Related Resources: heat transfer

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.

ALL calculators require a Premium Membership

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 < 10⁹
Nu = 0.68 + (0.670Ra^1/4 )/[1+(0.492/Pr)^9/16 ]^4/9

For Turbulent Flow - Ra is between > 10⁹
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) · H³ / v²

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_p⁴ - T_a⁴ )

Where:

q = Heat Dissipated (W)
h = Average heat transfer coefficient (W/m² - °C)
A = Area of plate (m²)
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 (m²/s)
a = Thermal Diffusivity (m²/s)
g = Gravity (9.80665 m/s² )
cte = Coefficient of thermal expansion (1/K)
ρ = Density calculated at film temperature T_f (kg/m³)
ρ_ref = Density at reference temperature T_ref of 20 °C (kg/m³)
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.