Formula Sheet: Convection

Fundamentals of Convection Heat Transfer

Newton's Law of Cooling

Convective Heat Transfer Rate:

\[q = h A_s (T_s - T_\infty)\]

• q = convective heat transfer rate (W or Btu/hr)
• h = convective heat transfer coefficient (W/m²·K or Btu/hr·ft²·°F)
• A_s = surface area exposed to convection (m² or ft²)
• T_s = surface temperature (K or °F)
• T_∞ = fluid free stream temperature (K or °F)

Convective Heat Flux:

\[q'' = \frac{q}{A_s} = h (T_s - T_\infty)\]

• q'' = convective heat flux (W/m² or Btu/hr·ft²)

Convection Coefficient from Temperature Gradient

\[h = -\frac{k_f}{(T_s - T_\infty)} \left(\frac{\partial T}{\partial y}\right)_{y=0}\]

• k_f = thermal conductivity of fluid (W/m·K or Btu/hr·ft·°F)
• ∂T/∂y = temperature gradient normal to surface at wall
• y = distance perpendicular to surface

Dimensionless Numbers in Convection

Reynolds Number

General Form:

\[\text{Re} = \frac{\rho V L_c}{\mu} = \frac{V L_c}{\nu}\]

• Re = Reynolds number (dimensionless)
• ρ = fluid density (kg/m³ or lb_m/ft³)
• V = fluid velocity (m/s or ft/s)
• L_c = characteristic length (m or ft)
• μ = dynamic viscosity (Pa·s or lb_m/ft·s)
• ν = kinematic viscosity (m²/s or ft²/s), where ν = μ/ρ

For Internal Flow (Circular Pipes):

\[\text{Re}_D = \frac{\rho V D}{\mu} = \frac{V D}{\nu}\]

• D = pipe diameter (m or ft)
• V = mean velocity (m/s or ft/s)

For External Flow (Flat Plate):

\[\text{Re}_x = \frac{\rho u_\infty x}{\mu} = \frac{u_\infty x}{\nu}\]

• u_∞ = free stream velocity (m/s or ft/s)
• x = distance from leading edge (m or ft)

Flow Regime Criteria:

• Internal pipe flow: Laminar if Re_D < 2300,="" turbulent="" if="">_D > 4000
• External flat plate flow: Laminar if Re_x < 5×10⁵,="" turbulent="" if="">_x > 5×10⁵

Prandtl Number

\[\text{Pr} = \frac{\nu}{\alpha} = \frac{c_p \mu}{k} = \frac{\mu c_p}{k}\]

• Pr = Prandtl number (dimensionless)
• α = thermal diffusivity (m²/s or ft²/s)
• c_p = specific heat at constant pressure (J/kg·K or Btu/lb_m·°F)
• k = thermal conductivity of fluid (W/m·K or Btu/hr·ft·°F)

The Prandtl number represents the ratio of momentum diffusivity to thermal diffusivity.

Nusselt Number

\[\text{Nu} = \frac{h L_c}{k}\]

• Nu = Nusselt number (dimensionless)
• h = convective heat transfer coefficient (W/m²·K or Btu/hr·ft²·°F)
• L_c = characteristic length (m or ft)
• k = thermal conductivity of fluid (W/m·K or Btu/hr·ft·°F)

The Nusselt number represents the ratio of convective to conductive heat transfer across the boundary.

Solving for Convection Coefficient:

\[h = \frac{\text{Nu} \cdot k}{L_c}\]

Grashof Number

\[\text{Gr}_L = \frac{g \beta (T_s - T_\infty) L_c^3}{\nu^2}\]

• 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)
• T_s = surface temperature (K or °R)
• T_∞ = fluid temperature (K or °R)
• L_c = characteristic length (m or ft)
• ν = kinematic viscosity (m²/s or ft²/s)

For Ideal Gases:

\[\beta = \frac{1}{T_f}\]

• T_f = film temperature (K or °R), where T_f = (T_s + T_∞)/2

Rayleigh Number

\[\text{Ra}_L = \text{Gr}_L \cdot \text{Pr} = \frac{g \beta (T_s - T_\infty) L_c^3}{\nu \alpha}\]

• Ra = Rayleigh number (dimensionless)
• α = thermal diffusivity (m²/s or ft²/s)

The Rayleigh number is used to characterize natural (free) convection.

Stanton Number

\[\text{St} = \frac{\text{Nu}}{\text{Re} \cdot \text{Pr}} = \frac{h}{\rho V c_p}\]

• St = Stanton number (dimensionless)

Peclet Number

\[\text{Pe} = \text{Re} \cdot \text{Pr} = \frac{V L_c}{\alpha}\]

• Pe = Peclet number (dimensionless)

Forced Convection - External Flow

Flow Over Flat Plate

Local Nusselt Number - Laminar (Re_x <>

\[\text{Nu}_x = 0.332 \text{Re}_x^{1/2} \text{Pr}^{1/3}\]

• Valid for Pr ≥ 0.6
• Uniform surface temperature condition

Average Nusselt Number - Laminar:

\[\overline{\text{Nu}}_L = 0.664 \text{Re}_L^{1/2} \text{Pr}^{1/3}\]

• Re_L = Reynolds number based on plate length L

Local Nusselt Number - Turbulent (Re_x > 5×10⁵):

\[\text{Nu}_x = 0.0296 \text{Re}_x^{4/5} \text{Pr}^{1/3}\]

• Valid for 0.6 ≤ Pr ≤ 60

Average Nusselt Number - Turbulent:

\[\overline{\text{Nu}}_L = 0.037 \text{Re}_L^{4/5} \text{Pr}^{1/3}\]

• Valid for 0.6 ≤ Pr ≤ 60
• Valid for 5×10⁵ ≤ Re_L ≤ 10⁷

Mixed Boundary Layer (Laminar + Turbulent):

\[\overline{\text{Nu}}_L = \left(0.037 \text{Re}_L^{4/5} - 871\right) \text{Pr}^{1/3}\]

• Valid when transition occurs at Re_x,c = 5×10⁵
• Use when flow begins laminar and transitions to turbulent

Flow Over Cylinder

Churchill-Bernstein Correlation:

\[\overline{\text{Nu}}_D = 0.3 + \frac{0.62 \text{Re}_D^{1/2} \text{Pr}^{1/3}}{[1+(0.4/\text{Pr})^{2/3}]^{1/4}} \left[1 + \left(\frac{\text{Re}_D}{282000}\right)^{5/8}\right]^{4/5}\]

• Valid for Re_D·Pr > 0.2
• Valid for wide range of Re_D and Pr
• D = cylinder diameter

Simplified Correlation for Gases (Pr ≈ 0.7):

\[\overline{\text{Nu}}_D = C \text{Re}_D^m \text{Pr}^{1/3}\]

• Constants C and m depend on Reynolds number range (see tables)

Flow Over Sphere

Whitaker Correlation:

\[\overline{\text{Nu}}_D = 2 + \left(0.4 \text{Re}_D^{1/2} + 0.06 \text{Re}_D^{2/3}\right) \text{Pr}^{0.4} \left(\frac{\mu_\infty}{\mu_s}\right)^{1/4}\]

• Valid for 0.71 < pr=""><>
• Valid for 3.5 <>_D <>
• Valid for 1.0 <>_∞/μ_s) <>
• D = sphere diameter
• μ_∞ = viscosity at free stream temperature
• μ_s = viscosity at surface temperature

For gases where μ_∞/μ_s ≈ 1:

\[\overline{\text{Nu}}_D = 2 + 0.4 \text{Re}_D^{1/2} + 0.06 \text{Re}_D^{2/3}\]

Flow Across Tube Banks

Zukauskas Correlation:

\[\overline{\text{Nu}}_D = C \text{Re}_{D,\text{max}}^m \text{Pr}^{0.36} \left(\frac{\text{Pr}}{\text{Pr}_s}\right)^{1/4}\]

• Re_D,max = Reynolds number based on maximum velocity
• Pr_s = Prandtl number at surface temperature
• Constants C and m depend on tube arrangement (in-line or staggered) and Re_D,max
• All properties evaluated at arithmetic mean of inlet and outlet temperatures except Pr_s

Maximum Velocity in Tube Banks:

For in-line arrangement:

\[V_{\text{max}} = \frac{S_T}{S_T - D} V\]

For staggered arrangement, if 2(S_D - D) <>_T - D):

\[V_{\text{max}} = \frac{S_T}{2(S_D - D)} V\]

Otherwise:

\[V_{\text{max}} = \frac{S_T}{S_T - D} V\]

• S_T = transverse pitch (spacing between tube centers perpendicular to flow)
• S_D = diagonal pitch
• D = tube outer diameter
• V = approach velocity

Forced Convection - Internal Flow

Mean Velocity and Reynolds Number

Mean (Average) Velocity:

\[V_m = \frac{\dot{m}}{\rho A_c} = \frac{\dot{V}}{A_c}\]

• V_m = mean velocity (m/s or ft/s)
• ṁ = mass flow rate (kg/s or lb_m/s)
• V̇ = volumetric flow rate (m³/s or ft³/s)
• A_c = cross-sectional area (m² or ft²)

Reynolds Number for Circular Tubes:

\[\text{Re}_D = \frac{\rho V_m D}{\mu} = \frac{V_m D}{\nu}\]

Reynolds Number for Non-Circular Ducts:

\[\text{Re}_{D_h} = \frac{\rho V_m D_h}{\mu} = \frac{V_m D_h}{\nu}\]

• D_h = hydraulic diameter

Hydraulic Diameter

\[D_h = \frac{4 A_c}{P}\]

• D_h = hydraulic diameter (m or ft)
• A_c = cross-sectional area of duct (m² or ft²)
• P = wetted perimeter (m or ft)

For Rectangular Duct (a × b):

\[D_h = \frac{2ab}{a+b}\]

For Annular Space (between two concentric tubes):

\[D_h = D_o - D_i\]

• D_o = outer diameter of annulus
• D_i = inner diameter of annulus

Entry Length

Hydrodynamic Entry Length - Laminar:

\[\frac{L_{h,\text{lam}}}{D} \approx 0.05 \text{Re}_D\]

Thermal Entry Length - Laminar:

\[\frac{L_{t,\text{lam}}}{D} \approx 0.05 \text{Re}_D \cdot \text{Pr}\]

Entry Length - Turbulent (both hydrodynamic and thermal):

\[\frac{L_{\text{turb}}}{D} \approx 10\]

• L_h = hydrodynamic entry length
• L_t = thermal entry length

Mean Temperature

Bulk (Mean) Temperature:

\[T_m = \frac{\int_0^r u(r) T(r) r \, dr}{\int_0^r u(r) r \, dr}\]

For energy balance calculations:

\[\dot{q} = \dot{m} c_p (T_{m,\text{out}} - T_{m,\text{in}})\]

• T_m = mean (bulk) temperature
• T_m,in = inlet mean temperature
• T_m,out = outlet mean temperature

Fully Developed Laminar Flow in Circular Tubes

Constant Surface Temperature:

\[\text{Nu}_D = 3.66\]

Constant Surface Heat Flux:

\[\text{Nu}_D = 4.36\]

• Valid only for fully developed flow (x > L_t)
• Re_D <>

Fully Developed Turbulent Flow in Circular Tubes

Dittus-Boelter Equation:

\[\text{Nu}_D = 0.023 \text{Re}_D^{4/5} \text{Pr}^n\]

• n = 0.4 for heating of fluid (T_s > T_m)
• n = 0.3 for cooling of fluid (T_s <>_m)
• Valid for 0.7 ≤ Pr ≤ 160
• Valid for Re_D ≥ 10,000
• Valid for L/D ≥ 10

Sieder-Tate Equation:

\[\text{Nu}_D = 0.027 \text{Re}_D^{4/5} \text{Pr}^{1/3} \left(\frac{\mu}{\mu_s}\right)^{0.14}\]

• Valid for 0.7 ≤ Pr ≤ 16,700
• Valid for Re_D ≥ 10,000
• μ = dynamic viscosity at bulk mean temperature
• μ_s = dynamic viscosity at surface temperature

Gnielinski Equation (more accurate for wider range):

\[\text{Nu}_D = \frac{(f/8)(\text{Re}_D - 1000) \text{Pr}}{1 + 12.7 (f/8)^{1/2} (\text{Pr}^{2/3} - 1)}\]

• Valid for 0.5 ≤ Pr ≤ 2000
• Valid for 3000 ≤ Re_D ≤ 5×10⁶
• f = Darcy friction factor

Petukhov Friction Factor (for use with Gnielinski):

\[f = (0.790 \ln \text{Re}_D - 1.64)^{-2}\]

• Valid for 3000 ≤ Re_D ≤ 5×10⁶

Developing Flow Correlations

Entry Region Correction for Turbulent Flow:

\[\overline{\text{Nu}}_D = \text{Nu}_{D,\text{fd}} \left[1 + \left(\frac{D}{L}\right)^{0.7}\right]\]

• Nu_D,fd = fully developed Nusselt number
• L = tube length
• Apply when L/D <>

Non-Circular Ducts

For non-circular ducts, replace D with D_h (hydraulic diameter) in the correlations above. The turbulent flow correlations remain reasonably accurate. For laminar flow, Nusselt numbers differ and depend on duct geometry.

Natural (Free) Convection

Vertical Flat Plate

Laminar Flow (10⁴ <>_L <>

\[\overline{\text{Nu}}_L = 0.59 \text{Ra}_L^{1/4}\]

Turbulent Flow (10⁹ <>_L <>

\[\overline{\text{Nu}}_L = 0.10 \text{Ra}_L^{1/3}\]

Churchill-Chu Correlation (entire range):

\[\overline{\text{Nu}}_L = \left\{0.825 + \frac{0.387 \text{Ra}_L^{1/6}}{[1+(0.492/\text{Pr})^{9/16}]^{8/27}}\right\}^2\]

• Valid for all Ra_L
• L = height of vertical plate

Horizontal Plate

Upper Surface of Hot Plate or Lower Surface of Cold Plate (10⁴ <>_L <>

\[\overline{\text{Nu}}_L = 0.54 \text{Ra}_L^{1/4}\]

Upper Surface of Hot Plate or Lower Surface of Cold Plate (10⁷ <>_L <>

\[\overline{\text{Nu}}_L = 0.15 \text{Ra}_L^{1/3}\]

Lower Surface of Hot Plate or Upper Surface of Cold Plate (10⁵ <>_L <>

\[\overline{\text{Nu}}_L = 0.27 \text{Ra}_L^{1/4}\]

• L = L_c = A_s/P for horizontal plates
• A_s = surface area
• P = perimeter

Inclined Plate

For a plate inclined at angle θ from vertical, replace g in Grashof and Rayleigh numbers with g cos θ for the upper surface of a hot plate or lower surface of a cold plate when θ <>

Vertical Cylinder

Use vertical flat plate correlations if:

\[\frac{D}{L} \geq \frac{35}{Gr_L^{1/4}}\]

• D = cylinder diameter
• L = cylinder height

Horizontal Cylinder

Churchill-Chu Correlation:

\[\overline{\text{Nu}}_D = \left\{0.60 + \frac{0.387 \text{Ra}_D^{1/6}}{[1+(0.559/\text{Pr})^{9/16}]^{8/27}}\right\}^2\]

• Valid for Ra_D <>
• D = cylinder diameter

Sphere

Churchill Correlation:

\[\overline{\text{Nu}}_D = 2 + \frac{0.589 \text{Ra}_D^{1/4}}{[1+(0.469/\text{Pr})^{9/16}]^{4/9}}\]

• Valid for Ra_D <>
• Valid for Pr ≥ 0.7
• D = sphere diameter

Enclosed Spaces

Vertical Rectangular Enclosure (between two vertical plates):

For aspect ratio H/L (height/spacing) between 2 and 10, and Pr <>

\[\overline{\text{Nu}}_L = 0.22 \left(\frac{\text{Pr}}{0.2 + \text{Pr}} \text{Ra}_L\right)^{0.28} \left(\frac{H}{L}\right)^{-1/4}\]

• Valid for 2 < h/l=""><>
• Valid for Pr <>
• Valid for 10³ <>_L <>
• L = spacing between plates
• H = height of plates

Horizontal Rectangular Enclosure (between horizontal plates):

For hot plate above cold plate (stable, conduction only):

\[\text{Nu}_L = 1\]

For hot plate below cold plate, Ra_L <>

\[\text{Nu}_L = 1\]

For hot plate below cold plate, 1708 <>_L <>

\[\overline{\text{Nu}}_L = 0.069 \text{Ra}_L^{1/3} \text{Pr}^{0.074}\]

• L = spacing between plates

Condensation Heat Transfer

Film Condensation on Vertical Plate

Average Heat Transfer Coefficient - Laminar Film:

\[\overline{h} = 0.943 \left[\frac{g \rho_l (\rho_l - \rho_v) h_{fg} k_l^3}{\mu_l (T_{sat} - T_s) L}\right]^{1/4}\]

• h̄ = average heat transfer coefficient (W/m²·K or Btu/hr·ft²·°F)
• ρ_l = density of liquid (kg/m³ or lb_m/ft³)
• ρ_v = density of vapor (kg/m³ or lb_m/ft³)
• h_fg = latent heat of vaporization (J/kg or Btu/lb_m)
• k_l = thermal conductivity of liquid (W/m·K or Btu/hr·ft·°F)
• μ_l = dynamic viscosity of liquid (Pa·s or lb_m/ft·s)
• T_sat = saturation temperature (K or °F)
• T_s = surface temperature (K or °F)
• L = vertical length of plate (m or ft)

Modified Latent Heat (accounting for subcooling):

\[h_{fg}^* = h_{fg} + 0.68 c_{p,l} (T_{sat} - T_s)\]

• h_fg* = modified latent heat
• c_p,l = specific heat of liquid
• Use h_fg* in place of h_fg in correlations for improved accuracy

Reynolds Number for Condensate Film:

\[\text{Re} = \frac{4 \dot{m}}{\mu_l P} = \frac{4 \Gamma}{\mu_l}\]

• Γ = condensate mass flow rate per unit width (kg/s·m or lb_m/s·ft)
• P = wetted perimeter

Laminar film: Re <>
Wavy laminar: 30 < re=""><>
Turbulent: Re > 1800

Film Condensation on Horizontal Tube

Single Horizontal Tube:

\[\overline{h} = 0.729 \left[\frac{g \rho_l (\rho_l - \rho_v) h_{fg} k_l^3}{\mu_l (T_{sat} - T_s) D}\right]^{1/4}\]

• D = tube outer diameter (m or ft)

Vertical Tier of N Horizontal Tubes:

\[\overline{h}_N = \frac{1}{N^{1/4}} \overline{h}_1\]

• h̄_N = average coefficient for N tubes
• h̄₁ = coefficient for single tube
• N = number of tubes in vertical tier

Boiling Heat Transfer

Pool Boiling Regimes

Pool boiling heat transfer depends on excess temperature ΔT_e = T_s - T_sat:

• Free convection boiling: ΔT_e <>
• Nucleate boiling: 5°C <>_e < 30°c="">
• Transition boiling: 30°C <>_e < 120°c="">
• Film boiling: ΔT_e > 120°C

Nucleate Pool Boiling

Rohsenow Correlation:

\[q'' = \mu_l h_{fg} \left[\frac{g(\rho_l - \rho_v)}{\sigma}\right]^{1/2} \left[\frac{c_{p,l}(T_s - T_{sat})}{C_{sf} h_{fg} \text{Pr}_l^n}\right]^3\]

• q'' = heat flux (W/m² or Btu/hr·ft²)
• σ = surface tension (N/m or lb_f/ft)
• C_sf = empirical constant depending on surface-fluid combination
• n = 1.0 for water, 1.7 for other fluids
• Pr_l = Prandtl number of liquid

Critical Heat Flux (Peak Heat Flux)

Zuber Correlation for horizontal surfaces:

\[q''_{\text{max}} = 0.149 h_{fg} \rho_v \left[\frac{\sigma g (\rho_l - \rho_v)}{\rho_v^2}\right]^{1/4}\]

• q''_max = critical (peak) heat flux
• Represents transition from nucleate to film boiling
• Operating above this flux leads to burnout

Film Boiling on Horizontal Cylinder

Bromley Correlation:

\[h = 0.62 \left[\frac{g k_v^3 \rho_v (\rho_l - \rho_v) h_{fg}^*}{\mu_v D (T_s - T_{sat})}\right]^{1/4}\]

• k_v = thermal conductivity of vapor
• ρ_v = density of vapor
• μ_v = dynamic viscosity of vapor
• D = cylinder diameter

\[h_{fg}^* = h_{fg} + 0.4 c_{p,v} (T_s - T_{sat})\]

Heat Exchangers

Overall Heat Transfer Coefficient

General Definition:

\[q = U A \Delta T_m\]

• U = overall heat transfer coefficient (W/m²·K or Btu/hr·ft²·°F)
• A = heat transfer area (m² or ft²)
• ΔT_m = appropriate mean temperature difference

Thermal Resistance Network (plane wall):

\[\frac{1}{UA} = \frac{1}{h_i A_i} + \frac{t}{k A_w} + \frac{1}{h_o A_o}\]

• h_i = inside convection coefficient
• h_o = outside convection coefficient
• A_i = inside surface area
• A_o = outside surface area
• A_w = wall mean area
• t = wall thickness
• k = wall thermal conductivity

For Cylindrical Tubes (based on outer area A_o):

\[\frac{1}{U_o A_o} = \frac{1}{h_i A_i} + \frac{\ln(r_o/r_i)}{2\pi k L} + \frac{1}{h_o A_o}\]

Or equivalently:

\[\frac{1}{U_o} = \frac{r_o}{r_i h_i} + \frac{r_o \ln(r_o/r_i)}{k} + \frac{1}{h_o}\]

• r_i = inner radius
• r_o = outer radius
• L = tube length

Including Fouling Resistances:

\[\frac{1}{UA} = \frac{1}{h_i A_i} + \frac{R''_{f,i}}{A_i} + \frac{t}{k A_w} + \frac{R''_{f,o}}{A_o} + \frac{1}{h_o A_o}\]

• R''_f,i = inside fouling factor (m²·K/W or hr·ft²·°F/Btu)
• R''_f,o = outside fouling factor

Log Mean Temperature Difference (LMTD) Method

Heat Transfer Rate:

\[q = U A \Delta T_{lm}\]

Log Mean Temperature Difference:

\[\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}\]

For Parallel Flow:

\[\Delta T_1 = T_{h,i} - T_{c,i}\] \[\Delta T_2 = T_{h,o} - T_{c,o}\]

For Counter Flow:

\[\Delta T_1 = T_{h,i} - T_{c,o}\] \[\Delta T_2 = T_{h,o} - T_{c,i}\]

• T_h,i = hot fluid inlet temperature
• T_h,o = hot fluid outlet temperature
• T_c,i = cold fluid inlet temperature
• T_c,o = cold fluid outlet temperature

For Complex Configurations (multi-pass, cross-flow):

\[\Delta T_m = F \cdot \Delta T_{lm,cf}\]

• F = correction factor (obtained from charts based on P and R)
• ΔT_lm,cf = LMTD for counter-flow arrangement

Correction Factor Parameters:

\[P = \frac{T_{c,o} - T_{c,i}}{T_{h,i} - T_{c,i}}\] \[R = \frac{T_{h,i} - T_{h,o}}{T_{c,o} - T_{c,i}} = \frac{\dot{m}_c c_{p,c}}{\dot{m}_h c_{p,h}}\]

Effectiveness-NTU Method

Heat Exchanger Effectiveness:

\[\varepsilon = \frac{q}{q_{\text{max}}}\]

• ε = heat exchanger effectiveness (dimensionless)
• q = actual heat transfer rate
• q_max = maximum possible heat transfer rate

Maximum Possible Heat Transfer:

\[q_{\text{max}} = C_{\text{min}} (T_{h,i} - T_{c,i})\]

• C_min = minimum heat capacity rate

Heat Capacity Rates:

\[C_h = \dot{m}_h c_{p,h}\] \[C_c = \dot{m}_c c_{p,c}\] \[C_{\text{min}} = \min(C_h, C_c)\] \[C_{\text{max}} = \max(C_h, C_c)\]

Capacity Rate Ratio:

\[C_r = \frac{C_{\text{min}}}{C_{\text{max}}}\]

Number of Transfer Units (NTU):

\[\text{NTU} = \frac{UA}{C_{\text{min}}}\]

Actual Heat Transfer:

\[q = \varepsilon C_{\text{min}} (T_{h,i} - T_{c,i})\]

Effectiveness Relations for Common Configurations:

Parallel Flow:

\[\varepsilon = \frac{1 - \exp[-\text{NTU}(1+C_r)]}{1 + C_r}\]

Counter Flow:

\[\varepsilon = \frac{1 - \exp[-\text{NTU}(1-C_r)]}{1 - C_r \exp[-\text{NTU}(1-C_r)]}\]

Counter Flow with C_r = 1:

\[\varepsilon = \frac{\text{NTU}}{1 + \text{NTU}}\]

Shell-and-Tube (one shell pass, 2, 4, 6... tube passes):

\[\varepsilon = 2\left\{1 + C_r + \sqrt{1+C_r^2} \frac{1+\exp[-\text{NTU}\sqrt{1+C_r^2}]}{1-\exp[-\text{NTU}\sqrt{1+C_r^2}]}\right\}^{-1}\]

Cross Flow (both fluids unmixed):

\[\varepsilon = 1 - \exp\left[\frac{\text{NTU}^{0.22}}{C_r}\left(\exp[-C_r \cdot \text{NTU}^{0.78}]-1\right)\right]\]

For C_r = 0 (phase change or C_max → ∞):

\[\varepsilon = 1 - \exp(-\text{NTU})\]

Energy Balance Relations

Hot Fluid:

\[q = \dot{m}_h c_{p,h} (T_{h,i} - T_{h,o}) = C_h (T_{h,i} - T_{h,o})\]

Cold Fluid:

\[q = \dot{m}_c c_{p,c} (T_{c,o} - T_{c,i}) = C_c (T_{c,o} - T_{c,i})\]

Overall Energy Balance:

\[\dot{m}_h c_{p,h} (T_{h,i} - T_{h,o}) = \dot{m}_c c_{p,c} (T_{c,o} - T_{c,i})\]

Additional Convection Considerations

Film Temperature

For property evaluation in external flow correlations:

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

• T_f = film temperature
• Fluid properties (ρ, μ, k, c_p, Pr) are evaluated at T_f

Bulk Mean Temperature

For internal flow correlations:

\[T_m = \frac{T_{m,i} + T_{m,o}}{2}\]

• T_m = bulk mean temperature
• Fluid properties are evaluated at T_m except where noted otherwise

Combined Convection and Radiation

When both convection and radiation occur simultaneously from a surface:

\[q_{\text{total}} = q_{\text{conv}} + q_{\text{rad}}\] \[q_{\text{total}} = h_{\text{conv}} A (T_s - T_\infty) + \varepsilon \sigma A (T_s^4 - T_{\text{sur}}^4)\]

Combined Heat Transfer Coefficient:

\[h_{\text{combined}} = h_{\text{conv}} + h_{\text{rad}}\]

Where the radiation heat transfer coefficient is linearized as:

\[h_{\text{rad}} = \varepsilon \sigma (T_s + T_{\text{sur}})(T_s^2 + T_{\text{sur}}^2)\]

• ε = emissivity of surface
• σ = Stefan-Boltzmann constant (5.67×10⁻⁸ W/m²·K⁴ or 0.1714×10⁻⁸ Btu/hr·ft²·R⁴)
• T_sur = surrounding surface temperature

Mixed Convection

When both forced and natural convection are significant, the combined effect can be estimated:

For assisting flow:

\[\text{Nu}^n = \text{Nu}_{\text{forced}}^n + \text{Nu}_{\text{natural}}^n\]

For opposing flow:

\[\text{Nu}^n = |\text{Nu}_{\text{forced}}^n - \text{Nu}_{\text{natural}}^n|\]

• Typically n = 3 for vertical surfaces
• Mixed convection is important when Gr/Re² ≈ 1

Thermal Boundary Layer

Boundary Layer Thickness

Hydrodynamic Boundary Layer Thickness - Laminar:

\[\delta \approx \frac{5x}{\text{Re}_x^{1/2}}\]

Thermal Boundary Layer Thickness - Laminar:

\[\delta_t \approx \frac{\delta}{\text{Pr}^{1/3}}\]

• δ = hydrodynamic boundary layer thickness
• δ_t = thermal boundary layer thickness
• x = distance from leading edge

Local Friction Coefficient

Laminar Flow over Flat Plate:

\[C_{f,x} = \frac{0.664}{\text{Re}_x^{1/2}}\]

Turbulent Flow over Flat Plate:

\[C_{f,x} = \frac{0.0592}{\text{Re}_x^{1/5}}\]

• C_f,x = local skin friction coefficient

Average Friction Coefficient

Laminar Flow over Flat Plate:

\[\overline{C}_f = \frac{1.328}{\text{Re}_L^{1/2}}\]

Turbulent Flow over Flat Plate:

\[\overline{C}_f = \frac{0.074}{\text{Re}_L^{1/5}}\]

• Valid for Re_L <>

Convection in Special Geometries

Annular Space

For fully developed laminar flow in annular region between concentric tubes:

Nusselt numbers depend on boundary conditions and diameter ratio:

\[D_r = \frac{D_i}{D_o}\]

• D_r = diameter ratio
• Values obtained from tables based on which surface is heated

Finned Surfaces

Fin Efficiency:

\[\eta_f = \frac{q_{\text{fin}}}{q_{\text{fin,max}}} = \frac{\tanh(mL)}{mL}\]

• η_f = fin efficiency
• m = (hp/kA_c)^(1/2)
• h = convection coefficient
• p = fin perimeter
• k = fin thermal conductivity
• A_c = fin cross-sectional area
• L = fin length

Overall Surface Efficiency:

\[\eta_o = 1 - \frac{A_f}{A_t}(1 - \eta_f)\]

• η_o = overall surface efficiency
• A_f = total fin surface area
• A_t = total heat transfer area (finned + unfinned)

Total Heat Transfer from Finned Surface:

\[q = \eta_o h A_t (T_b - T_\infty)\]

• T_b = base temperature

Compact Heat Exchangers

For compact heat exchangers with complex geometries, heat transfer and friction data are typically presented in terms of:

Colburn j-Factor:

\[j = \text{St} \cdot \text{Pr}^{2/3} = \frac{\text{Nu}}{\text{Re} \cdot \text{Pr}^{1/3}}\]

Friction Factor:

\[f = \frac{2 \Delta p}{\rho V^2} \frac{D_h}{L}\]

• j and f are provided as functions of Re_Dh in charts for specific geometries
• Δp = pressure drop