r ∞   r n  n 

X b
T̄ = A0 + B0 ln + An + Bn cos (nφ)
b n=1
b r
T − Tf αq”0 b
T̄ = k ⇒ T = Tf + T̄
αq”0 b k
∞ 
(  n  )
αq”0 b r X  r n b
T = Tf + A0 + B0 ln + An + Bn cos (nφ)
k b n=1
b r
∞ 
(  )
∂T αq”0 b B0 X An n−1 n −n−1
= + n nr − nBn b r cos (nφ)
∂r k r n=1
b
Boundary conditions
 ( ∞    )
αq”0 b X a n  a −n 
hf (T − Tf ) = B0 + n An − Bn cos (nφ)
k a n=1
b b
∞
( )
αq”0 X
hext (T − Tamb ) = B0 + n [An − Bn ] cos (nφ)
k n=1
Where
( 2 amb )(1−Bi1 ln(a/b))
 1+πBi T̄

 π[Bi1 +Bi2 (1−Bi1 ln(a/b))]
 n=0
1
An = 2[g10 (1)+Bi2 g1 (1)]
n=1

2 cos(nπ/2)

n>1

π(1−n2 )[gn 0 (1)+Bi g (1)]
2 n
(
A0 1−Bi1Biln(a/b)
1
n=0
Bn = a

2n n−Bi1
An b n+Bi1 n≥1
 a 2n  n − Bi 
1
gn (1) = 1 +
b n + Bi1
  a 2n  n − Bi 
0 1
gn (1) = n 1 +
b n + Bi1
Boundary conditions evaluated
      a −1      a −2  
αq”0 b a a 2
hf (T − Tf ) = B 0 + A1 − B1 cos (φ) + 2 A2 − B2 cos (2φ)
k a b b b b
αq”0
hext (T − Tamb ) = {B0 + [A1 − B1 ] cos (φ) + 2 [A2 − B2 ] cos (2φ)}
k
Where the coefficients are the following

1 + πBi2 T̄amb (1 − Bi1 ln (a/b))
A0 =
π [Bi1 + Bi2 (1 − Bi1 ln (a/b))]
Bi1
B 0 = A0
1 − Bi1 ln (a/b)
1
A1 = 0
2 [g1 (1) + Bi2 g1 (1)]
 a 2  1 − Bi 
1
B 1 = A1
b 1 + Bi1
2
A2 =
3π [g20 (1) + Bi2 g2 (1)]
 a 4  2 − Bi 
1
B 2 = A2
b 2 + Bi1
 a 2  1 − Bi 
1
g1 (1) = 1 +
b 1 + Bi1
 a 2  1 − Bi 
1
g10 (1) = 1 +
b 1 + Bi1
 a 4  2 − Bi 
1
g2 (1) = 1 +
b 2 + Bi1
 a 4  2 − Bi 
1
g20 (1) = 2 + 2
b 2 + Bi1