Link BD consists of a single bar 30 mm wide and 12 mm thick. Knowing that each pin has a10 mm diameter, determine the maximum value of the average normal stress in the link BD if θ = 15 degrees.

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Question

Water is pumped from a lower reservoir to a higher reservoir by a pump that provides 23 kW of useful mechanical power to the water. The free surface of the upper reservoir is 57 m higher than the surface of the lower reservoir. If the flow rate of water is measured to be 0.03 m³/s, determine the irreversible head loss of the system and the lost mechanical power during this process.

Solution

Step 1

Given;

Mechanical power by a pump, $P_m=23 kW$

Head difference between reservoirs, $H=57 m$

The flow rate of water, $Q=0.03 m^3/s$

Step 2

Now the mass flow rate of water, $\dot{m}=\rho .A.V=\rho .Q (Q=A.V)$

Here, A is the area of the pipe and V is the velocity of the flow. $\rho$ is the density of water=$1000 kg/m^3$

So, 

 $$\begin{gathered} \dot{m}=\rho .Q=1000\times 0.03 \\ m=30 kg/s \end{gathered}$$

Power required to deliver water into the higher reservoir, 

$$\begin{gathered} P=\dot{m}.g.H \\ P=30\times 9.81\times 57 \\ P=16775.1 W \\ P=16.775 kW \end{gathered}$$            ( g is the acceleration due to gravity=9.81 $m/s^2$ )

Now the lost mechanical power during this process;

 $$\begin{gathered} P_l=P_m-P \\ {P}_l=23-16.775 \\ {P}_l=\mathbf{6}\mathbf{.}\mathbf{225}\mathbf{ }\mathit{k}\mathit{W} \end{gathered}$$

and the irreversible head loss ($h_f$);

now, 

$$\begin{gathered} P_l=\dot{m}.g.h_f \\ 6.225\times {10}^3=30\times 9.81\times h_f \\ {h}_f=\mathbf{21}\mathbf{.}\mathbf{151}\mathbf{ }\mathit{m} \end{gathered}$$

Step 3

So the irreversible head loss, ${\mathit{h}}_{\mathbf{f}}\mathbf{=}\mathbf{21}\mathbf{.}\mathbf{151}\mathbf{ }\mathit{m}$ and the lost mechanical power during this process, $P_l=\mathbf{6}\mathbf{.}\mathbf{225}\mathbf{ }\mathit{k}\mathit{W}$