(a) Consider the system shown in Fig. 1(a). The two chambers initially have equal volumes of 28 litres and contain air (C_p = 1·005 kJ/kg-K and C_v = 0·717 kJ/kg-K) and hydrogen (C_p = 14·32 kJ/kg-K and C_v = 10·17 kJ/kg-K), respectively. The chamber

Question

(a) Consider the system shown in Fig. 1(a). The two chambers initially have equal volumes of 28 litres and contain air (C_p = 1·005 kJ/kg-K and C_v = 0·717 kJ/kg-K) and hydrogen (C_p = 14·32 kJ/kg-K and C_v = 10·17 kJ/kg-K), respectively. The chambers are separated by a frictionless piston which is non-heat-conducting. Both the gases are initially at 140 kPa and 40 °C. Heat is added to the air side until the pressure of both the gases reaches 280 kPa. All outside walls of the chambers are insulated except for the surface where heat is added to air. Calculate the final temperature of the air. (10 marks)

(b) What is 'choked flow' in a convergent-divergent nozzle? Explain, with diagram, the effect of pressure ratio on exit velocity of compressible gas in a convergent-divergent nozzle. (10 marks)

(c) An axial flow compressor with inlet and outlet angles of 40° and 15°, respectively has been designed for 50% reaction. The compressor has a pressure ratio of 6 : 1 and overall isentropic efficiency of 0·80, when inlet static temperature is 41 °C. The blade speed and axial velocity are constant throughout. Assuming a value of 210 m/s for blade speed, find the number of stages required if the work done factor is 0·88 for all the stages. Take Cp = 1·005 kJ/kg-K and γ = 1·4 for air. (10 marks)

(d) Hot water is flowing through a pipe made of cast iron having thermal conductivity of 52 W/m-°C, with an average velocity of 1·5 m/s. The inner and outer diameters of the pipe are 3 cm and 3·5 cm, respectively. The pipe passes through a 15 m long section of a basement whose temperature is 15 °C. The temperature of the water drops from 70 °C to 67 °C as it passes through the basement. The heat transfer coefficient on the inner surface of the pipe is 400 W/m²-°C. Determine the combined convection and radiation heat transfer coefficient at the outer surface of the pipe. (10 marks)

(e) (i) Define the total and spectral black body emissive powers. How are they related to each other? (5 marks)

(ii) Consider two identical bodies, one at 1000 K and the other at 1500 K. Which body emits more radiation in the shorter wavelength region? Which body emits more radiation at a wavelength of 20 μm? (5 marks)

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(a) Final temperature of the air

The hydrogen side is insulated and the piston is frictionless and non-conducting, so hydrogen is compressed reversibly and adiabatically (pV^γ = constant). Pressures on both sides are equal at all times.

Initial: p₁ = 140 kPa, T₁ = 313.15 K, V₁ = 28 L each. Final p₂ = 280 kPa.
γ_H = 14.32/10.17 = 1.408.
V_H2 = 28 × (140/280)^(1/1.408) = 28 × 0.5^0.710 = 17.11 L.
V_air2 = 56 − 17.11 = 38.89 L.

Air (ideal gas, same mass): T_air2 = T₁(p₂/p₁)(V_air2/V₁) = 313.15 × 2 × 38.89/28 = 869.8 K ≈ 596.6 °C.

(b) Choked flow and the effect of pressure ratio

Choked flow: when the back pressure is lowered so that the throat reaches sonic velocity (M = 1), the mass flow becomes the maximum possible for the given stagnation conditions. Further lowering of back pressure cannot raise the flow because pressure signals cannot travel upstream through the sonic throat. For air, the critical throat pressure ratio p*/p₀ = (2/(γ+1))^(γ/(γ−1)) = 0.528.

Effect of back-pressure ratio p_b/p₀ on exit velocity:
p_b/p₀ = 1: no flow.
Slightly below 1: subsonic flow throughout, the divergent part acts as a diffuser. Exit velocity and mass flow rise as p_b/p₀ falls.
First critical ratio: the throat becomes sonic and the flow chokes, the divergent part still decelerates subsonically.
Lower back pressure: supersonic flow after the throat ending in a normal shock in the divergent section, subsonic exit. The shock moves towards the exit as p_b falls.
Design ratio: fully isentropic supersonic flow, exit pressure equals back pressure and exit velocity reaches its design maximum for the area ratio.
Below design: under-expansion, exit conditions unchanged, further expansion occurs outside the nozzle. Above design but below the shock-at-exit value gives over-expansion with oblique shocks outside.
So exit velocity rises as p_b/p₀ falls until the design ratio, after which it stays constant, while mass flow stays fixed once choked.

(c) Number of stages of the axial compressor

For 50% reaction the triangles are symmetric (α₁ = β₂, α₂ = β₁) and U = Ca(tanβ₁ + tanβ₂):
Ca = 210/(0.8391 + 0.2679) = 189.7 m/s.
Work per stage = λ U Ca (tanβ₁ − tanβ₂) = 0.88 × 210 × 189.7 × 0.5712 = 20.02 kJ/kg.

Inlet: C₁ = Ca/cos15° = 196.4 m/s, T₀₁ = 314.15 + 196.4²/2010 = 333.3 K.
T₀₂s = 333.3 × 6^0.2857 = 556.2 K, actual rise ΔT₀ = (556.2 − 333.3)/0.80 = 278.5 K.
Total work = 1.005 × 278.5 = 279.9 kJ/kg.

N = 279.9/20.02 = 13.98, so 14 stages. (Using static inlet temperature in place of stagnation gives 13.2, which also rounds up to 14.)

(d) Combined outer heat transfer coefficient

Water at about 68.5 °C: ρ ≈ 1000 kg/m³, c_p ≈ 4.18 kJ/kg·K.
ṁ = 1000 × (π × 0.03²/4) × 1.5 = 1.060 kg/s.
Q = ṁc_pΔT = 1.060 × 4180 × 3 = 13 296 W.
LMTD with surroundings at 15 °C: ΔT_lm = (55 − 52)/ln(55/52) = 53.49 °C.
R_total = 53.49/13 296 = 4.023 × 10⁻³ K/W.
R_i = 1/(400 × π × 0.03 × 15) = 1.768 × 10⁻³ K/W.
R_wall = ln(3.5/3)/(2π × 52 × 15) = 3.15 × 10⁻⁵ K/W.
R_o = 4.023 × 10⁻³ − 1.768 × 10⁻³ − 0.031 × 10⁻³ = 2.223 × 10⁻³ K/W.
A_o = π × 0.035 × 15 = 1.649 m².
h_combined = 1/(R_o A_o) ≈ 273 W/m²·°C.

(e)(i) Total and spectral emissive power

Spectral black-body emissive power E_bλ is the energy emitted by a black body per unit time, per unit area, per unit wavelength at λ (W/m²·μm), given by Planck law E_bλ = C₁/[λ⁵(e^(C₂/λT) − 1)].
Total emissive power E_b is the energy emitted per unit time and area over all wavelengths (W/m²).
Relation: E_b = ∫₀^∞ E_bλ dλ = σT⁴ (Stefan–Boltzmann law), i.e. the total is the area under the spectral curve.

(e)(ii) Comparison of the two bodies

Wien law λ_max T = 2898 μm·K gives peaks at 2.90 μm (1000 K) and 1.93 μm (1500 K), so the 1500 K body emits more in the shorter-wavelength region.
At 20 μm: E_λ(1500)/E_λ(1000) = (e^0.719 − 1)/(e^0.480 − 1) ≈ 1.71, so the 1500 K body also emits more at 20 μm. A hotter black body emits more at every wavelength, since the Planck curves never cross.

(a) Consider the system shown in Fig. 1(a). The two chambers initially have equal volumes of 28 litres and contain air (C_p =…

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(a) Final temperature of the air

(b) Choked flow and the effect of pressure ratio

(c) Number of stages of the axial compressor

(d) Combined outer heat transfer coefficient

(e)(i) Total and spectral emissive power

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