Heat transfer during pasteurization of fruit pulps stored in containers with arbitrary geometries obtained through revolution of flat areas

Accepted Manuscript Heat transfer during pasteurization of fruit pulps stored in containers with arbitrary geometries obtained through revolution of flat areas

Wilton Pereira da Silva, Jair Stefanini Pereira de Ataíde, Maria Elieidy Gomes de Oliveira, Cleide Maria Diniz P. S. e Silva, Jarderlany Sousa Nunes PII:

S0260-8774(17)30345-X

DOI:

10.1016/j.jfoodeng.2017.08.012

Reference:

JFOE 8987

To appear in:

Journal of Food Engineering

Received Date:

16 February 2017

Revised Date:

23 May 2017

Accepted Date:

13 August 2017

Please cite this article as: Wilton Pereira da Silva, Jair Stefanini Pereira de Ataíde, Maria Elieidy Gomes de Oliveira, Cleide Maria Diniz P. S. e Silva, Jarderlany Sousa Nunes, Heat transfer during pasteurization of fruit pulps stored in containers with arbitrary geometries obtained through revolution of flat areas, Journal of Food Engineering (2017), doi: 10.1016/j.jfoodeng.2017.08.012

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ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 1

Heat transfer during pasteurization of fruit pulps stored in containers with

2

arbitrary geometries obtained through revolution of flat areas

3 4

Wilton Pereira da Silva*, Jair Stefanini Pereira de Ataíde, Maria Elieidy Gomes de

5

Oliveira, Cleide Maria Diniz P. S. e Silva, Jarderlany Sousa Nunes

6 7 8

Federal University of Campina Grande, PB, Brazil. *Corresponding

author: [email protected]

http://orcid.org/0000-0001-5841-6023

9 10

Abstract

11

Thermal diffusivity of papaya pulp, stored in metal container with arbitrary geometry

12

obtained through revolution of flat areas, was determined through optimization using

13

experimental data. To describe heat conduction during pulp pasteurization, the diffusion

14

equation in generalized coordinates was discretized and numerically solved, through the

15

finite volume method, with a fully implicit formulation. Temperature over time during

16

heating was measured by placing a thermocouple at the point of the container where the

17

equilibrium temperature occurs with greatest delay. Once the expression for thermal

18

diffusivity as a function of local temperature was known by optimization, it was

19

possible to determine, through simulation, the minimum time necessary for the pulp

20

stored in a new container, also with arbitrary geometry obtained through revolution of

21

flat areas, to come into thermal equilibrium with the pasteurization temperature.

22

Microbiological analysis performed before and after the second pasteurization showed

23

that there was a strong reduction of the total microorganisms. Since the thermal

24

equilibrium time was determined through simulation for the new container, the use of a

25

thermocouple for its experimental determination became unnecessary.

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Keywords: Food safety; heat conduction; numerical solution; complex geometries;

27

simulation; thermal diffusivity

28 29

List of symbols

30 31

Latin Letters

32

i, j – Indices for the position of points on the grid

33

J – Jacobian of the transformation

34

k – Thermal conductivity (Wm-1K-1)

35

N, S, E, W, NW, NE, SW, SE, P – Nodal points

36

S – Source term

37

t – Time in the physical domain (s)

38

T – Temperature (ºC)

39

x, y – Cartesian axes

40

x ξ , x η , yξ , y η – Derivatives of x and y with respect to  and  (m)

41

Δξ, Δη – Increment of position in the generalized axes ξ and η

42 43

Greek Letters

44

α ij – Components of the metric tensor

45

α – Thermal diffusivity ( m 2s -1 )

46

 – Dependent variable of the diffusion equation

47

 P ,  E ,  W ,  N ,  S ,  NE ,  NW ,  SE ,  SW – Dependent variables of the

48

discretized diffusion equation

49

Γ Φ , λ – Transport coefficients

2

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ξ, η – Curvilinear axes of the generalized coordinates system

51

τ – Time in the transformed domain (s)

52

χ 2 – Chi-square

53

Δ – Variation

54 55

Superscripts

56

0 – Previous time

57

P – Nodal point of the studied control volume

58 59

Subscripts

60

e, w, s, n – Boundaries of a control volume

61

i – Initial

62 63

1. Introduction

64

It is well known that the description of heat transfer requires the determination

65

of the thermophysical properties of the raw material used (Baïri et al.; 2007;

66

Ukrainczyk, 2009; Betta et al., 2009; Kiziltas et al., 2010; Tres et al., 2011; Silva et al.,

67

2011; Abakarov and Nuñez, 2013; Bhuvaneswari and Anandharamakrishnan, 2014;

68

Cho and Chung, 2016; Ohshima et al., 2016; Santos-González et al., 2016). One of

69

these properties is the thermal diffusivity (Carbonera et al., 2003; Lemmon et al., 2005;

70

Glavina et al., 2006; Kurozawa et al., 2008; Ruiz-Cabrera et al., 2014; Mohamed,

71

2015). However, it should be noted that, in the specific case of heating of food products,

72

many studies available in the literature consider that this property has a constant value

73

along the process, although various authors, such as Kurozawa et al. (2008), determine

74

the thermal diffusivity for products in thermal equilibrium at different temperatures.

3

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However, few studies consider that these properties are variable, for example, as a

76

function of the local temperature, during a heat transfer process in transient regime

77

(Mariani et al., 2009; Silva et al., 2011). In addition, products are generally stored in

78

containers with simple geometry, such as cylinders (Carbonera et al., 2003; Baïri et al.;

79

2007; Ukrainczyk, 2009; Betta et al., 2009; Tres et al., 2011).

80

Frequently, the use of heat aims to eliminate or reduce the levels of

81

microorganisms present in the foods and also denature enzymes, as in the case of

82

pasteurization (Plazl et al., 2006; Huang, 2007; Silva et al., 2014; Abbasnezhad et al.,

83

2016; Hong et al., 2016). To describe the heat transfer processes during the

84

pasteurization, the geometry and dimensions of the containers that contain the product

85

are important, as well as the knowledge on the thermophysical properties of the

86

container and the product inside it.

87

Regarding the determination of thermal diffusivity, it is important to highlight

88

the study of Ukrainczyk (2009), who estimated this property for pasty products stored in

89

long cylindrical containers, using the inverse method and a one-dimensional (1D)

90

numerical solution of the heat conduction equation. Betta et al. (2009) developed a

91

software to solve the equation of heat conduction in cylindrical coordinates, based on

92

the method of finite differences. To estimate thermal diffusivity using the inverse

93

method, these authors measured the temperature of the central point of products stored

94

in cylindrical containers over time, during the heating step of the pasteurization process.

95

The method proposed by the authors assumes a constant or variable heating

96

temperature, but is limited to two-dimensional (2D) cylindrical containers and to the

97

determination of thermal diffusivity with a constant value, not considering possible

98

variations of this parameter with the distribution of temperature inside the product

99

during the heating. On the other hand, Silva et al. (2011) conducted a study to determine

4

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the thermal diffusivity of tomato pulp, stored in cylindrical cans (2D) during the

101

pasteurization, assuming that this parameter varies with the local value of temperature.

102

For this, Silva et al. (2011) numerically solved the diffusion equation, in cylindrical

103

coordinates, for the boundary condition of the first kind. According to this study, the

104

best results were obtained assuming an exponential expression for thermal diffusivity,

105

increasing with the local value of temperature.

106

If the container geometry is a sphere, a cylinder or a parallelepiped, the

107

appropriate systems to analyze heat transfer are spherical, cylindrical and Cartesian

108

coordinates, respectively. This type of domain is sometimes referred in the literature as

109

regular geometry (Patankar, 1980). If the domain geometry is different of those above

110

mentioned, the domain is known as irregular, complex or arbitrary domain (Patankar,

111

1980; Farias et al., 2016). In this case, a boundary-fitted coordinate system (Tannehill et

112

al., 1997; Silva et al., 2009; Da Silva et al, 2014; Farias et al., 2016), also called

113

generalized coordinate system, facilitates studies on heat transfer in this domain. If the

114

arbitrary geometry is completely irregular, with no symmetry, a three-dimensional

115

equation in generalized coordinates must be used to describe a diffusion process. If

116

there is symmetry of revolution in the arbitrary geometry, a two-dimensional equation

117

may be enough to describe the diffusion. The literature consulted in the present study

118

allows to claim that the description of heat transfer in products stored in containers with

119

arbitrary geometry is scarce. In some studies, commercial software developed for the

120

study of computational fluid dynamics (CFD), such as CFX, Ansys, COMSOL

121

Multiphysics and Fluent, are used in the three-dimensional simulation of heat transfer

122

(Kiziltas et al., 2010; Bhuvaneswari and Anandharamakrishnan, 2014). However, due to

123

their complexity and the required computational effort, these programs are not normally

124

used for the determination of thermophysical parameters through algorithms of

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optimization. On the other hand, there are only few studies in the literature that propose

126

solutions of the diffusion equation in arbitrary domain, exploring possible symmetries

127

to reduce the computational effort required in the simulations. Nonetheless, it is

128

important to mention that this type of solution can be useful for the determination,

129

through optimization, of thermophysical parameters in heat transfer processes.

130

Silva et al. (2009) proposed a numerical solution of the diffusion equation for

131

geometries obtained through the revolution of arbitrary flat areas, to describe diffusion

132

phenomena. These authors discretized the diffusion equation in generalized coordinates,

133

presuming the boundary condition of the first kind. The researchers explored the

134

revolution symmetry of various containers, which reduced from three- to two-

135

dimensional the geometry to be considered in the solution of the equation. This

136

significantly decreased the computational effort in the solution of the equation, in

137

comparison to the typical three-dimensional solution in arbitrary domain. A similar

138

study was conducted by Da Silva et al. (2014) on drying of bananas, but in this case

139

considering the boundary condition of the third kind. On the other hand, according to

140

Farias et al. (2012), for solid and pasty products, with no heat source and no phase

141

change, in general, only conduction and radiation are involved in the heat transfer

142

processes. Many times, solely a conduction model is used to describe heat transfer and,

143

consequently, the parameters involved in these processes are considered as “apparent”.

144

In this context, the objectives of this paper are defined below.

145

This study aims to propose a model to determine the apparent thermal

146

diffusivity, as a function of the local temperature of pasty products stored in metal

147

containers with arbitrary geometry obtained through revolution of flat areas, using

148

experimental data. In addition, once the expression for the apparent thermal diffusivity

149

is known, this model can also be used to simulate heat transfer in products stored in new

6

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containers, with different arbitrary geometries obtained through revolution of flat areas,

151

without the need for acquisition of new experimental data. Thus, this study can be

152

useful for food industries, in terms of projects of packages for the storage of pasty

153

products subjected to pasteurization.

154 155

2. Material and Methods

156

In the present paper, the transient process of heat transfer was studied in the

157

pasteurization of papaya (Carica papaya), in pasty state, stored in metal containers with

158

arbitrary geometry obtained through revolution of flat areas. In this study, the following

159

assumptions are assumed for the mathematical model: (1) the product can be considered

160

as homogeneous and isotropic; (2) the distribution of temperature in the product is

161

axisymmetric; (3) the transport of heat in the product occurs through conduction; (4) the

162

boundary condition for heat transfer is of the first kind.

163 164

2.1. Diffusion equation in generalized coordinates

165

According to Silva et al. (2009), a revolution solid is generated by the rotation,

166

in relation to an axis (for example, y), of a flat area (defined by lines ξ and η , in

167

generalized coordinates), contained in the physical space xy, as is shown in Figure 1(a).

168

Hence, if an axisymmetric diffusion occurs in relation to the y-axis of the volume

169

generated by the revolution, there will be no flow perpendicular to the generating area

170

of this volume (axis  ). Thus, for a domain with these characteristics, the diffusion

171

equation can be written in only two dimensions, in generalized coordinates ξ and η

172

contained on the same xy plane (Silva et al., 2009), as follows:

173

 

 Φ      Φ     S       J     J    11      12  22     J         21   J

7

(1a)

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 174

Equation (1a) is known as two-dimensional diffusion equation in the

175

transformed domain ( ξ , η ), where:   and λ are transport coefficients;  is the

176

dependent variable; τ is the time; J is the Jacobian of the transformation from Cartesian

177

coordinates (x,y) to generalized coordinates ( ξ , η ) and S is the source term. The

178

expressions for the components of the metric tensor α ij and for the Jacobian J can be

179

viewed in Silva et al. (2009). For the use of the finite volume method (Patankar, 1980)

180

to discretize the Equation (1a) with a fully implicit formulation, such equation must be

181

integrated in the transformed space for each control volume Δξ Δη (from west (w) to

182

east (e) and from south (s) to north (n)), in the time interval Δτ (from τ to τ  Δτ ). This

183

integration results in Equation (1b):

184

185

  p  p  P0  0P  Jp 

       12 e J e e     11e J e  e    e  e   

186

     11w J ww   12 w J ww    w  w         21n J n n    22 n J n n    n  n  

187

    S p   21s J s s    22 s J s s   .   s  s  J p 

(1b)

188

It should be noted that the fully implicit formulation was chosen because of the

189

following reason: the obtained solution is unconditionally stable for any time interval

190

established (Farias et al., 2012). As additional information, before the integration that

191

resulted in Equation (1b), the partial derivatives of Equation (1a) were approximated as

192

follows:

f f f1  f10   ; f   1  f2   2 e 2    

8

w

and

f f   f3   3 n 3 s . The terms  

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 193

with zero in the superscript are evaluated in a time prior to the time of interest, while the

194

terms that do not have superscript are evaluated at the time of interest.

195

For complete discretization of Equation (1b), it is necessary to define the control

196

volume for which the partial derivatives of the dependent variable  in relation to ξ

197

and η will be calculated. Thus, the two-dimensional domain must be divided into

198

control volumes, which are differentiated from each other by their location on the

199

generated grid. The distinction between the types of control volumes is given by their

200

position and the number of faces in contact with the external medium. Therefore, for a

201

two-dimensional domain, there are 9 types of control volumes: internal, north, south,

202

east, west, northwest, northeast, southwest and southeast.

203 204

2.1.1. Discretization in generalized coordinates: control volume to the southeast.

205

As an example of the discretization of Equation (1), the control volume to

206

southeast, presented in the transformed domain through Figure 1(b), will be used

207

supposing boundary condition of the first kind. The nodal point P of this control volume

208

(as well as the nodal points of the neighboring control volumes) can be observed in the

209

grid fragment of the two-dimensional transformed domain. For this control volume, the

210

direct derivatives of the dependent variable  are given by the following equations:

211

    P  e ;  e  / 2 



 

w

P



W

      P s  N  P ; .  n   s  / 2

;

212

(2a-d)

213 The expressions for the cross derivatives are defined as follows: 214

215

 ne   e   s 2  ;  e 

 

w

 N   P   NW  W  s   sw  4 2  

9

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 216

(3a-d)

 ne   e  N   P   NW  W    sw    s   e 2 4 2   ;  n   s 

217 218 219 220 221

With the substitution of the partial derivatives given by Equations (2a-d) and (3a-d) in Equation (1b), the following algebraic equation is obtained:

Ap  P  AwW  An N  Anw NW  B

(4)

where:

 p        11w J w w   22 n J n n  2 11e J e e  2 22 s J s s  J p     

222

Ap 

223

1 1  12 w J w w   21n J n n ; 4 4

An   22 n J nn

 1 1  12 w J ww   21n J nn  4 4

(5a-b)

1 1  1 1  12 w J ww   21n J n n ; Anw   12 w J ww   21n J nn ; (6a-b)  4 4 4 4

224

Aw  11w J ww

225

0P  0P  S p   1 B    211e J e e  e  2 22 s J s s  s  12 w J w w ( s   sw )  JP  Jp   2

226

1 1 1   21n J n n ( ne   e )   12 e J e e ( ne   e  2 s )   21s J s s 2 e  ( s   sw ) (7) 2 2 2

227

The terms with zero in the superscript are evaluated in a time prior to the time of

228

interest, while the terms that do not have superscript are evaluated at the time of

229

interest. The subscripts “n”, “s”, “e”, “w”, “ne”, “se”, “nw” and “sw” represent the

230

interfaces north, south, east, west, northeast, southeast, northwest and southwest,

231

respectively, of any control volume considered.

232

The determination of the components of the metric tensor α11 , α12  α 21 , α 22

233

and of the Jacobian J assumes the knowledge on the metrics of the transformation, i.e.,

234

of the partial derivatives x ξ , x η , yξ and y η . Hence, expressions must be established

235

for these derivatives, for both the nodal point P in the control volume under analysis and 10

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 236

the interfaces to the north, south, east and west. Looking again at Figure 1, it can be

237

noted that the control volume with nodal point P is limited, in the transformed domain,

238

by the lines ξ = j and ξ = j+1 and by the lines η = i and η = i+1. It should be observed

239

that each intersection of the lines ξ and η has coordinates x and y in the physical

240

domain. Thus, it is possible to determine approximate expressions for the derivatives of

241

the Cartesian coordinates x and y of the nodal point P in relation to the generalized

242

coordinates ξ and η . For the coordinate x, for example:

243

xP 

xe  xw  xi , j 1  xi 1, j 1 xi , j  xi 1, j  1      2 2  

(8)

xP 

xn  xs  xi 1, j  xi 1, j 1 xi , j  xi , j 1  1      2 2  

(9)

244 245 246

Similar expressions can be obtained for the coordinate y. It must be observed

247

that, for the case of the control volume on the southeast boundary, the partial derivatives

248

on the internal interfaces (west and north) are calculated similar to the presented

249

derivatives, but not those in the boundaries east and south: xe ; ye ; xs and ys . For

250

these four derivatives, one can impose: xe  xP ; ye  yP ; xs  xP and ys  yP . With

251

a similar reasoning, discretized equations can be obtained for the other types of control

252

volumes (Silva et al., 2009). At the end of the discretization, a system of equations for

253

 was obtained (one equation for each control volume) and this system was solved for

254

each time step through the Gauss-Seidel method, with a tolerance of convergence of 10-

255

8.

256

the interface of two control volumes through the harmonic mean of the values obtained

257

for the nodal points of these control volumes (Silva et al., 2009; 2010; 2011).

On the other hand, it must be observed that the process parameter   is calculated on

11

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 258

In this work, the parameter   can be considered constant or variable,

259

depending on the local value of  in accordance with a function f, with two parameters

260

a and b, as seen in Equation (10):

261

   f ( , a , b ) .

(10)

262

Equation (10) gives the value of   on the nodal points of the control volumes.

263

However, as seen in Equations (5a-b), (6a-b) and (7), the value of this parameter at the

264

interface of two neighboring control volumes is required. For this, the following

265

equation was used (Patankar, 1980):

266



 fr 

  P Q

 f d Q  (1  f d ) P

,

(11)

267

in which fr represents the value of   on the interface fr of the control volume with

268

nodal point P and a control volume with nodal point Q. In Equation (11) fd is defined as

269

fd 

dP , d P  dQ

(12)

270

and dP and dQ are the distances from the interface “fr” to the nodal points P and Q,

271

respectively. At the nodal points,   should be calculated by an appropriate function

272

which relates that parameter with the value of  at each node as presented in Equation

273

(10).

274

Many expressions involving two fit parameters (a and b) were tested for the

275

apparent thermal diffusivity as a function of the local temperature (polynomials,

276

exponential, hyperbolic, constant, etc.), and both fitting parameters were determined

277

through optimizations, using experimental data of temperature measured by

278

thermocouple over time, through the robust method described by Silva et al. (2011).

279 280

2.2. Raw material and preparation of samples

12

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 281

The raw material was papaya (Carica papaya), obtained at the local market of

282

Campina Grande-PB, Brazil. Healthy ripe fruits, free from spots on the skin and with

283

firm pulp were selected. Initially, the entire bench, containers and instruments were

284

subjected to asepsis and sterilization. Gloves and masks were used to avoid undesirable

285

contaminations during the experimental procedure. The fruits were washed in running

286

water, sanitized in 0.2% (200 mg/L) sodium hypochlorite solution for 15 min and

287

washed again in running water. After asepsis, the fruits were peeled, cut and placed in a

288

multiprocessor (model Cadence Efficace Plus–600W), with no addition of water, to

289

obtain the pulp. Part of the pulp was placed in the freezer (with temperature of -18 ºC)

290

to be used in the physicochemical and microbiological analyses of the fresh papaya. The

291

other part was left on the bench to come into thermal equilibrium with the environment

292

(temperature of 22 ºC) before being subjected to the process of pasteurization.

293 294

2.2.1. Physicochemical and microbiological analyses of the samples

295

The physicochemical analysis for fresh pulp papaya was performed in triplicate

296

according to IAL (2005), Folch et al. (1957) and AOAC (2000). The microbiological

297

analysis (total count of microorganisms on the plate) was performed according to

298

Vanderzant and Spilttstoesser (1992) and also Brazil (2003). The count of

299

microorganisms, in log CFU/g, was made in a manual colony counter (Phoenix - model

300

CP608). These analyses were carried out at the Laboratory of Bromatology and Food

301

Microbiology of the Center of Education and Health of the Federal University of

302

Campina Grande, before and after the process of pasteurization, as a measure of the

303

efficiency of this conservation method.

304 305

2.2.2. Pasteurization

13

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 306

The process of pasteurization was performed at the Food Technology Laboratory

307

of the Center of Education and Health of the Federal University of Campina Grande.

308

The containers to store the pulp were obtained by putting together two identical metal

309

cups, made of stainless steel. In one of the cups, two holes were made to pass two

310

thermocouples (k-type). One thermocouple was positioned on the internal surface of the

311

metal cup, using a transparent acetic silicone adhesive (resistant to temperatures from -

312

50 to 150 ºC). The other thermocouple was positioned in the center of the upper part of

313

the same cup, as shown in Figure 2(a). Both cups, previously filled with papaya pulp

314

(Figure 2(b)), were united and glued with fast-drying liquid adhesive (Figure 2(c)). Both

315

holes to pass the thermocouples were sealed with silicone adhesive to improve their

316

fixation and avoid pulp leaking or even water entry during the process of pasteurization.

317

After that, the thermocouples were connected to the portable digital thermometer

318

(model TH – 095), which has two channels. In this assembly, the data are

319

instantaneously transmitted from the thermometer to a computer, through RS – 232

320

cable, and recorded in regular time intervals in a file with ‘.txt’ extension.

321

With the Etiel PP – 30 L pasteurizer/processor correctly installed, water was

322

placed in the internal tank to be heated to the temperature of 65 ºC. When this

323

temperature was achieved, the container filled with the pulp was placed in the

324

pasteurizer. The pasteurization procedure was performed four times and the water

325

temperature along the entire process was manually controlled, with the aid of probe

326

thermometer (model WT – 1), with reading capacity from -50 to 300 ºC. In the

327

experiments, the initial temperature of the product was the ambient temperature,

328

approximately 22 ºC.

329

The minimum heating time was established in such a way to guarantee that the

330

coldest point of container could come into thermal equilibrium with all other points of

14

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 331

the sample. The cooling step was performed immediately after the container with the

332

pulp was removed from the pasteurizer, by inserting it in a thermal box containing a

333

mixture of water and ice at temperature of approximately 1 ºC. The process was

334

considered as finished when the thermocouple located in the point of greatest delay of

335

cooling achieved the temperature of 22 ºC. Then, the pasteurized pulps were stored in

336

plastic containers with lids and placed in the freezer (-18 ºC) for microbiological

337

analysis.

338 339

3. Results and discussion

340

Physicochemical analysis performed in the pulp of fresh papaya presented

341

moisture content (%, w.b.), proteins (g/100 g), lipids (g/100 g) and total sugar (g/100 g)

342

of (86.4 ± 0.7), (0.36 ± 0.06), (0.89 ± 0.05) and (11.7 ± 0.6), respectively.

343

In this study, the two-dimensional diffusion equation was solved in generalized

344

coordinates, with the objective of studying the kinetics of heat transfer in ripe papaya

345

pulp, placed in containers with arbitrary geometry obtained through revolution of flat

346

areas,

347

     k / (  c p ) , S=0 and   T . In the previous expressions,  and k are

348

thermal diffusivity and conductivity, respectively, while  is the density and c p is the

349

specific heat at constant pressure of the papaya pulp. The simplification above

350

mentioned for   was considered reasonable for heating of papaya pulp because

351

empirical equations available in the literature (presented, for instance, by Fricke and

352

Becker, 2001) indicate that, for the main components of the pulp,  c p varies less than

353

1% between 22.4 and 65.4 oC (time interval of 2286 s). This type of simplification for

354

the diffusion equation, given by

considering

the

following

simplifications

in

Equation

(1a):

λ=1 ,

T  . (T )T  , is sometimes found in the 

15

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 355

literature as, for instance, in Mariani et al. (2008). To describe heat conduction in the

356

pulp using generalized coordinates, two-dimensional, non-orthogonal, structured grids

357

were created for the generating area of each container, using the software “2D Grid

358

Generation” (Silva, 2008). It was observed that grids with 24 x 32 control volumes are

359

sufficiently refined to solve the diffusion equation numerically, especially because they

360

explore the symmetry of revolution of the containers, and also the symmetry of the

361

generating areas of these containers. As to the refinement of the time interval,

362

preliminary studies indicated that the division of the total time of heating into 2000

363

steps was sufficient, since a relative tolerance of 10-4 was established for the parameters

364

to be determined through optimization.

365 366

3.1. Determination of apparent thermal diffusivity

367

The container used for the determination of the thermal diffusivity of papaya

368

pulp during the heating period, its symmetric half, generating area and also the

369

generated grid are presented in Figure 3. The part (d) of this figure highlights, through

370

the control volume to the southeast of the grid, the central point of the container, which

371

is the last one to achieve the equilibrium temperature, where the thermocouple was

372

placed for data acquisition. This figure also highlights the boundary conditions used for

373

the simulation. To determine the process parameters, the numerical solution was

374

coupled to an optimization algorithm described by Silva et al. (2011). The arithmetic

375

mean of four experimental data sets was used in the optimization process. Various

376

expressions for thermal diffusivity as a function of the local temperature were

377

investigated (Table 1). According to the obtained results, the best function to describe

378

the heating kinetics of the central point of the container was:

379

 (T )  b cosh(aT 2 ) ,

(13)

16

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 380

with a = 1.756x10-4 ºC-2 and b = 1.243x10-7 m2 s-1. For the thermal diffusivity given by

381

Equation (13), it was obtained: χ² = 13.625 and R² = 0.99952. Additionally, F-test

382

(Fisher-Snedecor) was calculated, with F = 1.401x105 indicating that, for 2 fitting

383

parameters and 88 experimental points, P(F) = 0.0. The least favorable result was

384

obtained for the apparent thermal diffusivity expressed by a constant value. This value,

385

obtained through optimization, was α = 1.37x10-7 m² s-1, which is a result close to those

386

obtained by Carbonera et al. (2003), Plazl et al. (2006) and Betta et al. (2009) for tomato

387

pulp. The statistical indicators obtained for this last result were χ² = 38.279 and R² =

388

0.99922. Despite a chi-square with a value approximately three times higher than that

389

obtained for the best result, this latter can also be considered as reasonable. In addition,

390

it can be observed that there is compatibility between the values obtained for constant

391

thermal diffusivity and the mean value of the variable thermal diffusivity expressed by

392

Equation (13): αaverage = 1.35x10-7 m² s-1.

393

Using the result of this research for the thermal diffusivity of papaya pulp, it was

394

obtained a range from 1.25x10-7 to 1.61x10-7 m2 s-1 corresponding to 22.4 and 65.4 oC,

395

respectively. The lower value is compatible with the value obtained by Kurozawa et al.

396

(2008) for the same fruit. These authors, using line heat source probe methodology,

397

determined an empirical equation for α(T), between 20 and 40 oC, which results in

398

1.15x10-7 m2 s-1 for the temperature of 22.4 oC. On the other hand, the value obtained in

399

the present article for α(22.4 oC), of 1.25x10-7 m2 s-1, is significantly lower than that

400

obtained for water at the same temperature (about 1.46x10-7 m2 s-1). According to

401

several researchers, including Azoubel et al. (2005), this type of result is explained by

402

the influence of the solid soluble content. These authors, using line heat source probe,

403

investigated the influence of the soluble solids concentration on the thermal diffusivity

17

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 404

of cashew juice at 30 oC. The researchers observed that, when the concentration varies

405

from 5.5 to 25 oBrix, α varies from 1.43x10-7 to 1.32x10-7 m2 s-1.

406

Using the expression given by Equation (13) for the apparent thermal diffusivity,

407

it was possible to describe the heating kinetics of the central point of the container

408

(Figure 4) and determine the distribution of temperature in the generating area predicted

409

by proposed mathematical model, at various times (Figure 5). As additional

410

information, these figures were obtained using the software developed in the present

411

study. The graph of Figure 4 demonstrates that the heating time of the central point of

412

the container with pulp was about 2300 s. In this figure, the standard deviation of the

413

simulated curve was approximately 0.4 oC, i.e., a little less than half degree Celsius.

414

Visually, it is possible to observe a good agreement between the experimental data and

415

the curve obtained through simulation, which allows to claim that the boundary

416

condition of the first kind is certainly adequate for the type of container used to store the

417

pulp (Plazl et al., 2006; Ukrainczyk, 2009; Betta et al., 2009; Silva et al., 2011; Silva et

418

al., 2014; Ruiz-Cabrera et al., 2014; Mohamed, 2015), and this fact was confirmed by

419

thermocouple placed in contact with the internal surface of the container.

420

Through an inspection of Figures 4 and 5, it is possible to observe the behavior

421

of temperature in the geometric center of the container with papaya pulp. It can be noted

422

that, at the beginning of the process, there is a certain delay in the heating of this point

423

and, consequently, its equilibrium temperature is achieved in a longer time, compared

424

with the points closer to the surface. The isothermal curves of Figure 5 also indicate that

425

the propagation of heat occurs, obviously, from the external surface to the center of the

426

container. Because of this, for many containers, the central point is selected so that its

427

transient state is monitored during the heating (Carbonera et al., 2003; Baïri et al.; 2007;

428

Ukrainczyk, 2009; Betta et al., 2009; Kiziltas et al., 2010; Tres et al., 2011; Silva et al.,

18

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 429

2011). However, depending on the generating area, the coldest point cannot be

430

determined so intuitively, and the model proposed in the present study allows to

431

determine this point through a simple simulation, as will be seen below, in a new

432

simulation, for another container.

433 434

3.2. Estimate of heating time using another container

435

Using the best result obtained for the apparent thermal diffusivity of papaya

436

pulp, given by  (T )  1.243 107 cosh(1.756 104 T 2 ) , a new simulation of the heating

437

of the product was performed for a new geometry, in order to determine the equilibrium

438

time of the coldest point. The new geometry and the two-dimensional, non-orthogonal,

439

structured grid of the symmetric half, with 24 x 32 control volumes, can be viewed in

440

Figure 6. With a simple initial simulation (with the same previous boundary conditions),

441

the developed software indicated that the coldest point is that whose control volume was

442

highlighted on the grid created in one of the symmetric halves of the generating area of

443

the container. Such point is located on the revolution axis (y-axis, west boundary), as

444

demonstrated in Figure 6(b). It is interesting to note that, for the new geometry chosen

445

for the container, the coldest point is not located in its geometric center. Nevertheless,

446

the heating kinetics of this point can be simulated through the proposed model, as will

447

be demonstrated below.

448

The graph in Figure 7 presents the evolution of temperature numerically

449

simulated for the control volume with greatest delay of heating, using the apparent

450

thermal diffusivity obtained through Equation (13). For this new geometry, it is noted

451

that the equilibrium temperature in the least favorable point is achieved in a time

452

slightly shorter than 1800 s. This information was used to perform the complete process

453

of pasteurization of the papaya pulp placed in the new container, but without the

19

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 454

acquisition of data using thermocouples, only relying on the heating time predicted by

455

the simulation presented in Figure 7. With the purpose of promoting a better

456

interpretation of the heating process for the new geometry, contour graphs with

457

isotherms were generated (Figure 8) for four times of the heating process. For the four

458

different times, it is possible to observe that the control volume with greatest delay in

459

the heating process is located in the west boundary of the generating area (where the y-

460

axis, of revolution, is located) and that it really must be located in the position indicated

461

in Figure 6(b).

462

As an additional information, the results obtained in the microbiological analysis

463

of fresh papaya pulp showed a count on plates of (6.38 ± 0.11), expressed in log CFU/g.

464

However, after pasteurization, the microbiological analysis was performed again and the

465

new count was (2.70 ± 0.01).

466 467

3.3. Overview of the results

468

Proposed model allowed to determine, for a new container with symmetry of

469

revolution, the coldest point and also the time necessary for the thermal equilibrium of

470

the product, both obtained through simulation. It constitutes an important result,

471

because it guarantees that the least favorable region during the heating of the product

472

achieves the temperature stipulated for the inactivation of the pathogenic agents,

473

without the need for direct measurement of the temperature in this point using

474

thermocouples. Thus, the model proposed in the present study allows to alter the

475

geometry of containers for the storage of pasty products, defining the heating time

476

through rapid computer simulations.

477

The apparent thermal diffusivity of a product is influenced by its composition,

478

temperature, density and specific heat, among others. Nevertheless, it is very common

20

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 479

to find in the literature various studies that consider the apparent thermal diffusivity

480

with a constant value, solving the equation of heat conduction with the diffusion term

481

given in the form of  2T (Lemmon et al., 2005; Glavina et al., 2006; Plazl et al.,

482

2006; Baïri et al., 2007; Huang, 2007; Ukrainczyk, 2009; Betta et al., 2009; Mohamed,

483

2015; Muramatsu et et al., 2017). Obviously, model proposed in the present study

484

allows to consider the thermal diffusivity with constant value and, for this case, the

485

result obtained in this study was compatible with others found in the literature for

486

similar products (Carbonera et al., 2003; Plazl et al., 2006; Kurozawa et al., 2008; Betta

487

et al., 2009). However, in the present study, it was observed that a constant value does

488

not express the apparent thermal diffusivity with accuracy in a transient process, as can

489

be observed through the statistical indicators obtained for this case. A simple

490

observation of Figure 5(a), for instance, indicates that after 100 s the outermost part of

491

the pulp, close to the container surface, had already reached the temperature of 65 ºC,

492

whereas the central region continued with 22.4 oC. Hence, one cannot expect that the

493

product, at such different temperatures, to have the same thermal diffusivity in different

494

points. Despite this observation, few studies in the literature, such as Mariani et al.

495

(2009) and Silva et al. (2011), determine expressions for α(T), in which T is the local

496

temperature during a transient process of heat transfer. However, it should be pointed

497

out that model proposed in these two studies was applied only to products stored in

498

cylindrical containers. In the present study, besides the apparent thermal diffusivity

499

being determined by an expression given as a function of the local temperature, the

500

container to store the pasty product can have any irregular geometry, provided that there

501

is symmetry of revolution. This generalization, obtained through simple simulations, is

502

an advance for food industries, in terms of projects of packages for the storage of pasty

503

products subjected to pasteurization.

21

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 504 505 506 507

4. Conclusion It was possible to conclude that the apparent thermal diffusivity of the papaya pulp must be given by increasing expressions, as a function of the local temperature.

508

Once the expression of thermal diffusivity as a function of the local temperature

509

was known, it was possible to determine the coldest point and also the time necessary

510

for the pulp to come into thermal equilibrium with a temperature previously defined,

511

through simulations. Hence, it became unnecessary the experimental determination of

512

this point and the use of thermocouple to measure this time of equilibrium every time a

513

container with new geometry is used to store the product. Thus, this study can be useful

514

for food industries, in terms of projects of packages for the storage of pasty products

515

subjected to pasteurization.

516

Regarding the microbiological analysis of papaya pulp stored in the new

517

container with symmetry of revolution, it was possible to claim that the pasteurization,

518

with the coldest point and the heating time determined through simulation, promoted a

519

significant reduction of the microorganisms.

520 521

Acknowledgment

522

The first author would like to thank CNPq (Conselho Nacional de

523

Desenvolvimento Científico e Tecnológico) for the support given to this research and

524

for his research grant (Processes Number 302480/2015-3 and 444053/2014-0).

525 526

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Numerical Modeling and Experimental Analysis of the Thermal Performance of a

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Compound

626

10.1016/j.applthermaleng.2016.10.100.

Parabolic

Concentrator,

26

Applied

Thermal

Engineering,

doi:

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 627 628

Silva,

W.

P.

Software

“2D

Grid

Generation”

(2008).

Available

at

, Access at October, 2, 2015.

629

Silva, W. P., Precker, J. W., Silva, D. D. P. S., Silva C. D. P. S., Lima, A. G. B. (2009).

630

Numerical simulation of diffusive processes in solids of revolution via the finite

631

volume method and generalized coordinates, International Journal of Heat Mass

632

Transfer, 52(21–22), 4976-4985.

633

Silva, W. P., Silva, C. M. D. P. S., Lins, M. A. A. (2011). Determination of expressions

634

for the thermal diffusivity of canned foodstuffs by the inverse method and

635

numerical simulations of heat penetration. International Journal of Food Science

636

and Technology, 46(4), 811–818.

637

Silva, W. P., Silva, C. M. D. P. S., Lins, M. A. A., Costa, W. S. (2014). Optimal

638

removal of experimental points to determine apparent thermal diffusivity of

639

canned products. International Journal of Food Engineering, 10(2), 223–231.

640 641

Tannehill, J.C.D., Anderson, A., Pletcher, R.H.

(1997). Computational Fluid

Mechanics and Heat Transfer. Second ed., Taylor & Francis, London.

642

Tres, M. V., Borges, G. R., Corazza, M. L., Zakrzevski, C. A. (2011). Determination of

643

Thermal Diffusivity of foods: Experimental Measurements and Numerical

644

Simulation (In Portuguese). Perspectiva, 35(131), 43–56.

645

Ukrainczyk, N. (2009). Thermal diffusivity estimation using numerical inverse solution

646

for 1 D heat conduction. International Journal of Heat and Mass Transfer, 52(25–

647

26), 5675 – 5681.

648

Vanderzant, C., Splittstoesser, D. F. (1992). Compendium of methods for the

649

microbiological examination of foods. 3th. ed. Washington: American Public

650

Health Association.

651

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ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 652

Figure and table caption

653

Figure 1. (a) Control volume with a nodal point P obtained by rotation about y of an

654

elementary cell of a two-dimensional structured grid in a vertical plane. (b)

655

Fragment of the grid in the transformed domain showing the control volume P to

656

southeast and its neighbors to the north (N), west (W) and northwest (NW).

657

Figure 2. (a) Positions of the thermocouples: internal surface and center of the upper

658

part of the cup; (b) Papaya pulp placed in both cups that compose the

659

container; (c) Container obtained by the junction of both cups with papaya

660

pulp.

661

Figure 3. (a) Container where the papaya pulp was stored; (b) Symmetric half of the

662

container; (c) Generating area of the symmetric half of the container; (d) Two-

663

dimensional grid generated with 24 x 32 control volumes, highlighting the point

664

where the thermocouple was placed.

665

Figure 4. Temperature versus time in the center of the container with papaya pulp.

666

Figure 5. Isotherms representing the distribution of temperature in the generating area of

667

the container with papaya pulp at the instant: (a) 100 s; (b) 200 s; (c) 400 s; (d)

668

600 s.

669

Figure 6. (a) New container to store papaya pulp; (b) Two-dimensional grid generated

670

with 24 x 32 control volumes for the symmetric half of the new container,

671

highlighting the control volume with greatest delay of heating.

672 673

Figure 7. Temperature versus time in the coldest point, obtained through numerical simulation for the new arbitrary geometry.

674

Figure 8. Isotherms representing the distribution of temperature in the generating area of

675

the new container with papaya pulp at the instant: (a) 100 s; (b) 200 s; (c) 400 s;

676

(d) 600 s.

28

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 677

Figure 1

678

679

(a)

680

681

(b)

682

Figure 1. (a) Control volume with a nodal point P obtained by rotation about y of an

683

elementary cell of a two-dimensional structured grid in a vertical plane. (b) Fragment of

684

the grid in the transformed domain showing the control volume P to southeast and its

685

neighbors to the north (N), west (W) and northwest (NW).

686 29

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 687

Figure 2

688

689 690

Figure 2. (a) Positions of the thermocouples: internal surface and center of the upper

691

part of the cup; (b) Papaya pulp placed in both cups that compose the container; (c)

692

Container obtained by the junction of both cups with papaya pulp.

693 694 695 696 697 698 699 700 701 702 703 704 705 706

30

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 707

Figure 3

708

709 710

Figure 3. (a) Container where the papaya pulp was stored; (b) Symmetric half of the

711

container; (c) Generating area of the symmetric half of the container; (d) Two-

712

dimensional grid generated with 24 x 32 control volumes, highlighting the point where

713

the thermocouple was placed.

714 715 716 717 718 719 720 721 722 723 31

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 724

Figure 4

725

726 727

Figure 4. Temperature versus time in the center of the container with papaya pulp.

728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 32

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 743

Figure 5

744

745

(a)

(b)

(c)

(d)

746

Figure 5. Isotherms representing the distribution of temperature in the generating area of

747

the container with papaya pulp at the instant: (a) 100 s; (b) 200 s; (c) 400 s; (d) 600 s.

748 749 750 751 752 753 754 755 756 757 758 759 760 761 762

33

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 763

Figure 6

764

765 766

Figure 6. (a) New container to store papaya pulp; (b) Two-dimensional grid generated

767

with 24 x 32 control volumes for the symmetric half of the new container, highlighting

768

the control volume with greatest delay of heating.

769 770 771 772 773 774 775 776 777 778 779 780

34

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 781

Figure 7

782

783 784

Figure 7. Temperature versus time in the coldest point, obtained through numerical

785

simulation for the new arbitrary geometry.

786 787 788 789 790 791 792 793 794 795 796 797 798 799 35

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 800

Figure 8

801

802 803

(a)

(b)

(c)

(d)

804

Figure 8. Isotherms representing the distribution of temperature in the generating area of

805

the new container with papaya pulp at the instant: (a) 100 s; (b) 200 s; (c) 400 s; (d) 600

806

s.

807 808 809 810 811 812 813 814 815 816 817 818 819

36

ACCEPTED Pasteurization of products stored inMANUSCRIPT containers with arbitrary geometries 820

Table 1

821 822

Table 1 – Functions for thermal diffusivity, parameters and statistical indicators. Rank

Function

a

b (m²s-1)

1

bcosh(aT²)

1.7563526x10-4

1.2433716x10-7

13.62506 0.9995167

2

be(aT²)

6.6771485x10-5

1.1695271x10-7

15.21006 0.9994777

3

bcosh(aT)

1.2118561x10-2

1.1635159x10-7

15.42183 0.9994757

4

aT²+b

8.9616890x10-12

1.1606156x10-7

15.53946 0.9994724

5

be(aT)

5.9826118x10-3

1.0314034x10-7

16.37410 0.9994482

6

bcosh( aT¹/²)

0.1169689

1.0217830x10-7

16.58376 0.9994557

7

aT+b

7.9637591x10-10

9.9493597x10-8

16.71353 0.9994436

8

be(aT¹/²)

7.9968914x10-2

7.9282870x10-8

17.05604 0.9994293

9

be(a/T)

-9.979097

1.7144248x10-7

19.27716 0.9993911

10

aT1/2+b

4.6032218x10-9

1.0587759x10-7

23.84062 0.9994333

11

b

-

1.3737640x10-7

38.27939 0.9992166

823 824 825 826 827 828 829 830 831

37

χ²

R²

ACCEPTED MANUSCRIPT  A model was proposed to determine thermal diffusivity of fruit pulps  Model was applied to products stored in containers with arbitrary geometry  Model enables to determine the coldest point during heating  The thermal equilibrium time during heating was predicted by simulation.