# Calculate the rise in temperature from directed radiation [closed]

For example I have a 1 watt laser and direct it to a sheet of metal (copper), if I were to direct it for say a time interval of 1 minute what would be the change in temperature? I can predict that it would depend on the size of the sheet as a larger sheet can dissipate more energy, the time exposed and the power of the laser, however I would appreciate someone walking me through.

• I will clarify that, because I have specified specific numbers on the question one should not assume it is for a homework assignment, the reason I provided numbers was because I thought it would make it easier for someone responding to formulate an answer. Commented Dec 15, 2014 at 18:41

## 3 Answers

The answer depends on many factors, but here are the basic bits of physics that play:

1. The power and wavelength of the laser
2. The reflectivity of the surface (function of wavelength of the laser)
3. The size of the focal spot
4. The thickness of the sheet
5. The thermal conductivity of the sheet

The reflectivity of the copper is a particularly important one. If you have a well polished sheet of copper, the emissivity might be as low as 0.03 (reference) or, if it became oxidized from heating, as high as 0.78. That obviously has a HUGE impact on the tempeature dependent solution (but interestingly, if the final heat loss mechanism is only radiative, then it would not affect the steady state solution - only how long it would take to get there).

Let's first compute the steady state solution - it is a little bit easier than the transient, and it gives us some of the principles. I will assume a sheet of thickness $t$, focal spot of diameter $d$ with power $P$, emissivity of the surface $\epsilon$. That means that power absorbed is $\epsilon P$. The way this heat leaves the focal spot area is through conduction into the metal, and then through heat loss at the surface (radiation, convection). Radiative heat loss goes as $\epsilon \sigma_B T^4$ per unit area, while convection depends on mass flow of air (which in turn depends on orientation...). All quite complicated - easy to be off by factors of 2x and greater with "reasonable" assumptions.

But let's assume 1 minute with 1 W laser (as in your question), and emissivity of 0.03 (polished copper). Then the total heat absorbed by the copper is 1.8 J. The specific heat capacity of copper is roughly 0.4 J/g/C. If the sheet of copper is 1 mm thick, and circular with an area of 100 cm^2, then the total mass is approximately $\rho V = 8960\cdot 10^{-3-2} kg = 90 g$. This means that the heat capacity of the copper sheet is about 36 J/C, and that on average it will be heated by 0.05°C. That is assuming that thermal conductivity is sufficient to get the heating evenly across the entire sheet, which is probably false; but it does suggest that radiative and convective heat losses are likely quite small on this time scale.

So let's look at the thermal conductivity. Because the sheet is much thinner than it is wide, I am going to treat as a 2D sheet - as though all the heat is applied over a circle of diameter $d$. The lateral thermal conductivity of the sheet can be estimated from

$$k_{2D} = k_{3D}*t = 400 * 10^{-3} = 0.4 W/K$$

A given heat flux $\Phi$ at a radius $r$ will cause a thermal gradient

$$\frac{dT}{dr} = \frac{\Phi}{2\pi r k_{2D}}$$

so with the numbers given above, we would expect

$$\frac{dT}{dr} = \frac{0.03}{2\pi r \cdot 0.4} = \frac{0.012}{r}$$

Obviously, you need some boundary conditions. If we assumed for a moment that the edge was kept at room temperature, the thermal profile (integrating from the edge inwards) would be of the form

$$T(r) = T_0 - 0.012\cdot \log\frac{r}{r_{outer}}$$

This means that for steady state, with the edge clamped at room temperature we would expect the temperature rise in the middle to be only about 0.03°C. That suggests that thermal conductivity of the copper is really quite effective for this kind of heat load. If you make the sheet much thinner, and the focal spot smaller, the answer will obviously change.

As you start playing around with these assumptions, the simplifications that worked above might break down - and you quickly end up with a tricky diffusion equation. But sometimes, before you get all carried away with those, it's a good idea to estimate the magnitudes. In this case, the magnitude of the temperature rise is quite small (with the values I chose).

• He just wanted the temperature rise, not a full blown temperature profile... way to go above and beyond I guess! Commented Dec 15, 2014 at 15:50
• @pentane - he wanted to "be walked through" the calculation. You can't get the local temperature without knowing something about the profile. But thanks. Commented Dec 15, 2014 at 15:50

Well it all depends on what assumptions you are able and willing to make.

For a start you can work out how much of the laser light is reflected. The reflected power will be something like (for normal incidence) $$\frac{P_{r}}{P_i} = \frac{(\eta_m - \eta_{vac})^{2}}{(\eta_{vac} + \eta_m)^2},$$ where $\eta_{vac}=377$ Ohms and $\eta_m$ is the impedance of the metal at the frequency you are interested in.

Next you can assume that what isn't reflected is transmitted into the metal. If the metal is thick enough, say several skin depths, then you can assume that all the transmitted power is absorbed.

If you can then assume that the metal is isolated and does not radiate or conduct away any heat, then you can use the specific heat capacity for the metal to work out by how much the temperature rises.

As you specify heating this for a minute I doubt that the assumption that the metal doesn't lose any heat by radiation will be a good one, though if you keep it in a vacuum you might get away with assuming no conduction/convection goes on. So you need to work out what the emissivity of the metal is as a function of temperature and factor that into a differential equation for how the temperature depends on time. It may even be that you find that after a minute the metal has reached some kind of equilibrium where it radiates away as much as is being absorbed. That will depend on the thickness, mass and emitting area I suppose.

There are so many variables here i think its nearly impossible to give an answer like how thick is the lazer, how far away is the lazer how reflective is the surface of the metal ......... the more variables there are the harder something is to predict if you wanted an answer it would be much more practical to get one by doing the experinment, probably not what you wanted to hear, sorry i couldn't be more helpful.

• Hmm, spoken like a true engineer. I think one can do a bit better than that before firing up the laser. I'm not clear why it should depend on how far the laser is away, unless it's so far that the beam doesn't entirely strike the metal surface. Commented Dec 15, 2014 at 14:24
• the further away the lazer is the the more energy is lost on the journey as heat ect and i'm fairly sure kinetic energy would have to be included somewhere in that as well, small as the effect would be Commented Dec 15, 2014 at 14:34
• Your post was very ztimulating! :-) Commented Dec 15, 2014 at 15:37