Subscript x Baseline equals v 1 left-parenthesis minus rho v 1 upper A 1 right-parenthesis"/>
Therefore,
which means that the x‐component of the force acting on the bolts is:
y‐component: With the assumption of no frictional forces, and considering that the exit section A1 is in atmosphere (gage pressure is null):
Due to gravity, the body force is given by the weight of the fluid inside the CV:
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 :
Overall,
which means that the y‐component of the force acting on the bolts is
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.
To understand the nature of flow forces and derive an analytical expression that can be useful to study most of the typical geometries of hydraulic control valves, the simplified case in Figure 3.17 is taken as reference. The figure represents a spool that varies the valve opening between the pump (1) supply and the work port (2). Specifically, the spool can move horizontally along the x axis, and in the position represented in the figure, it determines an effective opening area between the valve ports respectively at the sections indicated in figure with 1 (inflow) and 2 (outflow).
The figure considers the common situation of A1 ≫ A2 so that, per the conservation of mass, higher exit velocities are obtained. This is a very typical working condition of many hydraulic control valves, when low flow areas are implemented to regulate the actuator speed. The figure clearly shows how the flow changes its velocity and direction (therefore, its momentum) as it crosses the valve. Most of this momentum change is caused by the spool itself. The important question to be answered is: what is the force acting on the spool because of this momentum change? The answer is very important to understand how to size the valve actuation system, namely, the system that sets the position of the spool, which can be manually operated, electrically operated, or hydraulically operated. More details on the valve actuation systems will be provided in Chapter 8.
Figure 3.17 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
Assuming uniform flow at both inlet and outlet sections,
(3.45)
