<h2>
Answer:</h2>
(a) 5.223 m/s
(b) 0.314
<h2>
Explanation:</h2>
The mass flow rate (m⁺) of the turbine is related to the density of air(ρ), velocity of air (vₐ) and the cross-sectional area (A) of the turbine as follows;
m⁺ = ρ x vₐ x A -----------------(i)
<em>But;</em>
A = π x r²
<em>Substitute A = π x r² into equation (i) as follows;</em>
m⁺ = ρ x vₐ x π x r² --------------------(ii)
<em>But;</em>
The radius (r) of the turbine is the ratio of the velocity (vₓ) of the turbine to the angular velocity (ω) of its blades. i.e
r = vₓ / ω
Where;
ω = 2 π f [f = frequency i.e number of oscillations/revolutions per second of the blades of the turbine]
=> r = vₓ / (2 π f )
<em>Substitute the value r = vₓ / (2 π f ) into equation (ii) as follows;</em>
m⁺ = ρ x vₐ x π x [vₓ / (2 π f )]² ------------------(iii)
<em>(a) Now from the question,</em>
m⁺ = mass flow rate = 42,000 kg/s
ρ = density of air = 1.31 kg/m³
vₓ = 250km/h
= 250 x 1000/3600 m/s
= 69.44m/s
∴ vₓ = 69.44m/s
f = number of revolutions per second
= 15rpm
= 15 revolutions per minute
= 15/60 revolutions per second
∴ f = 0.25 revolutions per second
<em>Take π = 3.142 and substitute these values into equation (iii) as follows;</em>
m⁺ = ρ x vₐ x π x [vₓ / (2 π f )]²
42000 = 1.31 x vₐ x 3.142 x [69.44 / (2 x 3.142 x 0.25)]²
42000 = 1.31 x vₐ x 3.142 x [69.44 / 1.571]²
42000 = 1.31 x vₐ x 3.142 x [44.20]²
42000 = 1.31 x vₐ x 3.142 x [1953.64]
42000 = 8041.22 x vₐ
<em>Solve for vₐ;</em>
vₐ = 42000/8041.22
vₐ = 5.223 m/s
<em>Therefore, the average velocity of the air is </em><em>5.223m/s</em>
<em>(b) First, let's get the kinetic energy power (Pₓ) of the system as follows;</em>
Pₓ = KE / t
Where;
t = time
Kinetic energy (KE) = (1 / 2) x m x v²
=> Pₓ = (1 / 2) x m x v² / t
=> Pₓ = (1 / 2) x (m / t) x v² -----------------(iv)
Where;
m/t is the mass flow rate = m⁺ = 42000 kg/s
v = velocity of air = vₐ = 5.223 m/s
<em>Substitute these values into equation (iv) as follows;</em>
=> Pₓ = (1 / 2) x (42000) x (5.223)²
=> Pₓ = (1 / 2) x (42000) x (27.280)
=> Pₓ = 572880 W
=> Pₓ = 572.880 kW
Also, the power output (P₀) produced by the turbine is 180kW according to the question.
Now, let's calculate the conversion efficiency (η) of the turbine as follows;
Efficiency = Power Output / Kinetic Energy Power
η = P₀ / Pₓ
<em>Substitute the values of P₀ and Pₓ into the equation to give;</em>
η = 180 kW / 572.880kW
η = 0.314
<em>Therefore the conversion efficiency is </em><em>0.314</em>