*2.1. Membrane Plant Configuration*

The operational configuration assumed in developing the model for simulation of membrane plant is presented in this section. The plant's operation is supposed to be a continuous feed and bleed mode, as shown schematically in Figure 1. A summary of the notations is presented in Table 1. There are two important pumps in this configuration: the feed pump supplies the appropriate transmembrane pressure, while the recirculation pump supports the required cross-flow rate through the modules. At startup, the feed pump is used to fill the recirculation/module loop, and then the recirculation pump is started. After the system is stabilized, a small fraction of the flow is continually withdrawn as concentrate at a flow rate (*R*).

**Figure 1.** Schematic configuration of feed-and-bleed mode membrane system.


**Table 1.** Summary of system configuration notations.

## *2.2. Modeling of Membrane Modules*

In ultrafiltration, the pressure is also a significant factor affecting the flux behavior; therefore, the friction loss should be taken into consideration. The model proposed in this study is the improvement of our previous work [10]. In [10], rather than using incremental length step, the channel is subdivided into a number of segments by the protein concentration from the inlet (φi) to the outlet (φf). The concentration of the protein solution is expressed as a volume fraction. For each control volume, the transport properties, the permeate flux equation in [6], and friction loss formulae are applied. Then, the length of each increment and the pressure of the next segmen<sup>t</sup> can be obtained by mass balance. However, the results show that there are some discontinuous points due to discretization. Therefore, in this study, the chain rule of differentiation [12] is applied and a system of ordinary differential equations was developed. The derivatives are taken with respect to the concentration instead of the length. The mathematical form is discussed in detail as follows.

The overall mass balance and component material balance give:

$$\begin{aligned} F &= R + P\\ F\phi\_0 &= R\phi\_\mathbf{f} + P\phi\_\mathbf{p} \end{aligned} \tag{1}$$

Material balance around the mixing junction of the feed and recirculation stream provides:

$$F\phi\_0 + Q\phi\_\mathbf{f} = (F+Q)\phi\_\mathbf{i} \tag{2}$$

In Karasu's study [13], the author reconstituted various solutions with di fferent protein concentrations from the whey protein concentrated powder (ALACENTM, acquired from NZMP Ltd.) and measured their rheological properties. Therefore, the correlation equations from [13] can be used to estimate the properties of protein solution in the simulation.

The correlation equation for the viscosity of protein solution is:

$$
\mu\_{\rm m} = 8.94 \times 10^{-4} \exp(13.5482 \phi\_{\rm m}) \tag{3}
$$

in which μm, φ m are the viscosity and concentration of the mixture (solution), respectively.

The density of protein solution (ρm) can be approximated from the density of water (1000 kg/m3) and the density of dry protein powder (1360 kg/m3)

$$
\rho\_{\rm m} = 1000(1 - \phi\_{\rm m}) + 1360 \phi\_{\rm m} \tag{4}
$$

The permeate flux through the membrane is (obtained from [6])

$$\frac{J}{\mu} = 3.66 \times 10^{-7} \left(\frac{P}{\rho\_{\text{m}} \mu \nu^2}\right)^{0.27} \left(\frac{\rho\_{\text{m}} \mu d\_{\text{h}}}{\mu\_{\text{m}}}\right)^{0.52} \tag{5}$$

in which *P* is the transmembrane pressure, *u* is the cross-flow velocity, and *d*h is the hydraulic diameter of the flow channel.

From the shear-induced di ffusion or surface transport model, the permeate flux depends linearly on the membrane shear rate [14]. The increase of module height will decrease the shear rate at the membrane surface. In the experiment of this study, the height of the module channel remained constant. Therefore, in order to extend the application of the permeate flux equation to other systems which has di fferent module heights, the correction factor *d*h0/*d*h should be introduced as follows.

$$\frac{I}{\mu} = 3.66 \times 10^{-7} \left(\frac{P}{\rho\_{\rm m} \mu^2}\right)^{0.27} \left(\frac{\rho\_{\rm m} \mu d\_{\rm h}}{\mu\_{\rm m}}\right)^{0.52} \left(\frac{d\_{\rm h0}}{d\_{\rm h}}\right) \tag{6}$$

where *d*h0 is the value of hydraulic diameter of the membrane module used in the study [5].

The equation for permeate flux can be rewritten as:

$$J = 5.124 \times 10^{-9} \left(\frac{P}{\rho\_{\rm m} \mu^2}\right)^{0.27} \left(\frac{\rho\_{\rm m} \mu d\_{\rm h}}{\mu\_{\rm m}}\right)^{0.52} \left(\frac{\mu}{d\_{\rm h}}\right) \tag{7}$$

or in terms of shear rate at the membrane surface · γ = 6*u h* = 12*u d*h , where *h* is the channel height:

$$J = 4.27 \times 10^{-10} \left(\frac{P}{\rho\_{\rm m} \mu^2}\right)^{0.27} \left(\frac{\rho\_{\rm m} \mu d\_{\rm h}}{\mu\_{\rm m}}\right)^{0.52} \dot{\mathcal{V}} \tag{8}$$

The flow rate and cross-flow velocity at the entrance of the membrane module

$$Flow\_{\overline{1}} = F + Q \tag{9}$$

*Processes* **2020**, *8*, 4

$$\mu\_{\rm i} = \frac{F + Q}{A\_{\rm cross}} = \frac{F + Q}{h \times w} \tag{10}$$

where *w* × *h* is the cross-sectional area of the flow channel, *w* is the width, and *h* is the height of the membrane module.

The flow rate and velocity change along the length of the membrane module are determined by the mass balance with the assumption that there is no protein in the permeation, as:

$$Flow \times \phi\_{\rm m} = const \tag{11}$$

$$d(Flow \times \phi\_{\rm m}) = 0\tag{12}$$

After applying the product rule and then rearranging, we obtain

$$\frac{d(Flow)}{d\phi\_{\rm m}} = -\frac{Flow}{\phi\_{\rm m}}\tag{13}$$

$$
\mu = \frac{Flow}{h \times w} \tag{14}
$$

The mass balance for the element control volume along the length of the membrane:

$$d(Flow) = -f \times d(wz) = -fw \, dz \tag{15}$$

Incorporating with Equation (13), the differential equation for the length can be obtained:

$$dz = -\frac{d(Flow)}{w \, f} = \frac{Flow \times d\phi\_{\rm m}}{\phi\_{\rm m} \, w \, f} \tag{16}$$

The pressure loss is calculated by the Darcy–Weisbach equation in differential form as follows

$$dP = -4f\rho\_{\rm m}\frac{dz}{d\_{\rm h}}\frac{u^2}{2} \tag{17}$$

in which *f* is the friction factor

> *f* = 24Re for laminar flow in a rectangular channel (Re < 2000) [15]. *f* = 0.079 Re0.25 for turbulent flow (Re > 2000) (smooth pipes, Blasius correlation [16]).

Incorporating with Equation (16), the differential equation for the pressure can be obtained:

$$\frac{dP}{d\phi\_{\rm m}} = -4f\rho\_{\rm m}\frac{1}{d\_{\rm h}}\frac{\mu^2}{2}\frac{Flow}{\phi\_{\rm m}}\frac{1}{w\,f} \tag{18}$$

Finally, the system of ordinary equations for the membrane module system was developed as follows. ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

$$\begin{array}{l} \frac{d\langle hw\rangle}{d\phi\_{\rm m}} = -\frac{Flow}{\phi\_{\rm m}}\\ \frac{dL}{d\phi\_{\rm m}} = \frac{F\_{\rm kew}}{\phi\_{\rm m}w}\\ \frac{dP}{d\phi\_{\rm m}} = -4f\rho\_{\rm m}\frac{1}{h\_{\rm n}}\frac{u^{2}}{2}\frac{L\text{foc}}{\phi\_{\rm m}}\frac{1}{w^{3}}\\ \mu\_{\rm m} = 8.94 \times 10^{-4} \exp(13.5482\phi\_{\rm m})\\ \rho\_{\rm m} = 1000(1-\phi\_{\rm m}) + 1360\phi\_{\rm m}\\ f = 5.124 \times 10^{-9} \left(\frac{P}{\rho\_{\rm m}u^{2}}\right)^{0.27} \left(\frac{\rho\_{\rm m}uh\_{\rm h}}{\mu\_{\rm m}}\right)^{0.52} \left(\frac{u}{h\_{\rm h}}\right)\\ u = \frac{L\text{low}}{\rho\_{\rm k}^{\rm z}w}\\ f = \frac{2L}{\text{Ref}}\text{ if }\text{Re} < 2000\\ f = \frac{0.007}{\text{Re}^{0.25}}\text{ if }\text{Re} > 2000\end{array} \tag{19}$$

In the interval [φi, φf] and the initial condition

$$\begin{cases} \begin{aligned} \text{Flow}(\phi\_{\text{l}}) &= F + Q \\ z(\phi\_{\text{l}}) &= 0 \\ P(\phi\_{\text{l}}) &= P\_{\text{l}} \end{aligned} \end{cases} \tag{20}$$

The system of the ordinary equations can be solved by the Runge–Kutta 4th order method [17]. The solution is the flow rate, the length, and the pressure at the final concentration. After the total length was determined, the membrane area was obtained:

$$A\_{\text{membrane}} = w \times L \tag{21}$$

The total energy generated by the pumps:

$$\begin{array}{l} E\_{\text{pumps}} = E\_{\text{P}} + E\_{\text{Q}} \\ = F \times P\_{\text{F}} + \{ F \times (P\_{\text{i}} - P\_{\text{F}}) + Q \times (P\_{\text{i}} - P\_{\text{O}}) \} \\ = F \times P\_{\text{i}} + Q \times \Delta P\_{\text{drop}} \end{array} \tag{22}$$

In this equation,

*E*P, *<sup>E</sup>*Q are the power supplied by the feed pump and recirculation pump, respectively *P*F, *P*i, Po are the pressure at the outlet of the feed pump, at the inlet and outlet of the membrane module, respectively

<sup>Δ</sup>*P*drop is the pressure drop in the membrane module

For desirable feed flow rate, recirculation flow rate, and initial and final concentration, the total membrane area and energy usage are calculated from the simulation model with the governing equation of permeate flux. The total membrane area and energy usage are the important factors in the estimation of the total cost of the filtration plant, and the correlation to the total cost is discussed in the next section.
