A dynamic model for milk fouling in a plate heat exchanger Youcef Mahdi a, * , Abdelkader Mouheb b,1 , Lounes Oufer b,1 a Centre Universitaire de Me ´de ´a, Institut des Sciences et de la Technologie, Quartier Ain D’heb Me ´de ´a 26001, Algeria b Universite ´ des Sciences et de la Technologie Houari Boumediene, Faculte ´ de Ge ´nie Me ´canique et de Ge ´nie des Proce ´de ´s, Laboratoire des Phe ´nome `nes de Transfert, B.P. 32, El-Alia, Bab-Ezzouar, Algiers, Algeria Received 9 July 2007; received in revised form 27 November 2007; accepted 27 November 2007 Available online 22 January 2008 Abstract A two-dimensional dynamic fouling model for milk fouling in a plate heat exchanger (PHE) is proposed. Emphasis is placed on fouling prediction based on the hydrodynamic and thermodynamic performances of the PHE. A 12-channel PHE with counter-current flows is used in quantification of the milk deposition developed inside the channels. The aggre- gation rate of unfolded protein is found to increase exponentially with increasing wall temperature and is accompanied by a substantial reduction in the heat-transfer coefficient. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Milk; Plate heat exchanger; Modelling; Fouling; Pasteurization 1. Introduction Food products are subject to quite elaborate transformations in order to present qualities that are in con- formity with existing food safety regulations. Therefore, food producers and manufacturers always try to define the best lines of processing in order to obtain optimal sanitary and economic conditions. This is done by taking into account the physical and biochemical modifications of the raw materials along process produc- tion. For instance, the phenomenon of fouling constitutes one of the major encountered problems in the con- cerned processes [1]. This is because it generates limitations in the thermal performance of equipment which increases the cost of production but it can also have a considerable effect on the quality of the product as in the case of the dairy industry [2]. Fouling of plate heat exchangers (PHE) during milk processing is a major problem with a negative impact on operating costs and product quality. It also leads to a rise in pressure drop across the exchanger and to possible deterioration in product quality due to failure of the process fluid to reach the required temperature. Another serious problem associated with fouling is the cleaning of fouled surfaces by means of costly and 0307-904X/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2007.11.030 * Corresponding author. Tel./fax: +213 25 58 56 45. E-mail addresses: [email protected](Y. Mahdi), [email protected](A. Mouheb), [email protected](L. Oufer). 1 Tel./fax: +213 21 24 71 69. Available online at www.sciencedirect.com Applied Mathematical Modelling 33 (2009) 648–662 www.elsevier.com/locate/apm
Transcript
Available online at www.sciencedirect.com
Applied Mathematical Modelling 33 (2009) 648–662
www.elsevier.com/locate/apm
A dynamic model for milk fouling in a plate heat exchanger
a Centre Universitaire de Medea, Institut des Sciences et de la Technologie, Quartier Ain D’heb Medea 26001, Algeriab Universite des Sciences et de la Technologie Houari Boumediene, Faculte de Genie Mecanique et de Genie des Procedes,
Laboratoire des Phenomenes de Transfert, B.P. 32, El-Alia, Bab-Ezzouar, Algiers, Algeria
Received 9 July 2007; received in revised form 27 November 2007; accepted 27 November 2007Available online 22 January 2008
Abstract
A two-dimensional dynamic fouling model for milk fouling in a plate heat exchanger (PHE) is proposed. Emphasis isplaced on fouling prediction based on the hydrodynamic and thermodynamic performances of the PHE. A 12-channelPHE with counter-current flows is used in quantification of the milk deposition developed inside the channels. The aggre-gation rate of unfolded protein is found to increase exponentially with increasing wall temperature and is accompanied bya substantial reduction in the heat-transfer coefficient.� 2007 Elsevier Inc. All rights reserved.
Food products are subject to quite elaborate transformations in order to present qualities that are in con-formity with existing food safety regulations. Therefore, food producers and manufacturers always try todefine the best lines of processing in order to obtain optimal sanitary and economic conditions. This is doneby taking into account the physical and biochemical modifications of the raw materials along process produc-tion. For instance, the phenomenon of fouling constitutes one of the major encountered problems in the con-cerned processes [1]. This is because it generates limitations in the thermal performance of equipment whichincreases the cost of production but it can also have a considerable effect on the quality of the product as in thecase of the dairy industry [2].
Fouling of plate heat exchangers (PHE) during milk processing is a major problem with a negative impacton operating costs and product quality. It also leads to a rise in pressure drop across the exchanger and topossible deterioration in product quality due to failure of the process fluid to reach the required temperature.Another serious problem associated with fouling is the cleaning of fouled surfaces by means of costly and
0307-904X/$ - see front matter � 2007 Elsevier Inc. All rights reserved.
650 Y. Mahdi et al. / Applied Mathematical Modelling 33 (2009) 648–662
time-consuming techniques where environmentally offensive chemicals are employed. Given the economicimpact of fouling in milk heat exchangers, it is not surprising hat there is a considerable amount of literatureavailable on modelling of the fouling process. There seems to be an agreement that thermal denaturation ofwhey protein betalactoglobulin plays a major role in the fouling process, certainly when the temperature isbelow 90 �C. A secondary fouling process can also concern salts.
A number of authors have modelled fouling occurrence in PHE systems based on simplified 1D represen-tation of the process hydrodynamics [3–5]. For milk, another paper presented a mathematical model for com-plex PHE arrangements subjected to fouling by adapting a plug flow model with added axial convection andradial dispersion effects [6]. However, the presented models were based on 1D hydrodynamic performance ofthe PHE, which is a considerable simplification of the effect of plate geometry used in industry. In previousstudies such as those by Georgiadis, and Macchiatto (2000), a 2D model accounting for the hydrodynamicsof fluid flow was used to predict the temperature distribution of flow with higher accuracy than a one-dimen-sional model. The latter produced larger prediction errors for PHE with a larger plate aspect ratio. In theauthors best knowledge, the study of such problems by a specific 2D modelling including mass and energybalances such as described here in is nearly non-existent.
The aim of the present study is to apply the chemical reaction model in a 2D dynamic model to predict themilk deposit patterns on the plate surfaces with more accuracy. This approach is expected to pave the way toorganize and optimize the operating conditions for reducing the extra costs involved with fouling.
2. Mechanisms and mathematical model
In the present work, a formulation of the fouling problem occurring inside a plate heat exchanger allowingmilk pasteurization is proposed. A 2D mathematical model of the process is presented based on a dynamicbehaviour of the flowing fluid (milk). The considered model is based on chemical reaction, mass transferand various other factors taking place during thermal treatment of milk. The model is coupled with thedynamic models of a plate heat exchanger, hence resulting in a final model that includes a set of partial dif-ferential and algebraic equations. The analysis is carried out to allow parameter evaluation through a dynamicproblem optimization. At the end, the model is simulated using a chosen software program (Mathematica 5.0)designed to monitor the dynamic temperature profiles of fluid at different locations.
2.1. System description
The considered pasteurization process occurs in a plate heat exchanger recommended for the pasteurizationof milk because it offers an important convection coefficient and higher turbulence in comparison with otherclasses of heat exchangers. The process consists of heating milk to a given temperature during a certain time inorder to eliminate the pathological action of any possible bacteria. This is a necessary treatment that milkmust undergo before any further use. Milk flowing from a pre-heating tank enters the heat exchanger whereit is heated by hot steam in order to reach the pasteurization temperature.
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2.2. The mechanism of fouling
A schematic representation of the proposed fouling model is given in Fig. 1. When the milk is heated to atemperature greater than 65 �C, the betalactoglobulin protein is unsteady and becomes the precursor fordeposit formation according to two possible mechanisms [7,8]:
1. The betalactoglobulin natural protein (N-protein) experiences a denaturation process (change of structure)and becomes very reactive because of its SH bond (betalactoglobulin denatured form or D-protein).
2. An irreversible polymerisation reaction resulting in insoluble particles as aggregations noted protein A(betalactoglobulin aggregated form).
Usually, induction period is extremely short or even instantaneous in plate heat exchangers where intensemixing of fluid held due to high turbulence. Fouling decreases with increasing turbulence [1]. The thicknessand subsequently the volume of laminar sublayer decreases with increasing velocity and consequently theamount of foulant depositing on the heat-transfer surface decreases.
2.3. Causes of protein aggregation
The betalactoglobulin protein has a global structure held together by S–S bonds and one non-exposedinternal free SH group. When heated, the betalactoglobulin starts to unfold. The free thiol group is thereforeexposed and the molecule enters into an activated state making it possible to react with another betalactoglob-ulin molecule. Therefore, a radical chain may grow to form an aggregate that is able to deposit on the heat-transfer surface. It is known that the rate of deposition is related to the concentration of activated molecules inthe solution and may be calculated using the model of denaturation and aggregation of the betalactoglobulin[9,10].
The kinetics of the different reactions are well known [11]. A mass transfer process of the three forms of thebetalactoglobulin protein takes place between the fluid and the layer limit. The aggregated proteins formthe deposit but it is still essential to know the kinetics and the different physical and chemical parametersof the phenomena taking place in order to be able to quantify the extent of this deposit represented by thefouling resistance [9–11].
2.4. Mineral deposition
In addition to the above mentioned type of fouling takes place the calcium phosphate deposition whichpresents an inverse solubility relation which temperature. During the pre-heating process, the ionic productbecomes high with relation to the solubility concentration limits. Salts deposit in the form of crystals onthe surface of the heat exchanger [12]. Calcium phosphate may also precipitate in the core flow. It may thendeposit onto the casein micelle surface and/or onto the betalactoglobulin molecules contained in the milkserum phase. In all cases, it will in time ultimately form a deposit onto the stainless steel wall of the heatexchanger plates as shown on Fig. 1.
N D
D*N*
A
A* δT
Calcium phosphate
Fig. 1. Model of proteins and salts deposition on the heat exchanger surface.
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Fouling caused by milk constituents is a complex process in which both whey protein aggregation and cal-cium phosphate formation in the bulk fluid are to be accounted for. The whole fouling mechanism can bedescribed in the following steps:
1. Straight adsorption of a protein mono-layer even at room temperature.2. Formation of activated betalactoglobulin molecules in the bulk solution at temperatures higher than 65 �C.
The betalactoglobulin aggregates (tenths of nanometers) and calcium phosphate particles are formed.3. Continuous transport of these foulant particles formed in the bulk to the heated surface. However, some
activated molecules can be inactivated during this phase because some given reactions with other moleculesin the bulk can make the particles not enough active to create fouling.
4. Deposition of activated molecules by adsorption on the heat exchanger surface. Calcium ions entrapped inthe protein deposit may help to stabilize these structures.
5. At relatively high temperatures (above 85 �C), the main deposit component is calcium phosphate whichoffers an open network structure where small protein aggregates can be entrapped.
3. Mathematical model of fouling
3.1. Flow and energy equations
The heat exchanger is supplied with hot water with one channel per pass for a total of five passes. Milksupply is to five channels per pass for a total of five passes. The heat exchanger consists of 12 plates. Assumingthat the plate surface is flat and smooth (Fig. 2), the governing 2D flow equations include the continuity andthe momentum equations in the Cartesian coordinates, as given by the following equations:
ouoxþ ov
oy¼ 0; ð1Þ
ouotþ u
ouoxþ v
ouoy¼ � 1
qoPoxþ m
o2u
ox2þ ou2
oy2
� �; ð2Þ
ovotþ u
ovoxþ v
ovoy¼ � 1
qoPoyþ m
ov2
ox2þ ov2
oy2
� �; ð3Þ
Fig. 2. Control volume of fluid inside the channel in 2D.
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where m is the kinematic viscosity, q the density, P the pressure, t the time, and u and v are the velocity com-ponents along the x and y directions, respectively. The transient energy equation for a 2D constant property,incompressible flow can be given by:
TableKineti
NativeDenatuAggreg
ejqjCpj ¼oT j
ot
� �þ uj
oT j
ox
� �þ vj
oT j
oy
� �� �¼ Uj T pðj�1Þ þ T pj � 2T j
� �; ð4Þ
CppqpdpoT pj
oT¼ U jðT j þ T jþ1 � 2T pjÞ; ð5Þ
where Tj is the temperature of the fluid in channel j, Tpj is the temperature of plate j, Cpj and Cpp are respec-tively, the specific heats of fluid in channel j and plate p,qj and qp are the respective densities of fluid in channelj and plate p, ej is the distance between the plates, dp is the plate thickness, and Uj is the overall heat-transfercoefficient in channel j.
3.2. Mathematical model
The rate constant expression for the different reactions involved in the protein transformations is of theArrhenius-type form given by:
K ¼ K0 exp � EðRT Þ
� �; ð6Þ
where, values of K0 and E are given in Table 1 for different temperatures.Proteins react in both the fluid bulk and the thermal boundary layer in fluid milk. Native protein N is trans-
formed to denatured protein D, in a first order reaction. The denatured protein then reacts to give aggregatedprotein A in a second order reaction. Mass transfer between the fluid bulk and the thermal boundary layertakes place for each protein. Only the aggregated protein is deposited on the wall (M). The rate of depositionis proportional to the concentration of aggregated protein in the thermal boundary layer. The fouling resis-tance to heat transfer is proportional to the thickness of the deposit.
654 Y. Mahdi et al. / Applied Mathematical Modelling 33 (2009) 648–662
For the aggregated protein:
oCAj
otþ u
oCAj
oxþ v
oCAj
oy¼ kDO exp � ED
ðRT jÞ
� �C2
Dj þo
oxDA
oCNj
ox
� �� �þ o
oyDA
oCNj
oy
� �� �
� kmA
dTðCAj � C�ApÞ: ð9Þ
The mass balances for the various proteins in the boundary layer are as follows:
oC�Np
otþ u
oC�Np
oxþ v
oC�Np
oy¼ �kNO exp � EN
ðRT pÞ
� �C�Np þ
o
oxDN
oC�Np
ox
� �� �
þ o
oyDN
oC�Np
oy
� �� �� kmN
dTðC�Np � CNjÞ; ð10Þ
oC�Dp
otþ u
oC�Dp
otþ v
oC�Dp
oy¼ kNO exp � EN
ðRT pÞ
� �C�Np � kDO exp � ED
ðRT JÞ
� �C�2Dp
þ o
oxDD
oC�Dp
ox
� �� �þ o
oyDD
oC�Dp
oy
� �� �� kmD
dTðC�Dp � CDjÞ; ð11Þ
oC�Ap
otþ u
oC�Ap
oxþ v
oC�Ap
oy¼ kDO exp � ED
ðRT pÞ
� �C�2Dp þ
o
oxDA
oC�Ap
ox
� �� �
þ o
oyDA
oC�Ap
oy
� �� �� kmA
dTðC�Ap � CAjÞ � kwC�Ap; ð12Þ
oC�Mp
ot¼ kw
dTC�Ap: ð13Þ
Modelling the fouling phenomenon in the plate heat exchanger requires knowledge of several propertiesboth at bulk and at wall conditions. Moreover, the thickness of the hydrodynamic boundary layer, d, is relatedto that of the thermal boundary layer, dT, by the well known solution obtained by Eckert [13]:
dT
d¼ Pr1=3: ð14Þ
The use of the Von Karman integral theory can drive to this type of relation between dT and d knowing theboundary conditions along with the velocity and temperature profiles. Furthermore, the protein mass transfercoefficient, kmi, is related to the diffusion coefficient, Di, by:
kmi ¼Di
d; ð15Þ
where Di can be calculated as follows [14]:
Di ¼ 1:310�17 T j
lV 0:6i
ð16Þ
The mass of protein and salt material deposited on a plate heat exchanger, expressed in g/m2, can then becalculated as follows:
Masspðx; yÞ ¼kdBipðx; yÞqd
U 0
þ tks logI
LL
� �: ð17Þ
The overall heat-transfer coefficient for clean conditions, U0, is calculated according to the following equation[10]:
Nu ¼ hDe
k¼ 0:214ðRe0:662 � 3:2ÞPr0:4; ð18Þ
where
De ¼ 2ej ð19Þ
Y. Mahdi et al. / Applied Mathematical Modelling 33 (2009) 648–662 655
Flow rDiameWidthHeightThicknGap bConstaNativeWall r
1
U 0
¼ 1
hcþ 1
hfþ P j
kp: ð20Þ
The overall heat-transfer coefficient for the fouled conditions, U, is given by [1]:
U ¼ U 0
1þ Bi: ð21Þ
The rate of deposition is related to the rate of change of heat transfer, Bip, as follows [15]:
oBip
oT¼ bdT
oC�Mp
oT; ð22Þ
where b is a constant.The dynamic boundary layer thickness, d, can be calculated from the Sherwood number, Sh, by the relation
[16]:
d ¼ De
Sh; ð23Þ
where the Sherwood number is determined by:
Sh ¼ 0:214ðRe0:662 � 3:2ÞSc0:4: ð24Þ
Table 2 summarizes the methods by which physical and thermo-physical properties used in the calcula-tions are determined. Moreover, Table 3 gives the heat exchanger data and operating conditions used inthe study.
2tion of the thermo-physical properties used in the simulation [16]
al conductivity of milk (W/m �C) 0.00133T + 0.539911y of milk (kg/m3) 1033.7–0.2308T � 0.00246T2
ic viscosity of milk (Pa S) (�0.60445T + 0.947)10�3
c heat of milk (J/kg �C) 1.68T + 3864.2al conductivity of plate (W/m �C) 16.3c heat of plate (J/kg �C) 450y of plate (kg/m3) 7200al conductivity of deposit (W/m �C) 0.5y of deposit (kg/m3) 1030
3ing conditions and technical details of the PHE
eter Symbol Value
ate (kg/s) 0.074ter (m) De 0.0022of plates (m) l 0.1of plates (m) l 0.1ess of plates (m) Pj 8 � 10�4
etween plates (m) E 4 � 10�3
nt b 129protein diameter (m) DN 9.92 � 10�11
eaction rate (m/s) kw 10�7
656 Y. Mahdi et al. / Applied Mathematical Modelling 33 (2009) 648–662
3.3. Initial and boundary conditions
The plate heat exchanger was preceded at steady state with a non-fouling fluid. This is acceptable since pas-teurizers are pre-heated with hot water before the milk fluid is processed. The following initial conditions arethen imposed and given by:
The model described in this paper includes a set of integral partial differential and algebraic equations. Thesolution method is based on a two phase method of lines approach. In the first phase, the spatial dimensions arediscretized in terms of finite dimensional representations, leading to a reduction of the integral, partial differ-ential and algebraic equations into sets of differential algebraic equations with respect to time. The secondphase, the differential algebraic equations are integrated with respect to time by a modified Euler’s technique[4]. Hierarchical model construction is employed. Each channel is modelled as a sub-model. All the sub-modelsare eventually connected to a general model through appropriate boundary conditions. The plate heat exchan-ger thermal model is also defined in parallel and suitably connected with all the other sub-models to define thetemperatures required by the protein reaction scheme. Different numerical methods are used for discretization,according to the flow direction. A finite difference method with 40 elements is used if the flow is in the two direc-tions from zero to the exit of the channel. The plate heat exchanger thermal model is approximated by a secondorder centered finite difference method with 40 elements. This discretization schemes and orders where chosento obtain results. Parameter estimation analysis are performed and based on the solution of a dynamic optimi-zation problem. A regular, structured grid of plate domains in the 2D of hexahedral mesh elements was createdto divide the domain into 66,000 discrete elements. The domain was meshed using tetrahedral scheme.
The validation data were collected in an industrial plant with a capacity of approximately 2000 kg/h. Theplant consisted of a pre-heating section, a heating, and cooling sections. In the following, models and therelated variables are discussed. At first, the validation of the calculated values was done by comparing the
Y. Mahdi et al. / Applied Mathematical Modelling 33 (2009) 648–662 657
deposition rates with data from the literature, then, in turn, indirectly by comparison of the simulated resultswith temperatures measured in an industrial heating plant. The knowledge obtained from the experiments wasintegrated manually into the fuzzy systems. The operating time interval considered for both simulation andvalidation data was set to 8 h, since this was the approximative average operation time between two cleaningcycles.
4.2. Evolution of milk temperature
The calculate vector velocity pattern for milk in PHE is realised by simulations were performed withDx = 10 mm, Dy = 25 mm, and discretized by considering Dt = 0.5 s (<0.1 s), which satisfied the stability cri-terion for the explicit solution of the model. Since fluid milk flows upward through the plate surface from bot-tom left to top left, the model predicts a lesser amount of fluid flow on the right of plate surface, leading to anexcessive temperature rise.
Fig. 3 shows the milk temperature in the channel 4 of the PHE predicted after a eight (08) hours operation.The fluid enters the canal at 70 �C which then increases progressively the x and y two directions of the channelin an exponential fashion as the milk gains heat from the hot steam. The peaks in the temperature, recordedalong the height direction are the result of thermal exchange on both sides of the considered channel. The tem-perature is calculated while fixing the width. At the entry of the channel adjoining plate 4, milk enters by theorifice and creates a turbulence that generates a better thermal exchange on this side in relation to the otherside of the plate. This results in a more rapid progression of the temperature as opposed to the other side of thechannel. Once the pasteurization temperature is reached, the thermal exchange enters the two fluids is stopped.
4.3. Evolution of mass deposition
Fig. 4 shows the distribution of the foulant deposition from milk in the plates of the 4th channel. Thepotential foulant mass per unit area is calculated to be 16 g/m2 at the most. The peaks of fouling quantities,
Fig. 3. Simulated temperature of milk in side the 4th channel (�C).
Fig. 4. Fouling distribution of fluid milk in the plate adjacent to the 4th channel (g/m2).
658 Y. Mahdi et al. / Applied Mathematical Modelling 33 (2009) 648–662
in the height direction, are due to the thermal exchanges on both sides of the considered channel (channel 4)knowing that the activation of the fouling process depends mainly on temperature. Such heat exchangebecomes more important as the fluid advances in the channel. The results remain very similar to thoseobtained by other authors with different operating conditions but with similar type of geometry [7,17,18].The deposit is primarily controlled by the protein aggregation reaction. Once heated, the proteins aggregateand stick on the wall. It is believed that the deposit of salts remains a minor phenomenon at the pasteurizationtemperature of 90 �C. The faster progression of milk temperature in the entry side activates the process offouling more quickly through the deterioration of the proteins and other saturation of salts on this side incomparison with the side opposite of the plate. This result is more important deposit of the entry of the chan-nel of the side of the entry in relation to the contrary.
4.4. Evolution of the overall heat-transfer coefficient
The variation of the overall heat-transfer coefficient between the two fluids in the 4th channel of the heatexchanger is represented in Fig. 5. It is clearly established that the overall heat-transfer coefficient decreasesgradually along the length of the channel following the deposit formation on the plate walls which constitutesan additional layer resistant to heat exchange between the two fluids [13,19]. Fouling growth tends to be dom-inant over the entire plate surface leading to a decrease of the flow rate caused by a reduction of the sectioncrossing of flow. After a given time, the mass deposited does not affect any more the thermal exchange betweenthe two fluids a constant evolution of the exchange coefficient is observed. This is explained by the fact that,the value of the thickness of the deposit passes through a certain critical value after which the resistance of thefouling deposit remains appreciably constant.
Fig. 5. Evolution of the overall heat-transfer coefficient ratio inside the 4th channel.
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4.5. Evolution of the betalactoglobulin protein concentration
The form of the betalactoglobulin protein concentration along the exchanger is shown in Fig. 6. The pro-tein concentration decreases following the denaturation process induced by the increase in the milk tempera-ture during the heating operation. A reduction of 30% with respect to the initial concentration ofbetalactoglobulin is observed. The altered (active form) protein gives birth to aggregations that deposit onthe heat exchanger surface. From the 5th channel, the pasteurization temperature pasteurization (90 �C) is
Fig. 6. Evolution of the betalactoglobulin protein concentration in the heat exchanger.
Fig. 7. Evolution of the betalactoglobulin protein concentration inside the 11th channel.
660 Y. Mahdi et al. / Applied Mathematical Modelling 33 (2009) 648–662
reached, and the heating is stopped. Thus, it appears that the fouling is governed by the deposit of salts andnot by the aggregation of the proteins; the measure of the protein concentration is realized by Kjeldahl methodfor example.
The protein concentration profile along the 11th channel of the heat exchanger is shown in Fig. 7. The con-centration remains constant in the fluid along the channel. The reaction of denaturation is stopped and there isno more deposit of protein. This result confirms the one already observed (Fig. 6).
4.6. Effect of Reynolds number
The mass of deposit accumulated on the plate adjacent to the 4th channel of the heat exchanger surface as afunction of the Reynolds number is displayed in Fig. 8a,b and c. Observation of this figure shows that the massof foulant increases along the channel as mentioned. Furthermore, it also appears that the mass depositiondecreases with an increase in the Reynolds number. This result is in agreement with various previous findings[18–21] which is explained by the higher removal rate of foulant as shear forces increase. However, it should bereminded that higher Reynolds numbers values also mean higher pressure drops, which may be of consider-able impaction pumping costs.
5. Conclusions
A 2D fouling model, coupled with a 2D dynamic model using material balance equations for the variousinteractions and chemical reaction taking place during pasteurization of milk are applied to predict the per-formance of a plate heat exchanger subject to fouling.
The results showed fouling is highly dependent on the various process operating conditions. The differentparameters seem to affect the phenomenon more specifically in the flow rather than in the with direction. Themass of deposit depends mainly on milk temperature and time of processing. Moreover, it is observed that thefouling extent is strongly related to the Reynolds number value and is inversely proportional to the latter.
For a better control of the problem, which passes by a minimization of production expenses and an optimalquality of the product, a number of solutions are recommended. For instance, a good quality of the water used
Fig. 8. Foulant mass deposition (in g/m2) in the plate adjacent to the 4th channel for: (a) Re = 6000, (b) Re = 8000 and (c) Re = 10000.
Y. Mahdi et al. / Applied Mathematical Modelling 33 (2009) 648–662 661
is essential. Besides, a possible decrease in the pasteurization temperature is believed to allow an additionalenergy savings without affecting the product quality. Also, the use of food surfactants in the pasteurizer wouldprobably contribute to create a competition during the adsorption process between the surfactant and the pro-tein molecules which will avoid deposition on the heat exchanger surfaces. Finally, addition of an emulsifierwould eventually protect the proteins from aggregating.
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