1. Pipes and valves

$$C\_{\rm PV}^{\*} = \epsilon 000 (A\_{\rm membrane})^{0.42} \quad [\\$] \tag{29}$$

2. Instruments and controls

$$C^\*\_{\rm IC} = 1500 (A\_{\rm membrane})^{0.66} \quad [\\$] \tag{30}$$

3. Tanks and frames

$$C\_{\rm TF}^\* = 3100 (A\_{\rm membrane})^{0.53} \quad [\\$] \tag{31}$$

4. Miscellaneous

$$\mathcal{L}^\*\_{\text{MI}} = 8000 (A\_{\text{membrane}})^{0.57} \quad [\\$] \tag{32}$$

Annual Capital Cost

> The annual capital cost will be obtained from the capital cost based on the amortization factor, as

$$\mathcal{C}\_{\text{capital cost}} = \mathcal{C}^\*\_{\text{capital}} \times \left(\frac{A}{P}\right) \text{ [\\$/ year]} \tag{33}$$

For the plant design year of 20 years and the interest rate 8%, the amortization factor will be about 0.1.

#### **3. Formulizations of the Problem**

#### *3.1. Fix Parameters and Design Variables*

The optimization problem is the geometric design and operating condition of the membrane system. The operation is under steady-state conditions. In the design problem, some variables should meet the requirement of the process. These are inlet variables and usually are feed flow F, inlet concentration φ0, and outlet concentration φf. The protein is assumed to be entirely rejected by the membrane (φp = 0).

The decision variables investigated in the optimization problem were: channel geometry (width × length × height), the inlet pressure (*P*i), and recirculation flow rate (*Q*).
