Hydraulic Fluid Power - Andrea Vacca

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With the assumption of uniform flow,

integral Underscript italic upper C upper S Endscripts v Subscript x Baseline rho ModifyingAbove v With right-arrow dot d ModifyingAbove upper A With right-arrow equals v 1 left-parenthesis minus rho v 1 upper A 1 right-parenthesis

      Overall the x‐component of the momentum equation becomes

p Subscript 1 g Baseline upper A 1 plus upper R Subscript x Baseline equals v 1 left-parenthesis minus rho v 1 upper A 1 right-parenthesis

      Therefore,

StartLayout 1st Row 1st Column upper R Subscript x 2nd Column equals minus p Subscript 1 g Baseline upper A 1 minus rho v 1 squared upper A 1 equals minus 1 left-bracket italic b a r right-bracket dot 5 left-bracket italic c m squared right-bracket minus 870 left-bracket italic k g slash m cubed right-bracket dot 1.67 squared left-bracket m squared slash s squared right-bracket dot 5 left-bracket italic c m squared right-bracket 2nd Row 1st Column Blank 2nd Column equals minus 51.21 upper N EndLayout upper F Subscript x Baseline equals minus upper R Subscript x Baseline equals 51.21 upper N

      y‐component: With the assumption of no frictional forces, and considering that the exit section A1 is in atmosphere (gage pressure is null):

upper F Subscript upper S comma y Baseline equals upper R Subscript y

      Due to gravity, the body force is given by the weight of the fluid inside the CV:

upper F Subscript upper B comma y Baseline equals minus rho upper V Subscript elbow Baseline g

      The second term of the y‐component of the momentum equation can be written under the same assumption of uniform flow, and considering that vertical components of the velocity at the control surface are present only at the section A2 :

integral Underscript upper C upper S Endscripts v Subscript y Baseline rho ModifyingAbove v With right-arrow dot d ModifyingAbove upper A With right-arrow equals minus v 2 left-parenthesis rho v 2 upper A 2 right-parenthesis

      Overall,

upper R Subscript y Baseline equals minus upper F Subscript upper B comma y Baseline minus rho v 2 squared upper A 2 equals rho upper V Subscript elbow Baseline g minus rho v 2 squared upper A 2 upper R Subscript y Baseline equals 870 left-bracket italic k g slash m cubed right-bracket dot 70 left-bracket italic c m cubed right-bracket dot 9.81 left-bracket m slash s squared right-bracket plus 870 left-bracket italic k g slash m cubed right-bracket dot 2.78 squared left-bracket m squared slash s squared right-bracket dot 3 left-bracket italic c m squared right-bracket equals minus 1.43 upper N

      which means that the y‐component of the force acting on the bolts is

upper F Subscript y Baseline equals minus upper R Subscript y Baseline equals 1.43 upper N

      The orientation of the force on the bolts is therefore similar to the one indicated in the image above, with a horizontal component more pronounced than the vertical one.

      3.8.1 Flow Forces

      Another important application of the momentum equation is for the determination of the so‐called flow forces in hydraulic control valves.

      The flow forces act on the moving element (generally a spool or a poppet) of the valve, and they are generated by the flow through the component. As it will be mentioned in Chapter 8, the presence of flow forces can significantly affect the operation of hydraulic control valves as well as the design of the valve actuation mechanism.

Schematic illustration of the reference geometry for the analysis of flow forces.

      One possible way to answer this question consists in solving the pressure distribution at the spool walls, which is qualitatively shown in Figure 3.17. This approach requires a proper differential flow approach of analysis, where the governing equations are written for a differential fluid element and numerically integrated by mean of a computational fluid dynamics (CFD) tools. Numerical CFD techniques also allows studying more complex geometries that sometimes occurs in modern hydraulic control valves. However, the numerical CFD analysis can be time consuming, and it does not provide an analytical expression of the flow force. This analytical expression can be very useful to the valve designer to gain an intuitive understanding of the development of the flow forces, and more importantly it can be used if to formulate the proper controller parameters in case of hydraulic control valves using closed‐loop controls.

      An analytical expression can be easily calculated applying the momentum equation to a properly selected CV. By considering the annular shaped CV indicated in Figure 3.17, with constant fluid density, the equation becomes