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EXAMPLE 8.1-1: Conductance of a Cross-Flow Heat Exchanger EXAMPLE 8.1-1: Conductance of a Cross-Flow Heat Exchanger
A finned, circular tube cross-flow heat exchanger is shown in Figure 1. The width and height of
the front face of the heat exchanger are W = 0.2 m and H = 0.2 m, respectively. The fins are
made of copper with a thickness th
fin
= 0.33 mm and a fin pitch p
fin
= 3.18 mm. Ten rows of
tubes (N
t,rows
= 10) in two columns (N
t,col
= 2) are connected in series, as shown in Figure 1. The
vertical and horizontal spacing between adjacent tubes is s
v
= 25.4 mm and s
h
= 22 mm,
respectively. The length of the heat exchanger in the direction of the air flow is L

= 0.06 m. The
tubes are made of copper with an outer diameter D
out
= 1.02 cm and a wall thickness th = 0.9
mm. The roughness of the inner surface of the tube is e = 1.0
m.

Treated water enters the tube with mass flow rate
H
m = 0.03 kg/s and inlet temperature
T
H,in
=60°C. Clean dry air is forced to flow through the heat exchanger perpendicular to the tubes
(i.e., in cross-flow) with a volumetric flow rate
C
V = 0.06 m
3
/s. The inlet temperature of the air is
T
C,in
=20°C and the air is at atmospheric pressure.


Figure 1: Schematic of a plate fin heat exchanger.

a.) Determine the conductance of the heat exchanger.

The inputs are entered in EES:














"EXAMPLE 8.1-1: Conductance of a Cross-Flow Heat Exchanger"

$UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in


"Inputs"

D_out=1.02 [cm]*convert(cm,m)
"outer diameter of tube"

th = 0.9 [mm]*convert(mm,m)
"tube wall thickness"

N_t_row=10 [-]
"number of tube rows"

N_t_col=2 [-]
"number of tube columns"

H=0.2 [m]
"height of heat exchanger face"

W=0.2 [m]
"width of heat exchanger face"

L=0.06 [m]
"length of heat exchanger in air flow direction"

V_dot_C=0.06 [m^3/s]
"volumetric flow rate of air"

P=1 [atm]*convert(atm,Pa)
"atmospheric pressure"

T_C_in=convertTemp(C,K,20 [C])

"inlet air temperature"

T_H_in=convertTemp(C,K,60 [C])
"inlet water temperature"

m_dot_H=0.03 [kg/s]
"water flow rate"

s_v=25.4 [mm]*convert(mm,m)
"vertical separation distance between tubes"

s_h=22 [mm]*convert(mm,m)
"horizontal separation distance between tubes"

th_fin=0.33 [mm]*convert(mm,m)
"fin thickness"

p_fin=3.18 [mm]*convert(mm,m)
"fin pitch"

e=1.0 [micron]*convert(micron,m)


"roughness of tube internal surface"


The total thermal resistance between the water and the air,
R
tot
, is the inverse of the conductance
(
UA). The total resistance can be found by summing all of the resistances in series:


( )
,
1
tot
in
f in
cond
out
R
R
R
R
R
UA
=
=
+
+
+
(1)

where
R
in
is the convection resistance between the water and the inner surface of the tube,
R
f,in
is
the fouling resistance that occurs on the internal surface of the tube as a result of deposits that
accumulate from the flowing fluids (the fouling on the external surface is expected to be
negligible since there should be no build-up associated with clean dry air),
R
cond
is the resistance
to conduction through the tube wall, and
R
out
is the resistance between the air and the surface of
the plate fins and the outer tube surface (this resistance is due to both convection and conduction
resistance of the fins).

The resistance between the liquid and the inner surface of the tube can be represented as:


1
in
in
in
tube
R
h
D L =




where
in
h is the average convection coefficient between the water and the tube wall, D
in
is the
inner diameter of the tube


2
in
out
D
D
th
= ,

and L
tube
is the total length of the tube:

,
,
tube
t col
t row
L
N
N
W
=


"Internal flow through the tube"

D_in=D_out-2*th
"tube inner diameter"

L_tube=N_t_row*N_t_col*W
"total tube length"


The average convective heat transfer coefficient on the water-side can be determined using an
internal forced convection flow correlation, as explained in Chapter 5. The process is simplified
by use of the PipeFlow convection function that is available from the Function Info menu item in
the EES Options menu, as shown in Figure 2. The PipeFlow is the dimensional form of the
PipeFlow_ND function that was introduced in Chapter 5; the use of the PipeFlow function frees
us from having to compute fluid properties, Reynolds number, etc., which was necessary when
using the dimensionless version of the function.


Figure 2: Function information for internal flow convection relations in EES

The PipeFlow procedure is used to determine the average inside surface convection coefficient,
in
h ; note that the PipeFlow procedure provides other outputs, but they are optional and not
needed for this calculation. The value of
in
h is taken to be the average heat transfer coefficient
predicted for a constant temperature (as opposed to constant heat flux) boundary condition.

Also note that the determination of the heat exchanger conductance is necessarily an iterative
process when the outlet fluid temperatures are not known, since the heat transfer coefficients
depend on the outlet temperatures as a result of the temperature dependent properties. For
example, the temperature that should be provided to the PipeFlow function is an average of the
inlet and outlet water temperatures; the methods required to completely solve the heat exchanger
problem and therefore predict the outlet fluid temperatures are presented in Sections 8.2 and 8.3.
As a reasonable first guess, the average water temperature will be estimated as the average of the
inlet water and inlet air temperatures.














T_avg=(T_H_in+T_C_in)/2
"average temperature"

call PipeFlow('Water',T_avg,P,m_dot_H,D_in,L_tube,e/D_in:h_bar_T_H, &

h_bar_H_H ,DELTAP_H, Nusselt_bar_T_H, f_bar_H, Re_H)



"access correlations for internal flow through a tube"

h_bar_in=h_bar_T_H
"average heat transfer coefficient on water side"

R_in=1/(pi*D_in*L_tube*h_bar_in)
"resistance to convection on water-side"



The water-side heat transfer coefficient is
in
h = 3557 W/m
2
-K.

The fouling resistance on the internal surface of the tube can be expressed in terms of its fouling
factor,
"
,
f in
R
:



"
,
,
f in
f in
in
tube
R
R
D L =



The fouling factor can be estimated using an appropriate handbook reference or, more simply,
with the EES FoulingFactor function.

"Fouling resistance"

R``_f_in=FoulingFactor('Closed-loop treated water')
"fouling factor on inner surface of tube"

R_f_in=R``_f_in/(pi*D_in*L_tube)
"fouling resistance on inner surface of tube"



The resistance of the tube wall is probably not worth calculating (it is so small in comparison
with the others in Eq. (1)), but it is easy to do so. The resistance for a cylindrical tube was
derived in Section 1.2.4.



ln
2
out
in
cond
m
tube
D
D
R
k
L =


where k
m
is the conductivity of the tube, obtained using EES' built-in property routine.

"Conduction resistance"

k_m=k_('Copper',T_avg)
"tube conductivity"

R_cond=ln(D_out/D_in)/(2*pi*k_m*L_tube)
"tube resistance"





The resistance between the air and the outer surface of the finned tube can be expressed in terms
of an overall surface efficiency,
o , as discussed in Section 1.6.6.


,
1
out
o
out
s out
R
h A =
(1)

where A
s,out
is the sum of the total surface area of the fins (A
s,fin
) and the un-finned tube wall
surface (A
s,unfin
) and
out
h is the average heat transfer coefficient between the air and these
surfaces. We are assuming that the heat transfer coefficient for the entire surface exposed to air
is the same (i.e., the external surface of the tube and the fin surfaces). The overall surface
efficiency is related to the fin efficiency,
fin , as discussed in Section 1.6.6:

(
)
,
,
1
1
s fin
o
fin
s out
A
A
= (2)

The total fin area is simply the difference in the areas of the plates (both sides) and the cross-
sectional areas of the tubes.


2
,
,
,
2
4
out
s fin
t row
t col
fin
D
W
A
H L N
N
p =


The total un-finned tube wall surface is:


,
1
fin
s unfin
out
tube
fin
th
A
D L
p =



The total surface area is:


,
,
,
s out
s fin
s unfin
A
A
A
=
+



"External resistance"

A_s_fin=2*(W/p_fin)*(H*L-N_t_row*N_t_col*pi*D_out^2/4)
"total fin area"

A_s_unfin=pi*D_out*L_tube*(1-th_fin/p_fin)
"total un-finned area"

A_s_out=A_s_fin+A_s_unfin
"total air-side surface area"




In order to determine the fin efficiency, it is first necessary to estimate the heat transfer
coefficient on the air-side. The best method to determine
out
h is not apparent, since the flow of
the air through the heat exchanger core is actually very complex, combining aspects of internal
flow through the passage formed between adjacent fins with external flow over the tubes. The
heat transfer coefficient
out
h can be calculated using the techniques discussed for external flow
over a bare cylinder, as presented in Section 4.9.3. On the other hand, the fins provide channels
for the air flow, so perhaps
out
h should be calculated using the techniques discussed in Section
5.2.4 for internal flow in a rectangular channel. Here, we will estimate
out
h both ways and
compare the results. In EXAMPLE 8.1-2, we will also determine the value of
out
h using a
compact heat exchanger correlation that is based on experimen