2013 issn 2278 7763 local study of heat exchangers fouling

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International Journal of Advancements in Research & Technology, Volume 2, Issue2, February-2013

ISSN 2278-7763

Copyright © 2013 SciResPub.

LOCAL STUDY OF HEAT EXCHANGERS FOULING IN

LAMINAR FLOW REGIME: THE ENTROPY CRITERION OF

FOULING

Malahimi ANJORIN

1; Christophe AWANTO

1; Aristide C. HOUNGAN

1; Michel FEIDT

2

1 LEMA-EPAC, Université of Abomey-Calavi, 01 BP 2009 Cotonou Bénin; 2 LEMTA-CNRS 875-2, front of the forest of The Hague, LP 160 – 54504 Vandoeuvre France.

Email: [email protected]

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 pro-

cess. 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 loss-

es of pressure.

Fouling was simulated by frosting ice layers on the inner face of a cylindrical pipe. A local analysis allowed the precise localiza-

tion and the evaluation of the degradations of energy. The model is thermally validated.

Keywords : Heat exchangers ; pressure losses ; entropy criterion ; design ; fouling

1 INTRODUCTION

uring their operation, the heat exchangers overlap grad-

ually with no desired substances which cause the degra-

dation of the heat-transferring surface. A loss of effec-

tiveness 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 geome-

try, 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 present-

ed for the most used three types of exchangers in industry [2,

3]. The present research is about the study of the thermody-

namic 1D and 2D local models and focuses on the fouling due

to the phase change solidification of water in a stationary dy-

namic flow inside a cylindrical tube.

2 THEORETICAL FRAMEWORK

2.1 Assumptions of the problem

It is assumed that:

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 pro-

file 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.

Applying equation (1) to the elementary volume correspond-

ing to the length dz, the energy balance takes the form:

D

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( ) ( ) (2)

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; Tf:

the temperature at the moving interface. The subscript "d"

indicates the fouled state.

Let's define the dimensionless position z* and temperature

0 :

and

The solution of follows then an exponential

form

( ) ( ) (3)

Where C and 0 are given by

(

)

and

The expression of C varies with the fouling layer thickness

and the heat exchange coefficient at the solid-liquid moving

interface. It appears some difficulties to determine C since the

literature proposes only few correlations to calculate the heat

exchange coefficient h. In a first approximation, the Sied-

er-Tate correlation is used; that is:

For the laminar flow,

( ⁄ ) ⁄ (4)

And for the turbulent flow,

⁄ (5)

.

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 dissi-

pation depends on the fields of velocity and temperature.

The global balance [3] and the one-dimensional flow anal-

yses do not allow attaining the profile of the moving inter-

face. 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:

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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( )

(6)

∫

(7)

Equation of motion :

(

( ( )

*

) (8)

(

(

)

) (9)

Boundaryconditions: ( ) ( ) ( )

( (

)

* ( ) (10)

Equation of energy:

(

)

(

)

(

) +

((

)

(

)

(

)

*

(11)

( ) ( ) (12)

In the solid phase, the temperature profile is governed

by the equation (13), subject to the boundary conditions

(14) :

(

* (13)

( ) ( ) (14)

At the solid-liquid interface, considering infinitely low

rate ice sedimentation, the heat balance can be written

as:

( )

( ) (15)

The energy equation can be simplified. The T group ap-

proximately equals 1/10 as the fluid is incompressible; since

the terms

and (function of the viscous dissipation) are

in the same order of magnitude, the group

is neglected

next to [1]. Moreover, in parallel streamline flows as is the

case here, the radial component v of the velocity is zero. The

equation of energy takes then the form:

Let's define the following dimensionless variables:

(

) (

)

(16)

z* =

; =

;

;

=

(3)

The moving interface position and the ice thickness are re-

spectively given by (17) and (18):

(

( ))

(17)

The thickness of reduced ice = e/R results from

expression 17, that is to say

(

(

( )))

(18)

In the dimensionless coordinates, the energy equation

and the boundary conditions become:

( )

(

)

(19)

( )

( )

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( ) (20)

( ) 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)

With (

*

(24)

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.

Fig 6. Field of temperature according to the axial

position for Re = 1500 ; Tie = 30C ; z* = 1,72-206,7.

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Physical properties,

Temperatures, Dimensions

.

t0(1,j), t0(i,jmax+1), t(1,j), t(i,jmax+1)

Rf0=0, Viscous terms =0

J = 2 , jmax + 1

I=2 , imax+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 int (i)

Rf (i) = Rf int (i)

i = 1 , imax+1

J = 1 , jmax+1

. . .

Sc , Sp ,

End

Nonn

Oui

z (i)= Rf int (i)

(z (i)- Rf int (i))/ Rf (i)<

Fig 3. Flow chart of the program of clogging relating to the

two-dimensional model

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Fig 7. Field of temperature according to the axial posi-

tion for Re = 2000 ; Tie = 30C ; z* = 1,72-206,7.

Fig 8. Field of temperature according to the axial posi-

tion 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 con-

centric 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

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 denot-

ed .

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Concerning the second method, the ice thickness is evaluated by

considering the thermal resistance for cylindrical geometries with

fouling and clean conditions. Here, the ice thickness is denoted

.

1. Glycolée water vat 7. Pressure gauge

2. Group production of cold 8. Vat downstream of tranquillization and

3. Feed pump of power supply of the pump

4. electromagnetic Ratemeter 9. Thermostat ' ' Lauda' '

5. Vat upstream 10. Exchanger of heat

6. Tube transparent establishment of the mode 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 degra-

dations. Figures 12 to 15 present these degradations for Reyn-

olds numbers of 1500 and 2000, when the mass rate is con-

served. 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

deviation is due to the fact that the theoretical model allows only

for degradations due to friction. Toda & al. studied the solidifica-

tion 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 phenome-

non 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 calcu-

lated using the measured thermal data, do not permit to observe

a significant increase in degradations by theoretical losses of

pressure.

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Fig 12. Thermal degradations of energy and losses

of pressure for Re = 1500, .







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 me-

chanical pressure losses phenomena. The undertaken theo-

retical 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 de-

posit of ice. The cylindrical geometry was studied. The princi-

pal problem comes from the specificities of the deposit behav-

ior in relation with the flow regime and the temperature

. This is seen by the apparition of singularities at the outlet

of the vein and lead to a poor convergence between theoreti-

cal and experimental values of mechanical degradations. The-

oretical 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 loss-

es exceed those due to the thermal transfers.

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REFERENCES

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(1993).

[2] ANJORIN Mr., AWANTO C, FEIDT Mr., Influence of clogging on thermal,

mechanical and thermomechanical criteria of exchangers of heat. J Rech. Sci.

University of Benign, Togo, pp. 101-106 (1998).

[3] BOHNET Mr., BOTT T R., KARABELAS A. J., PILAVACHI P. A., SÉMÉRIA

R., VIDIL R. Fouling mechanisms theoretical and practical aspects, Proceed-

ings of the Eurotherm Seminar N 0 23, April 8-9, Grenoble, France, pp. 308,

(1992).

[4] TODA S. and Al. Laminar flow heat transfer in A tubes with internal solidifi-

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voisier Documentation. (1979).

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processes. Technique and Lavoisier Documentation, (1987).

2013 issn 2278 7763 local study of heat exchangers fouling

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