What is Thermal boundary layer?
When the free stream of fluid at temperature ‘`T_{\infty}`’ approaches to the plate at different temperature ‘`T_{s}`’ that is `T_{\infty}` ≠ `T_{S}`, then the thermal boundary layer is generated.
It is the region in which a temperature gradient (dT/dy) is present in the direction perpendicular to the flow of the free stream.
Thermal boundary layer generation:
The thermal boundary layer exists only when the temperature of the free stream and the surface of the plate are not equal.
Case 1: If `\mathbf{T_{\infty}>T_{S}}`:–
Following is the profile of the thermal boundary layer when the free steam temperature is higher than plate temperature.
![Thermal boundary layer for hot fluid flowing over a plate](https://mechcontent.com/wp-content/uploads/2021/10/Thermal-boundary-layer-1024x492.webp)
The free stream of fluid at temperature `T_{\infty}` touches the plate at the leading point.
1) In this profile the temperature of the fluid layer closed to the plate is almost equal to the plate temperature ( `T\ \approx\ T_{\infty}` ). As the layer nearest to the plate has negligible velocity therefore in this case the heat transfer is done by conduction.
2) As we move in a forward direction the temperature of lead also rises layer by layer.
3) At a distance y = `δ_{th}` the temperature of the fluid is almost equal to the temperature of the free stream.
4) Therefore the distance `δ_{th}` is known as boundary layer thickness at which,
`T-T_{s}=0.99(T_{\infty}-T_{s})`
5) Above the distance y = `δ_{th}` the temperature of the fluid remains constant and equals to the temperature of the free stream.
6) If we move ahead in the downstream direction (x-direction) then the thickness of the boundary layer also goes increasing due to the retardation of fluid particles in the downstream direction.
7) If we join points of `δ_{th}` from x = 0 to x = L then the curve generated is known as `δ_{th}` (x) curve.
8) The region under the `δ_{th}` (x) curve is known as the thermal boundary layer. While the region outside of the `δ_{th}` (x) curve is known as the outer flow region.
Boundary conditions for `\mathbf{T_{\infty}>T_{S}}`:
The boundary conditions for boundary layer region are,
a) At y = 0, T = Ts
b) At y = `δ_{th}`, (`T` – `T_{s}`) = 0.99 (`T_{\infty}` – `T_{s}`)
c) At y > `δ_{th}`, T = `T_{\infty}`
d) At x = 0, `δ_{th}` = 0
Case 2: If `\mathbf{T_{S}>T_{\infty}}`:-
The boundary layer region for surface temperature greater than free stream temperature can be plotted as,
![Thermal boundary layer for cold fluid flowing over a hot plate](https://mechcontent.com/wp-content/uploads/2021/10/Orange-and-White-Simple-Homemade-Food-BusinessRestaurant-169-Video-2021-10-26T074027.671-1-1024x456.webp)
1) As shown in the profile the temperature of the layer adjacent to the plate has a temperature equal to the plate temperature.
2) As we move in an upward direction the temperature of fluid decreases layer by layer.
3) At y = `δ_{th}` the temperature of the fluid is near equal to the free stream temperature.
Boundary conditions at `\mathbf{t_{S}\ > t_{\infty}}`:
In this case, the boundary conditions are given by,
a) At y = 0, T = `T_{s}`
b) At y = `δ_{th}`, (`T_{s}` – `T`) = 0.99(`T_{s}` – `T_{\infty}`)
c) At y > `δ_{th}`, `T` = `T_{\infty}`
d) At x = 0, `δ_{th}` = 0
Thermal boundary layer thickness for flat plate:
It is the perpendicular distance from the surface of the plate to the point in a fluid where the temperature gradient with respect to the height (dt/dy) becomes zero.
It is given by,
`\delta _{th}=\frac{\delta }{(Pr)^{\frac{1}{3}}}`
Where,
δ = Hydrodynamic boundary layer thickness
Pr = Prandtl number