# Steam Turbine Problem Solution Let's break down the problem step by step. --- ## **Given Data** - **Number of stages:** 4 - **Blades per stage:** 8 (stage 1), 14 (stage 2), 20 (stage 3), 36 (stage 4) - **Blockage ratio (β):** .5 (for all stages) - **Smallest blade diameter (D₁):** .5 m - **Largest blade diameter (D₄):** 2 m - **Drag coefficient (C_d):** .03 - **Lift coefficient (C_L):** .6 - **Boiler pressure (P₁):** 10 bar - **Boiler outlet temperature (T₁):** 800°C = 1073.15 K - **Turbine isentropic efficiency (η_turbine):** .87 - **Condenser pressure (P₂):** 50 kPa = .5 bar - **Electrical Power Output (W_elec):** 10 MW - **Generator efficiency (η_gen):** .97 --- ## **Step 1: Find Thermal Efficiency** ### **A. Find Actual Turbine Power Output** \[ W_{\text{turbine}} = \frac{W_{\text{elec}}}{\eta_{\text{gen}}} = \frac{10,000}{.97} = 10,309\, \text{kW} = 10.31\, \text{MW} \] ### **B. Find Enthalpies Using Steam Tables** #### **State 1 (Turbine Inlet: 10 bar, 800°C)** - At 10 bar, 800°C: - Superheated steam - From steam tables: \( h_1 \approx 3583 \text{ kJ/kg} \) \( s_1 \approx 7.014 \text{ kJ/kg·K} \) #### **State 2 (Turbine Exit: .05 MPa = .5 bar, s = s₁)** - At .5 bar: - \( s_f = 1.091 \) - \( s_g = 7.593 \) - \( h_f = 340.54 \) - \( h_g = 2643 \) - Isentropic expansion: \( s_2 = s_1 = 7.014 \) - Quality (\( x_2 \)): \[ x_2 = \frac{s_2 - s_f}{s_g - s_f} = \frac{7.014 - 1.091}{7.593 - 1.091} = \frac{5.923}{6.502} \approx .911 \] - Enthalpy at state 2s: \[ h_{2s} = h_f + x_2 (h_g - h_f) = 340.54 + .911 \times (2643 - 340.54) = 340.54 + .911 \times 2302.46 = 340.54 + 2098.34 = 2438.9\, \text{kJ/kg} \] #### **Turbine Actual Enthalpy Drop** \[ h_2 = h_1 - \eta_t (h_1 - h_{2s}) = 3583 - .87 \times (3583 - 2438.9) = 3583 - .87 \times 1144.1 = 3583 - 995.37 = 2587.6\, \text{kJ/kg} \] #### **Turbine Specific Work** \[ w_t = h_1 - h_2 = 3583 - 2587.6 = 995.4\, \text{kJ/kg} \] #### **Steam Mass Flow Rate** \[ \dot{m} = \frac{W_{\text{turbine}}}{w_t} = \frac{10,309}{995.4} = 10.36\, \text{kg/s} \] ### **C. Find Heat Supplied in Boiler** Assume feedwater enters boiler at condenser pressure and temperature. - At .5 bar: \( h_f = 340.54\, \text{kJ/kg} \) - Pump work (to be found below; for now, estimate \( h_4 \approx h_f \)) \[ q_{\text{in}} = h_1 - h_4 \approx 3583 - 340.54 = 3242.5\, \text{kJ/kg} \] ### **D. Thermal Efficiency** \[ \eta_{th} = \frac{W_{\text{turbine}} - W_{\text{pump}}}{\dot{m} \times q_{\text{in}}} \] But \( W_{\text{pump}} \) is small; for first approximation: \[ \eta_{th} \approx \frac{W_{\text{turbine}}}{\dot{m} \times q_{\text{in}}} = \frac{10,309}{10.36 \times 3242.5} = \frac{10,309}{33,608} = .307 = \boxed{30.7\%} \] --- ## **Step 2: Plant Layout and T-S Diagram** ### **A. Plant Layout** ``` [Boiler] --(h₁, 10 bar, 800°C)--> [Turbine] --(h₂, .5 bar)--> [Condenser] --(h₄, .5 bar, Sat. Liquid)--> [Pump] --(h₃, 10 bar)--> [Boiler] ``` ### **B. T-S Diagram** Draw a Rankine cycle on T-S axes: - **1 → 2:** Turbine expansion (isentropic, vertical), then real (slightly rightward) - **2 → 3:** Condensation (horizontal left) - **3 → 4:** Pump (almost vertical up) - **4 → 1:** Boiler heating (horizontal right, then vertical up for superheat) *(You can sketch this on paper or with a plotting tool)* --- ## **Step 3: Find Pump Power Consumption** ### **A. Pump Work per kg** \[ w_{pump} = v_f \cdot (P_1 - P_2) \] At .5 bar: \( v_f \approx .00101\, \text{m}^3/\text{kg} \) - \( P_1 = 10\, \text{bar} = 1,000\, \text{kPa} \) - \( P_2 = .5\, \text{bar} = 50\, \text{kPa} \) - \( \Delta P = 950\, \text{kPa} = 950,000\, \text{N/m}^2 \) \[ w_{pump} = .00101 \times 950,000 = 959.5\, \text{J/kg} = .96\, \text{kJ/kg} \] ### **B. Total Pump Power** \[ W_{pump} = \dot{m} \times w_{pump} = 10.36 \times .96 = 9.95\, \text{kW} \] --- ## **Step 4: Torque for Each Turbine Stage** The total turbine power (\( W_{\text{turbine}} \)) is distributed over 4 stages. Assume equal power per stage (approximate, unless velocity/enthalpy drops specified). \[ W_{stage} = \frac{W_{\text{turbine}}}{4} = \frac{10,309}{4} = 2,577\, \text{kW} \] ### **Blade Diameters** - **Stage 1:** \( D_1 = .5\, \text{m} \) - **Stage 2:** interpolate: \( D_2 = .5 + \frac{2-.5}{3} = 1.\, \text{m} \) - **Stage 3:** \( D_3 = .5 + 2 \times \frac{2-.5}{3} = 1.5\, \text{m} \) - **Stage 4:** \( D_4 = 2.\, \text{m} \) ### **Torque Formula** \[ \text{Power} = T \cdot \omega \] or \[ T = \frac{W_{stage}}{\omega} \] We need angular speed (\( \omega \)). Let's find it in next step and then compute \( T \). --- ## **Step 5: Find Turbine Rotational Speed (\( N \))** ### **A. Use Blade Velocity and Lift to Find Speed** The tangential force per stage: \[ F_{\text{lift}} = N_{blades} \cdot \frac{1}{2} \rho V^2 A \cdot C_L \cdot \beta \] But turbine total torque can be estimated as: \[ T = \frac{W_{stage}}{\omega} \] \[ \omega = \frac{2 \pi N}{60} \] ### **B. Assume Typical Steam Velocity at Blade** For high-pressure turbines, typical mean blade velocity (\( U \)) is \( 200-400\, \text{m/s} \). Let's estimate using stage 4 (largest) to ensure tip speed is reasonable. \[ U_{tip} = \pi D N / 60 \] Assume \( U_{tip} \leq 400 \) m/s Set \( D_4 = 2\, \text{m} \), solve for \( N \): \[ U_{tip} = \pi \times 2 \times N / 60 = .10472 N \] Set \( U_{tip} = 300\, \text{m/s} \) \[ N = 300 / .10472 \approx 2,865\, \text{rpm} \] ### **C. Use This Rotational Speed** \[ N = 2,870\, \text{rpm} \quad (\text{rounded}) \] \[ \omega = 2 \pi N / 60 = 2 \pi \times 2,870 / 60 = 301\, \text{rad/s} \] --- ## **Step 6: Compute Torque for Each Stage** \[ T_{stage} = \frac{W_{stage} \times 100}{\omega} \] \[ T_{stage} = \frac{2,577 \times 100}{301} = 8,561\, \text{Nm} \] **Summary Table** | Stage | Diameter (m) | W (kW) | N (rpm) | ω (rad/s) | T (Nm) | |-------|--------------|--------|---------|-----------|--------| | 1 | .5 | 2,577 | 2,870 | 301 | 8,561 | | 2 | 1. | 2,577 | 2,870 | 301 | 8,561 | | 3 | 1.5 | 2,577 | 2,870 | 301 | 8,561 | | 4 | 2. | 2,577 | 2,870 | 301 | 8,561 | --- # **Final Answers** ### **a) Thermal Efficiency** \[ \boxed{30.7\%} \] --- ### **b) Plant Layout and T-S Diagram** - **Layout:** See ASCII schematic above. - **T-S Diagram:** Standard Rankine cycle (sketch as described). --- ### **c) Pump Power Consumption** \[ \boxed{9.95\, \text{kW}} \] --- ### **d) Torque for Each Turbine Stage** \[ \boxed{8,561\, \text{Nm per stage}} \] --- ### **e) Turbine Rotational Speed** \[ \boxed{2,870\, \text{rpm}} \] --- ## **References** - Steam tables (IAPWS, standard engineering tables) - Thermodynamics textbooks (e.g., Çengel & Boles) --- ### **Note** - For more accurate torque distribution, use actual enthalpy drops per stage. - Blade force calculations can be further detailed if velocity triangles and steam properties at each stage are specified. --- **[No image included, as only text is supported. For T-S diagram, please draw a standard Rankine cycle as described.]**