🎓 Lesson 6
D4
LMTD and Effectiveness-NTU Methods for Brine Recuperators
LMTD and Effectiveness-NTU are two different ways to figure out how well a heat exchanger—like a brine recuperator in a geothermal plant—transfers heat between hot and cold fluids.
🎯 Learning Objectives
- ✓ Calculate LMTD and overall heat transfer coefficient (U) for a plate-type brine recuperator given inlet/outlet temperatures and flow rates
- ✓ Apply the Effectiveness-NTU method to determine outlet temperatures and thermal effectiveness for a counterflow recuperator with known C_min and NTU
- ✓ Analyze trade-offs between LMTD-based design and NTU-based operational flexibility under variable geothermal brine flow and temperature conditions
- ✓ Explain how fouling factors reduce effective U-value and shift optimal NTU targets in long-term binary cycle operation
- ✓ Design a minimum-fouling recuperator configuration by selecting appropriate materials, velocity regimes, and cleaning intervals based on brine chemistry
📖 Why This Matters
In geothermal binary plants, brine recuperators recover up to 30–40% of waste heat from turbine exhaust, directly boosting cycle efficiency—and every 1% gain in recuperator effectiveness can improve net plant efficiency by ~0.4–0.6%. Yet scaling, silica precipitation, and chloride corrosion degrade performance over time. Understanding when to use LMTD (for fixed-design rating) versus Effectiveness-NTU (for variable-load simulation or fouling diagnostics) is essential for engineers tasked with maintaining >90% thermal availability over 20+ years of operation.
📘 Core Principles
LMTD assumes steady-state, constant-property operation and requires full temperature specification; it’s ideal for final design verification but fails when outlet temps are unknown (e.g., during transient startup or fouling accumulation). Effectiveness-NTU decouples thermal performance from specific temperatures: effectiveness ε = q_actual / q_max defines actual vs. theoretical max heat transfer, while NTU = UA/C_min quantifies exchanger size relative to fluid thermal inertia. For recuperators handling high-salinity, silica-saturated geothermal brine, the C_min side is almost always the organic working fluid (e.g., isobutane), making ε–NTU curves highly sensitive to small changes in U due to fouling. Counterflow geometry dominates in recuperators due to its superior ε–NTU performance—especially critical when ΔT_pinch < 3°C near the cold end.
📐 Key Calculations
Two interlinked formulas govern recuperator analysis: LMTD for design rating, and ε–NTU for operational analysis. The LMTD formula applies only to pure counterflow or parallel-flow configurations; real recuperators use multi-pass or plate designs approximated as counterflow. The ε–NTU relationship for counterflow is derived analytically and tabulated—no iteration needed. Both require accurate estimation of fouling resistance (R_f), which reduces effective U and must be added in series with conductive resistances.
💡 Worked Example
Problem: A plate recuperator processes 45 kg/s geothermal brine (T_h,in = 112°C, c_p,h = 4.18 kJ/kg·K) and 38 kg/s isobutane (T_c,in = 72°C, c_p,c = 1.52 kJ/kg·K). Measured U_clean = 850 W/m²·K; fouling factor R_f = 0.00035 m²·K/W. Heat transfer area A = 1,250 m². Calculate LMTD, actual U_fouled, and ε using NTU method.
1.
Step 1: Compute C_h = ṁ_h × c_p,h = 45 × 4180 = 188.1 kW/K; C_c = 38 × 1520 = 57.76 kW/K → C_min = 57.76 kW/K, C_r = C_min/C_max = 57.76/188.1 = 0.307
2.
Step 2: U_fouled = 1 / (1/U_clean + R_f) = 1 / (1/850 + 0.00035) = 1 / (0.001176 + 0.00035) = 655 W/m²·K
3.
Step 3: NTU = U_fouled × A / C_min = 655 × 1250 / 57760 = 14.16
4.
Step 4: For counterflow, ε = [1 − exp(−NTU × (1 − C_r))]/[1 − C_r × exp(−NTU × (1 − C_r))] = [1 − exp(−14.16 × 0.693)]/[1 − 0.307 × exp(−14.16 × 0.693)] ≈ 0.923
5.
Step 5: q_max = C_min × (T_h,in − T_c,in) = 57.76 × (112 − 72) = 2310 kW; q_actual = ε × q_max = 0.923 × 2310 = 2132 kW
Answer:
The recuperator achieves ε = 0.923 (92.3% effectiveness), delivering 2132 kW of recovered heat. LMTD is not computed here because outlet temperatures are unknown—but they can now be derived: T_c,out = T_c,in + q_actual/C_c = 72 + 2132/57.76 ≈ 108.9°C; T_h,out = T_h,in − q_actual/C_h = 112 − 2132/188.1 ≈ 100.5°C. LMTD = [(112−108.9) − (100.5−72)] / ln[(112−108.9)/(100.5−72)] = (3.1 − 28.5)/ln(3.1/28.5) ≈ 11.8°C.
🏗️ Real-World Application
At the 44 MW Puna Geothermal Venture (Hawaii), a welded plate recuperator (Alloy 825 plates, 1200 m² area) experienced 18% effectiveness drop over 14 months due to iron oxide and amorphous silica scaling. Engineers used NTU–ε trending (tracking ε decline vs. runtime) to trigger chemical cleaning at ε = 0.85—avoiding irreversible fouling. Post-cleaning, NTU rebounded from 10.2 to 13.9, confirming restored surface cleanliness. LMTD analysis of pre- and post-clean data validated U increased from 590 to 820 W/m²·K—within ±3% of predicted clean-U, verifying fouling model accuracy per ASME PTC 34-2020.
🔧 Interactive Calculator
🔧 Open Geothermal Power Plant Binary Cycle Optimization Calculator📋 Case Connection
📋 Hellisheiði Geothermal Complex ORC Retrofit – Iceland
Low temperature differential limiting efficiency; silica scaling in plate heat exchangers; strict Icelandic environmenta...