Thermodynamics 1: cram pack

Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.

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1Formulas and methods

Ideal gas law

Ideal gas law: pV = nRT
Gas constant: R = 8.314 J/(mol·K)
Combined gas law: p₁V₁/T₁ = p₂V₂/T₂
Moles: n = m/M
Temperature: T[K] = T[°C] + 273.15

Energy balance

Sensible heat: Q = ṁ·c_p·ΔT
Latent heat: Q = ṁ·h_fg
Mixing: Σṁc_pT_in = Σṁc_pT_out
Water: c_p ≈ 4.18 kJ/(kg·K)

Heat engines & refrigerators

Efficiency: η = W/Q_H = 1 − Q_C/Q_H
Carnot efficiency: η_C = 1 − T_C/T_H (kelvin)
Refrigerator COP: COP_R = Q_C/W
Heat pump COP: COP_HP = Q_H/W = COP_R + 1
Energy balance: Q_H = W + Q_C

Gas power cycles

Isentropic, volume ratio: T₂/T₁ = r^k−1
Isentropic, pressure ratio: T₂/T₁ = r_p^(k−1)/k
Otto efficiency: η = 1 − 1/r^k−1
Diesel efficiency: η = 1 − (1/r^k−1)·(r_c^k − 1)/(k(r_c − 1))
Brayton efficiency: η = 1 − 1/r_p^(k−1)/k
Heat added: Otto: c_v(T₃ − T₂) Diesel, Brayton: c_p(T₃ − T₂)
Brayton net work: w = c_p[(T₃ − T₄) − (T₂ − T₁)]
Isentropic efficiencies: w_c = w_c,s/η_c, w_t = η_t·w_t,s
Cold air: k = 1.4, c_p = 1.005, c_v = 0.718 kJ/(kg·K)

2Practice problems

1.Ideal gas lawMedium

A tank of 5 L contains 110 g of He at 85 °C. Treat it as an ideal gas and find the pressure P.

m = 110 gM = 4.003 g/molV = 5 LT = 85 °CR = 8.314 J/(mol·K)

P = kPa

2.Ideal gas lawHard

Find the density ρ of He at 850 kPa and 190 °C, treating it as an ideal gas.

M = 4.003 g/molP = 850 kPaT = 190 °CR = 8.314 J/(mol·K)

ρ = kg/m³

3.Energy balanceMedium

Hot light oil cools water in a counter-flow heat exchanger with no heat loss. Find the water outlet temperature t_out.

ṁ_h = 2 kg/sc_p,h = 2.1 kJ/(kg·K)T_h: 140 → 110 °Cṁ_c = 3 kg/sc_p,c = 4.18 kJ/(kg·K)t_in = 30 °C

t_out = °C

4.Energy balanceHard

Hot light oil heats water in an exchanger that loses 5% of the heat released by the hot stream to the surroundings. Find the water flow rate ṁ_c needed.

ṁ_h = 2 kg/sc_p,h = 2.1 kJ/(kg·K)T_h: 160 → 100 °Ct: 25 → 75 °Cc_p,c = 4.18 kJ/(kg·K)loss = 5 %

ṁ_c = kg/s

5.Heat engines & refrigeratorsMedium

A heat engine works between a hot reservoir at 600 °C and a cold reservoir at 25 °C. Find the highest possible thermal efficiency η_max.

T_H = 600 °CT_L = 25 °C

η_max = %

6.Heat engines & refrigeratorsHard

A heat engine takes in 2500 kJ of heat from a reservoir at 300 °C, produces 1000 kJ of work and rejects the rest to a reservoir at 10 °C, per cycle. Find the total entropy generated S_gen per cycle.

T_H = 300 °CT_L = 10 °CQ_H = 2500 kJW = 1000 kJ

S_gen = kJ/K

7.Gas power cyclesMedium

An ideal Diesel cycle has a compression ratio r = 20 and a cutoff ratio r_c = v₃/v₂ = 2.2. At the start of compression the air is at 305 K. Use the cold-air-standard assumptions: air is an ideal gas with constant specific heats, k = 1.4, c_p = 1.005 kJ/(kg·K) and c_v = 0.718 kJ/(kg·K). Find the heat added q_in per kilogram of air, in kJ/kg.

r = 20r_c = 2.2T₁ = 305 Kk = 1.4c_p = 1.005 kJ/(kg·K)

q_in = kJ/kg

8.Gas power cyclesHard

An ideal Brayton cycle operates with a compressor pressure ratio of 14. Use the cold-air-standard assumptions: air is an ideal gas with constant specific heats, k = 1.4, c_p = 1.005 kJ/(kg·K) and c_v = 0.718 kJ/(kg·K). Find its thermal efficiency η in percent.

r_p = 14k = 1.4

η = %

3Solutions

1.Ideal gas law

Answer: P = 16365 kPa

Ideal gas law: PV = nRT, with n = m/M.

Moles

n = m/M = 110/4.003 = 27.48 mol

T = 85 + 273.15 = 358.1 K

Pressure

P = nRT/V = 27.48·8.314·358.1/0.005 = 16365 kPa

Answer

P = 16365 kPa

2.Ideal gas law

Answer: ρ = 0.8836 kg/m³

Ideal gas density: ρ = PM/(RT).

Rearrange the gas law

PV = (m/M)RT → ρ = m/V = PM/(RT)

T = 190 + 273.15 = 463.1 K

Density

ρ = 850·4.003/(8.314·463.1) = 0.8836 kg/m³

Answer

ρ = 0.8836 kg/m³

3.Energy balance

Answer: t_out = 40.05 °C

Exchanger energy balance: with no losses, ṁ_hc_p,h(T_h,in − T_h,out) = ṁ_cc_p,c(t_out − t_in).

Heat released by the hot stream

Q = 2·2.1·(140 − 110) = 126 kW

Heat picked up by the water

t_out = t_in + Q/(ṁ_cc_p,c) = 30 + 126/(3·4.18) = 40.05 °C

Answer

t_out = 40.05 °C

4.Energy balance

Answer: ṁ_c = 1.145 kg/s

Energy balance with losses: heat released = heat absorbed + heat lost.

Heat released

Q_h = 2·2.1·(160 − 100) = 252 kW

Heat that reaches the water

Q_c = (1 − 0.05)·252 = 239.4 kW

Water flow rate

ṁ_c = Q_c/(c_p,c·Δt) = 239.4/(4.18·50) = 1.145 kg/s

Answer

ṁ_c = 1.145 kg/s

5.Heat engines & refrigerators

Answer: η_max = 65.85 %

Second-law limits: no cycle working between reservoirs at absolute temperatures T_H and T_L can beat the reversible (Carnot) cycle: η_max = 1 − T_L/T_H, COP_R,max = T_L/(T_H − T_L) and COP_HP,max = T_H/(T_H − T_L). Energy is always conserved: Q_H = W + Q_L.

Absolute temperatures

T_H = 600 + 273.15 = 873.1 K

T_L = 25 + 273.15 = 298.1 K

Carnot efficiency

η_max = 1 − T_L/T_H = 1 − 298.1/873.1 = 65.85 %

Answer

η_max = 65.85 %

6.Heat engines & refrigerators

Answer: S_gen = 0.9357 kJ/K

Entropy generation: over a full cycle the working fluid has no net entropy change. The hot reservoir loses Q_H/T_H and the cold reservoir gains Q_L/T_L. Their sum is the entropy generated, which is zero for a reversible cycle and positive for every real one.

Heat rejected

Q_L = Q_H − W = 2500 − 1000 = 1500 kJ

Absolute temperatures

T_H = 300 + 273.15 = 573.1 K

T_L = 10 + 273.15 = 283.1 K

Entropy changes of the reservoirs

ΔS_H = −2500/573.1 = −4.362 kJ/K

ΔS_L = 1500/283.1 = 5.298 kJ/K

S_gen = 5.298 − 4.362 = 0.9357 kJ/K

Answer

S_gen = 0.9357 kJ/K

S_gen > 0, so the engine is possible but irreversible.

7.Gas power cycles

Answer: q_in = 1219 kJ/kg

Diesel cycle (compression-ignition engine): 1→2 isentropic compression, 2→3 heat addition at constant pressure, 3→4 isentropic expansion, 4→1 heat rejection at constant volume. With compression ratio r = v₁/v₂ and cutoff ratio r_c = v₃/v₂: T₂ = T₁·r^k−1, T₃ = r_c·T₂, q_in = c_p·(T₃ − T₂) and η = 1 − (1/r^k−1)·(r_c^k − 1)/(k·(r_c − 1)).

Temperatures

T₂ = 305·20^0.4 = 1011 K

T₃ = 2.2·1011 = 2224 K

Heat added at constant pressure

q_in = 1.005·(2224 − 1011) = 1219 kJ/kg

Answer

q_in = 1219 kJ/kg

8.Gas power cycles

Answer: η = 52.95 %

Brayton cycle (gas turbine): 1→2 isentropic compression, 2→3 heat addition at constant pressure, 3→4 isentropic expansion in the turbine, 4→1 heat rejection at constant pressure. With pressure ratio r_p = P₂/P₁: T₂ = T₁·r_p^(k−1)/k, T₄ = T₃/r_p^(k−1)/k and η = 1 − 1/r_p^(k−1)/k. Net work w = c_p·[(T₃ − T₄) − (T₂ − T₁)]. With isentropic efficiencies, the compressor needs w_s/η_c and the turbine gives η_t·w_s.

Thermal efficiency

η = 1 − 1/14^0.2857 = 1 − 1/2.126 = 52.95%

Answer

η = 52.95%

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