[Natural Convection]

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Concentric Cylinders — Natural Convection

Heat transfer across the annular gap between two concentric cylinders via effective conductivity.

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Inputs

Inner cylinder diameter (m)

Outer cylinder diameter (m)

Cylinder length (m)

Inner temperature (°C)

Outer temperature (°C)

Fluid Properties

FluidPressure (Pa)

Manual override (edit properties below)

ρkg/m³

CₚJ/(kg·K)

kW/(m·K)

μPa·s

νm²/s

Pr—

β1/K

Results

Modified Rayleigh Ra*c

——

k_eff / k

——

Effective conductivity

—W/m·K

Heat rate

—W

Concentric Cylinders — Natural Convection — schematic

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Concentric Cylinders — Natural Convection

Calculates the natural-convection heat transfer across the annular gap between two long concentric cylinders held at constant temperatures $T_{i}$ (inner) and $T_{o}$ (outer). The Raithby–Hollands method represents the buoyant flow in the gap as an enhanced conduction: heat is conducted through a stationary fluid of effective conductivity $k_{e f f} > k$ .

Gap Rayleigh Number

Properties are evaluated at the mean temperature $T_{f} = (T_{i} + T_{o}) /2$ . The characteristic length is the radial gap $b = (D_{o} - D_{i}) /2$ :

R a_{d} = \frac{g β ( T _{i} - T _{o} ) b ^{3} P r}{ν ^{2}}

Modified Rayleigh Number and Effective Conductivity

R a_{c}^{*} = \frac{[ ln ( D _{o} / D _{i} ) ] ^{4}}{b ^{3} ( D _{i}^{- 3/5} + D _{o}^{- 3/5} ) ^{5}} R a_{d}

\frac{k _{e f f}}{k} = 0.386 (\frac{P r}{0.861 + P r})^{1/4} (R a_{c}^{*})^{1/4}

The correlation is valid for $1 0^{2} \leq R a_{c}^{*} \leq 1 0^{7}$ ; below $R a_{c}^{*} = 100$ buoyancy is negligible and $k_{e f f} = k$ (pure conduction).

Heat Rate

Q = \frac{2 π k _{e f f} L ( T _{i} - T _{o} )}{ln ( D _{o} / D _{i} )}

References

Raithby, G. D. and Hollands, K. G. T., A General Method of Obtaining Approximate Solutions to Laminar and Turbulent Free Convection Problems, Advances in Heat Transfer, Vol. 11, Academic Press, pp. 265–315, 1975.
Incropera, F. P. and DeWitt, D. P., Fundamentals of Heat and Mass Transfer, eqs. 9.58–9.60.

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Frequently asked questions

Heat transfer across the annular gap between two concentric cylinders via effective conductivity. Enter your inputs and press Calculate — the worked solution shows every step of the method with your numbers substituted in.

When there is no fan, pump or significant draught — enclosures, outdoor surfaces on still days, passive heat sinks. If there is any imposed flow, compare the two mechanisms: when Gr/Re² is well below one, forced convection dominates and the forced-convection calculators are the right tools.

No — it computes the convective part only. At typical natural-convection coefficients of 2–10 W/m²·K, radiation from a painted or oxidised surface is often the same order of magnitude, so total heat loss should add a radiative term from the Radiation calculators.