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### Horizontal Concentric Cylinders Natural Convection Equations and Calculator

**Heat Transfer Engineering**

**Thermodynamics**

**Engineering Physics **

Natural Convection of a Horizontal Concentric Cylinders Equation and Calculator

Natural convection heat transfer between two concentric cylinders maintained at constant temperatures. The calculation is based on Rayleigh number and is valid for Rayleigh numbers below 10^{7}

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Preview: Natural Convection of a Horizontal Concentric Cylinders Calculator

Heat transfer between cylinders calculated from:

q_{c} = 2 · Π · L · k_{eff} · (T_{i} - T_{o} ) / ln(D_{o} /D_{i} )

Where:

k_{eff} / k = 0.386 · (Ra_{c})^{(1/4)} · (Pr / (0.861 + Pr))^{(1/4)}

valid between 10^{2} < Ra_{c} < 10^{7}

k_{eff} = < r below 10^{2} , k_{eff} = k

Ra_{c} = Ra_{d} · (ln(D_{o} /D_{i} ))^{4} / (d^{3} (D_{i}^{-3/5} + D_{o}^{-3/5} )^{5} )

Ra_{d}= g · cte · ρ^{2} · C_{p} · (T_{i} - T_{o}) · d^{3 }/ k · v

Film Temperature

T_{f} = ( T_{p} - T_{a} ) / 2

Fluid density at the film temperature is automatically calculated from the following relationship based on the perfect gas law:

ρ = ρ_{ref} (T_{ref} + 273) / (T_{f} + 273)

Prandtl Number

Pr = v / α

α = k / (ρ · Cp)

v = μ / ρ

Surface Area of Cylinder

A = Π · 2 · (D/2) · L + 2 · Π · (D/2)^{2}

Convective Heat Transfer

q_{conv} = h ·A · ( T_{c} - T_{a} )

Radiative heat transfer

q_{rad} = σ · A · e · ( T_{c}^{4} - T_{a}^{4} )

Total Heat Transfer

q_{tot} = q_{conv} + q_{rad}

Where:

T_{f} = Film temperature °C

T_{c} = Cylinder temperature °C

T_{a} = Ambient Temperature °C

T_{ref} = Reference Temperature °C

C_{p} = Specific Heat J/kg- °C

cte = Coefficient of thermal expansion (1/K)

k = Thermal Conductivity (W/m - °C)

μ = Dynamic Viscosity ( kg/m-s )

ρ = Density (kg/m^{3})

ρ_{ref} = Density (kg/m^{3})

Nu = Nusselt Number

D = Diameter of Cylinder (m)

L = Length of Cylinder (m)

A = Surface Area of Cylinder (m^{2} )

e = Emissivity

Pr = Prandtl number

Ra_{D} = Raleigh number - for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free convection or natural convection.

h = Average heat transfer coefficient (W/m^{2} - °C)

g = 9.81 (m/s^{2})

h = Average Heat Transfer (W/m^{2} - °C )

σ = .00000005678 0r 5.678 x 10^{-8}

q_{c} = Total uniform heat load on Cylinders (W)

**References **

Raithby, G. D., and K. G. T. Hollands, *A General Method of Obtaining Approximate Solutions to Laminar and Turbulent Free Convection Problems, *in T. F. Irvine and J. P. Hartnett, Eds., Advances in Heat Transfer, Vol. 11, Academic Press, New York, pp 265-315, 1975.

Incropera, De Witt., *Fundamentals of Heat and Mass Transfer *, 3rd ed., John Wiley & Sons, p563, eq 9.58-9.60, 1990.

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