Correction of cavitation with thermodynamic effect for a diaphragm pump in organic Rankine cycle systems

Diaphragm pumps are a sort of leakage-proof reciprocating pumps with low flow rate but high head and better efficiency, and can potentially find their applications in organic Rankine cycle (ORC) systems as the feed-pump of organic fluid to the evaporator. A diaphragm pump in an ORC system may suffer from cavitation in the pump suction chamber inevitably when the pump delivers an organic fluid. However, the cavitation performance of the pump has been a little known for organic fluids so far. In the article, the performance of a specific diaphragm pump was determined based on the existing performance charts provided by the pump manufactory in terms of pump rotating speed and inlet liquid pressure for cold water. The net positive suction head required (NPSHr) was predicted by involving thermodynamic effect in cavitation when the pump feeds the organic liquid R245fa to the evaporator in an ORC system at 480rpm rotational speed. The net positive suction head available (NPSHa) was calculated at 100 kPa and 141 kPa inlet liquid pressures, and the corresponding cavitation safety margins were addressed. The subcooling for the NPSHr and NPSHa as well as the safety margin were figured out. Two one-dimensional (1D) mechanical models for motion of the suction valve were built and solved at 480rpm and 100 kPa and 141 kPa inlet pressures. A preliminary experiment was performed to verify the analytical results. It turned out that the NPSHr is reduced to 2.02m from 3.02m NPSHr of cold water due to the thermodynamic effect in cavitation, and the corresponding subcooling is lowered to 8.28 ◦ C from 12.38 ◦ C. 100 kPa but 141 kPa inlet pressure can result in cavitation in the pump. The 1D mechanical models are subject to a rough spatial resolution for the flow field in the suction chamber, hence three-dimensional(3D) numerical simulations of the flow field are desirable. © 2020TheAuthor(s).PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).


Introduction
Cavitation is a phenomenon whereby a proportion of liquid suffers from vaporization when the absolute pressure in a local fluid field is as low as the saturated vapour pressure in a fluid flow system. When the cavitation occurs in a pump, the pump will experience performance degradation, noise, vibration and even potential mechanical damage. To prevent a pump from cavitation, a sufficient large net positive suction head (NPSH), or net positive suction head available (NPSHa), which is the total energy of a fluid at the pump inlet minus the saturated vapour pressure at the pump operating temperature, is required. In other words, a pump has a net positive suction head required (NPSHr), which is the total energy loss and pressure drop from the pump inlet to the impeller or pumping chamber. To avoid cavitation, NPSHa must be higher than NPSHr with a safety margin for a pump. Usually, for a rotodynamic pump, the NPSHa at the pump head * Corresponding author.
E-mail address: Zhibin.yu@glasgow.ac.uk (Z. Yu). has dropped off by 3% in comparison with the head under noncavitation condition is redeemed as the NPSHr of the pump. For a positive displacement pump, however, the counterpart at the pump mean flow rate has dropped off by 3% compared with the flow rate under non-cavitation condition is considered as the NPSHr of that pump Organic Rankine cycle (ORC) is the thermodynamic cycle which employs organic fluids are a sort of fluids with high molecular masses and low atmospheric boiling points. The first solar driven ORC system was created in 1960's by Israeli physicists. Since then, this power generation technique has found applications in heat recovery from lower temperature sources, namely biomass combustion, industrial waste heat, geothermal heat and so on. ORC research field is now well developed and an extensive body of ORC literature has been published. The latest reviews of ORC can be found in Chen et al. (2010), Quoilin et al. (2013), Tchanche et al. (2014), Colonna et al. (2015) and Park et al. (2018) to name a few.
Even though a laboratory-scale pumpless ORC system was proposed by swapping hot and cold sources periodically (Yamada et al., 2013;Gao et al., 2015), or the system was operated by using thermofluidic feed pumps (Richardson, 2016;Shu et al., 2018), in practical terms the addition of a mechanical feed pump remains convenient particularly for large-scale ORC systems (Usman et al., 2015;Rahbar et al., 2015). As a result, various studies have been devoted to different aspects of the ORC pump.
The hydraulic and cavitation performance of a variable-stroke axial piston swashplate pump was measured in a testing loop with organic fluids R11 and R113 at 1750-3000 rpm rotating speeds and 20-80 • C temperatures (Bollina, 1984). The flow rate versus NPSHa curves for R11 and R113 were obtained. Since the cavitation performance of the pump for water was not tested, one was unable to figure out the difference in cavitation performance between the organic fluids and the water. The hydraulic performance of a positive displacement pump with sliding vanes was tested with R11 and R113 and their blend mixture for ORC systems at different rotating speeds and pressure differences (Bala et al., 1985). The quantity of vapour in the pump with R11 was greater than R113. A positive displacement pump with sliding vanes was tested with organic fluid R236fa at 700-1200 rpm rotating speeds and 17-24 • C temperatures (Bianchi et al., 2016). Unfortunately, the pump total efficiency is 35% only. The pumping work in ORC systems was estimated analytically for 18 kinds of organic fluids (Borsukiewicz-Gozdur, 2013), and the work increases with decreasing critical temperature of the fluids. The performance match between expander (a scroll compressor in reverse mode) and feed pump (a reciprocating piston pump) was investigated experimentally by using organic fluid R123 in Yang et al. (2015), and the expander can operate in the whole range of torque when the pump rotational speed is less than 600 rpm (10 Hz) or the expander generates a smaller torque only due to cavitation in the pump caused from a faster rotational speed. To avoid the cavitation the liquid should be subject to a 20 • C subcooling at the pump inlet. The subcooling is the difference in the boiling temperature of a liquid from its actual temperature at the same pressure. In comparison, the subcooling is 13 • C for a reciprocating plunger pump when feeding R245fa in an ORC system (Chang et al., 2015).
The performance parameters of the diaphragm pumps, such as Hydra-Cell G03X, G10X, were measured in a simplified ORC system (without expander of turbine) with organic fluids R134a, R404 and a mixture NH 3 /H 2 O, respectively (Landelle et al., 2015(Landelle et al., , 2017. The measured Hydra-Cell G03X pump volumetric efficiency-pressure curve of R134a differs significantly from that of water provided by the pump manufacture. The NPSHr was 0.24 bar for R134a at nominal rotational speed, compared with 0.36 bar for water in the pump data sheet provided by the pump manufacture. The corresponding tested subcooling was 4.4 • C for that pump. For organic fluid R245fa, an empirical model was proposed to predict the non-cavitation performance of the diaphragm pump of Hydra-Cell D/G-10X pump in an ORC system (D'Amico et al., 2018).
The overall performance of three pumps such as multistage centrifugal pump, diaphragm pump, and roto-jet pump was measured in thermal test rig loop of R245fa in various rotational speeds (Yang et al., 2018). Three pumps share about the same overall efficiency, and the diaphragm pump is more suitable to a lower heat capacity ORC system, and the centrifugal pump is better for a higher heat capacity system, but the roto-jet pump is in between. The performance of a multistage centrifugal pump was tested in a thermal test rig loop of R245fa under various condensation conditions, and the condensation conditions affect the performance of the multistage centrifugal pump slightly (Yang et al., 2019). The ORC system was tested when a reciprocating plunger pump and a centrifugal pump were installed, respectively. The system with the plunger pump is more efficient 12.51% than that with the centrifugal pump, and the plunger pump is more suitable to the ORC system studied (Wang et al., 2020). A peripheral pump model PK70 with 0.8 kg/s flow rate and 65 m head was tested in transporting HEF-7100 organic fluid in an ORC system with regeneration and without regeneration in (17-57) • C and 2900 r/min Kaczmarczyk et al. (2016). The pump head is 41-43 m at (0.16-0.17) kg/s mass flow, and HEF-7100 vapour occurs in the pump. Transient CFD approach is launched to analyse fluid dynamics of the pumping system of a regenerative labscale micro-ORC system consisting of the liquid receiver, pipeline, external gear pump and mass flow meter (Casari et al., 2019), and the results indicated that cavitation phenomenon affects the pump and ORC system operation.
Based on the outcomes of the distinct studies mentioned above, most of them were focused on the pump overall performance to match a particular ORC system and cavitation in an ORC system can influence pump and system operation. The pump cavitation has been less known, especially in terms of thermodynamic effect in cavitation of organic fluids so far.
In the article, the cavitation performance of a diaphragm pump of G20-E model when delivering the organic fluid R245fa was investigated analytically. A NPSHr correction method was formulated based on the thermodynamic effect in cavitation, and one-dimensional(1D) motion of the suction valve was simulated numerically by using two mechanical models proposed to identify the cavitation at 100 kPa and 141 kPa pump inlet pressures along with preliminary experiments. The NPSHa values at those inlet pressures were calculated and cavitation safety margin was addressed, the subcooling for the NPSHr, NPSHa and safety margin was worked out. The study will provide useful information for both selection and safe operation of the diaphragm pump in ORC systems.

Pump operational parameter extraction
A positive displacement diaphragm pump of G20-E model, manufactured by Wanner Engineering Inc, was selected to circulate organic fluid R245fa in a small-scale ORC system, as shown in Fig. 1a. The pump liquid end structure is sketched conceptively in Fig. 1b. The pump mainly consists of eight mechanical components: drive crank, connecting rod, piston, piston casing, diaphragm, inlet and discharge valves and valve housing. The diaphragm is driven by the piston via the hydraulic oil in the left chamber of the diaphragm. When the piston moves to the left, the volume of the chamber between the two valves and the diaphragm is expanded, the pressure in the chamber is depressed, the suction valve is opened and the discharge valve is closed, drawing liquid into the chamber. In this period of time, the pump is in suction stroke. As the piston moves to the right, the volume of the chamber is reduced and the pressure is increased, the suction valve closes but the discharge valve opens, the working fluid present in the chamber is expelled out of the pump. In this process, the pump is in discharge stroke.
where n is pump rotating speed, rpm, p is pump nominal operational pressure, bar. The Q and NPSHr predicted with the models in Eq. (1) are compared with the data in the pump specification charts in Fig. 2. The two parameters are affected mainly by pump rotating speed. Note that these mean parameters in the pump specification charts were measured by using cold (around 20 • room temperature) water. The pump operational pressure in our experiments (p < 3.5 bar) has little influence the mean flow rate.
The mean flow rate is related to the flow induced by the periodic motion of the diaphragm in a diaphragm pump. The diaphragm is a flexible rubber material, therefore, fluid-structure interaction analysis or visualization experiments are required to completely understand its motion and deformation characteristics (van Rijswick et al., 2016). Since our aim in the paper is to explore the possibility of cavitation in the suction valve, the actual diaphragm motion is out of the scope of the paper. Here we just use an idealized diaphragm motion and deformation pattern to estimate the periodic flow rate through the suction valve in s = 4.98 mm working stroke measured.
The idealized diaphragm motion and deformation is demonstrated in Fig. 1c. In the motion, the diaphragm retains flat cone shape with two radii R 1 = 5.75 mm, R 2 = 15.75 mm and a variable cone height/displacement x, x ∈ [0, s]. As x = s/2, the diaphragm becomes a circular plane. The top surface with the radius R 1 = 5.75 mm is in motion with the same speed as the piston, v. The cone surface stroke is assumed to reduce to zero at R 2 = 15.75 mm from s at R 1 = 5.75 mm linearly. Thus, the stroke profile along the diaphragm is expressed as Taking the diaphragm shape in the beginning of suction stroke, i.e. t = 0 or ϕ = ωt = 0, as reference configuration, and neglecting the 2nd-order terms (Sahoo, 2006;Lee et al., 2015;Wu et al., 2019), the diaphragm displacement equation is written as The diaphragm moving velocity can be determined by taking the first derivative of x with respect to t, Since the suction stroke occurs in ϕ = [0, π], there is a flow rate in the suction pipe during this period; otherwise, the flow rate is zero in the pipe. The instantaneous flow rate in the suction pipe during ϕ = [0, π] is calculated by and the equation can be simplified into the following form /6. The mean flow rate in the suction pipe in the period of ϕ = [0, 2π ] should be calculated by If the diaphragm geometrical parameters, R 1 and R 2 , piston stroke s and rotational angular speed ω are known, the instantaneous and mean flow rates can be estimated with Eqs. (6) and (7), respectively.

Method for NPSHr correction
Usually, the NPSHr presented in Eq.
(1) is measured by using cold water in the pump manufacture's laboratory. This NPSHr is reduced when the pump delivers an organic fluid such as R245fa because of thermodynamic effect in cavitation process. The reduction effect on NPSHr in an ORC system is not commonly known.
The thermodynamic effect in cavitation was introduced in 1945 (Fisher et al., 1945). The thermodynamic effect results in heat transfer between a cavitating spot and its surrounding liquid. During the heat transfer, the surrounding liquid supplies an amount of heat via a heat flux to the cavitating spot to perpetuate the cavitation. The heat flux is driven by the difference in temperature from the surrounding liquid to the spot. This fact suggests the temperature of the surrounding liquid T l is higher than the temperature in the spot T v , i.e. there is a temperature depression from the liquid to the cavitating spot, ∆T (= T l − T v ). Ideally, the pressure in the cavitating spot is equal to the saturated vapour of the liquid at T v . Clearly, a vapour depression ∆p (=p v There is a link between ∆T and ∆p through a known slope of the saturated vapour pressure-liquid temperature curve, i.e. ∆p = (dp v /dT l ) ∆T . Since it is quite difficult to measure T v or p v (T v ), p v (T l ) has been adopted to calculate NPSHr in experiments for convenience. In reality, NPSHr should be based on p v (T v ). NPSHr correction means that a correction is applied to a p v (T l )-based NPSHr to obtain the p v (T v )-based NPSHr. The key thing in the correction is to find a temperature depression ∆T or pressure depression ∆p for a specific liquid pumped in terms of the similarity of cavitation for different liquids, i.e. thermodynamic factor.
Roughly, there are five correction methods for thermodynamic effect in cavitation, namely, B factor, B 1 factor, B t factor, Σ factor and NPSH formula-based approach. A comprehensive and critical review of these methods is present in Appendix. Σ factor method in Arakeri (1999) is adopted to perform NPSHr correction for the diaphragm pump here. In comparison with the other methods, Σ factor method can decide whether a pump needs cavitation correction or not based on the Σ factor value of the pumped liquid and the pump operating condition. A correlation curve of NPSHr correction can easily be generated in terms of a reduced liquid temperature. This method is straightforward and easily implemented in any applications. In the following, this method is adopted to predict NPSHr when the diaphragm pump delivers the organic fluid-R245fa.
The method relies on a cavitation criterion that is composed of a dimensional thermodynamic parameter Σ and an impeller cavitation performance variable Λ of centrifugal pump such as: In the boiling state, the liquid temperature remains unchanged without thermodynamic effect. In the cavitation state, the thermodynamic effect occurs and the temperature in the cavity is lower than the surrounding liquid during cavitation. According to this fact, a Σ and Λ-based method was developed in Arakeri (1999) to predict NPSHr (T l ) of hot water or hydrocarbons or cryogens. The procedure is as follows: (1) Determine an appropriate liquid temperature T * l with the condition Λ = Σ for a given pump and liquid pumped with the known V ∞ , σ l and D; (2) If the pump operating liquid temperature yields T l ≤ T * l , then there is no thermodynamic effect in cavitation, no NPSHr (T l ) correction is applied; (3) If the temperature is in line with T l > T * l , then NPSHr (T l ) correction is required. The NPSHr (T l ) experimental data in Salemann (1959), Stepanoff (1961) and Spraker (1965) was reprocessed by Arakeri (1999) and the correction ∆NPSHr was correlated to the reduced temperature ], T c is the critical temperature of the liquid. The ∆NPSHr − T R curves and the corresponding scattered experimental data are illustrated in Fig. 3.
The hot water data can be best fitted by a linear model, but those of the other liquids are better represented by a 3rd-order polynomial, the corresponding regression equations are read as The correction ∆NPSHr is plotted as a function of the reduced liquid temperature T R based on the NPSHr experimental data in Salemann (1959), Stepanoff (1961) and Spraker (1965) of water, butane, Freon-11 and methyl-alcohol. Noting ∆NPSHr = ∆p v /ρ l g − (∆p v /ρ l g) ref , the relationship between NPSHr (T l ) and NPSHr (T l ) ref is given by If one knows NPSHr (T l ) ref of a liquid at a temperature, then ∆NPSHr of the liquid at another temperature can be calculated, finally one can predict NPSHr (T l ) of that liquid at that temperature. Usually, NPSHr (T l ) ref is for cold water in industry, and ∆NPSHr is for a liquid other than water. Eq. (11) allows one to have NPSHr (T l ) of that liquid.
Since it yields ∆NPSHr ≥ 0, NPSHr (T l ) of that liquid is always smaller than NPSHr (T l ) ref of cold water. Nevertheless, a pump is subject to a lowered NPSHr when pumping hot water, hydrocarbons or cryogens or organic fluid compared with that when handling cold water.

NPSHr correction implementation and subcooling
The diaphragm pump lifts the organic fluid-R245fa in the experimental loop from T l = 8.4 • C (281.6 K), p 1 = 1.41 bar (141 kPa) to p 2 = 3.73 bar (373 kPa) at n = 480 rpm rotative speed. The characteristic liquid velocity in the suction chamber is V ∞ = 0.19 m/s, the characteristic diameter of the chamber is D = 18 mm. The head across the pump is H = (p 2 − p 1 ) /ρ l g = 17.1 m, where ρ l = 1382.4 kg/m 3 . According to Eq. (1), NPSHr (T l ) ref is 3.02 m at 480 r/min. The cavitation number is approximately The specific heat capacity c p , thermal diffusivity α, specific volume v l and latent heat L of the liquid R245fa, vapour pressure p v , specific volume v v of the vapour R245fa are best fitted by the following expressions in terms of liquid temperature T l ranged in 180 K and 420 K where the thermal conductivity of the liquid R245fa is a constant of k = 0.087 W/(m K) in the α expression. The thermodynamic parameter Σ of the liquid R245fa was calculated with the related parameters represented by Eq. (12) and exhibits the following regression equation Based on the condition Λ = Σ, the appropriate liquid tem- below the pump operating temperature T l = 281.6 K, shown in Fig. 4 where the Σ of water is illustrated for comparison; consequently, the thermodynamic effect must be taken into account when determining NPSHr (T l ).
Considering the critical temperature of the liquid R245fa T c = 427.01 K, the reduction temperature of the liquid is T R = 0.3696, and the corresponding NPSHr correction ∆NPSHr = 1.00 m is calculated with the second expression in Eq. (9). Eventually, when the pump transports the liquid R245fa, the corresponding NPSHr should be predicted by the following Next, the effect of inlet pressure on the pump cavitation performance will be examined. Considering two inlet pressures p 1 = 1.0 bar (100 kPa), 1.41 bar (141 kPa), the NPSHa for these inlet pressures are evaluated with The NPSHa values predicted with the two inlet pressures are tabulated in Table 1. The NPSHa at p 1 =100 kPa is 1.62 m only and smaller than NPSHr (T l ) = 2.02 m specified in Eq. (14). This means that the inlet tank is unable to supply significantly high pressure for the diaphragm pump to prevent cavitation in its suction valve chamber. Specially, this NPSHa value does not meet the NPSH criterion for non-cavitation operation used generally in the pump industry, namely (Deniau, 2009) where 0.5 m is a safety margin. Thus, the pump should suffer from cavitation problem at p 1 = 100 kPa. The NPSHa at p 1 =141 kPa is Table 1 NPSHa, NPSHr and subcooling of the diaphragm pump at two inlet pressures and 8.4 • C pumping temperature for liquid R245fa at 480 rpm rotating speed. 4.76 m and is higher than NPSHr (T l ) with a 2.64 m safety margin. As a result, p 1 =141 kPa is a more suitable inlet pressure for the diaphragm pump to avoid cavitation. Note that even though the NPSHr is reduced from that of cold water, the vapour pressure of the R245fa is 77.05 kPa at 8.4 • C (281.55 K) operating temperature, and much higher than 2.34 kPa of water at 20 • C temperature. Therefore, the pump inlet pressure for the liquid R245fa should be 77.05−2.34 = 75.71 kPa higher than the inlet pressure for cold water to prevent cavitation occurrence.
For refrigerants, cavitation can be suppressed by subcooling the fluid to bring the refrigerant saturation down (Sanger, 1968). The NPSHa and NPSHr in Table 1 and cavitation safety margins above can be converted into the subcooling available ∆T a and subcooling required ∆T r and subcooling safety margin by using the relationship ∆T r = (dT l /dp) ρ l gNPSHr (17) where dT l /dp is the R245fa vapour temperature-pressure curve slope at 8.4 • C operating temperature of the pump. If NPSHr in Eq. (17) is replaced with NPSHa and the cavitation safety margin, respectively, then the degree of subcooling can be determined with Eq. (17). The corresponding values of ∆T a and ∆T r are listed in Table 1. The 0.5 m and 2.64 m cavitation safety margins are for 2.05 • C and 10.82 • C subcooling, respectively.

1D analysis of suction valve motion
Cavitation can occur in the suction valve of a diaphragm pump if the valve is designed improperly or the inlet pressure is not high enough during operation of the pump. 1D analysis of suction valve motion may be helpful to identify cavitation issue during operation of a diaphragm pump. The suction valve structure of the diaphragm pump is illustrated in Fig. 5a, which includes four elements, i.e. valve seat, valve, spring and retainer, and its mechanical model is present in Fig. 5b.
Initially, the valve is closed, i.e. contacts with the seat due to the force generated by the spring in a pre-compressed length h 0 = −0.5 mm. As the diaphragm moves to the left gradually; the volume of the pumping chamber is increased, and the liquid pressure is depressed. The valve will open suddenly, i.e. moves off the seat as soon as the opening force due to the pressure difference across the valve is larger than the resistance generated by the spring. After the valve's opening reaches the maximum, the valve will be pulled back to the seat by the spring force. In this section, this process is simulated with 1D mechanical model to determine pressure drop during the valve opening and assess risk of cavitation.
Since experimental data of valve motion in the diaphragm pump is unavailable to us, two mechanical models have to be adopted to make sure a similar valve motion characteristic achieved with them. The first mechanical model is the 1st-order model in which both the drag force imposed on the valve by the fluid and the valve mass are neglected. The second is the 2ndorder valve motion equation with the two factors involved. In both the mechanical models, the fluid is considered incompressible and inviscid, and the fluid flow in front and rear of the valve and through the gap between the valve and the seat is 1D. The historical effects of fluid viscosity are taken into account with flow coefficient through the valve gap and drag coefficient of the valve body.
In the 1st-order mechanical model, the governing equations of valve motion consist of fluid continuity equation and force balance equation of the valve. The continuity equation for incompressible fluids reads as where q is the instantaneous volumetric flow rate in the suction pipe, q g is the flow rate through the gap between the valve and the seat, and q v is the flow rate along with the moving valve. q g and q v are expressed as where C q is the flow coefficient through the gap, d h is the seat diameter, d h = 18 mm, h is the valve lift, θ is the half of cone apex angle, ∆p is the pressure difference across the valve, ρ l is the density of the liquid pumped, d v1 is the diameter of the upper base of the valve, d v1 = 14 mm, v = ωdh/dϕ is valve moving velocity, where ω is the angular speed of diaphragm reciprocating motion, ω = πn/30, n is the rotational speed of the pump driver.
The force balance equation of the valve is the application of Newton's second law of motion to the valve. When the valve mass and drag force are ignored and the equation is written as where k is the spring stiffness. Substitute Eqs. (6), (18) and (19) into Eq. (20), the governing equation of the valve motion is obtained Eq. (20) is a 1st-order ordinary differential equation and can be solved for valve lift h numerically by using the 4th-order Runge-Kutta method (Griffiths and Smith, 1991) with the following initial condition The valve is subject to a pressure reduction in the gap at ϕ = 0 as the diaphragm starts to move. The pressure reduction can be obtained from Eqs. (20) and (22) as follows In Eq. (21), the flow/discharge coefficient C q through the gap between the valve and the seat is the key factor determining the valve opening. There is no such a flow coefficient measured on an actual valve in diaphragm pumps presently. Thus, based on the empirical correlation of hydraulic loss coefficient ξ for conical/poppet valves (Idelchik, 1966), the correlation of C q has to be proposed by employing the relationship C q = 1/ √ ξ , as such Finally, Eqs. (21), (22) and (24) form a 1st-order mechanical model for clarifying characteristics of the valve motion in the diaphragm pump.
If the valve mass and fluid drag force are considered, the force balance equation of the valve is expressed as where m v is the mass of the valve, F R is the drag force imposed by the pumped liquid, the drag force is related to the draft coefficient, C D , and the squared velocity difference between the flow in the gap and the valve itself, as well as the projection area of the valve, i.e.
where v g is the velocity of the pumped liquid through the gap, v is the valve velocity, d v2 is the lower base diameter of the valve, d v2 = 21 mm, the drag coefficient C D = 0.65 is after (White, 2011).
Combining Eqs. (18) and (19), (25) and (26), respectively, the governing equations for the valve motion are in the following forms To facilitate implementing the 4th-order Runge-Kutta method, Eq. (27) is decomposed into a set of three 1st-order ordinary To guarantee the solution of Eq. (28), the following initial conditions are needed Furthermore, the valve is subject to a pressure reduction in the gap at ϕ = 0 as the diaphragm starts to move with the maximum acceleration. Taking the derivative of the second expression in Eq. (28), we have Because of F R = 0 and h = 0 at ϕ = 0, Combining Eq. (29) and the last expression in Eq. (28) results in the pressure reduction After the governing equations for the valve motion Eq. (21) or (27) are solved, the pressure drop across the gap is available. The pressure at the gap outlet can be calculated at a known pump inlet pressure p 1 , and the expression is written as The equations above were coded in MATLAB 2018b. To obtain accurate solutions, the suction stroke ϕ = π(180 • ) is divided into 300 elements, and the governing equations are integrated over the elements by using the 4th-order Runge-Kutta method. The input data to the code is listed in Table 2 and the corresponding results are illustrated in Fig. 6.
In the figure, the results predicted with the 2nd-order mechanical model such as valve lift, velocity, force and pressures, exhibit a slightly pulsatile behaviour in high frequency during the opening process. The valve shows more delay in the valve lift and velocity during the opening and closing processes based on the 1st-order model than the 2nd-order model. However, the highest valve lift is very similar in both the models. The drag force is predicted with the 1st-order and 2nd-order mechanical models. minor in comparison with the spring force and the force induced caused by the pressure difference across the valve.
The minimum pressure at the outlet of the gap predicted by the 2nd-order model at ϕ = 0 • is the lowest, but still higher than the liquid-vapour pressure, suggesting cavitation may not occur. The minimum pressure predicted by the 1st-order mechanical model is always higher than the pressure predicted by the 2nd-order mechanical model at ϕ ≤ 130 • .
The fluid pressure at the gap outlet depends on flow rate (phase angle of the diaphragm). For the 2nd-order mechanical model, the fluid pressure is the lowest at ϕ = 0 • imposed by the initial condition Eq. (30). As ϕ > 0 • the pressure rises sharply, and then declines slowly until the maximum flow rate, and finally rises steadily. This trend suggests cavitation can occur at the beginning of the suction stoke and the maximum flow rate in the diaphragm pump. For the 1st-order mechanical model, the lowest pressure is predicted at the maximum flow rate only.
Based on the results, it is understood that even though the valve motion characteristics can be captured by using the 1stand 2nd-order mechanical models, the pressure drop across the gap is predicted by using the empirical flow coefficient. The local minimum fluid pressure in the gap, which is responsible for cavitation inception and development, cannot be captured, because the flow details through the gap cannot be resolved with the very simple 1D flow model. Nevertheless, CFD simulation in the pump suction chamber must be carried out when the diaphragm pump is handling liquid R245fa.

Preliminary experiment
A test rig, which is demonstrated in Fig. 7, was built to clarify the cavitation performance of a diaphragm pump in an ORC system. The rig is composed of the evaporator, expander, condenser and diaphragm pump. The liquid receiver accommodates the liquid R245fa discharged from the condenser and feeds it to the pump. The DaqVIEW was used to acquire experimental data from the temperature and pressure and flow rate sensors.
The expander (Air Squared: E15H022A-SH) works at 70 • C inlet temperature and 4 bar inlet pressure, 50 • C outlet temperature and 1 bar outlet pressure at 0.022 kg/s mass flow. The expander is directly coupled to a single-phase generator (Voltmaster: AB30L WEIPU) with any generated power being dissipated by three 400 W lamps.
The heating loop produces 93 • C water for the evaporator at 7 l/min flow rate, the cold loop temperature has a 3 • C variation between 7 • C and 10 • C with a cooling flow rate of 10 l/min.
The diaphragm pump (Hydra-cell G20 EDSPHFEHG) was connected to a three phase 6 pole motor (AEG: AM 80z AA 6). A frequency inverted was used to vary the pump supply frequency (AEG: AM 80z AA 6).
The monitored pressure and temperature time-history curves are illustrated in Fig. 8 at the pump inlet and outlet when the pump is running at 480 rpm under 1.41 bar mean inlet pressure condition. The pump outlet pressure seems noisy, and the inlet pressure, inlet and outlet temperature profiles exhibit a periodic oscillation (the periodicity ≈ 10 min). In Section 2.3, the 1.41 bar mean inlet pressure corresponds to a safety margin as high as 2.64 m, thus the oscillation is unlikely attributed to cavitation in the suction chamber.
The monitored temperature time-history curves at the inlet and outlet of the evaporator and condenser are plotted in Fig. 9a and b, and the pressure and temperature profiles at the inlet and outlet of the expander are also involved for an additional examination. Clearly, it is only the temperature profile at the condenser inlet that can exactly match the temperature profiles at the pump   inlet and outlet and the pressure profile at the pump inlet, respectively. Nevertheless, the temperature profile at the condenser inlet should be responsible for the pressure and temperature oscillation at the pump inlet and the temperature fluctuation at the pump outlet. Further work, including condenser inflow condition inspection and the pump performance measurements under various mean inlet pressures should be conducted.

Discussion
The performance of a specific diaphragm pump which is potentially applied to ORC systems was estimated based on the existing performance charts produced by the manufacture in terms of pump rotating speed and inlet liquid pressure when pumping cold water. In the article, the cavitation performance of a specific diaphragm pump was estimated by considering thermodynamic effect in cavitation of organic liquid R245fa based on the existing NPSHr correction method for centrifugal pumps but with a regression equation fitted. Such an analytical estimation for diaphragm pumps has not appeared in the literature so far. The study presented here can be meaningful in the selection of diaphragm pumps for ORC systems.
The subcooling estimated for the diaphragm pump-Hydra-Cell G20-E model is 8.28 • C and 12.38 • C respectively with and without the thermodynamic effect, as listed in Table 1. There is not directly experimental evidence to support these predictions presently. Fortunately, two experimental observations may be favourable to the estimations. For example, when pumping organic liquid R245fa, for a reciprocating plunger pump, the subcooling is 13 • C (Chang et al., 2015). For liquid R134a, the tested subcooling of the diaphragm pump-Hydra-Cell G03X model was 4.4 • C (Landelle et al., 2015(Landelle et al., , 2017.
In practice, except increasing organic fluid container pressure and subcooling, as common measures, a booter pump is often installed in front of the diaphragm pump to raise the inlet pressure of the pump to prevent cavitation.
The cavitation in positive displacement reciprocating pumps is related to liquid acceleration head and water hammer effect in the suction pipeline or the interaction between naturally pulsating reciprocating pumps and the piping system (Sahoo, 2006;Parry, 1986;Vetter and Schweinfuter, 1987;Wachel et al., 1989;Singh and Able, 1996). Whether the piping system is concerned in the cavitation depends on pump power density index (PDI) (Singh and Able, 1996) where n is strokes per minute per liquid chamber, s is stroke in inches, H is pump head rise in ft. If PDI is less than 0.02, piping system has negligible effect on the cavitation in the pump, so the liquid acceleration head and water hammer effect in the suction pipeline can be ignored (Singh and Able, 1996). For the diaphragm pump studied here, its PDI is 0.01, therefore the suction pipeline can be neglected in the mechanical model in Section 3.1. In Fig. 6a, the valve lift does not become zero when the suction stroke terminates (ϕ = 180 • ). This suggests there is a time lag in the suction valve closing. The models do exactly capture the lag effect.
High-speed camera-based flow visualization observations illustrated that cavitation occurs at the beginning of suction stroke owing to the sudden expansion of suction chamber, but it disappears after the chamber is filled with the liquid and its vapour bubbles collapse off the valve surface. This cavitation is an intrinsic property of reciprocating pumps and harmless to the valve function and pump performance (Opitz and Schlücker, 2010;Opitz et al., 2011). As the suction head is declined, the other cavitation can appear at the maximum flow rate and the vapour bubbles collapse at a low flow rate. This sort of cavitation is flow induced cavitation and can impair the valve function and pump performance, and it must be avoided in reciprocating pump operations (Opitz et al., 2011).  in suction stroke at the inlet pressure of p in = 84 kPa. For the 1st-order mechanical model, the pressure profile has the minimum at the maximum flow rate only, suggesting the flow induced cavitation there. For the 2nd-order mechanical model, however, the pressure profile is with two minimum pressures each at the beginning of suction stroke and at the maximum flow rate, indicating the cavitation due to expansion and flow induced cavitation in the stroke. This outcome seems in agreement with observations made in Opitz et al. (2011). In this context, the 2ndorder mechanical model is more significant than the 1st-order mechanical model. Note that the minimum pressure values may be overpredicted because the model is simple 1D. Nevertheless, 3D cavitating flow simulations in the suction chamber of the diaphragm pump are on demand to capture fluid flow details.
The flow coefficient through the gap between the valve and the vale seat is important to the mechanical models in Section 3.1. There is another flow coefficient based on the flow resistance factor ξ present in Vetter et al. (1989) where Re is Reynolds number of the gap, ν is liquid kinematic viscosity. Based on the 1st-order mechanical model, a comparison of the flow rate coefficients in Eqs. (24) and (34) is illustrated in Fig. 11. Clearly, the value predicted with the correlation of Eq. (34) seems to be too small. Honestly, the paper suffers from a few limitations. The compressibility of the liquid pumped is not considered. As a result, the diaphragm membrane can instantly respond to the motor rotation without dead angle. The discharge stroke is not involved in the paper, thus the time lag in the discharge valve closing, which can delay the suction valve to open, is not taken into account.
For the analytically approximate instantaneous and mean flow rate formulas, an idealized deformation and shape was assumed in the diaphragm in Section 2.1. Actually, the diaphragm likely experiences a complicated deformation pattern (Vetter et al., 1995) and local buckling (van Rijswick et al., 2016;Vetter et al., 1995). Additionally, the diaphragm was hydraulically driven, as show Fig. 1b. And an underlying assumption that the hydraulic oil Fig. 11. The flow coefficients determined by two empirical correlations based on Idelchik (1966) and Vetter et al. (1989), respectively. movement is the same as the piston without any oil leakage was held when the flow rate formulas were deduced. Nevertheless, the instantaneous and mean flow rate formulas provided in the paper need to be corrected by using experimental data in future.
The NPSHr predicted with thermodynamic effect in cavitation has not been validated with experimental observations. Hopefully, the validation will be available in near future.
The experimental data that are used to correct NPSHr are essentially measured from a variety of centrifugal pumps rather than from reciprocating pumps. Naturally, whether those data are applicable to reciprocating pumps is questionable and needs more experimental validations in future.

Conclusion
In the article, particular attention was devoted to NPSHr prediction by considering thermodynamic effect in cavitation when the pump feeds the organic liquid R245fa to the evaporator in an ORC system. A comprehensive and critical review was conducted on the thermodynamic effect and the corresponding correction methods for NPSHr of centrifugal pumps. Then the method in Arakeri (1999) was selected and updated to correct the NPSHr of cold water at 480 rpm pump rotational speed to obtain the NPSHr of the liquid R245fa. The 1st-and 2nd-order 1D mechanical models for motion of the suction valve were deduced and solved at 480 rpm and 100 kPa and 141 kPa inlet pressures to characterize the cavitation behaviour of the valve. A preliminary experiment was carried out to confirm the analysed results. The conclusions made include: (1) When the diaphragm pump delivers the 8.4 • C liquid R245fa at 480 rpm, its NPSHr is reduced to 2.02 m from 3.02 m NPSHr of cold water due to the thermodynamic effect in cavitation, and the corresponding subcooling is reduced to 8.28 • C from 12.38 • C. 100 kPa inlet pressure can lead to cavitation in the pump valve but 141 kPa cannot.
(2) The valve lift, velocity, force and pressures predicted with the 2nd-order mechanical model demonstrate a slightly pulsation in the opening process, while the in the valve lift and velocity based on the 1st-order model exhibit a more delay in the opening and closing processes.
(3) The highest valve lift is very close in both the 1st-and 2nd-order mechanical models; the drag force is minor in comparison with the spring force and the force caused by the pressure difference across the valve.
(4) The pressure profile at the outlet of the valve gap predicted by the 2nd-order mechanical model is with two minimums each at the beginning of suction stroke and at the maximum flow rate, showing the cavitation owing to the initial expansion and the latter cavitation induced by increasing flow rate in the stroke observed (Opitz et al., 2011). The 1st-order mechanical model, however, can capture the minimum pressure at the maximum flow rate only.
(5) The minimum pressure estimation in the 1st-and 2ndorder mechanical models relies completely on the very limited flow coefficients found in the literature, thus the minimum pressure values are inaccurate or overpredicted; consequently, both the models cannot predict the valve cavitation behaviour exactly.
Nevertheless, 3D cavitating flow simulations of organic fluids in the suction chamber of the diaphragm pump are desirable to accurately clarify the valve cavitation performance and capture the corresponding flow details in future.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements
The work was supported by EPSRC in the UK in terms of grants such as EP/N020472/1, EP/N005228/1, EP/P028829/1 and EP/R003122/1, the support was acknowledged indeed.

A.1. Thermodynamic effect
The thermodynamic effect came into a subject in 1945's (Fisher et al., 1945). In that pioneer work, the vapour pressuretemperature relationship of a liquid yields an exponential function, there is a thermal equilibrium between liquid and vapour under adiabatic conditions during cavitation, and the vapour is considered idealized gas, i.e.
where a and b are the constants for the vapour pressuretemperature relationship, p v is the saturated vapour of liquid, c p is the specific heat capacity of liquid, L is the latent heat of liquid, M is the molar mass of the vapour, R is the universal gas constant, R = 8.3145 J/mol K, T l is the liquid temperature pumped, W l is liquid mass considered, W v is the vapour mass, v l and v v are the specific volume of the liquid and vapour, respectively, v l = 1/ρ l and v v = 1/ρ v , ρ l and ρ v are the density of the liquid and vapour, ∆T (=T l − T v ) is temperature depression in a cavity compared with the temperature of the liquid around the cavity, is vapour depressions in a cavity compared with the temperature of the liquid around the cavity, T v is the vapour temperature.
are the liquid and vapour volumes, then, the ratio of the specific volume of liquid-vapour mixture v vl to the specific volume of liquid v l is expressed as where v vl is the specific volume of the vapour-liquid mixture, The parameter ∆p v v l is the loss of head owing to restriction of flow-through cross-section imposed by the cavity. The ratio v vl /v l was used to characterize the extent of cavitation or the similarity of cavitation. When a pump delivers different liquids at the same temperature, ∆p v v l should be in the following form for the same v vl /v l This is the first correlation of vapour pressure depression in terms of thermodynamics in cavitation. The pressure depression ∆p v v l is determined mainly by latent heat, specific heat and slope of the p v − T l curve of the liquid pumped.

A.2. B, B 1 and B t factors and their correlations
In Stahl and Stepanoff (1956), the heat balance expression in Eq. (A.1) was adopted to develop the criterion for thermodynam- the heat balance equation is rewritten as where B = V v /V l is the ratio of the vapour volume to the liquid volume, called the criterion B or B factor for thermodynamics in cavitation (Stahl and Stepanoff, 1956). B depends on thermodynamic properties (v v , v l , c p , L, dp v /dT l ) and temperature depression ∆T . In order to calculate the criterion B of various liquids pumped at a temperature T l , a 1.6 in (0.4 m) vapour pressure depression ∆p v , which means a depression in NPSHr, was given, and the corresponding temperature depression ∆T with a known p v − T l relationship, was estimated with ∆T = ∆p v /(dp v /dT l ). A series of cavitation experiments was conducted on a centrifugal pump with 38 mm suction nozzle diameter at 3470 rpm rotational speed and 1.22 m NPSHa when pumping water at temperatures 21, 82, 100, 121, 149 • C, respectively. B factor values at these temperatures were calculated and discussed (Stahl and Stepanoff, 1956). NPSHr at 3% head drop was measured in two centrifugal pumps (N s = 1200, 1600, n = 3585 rpm) handling water, butane, benzene +3% propane by weight, kerosene degasified, gasoline, and Freon 11 at various temperatures (Salemann, 1959). N s is specific speed of centrifugal pumps, N s = n √ Q /H 0.75 , Q is pump volume flow rate in US gallon/min, and H is pump head in ft. The NPSHr correction curves were generated for paraffin hydrocarbons and water in terms of vapour pressure and temperature. The NPSHr correction ∆NPSH is defined as the difference in NPSHr of a liquid at a pumping temperature from that of cold water (22 • C) NPSHr w , namely ∆NPSH = NPSHr − NPSHr w (A.5) In Stepanoff (1961), an empirical method was delivered for a liquid pumped at a known pumping temperature based on the criterion B by using two correlations, one is the B − p v curves at four suction specific speeds such as 6000, 7200, 9000 and 12 000 to cover the experimental data in Salemann (1959), one is Eq. (A.4). The procedure for determining ∆NPSH is rather straightforward, that is T l → p v → B → ∆T → ∆p v → ∆NPSHr, where ∆NPSHr = ∆p v /gρ l = ∆T dp v /dT l /gρ l , g is the acceleration due to the gravity.
Later, that method was updated in Stepanoff (1964) by using the Clausius-Clapeyron equation to estimate dp v /dT and by establishing analytical relation of ∆NPSHr with both p v and B 1 which the criterion B at ∆NPSHr 1 = 1 ft. the Clausius-Clapeyron equation reads as (Stepanoff, 1964) dp Substituting Eq. (A.6) for dp v /dT l in Eq. (A.4), the criterion B is rewritten as Under ∆p v1 v l /g = ∆NPSHr 1 condition, the criterion B becomes a specific numerical value denoted by B 1 It was found that the parameter B 1 ∆NPSHr 3/4 can be correlated with (p v v l /g) −3/4 in a linear relation for the experimental data in Salemann (1959). If the unit of ∆NPSHr and p v v l /g is in ft, then the correlation is expressed with In that case, the suction specific speed was excluded from the correlation and ∆NPSHr can be estimated very simply from Eq. (A.9) with known T , v v , v l , c p , L and B 1 .
In parallel, the thermodynamic criteria of cavitation were derived based on one-dimensional governing equations of inviscid fluid, including continuity, momentum and energy equations under cavitation and adiabatic conditions (Spraker, 1965) to obtain a general thermodynamic criterion. These equations are written as follows where ρ m is the density of a liquid-vapour mixture, ρ m = [(1 − y) v l + v v ] −1 , y is the mass percentage of vapour, y = W v /(W l + W v ), u m is the velocity of the mixture, ψ is enthalpy of the mixture, ψ = ψ l + yL, dψ l ≈ c p (dp v /dT l ) −1 dp v . Eliminating u m from Eq. (A.10), dp v is expressed with v l , v v , L, y and dp v /dT l Considering that v l , v v , L and dp v /dT l are constant, Eq. (A.10) is integrated from y = 0 to y to estimate pressure depression ∆p v Eq. (A.14) can be simplified to obtain an approximate relationship like Eq. (A.4). Eq. (A.14) is written as an exponential function, and c p (dp v /dT l ) −1 −v l ≈ c p (dp v /dT l ) −1 and ( The last term is in a 2nd-order small quantity owing to 0 < < 1, and it can be neglected naturally, then (dp v /dT l ) −1 was with dimension and was called the thermal cavitation parameter B t (Fisher et al., 1945). After the B is replaced with The vapour pressure depression is expressed as In Spraker (1965), it was assumed that V v /V l is the same for all centrifugal pumps and liquids during cavitation at 3% head drop point, hence the NPSHr correction ∆NPSHr(=∆p v v l /g) was a function of 1/B t . For pure liquids such as water, butane, methyl alcohol and Freon 11, ∆NPSHr can be correlated with ln (1/B t ).
For a mixture liquid, namely gasoline, crude oil and fuel oil, however, ∆NPSHr can be divided two parts, one part is correlated with ln (1/B t ) and the part is related to liquid temperature pumped (Spraker, 1965). Eq. (A.14) has been derived in Jacobs (1961) as well, but any ∆NPSHr correlations were not proposed even the suction performance of a single-stage submerged booster pump had been measured when pumping liquid hydrogen. Based on factor 1/B t and impeller Reynolds number, a correlation of NPSHr was proposed in Barenboim (1966) to fit the experimental NPSHr of various liquids in centrifugal pumps in Salemann (1959). Similarly, an empirical correlation of ∆NPSHr was developed for inducers in Shcherbatenko and Shapiro (1981) in terms factor 1/B t , Froude number, Weber number and Reynolds number of a wedge-shaped cavity in the blade suction side in the meanline cascade. The coefficients in the correlation were best fitted with the experimental data of nine inducers. Effects of liquid viscosity and thermodynamics on the cavitation performance of centrifugal pumps were discussed but any correlations were not put forward (Shcherbatenko, 1982).
Two linear empirical correlations of NPSHr were proposed by employing an integrated variable, for instance B −1 t (PrRe) 1/2 ( ρ lref /ρ l ) 4/3 (p v /p a ) Re m , ρ lref is density of the liquid at a reference temperature, p a is atmosphere pressure, in terms of the experimental data on suction performance of centrifugal pumps in Salemann (1959) and Spraker (1965). These correlations read as (Chivers, 1969-70) where a, b are empirical constant determined by curve fitting, m = 0.8 for the experimental data in Salemann (1959) and Spraker (1965).

A.3. ∆T and B correlations
In the thermal analysis of cavitation above, the thermodynamic or energy balance equation was concerned only, i.e. the fluid is stationary with heat conduction, thus the convective heat transfer process has been overlooked in flows. To take this process into account, it was assumed that the velocity of vapour removal from the cavity by entrainment is proportional to the velocity of fluid flow in turbulent regime and the temperature depression in the cavity is written as (Holl and Wisclicenus, 1961) where A v , A v are areas of vapour removal and heat supply through the liquid, Pe is the ratio of heat transfer by convection to that by conduction, i.e. the Peclet number, Pe = c p VD/kv l , V and D are the characteristic velocity of the liquid and the characteristic dimension of the pump impeller, k is the heat conductivity of the liquid, C is experimental constant. Once a ∆T is available, the pressure depression can be calculated via ∆p v = dp v /dT l ∆T . Another proposal of ∆T was raised in Acosta and Parkin (1961). The chief contribution is to include Reynolds number Re, Prandtl number Pr and Froude number Fr into the constant C , namely A similar piece of work was found in Holl et al. (1975), where the heat balance between the energy required in vaporization process and the heat transfer through the cavity boundary surface was established, such as where Q v is volume flow rate of the vapour, h f is the film heat transfer coefficient, A surf is the cavity surface area.
Expressing Q v with the flow rate coefficient C Q , and characteristic flow velocity V and pump impeller dimension D, Q v = C Q D 2 V , then ∆T can be written as the following based on Eq.
Eq. (A.25) is formulated in terms of nondimensional coefficients or numbers, that is where C A is area coefficient, , measuring the rate of heat transfer of a material from the hot end to the cold end, Nu is the Nusselt number, Nu = h f D/k, meaning the ratio of convective to conductive heat transfer at a boundary in a fluid. The empirical correlations of C A with D c /D, C Q with Re, Fr, D c /D and Nu with Re, Fr, Pr, D c /D were proposed according to previous cavitation tests on quarterand zero-calibre ogives in Holl and Wisclicenus (1961). Effects of liquid surface tension on C Q and Nu was clarified by introducing the Weber number, We = V √ D/ √ ς/ρ l , where ς is surface tension of the liquid, based on the experimental cavitation data in quarter-and zero-calibre ogives and Venturis (Holl et al., 1975).
A survey of empirical ∆T was made in Bonnin et al. (1981). A C Q empirical correlation was developed based on Venturi cavitation test in Fruman et al. (1999).
Experiments were performed in four-bladed inducer as pumping refrigerant R114 and B-factor was correlated to C Q , Re and Pr (Franc et al., 2004). The corresponding correlation reads as B = C Q Re 0.2 Pr 0.7 (A.27) where C Q = √ l/c, l is cavity length over a blade of the inducer, c is chord length of the blade.
Those studies suggest that it is impossible to predict ∆T and ∆p v accurately based on any theoretical thermodynamic or heat transfer models. Empirical correlations must be established and utilized by making use of existing experimental cavitation data to bridge the gap.

A.4. General B factor correlations
Refrigerant Freon 114 or R114 was applied to a specially designed Venturis to investigation into cavitation similarity in terms of thermodynamic effect expressed by measured vapour pressure and temperature depressions in Gelder et al. (1966). Firstly, a cavitation number for describing the similarity of developed cavitation in a Venturi was put forward based on the far field pressure p ∞ and velocity V ∞ of a liquid at the inlet of a Venturi, and the vapour pressure in the cavity p v (T v ), namely where σ v (T v ) is the cavitation number with thermodynamic effect at vapour temperature T v , σ l (T l ) is the cavitation number without thermodynamic effect at liquid temperature T l , σ l ( ∆p v is the vapour pressure depression due to the vapour pressure drop in the cavity, If σ v (T v ) of liquid flow systems keeps constant, then the developed cavitation in them with thermodynamic effect will have a similarity. Obviously, ∆p v must be known in advance to gain the σ v (T v ) in a liquid flow system. Secondly, the criterion B in Eq. (A.7) is in the following empirical correlation based on the experimental data for R114 (Gelder et al., 1966)  In Moor and Ruggeri (1968a), the exponent 0.16 for R114 in Eq. (A.29) is 0.15. In Moor and Ruggeri (1968b), the empirical correlation of B for both liquid-hydrogen and R114 is slightly different from Eq. (A.29), i.e.
where D and D ref are the inlet diameters of the current and reference Venturis, respectively.
Eq. (A.30) is applicable to centrifugal pumps as well. In industry, NPSHr is generally calculated by using the liquid-vapour pressure at a liquid pumping temperature T l . In this context, the NPSHr is expressed as (Ruggeri and Moor, 1969;Hord, 1974) where the pressure p ∞ and the pressure p ∞ and velocity V ∞ of the liquid at the inlet of a pump impeller. NPSHr (T l ) of a pump is very commonly measured in a pump manufacturer's laboratory by using cold water without thermodynamic effect during cavitation test. When the pump is used to transport hot water or hydrocarbon or refringent, the NPSHr (T l ) of the pump at that liquid pumped temperature can differ from that of cold water due to the thermodynamic effect in the cavitation of that liquid.
Eq. (A.28) states that the vapour temperature based σ v (T v ) is the factor for determining cavitation similarity with thermodynamic effect and should remain unchanged at a similar operating point in geometrically similar pumps. From Eq. (A.31), σ l (T l ) is expressed in terms of NPSHr (T l ) in the following form Substituting Eq. (A.32) into Eq. (A.28) for σ l (T l ), then σ v (T v ) in terms of NPSHr (T l ) and ∆p v arrives at under any other conditions with them. This is a drawback of the method even though it has found applications to hydrogen (Ruggeri and Moor, 1969;Meng, 1968) and butane (Ruggeri and Moor, 1969) and hot water (Kovich, 1972;Manabe and Miyagawa, 2006). The mechanism of head breakdown in cavitating inducers was associated with the Mach number of the liquid-vapour two-phase flow in the cavity (Jakobsen, 1964), especially, when the Mach number is unity, M = 1, the flow passages of inducer were choked. The Mach number is related to the characteristic velocity V ∞ , liquid sound speed a l , vapour sound speed a v , B factor, liquid specific volume v l and vapour specific volume v v , and expressed by using the following expression (Hord, 1974) M was used to correlate to experimental B factor by replacing V ∞ in Eq. (A.30), and the corresponding the correlation is written as (Hord, 1974) Since B factor has been involved in M expression, Eq. (A.38) is hard to be applied but also does not show any improvement in prediction accuracy (Hord, 1974).

A.5. Method of characteristic temperature depression ∆T +
A characteristic temperature depression-based approach was put forward in Zika (1984). According to Eq. (A.19), where factor B • is the volume ratio corresponding to a unit temperature drop. The ratio B/B • is with temperature unit and is named as characteristic temperature depression ∆T + , which is a function of liquid thermodynamic property only. It was illustrated that ∆T + (=B/B • ) data of water, benzene, butane, Freon 11 and methanol collapse onto a single curve in terms of their corrected vapour pressure p cor v (the actual vapour pressure minus the vapour pressure of water at 20 • C). This means that ∆T + can serve as the similarity parameter for cavitation with thermodynamic effect in a pump, i.e. the following expression is held It was hypothetical that after the vapour pressure depression ∆p v is replaced with NPSHr Eq. (A.40) is still true (Zika, 1984). Then NPSHr can be correlated to ∆T + with NPSHr dT l dp v Finally, NPSHr will be related to the ratio of ∆T + to dT l /dp v , that is Eq. (A.42) was applied to correlated NPSHr-p v curve of six centrifugal pumps tested in Spraker (1965). Since the number of examples is too few, the method of Zika (1984) does not exhibit a distinctly better accuracy than that in Stepanoff (1964). The approach does not seem to offer a resolution in correction of NPSHr of cold water to obtain the NPSHr of a liquid other than water.

A.6. Σ factor method
The compressibility of cavitation bubbles was analysed at the inlet of an inducer in terms of the Rayleigh-Plesset equation with thermodynamic effect; particularly, a criterion determining boiling driven by temperature increase and cavitation owing to pressure reduction in a pump impeller was established (Brennen, 1973). The criterion was described with a dimensional thermodynamic parameter Σ and an impeller cavitation performance variable Λ. The criterion is presented as follows In boiling state, the liquid temperature keeps unchanged, and there is not thermodynamic effect. In the cavitation state, however, the thermodynamic effect occurs and the liquid temperature is changed during cavitation. According to this fact, a Σ and Λ based method was developed in Arakeri (1999) to predict NPSHr (T l ) of hot water or hydrocarbons or refrigerans or cryogens. The method is in the following procedure: (1) Determine a appreciate liquid temperature T * l with the condition Λ = Σ for a given pump and liquid pumped with the known V ∞ , σ l and D; (2) If the pump operating liquid temperature yields T l ≤ T * l , then there is no thermodynamic effect in cavitation, NPSHr (T l ) has not a need of correction; (3) If the temperature is in line with T l > T * l , NPSHr (T l ) needs a correction. The NPSHr (T l ) experimental data in Salemann (1959), Stepanoff (1961) and Spraker (1965) were reprocessed by Arakeri (1999) and the correction ∆NPSHr was correlated to the reduced temperature T R [= ], T c is the critical temperature of the liquid. The ∆NPSHr − T R curves and the corresponding scattered experimental data are illustrated in If one knows NPSHr (T l ) ref of a liquid at a temperature, then ∆NPSHr of the liquid at another temperature can be figured out, finally one can predict NPSHr (T l ) of that liquid at that temperature. Usually, NPSHr (T l ) ref is for cold water in industry, and ∆NPSHr is for a liquid other than water, then Eq. (A.45) allows one to have NPSHr (T l ) of that liquid. Since it yields ∆NPSHr ≥ 0, NPSHr (T l ) of that liquid is always smaller than NPSHr (T l ) ref of cold water. This fact implies that a pump is subject to a lowered NPSHr when pumping hot water, hydrocarbons or refrigerants or cryogens compared with that when handling cold water.
Similarly, Z factor was proposed to describe thermodynamic effect in sheet cavitation over a hydrofoil (Kato, 1984), i.e.
Various factors for representing thermodynamic effect in cavitation were reviewed in Franc and Pellone (2007), Yoshida et al. (2008) and Yoshida et al. (2013). where C z is experimental constant. Additionally, based on (Eqn (A.43)) and letting ∧ = Σ and V ∞ = ωD, where ω is pump impeller angular speed, D is impeller diameter, the following dimensionless parameter is resulted (Ehrlich and Murdock, 2015) √ σ l = √ σ l decreases with increasing liquid temperature for inducers (Ehrlich and Murdock, 2015).

A.7. The Rayleigh-Plesset equation-based method
The formation and collapse of the individual cavitating bubbles around a cylindrical body with 1.5-caliber ogive nose in a water tunnel were studied by employing high-speed motion pictures up to 20,000 frames/second and the streamwise fluid pressure along the body wall was measured (Plessset, 1949). That fluid pressure profile was utilized to predict cavitating bubble growth and collapse in a variable fluid pressure field and compared with the observations made in Knapp and Hollander (1948), and the water temperature effect on the bubble growth and collapse was discussed quantitatively based on the conductive heat transfer equation between a spherical vapour bubble and its surrounding liquid. The work is a significant contribution to the very early work adhesive to constant fluid pressure condition in Rayleigh (1917).
The ideas about cavitating bubble growth and collapse and thermodynamic effect in Plessset (1949) were adopted and 1D numerical method for predicting cavitating bubble growth and collapse in a centrifugal pump impeller passage was put forwards in Grist (1986a) and Grist (1986b). The solved bubble growth and collapse 2nd-order differential equation i.e. Rayleigh-Plesset equation, has included bubble size change rate, number of bubbles, liquid surface tension, liquid thermal parameters, vapour and liquid volumetric flow rates, liquid pressure drop owing to cavity blockage, NPSH difference from cavitation inception, and pump non-cavitating head. The method can establish a relationship between bubble volumetric fraction and NPSH and predict pump head-NPSHa curve at different liquid temperatures for a specific centrifugal pump, but fail to develop empirical correlation for NPSHr based on experimental data of a variety of centrifugal pumps.

A.8. Empirical correlation for NPSH at cavitation inception
A theoretical model was built for estimating NPSH at cavitation inception in centrifugal pumps in Al- Arabi and Selim (2007). The NPSH at cavitation inception is written as where σ b is the cavitation number, v m1 is the meridional absolute velocity of fluid at impeller inlet, u 1 is the tip speed of impeller inlet, ∆v 1 is the change of the absolute velocity of fluid before and after the blade leading edge at impeller inlet. σ b was correlated to thermodynamic effect in cavitation by using the method in Hord (1974), pump flow rate ratio, rotational speed ratio, and liquid temperature, v m1 and ∆v 1 were related to pump flow rate ratio. The effect of non-condensable gas concentration on the cavitation threshold was considered. The model was validated by using 20 sets of experimental data of water at room or slightly high temperature (Al-Arabi and Selim, 2007). Whether the model is suitable for the liquids other than water is unknown.