Economic diameter of tunnel

The economic diameter of a tunnel (or penstock) is the diameter for which the total annual cost—the annual cost of construction (capital recovery + maintenance) plus the annual value of energy lost due to friction—is a minimum.

• A larger diameter gives lower velocity and smaller head loss, so less energy is lost, but construction cost (excavation, lining) is higher.
• A smaller diameter is cheaper to build but causes larger friction loss and loss of revenue every year.
• The economic diameter is the best balance between these two opposing costs.

Working it out

1. Annual construction cost. Cost of tunnel of length $L$ is roughly proportional to $D^2$ (excavation and lining). The annual cost is

$$\begin{aligned} C_1 = k_1 D^2 L \times (\text{capital recovery factor} + \text{O\&M rate}) \end{aligned}$$

2. Annual cost of energy lost. Head loss by Manning (for a full circular section $R = D/4$):

$$\begin{aligned} h_f &= \frac{n^2 V^2 L}{R^{4/3}}, \quad V = \frac{4Q}{\pi D^2} \\ \Rightarrow h_f &= 10.29\,\frac{n^2 Q^2 L}{D^{16/3}} \end{aligned}$$

Power lost $P_L = 9.81\,Q\,h_f\,\eta$ kW; annual energy lost $= P_L \times 8760 \times$ plant factor; its value at the energy rate $r$ gives

$$C_2 = \frac{k_2}{D^{16/3}}$$

3. Minimise total cost.

$$\begin{aligned} C &= C_1 + C_2 = k_1' D^2 + \frac{k_2}{D^{16/3}} \\ \frac{dC}{dD} &= 2k_1' D - \frac{16}{3}\frac{k_2}{D^{19/3}} = 0 \\ D_{eco} &= \left(\frac{8 k_2}{3 k_1'}\right)^{3/22} \end{aligned}$$

4. Graphical method (used in practice). Assume several trial diameters, compute for each the annual construction cost and annual loss cost, and plot them against $D$. The diameter at the lowest point of the total cost curve is the economic diameter.

 Annual cost
   ^   \                       / construction
   |    \                    /    cost
   |     \     total cost  /
   |      \__       ___ /
   |        \ ''-.-'' /
   |         \  min /'.
   |          '-..-'   '-. energy loss cost
   |              |       '--.__
   +--------------+---------------> D
                D_eco

5. Checks: velocity within the usual range (about 2–3 m/s in concrete-lined tunnels), minimum size for construction (about 2–2.5 m), and practical round size.