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Horizontal Concentric Cylinders Natural Convection Equations and Calculator

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 107

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Heat transfer between cylinders calculated from:
qc = 2 · Π · L · keff · (Ti - To ) / ln(Do /Di )

Where:

keff / k = 0.386 · (Rac)(1/4) · (Pr / (0.861 + Pr))(1/4)

valid between 102 < Rac < 107

keff = < r below 102 , keff = k

Rac = Rad · (ln(Do /Di ))4 / (d3 (Di-3/5 + Do-3/5 )5 )

Rad= g · cte · ρ2 · Cp · (Ti - To) · d3 / k · v

Film Temperature
Tf = ( Tp - Ta ) / 2

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

ρ = ρref (Tref + 273) / (Tf + 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
qconv = h ·A · ( Tc - Ta )

qrad = σ · A · e · ( Tc4 - Ta4 )

Total Heat Transfer

Where:

Tf = Film temperature °C
Tc = Cylinder temperature °C
Ta = Ambient Temperature °C
Tref = Reference Temperature °C
Cp = 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/m3)
ρref = Density (kg/m3)
Nu = Nusselt Number
D = Diameter of Cylinder (m)
L = Length of Cylinder (m)
A = Surface Area of Cylinder (m2 )
e = Emissivity
Pr = Prandtl number
RaD = 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/m2 - °C)
g = 9.81 (m/s2)
h = Average Heat Transfer (W/m2 - °C )
σ = .00000005678 0r 5.678 x 10-8
qc = 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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