**4. Results and Discussion**

#### *4.1. E*ff*ect of Recirculation Flow Rate on the Total Cost*

The relationship between the recirculation flow rate and the annual total cost is shown in Figure 2. Other design variables are held constant, inlet pressure TMP = 500 kPa, module height *h* = 5 mm, module width *w* = 1 m.

**Figure 2.** Effect of recirculation flow rate on the annual total cost of the plant.

As illustrated in the figure, the increase in the recirculation flow rate initially reduces the total cost of the membrane plant. However, further increasing the recirculation flow rate causes the total cost to increase. This might be explained by considering the permeate flux and power consumption. The rise in flow rate leads to an increase in the Reynolds number and the permeate flux, as demonstrated in the Equation (7). Consequently, the required membrane area can be reduced, which results in a decrease in capital investment and membrane replacement cost. Hence, the annual total cost of the plant is minimized. However, from the Equations (17) and (22), it can be obtained that the pressure drop increases rapidly in the turbulent flow regime and faster than the increase in permeate flux (the exponents of the velocity dependency are 1.75 and 1, respectively). The pressure loss is directly related to the power consumption of the pumps or the energy cost. Therefore, as the recirculation flow rate increases, the annual total cost decreases and reaches a minimum value, and then rises again with a further increase in flow rate.

#### *4.2. E*ff*ect of Module Inlet Pressure Operation*

Figure 3 demonstrates the relation between the operating pressure at the inlet and the annual total cost. Other design variables are held constant, recirculation flow rate *Q* = 35 m<sup>3</sup>/h, module height *h* = 15 mm, module width *w* = 1 m.

**Figure 3.** Effect of inlet pressure on the total cost of the plant.

As illustrated in the figure, the increase in the inlet pressure initially minimizes the annual total cost of the membrane plant. However, a further increase in pressure causes the annual total cost to increase. The explanation is as follows. The permeate flux benefits from the increase in pressure operation with the order of dependence of 0.26, as obtained from Equation (7). Hence, the required membrane area can be reduced, which results in a decrease in the capital investment and membrane replacement cost. Therefore, the annual total cost of the membrane plant is minimized. However, from Equations (22) and (27), it can be obtained that both the power consumption and the initial investment for the pumps rise much faster than the increase of permeate flux (the exponents of the pressure dependency are 1 and 0.4, respectively). Therefore, as the inlet operating pressure increases, the annual total cost decreases and reaches a minimum value, and then rises again with a further increase in operating pressure.

## *4.3. E*ff*ect of Module Height*

Figure 4 shows the effect of module height on the total cost. Other design variables are kept constant, inlet pressure TMP = 500 kPa, recirculation flow rate *Q* = 45 m<sup>3</sup>/h, module width *w* = 1 m.

From the figure, it is shown that the decrease in the height of the membrane module will decrease the total cost of the membrane plant. However, a further decrease in module height makes the total cost also increase. The reason is similar to the effect of the recirculation flow rate. The decrease in module height will increase the cross-flow velocity and thus increase the permeate flux. Therefore, the membrane area decreases, which leads to a decrease in the membrane replacement cost. Consequently, the annual total cost of the membrane plant will decrease. However, high velocity affects the pressure loss and hence, energy consumption. Therefore, the annual total cost initially decreases and reaches a minimum value, then rises again with a further decrease in module height.

**Figure 4.** Effect of module height on the total cost of the plant.

#### *4.4. E*ff*ect of Module Width*

Figure 5 shows the effect of module height on the total cost. Other design variables are kept constant, inlet pressure TMP = 500 kPa, recirculation flow rate *Q* = 35 m<sup>3</sup>/h, module height *h* = 10 mm.

**Figure 5.** Effect of module width on the total cost of plant.

Similar to the effect of the recirculation flow rate, it is shown that the decrease in the height of the membrane module will decrease the total cost of the membrane plant. However, a further decrease in module height makes the total cost also increase. The decrease in module width will increase the cross-flow velocity and shear rate, thus increase the permeate flux. Therefore, the membrane area decreases, which leads to a decrease of membrane replacement cost, and the annual total cost of the membrane plant will decrease. However, high velocity affects the pressure loss and hence, energy consumption. Therefore, the annual total cost initially decreases and reaches a minimum value, then rises again with a further decrease in module height.
