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Main >> Applications >> Sample problems >> Natural convection from the horizontal plate surface

Horizontal steel plate divides two air environments with different temperatures.

**Problem type:**

Plane problem of steady-state heat transfer.

**Geometry:**

Thickness of the plate *d* = 2 mm, plate length *l* = 1 m.

Plate surface area is 1 m^{2}.

**Given:**

temperature of the air below the plate *T*_{1} = 20°C,

temperature of the air above the plate *T*_{2} = -10°C

thermal conductivity of the plate λ_{plate} = 0.4 W/(m·K)

**Task:**

Calculate the temperatures of the lower and the upper 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 below of the plate, then through the plate toward air above). 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 upper and the lower surfaces of the plate equal to the average temperature of the plate which is:
*T*_{plate} = (*T*_{1}+*T*_{2})/2 = 5 °C.

This value is used for the convection coefficients calculation:

α_{1} = 1.9 W/(m^{2}·K) - for the lower surface,

α_{2} = 4.7 W/(m^{2}·K) - for the upper surface.

**Result:**

Temperature of the lower surface of the plate: *T*_{lower} = -1.2 °C.

Temperature of the upper surface of the plate: *T*_{upper} = -1.4 °C.

Heat flux: *F* = 40 W (plate surface area is 1 m^{2}).

The calculated plate surface temperature is quite different from the one we used to estimate convection coefficients. We can recalculate the convection coefficients and run the simulation again.