25.1 SECTION 25 HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN Determining Response Time of Pilot- Operated Solenoid-Energized Spool Valves in Hydraulic Systems 25.1 Hydraulic-System Reservoir and Heat Exchanger Selection and Sizing 25.12 Choosing Gaskets for Industrial Hydraulic Piping Systems 25.19 Computing Friction Loss in Industrial Hydraulic System Piping 25.26 Hydraulic-Cylinder Clearance for Damping End-of-Stroke Forces 25.29 Hydraulic System Pump and Driver Selection 25.32 Hydraulic Piston Acceleration, Deceleration, Force, Flow, and Size Determination 25.36 Hydropneumatic Accumulator Design for High Force Levels 25.39 Membrane Vibration in Hydraulic Pressure-Measuring Devices 25.41 Power Savings Achievable in Industrial Hydraulic Systems 25.42 Pneumatic-Circuit Analysis Using Various Equations and Coefficients 25.44 Air Flow Through Close-Clearance Orifices in Pneumatic Systems 25.49 Labyrinth Shaft Seal Leakage Determination 25.58 Pipe-Wall Thickness for Hydraulic Systems without Fluid Shock 25.67 Pipe-Wall Thickness for Hydraulic Systems with Fluid Shock 25.68 Allowable Stress in Hydraulic System Piping 25.68 Hydraulic Fluid Compressibility and System Bulk Modulus 25.69 Selection of Fluids for Oil Hydraulic and Control Systems 25.69 Effect of Trapped Air on Hydraulic System Bulk Modulus 25.71 Surge Pressure in Hydraulic Cylinders 25.72 Sizing a Hydraulic System Fluid Reservoir 25.72 Required Volume of Bladder-Type Accumulator 25.73 Determining Hydraulic Accumulator Demand Volume 25.74 Effective Force Developed by a Double- Acting Hydraulic Cylinder 25.74 Hydraulic Cylinder Oil Consumption and Extension Time 25.75 Hydraulic Cylinder Power Output 25.76 Selecting Hydraulic Motors and Pumps by Using Manufacturer’s Size Tables 25.76 DETERMINING RESPONSE TIME OF PILOT- OPERATED SOLENOID-ENERGIZED SPOOL VALVES IN HYDRAULIC SYSTEMS A pilot-operated solenoid-energized spool control valve in a hydraulic system has the dimensions, operating pressures, and performance given in Table 1. Pilot supply pressure is 100 lb/in 2 (689 kPa); main supply pressure is 500 lb/in 2 (3445 kPa). Find the maximum velocity of this valve, its acceleration time, and the total re- sponse time. Next, using the same dimensions and main operating pressure, find Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Source: HANDBOOK OF MECHANICAL ENGINEERING CALCULATIONS 25.2 DESIGN ENGINEERING TABLE 1 Dimensions and Operating Conditions* Pilot Spool Main spool Diameter, in d ϭ 0.25 D ϭ 2.5 Mass, lb-sec 2 /in m ϭ 0.0002 M ϭ 0.05 Stroke, in s ϭ 0.375 S ϭ 1.5 Land length, in — L ϭ 6.0 Radial clearance, in — C ϭ 0.0003 Coefficient of friction — F R ϭ 0.04 Solenoid force, lb (initial; final) F ϭ 1; 8.5 SOL — Back pressure, lb / in 2 — p B ϭ 20 Supply pressure, lb/in 2 p ϭ 100 P ϭ 500 Differential pressure, lb/in 2 ⌬p ϭ 70 (approx) ⌬P ϭ 450 (approx) Port area, in 2 a 0 ϭ 0.05 A M ϭ 1.2 Flow coefficient ƒ ϭ 0.6 ƒ ϭ 0.6 Viscosity, CP ϭ 80 ϭ 80 Density, lb-sec 2 /in 4 0.000,085 0.000,085 *SI values given in calculation procedure. the same unknowns when the pilot pressure is made equal to the main operating pressure i.e., 500 lb /in 2 (3445 kPa). As a further modification, a small actuating piston is placed at each end of the main spool, Fig. 3, to increase the longitudinal velocity for a given pilot-fluid flow rate. Trial and error would normally be used to calculate the most effective diameter for the actuating piston. In this procedure we will use a diameter d x ϭ 1.4 in (3.56 cm) for this small actuating piston. If the dimensions and operating pressures are unchanged, analyze the valve for the same unknowns as above. Calculation Procedure: 1. Compute the axial force on the main spool of this valve The forces acting on the main spool at maximum velocity are: Pilot backpressure, p B ; viscous damping force, D V ; and radial jet force P , Fig. 2. From the equation, rad P ϭ 2F ƒA ⌬P as r M where the symbols are as given, Table 2. Then, P ax ϭ 2(0.04)(0.6)(1.2)(450) ϭ 26 lb (115.6 N). converting to pressure by dividing by the area of the main spool valve end, we have 26/4.9 ϭ 5.3 lb / in 2 (36.5 kPa). 2. Compute the combined hydrodynamic resistance of the valve Provisionally, estimate that D V is equivalent to 3.2 lb/in 2 (22 kPa) and P B ϭ pilot- valve backpressure ϭ 20 lb/in 2 (138.8 kPa). The combined hydrodynamic resis- tance is then the sum of: Radial pressure, lb/in 2 (kPa) ϩ Viscous drag, lb/in 2 (kPa) ϩ Pilot-valve backpressure, lb/in 2 (kPa). Or combined hydrodynamic resistance ϭ 5.3 ϩ 3.2 ϩ 20 ϭ 28.5 lb / in 2 (196.4 kPa). Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN 25.3 TABLE 2 Valve Symbology* Pilot Actuating piston Main valve Spool Dimensions and Mass Diameter, in Cross-sectional area, in 2 Mass, lb-sec 2 /in Stroke: Intermediate Full Engagement (length in contact), in Land length, in (total) Spool-to-bore radial clearance, in d — m x s — — C d x a p — — S a l p — C D A s M — S — L C Solenoid Forces Initial, lb Gradient, lb/in Final, lb Ratio, A/B A B F SOL r — — — — — — — — Drag Forces Back pressure, psi Viscous drag, lb (or psi) Radial jet, lb Coefficient of friction Axial jet, lb Acceleration force, lb p B — — — — F ϭ ma p B d V P rad F R P ax — p B D V P rad F R P zx F ϭ Ma Oil pressure, flow, and port size Pressure: Supply, psi Pilot downstream, psi Differential, psi Port area, in 2 (effective orifice) Flow coefficient (0.55 to 0.70) Viscosity, centipoise Oil density, lb-sec 2 /in 4 Flow rate, in 3 /sec Oil velocity, in/sec (through port) Oil mass flow, lb-sec/in Oil-jet deflection angle, deg p p 1 ⌬p a 0 ƒ q — — — p p 1 ⌬p — ƒ q — — — P p 1 ⌬P A M ƒ — V 0 M ƒ a Valve response Acceleration time, sec Shifting velocity, in/sec Shifting time, sec (after energization) t a v p t T a v T T a V T *SI values given in calculation procedure. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN 25.4 DESIGN ENGINEERING 3. Calculate the pilot-valve flow rate The pilot-valve pressure differential, delta P ϭ 100 Ϫ 28.5 ϭ 71.5 lb/in 2 (492.6 kPa). Hence, the valve flow rate is, using the equation below 2⌬p q ϭ ƒa o Ί where q ϭ flow, in 3 /s (mL / s); ƒ ϭ flow factor, dimensionless, ranging from 0.55 to 0.70 depending on valve type; a o ϭ cross-sectional area, in 2 (cm 2 ); of the min- imum port opening—usually the drilled port hole; ⌬ p ϭ p Ϫ p 1 ϭ differential pressure, lb/in 2 (kPa) measured across the pilot inlet and outlet ports; p ϭ fluid mass density, lb-s 2 /in 4 , normally 0.000085 for oil. Substituting, q ϭ 0.6 (0.5)[(2)(71.5)/0.000085)] ϭ 40 in 3 /s (656 mL /s), using a value of ƒ ϭ 0.6 for 0.5 this valve. 4. Determine the maximum velocity of the main spool and the viscous damping force The maximum velocity of the main spool, Fig. 1, V ϭ (flow rate, in 3 /s/(area of spool end, in 2 ) ϭ 40 / 4.9 ϭ 8.2 in/s (20.8 cm/s). Knowing the velocity, we can find the damping force, D V , from D LV D ϭ V 6 C ϫ 6.9 ϫ 10 where D ϭ spool diameter, in (cm); L ϭ length of spool lands, in (cm); V ϭ main spool velocity, in / s (cm/s); mu ϭ absolute viscosity, centipoise; C ϭ spool-to-bore radial clearance, in (cm). If the temperature varies more than 30 to 50 degrees, it is nearly impossible to compute the viscous resistance. Substituting, D V ϭ 2.5 (6)(8.2)(80)/(0.0003)(6.9 ϫ 10 6 ) ϭ 3.05 lb/in 2 (21 kPa). Thus, the provisional estimate of D V ϭ 3.2 was close enough (within 4.9 percent) and recalculation is not necessary. 5. Find the accelerating pressure and acceleration time of the spool The forces acting upon the spool during acceleration are: p R , P ax , D V , and F, where F ϭ Ma. Assuming a mean value for initial port opening A M ϭ 0.4 in 2 (2.58 cm 2 ), then from P ϭ 2ƒA ⌬P cos ␣ as M where ␣ normally varies from 70 degrees at initial opening to 90 degrees at full opening. In calculations, use the axial jet pressure during initial opening, and the axial component of radial pressure during the remainder of travel. Substituting, P ax ϭ 2(0.6)(0.4)(450)(0.26) ϭ 56 lb (248.1 N). Then 56 / 4.9 ϭ 11.4 lb / in 2 (78.5 kPa), ␣ ϭ 75 deg; cos ␣ ϭ 0.26. Viscous drag will be the average: D V ϭ 3.2/2 ϭ 1.6 lb/in 2 (11 kPa). Backpres- sure is still p B ϭ 20 lb / in 2 (137.8 kPa). So the total is 11.4 ϩ 1.6 ϩ 20 ϭ 33 lb / in 2 (227.4 kPa). Therefore, accelerating pressure ϭ 100 Ϫ 33 ϭ 67 lb / in 2 (461.6 kPa). Con- verting to force, we have 67 (4.9) ϭ 328 lb (1441.2 N). The acceleration time, t a s ϭ MV /F ϭ 0.05 (8.2)/328 ϭ 0.0013 s. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN 25.5 FIGURE 1 Typical solenoid-energized pilot-operated spool valve. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN 25.6 DESIGN ENGINEERING FIGURE 2 Jet-force drag in pilot-operated spool valves. 6. Determine the main spool displacement and the energization time interval The displacement of the main spool during the acceleration period is negligible, being less than 1 percent of the total stroke. Time for the total stroke of 1.5 in (3.8 cm) is 1.5/8.2 ϭ 0.182 s, and the time interval from energization of the solenoid to completion of the main valve stroke, T ϭ 0.190 s. 7. Analyze the valve with the higher pilot pressure Much larger flow will pass through the pilot valve because of the higher pressure. Maximum velocity period: P ax ϭ 5.3 lb/in 2 (36.5 kPa), the same as before; D V ϭ 7.7 lb/in 2 (53.1 kPa)— a higher estimate, proportional to the anticipated velocity; p B ϭ 20.0 lb/in 2 (137.8 kPa), the same as before. The total is 33.0 lb / in 2 (227.5 kPa). The new ⌬ P ϭ 467 lb/in 2 (3217.6 kPa), and Q ϭ 4.7 (467) ϭ 102 in 3 /s 0.5 (1671.5 mL/s); V ϭ 102/4.9 ϭ 20.8 in / s (52.1 cm/s); D V ϭ 1.82 (20.8) ϭ 37.8 lb (168.1 N) ϭ 7.75 lb / in 2 (53.4 kPa), which proves out the assumption of 7.7 lb /in 2 (53.1 kPa). Accelerating time, t a ϭ (0.05)(20.8)/(2280) ϭ 0.0005 s. The 1.5-in (3.81-cm) stroke takes 1.5/20.8 ϭ 0.072 s. Total time ϭ 0.081 s. The flow rate of the pilot oil is more important than pressure intensity in ob- taining a fast-acting valve. A slightly larger pilot valve and enlarged porting have a marked effect on the operational speed of the main valve. Note that increasing the pilot pressure fivefold, from 100 lb/in 2 to 500 lb / in 2 (689 kPa to 3445 kPa) only doubles the speed of response, from 0.19 s to 0.08 s. Increasing the port area can result in an nearly proportional gain in speed, and no Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN 25.7 additional pressure is necessary, saving on pumping costs. Costs of producing a 0.375-in (9.52-mm) pilot spool are not much greater than those for a 0.25-in (6.35- mm) spool. The increase in capacity is 50 percent without the additional heat losses entailed by an increase in pressure. 8. Analyze the valve fitted with actuating pistons For the valve, Fig. 3, with the small actuating pistons, taking the summation of the viscous drag, ͚D V , and inserting the known optimum values in parentheses after the computed values, we have: 8.2 ϫ 4.9 ϫ 4 ϫ 80 ͚D ϭ V 6 0.0003 ϫ 6l9 ϫ 10 ϫ 1.4 2.5 ϫ 6 ϫϩ2 ϫ 1.5 ͩͪ 1.4 12800 15 ϭϩ3 ͩͪ 2075 ϫ 1.4 1.4 ϭ 60.5 lb (269.1 N) [59.0 lb optimum; 262.4 N] Introducing the value of ͚ D V in the equation, 3k 3(P ϩ ͚D ) 3 ax V a ϭϭ p 2k 2(p Ϫ p ) 2 B we have 3(26 ϩ 60.5) 22 a ϭϭ1.62 in (10.45 sq cm) [1.59 in ; 10.26 cm ] p 2(100 Ϫ 20) d ϭ 1.44 in (3.66 cm) [1.425 in, 3.62 cm]. With optimum a p ϭ 1.59 cu in (26.06 mL), piston velocity using 2⌬p q ϭ ƒa o Ί kk 23 v ϭ k Ϫ 1 23 Ί aa pp 22 kk kk 12 13 ϭϪ 23 Ί aa pp 80 (26 ϩ 59) v ϭ 4.6 Ϫis p Ί 2.53 4.0 ϭ 15.0 in/s (38.1 cm / s) The total time, T ϭ 0.100 ϩ 0.009 ϭ 0.109 s. Using a pilot pressure of 500 pst (3445 kPa), V ϭ 20.8 in / s (52.8 cm / s) and d ϭ 1.06 in (2.69 cm). Then: Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN 25.8 FIGURE 3 Piston-operated solenoid-energized spool valve. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN 25.9 TABLE 3 Effect of Adding Actuating Pistons* Pilot pressure p, psi Main valve diameter, in Maximum valve velocity, without piston in/sec Total shift time T, sec Maximum valve velocity, with piston in sec Piston diameter, in Total shift time T, sec 100 2.50 8.2 0.190 15.0 1.425 0.109 500 2.50 20.8 0.081 55.0 1.06 0.036 *SI values given in calculation procedure. 20.8 ϫ 4.9 ϫ 4.80 ͚DV ϭ 6 0.0003 ϫ 6.9 ϫ 10 ϫ 1.06 15 ϫϩ3 ϭ 258 lb ͩͪ 1.06 3(26 ϩ 258) 2 a ϭϭ0.886 in p 2(500 Ϫ 20) d ϭ 1.06 in z 480 284 v ϭ 4.6 Ϫ p Ί 0.784 0.61 ϭ 55 in/s T ϭ 0.036 s Table 3 and Fig. 4 depict the effect of actuating-piston area upon the spool shifting velocity and shifting time. Related Calculations. Pilot-operated flow-control valves are probably the most common valves used in industrial hydraulic systems. Speed of response of these valves is important during the design and operation of any hydraulic system. The procedure given here analyzes the speed of response of a typical valve in terms of the fluid flow rate, characteristic force-vs-airgap curve of the solenoid; shape, size, clearance, and displacement of each spool; and the fluid viscosity. The method given in this procedure relates the above parameters for the valve in Fig. 1. and can be applied to any other pilot-operated spool valve. And the procedure includes a special technique for a large spool valve, Fig. 3, actuated by a small auxiliary piston. In the sequence of operation of solenoid-energized pilot-operated spool valve, here is what happens. The solenoid is energized, the pilot spool moves quickly to the full open position, Fig. 5, and the main spool is shifted at a rate determined by the amount of fluid that can move through the pilot ports against these five resisting forces: (1) pilot system backpressure, lb/in 2 (kPa); (2) viscous damping force, lb (N); (3) radial jet force, lb (N); (4) axial jet force, lb (N); (5) acceleration force, lb (N). Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN 25.10 DESIGN ENGINEERING SI Values in./sec cm/sec in. 2 cm 2 5 12.7 1 6.45 10 25.4 2 12.9 15 38.1 3 19.4 20 50.8 4 25.8 25 63.5 5 32.3 FIGURE 4 Effect of varying piston diameter. FIGURE 5 Before energization and after full stroke of a solenoid-energized spool valve. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN