Temperature profiles and thermal resistance

• I
I heated two different materials with a laser beam for about 10 seconds and these are the measured temperature profiles:

From the various solutions to the general heat conduction equation, temperature rise seems to increase with increasing thermal resistance of the material. The total thermal resistance of a material of length ##L## (according to Carslaw & Jaeger) is:

$$\overline{R}=\intop_{0}^{L}\frac{dz}{K},$$

where ##K## is the thermal conductivity of the substance, which is itself confounded with density and heat capacity.

So, is it possible to argue that the sample with the red curve has a higher thermal resistance? If so, can we also say that it has a lower density and/or heat capacity?

Any explanation is greatly appreciated.

P. S. One of the features I observed is that the temperature did not decline immediately after the laser was turned off, but rather about a second later. Is this normal? (The data is extracted from a thermographic video so there could be some errors involved)

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jrmichler
Mentor
A good way to understand a problem of this type is to start by asking yourself some questions:
What happens if the material has high / low thermal conductivity?
What happens if the material has high / low absorbance at the laser beam frequency?
What happens if the material has high / low density?
What happens if the material has high / low specific heat?
What happens if the material has high / low transparency at the laser beam frequency?
Etc

Then try combinations:
What happens if the material has high specific heat and high density vs low specific heat and low density?
Etc

After you have a good qualitative understanding of what is happening, then you will know what equation(s) to use.

What happens if the material has high / low absorbance at the laser beam frequency?

I was unable to find an answer to this particular question. The material will have a certain absorptance at the wavelength of the laser, characterized by the absorption coefficient ##\alpha (\lambda)##.

But ##\alpha## has units of length-1, whereas thermal resistance is often given in K/W. I don't see how to connect the two, and I couldn't find an explanation in any textbooks. Do you know how they are related?

Also, as I mentioned, thermal resistance already takes into account the specific heat and density of the material. So, my question is if we can judge the relative thermal resistance from the plots I posted in my original post.

mjc123
Homework Helper
Absorbance and thermal resistance are not directly related. But absorbance is clearly related to the thing you are measuring - change in temperature. For all we know, the samples might have identical thermal resistances, but different absorbances. Or vice versa.

roam
jrmichler
Mentor
You need to figure this out, not look it up. Thought experiments are useful to understand the equations:
If the material has high absorbance, it will heat up quickly.
If the material has low absorbance, it will heat up slowly.
If the material has zero absorbance, it won't heat up at all.

Repeat for all of the variables. Are any other variables capable of affecting the rate of heating and maximum temperature? Is thermal resistance the only variable of interest? You need to understand what affects what before you understand which equation(s) to use. And you need to understand what affects what in order to justify your selection of equation(s) to your teacher / lab partner / coworker / boss.

roam
Absorbance and thermal resistance are not directly related. But absorbance is clearly related to the thing you are measuring - change in temperature. For all we know, the samples might have identical thermal resistances, but different absorbances. Or vice versa.

Hi mjc123,

This is interesting. Yes, absorbance should be related to how fast the laser heats the sample. But when the laser is turned off, what property determines how fast the temperature is lost?

As shown in my graph, the temperature declines shortly after we stop the irradiation. Do you think the rate of temperature loss is related to thermal resistance/conductivity?

You need to figure this out, not look it up. Thought experiments are useful to understand the equations:
If the material has high absorbance, it will heat up quickly.
If the material has low absorbance, it will heat up slowly.
If the material has zero absorbance, it won't heat up at all.

Repeat for all of the variables. Are any other variables capable of affecting the rate of heating and maximum temperature? Is thermal resistance the only variable of interest? You need to understand what affects what before you understand which equation(s) to use. And you need to understand what affects what in order to justify your selection of equation(s) to your teacher / lab partner / coworker / boss.

When you say that high absorbance means that the material heats up quickly, this seems to be true intuitively. I know absorbance in the form that is derived from the Beer-Lambert law:

$$A=\log_{10}\left(1/\mathcal{T}\right).$$

Do you know of any analytical formulas that can relate ##A## to the properties of the sample such as density or heat capacity?

mjc123
Homework Helper
Do you know of any analytical formulas that can relate A to the properties of the sample such as density or heat capacity?
No. Essentially you have to treat them as independent variables.
Do you think the rate of temperature loss is related to thermal resistance/conductivity?
What are the possible mechanisms for temperature loss? What properties do they depend on? Which, if any, do you think is the dominant mechanism for your sample?

roam
What are the possible mechanisms for temperature loss? What properties do they depend on? Which, if any, do you think is the dominant mechanism for your sample?

My samples are biological (plant) tissue. So I think the processes involved are conduction and evaporation. The flat parts of the graphs should correspond to phase change. By the time the laser turns off, it appears that there is no significant phase change taking place. I would say that temperature is mostly lost by conduction (heat flowing from the hotter parts to the cooler).

I believe a low conductivity means that the material heats up more quickly, and loses the temperature more slowly. But this doesn't seem to agree with my curves (and please correct me if I am wrong): the curve that heats up faster appears to also lose heat faster.

The thermal conductivity of a substance is itself equal to ##K=\kappa \rho C##, where ##\kappa## is the diffusivity. Is it possible to say which one of these three variables is mostly responsible for the steepness of the temperature decline?

mjc123
Homework Helper
I believe a low conductivity means that the material heats up more quickly, and loses the temperature more slowly. But this doesn't seem to agree with my curves (and please correct me if I am wrong): the curve that heats up faster appears to also lose heat faster.
Maybe it has a higher absorbance and a higher thermal conductivity?
How big is the laser spot compared to the size (area and depth) of the sample?

roam
Maybe it has a higher absorbance and a higher thermal conductivity?
How big is the laser spot compared to the size (area and depth) of the sample?

Yes, this could be possible.

But when the irradiation stops there is a very sharp decline in temperature, followed by a very gradual loss of temperature. In the interval where they are losing heat slowly, both curves seem to have the same rate of heat loss. Doesn't that imply that the samples have a very similar thermal conductivity?

Also, what is the reason for the transition from a sharp to a gradual decline?

The spot image size of the laser is slightly larger than the sample. But the sample is longe (about ~5 cm), so the light is completely absorbed in the top layers (extinction length is about 100 microns). The heat then redistributes by conduction.