Determination of Pressure Losses in Hydraulic Pipeline Systems by Considering Temperature and Pressure

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1 Paer received: UDC Paer acceted: Determination of Pressure Losses in Hydraulic Pieline Systems by Considering Temerature and Pressure Vladimir Savi 1,* - Darko Kneževi - Darko Lovrec - Mitar Jocanovi 1 - Velibor Karanovi 1 1 Faculty of Tehnical Sciences, University of Novi Sad, Serbia Faculty of Mechanical Engineering, Univesity of Banja Luka, Bosnia and Herzegovina Faculty of Mechanical Engineering, University of Maribor, Slovenia Generally acceted methods for calculating ressure losses within flat ielines as resented in literature and used in raxis, are based on the Reynolds number, which considers the viscosity and density of fluid, internal ie friction coefficient, ie geometry, and oil circulation velocity. Such an aroach contains serious inconsequentiality. Namely, the only nominal values for viscosity and density are considered in the calculation, which differs substantially from real conditions. This often leads to inaccurate calculations of ressure losses. A numerical model has been develoed in the work resented in the aer. It takes into account the actual changes in density and viscosity under the current oil ressure and temerature in order to overcome the above weaknesses of standard calculation rocedures. Such an aroach is novel and rovides new caacity for an accurate ressure dro analysis of advanced hydraulic systems. 009 Journal of Mechanical Engineering. All rights reserved. Keywords: hydraulic ieline, ressure losses, temerature, resure 0 INTRODUCTION Energy dissiation due to the frictional resistance of fluid flow within a hydraulic ieline system is reresented by any decrease in ressure. The ressure loss is determined according to the well-known Darcy-Weisbach equation: (1) d where: L ieline length, d diameter of the ie, v the fluid flow velocity, density of the hydraulic fluid and Re, d friction factor which deends on the Reynolds number Re, and ie roughness. In the cylindrical tubes, the friction factor is usually determined by using the Moody diagram (Fig. 1). This diagram deicts the Reynolds number against the friction factor, and describes the corresonding fluid regimes. The fluid flow through a ie can be either smooth or rough, deending on flow conditions. Various factors determine the nature of flow. In rincile, the flow can be stated as either laminar flow, i.e. steady, smooth flow, or turbulent flow, i.e. the flow is disturbed. From a ractical oint of view, the Reynolds number indicates if a flow is laminar or turbulent. In the case of turbulent flow, the friction factor deends on the Reynolds number, as in the case of laminar flow as well as on the coefficient of the relative roughness the relationshi between the absolute roughness of the ie s inner wall surface in contact with the fluid, and the diameter of the ie (/d), bearing in mind that the ie can be either erform as a smooth, rough or somewhere in-between liminal (transitional). Several tyes of emirical equations that can hel to determine the friction factor can be found in literature. Some fairly useful ones are given below [1] and []. a) Laminar flow area at Re < 000: =64/Re () b) Turbulent flow area within the smooth ie of a hydraulic system (according to Blasius []) at Re > 4000: =0.164/Re 0.5 () c) Transitional flow tye area within range 000 < Re < 4000:.9 Re (4) * Corr. Author's Address: University of Novi Sad, Faculty of Technical Sciences, Trg D. Obradovia 6, 1000 Novi Sad, Serbia, savicv@uns.ns.ac.yu 7

2 d) The Reynolds number value can be estimated using the following equation: Fig. 1. Moody diagram for the determination of flow regimes with regard to internal friction coefficient [] Re= vd/ (5) where [m /s] kinematic viscosity, d [m] ie diameter, v [m/s] average fluid flow velocity. Most of the authors [1] legitimately assume that all the hydraulic ieline systems are smooth because: a) Pies are roduced by recise drawing with low absolute roughness values 0.01 to 0.00 mm (average value mm). b) The alied Reynolds number is most often in the order of medium values, i.e. between 1000 and 0000 so the friction factor value is at the limit of what is considered smooth (the lowest curve in Moody's diagram reresents smooth hydraulic ies). Numerical verification of the aforementioned statement is based on the calculation within the following boundary values: a) mostly used hydraulic ielines range in diameter from 10 to 50 mm, so that the relative roughness /d is within the range of to Table 1. Deendency of Reynolds number of the ieline and fluid roerties Dimension which is at the lower end of the relative roughness, as deicted in Moody's diagram (see Fig. 1); b) usually fluid flow velocity v =1 to 6 m/s; c) common kinematic viscosity at oerating temerature (16 to 54) 10-6 m /s are borderline values of the otimum viscosity for the e.g. axis iston ums. For the borderline areas, the Reynolds number falls within the range indicated in Table 1. Number value Pie diameter [mm] Flow seed m/s Kinematic viscosity mm /s Reynolds number [ - ] Savi V. - Kneževi, D. - Lovrec, D. - Jocanovi, M. - Karanovi, V.

3 When analyzing the values of the Reynolds number, two conclusions can be drawn. The Reynolds number will increase with diameters d > 50 mm, flov seed v > 6.0 m/s, and kinematic viscosity < 16 mm /s, whereas the Reynolds number will decrease with d < 10 mm, v <1.0 m/s and < 54 mm /s. It follows, that the friction factor of the mineral hydraulic oil-flow through hydraulic ieline can be determined with satisfactory accuracy, using the Eqs. () to (4) and the ieline of the hydraulic system may be treated as smooth. 1 INFLUENCE OF TEMPERATURE AND PRESSURE CHANGES ON PHYSICAL- CHEMICAL PROPERTIES OF HYDRAULIC FLUID The characteristic hysical values of all hydraulic fluids which have a deciding influence on the loss of energy are density and viscosity of the fluid. Both are temerature (T) and ressure () deendent values:,t,,t..1 Density of Fluid Density is defined as the mass of fluid er unit volume: m (6) V where: m [kg] mass, V [m ] volume. The density of a hydraulic fluid is measured under atmosheric ressure at T = 88 K. Any increase in temerature results in increased fluid volume, whereas the fluid volume decreases according to any rise in ressure (and contrarily in the reverse order of temerature and ressure changes). Density changes as a consequence of temerature and ressure changes can be given in the form of a total differential: d d d dt d dt d T or in a rewritten form: d 1 -a dt d K (7) T from where results - volumetric temerature fluid exansion coefficient by constant ressure: 1 d (8) dt and K T isothermal comression module d d. (9) T.1.1 The Effect of Temerature on the Oil Density Using the law of mass conservation ( m V const. ) according Eqs. (8) and (9), two equivalent models can be exressed by the following equations: 1 dv (10) V dt and d V dv T. (11) Coefficient serves to define the relative change of fluid (volumetric) density. This is deendent on the mean molecular weight of the hydrocarbon. Figure shows a diagram of coefficient change, as drawn-u on the bases of the tested numerical values []. Based on the exerimentally verified Equation [4]: d const dt it is ossible to derive an equation which describes the relationshi between the hydraulic oil density for any temerature, and the atmosheric ressure, using the values that aly to the density at T = 88 K [4]: x T 88 (1) where the value of 15 = is evident on the diagram in Fig.. /10-4 [1/ C] T [ C] =a =100 bar =00 bar =00 bar =400 bar =500 bar Fig.. Changes in coefficient underconstant ressure Determination of Pressure Losses in Hydraulic Pieline Systems by Considering Temerature and Pressure 9

4 .1. The Effect of Pressure on the Hydraulic Oil density As mentioned above, the fluid volume reduces with any increase in ressure, while its density becomes greater at the same time. Comression module can be exressed in terms of the reviously-stated equations: d d V d T dv T so that any alteration in density, as a function of ressure change, for both the constant values of comression module K T and constant temerature, is worked out by the formula below: (1) whereby it follows that x x x x1. (14) Coefficient (K T ) change diagram modeled on the bases of the tested numeric values [] is resented in Fig.. [bar] K S [bar] [bar] T=15 C T=0 C T=50 C T=70 C T=100 C [bar] T=15 C T=0 C T=50 C T=70 C T=100 C Fig.. Change of isothermal (K T ) and adiabatic (K S ) comression module as a function of oerating ressure and oil temerature. Viscosity Viscosity reresents fluid resistance to shear or flow and is a measure of the adhesive/cohesive or frictional fluid s roerty. Used in hydraulics, there is a concet of dynamic viscosity, the value of which is determined by Newton s equation concerning arallel linear laminar flow. F dv (15) A dx where: F [N] is side force; A [m ] side force surface; dv/dx [1/s] velocity of angular deformation. Dynamic viscosity is exressed in Pascal seconds [Pa s], which corresonds to [kg/m s]. In technical ractice, the term kinematic viscosity is used, which is defined as the ratio between dynamic viscosity and fluid density:. (16)..1 Influence of Temerature Change on the Viscosity of the Hydraulic Oil The kinematic viscosity of industrial oils is measured under atmosheric ressure at 40 C. It is classified into VG viscosity classes according to the ISO standard. Hydraulic fluids of class ISO VG 46 are most frequently used in raxis. They are followed by fluids classed as VG, and less commonly by VG 68 or. Several formulae, derived for determining viscosity as a function of temerature, can be found in relevant literature, e.g. [6]: at be A - according to Reynolds (17) a - according to Slotte (18) b T C A 1 c v a b d T A - according to Walther (19) b TAc ae - according to Vogel. (0) The coefficients a, b, c, d contained in the Eqs. (17) to (0) are affected by the roerties of individual hydraulic fluids, with T A signifying the absolute temerature. The most accurate among them is Eq. (0) but the more frequently used one is Eq. (19), which also forms the basis for the ASTM viscose temerature diagram s design [5]. The measurement for the change of kinematic viscosity with temerature is commonly determined by the viscosity index (VI), a non-dimensional number. Oils of faster viscosity alteration are classified as HM oils 40 Savi V. - Kneževi, D. - Lovrec, D. - Jocanovi, M. - Karanovi, V.

5 (VI = 90), and those of slower alteration as HV oils (VI = 140). Figure 4 shows a diagram of viscosity alteration as a function of temerature and in relation to HM oils, 46, and 68. The diagram reresents the results of tests for oil s viscosity at 40 and 100 C [5]. Kinematic viscosity [mm /s] HM 68 HM 46 HM o Temerature T [ C] Fig. 4. The alteration of viscosity as a function of temerature for oils HM, 46 and 68.. The Effect of Pressure on the Viscosity of Hydraulic Oils Oil viscosity increases with any concurrent increase in ressure. The seed of this change deends on the molecular structure of the oil, meaning that the seed will be higher where the oil viscosity is lower. Numerical value of kinematic viscosity alteration as a function of the change in ressure can be worked out in two different ways: a) directly by kinematic viscosity calculation: 1 k (1) where: k 0.00 to 0.00 stands for the change of viscosity coefficient; b) indirectly, i.e. obtaining a dynamic viscosity value by first using the Barus equation: 0 e () where:,t coefficient of viscosity deending on ressure, 0 - value of dynamic viscosity under atmosheric ressure, and at an aroriate oerating temerature (see Fig. 5). The kinematic viscosity calculation drawn from this model is done in relation to the calculated value of dynamic viscosity and the density at the oerating temerature and ressure using the following formula: 0 e () x 1 which follows from Eqs. (14), (16) and (). [1/bar] T [ C] =a =500 bar Fig. 5. The alteration of the coefficient as a function of any change in temerature, and oerating ressure ranging from a to 500 bar for mineral oils with araffin base structure [5] NUMERICAL VALIDATION OF THE PROPOSED MODEL The correlation of temerature and ressure with changes in the key roerties of fluids, which should be taken into account when calculating ressure losses within hydraulic systems, is resented in Chaters 1 and. It remains unclear whether change in density and viscosity substantially effects ressure loss in hydraulic systems, and whether they should be considered while designing a hydraulic system; and if so, how to set u a general model for calculating the exact loss of ressure. It only makes sense to answer the question with regard to a reresentative model of the ieline comlying with defined criteria, as follows: - ieline diameters 15, 5 and 40 mm alternately; - ieline length 1 m; - velocity of flow 6 m/s; - tye of hydraulic mineral oil HM46, viscosity index 90 to 100; - oerating temerature of the hydraulic oil 50 C; - density of the hydraulic oil 875 kg/m at 15 C; - oerating ressure 100, 00, 00 and 400 bar; - the nature of the ressure change: adiabatic. Determination of Pressure Losses in Hydraulic Pieline Systems by Considering Temerature and Pressure 41

6 .1 Ste 1 - Calculation Model for Determining Oil Density at Oerating Temerature 50 C, and Atmosheric Pressure. The volumetric temerature exansion coefficient equals (Fig. ) = , which makes the fluid density at oerating temerature 50 C and under atmosheric ressure equal to: ( T 88) 85.6 kg/m Ste - Calculation Model for Determining Oil Density under Oerating Pressure The comression module for the adiabatic change of ressure (Fig. ) equals: a) 1 = 100 bar K S1 = bar; b) = 00 bar K S = 1800 bar; c) = 00 bar K S = 1900 bar; d) 4 = 400 bar K S4 = 0500 bar. Oil density under the defined ressures equals: kg/m kg/ m kg/m kg/m. The density change curve as a function of oerating temerature and temerature, is shown in Figure 6. Fig. 6. Density change as a function of ressure change at temerature 50 C. Ste - Calculation Model for Determining Viscosity at Oerating Temerature Kinematic viscosity of oil HM 46 is m /s at 40 C and m /s at 50 C..4 Ste 4 - Calculation Model for Determining Viscosity under Oerating Pressures Kinematic viscosity at 50 C and under atmosheric ressure equals m /s, whereas ressure coefficient is Therefore, viscosity of the determinate oerating ressures equals: - 1 = 100 bar: (1 k 1 ) m/s ; - 6 = 00 bar; m/s ; - 6 = 00 bar; m/s ; = 400 bar; m/s..5 Ste 5 - Calculation Model for Determining Pressure Loss a) Viscosity corrected in accordance with the level of oerating temerature. The starting arameters: 50 = m /s; = 875 kg/m ; v = 6 m/s; d = 5 mm; L = 1 m. b) Viscosity corrected in accordance with the level of oerating temerature and oerating ressure. The starting values: v = 6 m/s; d = 5 and 15 mm; L = 1 m. - a 0 = m /s; 0 = 875 kg/m ; - 1 = 100 bar 100 = m /s; 100 = kg/m ; - = 00 bar 00 = m /s; 00 = 86.9 kg/m ; - = 00 bar 00 = m /s; 00 = kg/m ; - 4 = 400 bar 00 = m /s; 400 = 870. kg/m. An examle of the calculation model for determining turbulent flow: ste 5a The Reynolds number equals: vd Re The friction coefficient equals: Re 4 Savi V. - Kneževi, D. - Lovrec, D. - Jocanovi, M. - Karanovi, V.

7 The ressure decrease within tube network equals: d 686 Pa. An examle of the calculation model for determining the flow during the liminal hase: ste 5b vd Re Re Pa. d An examle of the calculation model for determining laminar flow: ste 5c vd Re Re Pa. d Summary of the overall calculation for both ie diameters is shown in in Figure 7. ressure loss [Pa] oerating ressure [bar] d=15 mm d=5 mm d=40 mm Fig. 7. Pressure loss as a function of the oerating ressure level and the ie diameter 4 CONCLUSIONS Generally acceted methods for calculating ressure losses within flat ielines, as resented in technical literature and used in raxis are based on the Reynolds number, which considers viscosity and density of fluid, internal ie friction coefficient, ie geometry, and oil circulation seed. Technical literature treats the asect of density and viscosity alteration as a function of temerature, and ressure change as an isolated issue with only tangential links to the roblems of designing and calculating a hydraulic system. Nominal values for viscosity (at 40 C) and density (at 15 C and normal atmosheric ressure), which differ substantially from real conditions, are considered. Consequently, the derived values cannot be regarded as highly accurate, leading to inaccurate calculation of ressure losses (in some cases u to 00%). The numerical model, which takes into account the actual changes of density and viscosity at the current oil ressure and temerature in order to overcome the above weakness, is suggested in this aer. Such an aroach is novel and rovides a new caacity for an accurate ressure dro analysis of advanced hydraulic systems. 5 REFERENCES [1] Akers A., Gassman M., Smith R. Hydraulic Power System Analysis, CRC/Taylor & Francis, 006, ISBN [] Munson B.R., Young D. F., Okiishi T. H. Fundamentals of fluid mechanics, John Wiley & Sons, 006. ISBN [] Frauenstein, M. Directives about hydraulic fluids for design engineers, Mobil Oil AG, Hamburg,. Edition, (in German). [4] Knezevi, D. Effects of ga geometry in hydraulic comonents on hydraulic system efficiency, Ph.D. Thesis, University of Novi Sad, 007. (in Serbian). [5] Knezevi, D., Savi, V. Mathematical modelling of changing of dynamic viscosity asa function of temerature and ressure of mineral oils for hydraulic system, Facta Universitatis, Series: Mechanical Engineering, 007, vol. 4, no. 1, 006,. 7-4, Niš. [6] Stachowiak G.W., Batchelor A.W., Engineering Tribology, Butterworth- Heinemann, nd edition, 001, ISBN Determination of Pressure Losses in Hydraulic Pieline Systems by Considering Temerature and Pressure 4