cross-flow-hx

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).