Hydropower & Turbomachinery Engineering, Francis Reaction Turbine & Thoma Cavitation Law
Francis Turbine Specific Speed & Thoma Draft-Tube Formula
Calculate Metric Power Specific Speed Ns, Required Runner Suction Head NPSH_r (m), Maximum Safe Runner Elevation Above Tailwater Hs,max (m), and Cavitation Submergence Margin (m).
Set a high-specific-speed Francis hydro runner 1.2 m ABOVE tailwater at 1,000 m mountain altitude and draft-tube vacuum boils stainless runner blades into Swiss cheese!
**Francis Reaction Hydro Turbines** recover `10%–25%` of total net head `H` (`m`) inside a diverging **Conical/Elbow Draft Tube** that decelerates runner exit flow into static tailwater pressure—creating strong sub-atmospheric suction right at the runner throat! Classifying runner geometry by **Metric Power Specific Speed `N_s = (N * SQRT(P_{MW} * 1,000)) / H^1.25`** (`60–120` slow high-head Francis, `120–250` medium Francis, `250–400` fast low-head Francis; `> 400` Kaplan axial), **Dieter Thoma’s (`1924`) Cavitation Law** defines **Critical Cavitation Coefficient `sigma_c = NPSH_r / H`**! Subtracting **Required Suction Head `NPSH_r = sigma_c * H`** (`m`) and **Water Vapor Head `H_v`** (`m`) from **Altitude Barometric Head `H_{atm}`** (`m`) fixes **Maximum Safe Runner Setting Elevation Above Tailwater `H_{s,max} = H_{atm} - H_v - sigma_c * H`** (`m`—whenever `H_{s,max} < 0`, the powerhouse MUST be excavated **below minimum tailwater level**!)!
Verify on a `42` MW, `85` m Head Medium-Speed Francis Unit (`300` RPM) near Sea Level (`H_{atm} = 10.10` m)—Net_Head `H` = 85.0 m (B2), Synchronous_Speed `N` = 300 RPM (B3), Shaft_Power `P` = 42.0 MW (B4), Barometric_Head `H_{atm}` = 10.10 m (B5), `20°C` Vapor_Head `H_v` = 0.24 m (B6), Thoma_Sigma `sigma_c` = 0.088 (B7), and Installed_Runner_Setting `H_s` = +1.20 m above tailwater (B8): Metric Specific Speed `N_s` equals `238.2` (E2), `NPSH_r` equals `7.48` m (E3), Max Allowable Elevation `H_{s,max}` equals `+2.38` m (E4), leaving `+1.18` m positive anti-cavitation margin (E5)!
| Francis Turbine Thoma KPI (300 RPM, 42 MW, Hv=0.24 m, Installed Hs=+1.20 m) | Formula | Verified Output |
|---|---|---|
| Metric Power Specific Speed Ns | (RPM * SQRT(Power_MW * 1000)) / (Head_m ^ 1.25) | 238.2 medium Francis (486.6 axial/fast!) |
| Required Suction Head NPSHr (m) | Thoma_Crit_Sigma * Net_Head_H_m | 7.48 m (10.56 m @ sigma=0.22) |
| Max Runner Elevation Hs,max (m) | Barometric_m - Vapor_Hv_m - (sigma * H_m) | +2.38 m (-1.60 m MUST SUBMERGE!) |
| Net Submergence Safety Margin (m) | Max_Setting_Hs_max_m - Installed_Hs_m | +1.18 m safe (-2.80 m SEVERE CAVITATION!) |
Thoma (1924) Hydro-Turbine Cavitation Sigma & Draft-Tube Submergence Formula
Validate `Net_Head H` in `[5, 800]` m, `Runner_RPM` in `[50, 1,500]`, `Power` in `[0.1, 1,000]` MW, `Barometric_Head H_{atm}` in `[6.5, 10.6]` m, `Vapor_Head H_v` in `[0.05, 1.5]` m, `Thoma sigma_c` in `[0.02, 1.50]`, and `Installed H_s` in `[-30, 8]` m!
=LET(hn_m, B2, nr_p, B3, pm_w, B4, ha_m, B5, hv_m, B6, sc_t, B7, hs_i, B8, hm_x, ha_m - hv_m - (sc_t * hn_m), IF(OR(hn_m < 5, hn_m > 800, nr_p < 50, nr_p > 1500, pm_w < 0.1, pm_w > 1000, ha_m < 6.5, ha_m > 10.6, hv_m < 0.05, hv_m > 1.5, sc_t < 0.02, sc_t > 1.5, hs_i < -30, hs_i > 8), "Invalid Francis Input", ROUND(hm_x - hs_i, 2)))Low-Head High-Altitude Powerhouse (`H = 48` m, `H_atm = 9.2` m, `sigma = 0.22`) Cavitation Check
Re-apply the `+1.20` m setting to a `48` m head, `800` m altitude site (`B2 = 48`, `B5 = 9.2`, `B7 = 0.22`): Specific Speed E2 jumps to `486.6`, `H_{s,max}` E4 turns negative (**`-1.60` m**—requiring deep underground powerhouse submergence at least `1.60` m below minimum tailwater!), and Margin E5 plunges to **`-2.80` m**, triggering violent draft-tube vortex-rope surge and cavitation erosion!
=IF(ISTEXT(E5), E5, ROUND(B5 - B6 - (B7 * B2), 2))Try the example
Use the original values or change them to test your understanding.