Fluid Bulk Modulus: Comparison of Low Pressure Models

International Journal of Fluid Power 13 (2012) No. 1 pp. 7-16

FLUID BULK MODULUS: COMPARISON OF LOW PRESSURE MODELS

Hossein Gholizadeh, Richard Burton and Greg Schoenau

Department of Mechanical Engineering, University of Saskatchewan 57 Campus Drive, Saskatoon, Saskatchewan, Canada, S7N 5A9 Contact: [email protected]

ABSTRACT

Fluid bulk modulus is a fluid property that has been studied extensively over the past. The numerical value of this property depends on the operating conditions, the amount of entrained air/gas, and the way compression is applied and to some extent, the mathematical form it is defined. In a companion paper, an extensive review of fluid bulk modulus was presented. From this review, it was established that many models for fluid bulk modulus in the low pressure range (below critical pressure) have been forwarded. However, many of these models are based on assumptions that have not been explicitly defined. This paper considers these models and attempts to quantify the underlying assumptions. In addition some modification to these models are proposed in order to compare their prediction in the case where air/gas in entrained, for example. The paper concludes by categorizing the models into two groups and recommending the best model that can be used for each group. Finally some problems which observed in the models are discussed and future work for solving these problems presented.

Keywords: fluid bulk modulus models, hydraulic fluid, air/gas dissolving, adiabatic, isothermal, volumetric fraction of air/gas, critical pressure

1 Introduction

Fluid bulk modulus represents the resistance of a were not consistent. The objective of this paper is to liquid to compression and is the reciprocal of com- provide a summary of these models and the condi- pressibility (Manring, 2005). Bulk modulus is a funda- tions/assumptions upon which these were based. In mental and inherent property of liquids which repre- addition, the authors present some modifications to sents the change in density of the liquid as external these models which would allow a comparison to be pressure is applied to the liquid. It shows both the stiff- made for the same operating conditions. The paper will ness of the system and the speed of transmission of conclude by discussing some of the results and will pressure waves. Therefore, stability of servo-hydraulic present some guidelines on how best to choose the systems and efficiency of hydraulic systems is affected most appropriate formulation for a particular applica- by the value of compressibility (Hayward, 1963). tion. There have been many studies and publications on the topic of fluid bulk modulus. It is clear that the numerical value of this property depends on the operating 2 Models of the Effective Fluid Bulk conditions, the amount of entrained air/gas present, the Modulus way compression is applied and to some extent, the mathematical formulation. In a companion paper In practical hydraulic systems, fluid is a mixture of (Gholizadeh et al., 2011) an extensive review of the the basic fluid, dissolved air/gas, air/gas bubbles and research that has been published on this subject was sometimes also vapor (Kajaste et al., 2005). In addition presented. Many of these studies produced mathemati- to the composition of the fluid, operating pressure and cal and in some cases, experimental models to define temperature as well as the mechanical compliance of the operating behavior of fluid bulk modulus as a func- hydraulic components can affect the fluid bulk tion of pressure and temperature. It was evident from modulus. To account for the effects of these variables, these models that for similar conditions, the predictions different models have been proposed by different re- This manuscript was received on 16 June 2011 and was accepted searchers. Please note that in all of these models, the after revision for publication on 8 November 2011 term “fluid” means the homogeneous mixture of the

© 2012 TuTech 7 Hossein Gholizadeh, Richard Burton and Greg Schoenau liquid and air/gas. For the air/gas free fluid, the term An examination of Merrit’s equation shows that in liquid will be used. this model, the volumetric fraction of the entrained It was observed that different authors used different air/gas in the oil is always considered to be equal to the definitions for the volumetric fraction of the air/gas at volumetric fraction of the entrained air/gas at atmos- atmospheric pressure, which sometimes causes confu- pheric pressure and the effect of increasing pressure on sion and makes the comparison of the models difficult. the volumetric fraction of the entrained air/gas has not Therefore, adopting one of these definitions as the been considered. Since this has not been taken into “standard” definition was deemed necessary. In the account in this model, the effective bulk modulus value next section where appropriate, the volumetric fraction predicted in Merritt’s model will be lower than the of the entrained air/gas at atmospheric pressure used in actual effective bulk modulus. This also shows that various models will be changed to this standard defini- using the secant bulk modulus definition to find the tion. Thus, the following standard definition for the effective bulk modulus leads to the lower effective bulk “volumetric fraction of entrained air/gas at atmospheric modulus values. pressure (P0) and temperature 273°K” is adopted Nykanen et al. (2000) derived a two-phase model for V an air/gas-liquid mixture. In this model, the effect of g0 X 0 = (1) dissolving entrained air/gas has not been considered. The VVg + l 00 bulk modulus definition used to develop his model was Assume that unit volume of fluid is taken; therefore ⎛⎞∂P Ke0= ρ ⎜⎟ (6) VV+=1 ∂ρ gl00 ⎝⎠ XV= 0 g0 (2) This definition is not consistent with the standard 1−=X V definition of tangent bulk modulus in which ρ should 0 l0 be considered instead of considering ρ0. Moreover, in For each of the models introduced, the definition of order to find Vl based on the liquid bulk modulus, this parameter used by the various authors will be high- Nykanen et al. (2000) used the secant bulk modulus, lighted, and then where appropriate all of the models that is will be modified to follow this standard definition. It PP− 0 should be also noted that in those models which the KVll=− (7) VVll− effect of temperature on the volume of the entrained 0 air/gas has been neglected, the X0 and other parameters This definition is again in contrast with the gener- with the zero subscript, simply represent the value of ally accepted secant bulk modulus definition which that parameter at atmospheric pressure. uses initial volume of fluid in the numerator. His final Merritt (1967) defined the “effective bulk modulus” equation for bulk modulus was presented to be model for a liquid-gas mixture in a flexible container. ⎛⎞2 In his analysis, the following assumptions were made: ⎜⎟1 ⎛⎞PXn 1− secant bulk modulus was used to develop the model; ⎜⎟00X + gas bubbles were assumed to be uniformly distributed ⎜⎟P 0 PP− ⎜⎟⎝⎠ 1+ 0 throughout the liquid; solubility of the air/gas in the ⎜⎟ ⎝⎠K l K = liquid was not considered, air/gas was treated as a per- Nykanen 1 (8) fect gas, surface tension effects were neglected and the ⎛⎞P n 0 X liquid and gas assumed to have the same pressure and ⎜⎟P 0 1− X ⎝⎠ + 0 temperature. 2 nP ⎛⎞PP− Using these assumptions the effective bulk modulus 0 ⎜⎟1+ K l was defined as ⎝⎠K l

111Vg ⎛⎞ 11 Before comparing the models, it is important to =++0 ⎜⎟ − (3) K KKVKK⎜⎟ mention that from this point forward; all of the models ecl0 ⎝⎠ gl will be compared based on the same assumed condi- In Eq. 3, K represents the secant bulk modulus of tions of: P0 = 0.1 MPa, Kl = 1500 MPa, n = 1 (isother- g mal condition) and X = 0.1. the gas, however; instead of replacing the secant bulk 0 These conditions were arbitrary chosen just for modulus formula in Eq. 3, Merritt has replaced it with comparison purposes. But for practical conditions, the the tangent bulk modulus formula for the gas, which is real value of these parameters needs to be determined. K = nP g (4) Since for very small values of X0, the difference between the models was small, therefore, a larger value Assuming a rigid container; this model can be writ- was chosen for X in order to clearly show the differ- ten as 0 ences between the models. It should be noted that none K l of the following models consider the effect of tempera- K = (5) Merrit ⎛⎞K ture on the liquid bulk modulus (which is critical). 11+−X l 0 ⎜⎟ Since the models were compared in the low pressure ⎝⎠nP range (0 - 5 Mpa), the effect of pressure on the liquid It should be noted that this model is the same as the bulk modulus was neglected and the constant value for model proposed by Wylie (1983). the liquid bulk modulus was assumed.

8 International Journal of Fluid Power 13 (2012) No. 1 pp. 7-16 Fluid Bulk Modulus: Comparison of Low Pressure Models

⎛⎞1 ⎛⎞P n ⎜⎟0 ⎜⎟XX00+−(1 ) ⎜⎟⎝⎠P ⎝⎠ K = modified Nykanen 1 (13) XP⎛⎞n (1− X ) 00⎜⎟+ 0 nP⎝⎠ P Kl Cho et al. (2000) defined the effective bulk modulus

model for a liquid-gas mixture in a rigid container. The assumptions are the same as the Merritt’s model except that in this model, the definition of tangent bulk modulus has been used. The instantaneous total volume has been defined as the sum of the instantaneous volume of air/gas and liquid. The equation so derived was 1 ⎡⎤P-P n − 0 Comparison between the Nykanen and Merritt’s ⎢⎥⎛⎞P K Fig. 1: eXl + models ⎢⎥⎜⎟ Cho ⎝⎠P0 KK= ⎢⎥ Cho l 1 (14) P-P ⎢⎥n - 0 Figure 1 shows the difference between the Nykanen ⎛⎞P K X K ⎢⎥⎜⎟e l + Cho l and Merritt’s model plotted for the specified condi- PnP ⎣⎦⎢⎥⎝⎠0 tions. Merritt’s model is based on the standard definition of secant bulk modulus, but Nykanen’s model is Assuming that the oil bulk modulus is much larger based on the wrong definition of tangent bulk modulus. than the pressure, the term exp (-(P - P0) / Kl) can be The problems related to the Merritt model have already replaced by unity and the bulk modulus equation would been discussed. If the Nykanen model is plotted for be larger pressures, it can be observed that the effective ⎡⎤1 ⎛⎞P n bulk modulus does not converge to the specified liquid ⎢⎥+ X bulk modulus, because of the incorrect definition used ⎢⎥⎜⎟ Cho ⎝⎠P0 KK= ⎢⎥ in deriving this model. Cho l 1 (15) ⎢⎥ It is of interest to modify Nykanen’s model to be ⎛⎞P n X K ⎢⎥+ Cho l consistent with the definition of bulk modulus in which ⎜⎟ ⎣⎦⎢⎥⎝⎠PnP0 the final density is used, that is ⎛⎞∂P which the volumetric fraction of the air/gas at atmos- Ke = ρ ⎜⎟ (9) pheric pressure has been defined as ⎝⎠∂ρ Vg X = 0 Using Nykanen’s assumptions and equations, it can Cho (16) Vl be shown that 0 Equation 16 can be compared with the X (the stan- mm+ XX0gρρ+− l(1 0 ) 0 ρ ==gl 00 1 (10) dard definition adopted in this paper) used in Eq. 1 as VV+ P-P0 gl n − ⎛⎞P 0 K XXe+−(1 ) l X 0 ⎜⎟00 X = ⎝⎠P Cho (17) 1− X 0 which Vl has been found using the tangent bulk Replacing Eq. 17 in Eq. 15, the Cho model would be modulus definition for the liquid as ⎛⎞1 P-P00 P-P n -- ⎛⎞P 0 KK ⎜⎟XX+−(1 ) VVe==−ll()1 Xe (11) ⎜⎟00 ll0 0 ⎜⎟⎝⎠P ⎝⎠ K = Replacing Eq. 10 in Eq. 9, the modified Nykanen Cho 1 (18) model becomes: XP⎛⎞n (1− X ) 00⎜⎟+ 0 1 nP P K ⎛⎞P ⎝⎠ l n − ⎛⎞P K ⎜⎟0 XXe+−(1 ) l ⎜⎟00 In essence the Cho model is the same as the modi- ⎜⎟⎝⎠P ⎝⎠fied Nykanen model (Eq. 13). This was expected, since K = modified Nykanen P (12) 1 − the main differences between these two models were in K XP⎛⎞n (1− X ) el 00+ 0 the definition of the volumetric fraction of the air/gas ⎜⎟ (which was corrected) and in the way that the models nP⎝⎠ P Kl were derived. In the modified Nykanen model, the If it is assumed that Kl >> P, this model can be sim- density of the mixture of the air/gas and liquid was plified as used to derive the effective bulk modulus, while in the Cho model the volume of the mixture has been considered. Since the total mass of the air/gas and liquid is always constant, it is expected that these two models should give the same results.

International Journal of Fluid Power 13 (2012) No. 1 pp. 7-16 9 Hossein Gholizadeh, Richard Burton and Greg Schoenau

1 Unlike the Nykanen model, Cho and the modified 1+ ⎛⎞P n Nykanen models assume the true definition of tangent g Kl ⎜⎟1+ bulk modulus. Thus, as it can be seen from Fig. 2 ⎝⎠P0 K = Yu 1 (21) (which for comparison purposes, has been plotted to 1+ ⎛⎞P n X K higher pressure values (0 to 30 MPa)), the Cho and g Yu ⎛⎞l ⎜⎟11++−()cP1g⎜⎟ −− P 0 P g modified Nykanen models converge to the specified ⎝⎠PP00 ⎝⎠ n liquid bulk modulus at higher pressure values and as such are more consistent with what would be expected In this model, pressures have been expressed in dif- at higher pressures than with the Nykanen model. ferential pressure and in order to be comparable with Yu et al. (1994) developed a theoretical model the other models, the pressures in this equation are which was based on the definition of tangent bulk changed to absolute pressure. Thus every Pg in this modulus. The measurements taken in their experimen- equation is changed to P - P0. Eq. 22 shows the Yu tal work was based on the measurement of the velocity model where the pressures are expressed in absolute of sound because it was believed that the approach pressure. gave the isentropic (adiabatic) tangent bulk modulus K K = l (22) values. Yu 1 ⎛⎞P n ⎛⎞K The method used by Yu to derive the effective bulk 0 l 11+−−−XcPPYu⎜⎟() 1() 0 ⎜⎟ 1 modulus of the mixture of the air/gas and liquid is ⎝⎠PnP⎝⎠ similar to Merritt’s method. Since Yu has used the In Yu’s model, despite the fact that the pressure tangent bulk modulus definition, it is better not to use range is high (up to 30 MPa), the pressure dependence the “∆” notation as he did in his work; rather the nota- of liquid bulk modulus has not been considered. The tion “d” will be used in his derivatives. Using “∆” may unknown values of XYu, c1 and Kl were determined us- be interpreted as the secant bulk modulus definition. ing the identification method explained in their paper and were found to be (Yu et al. (1994)): n = 1.4, -6 -5 c1 = -9.307 × 10 , Kl = 1701 MPa and XYu = 4×10 .

Fig. 2: Comparison between the Nykanen, modified Nykanen and Cho models Using the tangent bulk modulus definition, the effective bulk modulus derived by Yu becomes ⎛⎞ 1 VdVggVdV⎛⎞ Fig. 3: Plot of the Yu model based on the parameters ob- =−⎜⎟ +−ll⎜⎟ (19) K V⎜⎟ V dP V V dP tained using the identification method Yu⎝⎠ g⎝⎠ l In this model, both air/gas compression and dissolv- The effective bulk modulus model proposed by Yu is ing effects have been considered and in order to include plotted as a function of pressure in Fig. 3 and is based on the dissolving effect of air/gas, a new constant named the parameter values obtained from the identification c1 was introduced by Yu. c1 was defined as the coeffi- method. This plot is different than that given in Yu’s cient of air/gas bubble volume variation due to the paper and the reason for this discrepancy is not known. variation of the ratio of the entrained and dissolved The identification method used by Yu is valid when air/gas content in oil. Since the mass of the entrained identified parameters are constant. For the constant tem- air is changing by considering the dissolving effect, the perature and pump operating conditions, Yu has consid- following polytropic equation was used to consider this ered that these parameters are fixed. However, XYu has effect been defined in a way to be a function of pressure. Yu

n n has defined XYu (in his paper shown by R) as the en- (VcVPPPPVg1g−−() 0) 0 = g (20) 00 trained air/gas content by volume in oil at atmospheric pressure, but mathematically has shown it as Vg found from Eq. 20 has been replaced in Eq. 19. Considering the above discussion, the Yu model be- Vg X = 0 comes: Yu (23) VVgl+

10 International Journal of Fluid Power 13 (2012) No. 1 pp. 7-16 Fluid Bulk Modulus: Comparison of Low Pressure Models

In Eq. 23, the variations of the liquid volume can be This equation is exactly the same as the LMS model neglected for the low pressure range but the final vol- (which will be discussed later) except that the effect of ume of gas (Vg) will change dramatically by increasing temperature on the volume of entrained air/gas has not pressure even in the low pressure range. Therefore, XYu been included here (which can be easily added). Another will be a function of pressure. difference to the LMS model is that in this equation the H It should be also noted that after a critical pressure atmospheric pressure P0 has been used instead of Pvap . in which all of the air/gas will dissolve in the liquid, the Yu has also provided the simplified form of his c value will be zero. Therefore the c value is also a 1 1 model by assuming c = 0 which means the air/gas function of pressure. Since these two parameters are 1 dissolving effect is neglected. Considering this assump- not constant and are a function of pressure, the validity tion, the Yu model will be reduced to: of the identification method may be suspect. K It is of interest to notice the way that the Yu model K = l Yu_reduced 1 (30) has been developed using Eq. 19. In the Yu model, Vg n ⎛⎞P 0 ⎛⎞Kl found from Eq. 20, is replaced in Eq. 19, but this was 11+−X Yu ⎜⎟⎜⎟ ⎝⎠PnP⎝⎠ done just for the term Vg / V, but not for the term - dVg / (VgdP). The term -dVg / (VgdP) in the Yu model Eq. 30 is the same as the modified Wylie model was replaced with the bulk modulus of air/gas as proposed by Kajaste et al. (2005). However, it should

−dVg 11 be noticed that the XYu in Eq. 30 must be replaced by == (24) VdP K nP Vg X gg X ==0 0 Yu 11(31) nn If Eq. 24 is treated mathematically, the Vg value ⎛⎞PP00 ⎛⎞ VVVXgl0g0⎜⎟++− ⎜⎟ ()1 found from Eq. 20 must be inserted in Eq. 24 and the 00⎝⎠PP ⎝⎠ new value for -dVg / (VgdP) will be found. This way will be similar to the method that LMS IMAGINE S.A. Therefore Eq. 30 would result in (2008) has derived the effective bulk modulus of the ⎛⎞1 P n ⎜⎟⎛⎞0 air/gas mixture and will be explained later. Before calcu- (1−+XX00 ) ⎜⎟ ⎜⎟⎝⎠P lating the modified -dVg / (VgdP) term, it is of interest to K = ⎝⎠ (32) apply a modification to Yu’s model and include “critical Yu_reduced 1 XPn (1− X ) pressure” instead of c . When P = P all the entrained 00⎛⎞ 0 1 C ⎜⎟+ air/gas dissolves in the oil and therefore the volume of nP⎝⎠ P Kl entrained air/gas would be equal to zero. This is the same as the Cho and modified Nykanen P 1 0 n models. It is therefore concluded that the Cho, modified VVcPPgg1C0=−−=()() 1 ( ) 0 (25) 0 Nykanen and Yu reduced models are the same model PC when the effect of air/gas dissolving in the liquid is not from which the constant c1 can be found as considered and the same definition for the volumetric 1 fraction of air/gas at atmospheric pressure is used. c = 1 (26) Ruan and Burton (2006) developed a model of fluid PC0− P effective bulk modulus which considers both the volu- Therefore the volumetric change of entrained metric compression and volumetric reduction of the air/gas in terms of the critical pressure is obtained as air/gas due to the air/gas dissolving in the oil. In their 1 model, after some critical pressure, all the air/gas com- ⎛⎞P n ⎛⎞PP− VV= 0C pletely dissolves in the oil and the effective bulk gg⎜⎟ 0 ⎜⎟ (27) ⎝⎠P ⎝⎠PPC0− modulus would be equal to the oil bulk modulus. They studied the fluid effective bulk modulus below this Therefore, for pressures below the critical pressure critical pressure and found that the critical pressure is −dVg 11 proportional to the square root of the volume of the =+ (28) entrained air/gas and the polytrophic constant. They VdPgC np P− P assumed an isothermal compression and used the poly- The term –dVg / (VgdP) in Eq. 28, is based on a tropic equation of ideal gas in order to find the volu- mathematical expansion of Eq. 27, for the case of air/gas metric variation of the entrained air/gas bubbles. They dissolving in the liquid. By finding the new term for -dVg included the effect of volumetric reduction of the / (VgdP) (Eq. 28) and using the standard definition of air/gas due to the air/gas dissolving in the oil in this air/gas content and also assuming that at lower pressures polytropic equation and derived a differential equation Vl=Vl0 , the modified Yu model can be found as to describe its behavior. Solving this differential equa-

KModified Yu = tion, the volumetric change of the entrained air/gas 1 below the critical pressure for the isothermal compres- ⎛⎞PXPPn ⎛⎞⎛⎞− sion (n = 1) of the entrained air/gas was found to be 1+ ⎜⎟00C⎜⎟⎜⎟ ⎝⎠PXPP1−− ⎛⎞22 ⎝⎠⎝⎠0C0 (29) PPP0C− VV= ⎜⎟ (33) ⎛⎞1 gg0P P22− P 11⎛⎞PXn ⎛⎞ ⎛⎞ PP− ⎝⎠C0 ++⎜⎟00 C1 ⎜⎟⎜⎟⎜⎟ ⎜⎟ KPliq⎝⎠⎝⎠1−− XPPnP 0 C 0 ⎝⎠ It is of interest to notice the difference between this ⎝⎠ equation and Eq. 27 in which both represent the volu-

International Journal of Fluid Power 13 (2012) No. 1 pp. 7-16 11 Hossein Gholizadeh, Richard Burton and Greg Schoenau metric variation of entrained air/gas below the critical Applying these modifications to the Ruan and Bur- pressure. Since both show different results for the same ton model, the modified Ruan and Burton model for the phenomena, it would be interesting to investigate which range of (P < PC) will essentially be the same as the one is more valid. modified Yu model. In Ruan and Burton’s work, the differential equa- KModified Ruan&Burton = tion which was used to find the volumetric variation of 1 entrained air/gas was based on the polytropic equation ⎛⎞PXPPn ⎛⎞⎛⎞− 1+ 00C in which the initial state of the entrained air/gas was ⎜⎟⎜⎟⎜⎟ ⎝⎠PXPP⎝⎠⎝⎠1−−0C0 assumed to be [P, V ]. Then by increasing pressure to (36) g ⎛⎞1 P + dP, the total change in the volume of the entrained 11⎛⎞PXn ⎛⎞ ⎛⎞ PP− ++⎜⎟00 C1 air/gas was considered to be the change in the volume ⎜⎟⎜⎟⎜⎟ ⎜⎟ KPliq⎝⎠⎝⎠1−− XPPnP 0 C 0 ⎝⎠ of air/gas due to the compression plus the change in the ⎝⎠ volume due to the dissolving of the air/gas in the oil. A comprehensive fluid bulk modulus model (LMS This cannot be valid, since one concept is based on Model) is used in a commercial software AMESim the volume compression and the other based on the developed by LMS IMAGINE S.A., (2008). Four cases remaining mass. The amount of air/gas left due to dis- have been considered in AMESim: solving is in fact a ratio of mass whereas the ideal gas • law is based on volumes. P > Psat: There is no vapor and all air/gas is dis- The total change in the volume of entrained air/gas solved • PPPH << is in fact the change in the compression volume times vap sat : There is no vapor and part of the the percentage of non-dissolved air/gas left. This ratio air/gas is dissolved and part entrained θ LH can be called which is the ratio of the number of • Pvap<

Since the way that XLMS has been defined is totally different from the proposed standard definition (Eq. 1), it was decided not to adjust the formula to change X 0.25 LMS to be consistent with the proposed standard definition X0. Instead, the way that LMS has derived the effective bulk modulus will be explained and the results will 0 interpreted with respect to the proposed standard way P Pressure (MPa) P 0 C (Eq. 1). In the LMS model, it has been assumed that the Fig. 4: The ratio of the number of moles of entrained air/gas to the original number of moles of entrained air/gas content and saturation pressure do not vary with air/gas () versus pressure (Henry’s law) time or position and the liquid density is independent of

12 International Journal of Fluid Power 13 (2012) No. 1 pp. 7-16 Fluid Bulk Modulus: Comparison of Low Pressure Models the temperature. It has been also assumed that when 1 ⎛⎞T ⎛⎞PXn ⎛⎞ air/gas is dissolved in a liquid, the air/gas molecules do 1+ ⎜⎟⎜⎟00⎜⎟ ⎝⎠273⎝⎠PX 1− not increase the volume but do increase the mass. For K = ⎝⎠0 LMS 1 the case that P > PC (recall, PC is the same as Psat), it is ⎛⎞ 11⎛⎞PXn T assumed that there is no entrained air/gas and all the + ⎜⎟00 ⎜⎟⎜⎟ air/gas is dissolved in the hydraulic liquid. The fluid Kliq⎝⎠PXPP273 1−− 0 C 0 ⎝⎠ (41) bulk modulus in this case (P > PC) would be equal to ⎛⎞ the liquid bulk modulus. PPC − H ⎜⎟ For the case that PPPvap<< C , it is assumed that ⎝⎠PPC0− just a volume fraction of air/gas (θ) is entrained and the ⎛⎞PP− ⎜⎟C +1 remainder of the air/gas which is dissolved in the liquid ⎝⎠nP causes an increase in the mass. This volumetric fraction of air/gas has been defined the same as Eq. 34 which Therefore, the modified Yu, modified Ruan and was already explained in modifying the Ruan and Bur- Burton and LMS models are essentially the same mod- ton model. els. However, in LMS model the effect of different If one looks at the way that LMS has derived the ef- operating temperature on the initial volume of en- fective bulk modulus, at first it may seem that this trained air/gas has been considered. Figure 5 shows the method is different from the way that the modified Yu plot of the LMS model with respect to pressure. Note and modified Ruan and Burton models were developed. that the critical pressure value was chosen to be 2 MPa But as it will be shown, essentially this model has been for comparison purpose. The actual value of the critical also derived the same way as the modified Yu and pressure needs to be determined experimentally. modified Ruan and Burton models and the same model Figure 5 reveals that the LMS model experiences a of the effective bulk modulus results. discontinuity at the critical pressure where the gas In LMS model, the effective bulk modulus model phase disappears. This discontinuity is related to the θ was derived considering the change in the density of function which is not continuous at the critical pres- the fluid as pressure increases. It was also assumed that sure. Since this discontinuity can be a source of diffi- as the air/gas is dissolved in the liquid, it will increase culties when applying numerical integration, another the mass of the liquid but not the liquid volume. The plot labeled in this paper as the “modified Henry’s total mass at pressure P and temperature T was found to law”, was proposed by LMS which smoothes the tran- be sition at P = PC and P = P0. This is shown in Fig. 6. mV=+ρ Vρ # ll00 gg 0 0 (38) Since as Eq. 38 shows, the total mass is always constant, including the effect of dissolved air/gas in the increase of the mass of the liquid would not affect the effective bulk modulus of the mixture. Thus it can be shown that dP dP KVe ==−ρ (39) ddVρ # Equation 39 shows that essentially the results calcu- lated using the density or volume method will be the FGS ypqx same. Therefore the same model form will result as the U us uyxq Bq xg modified Yu and modified Ruan and Burton models. ! " # In the LMS model, since the volumetric fraction of Pq q GPg air/gas is measured at 273 °K, the effect of operating at Fig. 5: Plot of the LMS model based on the specified condi- another temperature on the change in this initial volume tions has been considered. The effective tangent fluid bulk modulus formula derived by LMS at pressure P and The LMS model proposed a new θ based on this temperature T assuming constant liquid bulk modulus new curve, and was determined to be: and Kl >> P is given by: 5 234 1 θ =−()1yyyyy (1 + 5 + 15 + 35 + 70 ) T ⎛⎞P n 0 PP− (42) VVl0+ g0 θ ⎜⎟ 0 273 ⎝⎠P y = K = PP− LMS 1 (40) C0 VPTd⎛⎞n ⎛⎞θ V θ l0 0 g0 Figure 7 compares the LMS (with simple and modi- +−⎜⎟⎜⎟Vg0 Kl 273 ⎝⎠PnPdP⎝⎠ fied Henry’s law) and modified Nykanen models. The LMS model with the simple Henry’s law is approxi- Equation 40 can be simplified more by assuming mately the same as the modified Nykanen model up to K >> P. Simplifying Eq. 40 and writing it in the no- l the critical pressure. This behavior is inconsistent with menclature used in this manuscript, yields the physical behavior of the bulk modulus in that by increasing the density, the bulk modulus should in-

International Journal of Fluid Power 13 (2012) No. 1 pp. 7-16 13 Hossein Gholizadeh, Richard Burton and Greg Schoenau crease. As pressure increases, more of the air/gas is 3 Conclusion dissolved in the liquid and therefore it is expected that the LMS model would give a bulk modulus value Bulk modulus is one of the most important parame- which is greater than the modified Nykanen model. ters in fluid power applications because it reflects a 1 system stiffness. It is known that the presence of air/gas Modified Henry’s law in fluid has a substantial effect on the fluid bulk modulus. Since beyond the critical pressure, all the entrained air/gas is dissolved, the density and fluid bulk 0.75 modulus can be assumed the same as the liquid one and as a result of this the measuring and modeling methods for these high pressure systems is quite straightforward. θ 0.5 But in low pressure regions (below critical pressure) where the effect of entrained air/gas on the effective bulk modulus is substantial, it is important to be able to 0.25 measure or predict the effective bulk modulus. The main purpose of the paper was to consider and compare different theoretical models for this low pressure region and suggestions for improvement to the models have 0 P Pressure (MPa) P been forwarded. 0 C It was observed that different authors used different Fig. 6: Modified Henry’s law used in the LMS model definitions for the volumetric fraction of the air/gas at atmospheric pressure; therefore one of these definitions Another problem which is observed in the LMS was adopted as the “standard” definition to provide a model using the simple Henry’s law is related to the common base for comparison. For each of the models jump (discontinuity) in the bulk modulus value at the introduced, the definition of this parameter used by the critical pressure point. In addition, at the critical pres- authors was highlighted, and then where appropriate all sure, the derivative of the bulk modulus is also discon- of the models modified to follow this standard defini- tinuous. To compensate for these two problems, the tion. It was also shown that using the secant bulk mo- modified LMS has tried to smooth the transitions using dulus definition to find the effective bulk modulus the modified Henry’s law given by Eq. 42. Since at the leads to lower effective bulk modulus values (Merritt’s critical pressure point, all the air/gas suddenly disap- model) and using the tangent bulk modulus definition is pears and a transition from the two phase flow (ho- preferred. Some authors have used the wrong definition mogenous mixture of liquid and air/gas) to a single of tangent bulk modulus which has been observed and phase liquid (consisting of oil and dissolve air/gas) corrected. A summary of the investigated models and occurs, physically the discontinuity in the derivative of their definitions used to develop the models is pre- the bulk modulus when crossing the critical pressure sented in table 1. point would be expected. However, the appearance of a big jump at the critical pressure point, need to be inves- Table 1: Summary of the investigated models and tigated more. their definitions for developing the models As Fig. 7 shows, using the LMS model with the Volumetric Definition Volumetric modified Henry’s law, the prediction of the model fraction of Model of bulk variation of appears to deteriorate in the lower pressure regions (up air/gas modulus air/gas to 1.5 MPa). definition

# Vg 0 Merritt Secant Compression VV+ gl00

Non- Vg Nykanen standard Compression 0 VVgl0+ Tangent 0

Vg 0 Cho Tangent Compression V l0 # V g Compression 0 Yu Tangent and dissolve VV+ Gpuruqp Hwgq lg FGS Suyxq Bq xg Vg FGS Gpuruqp Bq xg Ruan& Compression 0 Tangent VV+ ! " # Burton and dissolve lg Pq q GPg V(gt) Compression 0 Fig. 7: Comparison of LMS models with the modified LMS Tangent and dissolve VVl(gt)+ Nykanen 00

14 International Journal of Fluid Power 13 (2012) No. 1 pp. 7-16 Fluid Bulk Modulus: Comparison of Low Pressure Models

In terms of dealing with the air/gas in the fluid, the This concern need to be addressed based on the physics models can be categorized in two groups: of what is really happening. (a) Models which just consider the volumetric It should be noted that using the modified Henry’s compression of the air/gas: law in the LMS model facilitates numerical integration The models by Merritt, Nykanen, Cho and Yu were (no discontinuity) but has not explained the physics introduced. After comparing and modifying these mod- associated with the jump of the original LMS model. els, it was found that the difference in some models Moreover, it is physically expected for the models related to the definition of the volumetric fraction of which consider both the compression and the dissolv- air/gas at atmospheric pressure and the way the effec- ing of the air/gas in the liquid (like the LMS model), to tive bulk modulus is defined. By considering the same always have bulk modulus values greater than those definition of bulk modulus and the volumetric fraction predicted by the models which just consider the com- of air/gas at atmospheric pressure, it was found that the pression of the air/gas (like the modified Nykanen modified Nykanen, modified Cho and reduced Yu model). This trend of what would physically be ex- models are essentially representing the same model. pected was not observed in the LMS model and hence Therefore, it is suggested that for fast acting hydraulic use of these modified versions cannot be recommended systems in which the rate of increase in pressure is such at present. This concern needs to be addressed based on that it does not allow for the air to dissolve in the oil, the physics of what is really happening and is a chal- this model is recommended lenge that the authors are working on. ⎛⎞1 Experimental results obtained by Ruan and Burton P n ⎜⎟⎛⎞0 (2006), clearly showed that the rate of increase in pres- ⎜⎟XX00+−(1 ) ⎜⎟⎝⎠P sure will change the critical pressure value and this will ⎝⎠ K (Only compression) = significantly affect the bulk modulus value. The effect e 1 (43) XP⎛⎞n (1− X ) of the rate of increase in pressure has not been consid- 00⎜⎟+ 0 ered in any of the previous mentioned models and nP⎝⎠ P K l hence it is another important parameter which needs to Figure 8 represents the plot of fluid bulk modulus be included in the future model. for both the isothermal and adiabatic compression of All the investigated models have just considered a the air and it is evident that there is a big difference in case that a fluid is at rest and should be used with cau- the fluid bulk modulus value for two extreme cases of tion in the evaluation of bulk modulus in hydraulic polytropic constants. The plot shows that depending on transmission lines. Traveling pressure waves in the the actual polytropic constant (which can be any value long transmission line would create alternative regions between isothermal (n = 1) and adiabatic (n = 1.4)), the of high and low pressures which would affect the re- fluid bulk modulus can be any curve between these two lease and dissolving process of the air. These factors curves. Consequently, it is essential to experimentally influence the amount of the mass of the released air, find the actual value of this polytropic value. which should be considered when modeling bulk modulus in transmission lines: Instantaneous line pres- ! sure, the time in which the fluid is subjected to the low !" pressure, maximum mass of releasable air, the initial mass of entrained air/gas, the agitation created by the traveling wave and the temperature of the fluid at the time of saturation with air/gas should also be considered (Baasiri, et al, 1983).

Nomenclature

K [MPa] C Secant bulk modulus of the container K Tangent effective bulk modulus [MPa] e K [MPa] e Secant effective bulk modulus K Comparison of K (only compression) for isother- g Secant bulk modulus of the air/gas [MPa] Fig. 8: e mal and adiabatic compression of air Kg Tangent bulk modulus of the air/gas [MPa]

Kl Secant bulk modulus of the liquid [MPa] (b) Models which consider both the volumetric Mass of fluid (entrained air/gas + liq- m [kg] compression of the air and the volumetric reduction of uid) at pressure P and temperature T the air due to the air dissolving into solution Mass of gas at pressure P and tem- m [kg] The modified Yu, modified Ruan and Burton and g perature T LMS models were investigated. Comparing the models, Mass of liquid at pressure P and tem- m [kg] it was found that these models are essentially the same. l perature T A common problem was found in these groups of mod- P Instantaneous pressure (absolute) [MPa] els in which the effective bulk modulus curve versus P Atmospheric pressure (absolute) [MPa] pressure experienced a big jump at the critical pressure. 0 PC Critical pressure (absolute) [MPa]

International Journal of Fluid Power 13 (2012) No. 1 pp. 7-16 15 Hossein Gholizadeh, Richard Burton and Greg Schoenau

Pg Instantaneous gauge pressure [MPa] Merritt, H. E. 1967. Hydraulic Control Systems. New H York, Wiley. Pvap High saturated vapor pressure [MPa] L Nykanen, T. H. A., Esque, S. and Ellman, A. U. Pvap Low saturated vapor pressure [MPa] 2000. Comparison of Different Fluid Models. Po- T Instantaneous temperature [°K] wer Transmission and Motion Control, PTMC Volume of fluid (entrained air/gas + 3 V0 ° [m ] 2000, Professional Engineering Publishing, Bath, liquid) at P0 and 273 K United Kingdom, pp. 101 - 110. Volume of fluid (entrained air/gas + 3 V [m ] liquid) at P and T Ruan, J. and Burton, R. 2006. Bulk Modulus of Air 3 Volume of entrained air/gas at P0 and [m ] Content Oil in a Hydraulic Cylinder. Proceedings of Vg 0 273°K 2006 ASME International Mechanical Engineering Total volume of air/gas (including Congress and Exposition, IMECE2006 - Fluid V 3 ()tg [m ] Power Systems and Technology Division, pp. 259- 0 both entrained and dissolved) at P0 ° 269. and 273 K 3 Vg Volume of entrained air/gas at P and T [m ] Wylie, E. B. and Streeter, V. L. 1983. Fluid Tran- V ° 3 l Volume of liquid at P and 273 K [m ] 0 0 sients. Ann Arbor, Michigan, USA, Feb Press. V Volume of liquid at P and T [m3] l Yu, J., Chen, Z. and Lu, Y. 1994. Variation of Oil Mass density of fluid (entrained -3 ρ0 [kgm ] Effective Bulk Modulus with Pressure in Hydraulic air/gas + liquid) at P and 273 °K 0 Systems. Journal of dynamic systems, measure- Mass density of fluid (entrained -3 ρ [kgm ] ment, and control, Vol. 116 (1), pp. 146 - 150. air/gas + liquid) at P and T

X Volumetric fraction of entrained 0 ° [-] air/gas at P0 and 273 K

n Polytropic index for air/gas content [-] The ratio of the number of moles of entrained air/gas at pressure P to the θ [-] number of moles of entrained air/gas Hossein Gholizadeh at pressure P0 Ph.D. Candidate in the Mechanical Engineer- ing Department, University of Saskatchewan, Canada. He Received his M.Sc. from Iran University of Science and Technology in References Tehran, Iran in 2003. His main research interest is in fluid power transmission and control. His current research Baasiri, M. and Tullis, J. P. 1983. Air Release During is focused on fluid bulk modulus and methods Column Separation. ASME Journal of Fluids Engi- of on-line measuring of effective bulk modulus

neering, Vol. 105 (1), pp. 113 - 118. Richard Burton Cho, B. H., Lee, H. W. and Oh J. S. 2000. Estimation P.Eng, Ph.D, FASME, Burton is a Professor of Technique of Air Content in Automatic Transmis- Mechanical Engineering, University of Sas- sion Fluid by Measuring Effective Bulk Modulus. katchewan He is involved in research pertain- ing to the application of intelligent theories to International Journal of Automotive Technology, control and monitoring of hydraulics systems, Vol. 3 (2), pp. 57 - 61. component design, and system analysis. He is a Fellow of ASME, a member of the executive Gholizadeh, H., Burton, R. and Schoenau. G. 2011. of ASME, FPST Division, and an active Fluid Bulk Modulus: a Literature Survey, Interna- member of FPNI. He is a reviewer for most Journals that contain fluid power topics. tional Journal of Fluid Power, Vol. 12 (3), pp. 5 - 16. Greg Schoenau Hayward, A. T. J. 1963. Research on Lubricants and Professor of Mechanical Engineering at the Hydraulic Fluids at National Engineering Labora- University of Saskatchewan. He was head of that Department from 1993 to 1999. He tory. Scientific Lubrication, Vol. 15 (3), pp. obtained B.Sc. and M. Sc. Degrees from the 112 - 124. University of Saskatchewan in mechanical engineering in 1967 and 1969, respectively. In Kajaste, J., Kauranne, H., Ellman, A. and Pietola, 1974 he obtained his Ph.D. from the Univer- M. 2005. Experimental Validation of Different sity of New Hampshire in fluid power control systems. He continues to be active in research Models for Bulk Modulus of Hydraulic Fluid. The in this area and in the thermal systems area as Ninth Scandinavian International Conference on well. He has also held positions in numerous Fluid Power, SICFP`05, Linköping, Sweden. outside engineering and technical organiza- tions. LMS IMAGINE, S.A., 2008. HYD Advanced Fluid Properties. LMS Imagine, S.A., Technical Bulletin n° 117, Rev 8B. Manring, N. D. 2005. Hydraulic Control Systems. New York, Wiley.

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