Page 428 - Design and Operation of Heat Exchangers and their Networks

Experimental methods for thermal performance of heat exchangers 411

By means of the Laplace transform, it is easy to obtain the real-time tem-
perature response:
ð
h p ffiffiffiffiffi i 0 p ffiffiffiffiffiffi
x
τ
x
x
0
θ B¼0 ^x, ^τð Þ ¼ e ^ ^ τ I 0 2 ^^ + e ^ x e τ I 0 2 ^τ dτ 0 (8.76)
0^ τ
τ
x
θ ^, ^Þ ¼ θ B¼0 ^x, ^τ B^xð Þ (8.77)
ð
and outlet fluid temperature variation
00
θ ^τðÞ ¼ θ B¼0 NTU, ^τ BNTUð Þ (8.78)
The governing equation system (8.72)–(8.75) is identical to
Eqs. (3.102)–(3.104) for B¼0. The solution for θ B¼0 ^x, ^τ B^xð Þ can be
found in Section 3.3.1 by replacing t h , x, and y in Eq. (3.118) with θ B¼0 ,
τ
x
^ x, and ^ B^, respectively.
8.3.2 Extended model
Two important improvements have been made to extend the fundamental
model. Howard (1964) added the axial heat conduction term into the energy
equation of the solid wall and solved the problem numerically. The other
improvement is to add the axial heat dispersion term into the energy equa-
tion of the fluid to take the effect of the nonuniform flow distribution into
account (Roetzel et al., 1993; Luo, 1997, 1998; Luo et al., 2001). Besides,
the inlet fluid temperature variation can be arbitrary. Thus, the mathematical
model for the single-blow testing technique can be extended as

2
C ∂t ∂t ∂ t αA
_
+ C ¼ A c D + ð t w tÞ (8.79)
L ∂τ ∂x ∂x 2 L
2
C w ∂t w ∂ t w + αA ð t t w Þ (8.80)
L ∂τ ¼ A c,w λ w ∂x 2 L
∂t
_
0
x ¼ 0 : Ct A c D ¼ t τðÞ (8.81)
∂x
∂t w
¼ 0 (8.82)
∂x
∂t
x ¼ L : ¼ 0 (8.83)
∂x
∂t w
¼ 0 (8.84)
∂x
τ ¼ 0 : t ¼ t w ¼ t 0 (8.85)