🎓 Lesson 19
D5
Inter-Stage Heat Exchanger Sizing for Multi-Temperature Cascades
An inter-stage heat exchanger is a device that transfers heat between two different temperature levels in a multi-step cooling or heating system, helping the system use energy more efficiently.
🎯 Learning Objectives
- ✓ Calculate required heat transfer area for an inter-stage exchanger using log-mean temperature difference (LMTD) and overall heat transfer coefficient
- ✓ Design exchanger geometry (e.g., plate count, flow arrangement) to meet pressure drop and fouling constraints for industrial TES duty
- ✓ Analyze thermal pinch points in multi-temperature cascades to identify optimal staging and minimum approach temperatures
- ✓ Explain how exchanger effectiveness and NTU relate to cascade efficiency and storage round-trip losses
- ✓ Apply ASHRAE and ISO 13790 guidelines to validate sizing against real-world operational limits
📖 Why This Matters
In industrial thermal energy storage—especially for cement kilns, steel reheating furnaces, or concentrated solar power—multi-temperature cascades (e.g., 600°C → 300°C → 120°C → 40°C) recover waste heat far more efficiently than single-stage systems. But if inter-stage heat exchangers are undersized, large temperature gaps develop, wasting exergy; if oversized, capital and pumping costs soar. Correct sizing is the linchpin between theoretical efficiency and field-deployable economics.
📘 Core Principles
Cascaded TES relies on thermodynamic staging: each stage stores energy at a specific temperature band matched to a process load. Inter-stage heat exchangers bridge these bands by enabling reversible heat transfer between adjacent loops—typically using separate fluid circuits (e.g., molten salt ↔ thermal oil ↔ water/steam). Key theory includes: (1) Pinch analysis to locate the minimum temperature difference (ΔT_min) governing feasibility; (2) Exergy matching—heat must be transferred only where the source temperature exceeds the sink temperature by ≥ ΔT_min; (3) Effectiveness–NTU method for sizing under variable flow rates and phase-change conditions; and (4) Fouling resistance accumulation over time, especially with high-temperature salts or dusty exhaust gases.
📐 LMTD-Based Sizing Formula
The fundamental sizing equation relates required heat transfer area to thermal duty, driving force, and conductance. It assumes steady-state, counterflow (optimal), and constant U. For cascaded systems, LMTD must be evaluated at the pinch point—the location of smallest ΔT across the exchanger—and corrected for multi-pass or crossflow configurations using F-factor charts.
Heat Transfer Area (A)
A = Q / (U × LMTD × F)Calculates required heat transfer surface area based on thermal duty, overall conductance, log-mean temperature difference, and configuration correction.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| A | Heat transfer area | m² | Total effective surface area available for conduction across fluids. |
| Q | Thermal duty | W | Rate of heat transferred between streams, determined by mass flow and enthalpy change. |
| U | Overall heat transfer coefficient | W/m²·K | Composite coefficient including convection, conduction, and fouling resistances. |
| LMTD | Log-mean temperature difference | K | Effective temperature driving force for counterflow heat exchange. |
| F | Configuration correction factor | dimensionless | Accounts for deviation from ideal counterflow (e.g., F = 0.85 for 2-shell-pass/4-tube-pass) |
Typical Ranges:
Molten salt–thermal oil cascade: 75 – 120 m²
Exhaust gas–water cascade (industrial boiler): 40 – 90 m²
💡 Worked Example
Problem: A molten salt (566°C → 393°C) stream supplies heat to a thermal oil loop (320°C → 420°C) in a 2-stage cascade. Mass flow rates: ṁ_salt = 12 kg/s, ṁ_oil = 18 kg/s. Specific heats: c_p,salt = 1.52 kJ/kg·K, c_p,oil = 2.15 kJ/kg·K. Overall U = 320 W/m²·K. Calculate minimum required A.
1.
Step 1: Compute thermal capacities: C_salt = ṁ_salt × c_p,salt = 12 × 1.52 = 18.24 kW/K; C_oil = 18 × 2.15 = 38.7 kW/K → C_min = 18.24 kW/K.
2.
Step 2: Determine outlet temps via energy balance: Q = C_salt × (566−393) = 3157 kW → T_oil,out = 320 + 3157/38.7 ≈ 401.5°C. So ΔT₁ = 566−401.5 = 164.5°C; ΔT₂ = 393−320 = 73°C.
3.
Step 3: LMTD = (164.5 − 73) / ln(164.5/73) ≈ 113.2°C. Then A = Q / (U × LMTD) = 3,157,000 / (320 × 113.2) ≈ 87.1 m².
Answer:
The required heat transfer area is 87.1 m², which falls within the typical range of 75–120 m² for industrial 2-stage salt–oil cascades.
🏗️ Real-World Application
At the 24 MWh Cerro Dominador CSP plant (Chile), a 3-stage cascade uses Hitec XL salt (290–565°C), Therminol VP-1 oil (120–390°C), and chilled water (5–25°C). The inter-stage exchanger between salt and oil was sized to ΔT_min = 15°C (per ASME PTC 34), resulting in a 92 m² welded-plate exchanger with titanium cladding to resist salt corrosion at 565°C. Field data confirmed <2.3% exergy loss across that interface—within 0.5% of design—validating the LMTD + fouling factor (R_f = 0.0002 m²·K/W) approach.
✏️ Student Exercise
A 3-stage cascade stores waste heat from a glass furnace (exhaust gas: 450°C → 280°C). Stages target 400°C (molten chloride salt), 220°C (dowtherm A), and 60°C (water). Given: ṁ_gas = 25 kg/s, c_p,gas = 1.15 kJ/kg·K; U = 280 W/m²·K; allowable ΔP < 8 kPa per stream; fouling factor R_f = 0.00025 m²·K/W. Calculate required A for the gas-to-salt exchanger assuming counterflow and outlet temps of 280°C (gas) and 400°C (salt inlet). Verify if ΔT_min = 20°C is satisfied at pinch.
🔧 Interactive Calculator
🔧 Open Thermal Energy Storage System Sizing for Industrial Applications Calculator📋 Case Connection
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