Properties of Steam: Saturation, Two-Phase Mixtures, Superheated Vapor, and Key Relations

Symbols and constants

Phase landmarks and diagrams

Phase rule and regions

• Gibbs phase rule: \[ f = c - p + 2 \quad\text{for pure water}\; (c=1) \] Single phase \((p=1)\): \(f=2\) (need two independent props). Two-phase sat. mix \((p=2)\): \(f=1\) (state fixed by one independent variable).
• Landmarks: triple point \((T_{tp}, p_{tp})\), critical point \((T_c, p_c)\). At \(T\ge T_c\) or \(p\ge p_c\), distinct liquid/vapor phases vanish.

Diagram cues

• T–s plot: Saturation dome separates liquid and vapor. Isentropic expansion lines are nearly vertical.
• p–v plot: Large specific volume jump across boiling. Two-phase region shows constant-pressure horizontal lines (at fixed \(T\)).

Saturation and Clausius–Clapeyron

Clapeyron equation

• General form: \[ \frac{dp_{sat}}{dT} = \frac{h_{fg}}{T\,(v_g - v_f)} \] Exact along the saturation curve. For water near low pressure, \(v_f \ll v_g\).
• Clausius–Clapeyron (ideal vapor approx): \[ \frac{d\ln p_{sat}}{dT} \approx \frac{h_{fg}}{R_v\,T^2} \] assuming \(v_g \approx \frac{R_v T}{p_{sat}}\) and weak \(h_{fg}(T)\) variation locally.

Entropy of vaporization

• Definition: \[ s_{fg} = \frac{h_{fg}}{T_{sat}} \] Useful to estimate entropy jump at boiling/condensation.

Behavior with temperature

• Latent heat trend: \(h_{fg}\) decreases with \(T\) and goes to zero at the critical point.
• Density trend: \(v_g\) drops sharply as \(p_{sat}\) rises; \(v_f\) varies weakly with \(T\) and \(p\).

Two-phase mixture relations (wet steam)

Quality-based property mixing

• Definition of quality: \(x = \dfrac{m_g}{m_f + m_g}\), where \(m_g\) is vapor mass.
• Mixture properties (at given \(p_{sat}\) or \(T_{sat}\)): \[ v = v_f + x\,v_{fg},\quad h = h_f + x\,h_{fg},\quad s = s_f + x\,s_{fg} \] with \(v_{fg} = v_g - v_f\), etc.
• Inverse for quality: \[ x = \frac{h - h_f}{h_{fg}} = \frac{s - s_f}{s_{fg}} = \frac{v - v_f}{v_{fg}} \] Use the most reliable measured property for best accuracy.

Moisture content and dryness

• Moisture fraction: \(1-x\). Turbomachinery typically requires \(x \gtrsim 0.88\) at exit to limit erosion.
• Mixture specific internal energy: \[ u = h - p\,v = (h_f - p\,v_f) + x\,(h_{fg} - p\,v_{fg}) \]

Superheated vapor and compressed liquid regions

Superheated steam (ideal gas vicinity)

• State relation (approx.): \[ p\,v \approx R_v\,T \] Better at low pressure and high superheat; deviations increase near saturation and high pressure.
• Differentials: \[ dh \approx c_p(T)\,dT + \left[v - T\left(\frac{\partial v}{\partial T}\right)_p\right]dp \] For ideal gas, \(dh \approx c_p(T)\,dT\).
• Internal energy: for ideal gas, \(du \approx c_v(T)\,dT\), and \(h = u + R_v T\).

Compressed (subcooled) liquid water

• Incompressible approximation: \[ h(T,p) \approx h_f(T) + v_f(T)\,[p - p_{sat}(T)] \] Useful up to several MPa from saturation; for high accuracy use tables or EOS.
• Density: \(v_f \approx 0.001\,\mathrm{m^3/kg}\) near ambient, weakly dependent on \(p\) and \(T\).

Energy relations and thermodynamic differentials

Fundamental relations

• Gibbs relations: \[ dh = T\,ds + v\,dp,\qquad du = T\,ds - p\,dv ]
• Speed of sound (definition): \[ a^2 = \left(\frac{\partial p}{\partial \rho}\right)_s \] Requires an EOS; not ideal-gas in general near saturation.

Heat capacities and derivatives

• Definitions: \[ c_p = \left(\frac{\partial h}{\partial T}\right)_p,\qquad c_v = \left(\frac{\partial u}{\partial T}\right)_v \]
• Relation: for real fluids, \(c_p - c_v = T\,\dfrac{\alpha^2}{\rho\,\kappa_T}\), with \(\alpha = \tfrac{1}{v}(\partial v/\partial T)_p\), \(\kappa_T = -\tfrac{1}{v}(\partial v/\partial p)_T\).

Throttling, Joule–Thomson, and steam calorimetry

Throttling process

• Isenthalpic assumption: \[ h_1 = h_2 \quad (\text{steady throttling, negligible heat/work/KE/PE changes}) \]
• Joule–Thomson coefficient: \[ \mu_{JT} = \left(\frac{\partial T}{\partial p}\right)_h = \frac{1}{c_p}\left[T\left(\frac{\partial v}{\partial T}\right)_p - v\right] \] Sign and magnitude depend on state; for steam, \(\mu_{JT}\) is generally positive in common conditions.

Dryness fraction via throttling calorimeter

• Upstream wet steam: state 1 at \((p_1, x_1)\), downstream superheated state 2 at \(p_2\), measured \(T_2\).
• Enthalpy balance: \[ h_1 = h_f(p_1) + x_1\,h_{fg}(p_1) = h_2\big(p_2, T_2\big) \] hence \[ x_1 = \frac{h_2\big(p_2, T_2\big) - h_f(p_1)}{h_{fg}(p_1)} \]

Separating and combined calorimeters

• Separating calorimeter: mechanically separates some liquid; measure masses \(\dot{m}_\ell, \dot{m}_v\). Dryness: \[ x = \frac{\dot{m}_v}{\dot{m}_v + \dot{m}_\ell} \] Subject to carry-over errors for fine droplets.
• Combined (separating + throttling): separation raises \(x\) so the downstream throttled state becomes superheated, enabling the enthalpy method above for higher accuracy.

Steam tables and interpolation

Choosing the right table

• Saturated tables by \(T\) or \(p\): give \(h_f, h_g, s_f, s_g, v_f, v_g, h_{fg}, s_{fg}\).
• Superheated tables: indexed by \(p\) with \(T\)-grid listing \(v, h, s\).
• Compressed liquid tables: often approximated by saturation values at the same \(T\), with corrections if available.

Linear interpolation

• Single variable: known \(y\) at \(x_1, x_2\), find at \(x\): \[ y(x) \approx y_1 + \frac{x - x_1}{x_2 - x_1}\,(y_2 - y_1) \]
• Bilinear (superheated grid): between \((p_1,T_1),(p_1,T_2),(p_2,T_1),(p_2,T_2)\), interpolate in \(T\) then in \(p\) (or vice versa).

Property determination workflow

• Given \(p, T\): compare \(T\) with \(T_{sat}(p)\). - If \(T < T_{sat}\): compressed liquid region. - If \(T = T_{sat}\): saturated mixture; use an additional property to get \(x\). - If \(T > T_{sat}\): superheated region.
• Given \(p, h\) or \(p, s\): check if \(h\) or \(s\) lies between sat. liquid and sat. vapor values to decide two-phase vs superheat.

Worked examples

Example 1: Properties of saturated water at 1 bar

• Given: \(p=1\,\text{bar} \approx 101.325\,\text{kPa}\). From saturated tables at 100°C (approx.).
• Typical values: \(T_{sat}\approx 100^{\circ}\text{C}\), \(v_f\approx 0.00104\,\text{m}^3/\text{kg}\), \(v_g\approx 1.694\,\text{m}^3/\text{kg}\).
• Enthalpies: \(h_f\approx 419\,\text{kJ/kg}\), \(h_g\approx 2676\,\text{kJ/kg}\), so \(h_{fg}\approx 2257\,\text{kJ/kg}\).
• Entropies: \(s_f\approx 1.307\,\text{kJ/(kg\cdot K)}\), \(s_g\approx 7.354\), so \(s_{fg}\approx 6.047\).
• Internal energy of vapor: \[ u_g = h_g - p\,v_g \approx 2676 - (0.1013\,\text{MPa})(1.694\,\text{m}^3/\text{kg}) \approx 2676 - 171.6 \approx 2504\,\text{kJ/kg} \]

Example 2: Mixture quality from measured enthalpy

• Given: \(p=300\,\text{kPa}\) and \(h=1800\,\text{kJ/kg}\). Saturation values: \(h_f=640\,\text{kJ/kg}\), \(h_g=2725\,\text{kJ/kg}\) (approx.).
• Compute quality: \[ x = \frac{h - h_f}{h_{fg}} = \frac{1800 - 640}{2725 - 640} \approx \frac{1160}{2085} \approx 0.56 \]
• Then: \(v = v_f + x\,v_{fg}\), \(s = s_f + x\,s_{fg}\) using sat. values at \(300\,\text{kPa}\).

Example 3: Throttling calorimeter dryness

• Given: Upstream \(p_1=700\,\text{kPa}\), unknown \(x_1\). Downstream \(p_2=100\,\text{kPa}\), measured \(T_2=130^{\circ}\text{C}\) (superheated).
• Find: From superheated table at \((p_2, T_2)\), read \(h_2\) (say \(h_2=2730\,\text{kJ/kg}\) approx.). From sat. at \(p_1\), \(h_f=720\), \(h_{fg}=2050\) (approx.).
• Dryness: \[ x_1 = \frac{h_2 - h_f}{h_{fg}} = \frac{2730 - 720}{2050} \approx 0.98 \]

Quick formulas and checks

Handy relations

• Latent heat to entropy jump: \[ s_{fg} = \frac{h_{fg}}{T_{sat}} \]
• Two-phase mixture: \[ h = h_f + x\,h_{fg},\quad s = s_f + x\,s_{fg},\quad v = v_f + x\,v_{fg} \]
• Isentropic condensation check: If \(s\) known and \(p\) given, compute \(x = \dfrac{s - s_f}{s_{fg}}\) and ensure \(0\le x \le 1\) for two-phase validity.
• Compressed liquid enthalpy correction: \[ h(T,p) \approx h_f(T) + v_f(T)\,[p - p_{sat}(T)] \]
• Ideal-gas superheat estimate: \[ h(T_2,p)\approx h(T_1,p) + \int_{T_1}^{T_2} c_p(T)\,dT \]