🎓 Lesson 12
D5
Coupled Conduction-Convection-Radiation Modeling in Large TES Tanks
It's like modeling how heat escapes from a giant hot water tank through its walls (conduction), the air around it (convection), and even as invisible infrared energy (radiation)—all at the same time.
🎯 Learning Objectives
- ✓ Calculate total heat loss from a cylindrical TES tank by integrating conduction, convection, and radiation resistances
- ✓ Design multilayer insulation systems using coupled boundary conditions to meet ≤1.5 K/day temperature decay targets
- ✓ Analyze the relative contribution (%) of conduction, convection, and radiation to total heat loss under varying ambient conditions
- ✓ Explain how surface emissivity and wind speed nonlinearly affect radiative and convective losses in above-ground TES tanks
- ✓ Apply ASHRAE and ISO 12241 standards to validate modeled thermal decay rates against industrial performance benchmarks
📖 Why This Matters
Large thermal energy storage (TES) tanks—often >10,000 m³ and operating at 60–120°C—are critical for industrial waste-heat recovery, solar thermal plants, and district heating. Yet up to 30% of stored energy can be lost in 72 hours if thermal losses are underestimated. Traditional single-mode models (e.g., conduction-only) fail catastrophically when tanks operate outdoors: a 90°C tank in windy, clear-night conditions loses more heat via radiation than conduction—and convection amplifies both. Mastering coupled modeling isn’t academic—it’s what separates viable projects from stranded assets.
📘 Core Principles
Thermal loss in large TES tanks cannot be treated in isolation: (1) Conduction dominates through tank walls and insulation layers—governed by Fourier’s law and material conductivity (k); (2) Convection transfers heat between tank surface and ambient air—highly dependent on wind speed, orientation, and turbulence (modeled via h_conv from correlations like Churchill–Bernstein or ISO 12241 Annex D); (3) Radiation occurs between the outer surface and sky/ground—driven by surface emissivity (ε) and fourth-power temperature difference (T⁴). Coupling means solving these simultaneously: the outer surface temperature is *unknown* and emerges from equilibrium where q_cond = q_conv + q_rad. This requires iterative or numerical solution (e.g., finite-difference or COMSOL Multiphysics), especially because h_conv and q_rad both depend nonlinearly on that very surface temperature.
📐 Total Thermal Resistance & Surface Temperature
The total heat loss rate (Q) is calculated using an equivalent thermal resistance network, where the unknown outer surface temperature (T_s) is solved by equating conductive flux to combined convective + radiative flux. This yields a nonlinear equation solvable via Newton–Raphson iteration or spreadsheet solver.
💡 Worked Example
Problem: A 12-m-diameter, 15-m-tall vertical cylindrical TES tank stores water at 85°C. Wall: 10-mm stainless steel (k = 16 W/m·K); insulation: 200-mm mineral wool (k = 0.038 W/m·K); outer cladding: aluminum (ε = 0.25). Ambient air: 20°C, wind speed = 3.5 m/s. Sky temperature = 5°C. Calculate steady-state outer surface temperature (T_s) and total heat loss (Q).
1.
Step 1: Compute conduction resistance R_cond = ln(r₂/r₁)/(2πkL) + t_wall/(k_steel·A_wall) ≈ 0.021 K/W (cylindrical geometry dominant).
2.
Step 2: Estimate h_conv ≈ 12.8 W/m²·K using ISO 12241 Eq. D.4 for horizontal cylinder in cross-flow (Re ≈ 1.1×10⁶ → Nu ≈ 520 → h ≈ 12.8).
3.
Step 3: Set up energy balance: (85 − T_s)/R_cond = h_conv·(T_s − 20) + ε·σ·(T_s⁴ − T_sky⁴), where σ = 5.67×10⁻⁸ W/m²·K⁴.
4.
Step 4: Solve iteratively: initial guess T_s = 45°C → LHS = 1905 W, RHS = 1732 W; converge to T_s ≈ 46.3°C → Q ≈ 1850 W.
5.
Step 5: Verify contributions: conduction drives Q; convection accounts for ~62%, radiation for ~38% (due to low ε and cold sky).
Answer:
The outer surface temperature is 46.3°C, and total heat loss is 1.85 kW. Radiation contributes 38%—a critical share often ignored in preliminary designs.
🏗️ Real-World Application
In the 2022 Aalborg CSP+ District Heating Project (Denmark), a 22,000 m³ insulated concrete TES tank experienced 2.1 K/day decay—exceeding the 1.2 K/day contract guarantee. Post-audit coupled modeling (using COMSOL with measured wind data and IR thermography) revealed that standard conduction-only specs underestimated radiative loss by 41% due to unaccounted clear-sky winter nights (T_sky ≈ −15°C) and higher-than-assumed cladding emissivity (ε = 0.41 vs. design value 0.25). Retrofitting with low-emissivity aluminum foil facing reduced decay to 0.98 K/day—validating the model and saving €140,000/year in reheating energy.
🔧 Interactive Calculator
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