Appunti completi del corso "Energy Systems LM"

Review of fluid properties

Pure fluids

Ideal cases

pV = nRT

pV = RT

The kinetic energy is stronger than the Van Der Waals forces

Andrew Curve (saturation curve)

Critical Point

T_cr

Subcooled Liquid

Saturated Liquid

2 phases

Saturated Liq + Vap

Vapor

Saturated Vapor

Isobar

Supercritical Fluids

Subcritical Fluids

P_cr

Review of fluid properties

Pure fluids

Ideal cases

pV = nRT

pV = RT

The kinetic energy is stronger than the Van Der Waals forces

T_cr

Saturated Liquid

Critical Point

Vapor

2 phases

Andrew Curve (saturation curve)

Subcooled Liquid

Isobar

Saturated Liq + Vap

Saturated Vapor

Supercritical Fluids

P_cr

Subcritical Fluids

Vapor phase

How to model the behavior of pure fluids, so how to calculate the dynamic properties? It depends where you are in the diagram:

T > 3T_cr, p < p_cr ideal gas model pv = RT (Equation of State)

T < 3T_cr or p > p_cr real gas (vapor) Z = v(p,T) / (RT/p) (compressibility factor)

The real gas effect increases if T is lower and p is higher.

Estimation of equation of state for real gases

• Tables and Diagrams with experimental data
• Best-Fit Function of experimental data
• General Equation of State, trying to model also the real gases

Ex: (p + n²a/v²)(V - nb) = nR_uT (Van Der Waals)

a = measure of the average attraction force between molecules

b = volume occupied by the molecules

Fundamental relations

T ds = du - vdp

(dH = dU + pdV + Vdp)

T ds = du + pdv

For ideal gases: C_v = _∂U/_∂T = _dU/_dT and C_v(T)

C_P = C_v + R

R = _Ru/_MN

u(T) = u(T_ref) + ∫_Tref^T C_v(T) dT = u(T_ref) + C_v(T-T_ref)

mean integral value: C_v = _{∫Tref^T Cv(T) dT}/_{T - Tref}

h(T) = h(T_ref) + ∫_Tref^T C_P(T) dT

Monoatomic Gases - Perfect Gases (C_v = const):

s(T,p) = s(T_ref, p_ref) + C_P ln_T/_Tref -R ln_P/_Pref

Biatomic or Triatomic Gases - Ideal Gases: we need to resolve the integrals dS = _{CP(T) dT}/ T - _{v dp}/_T

Liquid phase

To have an accurate estimate of v, h, u, etc. is required to use Best-Fit Equations.

To have a fairly accurate estimate the General Equation of State is enough.

In a defined range of T and p, we can make the assumption of incompressible fluid:

v = const

C = const

transformation from closest saturation liquid point

because the properties of saturated liquid are available

dl = du + pdv + vdp

for ideal liquids: u = U (T)

h_A = h_C + ∫_C^A(dupdpvdp) = h_C + v(p_A-P_C ) T = const v = const

h_A = h_B - ∫_A^B(dupdpvdp) = h_B - C(T_B-T_A) v = const p = const

Mass balance equation

The fundamental principle is the "Principle of Conservation of Mass" by Lavoisier. It's valid only if particles move at a speed much lower than the speed of light.

The mass of particles contained in a material control volume Ω(t) remain constant during time.

dM_Ω(t)/dt = 0

Material (Lagrangian) Control Volume: it follows the particles Ω(t)

Eulerian Control Volume: fixed in space, particles can cross it V

For t = t₀ : Ω(t₀) = V

∀t dM_Ω(t)/dt = d/dt ∫_Ω(t) ρ(x,t) dv = 0

∫_Ω(t) (∂ρ/∂t) dv + ∫_∂Ω(t) ρ V · η dA = 0

∫_{Ω(t0) = V} (∂ρ/∂t) dv + ∫_{∂Ω(t0) = ∂V} ρ V · η dA = 0

Reynolds Transport Theorem

∫_V (∂ρ/∂t) dV + ∫_Sin ρ V · η dA + ∫_Sout ρ V · η dA + ∫_SL ρ V · η dA