Boiling point of aqueous solutions of mate (Ilex Paraguariensis): Modeling and simulation of a batch evaporator

b i o s y s t e m s e n g i n e e r i n g 1 0 7 ( 2 0 1 0 ) 2 4 2 e2 5 0

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Research Paper

Boiling point of aqueous solutions of mate (Ilex Paraguariensis): Modeling and simulation of a batch evaporator Fla´vio Thihara Rodriguesa a, Lu´cio Cardozo-Filho b, E´verton Fernando Zanoelo c,* a

Federal University of Parana´, Polytechnic Center (DTQ/ST/UFPR), Graduate School of Food Technology, Jardim das Ame´ricas, 81531-990, Curitiba, Parana´, Brazil b State University of Maringa´, Center of Technology, Department of Chemical Engineering, Av. Colombo, 5790 Bloco D90 Campus Universita´rio 87020-900 e Maringa, Parana´, Brazil c Federal University of Parana´, Polytechnic Center (DTQ/ST/UFPR), Department of Chemical Engineering, Jardim das Ame´ricas, 81531-990, Curitiba, Parana´, Brazil

article info

Experiments were carried out to determine the boiling point of aqueous solutions of

Article history:

soluble powder extracts of mate (Ilex paraguariensis) using a simplified ebuliometer. The

Received 6 March 2010

consistency of the apparatus was checked based on a comparison between preliminary

Received in revised form

experimental results of this property and analogous data available in the literature for pure

19 August 2010

solvents (water, acetone and alcohol) and sucrose solutions at atmospheric pressure. The

Accepted 25 August 2010

effects of mass fraction of solids and pressure on the boiling point elevation (BPE) of

Available online 14 October 2010

solutions of mate extracts were investigated in the range of 0.1e0.5 and 24.0e91.4 kPa, respectively. In these operating conditions the boiling points were in the magnitude of 64e101
C, what means a maximum BPE close to 4
C. All these experimental results and the influence of investigated factors were correctly reproduced with two empirical and one thermodynamic model early reported in the literature. The tuned parameter of the ClausiuseClapeyron equation revealed a molar mass of solute equal to 138 kg kmol1. Experimental results of specific heat from 1528 to 4184 J kg1
C1 and apparent density varying between 260 and 1000 kg m3 were obtained for the investigated solutions of mate extracts in a calorimeter (method of mixtures) and a pycnometer at 25
C. An energy equation, a water and a global mass balance were presented to mathematically describe experiments of concentration of aqueous solutions of mate extracts in a batch evaporator at 64.5 and 91.4 kPa. ª 2010 Published by Elsevier Ltd on behalf of IAgrE.

1.

Introduction

The economic importance of mate (Ilex paraguariensis Saint Hilaire) crop is supported by the around US$ 1 billion involved in the global trade of this commodity marketed in more than

70 countries in all the continents (Halloy & Reid, 2003, chap. 11). In Brazil alone, there are 180,000 small farms and 750 small and large manufacturing companies that together employ 710,000 people that depend on this crop (EMBRAPA, 2005; Zanoelo, 2005).

* Corresponding author. Tel.: þ 55 41 33613202; fax: þ 55 41 33613674. E-mail address: [email protected] (E´. Fernando Zanoelo). 1537-5110/$ e see front matter ª 2010 Published by Elsevier Ltd on behalf of IAgrE. doi:10.1016/j.biosystemseng.2010.08.008

b i o s y s t e m s e n g i n e e r i n g 1 0 7 ( 2 0 1 0 ) 2 4 2 e2 5 0

Nomenclature A b Cp Dt DH D F H h Q M MW P R R2 r

2

area of heat transfer (m ) recycling ratio (J kg1
C1) specific heat (J kg1
C1) time variation (s) heat of water vaporization (J kg1) diameter of the spherical evaporator (m) mass flow rate (kg s1) enthalpy (J kg1) depth of solution in the spherical evaporator (m) apparent heat capacity of the calorimeter (J
C1) mass (kg) molecular weight (kg kmol1) pressure (kPa) gas constant (J K1 mol1) coefficient of correlation apparent density (kg m3)

Despite these positive statistics, recent official data (2005e2008) reveal that production of dry mate in Brazil (IBGE, 2009) and in Argentine (INYM, 2009) has approached a limit condition close to 230,000 and 234,000 tonne p.a., respectively. Since these countries are the leading global producers and exporters of mate, responsible for almost all the available amount of product for market, a world annual volume production of dehydrated mate not larger than approximately 500,000 t (Heck & De Mejia, 2007) is expected in a near future. A relevant aspect that contributes to this outlook of stagnant production is the traditional and almost exclusive (Bastos, Ishimoto, Marques, Ferri, & Torres, 2006; Esmelindro, Toniazzo, Waczuk, Dariva, & Oliveira, 2002) use of mate to produce a non-alcoholic beverage similar in taste and colour to the black and green tea obtained by the infusion of dry shoots from the Camellia sinensis bush (Zanoelo, Abitante, & Meleiro, 2008). As a consequence, despite the large number of bioactive compounds (e.g.; saponins, purine alkaloids and chlorogenic acids) (Bastos et al., 2006; Beninca´, Kaskantzis, & Zanoelo, 2009; Cardozo Ju´nior et al., 2007; Mazzafera, 1997; Zanoelo & Beninca´, 2009) that make the mate attractive for several applications in the food, cosmetic and pharmaceutical industry (Mazuchowski & Ru¨cker, 1993), a limited selection of commercial derivatives from this plant is currently available for selling (Cardozo Ju´nior et al., 2007). Among the new products from the dry leaves and branches of mate that may be obtained on a commercial scale, instantaneous soluble powder extracts is one that involves only few and well-known operations that have often been applied to other similar products, such as dehydrated soups and juices, powder egg, milk, coffee and chocolate (Teunou, Fitzpatrick, & Synnott, 1999; Vega, Goff, & Roos, 2005). Apart from a final stage of spray drying (Buffo, Probst, Zehentbauer, & Reineccius, 2002; Re´, 1998; Strumillo & Kudra, 1986; Vega & Roos, 2006), single- or multipleeffect evaporation under vacuum is always necessary (e.g. Moreira, Trugo, & De Maria, 2000; Pı´secky´, 2005) to concentrate the solution injected by the pressure nozzles located in the spray dryer chamber. However, the boiling point elevation (BPE) is a fundamental thermodynamic property required to correctly

t T U V X Y

243

time (s) temperature (
C) global coefficient of heat transfer (kJ s1 m2
C1) volume (m3) mass fraction of solid moisture content (dry basis)

Subscripts A water (solvent) B powder extract of mate (solute) e equilibrium i inlet or initial p pycnometer s solution t thermistor v vapour w water N ambient

design and simulate evaporators (Blackadder & Nedderman, 1982; McCabe, Smith, & Harriott, 1985). The main aim of this investigation was to experimentally determine the boiling point of aqueous solutions of powder soluble extracts of mate at different mass fractions of solid and pressures. The effect of these variables on BPE is well known and reported in the literature for similar products, such as liquid extracts of coffee (Telis-Romero, Cabral, Kronka, & Telis, 2002), but has not been reported for mate. Based on this set of experimental results, mathematical models were suggested to reproduce the influence of the aforementioned factors on boiling points. Experiments were also carried out to determine thermo physical properties (specific heat and apparent density) of aqueous solutions of mate at different powder concentration of solid extracts. These results were represented by mathematical expressions using a FORTRAN code to numerically solve a system of ordinary differential equations obtained to simulate the process of concentrating liquid extract of mate in a batch evaporator. Experiments of evaporation were performed to verify the reliability of these calculated results.

2.

Material and methods

2.1.

Experiments

The experiments to determine the boiling points were performed in a three necked round bottom flask made of glass with an internal diameter of approximately 0.06 m. An electric heating mantle equipped with a manual power regulator (Q321A23, Quimis Aparelhos Cientı´ficos, Diadema, Brazil) was used to heat transfer to a volume of approximately 60 ml of boiling liquid contained in the vessel. The lower and upper ends of a laboratory condenser jacket were connected to the central opening of the boiling vessel and to a centrifugal vacuum pump (Q355B, Quimis Aparelhos Cientı´ficos, Diadema, Brazil), respectively. An inlet trap and a filter were used to prevent liquid and vapour solvent back streaming into vacuum pump.

244

b i o s y s t e m s e n g i n e e r i n g 1 0 7 ( 2 0 1 0 ) 2 4 2 e2 5 0

Vacuum was controlled with a hand-operated needle valve and monitored with a pressure transducer (GDH12AN, Greisinger electronic GmbH, Regenstauf, German). A thermistor calibrated with a thermometer of reference (15-059-15 Ertco Thermometer, Thermo Fisher Scientific, Waltham, USA) and wired to an electronic device for direct readings of temperature (SD31, Watlow Electric Manufacturing Company, Winona, USA) was used to determine the boiling point within 0.2
C. A set of 40 readings of temperature of the liquid phase in a step time of 15 s defined the mean value for this thermodynamic property. Distilled water, ethyl alcohol (lot 34749, F. Maia Indu´stria e Come´rcio Ltda, Cotia, Brazil), acetone (lot 0708162, Vetec Quı´mica Fina Ltda, Duque de Caxias, Brazil) and sucrose solution were preliminary used to verify the reliability of the experimental procedure and apparatus. Soluble powder extracts of mate (I. paraguariensis) were from a mate manufacturing company (Lea˜o Ju´nior S.A., Curitiba, Brazil). Both distilled water and solid mixed to form aqueous solutions of sucrose and extracts of mate were weighed using a calibrated digital balance (Ohaus Adventurer, Toledo do Brazil Indu´stria de Balanc¸as Ltda, Sa˜o Bernardo do Campo, Brazil) with an uncertainty of 107 kg. The influences of mass fraction of powder and pressure on the boiling point of the solutions of mate were investigated from 0.1 to 0.5 and 24.0e91.4 kPa, respectively. Parameters for empirical and thermodynamic models (Barrow, 1988; Capriste & Lozano, 1988) were based on these experimental results to develop mathematical equations describing the effect of these factors. The simplex method of optimisation (Jenson & Jeffreys, 1977) was applied to minimise the sum of the absolute square differences between experimental and calculated boiling points of the mate solutions. Experiments to concentrate aqueous solutions of mate extracts were performed in the same aforementioned apparatus, but without using condensate recycling. The system was operated for no more than 5400 s at the conditions shown in Table 1. Sample solutions of approximately 1 ml were periodically (600 s intervals) removed from the evaporation vessel and placed in an oven (400/2ND, Nova E´tica, Vargem Grande Paulista, Brazil), where they were maintained for 24 h to determine the mass fraction of solid (X ) and water (1  X ). These results were also used to determine the mass of vaporised water in each consecutive time period (Dt) of concentration using Mv ¼

Mi ðX  Xi Þ X

where Mi is the inlet mass and Mv the mass of vapour, which necessary to experimentally estimate the global coefficient of heat transfer based on the arithmetic mean of values from U¼

Mv DHv DtAðTN  Ts Þ

(2)

where DHv is the change in vapour enthalpy, Dt is the time period, TN is ambient air temperature, Ts the temperature of the solution. The area of heat transfer A was calculated from A ¼ pDh

(3)

where h is the measured depth of solution in the spherical concentrator of internal diameter D (Spiegel, 1990). The classical method of mixtures (Mohsenin, 1980; Valentas, Rotstein, & Singh, 1997) was used to determine the specific heat of the aqueous solution of mate extracts. It involved the homogeneous mixture of a known mass of heated distilatted water and a solution of mate in a quasi-adiabatic vacuum-jacketed calorimeter with previously measured apparent heat capacity. The specific heat of the investigate solution Cps was obtained from an energy balance represented Cps ¼

Mw Cpw ðTw  Te Þ  QðTe  TN Þ Ms ðTe  Ti Þ

(4)

where Mw is the mass of water, Ms the mass of solution, Cpw is the specific heat of water, Tw, Te and Ti are water, equilibrium and initial temperatures respectively and Q is the apparent heat of the calorimeter. It assumed that all the heat from pure water was transferred to the mate solution and lost to the surrounding air at partial vacuum. A pycnometric method, often reported in the literature (Cansee, Watyotha, Thivavarnvongs, Uriyapongson, & Varith, 2008; Karimi et al., 2009; Moura, Vitali, & Franc¸a, 2001; Rahman, 1995; Valentas et al., 1997; Zanoelo, Beninca´, & Ribeiro, in press), was applied to experimentally estimate the apparent density of the aqueous solutions of mate extracts. Except for the mass (Mp) and volume (Vp) of the used pycnometer (Mp ¼ 2.21774  102 kg and Vp ¼ 5.405  105 m3), all the remaining details required to correctly reproduce the current experiments are presented in the literature (AOAC, 1970). However, the procedure necessary to obtain the desired property, which is basically the ratio of mass to volume of the aqueous solution of powder extract of mate is defined by

(1)

Fv Table 1 e Operating conditions to concentrate aqueous solutions of powder extracts of mate leaves in a batch evaporator. Variables Time of operation (s) Initial mass of aqueous solution (kg) Initial mass fraction of solid Initial mass fraction of water U (kJ s1 m2
C1) Initial depth of solution in the spherical evaporator (m) Absolute pressure (kPa) TN (
C)

Run 1 3000 0.1001 0.065 0.935 0.0572 0.0441

Fi

Run 2 5400 0.1008 0.049 0.951 0.0659 0.0442

Fβ Condensate

Vapour

F 91.4 248

63.3e65.6 147

F(1-β )

Fig. 1 e Schematic of a single-effect evaporator.

245

b i o s y s t e m s e n g i n e e r i n g 1 0 7 ( 2 0 1 0 ) 2 4 2 e2 5 0

a

b 110

80 70

100

TA (oC)

T (oC)

60 50

90

80

40 70

30

60

20 20

30

40

50

60

70

80

20

40

60

80

100

120

P (kPa)

Tt (oC)

Fig. 2 e (a) Thermistor calibration curve. (b) A comparison between experimental (symbol) and calculated (Antoine equation) (line) boiling point of water at pressures determined with the calibrated pressure transducer.

Ms Vp  Mrww

dMs ¼ Fi  Fð1  bÞ  Fv ¼ dt

(6)

where rs and rw are the apparent densities of the solution and water respectively. The influence of moisture content on the specific heat and apparent density was investigated in the range of 4.9e50% (wet basis). Controlled mixtures of soluble powder extract of mate and distilled water were necessary to prepare the aqueous solutions at different values of the investigated factor. The mean of three replicates defined these thermophysical properties at each condition of moisture content examined.

s dð1  XÞ Fi ð1  Xi Þ  Fð1  bÞð1  XÞ  Fv  ð1  XÞdM dt ¼ ðMs Þ dt

(7)

2.2.

where U is the global coefficient of heat transfer, Fv, Fi and F are mass flow rates of vapour, inlet and outlet, and Hv and H are the enthalpy of vapour and outlet solution. Eq. (9) is a classical algebraic expression required to estimate the enthalpy of the investigated solution as a function of the boiling point. The change of surface area along the time of integration of Eqs. (6)e(8) was based on a combination of Eqs. (10) and (3). The left-side term in Eq. (10) is the volume of solution in the spherical evaporation vessel with internal diameter D and depth h of solution (Spiegel, 1990).

rs ¼

(5)

Modeling

A schematic of a single-effect evaporator representing a continuous or a batch process is shown in Fig. 1. The latter operation happens when the recycle and bottom stream are identical (i.e. the recycling ratio b ¼ 1), which means that concentrated solutions are not removed from the evaporator. The global mass balance, the mass balance for water and the energy balance in the control volume depictured by the dashed circle in Fig. 1, are given by

a

Fv ¼

(8)

H ¼ Cps Ts

(9)

  ph2 3D Ms h ¼ 3 rs 2

b

120

UAðTN  Ts Þ þ Fi ðHi  HÞ  Ms dH dt ðHv  HÞ

100

(10)

116

112

TS (oC)

TA (oC)

80 60

108

104

40 100

20 0

96 Water

Ethanol Acetone

0.5

0.6

0.7

0.8

0.9

X

Fig. 3 e (a) A comparison between experimental (symbols) and calculated (Antoine equation) (bars) boiling point of pure solvents. (b) A comparison between experimental (symbols) and calculated results (solid line) (Eq. (13)) of boiling point of sucrose solutions at 91.4 kPa.

246

b i o s y s t e m s e n g i n e e r i n g 1 0 7 ( 2 0 1 0 ) 2 4 2 e2 5 0

a

b

112 108

110

100

TS (oC)

TS (oC)

104 100

90

80

96 70

92 88

60 0

0.2

0.4

0.6

0.8

0

0.2

X

0.4

0.6

0.8

X

Fig. 4 e A comparison between experimental (symbols) and calculated (line) (Eq. (14)) boiling point of aqueous solution of powder extract of mate at 91.4 kPa (a), 84.8 (a), 74.6 (a), 64.6 (b), 44.3 (b) and 24.0 (b) kPa.

The above system of algebraic-differential equations was solved using a numerical 4th order RungeeKutta method (Davis, 1984; Zill & Cullen, 1989) involving a FORTRAN computer program.

3.

Results and discussion

Fig. 2a reports a set of several temperature measurements simultaneously made with a thermistor and a calibrated mercury-in-glass thermometer (15-059-15 Ertco Thermometer, Thermo Fisher Scientific, Waltham, USA) placed in a heated bath. The correlation between these data is well described by a linear equation with an intercept equal to zero and a slope tuned on experimental results as shown below, where Tt is the thermistor temperature. T ¼ ð0:991  0:001ÞðTt  0:1Þ

(11)

The uncertainty propagation in temperature can be given by T ¼ 0:991Tt  ½0:991ð0:1Þ þ Tt 0:001

(12)

where the second right-side term is for a 90% confidence level. Based on this equation, the maximum uncertainty in

a

temperature approaches to 0.2
C when the highest value of boiling point is registered (i.e. w100
C). Fig. 2b shows a comparison between experimental and calculated boiling point of distilled water at different pressures sensed by a calibrated pressure transducer. All the experimental results were correctly reproduced by the Antoine vapourepressure correlation (Reid, Prausnitz, & Sherwood, 1977) that confirms the consistency of the transducer for readings of pressure. The Antoine equation was also applied to calculate the boiling point of pure water, ethyl alcohol and acetone at atmospheric pressure. The negligible difference between these estimated values and experimental results shown in Fig. 3a represents strong evidence of the reliability of the method and the apparatus to estimate this thermodynamic property. However, an additional test was carried out to check the consistency of the data involving solutions. Bubbles of superheated steam and heterogeneous mixtures are the major disturbances of boiling point measurements in this kind of system. Based on available results of this variable for a sucrose solution at 65.3 and 101.3 kPa (Oliveira, 2006), Eq. (13) below, was used to estimate the boiling point at the investigated atmospheric pressure (91.4 kPa). Only after this correction is applied, can a comparison between the experimental

b 110

112 108

100

TS (oC)

TS (oC)

104 100

90

80

96 70

92 88

60 0

0.2

0.4

X

0.6

0.8

0

0.2

0.4

0.6

0.8

X

Fig. 5 e A comparison between experimental (symbols) and calculated (line) (Eq. (15)) boiling point of aqueous solution of powder extract of mate at 91.4 kPa (a), 84.8 (a), 74.6 (a), 64.6 (b), 44.3 (b) and 24.0 (b) kPa.

247

b i o s y s t e m s e n g i n e e r i n g 1 0 7 ( 2 0 1 0 ) 2 4 2 e2 5 0

a

b

112 108

110

100

TS (oC)

TS (oC)

104 100

90

80

96 70

92

60

88 0

0.2

0.4

0.6

0.8

0

0.2

X

0.4

0.6

0.8

X

Fig. 6 e A comparison between experimental (symbols) and calculated (line) (Eq. (16)) boiling point of aqueous solution of powder extract of mate at 91.4 kPa (a), 84.8 (a), 74.6 (a), 64.6 (b), 44.3 (b) and 24.0 (b) kPa.

Ts ð91:4 kPaÞ ¼ Ts ð101:3 kPaÞ  ð101:3  91:4Þ 3:30  101
þ3:24  106 e12:778X

(13)

In Fig. 4 the experimental boiling points of aqueous solution of powder extract of mate at different mass fractions of solid and pressures are presented. A significant positive effect of these factors on the investigated property was noticed, in that the boiling point increased when both the independent variables were increased. This finding is in close agreement with analogous results reported in the literature for similar products, such as coffee extract (Telis-Romero et al., 2002). The boiling point of the aqueous solutions of mate extracts was an exponential function of the mass fraction of solid at the same pressure. Based on this evidence and on the reliability of the Du¨hring’s rule (Blackadder & Nedderman, 1982; McCabe et al., 1985) for the current case, a correlation for the boiling point of the investigated solution was suggested. According to Fig. 4, the obtained experimental results agree with the values of boiling point calculated from
Ts ¼ 2:734e1:653X þ 0:972e0:017X TA

(14)

where TA is the temperature of the water. An empirical expression, often reported in the literature (Capriste & Lozano, 1988), was also applied to reproduce the experimental values of boiling point
Ts ¼ TA þ 0:863X e4:055X P0:163

(15)

where P is pressure. Fig. 5 presents a comparison between these estimated results and the same full set of experimental data available in Fig. 4. Again, the empirical model was able to correctly describe the change of boiling point in the entire range of investigated mass fraction of solid and pressure. The empirical nature of the Eqs. (14) and (15) provides for a more consistent expression to calculate the boiling point of

aqueous solutions of mate extracts. In fact, the discrepancy between the results calculated with different correlations for solid concentrations higher than 50% (w/w) indicated their limited use. As a result, a semi-empirical model that involves only one tuned parameter was investigated

Ts ¼ TA þ

   RðTA Þ2 MWA X DHv MWB 1X

(16)

where MWA and MWB are the molecular weights of the water and solute (mate) respectively and R is the gas constant. Equation (16) described more than 99% (coefficient of determination, R2 ¼ 0.99) of the variations of boiling point due to changes in the investigated factors, such as observed in Fig. 6. A noteworthy aspect in this figure is that the boiling point determined from Eq. (16) at high values of mass fraction of solids was much higher than those shown in Figs. 4 and 5 and calculated from Eqs. (14) and (15), respectively.

5

4

BPE (oC)

and published results make sense. The good agreement between these data revealed in Fig. 3b validates the procedure and experimental apparatus adopted to determine the boiling point of solutions.

3

2

1

0 0

20

40

60

80

100

TA (oC) Fig. 7 e A comparison between calculated BPE of aqueous solution of mate extract (line) (Eq. (16)) and experimental BPE of coffee extract (symbols) available in the literature (Telis-Romero et al., 2002). Diamonds: X [ 0.524; Squares: X [ 0.332.

248

b i o s y s t e m s e n g i n e e r i n g 1 0 7 ( 2 0 1 0 ) 2 4 2 e2 5 0

a

b

800

600

3000

ρS (Kg m-3)

CpS (J Kg-1 oC-1)

4000

2000

400

200

1000

0

0 0

1

2

3

4

5

0

1

2

Y

3

4

5

Y

Fig. 8 e Effect of moisture content (dry basis) on specific heat (a) and apparent density (b) of solutions of powder mate extracts. Symbols: experimental results; Lines: calculated with Eqs. (17) (Cps) and (18) (rs).

The single-parameter in Eq. (16) represents the estimated average molecular weight of the solute (MWB ¼ 148 kg kmol1). It was not only close to the molecular weight of caffeine (MW ¼ 194 kg kmol1) found in large proportions in mate, but was also successfully used in Eq. (16) to reproduce experimental results for BPE of coffee extracts available in the literature (Telis-Romero et al., 2002) (see Fig. 7). This confirmed the resemblance in chemical composition between extracts of mate and coffee and supports the reliability of the experimental and calculated results of boiling point or BPE emerging from Eq. (16). The experimental effects of moisture content (dry basis) on the specific heat and apparent density of aqueous solutions of powder extracts of mate are shown in Fig. 8. Both the investigated thermo physical properties increase with moisture content. Over small moisture content ranges, the relation between these properties and the investigated factor is nearly linear. However, over high ranges of the independent variable, Fig. 8 reveals that the thermo physical properties increase markedly more slowly with moisture content. Two semi-empirical models already reported in the literature for leaves of mate (Zanoelo et al., in press), but with parameters tuned on results reported in Fig. 8, were suggested

a

to calculate the specific heat (Eq. (17)) and apparent density (Eq. (18)) Cps ¼

rA rB ð1 þ YÞ rB Y þ rA

(18)

Y¼

ð1  XÞ X

(19)

where Y is the moisture content (dry basis) and CpA, rA, CpB and rB are properties of pure water and the solute (CpA ¼ 4184 J kg1
C1, rA ¼ 1000 kg m3, CpB ¼ 1528 J kg1
C1, rB ¼ 260 kg m3). Fig. 8 presents a comparison between the experimental and calculated results of these properties for mixtures of these components and supports the reliability of both Eqs. (17) and (18). A residual difference of lower than 1% of both these calculated properties of water when compared with data available in the literature (Valentas et al., 1997) confirmed the consistency of the suggested models. From a practical point-of-view, the most important aspect of this investigation is the application of all these

1

0.8

X and (1-X )

0.8

X and (1-X)

(17)

rs ¼

b

1

CpB þ CpA Y 1þY

0.6

0.4

0.6

0.4

0.2

0.2

0

0 0

1000

2000

t (s)

3000

4000

0

2000

4000

6000

8000

10000

t (s)

Fig. 9 e Experimental (symbols) and calculated (lines) mass fractions of solid (X ) (diamonds) and water (1  X ) (squares) during the concentration of aqueous solutions of mate extracts at 91.4 (a) and 64.5 kPa (b) in a batch evaporator.

b i o s y s t e m s e n g i n e e r i n g 1 0 7 ( 2 0 1 0 ) 2 4 2 e2 5 0

experimental and calculated properties (boiling point, specific heat and apparent density) to correctly design and simulate evaporators. A final step towards this objective was the solution of the differential-algebraic system represented by Eqs. (6)e(10) at identical operating conditions applied to experimentally concentrate an aqueous solution of powder extracts of mate (see Table 1). Fig. 9 shows that without tuning any parameter on these experimental results, the evaporating model was able to predict the increase of solid concentration found experimentally. This cross-validation between independent experimental results and true model predictions strongly confirmed the reliability of the suggested models for boiling point, specific heat and apparent density of aqueous solutions of mate extracts.

4.

Conclusions

The negligible difference between the obtained experimental results of boiling point and data available in the literature for pure water, ethyl alcohol, acetone and sucrose solutions supports the reliability of the apparatus and procedure applied to experimentally determine this thermodynamic property. The observed positive effect of mass fraction of solid and pressure on the boiling point of aqueous solutions of mate extracts is confirmed in the literature for similar products. All the suggested mathematical models were able to correctly reproduce the experimental results of boiling point at the different conditions of the investigated factors. The increase of specific heat and apparent density of the solution of mate extracts with moisture content (d.b.), which has been extensively reported in the literature for foods and beverages, was well described by a combination of a zero-order and first-order polynomial function. The results for boiling point, specific heat and apparent density were successfully applied to mathematically reproduce the time variation of mass fraction of water and solid extract of mate in a spherical batch evaporator. The accurate prediction of experimental results in a laboratory spherical batch evaporator supports the use of the investigated properties and suggests the ability of the evaporating model to simulate, optimise and design industrial evaporators for mate liquors.

references

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