New Analytical Expresions of Concentrations in Packed Bed Immobilized-Cell Electrochemical Photobioreactor

1. Introduction

Nonlinear systems of equations usually present real-world physical applications. Over the past three decades, the search for efficient and reliable analytical asymptotic methods to solve these systems has intensified [1]. Of the methods have received a great attention are the perturbation method (PM) [4], homotopy perturbation method (HPM) [5,6,7,8,9,10,11], variational iteration method (VIM) [12,13,14,15,16], homotopy analysis method (HAM)[17,18,19,20], differential transform method (DTM) [21,22], Adomian decomposition method (ADM) [23,24], and Green’s function iterative method [25,26].

Two-phase flow and mass transport coupled with biochemical reactions co-occur in an operating photobioreactor. Therefore, a better understanding of the complicated transport mechanism will promote the application of electrochemical photobioreactors[28,29].This communication directs our attention to the analytical and numerical method of solving nonlinear equations in immobilized-cell photobioreactor [2,3]. A comparison study between Akbari-Ganji's method (AGM) [27,28,29] and the renowned Taylors series method (see [30,31] and the references therein are presented. We aim to derive approximate analytical expressions of the concentrations of substrate and product and both liquid and gas phases for various parameter values by using these two methods. This paper will present a profile of reliability, efficiency, and convergence of both approaches.

2. Mathematical Formulation of the Problems

An immobilized-cell photo bioreactor packed with transparent gel-granules containing immobilized PSB developed and modeled by Shirejini et al. [3] is illustrated in Figure 1(a).Figure 1(b) shows a schematic diagram of mass transfer in a single gel graule, such that the mass transfer of all reactants and products is dominated by diffusion. Using Fick’s law, the mass transport equations for substrate and the hydrogen inside the gel granules are given by[32]:

Figure 1. Schematic of the entrapped-cell photobioreactor.

Figure 1. Schematic of the entrapped-cell photobioreactor.

$${\mathrm{D}}_{\mathrm{granule}}^{\mathrm{S}}\left[\frac{1}{r^2}\frac{\mathrm{d}}{\mathrm{d}r}\left(r^2\frac{\mathrm{d}{\mathrm{C}}_{\mathrm{granule}}^{\mathrm{S}}}{\mathrm{d}r}\right)\right]={\varphi }_{\mathrm{granule}}^{\mathrm{S}}$$

(1)

$${\mathrm{D}}_{\mathrm{granule}}^{{\mathrm{H}}_2}\left[\frac{1}{r^2}\frac{\mathrm{d}}{\mathrm{d}r}\left(r^2\frac{\mathrm{d}{\mathrm{C}}_{\mathrm{granule}}^{{\mathrm{H}}_2}}{\mathrm{d}r}\right)\right]={\varphi }_{\mathrm{granule}}^{{\mathrm{H}}_2}$$

(2)

$${\varphi }_{\mathrm{granule}}^{\mathrm{S}}=\frac{\mu X_{\mathrm{cell}}}{Y_{X/S}^{*}}+m{X}_{\mathrm{cell}}, {\varphi }_{\mathrm{granule}}^{{\mathrm{H}}_2}=\frac{{\alpha }^{*}\mu X_{\mathrm{cell}}}{Y_{X/S}^{*}}+\beta X_{\mathrm{cell}}, {\mu }_{\max }=\frac{C_{\mathrm{granule}}^{\mathrm{S}}}{K_{\mathrm{S}}+C_{\mathrm{granule}}^{\mathrm{S}}}$$

(3)

where${\mathrm{D}}_{\mathrm{granule}}^{\mathrm{S}}$and ${\mathrm{D}}_{\mathrm{granule}}^{{\mathrm{H}}_2}$are the significant diffusion coefficients of glucose and hydrogen in the gelgranule,${\mathrm{C}}_{\mathrm{granule}}^{\mathrm{S}}$and ${\mathrm{C}}_{\mathrm{granule}}^{{\mathrm{H}}_2}$are the local concentrations of glucose and hydrogen inside the gel granule,${\varphi }_{\mathrm{granule}}^{\mathrm{S}}$ is the consumption rate of glucose, ${\varphi }_{\mathrm{granule}}^{{\mathrm{H}}_2}$is the generation rate of hydrogen. The parameter ${\alpha }^{*}$represents the growth associated kinetic constant for hydrogenproduction, $\beta$is the non-growth associated kinetic constant,$X_{\mathrm{cell}}$is the initial cell density,$Y_{X/S}^{*}$is the cell yield, and $K_{\mathrm{S}}$is a Monod constant. Furthermore, ${\mu }_{\max }$denotes the maximum specific growth rate, and $\mu$is the growth rate. The boundary conditions for the above system of Equations(1) and (2) are given by

$${\left.\frac{\mathrm{d}{\mathrm{C}}_{\mathrm{granule}}^{\mathrm{S}}}{\mathrm{d}r}\right|}_{r=0}={\left.\frac{\mathrm{d}{\mathrm{C}}_{\mathrm{granule}}^{{\mathrm{H}}_2}}{\mathrm{d}r}\right|}_{r=0}=0$$

(4)

$${\left.{\mathrm{C}}_{\mathrm{granule}}^{\mathrm{S}}\right|}_{r=R}={\mathrm{C}}_{\mathrm{l}}^{\mathrm{S}}, {\left.{\mathrm{C}}_{\mathrm{granule}}^{{\mathrm{H}}_2}\right|}_{r=R}={\mathrm{C}}_{\mathrm{g}}^{{\mathrm{H}}_2}$$

(5)

where ${\mathrm{C}}_{\mathrm{l}}^{\mathrm{S}}$and ${\mathrm{C}}_{\mathrm{g}}^{{\mathrm{H}}_2}$are bulk solutions, and $R$is the radius of catalyst. The source terms${\phi }^s,{\phi }^{H_2}$and ${\phi }^{C{O}_2}$are defined by:

$${\phi }^s=\alpha D_{granule}^s{\left(\frac{d{C}_{granule}^s}{dr}\right)}_{r=R},{\phi }^{H_2}=\alpha D_{granule}^{H_2}{\left(\frac{d{C}_{granule}^{H_2}}{dr}\right)}_{r=R},{\phi }^{C{O}_2}=\frac{M^{C{O}_2}}{2M^{H_2}}{\phi }^{H_2}$$

(6)

where $\alpha$is the specific area of the gel granule. The source terms of the liquid phase and gas phase are expressed by:

$$m_l^{•}=-{\phi }^s,m_g^{•}={\phi }^{H_2}+{\phi }^{C{O}_2}$$

(7)

By introducing the dimensionless parameters

$${\mathsf{\Phi }}_1=\frac{X_{\mathrm{cell}} {\mu }_{max}{R}^2}{Y_{\frac{X}{S}}^{*} {\mathrm{D}}_{\mathrm{granule}}^{\mathrm{S}}{K}_{\mathrm{S}}}, {\mathsf{\Phi }}_2=\frac{X_{\mathrm{cell}} {\mu }_{max}{R}^2{\mathrm{C}}_{\mathrm{l}}^{\mathrm{S}}}{Y_{\frac{X}{S}}^{*} {\mathrm{D}}_{\mathrm{granule}}^{\mathrm{S}}{K}_{\mathrm{S}}{\mathrm{C}}_{\mathrm{g}}^{{\mathrm{H}}_2}},{\gamma }_1=\frac{X_{\mathrm{cell}}m R^2}{ {\mathrm{D}}_{\mathrm{granule}}^{\mathrm{S}}{K}_{\mathrm{S}}},{\gamma }_2=\frac{X_{\mathrm{cell}}m R^2}{ {\mathrm{D}}_{\mathrm{granule}}^{\mathrm{S}}{\mathrm{C}}_{\mathrm{l}}^{\mathrm{S}}},{\gamma }_3=\frac{X_{\mathrm{cell}}m R^2{\mathrm{C}}_{\mathrm{l}}^{\mathrm{S}}}{ {\mathrm{D}}_{\mathrm{granule}}^{\mathrm{S}}{K}_{\mathrm{S}}{\mathrm{C}}_{\mathrm{g}}^{{\mathrm{H}}_2}},{\gamma }_4=\frac{X_{\mathrm{cell}}m R^2}{ {\mathrm{D}}_{\mathrm{granule}}^{H_2}{\mathrm{C}}_{\mathrm{g}}^{{\mathrm{H}}_2}}$$

$$\zeta =\frac{r}{R},u=\frac{C_{granule}^s}{{\mathrm{C}}_{\mathrm{g}}^s}, v=\frac{C_{granule}^{H_2}}{{\mathrm{C}}_{\mathrm{g}}^{H_2}} , {\alpha }_1=\frac{{\mathrm{C}}_{\mathrm{l}}^{\mathrm{S}}}{K_{\mathrm{S}}}$$

(8)

Equations (1) and (2),take the dimensionless forms:

$$\frac{1}{{\zeta }^2}\frac{d}{d\zeta }\left({\zeta }^2\frac{du\left(\zeta \right)}{d\zeta }\right)=\frac{({\phi }_1+{\gamma }_1)u\left(\zeta \right)+{\gamma }_2}{(1+{\alpha }_{1}u(\zeta \left)\right)}$$

(9)

$$\frac{1}{{\zeta }^2}\frac{d}{d\zeta }\left({\zeta }^2\frac{dv\left(\zeta \right)}{d\zeta }\right)=\frac{({\phi }_2+{\gamma }_3)u\left(\zeta \right)+{\gamma }_4}{(1+{\alpha }_{1}u(\zeta \left)\right)}$$

(10)

with the dimensionless boundary conditions:

$$\frac{du\left(\zeta \right)}{d\zeta }=\frac{dv\left(\zeta \right)}{d\zeta }=0 \mathrm{at} \zeta =0$$

(11)

$$u\left(\zeta \right)=v\left(\zeta \right)=1\mathrm{at}\zeta =1$$

(12)

The normalized steady-state source terms of liquid and gas phases is given by

$${\psi }_l=\frac{m_l^{•}R}{\alpha D_{granule}^s{C}_l^s}=-{\left(\frac{du}{d\zeta }\right)}_{\zeta =1}$$

(13)

$${\psi }_g=\frac{m_g^{•}R}{\alpha D_{granule}^{H_2}{C}_g^{H_2}}=\left(\frac{2+\omega }{2}\right){\left(\frac{dv}{d\zeta }\right)}_{\zeta =1}\mathrm{where}\omega =\frac{M^{C{O}_2}}{M^{H_2}}$$

(14)

3. Analytical expression of the concentrations using Akbari-Ganji’s method

In this section, we use AGM to derive explicit expressions for the concentrations of glucose and hydrogen. The AGM is a semi-analytical approach that has shown efficacy in solving nonlinear equations [33,34,35]. The AGM procedure begins by assuming a solution function with unknown constant coefficients, which are determined by solving a system of algebraic equations that is constructed from the differential equations and the initial conditions.

3.1. Concentration of Glucose (substrate)

Let

$$u\left(\zeta \right)=a_0+a_1\zeta +a_2{\zeta }^2$$

(15)

be a trial solution of Equation (9), where$a_0,a_1,a_2$are constants. The boundary conditions (11) and (12) imply

$$1=a_0+a_1+a_2,a_1=0.$$

(16)

Now define the function $F$by

$$F\left(\zeta \right):\frac{1}{{\zeta }^2}\frac{d}{d\zeta }\left({\zeta }^2\frac{du\left(\zeta \right)}{d\zeta }\right)-\frac{({\phi }_1+{\gamma }_1)u\left(\zeta \right)+{\gamma }_2}{(1+{\alpha }_{1}u(\zeta \left)\right)}=0$$

(17)

then

$$F(\zeta =1):2a_1+6a_2-\frac{({\phi }_1+{\gamma }_1)(a_0+a_1+a_2)+{\gamma }_2}{(1+{\alpha }_1(a_0+a_1+a_2\left)\right)}=0$$

(18)

Using Equation (16) in Equation (18), gives

$$a_2=\frac{{\phi }_1+{\gamma }_1+{\gamma }_2}{6(1+{\alpha }_1)},a_0=1-\frac{{\phi }_1+{\gamma }_1+{\gamma }_2}{6(1+{\alpha }_1)}$$

(19)

By substituting Equations (16) and (18) into Equation (15), the analytical expression for the substrate is

$$u\left(\zeta \right)=1+\frac{{\phi }_1+{\gamma }_1+{\gamma }_2}{6(1+{\alpha }_1)}({\zeta }^2-1)$$

(20)

3.2. Concentration of Hydrogen (product)

Assume the following solution to Equation (10):

$$v\left(\zeta \right)=b_0+b_1\zeta +b_2{\zeta }^2$$

(21)

From boundary conditions (11) and (12), we obtain:

$$1=b_0+b_1+b_2, \mathrm{and} b_1=0.$$

(22)

Define the G function by

$$G\left(\zeta \right):\frac{1}{{\zeta }^2}\frac{d}{d\zeta }\left({\zeta }^2\frac{dv\left(\zeta \right)}{d\zeta }\right)-\frac{({\phi }_2+{\gamma }_3)u\left(\zeta \right)+{\gamma }_4}{(1+{\alpha }_{1}u(\zeta \left)\right)}=0,$$

(23)

then

$$G(\zeta =1):2b_1+6b_2-\frac{({\phi }_2+{\gamma }_3)(a_0+a_1+a_2)+{\gamma }_4}{(1+{\alpha }_1(a_0+a_1+a_2\left)\right)}=0$$

(24)

Substituting Equation (22) into Equation (24) leads to

$$b_2=\frac{{\phi }_2+{\gamma }_3+{\gamma }_4}{6(1+{\alpha }_1)}, b_0=1-\frac{{\phi }_2+{\gamma }_3+{\gamma }_4}{6(1+{\alpha }_1)},$$

(25)

and substituting Equations (22) and (25) into Equation (16), the analytical expression for the product is

$$v\left(\zeta \right)=1+\frac{{\phi }_2+{\gamma }_3+{\gamma }_4}{6(1+{\alpha }_1)}({\zeta }^2-1)$$

(26)

3.3. Normalized steady-state source terms of liquid and gas phases

The analytical expressions of normalized steady-state source terms of liquid and gas phases aregiven by:

$${\psi }_l=-\frac{\left({\phi }_1+{\gamma }_1+{\gamma }_2\right)}{3(1+{\alpha }_1)}$$

(27)

$${\psi }_g=\frac{(2+\omega )}{2}\left(\frac{\left({\phi }_2+{\gamma }_3+{\gamma }_4\right)}{3(1+{\alpha }_1)}\right)$$

(28)

Remark 1. The analytical expressions given by Equations (20) and (26) for the concentrations of glucose and hydrogen, respectively, are identical to the expressions obtained by the Adomian decomposition method (ADM) [2] and the homotopy perturbation method [3].

4. Analytical expression of the concentrations using Taylor series method

The three-century-old Taylor series method (TSM) has been recently revived and exploited to accurately and efficiently solve many nonlinear differential equations representing nonlinear models in various sciences and engineering applications Equations [36,37]. In this section, we employ TSM to find the substrate and product concentrations.

4.1. Concentration of Glucose (substrate)

First, we assume that

$$u\left(0\right)=m$$

(29)

where$m$is an unknown constant to be determined. Steady state nonlinear reaction–diffusion equations can be written in the form:

$$\left(\zeta u^{\text{'}\text{'}}\left(\zeta \right)+2u^{\text{'}}\left(\zeta \right)\right)\left(1+{\alpha }_{1}u\left(\zeta \right)\right)=({\phi }_1+{\gamma }_1)\zeta u\left(\zeta \right)+{\gamma }_2\zeta$$

(30)

Taking the first three derivatives of Equation (30) with respect to $\text{'}\zeta \text{'}$ gives

$$\left(\zeta u^{\text{'}\text{'}\text{'}}\left(\zeta \right)+3u^{\text{'}\text{'}}\left(\zeta \right)\right)\left(1+{\alpha }_{1}u\left(\zeta \right)\right)+\left(\zeta u^{\text{'}\text{'}}\left(\zeta \right)+2u^{\text{'}}\left(\zeta \right)\right){\alpha }_{1}u\text{'}\left(\zeta \right)=({\phi }_1+{\gamma }_1)\left(u\left(\zeta \right)+\zeta u^{\text{'}}\left(\zeta \right)\right)+{\gamma }_2$$

(31)

$$\left(\zeta u^{\text{'}\text{'}\text{'}\text{'}}\left(\zeta \right)+4u^{\text{'}\text{'}\text{'}}\left(\zeta \right)\right)\left(1+{\alpha }_{1}u\left(\zeta \right)\right)+2\left(\zeta u^{\text{'}\text{'}\text{'}}\left(\zeta \right)+3u^{\text{'}\text{'}}\left(\zeta \right)\right){\alpha }_1{u}^{\text{'}}\left(\zeta \right)+\left(\zeta u^{\text{'}\text{'}}\left(\zeta \right)+2u^{\text{'}}\left(\zeta \right)\right){\alpha }_1{u}^{\text{'}\text{'}}\left(\zeta \right)=({\phi }_1+{\gamma }_1)\left(2u^{\text{'}}\left(\zeta \right)+\zeta u^{\text{'}\text{'}}\left(\zeta \right)\right)$$

(32)

$$\left(\zeta u^{\text{'}V}\left(\zeta \right)+5u^{\text{'}\text{'}\text{'}\text{'}}\left(\zeta \right)\right)\left(1+{\alpha }_{1}u\left(\zeta \right)\right)+3\left(\zeta u^{\text{'}\text{'}\text{'}\text{'}}\left(\zeta \right)+4u^{\text{'}\text{'}\text{'}}\left(\zeta \right)\right){\alpha }_1{u}^{\text{'}}\left(\zeta \right)+3\left(\zeta u^{\text{'}\text{'}\text{'}}\left(\zeta \right)+3u^{\text{'}\text{'}}\left(\zeta \right)\right){\alpha }_1{u}^{\text{'}\text{'}}\left(\zeta \right)+\left(\zeta u^{\text{'}\text{'}}\left(\zeta \right)+2u^{\text{'}}\left(\zeta \right)\right){\alpha }_1{u}^{\text{'}\text{'}\text{'}}\left(\zeta \right)=({\phi }_1+{\gamma }_1)\left(3u^{\text{'}\text{'}}\left(\zeta \right)+\zeta u^{\text{'}\text{'}\text{'}}\left(\zeta \right)\right)$$

(33)

For $\zeta =0$, Equations (31)–(33) give the following identities:

$$\begin{gathered} u^{\text{'}\text{'}}(0)=\frac{m({\phi }_1+{\gamma }_1)+{\gamma }_2}{3(1+{\alpha }_{1}m)}{u}^{\text{'}\text{'}\text{'}}\left(0\right)=0,u^{\text{'}\text{'}\text{'}\text{'}}\left(0\right)=\frac{\left(m({\phi }_1+{\gamma }_1)+{\gamma }_2\right)\left({\phi }_1+{\gamma }_1-{\alpha }_1{\gamma }_2\right)}{5({\alpha }_{1}m+1{)}^3},u^v(0)=0 \\ {u}^{vi}\left(0\right)=\frac{\left(m({\phi }_1+{\gamma }_1)+{\gamma }_2\right)\left({\phi }_1+{\gamma }_1-{\alpha }_1{\gamma }_2\right)\left(({\phi }_1+{\gamma }_1)(3-10{\alpha }_{1}m)-13{\alpha }_1{\gamma }_2\right)}{7({\alpha }_{1}m+1{)}^5}. \end{gathered}$$

(34)

The substrate concentration,using the Taylor’s series, is expressed by the expansion

$$u\left(\zeta \right)=u\left(0\right)+u^{\text{'}}\left(0\right)\frac{\zeta }{1!}+u^{\text{'}\text{'}}\left(0\right)\frac{{\zeta }^2}{2!}+u^{\text{'}\text{'}\text{'}}\left(0\right)\frac{3\zeta }{3!}+...=m+u_1\frac{{\zeta }^2}{2!}+u_2\frac{{\zeta }^4}{4!}+u_3\frac{{\zeta }^6}{6!}$$

(35)

where

$$u_1=\frac{m\left({\phi }_1+{\gamma }_1\right)+{\gamma }_2}{3\left({\alpha }_{1}m+1\right)},u_2=\frac{\left(m\left({\phi }_1+{\gamma }_1\right)+{\gamma }_2\right)\left({\phi }_1+{\gamma }_1-{\alpha }_1{\gamma }_2\right)}{5({\alpha }_{1}m+1{)}^3},u_3=\frac{\left(m({\phi }_1+{\gamma }_1)+{\gamma }_2\right)\left({\phi }_1+{\gamma }_1-{\alpha }_1{\gamma }_2\right)\left(({\phi }_1+{\gamma }_1)(3-10{\alpha }_{1}m)-13{\alpha }_1{\gamma }_2\right)}{7({\alpha }_{1}m+1{)}^5}$$

(36)

Using the boundary conditions$\zeta =1,u\left(\zeta \right)=1$in Equation (35) implies that

$$m+u_1\frac{1}{2!}+u_2\frac{1}{4!}+u_3\frac{1}{6!}=1$$

(37)

from which the unknown constant m can be obtained. For the fixed values of the parameters:

${\phi }_1=5,{\gamma }_1=0.1,{\gamma }_2=0.1$and ${\alpha }_1=5$, the numerical value of m is found to be$m=0.85803$and from Equation (36), we obtain $u_1=0.141015,u_2=0.001159$ and$u_3=-0.000207$, and thus from Equation (35), we obtain the analytical expression of the substrate expressed by

$$u\left(\zeta \right).=0.85803+0.141015\frac{{\zeta }^2}{2!}+0.001159\frac{{\zeta }^4}{4!}-0.000207\frac{{\zeta }^6}{6!}$$

(38)

4.2. Concentration of Hydrogen (product)

Similar to the approach in Section 4.1, we begin by assuming that

$$v\left(0\right)=l,$$

(39)

where l is an unknown constant to be determined and by direct differentiation of $v\left(\zeta \right)$, we obtain,

$$v^{\text{'}\text{'}}\left(0\right)=\frac{m({\phi }_2+{\gamma }_3)+{\gamma }_4}{3(1+{\alpha }_{1}m)}, v^{\text{'}\text{'}\text{'}}\left(0\right)=0 ,v^{\text{'}\text{'}\text{'}\text{'}}\left(0\right)=\frac{\left(m({\phi }_1+{\gamma }_1)+{\gamma }_2\right)\left({\phi }_2+{\gamma }_3-{\alpha }_1{\gamma }_4\right)}{5(1+{\alpha }_{1}m{)}^3}, v^v\left(0\right)=0,v^{vi}\left(0\right)=\frac{\left(m({\phi }_{`1}+{\gamma }_1)+{\gamma }_2\right)\left({\phi }_2+{\gamma }_3-{\alpha }_1{\gamma }_4\right)\left(({\phi }_1+{\gamma }_1)(1-{\alpha }_{1}m)-2{\alpha }_1{\gamma }_2\right)}{7(1+{\alpha }_{1}m{)}^5}.$$

(40)

Now, the product concentration is readily obtained using Taylor’s series

$$v\left(\zeta \right)=v\left(0\right)+v^{\text{'}}\left(0\right)\frac{\zeta }{1!}+v^{\text{'}\text{'}}\left(0\right)\frac{{\zeta }^2}{2!}+v^{\text{'}\text{'}\text{'}}\left(0\right)\frac{{\zeta }^3}{3!}+\dots$$

(41)

Using the boundary condition,$v\left(1\right)=1,$ in Equation (41) gives the numerical value of l. For the fixed values of the parameters${\phi }_1=1,{\gamma }_1=1,{\gamma }_2=1,{\phi }_2=5,{\gamma }_3=0.1,{\gamma }_4=0.1,{\alpha }_1=5$, and$m=0.915769$, we obtain $l=0.8563317$. The analytical expression for the product concentration takes on the form

$$v\left(\zeta \right)=0.8563317+0.1425122\frac{{\zeta }^2}{2!}+0.0011589\frac{{\zeta }^4}{4!}-0.0000581\frac{{\zeta }^6}{6!}$$

(42)

4.3. Normalized steady-state source terms of liquid and gas phases

The analytical expression of the normalized steady-state source terms of liquid phases, ${\psi }_l$, is obtained from the equation,

$$-{\psi }_l=u_1+u_2\frac{1}{6}+u_3\frac{1}{120}$$

(43)

Using, the values$u_1=0.141015,u_2=0.001159$and $u_3=-0.000207$, we obtain $-{\psi }_l=0.141207$. The analytical expression of normalized steady-state terms of gas phases, ${\psi }_g$, is given by

$${\psi }_g=\frac{2+\omega }{2}\left[v_1+v_2\frac{1}{6}+v_3\frac{1}{120}\right]$$

(44)

where ${\psi }_g=0.214057$ when $\omega =1.$

5. Numerical simulations and Discussion

Nonlinear equations in the immobilized-cell photobioreactor are analytically solved. The approximate analytical expressions of concentrations of glucose and hydrogen inside the gel and granule in addition to approximate analytical expressions of steady-state source terms of liquid and gas phases are derived using Taylor series (TSM) and Akbari-Ganji (AGM) methods.

The reaction-diffusion equations representing the packed bed photobioreactor with immobilized-cell were solved using the Adomian decomposition method (ADM) [2] and the homotopy perturbation method (HPM) [3]. Interstingly, the semi-analytical expressions of the concentrations of substrate and product obtained by the ADM, HPM, and AGM were identical (Equations (20) and (26)) for all values of parameters.

To examine the accuracy of the two proposed analytical approaches, we compared their results with numerical results obtained from the reliable MATLAB pdepe function (Appendix A) and with analytical results of other methods available in the literature. The approximate analytical and numerical concentrations of substrate and product for various parameters are summarized in Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6. Even though both methods gave satisfactory results, TSM is notably more accurate. The maximum relative error average is 0.6% for the TSM and 5% for the AGM. Comparisons of normalized steady-state source terms of both liquid and gas phases for various values of parameters${\phi }_1,{\phi }_2$ and ${\alpha }_1$ are given in Table 7 and Table 8.

Figures 2(a)-2(c) illustrate the behavior of the biodegradation of substrate for different values of the parameters. It is noticed that as any of the parameters ${\varphi }_1, {\gamma }_1, \mathrm{or}{ \gamma }_2$ decreases, the substrate concentration increases. In contrast, Figure 2(d) confirms a direct relationship between the parameter${\alpha }_1$ and the substrate concentration.

The effects of all parameters on the hydrogen production profiles are shown in Figures 3(a)-3(g),where it is noticed that the concentration of hydrogen increases the parameters ${\varphi }_2, {\gamma }_3, \mathrm{or}{\gamma }_4$ decreases or the parameter ${\alpha }_i$ increases. However, the concentration of hydrogen is independent of any of the parameters${\varphi }_1, {\gamma }_1, \mathrm{and}{\gamma }_2$.

Figure 2. Plot of substrate concentration for various values of parameters (Equations (20) and (35)).

Figure 2. Plot of substrate concentration for various values of parameters (Equations (20) and (35)).

Figure 3. Plot of product concentration,$v\left(\zeta \right)$, for various values of the parameters ${\varphi }_1, {\varphi }_2, {\gamma }_1, {\gamma }_2, {\gamma }_3, {\gamma }_4,$and $\alpha$ (Equations (26) and (41)).

Figure 3. Plot of product concentration,$v\left(\zeta \right)$, for various values of the parameters ${\varphi }_1, {\varphi }_2, {\gamma }_1, {\gamma }_2, {\gamma }_3, {\gamma }_4,$and $\alpha$ (Equations (26) and (41)).

Table 1. Comparison beteween numerical and analytical results for dimensionless concentration of substrate$u\left(\zeta \right)$ for various values of parameter ${\phi }_1$when ${\gamma }_1={\gamma }_2=0.1$and${\alpha }_1=5$.

$\mathit{\zeta }$	${\mathit{\phi }}_1=10$$\mathbf{and} \mathit{m}=0.728557$					${\mathit{\phi }}_1=15$$\mathbf{and} \mathit{m}=0.610154$
	Num.Equation (9)	AGMEq.(20)	TSMEq.(35)	Error % ofAGMEq.(20)	Error % of TSMEquation (35)	NumEquation (9)	AGMEq.(20)	TSMEq.(35)	Error % ofAGMEq.(20)	Error % of TSMEquation (35)
0	0.73	0.72	0.73	1.37	0.00	0.60	0.58	0.61	3.33	1.67
0.2	0.74	0.73	0.74	1.35	0.00	0.62	0.59	0.63	4.84	1.61
0.4	0.77	0.76	0.77	1.30	0.00	0.67	0.65	0.67	2.98	0.00
0.6	0.83	0.82	0.83	1.20	0.00	0.75	0.73	0.75	2.67	0.00
0.8	0.90	0.90	0.90	0.00	0.00	0.86	0.85	0.86	1.16	0.00
1	1.00	1.00	1.00	0.00	0.00	1.00	1.00	1.00	0.00	0.00
	Average error %			0.87	0.00	Average error %			2.50	0.55

Table 1. Comparison beteween numerical and analytical results for dimensionless concentration of substrate$u\left(\zeta \right)$ for various values of parameter ${\phi }_1$when ${\gamma }_1={\gamma }_2=0.1$and${\alpha }_1=5$.

$\mathit{\zeta }$	${\mathit{\phi }}_1=10$$\mathbf{and} \mathit{m}=0.728557$					${\mathit{\phi }}_1=15$$\mathbf{and} \mathit{m}=0.610154$
	Num.Equation (9)	AGMEq.(20)	TSMEq.(35)	Error % ofAGMEq.(20)	Error % of TSMEquation (35)	NumEquation (9)	AGMEq.(20)	TSMEq.(35)	Error % ofAGMEq.(20)	Error % of TSMEquation (35)
0	0.73	0.72	0.73	1.37	0.00	0.60	0.58	0.61	3.33	1.67
0.2	0.74	0.73	0.74	1.35	0.00	0.62	0.59	0.63	4.84	1.61
0.4	0.77	0.76	0.77	1.30	0.00	0.67	0.65	0.67	2.98	0.00
0.6	0.83	0.82	0.83	1.20	0.00	0.75	0.73	0.75	2.67	0.00
0.8	0.90	0.90	0.90	0.00	0.00	0.86	0.85	0.86	1.16	0.00
1	1.00	1.00	1.00	0.00	0.00	1.00	1.00	1.00	0.00	0.00
	Average error %			0.87	0.00	Average error %			2.50	0.55

Table 2. Comparison beteween numerical and analytical results for dimensionless concentration of substrate$u\left(\zeta \right)$ for various values of parameter ${\gamma }_1$when ${\phi }_1={\gamma }_2=0.1$and${\alpha }_1=5$.

$\mathit{\zeta }$	${\mathit{\gamma }}_1=7$$\mathbf{and} \mathit{m}=0.805191$					${\mathit{\gamma }}_1=15$$\mathbf{and} \mathit{m}=0.610124$
	Num.Equation (9)	AGMEq.(20)	TSMEq.(35)	Error % ofAGMEq.(20)	Error % of TSMEquation (35)	NumEquation (9)	AGMEq.(20)	TSMEq.(35)	Error % ofAGMEq.(20)	Error % of TSMEquation (35)
0	0.80	0.80	0.80	0.00	0.00	0.60	0.58	0.61	3.33	1.67
0.2	0.81	0.81	0.81	0.00	0.00	0.62	0.59	0.63	4.84	1.61
0.4	0.84	0.83	0.84	1.20	0.00	0.67	0.65	0.67	2.98	0.00
0.6	0.88	0.87	0.88	1.14	0.00	0.75	0.73	0.75	2.67	0.00
0.8	0.93	0.93	0.93	0.00	0.00	0.86	0.85	0.86	1.16	0.00
1	1.00	1.00	1.00	0.00	0.00	1.00	1.00	1.00	0.00	0.00
	Average error %			0.39	0.00	Average error %			2.50	0.55

Table 2. Comparison beteween numerical and analytical results for dimensionless concentration of substrate$u\left(\zeta \right)$ for various values of parameter ${\gamma }_1$when ${\phi }_1={\gamma }_2=0.1$and${\alpha }_1=5$.

$\mathit{\zeta }$	${\mathit{\gamma }}_1=7$$\mathbf{and} \mathit{m}=0.805191$					${\mathit{\gamma }}_1=15$$\mathbf{and} \mathit{m}=0.610124$
	Num.Equation (9)	AGMEq.(20)	TSMEq.(35)	Error % ofAGMEq.(20)	Error % of TSMEquation (35)	NumEquation (9)	AGMEq.(20)	TSMEq.(35)	Error % ofAGMEq.(20)	Error % of TSMEquation (35)
0	0.80	0.80	0.80	0.00	0.00	0.60	0.58	0.61	3.33	1.67
0.2	0.81	0.81	0.81	0.00	0.00	0.62	0.59	0.63	4.84	1.61
0.4	0.84	0.83	0.84	1.20	0.00	0.67	0.65	0.67	2.98	0.00
0.6	0.88	0.87	0.88	1.14	0.00	0.75	0.73	0.75	2.67	0.00
0.8	0.93	0.93	0.93	0.00	0.00	0.86	0.85	0.86	1.16	0.00
1	1.00	1.00	1.00	0.00	0.00	1.00	1.00	1.00	0.00	0.00
	Average error %			0.39	0.00	Average error %			2.50	0.55

Table 3. Comparison beteween numerical and analytical results for dimensionless concentration of substrate$u\left(\zeta \right)$ for various values of parameter ${\gamma }_2$when ${\gamma }_1={\phi }_1=0.1$and${\alpha }_1=5$.

$\mathit{\zeta }$	${\mathit{\gamma }}_2=5$$\mathbf{and} \mathit{m}=0.839877$					${\mathit{\gamma }}_2=7$$\mathbf{and} \mathit{m}=0.762254$
	Num.Equation (9)	AGMEquation (20)	TSMEq.(35)	Error % ofAGMEq.(20)	Error % of TSMEquation (35)	NumEquation (9)	AGMEq.(20)	TSMEq.(35)	Error % ofAGMEq.(20)	Error % of TSMEquation (35)
0	0.84	0.86	0.84	2.38	0.00	0.77	0.80	0.76	3.90	1.30
0.2	0.85	0.86	0.85	1.18	0.00	0.78	0.81	0.77	3.85	1.28
0.4	0.87	0.88	0.87	1.15	0.00	0.81	0.83	0.80	2.47	1.23
0.6	0.90	0.91	0.90	1.11	0.00	0.86	0.87	0.85	1.16	1.16
0.8	0.95	0.95	0.94	0.00	1.05	0.92	0.93	0.92	1.09	0.00
1	1.00	1.00	1.00	0.00	0.00	1.00	1.00	1.00	0.00	0.00
	Average error %			0.97	0.17	Average error %			2.09	0.83

Table 3. Comparison beteween numerical and analytical results for dimensionless concentration of substrate$u\left(\zeta \right)$ for various values of parameter ${\gamma }_2$when ${\gamma }_1={\phi }_1=0.1$and${\alpha }_1=5$.

$\mathit{\zeta }$	${\mathit{\gamma }}_2=5$$\mathbf{and} \mathit{m}=0.839877$					${\mathit{\gamma }}_2=7$$\mathbf{and} \mathit{m}=0.762254$
	Num.Equation (9)	AGMEquation (20)	TSMEq.(35)	Error % ofAGMEq.(20)	Error % of TSMEquation (35)	NumEquation (9)	AGMEq.(20)	TSMEq.(35)	Error % ofAGMEq.(20)	Error % of TSMEquation (35)
0	0.84	0.86	0.84	2.38	0.00	0.77	0.80	0.76	3.90	1.30
0.2	0.85	0.86	0.85	1.18	0.00	0.78	0.81	0.77	3.85	1.28
0.4	0.87	0.88	0.87	1.15	0.00	0.81	0.83	0.80	2.47	1.23
0.6	0.90	0.91	0.90	1.11	0.00	0.86	0.87	0.85	1.16	1.16
0.8	0.95	0.95	0.94	0.00	1.05	0.92	0.93	0.92	1.09	0.00
1	1.00	1.00	1.00	0.00	0.00	1.00	1.00	1.00	0.00	0.00
	Average error %			0.97	0.17	Average error %			2.09	0.83

Table 4. Comparison beteween numerical and analytical results for dimensionless concentration of product$v\left(\zeta \right)$ for various values of parameter ${\phi }_2$when ${\gamma }_1=1,{\gamma }_2=1,{\gamma }_3=0.1,{\gamma }_4=1,{\alpha }_1=5$and$m=0.915769$.

$\mathit{\zeta }$	${\mathit{\phi }}_2=20$$\mathbf{and} \mathit{l}=0.4444703$					${\mathit{\phi }}_2=35$$\mathbf{and} \mathit{l}=0.0320751$
	Num.Eq.(10)	AGMEq.(26)	TSMEq.(41)	Error% ofAGMEq.(26)	Error% ofTSMEquation (41)	NumEq.(10)	AGMEq.(26)	TSMEq.(41)	Error% ofAGMEq.(26)	Error% ofTSMEquation (41)
0	0.44	0.44	0.44	0.00	0.00	0.03	0.02	0.03	33.33	0.00
0.2	0.47	0.46	0.47	2.13	0.00	0.07	0.06	0.07	14.29	0.00
0.4	0.53	0.53	0.53	0.00	0.00	0.19	0.18	0.19	5.26	0.00
0.6	0.65	0.64	0.65	1.54	0.00	0.39	0.38	0.39	2.56	0.00
0.8	0.81	0.80	0.81	1.23	0.00	0.66	0.66	0.66	0.00	0.00
1	1.00	1.00	1.00	0.00	0.00	1.00	1.00	1.00	0.00	0.00
	Average error %			0.82	0.00	Average error %			9.24	0.00

Table 4. Comparison beteween numerical and analytical results for dimensionless concentration of product$v\left(\zeta \right)$ for various values of parameter ${\phi }_2$when ${\gamma }_1=1,{\gamma }_2=1,{\gamma }_3=0.1,{\gamma }_4=1,{\alpha }_1=5$and$m=0.915769$.

$\mathit{\zeta }$	${\mathit{\phi }}_2=20$$\mathbf{and} \mathit{l}=0.4444703$					${\mathit{\phi }}_2=35$$\mathbf{and} \mathit{l}=0.0320751$
	Num.Eq.(10)	AGMEq.(26)	TSMEq.(41)	Error% ofAGMEq.(26)	Error% ofTSMEquation (41)	NumEq.(10)	AGMEq.(26)	TSMEq.(41)	Error% ofAGMEq.(26)	Error% ofTSMEquation (41)
0	0.44	0.44	0.44	0.00	0.00	0.03	0.02	0.03	33.33	0.00
0.2	0.47	0.46	0.47	2.13	0.00	0.07	0.06	0.07	14.29	0.00
0.4	0.53	0.53	0.53	0.00	0.00	0.19	0.18	0.19	5.26	0.00
0.6	0.65	0.64	0.65	1.54	0.00	0.39	0.38	0.39	2.56	0.00
0.8	0.81	0.80	0.81	1.23	0.00	0.66	0.66	0.66	0.00	0.00
1	1.00	1.00	1.00	0.00	0.00	1.00	1.00	1.00	0.00	0.00
	Average error %			0.82	0.00	Average error %			9.24	0.00

Table 5. Comparison beteween numerical and analytical results for dimensionless concentration of product$v\left(\zeta \right)$ for various values of parameter ${\gamma }_3$when ${\gamma }_1=1,{\gamma }_2=1,{\phi }_1=0.1,{\gamma }_4=1,{\alpha }_1=10$and$m=0.954173$.

$\mathit{\zeta }$	${\mathit{\gamma }}_3=40$$\mathbf{and} \mathit{l}=0.39268025$						${\mathit{\gamma }}_3=65$$\mathbf{and} \mathit{l}=0.01502483$
	Num.Eq.(10)	AGMEq.(26)	TSMEq.(41)	Error % ofAGMEq.(26)	Error % of TSMEquation (41)	NumEq.(10)		AGMEq.(26)	TSMEq.(41)	Error % ofAGMEq.(26)	Error % of TSMEquation (41)
0	0.39	0.39	0.39	0.00	0.00	0.01		0.01	0.01	0.00	0.00
0.2	0.42	0.42	0.42	0.00	0.00	0.05		0.05	0.05	0.00	0.00
0.4	0.49	0.49	0.49	0.00	0.00	0.18		0.17	0.18	5.55	0.00
0.6	0.62	0.61	0.62	1.61	0.00	0.38		0.37	0.38	2.63	0.00
0.8	0.79	0.79	0.79	0.00	0.00	0.66		0.66	0.66	0.00	0.00
1	1.00	1.00	1.00	0.00	0.00	1.00		1.00	1.00	0.00	0.00
	Average error %			0.27	0.00	Average error %				1.36	0.00

Table 5. Comparison beteween numerical and analytical results for dimensionless concentration of product$v\left(\zeta \right)$ for various values of parameter ${\gamma }_3$when ${\gamma }_1=1,{\gamma }_2=1,{\phi }_1=0.1,{\gamma }_4=1,{\alpha }_1=10$and$m=0.954173$.

$\mathit{\zeta }$	${\mathit{\gamma }}_3=40$$\mathbf{and} \mathit{l}=0.39268025$						${\mathit{\gamma }}_3=65$$\mathbf{and} \mathit{l}=0.01502483$
	Num.Eq.(10)	AGMEq.(26)	TSMEq.(41)	Error % ofAGMEq.(26)	Error % of TSMEquation (41)	NumEq.(10)		AGMEq.(26)	TSMEq.(41)	Error % ofAGMEq.(26)	Error % of TSMEquation (41)
0	0.39	0.39	0.39	0.00	0.00	0.01		0.01	0.01	0.00	0.00
0.2	0.42	0.42	0.42	0.00	0.00	0.05		0.05	0.05	0.00	0.00
0.4	0.49	0.49	0.49	0.00	0.00	0.18		0.17	0.18	5.55	0.00
0.6	0.62	0.61	0.62	1.61	0.00	0.38		0.37	0.38	2.63	0.00
0.8	0.79	0.79	0.79	0.00	0.00	0.66		0.66	0.66	0.00	0.00
1	1.00	1.00	1.00	0.00	0.00	1.00		1.00	1.00	0.00	0.00
	Average error %			0.27	0.00	Average error %				1.36	0.00

Table 6. Comparison beteween numerical and analytical results for dimensionless concentration of product$v\left(\zeta \right)$ for various values of parameter ${\gamma }_4$when ${\gamma }_1=1,{\gamma }_2=1,{\gamma }_3=0.1,{\phi }_1=1,{\alpha }_1=5$and $m=0.915769$.

$\mathit{\zeta }$	${\mathit{\gamma }}_4=10$$\mathbf{and} \mathit{l}=0.70248543$					${\mathit{\gamma }}_4=20$$\mathbf{and} \mathit{l}=0.41040946$
	Num.Eq.(10)	AGMEq.(26)	TSMEq.(41)	Error % ofAGMEq.(26)	Error % of TSMEquation (41)	NumEq.(10)	AGMEq.(26)	TSMEq.(41)	Error % ofAGMEq.(26)	Error % of TSMEquation (41)
0	0.70	0.72	0.70	2.86	0.00	0.41	0.44	0.41	7.32	0.00
0.2	0.71	0.73	0.71	2.82	0.00	0.43	0.46	0.43	6.98	0.00
0.4	0.75	0.76	0.75	1.33	0.00	0.51	0.53	0.51	3.92	0.00
0.6	0.81	0.82	0.81	1.23	0.00	0.63	0.64	0.63	1.59	0.00
0.8	0.90	0.90	0.90	0.00	0.00	0.80	0.80	0.80	0.00	0.00
1	1.00	1.00	1.00	0.00	0.00	1.00	1.00	1.00	0.00	0.00
	Average error %			1.37	0.00	Average error %			3.30	0.00

Table 6. Comparison beteween numerical and analytical results for dimensionless concentration of product$v\left(\zeta \right)$ for various values of parameter ${\gamma }_4$when ${\gamma }_1=1,{\gamma }_2=1,{\gamma }_3=0.1,{\phi }_1=1,{\alpha }_1=5$and $m=0.915769$.

$\mathit{\zeta }$	${\mathit{\gamma }}_4=10$$\mathbf{and} \mathit{l}=0.70248543$					${\mathit{\gamma }}_4=20$$\mathbf{and} \mathit{l}=0.41040946$
	Num.Eq.(10)	AGMEq.(26)	TSMEq.(41)	Error % ofAGMEq.(26)	Error % of TSMEquation (41)	NumEq.(10)	AGMEq.(26)	TSMEq.(41)	Error % ofAGMEq.(26)	Error % of TSMEquation (41)
0	0.70	0.72	0.70	2.86	0.00	0.41	0.44	0.41	7.32	0.00
0.2	0.71	0.73	0.71	2.82	0.00	0.43	0.46	0.43	6.98	0.00
0.4	0.75	0.76	0.75	1.33	0.00	0.51	0.53	0.51	3.92	0.00
0.6	0.81	0.82	0.81	1.23	0.00	0.63	0.64	0.63	1.59	0.00
0.8	0.90	0.90	0.90	0.00	0.00	0.80	0.80	0.80	0.00	0.00
1	1.00	1.00	1.00	0.00	0.00	1.00	1.00	1.00	0.00	0.00
	Average error %			1.37	0.00	Average error %			3.30	0.00

Table 7. Comparison between numerical and analytical normalized steady-state source terms of liquid phase $-{\psi }_l$ for various values of parameter ${\phi }_1$and ${\alpha }_1$when ${\gamma }_1={\gamma }_2=0.1$.

${\mathit{\phi }}_1$	${\mathit{\alpha }}_1=0.1$					${\mathit{\alpha }}_1=0.5$
	Num.Eq.(13)	AGMEq.(27)	TSMEq.(43)	Error% ofAGMEq.(27)	Error% ofTSMEquation (43)	Num.Eq.(13)	AGMEquation (27)	TSMEq.(43)	Error% ofAGMEq.(27)	Error%ofTSMEquation (43)
0	0.06	0.06	0.06	0.00	0.00	0.04	0.04	0.04	0.00	0.00
0.1	0.09	0.09	0.09	0.00	0.00	0.07	0.07	0.07	0.00	0.00
0.5	0.20	0.21	0.21	5.00	5.00	0.15	0.15	0.15	0.00	0.00
1	0.34	0.37	0.34	8.82	0.00	0.26	0.27	0.26	3.85	0.00
5	1.25	1.56	1.33	24.8	6.40	1.00	1.15	1.01	15.00	1.00
10	2.08	3.09	2.42	48.56	16.35	1.74	2.27	1.90	30.46	9.19
50	5.64	15.2	5.23	169.7	7.27	4.81	11.15	5.22	131.8	8.52
100	7.75	30.4	5.66	291.7	26.97	7.31	22.27	5.66	204.6	22.57
	Average error %			68.68	7.75	Average error %			48.22	5.16

Table 7. Comparison between numerical and analytical normalized steady-state source terms of liquid phase $-{\psi }_l$ for various values of parameter ${\phi }_1$and ${\alpha }_1$when ${\gamma }_1={\gamma }_2=0.1$.

${\mathit{\phi }}_1$	${\mathit{\alpha }}_1=0.1$					${\mathit{\alpha }}_1=0.5$
	Num.Eq.(13)	AGMEq.(27)	TSMEq.(43)	Error% ofAGMEq.(27)	Error% ofTSMEquation (43)	Num.Eq.(13)	AGMEquation (27)	TSMEq.(43)	Error% ofAGMEq.(27)	Error%ofTSMEquation (43)
0	0.06	0.06	0.06	0.00	0.00	0.04	0.04	0.04	0.00	0.00
0.1	0.09	0.09	0.09	0.00	0.00	0.07	0.07	0.07	0.00	0.00
0.5	0.20	0.21	0.21	5.00	5.00	0.15	0.15	0.15	0.00	0.00
1	0.34	0.37	0.34	8.82	0.00	0.26	0.27	0.26	3.85	0.00
5	1.25	1.56	1.33	24.8	6.40	1.00	1.15	1.01	15.00	1.00
10	2.08	3.09	2.42	48.56	16.35	1.74	2.27	1.90	30.46	9.19
50	5.64	15.2	5.23	169.7	7.27	4.81	11.15	5.22	131.8	8.52
100	7.75	30.4	5.66	291.7	26.97	7.31	22.27	5.66	204.6	22.57
	Average error %			68.68	7.75	Average error %			48.22	5.16

Table 8. Comparison between numerical and analytical normalized steady-state source terms of gas phase${\psi }_g$ for various values of parameter ${\phi }_2$and ${\alpha }_1$when ${\gamma }_1={\gamma }_2=1,{\gamma }_3={\gamma }_4=0.1,{\phi }_1=1$and $\omega =1$.

${\mathit{\phi }}_2$	${\mathit{\alpha }}_1=0.1$						${\mathit{\alpha }}_1=0.5$
	Num.Eq.(14)	AGMEq.(28)	TSMEq.(44)	Error % ofAGMEq.(28)	Error % of TSMEquation (44)	Num.Eq.(14)		AGMEquation (28)	TSMEquation (44)	Error % ofAGMEq.(28)	Error % of TSMEquation (44)
0	0.08	0.09	0.09	12.50	12.50	0.06		0.07	0.06	16.67	0.00
0.1	0.12	0.14	0.12	16.67	0.00	0.10		0.10	0.10	0.00	0.00
0.5	0.27	0.32	0.28	18.52	3.70	0.22		0.23	0.22	4.54	0.00
1	0.47	0.54	0.47	14.87	0.00	0.37		0.40	0.37	8.11	0.00
5	2.02	2.36	2.03	16.83	0.49	1.58		1.73	1.60	9.49	1.27
10	3.96	4.64	3.98	17.17	0.50	3.10		3.40	3.14	9.68	1.29
50	19.5	22.8	19.55	17.21	0.41	15.26		16.73	15.42	9.63	1.05
100	38.9	45.5	39.02	17.19	0.41	30.46		33.40	30.78	9.65	1.05
	Average error %			16.37	2.25	Average error %				8.47	0.58

Table 8. Comparison between numerical and analytical normalized steady-state source terms of gas phase${\psi }_g$ for various values of parameter ${\phi }_2$and ${\alpha }_1$when ${\gamma }_1={\gamma }_2=1,{\gamma }_3={\gamma }_4=0.1,{\phi }_1=1$and $\omega =1$.

${\mathit{\phi }}_2$	${\mathit{\alpha }}_1=0.1$						${\mathit{\alpha }}_1=0.5$
	Num.Eq.(14)	AGMEq.(28)	TSMEq.(44)	Error % ofAGMEq.(28)	Error % of TSMEquation (44)	Num.Eq.(14)		AGMEquation (28)	TSMEquation (44)	Error % ofAGMEq.(28)	Error % of TSMEquation (44)
0	0.08	0.09	0.09	12.50	12.50	0.06		0.07	0.06	16.67	0.00
0.1	0.12	0.14	0.12	16.67	0.00	0.10		0.10	0.10	0.00	0.00
0.5	0.27	0.32	0.28	18.52	3.70	0.22		0.23	0.22	4.54	0.00
1	0.47	0.54	0.47	14.87	0.00	0.37		0.40	0.37	8.11	0.00
5	2.02	2.36	2.03	16.83	0.49	1.58		1.73	1.60	9.49	1.27
10	3.96	4.64	3.98	17.17	0.50	3.10		3.40	3.14	9.68	1.29
50	19.5	22.8	19.55	17.21	0.41	15.26		16.73	15.42	9.63	1.05
100	38.9	45.5	39.02	17.19	0.41	30.46		33.40	30.78	9.65	1.05
	Average error %			16.37	2.25	Average error %				8.47	0.58

7. Conclusions

The objective of this research is multi-folded. First, we successfully employed two widely used analytical methods (AGM and TSM) to solve two reaction-diffusion equations representing the packed bed photobioreactor with immobilized-cell, and derive simple semi-analytical expressions of the substrate and product concentrations and analytical expressions of steady-state source terms of liquid and gas phases. Second, we studied the effect of the reaction-diffusion parameters on the concentrations of substrate and product. Third, we added to the literature some useful information about the exploitation of some of the widely used methods, in particular the AGM and TSM. As both methods appear to be effective and reliable in solving nonlinear systems, it is noticed that the AGM is more or less a modification of the Adomian decomposition method and is likely to involve some tedious algebraic computations. The TSM, on the other hand, involves less algebraic computations, gives more accurate results, and guarantees convergence if the conditions of Taylor’s theorem are met.