Appunti completi del corso "Energy Systems LM"

Review of Fluid Properties

Pure Fluids

Ideal Gases

pV = nRT

pV = RT

The kinetic energy is stronger than the Van Der Waals forces

Subcooled Liquid

Saturated Liquid

Critical Point

Vapor

2 phases

Saturated Liq. + Vap.

Saturated Vapor

Andrew Curve (saturation curve)

Supercritical Fluids

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 > 3 T_cr, p < p_cr ideal gas model
• T < 3 T_cr, p > p_cr real gas (vapor)

p v = R T (Equation of State)

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) = n R_u T (Van Der Waals)

• a = measure of the average attraction force between molecules
• b = volume occupied by the molecules

∫_V ∂ρ/∂t dV + ∫_Sin ρ V_r v dA + ∫_Sout ρ V_r v dA + ∫_SL ρ V_η dA + ∫_SR ρ V_η dA + ∫_SP ρ V_r v dA = 0

ṁ_in ṁ_out

∫_V ∂ρ/∂t dV − ṁ_in + ṁ_out + ∫_Sη ρ V_r v dA + ∫_SP ρ V_r v dA = 0

• Steady Flow Condition:

  for every property of the fluid (p, h,...):∂ϕ/∂t = 0 (for ∀ point of V)

  ∑_IN ṁ_i = ∑_OUT ṁ_i

• Steady Average Flow Condition:

  ∂ϕ̅/∂t = 0 where ϕ̅ = 1/T ∫_t^t+T ϕ(t) dt

  ∑_IN ṁ̅_i = ∑_OUT ṁ̅_i

Examples without chemical reactions:

Valve

• Mass Balance: ̅m₁ = ̅m₂
• Atomic Balance: A_{i, T} = A_{i,2, T}
• N_{i, K} = N_{i, K} (no chemical reactions)
• Energy Balance: Q̇_IN - Ẇ_OUT + ̅m₁h_TOT,1 = ̅m₂h_TOT,2 = 0 (y adiabatic)

h_TOT,1 = h_TOT,2

h₁ + V₁²/2 g z₁ + e_CH,1 = h₂ + V₂²/2 g z₂ + e_CH,2

h₁ = h₂

Turbine / Compressor

• Mass Balance: ̅m₁ = ̅m₂
• Atomic Balance: A_{i, T} = A_{i,2, T}
• N_{i, K} = N_{i, K} (no chemical reactions)
• Energy Balance: Ẇ_OUT = ̅m₁(h₁ + V₁²/2) - ̅m₂(h₂+ V₂²/2)

BLADE POWER (power extracted from the fluid)

MECHANICAL POWER: Ẇ_mec = Ẇ_OUT η_mec

ELECTRICAL POWER: Ẇ_el = Ẇ_mec η_el

Mixtures

In general the stream of reactants R can be a mixture of species, how can we determine the enthalpy and the chemical energy of a mixture?

1. h_TOTR = h_R(T,p) + V_R² / 2 + gz_R + e_CNR

• Chemical Energy

e_CNR = ^mΣ_k=1 Y_R,k e_CNK

• Enthalpy

h_R(T,p)

• if R is a pure fluid: h(T,p) is computed using the relations of that fluid
• if R is a mixture:
• real mixture: molecules of different types tend to interact, with forces

h_R = ⁿΣ_k=1 y_nk h_k(T,p,x_nι) ↳ concentration of the other molecules

• ideal mixture: there are no forces between different molecules

h_R = ⁿΣ_k=1 y_nk h_k(T,p⁰_k) ↳ partial pressure

• ideal mixture of ideal gases:

h_R = ⁿΣ_k=1 y_nk h_k(T)

Example:

h_TOTR = ^mΣ_k=1 y_nk C_p,k (T - T_ref) + V_R² / 2 + gz_R + e_CNR

C_p,R = C_p of the mixture

Parallel Flow

the slope of the lines depends on the mass flow rate of the fluid:

_Q̇ = ṁ_HOT C_{P HOT} (T₁ - T)

T = T₁ - _Q̇ / ṁ_HOT C_{P HOT}

dT / d_Q̇ = - 1 / ṁ_HOT C_{P HOT}

Counter Flow

To increase the heat transferred, instead of increasing the length of the fins, is better to increase the density of the fins.

In order to optimize the heat transfer we have to find the right combination of length and density.

Design Procedure of Direct Transfer Heat Exchanger

Problem to be solved:

• Input data: design specifications:

Determine the best HX type and its design (area, n° of tubes, diameter, thickness and length of the tubes and of the shell)

Steps:

• Determine the heat to be transferred Q̇ = ṁₕₒₜ Cₚₕₒₜ (Tₕₒₜ ₍ᵢₙ₎ - Tₕₒₜ ₍ₒᵤₜ₎)
• Select the construction type
• Set the desired speed for the two fluids → h
• Evaluate the heat transfer area A using the heat transfer rate equation
• Work out the geometrical parameters of the HX. These variables cannot be selected independently, but the combination must satisfy the following equations:
  • Aᵢ = π Dᵢₜ Lᵢ Nₚₐₛₛₑₛ
  • Sᵥ = π Dᵢₜ² ṁ̇ / ρ V Nₜ
• Preliminary mechanical design or check
• Rating calculation/simulation CFD: We fix the flow rate and the temperatures at the inlet and we fix the geometry, and we calculate the outlet temperatures. With the software we look for the most accurate estimate of Q̇ and the outlet temperatures, because we need to meet the specifications

This situation can lower the efficiency of the engine.

How do we solve this problem, so how can we decrease ^w?

• reducing the area of the HX_w = ^w (̇ (̇)^w_min (^w_hot - ^w_cold))
• reducing the flow rate of air reducing the rotational speed of the fan, or the flow rate of water with a bypass valve_w = ^w (̇ (̇)^w_min (^w_hot - ^w_cold))
• recycle across HX_w = ^w (̇ (̇)^w_min (^w_hot - ^w_cold))

In the previous example let's assume to decrease the flow rate of water:

̇^w_H2O = 0.6 ̇_H2O

_w = ^w (̇ (̇)^w_min (^w_hot - ^w_cold))

(̇)^w_hot: 0.6

(̇)_cold: 0.5

(̇)^w_min = (̇)^w_min

NTU^w = U^w A^w / (̇)^w_min = NTU^s? → NTU^w = NTU^s

̇^w has decreased, so ^w will decrease, but U mainly depends on h_air, so U^w = U^s

C_r^w = 0.83 ≠ C^s → ^w ≠ ^s