Correlating albedo with temperature?

[AppleiPad556]

Homework Statement

Hi!

For my Physics IB Extended Essay, I’m interested in exploring the correlation between the colour of a surface (using albedo) and the temperature of the surface (after reading articles about white roofs and the urban heat island). Currently I’m hoping to execute an experiment and correlate the data points with an equation, and I have a few questions that I hope to gain more insight on below.

Relevant Equations

General formula for Albedo: $\alpha = \frac{\text{total scattered power}}{\text{total incident power}}$
Stefan-Boltzmann equation: $P = \varepsilon\sigma A(T^4-T_s^4)$
Combining the two: $$T = \sqrt[4]{\frac{\alpha P_{incident}}{\varepsilon\sigma A}+T_s^4}$$

1. As mentioned above, I wanted to devise an equation that can relate the colour of a surface to the temperature on the surface. I tried using the general definition of albedo and combining it with the Stefan-Boltzmann equation (see above, Relevant Equations). However this means that the higher the albedo value, the higher the temperature, which goes against the fact that higher albedo has more energy reflected (which should cause lower temperature).
   Is this correct, or did I misuse these equations (am I able to relate the two powers together?), and are there other equations or values that might work better in this case?
2. Along these lines, is albedo a good value to use in my case? I chose albedo originally as it appeared to have a good correlation with energy and colour, but it seems that albedo tends to be used more for planetary bodies instead of specific surfaces. Is there something else that I can use that can relate colour and reflected energy, or is albedo okay for this situation?
3. With the lack of laboratories/detailed equipment, my current plan for the experiment is to create model structures with the top of the structure being flat with different-coloured construction paper. With an infrared lightbulb as a heat source, I would fill the structure with water and use a thermometer to calculate the energy reflected (power from bulb – energy in water), and I can also use an infrared thermometer to determine the temperature of the roof itself. This experiment can be easily done at home, but is it too simple to get accurate data? Are there better materials that I can substitute that are readily available?
4. Thinking about my discussion, another goal I wanted to achieve in this experiment was to see if I could also mix in surface area of the roof into the relationship and to try to see how accurate it is using real case studies of cool roofs. I can factor this into my experiment (keep volume but change dimensions of structure), and it seems to be included in the Stefan-Boltzmann equation. Is this too ambitious? I am aware that there are a plethora of factors that need to be factored into the real world, and this is only something that I want to use as further exploration.

I’d appreciate any feedback and advice (or suggestions on the topic) you can give me! Sorry if this is in the wrong forum.
Thank you for reading, and have a good day,
Eric

 

[Bystander]

Emissivity, the epsilon in S-B, is the variable/property in which you're interested; range is from 0 to 1, and exhibits strong and strongly variable temperature dependence. Tabulations are available in Rohsenow & Hartnett, Handbook of Heat Transfer, generally unreliable, since surface history is a MAJOR part/contributor at ordinary temperatures, and only approaches 1 at "black-body" temperatures, 10,000 K or greater.

 

[haruspex]

Combining the two: $$T = \sqrt[4]{\frac{\alpha P_{incident}}{\varepsilon\sigma A}+T_s^4}$$

However this means that the higher the albedo value, the higher the temperature

You can't combine them like that. The two P's both stand for power, but do not refer to the same power.
In your Stefan-Boltzmann equation, P is radiated power, i.e. some portion of power previously absorbed. You seem to have equated this is to scattered (reflected) power, which is the power that was not absorbed in the first place.
If you assume steady state, radiated power + reflected power = incident power.

Also, all three vary according to wavelength, meaning the Stefan-Boltzmann equation is really an integral.
Roughly speaking, at a given wavelength, albedo = 1-emissivity. See e.g. https://www.physicsforums.com/threads/connection-between-emissivity-and-albedo.590569/. But the incoming radiation is typically at much shorter wavelengths than a black body at Earth's temperature would emit, so the two will not cancel.

I assume Ts in your equation is the temperature of space. You can safely ignore it.

 

[256bits]

just by chance I came across this, which follows up on the previous posts.
it's a cheat in a way, but you still have to understand the formulation.

Spoiler

https://plus.maths.org/content/climate-modelling-made-easy

 

[AppleiPad556]

Thank you all for the prompt replies! I appreciate your time helping me out.

Emissivity, the epsilon in S-B, is the variable/property in which you're interested; range is from 0 to 1, and exhibits strong and strongly variable temperature dependence.

Thank you for clarifying! Is there a way to measure emissivity, or is it difficult to do so; or is this information already common (hence the tabulations in the book)? In terms of my experiment, if the material of the roof is kept constant but only the colour changes, would emissivity be a reliable figure to compare the effects of colour? (as emissivity is a property of the material)

Tabulations are available in Rohsenow & Hartnett, Handbook of Heat Transfer, generally unreliable, since surface history is a MAJOR part/contributor at ordinary temperatures, and only approaches 1 at "black-body" temperatures, 10,000 K or greater.

I'm a bit lost with what you mean by surface history. What can I do/what assumptions should I make if the values are generally unreliable?

You can't combine them like that. The two P's both stand for power, but do not refer to the same power.
In your Stefan-Boltzmann equation, P is radiated power, i.e. some portion of power previously absorbed. You seem to have equated this is to scattered (reflected) power, which is the power that was not absorbed in the first place.
If you assume steady state, radiated power + reflected power = incident power.

I thought so, thank you for clarifying the two powers! When is an object at a steady state, and when can I assume it is so?

Also, all three vary according to wavelength, meaning the Stefan-Boltzmann equation is really an integral.
Roughly speaking, at a given wavelength, albedo = 1-emissivity. See e.g. https://www.physicsforums.com/threads/connection-between-emissivity-and-albedo.590569/. But the incoming radiation is typically at much shorter wavelengths than a black body at Earth's temperature would emit, so the two will not cancel.

I did not know that! Could you explain to me more about this?

I assume Ts in your equation is the temperature of space. You can safely ignore it.

My bad, I should have clarified this; this equation is actually the net rate at which energy leaves the body (according to my textbook and this site), taking into consideration the rate at which the body radiates ($\varepsilon \sigma AT^4$) and the rate it absorbs ($\varepsilon \sigma AT_s^4$). Does this change anything?

just by chance I came across this, which follows up on the previous posts.

Thank you for linking this! I took a read and it does make sense; do you think this can still apply for bodies on Earth? It also goes back to the emissivity question that I asked above.

 

[Bystander]

I'm a bit lost with what you mean by surface history.

"Mirror/specular finishes" range from anywhere from hundredths (old and oxidized, not visibly) to thousandths (lots of polishing); textures, matte vs. gloss, which white?

 

[haruspex]

1. Is there a way to measure emissivity, or is it difficult to do so; or is this information already common (hence the tabulations in the book)? In terms of my experiment, if the material of the roof is kept constant but only the colour changes, would emissivity be a reliable figure to compare the effects of colour? (as emissivity is a property of the material)

2. When is an object at a steady state, and when can I assume it is so?

3. Could you explain to me more about this?

4. this equation is actually the net rate at which energy leaves the body (according to my textbook and this site), taking into consideration the rate at which the body radiates ($\varepsilon \sigma AT^4$) and the rate it absorbs ($\varepsilon \sigma AT_s^4$). Does this change anything?

1. You did not specify the context of your qn in post #1. I took it to be re Earth, or some arbitrary planet, but it now looks like it is roofs of buildings.
The visible colour, green versus red say, as opposed to shade, might not be that relevant. Clearly a lighter shade is more reflective. But that is just the visible part if the spectrum. The infrared is also important...

3. ... the sun's power peaks in the visible part of the spectrum (no coincidence, of course). The Earth, being much cooler has peak radiative power up the far end of infrared (IR-C).
http://agron-www.agron.iastate.edu/courses/Agron541/classes/541/lesson09a/9a.4.html
This is the basis of the Greenhouse effect, that the atmosphere let's in all that power in visible and near IR, but blocks what Earth radiates.
A similar effect arises with emissivity. If a surface reflects much of the visible and near IR but happily radiates in far IR then it can provide "radiative cooling".

2. Re Earth, steady state would normally be assessed as an average over centuries. E.g. just consider 24 hours at one location. During the day, there is a net influx of energy, but overnight a net efflux.
There are satellites in orbit which measure these accurately enough to calculate the imbalance that is presently warming Earth...

4. ... If using the equation with the incoming radiation term then the P on the left is the power imbalance, i.e. the net inflow of energy. Mostly when the S-B equation is quoted P just refers to the emitted power, so no Ts term.