Effect of Prandtl Number on Thermal Boundary Layer

[sanka]

I'm currently having a bit of an intuitive problem understanding the Prandtl number effect on boundary layers and I'm hoping that someone can explain it better than what I've read in some heat transfer books.

According to various HT books, a low Prandtl number means that heat diffuses quickly (predominantly via the conduction mechanism) whereas momentum does not diffuse as quickly. This results in a large thermal BL relative to the hydrodynamic BL. I am having trouble understanding why this occurs? Why is the thermal BL "thick"? My current thinking is that if the heat (thermal energy) diffuses/spreads through the fluid quickly (through molecular collisions between adjacent fluid laminae) the free-stream temperature would be approached more quickly, resulting in a thinner thermal BL. Basically,I would have thought that if heat diffuses quickly through the fluid that the thermal BL would be "thin" as the heat quickly reaches the free-stream temperature.

I know my understanding is flawed but if someone can explain to me why I'm wrong, I would appreciate it!

Cheers

 

[bigfooted]

In a boundary layer, inertia U*U/L needs to balance with the viscous term nu*U/(L*L). In order for the viscous term to survive at high Reynolds numbers (we know it cannot disappear at high Reynolds numbers ), the length scale of the viscous term cannot be L, but should have its own lengthscale, δ,, which is the boundary layer thickness.
Therefore, the boundary layer thickness decreases for increasing velocity

The same analysis for a thermal boundary layer: again, the thermal boundary layer may not vanish for high Reynolds numbers, so there must be a balance between thermal convection and diffusion:
$\frac{UT}{L} = \frac{T}{Re \cdot Pr} \frac{1}{\delta^2}$
So: when the Prandtl number is high, the thermal boundary layer thickness decreases

So: when the Reynolds number is high, inertia governs the flow, not viscosity. Therefore, the region where viscosity is important (the boundary layer) becomes smaller
When the Prandtl number is high, viscous/momentum diffusion governs the flow, not thermal diffusion. Therefore, the region where thermal diffusion is important (the thermal boundary layer) becomes smaller.

Note that usually, Pr is fixed for a certain gas, e.g. Pr=0.7 for air and 7 for water. So you can have a thick thermal boundary layer for air and a thin one for water.

 

[sanka]

Thanks for the explanation, seems to make more sense now.

Just have one more question regarding the BL topic.
For a given fluid (e.g. Air), the Prandtl number will be constant provided that the temp. and pressure of the fluid remain relatively constant (e.g. Pr=0.7 for atmospheric air). So, suppose we have a forced-convection scenario where a fan is blowing air over a heated plate, what happens to the thermal BL when the fan speed is increased?

I know that the velocity of the air flow increases and thus, the Reynolds number will correspondingly increase. This implies that inertial forces will dominate and the region where viscous forces are felt becomes smaller (hydro BL becomes smaller). What effect does all this have on the Thermal BL?

Thanks

 

[Chestermiller]

Thanks for the explanation, seems to make more sense now.

Just have one more question regarding the BL topic.
For a given fluid (e.g. Air), the Prandtl number will be constant provided that the temp. and pressure of the fluid remain relatively constant (e.g. Pr=0.7 for atmospheric air). So, suppose we have a forced-convection scenario where a fan is blowing air over a heated plate, what happens to the thermal BL when the fan speed is increased?

I know that the velocity of the air flow increases and thus, the Reynolds number will correspondingly increase. This implies that inertial forces will dominate and the region where viscous forces are felt becomes smaller (hydro BL becomes smaller). What effect does all this have on the Thermal BL?

Thanks

When it is windy outside, my skin feels cooler than when there is no wind. This means that the thermal boundary layer is thinner when it is windy, so that the temperature gradient near my skin is higher, and the heat transfer rate is higher.