Heat transfer coeffcient expression

[Sahil Dev]

Homework Statement

Getting an expression for the convective heat tarnsfer coefficient.

Homework Equations

See attached images

The Attempt at a Solution

I have been reading about the boundary layers.

Here is an extract of my notes corresponding to thermal boundary layer and consequently my question.

Consider flow over an isothermal plate as shown in the figure.

As seen in the figure above, at the leading edge, the temperature profile is uniform with T(y) = T∞. However, the fluid particles that come into contact with the plate, achieve thermal equilibrium at the plate’s surface temperature.
In turn, these particles exchange energy with those in the adjoining fluid layer and temperature gradients develop in the fluid. The region of the fluid in which these temperature gradients exist is the thermal boundary layer.
The thickness δst of the thermal boundary layer is typically defined as the value of y for which the ratio:

With increased distance from the leading edge, the effects heat transfer penetrates further into the free stream and the thermal boundary layer grows.
The relationship between the conditions in the boundary layer and the convection heat transfer coefficient may readily be demonstrated.
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. That is;

This expression is appropriate because at the surface, there is no fluid motion and heat transfer occurs by conduction.
Recalling, the Newton’s law of cooling, we see that;

And combining this with the expression of conduction transfer as just above;

QUESTION:

1) I do not follow how the above equations: one corresponding to heat transfer through conduction) and the other corresponding to heat transfer through convection, can eb combned to give the heat transfer coefficient?
2) I do not see the logic in combining conductive and convective heat transfer related equations to give the heat trasnfer coefficient: h.

Please can anyone help?

 

[Chestermiller]

Homework Statement

Getting an expression for the convective heat tarnsfer coefficient.

Homework Equations

See attached images

The Attempt at a Solution

I have been reading about the boundary layers.

Here is an extract of my notes corresponding to thermal boundary layer and consequently my question.

Consider flow over an isothermal plate as shown in the figure.

View attachment 84992

As seen in the figure above, at the leading edge, the temperature profile is uniform with T(y) = T∞. However, the fluid particles that come into contact with the plate, achieve thermal equilibrium at the plate’s surface temperature.
In turn, these particles exchange energy with those in the adjoining fluid layer and temperature gradients develop in the fluid. The region of the fluid in which these temperature gradients exist is the thermal boundary layer.
The thickness δst of the thermal boundary layer is typically defined as the value of y for which the ratio:
View attachment 84993

With increased distance from the leading edge, the effects heat transfer penetrates further into the free stream and the thermal boundary layer grows.
The relationship between the conditions in the boundary layer and the convection heat transfer coefficient may readily be demonstrated.
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. That is;
View attachment 84994

This expression is appropriate because at the surface, there is no fluid motion and heat transfer occurs by conduction.
Recalling, the Newton’s law of cooling, we see that;
View attachment 84995

And combining this with the expression of conduction transfer as just above;

View attachment 84996

QUESTION:

1) I do not follow how the above equations: one corresponding to heat transfer through conduction) and the other corresponding to heat transfer through convection, can eb combned to give the heat transfer coefficient?
2) I do not see the logic in combining conductive and convective heat transfer related equations to give the heat trasnfer coefficient: h.

Please can anyone help?

The sentence that seems to be causing some of your confusion is the following:

"This expression is appropriate because at the surface, there is no fluid motion and heat transfer occurs by conduction."

The reality of the situation is that, irrespective of whether the fluid is in motion and shearing, the heat flux at a distance y from the wall is always occurring exclusively by conduction, and the flux is given by:

$$q(y)=-k_f\frac{\partial T}{\partial y}$$

But, if you want the heat flux at the wall, you need to evaluate the partial derivative at y = 0.

Chet

 

[Sahil Dev]

Sorry for the late reply. I was concentarting on another subject during the past fw weeks.

1) How can you say :

The reality of the situation is that, irrespective whether the fluid is in motion and shearingm the ehat flux at a distance y from the wall is always occurring exclusively by conduction ?

The convection heat transfer mode is sustained both by random molecular motion and by the bulk motion of the fluid within the boundary layer. The contribution due to random molecular motion (diffusion / conduction) dominates near the surface where the fluid velocity is low. In fact, at the interface between the surface and the fluid the fluid velocity is zero, and heat is transferred by this (conduction) mechanism only.

The contribution due to bulk fluid motion originates rom the fact that the boundary layer grows as the flow progresses in the x- direction.

So, there is a heat transfer due to bulk fluid motion (advection) and conduction. I do not agrree wit you when yoiu say that the transfer is through pure conduction.

2) On further reading, I found that the convection heat transfer appears as a boundary condition in solution od conduction problems.


Consiuder the following problem;

We can solve the above as below;

We solve (1) aby appplying boundary conditions 2,3, 4

Theerfore convective heat tarnsfer is applied as a boundary condition in conduction problems.

 

[Chestermiller]

In the example you cited, it is inherently assumed that the heat transfer coefficient is constant, the boundary layer thickness is constant, and the flow is turbulent. This certainly qualifies as convectively enhanced heat transfer. In the free stream turbulent flow, tiny hotter and colder parcels of fluid are intermittently brought in and out of contact with one another, which results in very rapid conductive thermal equilibration of the parcels. So, even here, conduction is a major part of what is happening.

In the example you cited in post #1, the flow regime appears to be laminar flow heat transfer over a flat plate. If the plate is hotter than the fluid, then as the fluid flows over the plate, colder outer layers of fluid are flowing faster than slower inner layers of fluid. This convective shear increases the vertical temperature gradients within the fluid in the y direction at all locations. This results in enhanced conductive heat flux in the y direction at all locations, both close to the wall and away from the wall. However, since the velocity gradients are greater near the wall, the enhancement in this region is more effective. In any event, vertical conduction is occurring in the y direction at all y locations, but this is enhancement is associated with sharpening of the temperature gradients as a result of the shear flow.

Hope this helps.

Chet