Thermodynamic Cycles and Applications

Carnot Cycle (Theoretical Maximum)

The Carnot cycle consists of two isothermal and two adiabatic (isentropic) processes.

Efficiency: $\eta_{Carnot} = 1 - \frac{T_L}{T_H}$

where TL and TH are the absolute temperatures (Kelvin) of the cold and hot reservoirs.

Key Point: No real heat engine can be more efficient than a Carnot engine operating between the same temperatures. The Carnot efficiency sets the upper bound.

Power Cycles

Rankine Cycle (Steam Power Plants)

Components: Pump → Boiler → Turbine → Condenser

Efficiency: $\eta = \frac{W_{net}}{Q_{in}} = \frac{(h_3 - h_4) - (h_2 - h_1)}{h_3 - h_2}$

Improvements: Superheating, reheating, regeneration (feedwater heating)

Otto Cycle (Gasoline Engines)

Ideal processes: Two isentropic + two isochoric (constant volume)

Efficiency: $\eta_{Otto} = 1 - \frac{1}{r^{k-1}}$

where r = V₁/V₂ = compression ratio, k = specific heat ratio (1.4 for air)

Example: r = 8, k = 1.4: η = 1 - 1/8^0.4 = 1 - 1/2.297 = 0.565 = 56.5%

Diesel Cycle

Ideal processes: Two isentropic + one isobaric (constant pressure) + one isochoric

$\eta_{Diesel} = 1 - \frac{1}{r^{k-1}} \cdot \frac{r_c^k - 1}{k(r_c - 1)}$

where rc = cutoff ratio (V₃/V₂).

The Diesel cycle is more efficient than the Otto cycle at the same compression ratio, and diesel engines operate at higher compression ratios.

Brayton Cycle (Gas Turbines)

Components: Compressor → Combustor → Turbine

$\eta_{Brayton} = 1 - \frac{1}{r_p^{(k-1)/k}}$

where rp = P₂/P₁ = pressure ratio.

Refrigeration Cycles

Vapor-Compression Refrigeration

Components: Compressor → Condenser → Expansion valve → Evaporator

Coefficient of Performance (COP)

Refrigerator (cooling): $COP_R = \frac{Q_L}{W_{net}} = \frac{h_1 - h_4}{h_2 - h_1}$

Heat Pump (heating): $COP_{HP} = \frac{Q_H}{W_{net}} = COP_R + 1$

Carnot COP: $COP_{R,Carnot} = \frac{T_L}{T_H - T_L}$

Combustion

Complete Combustion of Hydrocarbons

$C_x H_y + (x + y/4)O_2 \rightarrow xCO_2 + (y/2)H_2O$

Air-fuel ratio: AFR = mass of air / mass of fuel

• Stoichiometric AFR for gasoline ≈ 14.7:1
• Lean mixture: excess air (AFR > stoichiometric)
• Rich mixture: excess fuel (AFR < stoichiometric)

Combustion Products

• Complete: CO₂ and H₂O
• Incomplete: includes CO, unburned hydrocarbons
• NOx: formed at high temperatures
• Particulates: soot, ash

Psychrometrics (Air-Water Vapor Mixtures)

Key Properties

$\omega = 0.622 \frac{P_v}{P_{atm} - P_v}$

Psychrometric Chart

Plots humidity ratio vs. dry-bulb temperature. Lines for:

• Constant relative humidity (curves)
• Constant wet-bulb (diagonal lines)
• Constant enthalpy (approximately parallel to wet-bulb lines)
• Constant specific volume (steep diagonal lines)