Forces due to Flow Through Expanding or Reducing Pipe Bends

Forces due to Flow Through Expanding or Reducing Pipe Bends Let us consider, a fluid flow through an expander shown in Fig. 11.1a below. The expander is held in a vertical plane. The inlet and outlet velocities are given by V1 and V2 as shown in the figure. The inlet and outlet pressures are also prescribed as p1 and p2. The velocity and pressure at inlet and at outlet sections are assumed to be uniform. The problem is usually posed for the estimation of the force required at the expander support to hold it in position. Fig 11.1a   Flow of a fluid through an expander For the solution of this type of problem, a control volume is chosen to coincide with the interior of the expander as shown in Fig. 11.1a. The control volume being constituted by areas 1-2, 2-3, 3-4, and 4-1 is shown separately in Fig.11.1b. The external forces on the fluid over areas 2-3 and 1-4 arise due to net efflux of linear momentum through the interior surface of the expander. Let these forces be Fx and Fy. Since the control volume 1234 is stationary and at a steady state, we apply Eq.(10.18d) and have for x and y components     (11.1a) and (11.1b) or, (11.2a) and (11.2b) where  = mass flow rate through the expander. Analytically it can be expressed as where A1 and A2 are the cross-sectional areas at inlet and outlet of the expander and the flow is considered to be incompressible. M represents the mass of fluid contained in the expander at any instant and can be expressed as    where  is the internal volume of the expander.   Thus, the forces Fx and Fy acting on the control volume (Fig. 11.1b) are exerted by the expander. According to Newton’s third law, the expander will experience the forces Rx (= − Fx) and Ry ( = − Fy) in the x and y directions respectively as shown in the free body diagram of the expander. in fig 11.1c. Fig 11.1b   Control Volume Comprising the fluid contained in the expander at any instant Fig 11.1c    Free Body Diagram of the Expander The expander will also experience the atmospheric pressure force on its outer surface. This is shown separately in Fig. 11.2. Fig 11.2     Effect of atmospheric pressure on the expander From Fig.11.2 the net x and y components of the atmospheric pressure force on the expander can be written as The net force on the expander is therefore, (11.3a) (11.3b)  or, (11.4a) (11.4b) Note: At this stage that if Fx and Fy are calculated from the Eqs (11.2a) and (11.2b) with p1 and p2 as the gauge pressures instead of the absolute ones the net forces on the expander Ex and Ey will respectively be equal to −Fx and −Fy.