Cheatsheet: Convection

1. Convection Fundamentals

1.1 Basic Definitions

1.2 Newton's Law of Cooling

1.3 Thermal Boundary Layer

• Region where temperature gradients exist near solid surface
• Velocity boundary layer: region where velocity gradients exist
• Boundary layer thickness (δ_t): distance from surface where (T_s - T)/(T_s - T_∞) = 0.99
• Laminar boundary layer: smooth, orderly flow with Re <>_critical
• Turbulent boundary layer: chaotic flow with Re > Re_critical

2. Dimensionless Numbers

2.1 Key Dimensionless Parameters

2.2 Variable Definitions

• ρ = fluid density (kg/m³)
• V = characteristic velocity (m/s)
• L = characteristic length (m)
• μ = dynamic viscosity (Pa·s or N·s/m²)
• ν = kinematic viscosity (m²/s)
• α = thermal diffusivity = k/(ρc_p) (m²/s)
• k = thermal conductivity (W/(m·K))
• c_p = specific heat at constant pressure (J/(kg·K))
• g = gravitational acceleration = 9.81 m/s²
• β = volumetric thermal expansion coefficient (1/K); β = 1/T for ideal gases

3. External Forced Convection

3.1 Flow Over Flat Plate

3.1.1 Critical Reynolds Number

• Re_x,c = 5×10⁵ (transition from laminar to turbulent)
• Re_x = Vx/ν where x = distance from leading edge

3.1.2 Local Nusselt Number

3.1.3 Average Nusselt Number

3.2 Flow Across Cylinders

3.2.1 Churchill-Bernstein Correlation

3.2.2 Simplified Correlation for Cylinders

• Nu_D = CRe_D^mPr^1/3
• Re_D = VD/ν where D = cylinder diameter
• Properties evaluated at film temperature: T_f = (T_s + T_∞)/2

3.3 Flow Across Sphere

3.4 Flow Across Tube Banks

3.4.1 Configurations

• Aligned (in-line): tubes directly behind each other
• Staggered: tubes offset in successive rows
• S_T = transverse pitch (center-to-center spacing perpendicular to flow)
• S_L = longitudinal pitch (center-to-center spacing parallel to flow)
• S_D = diagonal pitch in staggered arrangement

3.4.2 Maximum Velocity Location

• Aligned: V_max = [S_T/(S_T - D)]V
• Staggered: V_max = [S_T/(S_T - D)]V or V_max = [S_T/(2(S_D - D))]V (use larger value)

3.4.3 Zukauskas Correlation

• Nu_D = C₁Re_D,max^mPr^0.36(Pr/Pr_s)^1/4
• Re_D,max = V_maxD/ν
• N_L ≥ 20 rows (apply correction factor C₂ for N_L <>
• Properties at T_∞ except Pr_s at T_s

4. Internal Forced Convection

4.1 Flow Regimes in Pipes/Tubes

4.2 Entrance Lengths

4.3 Mean Temperature

• Bulk mean temperature: T_m = ∫(ρVc_pT)dA_c / ∫(ρVc_p)dA_c
• For constant properties: T_m = ∫(VT)dA_c / ∫(V)dA_c
• Use T_m for property evaluation in internal flow correlations

4.4 Laminar Flow in Circular Tubes

4.4.1 Fully Developed Nusselt Numbers

4.4.2 Developing Flow (Sieder-Tate)

• Nu_D = 1.86(Re_DPr·D/L)^1/3(μ/μ_s)^0.14
• Valid for: 0.48 < pr="">< 16,700="" and="" 0.0044=""><>_s) <>
• Properties at bulk mean temperature except μ_s at surface temperature

4.5 Turbulent Flow in Circular Tubes

4.5.1 Dittus-Boelter Equation

4.5.2 Gnielinski Equation

• Nu_D = [(f/8)(Re_D - 1000)Pr] / [1 + 12.7(f/8)^1/2(Pr^2/3 - 1)]
• f = friction factor = (0.790ln(Re_D) - 1.64)^-2 (smooth tubes)
• Valid for: 3000 <>_D < 5×10⁶;="" 0.5="" ≤="" pr="" ≤="">
• More accurate than Dittus-Boelter for wider range

4.5.3 Petukhov Equation

• Nu_D = [(f/8)Re_DPr] / [1.07 + 12.7(f/8)^1/2(Pr^2/3 - 1)]
• Valid for: 10⁴ <>_D < 5×10⁶;="" 0.5="" ≤="" pr="" ≤="">

4.6 Noncircular Tubes

• Use hydraulic diameter: D_h = 4A_c/P where A_c = cross-sectional area, P = wetted perimeter
• Re_Dh = VD_h/ν
• Laminar fully developed Nu varies with geometry (not 3.66 or 4.36)

4.7 Heat Transfer Rate Calculations

4.7.1 Constant Surface Temperature

• q = hA_sΔT_ln
• ΔT_ln = [(T_s - T_m,i) - (T_s - T_m,o)] / ln[(T_s - T_m,i)/(T_s - T_m,o)]
• T_m,o = T_s - (T_s - T_m,i)exp(-πDL·h/(ṁc_p))

4.7.2 Constant Surface Heat Flux

• q = q"πDL = ṁc_p(T_m,o - T_m,i)
• T_m(x) = T_m,i + (q"P/ṁc_p)x
• Linear temperature rise along tube length

5. Natural Convection

5.1 Governing Parameters

5.2 Vertical Flat Plate

5.2.1 Correlations

5.2.2 Characteristic Length

• L = height of vertical plate
• Properties evaluated at film temperature: T_f = (T_s + T_∞)/2

5.3 Inclined Flat Plate

• Replace g with g·cos(θ) for surfaces inclined at angle θ from vertical
• Valid for upper surface of cold plate or lower surface of hot plate
• Valid for θ < 60°="" from="">

5.4 Horizontal Flat Plate

5.4.1 Upper Surface of Hot Plate or Lower Surface of Cold Plate

5.4.2 Lower Surface of Hot Plate or Upper Surface of Cold Plate

• Nu_L = 0.27Ra_L^1/4 for 10⁵ <>_L <>

5.4.3 Characteristic Length

• L = A_s/P where A_s = surface area, P = perimeter

5.5 Long Horizontal Cylinder

5.5.1 Churchill-Chu Correlation

• Nu_D = [0.60 + 0.387Ra_D^1/6/[1 + (0.559/Pr)^9/16]^8/27]²
• Valid for: Ra_D <>
• Characteristic length: L = D (cylinder diameter)

5.5.2 Simplified Correlations

5.6 Sphere

• Nu_D = 2 + 0.589Ra_D^1/4/[1 + (0.469/Pr)^9/16]^4/9
• Valid for: Pr ≥ 0.7, Ra_D ≤ 10¹¹
• Characteristic length: L = D (sphere diameter)

5.7 Vertical Cylinder

• Use vertical plate correlations if D ≥ 35L/Gr_L^1/4
• Otherwise use specialized cylinder correlations

5.8 Enclosed Spaces

5.8.1 Vertical Rectangular Cavity

• Nu_L = 0.22(Pr/(0.2 + Pr)·Ra_L)^0.28(H/L)^-1/4
• Valid for: 2 < h/l="">< 10,="" pr="">< 10⁵,="" 10³=""><>_L <>
• L = spacing between vertical walls, H = height

5.8.2 Horizontal Rectangular Cavity (Heated from Below)

6. Condensation and Boiling

6.1 Film Condensation

6.1.1 Vertical Plate (Nusselt Theory)

6.1.2 Film Reynolds Number

• Re_f = 4Γ/μ_l where Γ = ṁ/P = mass flow rate per unit width
• Γ = ρ_lVδ for film of thickness δ

6.1.3 Modified Latent Heat

• h'_fg = h_fg + 0.68c_p,l(T_sat - T_s)
• Accounts for sensible cooling of condensate film

6.1.4 Horizontal Tube

• h = 0.729[ρ_l(ρ_l - ρ_v)gh'_fgk³_l/(μ_lD(T_sat - T_s))]^1/4
• For N horizontal tubes in vertical tier: multiply by N^-1/4

6.1.5 Property Evaluation

• Liquid properties at film temperature: T_f = (T_sat + T_s)/2
• ρ_v and h_fg at T_sat

6.2 Pool Boiling

6.2.1 Boiling Regimes

6.2.2 Rohsenow Correlation (Nucleate Boiling)

• q" = μ_lh_fg[g(ρ_l - ρ_v)/σ]^1/2[c_p,l(T_s - T_sat)/(C_s,fh_fgPr_lⁿ)]³
• C_s,f = empirical constant (surface-fluid combination)
• n = 1.0 for water, 1.7 for other fluids
• σ = surface tension (N/m)

6.2.3 Critical Heat Flux (Zuber)

• q"_max = 0.149h_fgρ_v[σg(ρ_l - ρ_v)/ρ²_v]^1/4
• Maximum heat flux in nucleate boiling before transition to film boiling

6.2.4 Film Boiling on Horizontal Cylinder

• h = 0.62[ρ_v(ρ_l - ρ_v)gh'_fgk³_v/(μ_vD(T_s - T_sat))]^1/4
• Properties of vapor at film temperature: T_f = (T_s + T_sat)/2
• h'_fg = h_fg + 0.80c_p,v(T_s - T_sat)

6.3 Flow Boiling

• Occurs inside tubes with forced flow
• Two-phase flow: complex function of flow quality, mass flux, geometry
• Heat transfer coefficient depends on both convective and nucleate boiling contributions
• Flow patterns: bubbly, slug, annular, mist flow

7. Heat Exchangers

7.1 Heat Exchanger Types

7.2 Overall Heat Transfer Coefficient

7.2.1 Based on Inner Surface Area

• 1/(U_iA_i) = 1/(h_iA_i) + R"_f,i/A_i + ln(r_o/r_i)/(2πkL) + R"_f,o/A_o + 1/(h_oA_o)

7.2.2 Based on Outer Surface Area

• 1/(U_oA_o) = 1/(h_oA_o) + R"_f,o/A_o + ln(r_o/r_i)/(2πkL) + R"_f,i/A_i + 1/(h_iA_i)

7.2.3 Fouling Factors

• R"_f = fouling thermal resistance (m²·K/W)
• Account for deposit buildup on heat transfer surfaces
• Typical values: 0.0001-0.001 m²·K/W depending on fluid

7.3 Log Mean Temperature Difference (LMTD) Method

7.3.1 Heat Transfer Rate

• q = UAΔt_lm
• q = ṁ_hc_p,h(T_h,i - T_h,o) = ṁ_cc_p,c(T_c,o - T_c,i)

7.3.2 LMTD Calculation

• Δt_lm = (ΔT₁ - ΔT₂)/ln(ΔT₁/ΔT₂)
• Parallel flow: ΔT₁ = T_h,i - T_c,i, ΔT₂ = T_h,o - T_c,o
• Counter flow: ΔT₁ = T_h,i - T_c,o, ΔT₂ = T_h,o - T_c,i

7.3.3 Correction Factor for Complex Configurations

• q = UAFΔT_lm,CF where F = correction factor
• ΔT_lm,CF = LMTD for counter-flow arrangement
• F depends on dimensionless parameters P and R
• P = (T_c,o - T_c,i)/(T_h,i - T_c,i) = temperature effectiveness
• R = (T_h,i - T_h,o)/(T_c,o - T_c,i) = heat capacity rate ratio
• F obtained from charts for specific configurations (shell-and-tube, cross-flow)

7.4 Effectiveness-NTU Method

7.4.1 Heat Capacity Rates

• C_h = ṁ_hc_p,h, C_c = ṁ_cc_p,c
• C_min = minimum of C_h and C_c
• C_max = maximum of C_h and C_c
• C_r = C_min/C_max

7.4.2 Maximum Possible Heat Transfer

• q_max = C_min(T_h,i - T_c,i)

7.4.3 Effectiveness

• ε = q/q_max
• ε = C_h(T_h,i - T_h,o)/[C_min(T_h,i - T_c,i)]
• ε = C_c(T_c,o - T_c,i)/[C_min(T_h,i - T_c,i)]

7.4.4 Number of Transfer Units

• NTU = UA/C_min

7.4.5 Effectiveness Relations

7.5 Special Cases

7.5.1 Phase Change (C_r = 0)

• Condensers and evaporators: one fluid isothermal (infinite heat capacity)
• ε = 1 - exp(-NTU) for all configurations

7.5.2 Balanced Counter Flow (C_h = C_c)

• Counter flow with C_r = 1
• Maximum effectiveness for given NTU
• T_h,o = T_c,o at infinite NTU

8. Practical Correlations and Tips

8.1 Property Evaluation Guidelines

8.2 Typical Convection Coefficient Ranges

8.3 Common Fluid Properties at 300 K

8.4 Selection Guide: LMTD vs Effectiveness-NTU

8.5 Enhancement Techniques

• Surface roughness: increases turbulence, enhances h but increases pressure drop
• Extended surfaces (fins): increases surface area, most effective for low h (gas-side)
• Flow inserts: twisted tapes, wire coils promote swirl and mixing
• Surface coatings: promote dropwise condensation or nucleate boiling

8.6 Boundary Layer Approximations

• Thermal boundary layer thinner than velocity boundary layer when Pr > 1 (liquids)
• Thermal boundary layer thicker than velocity boundary layer when Pr < 1="" (liquid="">
• Boundary layers approximately equal thickness when Pr ≈ 1 (gases)

8.7 Key Assumptions in Correlations

• Steady-state conditions
• Constant fluid properties (evaluated at reference temperature)
• Negligible radiation effects (unless specified)
• Negligible viscous dissipation
• Developed flow (unless entrance region correlation)
• Smooth surfaces (unless roughness specified)