Page 430 - Design and Operation of Heat Exchangers and their Networks

Experimental methods for thermal performance of heat exchangers 413

1 dθ 0 dθ w
e
e
e
x ¼ 0 : θ ¼ θ sðÞ, ¼ 0 (8.93)
e
Pedx dx
dθ dθ w
e
e
x ¼ 1 : ¼ ¼ 0 (8.94)
dx dx
Substitution of Eq. (8.91) into Eq. (8.92) yields

d θ d θ s + NTU d θ s + NTUdθ
3 e
e
2 e
4 e
Pe Pe sB + NTUÞ + +Pe
ð
dx 4 dx 3 K w dx 2 K w dx
Pe 2
+ s B + sBNTU + sNTU θ ¼ 0 (8.95)
e
K w
This is an ordinary differential equation of fourth order whose eigenfunction is

s + NTU s + NTU
4 3 2
r Per Pe sB + NTUð Þ + r +Pe r
K w K w
Pe 2
+ s B + sBNTU + sNTU ¼ 0 (8.96)
K w
The four complex roots of Eq. (8.96), r i (i¼1, 2, 3, 4), can be found ana-
lytically and explicitly (Anon., 1979). Thus, we obtain the solution in the
Laplace domain as
4
X r i x
e
θ ¼ c i e (8.97)
i¼1
4 2
X 1 r i r i x
θ w ¼ sB + NTU + r i c i e (8.98)
e
NTU Pe
i¼1
The coefficients c i in Eqs. (8.97), (8.98) are determined with the bound-
ary conditions. Substituting Eqs. (8.97), (8.98) into Eqs. (8.93), (8.94),we
have
4
X
r i 0
e
1 c i ¼ θ (8.99)
Pe
i¼1
4 3
X r
2 i
ð sB + NTUÞr i + r c i ¼ 0 (8.100)
i Pe
i¼1