Question

In an old house, a fireplace in the first floor keeps its temperature at a cozy 70°F, while there is no heating in the two floors above. Assume that the heat flows vertically. What would be the temperature (in °F) in the second and in the third floors if the R-value of each floor surface is 8, and the R-value of the roof is 12? The outside temperature is 23°F.

second floor =? °F

third floor =? °F

Answer 1

The rate of heat flow is equal to [ T(h)-T(l) ]/R,where T(h) is the temperature at hotter end while T(l) is the temperature at colder end.

Consider that 2nd floor is at temperature T_{​​​​​​}​​​​​ while the 3rd floor is at temperature t.

Rate of heat flow into 2nd floor is (70-T)/8

Rate of heat exit from 2nd floor is (T-t)/8

Since,there is steady state,rate of heat coming in=rate of heat exit

So,(70-T)/8=(T-t)/8 which implies , 70-T=T-t which implies

2T= 70+t

So,T=(70+t)/2...................(eqn 1)

Similarly,rate of heat flow into 3rd floor is (T-t)/8 while rate of heat exit from 3rd floor is (t-23)/12. Again,equating rate of heat coming in to rate of heat exit,

(T-t)/8=(t-23)/12 which implies 3T=5t-46

So T=(5t-46)/3...........eqn2

Equating eqn1 and eqn2,(70+t)/2=(5t-46)/3

Solving this we get,t=43.14 °F

Putting this value into eqn 1, we get T=56.57 °F

So, temperature of 2nd floor=56.57°F and temperature of 3rd floor=43.14°F.