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International Journal of Advancements in Research & Technology, Volume 2, Issue2, February-2013
ISSN 2278-7763
LOCAL STUDY OF HEAT EXCHANGERS FOULING IN
LAMINAR FLOW REGIME: THE ENTROPY CRITERION OF
FOULING
Malahimi ANJORIN1; Christophe AWANTO1; Aristide C. HOUNGAN1; Michel FEIDT2
1
2
LEMA-EPAC, Université of Abomey-Calavi, 01 BP 2009 Cotonou Bénin;
LEMTA-CNRS 875-2, front of the forest of The Hague, LP 160 – 54504 Vandoeuvre France.
Email: hounaris@yahoo.fr
ABSTRACT
During their operation, the heat exchangers undergo fouling process. This phenomenon results in a reduction of heat exchange
and an increase in the pressure losses. Usually, the design of these equipments gives preference to the heat exchange process. Recent studies pointed out the difficulties to assess the fouling process regarding the evolution of the heat transfer alone.
This study, basing on the second principle of thermodynamics, proposes local 1D and 2D models where the entropy criterion of
fouling is used. It combines the two immediate consequences of the phenomena namely the thermal degradations and the losses of pressure.
Fouling was simulated by frosting ice layers on the inner face of a cylindrical pipe. A local analysis allowed the precise localization and the evaluation of the degradations of energy. The model is thermally validated.
Keywords : Heat exchangers ; pressure losses ; entropy criterion ; design ; fouling
1 INTRODUCTION
D
uring their operation, the heat exchangers overlap gradually with no desired substances which cause the degradation of the heat-transferring surface. A loss of effectiveness then is observed. An investigation carried out by the
"Ecole Centrale de Paris" and reported by Anjorin [1], shows
that the most common failures of heat exchangers are caused
by the fouling phenomenon.
Fouling involves complex phenomena and has an economic
cost. As a result of the modification in the apparatus geometry, the fouling thermal resistance reduces the heat exchange
and increases the losses of pressure.
During the design of heat exchangers, thermal phenomena
are generally the leading factor. Rene and Lalande [2] used
the loss of pressure ratio, a partial mechanical criterion for the
heat treatment of milk.
A global analysis of the entropic criterion, combining the two
phenomena influencing fouling process, was already presented for the most used three types of exchangers in industry [2,
3]. The present research is about the study of the thermodynamic 1D and 2D local models and focuses on the fouling due
to the phase change solidification of water in a stationary dynamic flow inside a cylindrical tube.
a)
the solid-liquid interface is at a constant temperature
which is the temperature of solidification;
b) the inlet temperature is uniform and the velocity profile is established;
c) the wall temperature is constant;
d) the frosted solid phase is homogeneous and isotropic;
the effects related to the inlet and the flow are neglected.
2.2 The one-dimensional flow model
The determination of the temperature field in the tube informs
on the evolution of the ice sediment thickness and the field of
entropy. The principal equation governing the phenomena
along the axial direction z (fig 1) is the conservation of the
total energy given by:
( )
(
)
(1)
Φ(z) and Φ(z+dz) are the heat flux respectively at the z and the
z+dz coordinates. ΦRf is the heat flux along the radial direction
across the tube wall.
2 THEORETICAL FRAMEWORK
Applying equation (1) to the elementary volume correspond-
2.1 Assumptions of the problem
It is assumed that:
ing to the length dz, the energy balance takes the form:
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International Journal of Advancements in Research & Technology, Volume 2, Issue2, February-2013
ISSN 2278-7763
(
)(
)
(2)
Where C and 0 are given by
(
)
and
̇
where V represents the output velocity; a is the cross section
area; h is the heat transfer coefficient at the interface; e: the
thickness of the ice layer; d: the inner diameter of the tube; T f:
The expression of C varies with the fouling layer thickness
the temperature at the moving interface. The subscript "d"
and the heat exchange coefficient at the solid-liquid moving
indicates the fouled state.
interface. It appears some difficulties to determine C since the
Let's define the dimensionless position z* and temperature
literature proposes only few correlations to calculate the heat
0 :
exchange coefficient h. In a first approximation, the Sieder-Tate correlation is used; that is:
and
For the laminar flow,
(
⁄ )
⁄
(4)
The solution of follows then an exponential
And for the turbulent flow,
⁄
form
( )
(
)
(5)
(3)
.
Fig 1. Geometry of clogged control. .
Figure 2. Degradations of thermal and mechanical energy for
2.3 The two-dimensional flow model
All tables and figures will be processed as images. You
need to embed the images in the paper itself. Please don’t
send the images as separate files. The major consequence
of the heat exchangers fouling is the dissipation of part of
the thermal and the mechanical energy. This energy dissipation depends on the fields of velocity and temperature.
The global balance [3] and the one-dimensional flow analyses do not allow attaining the profile of the moving interface. The problem now is to consider the radial growth of ice
during a liquid-solid phase change of water, on the inner
face of a cylindrical pipe at constant wall temperature. For
this 2D flow model, the following additional assumptions are
made:
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a) the flow is axis-symmetric;
b) the mass flow is conserved in the pipe;
c) the axial conduction is much lower than radial;
the solid conductivity coefficient is constant
2.3.1. Governing equations
In the liquid phase, the phenomenon is described by the
following relations:
Continuity equation:
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International Journal of Advancements in Research & Technology, Volume 2, Issue2, February-2013
ISSN 2278-7763
(
)
(
(6)
∫
(7)
)
(
)
(15)
The energy equation can be simplified. The
T group ap-
proximately equals 1/10 as the fluid is incompressible; since
and  (function of the viscous dissipation) are
the terms
Equation of motion :
in the same order of magnitude, the group
(
(
(
)
*
is neglected
next to  [1]. Moreover, in parallel streamline flows as is the
)
(8)
case here, the radial component v of the velocity is zero. The
equation of energy takes then the form:
(
(
Boundaryconditions: (
(
(
( ) *
)
(
Let's define the following dimensionless variables:
)
)
(
)
)
(9)
(
z* = ;
)
=
( )
;
;
(
=
)
(
(
)
(16)
(10)
Equation of energy:
(
)
)
)
(
)+
The moving interface position and the ice thickness are respectively given by (17) and (18):
(( )
(
)
( )
(
(
)
) *
(11)
(12)
In the solid phase, the temperature profile is governed
by the equation (13), subject to the boundary conditions
(14) :
(
(
*
)
(
(
)
(17)
)
The thickness of reduced ice
expression 17, that is to say
= e/R results from
(13)
(
)
(14)
At the solid-liquid interface, considering infinitely low
(
(
(
))
(18)
)
In the dimensionless coordinates, the energy equation
and the boundary conditions become:
rate ice sedimentation, the heat balance can be written
as:
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(
)
(
)
(19)
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International Journal of Advancements in Research & Technology, Volume 2, Issue2, February-2013
ISSN 2278-7763
With
(
(
)
and
)
(21)
.2.3.2. Local entropy generation in the liquid phase
The local entropy generation is calculated by the relation:
̇ (
)
((
)
(
) *
( )
(22)
This is also expressed by the non-dimension equation:
̇
(
)
(
(
))
((
)
( ) *
(23)
(
(
*
))
The energy equation Erreur ! Source du renvoi introuvable.)
is solved numerically by the finite differences method. The
solution of the resulting tri-diagonal system is obtained using
the algorithm of Thomas [1] (fig 3 following page).
The interface profile and the field of temperature in the
stream are influenced by the Reynolds number and the inlet
temperature; this last variable is dominating as one can on
figures 4 to 8. The profile of temperature is never established
since the thickness of the boundary layer varies according to
axial coordinate Z* as one moves away from the center of the
stream.
Fig 9 shows that thermal degradations increase much more
quickly at the inlet of the vein and reach a higher level at the
water-ice interface. Progressively as one moves away from
the inlet, this level decreases. Thermal degradations do not
occur at the heart of the flow.
The phenomenon is reversed in regard to the pressure
losses. The degradations are weak in the heart of the flow
and at the inlet of the vein; they progressively and become
more intense at the outlet side of the tube (fig. 10).
Fig4. Profile of the interface for Re = 1500, 2000 ; Tie = 30C, 50C.
Fig 5. Field of temperature according to the axial
position for Re = 1500 ; Tie = 50C ; z* = 1,72-206,7.
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(24)
(20)
)
(
(
Fig 6. Field of temperature according to the axial
position for Re = 1500 ; Tie = 30C ; z* = 1,72-206,7.
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International Journal of Advancements in Research & Technology, Volume 2, Issue2, February-2013
ISSN 2278-7763
Physical properties,
Temperatures, Dimensions
.
t0(1,j), t0(i,jmax+1), t(1,j),
t(i,jmax+1)
Rf0=0, Viscous terms =0
I=2 , imax+1
J = 2 , jmax + 1
Calculation of A0, B0, C0, D0
Call Algorithm of THOMAS
t0(i,j)
Rf0 (i)
Z (i) = Rf0 (i)
i = 2 , imax + 1
J = 2 , jmax + 1
Calculation of A, B, C, D
Call Algorithm of THOMAS
t (i , j)
Rf
z (i)= Rf int
(i)
No
nn
int
(i)
(z (i)- Rf int (i))/ Rf
(i)<
O
ui
= Rf int (i)
Rf (i)
i = 1 , imax+1
J = 1 , jmax+1
.
.
Sc
.
,
Sp
,
End
Fig 3. Flow chart of the program of clogging relating to the
two-dimensional model
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International Journal of Advancements in Research & Technology, Volume 2, Issue2, February-2013
ISSN 2278-7763
Fig 7. Field of temperature according to the axial position for Re = 2000 ; Tie = 30C ; z* = 1,72-206,7.
Fig 8. Field of temperature according to the axial position for Re = 2000 ; Tie = 50C ; z* = 1,72-206,7.
Fig 9. Profile of the entropy due to the transfers of heat
for Re = 1500; Tie = 50C.
Figure 10. Profile of the entropy due to the losses of
pressure for Re = 1500; Tie = 50C.
3. EXPERIMENTAL SETUP AND RESULTS
3.1. Layout
The experimental setup consists essentially of two loops: the
cooling circuit represents the primary loop and is connected to the
glycol-water chiller of the laboratory; the secondary loop is fed
with the tap water to be frozen. The double pipe heat exchanger
in study is made of an outer polymethylmethacrylate (altuglas)
tubing of 72 mm in diameter and an inner 35 mm diameter concentric copper pipe. The primary fluid flows through the annular
space between the inner and outer tubes while the secondary
fluid flows through the inner tube. 44 thermocouples are soldered
inside the inner tube and connected to a data
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acquisition unit. They measure the wall temperature. The inlet
and outlet temperatures of the secondary fluid are measured
using 2 platinum sensors with a high precision thermometer. Two
pressure plugs located respectively 40 cm from the inlet and
50 cm from the outlet are used.
3.2. Ice deposit thickness
The ice thickness has been evaluated using 2 methods: for the
first method, the cylindrical pipe is assimilated to a plate and the
series resistance approach is used. The thickness is denoted
.
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International Journal of Advancements in Research & Technology, Volume 2, Issue2, February-2013
ISSN 2278-7763
Concerning the second method, the ice thickness is evaluated by
considering the thermal resistance for cylindrical geometries with
1.
2.
3.
4.
5.
6.
fouling and clean conditions. Here, the ice thickness is denoted
.
Glycolée water vat
Group production of cold
Feed pump
electromagnetic Ratemeter
Vat upstream
Tube transparent establishment of the mode
7. Pressure gauge
8. Vat downstream of tranquillization and
of power supply of the pump
9. Thermostat ' ' Lauda' '
10. Exchanger of heat
11. Vein test
Figure 11. General outline of the loop
3.3. Degradation of energy
The experimental determination of the ice deposit thickness
makes it possible to calculate the thermal and mechanical degradations. Figures 12 to 15 present these degradations for Reynolds numbers of 1500 and 2000, when the mass rate is conserved. One can see that at the beginning of the experimentation,
mechanical degradations are almost non-existent. The influence
of the pressure losses becomes significant after a period, which
marks a level in the increase of mechanical degradations.
The theoretical and experimental values of the pressure losses
degradations seem to agree at the beginning. The significant
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deviation is due to the fact that the theoretical model allows only
for degradations due to friction. Toda & al. studied the solidification of water in a vertical cylindrical pipe; according to them, it
appears singularities at the outlet of the vein so that the profile of
the interface undergoes a negative jump of slope. This phenomenon has been observed during the present tests, in the form of a
jet accompanied with air bubbles at the outlet of the stream. The
low values of the ice deposit thicknesses, which have been calculated using the measured thermal data, do not permit to observe
a significant increase in degradations by theoretical losses of
pressure.
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International Journal of Advancements in Research & Technology, Volume 2, Issue2, February-2013
ISSN 2278-7763
Fig 12. Thermal degradations of energy and losses
of pressure for Re = 1500,
.
Fig 14. Thermal degradations of energy and losses
of pressure for Re = 2000,
.
Fig 13. Thermal degradations of energy and losses
of pressure for Re = 1500,
.
Fig 15. Thermal degradations of energy and losses
of pressure for Re = 2000,
.
.
4
CONCLUSION
The heat exchangers fouling have been modeled from the
thermodynamic point of view by using the entropic criterion of
fouling. This permitted to combine the thermal and the mechanical pressure losses phenomena. The undertaken theoretical study shows that the criterion depends primarily on the
flow regime and the wall and inlet temperatures of the fluid in
the vein.
In the experimentation, the fouling was simulated with a deposit of ice. The cylindrical geometry was studied. The principal problem comes from the specificities of the deposit behavior in relation with the flow regime and the temperature
Copyright © 2013 SciResPub.
This is seen by the apparition of singularities at the outlet
of the vein and lead to a poor convergence between theoretical and experimental values of mechanical degradations. Theoretical thermal degradations agree in a satisfactory way with
the two selected dynamic cases.
The low values of ice thickness calculated using the thermal
measurements did permit to show through experiments that
for a given level of fouling, degradations due to pressure losses exceed those due to the thermal transfers.
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International Journal of Advancements in Research & Technology, Volume 2, Issue2, February-2013
ISSN 2278-7763
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(1992).
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