Respuesta :
Answer:
Power output, [tex]P_{out} = 178.56 kW[/tex]
Given:
Pressure of steam, P = 1400 kPa
Temperature of steam, [tex]T = 350^{\circ}C[/tex]
Diameter of pipe, d = 8 cm = 0.08 m
Mass flow rate, [tex]\dot{m} = 0.1 kg.s^{- 1}[/tex]
Diameter of exhaust pipe, [tex]d_{h} = 15 cm = 0.15 m[/tex]
Pressure at exhaust, P' = 50 kPa
temperature, T' = [tex]100^{\circ}C[/tex]
Solution:
Now, calculation of the velocity of fluid at state 1 inlet:
[tex]\dot{m} = \frac{Av_{i}}{V_{1}}[/tex]
[tex]0.1 = \frac{\frac{\pi d^{2}}{4}v_{i}}{0.2004}[/tex]
[tex]0.1 = \frac{\frac{\pi 0.08^{2}}{4}v_{i}}{0.2004}[/tex]
[tex]v_{i} = 3.986 m/s[/tex]
Now, eqn for compressible fluid:
[tex]\rho_{1}v_{i}A_{1} = \rho_{2}v_{e}A_{2}[/tex]
Now,
[tex]\frac{A_{1}v_{i}}{V_{1}} = \frac{A_{2}v_{e}}{V_{2}}[/tex]
[tex]\frac{\frac{\pi d_{i}^{2}}{4}v_{i}}{V_{1}} = \frac{\frac{\pi d_{e}^{2}}{4}v_{e}}{V_{2}}[/tex]
[tex]\frac{\frac{\pi \times 0.08^{2}}{4}\times 3.986}{0.2004} = \frac{\frac{\pi 0.15^{2}}{4}v_{e}}{3.418}[/tex]
[tex]v_{e} = 19.33 m/s[/tex]
Now, the power output can be calculated from the energy balance eqn:
[tex]P_{out} = -\dot{m}W_{s}[/tex]
[tex]P_{out} = -\dot{m}(H_{2} - H_{1}) + \frac{v_{e}^{2} - v_{i}^{2}}{2}[/tex]
[tex]P_{out} = - 0.1(3.4181 - 0.2004) + \frac{19.33^{2} - 3.986^{2}}{2} = 178.56 kW[/tex]
The power output of the Turbine is; 225.69 kW
What is the Power Output?
We are given
Pressure of steam; P = 1400 kPa
Temperature of steam at state 1; T = 350°C
Diameter of pipe; d₁ = 8 cm = 0.08 m
Mass flow rate; m' = 0.1 kg/s
Diameter of exhaust pipe; d₂ = 15 cm = 0.15 m
Pressure at exhaust; P' = 50 kPa
Temperature at state 2; T' = 100°C
Area; A = πd²/4
A = π * 0.08²/4
A = 0.0016π m²
We can find the find initial velocity from the formula;
v₁ = m' * V₁/A
v₁ = (0.1 * 0.2004/(0.0016π))
v₁ = 3.986 m/s
From equation of compressible fluid, we know that;
(A₁ * v₁)/V₁ = (A₂ * v₂)/V₂
A₂ = πd₂²/4
A₂ = π * 0.15²/4
A₂ = 0.005625π m²
(0.0016π * 3.986)/0.2004 = (0.005625π * v₂)/3.4181
Solving for v₂ gives;
v₂ = 19.33 m/s
Finally power output is gotten from the expression;
P_out = -m'(H₂ - H₁) + (v₂² - v₁²)/2
P_out = -0.1(2682.6 - 3150.7) + (19.33² - 3.986²)/2
P_out = 225.69 kW
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