Question 2 of 6: Transmission main with a control valve

Question 2: Transmission main with a control valve (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. $L=5{,}000\text{ m}$, $C=120$, $D=0.45\text{ m}$, $H_{u/s}=95\text{ m}$, $E_s=0.45\text{ m}^{5/2}/\text{s}$; valve law $Q=\tau E_s\sqrt{\Delta H_{\text{valve}}}$.

Find. (a) the downstream hydraulic grade line $H_{d/s}$ for $Q=0.2$, $\tau=0.8$; (b) the discharge when $\tau$ drops to 0.6 with $H_{d/s}$ fixed.

Approach. Local losses and velocity head are negligible (Note 6), so the reservoir-to-reservoir energy balance is $H_{u/s}-H_{d/s}=h_{f,\text{pipe}}(\text{full }5{,}000\text{ m})+\Delta H_{\text{valve}}$, with the valve drop from the valve law $\Delta H_{\text{valve}}=(Q/\tau E_s)^2$.

1. (a) Valve head drop. Rearranging the valve law, $$\Delta H_{\text{valve}}=\left(\frac{Q}{\tau E_s}\right)^2=\left(\frac{0.2}{0.8\cdot0.45}\right)^2=0.309\ \text{m}.$$
2. (a) Pipe friction over 5,000 m. With $k=L/(0.278\,C\,D^{2.63})^{1.852}=369.1$ (SI), $$h_{f,\text{pipe}}=kQ^{1.852}=369.1\,(0.2)^{1.852}=18.74\ \text{m}.$$
3. (a) Downstream grade line. Energy balance A→B: $$H_{d/s}=H_{u/s}-h_{f,\text{pipe}}-\Delta H_{\text{valve}}=95-18.74-0.309 \;\Rightarrow\; \boxed{H_{d/s}=75.95\ \text{m}.}$$
4. (b) Re-close the valve. Now $\tau=0.6$ and $H_{d/s}=75.95$ m is held, so the available head is $95-75.95=19.05$ m and must again be split between pipe friction and the valve: $$kQ^{1.852}+\left(\frac{Q}{0.6\cdot0.45}\right)^2 = 19.05.$$
5. (b) Solve for the discharge. Iterating (the friction term dominates), $$\boxed{Q = 0.199\ \text{m}^3/\text{s}.}$$ The friction loss is 18.5 m and the valve now drops 0.54 m.

Notice how little the discharge changes: tightening the valve from $\tau=0.8$ to $0.6$ moves $Q$ only from 0.200 to 0.199 m³/s. This 5,000 m / 450 mm main is friction-dominated — the pipe eats ~18.7 m of the 19 m budget while the valve contributes well under 1 m — so throttling the valve is an ineffective flow control here. The valve law text in part (b) says “held at the level computed in (b)”; this is a typo for the level computed in part (a), which is the value used above.