Thermal Stratification Analysis
Theory: buoyancy suppresses mixing
Overview
I assumed hot water in a storage tank is well mixed. Isn't it?
In a well-designed tank, hot water sits on top and cold below, separated by a thin layer where temperature changes sharply — the thermocline. Hot water is lighter and stays up by buoyancy, resisting mixing. When hot water is drawn from the top, cold water enters at the bottom and the boundary simply rises. Whether this stratification survives changes how much hot water the same tank delivers. Let's check the numbers.
A one-dimensional model
$u$ is the upward speed from drawing, $D_{eff}$ the effective vertical diffusivity. Ideally it's just water's thermal diffusivity $\alpha$, but inlet momentum, heat along the walls and convective mixing add $D_{mix}$.
What breaks and what protects stratification
The Richardson number $Ri$ compares buoyancy (density difference) with inlet momentum; above 1, buoyancy wins and stratification survives. Even undisturbed, conduction alone thickens the thermocline $\delta$ with the square root of time.
Summer stratification in lakes
In nature, thermal stratification is well known in lakes. In summer a lake divides into a light, sun-warmed surface layer and a cold, heavy deep layer, with a thermocline between. The deep water doesn't mix with the surface, so oxygen isn't replenished and fish may be unable to live there. In autumn the surface cools and grows heavy, and the layers overturn all at once. Engineered hot-water tanks and thermal stores deliberately exploit this natural mechanism.
Worked example: a 200 L hot-water tank
1.5 m tall, initially all 60 °C. Water is drawn from the top at 10 L/min while 15 °C water enters at the bottom (one tank volume in 20 minutes). The volume delivered above 45 °C is compared as a fraction of tank volume. Stored heat is 10.4 kWh.
Mixing and usable hot water
| Vertical mixing | Delivered above 45 °C |
|---|---|
| Conduction only (α) | 99.5% |
| 10 × α | 98.3% |
| 100 × α | 94.8% |
| 1000 × α | 85.7% |
| Fully mixed | 40.5% |
Thermocline growth during standby
| Standby time | Thermocline thickness (10–90%) |
|---|---|
| 1 h | 8 cm |
| 6 h | 21 cm |
| 24 h | 41 cm |
Inlet design and Richardson number
| Inlet | Inlet velocity | Ri |
|---|---|---|
| 15 mm pipe | 0.94 m/s | 0.003 |
| 100 mm diffuser (radial) | 0.053 m/s | 5.6 |
Same stored heat, but mixing cuts the usable hot water to less than half.
Fully mixed, the whole tank is gradually diluted by cold water and drops below 45 °C after 40% of its volume. Stratified, the cold water pools at the bottom and the 60 °C water above stays undiluted almost to the end. The energy is the same, but how much heat remains at a usable temperature (exergy) differs. The main destroyer of stratification is inlet momentum: a jet from a thin pipe gives Ri = 0.003 and stirs the cold water into its surroundings. A diffuser that slows it about eighteenfold gives 5.6, and the cold water spreads gently along the bottom.
Analysis tips
- Refine vertically to resolve the thermocline (ten or more points within it).
- Numerical diffusion can smear stratification; use higher-order convection schemes.
- Use turbulence models that account for buoyancy damping; standard k-ε tends to over-predict mixing.
- Include wall conduction and heat loss; metal walls are shortcuts for vertical heat flow.
- Model inlets and outlets as built; they decide the quality of stratification.
“A bigger tank, still not enough hot water”
At one facility, complaints about insufficient hot water led to a larger tank, but improvement was smaller than hoped. Investigation showed cold water jetting in through a thin bottom pipe, constantly stirring the lower half so little stratification formed. Just fitting a diffuser to spread the flow sideways greatly increased the volume delivered above 45 °C from the same tank. Temperature layering, not tank size, governed performance.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Modelling the tank as fully mixed | Usable volume underestimated | Model stratification |
| Coarse vertical grid | Thermocline smeared | Refine thermocline |
| First-order upwind | Numerical diffusion | Higher-order scheme |
| Ignoring buoyancy in turbulence | Mixing overestimated | Add buoyancy terms |
| Simplified inlet | Mixing missed | Model actual geometry |
I'd like to learn related topics.
Try the molten-salt thermal storage tool and natural convection tool. Related pages include thermal plumes, mixed convection, natural convection–conduction coupling and conjugate heat transfer.
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