1+k2)) + R5[A]((k4)/(k1+k2)) (24)
BOD ) (R3 - R1)A[0]e-(k1+k2)t + (R2 - R4)A[0](k3/(k1+k2))e-k3t + R5A[0](k4/(k1+k2))e-k4t (25) and
COD ) (1 + R3 - R1)A[0]e-(k1+k2)t + (R2 - R4)A[0](k3/(k1+k2))e-k3t + R5A[0](k4/(k1+k2))e-k4t (26)
Integrating eq 20 then yields the solution for lump C:
[C] ) CT[0] - [A] - [BL] - [BS]
The corresponding COD is obtained by adding the concentration of lump A to the BOD:
Subsequently substituting eq 5 into eqs 23 and 24 expresses the BOD and the COD of wastewater as functions of wet oxidation time t, respectively:
R4A[0]((k3-k4)/(k1+k2)) (19)
1226
assumptions, A[0] ) COD0 - BOD0, CT[0] ) COD0, and BL[0] + BS[0] ) BOD0, where BOD0 is the initial BOD at t ) 0. Adding eq 9 to eq 16 then expresses the BOD exclusively as a function of lump A:
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 37, NO. 6, 2003
where A[0] ) COD0 - BOD0. The analytical solution given by eqs 25 and 26 may be used conveniently by people with or without the numerical experience to determine the BOD, the COD, and the ratio of the BOD vs the COD in wastewater after wet oxidation.
Literature Cited (1) Verenich, S.; Kallas, J. Environ. Sci. Technol. 2002, 36, 33353339. (2) Hildebrand F. B. Advanced Calculus for Applications, 2nd ed.; Prentice-Hall: New York, 1976.
S. Qi* 4111 Newmark CEL Department of Civil & Environmental Engineering University of Illinois Urbana, Illinois 61801
K. J. Hay U.S. Army Engineering Research and Development Center Construction Engineering Research Lab Champaign, Illinois 61826 ES020910B
