Constitutive relationships for solids, incompressible fluids, ideal gases and pure substances Model Equation of State Specific Internal Energy Incompressible Liquid or Solid 𝜌 = 𝑐𝑜𝑛𝑠𝑡 𝑢 = 𝑢0 + 𝑐(𝑇 − 𝑇0 ) Ideal Gas* 𝑃𝑉 = 𝑚𝑅𝑇 or 𝜌 = 𝑅𝑇 𝑢 = 𝑢0 + 𝑐𝑣 (𝑇 − 𝑇0 ) ℎ = ℎ0 + 𝑐𝑝 (𝑇 − 𝑇0 ) 𝜌 = 𝜌(𝑃, 𝑇) (single-phase) 𝑢 = 𝑢(𝑃, 𝑇) (single-phase) ℎ = ℎ(𝑃, 𝑇) (single-phase) 𝑠 = 𝑠(𝑃, 𝑇) (single-phase) 𝑢 = 𝑢𝑓 (𝑃)(1 − 𝑥) + 𝑢𝑔 (𝑃)𝑥 (two-phase) ℎ = ℎ𝑓 (𝑃)(1 − 𝑥) + ℎ𝑔 (𝑃)𝑥 (two-phase) 𝑠 = 𝑠𝑓 (𝑃)(1 − 𝑥) + 𝑠𝑔 (𝑃)𝑥 (two-phase) Pure Substance** 𝑃 1 𝜌 1−𝑥 𝑥 + 𝜌 (𝑃) (𝑃) 𝑓 𝑔 =𝜌 (two-phase) Specific Enthalpy ℎ = ℎ0 + 𝑐(𝑇 − 𝑇0 ) + 𝑃 − 𝑃0 𝜌 Specific Entropy 𝑠 = 𝑠0 + 𝑐 ln 𝑠 = 𝑠0 + 𝑐𝑝 ln * Relation of universal gas constant R and specific ideal gas constant R: R=R/A, where A is the molecular weight; 𝑐𝑝 = 𝑐𝑣 + 𝑅 ** Phases present to be determined from phase (P-T) diagram 𝑇 𝑇0 𝑇 𝑃 − 𝑅 ln 𝑇0 𝑃0

Constitutive Relationships for Solids, Incompressible Fluids, Ideal Gases and Pure Substances

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