Let's analyze and solve the problem step by step. --- # Given: - Efficiency of each Pelton turbine:97%** Power output per turbine: ** MW** - Total flow rate: **4 m³/s** - of water: **83 m/s** - Water jets drive the turbines with multiple jets (number of jets per turbine not specified directly, but implied to be consistent- Water deflects by: **150 degrees** (for force estimation) - Temperature or other parameters are not given, so focus on fluid dynamics and energy principles. --- # Part (a): Determine the maximum head (h_t) provided by one turbine ### Step 1: Convert power to useful hydraulic power Since efficiency (η) = 97%, the hydraulic power (P_hydraulic) supplied to the turbine is: \[ P_{hydraulic} = \frac{P_{output}}{\eta} \] \[ P_{hydraulic} = \frac{55 \text{ MW}}{.97} \approx 56.7 \text{ MW} \] ### Step 2: Calculate the hydraulic power using flow rate and head Hydraulic power can be expressed as: \[ P_{hydraulic} = \rho \cdot g \cdot Q \cdot h_t \] Where: - \(\rho\) = density of water = **100 kg/m³** - \(g\) = acceleration due to gravity = **9.81 m/s²** - \(Q\) = flow rate = **4 m³/s** - \(h_t\) = head in meters (unknown, to find) ### Step 3: Solve for \(h_t\) \[ h_t = \frac{P_{hydraulic}}{\rho \cdot g \cdot Q} \] \[ h_t = \frac{56.7 \times 10^6}{100 \times 9.81 \times 4} \] \[ h_t \approx \frac{56.7 \times 10^6}{39,240} \] \[ h_t \approx 1444.8 \text{ meters} \] --- # **Answer for Part (a):** **Maximum head \(h_t\) provided by one turbine is approximately \(\boxed{1445 \text{ meters}}\).** --- # Part (b): Estimate the force (kN) exerted by water on the turbine blade when the water is deflected by 150 degrees ### Step 1: Understand the problem The force exerted by the water depends on the change in momentum of water as it strikes and deflects off the turbine blades. ### Step 2: Calculate the initial momentum flux The velocity of water jets: \[ V = 83 \text{ m/s} \] Flow rate: \[ Q = 4 \text{ m}^3/\text{s} \] Number of jets isn't specified explicitly, but since the problem asks for the force exerted by water on one turbine, assume the total flow rate applies to all jets combined. The mass flow rate: \[ \dot{m} = \rho \times Q = 100 \times 4 = 400 \text{ kg/s} \] ### Step 3: Determine change in velocity components Water strikes the blades with velocity \(V\). After deflection by \(\theta = 150^\circ\), the change in velocity component in the direction of the jet is: \[ \Delta V = V (\cos 150^\circ - \cos ^\circ) \] But more straightforwardly, the force relates to the change in momentum flux: \[ F = \dot{m} \times \Delta V \] Since the water is deflected by 150 degrees, the change in velocity component along the original direction: \[ \Delta V = V (1 - \cos 150^\circ) \] \[ \cos 150^\circ = -\cos 30^\circ = -.866 \] \[ \Delta V = 83 \times (1 - (-.866)) = 83 \times 1.866 \approx 154.9 \text{ m/s} \] ### Step 4: Calculate the force \[ F = \dot{m} \times \Delta V = 400 \times 154.9 \approx 619,600 \text{ N} \] Convert to kN: \[ F \approx 620 \text{ kN} \] --- # **Final answer for Part (b):** **The force exerted by the water on one turbine blade when deflected by 150 degrees is approximately \(\boxed{620 \text{ kN}}\).** --- **Summary:** - Part (a): Maximum head \(h_t \approx 1445 \text{ meters}\) - Part (b): Force exerted \( \approx 620 \text{ kN} \) Let me know if you'd like further clarification!