Cheatsheet: Heat Transfer

1. Conduction Heat Transfer

1.1 Fourier's Law

1.2 Thermal Resistance

1.3 Heat Transfer Rate Through Resistances

• q = ΔT_overall/R_total
• Temperature drop across any layer: ΔT = qR
• For composite walls: q = (T₁ - T₂)/(R₁ + R₂ + R₃)

1.4 Critical Radius of Insulation

• Adding insulation increases heat transfer if outer radius < critical="">
• Maximum heat transfer occurs at r = r_c

1.5 Heat Conduction with Heat Generation

1.6 Extended Surfaces (Fins)

1.6.1 Fin Equations

1.6.2 Common Fin Configurations

• Rectangular fin: P = 2(w + t), A_c = wt
• Cylindrical pin fin: P = πD, A_c = πD²/4
• For very long fins (L → ∞): η_f → 1/(mL)
• Overall surface efficiency: η_o = 1 - (A_f/A_total)(1 - η_f)

2. Convection Heat Transfer

2.1 Newton's Law of Cooling

2.2 Dimensionless Numbers

2.3 Forced Convection - Internal Flow

2.3.1 Laminar Flow in Circular Tubes (Re <>

2.3.2 Turbulent Flow in Circular Tubes (Re > 4000)

• Valid for 0.5 < pr="">< 2000="" and="" 3000="">< re=""><>
• Evaluate properties at bulk mean temperature T_m = (T_in + T_out)/2

2.4 Forced Convection - External Flow

2.4.1 Flow Over Flat Plate

2.4.2 Flow Across Cylinder

2.4.3 Flow Across Sphere

• Nu = 2 + (0.4·Re^0.5 + 0.06·Re^(2/3))·Pr^0.4·(μ/μ_s)^0.25
• Valid for 3.5 < re="">< 80000,="" 0.7="">< pr=""><>

2.5 Natural Convection

2.5.1 Vertical Plates and Cylinders

2.5.2 Horizontal Plates

2.5.3 Horizontal Cylinder

• Nu = [0.6 + 0.387·Ra^(1/6)/[1+(0.559/Pr)^(9/16)]^(8/27)]² for 10⁻⁵ < ra=""><>

2.6 Heat Exchanger Correlations

• Properties evaluated at film temperature: T_f = (T_s + T_∞)/2 for external flow
• For liquids: β ≈ 1/T_f (ideal gas approximation)
• Characteristic length L varies by geometry: plate length, cylinder diameter, etc.

3. Radiation Heat Transfer

3.1 Fundamental Laws

3.2 Surface Properties

• Energy balance: α + ρ + τ = 1
• For opaque surfaces: τ = 0, so α + ρ = 1
• Kirchhoff's Law: α = ε (for surfaces at same temperature and wavelength)
• Graybody: ε and α independent of wavelength

3.3 View Factors

3.3.1 Definition and Properties

• F_ij = fraction of radiation leaving surface i that is intercepted by surface j
• Reciprocity: A_i·F_ij = A_j·F_ji
• Summation rule: Σ F_ij = 1 (for j = 1 to N)
• For flat/convex surface to itself: F_ii = 0

3.3.2 Common View Factor Relations

3.4 Radiation Exchange Between Surfaces

3.4.1 Two Black Surfaces

• q_12 = A_1·F_12·σ(T_1⁴ - T_2⁴)

3.4.2 Two Gray Diffuse Surfaces

3.5 Radiation with Convection

• Combined heat transfer: q_total = q_conv + q_rad
• q_total = hA(T_s - T_∞) + εAσ(T_s⁴ - T_surr⁴)
• Radiation heat transfer coefficient: h_r = εσ(T_s + T_surr)(T_s² + T_surr²)
• Linearized form: q_rad = h_r·A(T_s - T_surr)

3.6 Radiation Shields

• For N shields between two surfaces: q_with_shields = q_no_shields/(N+1) if all ε equal
• Single shield between infinite parallel plates: q = σ(T_1⁴ - T_2⁴)/[1/ε₁ + 1/ε₂ + 2/ε_shield - 2]
• Shields most effective when highly reflective (low ε)

4. Heat Exchangers

4.1 Heat Exchanger Types

• Parallel flow: both fluids enter same end, flow in same direction
• Counter flow: fluids enter opposite ends, flow in opposite directions
• Cross flow: fluids flow perpendicular to each other (mixed or unmixed)
• Shell-and-tube: one fluid in tubes, other in shell around tubes

4.2 Log Mean Temperature Difference (LMTD) Method

• P = (t₂ - t₁)/(T₁ - t₁) for tube-side effectiveness
• R = (T₁ - T₂)/(t₂ - t₁) for heat capacity ratio
• F = 1 for pure counter flow and parallel flow (double pipe)

4.3 Effectiveness-NTU Method

4.3.1 Effectiveness Relations

4.4 Overall Heat Transfer Coefficient

• Based on inside area: 1/(U_i·A_i) = 1/(h_i·A_i) + R_wall + A_i/(h_o·A_o)
• Based on outside area: 1/(U_o·A_o) = A_o/(h_i·A_i) + R_wall + 1/(h_o·A_o)

4.5 Heat Exchanger Analysis

• Energy balance (no phase change): q = ṁ_h·c_p,h(T_h,in - T_h,out) = ṁ_c·c_p,c(T_c,out - T_c,in)
• Use LMTD method when inlet and outlet temperatures known
• Use ε-NTU method when outlet temperatures unknown
• Counter flow most efficient (highest ε for given NTU and C_r)
• For C_r = 0: all configurations have same effectiveness

5. Transient Heat Conduction

5.1 Lumped Capacitance Method

5.1.1 Applicability

• Biot number: Bi = hL_c/k < 0.1,="" where="" l_c="V/A_s" (characteristic="">
• Assumes uniform temperature throughout body at any instant
• Internal resistance negligible compared to external resistance

5.1.2 Temperature Response

5.2 Semi-Infinite Solid

• Surface heat flux: q_s" = k(T_s - T_i)/√(παt)
• Penetration depth: δ ≈ 2√(αt) (where temperature change is ~1% of initial)

5.3 Finite-Sized Bodies

5.3.1 One-Dimensional Solutions (Charts/Heisler Charts)

• Requires Bi > 0.1 (lumped capacitance not valid)
• Fourier number: Fo = αt/L_c² (dimensionless time)
• For plane wall: L_c = L (half-thickness)
• For cylinder: L_c = r_o (radius)
• For sphere: L_c = r_o (radius)

5.3.2 Approximate Solutions (Fo > 0.2)

5.3.3 Multi-Dimensional Solutions

• Product solution method: θ(x,y,z,t) = θ_wall(x,t)·θ_wall(y,t)·θ_wall(z,t)
• Short cylinder: θ = θ_cylinder·θ_plane_wall
• Rectangular bar: θ = θ_wall,x·θ_wall,y
• Total heat transfer: Q/Q_max = 1 - product of (1 - Q_i/Q_max,i) for each dimension

5.4 Key Dimensionless Groups

6. Boiling and Condensation

6.1 Pool Boiling

6.1.1 Boiling Regimes

• Free convection: ΔT_e < 5°c,="" bubbles="" not="">
• Nucleate boiling: 5°C < δt_e="">< 30°c,="" bubbles="" form="" at="" nucleation="" sites,="" high="" heat="">
• Critical heat flux (CHF): maximum heat flux, transition to film boiling
• Film boiling: ΔT_e > 120°C, vapor film blankets surface, lower heat transfer

6.1.2 Nucleate Boiling Correlation

• Rohsenow correlation: q" = μ_l·h_fg[g(ρ_l - ρ_v)/σ]^0.5[c_p,l·ΔT_e/(C_sf·h_fg·Pr_l^n)]³
• C_sf = surface-fluid constant (0.013 for water-copper)
• n = 1.0 for water, 1.7 for other fluids
• σ = surface tension, h_fg = latent heat of vaporization

6.1.3 Critical Heat Flux

• Zuber correlation (horizontal surfaces): q"_max = 0.149·ρ_v·h_fg[σg(ρ_l - ρ_v)/ρ_v²]^0.25
• Burnout point: surface temperature increases dramatically if exceeded

6.1.4 Film Boiling

• Bromley correlation (horizontal cylinder): h = 0.62[k_v³ρ_v(ρ_l - ρ_v)gh'_fg/(μ_v·D·ΔT_e)]^0.25
• h'_fg = h_fg + 0.4c_p,v·ΔT_e (corrected latent heat)
• Combined radiation and convection: h_total = h_conv + (3/4)h_rad

6.2 Flow Boiling

• Occurs inside tubes with forced convection
• Quality x = vapor mass fraction = (h - h_f)/h_fg
• Two-phase flow patterns: bubbly, slug, annular, mist flow
• Higher heat transfer coefficients than pool boiling due to forced convection

6.3 Film Condensation

6.3.1 Laminar Film on Vertical Plate

• Nusselt theory: h = 0.943[k_l³ρ_l(ρ_l - ρ_v)gh'_fg/(μ_l·L·ΔT)]^0.25
• h'_fg = h_fg + 0.68c_p,l·ΔT (corrected latent heat)
• ΔT = T_sat - T_s (temperature difference between saturation and surface)
• Valid for Re_f < 30,="" where="" re_f="4Γ/μ_l" and="" γ="">

6.3.2 Turbulent Film on Vertical Plate

• Re_f > 1800: h = 0.0077·Re_f^0.4·k_l³ρ_l(ρ_l - ρ_v)g/(μ_l²)]^(1/3)

6.3.3 Laminar Film on Horizontal Tube

• Single tube: h = 0.725[k_l³ρ_l(ρ_l - ρ_v)gh'_fg/(μ_l·D·ΔT)]^0.25
• N horizontal tubes (vertical tier): h_N = h_1/N^0.25, where h_1 = single tube coefficient

6.4 Dropwise Condensation

• Occurs when condensate forms droplets instead of film
• Heat transfer coefficient 5-10 times higher than film condensation
• Difficult to maintain in practice (requires surface coating)

6.5 Key Properties and Parameters

7. Thermal Properties and Data

7.1 Thermal Conductivity Ranges (W/m·K at 300K)

7.2 Convection Coefficient Ranges (W/m²·K)

7.3 Emissivity Values (Total, Normal, 300K)

7.4 Fouling Resistances (m²·K/W)

7.5 Common Physical Constants

7.6 Temperature-Dependent Properties

• Thermal conductivity k: increases with T for gases, decreases with T for liquids, varies for solids
• Viscosity μ: decreases with T for liquids, increases with T for gases
• For gases: ideal gas approximation β = 1/T_f (T in Kelvin)
• Always evaluate properties at appropriate reference temperature (bulk, film, or surface)

8. Problem-Solving Strategies

8.1 Heat Transfer Rate Equations

8.2 Systematic Approach

• 1. Identify heat transfer modes involved (conduction, convection, radiation)
• 2. Draw thermal circuit with resistances if applicable
• 3. Determine if steady-state or transient analysis required
• 4. Calculate dimensionless numbers (Re, Pr, Nu, Gr, Ra, Bi, Fo) to identify regime
• 5. Select appropriate correlations based on geometry and flow conditions
• 6. Evaluate fluid properties at correct reference temperature
• 7. Check assumptions and validity ranges of correlations used
• 8. Solve for unknown using energy balance or rate equations

8.3 Common Analysis Types

8.4 Units and Conversions

• Temperature: T(K) = T(°C) + 273.15
• Heat flux: 1 W/m² = 0.3171 Btu/h·ft²
• Thermal conductivity: 1 W/m·K = 0.5778 Btu/h·ft·°F
• Heat transfer coefficient: 1 W/m²·K = 0.1761 Btu/h·ft²·°F
• Always use absolute temperature (K) for radiation calculations