### Isothermal Vertical Plate Natural Convection Equation and Calculator

**Heat Transfer Engineering**

**Thermodynamics**

**Engineering Physics **

Isothermal Vertical Plate Natural Convection Equation and Calculator

Radiative and Convective heat transfer from an isothermal vertical plate. The average convective heat transfer coefficient is calculated assuming natural convection conditions. The convection calculation automatically switches between laminar and turbulent convection correlations based on Rayleigh number. The plate transfers heat from only one of its sides. It is assumed to be insulated on the other. equation and calculator.

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Preview: Isothermal Vertical Plate Natural Convection Calculator

Fluid Properties at Film Temperature , Except for Density. (Defaults are for Air at 20 °C).

To account for convection only, make the emissivity equal to 0. Similarly, to account for radiation only, make the thermal conductivity = 0.

Calculations is based on Nusselt number correlations. The heat flow (q) from the Circular Rod is calculated as:

q = h · A · (T_{p} - T_{a} )

h = Nu · k / H

The Nusselt number is calculated as:

For Laminar Flow - Ra < 10^{9}

Nu = 0.68 + (0.670Ra^{1/4} )/[1+(0.492/Pr)^{9/16} ]^{4/9}

For Turbulent Flow - Ra is between > 10^{9}

Nu = {0.825 + (0.387Ra^{1/6} )/[1+(0.492/Pr)^{9/16} ]^{8/27} }^{2}

Ra = Gr · Pr

Pr = v / a

a = k / (
ρ Cp)

Gr = g ·cte ( T_{p} - T_{a}) · H^{3} / v^{2}

Film temperature at the reference temperature of 20 °C.

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

Fluid density at at film temperature using perfect gas law:

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

Radiative heat transfer

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

Where:

q = Heat Dissipated (W)

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

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

T_{p} = Plate temperature °C

T_{a} = Ambient fluid temperature °C

T_{f} = Film temperature °C

Nu = Nusselt Number

k = Thermal Conductivity of Fluid (W/m - °C)

H = Height of plate (m)

W = Width of plate (m)

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

Pr = Prandtl number

Gr = Grashof number

v = Kinematic viscosity (m^{2}/s)

a_{} = Thermal Diffusivity (m^{2}/s)

g = Gravity
(9.80665 m/s^{2} )

cte
= Coefficient of thermal expansion (1/K)

ρ =
Density calculated at film temperature T_{f} (kg/m^{3})

ρ_{ref} =
Density at reference temperature T_{ref} of 20 °C (kg/m^{3})

T_{ref} = 20 °C

Q_{rad} = Radiative heat transfer (W)

e = Emissivity
of surface

σ =
Stefan-Boltzman constant =
5.670367 × 10^{-8} kg- s^{-3} K^{-4}

**References **

Churchill, S. W. and H. H. S. Chu, *Correlating Equations for Laminar and Turbulent Free Convection from a Vertical Plate *, Int. J. Heat and Mass Transfer , 18, 1323, 1975

Incropera, De Witt., *Fundamentals of Heat and Mass Transfer *, 3rd ed., John Wiley & Sons, p543, eq.9.26 & 9.27, 1990.

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