Approximate Analytical Solutions of Biofilm Reactor Problem in Applied Biotechnology

Abstract

Dueck’s mathematical model of biofilm is discussed. This model is also generalized to cover all substratum types. This system is based on a nonlinear reaction–diffusion equation that includes a nonlinear term related to square law of microbial death rate. In this paper, the analytical expression of the substrate consumption and flux in a biofilm is obtained by solving the nonlinear differential equation using the Taylor series and the modified Adomian decomposition method. Comparing the results obtained with the numerical method and limiting case results shows the validity and usefulness of these techniques.

Appendices

APPENDIX A

Analytical solution of nonlinear reaction Eq. (11)using the TSM method. The steady-state nonlinear equation (11) can be written as follows:

$$S{\kern 1pt} ''(x){{\left[ {1 + S(x)} \right]}^{2}} = \delta {{\left[ {S(x)} \right]}^{2}}.$$

(A1)

By differentiating the above Eq. (A1) successively with respect to “x”, we get the following results.

$$\begin{gathered} S{\kern 1pt} '''(x){{\left[ {1 + S(x)} \right]}^{2}} + 2\left[ {1 + S(x)} \right]S{\kern 1pt} '(x)S{\kern 1pt} ''(x) \\ = \,\,2\delta S(x)S{\kern 1pt} '(x), \\ \end{gathered} $$

(A2)

$$\begin{gathered} S{\kern 1pt} ''''(x){{\left[ {1 + S(x)} \right]}^{2}} + 4S{\kern 1pt} '''(x)\left[ {1 + S(x)} \right]S{\kern 1pt} '(x) \\ + \,\,2{{(S{\kern 1pt} ''(x))}^{2}}\left[ {1 + S(x)} \right] + 2{{(S{\kern 1pt} '(x))}^{2}}S{\kern 1pt} ''(x) \\ = \,\,2\delta (S(x)S{\kern 1pt} ''(x) + {{(S{\kern 1pt} '(x))}^{2}}), \\ \end{gathered} $$

(A3)

$$\begin{gathered} S{\kern 1pt} '''''(x){{\left[ {1 + S(x)} \right]}^{2}} + 6S{\kern 1pt} ''''(x)S{\kern 1pt} '(x)\left[ {1 + S(x)} \right] \\ + \,\,12S{\kern 1pt} '''(x)S{\kern 1pt} ''(x)\left[ {1 + S(x)} \right] \\ + \,\,8S{\kern 1pt} '(x){{(S{\kern 1pt} ''(x))}^{2}} + 2{{(S{\kern 1pt} '(x))}^{2}}S{\kern 1pt} '''(x) \\ = \,\,6\delta S(x)S{\kern 1pt} ''(x) + 2\delta S(x)S{\kern 1pt} '''(x). \\ \end{gathered} $$

(A4)

The boundary condition (12) can be written as

$$S{\kern 1pt} '(0) = 0.$$

(A5)

Using this condition in the above Eqs. (A1) to (A4), we get

$$S{\kern 1pt} ''(0) = \frac{{\delta {{{(S(0))}}^{2}}}}{{{{{(S(0) + 1)}}^{2}}}},$$

(A6)

$$S{\kern 1pt} '''(0) = 0,$$

(A7)

$$S{\kern 1pt} ''''(0) = \frac{{2{{\delta }^{2}}{{{(S(0))}}^{3}}}}{{{{{(S(0) + 1)}}^{5}}}},$$

(A8)

$$S{\kern 1pt} '''''(0) = 0,$$

(A9)

$$S{\kern 1pt} ''''''(0) = \frac{{2{{\delta }^{3}}{{{(S(0))}}^{4}}\left[ {5 - 6S(0))} \right]}}{{{{{(S(0) + 1)}}^{8}}}}.$$

(A10)

The concentration of substrates using the Taylor series is

$$\begin{gathered} S(x) = S(0) + S{\kern 1pt} '(x)\frac{x}{{1!}} \\ + \,\,S{\kern 1pt} ''(x)\frac{{{{x}^{2}}}}{{2!}} + S{\kern 1pt} '''(x)\frac{{{{x}^{3}}}}{{3!}} + ...\,\,. \\ \end{gathered} $$

(A11)

Substituting Eqs. (A5) to (A10) into Eq. (A11), we obtain Eq. (19) in the text.

APPENDIX B

Analytical solution of nonlinear reaction Eq. (14)using the boundary conditions ((15) and (16)) by the modified Adomian decomposition method. In this Appendix, we indicate how Eq. (24) is derived. In order to solve Eq. (14) by means of the modified Adomian decomposition method [46]. We write Eq. (14) in the operator form

$$L\left( S \right) = \delta N(S),$$

(B1)

where $L$ and $N$ are linear and nonlinear terms of Eq. (14). Here, $N(S) = {{\left( {\frac{S}{{1 + S}}} \right)}^{2}}.$ The operator for spherical substratum is defined as [46]

$$L(.) = {{x}^{{ - 1}}}\frac{d}{{dx}}{{x}^{0}}\frac{d}{{dx}}{{x}^{1}}(.).$$

(B2)

Applying the inverse operator [46]

$${{L}^{{ - 1}}}(.) = {{x}^{{ - 1}}}\int\limits_1^{x - m} {{{x}^{0}}} \int\limits_0^x {x(.)} dxdx,$$

(B3)

on both sides of Eq. (B1) yields

$$S\left( x \right) = Ax + B + \delta {{L}^{{ - 1}}}{{\left( {\frac{S}{{1 + S}}} \right)}^{2}},$$

(B4)

where $A$ and $B$ are constants of integration. Let

$$S\left( x \right) = \sum\limits_{i = 0}^\infty {{{S}_{i}}} \left( x \right),$$

(B5)

$$N\left( {S(x)} \right) = \sum\limits_{i = 0}^\infty {{{A}_{i}}} .$$

(B6)

In view of Eqs. (B4) to (B6) gives

$$\sum\limits_{i = 0}^\infty {{{S}_{i}}\left( x \right)} = Ax + B + \delta {{L}^{{ - 1}}}\sum\limits_{i = 0}^\infty {{{A}_{i}}} .$$

(B7)

The zero elements is defined as

$${{S}_{0}}\left( x \right) = Ax + B,$$

(B8)

and the remaining components as the recurrence relation ${{S}_{{i + 1}}}\left( x \right) = \delta {{L}^{{ - 1}}}{{A}_{i}},$ $i \geqslant 0$

where ${{A}_{i}}(x) = \frac{1}{{i!}}\frac{{{{d}^{i}}}}{{d{{\lambda }^{i}}}}N$${{\left( {\sum\limits_{n = 0}^\infty {{{\lambda }^{i}}{{S}_{i}}} } \right)}_{{\lambda = 0}}}$ are the Adomian polynomials of ${{S}_{1}},{{S}_{2}},...{{S}_{i}}$. We can find ${{A}_{i}}$ as follows:

$${{A}_{0}} = N\left( {{{S}_{0}}} \right) = {{\left( {\frac{{{{S}_{0}}}}{{1 + {{S}_{0}}}}} \right)}^{2}},$$

(B9)

$${{A}_{1}} = {{\left[ {\frac{d}{{d\lambda }}N\left( {{{S}_{0}} + \lambda {{S}_{1}}} \right)} \right]}_{{\lambda = 0}}} = \frac{{2{{S}_{0}}{{S}_{1}}}}{{{{{(1 + {{S}_{0}})}}^{3}}}}.$$

(B10)

The remaining polynomials can be easily generated and so on

$${{S}_{0}}(x) = {{S}_{L}},$$

(B11)

$${{S}_{1}}(x) = \frac{\delta }{6}{{\left( {\frac{{{{S}_{L}}}}{{1 + {{S}_{L}}}}} \right)}^{2}}\left[ {{{{(x - m)}}^{2}} - 1} \right],$$

(B12)

$$\begin{gathered} {{S}_{2}}(x) = \frac{{{{\delta }^{2}}S_{L}^{3}}}{{3{{{(1 + {{S}_{L}})}}^{5}}}}\left[ {\frac{{{{x}^{4}}}}{{20}} - \frac{{m{{x}^{3}}}}{6} + \frac{{({{m}^{2}} - 1){{x}^{2}}}}{6}} \right. \\ \left. { - \,\,\frac{{({{m}^{2}} - 10)mx}}{{30}} - \frac{{({{m}^{4}} + 8{{m}^{2}} + 2m - 7)}}{{60}}} \right]. \\ \end{gathered} $$

(B13)

Adding Eqs. (B11) to (B13), we get the concentration of substrate in the biofilm (Eq. (24)).

NOTATION

L f	biofilm thickness, m
S f	concentration of substrate in the biofilm, g/m3
S 1	concentration of substrate outside the biofilm, g/m3
D f	diffusion coefficient within the biofilm, m2/s
z	distance from biofilm, m
ψ	flux of substrate, g/m2 d
b	microbial death constant, m3/g d
K	Michaelis-Menten constant, g/m3
X f	position of an occupied grid cell above the substratum within the grid, g/m3
q	rate constant of substrate consumption, d–1
Y	yield per unit amount of substrate consumed in biofilm, g/m3
R	radius of the sphere, m
S	dimensionless concentration of substrate in the biofilm
S L	dimensionless concentration of substrate outside the biofilm
x	dimensionless distance from the biofilm
δ	dimensionless biofilm thickness
m	dimensionless constant

SUBSCRIPTS AND SUPERSCRIPTS

f	film
L, 1	solution