User:Nidhi baranwal

Thermal Boundary Layer

Thermal boundary layer development on a flat plate.

A thermal boundary layer develops when a fluid at a specified temperature flows over a surface of different temperature. The fluid particles that come into contact with the plate reach thermal equilibrium with the plate's surface and assume the same surface temperature T_s. These fluid particles then exchange energy with the particles at the adjoining layer. As a result, a temperature profile develops in the flow field that ranges from T_s to T_∞ (free stream temperature). The flow region over the surface in which the temperature variation in the direction normal to the surface is significant is the thermal boundary layer.

The thickness of the thermal boundary layer δ_t at any location along the surface is defined as the distance from the surface at which the temperature difference T – T_s = 0.99(T_∞ – T_s). The thickness of the thermal boundary layer increases in the flow direction, since the effects of heat transfer are felt at greater distances from the surface further downstream.

At any distance x from the leading edge, the local surface heat flux may be obtained by applying Fourier's law to the fluid at y = 0. The expression used below for calculating surface heat flux is appropriate, because at the surface, there is no fluid motion and energy transfer occurs only by conduction.

$q_{s}=-k_{f}\left[{\partial T \over \partial x}\right]_{y=0}$

where q_s has been used for surface heat flux.

The thickness of the hydrodynamic boundary layer (velocity boundary layer) is normally defined as the distance from the solid body at which the viscous flow velocity is 99% of the free stream velocity. The ratio of the two thicknesses is governed by the Prandtl number. If the Prandtl number (Pr) is 1, the two boundary layers (hydrodynamic and thermal) are the same thickness. If the Pr > 1, the thermal boundary layer is thinner than the velocity boundary layer. If the Pr < 1, which is the case for air at standard conditions, the thermal boundary layer is thicker than the velocity boundary layer.

Continuity equation

${\partial u \over \partial x}+{\partial \upsilon \over \partial y}=0$

Boundary layer energy equation

$u{\partial T \over \partial x}+v{\partial T \over \partial y}={\alpha }{\partial ^{2}T \over \partial y^{2}}$

Thick thermal boundary layer

Plot showing the relative thickness in the Thermal boundary layer versus the Velocity boundary layer (in blue) for various Prandtl Numbers. For Pr =1, the two are equal.

From scale analysis

In this case, the δ_t layer is thick as compared to hydrodynamic boundary layer thickness (δ) measured at the same L. The velocity outside the hydrodynamic boundary layer and inside the thermal boundary layer is U_∞. From continuity eq., the v scale in the same region is v∼U_∞δ/L. This means that the second term on the left side of boundary layer energy equation is of order

$v\left[{\frac {{\Delta }T}{\delta _{t}}}\right]$ ∼ $U_{\infty }\left[{\frac {{\Delta }T}{L}}\right]\left[{\frac {\delta }{\delta _{t}}}\right]$

in which δ/δ_t <<< 1. The second term, (v ΔT)/δ_T, is therefore δ/δ_T times smaller than the first,(u ΔT)/L, and the entire left side of boundary layer energy equation is dominated by the scale U_∞ΔT/L. By convection and conduction balance from energy equation is (U_∞ΔT)/L∼(α ΔT)/δ_T²,

${\frac {\delta _{T}}{L}}$ ∼ $Pr^{-1/2}$ ${Re_{L}}^{-1/2}$

the interesting fact is that the relative size of δ_T and δ depends on the Prandtl number Pr = ν/α.

${\frac {\delta _{T}}{\delta }}$ ~ $Pr^{-1/2}>>1$

The assumption δ/δ_t <<< 1 is valid only when Pr^1/2 << 1.

The thermal boundary layer is much thicker for liquid metals and much thinner for oils relative to the velocity boundary layer. Heat diffuses very quickly in liquid metals (Pr << 1) and very slowly in oils (Pr >> 1) relative to momentum. The Prandtl numbers of gases are about 1, which indicates that both momentum and heat dissipate through the fluid at about the same rate.

Thin thermal boundary layer

From scale analysis

In this case, the thermal boundary layer thickness δ_T layer is smaller than the hydrodynamic boundary layer thickness (δ) measured at the same L. Geometrically, the scale of u in the δ_T layer is not U_∞ but

u ~ U_∞ ${\frac {\delta _{T}}{\delta }}$

substituting this scale into the convection ~ conduction balance yields

${\frac {\delta _{T}}{L}}$ ~ $Pr^{-1/3}{Re_{L}}^{-1/2}$

which means that

${\frac {\delta _{T}}{\delta }}$ ~ $Pr^{-1/3}<<1$

Therefore, the thin thermal boundary layer thickness assumption is valid for Pr^(1/3) >> 1 fluids.

Comparison between exact solution and solution obtained using scale analysis

When Pr << 1 (thick boundary layer), exact solution from integral solution gives

${\frac {\delta _{T}}{\delta }}$ = $0.577Pr^{-1/2}$

when Pr >> 1, i.e. in case of thin boundary layer

${\frac {\delta _{T}}{\delta }}$ = $0.976Pr^{-1/3}$

• The coefficient in exact solution is order of unity.
• Clearly scaling analysis gives nearly correct result compared to more complicated and expensive analysis.

See also

• Prandtl number
• Boundary layer
• Boundary layer separation
• Boundary-layer thickness