1. Introduction
Facing the rapid growth of the global economy and the improvement of the people’s living standards, the demand for water and energy is becoming more and more intense, which promotes the agenda of energy structure optimization tending to be more cost-effective and energy-efficient. In addition to chemical engineering and office buildings, humidifiers are also widely used in people’s daily lives for regulating air humidity. Among them, the packed bed humidifier has tremendous applications thanks to its high energy effectiveness and large contact area [
1]. Packings such as wooden slates [
2], corrugated aluminum sheets [
3], polypropylene [
4], and ceramic foam [
5] have shown great humidification performance. Designers expect to achieve maximum humidification effectiveness with minimum costs. However, the relationship between humidification capacity and the input cost of the humidifier is proverbially a positive correlation in the physical process. Therefore, it is significant to optimize the overall performance of a packed bed humidifier referred to thermodynamics and economics.
In recent years, many optimization methods were developed and generally can be divided into parameter analysis and mathematical algorithm [
6]. The parameters affecting the thermodynamic performance of the packed bed humidifier were highlighted in detail by Xu [
5], and many remarkable results were found according to the univariate optimization. It was shown that the independent influences of the liquid–gas ratio, operating pressure, inlet water temperature, and inlet air enthalpy on the thermodynamic performance are important. In addition to the above parameters, the effects of inlet air temperature, inlet relative humidity, inlet wet bulb temperature of the humid air, and the specific surface area of the packing were investigated through theory and experiment at on-design and off-design conditions [
7,
8]. The main influencing factors, including the liquid–gas ratio, the inlet air temperature, the inlet water temperature, and the specific surface area of the packing, were revealed. It can be summarized that the influencing parameters mainly contain thermodynamic parameters and geometric parameters, while the researches on geometric parameters commonly refer to the packing. It was demonstrated that the size and material of the packing have a significant influence on the thermodynamic and economic performance of the packed bed humidifier [
9,
10]. For instance, packings such as corrugated aluminum sheets with length × width × height of 650 × 650 × 650 mm
3 [
3] and corrugated wire mesh with 0.15 m high, 0.39 m diameter [
8] showed gratifying thermodynamic and economic performance. The maximum humidification capacity could be reached 15 kg/h utilizing corrugated aluminum sheets, and the maximum values of unit humidification capacity of volume and unit humidification capacity of cost with corrugated wire mesh presented as 3.82 × 10
–2 kgs
−1m
−3 and 12.74 kg
$−1, respectively. Moreover, it should be found that the degree of univariate optimization is limited for simultaneously improving the thermodynamic and economic performance of the humidifier, subject to the finite-size and finite-time process.
Furthermore, in order to reduce the residual irreversibilities of the humidification process, which is subject to the second law, the entropy generation minimization method was employed for thermodynamic optimization [
11,
12]. Thermodynamic optimization means employing some measures to change the flow distribution in the humidification process to make the distributions of the driving forces of the heat and mass transfer process more uniform, such that the energy effectiveness can be improved. For this idea, the water extraction/injection and air extraction/injection between dehumidifier and humidifier in the humidification–dehumidification desalination system were implemented to improve the system performance [
13,
14]. It was proved that the extraction direction from the dehumidifier to the humidifier was the preferred measure for both the water stream and the air stream. For water extraction/injection, the gained output ratio of single extraction was approximately four times that of zero extraction [
13]. For air extraction/injection, the gained output ratio was elevated by 91% and 112% with single and double extraction systems, respectively, compared with that of the base system [
14]. Originally, Chen [
15,
16] successively studied the influences of extraction/injection from both the water side and air side on the thermodynamic performance of the packed bed humidifier. It was found that single water injection and air extraction for the single humidifier are more desirable measures compared with zero and single water extraction and air injection. The energy effectiveness of the packed bed humidifier increased, respectively, by 19.74% with single water injection and 26.32% with air extraction than that of zero measure. Furthermore, it was shown that the liquid–gas ratio and inlet relative humidity of humid air are the main single factors affecting thermodynamic optimization.
For a different parameter analysis (generally for single-objective optimization), mathematical algorithms come in handy when optimization involves multi-objective optimization. Many popular methods, such as the response surface method (RSM) [
9], genetic algorithm (GA) [
17], particle swarm algorithm [
18], etc., can be used to analyze and solve the complex physical problems with multiple objectives that need to be optimized, affected by multiple factors. Among them, RSM is a well-developed experimental design method, and GA is a computational method to search for the optimal solution by following the natural evolution process based on the genetic mechanism of Darwin’s bio evolution theory. Moreover, it is verified that GA is a more complete and sounder model than particle swarm algorithm. A more accurate correlation for the temperature prediction of solar panels was established based on GA, while the independent factors and interaction factors between solar radiation, ambient temperature, wind velocity, and ambient relative humidity were considered [
17]. It was found that the prediction accuracy of the solar panel temperature was improved by approximately 2–4 times. For further improvement, combination optimizations between RSM and GA [
19] and RSM and the particle swarm algorithm [
20] were developed to improve the experimental design. It was demonstrated that most performance parameters after multi-objective optimization were better than those without optimization. The effects of pitch, diameter, pipe length, and inlet velocity on the separation efficiency and pressure drop were investigated based on the RSM and GA [
21]. The optimization results showed that the optimization degree of the separation efficiency had a certain limitation at the optimal points, while the pressure drop can be reduced by approximately 50%. Recently, Shahverdian [
22] proposed a dynamic multi-objective optimization method in order to improve the power output and simultaneously reduce the cooling water consumption of a photovoltaic module. It was found that the temperature difference between the average and maximum value in a year evaluated by the novel method decreased, respectively, by 79.79% and 54.53%, compared with no-cooling and a constant water flow of 0.1 LPM. These achievements obtained by multi-objective optimization provide a robust foundation for optimizing the overall performance of the packed bed humidifier.
From the literature survey, it was found that the improvement of thermodynamic performance of the packed bed humidifier is generally at the expense of input cost, while the performance optimization is subject to the univariate optimization (single-objective optimization). In addition, the overall optimization of thermodynamic and economic performance for the packed bed humidifier is missing. Moreover, a previous physical model can become too complicated and time-consuming in the optimization process. To simplify the optimization process, save time cost, and improve overall performance simultaneously, in this study, the combination optimization of RSM and GA is employed to assess and optimize the thermodynamic and economic performance of a packed bed humidifier filled with corrugated wire mesh. Surrogate models for quickly searching the optimal thermodynamic and economic performances are established. Furthermore, the influences of independent and interactive factors on designated objective functions are addressed, and an explicit influence degree of the important parameters is given based on the analysis of variance. Finally, a series of Pareto-optimal points for possible optimal thermodynamic and economic performance is given with optimal parameter combination, which can provide significant references for the designer of packed bed humidifiers.
4. Discussion
According to the experimental design, it is found from
Table 4 that it is extremely difficult for the thermodynamic and economic performance of the packed bed humidifier to be simultaneously optimal as expected at the same condition, based on the simplified orthogonal combination. Furthermore, it is expected that there is still great potential for the UHCV and UHCC to have a higher value through multi-objective optimization.
Based on the ANOVA results, it is found from
Table 5 that the F-value of 2387.81 for UHCV and 414.45 for UHCC illustrate that the models are extremely significant. In addition, it is demonstrated that the
p-value of both UHCV and UHCC is less than 0.0001, which shows that the regression models are effective, among which the UHCV model has better accuracy. Moreover, according to the values of R-Squared and Adj R-Squared, it is clear each of the models has a great fitting quality. It is also found that the value of Pred R-Squared is in reasonable agreement with the Adj R-Squared value both for UHCV and UHCC, and the differences are both less than 0.2, which confirmed this fact again. Furthermore, the Adeq Precision value of 214.256 for UHCV and 84.571 for UHCC are both greater than 4, which are satisfactory and an adequate signal to navigate the experimental design.
From the results in
Table 6 and
Table 7, it is said that a
p-value less than 0.05 indicates the relevant model term is significant. Therefore, it is summarized that parameters A, B, D, E, AD, AE, CE, A
2, and B
2 have significant influences on UHCV, while parameters A, B, C, D, E, AB, AD, A
2, and B
2 are significant model terms for UHCC. Here, it is found that in addition to the independent parameters, the interactive parameters such as AB, AD, AE, and CE also have significant impacts on the thermodynamic and economic performance. This is the reason why multi-objective optimization on the overall performance of the packed bed humidifier is developed in this study.
Through model diagnosis, it is shown from
Figure 4 that most design points inhabit near the diagonal, and the relationship between predicted values and actual values is linear, which confirms that the proposed models can be employed to predict the thermodynamic and economic performance of the packed bed humidifier.
4.1. Thermodynamic Performance Analysis
Figure 5 exhibits the case that the other independent parameters are constant in the middle value of the variable range. The relationship between liquid–gas ratio and UHCV is a quadratic parabola correlation, while there is a linear relationship between inlet water temperature and UHCV, as shown in
Figure 5a. Therefore, a significant interaction influence appears between these two parameters. It is found that higher values of the liquid–gas ratio and the inlet water temperature will lead to a larger UHCV evidently. This is because either increasing the inlet water temperature or increasing the liquid–gas ratio will be beneficial to the heat load of the hot side. Then, the moisture absorption capacity of humid air will rise sharply with the increase in its temperature, which will cause a higher value of the UHCV, as shown in
Figure 5b. It is observed that a maximum value of 0.11 kgs
−1m
−3 for the UHCV appears at the conditions of the liquid–gas ratio of 5 and inlet water temperature of 90 °C.
4.2. Economic Performance Analysis
It is highlighted that
Figure 6 is plotted when the other parameters are set as the middle value. From
Figure 6a, it is found that there is a slight linear relationship between the inlet water temperature and response UHCC, while the influence relationship between the liquid–gas ratio and UHCC is a quadratic parabola. The difference is that the impact trend of these two variables on UHCC is the opposite. It can be seen that the increase in inlet water temperature can enhance the value of the UHCC, while a higher value of the liquid–gas ratio will rapidly reduce the value of the UHCC. It is found that a maximum UHCC of 15.51 kg
$−1 emerges at the conditions of the liquid–gas ratio of 1 and inlet water temperature of 90 °C. That is because increasing the inlet water temperature will not only improve the moisture absorption capacity of the humid air but also increase the energy consumption of the electric heater. Therefore, the comprehensive improvement effect is not obvious. However, the decrease in the liquid–gas ratio will greatly reduce the power consumption of the water pump and electric heater. Simultaneously, since the spray water temperature is unchanged, the moisture absorption of the humid air is relatively small compared to the cost increase. As a result, the interaction of parameter AD shows a sloping surface on the UHCC from the front view in
Figure 6b.
4.3. Multi-Objective Optimization Analysis
As analyzed in
Section 4.1 and
Section 4.2, it is summarized that it is laborious to obtain the desirable thermodynamic and economic performance simultaneously for the packed bed humidifier based on BBD. To overcome this problem, a multi-objective optimization is implemented to predict the best performance for the packed bed humidifier employing the Non-dominated Sorting Genetic Algorithm-II. After natural selection, as shown in
Figure 7, it can be found that the optimal points obtained are all located in the upper right of the experimental values, which indicates that the overall performance of the packing bed humidifier can be substantially improved.
It is found from
Figure 8 that population B (inlet air temperature), C (inlet relative humidity), D (inlet water temperature), and E (specific surface area) have a favorite habitat around 16 °C, 0.6, 90 °C, and 700 m
2m
−3, respectively, while population A (liquid–gas ratio) is randomly distributed in the specific range. Additionally, a higher liquid–gas ratio is beneficial to thermodynamic performance, while a lower value of liquid–gas ratio can improve the economic performance in combination with the optimal parameter group.
4.4. Verification of Optimal Solution
It was shown that the optimal performance obtained through multi-objective optimization is an optimization interval. Here, in order to verify the accuracy of the optimal values, two Pareto-optimal points recorded as point 1 (0.032 kgs
−1m
−3 for UHCV, 15.99 kg
$−1 for UHCC) and point 2 (0.11 kgs
−1m
−3 for UHCV, 11.74 kg
$−1 for UHCC) are selected as the comparison objects as shown in
Figure 7. The corresponding optimal parameter groups are, respectively, 1, 11.62 °C, 0.6, 90 °C, and 700 m
2m
−3, and 5, 16.71 °C, 0.6, 90 °C, and 700 m
2m
−3, successively. Through introducing the optimal groups into the established mathematical model, the actual performances are obtained as 0.031 kgs
−1m
−3 for UHCV, 15.86 kg
$−1 for UHCC corresponding to point 1, and 0.11 kgs
−1m
−3 for UHCV, 11.60 kg
$−1 for UHCC corresponding to point 2, respectively. It was found that the errors between predicted and actual values are, respectively, +3.22% and +0.82% at point 1 based on the actual value, while +0.63% and +1.21% are at point 2, which demonstrate that the predicted optimal results are desirable and reliable. Furthermore, through multi-objective optimization, the value of UHCV increases by 56% compared with the result of group 22 with the same UHCC, and the value of UHCC raises by 6.55% compared with the value of group 26 with the same UHCV, respectively.
Furthermore, the computational time to find the optimization results using a mathematical model and regression model is extremely different. As shown in
Figure 9, the quickest computational time for optimization results employing the original mathematical model is 69,000 s, through the simplified orthogonal method, while the computational time to search the optimization results drops to only 10 s for the novel regression model through genetic algorithm. Therefore, the established regression model can give the optimization results more quickly, which justifies the usefulness and robustness of the surrogate model.
Although the proposed method is time-saving, efficient, and accurate, it also has some problems, such as premature convergence and nonstandard coding, especially the specific design conditions given in
Table 2. Maybe the coming genetic algorithm-III can improve this limitation.