### Horizontal Heated Plate Facing Up Equation and Calculator

**Heat Transfer Engineering**

**Thermodynamics**

**Engineering Physics **

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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Preview: Horizontal Plate Isothermal Heated Natural Convection Calculator

Calculations is based on Nusselt number correlations.

q = h · A · ( T_{p} - T_{a} )

Film Temperature

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

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

Grashof Number

Gr = g · cte · ( T_{p} - T_{a} ) · ( A / (2W + 2L) )^{3} / v^{2}

Rayleigh Number

Ra = Gr · Pr

The Nusselt number is calculated as:

For Ra values between 2 x 10^{4} and 10^{6}

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

q_{conv} = h ·A · ( T_{p} - T_{a} )

Radiative heat transfer

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

Total Heat Transfer

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

Where:

T_{f} = Film temperature °C

T_{p} = Plate 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

W = Width of Plate (m)

L = Length of Plate (m)

a = Area of plate (m^{2} )

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/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_{conv} =
Convective Heat Transfer (W)

q_{rad} = Radiative Heat Transfer (W)

q_{tot} = 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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