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 nu- merical 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 be- tween 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 vol- ume of air/gas and liquid.