The impact of pre-compression and temperature on perfluoroalkoxy alkane springs: Insights into spring relaxation

Abstract

A predictive model was proposed for determining the high-temperature free height of perfluoroalkoxy alkane springs in air-operated double-bellow pumps with the aim of investigating their relaxation. The model incorporates classical spring deformation theory and considers the material, structure, and real-world operating conditions of the perfluoroalkoxy alkane springs. Experimental validation of the model is also conducted. This study examines the effects of varying temperatures and pre-compression values on the relaxation of perfluoroalkoxy alkane springs’ free height. It collects relaxation curves under different temperatures and various pre-compression conditions. The results indicate that spring relaxation increases with higher temperatures when there is no pre-compression. Furthermore, increasing pre-compression at the same temperatures leads to greater spring relaxation. Pre-compression has a more significant impact on spring relaxation. By comparing the experimental data with the simulated curve generated by the model, it is evident that the predicted spring free height relaxation closely aligns with the actual measurements. This verification demonstrates the effectiveness and accuracy of the proposed model in evaluating the relaxation of perfluoroalkoxy alkane springs’ free height. Moreover, the model provides a valuable tool for predicting the lifespan of similar perfluoroalkoxy alkane springs in engineering applications.

Introduction

The perfluoroalkoxy alkane (PFA) springs serve as crucial constituents in essential components used in the air-operated double-bellow pumps, which find extensive application in the production of semiconductors, photovoltaic solar cells, LEDs, flat-panel displays, and electronics. These springs fulfill indispensable roles, including mechanical support, energy storage, and valve control. During their service, the performance of these springs is affected by the environment, such as high temperatures, potentially compromising the proper operation of equipment. Consequently, it becomes imperative to gain a comprehensive understanding and predict of the behavior and properties of springs under service process circumstances.

The selection of plastic springs that provide stability is essential due to the need for cleaning standards. Extensive research has been conducted by scholars from various countries on these polymer materials. Wang et al. ¹ systematically studied the influence of temperature on the tensile properties and viscoelasticity of polytetrafluoroethylene woven fabrics through experimental studies. Tavares et al. ² proposed a spring element model that takes into account the dynamic effects of fiber failure in composite materials, which was implemented in a parallel environment for better performance in predicting complex mechanisms associated with longitudinal tensile failure. Bouzid and Benabdallah ³ investigated the relaxation deformation of PTFE gaskets due to thermal cycling, gasket stress level and holding time. Yang et al. ⁴ put forward a stiffness prediction model based on composite mechanics and energy method to accurately predict the stiffness of composite leaf springs, and verified it by finite element method, existing theoretical methods and bench tests. Baniasadi et al. ⁵ offered an analytical solution to accurately simulate the mechanical response of a smart helical spring with variable stiffness, considering the effects of time and temperature. A method for predicting the fatigue life of springs was proposed by Shi et al., ⁶ based on Walker model. This model allows for simultaneous consideration of residual stress, crack closure effect and load eccentricity. Wróbel ⁷ posited a static dynamic model of polymer ring spring, analyzed the effect of friction on its performance characteristics, and verified the correctness of the model by design experiments. Foard et al. ⁸ studied the mathematical modeling and fabrication of polymer composite Belleville springs, as well as their potential applications. Their study successfully achieved the prediction of complex stiffness changes through progressive action, with theoretical predictions aligning closely with experimental results. Ni et al. ⁹ investigated the impact of different operating frequencies on the dynamic characteristics of PFA springs through compression cycle experiments under uniaxial sinusoidal excitation and analyzed the variation pattern of the spring hysteresis coefficient. Most of what the researchers have achieved are properties of other materials that are not PFA. At the same time, they did not study the relaxation behavior of spring components at high temperatures.

The primary focus of this paper encompasses several key aspects. Firstly, an experimental investigation is conducted to measure the elastic modulus of PFA materials at various temperatures, and subsequently, a fitted relationship between the elastic modulus and temperature is established. Additionally, a predictive model for the free height of PFA springs at elevated temperatures is proposed, followed by the design of a high-temperature heating experiment to analyze the relaxation mechanism of the spring's free height. The obtained experimental results validate the accuracy of the prediction model and offer a viable approach for assessing the lifespan of PFA springs.

Springs and modeling

Spring application and parameters

The spring serves as the primary component of an air-operated double-bellow pump, as depicted in Figure 1. The assembly of a check valve, consisting of one valve body and one valve disk, is its primary responsibility. In the operational process illustrated in Figure 1(a), upon activation of the pressurization device, the bellows located on the left side of the pump undergo compression. As the internal pressure rises, the upper left valve opens, simultaneously compressing the connecting spring, while the bottom left valve reverses its position and closes to prevent the ingress of liquid. Subsequently, the liquid contained within the bellows is pumped to the processing cell via an online heater and clean-up filter before being returned to the bellows on the right side. Similarly, when pressurized by a compressor on its right side, the liquid is pumped into the left side, as shown in Figure 1(b).

Figure 1.

The application of spring in cleaning pump and its main working process.

In technical discourse, the models of air-operated double-bellow pumps can be categorized according to their maximum flow rates of 15 and 30 lpm. Consequently, those models are denoted as Model-15 and Model-30, respectively, and their dimension parameters are shown in Table 1. In order to function properly, the spring must be submerged in a cleaning solution for an extended duration, necessitating its ability to withstand acid, alkali, and corrosion. Moreover, the use of a metal spring is deemed inappropriate due to cleanliness criteria, thus making it imperative to select a plastic spring that offers stability. However, temperature variations have a significant impact on plastic materials. Hence, it is essential to accurately characterize the relaxation behavior of the spring's free height under high-temperature working conditions to ensure its long-term immersion reliability.

Table 1.

Dimension parameters of springs.

	Model-15	Model-30
Section width(a)	2±0.1 mm	3±0.1 mm
Section height(b)	1.55±0.1 mm	4±0.1 mm
Pitch(t)	3.7±0.1 mm	9.5±0.1 mm
Outer diameter(D)	15±0.1 mm	27±0.1 mm
Effective coils(n)	6	2
Helix angle(α)	4.5°	6.7°

Spring materials

PFA materials exhibit corrosion resistance, high-temperature stability, low friction, and excellent electrical insulation properties, making them suitable for various demanding environments, such as the manufacture of pipes, valves, accessories, linings, wire insulation, connectors, and seals. The AP-231SH variant manufactured by Shanghai VALQUA, LTD, was employed for the springs in this study. The material properties of PFA are detailed in Table 2.

Table 2.

Mechanical properties of perfluoroalkoxy alkane (PFA).

Performance parameters	Value
Density(kg/m³) Poisson's ratio Melting point(°C) Heat deflection temperature(°C) Coefficient of linear expansion(10−5/°C）	2170 0.46 290 75 12

Similar to other polymer materials, PFA exhibits temperature-dependent fluctuations in its elastic modulus, lacking a clearly defined governing principle for these variations. ¹⁰ To establish a correlation between the elastic modulus of PFA and temperature, the elastic modulus of PFA at various temperature points was measured through a compressive elastic modulus test, as shown in Figure 2.

Figure 2.

Experimental setup for the elastic modulus of perfluoroalkoxy alkane (PFA).

A cylindrical sample of PFA with a diameter of 20 mm and a length of 30 mm is carefully prepared and placed in a heating chamber. The fixtures are slightly adjusted to ensure that the distance between them surpasses the length of the sample, thereby guaranteeing appropriate placement. The PFA samples are meticulously positioned between the fixtures to ensure contact and parallel alignment. The initial height of the sample is designated as H₀.

For the testing procedure, a laboratory temperature setting ranging from 20 to 250 °C is selected, with 16 evenly distributed temperature gradients. The hydraulic system is employed to apply a load of F = 150 N at a loading rate of 1 mm/min over a duration of 10 min. Throughout the compression process, the workpiece undergoes a transformation in its shape. This transformation is influenced by the temperature-dependent characteristics of the material's elastic modulus, resulting in variations in the degree of deformation at different temperatures compared to the final stable state achieved after compression.

To capture this information, sensors are employed to continuously record measurements of the section area A and deformation data H. These recorded measurements are consistently outputted and serve as the foundation for computing stress-strain relationships, ultimately generating the stress–strain curve. By calculating the slope of each curve in the linear region, the elastic modulus of PFA material at different temperatures can be obtained, as shown in Table 3.

Table 3.

Elastic modulus of perfluoroalkoxy alkane (PFA) materials at different temperatures.

Temperature/°C	20	30	40	50	60	70	80	90
E/Mpa	725.8	455.5	289.6	194.9	136.1	104.4	88.73	77.05
Temperature/°C	110	130	150	170	190	210	230	250
E/Mpa	62.94	48.74	36.14	29.53	28.49	26.38	19.1	18.52

The collected data were subsequently subjected to curve fitting using MATLAB. In principle, augmenting the number of fittings has the potential to yield a more accurate curve; nevertheless, it may also introduce greater fluctuations. Consequently, as a customary practice, it is generally advised to limit the number of fittings no more than 7. ¹¹ Taking into account the precision and computational simplicity, a sixth-degree polynomial fitting was employed to analyze the experimental data, and the results are shown in Figure 3.

Figure 3.

Thermal states of the spring at high temperature.

According to the findings presented in Figure 4, PFA exhibits a higher elastic modulus at lower temperatures but experiences a notable decrease in elasticity at higher temperatures. The influence of temperature on the elastic modulus of the material is substantial. In general, the elastic modulus decreases as temperature increases; however, this relationship is non-linear, which aligns with the conclusion reached by Abdel-Wahab et al. ¹⁰ It is important to highlight that there is a significant decrease in the elastic modulus of the material as temperature rises from 20 to 80 °C. Subsequently, between 80 and 250 °C, the decline in the elastic modulus becomes more gradual.

Figure 4.

The results of elastic modulus data curve fitting.

Based on the fitting results, the relationship between elastic modulus and temperature was ultimately determined.

$$\mathrm{E}=1.7101\times {10}^{-10}{\mathrm{T}}^6-1.6188\times {10}^{-7}{\mathrm{T}}^5+6.169\times {10}^{-5}{\mathrm{T}}^4-1.2094\times {10}^{-2}{\mathrm{T}}^3+1.2887{\mathrm{T}}^2-71.745\mathrm{T}+1731.1$$ (1)

The elastic modulus of the spring material is denoted as E(T) in the following formula.

Modeling

Analysis of spring free height relaxation at high temperature

The thermal states of the spring, including both the initial and subsequent states, are illustrated in Figure 5. The initial free height of the spring during the spring's non-heated state is indicated as H₀, whereas the free height of the spring after heating is labeled as H. The resulting alteration in the spring's free height is represented as f.

$$f=H_0-H$$ (2)

Figure 5.

Free height relaxation tests within a high-temperature test chamber: (a) heating experiments on a chamber named model GW030; (b) An image dimensional gauge named KEYENCE IM-8020.

The alteration in the free height of the spring at elevated temperatures is influenced by several factors. These factors, as per the deformation theory proposed by previous researchers for circular section springs, encompass the axial load applied to the spring, its structural parameters, the rectangular section of the machined spring, and the temperature-dependent elastic modulus of the spring material. Consequently, a mathematical expression can be employed to articulate the deformation of the spring's free height.

$$f(T)=f[F,E(T),\alpha ,a,b,D,T]$$ (3)

where F is the load on the spring, T is the temperature variable, E is the elastic modulus, and $\alpha$ is the helix angle of the spring. a is the length of the spring section, b is the width of the spring section, and D is the outer diameter r of the spring.

Free high relaxation model of spring at high temperature

By consulting the spring manual and utilizing established theories of deformation for cylindrical helical compression springs, a precise relationship between deformation and load can be elucidated for a cylindrical spiral compression spring. ¹²

$$f={\displaystyle \frac{\pi F{D}^{3}n}{4\cos \alpha }\left({\displaystyle \frac{{\cos }^2\alpha }{G{I}_p}+{\displaystyle \frac{{\sin }^2\alpha }{EI}}}\right)}$$ (4)

where n is the number of effective coils; G is the shear modulus of elasticity for the spring material; E is the elastic modulus; I_p is its polar moment of inertia; I is the moment of inertia.

Figure 6 displays a schematic diagram of the rectangular cross-section spring utilized in the experiment, with the length and width of the rectangular section denoted as a and b, respectively. In the context of Model-15 pumps where a > b, the determination of I_p and I values can be elucidated as follows ¹² :

$$I_p=k_{1}a{b}^3$$ (5)

$$I={\displaystyle \frac{a{b}^3}{12}}$$ (6)

Figure 6.

Three-dimensional diagram of Model-15 and Model-30 springs.

In the context of Model-30 pumps, where a < b, the determination of I_p and I values can be elucidated as follows ¹² :

$$I_p=k_1{a}^{3}b$$ (7)

$$I={\displaystyle \frac{a^{3}b}{12}}$$ (8)

The inclusion of the spring curvature correction factor, denoted as k₁, is an essential parameter in the theoretical framework accounting for the relaxation of the spring's free height at elevated temperatures. The specific value of k₁ can be ascertained by consulting in Table 4. ¹² By cross-referencing the values of a and b for springs of varying dimensions and consulting the aforementioned table, the corresponding k₁ values for Model-15 and Model-30 springs can be acquired. For Model-15 springs, the k₁ value is determined to be 0.1771, whereas for Model-30 springs, it is established as 0.1800.

Table 4.

The spring curvature correction factor of k₁ in different cases.

$\frac{b}{a}/\frac{a}{b}$	$k_1$	$\frac{b}{a}/\frac{a}{b}$	$k_1$	$\frac{b}{a}/\frac{a}{b}$	$k_1$	$\frac{b}{a}/\frac{a}{b}$	$k_1$
1.00 1.05 1.10 1.15 1.20 1.30	0.1406 0.1474 0.1504 0.1602 0.1661 0.1771	1.40 1.50 1.60 1.70 1.80 1.90	0.1869 0.1958 0.2037 0.2109 0.2174 0.2233	2.0 2.25 2.50 2.75 3.00 3.50	0.2287 0.2401 0.2494 0.2570 0.2633 0.2733	4.00 4.50 5.00 5.50 7.00 10.00	0.2808 0.2866 0.2914 0.2946 0.3010 0.3123

The shear modulus of the spring material can be expressed as G.

$$G={\displaystyle \frac{E}{2(1+\mu )}}$$ (9)

where $\mu$ is Poisson's ratio of PFA.

The derivation of the characteristic model for the free height of the Model-15 spring at different temperatures can be achieved by employing substitution and simplification techniques, utilizing equations (2), (4), (5), (6), and (9).

$$f={\displaystyle \frac{(1+\mu )\pi F{D}^{3}n\cos \alpha }{2E(T)k_{1}a{b}^3}+{\displaystyle \frac{\pi F{D}^{3}n{\sin }^2\alpha }{3\cos \alpha E(T)a{b}^3}}}$$ (10)

Similarly, the derivation of the characteristic model for the free height of the Model-30 spring at different temperatures can be obtained by employing substitution and simplification techniques, utilizing equations (1), (4), (7), (8), and (9).

$$f={\displaystyle \frac{(1+\mu )\pi F{D}^{3}n\cos \alpha }{2E(T)k_1{a}^{3}b}+{\displaystyle \frac{\pi F{D}^{3}n{\sin }^2\alpha }{3\cos \alpha E(T)a^{3}b}}}$$ (11)

Experiment

To conduct the free height relaxation tests, the springs should be subjecting to heating within a high-temperature test chamber. Specifically, the experiments utilized Model GW030 as the designated high-temperature test chamber, as seen in Figure 5(a). The upper section of the chamber was equipped with both a load sensor and bracing pieces on both sides. The environmental chamber was centrally positioned within the apparatus, with the spring securely affixed between the lower and pressure platforms for the purpose of heating. A specifically designed testing setup was utilized to evaluate the free height of the springs. This measurement setup included a computer and an image dimensional gauge, as depicted in Figure 5(b). In order to accurately measure the free height of the springs, a KEYENCE IM-8020 image dimensional gauge manufactured by the Japanese company was employed. The spring specimen was positioned on the platform below an observation port, with its controller situated at the bottom. All spring images that were observed and corresponding free height data were displayed on an external computer screen.

The primary aim of this study was to examine the relaxation behavior of springs that are appropriate for cleaning pumps under varying temperature gradients. Four temperature gradients, specifically 90, 120, 150, and 180 °C, were chosen based on real-life operational conditions. The two types of springs were divided into groups of four and subjected to a 7-h test at different temperature gradients.

The adjustment of the opening pressure of a check valve and the pre-compression of its spring are occasionally necessary to accommodate diverse operational circumstances and demands. In order to investigate the influence of pre-compression on the relaxation behavior of the spring's free height, five sets of Model-30 springs were chosen and individually pre-compressed by 5, 7, 9, 11, and 13 mm, respectively. Subsequently, these springs underwent a 7-h heating experiment to observe their behavior.

Results and discussion

Simulations and experimental analyses were conducted based on a cutting force model and an experimental scheme. The outcomes of these analyses are presented in Tables 5–6 and Figures 7–9. Additionally, Tables 5–6 record the initial free height of the spring and its height after heating as separate entities. The relaxation amount, which refers to the change in free height, is subsequently calculated. The experimental data of Model-15 and Model-30 springs at different temperatures without precompression is displayed in Table 5, while Table 6 presents the experimental data of Model-30 springs with fixed temperatures under varying precompression amounts.

Table 5.

Experimental data of Model-15 and Model-30 springs at different temperatures without precompression.

	Group	Initial	90 °C	120 °C	150 °C	180 °C	Variation(mm)
Model-15 spring (mm)	15–1 15–2 15–3 15–4	21.856 20.603 20.603 20.782	20.851	18.533	17.729	17.590	1.0005 2.07 2.874 3.192
Model-30 spring (mm)	30–1 30–2 30–3 30–4	37.960 37.723 37.871 38.164	37.733	37.286	37.169	37.083	0.227 0.437 0.702 0.909

Table 6.

Experimental data of Model-30 springs with fixed temperatures under varying precompression amounts.

	Group	Initial	5 mm	7 mm	9 mm	11 mm	13 mm	Variation(mm)
Model-30 spring (mm)	30–5 30–6 30–7 30–8 30–9	37.059 36.889 36.981 37.030 37.108	34.422	33.522	32.329	30.803	30.520	2.637 3.367 4.652 6.227 6.588

Figure 7.

Temperature-induced experiments and simulations conducted on Model-15 spring without precompression (Heating temperature of 90, 120, 150, and 180 °C; heating time of 7h).

Figure 9.

Variation of precompression amount for Model-30 spring at a fixed temperature (Fixed temperature of 90 °C; Pre-compression values of 5, 7, 9, 11, and 13mm; heating time of 7h).

In order to assess and compare the predicted model with the experimentally obtained changes in spring-free height, a MATLAB script was utilized. The comparison between the simulation and experiment is visually depicted in a comparison diagram. Figure 7 to 9 illustrate various scenarios: namely: the temperature-induced experiments and simulations conducted on the Model-15 spring without precompression; the temperature-induced experiments and simulations performed on the Model-30 spring without precompression; and the variation of precompression amount through experiments and simulations carried out on the Model-30 spring at a fixed temperature.

The orange bar charts in these figures represent the experimental values, while the blue bar charts represent the simulation values. Additionally, the solid orange line represents the experimental curve, while the dashed blue line represents the simulation curve.

Based on the trend curves presented in Figures 8 and 9, it can be inferred that the fluctuation in the free height of the springs escalates as the heating temperatures rise. This phenomenon can be attributed to the occurrence of the thermal expansion of PFA materials at elevated temperatures, whereby the linear expansion coefficient of the polymer material augments with the temperature increment. ¹³ Consequently, when the spring is subjected to high temperatures, the material experiences thermal expansion, leading to the enlargement of the spring coil's dimensions and consequently amplifying the relaxation of the spring's free height. Secondly, based on the preceding experimental findings, it is evident that the elastic modulus of PFA material diminishes as temperature rises. This implies that, under identical stress conditions, materials subjected to higher temperatures will exhibit greater strain, ¹⁴ resulting in the spring deformation and the relaxation of its free height. Consequently, elevated temperatures induce temperature variations within the spring and discrepancies in the thermal expansion rates of its individual coils, leading to thermal stress. ¹⁵ These thermal stresses contribute to an uneven distribution of stress, resulting in coil deformation and relaxation, ultimately leading to an increase in free height relaxation.

Figure 8.

Temperature-induced experiments and simulations performed on Model-30 spring without precompression (Heating temperature of 90, 120, 150, and 180 °C; heating time of 7h.

The increase in the pre-compression amount of the spring is found to result in an increase in the relaxation of its free height, as observed in Figures 7 and 8, while maintaining a constant heating temperature. This can be attributed to a reduction in the initial gap between the coils of the spring due to the increased precompression, leading to a greater deformation and force on the spring when exposed to external forces or thermal expansion. As a result, the spring becomes more prone to free height relaxation. Moreover, it is evident that a spring that has been pre-compressed exhibits a significantly higher level of free height relaxation in comparison to an uncompressed spring, owning to its partially compressed state and increased susceptibility to further expansion and relaxation under high-temperature conditions.

In the analysis of the bar graph presented in Figures 7 to 9, experimental errors for the Model-15 springs are determined to be 0.162, 0.41, 0.146, and 0.266 mm at temperatures of 90, 120, 150, and 180 °C, respectively. Similarly, for the Model-30 springs, the measured errors are 0.085, 0.007, 0.027, and 0.016 mm at the same temperatures. Furthermore, the simulation error between the predicted and actual free height relaxations of the model studied in this paper, without precompression, is found to be less than 0.5 mm. This result serves as evidence for the accuracy of our proposed prediction model for free height relaxation of spring.

When the temperature is maintained at a constant value of 70 °C and the pre-compression value varies between 5 and 13 mm, the spring's free height experiences experimental errors of 0.637, 0.921, 0.719, 0.207, and 1.24 mm respectively. The simulated curve consistently demonstrates a slightly elevated trend in comparison to the experimental curve, which can to attributed to two primary factors: firstly, the inherent complexities inherent within the experiment itself; secondly, the spring's elastic properties may exhibit nonlinearity when subjected to pre-compression force. ¹⁶ As a result, the relaxation behavior of the spring no longer strictly adheres to a simple linear elastic model, but necessitates the consideration of more intricate nonlinear behaviors, leading to notable errors.

Conclusion

In this study, the performance of PFA springs at high temperatures was investigated. The results show that the pre-compressed spring relaxes more easily than the uncompressed spring at the same temperature. The degree of precompression has significant influence on the free relaxation and high relaxation of PFA springs at high temperature. In addition, the prediction model of free height relaxation of PFA cylindrical helical compression springs with rectangular section is established. The model takes into account material properties, structural characteristics, and actual working conditions. In the temperature range of 90–180 °C, the error between the predicted results and the experimental results is less than 0.5 mm, which verifies the accuracy of the model. These results provide an important theoretical basis for us to deeply understand and optimize the performance of springs at high temperatures.

Author biographies

Lihua He, an associate professor, obtained his doctorate degree from Hunan University in 2017. Currently, he is working at Hangzhou Dianzi University where his research primarily focuses on high-performance manufacturing of ultra-clean fluid control components and ultra-precision machining technology.

Xinyu Gao obtained his bachelor's degree from Liaoning Technical University in 2019. Currently, he is pursuing a master's degree at Hangzhou Dianzi University, specializing in the research of properties of PTFE complicated parts.

Jing Ni, a professor, obtained his PhD from Zhejiang University in 2006. He joined Hangzhou Dianzi University in 2006 and currently serves as the dean of the School of Mechanical Engineering at Hangzhou Dianzi University. His primary research focuses on high-performance manufacturing processes, software development, and equipment innovation.

Zhi Cui obtained his master's degree from Hangzhou Dianzi University in 2021 and is currently pursuing a PhD at the same institution. His primary research focuses on high-performance manufacturing technology and precision manufacturing technology for complex components.

Jingbo Sun obtained a master's degree from Hangzhou Dianzi University in 2022 and is currently employed at Zhejiang Fute Technology Co., LTD.

Zefei Zhu, a professor, obtained his PhD degree from Zhejiang University in 1999. Currently employed at Hangzhou Dianzi University, he primarily focuses on conducting advanced research in the fields of fluid mechanics, mechanical dynamics, and fluid machinery.