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Scientific Reports volume 14, Article number: 29139 (2024 ) Cite this article floating cryogenic ball valve
The paper proposes the full linearization of nonlinear characteristics of flow control valves via variable gear ratio design. The inherent nonlinear function of the normalized valve capacity coefficients to the percentages of valve opening is derived from flow experiments. In analysis, it is found that the derivative of the inverse function is the reciprocal of the gear ratio, which is used to generate the corresponding driving/driven gear profiles. The rotation of the driven stem of the flow control valve from a driving knob is altered by the variable gear ratio to achieve fully linearized flow characteristics. The experimental results confirm the fully linearized characteristics of a conventional nonlinear flow control valve with correlation to the linearity with R-squared = 0.9984.
A flow control valve is a mechanical mechanism of adjustable restriction on a fluid flowing through a pipeline. Typically, the mechanism in opening the restriction passage is altered with different degrees so as to vary amount of fluid flow. For example, a simple on/off valve yields the full flow rate at fully opened position and the null flow rate at fully closed position. Essentially, the volume flow rate should be regulated by the flow control valve with an effortless adjustment. The ideal characteristic of a flow control valve is that the valve capacity coefficient should be varied linearly or evenly proportional to the percentage of valve opening. Technically, the flow control valve has a linear valve characteristic function. In practice, the relation of the valve capacity coefficient to the percentage of valve opening is nonlinear1,2. Some of valves are quick opening where the change in flow rate is high at early opening position and some of valves are equal percentage where the change in the flow rate is high at later opening position3. In CFD analysis4, the valve capacity coefficient of the ball valve increases exponentially over percentages of valve opening. This property yields control of flow with difficulty during adjustment. There have been research attempts to improve the characteristics of the flow control valve in different approaches. The modification of the cross-sectional opening area of a ball valve was studied analytically and experimentally in generic nonlinear flow characteristics to extend the design for linearity5. The various shapes of ball hole were investigated to find feasibility on flow characteristics of valves close to linear one6. In the research work7, the V-port shape in a ball valve with angles of 30 degree and 60 degree yielded proportional flow rate to a small range of valve opening. This result had good agreement with the CFD analysis of the triangular opening ball valve8,9. Up to now, the potential shapes yielding the linear valve characteristic have not found yet for a full range of valve opening. The modification of the cross-sectional opening area may not be effective for a linear flow control valve in practice. Therefore, a specific modification on the valve mechanism for a given valve is still undergone to be investigated. For example, a spool opening profile based on three-dimensional ball helix in a ball valve was proposed for linear variation of opening area to regulate cryogenic fuels in a spacecraft10. In the research attempt11, the various curve combined spool profile was also proposed to achieve the linear flow characteristic. However, this design cannot be applied, in general, to the opening mechanisms of conventional flow control valves. This motivation is challenging to be overcome in this study.
Those actions of passive control, mentioned above, were done by fixed control laws. On the other hand, the action of active control is done by variable control laws supported by measurement devices. Comparatively, the active control law is flexible, versatile and efficient. However, the costs of real-time analysis and implementation are high. In the research work of compressible flow12, the averaging pulse width modulation control of on–off valves in multichannel was proposed to linear flow characteristics of airflow. In fact, the closed-loop control with a PWM signal was implemented for flow control13. However, the transient duration to reach the desired flow rates was required in this case. Without anything done on the valve mechanism14, the neural network model of valve characteristic is embedded as a flow meter to be a part of the flow control valve in automatically regulating the volume flow rate at desired one. The data-driven approach leveraged machine learning and artificial intelligence to models and control strategies based on data rather than purely analytical methods. In the research15, the nonlinear characteristics were analyzed by using control Lyapunov functions with PI control to stably enlarge the operating range. It was found that the nonlinear valve controls of flow were implemented on strong nonlinearity of flow characteristics to obtain the desired volume flow rates with considerable effort of control.
The novel passive control, which is developed in this work for conventional nonlinear incompressible flow control valves, has not been published in literature. The proposed methodology is to cancel nonlinearity of the flow control valves by applying the linearization of nonlinear valve characteristics via variable gear ratio design. The original percentage of valve opening, which is coupled with the new percentage of valve opening via a driving gear and a driven gear, is altered according to the determined variable gear ratio such a way that the linearized characteristic of the flow control valve is obtained. Consequently, the nonlinear flow control valve can be implemented as the linearized flow control valve in flow control design and implementation.
Figure 1a shows the experimental setup of water flow measurement in assessing the valve characteristic of a conventional bronze globe-type flow control valve of ½ inch in diameter, which is used in the testing work. A control stem of the flow control valve is driven by a DC stepper motor with a step angle resolution of 1.8° for accurate and repeatable control in the percentage of valve opening. This manipulation yields that the percentage of valve opening can be adjusted with the smallest magnitude of 0.5%. A 0.75 kW centrifugal pump with a normal flow rate of 120 L/min is used to supply water from a 90 L supply tank through a 304 stainless steel pipe of ½ inch in diameter at various flow rates. The bypass valve and the regulating valve are adjusted so that different volume flow rates can be obtained. In each run, the return valve is partially closed in such a way that the pressure drops across a flow control valve can be developed entirely with full flow in the testing pipe under various flow rate conditions. The measuring points of two pressure transducers (PTs) with a range of 0–16 bar and an accuracy of 1.0% full scale are located at the upstream position and the downstream position of the flow control valve. The Arduino Uno R3 board with ATMega328P processor is used to acquire the measurements of water pressure and command the DC stepper motor with PWM signals to obtain the desired percentages of valve opening for each test. The actual flow rates of water are measured by the water flow meter (FT) with the measuring range from 0.5 L/min to 50 L/min and accuracy of 2%. To validate the viability of the proposed methodology, the installation of the developed gears are presented, as shown in Fig. 1b, where the shaft of the driving gear is similarly driven by the DC motor for accurate and precise percentage of valve opening while the driven gear is coupled with the original control stem of the valve. The water flows through the flow control valve, which is embedded without and with the developed gears, so as to obtain the comparative results of the valve characteristics in both two cases.
Test rig: (a) flow measurement devices and (b) installation of proposed gears.
A flow control valve has an important mechanical function as a variable restriction passage of fluid flow under a turbulent flow process when it is operated between fully open position and fully closed position. Physically, it throttles the fluid stream having a greater upstream pressure than downstream pressure. The amount of flow work, which is dissipated with the pressure drop across a flow control valve, is proportional to the kinetic energy in term of a volume flow rate. For incompressible fluids, the flow characteristic of a flow control valve is governed by:
where \(Q\) is the volume flow rate, \(C_{v}\) is the valve capacity coefficient, \(\gamma\) is the specific gravity, and \(\Delta P\) is the pressure drop between the upstream pressure and the downstream pressure through a flow control valve.
It can be seen from Eq. (1) that the volume flow rate is dependent on not only the pressure drop across the flow control valve in a flow process but also the valve capacity coefficient from the functional opening mechanism of the flow control valve. In fact, the valve capacity coefficient is the amount of volume flow rate of water that flows through the flow control valve under the pressure drop of 1 unit. Therefore, the valve capacity coefficient in Eq. (1) can be interpreted as a sizing factor of water flow through the flow control valve.
It should be noted that this valve characteristics under various types of incompressible fluids can be considered with Eq. (1) from assessing the corresponding specific gravity if the valve capacity coefficient for water is known. Without loss of generality, the analysis and experimental investigation in this study are done on water where \(\gamma = 1\) .
With the inherent opening mechanism of the flow control valve, the valve capacity coefficient is experimentally determined for an empirically descriptive model. It is observed that the valve capacity coefficient is not constant but it is dependent on the percentage of valve opening in the opening mechanism of the flow control valve. The valve capacity coefficient can be defined as a function of the percentage of valve opening 5,16,17 in Eq. (2).
where \(p\) is the percentage of valve opening in the range between zero and one.
It is a fact that the upstream pressure and downstream pressure are not expected to be altered in a flow process so as to regulate the volume flow rate. On the other hand, the valve capacity coefficient is the characteristic variable, which is practically adjusted by changing the percentage of valve opening so that the volume flow rate is varied accordingly, when the flow control valve is operated via a valve mechanism.
For each percentage of valve opening, the inherent value of valve capacity coefficient can be determined from flow experiments under various pressure drops before choking behavior as the fluid flowing through the flow control valve is investigated. The slope of the graph between the volume flow rates and the square-rooted pressure drops indicates the valve capacity coefficient according to Eq. (1). The empirical model of the valve capacity coefficient is defined from the plots of the valve capacity coefficient against the corresponding percentages of opening. Figure 2a shows typical relationships between the valve capacity coefficient and the percentage of valve opening in common flow control valves.
Characteristics of flow control valves.
The valve characteristics of the conventional flow control valves are nonlinear, as depicted by the dashed lines. For example, one characteristic from valve manufacturers is quick opening. The valve capacity coefficient of this type increases substantially during the early stage of opening. After that, it increases slightly. To influence on volume flow rate, this quick opening characteristic causes high change in volume flow rate once the flow control valve is operated at the early percentage of valve opening while slight change or no change in the volume flow rate takes place afterward to the end of valve opening. Another characteristic is equal percentage where the valve capacity coefficient increases at low rate during the early stage of opening but it increases exponentially during nearly full opening. In this case, the fluid flows through the flow control valve at low volume flow rate when the flow control valve is operated at the beginning percentage of valve opening whereas, at the ending percentage of valve opening, the flow with high change in the volume flow rate occurs. As depicted in Fig. 2b, those nonlinear characteristics with those high sensitivities within a narrow range on the percentage of valve opening are not effective in use to adjust the valve mechanism in controlling the volume flow rate accurately. The ideal characteristic for the flow control valve, which is indicated in the solid line, is linear where the valve capacity coefficient is constantly proportional to the percentage of valve opening. In other words, the volume flow rate can be regulated uniformly on the whole range of the percentage of valve opening. The larger the percentage of valve opening, the higher the volume flow rate. To overcome the inherent drawback of flow control, the novel methodology of this work in the next section is proposed to modify the nonlinear characteristics of the flow control valves to be the linearized characteristics via variable gear ratio design.
The section is to explain the implementation of variable gear ratio design to modify the nonlinear characteristics of the flow control valves so that the linearized characteristics of the flow control valves can be obtained.
Without loss of generality, define the normalized value of the valve capacity coefficient in the range between zero and one where Eq. (2) is divided by the maximum valve capacity coefficient \(C_{v,\max }\) for the following non-dimensional analysis.
The function \(f^{ - 1}\) is proposed to be an inverse function of \(f\) where the normalized valve capacity coefficient can be mapped to one and only one percentage of valve opening such that Eq. (3) is valid. For example, the n-power function, such as \(c_{v} = p^{\frac{1}{n}}\) and \(c_{v} = p^{n}\) with \(n > 1\) , can be used to express the relationships between the normalized valve capacity coefficient and the percentage of valve opening for the quick opening characteristic and the equal percentage characteristic, respectively. Therefore, the corresponding inverse functions can be defined as \(p = c_{v}^{n}\) and \(p = c_{v}^{\frac{1}{n}}\) , respectively.
With the property mentioned above, it should be noted that
To explain implementation of the variable gear design, the percentage of valve opening is mechanically adjusted by turning the shaft mechanism of the flow control valve. It should be remarked that the concept can be implemented for other mechanisms in changing percentage of valve opening. In this work, the positioning shaft of the flow control valve is driven by a gear. Therefore, the percentage of valve opening is proportional to the angular displacement of the gear \(p\) , which is driven by a driving gear with different angular displacement \(u\) in the range between zero and one, which is regarded as the percentage of valve opening of \(u\) according to the gear ratio \(G\) .
To determine the gear ratio \(G\) , it is proposed to define the relation of the driving angular displacement to the driven angular displacement by applying the inverse function of \(f\) as:
In order to find the gear ratio between two gears of the driven shaft and driving shaft, the function of the driven angular displacement in term of the driving angular displacement is taken by the derivative with respect to the driving angular displacement as:
To find the relationship between Eq. (6) and the gear ratio \(G\) , the derivatives with respect to time are taken to the numerator and the denominator of Eq. (6). Accordingly, the ratio of the angular speed of driven gear to the angular speed of driving gear is inverse of gear ratio between two gears. Therefore, the variable gear ratio is governed by Eq. (7).
The profiles of the driving gear and the driven gear can be formed from the variable gear ratios, which is expressed as the inverse of derivative in Eq. (6), corresponding to both the driving angular displacement \(u\) and driven angular displacement \(p\) , which is determined by using Eq. (5) when the corresponding \(u\) is given.
To confirm the linearized characteristic of the flow control valve, Eq. (5) is applied to Eq. (3). With the property in Eq. (4), the result of the function is equal to the driving angular displacement. Therefore, the normalized valve capacity coefficient with respect to the driving angular displacement can be expressed in a linear function with unity slope, as shown in Eq. (8).
By applying the definition of Eq. (3) to Eq. (8), the valve capacity coefficient can be obtained as:
Equation (9) presents the linearized characteristics of the flow control valve by using variable gear design. The flow control valve is adjusted by changing the driving angular displacement \(u\) . The driving gear turns the driven gear with the driven angular displacement \(p\) according to the variable gear ratio in Eq. (7) in such a way that the normalized valve capacity coefficient is proportional to the driving angular displacement in Eq. (8). Also, the valve capacity coefficient is proportional to the driving angular displacement in Eq. (9). As the result, the nonlinear characteristics of the flow control valve to the percentage of valve opening become the linearized characteristics of the flow control valve to the driving angular displacement.
In the experimental rig of flow measurement, as shown in Fig. 3, a globe valve is used in tests as a common type of industrial flow control valve in flow processes where a typical nonlinear characteristic of the globe valve can be represented as a challenge for the viability of linearization design in this study. Additionally, the globe valve is required to turn with five revolutions from fully closed position to fully open position. Therefore, a planetary gear with gear ratio of 1:5 is inserted between the developed gear and the control stem of the globe valve in order to drive the control stem of the globe valve from five revolutions to one revolution. As mentioned in section "Experimental setup", the regulating valve and bypass valve are mainly adjusted to obtain the volume flow rates of water flowing through the flow control valve while the return valve is adjusted to obtain the pressure drops across the flow control valve under each state of the opening valve. The flow control valve is opened partially according to the percentage of valve opening, as mentioned in section "Characteristics of valves in flow control".
Experimental rig of flow measurement.
Figure 4 shows the experimental results on plots of the volume flow rates against the pressure drops at different percentages of valve opening. It can be observed that the volume flow rate linearly increases with respect to the square-rooted pressure drops for each percentage of valve opening before choking effects take place. Therefore, the slopes of the graphs yield the valve capacity coefficients for given percentages of valve opening, as defined in Eq. (1).
Flow control valve performance of volume flow rate under pressure drops.
Figure 5a presents the graphical relationships between the valve capacity coefficients and the corresponding percentages of valve opening, which are derived from Fig. 4. The nonlinear characteristic of the flow control valve in this experiment is quick opening. In this type, it can be observed that the flow control valve yields high volume flow rates at the early stage of valve opening. After that, the full volume flow rate is almost reached. Therefore, the further adjustment of the flow control valve in percentage of valve opening is not useful in regulating the volume flow rate. On the other hand, the fine adjustment of the flow control valve in the early stage of valve opening is significantly required for accuracy of volume flow rates. With this manner, it is not effective to adjust the flow control valve in practical use.
Nonlinear valve characteristics of globe valve: (a) Valve capacity coefficients, and (b) their normalized values.
To implement the linearization via variable gear ratio design, the valve capacity coefficients are normalized in a range between zero and one according to Eq. (3). Figure 5b illustrates the plots of the normalized values of the valve capacity coefficients against the percentages of valve opening, as presented by the symbols of circles.
In Fig. 5b, the inverse function, which is shown as the solid line, is obtained by fitting a curve to data of exchanging the values of function coordinates from \(c_{v}\) to \(p\) and from \(p\) to \(c_{v}\) , as depicted by the symbols of squares.
It is found that the corresponding inverse function of the fitted curve can be defined as:
with \(a = 0.3\) and \(b = 17.52\) for the best fitted curve of the inverse function.
To obtain the variable gear ratio, the inverse function of the percentage of valve opening with respect to the angular displacement of driving gear is defined according to Eq. (10) as:
The variable gear ratio is determined by using Eq. (7) as:
From Eqs. (12) and (11), respectively, the variable gear ratios and the corresponding percentages of valve opening or the angular displacement of driven gear are plotted against the angular displacement of driving gear, as shown in Fig. 6a. According to the gear ratios in Fig. 6a, the gear profiles of the driving gear and the driven gear are presented by mean pitch circles of both two gears, as shown in Fig. 6b.
Linearization of valve characteristic: (a) variable gear ratio and (b) gear profiles at position when \(p = 0\) and \(u = 0\) .
The proposed gears are installed to linearize the nonlinear characteristics of the globe-type flow control valve. In Fig. 4b, the original characteristics of the flow control valve is quick opening. When the flow control valve is opened at early stage, the sensitivity of the flow rate to the percentage of valve opening is high. As illustrated in Fig. 6a, the high gear ratio from the design is implemented in order to decrease the rotational speed of the driven gear. The percentages of valve opening are slightly obtained when the driven gear is turned while the angular displacement of the driving gear performs as the percentage of valve opening in the adjustment. On the other hand, the sensitivity of the flow rate to the percentage of valve opening is low, when the flow control valve is turned at the final stage. The low gear ratio is applied to increase the rotational speed of the driven gear. The percentages of valve opening are considerably gained. Therefore, the original rate of change in the percentage of valve opening is adjusted to be the new rate via the determined variable gear ratio so that the linearized characteristic of the flow control valve is obtained, as shown in Fig. 7. Figure 7a shows the valve performance on the volume flow rates under various pressure drops at different angular displacements of driving gear. Accordingly, the linearized valve characteristics is shown in Fig. 7b. According to the analytical proof in Eq. (8), the plots of the normalized valve capacity coefficient with respect to the angular displacement of the driving gear is significantly close to the linear function of unity slope with the coefficient of performance R2 of 0.9984. Those experimental results with R2 close to unity confirm that the proposed methodology yields the linearization design for nonlinear valve characteristics via viable gear ratio design.
Valve characteristics with variable gear ratio: (a) valve performance and (b) linearized characteristics.
It is observed in practice that the nonlinear characteristics of the flow control valve is fully linearized by using the variable gear mechanism, which does not require special maintenance and works along with the original flow control valve mechanism in the experiments. The linearized characteristics cause less effort and higher stability than the nonlinear characteristics in order to regulate the flow rate under different fluid types and operating conditions. Therefore, it is advantageous to use the linearized flow control valve for the control systems, which have been implemented in flow processes.
The linearization of nonlinear characteristics of flow control valves via variable gear ratio design is presented analytically. A conventional nonlinear flow control valve is studied in flow experiments to derive the inherent nonlinear function of the normalized valve capacity coefficients to the percentages of valve opening. With the proposed design, the quick-opening relation of the percentage of valve openings to angular displacements of driving gear is defined as the inverse function. The variable gear ratio is determined from the reciprocal of the derivative of the inverse function. To generate gear profiles, the obtained variable gear ratio is determined from the angular displacements of the driving gear and the corresponding percentage of valve opening at the driven gear, accordingly. The experimental results confirm the fully linearized characteristics of the nonlinear flow control valve by adjusting the angular displacement of driving gear through the driven gear on the valve shaft according to the determined variable gear ratio with correlation to the linearity with R-squared = 0.9984.
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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School of Manufacturing Systems and Mechanical Engineering, Sirindhorn International Institute of Technology, Thammasat University, Thammasat Rangsit Post Office, P.O. Box 22, Pathum Thani, 12121, Thailand
Thitipun Thongnueakhaeng, Tawat De Haas & Thananchai
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Thitipun Thongnueakhaeng and Tawat De Haas: Investigation, Data curation. Thananchai Leephakpreeda: Conceptualization, Formal analysis, Investigation, Data curation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing—original draft, Writing—review & editing.
The authors declare no competing interests.
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Thongnueakhaeng, T., De Haas, T. & Leephakpreeda, T. Full linearization of nonlinear flow control valves via variable gear ratio design. Sci Rep 14, 29139 (2024). https://doi.org/10.1038/s41598-024-81002-z
DOI: https://doi.org/10.1038/s41598-024-81002-z
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