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INSTITUTE OF PHYSICS PUBLISHING
Eur. J. Phys. 26 (2005) N1–N11
EUROPEAN JOURNAL OF PHYSICS
doi:10.1088/0143-0807/26/3/N01
NOTE
Experimental apparatus for measuring
heat transfer coefficients by the Wilson
plot method
José Fernández-Seara1, Francisco José Uhı́a1,
Jaime Sieres1 and Antonio Campo2
1
Area de Máquinas y Motores Térmicos, Escuela Superior de Ingenieros Industriales,
Campus Lagoas-Marcosende, No 9, 36200 Vigo, Spain
2 Mechanical Engineering Department, The University of Vermont, Burlington, VT 05405, USA
E-mail: [email protected], [email protected] and [email protected]
Received 5 November 2004, in final form 19 January 2005
Published 5 April 2005
Online at stacks.iop.org/EJP/26/N1
Abstract
The Wilson plot is a technique to estimate the film coefficients in several
types of heat transfer processes and to obtain general heat transfer correlations.
This method is an outstanding tool in practical applications and in laboratory
research activities that involve analysis of heat exchangers. Moreover, the
application of this method is simple enough to be taught in laboratory practices
for students at university and doctoral level of physics and engineering.
Therefore, an experimental apparatus has been designed and built in our
laboratory that allows the students to carry out experiments based on the
application of the Wilson plot method. In this note, the principles of the method
are explained, the experimental apparatus is described and representative results
of the experimental data taken from the apparatus and the application of the
Wilson plot method are shown.
1. Introduction
Convection is one of the three basic mechanisms of heat transfer. It consists of energy transfer
between a surface and a fluid moving over the surface due to the existence of a temperature
difference between the surface and the fluid. The heat transfer in the fluid is due to combined
effects of conduction and bulk fluid motion. This is a very complex mechanism that involves
the fluid flow established over the surface and the transfer of energy through it. The analytical
solution of a convection problem requires determination of the flow field and the temperature
distribution in the fluid. It can be accomplished by solving the mass, momentum and energy
conservation equations taking into account the surface geometry and the fluid properties.
c 2005 IOP Publishing Ltd Printed in the UK
0143-0807/05/030001+11$30.00 N1
N2
Note
However, this type of approach is very complex and solutions have only been found for simple
surface geometries and under several assumptions.
Most of the actual heat transfer processes usually involve complex geometries and flow
configurations and the analytical solutions are not useful. Therefore, a more practical approach
has been developed based on Newton’s law of cooling, given by equation (1):
q = Ah(Ts − Tf ).
(1)
Newton’s law of cooling states that the heat flow by convection (q) is obtained by taking
into account the surface area (A), an average convection heat transfer coefficient (h) and the
temperature difference between the surface (Ts ) and the fluid (Tf ). Under this approach the
convection problems are reduced to obtaining the convection heat transfer coefficients.
The heat transfer coefficients are commonly calculated from empirical correlations based
on non-dimensional analysis and experimental data. These empirical correlations are obtained
by correlating measured heat transfer coefficients by means of appropriate dimensionless
numbers. For instance, the forced convection correlations are commonly expressed as a
function of the Nusselt (Nu), Prandtl (Pr) and Reynolds (Re) numbers with a functional form
as expressed in equation (2). The coefficient C and the exponents m and n must be determined
based on the experimental data;
N u = CRem P r n .
(2)
The experimental data for a given flow configuration and surface geometry are usually obtained
by measuring the heat transfer area and the surface and fluid temperatures for an imposed heat
flow. Then, the heat transfer coefficient is calculated from equation (3). Analysing the process
for different conditions and fluids the unknown parameters C, m, n, can be adjusted to measured
data and a general correlation is obtained for the regarded geometry.
h=
q
.
A(Ts − Tf )
(3)
However, the main difficulty of this method is the measurement of the surface temperature. It
is even more complicated if the heat transfer surface is not accessible, as happens within heat
exchangers. Heat exchangers are the devices that facilitate heat transfer between two or more
fluids at different temperatures. They are used in a wide variety of industrial, commercial and
household applications. Therefore, any alternative method to calculate the convective heat
transfer coefficients (film coefficients) in heat exchangers is of special interest because of its
practical applications.
The Wilson plot method represents a suitable technique to estimate the film coefficients in
several types of heat exchangers and to obtain general heat transfer correlations. This method
is an outstanding tool in practical applications and in laboratory research activities that involve
analysis of heat exchangers. Moreover, the application of this method is simple enough to
be lectured in laboratory practices for students at university and doctoral level of physics and
engineering.
Therefore, taking into account the reasons cited above, an experimental apparatus has
been designed and built in our laboratory. It allows the students to carry out experiments
based on the application of the Wilson plot method.
The objective of this note is to expose the fundamentals of the Wilson plot method, to
describe the apparatus built and to show the results obtained from a representative laboratory
practice carried out by students within different courses that include the teaching of heat
transfer.
Note
N3
2. The Wilson plot method
This method was proposed by Wilson [1] to evaluate the film coefficients in shell and tube
condensers for the case of a vapour condensing outside the tubes by means of a cooling liquid
flowing inside the tubes. It is based on the separation of the overall thermal resistance into
the inside convective thermal resistance and the remaining thermal resistances involved in the
global process. The basic principles of heat transfer teach that the overall thermal resistance
is the sum of the thermal resistances in series corresponding to each one of the different heat
transfer processes.
Therefore, the overall thermal resistance of the condensation process in shell and tube
condensers (Rov) can be expressed as the sum of the thermal resistances corresponding to
external convection (Ro ), the external fouling film (Rf,o ), the tube wall (Rt ), the internal
fouling film (Rf,i ) and the internal convection (Ri ), as shown in equation (4);
Rov = Ro + Rf,o + Rt + Rf,i + Ri .
(4)
On the other hand, the overall thermal resistance can also be written as a function of the overall
heat transfer coefficient referred to the inner or outer surface of the tubes and the corresponding
area. Equation (5) expresses the overall heat transfer resistance as a function of the overall
heat transfer coefficient referred to the outer surface (Uo ) and the outer surface area (Ao ).
1
Rov =
.
(5)
Uo Ao
The thermal resistances in each one of the heat transfer processes in series from the condensing
vapour to the cooling liquid considered in equation (4) are obtained from equations (6), (7),
(8), (9) and (10), respectively.
Ro =
1
ho Ao
1
hf,o Ao
(7)
ln(do /di )
2π kt Lt
(8)
1
hf,i Ai
(9)
Rf,o =
Rt =
Rf,i =
Ri =
(6)
1
hi Ai
(10)
where ho and hi are the internal and external convection coefficients, hf,o and hf,i are the
fouling factors of the fouling films on the outer and inner tube surfaces, do and di are the outer
and inner tube diameters, kt is the tube thermal conductivity, Lt is the tube length and Ai is
the inner tube surface area.
Taking into account the specific conditions of the condensation process and the equations
indicated above, if the cooling liquid flow is modified then the change in the overall thermal
resistance is mainly due to the in-tube film coefficient variation, meanwhile the remaining
thermal resistances stay nearly constant. Therefore, the thermal resistances due to the outside
tubes convection process, the outer and inner fouling films and the tube wall can be considered
constant, as indicated in equation (11), where C1 is the constant.
Ro + Rf,o + Rt + Rf,i = C1 .
(11)
N4
Note
The film heat transfer coefficient for a turbulent flow in circular tubes can be obtained from
equation (12), according to the correlation proposed by Dittus–Boelter [2] with the exponent
of the Prandtl number assumed known and equal to 0.4.
m
0.4 kl
(12)
hi = CRe Pr
di
where kl is the cooling liquid thermal conductivity and Re and Pr are the cooling liquid
Reynolds and Prandtl numbers at the bulk temperature, respectively.
Moreover, if the effects of the variation of the liquid mass flow on the fluid properties are
neglected, then the internal convection coefficient will be proportional to Rem , according to
equation (12), and the inner convection thermal resistance will be proportional to 1/Rem , as
shown by equation (13), where C2 is constant.
1
Ri = C2 m .
(13)
Re
Furthermore, taking into account equations (4), (11) and (13), the overall thermal resistance
is obtained as a linear function of 1/Rem, according to equation (14). At a glance, expression
(14) shows that C1 is the intercept of the regression line and the overall thermal resistance axis
and C2 is the slope of the straight line.
1
Rov = C1 + C2 m .
(14)
Re
On the other hand, the overall thermal resistance and the cooling liquid Reynolds number
can be obtained from the measured experimental data. The overall thermal resistance can be
calculated from the well-known equation (15) by experimentally measuring the inlet (Tl,in )
and the outlet (Tl,out ) cooling liquid temperatures and the vapour condensation temperature
(Tv ) at various mass flow rates (ml ) of the cooling liquid. The logarithmic mean temperature
difference (Tlm ) and the heat flow (q) are obtained according to equations (16) and (17),
respectively. The cooling liquid Reynolds number is obtained from equation (18) for the
corresponding mass flow rate. In equations (16), (17) and (18), Cpl is the specific heat of
the cooling liquid, ρl is its density, µl is the dynamic viscosity and vl is the velocity inside
the tube.
Tlm
(15)
Rov =
q
Tlm =
(Tv − Tl,in ) − (Tv − Tl,out )
ln[(Tv − Tl,in )/(Tv − Tl,out )]
q = ml Cpl (Tl,out − Tl,in )
(16)
(17)
ρl vl di
.
(18)
µl
Therefore, if the value of the exponent m is assumed, then the experimental values of the
overall thermal resistance can be represented as a function of the experimental values of
1/Rem and the equation of the straight line that fits the experimental data can be obtained by a
simple linear regression. Then, the values of the constants C1 and C2 are obtained according
to equation (14). Once the constants C1 and C2 are determined, then the external and internal
convection coefficients for a given flow rate and the unknown parameter C in equations (2)
and (12) can be calculated from equations (19), (20) and (21), respectively.
1
(19)
ho =
[C1 − (Rf,o + Rt + Rf,i )]Ao
Re =
Note
N5
m
hi =
Re
C2 Ai
(20)
C=
1
kl 0.4 .
C2 di Pr Ai
(21)
The method exposed above is known as the Wilson plot method. As a result of the Wilson
method, the mean value of the convection coefficient outside the tubes and the convection
coefficient inside the tubes as a function of the cooling liquid mass flow (or velocity) are
obtained. Moreover, the coefficient of the general dimensionless correlation is also calculated,
thus the general correlation is determined.
However, the Wilson method requires the assumption of the exponents of the Reynolds
and the Prandtl dimensionless numbers. Therefore, a modification of the method is commonly
applied to overcome the assumption of the value of the Reynolds exponent. This variation is
known as the modified Wilson plot method.
The modified Wilson plot method [3, 4] is based on the described Wilson plot method but
takes into account a second linear equation obtained by applying logarithms to both sides of
equation (14) and an iteration procedure.
Equation (22) is obtained after applying logarithms to both sides of equation (14) and
rearranging it. It results in a linear equation of the term ln[1/(Rov − C1 )] as a function of
ln(Re), with the term ln(1/C2 ) being the intercept between the straight line and the vertical
axis and the parameter m the slope of the straight line.
1
1
= ln
+ m ln(Re).
(22)
ln
Rov − C1
C2
The iteration procedure for obtaining the value of the exponent m consists of:
1. The application of the Wilson plot method assuming the value of the exponent m. Then
the values of C1 and C2 are obtained.
2. Taking into account the values of the overall thermal resistance and the Reynolds number
obtained from the measured experimental data, the values of the term ln[1/(Rov − C1 )]
are plotted as a function of ln(Re). Then, the equation of the straight line that fits the
experimental data is obtained by a simple linear regression.
3. The value of the parameter m is obtained from the equation of the straight line since m is
the slope of the line, according to equation (22).
4. If the value of m obtained is equal to the value initially assumed, then the process is
finished and the value of the exponent m is determined. Otherwise, the iteration process
is repeated from step 1 taking a new value of m.
Therefore, after applying the modified Wilson plot method the mean value of the
convection coefficient outside the tubes and the convection coefficient inside the tubes as
a function of the cooling liquid mass flow (or velocity) are obtained. Moreover, the coefficient
C and the exponent of the Reynolds number m of the general dimensionless correlation are also
obtained, thus the general correlation is determined assuming only the value of the exponent
of the Prantdl number, n.
3. Experimental apparatus description
An experimental apparatus has been designed and built in our laboratory in order to lecture
on the practical application of the Wilson plot method.
The schematic diagram of the experimental apparatus is shown in figure 1. The test
section is composed of an interchangeable copper tube placed inside a small recipient made of
N6
Note
Figure 1. Schematic diagram of experimental apparatus.
Table 1. Average estimated per cent measurement uncertainties.
vl
Tlm
q
Rov
Re
1/Re0.8
3.63
0.56
1.69
1.78
3.75
3
transparent methacrylate. The dimensions of this recipient are: 400 mm long, 100 mm wide
and 400 mm high. The tube is located at the upper part of the recipient. The recipient is filled
with water upto a level below the tube. The water is heated by an electric heater placed at the
bottom of the recipient. The vapour generated from the water fills the recipient and condenses
on the outside of the copper tube. The vapour pressure inside the recipient is kept close to
atmospheric pressure by means of a relief valve. The tube is internally cooled by circulating
cooling water through it. The cooling water is forced through the tube by a pump from a small
reservoir recipient and after going through the tested tube is cooled in a fan-coil and returned
to the reservoir recipient. Therefore, the cooling water circulates in a closed loop removing
the heat of condensation of the vapour that condenses over the tested tube and delivering it in
the fan-coil to the atmosphere.
The mass flow of the cooling water through the tested tube is regulated by means of two
hand valves which allow the recirculation of water from the outlet of the pump to the reservoir
recipient. Moreover, the apparatus is equipped with an electric power regulator that allows
control of the heating power delivered by the electrical resistance and with a water level sensor
which cuts off the electrical power to the resistance for low levels of water.
In order to obtain the experimental data required to apply the Wilson plot method, the
experimental apparatus has been equipped with a data acquisition system. The condensation
temperature of the vapour and the temperatures of the cooling water at the inlet and outlet of
the tested tube are measured by means of PT100 sensors. The values of the three temperatures
are shown on digital displays. The flow rate of the cooling water is measured with a variable
area flow meter. The water flow is also shown on a digital display. The accuracy of the
Pt100 sensors is estimated in ±0.3 ◦ C, whereas the accuracy of the liquid flow meter is ±2%.
Table 1 shows the estimated per cent uncertainty values of the main results for the measured
data. The uncertainty estimates given in table 1 are obtained from the root-sum squares of the
dependent uncertainty parameters.
Note
N7
Table 2. Experimental data used to apply the Wilson plot method.
vl (m s−1)
0.31
0.45
0.73
1.06
1.23
1.49
1.76
(◦ C)
100.1
45.7
73.0
100.2
46.8
70.0
100.2
47.0
67.1
100.2
48.1
65.5
100.2
48.4
65.3
100.2
49.3
63.9
100.2
49.9
62.7
Tv
Tl,in (◦ C)
Tl,out (◦ C)
Table 3. Values of the calculated magnitudes to apply the Wilson plot method.
vl (m s−1)
0.31
0.45
0.73
1.06
1.23
1.49
1.76
Tl,m (◦ C)
Tlm (◦ C)
q (W)
Re
Pr
Rov (K W−1)
59.4
39.2
644.1
2839.3
3.310
0.061
58.4
40.7
804.1
4109.3
3.356
0.051
57.0
42.4
1129.9
6553.9
3.424
0.038
56.8
42.8
1406.8
9420.3
3.433
0.030
56.8
42.8
1589.2
10 966.2
3.433
0.027
56.6
43.1
1670.3
13 280.1
3.444
0.026
56.3
43.6
1724.9
15 599.0
3.460
0.025
4. Experimental procedure and results
This section describes the experimental methodology used and the results obtained from a
practical lecture carried out by students.
The test tube used in the practice is a copper tube with inner/outer diameters of
4.88/6.40 mm and with a length of 380 mm. The tube is set up for the first time; therefore the
effects of the fouling film can be neglected.
The experiment consists in varying the cooling water flow through the tested tube by
manually regulating the two hand valves. The water flow has been adjusted to obtain water
velocities in the tube in steps of 0.25 m s−1 approximately and from 0.25 to 1.75 m s−1
(Reynolds numbers from 2300 to 15 600, approximately). Every time that the water flow is
varied, it is necessary to wait until the system is stabilized, then the temperatures are read from
the digital displays. Therefore, in this case seven different experimental conditions are being
measured. The values of the measured data are shown in table 2.
From the experimental data in table 2, the following parameters are obtained: the mean
temperature of the water inside the tube (Tl,m ), the logarithmic mean temperature difference
(Tlm ) according to equation (16), the heat flow transferred taking into account an energy
balance in the water stream (q), the overall thermal resistance (Rov) and the dimensionless
Reynolds (Re) and Prandtl (Pr) numbers. The values of the magnitudes cited above are shown
in table 3.
Before applying the Wilson plot method, the data in tables 2 and 3 should be carefully
analysed in order to guarantee that the assumptions considered are satisfied. The condensation
temperature of the vapour (table 2) and the mean cooling water temperature (table 3) remain
nearly constant, so the external surface temperature of the tube and the thermal resistance of
the convection process outside the tube can be taken as constant. Therefore, the variation of the
overall thermal resistance can be considered due to the variation of the thermal resistance of the
inside convection process since the tube wall and the outside convection thermal resistances
can be taken as constant. Thus, the first assumption expressed by equation (11) is fulfilled.
The values of the Prandtl number shown in table 3 remain nearly constant, so the inside
convection coefficient and thermal resistance depend only on the Reynolds number and
therefore the second assumption of the method considered in equation (13) is also satisfied.
N8
Note
Figure 2. Wilson plot.
4.1. Results of the Wilson plot method
The Wilson plot method is applied considering the Dittus–Boelter correlation as a general
functional form for the internal convection heat transfer coefficient. The exponents of the
Prandtl and Reynolds numbers are taken as 0.4 and 0.8, respectively. Then, the internal
convection coefficient can be expressed according to equation (23), as a function of the
unknown coefficient C, according to equation (12):
0.8
0.4 kl
.
(23)
hi = CRe Pr
di
Taking into account the experimental data in table 3, the values of the overall thermal resistance
are represented as a function of the values of 1/Rem in figure 2. It can be seen that the
experimental points obtained are practically aligned; therefore it can be concluded that the
method is well established. Figure 2 also shows the straight line that fits the experimental data
as well as the equation obtained by linear regression (equation (24)),
1
(24)
Rov = 0.0113 + 29.252 0.8 .
Re
From equation (24), the values of the constants C1 and C2 are obtained (C1 = 0.0113,
C2 = 29.252). Then, the external convection coefficient is obtained from equation (19)
(ho = 11 892.3 W m−2 K−1), the internal convection coefficient is obtained from equation (20)
as a function of the Reynolds number (hi = 5.868Re0.8 W m−2 K−1) and the coefficient C of
the general correlation is obtained from equation (21) (C = 0.0271). So, the general correlation
for the calculation of the inside convection coefficient into the experimental range is given by
equation (25),
(25)
Nu = 0.0271Re0.8 Pr0.4 .
4.2. Results of the modified Wilson plot method
The modified Wilson plot method is applied according to the iteration procedure indicated
in section 2. An initial value of 0.8 has been considered for the exponent of the Reynolds
number. The value of the exponent of the Reynolds number obtained from the iteration process
is 0.743. Figure 3 shows the plot of the overall thermal resistance as a function of 1/Re0.743
based on the experimental data in table 3. Equation (26) is the equation of the straight line
obtained by linear regression from the experimental points represented in figure 3,
1
(26)
Rov = 0.0093 + 19.27 0.743 .
Re
Note
N9
Figure 3. Modified Wilson plot 1.
Figure 4. Modified Wilson plot 2.
Figure 4 shows the values of the term ln[1/(Rov − C1 )] plotted as a function of ln(Re) taking
into account the values of the overall thermal resistance and the constant C1 obtained from
equation (26). Equation (27) is the equation of the straight line obtained by simple linear
regression that fits the experimental points represented in figure 4. The slope of the straight
line shown in figure 4 (equation (27)) is given by the coefficient of the term ln(Re), which is
equated to the guess value when the iteration process is fulfilled;
1
= −2.947 + 0.7427 ln(Re).
(27)
ln
Rov − C1
The results of the modified Wilson plot method are taken from equation (26). The values of
the constants C1 and C2 are: C1 = 0.0093 and C2 = 19.27. The external convection coefficient
is obtained from equation (19) (ho = 14 533.3 W m−2 K−1), the internal convection coefficient
is obtained from equation (20) as a function of the Reynolds number and the value of the
exponent obtained (hi = 8.908Re0.743 W m−2 K−1) and the coefficient C of the general
correlation is obtained from equation (21) (C = 0.0411). The general correlation for the
calculation of the internal convection coefficient into the experimental range is now given
by equation (28), taking into account the new values of the coefficient C and the Reynolds
exponent, m.
Nu = 0.0411Re0.743 Pr0.4 .
(28)
N10
Note
Figure 5. Comparison of the internal film coefficients calculated by Wilson and the modified
Wilson methods with film coefficients calculated by Dittus–Boelter and Gnielinski correlations.
5. Results analysis
Finally, the students should compare and analyse the results obtained from their experimental
work and the application of the Wilson and modified Wilson plot methods. Moreover, the
results obtained should also be compared to other general correlations commonly found in the
general textbooks on heat transfer.
An example of the comparison between the results of the internal convection heat transfer
coefficient calculated from the general correlations obtained in sections 4.1 and 4.2 is shown
in figure 5. Moreover, also represented in figure 5 are the heat transfer coefficients obtained
from the well-known correlations of Dittus–Boelter [2] and Gnielinski [5] for the range of the
experimental data.
The results in figure 5 show that the values of the convection coefficient obtained from
the Wilson plot method are slightly higher than the values obtained from the modified Wilson
method and from the Dittus–Boelter and Gnielinski correlations. However, the values obtained
from the modified Wilson plot method are close to the values obtained from the Gnielinski
correlation. It is also noteworthy that the slope of the straight line obtained from the modified
Wilson method is similar to the slope of the line obtained from the Dittus–Boelter correlation,
as can be observed in figure 5.
6. Heat transfer process visualization
The processes of vapour generation and its condensation on the tested tube take place inside a
recipient made of transparent methacrylate. The objective is to allow the visualization of the
heat transfer mechanisms happening in both processes. The visualization of these mechanisms
is a very attractive result for the students. It is interesting to note the detailed visualization of
the heating and boiling processes of water by means of the electrical resistance. At the first
stage of the water heating, the natural convection mechanism in the water is perfectly observed.
After that, the regimes of the natural convection boiling and the nucleate boiling within the
pool boiling process are perfectly distinguished and can be clearly explained to the students.
Moreover, the different regimes can be maintained by controlling the electrical power supplied
to the electrical resistance. The condensation process of the vapour on the tested tube surface
is also perfectly shown. The characteristic properties of the film-wise condensation mode as
well as the condensate dripping from the tube are clearly visualized. The heat transfer process
Note
N11
Figure 6. Visualization of the nucleate boiling regimen on the electric heater and the film-wise
condensation and condensate dripping from the tested tube.
visualization is illustrated in figure 6, where the nucleate boiling regimen in the water as well
as the film-wise condensation on the tested tube and the condensate dripping from the tube
can be seen.
7. Conclusions
1. A simple experimental apparatus has been designed and built, which is used in practical
lectures on the Wilson plot method for students at university and doctoral levels.
2. The practical lectures consist of a theoretical exposition on the Wilson plot method, the
experimental work carried out by the students using the apparatus and the analysis of the
results obtained.
3. The experimental apparatus allows the visualization of the heat transfer mechanisms that
take place in the processes of vapour generation and its condensation on the tested tube.
The visualization of the results of these mechanisms is very helpful from the viewpoint
of teaching and very attractive and interesting for the students.
References
[1] Wilson E E 1915 A basis for rational design of heat transfer apparatus Trans. ASME 37 47–82
[2] Dittus F W and Boelter L M K 1930 Publication on Engineering vol 2 (Berkeley, CA: University of California
Press) p 443
[3] Shah R K 1990 Assessment of modified Wilson plot techniques used for obtaining heat exchanger design data
Heat Transfer 1990: Proc. 9th Int. Heat Transfer Conf. vol 5 pp 51–6
[4] Briggs D E and Young E H 1969 Modified Wilson plot techniques for obtaining heat transfer correlations for
shell and tube heat exchangers Chem. Eng. Proc. Symp. vol 65 pp 35–65
[5] Gnielinski V 1976 Int. Chem. Eng. 16 359