Economic diameter of penstock

Economic diameter

The economic diameter of a penstock is the diameter for which the total annual cost (annual cost of the penstock + annual value of energy lost in friction) is minimum.

• A larger diameter gives lower velocity and lower head loss (more energy), but costs more steel, excavation, supports and transport.
• A smaller diameter is cheaper to build, but head loss $h_f = \dfrac{8fLQ^2}{\pi^2 g D^5}$ is high, so energy is lost every year.

Cost
 |T                        T
 | T                     T
 |  T                  T
 |   T     min       T
 |     T T  *  T T        C
 |E         :        C
 | E        :    C
 |   E      : C
 |     E   C:
 |       C  E
 |   C      :  E   E   E
 +----------:-------------> D
         D_econ
 T = total, C = penstock cost,
 E = value of energy lost

Method of determination (analytical/graphical)

1. Fix data: design discharge Q, gross head H, penstock length L, friction factor f, energy rate, interest rate, life, steel price.
2. Assume trial diameters $D_1, D_2, \dots$ (e.g. velocity 2–6 m/s).
3. Penstock cost: for each D, find shell thickness $t = \dfrac{pD}{2\sigma\eta_j}$ + corrosion allowance (p includes water hammer), steel weight $= \pi D t L\rho_s$, then capital cost (steel + fabrication + supports + excavation). Annual cost = capital × capital recovery factor + O&M.
4. Energy loss cost: $h_f = \dfrac{8fLQ^2}{\pi^2 g D^5}$; power lost $= 9.81Qh_f\eta$ kW; annual energy lost $=$ power × hours of operation (× plant factor); its value = energy × tariff.
5. Total annual cost = penstock annual cost + value of lost energy.
6. Plot the three curves against D; the D at the lowest total cost is the economic diameter. Analytically, set $\dfrac{d(\text{total cost})}{dD} = 0$.
7. Round up to a standard size and check velocity, water hammer, and that head loss is usually 2–5% of gross head.

Empirical formulas (for preliminary design)

With P = rated power in kW, H = head in m, D in m:

• Sarkaria (1979): $D = 0.62\dfrac{P^{0.35}}{H^{0.65}}$
• USBR: $D = 0.176\left(\dfrac{P}{H}\right)^{0.466}$
• Fahlbusch (1987): $D = 0.52\,H^{-0.17}\left(\dfrac{P}{H}\right)^{0.43}$
• Velocity method: $D = \sqrt{4Q/(\pi V)}$ with economic velocity about 3–5 m/s.

These give a first estimate, which is refined by the cost analysis above.