Influence of flow regime and thermal power on residence time distribution in tubular Joule Effect Heaters

Journal of Food Engineering 95 (2009) 489–498

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Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng

Influence of flow regime and thermal power on residence time distribution in tubular Joule Effect Heaters L. Fillaudeau a, K. Le-Nguyen b, C. André b,* a b

LISBP (CNRS UMR5504, INRA UMR792, INSA), Toulouse, France HEI, Chem. Eng. Dpt., HEI, 13, rue de Toul, 59046 LILLE Cedex, France

a r t i c l e

i n f o

Article history: Received 18 November 2008 Received in revised form 13 May 2009 Accepted 5 June 2009 Available online 11 June 2009 Keywords: RTD Joule Effect Heater Food process Flow regime Heat treatment Newtonian fluid

a b s t r a c t To improve treatment homogeneity in tubular Joule Effect Heater (JEH), geometric modifications could be used even in laminar regime inducing flow perturbation and mixing.As a response variable, residence time distribution (RTD) is an important parameter and it has been commonly used in determining the performances of industrial heat exchangers.In present work, our objectives were (i) to investigate the impact of processing conditions (flow regime, heat flux) on RTD in an industrial JEH equipped with smooth and modified tubes, (ii) to contribute to the estimation of treatment homogeneity versus global energetic performances of heat exchanger and (iii) to validate a general reactor model.Analytical solution and systemic analysis of RTD signals were reported. The evolutions of mean reduced variance, b2 against efficiency number, Eff for smooth (b2 = 0.00129  Eff  0.0300, R2 = 0.992) and modified (b2 = 0.000547  Eff  0.0169, R2 = 0.979) tubes exhibited a similar and linear relationship. Under the conditions investigated (38 < Re < 10,000, 4 < Pr < 950 with Newtonian fluids), treatment homogeneity was significantly improved by modified geometry and strong interactions between heat transfer and hydrodynamics. A significant decrease in reduced variance under both laminar ðb2ST ¼ 0:1054  Expð0:00518  P=ðq  Q ÞÞ, b2MT ¼ 0:0661  Expð0:00342  P=ðq  Q ÞÞÞ and turbulent ðb2ST ¼ 0:00624  Expð0:00447  P=ðq  Q ÞÞ, b2MT ¼ 0:00108  Expð0:00195  P=ðq  Q ÞÞÞ regimes was observed versus heat energy.However geometric modification and heat treatment affected the residence time distribution and specifically reduced variance, b2 within same order of magnitude.Systemic analysis of experimental data enabled to evaluate two reactor models:Dispersed Plug Flow (DPF) and Plug Flow (PF) + 2 Continuous Stirred Tank Reactor (CSTR) with and without convolution and with 1 or 2 degrees of freedom.Second model could be considered as the most accurate model to predict RTD in JEH with an accurate degree of confidence for residence time  0:73 and reduced variance estimation (s = 0.995  ts R2 = 0.64, error < 3% and b2 ¼ 0:3119  b2exp R2 ¼ 0:98) and a simplified model with only 1 degree of freedom can be used. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction In the food industry, heat treatment remains the oldest and the most frequently used process (heating, pasteurisation, sterilisation, cooking and cooling) and heat exchangers stand as fundamental equipment. In spite of great improvement in conventional technologies over the last few decades, heat treatment remains a complex operation (reduction of heat transfer coefficient, impact and kinetics of fouling mechanism, heat treatment homogeneity and flow pattern). Enhancing the performance of heat exchangers is at the heart of improving exchanger efficiency and thus at the core of energy optimization of industrial process (Harion et al., 2000). Many studies are and were devoted to the increase of heat transfer and mix* Corresponding author. E-mail address: [email protected] (C. André). 0260-8774/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2009.06.010

ture/flow pattern modification in food process, implementing different technological option. Vashitsh et al. (2008) reported a review on the potential applications of curved geometries whose chaotic advection (impact of geometrical modification on flow pattern, heat transfer). To improve treatment homogeneity in tubular Joule Effect Heater (JEH), geometric modifications could be used even in laminar regime inducing flow perturbation and mixing. As a response variable, residence time distribution (RTD) is an important parameter and it has been commonly used in determining the performance of industrial heat exchangers (Pinheiro Torres and Oliviera, 1998; Roetzel and Balzereit, 2000; Sancho and Rao, 1992). RTD analyse provides information about the degree of mixing, cooking and shearing which play an important role in the final product quality. RTD are used for scale-up and improving equipment design. RTD analyse provides information about the degree of mixing, cooking and shearing which play an important role in the final product

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L. Fillaudeau et al. / Journal of Food Engineering 95 (2009) 489–498

Nomenclature List of symbols a, b, c, d coefficient C concentration (mol/l) Cp specific heat (J/kg K1) Da Darcy number (/) axial dispersion coefficient (m2/s) Dax hydraulic diameter (m) dh e space (m) E RTD function (s1) or (/) Eff efficiency number (/) F cumulative RTD function (/) G Laplace transform of E Gr Grasshof number (/) Gz Graetz number (/) j Colburn number (/) L length (m) n exponents (/) P heat power (W) Peclet number calculated with the pipe length (/) PeL Pr Prandtl number (/) Nu Nusselt number (/) Q volume flow rate (m3/s) Re Reynolds number (/) s skewness (s3) t time (s) mean residence time (s) tS T temperature (°C) U mean fluid velocity (m/s) V volume (m3)

quality. RTD are used for scale-up and improving equipment design. However, RTD of real processes arise from a complex interaction between the velocity profile, diffusion, turbulence, heat transfer, etc. For the analysis of experimental results, the non-ideality of the tracer and its detectability must also be considered. RTD are often described on the basis of simple models, dispersed plug flow (DPF) or a cascade of N continuous stirred tank reactors (CSTR), whereby the fitting is not always good. The assumption under these models only roughly characterizes the real existing processes. Harion et al. (2000) reports the mixing and heat transfer increase in a tube with alternate successive deformations. This geometry from a cylindrical tube, imposes on the fluid flow radial contractions and stretching keeping a constant (or quasi constant) section. The deformation based on alternate elliptic forms is a function of the ondulations amplitude, the streamwise wavelength and the direction of the principal axes of deformation. Their results lead to an efficiency factor (ratio between pressure drop and heat transfer coefficient increases) of about 1.2 for Re < 3000. Castelain et al. (1997,2006) and Chagny et al. (2000) investigated the efficiency of chaotic advection regime in a twisted duct flow. A configuration representing a three dimensional steady open flow consisted of helical and chaotic mixers made of identical bends. Each bends consists of a 90° curve stainless-steel tube of circular cross section (;int/ext = 23/25 mm). The mean radius of curvature of the bends is 126.5 mm, which yields a mean curvature ratio of 0.18. Hydrodynamic (friction curve and residence time distribution) as well as heat transfer performances are reported and compared with Newtonian and non-Newtonian fluids. The results show that at low Reynolds numbers, heat transfer is higher and heating more homogeneous for chaotic advection flow, with no increase in energy expenditure. Overall heat transfer coefficient reaches a

x, y z X, Y

inlet and outlet normalised signals axial direction (m) Laplace transform of x and y

Greek symbols plug flow contribution (/) n geometrical factor (/) reduced variance (/) b2 h reduced time (/) DP pressure drop (bar) l viscosity (Pa s) q fluid density (kg/m3) r2 variance (s2) Cj moment of order j (sj) 0 C centred moment of order j (sj) s mean holding time (s)

a

Indices CSTR exp FC in PF lam m MC out tran turb

constant stirred tank reactor experimental forced convection inlet plug flow laminar modification mixed convection outlet transitory turbulent

maximum at Reynolds number around 250. At high Reynolds numbers (Re > 1000), the configuration has no influence on heat transfer. The experimental evaluation of the residence time distribution with the use of a plug flow model with axial dispersion part exchanging mass with a stagnant region, has allowed the determination of an effective axial dispersion coefficient with two configurations in the case of pseudoplastic fluid and for Re < 300. André et al. (2007) and Fillaudeau et al. (2001) investigated the impact of processing conditions on RTD in industrial JEH with smooth and modified tubes. A geometrical modification was selected in agreement with food applications considering pressure drop increase, heat transfer coefficient increase, propensity to fouling and cleaning efficiency (Lefebvre, 1998). This motive was validated with model and real product at industrial scale and patented. RTD with both geometries (smooth circular and modified tubes) on a complete industrial exchanger (L = 1.40 and 1.50 m, ;int/ext = 18/ 20 and 23/25 mm) was studied versus flow regime (80 < Re < 2000) and a general reactor model was validated. The analysis demonstrates that (i) flow regime improve treatment homogeneity by increasing the plug flow contribution and reducing the value of reduced variance, (ii) the earliest transition from laminar to turbulent flow regimes (shifting in critical Reynolds number) due to geometrical modifications and nominal tube diameter induce a reduced variance decrease. In present work, our objectives were (i) to investigate the impact of processing conditions (flow regime, heat flux) on RTD in an industrial Joule Effect Heater (JEH) equipped with smooth and modified tubes (L = 1.40 m, ;int/ext = 18/20 mm), (ii) to validate a general reactor model and (iii) to contribute to estimate treatment homogeneity versus flow and heating conditions. RTD is investigated following the methodology described by Thereska (1998). In a first step, analytic solution of RTD signals are reported and dis-

L. Fillaudeau et al. / Journal of Food Engineering 95 (2009) 489–498

cussed versus global energetic performances of heat exchanger. In a second step, a systemic analysis of experimental data leads to evaluate two reactor models (DPF model, PF + 2CSTR model) with and without convolution and with 1 or 2 degrees of freedom.

2. Theory

491

Turbulent flow ðForced convectionÞ NuFC ¼

Da=32  ðRe  1000Þ  Pr   1 þ 12:7  ðDa=32Þ1=2  Pr2=3  1  2=3 ! dh  1þ L

from Gnielinski ð1976Þ with 2300 2.1. Friction curves Sizing of heat transfer equipment requires the knowledge of both the heat transfer area and the pressure drop. For heat exchangers, performance is described by empirical correlations between dimensionless numbers, such as the Darcy, Reynolds or Prandtl numbers. Isothermal flow of Newtonian and non-Newtonian fluids in relatively simple geometries has been studied extensively. Shah and London (1978) gave a complete overview of the analytical solutions obtained in laminar flow and semi-empirical correlations for transition and turbulent flow regimes. Classical definition and correlation are reported in Appendix A. 2.2. Thermal performances and heat transfer correlations The estimation of mean, average peripherally local and local peripherally Nusselt number, Nu through theoretical or semiempirical correlations, f(Re, Pr, Ra, Gr, Ri, . . .) is fundamental to size heat exchanger and to define mean or local operating condition like wall overheating. The thermal boundary condition is a set of specification describing temperature and/or heat flux conditions at the inside wall of the duct. The peripherally average heat transfer flux s strongly dependent on the thermal boundary condition in the laminar flow regime, while very much less dependant in the turbulent flow regime for fluids with Pr > 1. A large variety of thermal boundary conditions can be specified for the temperature problem. Generally, these boundary conditions are not clearly and consistently defined in the literature, and therefore these highly sophisticated results are difficult to interpret by a designer (Shah and London, 1978; Schlunder, 1983). The thermal boundary condition of approximately constant axial heat rate per unit duct length is realised in many practical applications: electric resistance heating (JEH), nuclear heating, and in counter flow heat exchanger with equal thermal capacity rates. In addition, mixed convection in a horizontal tube with constant heat flux was observed by Petukov and Polyakov (1967) from experimental results with Newtonian fluids. They pointed out the increase of heat transfer coefficient in presence of mixed convection. Numerical simulation of fully developed laminar mixed convection by Muzzio and Parolini (1994) shows that the natural convection can affect both mass flow rate and heat transfer to a great extent. Furthermore the large angular variation of the Nusselt number illustrated the non-uniformity of wall temperature in a cross section. Lefebvre (1998) demonstrated this fact from experimental measurements. This effect was characterized by the temperature difference between the top and the bottom of a cross section measured by infrared thermographs. These authors show that an increase of the flow rate leads to an intermittent phenomenon which is occurring with fluctuation with time of this difference. The amplitude of these oscillations depends on the test section position and the fluid velocity. Numerical simulation carried out by Abid et al. (1994a,b) and Lefebvre (1998) concluded that mixed convection could lead to a stratification of temperature field in correlation with secondary flow in a cross section. In agreement with literature overview from Shah and London (1978) and Schlunder (1983) heat transfer correlation used to estimate the mean Nusselt number along JEH heat exchanger for Newtonian fluids in laminar and turbulent flow are described by Eqs. (1) and (2).

< Re < 10Eþ6 ; 0:6 < Pr < 2000; 0 < dh=L > 1

ð1Þ

Laminarflow ðForced convectionÞ "  10=9 #3=10 55 NuFC ¼ 5:364  1 þ 1 Gz from Churchill and Ozoe (1973), 8Gz

ð2Þ

2.3. Residence time distribution Flow patterns in continuous systems are usually too complex to be experimentally measured or theoretically predicted from solutions of the Navier–Stokes equation or statistical mechanical considerations. The residence time of an element of fluid is defined as the time elapsed from its entry into the system until it reaches the exit (Villermaux, 1993). RTD theory and classical models are reported in Appendix B (André et al., 2007). 3. Experimental 3.1. Experimental set-up and injection device Fig. 1 displays the experimental set-up which consists of a 600 l agitated vessel, a volumetric feed pump (Albin, MR 25I2-207548) and a tubular Joule Effect Heater. The JEH is a horizontal tubular heat exchanger delivered by the French company Actini (P = 20 kW, U = 18 V, I = 600 A). Table 1 displays the general characteristics of smooth tube (ST) and modified tubes (MT). The JEH consists of 12 tubes of 1.40 m length with 1.20 m heating length. Tubes are connected each other with 180° junctions (six horizontal, five vertical). Smooth tubes were regular circular straight tubes with an internal diameter of 18 mm and 1 mm wall thickness. Modified tubes are made from smooth tubes; the geometrical modifications consist in three pinches on a section with a 120° angle between them. This pattern is repeated along the heating length with a regular space (140 mm) and an alternated angle (60°). For each pinching, the tube wall was pushed inside the tube with a depth of 4 mm on a length of 25 mm and 13 mm width. An injection and detection device was developed to realise homogenous injections of tracer even in a laminar flow. Tracer was injected by applying a backpressure on a ceramic microfiltration membrane (19 channels, diameter 2.5 mm, permeable length L = 2 cm, Fig. 1). The objective is to realise an ideal pulse (injection time <5% mean residence time) (Villermaux, 1993), and the tracer was quantified by electrical conductivity and temperature measurements at the inlet and outlet of heat exchanger. Injected volume and tracer quantity were controlled at each injection. Experimental measurements were: flow-rate, temperature, electric conductivity, differential and relative pressure, and heating power (0 < P < 13 kW) and electrical conductivity. The flow rate was measured using electromagnetic flow-meter (KHRONE, type X1000/6, precision ±1%), the temperatures by use of platinum resistance probes (Pt 100 X) placed at the entrance and exit of each exchange zone (precision ±0.5 °C) and the pressure with relative pressure sensors (JPB, type TB233, precision 0,2%). Differential pressure (Bayley Fischer Porter: 0–75 mBar and 0–900 mBar and Schlumberger: 0–750 mBar and 0–2 Bar) was measured to

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L. Fillaudeau et al. / Journal of Food Engineering 95 (2009) 489–498

P1

T1 T

SI

QP : P1, P2 : DP : T1,T2 : SI : C1, C2 :

Flow-meter Relative pressure sensors Differential pressure sensor Température gauges (Pt100) Injection device Electrical conductimeter

Pi

Pi : Injection pressure Pf : Flow pressure Patm : Room Pressure Q : Flowrate

C1

Patm

F QP DP P2

T2 T

C2

Tubular Joule Effect Heater

Electrical Conductimeter

Pf Q

8 cm

4 cm 5cm

15cm

Fig. 1. Experimental set-up and injection device (SI) – image of smooth (ST) and modified (MT) tubes.

Table 1 Geometric characteristics of smooth (ST) and modified (MT) tubes. Tube

L (mm)

dh (mm)

nb

e (mm)

Lm/lm/em (mm)

V (l)

DV (%)

ST 18/20 MT 18/20

1400 1400

17.91 17.91

0 9

– 140

–/–/– 25/13/4

4.7848 4.6564

– 2.75%

establish the friction curve. Electrical conductivity was monitored with two sensors at the inlet and outlet of the exchanger (Stratos, type 9117/93, no. 31308, K = 0.3790 and type 9111/93, no. 31403, K = 0.2340, range 0.2 lS to 1000 mS, precision ±1%). All signals were electrically conditioned (module SCX-1) and collected using a data acquisition card (AT-MOI-16E-10). A software driver NiDAQ made the configuration and control of data acquisition system possible. Measurements were saved on a PC (PC Pentium 200 MHz) with Labview software. 3.2. Fluids and experimental conditions Friction curves and RTD experiments were conducted at room temperature (21.7 °C ± 3.2) with water and sucrose solutions. For friction curves determination, the flow rate ranged between 100 and 7000 l h1, whereas for RTD experiments, the flow-rates remained close to 200 l h1. For RTD, non-fouling model fluids, water and a sucrose solution (64%, wt/wt) were used. Their physical properties (q, Cp, k, l) were measured and compared with the literature (Norrish, 1967; Weast, 1983–1984, http://www.associationavh.com/fr/feuilles.html). Electrical conductivities (r) were adjusted by sodium chloride adjunction and plotted versus concentration and temperature. For RTD, laminar and turbulent flow regimes and smooth and modified tube geometries were scrutinised versus heat power (0 up to 10 kW, experiences no. 1–9) in accordance with established friction curves and corresponding to the industrial practices. The

measured conductivities are converted into concentration values taking into account temperature increase along JEH. Experimental results where the mean holding time, s differed significantly from the mean residence time, tS obtained from RTD curve (above 10%) were rejected. Average experimental conditions (from at least three runs) are reported in Table 2. Dimensionless numbers were calculated at inlet, outlet and average bulk temperature respectively for inlet and outlet Reynolds and Nusselt numbers. In our experimental condition, mixed convection does not occur and Eq. (8) could be neglected.

4. Results and discussion 4.1. Analytical solution: friction curves, RTD formulation and heat performances Friction curves (Fig. 2) exhibit a classical shape for industrial tubular heat exchanger including 180° junctions between tubes (André et al., 2007). In our condition, Churchill’s equation (Appendix A) is simplified and modelled by three parameters (n, a and b with n1 = 1 and n2 = 2) which are experimentally identified in laminar and turbulent regimes and displayed in Table 3.

 1=2 Da ¼ Da2lam þ Da2turb

ð3Þ

Inlet and outlet Reynolds numbers (Table 2, Fig. 2) indicate that laminar and turbulent regimes are investigated separately. Outlet Reynolds number in laminar flow increase above the first critical Reynolds numbers, Rec1 without reaching turbulent flow. Following Thereska (1998), experimental inlet and outlet RTD signals, x(t) and y(t) are formulated. The measured conductivity values were converted into concentration values. The concentration profiles were obtained as a function of time and Fig. 3 shows typical inlet and outlet normalised signals.

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L. Fillaudeau et al. / Journal of Food Engineering 95 (2009) 489–498 Table 2 Average operating conditions from experimental data (n: number of experiments).

Laminar

Turbulent

Modified tubes

Laminar

Turbulent

n

Q (L/h)

V (l)

TIn (°C)

TOut (°C)

P (kW)

s (s)

ReIn(/)

ReOut (/)

NuFC(/)

1 2 3 4 5 6 7 8 9

4 3 4 4 4 4 4 4 4

202.3 199.5 198.6 199.3 199.1 201.2 199.5 203.7 202.8

4.78 4.78 4.78 4.78 4.78 4.78 4.78 4.78 4.78

19.3 20.0 20.7 20.8 18.4 18.3 18.4 19.0 21.0

19.5 30.9 47.7 65.0 18.6 27.4 42.4 63.0 77.3

0 2.2 5.6 9.3 0 2.1 5.5 10.1 12.8

85.1 86.3 86.8 86.5 86.5 85.6 86.3 84.6 84.9

38 39 41 42 3874 3898 3874 4008 4201

38 88 196 250 3884 4860 6514 8725 10,254

5.00 4.98 4.97 4.97 13.5 14.3 15.3 17.1 18.0

1 2 3 4 5 6 7 8 9

4 4 4 4 5 4 4 4 4

202.1 199.7 200.6 199.2 199.4 197.9 199.0 194.0 200.9

4.65 4.65 4.65 4.65 4.65 4.65 4.65 4.65 4.65

21.3 20.5 20.4 23.1 19.0 17.9 18.4 17.5 18.3

21.1 30.6 47.0 65.7 19.2 27.5 43.3 63.4 74.2

0 2.1 5.5 9.0 0 2.3 5.7 10.1 12.7

82.9 83.9 83.6 84.15 84.7 86.1 84.2 85.6 83.3

43 41 41 49 3934 3796 3866 3683 3892

43 88 196 262 3952 4786 6597 8357 9758

5.00 4.99 4.98 4.97 23.3 24.3 27.0 29.0 31.6

RTD Turbulent

RTD Laminar

10.00 3900 3900 (5) 4900 (6) 6500 (7) 8700 (8) 10000 (9)

38 38 (1) 90 (2) 200 (3) 250 (4)

Da. [/]

1.00 Rec 1

Rec 2

0.10 Friction curve (Smooth tube) Friction curve (Modified tubes) Rec1 and Rec2 RTD : Re min-max (Ref)

0.01 1

100

1000

10000

100000

Re. [/] Fig. 2. Friction curves of ST and MT 18/20, min and max Reynolds numbers for RTD experiments.

Table 3 Friction curve parameters and critical Reynolds numbers. Laminar

Tube ST18/20 MT18/20

Turbulent

8.n

Rec1

a

b

Rec2

64 53.8 75.1

<2000 200 100

0.316 0.25 0.85

0.25 0.20 0.25

>4000 1500 500

0:1054  Expð0:00518  P=ðq  Q ÞÞ, b2MT ¼ 0:0661ð0:00342  P=ðq Q ÞÞÞ and turbulent ðb2ST ¼ 0:00624  Expð0:00447 P=ðq  Q ÞÞ, b2MT ¼ 0:00108  Expð0:00195  P=ðq  Q ÞÞÞ regimes versus heat energy. The performance of heat exchangers depends on the flow pattern, a hydraulic accident can break and renew the boundary layer of fluid, and consequently, higher heat transfer performances may be expected. André et al. (2007) studied the benefits of geometrical modifications in tubular JEH which reduces the dispersion under isothermal condition. Irreducible coupling between heat transfer and fluid mechanics results in a complex flow pattern along modified and smooth tubular JEH. Flow patterns, however, deduced from experimental measurements or theoretical prediction (Navier– Stokes), in conjunction with the equation of conservation of energy (heat transfer) may hardly be considered. The global energetic performance of a heat exchanger is defined as the ratio of the degraded mechanical energy flux to the transferred enthalpy flux. Heat transfer performance is characterized by the Colburn factor, j (Eq. (4)) and mechanical performances through the Darcy number, Da. The dimensionless form of mechanical and heat transfer values are plotted against their Reynolds (Chagny et al., 2000; Yan and Sheen, 2000; Peng and Ling, 2008), as well as their ratio, the efficiency number, Eff (Eq. (5)) or 1/Eff. Efficiency number shows the deviation from the Chilton–Colburn analogy. Hydrodynamic and heat performances of both smooth and modified JEH are described by Lefebvre (1998) and André

5.0E-02

1.0 0.9

Distribution function, x(t)/10 and y(t), [1/s]

Table 4 reports mean holding time,s, mean residence time (moment of 1st order), ts and reduced variance (centred moment of 2nd order), b2 taking into account inlet signal or not. Even if a good agreement is observed between mean holding time, s and mean residence time, ts; the differences are close in turbulent regime (<2%) but slightly higher in laminar regime (<7%). Mean residence time and the dispersion of inlet signals are negligible compared to outlet signals (tsIn/tsOut < 3.8%). Therefore, we may assume that the inlet signal acts as a Dirac function (pulse injection). This assumption will be further discussed by comparing RTD treatment with and without convolution. In Fig. 4, experimental reduced variance, b2exp is analysed versus heat exchanger power, P/(q  Q) (kJ/kg). There is a significant decrease in reduced variance under both the laminar ðb2ST ¼

4.0E-02

0.8 x(t). Inlet (distribution) [1/s] y(t). Outlet (distribution) [1/s] X(t). Inlet (cumulative) [/] Y(t). Outlet (cumulative) [/]

3.0E-02

2.0E-02

0.7 0.6 0.5 0.4 0.3

1.0E-02

0.2 0.1

0.0E+00 -50

0.0 0

50

100

150

200

Cumulative distribution function, X(t) and Y(t), [/]

Smooth tubes

Ref.

250

Time. [s] Fig. 3. Experimental distribution x(t), y(t) and cumulative X(t), Y(t) functions (Modified tubes, laminar regime, experiment 4).

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L. Fillaudeau et al. / Journal of Food Engineering 95 (2009) 489–498

Table 4 Mean holding time, s, residence time, ts and reduced variance, b2, from selected experiments.

Smooth tubes

Laminar

Turbulent

Modified tubes

Laminar

Turbulent

Ref.

s (s)

tsout (s)

b2out (/)

ts (s)

b2exp (/)

1 2 3 4 5 6 7 8 9

85.1 86.3 86.8 86.5 86.5 85.6 86.3 84.6 84.9

96.45 95.02 94.17 100.08 92.51 91.10 91.50 88.98 90.01

0.09664 0.08488 0.06852 0.04294 0.00591 0.00501 0.00381 0.00258 0.00205

93.61 92.37 91.64 97.90 89.05 88.51 87.80 87.34 86.94

0.10394 0.08955 0.07208 0.04569 0.00621 0.00522 0.00419 0.00269 0.00204

1 2 3 4 5 6 7 8 9

82.9 83.9 83.6 84.15 84.7 86.1 84.2 85.6 83.3

91.22 94.25 88.82 95.98 89.90 90.63 89.86 91.75 87.09

0.06039 0.04342 0.04373 0.03939 0.00238 0.00199 0.00198 0.00171 0.00127

86.20 89.41 88.12 92.11 84.32 85.20 84.91 86.23 83.59

0.06614 0.04837 0.03840 0.04253 0.00132 0.00109 0.00090 0.00070 0.00049

et al. (2007), and established correlations were used to determine the Colburn factor and efficiency number.

j¼

Nu

ð4Þ

Re  Pr1=3 Da Eff ¼ j

ð5Þ

Fig. 5 shows that the changes of mean reduced variance, b2 against efficiency number, Eff exhibit a similar and linear relationship. It highlights that treatment homogeneity is significantly improved by modified geometry and strong interactions between heat transfer and hydrodynamics under investigated conditions (38 < Re < 10,000, 4 < Pr < 950 with Newtonian fluids). Geometrical modification and heat treatment affect the RTD and the reduced variance with same order of magnitude. 4.2. Systemic analysis of RTD: comparison between DPF and PF + 2 CSTR models André et al. (2007) showed that both, a DPF model and a plug reactor in series with 2 CSTR may be used to describe RTD in similar tubular JEH (six tubes, five vertical junctions) under isothermal conditions with Newtonian fluids. Several other models have been proposed as reported by Ham and Platzer (2004). Our purpose is to evaluate the validity of the DPF and PF + 2CSTR models, to under-

stand the hydraulic and thermal interactions occurring in relation to treatment homogeneity of liquid food products in JEH. Table 4 displays the experimental conditions (ref. no. 1–9 with smooth and modified tubes), a data treatment, with and without convolution and with 1 or 2 degrees of freedom is investigated for model no. 1, DPF and 2, PF + 2CSTR (Table 5). When calculating with 1 degree of freedom, the mean residence time is considered as an experimental and known parameter. For each trial, curves fitting of both models and parameters identification (s, PeLor a) is realised by minimizing the sum of square residuals (SSR) with DTS Progepi software V4.2 (Leclerc et al., 1995) or Excel (MicrosoftÒ, Office Excell 2003/SP3). Reduced variance, b2 is calculated from identified parameters. The DPF model (Appendix B) has been widely applied to describe the flow in a tube, and is the most often selected to simulate flow in holding tubes in aseptic processes (Pinheiro Torres and Oliviera, 1998; Roetzel and Balzereit, 1997). In order to obtain a better agreement, a second model (Fig. 6) defined by a cascade of a plug flow reactor in series with 2 CSTR was investigated. This model was chosen because it is simple and in close correlation to the physical structure of the process. The corresponding expression of the transfer function G(s) and E(t) are formulated by Eqs. (6) and (7).

1.0000

Mean reduced variance, β², [/]

Experimental reduced Variance, βexp² [/]

1 Smooth tubes - Turbulent Smooth tubes - Laminar Modified tubes - Turbulent Modified tubes - Laminar

0.1

0.01

0.001

Smooth tubes Modified tubes

0.1000

0.0100 Smooth tubes y = 1.29E-03x - 3.00E-02, R 2 = 9.92E-01

0.0010 Modified tubes y = 5.47E-04x - 1.69E-02, R 2 = 9.79E-01

0.0001 0.0001

10 0

50

100

150

200

250

100

1000

Efficiency number. Da/j [/]

P/(ρ.Q), [kJ/kg] Fig. 4. Experimental reduced variance, b2exp versus heat power, P/Q.

Fig. 5. Evolution of mean reduced variance, b2 versus efficacy number, Da/j for JEH with smooth and modified tubes.

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L. Fillaudeau et al. / Journal of Food Engineering 95 (2009) 489–498 Table 5 Data treatment and assumptions in order to identify parameters with model no. 1 and 2.

Experiments

Mean residence time (s) Variance (s2)

With convolution

No convolution

ts = tsout-tsinlet

tsout

r2exp ¼ r2out r2inlet

r2out

Model no. 1: DPF

1 Degree of freedom 2 Degrees of freedom

s = ts, Pe s, Pe

s = tsout, Pe s, Pe

Model no. 2: PF + 2CSTR

1 Degree of freedom 2 Degrees of freedom

s = ts, a s, a

s = tsout,, a s, a

exp ðs  sPF Þ GðsÞ ¼   2 1 þ ss2CSTR  2   2 2  ðt  sPF Þ  ðt  sPF Þ  exp Eðt Þ ¼ Hðt  sPF Þ 

sCSTR

ð6Þ

Fig. 6. Reactor model, cascade of 1 PF reactor in series with 2 CSTR.

With Van der Laan’s relations (Appendix B), a simple relation between a and b2 is established.

G0 ð0Þ ¼ C1 ¼ ðsPF þ sCSTR Þ ¼ s G00 ð0Þ ¼ C20 ¼ s2 þ

ð7Þ

sCSTR

ð8Þ

s2CSTR J

Table 6 Parameters (s, PeL) and reduced variance, b2, identified from DPF model (no. 1) with or without convolution, with 1 or 2 degrees of freedom. With convolution

None convolution

2 Parameters

Smooth tubes

Laminar

Turbulent

Modified tubes

Laminar

Turbulent

1 Parameter

Ref.

s (s)

PeL

b

1 2 3 4 5 6 7 8 9

82.06 83.17 81.87 89.39 87.05 86.89 86.95 86.38 86.07

44 59 77 100 551 580 737 951 1231

1 2 3 4 5 6 7 8 9

78.97 83.81 80.56 85.21 82.85 84.72 84.40 86.11 83.53

60 52 69 87 901 1408 1480 1844 2259

2

2 Parameters 2

1 Parameter PeL

b2

0.0392 0.0303 0.0290 0.0200 0.0035 0.0032 0.0025 0.0021 0.0017

32 43 46 62 446 510 661 857 1091

0.0698 0.0506 0.0468 0.0340 0.0045 0.0039 0.0030 0.0023 0.0018

0.0309 0.0342 0.0284 0.0221 0.0024 0.0019 0.0018 0.0015 0.0012

43 54 60 65 787 985 1018 1244 1646

0.0511 0.0402 0.0357 0.0328 0.0025 0.0020 0.0020 0.0016 0.0012

s (s)

PeL

b

0.0752 0.0529 0.0457 0.0327 0.0048 0.0042 0.0034 0.0025 0.0019

84.59 85.11 84.67 91.54 90.32 89.24 89.90 87.91 89.19

55 70 73 104 582 635 808 972 1189

0.0523 0.0496 0.0384 0.0337 0.0024 0.0016 0.0015 0.0011 0.0009

82.09 88.36 83.19 88.73 89.07 89.83 89.07 91.08 86.64

68 62 74 94 845 1072 1110 1329 1708

PeL

b

0.0495 0.0359 0.0273 0.0208 0.0036 0.0035 0.0027 0.0021 0.0016

30 41 47 65 424 478 596 816 1047

0.0358 0.0417 0.0309 0.0241 0.0022 0.0014 0.0013 0.0011 0.0009

42 44 56 63 822 1279 1358 1844 2259

2

Table 7 Parameters (s, Pe) and reduced variance, b2 identified from PF + 2CSTR model (no. 2) with or without convolution, with 1 or 2 degrees of freedom. With convolution

None convolution

2 Parameters

Smooth tubes

Laminar

Turbulent

Modified tubes

Laminar

Turbulent

1 Parameter

2 Parameters

1 Parameter

Ref.

s (s)

a

b2

a

b2

s (s)

a

b2

a

b2

1 2 3 4 5 6 7 8 9

90.29 90.21 87.86 95.05 89.21 88.97 88.78 87.95 87.43

0.65 0.70 0.73 0.76 0.89 0.90 0.91 0.92 0.93

0.0608 0.0464 0.0365 0.0288 0.0056 0.0053 0.0042 0.0033 0.0026

0.63 0.68 0.70 0.74 0.90 0.90 0.91 0.93 0.94

0.0689 0.0513 0.0451 0.0337 0.0053 0.0046 0.0037 0.0027 0.0021

93.29 92.96 91.15 97.83 92.86 91.67 91.95 89.68 90.81

0.66 0.70 0.73 0.77 0.89 0.90 0.91 0.92 0.92

0.0582 0.0451 0.0372 0.0276 0.0057 0.0053 0.0042 0.0034 0.0030

0.64 0.69 0.71 0.75 0.90 0.90 0.91 0.93 0.93

0.0657 0.0487 0.0433 0.0319 0.0052 0.0047 0.0037 0.0028 0.0022

1 2 3 4 5 6 7 8 9

85.64 91.49 86.86 91.04 84.09 85.95 85.60 87.18 84.45

0.70 0.68 0.71 0.74 0.93 0.93 0.93 0.94 0.95

0.0462 0.0527 0.0407 0.0327 0.0027 0.0022 0.0021 0.0017 0.0014

0.68 0.69 0.72 0.74 0.93 0.94 0.94 0.95 0.96

0.0508 0.0475 0.0382 0.0345 0.0028 0.0016 0.0015 0.0012 0.0009

89.53 96.73 89.96 95.21 90.00 91.59 90.72 92.67 87.92

0.70 0.68 0.72 0.75 0.91 0.92 0.92 0.93 0.94

0.0437 0.0501 0.0392 0.0310 0.0041 0.0031 0.0031 0.0025 0.0020

0.69 0.70 0.73 0.75 0.92 0.93 0.93 0.94 0.94

0.0471 0.0448 0.0366 0.0320 0.0032 0.0023 0.0024 0.0017 0.0015

L. Fillaudeau et al. / Journal of Food Engineering 95 (2009) 489–498

An advantage of this model is that when the mean residence time is known, only one parameter a is used. The reduced variance (Eq. (10)) is calculated from the plug flow reactor contribution in term of residence time (Eq. (9)).

sPF sPF þ sCSTR ð1  aÞ2

a¼

b2 ¼

ð9Þ ð10Þ

2

Tables 6 and 7 summarized the identified parameters (s, PeL and a) and reduced variance, b2 for the DPF and PF + 2CSTR models. Models and treatment methods (with and without convolution,

3.5E-02

y(t), DPF and PF+2CSTR models,[1/s

496

y(t)

3.0E-02

y(t) from PF+2CSTR model

2.5E-02

y(t) from DPF model

2.0E-02 1.5E-02 1.0E-02 5.0E-03

0.0E+00 0

50

100

150

200

t,[s] Fig. 9. Comparison between experimental, DPF and PF + 2CSTR models (without convolution, 1 degree of freedom) for MT 18/20 in laminar flow, Rein = 41 and P = 5.54 kW).

100

τ-Mod1-WC-2p τ-Mod1-NC-2p τ-Mod2-WC-2p τ-Mod2-NC-2p Linéaire ( τ-Mod2-WC-2p) Linéaire ( τ-Mod2-NC-2p) Linéaire ( τ-Mod1-WC-2p) Linéaire ( τ-Mod1-NC-2p)

τ, [s]

95

90

τ-Mod1-WC-2p : y = 0,9515x, R² = -1,182

85 τ-Mod1-NC-2p: y = 0,9515x, R² = -0,987

y=x

80

τ-Mod2-WC-2p: y = 0,995x, R² = 0,6398 τ-Mod2-NC-2p: y = 0,998x, R² = 0,6056

75

75

80

85

90

ts or ts

out ,

95

100

105

[s]

Fig. 7. Identified mean residence time, s with models no. 1 and no. 2 versus experimental mean residence time, ts or tsout (WC: with convolution, NC: none convolution, 2P: 2 degrees of freedom).

A

1.0000

β². [/]

0.1000

β²-Mod1-WC-1p 0.8451 y = 0.4673x R2 = 0.9826

β²-Mod1-WC-1p β²-Mod1-NC-1p β²-Mod1-WC-2p β²-Mod1-NC-2p Puissance ( β²-Mod1-WC-1p) Puissance ( β²-Mod1-NC-1p) Puissance ( β²-Mod1-WC-2p) Puissance ( β²-Mod1-NC-2p)

β²-Mod1-NC-1p 0.9353 y = 0.6281x R2 = 0.9952

0.0100 β²-Mod1-WC-2p 0.7736 y = 0.2639x R2 = 0.9697

0.0010

β²-Mod1-NC-2p 0.833 y = 0.3036x R2 = 0.9849

y=x

0.0001 0.0001

0.0010

0.0100

0.1000

1.0000

β²out or β²exp . [/]

B

1.0000

β². [/]

0.1000

β²-Mod2-WC-1p y = 0.4322x 0.8209 R2 = 0.9847

β²-Mod2-WC-1p β²-Mod2-NC-1p β²-Mod2-WC-2p β²-Mod2-NC-2p Puissance ( β²-Mod2-WC-2p) Puissance ( β²-Mod2-NC-2p) Puissance ( β²-Mod2-WC-1p) Puissance ( β²-Mod2-NC-1p)

β²-Mod2-NC-1p 0.8742 y = 0.5153x R2 = 0.9929

0.0100 β²-Mod2-WC-2p 0.7345 y = 0.3119x R2 = 0.9759

0.0010

β²-Mod2-NC-2p 0.7772 y = 0.3668x R2 = 0.9839

y=x

0.0001 0.0001

and 1 and 2 degrees of freedom) are discussed considering experimental results based on mean residence time and reduced variance. Fig. 7 displays the identified mean residence time, s of both models, considering 2 degrees of freedom versus the experimental mean residence time, ts or tsout. The DPF model results in a poor correlation with and without convolution. This point is strengthened under laminar flow. The PF + 2CSTR results in an accurate correlation under treatments with and without convolution, where the experimental and identified mean residence time are almost equal (difference <3%). This observation enables to consider that fitting of second model in assuming 1 degree of freedom will lead to similar results. In Fig. 8, the identified reduced variance, b2 with models no. 1 (A) and no. 2 (B) versus experimental reduced variance, b2out or b2exp are plotted, where convolution and degree of freedom are considered. The DPF model fails to show the difference in reduced variance between ST and MT versus heat power. The analysis of RTD curves, under laminar regime, was inaccurate and inconsistent with DPF model as shown in Fig. 9. The analysis of the Peclet number, PeL versus the dimensionless Reynolds Schmidt product, ReSc and the calculation of the axial dispersion coefficient, Dax is ineffective as DPF model results in a poor fit with the mean residence time, as seen in Fig. 6. The difference between mean holding time, s and mean residence time of the DPF model, ts were not explained by the Peclet number, PeL as it often exceeded the value of 50. The PF + 2CSTR model and experimental data analysis resulted in a close fit for mean residence time and variance. This model clearly distinguishes the reduced variance, b2 in ST and MT versus heating power, in term of distribution. Experimental data comparison with 1 or 2 degrees of freedom, with model estimation of mean residence time result in a similar fit. Convolution needs to be considered under the turbulent flow regime (lowest reduced variance) even through the inlet signal may theoretically neglected (tsin/tsout < 3%). The PF + 2CSTR model, in comparison with DPF model, is more accurate in describing RTD. Dispersion in JEH under investigated heat and flow conditions where a simplified model with only 1 degree of freedom may be used. Data treatment with convolution leads to a better estimation of reduced variance but may be neglected if inlet signal does not exceed 1% of mean residence time. 5. Conclusion

0.0010

0.0100

0.1000

1.0000

β²out or β²exp . [/] 2

Fig. 8. Identified reduced variance, b with models no. 1 (A) and no. 2 (B) versus experimental reduced variance, b2out or b2exp (WC: with convolution, NC: none convolution, 1P or 2P: 1 or 2 degrees of freedom).

The experimental evaluation of the residence time distribution in a tubular Joule effect heater equipped with smooth and modified tube has allowed quantifying treatment homogeneity versus global energetic performances of heat exchanger. Analytic solutions and

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L. Fillaudeau et al. / Journal of Food Engineering 95 (2009) 489–498

systemic analysis of RTD signals are reported and discussed. The evolutions of mean reduced variance, b2 against efficiency number, Eff with smooth and modified tubes exhibit a similar and linear relationship. Treatment homogeneity is significantly improved by modified geometry and strong interactions between heat transfer and hydrodynamics but geometrical modification and heat treatment affect the residence time distribution and specifically reduced variance within same order of magnitude. The obtained results confirm that simple DPF model is not adapted to the small Reynolds numbers. The proposed reactor model (PF + 2CSTR) is closely correlated to the physical structure of the process and is able to predict RTD in JEH with an accurate degree of confidence within investigated conditions (38 < Re < 10,000, 4 < Pr < 950 with Newtonian fluids). A simplified model with only 1 degree of freedom can be used. Convolution will lead to a better estimation of reduced variance but could be neglected if inlet signal does not exceed 1% of mean residence time. In consequence, the knowledge of operating conditions (flow and heat transfer) enable to evaluate the mean residence time, the efficiency number whose reduced variance and plug flow contribution may be deduced. These results are encouraging for the use of accurate cooking, pasteurisation or sterilisation values in adjusting time–temperature treatment without over-evaluated security factor for residence time and associated dispersion.

Appendix B The distribution of these times is called the RTD function of the fluid E, or E-curve, and represents the fraction of fluid leaving the system at each time (Villermaux, 1993). We define x(t) and y(t): experimental inlet and outlet normalised signals. The mathematical relation between x(t) and y(t) is described by Eq. (16). This product of convolution can be replaced in Laplace domain by a simple product and X(s), Y(s) and G(s) are the Laplace transforms of x(t), y(t) and E(t), Eq. (17).

yðtÞ ¼

Z

t

EðuÞ  xðt  uÞ  du

ð16Þ

0

Y ðsÞ ¼ GðsÞ  X ðsÞ

ð17Þ

Each distribution function can be characterized by a set of moments and centred moments of order j (Villermaux, 1993; Ham and Platzer, 2004). The function E(t) is characterized by the mean residence time, tS (Eq. (18)) which is moment of first order, C1. Mean holding time, s is calculated as the ratio between the volume of the corresponding test section and the flow rate (Eq. (19)). For such experimental set-up, tS and s are equal. If not, experimental results must be rejected. It is an indication that a channelling in the fluid circuit has occurred.

Z

C1 ¼ t S ¼

1

t  EðtÞ  dt

ð18Þ

0

V Q

Appendix A

s¼

Da is the Darcy number and Re is the classical Reynolds numbers defined as follows:

Variance (r2) and reduced variance (b2 corresponding to the 0 centred moments of second order, C2 are defined by Eqs. (20) and (21):

Da ¼

2  d h DP 

ð11Þ

q  U2 L q  U  dh Re ¼ l

C2 ¼ r 2 ¼ b2 ¼

8n Re

ð13Þ

Z

1

ðt  t S Þ2  EðtÞ  dt ¼

0

ð12Þ

The friction curve is the representation of Da against Re. The relationship between the friction factor and the Reynolds numbers for laminar isothermal flow of Newtonian fluids in cylindrical ducts is given by Eq. (13). The parameter, n which is the product of the Reynolds number and Darcy stand as the geometrical parameter. It may be theoretical (simple geometries, e.g., circular ducts n = 8, infinite parallel plate n = 12, square duct n = 7.113), semi-theoretical or experimental (Churchill, 1977).

Dalam ¼

ð19Þ

Z 0

1

t 2  EðtÞ  dt  t 2S

r2

ð20Þ ð21Þ

t 2s 0

The centred moment of third order, C3 estimates the skewness which is the deviation from a symmetrical distribution (s = 0). A left-skew distribution exists for s < 0 and a right skew for s > 0.

C30 ¼ s ¼

Z

1

ðt  t S Þ3  EðtÞ  dt

ð22Þ

0 0

The centred moment of fourth order, C4 evaluates the spreading of the RTD. 0

C4 ¼

Z

1

ðt  t S Þ4  EðtÞ  dt

ð23Þ

0

For transition and turbulent flow regime, numerous semiempirical correlations could be used (e.g. Blassius) with the following expression (a, b, c and d: constant coefficient):

Daturb ¼ a  Reb

and Datrans ¼ c  Red

ð14Þ

The friction curve could be described using a unique expression based on Churchill’s model (Eq. (15)). We can use this expression for a laminar, transitory or a turbulent flow regime (Churchill, 1977). This expression is based on the sum of the three regimes contributions as follows:

Da ¼

 1=n2 n2=n1 n1 ðDaÞn1 þ ðDaÞn2 tur þ ðDaÞtran lam

ð15Þ

Critical Reynolds numbers, Rec1 and Rec2, transition from laminar to transitory regimes and from transitory to turbulent regimes, were identified when difference between experimental and modelled Darcy numbers exceeds 10%.

In theory, all moments and centred moments of order j can be calculated. However, the quality of signals often leads to non sig0 0 nificant or reliable results for C3 and C4 . In terms of the type of flow encountered in continuous reactor, there are two limiting cases, the ideal plug flow reactor (PFR) and the continuous stirred tank reactor (CSTR). In the PFR, the velocity profile at a given cross section is flat without axial mixing of fluid elements, which therefore have the same residence time. In the CSTR, the fluid is perfectly stirred, which leads to a broad RTD. The flow pattern in a real reactor does not conform to the PFR or CSTR but in some cases, it even converges (Roetzel and Balzereit, 2000). To consider the deviation from ideal flow behaviour, simple models such as cascade of N continuous stirred tank reactors (CSTR) or a dispersed plug flow (DPF) have been extensively used to model RTD experiments in a large variety of flow system (Castelain et al., 1997; Pinheiro Torres and Oliviera, 1998,,; Roetzel and Balzereit, 2000; Tang and Jess, 2004).

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This model considers mass transport in the axial direction as a diffusion-like process (analogous to Fick’s diffusion law) in terms of an effective longitudinal diffusivity, Dax (axial dispersion coefficient), which is superimposed to the plug flow. The concentration C of the tracer injected uniformly over the cross section at the inlet of the system is given as a function of time, t and streamwise coordinate, z by Eq. (24). The axial dispersion coefficient is assumed to be independent of the concentration and the position. The first assumption is usually verified if molecular diffusion can be neglected and the second one can be used when a fully developed flow is obtained at the inlet and outlet of the process.

@C @2C @C ¼ Dax  2  U  @t @z @z

ð24Þ

The transfer function, G of the plug flow with axial dispersion model, when dispersion extends before the inlet detector and after the outlet detector, is easily found from Eq. (24). This choice for the boundary conditions (called ‘‘open–open”) leads to the following expression (Eq. (25)).

1 EðtÞ ¼  2 With PeL ¼

tS

s



PeL pst

¼1þ

LU Dax

1=2

PeL  ðs  t Þ2  exp  4st

2 r2 2 8 ; b2 ¼ 2 ¼ þ PeL PeL Pe2L ts

! ð25Þ ð26Þ ð27Þ

PeL is calculated with the pipe length and is a dimensionless measure of the axial dispersion, Dax. The DPF model was supposed to yield the best agreement between numerical and experimental results. Furthermore, it has the advantage of requiring the estimation of only one parameter (Eq. (20)), the Peclet number (PeL). References Abid, C., 1994. Etude des instabilités thermiques dans un écoulement ouvert horizontal, soumis à un phénomène de convection mixte. Ann. Phys. Fr. 19, 739–741. Abid, C., Panini, F., Ropke, A., Veyret, D., 1994. Etude de la convection mixte dans un conduit cylindrique. Approches analytique/numérique et détermination expérimentales de la température de paroi par thermographie infrarouge. Int. J. Heat Mass Transfer 37 (1), 91–101. André, C., Boissier, B., Fillaudeau, L., 2007. Residence time distribution in tubular Joule effect heaters with and without geometric modifications. Chem. Eng. Technol 30 (1), 33–40. Castelain, C., Mokrani, A., Legentilhomme, P., Peerhossaini, H., 1997. Residence time distribution in twisted pipe flows: helically coiled system and chaotic system. Exp. Fluids 22, 359–368. Castelain, C., Legentilhomme, P., 2006. Residence time distribution of a purely viscous non-Newtonian fluid in helically coiled or spatially chaotic flows. Chem. Eng. J. 120, 181–191. Chagny, C., Castelain, C., Peerhossain, H., 2000. Chaotic heat transfer for heat exchanger design and comparison with a regular regime for a large range of Reynolds numbers. Appl. Therm. Eng. 20, 1615–1648. Churchill, S.W., 1977. Friction-factor equation spans all fluid-flow regimes. Chem. Eng., 91–92.

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