Читать книгу Hydraulic Fluid Power - Andrea Vacca - Страница 68

The force components acting on the elbow can be found by applying the momentum Eq. (3.43) on the CV that includes the elbow. The CV is conveniently chosen to include control surfaces (CS) where the fluid velocity is known. In particular, the CS includes the entrance and exit surfaces, A₁ and A₂, as well as surfaces where there is no flow (therefore with null fluid velocity).

The velocities v₁ and v₂ can be calculated from the flow rates:

The momentum Eq. (3.43) can be solved separately in both the x and y components, considering the CV shown in the above figure.

x‐component: For the calculation of the horizontal component of the total force, only the surface force term has to be considered. Neglecting frictional effects in the elbow, the only force component is given by the pressure force on the area A₁ and the force exerted by the elbow walls. Considering the elbow in atmosphere, it is convenient to use gage pressure instead of absolute pressure:

where R_x = − F_x, meaning that the surface force seen by the fluid has equal but with opposite sign with respect to the force that the fluid exerts to the containing walls.

The x‐component of the second member of Eq. (3.43) can be written by considering the assumption of stationary conditions, which implies that the momentum does not change over time. Therefore, the contribution from the control surfaces is the only one different from zero. Indicating the scalar horizontal velocity component with v_x, the terms at the second member of the momentum equation becomes

There is only one section crossed by the flow where the u velocities are not null, namely, the entrance section A₁. Therefore, the CS term above reduces to the only section A₁. With the assumption of uniform flow,

Overall the x‐component of the momentum equation becomes

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.