Research on Performance Prediction of Elbow Inline Pump Based on MSCSO-BP Neural Network

Abstract

The vertical inline pump, a single-stage centrifugal pump with a bent elbow inlet, is widely used in marine engineering. The unique water inlet passage combined with uneven inflow at the impeller inlet tends to form an inlet vortex and secondary flow area, which reduces performance and causes vibration. To predict the performance of the elbow inline pump, this study uses spline curve fitting for the centerline and cross-sectional shape of the elbow passage. With four elbow inlet variables from experimental design as the input layer and targeting efficiency under pump operating conditions, a pump performance prediction model based on an improved sand cat swarm optimization algorithm combined with a BP neural network (MSCSO-BP) is proposed. Six test functions are used to effectively test the improved sand cat swarm optimization algorithm. The results show that compared to the unimproved algorithm, the improved algorithm has significantly faster convergence speed, shorter parameter optimization time, and higher accuracy. For more demanding multidimensional test functions, the improved optimization algorithm can more accurately find the optimal solution, enhancing the prediction accuracy and generalization ability of inline pump performance. This provides a more effective engineering solution for the design and optimization of inline pumps.

1. Introduction

Vertical inline pumps, as core power units in marine engineering fluid transport systems, are crucial for the operational stability of key processes like seawater circulation cooling, ballast water treatment, and oil-gas transportation. Widely used in deep-sea platforms, ship propulsion systems, and marine resource development facilities, these pumps are preferred in confined spaces due to their compact structure and strong spatial adaptability. However, the inherent conflict between harsh marine operating conditions and limited installation space inevitably induces fluid curvature in the inlet flow passages. This results in increased inlet losses, distorted impeller inflow patterns, and ultimately restricted system energy efficiency and equipment lifespan.

Research indicates that the hydraulic design of the inlet elbow has a decisive impact on the uniformity of the internal flow field of the pump. Optimizing the inlet of the elbow can improve the flow characteristics of the fluid as it enters the impeller, reducing energy and hydraulic impact losses and enhancing the performance and efficiency of the pipeline system [1]. On the contrary, an unreasonable bending structure will trigger flow separation and secondary flow effects, exacerbating the velocity gradient and pressure fluctuation at the impeller inlet, which may induce cavitation vibration and hydraulic impact, threatening the long-term stable operation of the system [2,3,4,5,6,7,8,9]. Although many scholars have revealed the regulatory laws of different inflow channels on pump performance through numerical simulation and experimental methods, there are still two key challenges in actual engineering applications: firstly, traditional empirical formulas struggle to accurately predict multiphase flow coupling effects in complex curved flow passages; secondly, existing optimization methods exhibit modeling errors in capturing the nonlinear mapping between inlet flow parameters and performance metrics, resulting in insufficient prediction accuracy.

With the advancement of metaheuristic intelligent algorithms, which aim to enhance model accuracy, the Backpropagation (BP) neural network has emerged as a prominent representative among commonly used artificial neural network models. It has demonstrated exceptional performance in various domains, including fault diagnosis, energy efficiency prediction, and optimization [10,11,12,13,14,15]. However, traditional BP neural network training still suffers from significant errors in training accuracy, easily falling into local optimal solutions when solving complex problems, and has a slow convergence rate. To enhance the accuracy and convergence speed of the BP neural network, researchers have attempted to apply intelligent algorithms to fit high-precision network structures, achieving the goal of performance prediction. Specifically, Zhang Yiming [16] combined an improved Sparrow Search Algorithm (SSA) with a hyperparameter-optimized BP neural network to accurately estimate the operational performance of electric cooling pumps. The proposed ISSA-BPNN prediction model’s estimation accuracy and reliability are far higher than existing theoretical models. Wang Zheng [17] introduced the Maximum Entropy Criterion (MCC) algorithm to replace the traditional BP neural network prediction method, reducing prediction errors and improving the prediction accuracy of the approximation model. Kuang-Hua Fuh [18] used an error distribution function to improve the BP neural network model, addressing local optimal solutions while accelerating network convergence. Li Bin [19] optimized the BP neural network based on genetic algorithms to predict the influence of fluid parameters on the separation efficiency of a petroleum swirl separator. The results showed a significant improvement in prediction accuracy, with a maximum increase of over 50%. Gonggui Chen [20] designed an improved BP neural network prediction model combining deep learning algorithms and Sparrow Search Algorithm (SSA) to handle nonlinear wind speed prediction. Li Chenghua [21] proposed an approach that adopts the Singular Value Decomposition (SVD) technique to reduce dimensionality and construct latent semantics between items, improving the traditional BP neural network’s performance and accuracy. In the field of pump performance optimization, recent studies have shown the potential of neural network-based approaches. Yonglin Kuang [22] proposed a neural network model for predicting the performance of turbomolecular pumps, which are critical for maintaining high-vacuum environments in applications such as semiconductor manufacturing and space simulation. Similarly, Yuqing Wang [23] presented an optimization method for centrifugal pump cavitation performance using an improved BP neural network algorithm, demonstrating significant advantages in enhancing pump efficiency and reliability.

While these algorithm–BP hybrids markedly improve prediction accuracy, balancing global search efficiency and local optimization precision remains challenging in high-dimensional, strongly nonlinear elbow flow optimization. Recent advances in bio-inspired metaheuristic algorithms have demonstrated unique advantages in complex engineering optimization through their swarm intelligence mechanisms. Among these, the Sand Cat Swarm Optimization (SCSO) algorithm, which mimics sand cats’ survival strategies, dynamically balances exploration and exploitation phases [24]. Its robust global search capability and simplicity have proven effective in enhancing data prediction, classification, and fault detection models [25,26,27,28,29]. Based on this, our study analyzes the relationship between elbow inlet parameters and efficiency in vertical inline pumps. By combining an improved Sand Cat Swarm Algorithm with the BP neural network, we aim to build a prediction model for elbow inlet parameters and efficiency performance. Experimental verification will be conducted to achieve early performance prediction for elbow inline pumps, providing an effective foundation for optimizing elbow inlet design using intelligent algorithms.

2. Computational Model

2.1. Model Pump

The model pump used in this study is shown in Figure 1, with a speed of n = 2900 r/min, a design flow rate of 55 m³/h, a head of 7.8 m, an impeller outer diameter of 94.7 mm, five blades, an impeller inlet diameter of 80.67 mm, and a connection method between the pump body and the motor without a fixed bracket. The impeller material is 316L stainless steel. The stainless-steel impeller is made using indirect 3D printing technology.

2.2. Computational Domain Mesh Generation

Using Creo 6.0 3D software, detailed 3D modeling of the inline pump was conducted. The constructed 3D model of the fluid domain includes five parts: inlet pipe, elbow pipe, impeller, volute, and outlet pipe.

ANSYS Meshing 2021 R1 was used to perform grid generation on the computational model. During grid generation, dislocations and negative grids are prone to occur in the elbow inlet computational domain and at the volute tongue. To address this, a hybrid grid approach was used, combining tetrahedral and hexahedral grids with local encryption applied to relevant parts of the tongue. The water body grid is shown in Figure 2, and the finite volume method was used to discretize the governing equations. The SIMPLEC algorithm on a collocated grid was adopted for the coupling of velocity and pressure.

By setting different maximum cell sizes, a total of five grids with different sizes were generated, and grid independence verification was performed as shown in Figure 3. Using the head values under rated conditions of the inline pump as verification standards, when the total number of grids in the computational domain exceeds 8 million, the computational head tends to stabilize. Also, the y+ contours of impeller and volute from 10,062,836 grid configuration show that most regions’ y+ is in the reasonable range between 20 and 100 (as shown in Figure 4). Taking into account time cost and simulation accuracy, a total grid count of 10,062,836 was selected for subsequent calculations.

2.3. Numerical Simulation Settings

ANSYS CFX 2021 R1 software was used to perform transient numerical simulations to obtain pump performance. The medium is water at 25 °C, and the continuity equation and Reynolds-averaged Navier–Stokes (RANS) Equations (1) and (2) were used as the governing equations. When considering an absolute reference frame expressed in rectangular coordinates, the essential control equations that delineate the fluid’s flow state encompass the following.

$${\displaystyle \frac{\partial {\rho }_m}{\partial t}}+{\displaystyle \frac{\partial \left({\rho }_m{u}_j\right)}{\partial x_j}}=0.$$

(1)

$${\displaystyle \frac{\partial \left({\rho }_m{u}_j\right)}{\partial t}}+{\displaystyle \frac{\partial \left({\rho }_m{u}_i{u}_j\right)}{\partial x_j}}=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_i}}\left(u_m{\displaystyle \frac{\partial u_i}{\partial x_j}}\right).$$

(2)

In the above two equations, ${\rho }_m$ is the liquid density, $u$ is the velocity, $p$ is the pressure, $t$ is the time, $x$ is the spatial coordinate, $\mu$ is the dynamic viscosity, and $i$ and $j$ represent the direction component of the coordinate axis and comply with the summation convention in the tensor.

The SST k-ω turbulence model was chosen because it can more accurately describe the behavioral characteristics of turbulence at different scales. The formulas are shown in Equations (3) and (4). The SST k-ω turbulence equations, comprising the kinetic energy and specific dissipation rate equations, are given by Equations (5) and (6).

$${\displaystyle \frac{\partial (u_i)}{\partial x_i}}=0.$$

(3)

$$\rho {\displaystyle \frac{\partial (u_i)}{\partial t}}+{\displaystyle \frac{\partial (u_i)}{\partial x_j}}=\rho F_i-{\displaystyle \frac{\partial p}{\partial x_i}}+\mu {\displaystyle \frac{{\partial }^2{u}_i}{\partial x_i\partial x_j}}.$$

(4)

In the above two equations, u_i is the instantaneous velocity in the i-direction, x_i is the coordinate, ρ is the fluid density, p is the fluid pressure, and F_i is the dynamic viscosity.

$${\displaystyle \frac{\partial (\rho k)}{\partial t}}+{\displaystyle \frac{\partial (\rho u_{j}k)}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\sigma }_k{\mu }_t\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]-\rho \epsilon -\beta *\rho k\omega .$$

(5)

$${\displaystyle \frac{\partial (\rho \omega )}{\partial t}}+{\displaystyle \frac{\partial (\rho u_j\omega )}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]-\alpha {\displaystyle \frac{\omega }{k}}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}-\beta *\rho k{\omega }^2.$$

(6)

In the above two equations, ρ is the fluid density, t is time, x_j is the spatial coordinate, u_i is the velocity component, μ is the dynamic viscosity, μ_t is the turbulent viscosity, σ_k is the effective function of k, ε is the dissipation rate of k, and β* is the coupling coefficient between k and ω. α is the model parameter, σ_ω is the effective function of ω, and β is the coupling coefficient of the dissipation rate of ω.

Additionally, the model excels in analyzing near-wall regions and captures flow separation phenomena with high accuracy. The boundary conditions are set as follows: the inlet boundary adopts a flow rate inlet setting, the outlet boundary is set as a pressure outlet with a specific pressure value of 1.01 × 10⁵ Pa. Transient rotor–stator interfaces were applied between rotating domain and stationary domain to catch the unsteady rotor–stator interaction. The initial number of time-steps was 120 with 10 revolutions, which meant 30 degrees per time-step. In the final 10 revolutions, 6 degrees per time-step was used to achieve the stable and accurate unsteady numerical results. The convergence criteria were maximum residual with 10⁻⁴ target. Therefore, in total, there were 20 revolutions with time of 0.8275 s.

3. Experimental Verification

To verify the reliability of the prediction results, this study conducted a pump performance experimental verification on the model pump. The experiment was conducted on a closed-loop test bed located at the National Pump Product Quality Inspection Center in Zhejiang (Figure 5). During the experimental process, measurement errors were strictly controlled to ensure that the error range of head and efficiency was within 0.5%, while the uncertainty of flow measurement was maintained at 3%.

As depicted in Figure 5, the pressure sensors are positioned at the inlet and outlet pipes of the inline pump. An electromagnetic flow meter is utilized and the rotational speed of the inline pump is 2900 r/min. To evenly cover the operating range of the inline pump from 0 to 100 m³/h, a total of seven operating points were selected and recorded. To ensure the accuracy of the test data, five repeated tests were conducted for each of these seven operating points. The final test results were calculated based on the average value of the five test datasets.

The comparison between the experimental test and numerical simulation (CFD) performance is presented in Figure 6. Both the head curves and efficiency curves show a good match. Particularly at the design point, which is at 55 m³/h, as shown in Figure 6, the efficiency of the experimental test and the numerical simulation is 68.5% and 68.9%, respectively, and the error between the CFD and test efficiency is only 0.4%. The two efficiency curves from 31.5 m³/h to 63 m³/h are almost identical. Only the efficiency error at the high flow rate is around 3%. It is evident from the figure that the pump performance test results and the numerical simulation results are in good agreement, indicating a high degree of reliability in the numerical calculations near the optimal operating point, which means this numerical model can be used in the training of optimization algorithms.

4. Optimization Algorithm Improvement and Prediction Model Setting

4.1. Theoretical Foundation of Sand Cat Swarm Optimization Algorithm

The Sand Cat Swarm Optimization (SCSO) algorithm, proposed by Amir Seyyedabbasi and Farzad Kiani in 2022 [24], is a novel intelligent algorithm. It primarily mimics two hunting behaviors of sand cats: exploration and attack. When the control parameter |R| is less than or equal to 1, the sand cats switch to an attacking mode; otherwise, they persist in their search for food. This mathematical model is formulated as follows:

$$Pos(t+1)=\left\{\begin{array}{c}Po{s}_b(t)-r\cdot Po{s}_{rnd}\cdot \cos (\alpha )\\ r\cdot (Po{s}_{bc}(t)-rand(0,1)\cdot Po{s}_c(t))\end{array}\begin{array}{c}\left|R\right|\\ \left|R\right|\end{array}\right.\begin{array}{c}\le 1;\\ >1\end{array}.$$

(7)

Pos_bc represents the best candidate position, Pos_c is the sand cat’s current position, r is the sensitivity range, Pos_b is the best position found, Pos_rnd is a random position, α indicates the random movement angle of the sand cat, and (t) represents the position at the (t)-th iteration.

4.2. Improvement Strategy

To increase the search range of the sand cats, the search, attack, and determination phases are improved using the Levy flight strategy, triangular walk strategy, and mirror imaging reverse learning, respectively (see Figure 7 and Figure 8.).

$$Po{s}_{new}=po{s}_b(t)+(po{s}_b(t)-pos(t))\times C\times Levy,$$

(8)

$$L_1=po{s}_b(t)-po{s}_c(t),$$

(9)

$$\overrightarrow{L_2}=rand()\times \overrightarrow{L_1},$$

(10)

$$\beta =2\times pi\times rand(),$$

(11)

$$P={L_1}^2+{L_2}^2-2\times L_1\times L_2\times \cos (\beta ),$$

(12)

$$Po{s}_{new}=po{s}_b(t)+r\times P.$$

(13)

4.3. Testing and Discussion on the Improved Sand Cat Swarm Algorithm

4.3.1. Introduction to Test Functions

Verifying the effectiveness and performance of optimization algorithms is a critical task in the field of optimization. For this purpose, test functions play a crucial role. These functions provide a standardized problem domain for optimization algorithms, enabling different algorithms to be compared and evaluated under the same conditions. They should cover various types of problems, such as unimodal functions, multimodal functions, and high-dimensional functions, to ensure that the algorithms can work effectively in diverse situations.

Test functions can also be used to evaluate new optimization algorithms or improvements to existing ones. By testing new algorithms on these functions and comparing their performance with that of existing algorithms, the effectiveness and performance of the new algorithms can be assessed, thereby promoting the progress and development of optimization algorithms. Testing on different types of base functions can evaluate the robustness of improved algorithms. If the improved algorithm performs well on both unimodal and multimodal basis functions, it indicates good applicability and the ability to handle different types of optimization problems. Therefore, for comprehensive algorithm effectiveness testing, standard test functions include both unimodal and multimodal test functions. In this study, six test functions, as shown in

Table 1. Six types of test functions.

Function Type	Test Function	dim	Range	Objective
Unimodal	$F_4(x)={\max }_i|x_i|$	30	[−100, 100]	0
	$F_5(x)={\displaystyle \sum _{i=2}^n\left[100{\left(x_i-x_{i-1}^2\right)}^2+{(x_{i-1}-1)}^2\right]}$	30	[−30, 30]	0
Multimodal	$F_1(x)={\displaystyle \sum _{i=1}^n-}{x}_i\cdot \sin \left(\sqrt{|x_i|}\right)$	10	[−50, 50]	−350
	$F_2(x)={\displaystyle \sum _{i=1}^n|}{x}_i|+{\displaystyle \prod _{i=1}^n|}{x}_i|$	30	[−10, 10]	0
	$F_3(x)={\left({\displaystyle \frac{1}{500}}+{\displaystyle \sum _{j=1}^{25}\frac{1}{1+{\displaystyle \sum _{i=1}^n{(x_i-a_{ij})}^6}}}\right)}^{-1}$	2	[−65, 65]	1
	$\begin{array}{l}{F}_6(x)={\left(x_2-{\displaystyle \frac{x_1^2\cdot 5.1}{4{\pi }^2}}+{\displaystyle \frac{5x_1}{\pi }}-6\right)}^2\\ +10\left(1-{\displaystyle \frac{1}{8\pi }}\right)\cos (x_1)+10\end{array}$	2	[−5, 20]	0.4

, are used to test the effectiveness of the optimization algorithm. To explore the self-parameter optimization and convergence performance of the improved sand cat swarm algorithm, a comparison is made between the traditional Sand Cat Swarm Optimization (SCSO) and the improved Sand Cat Swarm Optimization (MSCSO).

Through comparative testing on six standard test functions (for details, see Table 1), the algorithm parameters are uniformly set. In order to save computation time, the population size is set to 30, the maximum number of iterations is set to 80, and 20 experimental repetitions are conducted to ensure the reliability of the experimental results.

4.3.2. Test Results and Analysis

As depicted in Figure 9, the horizontal axis denotes the number of iterations, whereas the vertical axis signifies the optimal value. The slope of the curve reflects the convergence speed, with a steeper slope indicating a faster convergence rate.

Upon conducting a comprehensive analysis, it was discovered that irrespective of whether the test function is unimodal or multimodal, the convergence curve of the MSCSO algorithm exhibits steeper characteristics, and the number of iterations is relatively smaller. In the case of the F1 test function, the SCSO algorithm failed to attain the optimal value, even upon convergence. For the multimodal benchmark functions F3 and F6, the MSCSO algorithm demonstrated a greater capability in accurately pinpointing the optimal value with a smaller margin of error. Overall, in contrast to the SCSO algorithm, the convergence speed of the MSCSO algorithm has been significantly enhanced, the time required for parameter optimization has been reduced, and the accuracy has also been considerably improved. Hence, it can identify the optimal solution more rapidly and precisely than the SCSO algorithm.

4.4. BP Neural Network Optimization

BP (Backpropagation) neural network is a common type of artificial neural network, also known as a feedforward neural network with backpropagation. It was proposed in the 1990s and further developed and refined by multiple scientists in the following years [30]. BP neural network is trained by continuously adjusting the connection weights and bias terms to minimize the error between the network’s output and the expected output. Specifically, the training process consists of two stages: forward propagation of signals and backward propagation of errors. A schematic diagram of the neural network is shown in Figure 10.

The essence of utilizing the improved Sand Cat Swarm Optimization (MSCSO) algorithm to optimize the BP neural network is to mimic the behavior of sand cats in searching for food. Individuals within the sand cat swarm seek optimal solutions based on their own positions. The sand cats update their positions based on their current location and the surrounding environment, which is analogous to adjusting the weights and thresholds in the neural network to reduce prediction errors of the traditional BP neural network. Leveraging the search strategies and adaptability of the MSCSO algorithm, it attempts different combinations of weights and thresholds to find the optimal parameters.

4.5. Prediction Model Setting

Uniform testing can ensure that the sample distribution in the training set and test set is relatively uniform, reducing model bias or overfitting issues caused by skewed training data. This approach helps the network learn a wider range of features and patterns. Using the uniform test design method, 80 design sets were generated, far exceeding the number of systems in the BP neural network prediction model under the improved sand cat swarm algorithm.

In this study, the input variables of the prediction model were determined as the geometric parameters of the elbow shape, including DS-C, DS-D, DS-G, and LA. These parameters are key geometric features that determine the elbow shape, directly influencing the fluid flow characteristics (such as velocity, and pressure distribution) entering the impeller. Moreover, they affect the uniformity and stability of the inflowing fluid, significantly impacting the pump’s efficiency and performance. Optimizing these parameters can reduce flow distortion, improving the pump’s efficiency and operational stability.

As mentioned above, the input variables of the prediction model are determined as DS-C, DS-D, DS-G, and LA of the elbow-shaped bend (shown in Figure 11), and the prediction target is the efficiency at the pump’s design point. The elbow-shaped streamline is determined based on a 6-degree B-spline curve fit of the midpoint of the cross-section, and the input parameters are selected based on the significance of the experimental design.

Using Matlab code was written to train the network model using 70% of the total 80 sets of data as samples. The number of hidden layer nodes in the prediction model was determined to be nine based on the minimum mean square error. The datasets for testing and validating the neural network each accounted for 15% of the sample data. The relevant settings for the neural network are as follows: the training function employs the Levenberg–Marquardt (L–M) optimization algorithm Trainbr, the node transfer function uses the Logsig function and the linear function Pureline, the number of training iterations is set to 1000, the learning rate is 0.01, and the minimum training target error is 0.00001.

5. Analysis and Discussion

5.1. Evaluation and Verification of Predictive Models

To be able to objectively evaluate the prediction error of a predictive model, we use the following metrics to test the model’s accuracy.

1.

Mean Squared Error (MSE)

$$MSE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n(T_i}-O_i{)}^2,$$

(14)

where T_i represents the predicted efficiency value of the inline pump at a specific time; O_i represents the efficiency value calculated by CFD numerical simulation; and n represents the number of samples.

2.

Root Mean Squared Error (RMSE)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n(T_i}-O_i{)}^2},$$

(15)

3.

Correlation Coefficient (R)

$$R=\sqrt{1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{(O_i-f_i)}^2}}{{\displaystyle \sum _{i=1}^n{(O_i-\overline{O})}^2}}}},$$

(16)

where f_i represents the actual efficiency value of the inline pump’s calculation; and $\overline{O}$ represents the average efficiency value calculated by numerical simulation across all time points.

4.

Mean Absolute Error (MAE)

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|T_i-O_i\right|}$$

(17)

There is a certain significance in using both MSE and RMSE as the objectives of the optimization function. Considering both MSE and RMSE as the optimization targets can comprehensively evaluate the predictive accuracy and stability of the model. The closer the determination coefficient R² is to 1, the stronger the applicability of the model and the higher the prediction accuracy. By optimizing the process to minimize both MSE and RMSE, we can ensure that the model has high predictive accuracy and stability. The accuracy of the prediction models of the two optimization algorithms before and after optimization is evaluated using these assessment indicators, as shown in

Table 2. Comparison of the effectiveness of the model before and after optimization.

Algorithm Prediction Categories	MSE (%)		RMSE (%)		MAE (%)
	Training Set	Test Set	Training Set	Test Set	Training Set	Test Set
MSCSO-BP	2.777	0.896	1.666	0.946	1.805	1.934
SCSO-BP	3.401	4.739	1.844	2.177	2.156	1.988
BP	5.756	6.63	2.399	2.575	2.872	2.984

.

An examination of the data presented in Table 2 reveals that the MSE values for the training sets of the MSCSO-BP, SCSO-BP, and BP prediction models are 2.777%, 3.401%, and 5.756%, respectively. In the case of the test sets, the MSE values are 0.896%, 4.739%, and 6.63%, respectively. When it comes to the MAE, the training sets of the MSCSO-BP, SCSO-BP, and BP prediction models exhibit values of 1.805%, 2.156%, and 2.872%, respectively. For the test sets, the MAE values are 1.934%, 1.988%, and 2.984%, respectively. Among the three evaluation criteria, the MSCSO-BP prediction model has smaller error values, which initially indicates that the optimized backpropagation neural network model is more suitable for multi-parameter performance prediction.

As depicted in Figure 12, the determination coefficient (R-squared) of the MSCSO-BP prediction model, whether for the training set, test set, or validation set, exhibits a correlation coefficient R value that is bigger than that of SCSO and closer to 1. Both the SCSO-BP and BP prediction models show varying degrees of deviation from local samples.

The BP prediction model has a steeper slope in the linear regression for the validation and test sets, with instances of data points being marginalized. In comparison to the BP prediction model, although the SCSO-BP prediction model demonstrates some improvement on the validation set, it has an even steeper slope for the regression line on the test set. When data are randomly partitioned, the prediction model of the improved algorithm is more likely to approach the true values, further indicating that the MSCSO-BP prediction model is more suitable.

We selected 35 samples from the 70% training dataset (56 samples) and 12 test samples for further investigation. An analysis of the predictions generated by the MSCSO-BP neural network indicates that, whether for the training set or the test set (as shown in Figure 13a,b), when comparing the CFD values with the predicted efficiencies of the BP neural network before and after optimization, it can be clearly seen that the network structure optimized by the Sand Cat Swarm Optimization (MSCSO) algorithm demonstrates a higher level of prediction accuracy, with the error range being effectively controlled within 5%. Upon further examination of the sample errors in the test set, it is evident that, in comparison with the unoptimized SCSO-BP neural network, the error range of the BP neural network enhanced with MSCSO is significantly reduced and the error fluctuations become more stable. In light of these observations, it can be concluded with confidence that the efficiency prediction model trained utilizing the MSCSO-BP neural network is highly accurate and reliable.

In summary, whether in the training phase or the testing phase, the evaluation metrics of the Sand Cat Swarm Optimization (MSCSO) improved Backpropagation Neural Network prediction model based on the wandering strategy are superior to the SCSO-BP prediction model, demonstrating higher prediction accuracy and better prediction performance.

5.2. Analysis of Elbow Inlet Optimization Results Based on CFD

5.2.1. Elbow Inlet Profile Comparison

One model from optimization is chosen for the further CFD analysis. Figure 14 shows the comparison of the inlet profiles before and after optimization, the main changes are reflected in the transition angle between the flow channel profile and the vertical cross-section, which has become larger. Although the LA distance has become longer, it still complies with the design principle of linear gradual change for the inlet cross-section and relevant design specifications.

The distance from the centerline of the impeller to the bottom plate of the inflow channel has hardly changed, indicating that the optimized model has a relatively small impact on the axial distribution uniformity of the outlet water flow and does not cause greater hydraulic losses.

The elbow inlet profile changes mainly in two areas. At the throat, where velocity gradients are most likely, the optimized model’s middle cross-section shows a slightly larger curvature radius near the first throat compared to the original model, while the second throat is nearly unchanged. At the inlet cross-section, the optimized model’s upper and lower contour lines have a larger vertical angle than the original model.

5.2.2. Velocity Comparison

In the optimized elbow inlet, the streamlined transition between the bend and the preceding and subsequent flow components is smoother, particularly near the second throat, where the curvature of the bend is gentler. Due to the sharp curvature of the original model at the first bend of the inlet profile, it generated a certain degree of impact loss between the inlet flow channel and the bend wall near the inlet, resulting in low-velocity flow regions, as depicted in Figure 15. This impact loss, which causes the velocity gradient near the second throat in the original model, was significantly higher than that in the optimized model, which negatively affected the flow state of the elbow inlet flow channel and caused an uneven distribution of the pipeline. In the optimized model, the velocity gradient in the bend decreases, and the velocity distribution in the middle cross-section becomes more uniform, leading to a better flow state into the impeller and an overall increase in efficiency.

As shown in Figure 15, the original model was prone to large velocity gradients in the throat area at high flow rates. In the optimized model, the velocity gradient near the throat is smaller, and the inlet flow velocity bias into the impeller is also reduced.

The increase in maximum velocity in the throat region has a smaller impact on pump performance compared to the improvements in pressure and velocity gradients. The localized effect of elevated throat velocity may cause minor energy losses or flow disturbances, but it does not substantially alter the overall flow conditions at the impeller inlet. In contrast, improved pressure gradients reduce total pressure losses in the pipeline, enhancing energy utilization at the pump inlet. Similarly, reduced velocity gradients promote a more uniform flow field at the impeller inlet, minimizing impact losses and turbulence-induced losses, thereby directly boosting impeller efficiency.

5.2.3. Pressure Comparison

As depicted in Figure 16, there are notable differences in the static pressure distribution of the suction chamber before and after optimization, particularly at the inlet, outlet, location 1, location 2, and location 4. Following the optimization process, the static pressure gradient was reduced, and the area of high-pressure regions at locations 1, 2, and 4 was decreased. Although the static pressure distribution at locations 1 and 3 became less uniform, the static pressure gradient was reduced. Moreover, the static pressure gradient in the optimized elbow inlet flow channel at location 5 became gentler.

5.2.4. Efficiency Comparison

The comparison of the best efficiency point performance before and after optimization is shown in

Table 3. Comparison of best efficiency point performance before and after optimization.

Q (m3/h)	H (m)		η (%)
	Before	After	Before	After
55	8.05	8.45	68.9	72.4

. At this flow rate (55 m³/h), the optimized design consistently outperforms the original one in both head and efficiency. It achieves higher head values, with a 0.4 m improvement, and efficiency improves from 68.9% to 72.4% with a 3.5% increase. These results align with velocity and pressure comparison analysis.

6. Conclusions

This study proposes a modified sand cat swarm optimization (MSCSO) algorithm and its application in pump performance prediction. The main contributions are summarized as follows.

• Enhanced MSCSO Algorithm: By integrating an improved wandering strategy and lens-imaging opposition-based learning, the MSCSO algorithm demonstrates superior convergence speed, reduced optimization time, and enhanced accuracy compared to the original SCSO, particularly in complex multidimensional problems. Experimental validation reveals an 81.1% reduction in test set MSE and a 2.7% decrease in MAE, with error fluctuations stably confined within 5%.
• Validation of Numerical Simulation Reliability: A dedicated pump performance test rig was established to verify numerical simulation consistency. The close agreement between experimental and simulated results (with errors controlled below 5%) confirms the credibility of numerical data near optimal operating points, thereby facilitating the construction of an effective sample space for predictive models.
• MSCSO-BP Hybrid Model for Efficiency Prediction: The integration of MSCSO with a BP neural network significantly improves prediction accuracy for vertical inline pumps with elbow-shaped inlets. Compared to SCSO-BP and conventional BP models, the MSCSO-BP model achieves R² values closer to 1 and exhibits stronger generalization capabilities. A comprehensive error reduction exceeding 50% across training and test sets underscores its superiority in multi-parameter optimization, establishing a robust foundation for high-efficiency inline pump design.

In conclusion, the MSCSO-BP neural network model delivers high-precision performance predictions for elbow inline pumps. The enhanced MSCSO algorithm exhibits strong global search capabilities and rapid convergence, effectively avoiding local optima. It is particularly well-suited for addressing non-uniform inlet flow distributions and complex nonlinear multi-parameter optimization challenges. With strong adaptability and the ability to handle intricate inlet flow conditions, its generalization makes it a valuable tool for pump design and optimization in marine and other engineering applications. However, the prediction model’s performance heavily relies on the quality and representativeness of input data, and parameter selection difficulties may impact its performance and convergence speed. Currently tailored for specific inlet model optimization, its application to other models requires precise numerical models and extensive CFD and experimental data validation. Future research could optimize the algorithm for real-time performance, explore its broader application in complex fluid dynamics problems, pump geometries. This study provides a robust methodological framework for pump performance prediction and fluid machinery optimization design in diverse engineering contexts.