What would need to be the temperature difference between the inner and the outer surface of a copper pipe with thermal conductivity , and with radii and , if through the outer surface of the pipe is supposed to radially flow an amount of heat per second ?
Details and assumptions:
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First we should define the length of the pipe l , which gives the area surface S = 2 π r 2 l , so
l = 2 π r 2 S = π ⋅ 1 2 5 0 0 0 = 1 3 2 . 6 cm .
The heat amount per second Q ∗ = τ Q ( 1 ) from q = l ⋅ τ Q = − λ 2 π x d x d t
is equal to Q ∗ = q ⋅ l .
So according to ( 1 ) , we get the expression q = 2 π λ lg r 1 r 2 t 1 − t 2 ,
⟹ Δ t = 2 π λ l Q ∗ ( lg r 2 − lg r 1 ) = 2 π ⋅ 0 . 9 ⋅ 1 3 2 . 6 6 0 0 0 ( 4 . 0 9 4 − 3 . 9 1 2 ) = 1 . 4 6 ∘ C .