European Conference on Computational Fluid Dynamics
ECCOMAS CFD 2006
P. Wesseling, E. Oñate, J. Périaux (Eds)
© TU Delft, The Netherlands, 2006
DETERMINATION OF THE HEAT TRANSFER COEFFICIENT
DISTRIBUTION ON THE LONGITUDINAL FINNED TUBES IN
STAGGERED ARRANGEMENT USING INVERSE AND CFD
METHOD
A. Cebula*, T. Sobota†
*†
Cracow University of Technology,
Al. Jana Pawła II 37, 31-864 Cracow,
Poland
*e-mail: acebula@pk.edu.pl
†
e-mail: tsobota@mech.pk.edu.pl
Key words: heat transfer coefficient, heat exchanger, CFD, convection
Abstract. This paper presents results of an experimental study on heat exchanger consisting
of finned tube.The presented method for determining the local heat transfer coefficient on
external tube surfaces is characterized by very high accuracy and can be applied to determine
the spatial heat transfer distribution on objects with a complex shape. As a results of
experimental and numerical investigations were obtained distribution of the local heat
transfer coefficient on the surface of the fined tube for various Reynolds numbers and the
temperature distribution on their surfaces too.
1 INTRODUCTION
Local heat transfer from a cylinder in tube bank has been extensively studied. In many
applications such as design of heat exchangers, detailed information regarding the
circumferential and longitudinal variation of heat transfer to a cylinder is required. There are
many different methods for measuring local heat transfer like techniques using liquid crystals,
thermal paints, heater foils, naphthalene heat-mass transfer analogy [1-5]. All techniques have
been widely used with considerable success but they have certain difficulties.
An alternative method to obtain the local convective heat transfer coefficient is the inverse
procedure. This paper presents results of an experimental study on heat exchanger consisting
of finned tube. Tubes have two longitudinal fins which join neighboring tubes, creating
membrane panels. The local and mean heat transfer coefficients on the tube circumference
were determined using the inverse heat transfer method. Using the inverse methods [6,7], the
convective heat-transfer coefficient was determined very exactly, reproducing the real twodimensional temperature distribution in the cross-section of the tube. In this way, the
circumferential heat flow in the tube caused by the non-uniform heat-transfer coefficient was
taken into consideration. In many past examinations that phenomenon was disregarded,
assuming heat flow in the tube in radial direction only. Such an incorrect assumption can lead
to incorrectly determining heat transfer coefficients, which can differ from real values by even
A. Cebula, T. Sobota
tens of percents.
Determination of the space-variable heat transfer coefficient on a complex shape surface
requires the solution of the nonlinear inverse heat conduction problem [8-10]. The unknown
parameters associated with the solution are selected to achieve the closest agreement in a least
squares sense between the computed and measured temperatures using the Gauss-Newton
method in conjunction with the singular value decomposition or modified Gram-Schmidt
methods. Hensel and Hills [6] approached the two-dimensional steady-state inverse heat
conduction problem using the linear least-squares method. Linearization of the least squares
problem is accomplished by assuming unknown temperatures [6] or temperatures and heat
fluxes [7, 8] on the boundary.
The boundary is divided into a large number of elements, and temperatures or heat fluxes
are assumed to be constant over each element. Having determined the boundary values of
temperature and heat flux from the solution of the IHCP, the convective heat transfer
coefficients are determined from Newton`s Law of Cooling. Numerical [6,7] and
experimental tests demonstrated that spatial distribution of the heat transfer coefficient can be
estimated with satisfactory accuracy if the division of the boundary into elements is very fine.
If the number of segments on the boundary is too small, then the constant value of
temperature or heat flux over an element can not be assumed. In order to solve overdetermined IHCP the number of interior temperature measurement points should be greater
than the number of components of boundary heat transfer coefficient or than the number of
unknown parameters. In this paper techniques are considered in which the distribution of heat
transfer coefficient is deduced from internal temperature measurements. The parameters in the
function describing the spatial variation of the heat transfer are investigated. The thermal
conductivity of the solid k(T) may be temperature-dependent.
The experimental results reported herein are among the first that show the variation of the
local heat transfer coefficients over the circumference of the finned tube. Most data reported
previously were acquired for smooth tubes at low temperatures. The main advantage of the
method is that it does not require any knowledge of, or solution to, the complex fluid flow
field. It should be noted that determining unknown steady distribution of heat transfer
coefficients by using the developed method is inexpensive, since it requires only one fluid
temperature probe and a few thermocouples for temperature measurements inside the solid.
2 MATHEMATICAL FORMULATION OF THE INVERSE HEAT CONDUCTION
The temperature distribution in the body is governed by nonlinear partial differential equation
∇ ⋅ ⎣⎡ k (T ) ∇T ⎦⎤ = 0.
(1)
The unknown boundary condition of the third kind is prescribed on the outer surface of the
body (Fig. 1)
k (T )
∂T
∂n
(
= h ( rs ) T∞ − T
s
where rs represents points on the boundary, s.
s
)
(2)
A. Cebula, T. Sobota
Figure 1: Location of temperature sensors and boundary discretization; h1, ..., hn represent unknown heat transfer
coefficients
In addition to the unknown boundary conditions the internal temperature fi measurements are
included in the analysis:
T ( ri ) = f i , i = 1, ..., m, m ≥ n
(3)
The objective of the present approach is to determine the spatial distribution of the heat
transfer coefficient h(s) based on measured temperatures at m interior locations. The way of
solving this problem is presented in the paper. In the approach, the problem of determining
space-variable heat transfer coefficient is formulated as a parameter estimation problem by
selecting the functional form h = h(s, x1, .., xn) for the heat transfer coefficient. In this case
parameters x1, .., xn have no physical meaning. There are n parameters in x = (x1, ..., xn)T to be
determined such that the computed temperatures Ti agree with the experimentally acquired
temperatures fi , i=1, ..., m in the least-squares sense. A standard procedure is to take more
temperature measurements than the number of unknown parameters xi. The least-squares
method is used to determine x1, ..., xn when m > n. To measure how well the calculated
temperature agree with data, the sum of squares is used
(
S = f − Tm
) (f − T ) .
T
(4)
m
The Levenberg-Marquardt method [11] is used to determine x* for which the sum S becomes
minimum.
The method performs the k-th iteration as
x(
where
k +1)
= x( ) + δ( ) ,
k
k
(5)
A. Cebula, T. Sobota
( )
k
k
δ ( ) = ⎡⎢ J (m )
⎣
T
k
k
J (m ) + µ ( ) I n ⎤⎥
⎦
−1
(J( ) )
k
T
⎡⎣f − Tm ( x ) ⎤⎦ ,
k = 0, 1, ... ,
(6)
and
Jm =
∂Tm ( x )
∂xT
⎡⎛ ∂Ti ( x ) ⎞ ⎤
= ⎢⎜
⎟⎥
⎢⎣⎝⎜ ∂x j ⎠⎟ ⎥⎦ mxn
i = 1,K , n
j = 1,K , m
(7)
where
J – Jacobian,
I – identity matrix,
µ – Levenberg-Marquardt parameter,
The value of the parameter µ(k) → 0 when x(k) → x*. In the proximity of minimum x* the
iteration step in the Levenberg-Marquardt method is almost the same as in the Gauss-Newton
method. The computation programs for solving the non-linear least squares problem by the
Levenberg-Marquardt method are described in [13] and in the IMSL Library [12].
3 EXPERIMENTAL APPARATUS AND TECHNIQUE
In the present experiment a wind tunnel of 270 x 300 mm cross-section and a radial fan is
used.A bank of finned tubes (called membrane panels) in staggered arrangement is
schematically shown in Fig. 2. The fan output is controlled by an inverter. In the first part of
the aerodynamic channel, there is a chamber in which the heating banks are placed. Each bank
consists of 36 tubes made of K18 boiler steel, having a length of 300 mm and a 24.70 mm
outer diameter. The tubes are arranged in staggered order, in rows of five tubes each, with a
transverse pitch S1 = 53.25 mm and longitudinal pitch S2 = 31.00mm. Each tube was fitted
with a heating element supplied from 380 V AC mains. The stand was arranged with
measuring instruments for air temperature, static and dynamic pressure, tube wall
temperature, and power consumed by the heating elements. The temperature at the inlet Tin
and outlet Tout of the tube bank was measured by NiCr-Ni thermocouples. The central tube
was fitted at the middle of its height with 9 thermocouples (Fig. 2b). They supplied data for
recording the temperature of the tube surface. The UPN-100 data-logger is provided with
software for data acquisition by an PC computer. Heat loses from the duct and heated tube
bundle were minimised by insulating the outside surface of the duct with fiberglass mats. The
heat transfer coefficient on the outer surface of the finned tube was determined by measuring
the electric power Q& e supplied to the heater placed inside the tube of the inner diameter din
and the height ht. The heat transfer rate generated by the electric heater at the interior of the
investigated tube is transferred to the air through its the inner and outer surfaces. The heat flux
at the tube inner surface is not uniform due to circumferential heat conduction at the tube
wall.
A. Cebula, T. Sobota
5
1,0 mm
1,0 mm
6
4
11,2 mm
2
7
8
24,7
mm
1
3
10.5
o
45
o
b)
a)
62,0 mm
Figure 2: Cross section of: a) the tube bank, b) the finned tube showing locations of nine thermocouples
The same heat flow rate is generated at each tube row. Hence, the air temperature changes
along its flow path can be assumed as linear. The air temperature flowing around the
investigated tube (Fig. 2b) was computed using the formula
Ta =
( Lb − Lt )
Lb
Tin +
( 215 − 93)
Lt
93
Tout =
Tin +
Tout = 0.5674Tin + 0.4326Tout ,
Lb
215
215
(8)
where
Lb – bundle length, m,
Lt – distance of the investigated tube from the bundle inlet, m.
The local heat transfer coefficients were determined on the basis of the balance of heat
energy supplied by the heating elements placed in the tubes and the power absorbed by air
flowing around the exchanger tubes surface. The heat flux on the internal tube surface is
calculated from the equation
q& =
Q& e
.
(πdin ht )
(9)
The temperature distribution in the tube cross section was calculated using the control
volume method. Owing to the symmetry of the system, the calculus was restricted to one half
of the tube cross section. That half of the cross section was divided into control volumes, for
which 75 heat balance equations were obtained. The distribution of the convective heat
transfer coefficient h was determined by the Levenberg-Marquardt method, using nine
temperature values measured on the finned tube. The thermocouples were located 0.5 mm
under the tube surface, at the nodes no. 1, 4, 7, 9, 11, 13, 15, 18, 21 (Fig. 3). The distribution
of the local convective heat transfer coefficient on the surface of the finned tube was
A. Cebula, T. Sobota
approximated by the function:
4
⎡ H ⎤
π⎥
h ( H ) = x0 + ∑ xi ⋅ cos ⎢i
H
i =1
⎣ c ⎦
H
Hc
(10)
– coordinate, measured from point 1, m
– extended length of tube semi-circumference between points 1 and 21(Fig. 3), m
Figure 3: Control volume arrangement on finned tube
At each iteration step the set of balance equations system of control volumes was solved using the
Gauss elimination method.
3 EXPERIMENTAL RESULTS
The measured and calculated temperature distribution on the fined tube
semicircumference for the two values of the Reynolds number is shown in Fig. 4. The axis of
abscissae represent half length of developed circumference measured from point 1 to 21
(Fig.3) in a direction of flow. The length of developed semicircumference is Hc = 73,29 mm.
The changes of the heat transfer coefficient for various Reynolds numbers are depicted in
Fig.5. It follows from Fig. 5 that high values of h heat transfer coefficient occur at the front
of the fin. It decreases gradually since it reaches first minimum at the base of fin.
Heat transfer results show a peak on cylindrical part at point ϕ = 90°. Second minimum occur
at the base of rear fin. At these locations a large reduction in h is due the wakes. Flow in the
wake region is characterised by low air velocities what corresponds to the low value of heat
transfer coefficient.
A. Cebula, T. Sobota
T [oC]
99
tube
fin
T [o C]
56
fin
98
55
97
54
fin
tube
fin
53
96
52
95
51
94
50
93
49
92
48
91
47
calculated temperature
measured temperature
90
89
a)
calculated temperature
measured temperature
46
45
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 H[m]
0o 30o 60o 90o 120o 150o 180o
0.01 0.02 0.03 0.04 0.05 0.06 0.07 H[m]
b) 0
0o 30o 60o 90o 120o 150o 180o ϕ
ϕ
a)
tube
fin
280
260
240
220
200
180
160
140
120
100
80
60
40
20
0
fin
Re = 43340
Re = 30927
Re = 16351
Re = 5018
h[W/m2K]
h [W/m2K]
Figure 4: Temperature distribution on the surface of finned tube panel a) Re = 5018 b) Re = 43340
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 H[m]
0 30 60 90 120 150 180 ϕ
o
o
o
o
o
o
o
b)
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
0
fin
0
tube
fin
0.01 0.02 0.03 0.04 0.05 0.06 0.07 H[m]
0o 30o 60o 90o 120o 150o 180o ϕ
Figure 5: Distribution of the local heat transfer coefficient on the surface of the fined tube a) for various
Reynolds numbers, b) for Re = 44960
4 NUMERICAL RESULTS
In order to validate the inverse method for obtaining local heat transfer coefficient,
numerical simulations of the flow in the heat exchanger which consists of finned tube were
carried out. Numerical simulations were conducted using commercial software Fluent.
Because of the symmetry of the tube bank geometry, only a portion of the domain needs to be
modeled. The computational domain is shown in Figure 6. The temperature and velocity are
obtained Fig.7. The comparison of the distribution of the local heat transfer coefficient
A. Cebula, T. Sobota
obtained from the numerical simulation using the program Fluent [14] and from the inverse
method is shown in Fig. 8.
v = 3,6 m/s
t = 22,4 oC
q = 5080 W/m2
q = 5080 W/m2
Figure 6: Computational domain and boundary condition
From an inspection of the results shown in Fig.8 it follows that both approaches give similar
results. The distributions of the heat transfer coefficient depicted in Fig. 8a and 8b show
similar behaviour, although noticeable differences are observed in the values of h(ϕ). The
choice of a turbulence model has remarkable influence on the obtained distribution h(ϕ) and
the mean value h of the heat transfer coefficient (Table 1). The value of hF = 57.7 W/(m2K)
obtained from numerical calculation using k-ε turbulence model is close to the value h =
54.8 W/(m2K) obtained from the inverse method.
a)
b)
Figure 7: Velocity in m/s (a) and temperature in oC (b) distribution in a channel formed by two adjacent
membrane panels
a)
tube
fin
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
fin
h[W/m 2K]
h [W/m 2K]
A. Cebula, T. Sobota
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 H[m]
280
260
240
220
200
180
160
140
120
100
80
60
40
20
0
b) 0
fin
tube
fin
0.01 0.02 0.03 0.04 0.05 0.06 0.07 H[m]
0o 30o 60o 90o 120o 150o 180o ϕ
0o 30o 60o 90o 120o 150o 180o ϕ
Figure 8: Distribution of heat transfer coefficient on membrane panel surface for Re = 33000
a) numerical simulation, k-ε turbulence model, hF = 139.9 W/(m2K), b) inverse method h =125.8 W/(m2K)
Mesh
55.0·103
cells
Turbulence
model
S-A
k-ε
RSM
Heat transfer
coefficient
hF ,W/(m2K)
(Fluent)
49.5
57.7
44.8
Heat transfer
coefficient
h , W/(m2K)
(Measurement)
54.8
hF – h
2
100%·
W/(m K)
-5.3
2.9
-10.0
hF − h
h
9.7
5.3
18.2
Table 1 : Computation results for the membrane surface heat exchanger; w = 3.6 m/s, Tin = 22.4 oC, q& = 5080
W/m2, Tu=3%
S-A
k-ε
RSM
– Spalart-Allmaras turbulence model,
– k-ε turbulence model,
– Reynolds stress turbulence model
5 CONCLUSION
The results of an experimental studies and numerical calculations of the finned tube heat
exchanger were presented in the paper. The experimental data were obtained from the
measurements of the finned tube temperature on the outer surface, air velocity and heat on the
inner surface of the tube. The distribution of local heat transfer coefficient on the outer
surface of the finned tube was approximated by the trigonometric polynomial of the fourth
A. Cebula, T. Sobota
kind. And the local heat transfer coefficient was calculated using least squares method with
Levenberg-Marquardt algorithm. The average heat transfer coefficient was calculated using
CFD software – FLUENT. The comparison of the local and average heat transfer coefficient
was shown.
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