Question 3: Assume steady state for this problem and all properties are in SI un

Question

Question 3: Assume steady state for this problem and all properties are in SI units. You may assume leaf, sky and earth to be black bodies. Consider a flat orange leaf (A is the area of the leaf) parallel to the ground. The top surface of the leaf radiates to the sky at Tsky. The bottom surface of the leaf radiates to the ground which is at temperature Tground. The leaf also exchange heat by convection with air (temperature and convective heat transfer coefficient of air are: Tair, hair) from both top and bottom surface. Write the energy balance and boundary conditions for following cases: (20 points) A) If the leaf is thin so that entire leaf (including top and bottom surfaces) are at one temperature (Tleaf) considered as a 2-d slab. activity (q) B) If the leaf has a finite thickness (I) and thermal conductivity (k), and can be C) Same as (B) with additional heat generation source in the leaf due to metabolic

Explanation / Answer

SOLUTION:

(A) Energy in- energy out= 0

Let the leaf temperature =Tl

so, different terms are,

radiative loss to sky=sigma*A*(Tl^4-Tsky^4)

radiative loss to ground=sigma*A*(Tl^4-Tground^4)

Convective loss from both side= 2*h*A(Tl-Tair)

Now, sum of all these three terms are zero, solving by trial and error, we get Tl

(B) If the leaf has a finite thickness l and the thermal conductivity K and can be considered as a 2-d slab , then

Heat transfer by the 2-d slabe=K*A*(Tl-Tair)

Now all the four equations , equate to zero then we get Tl

sigma*A*(Tl^4-Tsky^4)+sigma*A*(Tl^4-Tground^4)+ 2*h*A(Tl-Tair)+K*A*(Tl-Tair)=0

(C) If there is additional heat generation source Qgen.then

Energy in- energy out +Qgen.= 0

sigma*A*(Tl^4-Tsky^4)+sigma*A*(Tl^4-Tground^4)+ 2*h*A(Tl-Tair)+K*A*(Tl-Tair) +Qgen.=0

solving by trial and error, we get Tl

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