Page 58 - Design and Operation of Heat Exchangers and their Networks

Basic thermal design theory for heat exchangers 45

2.1.4 General energy equations for steady-state and dynamic
analysis of heat exchangers

For general flow processes, the energy equation can be expressed as
(Whitaker, 1977)

∂T ∂p
ð
ρc p + V rT ¼r λrTÞ + βT + V rp + μΦ + s (2.90)
∂τ ∂τ
in which T is the thermodynamic temperature, s the volumetric heat source,
Ф the dissipation function, and β the thermal expansion coefficient:

1 ∂v
β (2.91)
v ∂T
p
Since the flow in a heat exchanger is usually a low-velocity flow (Mach
number Ma<0.3), the pressure variation would not be very large; therefore,
even for gases, the pressure term in Eq. (2.90) can be omitted. Furthermore,
if the Peclet number is not very high, the viscous dissipation in low-velocity
flow can be neglected. Although the real velocity distribution in a heat
exchanger is three-dimensional and could be very complicated, there is a
main flow direction for each fluid. The main assumption to be used is that
the fluid is completely mixed in the lateral direction but the axial mixing is
negligible. Therefore, the flow velocity and temperature over the section
perpendicular to the main flow direction are uniform. Such a flow pattern
is called plug flow. By integrating Eq. (2.90) over the flow passage section
perpendicular to the main flow direction and using the average values of the
temperature and fluid properties over the cross-sectional area, the energy
equation can be simplified to

Z ðð
∂t _ ∂t ∂ ∂t
A c ρc p + C ¼ A c λ + qdP + sdA c (2.92)
∂τ ∂x ∂x ∂x P A c
Ð
The term P qdP is the convective heat transfer from the solid wall of the
exchanger to the fluid per unit length along the flow direction:
Z Z
qdP ¼ α t w tÞdP (2.93)
ð
P P
in which P is the wetted perimeter of the flow channel at x. If a flow channel
consists of N w walls and the wall temperature and convective heat transfer
coefficient of each wall are uniformly distributed along its wetted perimeter,
then the convective heat transfer term can be expressed as
Z N w
X

qdP ¼ α j P j t w, j t (2.94)
P j¼1