Enthalpy

Enthalpy (H) is a thermodynamic state function defined as H = U + pV for a system, where U is internal energy, p pressure and V volume. For specific (per unit mass) quantities: h = u + pv. Enthalpy is particularly useful in open (flow) systems because it naturally includes flow work (pv) so that energy transport by mass is expressed as ṁ h. For simple compressible substances, the differential form is dh = T ds + v dp (closed system reversible relation). For ideal gases with constant composition and negligible potential/kinetic energy, specific enthalpy depends only on temperature: h = h(T) and dh = cp dT (if cp is constant). For constant-pressure processes, Δh = ∫ cp dT. For incompressible substances (approximate liquids), specific volume v ≈ constant, so Δh ≈ v Δp (pressure contribution) plus any temperature-dependent internal energy change. Energy balances for steady-flow devices commonly use ṁ (h + (V^2)/2 + gz) terms.

Governing FormulaKey relations and formulas (SI units): - Definition: H = U + pV ; specific: h = u + pv - Ideal gas: p v = R T (specific R), h = h(T), dh = cp dT - Specific heats: cp - cv = R - Enthalpy change (constant cp): Δh = cp (T2 - T1) - Incompressible liquid (constant v): Δh ≈ v Δp + cp ΔT (often v Δp term dominates pressure effect) - Steady-flow enthalpy power: Power = ṁ (h_in - h_out) - Differential form (Gibbs relation): dh = T ds + v dp - Mixing (no heat loss): T_final = (Σ m_i cp_i T_i) / (Σ m_i cp_i) (if cp constant and no phase change) (Use SI units: J, kg, m, s, K, Pa; specific enthalpy in J/kg.)

Knowledge Check

10 Questions

1.Which of the following is the correct definition of enthalpy for a closed system (total quantities)?

2.For a pure ideal gas with fixed composition, which thermodynamic variable(s) determine the specific enthalpy h?

3.For an ideal gas the relation between specific heats is cp - cv = R. Given cp = 1005 J·kg⁻¹·K⁻¹ for air and R = 287 J·kg⁻¹·K⁻¹, what is cv (J·kg⁻¹·K⁻¹)?

4.2.00 kg of air (take cp = 1005 J·kg⁻¹·K⁻¹) is heated from 300 K to 600 K at constant pressure. What is the total enthalpy increase (in J)?

5.Estimate the change in specific enthalpy (Δh) of liquid water when its pressure increases by 5.0 MPa, assuming incompressible behavior with density ρ = 1000 kg·m⁻³ (specific volume v = 1.0×10⁻³ m³·kg⁻¹) and negligible temperature change.

6.A steady-flow turbine operates with a mass flow ṁ = 1.50 kg·s⁻¹. If the specific enthalpy falls from 420 kJ·kg⁻¹ at inlet to 300 kJ·kg⁻¹ at outlet (neglect kinetic and potential effects), what is the ideal shaft power output in kW?

7.You mix 3.00 kg of water at 80.0 °C with 2.00 kg of water at 20.0 °C. Take cp = 4184 J·kg⁻¹·K⁻¹ and assume no heat loss and no phase change. What is the equilibrium temperature (°C)?

8.Why is specific enthalpy used in the energy balance for open systems (flow processes)?

9.An ideal gas has specific enthalpy h_ref = 250 kJ·kg⁻¹ at T_ref = 300 K and constant cp = 1000 J·kg⁻¹·K⁻¹. What is the specific enthalpy at T = 400 K (in kJ·kg⁻¹)?

10.Which differential relation (Gibbs relation for a simple compressible system) correctly expresses dh in terms of entropy s and pressure p?