# How to formulate surface temperature of a material exposed to sunlight and wind?

I am currently searching for a model or an approach (reference, course, demonstration) to formulate and estimate surface temperature of a known material (for example, concrete, metal), exposed to sunlight/wind, over a given period of time (day, week). The objective would be to be able to give an estimated time serie of the surface temperature based on local weather data.

• I consider the material homogenous, and the surface whose temperature I want to estiamte is flat ;
• I consider only one face (the studied face) to be exposed to sunlight/wind/other weather conditions (others faces are exposed to air temperature) ;
• I suppose I do not know the exact thickness or shape of my material (let's say a block of concrete)
• However, I know that my block of material is thick enough to not be considered as an infinitely small sheet. For example, I consider 2meters thick concrete. Semi-infinite thickness hypothesis can be assumed, if practical.
• I consider there isn't any rain/snow/ice

The data/parameters I have :

• Any thermodynamic properties of the material itself
• Time series of ambient air temperature, sunlight intensity (therefore if sunny or cloudy), windspeed.
• Position of the sun, and incidence angle of sunlight on my surface at any given time.

I have started researching papers/courses about estimation of the surface temperature with similar inputs, finding some papers presenting specific applications and hypothesis valid in only certain scenarios or with specific materials, or only valid for maximum temperature. I am still searching for a more generic expression of the surface temperature (or if possible, temperature model in the first centimeters below the surface), considering I have access to properties of the material and significant weather data.

• Incropera and DeWitt's textbook on heat transfer is an excellent resource for all aspects of this type of problem. You would formulate an energy balance in the material, set the boundary conditions, and solve for the temperature profile either analytically or, more likely, numerically. – Chemomechanics Jan 8 at 18:43
• Welcome to the site. You need to improve and formulate a clear question, Maybe you want to ask - how should you model a temperature dependence of a concrete layer illuminated by sun? – jaromrax Jan 8 at 18:45
• Are you willing to assume that the solid is semi-infinite in thickness? Are you able to specify the heat flux as a function of time at the surface? – Chet Miller Jan 8 at 20:19
• @Chemomechanics Thank, I'll be lloking for it. I don't have many boundary conditions, except ambient temperatures I guess ? – NMZ Jan 8 at 21:36
• @jaromrax Better this way ? – NMZ Jan 8 at 21:37

The semi-infinite approximation will hold as long as the Fourier number

$$Fo = \frac{\alpha t}{L^2} < 0.05$$

where $$\alpha$$ is the slab thermal diffusivity, $$L$$ is the slab thickness, and $$t$$ is time. So, for times $$t <\frac{0.05L^2}{\alpha}$$, we can make the semi-infinite approximation.

You said you consider 2 m of concrete. I estimate $$\alpha = \frac{k}{\rho c_p} = 5 \times 10^{-7} m^2/s$$, thus the semi-infinite approximation will only be valid for $$t = \frac{0.05\times2^2}{5\times10^{-7}}=400,000$$ seconds or ~ 4.5 days. You said you wanted to track the time for days/weeks, so I don't think we can apply the semi-infinite approximation for your purpose.

If you still want to use the semi-infinite approximation, I suggest using the form derived in Carslaw and Jaeger, Chapter 2:

$$\frac{T(x,t)-T_\infty}{T_0-T_\infty} = erf(\frac{\zeta}{2})+exp(\beta \zeta + \beta^2) \times erfc(\frac{\zeta}{2} + \beta)$$

where $$\zeta = \frac{x}{\sqrt{\alpha t}}$$, $$\beta = \frac{\bar{h}\sqrt{\alpha t}}{k}$$. If you want the surface temperature, simply set $$x=0$$ and then you have it as a function of time.

You can use the effective heat transfer coefficient $$\bar{h} = \bar{h}_{conv} + \bar{h}_{rad}$$ which combines both convection and radiation heat transfer coefficients (which can change with time).

Given that this solution is only valid for a 2 meter slab of concrete for 2 hours, however, I'm not sure if it's what you really want.

A 2 meter concrete slab will reach steady state in about $$t = \frac{1.5*2^2}{5 \times 10^{-7}} = 12,000,000$$ seconds or 138 days, so I also don't think steady-state solutions are useful to you.

In the regime that you're interested in (days/weeks) I suggest using series solutions or one-term approximations found in any heat transfer text. The series solutions are valid for all times. You can use an effective $$\bar{h} = \bar{h}_{conv} + \bar{h}_{rad}$$ in your boundary condition, which may change with time.

• Thank you for your explanation on semi-infinite hypothesis. I see it could work if I was to make a one day simulation, since 400,000s ~ 4.5 days. For the formula from Carslaw and Jaeger, doesn't this expression require $T_\infty$ to be constant ? – NMZ Jan 9 at 13:27
• Also, you mentionned series or one-term approximations: sorry for being clueless but most papers I've been looking at use empirical approximations which may not be applicable when you change the material/the conditions. Do you have any reference that could be useful for a beginner like me ? – NMZ Jan 9 at 13:30
• @NMZ $T_\infty$ may change with time. You may have a slow change (e.g., the ambient air temperature may change by 30 ° C over a period of 24 hours). This is hard to predict but a rough estimate would be to use use weather forecasts. – Drew Jan 9 at 13:45
• @NMZ I recommend not looking at papers, as these are probably kind of advanced and your problem is relatively simple. Heat transfer is an established subject in engineering curricula, and beginner textbooks like Incropera's chapter on transient conduction is a good place to start. You'll find the general equations and solutions in there. That semi-infinite solution I posted is simply a solution to the heat equation witih convective boundary conditions and small Fourier number. – Drew Jan 9 at 13:48
• Thank you for you feed back. I assumed that those daily variations were too high/fast for $T_\infty$ in this hypothesis. I will try this since I have access to ambient air temperature close to my slab. Thank you for the reference, I will try and find what may be used for my surface temperature approximation. – NMZ Jan 9 at 14:01