# Rate of conduction heat transfer with different hot-side and cold-side areas

• emtae55
In summary: Summary:: When we measure a rate of conduction heat transfer, we assume that the hot side and the cold side's area are the same. However, if the areas are different, we would need to calculate the resistance by making assumptions about the uniformity and directionality of heat flow. This can be done by calculating the resistance of a differential cross-sectional volume and integrating it over the length of the object. For more complex geometries, the 2D or 3D heat conduction equation must be solved.
emtae55
When we measure 'the rate of conduction heat transfer'=Q , we assume that the hot side and the cold side's area are same. But if the both side's area is different to each other, how can i know the rate of conduction heat transfer?
like below figure.

Would you like to help me?? Thanks.

Last edited:
emtae55 said:
Summary:: When we measure a thermal conductivity, we assume that the hot side and the cold side's area are same. But if the both side's area is different to each other, how can i know the thermal conductivity?

When we measure a thermal conductivity, we assume that the hot side and the cold side's area are same. But if the both side's area is different to each other, how can i know the thermal conductivity?
Would you like to help me?? Thanks.
Please provide a diagram illustrating the kind of geometric arrangement you are referring to.

emtae55 said:
Summary:: When we measure a thermal conductivity, we assume that the hot side and the cold side's area are same. But if the both side's area is different to each other, how can i know the thermal conductivity?
...
Thermal conductivity remains the same for same material.
In your case, what changes is the heat flux, which dependds on cross section or area.

https://en.m.wikipedia.org/wiki/Heat_flux

russ_watters
Chestermiller said:
Please provide a diagram illustrating the kind of geometric arrangement you are referring to.
I edited it.

As said, conductivity is a property of the material, not iys geometry.
So you prob. meant conductance or resistance.
Just calculate the reisitance of a differential cross-sectional volume & integrate.

russ_watters
rude man said:
As said, conductivity is a property of the material, not iys geometry.
So you prob. meant conductance or resistance.
Just calculate the reisitance of a differential cross-sectional volume & integrate.
@Chestermiller is the subject matter expert here, but it seems to me that you would need to make some assumptions of uniformity and directionality for the heat flow before a scalar figure for resistance across a cross-section becomes meaningful.

For a reasonably long and thin object, such assumptions may be reasonable, but it is good to be aware that one is making them.

etotheipi, Chestermiller and russ_watters
jbriggs444 said:
@Chestermiller is the subject matter expert here, but it seems to me that you would need to make some assumptions of uniformity and directionality for the heat flow before a scalar figure for resistance across a cross-section becomes meaningful.

For a reasonably long and thin object, such assumptions may be reasonable, but it is good to be aware that one is making them.
For the figure the OP provided, what i suggested would be appropriate.

Of course, you can be infInitely picky, such as heat lost/gained along the figure - but this is supposed to be introductory physIcs. Sturm-Liouville is not yet encountered, for example.

Oh, I made a mistake. What i meant was 'not' a conductivity but 'the rate of conduction heat transfer'
Sorry :(
I'll edit my subject.

I agree with @jbriggs444. If the cross sectional area were changing very gradually (like toward the right side of the figure, but not like toward the left side of the figure in post #1), you could write $$Q=kA(x)\frac{dT}{dx}$$where Q is the rate of heat flow (independent of x) and A(x) is the local cross sectional area. This would integrate to $$Q=k\frac{\Delta T}{\int_0^L{\frac{dx}{A(x)}}}$$

If the geometry does not vary gradually, one would have to solve the 2D or 3D heat conduction equation, a partial differential equation.

etotheipi

## 1. What is the rate of conduction heat transfer?

The rate of conduction heat transfer is the amount of heat energy transferred through a material per unit time, due to a difference in temperature between two points.

## 2. How does the hot-side and cold-side area affect the rate of conduction heat transfer?

The rate of conduction heat transfer is directly proportional to the temperature difference between the hot-side and cold-side areas. A larger temperature difference will result in a higher rate of heat transfer.

## 3. What is the relationship between the material's thermal conductivity and the rate of conduction heat transfer?

The rate of conduction heat transfer is directly proportional to the material's thermal conductivity. A material with a higher thermal conductivity will allow for a higher rate of heat transfer.

## 4. How does the thickness of the material affect the rate of conduction heat transfer?

The rate of conduction heat transfer is inversely proportional to the thickness of the material. A thicker material will result in a lower rate of heat transfer.

## 5. What other factors can affect the rate of conduction heat transfer?

Other factors that can affect the rate of conduction heat transfer include the surface area of the material, the type of material, and the presence of any insulating layers or barriers.

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