Evaluation of Sodium Alginate as a Drag Reduction Agent in Flowing Fluids: Effects of Concentration, Temperature, and Flow Rate

Evaluation of Sodium Alginate as a Drag Reduction Agent in Flowing Fluids: Effects of Concentration, Temperature, and Flow Rate | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Evaluation of Sodium Alginate as a Drag Reduction Agent in Flowing Fluids: Effects of Concentration, Temperature, and Flow Rate Zhensong Cheng, Yunxia Chang, Xiaodong Dai, Xin Zhang, Xudong Wang, and 3 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7356460/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 10 You are reading this latest preprint version Abstract Sodium alginate, a natural polymer, is widely utilized in various fields due to its exceptional drag reduction properties. This study systematically investigates the effects of sodium alginate concentration, flow rate, and temperature on the drag reduction rate. Utilizing Response Surface Methodology (RSM), a series of experiments were designed to optimize and analyze the influence of these factors. The results indicate that sodium alginate concentration, flow rate, and temperature have a significant impact on drag reduction performance, with temperature primarily affecting the intermolecular interactions and the contraction of polymer chains. A multi-factor mathematical model was developed using RSM to describe the relationship between the drag reduction rate and the parameters of concentration, flow rate, and temperature of sodium alginate. Validation of the model against experimental data revealed a strong correlation, confirming that the model accurately characterizes the drag reduction behavior of sodium alginate under varying operational conditions. These findings highlight that optimizing the conditions for sodium alginate application can significantly enhance fluid flow efficiency, providing a theoretical foundation for its use in industrial and environmental engineering applications. Physical sciences/Engineering Physical sciences/Materials science Physical sciences/Mathematics and computing Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 1. Introduction The application of fluid mechanics in industrial transport systems holds significant economic implications, particularly in reducing energy consumption and improving transportation efficiency. One of the primary sources of energy loss in fluid transportation systems is frictional resistance, making the reduction of flow resistance a central focus for enhancing pipeline efficiency [ 1 ] . The phenomenon of drag reduction was first observed by Toms (1947) during the mechanical degradation of polymer substances in pipe flow, a finding that was officially presented at the First International Rheological Conference in 1948[2]. Several methods for drag reduction are commonly employed, including pipeline design optimization, coating applications, fluid velocity control, and the use of drag-reducing additives [ 3 – 6 ] . Among these, the use of drag-reducing agents stands out for its cost-effectiveness and minimal requirement for large-scale modifications to the pipeline infrastructure. Drag-reducing agents are applicable to a wide range of fluids and pipeline systems, including those transporting oil, natural gas, and water, thus making them an appealing solution for industrial applications [ 7 , 8 ] . As a result, drag reduction in turbulent flow remains a key area of research. Traditionally, drag-reducing agents have been synthetic chemicals, which are effective in reducing frictional losses [ 9 , 10 ] . However, concerns regarding environmental pollution, resource consumption, and health risks have sparked interest in the development of natural drag-reducing agents. Among the natural polymers studied, xanthan gum and guar gum have shown promise in reducing water flow resistance. Edomwonyi Otu et al. [ 11 ] evaluated these biopolymers as drag reducers in water flow and highlighted their effectiveness in reducing drag in 12-mm diameter pipes. This study underscores the potential of biopolymers in diverse fluid systems. Singh et al. [ 12 ] discussed grafted polysaccharides as effective turbulent drag reducers and flocculants, identifying grafted starch as particularly beneficial for industrial wastewater applications. This finding further supports the potential of polysaccharide-based biopolymers for natural drag reduction. Marhefka et al. [ 13 ] expanded on this by investigating the drag-reducing properties of aloe vera. Their study found that aloe vera's drag reduction performance was superior to many synthetic polymers, demonstrating advantages in terms of stability and performance. Xie et al. [ 14 ] proposed yam mucilage as an environmentally friendly alternative, emphasizing its potential under turbulent conditions. The mucilage's high shear resistance and natural origin make it a promising candidate for sustainable flow enhancement. Shi et al. [ 15 ] conducted supplementary studies on yam mucilage, further elucidating its drag-reducing capabilities and identifying factors influencing its effectiveness Similarly, dos Santos et al. [ 16 , 17 ] explored diutan gum, a natural polymer from Shewanella species, noting its stability and effectiveness under turbulent flow conditions. Salehudin et al. [ 18 ] synthesized carboxymethyl cellulose from coconut residue via an alkali-catalyzed reaction. The performance of the synthesized carboxymethyl cellulose was evaluated in a water injection system, examining its effectiveness at various concentrations and flow rates. Another promising natural polymer, sodium alginate, has also been investigated for its drag-reducing properties, particularly in turbulent flow within millimeter-scale pipes. Cheng et al. [ 19 ] confirmed the effectiveness and elasticity of sodium alginate in resisting shear forces, addressing concerns related to the irritancy and toxicity of its synthetic counterparts. Their findings highlighted sodium alginate's potential for resolving common stability issues faced by natural polymers under flow conditions. However, temperature sensitivity remains a major challenge for natural drag-reducing agents, as temperature variations can affect their physical properties—including viscosity, molecular structure, and solubility—directly influencing their drag-reduction performance [ 20 ] . For instance, under high-temperature conditions, natural drag-reducing agents may degrade or lose viscosity, reducing their effectiveness. Consequently, investigating the performance of natural drag-reducing agents at varying temperatures is essential for broadening their application in industrial processes [ 21 ] . While sodium alginate has demonstrated promising shear resistance and drag-reducing performance under turbulent conditions, its behavior under varying temperature conditions requires further exploration. This study aims to analyze the effects of temperature, concentration, and flow rate on the drag-reducing properties of sodium alginate solutions. 2. Experimental setup and fluid preparation The experimental setup was adapted from the original design developed by the research group (as shown in Fig. 1 [ 22 ] ). The testing system consists of several critical components: Liquid Storage: A 10L transparent liquid storage tank is used to store the experimental fluid. Circulation System: A peristaltic pump circulates the liquid through the flow system. The system includes a 0.477-meter-long test section with an internal diameter of 2.23 mm. The inlet section's length satisfies the requirement of being 138 times the pipe diameter [ 23 ] . Frictional Resistance Measurement: To measure the frictional resistance in the straight pipe flow, a differential pressure sensor was installed. This sensor has a maximum measurable pressure difference of 100 kPa and an accuracy of ± 0.055%. Temperature and Solution Control: Given the limited research on the biodegradability of sodium alginate, the preparation process was completed quickly to avoid prolonged storage and to ensure solution stability. A magnetic stirrer and thermostatic water bath were used to control the temperature and accelerate the dissolution of sodium alginate. The water bath's temperature range extends from room temperature to 95°C, with a control accuracy of ± 0.5°C. Additionally, aluminum foil was used to cover the beaker, preventing solution evaporation. Data Acquisition: The experimental data acquisition system, based on the LabVIEW platform, enabled real-time monitoring of both temperature and pressure differential data. Data was collected every second and transmitted in real time to the controller for storage and display. Sodium alginate is a natural polysaccharide composed of α -L-guluronic acid (G) and its stereoisomer, β -D-mannuronic acid (M) [ 24 , 25 ] . The sodium alginate used in this study was sourced from Fuchen (Tianjin) Chemical Co., Ltd. To prepare the sodium alginate solution, sodium alginate was dissolved in a thermostatic water bath with low-speed stirring for 60 minutes, ensuring complete dissolution while the temperature was gradually increased to the experimental condition. After stirring, the solution was maintained at a constant temperature for 1 hour to ensure full dissolution of the polymer and accurate experimental results. At the start of the experiment, deionized water was introduced into the test section to calibrate the system. The friction factor ( f ) measured during the flow of deionized water was used as the calibration baseline, ensuring both the accuracy and stability of the experimental system. Following calibration, the sodium alginate solution was pumped into the testing system using the peristaltic pump, with the flow rate precisely set to the target value. As the solution passed through the test section, the pressure differential gradually stabilized. Data recording commenced once a steady state was achieved. After the test, all solutions were directed into a waste container for disposal to prevent reuse, ensuring the stability of the solution and the reliability of the experimental results. 3. Results and analyses 3.1 Drag reduction flow analysis The measured differential pressure data are used to calculate the Darcy friction factor f , which is an essential parameter in determining the flow resistance in the pipe. The conversion from pressure differential to f is done using the following equation: \(\:\text{f=}\frac{\text{2}\text{d∆p}}{\text{lρ}{\text{U}}_{\text{b}}^{\text{2}}}\) (1) Where: Δ p is the flow pressure difference in the test section (Pa), d is the diameter of the pipe (m), ρ is the density of the solution (kg/m³), U b is the average flow velocity in the test pipe (m/s), l is the length of the test section (m). The drag reduction rate DR is calculated using the following equation: \(\:\text{DR=}\frac{{\text{Δ}\text{p}}_{\text{s}}-{\text{Δ}\text{p}}_{\text{p}}}{{\text{Δ}\text{p}}_{\text{s}}}\) (2) In the equation, Δ p s represents the pressure difference of the flowing tap water at a specified flow rate; Δ p p represents the flow pressure difference of the sodium alginate solution under the same flow conditions. By comparing these values, the drag reduction rate can be calculated, providing a quantitative description of the drag-reducing effects of the additive at different temperatures. The flow rate of the solution is characterized by the average flow Reynolds number ( Re ), which is a dimensionless quantity that helps determine the flow regime (laminar, transitional, or turbulent). The Reynolds number is calculated using the following equation: \(\:\text{Re}\text{=}\frac{{\text{U}}_{\text{b}}\text{ρ}\text{d}}{\text{μ}}\) (3) The dynamic viscosity of the solution ( µ ) of the solution is a crucial parameter for characterizing flow behavior. The solvent density and viscosity are referenced under the same experimental conditions as the test fluid, as outlined in previous studies [ 26 , 27 ] . The values for the dynamic viscosity of tap water ( µ ) at different temperatures are provided in Table 1 . Table 1 Dynamic viscosity of tap water at experimental temperatures t /℃ 25 30 40 50 µ × 10 − 3 /pa•s 0.8937 0.8007 0.6560 0.5494 \(\:{\text{U}}_{\text{b}}\text{=}\frac{\text{4}\text{Q}}{\text{ρπ}{\text{d}}^{\text{2}}}\) (4) Q is the volumetric flow rate of the solution, measured in ml/min. $$\:\text{f=}\text{0.3164}{\text{Re}}^{\text{-}\text{0.25}}$$ 5 Figure 2 presents the relationship between the friction factor and Reynolds number (or flow rate) for deionized water at 30°C. The data reveal a strong correlation between the measured friction factor and the empirical formula for turbulent flow in smooth pipes, specifically the Blasius formula [ 28 , 29 ] . The calculated values using the Blasius equation align well with the experimental data, remaining within a ± 3% error margin. This comprehensive analysis demonstrates a high degree of consistency between the experimental measurements and the theoretical predictions, thus confirming the reliability and accuracy of the experimental results. 3.2 Single Factor Experiment on Drag Reduction Flow of Sodium Alginate Solutions 3.2.1 Effect of Concentration Gasljevic et al. [ 30 ] identified the "pipe diameter effect" concerning drag-reducing fluids, demonstrating that varying pipe sizes can yield different drag reduction results, even when the same solution is utilized. Previous studies [ 31 ] prepared a sodium alginate solution at 21°C and assessed its drag reduction properties in a circular pipe with an internal diameter of 1.85 mm. In this study, the internal diameter of the circular pipe used is 2.23 mm, which is comparable to the diameter tested in earlier research. Consequently, experiments were conducted within the concentration range corresponding to the maximum observed drag reduction rate from prior studies (750–1250 ppm). Sodium alginate solutions at concentrations of 700, 900, 1100, 1300, and 1500 ppm were prepared for testing. All experiments were performed once the solutions were equilibrated to room temperature (25°C), and the comparative results are presented in Fig. 3 . Figure 3 illustrates that, due to variations in test pipe diameter and temperature relative to previous studies, the drag reduction rate initially increases and subsequently decreases with increasing concentration of the sodium alginate solution. The maximum drag reduction rate is observed within a concentration range of 1200–1400 ppm, with 1300 ppm selected as the midpoint concentration for subsequent response surface analysis. 3.2.2 Effect of Temperature After establishing the concentration range and keeping all other conditions constant, three temperatures (30°C, 40°C, and 50°C) were selected for comparison. In alignment with the findings of Rahaman et al. [ 32 ] , it was observed that the drag reduction rate of natural drag reducers exhibits an extremum as a function of temperature, although the distribution pattern is complex. Consequently, a concentration of 1300 ppm was chosen for the temperature effect experiments across different flow conditions, with the results illustrated in Fig. 3 . Figure 4 clearly demonstrates that at a concentration of 1300 ppm, all three flow rates achieve a maximum drag reduction rate ( DR ) at 40°C. Consequently, 40°C was selected as the midpoint temperature for the response surface analysis. However, it is also evident that under high flow conditions, the influence of temperature on the drag reduction rate of the sodium alginate solution diminishes. 3.2.3 Effect of Flow Rate After establishing fixed concentration and temperature ranges, drag reduction comparison experiments were conducted under varying flow conditions, with the results depicted in Fig. 5 . It is evident that in the turbulent flow regime, the drag reduction rate ( DR ) increases with rising flow rate, with no extremum observed within the tested range. Consequently, a flow rate of 700 ml/min was chosen as the midpoint flow rate for the response surface analysis. 3.3 Response surface methodology (RSM) results and analysis Building on the results from the univariate experiments, a response surface design experiment was conducted using the Box-Behnken central composite experimental design principle. Three factors that significantly influence the drag reduction rate ( DR ) were selected: concentration ( C ), flow rate ( Q ), and solution temperature ( t ). A three-factor, three-level response surface analysis was conducted, with the drag reduction rate ( DR ) of the sodium alginate solution serving as the response variable, as detailed in Table 2 . Table 2 Variables and levels in central composite design. Factor Lowest level (-1) Median (0) Highest level (+ 1) Q (ml/min) 900 1300 1700 C (ppm) 600 700 800 t (℃) 30 40 50 Table 3 summarizes the experimental design generated by the Design-Expert software, which necessitates a total of 17 distinct experimental groups to meet statistical requirements. A quadratic response surface regression analysis, as recommended by the Design-Expert software, was performed, resulting in the following ternary quadratic response surface regression model represented in Eq. ( 6 ). Table 3 Process variables and their levels in the Box-Behnken design, along with the corresponding response values for the drag reduction rate ( DR ) of sodium alginate. A: C (%) B: Q (ml/min) C: t (℃) DR (%) 1300 800 30 22.49 900 800 40 19.46 1700 800 40 22.49 1700 700 30 22.86 1300 700 40 19.93 1300 700 40 19.93 1300 700 40 19.93 900 600 40 16.58 1300 600 30 19.95 1300 700 40 19.93 1700 600 40 17.95 1700 700 50 16.28 900 700 50 15.51 900 700 30 18.68 1300 800 50 21.07 1300 700 40 19.93 1300 600 50 16.26 $$\:\text{DR=}\text{25.51+0.0238}\text{C}\text{-0.0737}\text{Q}\text{+4.06}\text{×}{\text{10}}^{\text{-3}}\text{t}\text{+1.0}\text{0}\text{×}{\text{10}}^{\text{-5}}\text{CQ}\text{-2.13×}{\text{10}}^{\text{-4}}\text{Ct+}\text{5.6}\text{7}\text{×}{\text{10}}^{\text{-}\text{4}}\text{Qt-}\text{7.56}\text{×}{\text{10}}^{\text{-6}}{\text{C}}^{\text{2}}\text{+}\text{4.1×}{\text{10}}^{\text{-5}}{\text{Q}}^{\text{2}}\text{-3.}\text{88}\text{×}{\text{10}}^{\text{-3}}{\text{t}}^{\text{2}}$$ 6 Table 4 Analysis of variance (ANOVA) for the regression equation of the drag reduction rate ( DR ) of sodium alginate as a function of three variables. Source Sum of Squares df Mean Square F-value p-value Model 78.12 9 8.66 22.07 0.0002 Significant A-Cs 10.93 1 10.95 27.92 0.0011 B-Q 27.27 1 27.20 69.34 < 0.0001 C-t 27.60 1 27.55 70.21 < 0.0001 AB 0.6889 1 0.6913 1.76 0.2255 AC 2.91 1 2.90 7.39 0.0293 BC 1.29 1 1.29 3.29 0.1120 A² 6.16 1 6.08 15.50 0.0054 B² 0.6737 1 0.7028 1.79 0.2301 C² 0.6322 1 0.6091 1.55 0.2436 Residual 2.73 7 0.3899 Lack of Fit 2.73 3 0.9099 Pure Error 0.0000 4 0.0000 Cor Total 80.85 16 To evaluate the goodness of fit of the response surface regression model and the effects of each factor on the response variable, an analysis of variance (ANOVA) was conducted. Table 4 presents the results of the ANOVA for the fitted model. According to the ANOVA results, the total sum of squares for the model is 78.12, with 9 degrees of freedom, a mean square of 8.66, an F-value of 22.07, and a p-value of 0.0002. These findings collectively indicate that the overall model possesses significant explanatory power regarding the response variable, suggesting that both the main effects and interaction effects included in the model effectively account for the variability observed in the response variable. In the model, the three main effect factors (A- C , B- Q , and C- t ), along with the interaction term AC and the quadratic term A², all exert significant effects on the response variable. Conversely, the P-values for the factors AB, BC, B², and C² exceed 0.05, indicating that these factors do not have significant effects on the response variable. Consequently, consideration may be given to removing these factors from the model. The optimized results are presented in Eq. (7). The calculated coefficient of determination ( R 2 ) value is 0.9662, indicating an excellent fit of the model to the experimental data, with robust explanatory power and predictive capability. This suggests that the model effectively describes the variability in the experimental data, thereby validating its reliability and accuracy. \(\:\text{DR}\text{=}\text{-13.21+0.0311}\text{C}\text{+0.0185}\text{Q}\text{+0.0913}\text{t}\text{-2.13×}{\text{10}}^{\text{-4}}\text{Ct}\text{+}\text{7.59}\text{×}{\text{10}}^{\text{-6}}{\text{C}}^{\text{2}}\) (7) In this study, the three key assumptions related to Response Surface Methodology (RSM) analysis were validated. First, the normal probability plot of the residuals (Fig. 6 A) was examined, which revealed that the residual data points are symmetrically distributed around the reference line, indicating that the residuals satisfy the normal distribution assumption. Second, Fig. 6 B illustrates that the residual values are confined within the range of ± 4.82. Additionally, as the predicted values increase, there is no significant divergence in the variance of the experimental data, thereby confirming the validity of the assumption of constant variance. Finally, the randomness of the data was assessed by analyzing the correlation between the run order and the residuals. The results presented in Fig. 6 C show no apparent temporal correlation trend between the residuals and the run order, demonstrating that the experimental data meet the randomness assumption. In summary, the drag reduction rate testing data satisfy the three fundamental assumptions of response surface methodology analysis, indicating that the testing methods and data used in this study conform to the analytical requirements. Figure 7 A illustrates the response surface for the drag reduction rate ( DR ) in relation to flow rate Q and the concentration of the drag-reducing agent C at a testing temperature of 40°C. It is evident that, at a constant flow rate, the drag reduction rate DR initially increases and subsequently decreases with increasing concentrations of sodium alginate, indicating the existence of an optimal concentration for drag reduction. This behavior suggests that the long-chain structure of the polymer in the sodium alginate solution enhances the solution's elasticity, allowing a portion of the energy from turbulent vortices to be stored in the polymer chains in an elastic form. As a result, the kinetic energy of the turbulent vortices is reduced, leading to a decrease in the energy dissipated by these vortices and resulting in drag reduction. However, as the concentration of the solution increases, the entanglement between sodium alginate molecular chains intensifies, and the intermolecular interactions become progressively stronger. This leads to an increase in solution viscosity, which ultimately diminishes the drag reduction effect. Figure 7 B illustrates that the optimal point for the drag reduction rate increases with the level of turbulence (i.e., as the flow rate increases), along with a corresponding rise in the concentration of sodium alginate. This observation indicates that higher concentrations of sodium alginate contribute to the suppression of turbulent vortex kinetic energy. However, intense turbulence can disrupt the intermolecular interactions within the sodium alginate solution, resulting in the maximum drag reduction rate being achieved under conditions of high concentration and high flow. Figure 8 A and 8 B illustrate the response surfaces for the drag reduction rate ( DR ) in relation to temperature t and the concentration of the drag-reducing agent C at a flow rate Q of 700 mL/min. It is evident from the figures that the drag reduction rate continues to decrease with increasing temperature. Although it is generally observed that the viscosity of liquids decreases with rising temperature, the dissolution behavior of sodium alginate, as a polymer in water, is relatively complex. Higher temperatures may enhance the thermal motion of the polymer chains, potentially increasing intermolecular interactions. This could lead to the contraction of the long-chain structures that facilitate drag reduction and ultimately weaken the intermolecular interactions, which is unfavorable for achieving effective drag reduction. Typically, the relationship between the drag reduction rate of polymer solutions and the Reynolds number is not monotonically increasing; rather, there exists a critical Reynolds number at which maximum drag reduction is achieved. When the Reynolds number exceeds this critical value, the long chains of the polymers in the solution may rupture and degrade [ 33 ] , resulting in a decrease in the drag reduction rate. As illustrated in Fig. 9 A and Fig. 9 B, the Reynolds number corresponding to the tested flow rate has not reached this critical threshold; therefore, the effects of polymer chain rupture and degradation on the drag reduction rate do not need to be considered at this stage. Based on previous analyses, as the temperature increases, the long chain structure of sodium alginate contracts, requiring sufficient shear force to extend the polymer chains. Consequently, under consistent flow conditions, an increase in temperature leads to a reduction in the drag reduction rate. To verify the applicability of Eq. ( 6 ) and Eq. (7), experimental data within the ranges of drag-reducing agent concentrations from 500 to 1700 ppm, volumetric flow rates from 550 to 900 mL/min, and temperatures from 25 to 50°C were compared with the predicted data from the correlation equations. Figure 10 A presents a comparison of the predicted data from Eq. (7) with the experimental data, showing a good correlation, as all data points are distributed within ± 20%. Although Eq. (7) has simplified Eq. ( 6 ), it is evident from Fig. 10 B that the vast majority of data still falls within the ± 20% range, and the data points are clustered without any divergence trend, indicating that Eq. (7) can effectively predict the experimental data. To verify the applicability of Eq. ( 6 ) and Eq. (7), experimental data were collected within the ranges of drag-reducing agent concentrations from 500 to 1700 ppm, volumetric flow rates from 550 to 900 mL/min, and temperatures from 25 to 50°C. These data were then compared with the predicted values obtained from the correlation equations. Figure 10 A displays a comparison of the predicted data from Eq. (7) with the experimental data, revealing a strong correlation, as all data points fall within ± 20%. While Eq. (7) simplifies Eq. ( 6 ), Fig. 10 B shows that the majority of data points still reside within the ± 20% range, demonstrating a clustered distribution without any discernible divergence trend. This indicates that Eq. (7) is capable of effectively predicting the experimental data. Conclusion This study rigorously investigated the drag reduction characteristics of polymer solutions, specifically focusing on sodium alginate under varying flow conditions, concentrations, and temperatures. The experimental results demonstrate that flow rate, concentration, and temperature are key factors influencing the drag reduction rate (DR). Analysis using Response Surface Methodology (RSM) confirmed a strong correlation between the experimental data and the theoretical models, particularly when the critical Reynolds number had not been exceeded. This indicates that polymer chain rupture and degradation did not significantly impact the drag reduction rate. Additionally, the study revealed that, under consistent flow conditions, increasing temperature leads to the contraction of the polymer chains, necessitating greater shear forces to extend the chains favorable for drag reduction, ultimately resulting in a decrease in the drag reduction rate. By comparing the actual experimental data with the predicted values derived from Eq. ( 6 ) and Eq. (7), the results further validate the applicability of these equations. Notably, Eq. (7) effectively predicts the experimental data with a high degree of agreement, within ± 20%. These findings provide a critical theoretical foundation for optimizing the use of sodium alginate in industrial applications. Declarations Funding Dongying Science Development Fund, 2023SS019 Dongying Science Development Fund, 2023SS017 Author Contribution Z.C. wrote the main manuscript text; X.Z. and X.D. prepared figures; Y.C. performed experiment; X.W. and X.S. revised the manuscript. 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18:17:36","extension":"xml","order_by":57,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":123898,"visible":true,"origin":"","legend":"","description":"","filename":"e26cf91618ec46aaac9815ef9f6b1b6c1structuring.xml","url":"https://assets-eu.researchsquare.com/files/rs-7356460/v1/43339200051236278c8ec37c.xml"},{"id":94586764,"identity":"11fe4df6-b815-4413-b4f1-de3d35ea2de3","added_by":"auto","created_at":"2025-10-28 18:17:21","extension":"html","order_by":58,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":135753,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-7356460/v1/dfa93ef5d19697af566e1eed.html"},{"id":94587192,"identity":"0bab4177-e073-49b0-a3cb-2e6b02ef3feb","added_by":"auto","created_at":"2025-10-28 18:17:44","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":181138,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic of microtube-flow experimental apparatus.\u003c/p\u003e","description":"","filename":"Fig.1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7356460/v1/d5b230eaa20ec92dae792808.jpg"},{"id":94587135,"identity":"c8c2a482-71c3-4c2a-a618-cb2a884f483a","added_by":"auto","created_at":"2025-10-28 18:17:40","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":925643,"visible":true,"origin":"","legend":"\u003cp\u003eWater calibration data in straight pipe at various Reynolds Number.\u003c/p\u003e","description":"","filename":"Fig.2.tif.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7356460/v1/b7f9a372d5359718c08da0af.jpg"},{"id":94586533,"identity":"4917820a-0742-43bd-9d32-dc1fd1b15707","added_by":"auto","created_at":"2025-10-28 18:17:00","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":928539,"visible":true,"origin":"","legend":"\u003cp\u003eVariation of drag reduction rate with concentration (\u003cem\u003eC\u003c/em\u003e) for codium alginate solutions at four different flow rates.\u003c/p\u003e","description":"","filename":"Fig.3.tif.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7356460/v1/467f6a5bd59cbb91f9c57dd6.jpg"},{"id":94587284,"identity":"fcc4c62a-acdc-485c-b9d2-99538d863b4e","added_by":"auto","created_at":"2025-10-28 18:18:01","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":928539,"visible":true,"origin":"","legend":"\u003cp\u003eEffect of temperature on the drag reduction rate of sodium alginate solutions at various flow rates.\u003c/p\u003e","description":"","filename":"Fig.4.tif.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7356460/v1/f519de7badbfd9bfd1cd17d7.jpg"},{"id":94587389,"identity":"f104e3a0-7895-41d4-9b06-54c8a7da0bfb","added_by":"auto","created_at":"2025-10-28 18:18:14","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":883758,"visible":true,"origin":"","legend":"\u003cp\u003eVariation of drag reduction (\u003cem\u003eDR\u003c/em\u003e) with flow rate (\u003cem\u003eQ\u003c/em\u003e) and Reynolds Number (\u003cem\u003eRe\u003c/em\u003e) for sodium alginate solutions at 40°C and 1300 ppm concentration.\u003c/p\u003e","description":"","filename":"Fig.5.tif.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7356460/v1/5f8fcb518e6764081627d1b5.jpg"},{"id":94587440,"identity":"8183db29-6ff5-4134-b152-cfbf51b9a74e","added_by":"auto","created_at":"2025-10-28 18:18:20","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":149537,"visible":true,"origin":"","legend":"\u003cp\u003ePlots illustrating the three assumptions of ANOVA. (A) The normal probability plot assesses whether the residuals follow a normal distribution. (B) The constant variance plot demonstrates the relationship between predicted values and residuals. (C) The plot of residuals versus run order examines potential time-dependent trends.\u003c/p\u003e","description":"","filename":"Fig.6.png","url":"https://assets-eu.researchsquare.com/files/rs-7356460/v1/ac25f1467d2bbd49b940f780.png"},{"id":94587376,"identity":"ee763a97-0d39-423c-a365-1df6353ce2b5","added_by":"auto","created_at":"2025-10-28 18:18:13","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":342247,"visible":true,"origin":"","legend":"\u003cp\u003eResponse surface plot of drag Reduction Rate for sodium alginate solutions at different flow rates and concentrations.\u003c/p\u003e","description":"","filename":"Fig.7.png","url":"https://assets-eu.researchsquare.com/files/rs-7356460/v1/3cbe60dc1da1a264219b8b01.png"},{"id":94587603,"identity":"45a292c7-e2dc-491b-9f01-ce98efafdaab","added_by":"auto","created_at":"2025-10-28 18:18:25","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":403527,"visible":true,"origin":"","legend":"\u003cp\u003eResponse surface plot of drag reduction rate for sodium alginate solutions at different temperatures and concentrations\u003c/p\u003e","description":"","filename":"Fig.8.png","url":"https://assets-eu.researchsquare.com/files/rs-7356460/v1/80590e6bc488844840758f9f.png"},{"id":94587256,"identity":"151da414-9615-4051-a122-d168235d92ed","added_by":"auto","created_at":"2025-10-28 18:17:56","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":349831,"visible":true,"origin":"","legend":"\u003cp\u003eResponse surface plot of drag reduction rate for sodium alginate solutions at different temperatures and flow rates\u003c/p\u003e","description":"","filename":"Fig.9.png","url":"https://assets-eu.researchsquare.com/files/rs-7356460/v1/7a41d48e685ae50841a93531.png"},{"id":94587029,"identity":"36e6efd5-69f0-4caa-83f2-f11d53911de2","added_by":"auto","created_at":"2025-10-28 18:17:35","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":149831,"visible":true,"origin":"","legend":"\u003cp\u003ePrediction of drag reduction rate by different models. (A) Comparison of predicted and experimental data using Eq. (6). (B) Comparison of predicted and experimental data using Eq. (7).\u003c/p\u003e","description":"","filename":"Fig.10.png","url":"https://assets-eu.researchsquare.com/files/rs-7356460/v1/11e10a6ea375c1841bd0263b.png"},{"id":94598502,"identity":"227268b9-1352-4e93-b66e-a79783b8666c","added_by":"auto","created_at":"2025-10-28 18:54:00","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":6032483,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7356460/v1/49543ed2-1a66-47c6-9f7e-f57d04925592.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Evaluation of Sodium Alginate as a Drag Reduction Agent in Flowing Fluids: Effects of Concentration, Temperature, and Flow Rate","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eThe application of fluid mechanics in industrial transport systems holds significant economic implications, particularly in reducing energy consumption and improving transportation efficiency. One of the primary sources of energy loss in fluid transportation systems is frictional resistance, making the reduction of flow resistance a central focus for enhancing pipeline efficiency\u003csup\u003e[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]\u003c/sup\u003e. The phenomenon of drag reduction was first observed by Toms (1947) during the mechanical degradation of polymer substances in pipe flow, a finding that was officially presented at the First International Rheological Conference in 1948[2]. Several methods for drag reduction are commonly employed, including pipeline design optimization, coating applications, fluid velocity control, and the use of drag-reducing additives\u003csup\u003e[\u003cspan additionalcitationids=\"CR4 CR5\" citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]\u003c/sup\u003e. Among these, the use of drag-reducing agents stands out for its cost-effectiveness and minimal requirement for large-scale modifications to the pipeline infrastructure. Drag-reducing agents are applicable to a wide range of fluids and pipeline systems, including those transporting oil, natural gas, and water, thus making them an appealing solution for industrial applications\u003csup\u003e[\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]\u003c/sup\u003e. As a result, drag reduction in turbulent flow remains a key area of research.\u003c/p\u003e\u003cp\u003eTraditionally, drag-reducing agents have been synthetic chemicals, which are effective in reducing frictional losses\u003csup\u003e[\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]\u003c/sup\u003e. However, concerns regarding environmental pollution, resource consumption, and health risks have sparked interest in the development of natural drag-reducing agents. Among the natural polymers studied, xanthan gum and guar gum have shown promise in reducing water flow resistance. Edomwonyi Otu et al.\u003csup\u003e[\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]\u003c/sup\u003e evaluated these biopolymers as drag reducers in water flow and highlighted their effectiveness in reducing drag in 12-mm diameter pipes. This study underscores the potential of biopolymers in diverse fluid systems. Singh et al.\u003csup\u003e[\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]\u003c/sup\u003e discussed grafted polysaccharides as effective turbulent drag reducers and flocculants, identifying grafted starch as particularly beneficial for industrial wastewater applications. This finding further supports the potential of polysaccharide-based biopolymers for natural drag reduction. Marhefka et al.\u003csup\u003e[\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]\u003c/sup\u003e expanded on this by investigating the drag-reducing properties of aloe vera. Their study found that aloe vera's drag reduction performance was superior to many synthetic polymers, demonstrating advantages in terms of stability and performance. Xie et al.\u003csup\u003e[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]\u003c/sup\u003e proposed yam mucilage as an environmentally friendly alternative, emphasizing its potential under turbulent conditions. The mucilage's high shear resistance and natural origin make it a promising candidate for sustainable flow enhancement. Shi et al.\u003csup\u003e[\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]\u003c/sup\u003e conducted supplementary studies on yam mucilage, further elucidating its drag-reducing capabilities and identifying factors influencing its effectiveness Similarly, dos Santos et al.\u003csup\u003e[\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]\u003c/sup\u003e explored diutan gum, a natural polymer from Shewanella species, noting its stability and effectiveness under turbulent flow conditions. Salehudin et al.\u003csup\u003e[\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]\u003c/sup\u003e synthesized carboxymethyl cellulose from coconut residue via an alkali-catalyzed reaction. The performance of the synthesized carboxymethyl cellulose was evaluated in a water injection system, examining its effectiveness at various concentrations and flow rates.\u003c/p\u003e\u003cp\u003eAnother promising natural polymer, sodium alginate, has also been investigated for its drag-reducing properties, particularly in turbulent flow within millimeter-scale pipes. Cheng et al.\u003csup\u003e[\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]\u003c/sup\u003e confirmed the effectiveness and elasticity of sodium alginate in resisting shear forces, addressing concerns related to the irritancy and toxicity of its synthetic counterparts. Their findings highlighted sodium alginate's potential for resolving common stability issues faced by natural polymers under flow conditions. However, temperature sensitivity remains a major challenge for natural drag-reducing agents, as temperature variations can affect their physical properties\u0026mdash;including viscosity, molecular structure, and solubility\u0026mdash;directly influencing their drag-reduction performance\u003csup\u003e[\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]\u003c/sup\u003e. For instance, under high-temperature conditions, natural drag-reducing agents may degrade or lose viscosity, reducing their effectiveness. Consequently, investigating the performance of natural drag-reducing agents at varying temperatures is essential for broadening their application in industrial processes\u003csup\u003e[\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eWhile sodium alginate has demonstrated promising shear resistance and drag-reducing performance under turbulent conditions, its behavior under varying temperature conditions requires further exploration. This study aims to analyze the effects of temperature, concentration, and flow rate on the drag-reducing properties of sodium alginate solutions.\u003c/p\u003e"},{"header":"2. Experimental setup and fluid preparation","content":"\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe experimental setup was adapted from the original design developed by the research group (as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e\u003csup\u003e[\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]\u003c/sup\u003e). The testing system consists of several critical components:\u003c/p\u003e\u003cp\u003e\u003col style=\"list-style-type:lower-roman;\"\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eLiquid Storage: A 10L transparent liquid storage tank is used to store the experimental fluid.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eCirculation System: A peristaltic pump circulates the liquid through the flow system. The system includes a 0.477-meter-long test section with an internal diameter of 2.23 mm. The inlet section's length satisfies the requirement of being 138 times the pipe diameter\u003csup\u003e[\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]\u003c/sup\u003e.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eFrictional Resistance Measurement: To measure the frictional resistance in the straight pipe flow, a differential pressure sensor was installed. This sensor has a maximum measurable pressure difference of 100 kPa and an accuracy of \u0026plusmn;\u0026thinsp;0.055%.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eTemperature and Solution Control: Given the limited research on the biodegradability of sodium alginate, the preparation process was completed quickly to avoid prolonged storage and to ensure solution stability. A magnetic stirrer and thermostatic water bath were used to control the temperature and accelerate the dissolution of sodium alginate. The water bath's temperature range extends from room temperature to 95\u0026deg;C, with a control accuracy of \u0026plusmn;\u0026thinsp;0.5\u0026deg;C. Additionally, aluminum foil was used to cover the beaker, preventing solution evaporation.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eData Acquisition: The experimental data acquisition system, based on the LabVIEW platform, enabled real-time monitoring of both temperature and pressure differential data. Data was collected every second and transmitted in real time to the controller for storage and display.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003c/p\u003e\u003cp\u003eSodium alginate is a natural polysaccharide composed of \u003cem\u003eα\u003c/em\u003e-L-guluronic acid (G) and its stereoisomer, \u003cem\u003eβ\u003c/em\u003e-D-mannuronic acid (M)\u003csup\u003e[\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]\u003c/sup\u003e. The sodium alginate used in this study was sourced from Fuchen (Tianjin) Chemical Co., Ltd. To prepare the sodium alginate solution, sodium alginate was dissolved in a thermostatic water bath with low-speed stirring for 60 minutes, ensuring complete dissolution while the temperature was gradually increased to the experimental condition. After stirring, the solution was maintained at a constant temperature for 1 hour to ensure full dissolution of the polymer and accurate experimental results.\u003c/p\u003e\u003cp\u003eAt the start of the experiment, deionized water was introduced into the test section to calibrate the system. The friction factor (\u003cem\u003ef\u003c/em\u003e) measured during the flow of deionized water was used as the calibration baseline, ensuring both the accuracy and stability of the experimental system. Following calibration, the sodium alginate solution was pumped into the testing system using the peristaltic pump, with the flow rate precisely set to the target value. As the solution passed through the test section, the pressure differential gradually stabilized. Data recording commenced once a steady state was achieved. After the test, all solutions were directed into a waste container for disposal to prevent reuse, ensuring the stability of the solution and the reliability of the experimental results.\u003c/p\u003e"},{"header":"3. Results and analyses","content":"\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\u003ch2\u003e3.1 Drag reduction flow analysis\u003c/h2\u003e\u003cp\u003eThe measured differential pressure data are used to calculate the Darcy friction factor \u003cem\u003ef\u003c/em\u003e, which is an essential parameter in determining the flow resistance in the pipe. The conversion from pressure differential to f is done using the following equation:\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Taba\" border=\"1\"\u003e\u003ccolgroup cols=\"2\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{f=}\\frac{\\text{2}\\text{d∆p}}{\\text{l\u0026rho;}{\\text{U}}_{\\text{b}}^{\\text{2}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e(1)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere: Δ\u003cem\u003ep\u003c/em\u003e is the flow pressure difference in the test section (Pa),\u003cem\u003ed\u003c/em\u003e is the diameter of the pipe (m),\u003cem\u003eρ\u003c/em\u003e is the density of the solution (kg/m\u0026sup3;),\u003cem\u003eU\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e is the average flow velocity in the test pipe (m/s),\u003cem\u003el\u003c/em\u003e is the length of the test section (m).\u003c/p\u003e\u003cp\u003eThe drag reduction rate \u003cem\u003eDR\u003c/em\u003e is calculated using the following equation:\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Tabb\" border=\"1\"\u003e\u003ccolgroup cols=\"2\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{DR=}\\frac{{\\text{\u0026Delta;}\\text{p}}_{\\text{s}}-{\\text{\u0026Delta;}\\text{p}}_{\\text{p}}}{{\\text{\u0026Delta;}\\text{p}}_{\\text{s}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e(2)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eIn the equation, Δ\u003cem\u003ep\u003c/em\u003e\u003csub\u003es\u003c/sub\u003e represents the pressure difference of the flowing tap water at a specified flow rate; Δ\u003cem\u003ep\u003c/em\u003e\u003csub\u003ep\u003c/sub\u003e represents the flow pressure difference of the sodium alginate solution under the same flow conditions. By comparing these values, the drag reduction rate can be calculated, providing a quantitative description of the drag-reducing effects of the additive at different temperatures.\u003c/p\u003e\u003cp\u003eThe flow rate of the solution is characterized by the average flow Reynolds number (\u003cem\u003eRe\u003c/em\u003e), which is a dimensionless quantity that helps determine the flow regime (laminar, transitional, or turbulent). The Reynolds number is calculated using the following equation:\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Tabc\" border=\"1\"\u003e\u003ccolgroup cols=\"2\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{Re}\\text{=}\\frac{{\\text{U}}_{\\text{b}}\\text{\u0026rho;}\\text{d}}{\\text{\u0026mu;}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e(3)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe dynamic viscosity of the solution (\u003cem\u003e\u0026micro;\u003c/em\u003e) of the solution is a crucial parameter for characterizing flow behavior. The solvent density and viscosity are referenced under the same experimental conditions as the test fluid, as outlined in previous studies\u003csup\u003e[\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e, \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e]\u003c/sup\u003e. The values for the dynamic viscosity of tap water (\u003cem\u003e\u0026micro;\u003c/em\u003e) at different temperatures are provided in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eDynamic viscosity of tap water at experimental temperatures\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"8\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e\u003cp\u003e\u003cem\u003et\u003c/em\u003e/℃\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003e25\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003e40\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colspan=\"2\" nameend=\"c7\" namest=\"c6\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colspan=\"1\" nameend=\"c8\" namest=\"c8\"\u003e\u0026nbsp;\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e\u003cp\u003e\u003cem\u003e\u0026micro;\u003c/em\u003e\u0026thinsp;\u0026times;\u0026thinsp;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e/pa\u0026bull;s\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e0.8937\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.8007\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0.6560\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colspan=\"2\" nameend=\"c7\" namest=\"c6\"\u003e\u003cp\u003e0.5494\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colspan=\"1\" nameend=\"c8\" namest=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colspan=\"5\" nameend=\"c7\" namest=\"c3\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{U}}_{\\text{b}}\\text{=}\\frac{\\text{4}\\text{Q}}{\\text{\u0026rho;\u0026pi;}{\\text{d}}^{\\text{2}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e(4)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003cem\u003eQ\u003c/em\u003e is the volumetric flow rate of the solution, measured in ml/min.\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:\\text{f=}\\text{0.3164}{\\text{Re}}^{\\text{-}\\text{0.25}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e presents the relationship between the friction factor and Reynolds number (or flow rate) for deionized water at 30\u0026deg;C. The data reveal a strong correlation between the measured friction factor and the empirical formula for turbulent flow in smooth pipes, specifically the Blasius formula\u003csup\u003e[\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e]\u003c/sup\u003e. The calculated values using the Blasius equation align well with the experimental data, remaining within a\u0026thinsp;\u0026plusmn;\u0026thinsp;3% error margin. This comprehensive analysis demonstrates a high degree of consistency between the experimental measurements and the theoretical predictions, thus confirming the reliability and accuracy of the experimental results.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\u003ch2\u003e3.2 Single Factor Experiment on Drag Reduction Flow of Sodium Alginate Solutions\u003c/h2\u003e\u003cdiv id=\"Sec6\" class=\"Section3\"\u003e\u003ch2\u003e3.2.1 Effect of Concentration\u003c/h2\u003e\u003cp\u003eGasljevic et al.\u003csup\u003e[\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e]\u003c/sup\u003e identified the \"pipe diameter effect\" concerning drag-reducing fluids, demonstrating that varying pipe sizes can yield different drag reduction results, even when the same solution is utilized. Previous studies\u003csup\u003e[\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]\u003c/sup\u003e prepared a sodium alginate solution at 21\u0026deg;C and assessed its drag reduction properties in a circular pipe with an internal diameter of 1.85 mm. In this study, the internal diameter of the circular pipe used is 2.23 mm, which is comparable to the diameter tested in earlier research. Consequently, experiments were conducted within the concentration range corresponding to the maximum observed drag reduction rate from prior studies (750\u0026ndash;1250 ppm). Sodium alginate solutions at concentrations of 700, 900, 1100, 1300, and 1500 ppm were prepared for testing. All experiments were performed once the solutions were equilibrated to room temperature (25\u0026deg;C), and the comparative results are presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e illustrates that, due to variations in test pipe diameter and temperature relative to previous studies, the drag reduction rate initially increases and subsequently decreases with increasing concentration of the sodium alginate solution. The maximum drag reduction rate is observed within a concentration range of 1200\u0026ndash;1400 ppm, with 1300 ppm selected as the midpoint concentration for subsequent response surface analysis.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec7\" class=\"Section3\"\u003e\u003ch2\u003e3.2.2 Effect of Temperature\u003c/h2\u003e\u003cp\u003eAfter establishing the concentration range and keeping all other conditions constant, three temperatures (30\u0026deg;C, 40\u0026deg;C, and 50\u0026deg;C) were selected for comparison. In alignment with the findings of Rahaman et al.\u003csup\u003e[\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e]\u003c/sup\u003e, it was observed that the drag reduction rate of natural drag reducers exhibits an extremum as a function of temperature, although the distribution pattern is complex. Consequently, a concentration of 1300 ppm was chosen for the temperature effect experiments across different flow conditions, with the results illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e clearly demonstrates that at a concentration of 1300 ppm, all three flow rates achieve a maximum drag reduction rate (\u003cem\u003eDR\u003c/em\u003e) at 40\u0026deg;C. Consequently, 40\u0026deg;C was selected as the midpoint temperature for the response surface analysis. However, it is also evident that under high flow conditions, the influence of temperature on the drag reduction rate of the sodium alginate solution diminishes.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec8\" class=\"Section3\"\u003e\u003ch2\u003e3.2.3 Effect of Flow Rate\u003c/h2\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eAfter establishing fixed concentration and temperature ranges, drag reduction comparison experiments were conducted under varying flow conditions, with the results depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. It is evident that in the turbulent flow regime, the drag reduction rate (\u003cem\u003eDR\u003c/em\u003e) increases with rising flow rate, with no extremum observed within the tested range. Consequently, a flow rate of 700 ml/min was chosen as the midpoint flow rate for the response surface analysis.\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\u003ch2\u003e3.3 Response surface methodology (RSM) results and analysis\u003c/h2\u003e\u003cp\u003eBuilding on the results from the univariate experiments, a response surface design experiment was conducted using the Box-Behnken central composite experimental design principle. Three factors that significantly influence the drag reduction rate (\u003cem\u003eDR\u003c/em\u003e) were selected: concentration (\u003cem\u003eC\u003c/em\u003e), flow rate (\u003cem\u003eQ\u003c/em\u003e), and solution temperature (\u003cem\u003et\u003c/em\u003e). A three-factor, three-level response surface analysis was conducted, with the drag reduction rate (\u003cem\u003eDR\u003c/em\u003e) of the sodium alginate solution serving as the response variable, as detailed in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eVariables and levels in central composite design.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eFactor\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eLowest level (-1)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eMedian (0)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eHighest level (+\u0026thinsp;1)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cem\u003eQ\u003c/em\u003e(ml/min)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e900\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e1300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e1700\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cem\u003eC\u003c/em\u003e(ppm)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e600\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e700\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e800\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cem\u003et\u003c/em\u003e(℃)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e summarizes the experimental design generated by the Design-Expert software, which necessitates a total of 17 distinct experimental groups to meet statistical requirements. A quadratic response surface regression analysis, as recommended by the Design-Expert software, was performed, resulting in the following ternary quadratic response surface regression model represented in Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e6\u003c/span\u003e).\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eProcess variables and their levels in the Box-Behnken design, along with the corresponding response values for the drag reduction rate (\u003cem\u003eDR\u003c/em\u003e) of sodium alginate.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eA:\u003cem\u003eC\u003c/em\u003e (%)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eB:\u003cem\u003eQ\u003c/em\u003e(ml/min)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eC: \u003cem\u003et\u003c/em\u003e(℃)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003e\u003cem\u003eDR\u003c/em\u003e(%)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e800\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e22.49\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e900\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e800\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e19.46\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1700\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e800\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e22.49\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1700\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e700\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e22.86\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e700\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e19.93\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e700\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e19.93\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e700\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e19.93\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e900\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e600\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e16.58\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e600\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e19.95\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e700\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e19.93\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1700\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e600\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e17.95\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1700\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e700\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e16.28\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e900\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e700\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e15.51\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e900\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e700\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e18.68\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e800\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e21.07\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e700\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e19.93\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e600\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e16.26\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:\\text{DR=}\\text{25.51+0.0238}\\text{C}\\text{-0.0737}\\text{Q}\\text{+4.06}\\text{\u0026times;}{\\text{10}}^{\\text{-3}}\\text{t}\\text{+1.0}\\text{0}\\text{\u0026times;}{\\text{10}}^{\\text{-5}}\\text{CQ}\\text{-2.13\u0026times;}{\\text{10}}^{\\text{-4}}\\text{Ct+}\\text{5.6}\\text{7}\\text{\u0026times;}{\\text{10}}^{\\text{-}\\text{4}}\\text{Qt-}\\text{7.56}\\text{\u0026times;}{\\text{10}}^{\\text{-6}}{\\text{C}}^{\\text{2}}\\text{+}\\text{4.1\u0026times;}{\\text{10}}^{\\text{-5}}{\\text{Q}}^{\\text{2}}\\text{-3.}\\text{88}\\text{\u0026times;}{\\text{10}}^{\\text{-3}}{\\text{t}}^{\\text{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eAnalysis of variance (ANOVA) for the regression equation of the drag reduction rate (\u003cem\u003eDR\u003c/em\u003e) of sodium alginate as a function of three variables.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSource\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eSum of Squares\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003edf\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eMean Square\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003eF-value\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003ep-value\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cb\u003eModel\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e78.12\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" 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colname=\"c3\"\u003e\u003cp\u003e1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e27.55\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e70.21\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e\u0026lt;\u0026thinsp;0.0001\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eAB\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.6889\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.6913\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e1.76\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e0.2255\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eAC\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e2.91\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e2.90\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e7.39\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e0.0293\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eBC\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e1.29\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e1.29\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e3.29\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e0.1120\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eA\u0026sup2;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e6.16\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e6.08\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e15.50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e0.0054\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eB\u0026sup2;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.6737\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.7028\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e1.79\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e0.2301\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eC\u0026sup2;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.6322\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.6091\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e1.55\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e0.2436\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cb\u003eResidual\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e2.73\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.3899\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLack of Fit\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e2.73\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.9099\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ePure Error\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.0000\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.0000\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cb\u003eCor Total\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e80.85\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e16\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eTo evaluate the goodness of fit of the response surface regression model and the effects of each factor on the response variable, an analysis of variance (ANOVA) was conducted. Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e presents the results of the ANOVA for the fitted model. According to the ANOVA results, the total sum of squares for the model is 78.12, with 9 degrees of freedom, a mean square of 8.66, an F-value of 22.07, and a p-value of 0.0002. These findings collectively indicate that the overall model possesses significant explanatory power regarding the response variable, suggesting that both the main effects and interaction effects included in the model effectively account for the variability observed in the response variable.\u003c/p\u003e\u003cp\u003eIn the model, the three main effect factors (A-\u003cem\u003eC\u003c/em\u003e, B-\u003cem\u003eQ\u003c/em\u003e, and C-\u003cem\u003et\u003c/em\u003e), along with the interaction term AC and the quadratic term A\u0026sup2;, all exert significant effects on the response variable. Conversely, the P-values for the factors AB, BC, B\u0026sup2;, and C\u0026sup2; exceed 0.05, indicating that these factors do not have significant effects on the response variable. Consequently, consideration may be given to removing these factors from the model. The optimized results are presented in Eq.\u0026nbsp;(7). The calculated coefficient of determination (\u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e) value is 0.9662, indicating an excellent fit of the model to the experimental data, with robust explanatory power and predictive capability. This suggests that the model effectively describes the variability in the experimental data, thereby validating its reliability and accuracy.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Tabd\" border=\"1\"\u003e\u003ccolgroup cols=\"2\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{DR}\\text{=}\\text{-13.21+0.0311}\\text{C}\\text{+0.0185}\\text{Q}\\text{+0.0913}\\text{t}\\text{-2.13\u0026times;}{\\text{10}}^{\\text{-4}}\\text{Ct}\\text{+}\\text{7.59}\\text{\u0026times;}{\\text{10}}^{\\text{-6}}{\\text{C}}^{\\text{2}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e(7)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eIn this study, the three key assumptions related to Response Surface Methodology (RSM) analysis were validated. First, the normal probability plot of the residuals (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eA) was examined, which revealed that the residual data points are symmetrically distributed around the reference line, indicating that the residuals satisfy the normal distribution assumption. Second, Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eB illustrates that the residual values are confined within the range of \u0026plusmn;\u0026thinsp;4.82. Additionally, as the predicted values increase, there is no significant divergence in the variance of the experimental data, thereby confirming the validity of the assumption of constant variance. Finally, the randomness of the data was assessed by analyzing the correlation between the run order and the residuals. The results presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eC show no apparent temporal correlation trend between the residuals and the run order, demonstrating that the experimental data meet the randomness assumption. In summary, the drag reduction rate testing data satisfy the three fundamental assumptions of response surface methodology analysis, indicating that the testing methods and data used in this study conform to the analytical requirements.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003eA illustrates the response surface for the drag reduction rate (\u003cem\u003eDR\u003c/em\u003e) in relation to flow rate \u003cem\u003eQ\u003c/em\u003e and the concentration of the drag-reducing agent \u003cem\u003eC\u003c/em\u003e at a testing temperature of 40\u0026deg;C. It is evident that, at a constant flow rate, the drag reduction rate \u003cem\u003eDR\u003c/em\u003e initially increases and subsequently decreases with increasing concentrations of sodium alginate, indicating the existence of an optimal concentration for drag reduction. This behavior suggests that the long-chain structure of the polymer in the sodium alginate solution enhances the solution's elasticity, allowing a portion of the energy from turbulent vortices to be stored in the polymer chains in an elastic form. As a result, the kinetic energy of the turbulent vortices is reduced, leading to a decrease in the energy dissipated by these vortices and resulting in drag reduction. However, as the concentration of the solution increases, the entanglement between sodium alginate molecular chains intensifies, and the intermolecular interactions become progressively stronger. This leads to an increase in solution viscosity, which ultimately diminishes the drag reduction effect.\u003c/p\u003e\u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003eB illustrates that the optimal point for the drag reduction rate increases with the level of turbulence (i.e., as the flow rate increases), along with a corresponding rise in the concentration of sodium alginate. This observation indicates that higher concentrations of sodium alginate contribute to the suppression of turbulent vortex kinetic energy. However, intense turbulence can disrupt the intermolecular interactions within the sodium alginate solution, resulting in the maximum drag reduction rate being achieved under conditions of high concentration and high flow.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003eA and \u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003eB illustrate the response surfaces for the drag reduction rate (\u003cem\u003eDR\u003c/em\u003e) in relation to temperature t and the concentration of the drag-reducing agent \u003cem\u003eC\u003c/em\u003e at a flow rate \u003cem\u003eQ\u003c/em\u003e of 700 mL/min. It is evident from the figures that the drag reduction rate continues to decrease with increasing temperature. Although it is generally observed that the viscosity of liquids decreases with rising temperature, the dissolution behavior of sodium alginate, as a polymer in water, is relatively complex. Higher temperatures may enhance the thermal motion of the polymer chains, potentially increasing intermolecular interactions. This could lead to the contraction of the long-chain structures that facilitate drag reduction and ultimately weaken the intermolecular interactions, which is unfavorable for achieving effective drag reduction.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eTypically, the relationship between the drag reduction rate of polymer solutions and the Reynolds number is not monotonically increasing; rather, there exists a critical Reynolds number at which maximum drag reduction is achieved. When the Reynolds number exceeds this critical value, the long chains of the polymers in the solution may rupture and degrade\u003csup\u003e[\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e]\u003c/sup\u003e, resulting in a decrease in the drag reduction rate. As illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003eA and Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003eB, the Reynolds number corresponding to the tested flow rate has not reached this critical threshold; therefore, the effects of polymer chain rupture and degradation on the drag reduction rate do not need to be considered at this stage. Based on previous analyses, as the temperature increases, the long chain structure of sodium alginate contracts, requiring sufficient shear force to extend the polymer chains. Consequently, under consistent flow conditions, an increase in temperature leads to a reduction in the drag reduction rate.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eTo verify the applicability of Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e6\u003c/span\u003e) and Eq.\u0026nbsp;(7), experimental data within the ranges of drag-reducing agent concentrations from 500 to 1700 ppm, volumetric flow rates from 550 to 900 mL/min, and temperatures from 25 to 50\u0026deg;C were compared with the predicted data from the correlation equations. Figure\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003eA presents a comparison of the predicted data from Eq.\u0026nbsp;(7) with the experimental data, showing a good correlation, as all data points are distributed within \u0026plusmn;\u0026thinsp;20%. Although Eq.\u0026nbsp;(7) has simplified Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e6\u003c/span\u003e), it is evident from Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003eB that the vast majority of data still falls within the \u0026plusmn;\u0026thinsp;20% range, and the data points are clustered without any divergence trend, indicating that Eq.\u0026nbsp;(7) can effectively predict the experimental data.\u003c/p\u003e\u003cp\u003eTo verify the applicability of Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e6\u003c/span\u003e) and Eq.\u0026nbsp;(7), experimental data were collected within the ranges of drag-reducing agent concentrations from 500 to 1700 ppm, volumetric flow rates from 550 to 900 mL/min, and temperatures from 25 to 50\u0026deg;C. These data were then compared with the predicted values obtained from the correlation equations. Figure\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003eA displays a comparison of the predicted data from Eq.\u0026nbsp;(7) with the experimental data, revealing a strong correlation, as all data points fall within \u0026plusmn;\u0026thinsp;20%. While Eq.\u0026nbsp;(7) simplifies Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e6\u003c/span\u003e), Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003eB shows that the majority of data points still reside within the \u0026plusmn;\u0026thinsp;20% range, demonstrating a clustered distribution without any discernible divergence trend. This indicates that Eq.\u0026nbsp;(7) is capable of effectively predicting the experimental data.\u003c/p\u003e\u003c/div\u003e"},{"header":"Conclusion","content":"\u003cp\u003eThis study rigorously investigated the drag reduction characteristics of polymer solutions, specifically focusing on sodium alginate under varying flow conditions, concentrations, and temperatures. The experimental results demonstrate that flow rate, concentration, and temperature are key factors influencing the drag reduction rate (DR).\u003c/p\u003e\u003cp\u003eAnalysis using Response Surface Methodology (RSM) confirmed a strong correlation between the experimental data and the theoretical models, particularly when the critical Reynolds number had not been exceeded. This indicates that polymer chain rupture and degradation did not significantly impact the drag reduction rate. Additionally, the study revealed that, under consistent flow conditions, increasing temperature leads to the contraction of the polymer chains, necessitating greater shear forces to extend the chains favorable for drag reduction, ultimately resulting in a decrease in the drag reduction rate. By comparing the actual experimental data with the predicted values derived from Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e6\u003c/span\u003e) and Eq.\u0026nbsp;(7), the results further validate the applicability of these equations. Notably, Eq.\u0026nbsp;(7) effectively predicts the experimental data with a high degree of agreement, within \u0026plusmn;\u0026thinsp;20%. These findings provide a critical theoretical foundation for optimizing the use of sodium alginate in industrial applications.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eFunding\u003c/h2\u003e\u003cp\u003eDongying Science Development Fund, 2023SS019\u003c/p\u003e\u003cp\u003eDongying Science Development Fund, 2023SS017\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eZ.C. wrote the main manuscript text; X.Z. and X.D. prepared figures; Y.C. performed experiment; X.W. and X.S. revised the manuscript. J.L. and G.L. performed the validation. All authors reviewed the manuscript.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eTo datasets generated and analyzed during the current study are not publicly available due to the policy of Shandong Institute of Petroleum and Chemical Technology, but are available from the corresponding author upon reasonable request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eNiazi, M., Ashrafizadeh, S. N., Hashemabadi, S. H. \u0026amp; Karami, H. CFD simulation of drag-reducing fluids in a non-Newtonian turbulent pipe flow. \u003cem\u003eChem. Eng. Sci.\u003c/em\u003e \u003cb\u003e285\u003c/b\u003e, 119612 (2024).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eToms, B. A. Some observations on the flow of linear polymersolutions through straight tubes at large Reynolds numbers. In Proc. 1st Intl Congr. 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An experimental study on friction reducing polymers in turbulent pipe flow. \u003cem\u003eOcean Eng.\u003c/em\u003e \u003cb\u003e274\u003c/b\u003e, 114039 (2023).\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-7356460/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7356460/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eSodium alginate, a natural polymer, is widely utilized in various fields due to its exceptional drag reduction properties. This study systematically investigates the effects of sodium alginate concentration, flow rate, and temperature on the drag reduction rate. Utilizing Response Surface Methodology (RSM), a series of experiments were designed to optimize and analyze the influence of these factors. The results indicate that sodium alginate concentration, flow rate, and temperature have a significant impact on drag reduction performance, with temperature primarily affecting the intermolecular interactions and the contraction of polymer chains. A multi-factor mathematical model was developed using RSM to describe the relationship between the drag reduction rate and the parameters of concentration, flow rate, and temperature of sodium alginate. Validation of the model against experimental data revealed a strong correlation, confirming that the model accurately characterizes the drag reduction behavior of sodium alginate under varying operational conditions. These findings highlight that optimizing the conditions for sodium alginate application can significantly enhance fluid flow efficiency, providing a theoretical foundation for its use in industrial and environmental engineering applications.\u003c/p\u003e","manuscriptTitle":"Evaluation of Sodium Alginate as a Drag Reduction Agent in Flowing Fluids: Effects of Concentration, Temperature, and Flow Rate","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-28 16:35:52","doi":"10.21203/rs.3.rs-7356460/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-11-19T11:50:22+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-11-19T10:03:35+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-11-05T13:49:29+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"108765001044190687896973996988332761307","date":"2025-10-27T07:07:03+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"190989048979749436996416846368052678117","date":"2025-10-18T08:13:04+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-10-14T04:54:15+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-08-20T11:00:43+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-08-18T06:33:52+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-08-14T09:08:36+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-08-12T13:41:14+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"0b6b09e4-66bb-416e-98c4-97dcaca4f7bf","owner":[],"postedDate":"October 28th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[{"id":56909320,"name":"Physical sciences/Engineering"},{"id":56909321,"name":"Physical sciences/Materials science"},{"id":56909322,"name":"Physical sciences/Mathematics and computing"}],"tags":[],"updatedAt":"2025-11-19T11:53:48+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-28 16:35:52","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7356460","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7356460","identity":"rs-7356460","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}