Economic diameter of penstock by graphical method

Economic diameter

The economic diameter of a penstock is the diameter for which the total annual cost (the annual fixed charges on the penstock plus the annual value of the power lost in friction) is the minimum. A small diameter is cheap but gives a high velocity and a high head loss; a large diameter is costly but gives a low head loss.

Graphical method

1. Take the design discharge $Q$, the gross head and the penstock length. Select a range of trial diameters $d_1, d_2, \dots, d_n$.
2. For each diameter, find the velocity $v = 4Q/\pi d^2$ and the friction head loss $h_f = fLv^2/2gd$ (or use the Manning or Hazen-Williams formula).
3. Annual cost of the penstock (curve 1): the thickness $t = pd/2\sigma\eta$ gives the steel weight and cost; add fabrication, erection and anchors, and multiply by the annual charge rate (interest + depreciation + maintenance, say 8 to 12 %).
4. Annual cost of the power lost (curve 2): $P_{loss} = \gamma Q h_f \eta_{overall}$ (kW), converted to energy per year and multiplied by the tariff (cost per kW-year or per kWh).
5. Plot the diameter on the x-axis and the annual cost on the y-axis for both curves, then add them to get the total annual cost curve (curve 3).
6. The diameter at the lowest point of the total curve is the economic diameter (it is also where the slopes of curves 1 and 2 are equal and opposite).

 Annual cost
   ^
   |  T                                 T
   |   T                              T
   |  L  T                         T      P
   |   L    T                   T      P
   |    L      T     T     T       P
   |     L           :          P
   |      L          :       P
   |        L        :    P
   |          L      : P
   |            L   P:
   |              X  :
   |            P   L:
   |         P       :L L
   |      P          :    L L L
   |   P             :          L L L L
   +-----------------+-----------------------> d
                 d_economic

   P = annual cost of the penstock (rises with d)
   L = annual cost of the power lost in friction (falls with d)
   T = total annual cost = P + L (minimum at d_economic)

Note: since the cost of the power lost falls steeply with the diameter while the penstock cost rises, their sum has one clear minimum, which is read off the plot.