Formula Sheet: Heat Transfer

Conduction Heat Transfer

Fourier's Law of Heat Conduction

One-Dimensional Steady-State Conduction: \[q = -kA\frac{dT}{dx}\]

• q = heat transfer rate (W or Btu/h)
• k = thermal conductivity (W/m·K or Btu/h·ft·°F)
• A = cross-sectional area perpendicular to heat flow (m² or ft²)
• dT/dx = temperature gradient in direction of heat flow (K/m or °F/ft)
• Negative sign indicates heat flows from high to low temperature

Heat Flux (q"): \[q'' = \frac{q}{A} = -k\frac{dT}{dx}\]

• q" = heat flux (W/m² or Btu/h·ft²)

Plane Wall Conduction

Single Layer Plane Wall: \[q = \frac{kA(T_1 - T_2)}{L}\]

• L = wall thickness (m or ft)
• T₁, T₂ = temperatures at surfaces (K or °F)

Thermal Resistance (Plane Wall): \[R_{cond} = \frac{L}{kA}\]

• R_cond = conductive thermal resistance (K/W or °F·h/Btu)
• Units: °C/W, K/W, or °F·h/Btu

Multi-Layer Plane Wall (Series Resistances): \[q = \frac{\Delta T_{overall}}{R_{total}} = \frac{T_1 - T_{n+1}}{\sum_{i=1}^{n}\frac{L_i}{k_iA}}\] \[R_{total} = \sum_{i=1}^{n}R_i = \sum_{i=1}^{n}\frac{L_i}{k_iA}\]

• n = number of layers
• Assumes perfect thermal contact between layers

Cylindrical Coordinates (Radial Conduction)

Hollow Cylinder (Radial Heat Flow): \[q = \frac{2\pi Lk(T_1 - T_2)}{\ln(r_2/r_1)}\]

• L = length of cylinder (m or ft)
• r₁ = inner radius (m or ft)
• r₂ = outer radius (m or ft)
• T₁ = temperature at inner surface
• T₂ = temperature at outer surface

Thermal Resistance (Cylindrical): \[R_{cyl} = \frac{\ln(r_2/r_1)}{2\pi Lk}\] Multi-Layer Cylindrical Wall: \[q = \frac{2\pi L(T_{inner} - T_{outer})}{\sum_{i=1}^{n}\frac{\ln(r_{i+1}/r_i)}{k_i}}\]

Spherical Coordinates (Radial Conduction)

Hollow Sphere: \[q = \frac{4\pi kr_1r_2(T_1 - T_2)}{r_2 - r_1}\]

• r₁ = inner radius
• r₂ = outer radius

Thermal Resistance (Spherical): \[R_{sph} = \frac{r_2 - r_1}{4\pi kr_1r_2}\]

Critical Radius of Insulation

Critical Radius for Cylinder: \[r_{cr} = \frac{k}{h}\]

• r_cr = critical radius (m or ft)
• h = convective heat transfer coefficient (W/m²·K or Btu/h·ft²·°F)
• Adding insulation increases heat transfer if outer radius <>_cr
• Adding insulation decreases heat transfer if outer radius > r_cr

Critical Radius for Sphere: \[r_{cr} = \frac{2k}{h}\]

Contact Resistance

Thermal Contact Resistance: \[R_{contact} = \frac{1}{h_cA}\] \[q = \frac{T_1 - T_2}{R_{contact}} = h_cA(T_1 - T_2)\]

• h_c = contact conductance (W/m²·K or Btu/h·ft²·°F)
• Accounts for imperfect contact between solid surfaces

Fins and Extended Surfaces

Fin Efficiency: \[η_f = \frac{q_{actual}}{q_{ideal}} = \frac{q_{actual}}{hA_f(T_b - T_\infty)}\]

• η_f = fin efficiency (dimensionless)
• A_f = total fin surface area
• T_b = base temperature
• T_∞ = fluid temperature

Fin Parameter (m): \[m = \sqrt{\frac{hP}{kA_c}}\]

• P = perimeter of fin cross-section (m or ft)
• A_c = cross-sectional area of fin (m² or ft²)

Long Fin (Infinite Length) - Heat Transfer Rate: \[q_f = \sqrt{hPkA_c}(T_b - T_\infty)\] Long Fin Efficiency: \[η_f = \frac{\tanh(mL)}{mL}\]

• L = fin length
• Valid for fin with insulated tip approximation

Straight Rectangular Fin (Insulated Tip): \[q_f = \sqrt{hPkA_c}(T_b - T_\infty)\tanh(mL)\] Fin with Convection at Tip: \[q_f = \sqrt{hPkA_c}(T_b - T_\infty)\frac{\sinh(mL) + (h/mk)\cosh(mL)}{\cosh(mL) + (h/mk)\sinh(mL)}\] Overall Fin Effectiveness: \[\varepsilon_f = \frac{q_{with\,fin}}{q_{without\,fin}} = \frac{q_f}{hA_b(T_b - T_\infty)}\]

• ε_f = fin effectiveness (dimensionless)
• A_b = base area (area of fin at attachment)
• Fin is beneficial if ε_f > 1

Total Heat Transfer from Finned Surface: \[q_{total} = η_f hA_f(T_b - T_\infty) + hA_b(T_b - T_\infty)\]

• A_b = base area not covered by fins

Overall Surface Efficiency: \[η_o = 1 - \frac{A_f}{A_{total}}(1 - η_f)\]

• A_total = total heat transfer area (fins + base)

Convection Heat Transfer

Newton's Law of Cooling

Convective Heat Transfer: \[q = hA(T_s - T_\infty)\] \[q = hA\Delta T\]

• h = convective heat transfer coefficient (W/m²·K or Btu/h·ft²·°F)
• T_s = surface temperature
• T_∞ = fluid bulk temperature (far from surface)

Convective Thermal Resistance: \[R_{conv} = \frac{1}{hA}\]

Dimensionless Numbers for Convection

Reynolds Number: \[Re = \frac{\rho VL}{\mu} = \frac{VL}{\nu}\]

• Re = Reynolds number (dimensionless)
• ρ = fluid density (kg/m³ or lb_m/ft³)
• V = fluid velocity (m/s or ft/s)
• L = characteristic length (m or ft)
• μ = dynamic viscosity (Pa·s or lb_m/ft·s)
• ν = kinematic viscosity = μ/ρ (m²/s or ft²/s)
• Indicates ratio of inertial to viscous forces

Prandtl Number: \[Pr = \frac{c_p\mu}{k} = \frac{\nu}{\alpha}\]

• Pr = Prandtl number (dimensionless)
• c_p = specific heat at constant pressure (J/kg·K or Btu/lb_m·°F)
• α = thermal diffusivity = k/(ρc_p) (m²/s or ft²/s)
• Ratio of momentum diffusivity to thermal diffusivity

Nusselt Number: \[Nu = \frac{hL}{k}\]

• Nu = Nusselt number (dimensionless)
• L = characteristic length
• k = thermal conductivity of fluid
• Ratio of convective to conductive heat transfer

Grashof Number (Natural Convection): \[Gr = \frac{g\beta(T_s - T_\infty)L^3}{\nu^2}\]

• Gr = Grashof number (dimensionless)
• g = gravitational acceleration (9.81 m/s² or 32.2 ft/s²)
• β = coefficient of thermal expansion (1/K or 1/°R)
• For ideal gas: β = 1/T_film (absolute temperature)
• Ratio of buoyancy to viscous forces

Rayleigh Number: \[Ra = Gr \cdot Pr = \frac{g\beta(T_s - T_\infty)L^3}{\nu\alpha}\]

• Ra = Rayleigh number (dimensionless)
• Used for natural convection correlations

Stanton Number: \[St = \frac{Nu}{Re \cdot Pr} = \frac{h}{\rho Vc_p}\]

• St = Stanton number (dimensionless)

Peclet Number: \[Pe = Re \cdot Pr = \frac{VL}{\alpha}\]

• Pe = Peclet number (dimensionless)

Forced Convection - External Flow

Flat Plate - Laminar Flow (Re_x <> \[Nu_x = 0.332Re_x^{1/2}Pr^{1/3}\]

• Local Nusselt number at position x
• Valid for Pr ≥ 0.6

Flat Plate - Average Laminar (0 to L): \[Nu_L = 0.664Re_L^{1/2}Pr^{1/3}\]

• Average Nusselt number over length L
• Valid for Re_L <>

Flat Plate - Turbulent Flow (Re_x > 5×10⁵): \[Nu_x = 0.0296Re_x^{4/5}Pr^{1/3}\]

• Valid for 5×10⁵ <>_x < 10⁷="" and="" 0.6="" ≤="" pr="" ≤="">

Flat Plate - Average Turbulent: \[Nu_L = 0.037Re_L^{4/5}Pr^{1/3}\]

• Valid for 5×10⁵ <>_L <>

Flat Plate - Mixed (Laminar + Turbulent): \[Nu_L = (0.037Re_L^{4/5} - 871)Pr^{1/3}\]

• Valid when flow transitions from laminar to turbulent
• Assumes transition at Re = 5×10⁵

Flow Across Cylinder (Average): \[Nu_D = CRe_D^mPr^{1/3}\]

• D = cylinder diameter
• C, m = constants depending on Re_D range (from tables)
• Evaluate properties at film temperature T_film = (T_s + T_∞)/2

Churchill-Bernstein Correlation (Cylinder): \[Nu_D = 0.3 + \frac{0.62Re_D^{1/2}Pr^{1/3}}{[1+(0.4/Pr)^{2/3}]^{1/4}}\left[1+\left(\frac{Re_D}{282000}\right)^{5/8}\right]^{4/5}\]

• Valid for Re_D × Pr > 0.2
• Covers wide range of Re and Pr

Flow Across Sphere: \[Nu_D = 2 + (0.4Re_D^{1/2} + 0.06Re_D^{2/3})Pr^{0.4}\left(\frac{\mu}{\mu_s}\right)^{1/4}\]

• Valid for 3.5 <>_D < 8×10⁴="" and="" 0.7="">< pr=""><>
• μ = viscosity at T_∞
• μ_s = viscosity at T_s

Forced Convection - Internal Flow

Hydraulic Diameter (Non-Circular Ducts): \[D_h = \frac{4A_c}{P}\]

• D_h = hydraulic diameter (m or ft)
• A_c = cross-sectional flow area
• P = wetted perimeter
• For circular pipe: D_h = D

Entry Length - Laminar: \[L_h \approx 0.05Re_D D\]

• L_h = hydrodynamic entry length

Thermal Entry Length - Laminar: \[L_t \approx 0.05Re_D Pr \cdot D\]

• L_t = thermal entry length

Fully Developed Laminar Flow (Circular Tube, Constant Surface Temperature): \[Nu_D = 3.66\]

• Valid for L/D > 10 and Re_D <>

Fully Developed Laminar Flow (Circular Tube, Constant Heat Flux): \[Nu_D = 4.36\] Laminar Flow - Developing (Sieder-Tate): \[Nu_D = 1.86\left(\frac{Re_D Pr \cdot D}{L}\right)^{1/3}\left(\frac{\mu}{\mu_s}\right)^{0.14}\]

• Valid for simultaneously developing thermal and velocity boundary layers
• Valid for Re_D <>

Turbulent Flow (Dittus-Boelter): \[Nu_D = 0.023Re_D^{4/5}Pr^n\]

• n = 0.4 for heating (T_s > T_bulk)
• n = 0.3 for cooling (T_s <>_bulk)
• Valid for Re_D > 10,000 and 0.7 ≤ Pr ≤ 160
• Valid for L/D > 10 (fully developed)

Turbulent Flow (Sieder-Tate): \[Nu_D = 0.027Re_D^{4/5}Pr^{1/3}\left(\frac{\mu}{\mu_s}\right)^{0.14}\]

• Valid for 0.7 ≤ Pr ≤ 16,700 and Re_D > 10,000

Turbulent Flow (Gnielinski): \[Nu_D = \frac{(f/8)(Re_D - 1000)Pr}{1 + 12.7(f/8)^{1/2}(Pr^{2/3} - 1)}\]

• f = Darcy friction factor
• Valid for 3000 <>_D < 5×10⁶="" and="" 0.5="" ≤="" pr="" ≤="">
• More accurate for transition and turbulent flow

Darcy Friction Factor (Smooth Tubes, Turbulent): \[f = (0.790\ln Re_D - 1.64)^{-2}\]

• Petukhov correlation
• Valid for 3000 <>_D <>

Natural Convection

Vertical Plate (Laminar, 10⁴ <>_L <> \[Nu_L = 0.59Ra_L^{1/4}\] Vertical Plate (Turbulent, 10⁹ <>_L <> \[Nu_L = 0.10Ra_L^{1/3}\] Vertical Plate (Churchill-Chu, All Ra): \[Nu_L = \left[0.825 + \frac{0.387Ra_L^{1/6}}{[1+(0.492/Pr)^{9/16}]^{8/27}}\right]^2\]

• Valid for entire range of Ra_L

Horizontal Plate (Hot Surface Up or Cold Surface Down): \[Nu_L = 0.54Ra_L^{1/4}\quad (10^4 < ra_l="">< 10^7)\]="" \[nu_l="0.15Ra_L^{1/3}\quad" (10^7="">< ra_l="">< 10^{11})\]="">

• L = A_s/P where A_s is surface area and P is perimeter

Horizontal Plate (Hot Surface Down or Cold Surface Up): \[Nu_L = 0.27Ra_L^{1/4}\quad (10^5 < ra_l="">< 10^{10})\]="">Horizontal Cylinder: \[Nu_D = \left[0.60 + \frac{0.387Ra_D^{1/6}}{[1+(0.559/Pr)^{9/16}]^{8/27}}\right]^2\]

• Valid for 10⁻⁵ <>_D <>

Sphere: \[Nu_D = 2 + \frac{0.589Ra_D^{1/4}}{[1+(0.469/Pr)^{9/16}]^{4/9}}\]

• Valid for Ra_D < 10¹¹="" and="" pr="" ≥="">

Condensation

Film Condensation on Vertical Plate (Laminar): \[h = 0.943\left[\frac{g\rho_L(\rho_L - \rho_v)kh_{fg}^3}{4\mu_L(T_{sat} - T_s)L}\right]^{1/4}\]

• ρ_L = liquid density
• ρ_v = vapor density
• h_fg = latent heat of vaporization
• μ_L = liquid dynamic viscosity
• T_sat = saturation temperature
• T_s = surface temperature
• L = plate length (vertical dimension)

Modified Latent Heat (Accounts for Sensible Heat): \[h_{fg}' = h_{fg} + 0.68c_{p,L}(T_{sat} - T_s)\]

• Use h'_fg in place of h_fg for better accuracy

Film Condensation on Horizontal Tube: \[h = 0.725\left[\frac{g\rho_L(\rho_L - \rho_v)kh_{fg}^3}{\mu_L(T_{sat} - T_s)D}\right]^{1/4}\]

• D = tube outer diameter

Boiling

Pool Boiling - Rohsenow Correlation (Nucleate Boiling): \[q'' = \mu_L h_{fg}\left[\frac{g(\rho_L - \rho_v)}{\sigma}\right]^{1/2}\left[\frac{c_{p,L}(T_s - T_{sat})}{C_{sf}h_{fg}Pr_L^n}\right]^3\]

• σ = surface tension (N/m or lb_f/ft)
• C_sf = surface-fluid constant (from tables)
• n = constant (typically 1.0 for water, 1.7 for other fluids)
• Pr_L = Prandtl number of liquid

Critical Heat Flux (Peak Boiling): \[q''_{max} = C h_{fg}\rho_v\left[\frac{\sigma g(\rho_L - \rho_v)}{\rho_v^2}\right]^{1/4}\]

• C ≈ 0.149 for large horizontal surfaces
• Transition point from nucleate to film boiling

Radiation Heat Transfer

Fundamental Radiation Laws

Stefan-Boltzmann Law (Blackbody Emission): \[E_b = \sigma T^4\]

• E_b = blackbody emissive power (W/m² or Btu/h·ft²)
• σ = Stefan-Boltzmann constant = 5.67×10⁻⁸ W/m²·K⁴ or 0.1714×10⁻⁸ Btu/h·ft²·°R⁴
• T = absolute temperature (K or °R)

Blackbody Radiation Heat Transfer: \[q = A\sigma T^4\] Real Surface Emissive Power: \[E = \varepsilon E_b = \varepsilon\sigma T^4\]

• ε = emissivity (dimensionless, 0 ≤ ε ≤ 1)
• ε = 1 for blackbody

Planck's Law (Spectral Distribution): \[E_{b,\lambda} = \frac{C_1}{\lambda^5[e^{C_2/\lambda T} - 1]}\]

• E_b,λ = spectral blackbody emissive power (W/m²·μm)
• λ = wavelength (μm)
• C₁ = 3.742×10⁸ W·μm⁴/m²
• C₂ = 1.439×10⁴ μm·K

Wien's Displacement Law: \[\lambda_{max}T = 2898\,\mu m \cdot K\]

• λ_max = wavelength at maximum emission (μm)
• Identifies peak wavelength for given temperature

Radiative Properties

Absorptivity (α): \[\alpha = \frac{G_{absorbed}}{G_{incident}}\]

• α = absorptivity (fraction of incident radiation absorbed)

Reflectivity (ρ): \[\rho = \frac{G_{reflected}}{G_{incident}}\]

• ρ = reflectivity (fraction of incident radiation reflected)

Transmissivity (τ): \[\tau = \frac{G_{transmitted}}{G_{incident}}\]

• τ = transmissivity (fraction of incident radiation transmitted)

Conservation of Incident Radiation: \[\alpha + \rho + \tau = 1\]

• For opaque surfaces: τ = 0, so α + ρ = 1

Kirchhoff's Law: \[\alpha = \varepsilon\]

• Valid for surfaces in thermal equilibrium
• Absorptivity equals emissivity at same wavelength and temperature

Gray Surface Assumption: \[\alpha_\lambda = \varepsilon_\lambda = constant\]

• Properties independent of wavelength

View Factor (Configuration Factor)

Definition: \[F_{ij} = \frac{Q_{i \to j}}{A_i E_{b,i}}\]

• F_ij = view factor from surface i to surface j (dimensionless)
• Fraction of radiation leaving surface i that directly strikes surface j

Reciprocity Relation: \[A_iF_{ij} = A_jF_{ji}\] Summation Rule (Enclosure): \[\sum_{j=1}^{N}F_{ij} = 1\]

• Sum of all view factors from surface i to all surfaces (including itself) equals 1

View Factor for Flat or Convex Surface to Itself: \[F_{ii} = 0\] View Factor for Concave Surface to Itself: \[F_{ii} > 0\]

Radiation Exchange Between Surfaces

Radiation Exchange Between Two Blackbodies: \[q_{12} = A_1F_{12}\sigma(T_1^4 - T_2^4)\]

• Net heat transfer from surface 1 to surface 2

Radiosity (J): \[J = \varepsilon E_b + \rho G = \varepsilon\sigma T^4 + \rho G\]

• J = radiosity = total radiation leaving surface per unit area (W/m²)
• G = irradiation = incident radiation per unit area (W/m²)

Net Radiation Heat Transfer from Surface: \[q_i = \frac{E_{b,i} - J_i}{(1-\varepsilon_i)/(\varepsilon_i A_i)}\]

• Surface resistance term: (1-ε_i)/(ε_iA_i)

Radiation Exchange Between Surfaces (Network Method): \[q_{ij} = \frac{J_i - J_j}{1/(A_iF_{ij})}\]

• Space resistance: 1/(A_iF_ij)

Two-Surface Enclosures

General Formula for Two Gray, Diffuse, Opaque Surfaces: \[q_{12} = \frac{\sigma(T_1^4 - T_2^4)}{\frac{1-\varepsilon_1}{\varepsilon_1 A_1} + \frac{1}{A_1F_{12}} + \frac{1-\varepsilon_2}{\varepsilon_2 A_2}}\] Large Parallel Plates (A₁ = A₂ = A, F₁₂ = 1): \[q_{12} = \frac{A\sigma(T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1}{\varepsilon_2} - 1}\] Small Object in Large Enclosure (A₁ < a₂,="" f₁₂="" ≈=""> \[q_{12} = A_1\varepsilon_1\sigma(T_1^4 - T_2^4)\]

• Simplifies because small object sees only the enclosure

Long Concentric Cylinders or Spheres: \[q_{12} = \frac{A_1\sigma(T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1-\varepsilon_2}{\varepsilon_2}\left(\frac{A_1}{A_2}\right)}\]

• For cylinders: A = 2πrL
• For spheres: A = 4πr²

Radiation Shields

Single Shield Between Two Parallel Plates: \[q_{with\,shield} = \frac{q_{without\,shield}}{2}\]

• Assumes all surfaces have same emissivity

N Shields: \[q_{with\,N\,shields} = \frac{q_{without\,shields}}{N+1}\] Heat Transfer with Shield (Different Emissivities): \[q = \frac{A\sigma(T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1}{\varepsilon_2} - 1 + \frac{1}{\varepsilon_{s1}} + \frac{1}{\varepsilon_{s2}} - 1}\]

• ε_s1, ε_s2 = emissivities of shield sides 1 and 2

Gas Radiation

Mean Beam Length (L_m):

• Characteristic length for gas radiation
• Sphere (diameter D): L_m = 0.65D
• Infinite cylinder (diameter D): L_m = 0.95D
• Semi-infinite cylinder (height = diameter): L_m = 0.60D
• Cube (side L): L_m = 0.66L
• General volume: L_m = 3.6V/A_s (V = volume, A_s = surface area)

Radiation from Gas to Surface: \[q = A\sigma(T_g^4\varepsilon_g - T_s^4\alpha_g)\]

• ε_g = gas emissivity at T_g
• α_g = gas absorptivity for radiation from surface at T_s
• ε_g and α_g obtained from charts (functions of temperature, partial pressure, and L_m)

Combined Heat Transfer Modes

Overall Heat Transfer Coefficient

Composite Wall (Plane): \[q = UA\Delta T_{overall}\] \[\frac{1}{UA} = R_{total} = \frac{1}{h_1A} + \sum\frac{L_i}{k_iA} + \frac{1}{h_2A}\]

• U = overall heat transfer coefficient (W/m²·K or Btu/h·ft²·°F)
• h₁, h₂ = inside and outside convection coefficients

Overall Heat Transfer Coefficient (Based on Inside Area): \[\frac{1}{U_iA_i} = \frac{1}{h_iA_i} + \sum\frac{R_{cond,i}}{1} + \frac{1}{h_oA_o}\] Overall Heat Transfer Coefficient (Based on Outside Area): \[\frac{1}{U_oA_o} = \frac{1}{h_iA_i} + \sum R_{cond,i} + \frac{1}{h_oA_o}\] Cylindrical System: \[\frac{1}{U_iA_i} = \frac{1}{h_iA_i} + \frac{A_i\ln(r_o/r_i)}{2\pi Lk} + \frac{A_i}{h_oA_o}\]

• Can also be based on outer area or mean area

Log Mean Temperature Difference (LMTD)

Parallel Flow or Counter Flow Heat Exchanger: \[q = UA\Delta T_{lm}\] \[\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}\]

• ΔT_lm = log mean temperature difference

Parallel Flow:

• ΔT₁ = T_h,in - T_c,in (at inlet)
• ΔT₂ = T_h,out - T_c,out (at outlet)

Counter Flow:

• ΔT₁ = T_h,in - T_c,out
• ΔT₂ = T_h,out - T_c,in

LMTD Correction Factor Method: \[\Delta T_m = F \cdot \Delta T_{lm,cf}\]

• F = correction factor for configuration (from charts)
• ΔT_lm,cf = LMTD for counter-flow
• F = 1 for true counter-flow
• F < 1="" for="" other="" configurations="" (cross-flow,="">

Effectiveness-NTU Method

Heat Capacity Rate: \[C = \dot{m}c_p\]

• C = heat capacity rate (W/K or Btu/h·°F)
• ṁ = mass flow rate

Minimum and Maximum Heat Capacity Rates: \[C_{min} = \min(C_h, C_c)\] \[C_{max} = \max(C_h, C_c)\] Maximum Possible Heat Transfer: \[q_{max} = C_{min}(T_{h,in} - T_{c,in})\] Effectiveness (ε): \[\varepsilon = \frac{q_{actual}}{q_{max}} = \frac{C_h(T_{h,in} - T_{h,out})}{C_{min}(T_{h,in} - T_{c,in})} = \frac{C_c(T_{c,out} - T_{c,in})}{C_{min}(T_{h,in} - T_{c,in})}\] Actual Heat Transfer: \[q = \varepsilon C_{min}(T_{h,in} - T_{c,in})\] Number of Transfer Units (NTU): \[NTU = \frac{UA}{C_{min}}\] Heat Capacity Rate Ratio: \[C_r = \frac{C_{min}}{C_{max}}\] Effectiveness Relations (Parallel Flow): \[\varepsilon = \frac{1 - e^{-NTU(1+C_r)}}{1 + C_r}\] Effectiveness Relations (Counter Flow): \[\varepsilon = \frac{1 - e^{-NTU(1-C_r)}}{1 - C_re^{-NTU(1-C_r)}}\quad (C_r < 1)\]="" \[\varepsilon="\frac{NTU}{1" +="" ntu}\quad="" (c_r="1)\]">Effectiveness (One Fluid Condensing or Boiling, C_r = 0): \[\varepsilon = 1 - e^{-NTU}\]

• Valid when C_max → ∞ (phase change fluid)

Transient Heat Conduction

Lumped Capacitance Method

Biot Number: \[Bi = \frac{hL_c}{k}\]

• Bi = Biot number (dimensionless)
• L_c = characteristic length = V/A_s (volume/surface area)
• For sphere: L_c = r/3
• For cylinder: L_c = r/2
• For slab: L_c = L/2 (half-thickness)
• Lumped capacitance valid when Bi <>

Temperature Response (Lumped Capacitance): \[\frac{T(t) - T_\infty}{T_i - T_\infty} = e^{-t/\tau} = e^{-\frac{hA_s}{\rho Vc_p}t}\]

• T(t) = temperature at time t
• T_i = initial uniform temperature
• T_∞ = surrounding fluid temperature
• τ = time constant = ρVc_p/(hA_s)

Time Constant: \[\tau = \frac{\rho Vc_p}{hA_s}\] Total Energy Transfer (up to time t): \[Q = \rho Vc_p(T_i - T(t))\]

Semi-Infinite Solid

Constant Surface Temperature: \[\frac{T(x,t) - T_s}{T_i - T_s} = \text{erf}\left(\frac{x}{2\sqrt{\alpha t}}\right)\]

• erf = error function
• α = thermal diffusivity = k/(ρc_p)
• x = distance from surface

Surface Heat Flux (Constant Surface Temperature): \[q''_s(t) = \frac{k(T_s - T_i)}{\sqrt{\pi\alpha t}}\] Constant Surface Heat Flux: \[T(x,t) - T_i = \frac{q''_s}{k}\left[2\sqrt{\frac{\alpha t}{\pi}}e^{-x^2/(4\alpha t)} - x\cdot\text{erfc}\left(\frac{x}{2\sqrt{\alpha t}}\right)\right]\]

• erfc = complementary error function = 1 - erf

One-Dimensional Transient Conduction (Heisler Charts)

Fourier Number: \[Fo = \frac{\alpha t}{L_c^2}\]

• Fo = Fourier number (dimensionless time)
• Characterizes degree of penetration of heat into solid

Plane Wall (Thickness 2L): \[\frac{T(x,t) - T_\infty}{T_i - T_\infty} = \theta(x,t) = f(Bi, Fo, x/L)\]

• Use Heisler charts or analytical series solutions
• Centerline temperature (x = 0): θ₀(t) = f(Bi, Fo)
• Position factor: θ(x,t)/θ₀(t) = f(x/L, Bi)

Infinite Cylinder (Radius r₀): \[\frac{T(r,t) - T_\infty}{T_i - T_\infty} = f(Bi, Fo, r/r_0)\]

• L_c = r₀ for Bi calculation

Sphere (Radius r₀): \[\frac{T(r,t) - T_\infty}{T_i - T_\infty} = f(Bi, Fo, r/r_0)\]

• L_c = r₀ for Bi calculation

Multidimensional Systems (Product Solution): \[\theta(x,y,z,t) = \theta_{plane\,wall,x}(x,t) \cdot \theta_{plane\,wall,y}(y,t) \cdot \theta_{plane\,wall,z}(z,t)\]

• For semi-infinite cylinder: multiply plane wall and infinite cylinder solutions
• For short cylinder: multiply two infinite cylinder solutions
• For rectangular bar: multiply three plane wall solutions

Analytical Solution (One-Term Approximation)

Valid for Fo > 0.2: Plane Wall: \[\frac{T(x,t) - T_\infty}{T_i - T_\infty} = C_1e^{-\lambda_1^2 Fo}\cos\left(\lambda_1\frac{x}{L}\right)\] Infinite Cylinder: \[\frac{T(r,t) - T_\infty}{T_i - T_\infty} = C_1e^{-\lambda_1^2 Fo}J_0\left(\lambda_1\frac{r}{r_0}\right)\] Sphere: \[\frac{T(r,t) - T_\infty}{T_i - T_\infty} = C_1e^{-\lambda_1^2 Fo}\frac{\sin(\lambda_1 r/r_0)}{\lambda_1 r/r_0}\]

• C₁, λ₁ = constants from tables (functions of Bi)
• J₀ = Bessel function of first kind, order zero

Heat Exchangers

Energy Balance

Hot Fluid: \[q = \dot{m}_hc_{p,h}(T_{h,in} - T_{h,out}) = C_h(T_{h,in} - T_{h,out})\] Cold Fluid: \[q = \dot{m}_cc_{p,c}(T_{c,out} - T_{c,in}) = C_c(T_{c,out} - T_{c,in})\] Energy Balance: \[C_h(T_{h,in} - T_{h,out}) = C_c(T_{c,out} - T_{c,in})\]

Heat Exchanger Types

• Parallel Flow: Both fluids enter at same end, flow in same direction
• Counter Flow: Fluids enter at opposite ends, flow in opposite directions
• Cross Flow: Fluids flow perpendicular to each other
• Shell-and-Tube: One fluid in tubes, other fluid in shell around tubes

Fouling Factor

Overall Heat Transfer with Fouling: \[\frac{1}{U} = \frac{1}{h_i} + R''_{f,i} + \frac{t_{wall}}{k_{wall}} + R''_{f,o} + \frac{1}{h_o}\]

• R"_f,i = inside fouling resistance (m²·K/W or h·ft²·°F/Btu)
• R"_f,o = outside fouling resistance
• Fouling resistances obtained from tables

Effectiveness Relations for Common Configurations

Cross Flow (Both Fluids Unmixed): \[\varepsilon = 1 - \exp\left[\frac{NTU^{0.22}}{C_r}(e^{-C_r \cdot NTU^{0.78}} - 1)\right]\] Cross Flow (One Fluid Mixed, Other Unmixed):

• C_max mixed: \(\varepsilon = \frac{1}{C_r}\left[1 - e^{-C_r(1-e^{-NTU})}\right]\)
• C_min mixed: \(\varepsilon = 1 - e^{-\frac{1}{C_r}(1-e^{-C_r \cdot NTU})}\)

Shell-and-Tube (One Shell Pass, Even Number Tube Passes): \[\varepsilon = \frac{2}{\left[1 + C_r + \sqrt{1+C_r^2}\frac{1+e^{-NTU\sqrt{1+C_r^2}}}{1-e^{-NTU\sqrt{1+C_r^2}}}\right]}\]

Thermal Properties and Relations

Thermal Diffusivity

\[\alpha = \frac{k}{\rho c_p}\]

• α = thermal diffusivity (m²/s or ft²/s)
• Measure of how quickly temperature changes propagate through material

Film Temperature

\[T_{film} = \frac{T_s + T_\infty}{2}\]

• Used to evaluate fluid properties for external convection correlations

Bulk Mean Temperature

\[T_m = \frac{T_{in} + T_{out}}{2}\]

• Used to evaluate fluid properties for internal flow

Energy Equation for Fluid Flow

Heat Transfer to/from Fluid in Duct: \[q = \dot{m}c_p(T_{out} - T_{in})\] Constant Surface Temperature: \[\frac{T_s - T_{m,out}}{T_s - T_{m,in}} = e^{-\frac{hA_s}{\dot{m}c_p}}\]

• T_m = bulk mean temperature
• A_s = heat transfer surface area

Constant Heat Flux: \[T_{m,out} = T_{m,in} + \frac{q''P}{\dot{m}c_p}L\]

• P = perimeter
• L = length

Thermal Resistance Network

Series Resistances

\[R_{total} = R_1 + R_2 + R_3 + ... + R_n\] \[q = \frac{\Delta T_{overall}}{R_{total}}\]

Parallel Resistances

\[\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}\]

Combined Convection and Radiation

Parallel Combination: \[h_{total} = h_{conv} + h_{rad}\] \[q_{total} = (h_{conv} + h_{rad})A(T_s - T_\infty)\] Radiation Heat Transfer Coefficient (Linearized): \[h_{rad} = \varepsilon\sigma(T_s + T_{surr})(T_s^2 + T_{surr}^2)\]

• Allows radiation to be treated similar to convection
• Valid for small temperature differences

Additional Important Relations

Thermal Penetration Depth

\[\delta_t \approx \sqrt{\alpha t}\]

• Approximate depth of temperature penetration in transient conduction

Heat Generation

Volumetric Heat Generation: \[q_{gen} = \dot{q}V\]

• q̇ = volumetric heat generation rate (W/m³ or Btu/h·ft³)
• V = volume

Plane Wall with Uniform Heat Generation: \[T(x) - T_s = \frac{\dot{q}L^2}{2k}\left[1 - \left(\frac{x}{L}\right)^2\right]\]

• Maximum temperature at centerline (x = 0)

Cylinder with Uniform Heat Generation: \[T(r) - T_s = \frac{\dot{q}r_0^2}{4k}\left[1 - \left(\frac{r}{r_0}\right)^2\right]\]

Average vs. Local Convection Coefficient

\[\bar{h} = \frac{1}{L}\int_0^L h(x)dx\]

• h̄ = average heat transfer coefficient over length L
• h(x) = local heat transfer coefficient at position x