Question

For flow of fluid over a heated plate, the following fluid properties are known viscosity = 0.001 Pa.s ; specific heat at constant pressure = 1 kJ/kg.K ; thermal conductivity = 1 W/m.k. The hydrodynamic boundary layer thickness at a specified location on the plate is 1 mm. The thermal boundary layer thickness at the same location is

1. 0.001 mm
2. 0.01 mm
3. 1 mm
4. 1000 mm

Answer 1

Correct Answer - Option 3 : 1 mm

Concept:

Prandtl number Pr is defined as the ratio of momentum diffusivity to thermal diffusivity.

\(Pr = \frac{\nu }{\alpha } = \frac{{momentum\;diffusivity}}{{thermal\;diffusivity}}\)

\(Pr = \frac{{μ {C_p}}}{K} = \frac{{\left( {\frac{μ }{\rho }} \right)}}{{\left( {\frac{K}{{\rho {C_p}}}} \right)}}\)

In another way, we can define Prandtl number as, the ratio of the rate that viscous forces penetrate the material to the rate that thermal energy penetrates the material.

\(\frac{δ }{{{δ _T}}} = {\left( {Pr} \right)^{1/3}}\;\)

where δ is hydrodynamic boundary layer thickness and δT is thermal boundary layer thickness.

Calculation:

Given:

μ = 0.001 Pa.s, C_p = 1 kJ/kg.k, K = 1 W/m.k

Since, Prandtl number, \(Pr = \frac{{μ Cp}}{K}\) 

So, \(Pr = \frac{{0.001 \times 1000}}{1} = 1\)

Since \(\frac{\delta }{{{\delta _T}}} = {\left( {Pr} \right)^{\frac{1}{3}}} = {\left( {1} \right)^{\frac{1}{3}}} = 1\)

∴ Thermal boundary layer thickness at the same location = 1mm