Conduction
Heat transfer through a solid or stationary fluid due to a temperature gradient:
- Fourier's law: q = -k * A * (dT/dx) (k = thermal conductivity, W/(m*K))
- Plane wall: Q = k*A*(T1 - T2)/L
- Thermal resistance (plane wall): R = L/(k*A)
- Cylindrical wall: Q = 2*pi*k*L*(T1 - T2)/ln(r2/r1)
- Composite walls: resistances in series add, parallel resistances combine as 1/R_total = 1/R1 + 1/R2
- Contact resistance: R_contact = 1/(h_c*A) at interfaces between materials
Convection
Heat transfer between a surface and a moving fluid:
- Newton's law of cooling: Q = h*A*(T_s - T_inf) (h = convection coefficient, W/(m^2*K))
- Thermal resistance: R_conv = 1/(h*A)
- Nusselt number: Nu = h*L/k_fluid (dimensionless convection coefficient)
- For forced convection: Nu depends on Re and Pr (Prandtl number)
- For natural convection: Nu depends on Gr (Grashof) and Pr, or Ra (Rayleigh)
- Prandtl number: Pr = nu/alpha = mu*c_p/k (ratio of momentum to thermal diffusivity)
Radiation
Heat transfer via electromagnetic waves; no medium required:
- Stefan-Boltzmann law: Q = epsilon*sigma*A*T^4 (sigma = 5.67 x 10^-8 W/(m^2*K^4))
- Net radiation between two surfaces: Q = sigma*A*F*(T1^4 - T2^4)
- Emissivity (epsilon): ratio of actual emission to blackbody emission (0 to 1)
- Blackbody: perfect emitter and absorber, epsilon = 1
- View factor (F): fraction of radiation leaving one surface that reaches another
- Kirchhoff's law: absorptivity = emissivity at thermal equilibrium
Heat Exchangers
- Energy balance: Q = m_dot_h*c_p_h*(T_h_in - T_h_out) = m_dot_c*c_p_c*(T_c_out - T_c_in)
- LMTD method: Q = U*A*LMTD
- LMTD = (delta_T1 - delta_T2) / ln(delta_T1/delta_T2)
- Parallel flow: both fluids enter at the same end
- Counter flow: fluids enter at opposite ends (more effective)
- Effectiveness-NTU method: useful when outlet temperatures are unknown
- NTU = U*A/C_min, effectiveness = Q_actual/Q_max
Fins and Extended Surfaces
- Purpose: increase surface area to enhance heat transfer
- Fin efficiency: eta_f = Q_actual / Q_if_entire_fin_at_base_temperature
- Fin effectiveness: ratio of fin heat transfer to heat transfer without fin
- Fins are most effective with low h, high k materials, and thin, closely spaced designs
- Overall surface efficiency: eta_o = 1 - (A_f/A_total)*(1 - eta_f)
Thermal Resistance Networks
- Analogy to electrical circuits: Q is analogous to current, delta_T to voltage, R_th to resistance
- Series: R_total = R1 + R2 + R3 + ...
- Parallel: 1/R_total = 1/R1 + 1/R2 + ...
- Overall heat transfer coefficient: U = 1/(R_total * A)
- Common network: convection (inner) + conduction (wall) + convection (outer)