ces

ira

har

83-8

idad

; acc24

processes. Thus, the annular space is an important geomet-ric conguration in the design of many uid-ow and heat

David and Filip (1996) derived algebraic equations forthe pressure drop predictions. Kozicki et al. (1966)extended the RabinowitschMooney equation for Power-Law, Bingham and Rabinowitsch uids in ducts of arbi-trary cross section (circular, slit, concentrically annular,rectangular, elliptical and isosceles triangular ducts); they

* Corresponding author. Fax: +55 17 32212299.E-mail address: javier@ibilce.unesp.br (J. Telis-Romero).

Journal of Food Engineering 781. Introduction

Laminar ow of non-Newtonian uids in annuli hasmany industrial applications such as the drilling of oil wellsand the extrusion of molten plastic and polymer solutions.Annular geometries are also found in industries leadingwith the transport of slurries and suspensions such as pro-cessed foodstu, sewage and other industrial waste, syn-thetic bers and even blood. Of particular interest for thefood industry is the ow of non-Newtonian liquids in con-centric annular ducts used as regenerators in pasteurization

transfer devices, such as the cored-cylindrical extruders andthe double-pipe and triple-pipe heat exchangers.

Fredrickson and Bird (1958a) were perhaps one of therst to theoretically analyze the isothermal and laminarfully developed ow of an incompressible, inelasticPower-Law uid in concentric annuli. In a subsequentpaper, Fredrickson and Bird (1958b) showed how their pre-vious results could be interpreted in terms of the frictionfactors, usually used in engineering process design. Hanksand Larsen (1979) simplied the solution of Fredricksonand Bird (1958a), while Prasanth and Shenoy (1992) andAbstract

Laminar axial ow of a pseudoplastic uid food (soursop juice) in annular ducts has been experimentally investigated. In the rst partof the manuscript, the rheological behavior of soursop juice, being essential for the annular ow analysis, was completely determinedfrom 9.3 to 49.4 Brix and temperatures from 0.4 C to 68.8 C, using a rotational rheometer equipped with coaxial cylinders. In orderto test the adequacy of the rheology results, pressure loss data in the laminar pipe ow were collected and then experimental and the-oretical friction factors were compared, showing excellent agreement, which indicated the reliability of the Power-Law model for describ-ing the soursop juices. In the second part, pressure loss in annular regions was measured and used to estimate friction factors, which werethen compared to those resulted from analytical and semi-analytical equations. The principal contributions of this article are to provide areview on the determination of friction factors-Reynolds number of pseudoplastic uids in annuli, and also supply extensive new exper-imental data on the rheological properties and pressure loss of an important shear-thinning uid food, which is of particular interest forthe food engineering process design. 2006 Elsevier Ltd. All rights reserved.

Keywords: Soursop juice; Rheology; Power-Law; Friction factor; Laminar ow; Concentric annuliLaminar ow of soursop juiFriction factor

A.C.A. Gratao a, V. Silvea Departamento de Engenharia de Alimentos, Faculdade de Engen

Campinas, 130b Departamento de Engenharia e Tecnologia de Alimentos, Univers

Received 29 July 2005Available online0260-8774/$ - see front matter 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.jfoodeng.2006.01.006through concentric annuli:and rheology

Jr. a, J. Telis-Romero b,*

ia de Alimentos, Universidade Estadual de Campinas, C.P. 6121,

62 SP, Brazil

e Estadual Paulista, Sao Jose do Rio Preto, 15054-000 SP, Brazil

epted 4 January 2006February 2006

www.elsevier.com/locate/jfoodeng

(2007) 13431354

oodNomenclature

a geometric constant dened by Kozicki, Chou,and Tiu (1966) (dimensionless), Eq. (11)

dvzdr local shear rate (s

1), Eq. (1)D tube diameter (m)Dh equivalent diameter (m)Ea ow activation energy (J mol

1), Eq. (20)f Fanning friction factor (dimensionless)fexp friction factor obtained from experimental

pressure loss data (dimensionless)ftheo Fanning friction factor obtained theoretically

(dimensionless)K consistency index (Pa sn), Eq. (1)L tube length (m)n ow behavior index (dimensionless), Eq. (1)n average ow behavior index (dimensionless)r radial coordinate in cylindrical systems (m)r* dimensionless radial coordinate (r/R)R internal radius of the outer cylinder for two

coaxial cylinders (m)Ri external radius of the inner cylinder for two

1344 A.C.A. Gratao et al. / Journal of Falso presented a generalization of the Fanning friction fac-tor-Reynolds number based on a numerical determinationof the velocity proles, which requires the determination oftwo geometric parameters. Tuoc and Mcgiven (1994) alsoproposed a new theoretical derivation for the friction fac-tor applicable to all time-independent non-Newtonian u-ids, which is identical to the expression proposed byDelplace and Leuliet (1995) for cylindrical ducts of arbi-trary cross-section (concentric annuli, isosceles triangularand elliptical ducts). Vaughn and Bergman (1966), Russeland Christiansen (1974) and Ilicali and Engez (1996) pro-posed similar design methods to predict pressure loss infully developed annular ow of Power-Law uids frompipe ow data, while the laminar entrance region was the-oretically studied by Tiu and Bhattacharyya (1973), Nouar,Ouldrouis, Salem, and Legrand (1995) and Maia andGasparetto (2003).

In the last 10 years, purely numerical procedures havebeen used to solve the complete problem of uid owand heat transfer of non-Newtonian uids in concentricand eccentric annular geometries with and withoutcentre-body rotation (Escudier, Oliveira, & Pinho, 2002;Escudier, Oliveira, Pinho, & Smith, 2002; Fang, Manglik,& Jog, 1999; Kaneda, Yu, Ozoe, & Churchill, 2003; Mang-lik & Fang, 2002; Nouar, Benaouda-Zouaoui, & Desaubry,

coaxial cylinders (m)Rh hydraulic radius (m)Reg generalized Reynolds number (dimensionless),

Eq. (7)Remr MetznerReed Reynolds number (dimension-

less), Eq. (4)RemrRh Reynolds number based on the mean hydraulicradius (dimensionless)

Ren,j Reynolds number dened by Fredrickson andBird (1958b) (dimensionless), Eq. (14)

T temperature (C or K)vz local axial ow velocity (m s

1)vz average axial ow velocity (m s

1)xs content of soluble solids (Brix)

Greek lettersDP pressure drop (Pa)/ function dened in Eq. (8) (dimensionless)c function dened by Fredrickson and Bird

(1958a), Eq. (16)g0 model parameter (Pa s

n), Eq. (20)j annulus aspect ratio (dimensionless)k location of maximum uid velocity in an

annulus (dimensionless), Eq. (17)q uid density (kg m3)srz local shear stress (Pa), Eq. (1)

Engineering 78 (2007) 134313542000; Soares, Naccache, & Mendes, 2003; Viana, Nasci-mento, Quaresma, & Macedo, 2001), which is very interest-ing from a theoretical point of view, but not easilyapplicable. On the other hand, few studies report experi-mental pressure loss and friction factors-Reynolds numberdata (Escudier, Gouldson, & Jones, 1995; Ilicali & Engez,1996; Tuoc & Mcgiven, 1994; Vaughn, 1963; Vaughn &Bergman, 1966). Also, since the early experimental workof Tiu and Bhattacharyya (1974), extensive attention hasbeen undertaken to determine experimental velocity pro-les (Escudier et al., 1995; Nouri, Umur, & Whitelaw,1993; Nouri & Whitelaw, 1994).

The soursop juice was chosen as the uid test in thismanuscript due to its rheological behavior (shear-thinninguid) and its important potential for the international mar-ket. Soursop (Annona muricata L.), known as graviola inBrazil and guanabana in Mexico, is a popular fruit tree cul-tivated throughout the tropical regions of the world andprized for its very pleasant, sub-acid, aromatic, juicy eshand distinctive avor. The soursop pulp is widely usedfor manufacturing various juice blends, nectars, syrups,shakes, jams, jellies, preserves and ice creams (Ledo,1996; Umme, Salmah, Jamilah, & Asbi, 1999); it is also araw material for powders, fruit bars and akes (Umme,Bambang, Salmah, & Jamilah, 2001). Like most tropical

t parameter dened in Eq. (8) (dimensionless)n geometric parameter (dimensionless), Eq. (9)Xp function of the volume rate of ow dened by

Fredrickson and Bird (1958a), Eq. (15)

oodfruits, the soursop has a great potential for exportation andit is able to compete in the international market, either aspuree, juice or as mixtures with other juices (Jaramillo-Flo-res & Hernandez-Sanchez, 2000). Its pulp has been alreadyfound in European, North American and Brazilianmarkets.

The rheological parameters of soursop juice are essentialsubsidies for the evaluation of friction factors. Actually,the determination of rheology is totally concerned withthe modeling and optimization of the unit processes. In thefood industry, juices are the most important liquid deriva-tives of fruits and their ow characteristics have beenan extensive topic of study in the last years (Cepeda & Vil-lara`n, 1999; Giner, Ibarz, Garza, & Xhian-Quan, 1996;Ibarz, Garvin, & Costa, 1996). However, to the authorsknowledge, no studies on the rheological behavior of sour-sop juice have been reported. Therefore, the design of unitoperations, such as pumping, mixing, heat exchange andevaporation are hindered due to the inexistence of simplemathematical equations that express the temperature andconcentration dependence of the rheological parameters.Many uids of interest in industrial practice are non-New-tonian in nature and dierent empirical equations are avail-able for the description of their rheological behavior, suchas the HerschelBulkley model (Barbosa Canovas & Peleg,1983; Saenz & Costell, 1986; Saravacos & Kostaropoulos,1995) and the Ostwaldde Waele model, also known asthe Power-Law, which is one of the most extensively usedto describe the rheological behavior of fruit juices (Rao,Cooley, & Vitali, 1984). The two-parameter Power-Lawmodel is often sucient for industrial purposes and pro-vides a reasonable representation of many practicalshear-thinning uids.

According to the facts described previously, the princi-pal aim of this manuscript is to provide extensive newexperimental data on pressure loss and friction factors ofa pseudoplastic uid food owing through a concentricannulus in laminar, fully developed regime; the currentwork also summarizes dierent methodologies to deter-mine the friction factors-Reynolds number. In addition, acomplete rheological characterization of soursop juicewas obtained, and simple equations were established tocorrelate the rheological data under dierent temperature(0.4 C to 68.8 C) and concentration (9.349.4 Brix) con-ditions. Finally, the adequacy of the Power-Law model fordescribing the soursop juices was veried through theexperimental values of pressure loss in the laminar pipeow of soursop juices.

2. Materials and methods

2.1. Raw material: soursop juices

All the experimental measurements were made withsamples prepared from the same batch of concentrated

A.C.A. Gratao et al. / Journal of Fsoursop juice (51.2 Brix and 2.8% pulp content), producedfrom soursop fruit (Annona muricata L.) in a single-stageT.A.S.T.E. evaporator and stored at 17 C. In orderto acquire dierent water contents, the concentrated juicewas diluted with distilled water to obtain juices containing9.3, 19.1, 24.3, 29.4, 34.6, 38.8, 44.6 and 49.4 Brix. Thedensity of soursop juice at dierent temperatures (0.468.8 C) and concentrations (9.349.4 Brix) was obtainedin triplicate by pycnometry (Constenla, Lozano, & Crap-iste, 1989).

2.2. Rheological measurements and ow characterization

Rheological measurements were carried out using aRheotest 2.1 (VEB-MLW Prufgerate-Werk, Germany)Searle type rheometer, equipped with a coaxial cylindersensor system (radii ratio of 1.06). The instrument can beoperated at 44 dierent speeds, which are changed stepwisewith a selector switch. The speed of the rotating cylindervaried from 0.028 to 243 rpm. A thermostatic bath (modelMa-184, Marconi) containing ethyl alcohol was used tocontrol the working temperature within the range 0.468.8 C. Shear stress values at the surface of the internalcylinder were obtained by multiplying torque readings bythe rheometer constant, whereas shear rate values wereevaluated according to Krieger and Elrod (1953). Thewidely known empirical expression for the stress tensor,the Power-Law model, was used for describing the owbehavior of soursop juices. For the Power-Law model thelocal shear stress depends on the local shear rates as follows(Bird, Stewart, & Lightfoot, 2002):

srz K dvzdr

n1

dvzdr

1

where srz (Pa) is the local shear stress, K (Pa sn) is the con-

sistency index, n (dimensionless) is the ow behavior indexand dvz/dr (s

1) is the local shear rate.

2.3. Pressure drop measurements in pipe ow: theapparatus

The apparatus specied in full details by Telis-Romero,Telis, and Yamashita (1999) was used to measure pressureloss during laminar pipe ow of soursop juices. It consistsof a heat transfer, circular section, which is submerged ina large thermostatic bath (model MA-184, MarconiEquipamentos para Laboratorio Ltda., SP, Brazil) contain-ing water at constant temperature. Flow experiments werecarried out during the heating of the samples by the solutionin the thermostatic bath. The equipment was made with twohorizontal steel circular tubes with nominal diameters of 3/4in and 1 1

2in (Schedule 40), connected to a stainless steel

cylindrical tank with a capacity of 270 L. The total lengthof the section was 1.2 m providing a maximum length-to-diameter ratio (L/D) of 54.8. A distance of 1.50 m providedthe developing length of the ow regime for all experimentaltests. Dierential pressure transmitters (model LD-301,

Engineering 78 (2007) 13431354 1345Smar Equipamentos Industriais Ltda., SP, Brazil) con-nected to pipes were used to measure static pressure along

UserNota adhesivaMateria prima Jugos de Guanabana

UserNota adhesivaUnmarked definida por User

UserNota adhesivaFue Obtenido Por Picnometra la densidad, adems que son las mismas muestras preparados en una sola etapa

the end of the test section. The less concentrated soursop

surements in pipe ow, but the circular pipes were replaced

oodjuices were pumped by means of a peripheral pump (modelP-500, KSB Bombas Hidraulicas S.A., SP, Brazil), while themost concentrated used a gear pump (model Triglav, KSBBombas Hidraulicas S.A., SP, Brazil). A static mixer wasplaced at the end of the equipment to homogenize the naltemperature of the juices. A ow meter (model LD100,MLW Prufgerate-Werk, Germany) was used to initiallyadjust the desired ow rate in each experiment, but exactmeasures were obtained by weighing uid samples collectedat determined time intervals. A HP data logger model75.000-B, an interface HP-IB and a HP PC running a dataacquisition and control program written in IBASIC moni-tored temperatures and pressures. The tested samples weresoursop juices containing 9.3, 24.3 and 34.6 Brix. The aver-age ow velocities were varied from 0.05 to 2.50 m s1,totalizing one hundred experimental values of pressure lossfor each sample.

2.4. Evaluation of friction factors in pipe ow

The friction factor for an incompressible uid movingin a straight pipe of uniform cross section may be writtenin terms of pressure loss, as given by Eq. (2). The quantityfexp calculated from experimental data on pressure loss issometimes called the Fanning friction factor (Bird et al.,2002),

fexp DPD2qv2z L

2

in which q (kg m3) is the uid density, vz (m s1) is the

average axial ow velocity, D (m) is the tube diameterand DP (Pa) is the pressure drop observed in a length L(m) of the tube. For the fully developed laminar pipe owof Power-Law uids, the friction factor is given by an anal-ogous expression of the well-known dimensionless form ofthe HagenPoiseuille equation (Darby, 2001):

ftheo 16Remr 3

in which ftheo is the friction factor estimated theoretically.By using the Power-Law model for simple ducts, such asthe circular pipe, it is possible to analytically solve themomentum equation and to obtain the generalizedReynolds number dened by Metzner and Reed (1955):

Remr qv2nz D

n

8n1K

4n

1 3n n

4

2.5. Pressure drop measurements in annular ow: the

apparatusthe equipment. Temperature transducers (model TT-302,Smar Equipamentos Industriais Ltda., SP, Brazil) wereused to measure the temperature at the beginning and at

1346 A.C.A. Gratao et al. / Journal of FThe apparatus used to measure pressure drop in theannular region was practically the same described for mea-in which j is the annulus aspect ratio (Ri/R), Ri is the exter-nal radius of the inner cylinder and R is the internal radiusof the outer cylinder for two coaxial cylinders.

For annular geometries, there is no simple analyticalvelocity prole and a simple theoretical expression for Rey-nolds number cannot be easily found. Thus, one of thecommonest used relations for evaluation of friction factorfor non-Newtonian ow in ducts of arbitrary cross sectionis based on the mean hydraulic radius concept (Birdet al., 2002). In that case, the friction factor is obtainedfrom the expressions for laminar ow of non-Newtonianuids in cylindrical pipes, given by Eqs. (3) and (4), usingthe hydraulic radius (4Rh) or the equivalent diameter(Dh) instead of the pipe diameter (D). In this work, theRemr will be called RemrRh when the mean hydraulic radiusconcept is used.

There have been a few attempts to develop correlationsfor predicting the isothermal friction factors for variousnon-Newtonian uid ows in non-circular ducts. Thesemi-theoretical expressions proposed by Kozicki et al.(1966), Tuoc and Mcgiven (1994) and Delplace and Leuliet(1995) were used to generalize the Reynolds number.According to these authors, the Fanning friction factorcan be calculated through:

2nby two sections of horizontal steel coaxial cylinders withdierent annulus aspect ratios. The external diameter ofthe inner cylinder was xed at 0.0140 m, while the internaldiameter of the outer cylinder was 0.0381 m and 0.0590 m,thus providing the respective values of 0.0060 m and0.0113 m for the hydraulic radius, and the values of0.3675 and 0.2373 for the annulus aspect ratio (j). Thetotal length of the test section was 1.2 m. According toTiu and Bhattacharyya (1974), a distance of 0.15 m is suf-cient for the development of the ow regime; hence, theannular ducts were extended at each end by 0.15 m. Theexperiments were conducted during the heating of the sam-ples by owing water in the internal cylinder between33.8 C and 52.4 C. The water was pumped by a centrifu-gal pump (model C-1010, KSB Bombas Hidraulicas S.A.,SP, Brazil). The wall surface of the outer cylinder was insu-lated with glass wool to minimize the heat exchange withthe environment.

2.6. Evaluation of friction factors in annular ow

For the annulus, the fexp is dened as given by Eq. (5),which is identical to Eq. (2), with D replaced by Dh (Birdet al., 2002):

fexp DPR1 j2

qv2z L1 j DPDh2qv2z L

5Engineering 78 (2007) 13431354ftheo Reg 6

soodwhere Reg is the generalized Reynolds number, which iswritten in the same form as the Remr dened by Eq. (4)for circular ducts:

Reg qv2nz D

nh

K/nnnjn1 7

The function /(n) is described by the hyperbolic formgiven by Eq. (8), and the geometrical parameter n(j), givenby Eq. (9), is the product of the friction factor and the Rey-nolds number for a Newtonian uid under laminar owconditions in concentric annuli, which is easily obtainedtheoretically.

/n tn 1t 1n 8

nj 81 j2

1j2ln j 1 j2

9

For the ow of Power-Law uids in concentricallyannular ducts, the relations derived by Kozicki et al.(1966) are given by Eqs. (10) and (11), which are expressedin terms of two parametric constants (a and n), character-istics of the shape of the ow cross section. The constanta(j) is obtained in a similar fashion to n(j), by normalizingthe wall shear stress for laminar Newtonian ows in anannular duct to the wall shear stress in a circular tube ofthe same hydraulic diameter. For Tuoc and Mcgiven(1994) and Delplace and Leuliet (1995), the function t isgiven by Eq. (12). One notes that for the circular cross sec-tion a = 1/4, and the functions /(n) = (3n + 1)/4n, t = 3and n = 8.

t n8a

1 10

a 1 j2

4 1 1j22 ln1=j 1 ln 1j

2

2 ln1=j

h in o 11t 24

n12

Fredrickson and Bird (1958b) suggested an expressionfor the Reynolds number that requires a relatively simplenumerical integration. They derived from the momentumequation an expression for the friction factor of a pseudo-plastic uid in annuli, valid for axial ow in annulus oflength L formed by cylindrical surfaces at r = jR andr = R:

ftheo 16Ren;j 13

Ren;j qv2nz D

n

2n3K

1 j1 j2n1" #

Xpn 14

The functions Xp(n,j), c(n,j) and k(n,j) were denedaccording to Eqs. (15)(17), respectively. Fredrickson andBird (1958b) tabulated these functions for 0.1 6 n 6 10

A.C.A. Gratao et al. / Journal of Fand 0.01 6 j 6 0.9.3. Results and discussion

3.1. Density of soursop juice

The tting of Eq. (19) to the experimental soursop juicedensity was satisfactory (R2 = 0.999, RMS = 0.2%, v2 = 3and SSR = 147). The developed model was parameterizedwith a sub-set of the total experimental data, and theremaining data were used to conrm the reliability ofEq. (19). The absolute relative errors between predicteddensity and the remaining experimental data were calcu-lated and presented a maximum value of 4.2%, with aminimum of 0.1% and an average of 2.2%. The RMS valuewas 2.5%.

q 981:4 4:5xs 0:23T 19

in which x (Brix) is the content of soluble solids and T2.7. Calculus and data analysis

The numerical integrations were performed by adaptiveLobatto quadrature using the software MatLab 6.1 (TheMathWorks Inc., 2001). All tted functions wereperformed using Nonlinear Estimation Procedures fromMatLab 6.1 (The MathWorks Inc., 2001), which solvesnonlinear curve tting problems in the least square sense.The adequacy of the tted functions was evaluated by thecorrelation coecient (R2), the magnitude of the root meansquare (RMS) calculated according to Gabas, Menegalli,Ferrari, and Telis-Romero (2002), the chi-square valuesof t (v2) and the sum of squares of the dierence betweendata and t values (SSR). The performance of the frictionfactor models was assessed by the RMS values and theabsolute relative errors between observed and predictedvalues, as given by Telis-Romero, Cabral, Gabas, and Telis(2001).Xpn; j Z 1jjk2 r2j1=n1r1=n dr 15

cn; j 1=n 21 j1=n2

Xpn; j 16Z kj

k2

r r

1=ndr

Z 1k

r k2

r

1=ndr 0 17

in which k (n,j) represents the local of maximum uidvelocity in the annular region. Engez (1995) (cited by Ilicali& Engez, 1996) used the charts supplied by Fredricksonand Bird (1958a) to t an expression to the c (n,j) data, va-lid for jP 0.2, which is given by Eq. (18).

cn; j 0:5017 0:0166n

j 0:497 0:0219n

18

Engineering 78 (2007) 13431354 1347(C) is the temperature.

3.2. Flow behavior of soursop juices

Rheograms of soursop juice were obtained in the rangeof shear rates from 0.7 s1 to 529.5 s1. In the testedranges the samples behaved as pseudoplastic uids, andthe Power-Law model was satisfactorily tted to the exper-imental data, with 0.991 6 R2 6 0.999 and 0.76 6RMS(%) 6 2.50. The HerschelBulkley model was also t-ted to the experimental data and provided good statisticalresults (R2 0.999 and RMS 1.00%) since it is a three-parameter model, though the yield stress values were nega-tive, which is meaningless from a physical standpoint.Polizelli, Menegalli, Telis, and Telis-Romero (2003) alsoobtained negative yield stress values in the rheologicalcharacterization of aqueous solutions of sucrose and xan-than gum. The experimental shear rate and shear stressfor the soursop juices having 29.4 Brix are presented inFig. 1; similar rheograms at the same temperatures wereobtained for the other samples. The shear rates were nearly

1348 A.C.A. Gratao et al. / Journal of Foodconstant as the temperature was increased from 0.4 to68.8 C.

The tting of Eq. (1) to the experimental data permittedthe evaluation of K and n, which are presented in Table 1.The Power-Law is a very simple empirical model exten-sively used for engineering calculations due to its simplicityof having only two parameters. It has been used fordescribing many liquid foods, such as apple, peach andpear purees (Saravacos, 1970), orange juice concentrate(Crandall, Chen, & Carter, 1982), guava puree (Vitali &Rao, 1982), low-pulp concentrated orange juice (Vitali &Rao, 1984), lemon juice (Saenz & Costell, 1986), concen-trated raspberry juice (Ibarz & Paga`n, 1987), aqueous solu-tion of carboxymethylcellulose (CMC) (Nouar et al., 2000),sucrose-CMC model solution (Berto, Gratao, Silveira, &Vitali, 2003) and many others.

1 10 100

100

ln

rz (P

a)

ln dvz /dr (s-1)

Fig. 1. Rheograms of soursop juice having 29.4 Brix at several temper-atures. Experimental values: (j) 0.4 C, (s) 8.3 C, (m) 19.7 C, (+) 28.4

C, (h) 37.8 C, (d) 48.2 C, (n) 59.2 C and ( ) 68.8 C. Predictedvalues: () Power-Law model, Eq. (1).3.3. Eect of temperature and concentration on the

rheological parameters

The temperature eect on the consistency index wasgiven by an Arrhenius-type relationship, according to Eq.(20). As expected, K exponentially decreased with the risein temperature, being the reduction more accented for theconcentrated samples.

K g0 expEa

8:314T

20

where g0 is a model parameter (Pa sn), Ea is the ow activa-

tion energy (J mol1), 8.314 J mol1 K1 represents theideal gas constant and T is the absolute temperature (K).In Table 2, g0 and Ea are presented together with the resultsof R2, SSR, RMS (%) and v2. The activation energy was al-most constant for all the tested samplesits mean valuewas 11.39 with a standard deviation of 0.40while g0 in-creased with the increase in the content of soluble solids.

The rheological properties of most of the liquid foodsexhibit substantial changes during the processing stagesbecause of their dependence upon temperature and concen-tration. That is the case of orange juice (Crandall et al.,1982; Rao et al., 1984; Telis-Romero et al., 1999), fruitspurees (Guerrero & Alzamora, 1998), claried fruit juices(Ibarz, Gonzalez, & Esplugas, 1994; Ibarz, Gonzalez,Esplugas, & Vicente, 1992), concentrated milk (Velez-Ruiz& Barbosa-Canovas, 1998) and coee extract (Telis-Romero et al., 2001). In general the viscosity of liquidsdecreases with increase in temperature and a measure ofthe temperature inuence on the rheological parametersis usually obtained from the Arrhenius type equation(Berto et al., 2003; Ibarz et al., 1996; Kaya, 2001; Sarava-cos, 1970), while the concentration eect is generally writ-ten in terms of power-type or exponential relations (Vitali& Rao, 1982).

For the description of the ow behavior index of sour-sop juices as a function of temperature, the linear, power-type and exponential models were tested, but they didnot provide good ttings. For each concentration, a smalluctuation of the ow behavior index values was observedwhen the temperature raised from 0.4 C to 68.8 C; asexpected, the ow behavior index slightly increased withthe rise in temperature. In Fig. 2, the ow behavior indexis presented as a function of temperature for several con-centrations, along with the predictions by the linear model,which displayed the best statistical results.

Adorno (1997) obtained a similar uctuant behavior forn in the rheological characterization of various Braziliantropical fruit juices (mango, passion fruit, papaya and pinkguava). Furthermore, Adorno (1997) did not nd goodmodels to describe n as a function of temperature and con-centration. For fruit juices, the ow behavior index usuallydecreases with the increase in concentration of soluble sol-ids; also, n slightly increases as the temperature rises. Gen-

Engineering 78 (2007) 13431354erally, these variations are quite small and that is why someauthors adopt n as a constant (Vitali, 1981). Vitali and Rao

Table 1Rheological properties (K and n) of soursop juices

Temperature (C) Content of soluble solids, xs (Brix)

9.3 19.1 24.3 29.4 34.6 39.8 44.6 49.4

Consistency index, K (Pa sn)

0.4 2.29 19.25 44.93 65.41 115.86 176.83 258.82 335.938.3 1.99 17.75 35.56 59.49 96.99 145.45 216.63 290.9819.7 1.58 14.36 28.93 48.52 86.60 124.35 185.33 241.4628.4 1.45 11.82 28.20 42.38 69.91 106.02 158.47 203.1137.8 1.19 10.43 23.47 35.59 64.72 96.98 141.41 182.8648.2 1.11 8.88 19.32 34.20 58.28 78.42 119.20 157.9559.2 0.94 7.74 17.43 28.568.8 0.81 6.94 15.96 23.8

Flow behavior index, n (Dimensionless)

0.4 0.413 0.308 0.299 0.28.3 0.402 0.337 0.306 0.219.7 0.422 0.323 0.295 0.328.4 0.417 0.325 0.315 0.237.8 0.408 0.325 0.303 0.348.2 0.423 0.338 0.320 0.3

0.30.3

A.C.A. Gratao et al. / Journal of Food Engineering 78 (2007) 13431354 134959.2 0.444 0.331 0.31368.8 0.435 0.341 0.326(1982) observed nearly constant values of n for their entiresamples (guava puree from 9.8 to 16 Brix) over a temper-

Table 2Model parameters (g0 and Ea) and statistical results (R

2, SSR, RMS and v2) o

xs (Brix) g0 (Pa sn) Ea (kJ mol1) R2

9.3 0.014 11.59 0.919.1 0.102 11.98 0.924.3 0.238 11.83 0.929.4 0.522 11.02 0.934.6 0.972 10.84 0.938.9 1.081 11.54 0.944.6 1.876 11.17 0.949.4 2.447 11.18 0.9

270 280 290 300 310 320 330 340 3500.24

0.26

0.28

0.30

0.32

0.34

0.36

0.38

0.40

0.42

0.44

0.46

Flow

beh

avio

r ind

ex

Temperature (K)Fig. 2. Flow behavior index of soursop juices as a function of temperaturefor various concentration. Experimental values: (j) 9.3 Brix, (s)19.1 Brix, (m) 24.3 Brix, ( ) 29.4 Brix, (h) 34.6 Brix, (d) 38.9 Brix,(n) 44.6 Brix and (+) 49.4 Brix. Predicted values: () Linear model.6 49.83 70.99 111.75 145.097 42.91 65.00 96.41 129.35

84 0.270 0.271 0.264 0.26285 0.288 0.269 0.266 0.26200 0.287 0.279 0.273 0.26698 0.281 0.277 0.284 0.27410 0.291 0.303 0.273 0.28215 0.301 0.287 0.278 0.28309 0.298 0.314 0.299 0.28914 0.322 0.304 0.294 0.288ature range of 2560 C; its mean value was 0.430 with astandard deviation of 0.031. In the present manuscript, ncan also be considered almost constant as the temperaturerises from 0.4 C to 68.8 C (Fig. 2). The mean values of theow behavior index n are presented in Table 3, alongwith the standard deviations (sd) for each studiedconcentration.

Finally, the concentration eect on the consistency andow behavior indices was satisfactorily given by power-type models. The exponential models were also tested

btained from the tting of Eq. (20) to the consistency index data

SSR RMS (%) v2

96 0.01 1.1 0.093 1.02 1.0 0.282 12.55 1.9 7.093 11.12 1.5 1.791 39.41 1.4 6.895 57.96 1.2 10.696 92.14 1.0 15.496 132.29 1.1 22.1

Table 3Mean values of the ow behavior index n and standard deviations (sd)for each studied concentration of soursop juices

xs n sd

9.3 0.421 0.01419.1 0.329 0.01124.3 0.310 0.01029.4 0.302 0.01234.6 0.292 0.01538.9 0.288 0.01744.6 0.279 0.01249.4 0.276 0.011

by Telis-Romero et al. (2001). In addition, these results val-idate the experimental apparatus for pressure loss measure-ments. On the other hand, we may say that the alteration ofthe velocity prole due to the sample heating, and hencethe decrease of the consistency index, was not so expres-sive, because the theoretical friction factor analyses arebased on the assumption of isothermal ow, and the exper-iments were conducted during the heating of the samples.The bulk inlet temperatures were about 19.9 0.5 C,which resulted in bulk outlet temperatures of 30.3 3.4 C. These variations of approximately 10 C betweenthe inlet and outlet resulted in a decrease of the consistencyindex of about 15% during the experiments.

3.5. Friction factors in annular ow

The pressure loss measurements in the laminar annularow of soursop juices permitted the evaluation ofexperimental friction factors (fexp) according to Eq. (5).The ranges attained in the experiments were: 0.01 6 fexp 6

oodand provided good statistical results, but not the bestones. As expected, the consistency coecient decreased,and the ow behavior index slightly increased as the con-centration diminished. In fact, for engineering processdesign, it is very useful to obtain a simple correlationdescribing the combined eect of temperature and concen-tration. Therefore, a single equation was established todescribe the rheological parameters (K and n) of soursopjuices as functions simultaneously dependent on tempera-ture and concentration.

K 3 105 exp 11; 203RT

xs 2:915 21

n 0:407 1:1 1003T xs0:260 22

The tting of Eq. (21) to the consistency coecientdata presented the following statistical results: R2 =0.998, RMS = 13.0%, SSR = 786 and v2 = 12.9, whilethe tting of Eq. (22) to the ow behavior data pre-sented R2 = 0.938, RMS = 3.6%, SSR = 8.4 103 andv2 = 1.4 104. In Eq. (21), the correspondent termto the temperature eect (Ea = 11.20 kJ mol

1) is similarto that of the Arrhenius type equation. The higher theow activation energy is, the higher the temperature eectis.

For the evaluation of friction factors and Reynoldsnumbers in the following sections, Eq. (19) was used fordetermining the soursop juice density, Eq. (20) was usedfor determining the consistency index, together with theparameters presented in Table 2, and Eq. (22) wasemployed for estimating the ow behavior index. Density,consistency and ow behavior indices were evaluated at theaverage temperature between the initial and nal condi-tions attained by the samples during the ow experiments.

3.4. Friction factors in pipe ow

The pressure loss experiments in pipe ow permittedthe evaluation of friction factors according to Eq.(2). The ranges of the experiments were: 0.01 6 fexp 6153, 568 6 DP 6 160,044 (Pa) and 1017.5 6 q 6 1133.9(kg m3). The experimental friction factors calculatedfrom Eq. (2) were compared to those estimated by Eq.(3), with the Reynolds number evaluated according toEq. (4). The results are presented in Fig. 3. The consis-tency index of the samples varied from 1.47 Pa sn to78.32 Pa sn, and the ow behavior index ranged from0.292 to 0.412, while the MetznerReed Reynolds number(Remr) assumed values from 0.11 to 2259, demonstratingthat the experiments were conducted only in the laminarregime. Metzner and Reed (1955) indicate that stable lam-inar ow of time-independent non-Newtonian uidsextends to a value of Remr of 20002500. The maximumabsolute relative errors between the analytical results(ftheo) and the experimental friction factors (fexp) was

1350 A.C.A. Gratao et al. / Journal of F46.6%, with a minimum error of 0.1% and an averageof 7.4%, while the RMS value was 10.4%.The experimental friction factors and Reynolds numbers(Remr) of soursop juices in the laminar pipe ow were alsosubmitted to nonlinear regression analyses, resulting in Eq.(23). The model parameters obtained in Eq. (23) were verysimilar to the theoretical values in Eq. (3), and the ttingresults were very satisfactory, with R2 = 0.997, RMS =10.4%, SSR = 137 and v2 = 0.5.

f 16:2Remr1:0023

The good agreement between friction factors calculatedfrom experimental data on pressure loss and those esti-mated from analytical equations can be taken as an indic-ative of reliability of the adopted rheological model todescribe the ow behavior of soursop juices, as suggested

0.1 1 10 100 1000

0.01

0.1

1

10

100 9.3 Brix 24.3 Brix 34.6 Brix Predicted Values, Eq. (3)

Fric

tion

fact

or in

pip

e flo

w

Remr

Fig. 3. Experimental friction factors calculated by Eq. (2) and predictedvalues by Eq. (3) for the fully developed laminar pipe ow of soursopjuices.

Engineering 78 (2007) 13431354157, 571 6 DP 6 149,073 (Pa) and 1017.7 6 q 6 1131.9(kg m3). The theoretical friction factors (ftheo) were evalu-

ated according to the following methods: (i) the meanhydraulic radius concept, using Eqs. (3) and (4) with Dh;(ii) the generalization of the Fanning friction factor-Reynolds number proposed by Kozicki et al. (1966), inwhich ftheo is evaluated by Eq. (6), and the Reynolds num-

ber is obtained from Eqs. (7)(9), with t and a performedrespectively by Eqs. (10) and (11); (iii) the derivations ofTuoc and Mcgiven (1994) and Delplace and Leuliet(1995), with t estimated from Eq. (12), and nally, (iv)the expressions proposed by Fredrickson and Bird(1958b), where ftheo is evaluated by Eq. (13), with the Rey-nolds number computed from Eq. (14), and the functionsXp(n,j) and k(n,j) obtained from Eqs. (15) and (17),respectively. When evaluating the Reynolds numbers, Kvalues ranged from 1.49 to 73.78 Pa sn, and n values variedfrom 0.294 to 0.412. In Table 4, the RMS values for eachtested methodology are presented, along with the absoluterelative errors between the fexp and the ftheo estimated fromEqs. (3), (6) and (13). The agreement between the experi-ments and the calculations is clearly satisfactory.

The fexp values were also submitted to a nonlinearregression analysis, and the obtained models are given byEqs. (24)(26).

Table 5Pressure loss, friction factors and Reynolds numbers for the fully developed laminar ow of soursop juices in concentric annuli

j mz DP fexp(Eq. (5))

(i) (ii) (iii) (iv)

ftheo(Eq. (3))

RemrRh(Eq. (4))

ftheo(Eq. (6))

Reg(Eq. (7))

ftheo(Eq. (6))

Reg(Eq. (7))

ftheo(Eq. (13))

Ren,j(Eq. (14))

n 0:4920.237 0.5 1501 0.111 0.086 186.8 0.103 225.5 0.103 225.5 0.102 157.1

1.0 1974 0.036 0.028 570.3 0.034 688.2 0.034 688.3 0.033 479.41.5 2330 0.019 0.015 1,064.9 0.018 1,285.5 0.018 1,285.7 0.018 895.62.0 2710 0.012 0.010 1,682.1 0.011 2,030.5 0.011 2,030.8 0.011 1,414.72.5 3003 0.009 0.007 2,375.2 0.008 2,867.7 0.008 2,868.1 0.008 1,997.9

0.367 0.5 3460 0.136 0.112 142.9 0.136 173.9 0.136 173.9 0.135 118.51.0 4533 0.045 0.037 437.3 0.044 531.9 0.044 531.9 0.044 362.71.5 6498 0.028 0.019 825.8 0.024 1,004.8 0.024 1,004.8 0.023 685.02.0 6705 0.017 0.012 1,295.3 0.015 1,576.2 0.015 1,576.3 0.015 1,074.62.5 7109 0.011 0.009 1,855.3 0.010 2,257.4 0.010 2,257.5 0.010 1,539.0

n 0:3100.237 0.5 20,067 1.386 0.981 16.3 1.143

1.0 20,703 0.358 0.301 53.1 0.3511.5 22,654 0.174 0.155 103.0 0.1812.0 25,763 0.111 0.096 166.9 0.1122.5 28,037 0.077 0.067 240.0 0.078

0.367 0.5 33,801 1.250 1.212 13.2 1.4201.0 41,077 0.380 0.372 43.0 0.4361.5 83,873 0.345 0.189 84.5 0.2222.0 64,203 0.148 0.118 135.9 0.1382.5 61,028 0.090 0.081 198.6 0.094

n 0:2920.237 0.5 36,499 2.419 2.195 7.3 2.530

1.0 44,913 0.744 0.662 24.2 0.7631.5 50,254 0.370 0.337 47.5 0.388

85

709

Table 4RMS values and absolute relative errors between fexp obtained from Eq.(5) and ftheo calculated from Eqs. (3), (6) and (13)

Testedmethodologies

% Maximumerror

% Minimumerror

% Averageerror

% RMS

(i) Eqs. (3)and (4) Rhconcept

60.33 0.05 17.53 19.8

(ii) Eqs. (6)(11) 53.50 0.03 7.31 10.3(iii) Eqs. (6)(9)and (12)

53.50 0.03 7.31 10.3

(iv) Eqs. (13)(17) 53.82 0.06 7.41 10.5

A.C.A. Gratao et al. / Journal of Food Engineering 78 (2007) 13431354 13512.0 56,833 0.235 0.206 77.5 0.232.5 60,778 0.161 0.143 112.0 0.16

0.367 0.5 78,804 2.797 2.663 6.0 3.081.0 94,430 0.838 0.802 19.9 0.931.5 129,449 0.511 0.404 39.6 0.46

2.0 128,918 0.286 0.249 64.2 0.2892.5 133,111 0.189 0.169 94.4 0.19720.3 1.143 20.4 1.128 14.266.2 0.351 66.3 0.346 46.2128.5 0.181 128.5 0.179 89.6208.3 0.112 208.3 0.110 145.2299.5 0.078 299.5 0.077 208.8

16.6 1.420 16.6 1.410 11.354.2 0.436 54.2 0.433 37.0106.4 0.222 106.4 0.220 72.6171.2 0.138 171.2 0.137 116.7250.2 0.094 250.2 0.094 170.6

9.2 2.530 9.2 2.496 6.430.5 0.763 30.5 0.753 21.259.9 0.388 59.9 0.383 41.797.8 0.238 97.8 0.235 68.2141.3 0.165 141.3 0.162 98.5

7.7 3.087 7.7 3.066 5.225.4 0.930 25.4 0.924 17.350.4 0.469 50.4 0.465 34.4

81.7 0.289 81.7 0.287 55.7120.2 0.197 120.2 0.195 82.0

f 19:6RemrRh0:97R2 0:993; RMS 18:9%;

SSR 366 and v2 1:2 24f 17:1Ren;j0:97

R2 0:994; RMS 18:5%;

SSR 326 and v2 1:1 25f 2:3nReg0:97

R2 0:994; RMS 18:2%;

SSR 329 and v2 1:1 26The measured DP, the computed experimental friction

factors (Eq. (5)), the theoretical friction factors (Eqs. (3),(6), (13)) and the Reynolds numbers (Eqs. (4), (7), (14))for the fully developed laminar ow of soursop juices inconcentric annuli with j = 0.237, 0.367 and n 0:421(9.3 Brix), 0.310 (24.3 Brix), 0.292 (34.6 Brix) are pre-

1352 A.C.A. Gratao et al. / Journal of Foodsented in Table 5 for some of the experimental ow ratestested. It was observed an increase in the friction factorsand a decrease in the Reynolds numbers by increasingthe annulus aspect ratio. Obviously, in the highly shearthinning ows n 0:292, the Reynolds numbers weresmaller. The experimental results are consistent with previ-ous works for non-Newtonian uid ow in annulargeometries.

In Fig. 4, the experimental friction losses obtained fromEq. (5) together with the theoretical friction factors esti-mated by Eq. (13) are presented; the other used correla-tions for estimation of friction factors-Reynolds numberprovided very similar graphics. Concluding remarks onthe RMS values, on tting results and on the absolute rel-ative errors reveal that the expressions proposed by Fred-rickson and Bird (1958b), Kozicki et al. (1966), Tuoc andMcgiven (1994) and Delplace and Leuliet (1995) displayedbetter results than the hydraulic radius concept. Besides,the combination of Eqs. (5), (13) and (14) for n = 1 leads

0.1 1 10 100 1000

0.01

0.1

1

10

100

Fric

tion

fact

or in

ann

ular

flow

Ren,

9.3 Brix 24.3 Brix 34.6 Brix Predicted values, Eq. (13)

Fig. 4. Experimental friction losses estimated by Eq. (5) and predicted

values from Eq. (13) for the fully developed laminar ow of soursop juicesin annular regions.to the exact solution for laminar, Newtonian ow in ann-uli, while the hydraulic radius concept does not. It is clearthat all the theoretical friction factors are in excellentagreement with the experimental data; moreover, all thetested methodologies provided practically identical frictionfactors among themselves, although the methods of Koz-icki et al. (1966), Tuoc and Mcgiven (1994) and Delplaceand Leuliet (1995) predicted higher values of the Reynoldsnumber than the expressions of Fredrickson and Bird(1958b), which have the great inconvenience of requiringnumerical integration techniques.

The equations of Kozicki et al. (1966), Tuoc andMcgiven (1994) and Delplace and Leuliet (1995) displayedpractically the same results and turned out to be excellent,principally because the relation f-Reynolds is explicitlyevaluated, eliminating extensive and laborious calculations;however, the expressions of Kozicki et al. (1966) requirethe determination of two geometric constants (a and n),while the denitions of Tuoc and Mcgiven (1994) and Del-place and Leuliet (1995) have the great advantage ofinvolving the estimation of only one geometrical parameter(n) to dene the generalized Reynolds number.

Regarding Eq. (18) dened by Engez (1995), if it is usedfor determining the function Xp(n,j), the friction factorsare nearly equal to those ones found when estimatingXp(n,j) by Eq. (15); therefore, the expressions proposedby Fredrickson and Bird (1958b) together with the ttingequation dened by Engez (1995) also provide an explicitand outstanding method for determining the friction fac-tors of pseudoplastic uids in annuli ow, though it isimportant to notice that this methodology is valid onlyfor jP 0.2; the method consists of evaluating the frictionfactor by Eq. (13), the Reynolds number by Eq. (14), andthe functions Xp(n,j) and c(n,j) by Eqs. (16) and (18),respectively.

4. Conclusions

In the rst part of the manuscript, the rheologicalbehavior of soursop juice was experimentally determined,and the Power-Law model was satisfactorily tted to thedata in the tested ranges (9.349.4 Brix and 0.468.8 C).The combined eect of temperature and concentration onconsistency coecient and ow behavior index was welldescribed by Eqs. (21) and (22), respectively, which arequite useful for engineering applications. Although, we rec-ommend the use of Eq. (20), together with the parameterspresented in Table 2, for determining the temperaturedependence of consistency index, because Eq. (20) seemedto be more accurate than Eq. (21). In addition, experimen-tal values of friction factors in the fully developed, laminar,pipe ow of soursop juices were reported, and the excellentagreement between analytical and experimental frictionlosses conrmed the veracity of the Power-Law model fordescribing the ow behavior of soursop juices in the tested

Engineering 78 (2007) 13431354ranges, and also the experimental apparatus for pressureloss measurements was validated. In the second part, the

oodpressure loss data and the friction factors-Reynolds num-bers of soursop juices owing in annular geometries arepresented. All the tested methodologies for determiningfriction factors versus Reynolds showed to be in goodagreement with experimental data. From the analyses oftting and statistical results, we concluded that the simplestand most accurate method to calculate the friction factorsfor the annular ow of a pseudoplastic uid is: determiningthe friction factor from Eq. (6), the Reynolds number fromEq. (7), and the functions /(n), n(j) and t from Eqs. (8), (9)and (12), respectively. Besides, the proposed empiric corre-lation given by Eq. (26), along with the Eqs. (7)(9) and(12), may be used for manufacturers of soursop juice andsimilar products when dealing with practical applicationsinvolving fully developed laminar ow in concentric annu-lar systems, as in the pasteurization processes in double-pipe and triple-pipe heat exchangers.

Acknowledgements

The authors would like to thank the National Councilfor Scientic and Technological Development, CNPq,and Sao Paulo State Research Fund Agency, FAPESP(Proc. 2002/02461-0), for their nancial support.

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Laminar flow of soursop juice through concentric annuli: Friction factors and rheologyIntroductionMaterials and methodsRaw material: soursop juicesRheological measurements and flow characterizationPressure drop measurements in pipe flow: theapparatusEvaluation of friction factors in pipe flowPressure drop measurements in annular flow: the apparatusEvaluation of friction factors in annular flowCalculus and data analysis

Results and discussionDensity of soursop juiceFlow behavior of soursop juicesEffect of temperature and concentration on the rheological parametersFriction factors in pipe flowFriction factors in annular flow

ConclusionsAcknowledgementsReferences