**1. Introduction**

The performance of cross-flow ultrafiltration is greatly influenced by the behavior of permeate flux. The flux declination depends on many factors, including solution properties, membrane characteristics, and operating conditions. Currently, most research has focused on improving membrane performance in terms of permeability and selectivity [1,2]. Only a few studies have paid attention to how the membrane module is configured and operated [3].

The selection of a specific membrane module geometry for a specific application is a ffected by several factors: the fabrication method, energy consumption, and the possibility of fouling [4]. Manufacturers usually sugges<sup>t</sup> the geometric design of the membrane module from the fabrication aspect [4]. There is little evidence that energy e fficiency is primarily considered in their approach. Presently, with an increase in energy costs and a decrease in membrane price, energy e fficiency should be given greater emphasis in the design of membrane module geometry. Therefore, a module design methodology which includes energy consideration is necessary.

In addition, the determination of operating conditions is normally derived from experience, obtained from the handbook or by recommendation from the membrane manufacturer. However, the performance of the membrane system, which is governed by the permeate flux equation, varies

considerably between di fferent situations and applications. In this manner, a general methodology for the design and operation scheme should be studied for any specific application.

In our earlier work [5], a simple combined model, which simultaneously considers pore blockage and cake filtration, was proposed and proved its potential for estimation of flux decline in cross-flow ultrafiltration of protein solution. In our other study [6], the steady-state permeate flux was predicted and correlated to operating conditions from the model. The methodology can be generalized for any particular process to determine the steady state operation equations of the membrane for protein separation. From the operation equation, an optimal design for any certain application can be obtained. This design is dependent on various factors, such as plant capacity, desired recovery, and the energy cost at the specific location. The optimal design will give additional information for supporting the assessment of existing membrane modules and the fabrication and operation decisions of the new membrane system.

However, there are few studies on the optimization of membrane processes and cost estimation. For example, in the study of Wiley, Fell, and Fane [4], membrane module design for brackish water desalination was optimized. However, the configuration is single-pass operation, and the cost only included the membrane cost and energy cost. In the study of Sethi and Wiesner [7], a cost model for the removal of natural organic matter was developed, but the optimization was not conducted. In membrane technology, the feed and bleed configuration, which merges the batch and the single-pass operations, is widely used for continuous full-scale operation [8,9]. In our previous study [10], a simulation and optimization model of a feed and bleed membrane system was investigated. The model was solved by a simple discretization method, and some discontinuous points of the objective function appeared. Moreover, the fixed cost was assumed independently of the system size. Therefore, in this study, the optimization of a membrane module for protein ultrafiltration is considered, in which the geometric design and operating conditions are variables. A system of ordinary di fferential equations is developed and numerically solved to improve the accuracy of the model. The objective function is the annual cost. The cost consists of various types of capital investments depending on the plant scale and an operating expense associated with energy consumption. The core facilities of the membrane plant are classified into pumps, valves and pipes, instruments and controls, vessels and frames, and miscellaneous. The cost of each category is individually correlated to plant scale, specifically the membrane area. A model incorporating the pressure loss is introduced to simulate the membrane plant and calculate the membrane area and energy usage, which are directly related to the total cost. In the optimization problem, the channel dimensions (width × length × height) and operating conditions (the inlet pressure and recirculation flow rate) were considered as decision variables.

Among various optimization techniques, evolutionary algorithms which are a broad class of population-based metaheuristic optimization techniques inspired by the evolution of species in nature, have been e ffectively utilized in di fferent fields to solve numerous optimization problems [11]. Evolutionary algorithms, including the genetic algorithm, have some advantages such as being derivative-free and having durable robustness, flexibility, and the ability to discover the global optimum. Therefore, the genetic algorithm is used in this study to find the most cost-e ffective design and operation of the membrane module.

#### **2. Process Configuration and Model Calculation**

In this section, the configuration of the plant operation is introduced. The system of ordinary di fferential equations for modeling the membrane plant is developed. The results from the simulation are essential for the estimation of the membrane area and energy usage, which are two significant factors a ffecting the total cost of the plant.
