| name | cross-flow-hx |
| description | Cross-flow heat exchanger design — NTU-effectiveness, LMTD correction factor F, unmixed/mixed flow configurations, fin array, bank of tubes, TEMA, automotive radiator design, pressure drop. |
| metadata | {"priority":7,"promptSignals":{"phrases":["cross flow heat exchanger","crossflow HX","NTU effectiveness","LMTD correction","automotive radiator","cross flow"],"minScore":3}} |
Cross-Flow Heat Exchanger Design — Complete Skill
Configuration Types
Unmixed-unmixed (both fluids confined to channels):
ε = 1 - exp(-NTU^0.22/C_r × [exp(-C_r × NTU^0.78) - 1])
C_r = C_min/C_max; NTU = UA/C_min
Unmixed-mixed (one fluid transversely unmixed, one well-mixed):
C_min = unmixed side: ε = (1/C_r) × {1 - exp[-C_r × (1 - exp(-NTU))]}
C_min = mixed side: ε = 1 - exp{-(1/C_r) × [1 - exp(-C_r × NTU)]}
Single pass crossflow (automotive radiator):
Typically: air side unmixed (finned channels), liquid side mixed (tube circuits parallel in manifold)
NTU-Effectiveness Method
Effectiveness:
ε = q_actual / q_max
q_max = C_min × (T_hot_in - T_cold_in)
Number of Transfer Units:
NTU = UA / C_min [dimensionless]
Heat capacitance rate:
C_hot = ṁ_hot × c_p_hot [W/K]
C_cold = ṁ_cold × c_p_cold [W/K]
C_min = min(C_hot, C_cold); C_max = max
Capacity ratio: C_r = C_min/C_max (0 ≤ C_r ≤ 1)
C_r = 0: one fluid isothermal (condensing/boiling) → ε = 1 - exp(-NTU) for all configurations
ε-NTU inversion to find NTU:
For unmixed-unmixed: solve iteratively from ε equation
For C_r → 0: NTU = -ln(1 - ε)
LMTD Method with Correction Factor
LMTD for counterflow reference:
LMTD_cf = (ΔT₁ - ΔT₂) / ln(ΔT₁/ΔT₂)
ΔT₁ = T_h,in - T_c,out; ΔT₂ = T_h,out - T_c,in
Cross-flow correction factor F:
Q = U × A × F × LMTD_cf [F ≤ 1 for all non-counterflow configurations]
F for single-pass cross-flow (TEMA charts):
P = (T_cold,out - T_cold,in) / (T_hot,in - T_cold,in) [temperature efficiency, cold side]
R = (T_hot,in - T_hot,out) / (T_cold,out - T_cold,in) [heat capacity ratio]
F from TEMA chart (Bowman-Mueller-Nagle)
Approximate F formula (cross-flow, both unmixed):
F ≈ (1 - R²)^0.5 × [ln((1-P)/(1-RP))] / {√(R²+1) × ln[(2-P(R+1-√(R²+1)))/(2-P(R+1+√(R²+1)))]}
F should not drop below 0.75 (sharp slope → uncertain ΔT driving force)
Tube Bank in Cross-Flow
Heat Transfer (Bare Tubes)
Churchill-Bernstein correlation (cylinder cross-flow):
Nu_D = 0.3 + [0.62 × Re_D^(1/2) × Pr^(1/3)] / [1 + (0.4/Pr)^(2/3)]^(1/4) × [1 + (Re_D/282000)^(5/8)]^(4/5)
Valid: Re × Pr > 0.2
Zukauskas correlation (tube banks):
Nu_D = C₁ × C₂ × Re_D_max^m × Pr^0.36 × (Pr/Pr_s)^0.25
Re_D_max = ρ × V_max × D / μ [V_max at narrowest gap]
V_max = V_∞ × (S_T / (S_T - D)) [S_T = transverse pitch]
Constants (C₁, C₂, m) from bank geometry table (5 or more rows; short bank correction otherwise)
Bank row correction for < 10 rows:
h_N = F_N × h_∞ [F_N < 1 for few rows; from table]
| N_rows | F_N (staggered) | F_N (inline) |
|---|
| 1 | 0.68 | 0.64 |
| 2 | 0.75 | 0.76 |
| 4 | 0.89 | 0.90 |
| 6 | 0.95 | 0.95 |
| 10+ | 1.00 | 1.00 |
Pressure Drop (Tube Bank)
Zukauskas ΔP:
ΔP = N_L × χ × f × ρ × V_max² / 2 [N_L = number of tube rows in flow direction; f = friction factor from chart]
χ = correction factor for tube bundle arrangement (staggered/inline, pitch ratio)
Inline vs. staggered:
Staggered: higher h (30–50%) but higher ΔP vs. inline
f_staggered > f_inline for same Re
Finned Surface Cross-Flow (Automotive Radiator)
Fin Efficiency
For constant cross-section fins (rectangular):
η_fin = tanh(mL_c) / (mL_c)
m = √(h_P / (k_fin × A_c)) [m = fin parameter; P = fin perimeter; A_c = cross-section area]
L_c = L + t/2 (corrected length for rectangular fin tip)
For annular fin:
m = √(2h / (k_fin × t)) [t = fin thickness]
η_fin from chart or Bessel function solution
Overall Surface Efficiency
η_overall = 1 - (A_fin/A_total) × (1 - η_fin)
A_total = A_fin + A_bare [total surface area]
Overall Heat Transfer Coefficient (Finned)
1/UA = 1/(η_o h_o A_o) + R_wall + 1/(η_i h_i A_i)
η_o = overall surface efficiency (air side, outer)
h_i = internal (liquid) side HTC
A_o, A_i = outer and inner areas
Automotive Radiator Sizing
Frontal area calculation:
A_front = Q / (j_c × C_min/A_front) [iterative; j_c = Colburn j factor]
j_c = St × Pr^(2/3) = h / (ρ c_p V) × Pr^(2/3)
Core depth: L_c = 30–100 mm (passenger car); 100–200 mm (heavy duty)
Fin pitch: 10–20 fins/inch (passenger car); finer → lower ΔP_air
Tube OD: 10–20 mm flat tubes; multi-port aluminum extrusions (MCHX)
Air-side pressure drop:
ΔP_air = f_Fanning × (L_c/D_h) × ρ × V_fr² / 2 + (A_c/A_front - 1) × ρ × V_fr² / 2 [contraction + core + expansion]
Standards
| Standard | Scope |
|---|
| TEMA | Shell and tube heat exchangers |
| ASME PTC-12.1 | Feedwater heater performance test |
| SAE J814 | Automotive radiator coolant standard |
| VDI Heat Atlas | Comprehensive cross-flow correlations |
| ASHRAE Handbook | HVAC heat exchanger design data |
Output
Provide: HX configuration (unmixed-unmixed/mixed-unmixed), ε-NTU analysis (ε [-], NTU, C_r), LMTD_cf [°C], correction factor F, actual heat transfer rate Q [kW], required UA [W/K], tube bank geometry (D [mm], S_T, S_L, N rows), Re_max and Nu_D, overall h [W/(m²·K)] on both sides, fin efficiency η_fin [%], overall surface efficiency η_o [%], total area A_total [m²], core dimensions (frontal area [m²], depth [mm]), air-side ΔP [Pa], tube-side ΔP [kPa], and applicable standard (TEMA, SAE J814).
