Natural convection from the vertical plate surface

QuickField simulation example

Vertical steel plate divides two air environments with different temperatures.

Problem Type
Plane problem of steady-state heat transfer.

Geometry

Height of the plate L = 1000 mm, thickness of the plate d = 2 mm.

Given
temperature of the air on the left side of the plate T₁ = 20°C,
temperature of the air on the right side of the plate T₂ = -10°C
thermal conductivity of the plate λ_plate = 0.4 W/(m·K)

Task
Calculate the temperatures of the right and the left surfaces of the plate and the heat flux passing through the surfaces.

Solution
The heat flux flows from hot to cold areas (in our case from the left surface of the plate then through the plate toward air on the right). Heat flux in the plate is regulated by the plate thermal conductivity. The heat exchange between air and the plate is regulated by natural convection.
We estimate that the temperatures of the left and the right surfaces of the plate equal to the average temperature of the plate which is
T_plate = (T₁+T₂)/2 = 5 °C.
This value is used for the convection coefficients calculation*:
α₁ = 3.6 W/(m²·K) for the left surface,
α₂ = 3.7 W/(m²·K) for the right surface.

Result
Temperature of the left surface of the plate: T_left = 4.9 °C.
Temperature of the right surface of the plate: T_right = 4.6 °C.
Heat flux: f = 54 W (plate surface area is 1 m²).

^*Remark: the formulas of the similarity theory relate to the big plates and gives the average value of the convection coefficient. This approach is very approximate by its nature, so there is no reason to make iterations to calculate the accurate convection coefficient value related to actual surfaces temperatures.

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