### Horizontal Heated Plate Facing Up Equation and Calculator

Horizontal Heated Plate Facing Up Isothermal Natural Convection Equation and Calculator

Average heat transfer coefficient and plate temperature for an isothermal (constant temperature) heated horizontal flat plate's upper surface or a cooled plate's lower surface in a natural convection environment. The convection calculation automatically switches between calculation methods based on Rayleigh number.

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Calculations is based on Nusselt number correlations.
q = h · A · ( Tp - Ta )

Film Temperature
Tf = ( Tp - Ta )

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

Grashof Number
Gr = g · cte · ( Tp - Ta ) · ( A / (2W + 2L) )3 / v2

Rayleigh Number
Ra = Gr · Pr

The Nusselt number is calculated as:

For Ra values between 2 x 104 and 106
Nu = 0.54 · (Gr · Pr)0.25

For Ra values between 10 6 and 10 11
Nu = 0.15 · (Gr · Pr)0.33

h is Average heat transfer coefficient
h = Nu · k / ( A / (2W + 2L) )

Convective Heat Transfer
qconv = h ·A · ( Tp - Ta )

qrad = σ · A e · ( Tp4 - Ta4 )

Total Heat Transfer

Where:

Tf = Film temperature °C
Tp = Plate 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
W = Width of Plate (m)
L = Length of Plate (m)
a = Area of plate (m2 )
e = Emissivity
Pr = Prandtl number
Ra = 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
qconv = Convective Heat Transfer (W)
qtot = Total uniform heat load on plates (W)

References

Lloyd, J. R. and W. R. Moran, Natural Convection Adjacent to Horizontal surface of Various Planforms , ASME Pap. 74-WA/HT-66.

Fujii, T. and H. Imura, Natural Convection Heat Transfer from a Plate with Arbitrary Inclination , Int. J. Heat Mass Transfer, vol. 15, p. 755, 1972.

Holman, J.P., Heat Transfer , 7th ed., McGraw Hill Book Company, New York, 1990. p. 333 - 352

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