Page 427 - Design and Operation of Heat Exchangers and their Networks

410 Design and operation of heat exchangers and their networks

8.3.1 Fundamental model
In the fundamental mathematical model of the single-blow problem, it is
assumed that the flow is a steady one-dimensional plug flow. The heat trans-
fer surface (test core) can be considered as a porous medium. The axial heat
conduction in the wall material and heat loss to the environment are neg-
ligible. The properties of the fluid and wall material are constant. Before the
test, the fluid and wall material have the same uniformly distributed temper-
ature t 0 .At τ¼0, the inlet fluid temperature undergoes a sudden step change
Δt. According to the earlier physical model, the energy equations for the
fluid and test core can be written as
C ∂t ∂t αA
+ C _ ¼ ð t w tÞ (8.68)
L ∂τ ∂x L
αA
C w ∂t w
¼ ð t t w Þ (8.69)
L ∂τ L
(8.70)
τ ¼ 0 : t ¼ t w ¼ t 0
x ¼ 0 : t ¼ t 0 + Δt (8.71)
_
in which C is the thermal capacity rate of the fluid, C the thermal capacity of
the fluid mass in the test core, C w the thermal capacity of the solid material
of the test core, α the heat transfer coefficient, A the heat transfer area, and L
the length of the test core.
By introducing the dimensionless parameters,
αA C t t 0 t w t 0 x
x
NTU ¼ , B ¼ , θ ¼ , θ w ¼ , ^¼ NTU ,
C _ C w Δt Δt L
C _
τ τ
^¼ NTU
C w
τ
x
we include the variable of interest α into ^ and ^ and get the dimensionless
governing equation system as follows:
∂θ ∂θ
B + ¼ θ w θ (8.72)
τ
∂^ ∂^
x
∂θ w (8.73)
τ
∂^ ¼ θ θ w
τ (8.74)
^¼ 0 : θ ¼ θ w ¼ 0
^ x ¼ 0 : θ ¼ 1 (8.75)