The following assumptions were made in the development of the design equations:

1) The flow rate of the produced water in the MBR is constant throughout.

2) There is no concentration gradient of contaminant in the MBR.

3) The MBR is operated at isothermal, isobaric and steady state condition.

4) There is no diffusion/dispersion of contaminants in the MBR.

These assumptions are made to simplify the model and avoid complexity arising from flow rate fluctuation, concentration gradients, density gradients, and diffusivity.

Applying the mass conservative principle to the bioreactor in Figure 1 based on the above assumptions gives the overall component balance as:

F i o C i o = F i C i + R i V M + d N i d t (1)

where: V_M = MBR volume (m³), C i o , C i = influent and effluent oxygen concentration (kg/m³), t = Time (hr), F i o , F i = influent and effluent flow rate of oxygen (m³/hr), R_i = Rate of transfer or uptake (kg/m³hr), i = oxygen, contaminants.

In aerobic process, an uptake of oxygen is necessary for the survival of the biomass and products uniformity, thus the rate of oxygen transfer [10] can be evaluated using the mass transfer equation expressed as:

R O 2 = Δ J = ( J o − J ) = K L a ( C O 2 * − C O 2 ) (2)

From Equation (1) with respect to oxygen transfer rate in MBR is expressed as:

F O 2 ( o ) C O 2 ( o ) = F O 2 C O 2 + R O 2 V + d N O 2 d t (3)

where R O 2 = K L a ( C O 2 * − C O 2 ) and

d N O 2 d t = V d C O 2 d t (4)

Combining Equations (3) and (4) gives

V d C O 2 d t = F o C O 2 ( o ) − F C O 2 + K L a ( C O 2 * − C O 2 ) V (5)

For constant flow rate, Equation (5) reduces to

V d C O 2 d t = F ( C O 2 ( o ) − C O 2 ) + K L a ( C O 2 * − C O 2 ) V (6)

At steady state operation, rearranging Equation (6) reduces to

V = F ( C O 2 ( o ) − C O 2 ) K L a ( C O 2 * − C O 2 ) (7)

where: K_L = Oxygen mass transfer coefficient (m/hr), a = Interfacial area of mass transfer (m²/m³), C O 2 * = Saturated oxygen concentration (kg/m³), J_o and J = inflow and outflow flux in the permeate zone (kg/m² hr).

The height of cylindrical shaped reactor is given as:

H = 4 V π D 2 (8)

substituting Equations (7) in (8), we have

H = 4 F ( C O 2 ( o ) − C O 2 ) π D 2 K L a ( C O 2 ∗ − C O 2 ) (9)

The saturated oxygen concentration can be evaluated using equation:

C O 2 * = P O 2 H (10)

where: P O 2 = Partial pressure of oxygen (bar), H = Henry’s law constant (bar∙m³/kg)

K L a can be evaluated for non-coalescing fluid [10] as:

K L a = 0.02 ( P V ) 0.7 ( v g p p o ) 0.2 (11)

where: v_g = superficial gas velocity (m/hr), p_o = Reference pressure of 1 bar, p = Pressure in the system (bar), P V = Input power per unit Volume (kW/m³) and it ranges from (0.5 to 10).

The substrate consumption rate is expressed as [11] :

− R s = − d C s d t = K S K m + S (12)

Substituting Equation (12) into (1), the MBR volume base on substrate consumption at steady state and constant flow rate, after simplification gives:

V M = F c ( o ) C ( o ) x c ( K m + S K S ) (13)

The substrate rate constant is expressed as:

k = μ max Y (14)

Combining Equations (13) and (14) gives:

V M = F c ( o ) C ( o ) x c ( K m + S μ max S ) Y (15)

where: S = Substrate concentration (kg/m³), F_c = Flow rate of contaminant (m³/hr), C_(o) = Initial concentration of contaminant (kg/m³), x_c = Contaminant conversion, Y = Biomass yield, K_m = Monod constant.

The evaluation of biomass production in this study was done in terms of mixed liquor suspended solid (MLSS). The performance equations of MBR depend on the production yield of biomass in the process capable of treating the produced water. According to (zaerpour, 2014) biomass production given by [12]

X ( M L S S ) = F Y ( S o − S ) 1 + k d S R T × ( S R T ) (16)

The hydraulic retention time (HRT) was calculated according to:

H R T = V M F (17)

where: HRT = Hydraulic retention time (hours), V_M = Volume of MBR, F = Flow rate.

Solids retention time (SRT) was calculated according to:

S R T = 1 μ (18)

The kinetic of biological activities in the MBR was expressed with respect to the substrate consumption and endogenous decay of the produced organism according to (Zaerpour 2014) [12] as:

μ = μ m S K m + S − k d (19)

where: μ = Specific growth rate of nitrifying bacteria (kg new cells/kg cells day), μ_m = Maximum specific growth rate of nitrifying bacteria (kg new cells/kg cells day) S = Nitrogen (substrate) concentration (g/m³), K_m = Monod constant (kg/m³), k_d = Endogenous decay coefficient for nitrifying organisms (kg MLSS/ kg MLSS day).

Combining Equations (18) and (19) gives an expression for SRT as

S R T = K m + S μ m S − 1 k d (20)

In terms of substrate conversion, Equation (20) can be further expressed as

S R T = K m + μ m ( S o − x s ) μ m ( S o − x s ) − 1 k d (21)

The concentration of oxygen at saturation point was calculated using the expression:

C O 2 * = P O 2 H (22)

where C^* in the case of oxygen transfer, is measured in terms of solubility of O₂.

The Henry’s law constant at 1 atm pressure of oxygen and ambient temperature of 25˚C is 24.2 atm∙m³/kg while the solubility is 8.69 × 10⁻³ kg/m³.

The Monod constant, maximum specific growth rate constant, substrate utilization rate constant and endogenous decay coefficient for nitrifying organisms were obtained from (zaerpour, 2014 and Kwon et al., 2008) [12] [13] as presented in Table 1.

Table 2 shows the initial concentration of contaminants (COD, oil/grease, TSS, TOC and TDS) between 95% to 99% conversion) in Nigerian Agip Oil Company produced water, Port Harcourt.

The substrate specific rate is an important parameter in the design of MBR. Increasing or decreasing of substrate rate can affect the production of nitrogenous organisms that utilizes the produced water contaminant and thereby affecting its concentration in the produced water at the exit stream. Figures 2-4 show the profile of MBR volume, length and hydraulic retention time (HRT) versus conversion of produced water oil/grease contaminant at varying rate constants.

The effect of rate constant on the volume of MBR is shown in Figure 2. As the conversion increased, the volume also increased at various specific rate constant. At the rate constants investigated (2 to 8 h⁻¹), the MBR volume required for the conversion of oil contaminant in produced water varied. Thus, at 95% to 99%

oil reduction, the volume of MBR increased from 1.45 to 6.95 m³ at rate constant of 2 h⁻¹, 0.73 to 3.48 m³ at 4 h⁻¹, 0.48 to 2.32 m³ at 6 h⁻¹ and 0.36 to 1.74 m³ at 8 h⁻¹. However, the volume decreased with increase in specific rate constant. Indicating that, substrate rate constant influenced MBR performance and the optimum operational volume should be tailored to achieve high percentage removal of oil contaminants in produced water.

The effect of rate constant on the height of MBR is shown in Figure 3. In similar profile like the volume, the conversion of oil contaminated produced water increased with increase in height, while increasing the specific rate constant for substrate utilization resulted in decrease in the height. Thus, at 95% to 99% oil reduction, the height of MBR increased from 1.28 to 6.15 m at rate constant of 2 h⁻¹, 0.64 to 3.67 m at 4 h⁻¹, 0.43 to 2.05 m at 6 h⁻¹ and 0.32 to 1.54 m at 8 h⁻¹. This implies that, higher percentage removal of oil contaminant could be obtained when the substrate rate constant is controlled.

The variation of rate constant with HRT was also investigated as shown in Figure 4. Similarly, the conversion of oil contaminated produced water increased with increase in HRT, while the HRT decreased with increase in the specific rate constant. At 95% to 99% oil reduction, the HRT increased from 2.91 to 13.96 hr at rate constant of 2 h⁻¹, 1.45 to 6.95 hr at 4 h⁻¹, 0.97 to 4.63 hr at 6 h⁻¹ and 0.73 to 3.48 hr at 8 h⁻¹. However, hydraulic retention time as an important parameter in the design of MBR for the treatment of produced water needs to be sufficient for high removal contaminants. (Zaerpour 2014) reported a range of HRT obtained from 13 MBRs in the United States to be from 2 to 8.1 hr [12] , while Naghizadeh et al. (2013) obtained a range of HRT from 4 to 12.8 hr at different runs and conditions [11] .

Figures 5-7 show the effect of P/V on the conversion of Chemical Oxygen Demand (COD) relative to MBR size and Hydraulic Retention Time (HRT). The rate of energy supplied per reactor volume was investigated following range of values recommended by (Badrnezhad and Beni, 2013) [10] . Figure 5 shows the conversion of COD at varying P/V. COD conversion increased with increase in the volume but decreases with increase in the ratio of energy supplied per volume. Thus, at 95% to 99% COD conversion, the corresponding volume of MBR were 0.34 to 5.91 m³ at 4 kW/m³, 0.25 to 4.44 m³ at 6 kW/m³, 0.21 to 3.64 m³ at 8 kW/m³ and 0.18 to 3.11 m³ at 10 kW/m³. These values were in agreement with values obtained by (Janus 2013) [16] with optimum energy/volume of 9.5 kW/m³.

These showed that energy supply per volume of reactor was essential for the determination of the overall performance of MBR operations. However, the removal of COD using MBR has been reported with high percentage conversion [10] [11] [17] .

The effects of energy supplied per reactor volume on conversion of COD with respect to height is shown in Figure 6. The conversion of COD increased with

increase in height at a fixed rate of energy supply per volume, while the increase in the rate of energy supply per volume resulted in the decrease of height at a fixed COD percentage removal. Thus at 95% to 99% COD conversion, the height increased from 0.30 to 5.22 m at 4 kW/m³, 0.22 to 3.93 m at 6 kW/m³, 0.18 to 3.21 m at 8 kW/m³ and 0.16 to 2.75 m at 10 kW/m³. The results showed that the operation of MBR with moderate height capable of removing high percentage of COD required high rate of energy supply per volume.

The effect of rate energy supply per volume on HRT is depicted as shown in Figure 7. The conversion of COD in produced water increased with increase in HRT at fixed energy per volume, while HRT decreased with increase in the rate of energy per volume at fixed COD conversion. At 95% to 99% conversion of COD, the HRT increased from 0.42 to 7.38 hr at 4 kW/m³, 0.32 to 5.56 hr at 6 kW/m³, 0.26 to 4.54 hr at 8 kW/m³ and 0.22 to 3.89 hr at 10 kW/m³. These again, showed that hydraulic retention time influenced the rate of energy supply per volume of MBR produced water treatment process.

Figure 8 shows the MBR volume required for the conversion of contaminants in produced water. At every fixed percentage removal, the MBR volume required to treat the investigated contaminant increased in the following order: oil/grease < TSS < TOC < TDS. However, the contaminants’ conversion increased exponentially as volume increased. For the four contaminants studied, the corresponding volume required to remove 95% to 99% of the contaminants were: 0.36 to 1.74 m³ for oil/grease, 0.60 to 2.94 m³ for TSS, 0.66 to 3.25 m³ for TOC and 1.10 to 5.29 m³ for TDS.

Although, the concentration of contaminants in oil field produced water may not be the same, it is reasonable that when designing a MBR for multicomponent treatment, the size should be considered based on the contaminants

with highest concentrations. However, various works on treatment of produced water contaminants have been studied with various percentage conversions (removal) as recommended by (Kwong et al., 2008; Sharrer et al., 2010 Badrnezhad and Beni, 2013; Naghizadeh et al., 2013 and Mousa and Arafat, 2015) [10] [11] [13] [18] [19] .

The variation of conversion of contaminants with height is shown in Figure 9. Thus, at every fixed percentage removal, the MBR height required to treat the produced water contaminants increased in the following order: oil/grease < TSS < TOC < TDS. The corresponding height required to remove 95% to 99% of the contaminants were: 0.32 to 1.54 m for oil/grease, 0.53 to 2.60 m for TSS, and 0.58 to 2.87 m for TOC and 0.98 to 4.68 m for TDS. Likewise, the design of MBR for treatment multi-component contaminants should be considered with reference to the contaminants with highest concentration.

The variation of the HRT with conversion is shown in Figure 10. In a similar manner, the conversion of contaminants increased with increased in HRT. The

highest HRT was observed in TDS while the lowest was in oil/grease. This was because of the variation in the composition of contaminants in the produced water. At 95% to 99% conversion, the HRT increased from 0.45 to 2.17 hr for oil/grease, 0.75 to 3.68 hr for TSS, 0.82 to 4.06 hr for TOC and 1.38 to 6.62 hr for TDS.

The capacity of the membrane to trap and keep hold of the solid contaminant at a substantial period depends on the substrate loading and the pore size of the membrane.

The effects of initial loading of substrate on SRT is shown in Figure 11. Increased in substrate loading resulted in the increase in retention time of the solid at a fixed substrate conversion. Also, the conversion of substrate by micro-organisms increased as the solid retention time increased. However, excess loading of the substrates could promote the fouling of the membrane thereby reducing the performance of the treatment process. At substrate conversion between 0% and 50%, the SRT obtained were 58.86, 82.67 and 138.22 days at initial loading of 0.8, 1.0 and 1.2 kg/m³ respectively. However, different SRT values had been obtained using MBR: 100 days by kwong et al. (2008) [13] and 24.7 days by Sharrer et al. (2010) [18] .

The effect substrate loading on mixed liquor suspended solids (MLSS) production is depicted in Figure 12. The concentration of substrate in the membrane affects the growth of micro-organisms and hence, their activities in decomposing contaminants. Thus, increase in the initial loading of substrate concentration increased the population of the organisms. Though, between zero and about 1 hour, there was a lag in biomass growth, which could be attributed to process of microbes’ adaptation. MLSS concentration changed the kinetics of the treatment process and it was influenced by the concentration and type of nutrient’s source [11] . However, the biomass concentration at the initial substrate loading of 0.8, 1.0 and 1.2 kg/m³ are 0.37, 0.46 and 0.55 at HRT of 4.35 hr.

The volume of MBR required for the removal of produced water contaminants between 95% to 99% at 1.0 kg/m³ initial substrate loading is shown in Figure 13. The MRB size required to remove the contaminants from produced water increased as the MLSS concentration was increased. However, with the produced MLSS concentration, oil/grease required smaller volume of MBR as compared with TDS.