Coupled heat and mass transfer modelling in convective drying of biomass at particle-level: Model validation with experimental data

Journal Pre-proof Coupled heat and mass transfer modelling in convective drying of biomass at particlelevel: Model validation with experimental data Gabriele A. Nagata, Thiago V. Costa, Maisa T.B. Perazzini, Hugo Perazzini PII:

S0960-1481(19)31621-0

DOI:

https://doi.org/10.1016/j.renene.2019.10.123

Reference:

RENE 12492

To appear in:

Renewable Energy

Received Date: 15 May 2019 Revised Date:

20 October 2019

Accepted Date: 21 October 2019

Please cite this article as: Nagata GA, Costa TV, Perazzini MTB, Perazzini H, Coupled heat and mass transfer modelling in convective drying of biomass at particle-level: Model validation with experimental data, Renewable Energy (2019), doi: https://doi.org/10.1016/j.renene.2019.10.123. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Experimental approach

60

Temperature, T [°C]

55 50 45 40 35

Experimental data (Tg = 55°C) Non-isothermal model (Tg = 55°C) Experimental data (Tg = 40°C) Non-isothermal model (T g = 40°C)

30

∂X ∂t

h

3 R X

25

T

ρc

X

T

0

dX ∆H dt c

D && 6Bi! ∙ #$ ! ' exp R

R!

,

λ+!

1 Bi .Bi

1/

0

$

D && ' exp R!

Acai-berry residue X X D

X X

&&

10

20

30

40

50

Time, t [min]

λ+! D &&t

∞

62 38+

λ!3 4λ!3

Bi! Bi .Bi

1/5

exp 6

Ea D exp , 0 R .T 273.15/

Theoretical approach

λ!3

D && $ ! t'7 R

λ!!D && t R!

,

λ!!

1 Bi .Bi

0.55

1/

01

Non-isothermal model (Tg = 55°C) Isothermal model (diffusive model) (Tg = 55°C) Non-isothermal model (Tg = 40°C) Isothermal model (diffusive model) (Tg = 40°C)

0.50

Moisture content, X [kg/kg]

dT dt

0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0

500

1000

1500

2000

2500

Time, t [min]

3000

3500

4000

1 1 2 3 4 5 6 7 8 9

Coupled heat and mass transfer modelling in convective drying of biomass at particle-level: model validation with experimental data Gabriele A. Nagata, Thiago V. Costa, Maisa T. B. Perazzini and Hugo Perazzini* Institute of Natural Resources, Federal University of Itajubá, Itajubá – MG, Brazil, P.O. Box: 50, zip code: 37500-903. E-mail: [email protected]*Corresponding Author

Abstract

10

Acai berry waste (13.40 mm diameter, 1250 kg·m−3 specific mass, 1850 J·kg−1·°C−1 specific heat

11

and 0.91 sphericity) is a large-scale agroindustrial by-product that can be used as source of biomass

12

for thermochemical conversion. As this material has high initial moisture content, prior thermal

13

treatment using drying is necessary. In this work, mathematical modelling of the single particle

14

drying kinetics of acai berry waste was performed. The experimental conditions were: temperature

15

of 42 ºC and 57 ºC and gas velocity of 1.8 m·s‒1. Non-isothermal model consisted of differential

16

equations based on macroscopic energy balance and on the differential form of the analytical

17

solution of the diffusive model. The estimated values of h were 47.78 W·m‒2·°C‒1 at 42 °C and

18

42.55 W·m‒2·°C‒1 at 57 °C and for D0 and Ea were 1.39×10‒4 m2·min‒1 and 26.88 kJ·mol‒1 at 42 °C,

19

respectively, and 2.94×10‒4 m2·min‒1 and 25.05 kJ·mol‒1 at 57 °C, respectively. The optimum value

20

of Bim was equal to 2.5. The results showed that the non-isothermal model gave good predictions

21

for both temperature and moisture content of the solid during drying. It was verified that there was

22

not a significative dependence of the moisture diffusion with the heating rate of the solid.

23

Keywords: bio-residue; effective diffusivity; heat transfer coefficient; moisture content; solid

24

waste; temperature.

25 26 27 28 29 30

2 31

1. Introduction

32

One of the greatest challenges in every urban center is to minimize the utilization of fossil

33

fuels in typical daily activities. Proposing the use of agroindustrial solid wastes as biomass in the

34

generation of renewable energy through its combustion and gasification, as well as in the

35

production of biofuels by pyrolysis processes, is a demanding task.

36

One of the most important properties of a given biomass is its moisture content [1,2]. High

37

levels of water concentration reduce the efficiency of the thermochemical conversion process due to

38

the reduction of both heating value of biomass [3] and conversion temperature [4]. In addition, high

39

moisture content in biomass increases the tendency of spontaneously combustion [5]. Drying can be

40

seen as an important biomass pre-thermal treatment, employed in order to minimize such problems,

41

but also to improve operation of the conversion process [6], to prevent organic matter loss due to

42

microbial and fungal activity [7], to improve energy quality and to minimize the emissions [4] and

43

to reduce the grinding energy consumption [8]. On the other hand, drying is seeing as one of the

44

most energy-intensive operation, consuming up to 50% more energy to evaporate water than the

45

latent heat [7]. Besides typical heat losses in the exhaust air and other inefficiencies of the drying

46

systems, additional thermal energy is required to break moisture bonds and release bound moisture

47

[9], which is a characteristic of biomass drying [7]. In turn, a drying process with high energy

48

efficiency and low energy consumption is essential for both the viability of the operation and the

49

reduction of operating costs. Keeping track of the process variables is a key step for an efficient

50

biomass drying. Final properties and quality of the dried product (fuel) in terms of the water content

51

must be assured during drying. For this sake, key variables such as moisture content and

52

temperature of the biomass have to be accurately monitored and predicted, which may be

53

accomplished by means of mathematical modelling and simulation studies.

54

There are several approaches presented in literature to model a given drying problem, such

55

as the characteristic drying rate curve (CDRC) [10], the drying kinetics models (empirical or semi-

56

empirical, frequently known as thin-layer drying equations) [11] and the distributed parameter

3 57

models, including coupled heat and mass transfer models [12]. Apart from the empirical or semi-

58

empirical drying kinetics models used to simply correlate the moisture content over time in drying

59

of a given biomass or solid waste in a greenhouse dryer [13], forced convection solar dryer [14],

60

solar tunnel dryer [15], cabinet dryer [16] and vibro-fluidized bed dryer [5], as well as the CDRC

61

approach to predict drying kinetics data in solar dryer [17] and tray dryer [2], more rigorous models

62

derived from heat and mass conservation laws are also interesting, since they help in optimization

63

of the process in order to determine its viability regarding to economic benefits. Besides their

64

application in solar drying, distributed parameter models were applied with success in other drying

65

technologies, such as in conveyor-belt dryer [18], fluidized [19] and fixed bed dryer [20], and

66

vacuum oven dryer [21]. Those mathematical models are an essential tool to predict air temperature

67

and moisture evaporation rate to optimize operating conditions for an efficient use of drying

68

technologies [22].

69

Several works presented in literature that deals with coupled heat and mass transfer

70

modelling of biomass drying use the approach of thin-layer or deep bed drying. This preference can

71

be justified due to the high heterogeneous characteristics of a given biomass, which can be

72

constituted of different components in terms of geometry, shape and internal structure. However, if

73

the shape of the particle is known, the findings obtained in a very thin product layers, normally one

74

particle size, are interesting for the application at a deep bed scale, for equipment simulation and

75

design [23]. Furthermore, analysis of heat and mass transfer at particle-scale is a key point for

76

grasping drying behaviors in the granular wet material [24].

77

Acai berry waste is a large-scale agroindustrial by-product that can be used as source of

78

biomass for thermochemical conversion. Drying of this bio-waste is necessary in order to improve

79

the efficiency of the thermochemical conversion processes by increasing the combustion yield and

80

minimizes water vapor condensation, as it has an initial moisture content of about 40% (wet basis).

81

As drying of biomass is a complex and highly-cost process, performing mathematical modelling is

82

very important to warrant careful assessment of the operating conditions, which may contribute to

4 83

the improvement of the drying process and to the determination of the optimal operating conditions

84

to be used in industrial scale with low energy consumption.

85

This work presents a study about the mathematical modelling of the drying kinetics of acai

86

berry waste (fruit stone) under inlet air temperatures of 42 ºC and 57 ºC and velocity of 1.8 m·s−1,

87

once moderate drying conditions contribute to the minimization of the emissions of volatile organic

88

compounds to ambient air. High range of temperature employed is essential to deepen the

89

understanding of the coupling between heat and mass transfer established in a given drying problem

90

for different inlet air condition used. Such analysis contributes with valuable information regarding

91

to the formulation of energy and mass balance equations of the mathematical model, which in this

92

work consisted of a group of differential ordinary equations to model the heating and the drying

93

rate. To validate the proposed model, experiments were performed in a tunnel dryer at particle level,

94

once the results can be applied to the design and optimization of other drying systems at industrial

95

scale. It is expected that the conclusions drawn can contribute with reliable information regarding to

96

industrial implementation of thermal treatment of acai berry solid wastes by means of drying

97

technologies.

98 99

2. Mathematical modelling

100

It is well stated in literature the complexities involved in convective drying of a porous solid

101

regarding to transport phenomena. Simultaneous heat and mass transfer in a heterogeneous system

102

is a complex combination of different mechanisms of momentum, heat and mass transfer, such as

103

conduction thorough solid, liquid and gaseous phases, convection in the pores occupied by gas or

104

liquid and enthalpy transfer by evaporation and condensation cycles [25]. In the sense to describe

105

such fundamentals, several theories have been proposed by literature. Among them, two can be

106

highlighted: Luikov’s theory [26] and Whitaker’s theory [27].

107

Luikov’s theory is based on the irreversible thermodynamics principles and suggests that

108

drying is described in terms of pressure, temperature and moisture gradients. As a result, a group of

5 109

coupled partial differential equations is used to describe the phenomena established in drying. For a

110

system at constant pressure, simultaneous heat and moisture transfer in a rigid porous medium can

111

be described by the following equations in terms of effective transport parameters [28]: ∂X = ∇. D ∇X + ∇. D δ ∇T ∂t

(1)

∂T ∆H ϕ ∂X = ∇. α ∇T + ∂t c ∂t

(2)

112

An important aspect of this group of equations is the coupling between them, representing

113

the effects of the temperature gradient on the moisture transfer and of the moisture gradient and the

114

latent heat of evaporation on the heat diffusion.

115

Whitaker proposed a more detailed theory in which transport equations are described for

116

each phase (solid, liquid and gas) in macroscopic and microscopic levels based in the local average

117

volume behavior. Perré [29], in his work, presented a summary of the Whitaker’s theory. From

118

some simplifications, the author showed that the Whitaker’s theory can be described based on a

119

model with two dependent variables (moisture and temperature). The differential equations of

120

moisture and energy are shown according to Equations (3) and (4), respectively [29]. ρ

∂X = ∇. ρ ∂t

∇w +

∇ρ − ρ !" #

∂ ε ρ h + ε ρ h + ρ" h" + ε ρ h # ∂t = ∇. k ∇T + h ρ

∇w + h

∇ρ − h ρ !"

(3)

(4)

121

Equation (3) indicates that moisture migration is due to the contribution of different

122

mechanisms of mass transfer, namely vapor diffusion, bound water diffusion and mass convection.

123

Equation (4), described in terms of the enthalpies of each phase, indicates the contribution of a

124

temperature gradient, the moisture migration by diffusion and convection due to the mass flux.

125

In both presented theories, it is possible to note the necessity to consider the thermal energy

126

contribution in moisture migration, which makes the solution of the drying problem very difficult

127

due to the coupling between the transport parameters, sometimes of difficult estimation, and due to

6 128

the complex resolution of the partial differential equations due to their non-linear characteristic. All

129

these facts lead to the use of simple mathematical models, but based on a well-founded theory,

130

which are more attractive for engineering purposes [30,31], such as in developing equations for use

131

in energy analysis and optimization of drying processes. Thus, seeking for a simplification of the

132

presented theories, the main assumption is the one that considers the moisture transfer in vapor

133

phase through the stagnant air inside the pores of the solid material, i.e., the water vapor does not

134

flow as single component [32], and the other is the isothermal condition.

135

Initially assuming that the isothermal approach can be satisfied, it is not necessary to make

136

distinctions between moisture in liquid and vapor phases [29], as occurs in Whitaker’s theory

137

(

138

neglected, which leads to a simplification of Equations (1) and (3), as well as Equations (2) and (4)

139

are not important to describe the drying problem. As a result, a simple microscopic model with only

140

one dependent variable is necessary to fit drying kinetics data, considering the volume of a biomass

141

particle as the interest system to be modeled:

=

). The contribution of the temperature gradients in moisture transfer can also be

∂X = ∇. D ∇X ∂t

(5)

142

Equation (5) is known as Fick’s second law equation of diffusion, which combines the mass

143

conservation equation with Fick’s first law of diffusion, and written in terms of an effective

144

diffusive parameter (D ), which is expected to account the effects of both liquid and vapor

145

transport. Analytical solutions of Equation (5) can be found in the work of Crank [33], for different

146

boundary conditions. The main assumptions applied to Equation (5) for an isothermal approach are

147

the following [34-36]:

148

1. The temperature of the solid remains constant during drying;

149

2. The effective thermal conductivity of the solid is infinite;

150

3. The difference of the temperature between the material to be dried to that for the gas is

151 152

very small; 4. The internal resistance to heat conduction is very small, or

7 153

5. The heat transfer rate inside the material is higher than the moisture transfer.

154

If the isothermal condition is valid, all these assumptions can be considered for the

155

formulation of a macroscopic energy balance. In this approach, it is assumed that the energy

156

transfer between solid and fluid phases is governed by the external control and that the temperature

157

profiles inside the particle are considered flat. Thus, the variation of temperature with time is more

158

important than its variation in space. Taking the spherical particle as the control volume, the

159

temperature variation of the single particle during drying can be modeled as [37]: ρc

160 161

dT dX = ha T − T# − ρ *− + ∆H dt dt

(6)

The heat transfer coefficient, h, was calculated according to the correlation proposed by Whitaker for convection heat transfer around a sphere [38], presented in Equation (7). 3 4 hD μ 3/; 4 6 :,; = 2 + *0,4Re2 + 0,06Re2 + ∙ Pr < > k μ

(7)

162

Under the operating conditions studied in this work, Equation (7) gave the following values

163

for h: 47.78 W·m‒2·°C‒1, for T = 42 °C, and 42.55 W·m‒2·°C‒1 for T = 57 °C. The properties of the

164

humid air were obtained from the correlations proposed by Tsilingiris [39] evaluated at film

165

temperature, which was approximated as the average of the bulk air and particle temperatures, once

166

the temperature of the solid reached the inlet air temperature asymptote very fast. μ was evaluated

167

at the average temperature of the solid recorded during drying.

168 169

The latent heat of vaporization, ∆H , was obtained according to the following relationship, among several correlations previously tested [40]: 3/6

374.14 − T + 273.15 ∆H = 2500 A E 374.14

170 171

(8)

The specific heat of the solid, c , treated as a function of the average spatial moisture content, X, was determined according to the following fashion: c

=c

F

+ 4187 ∙ X

(9)

8 172

The average moisture content is defined as: X=

3 I H X ∙ r 4 dr R6 :

(10)

173

The mass balance over the particle volume, developed to obtain X as a function of the time

174

under the isothermal approach, considers the moisture gradient transport only by molecular

175

diffusion. Assuming constant solid properties, the effective moisture diffusivity independent of

176

moisture content and unidirectional water flux, the diffusive model (Equation 5) can be written as: ∂X D ∂ ∂X = 4 J K ∂t r ∂r ∂r

177 178

(11)

Equation (11) is subjected to the following initial (Equation 12) and boundary conditions (Equations 13 and 14): X r, 0 = X: , t = 0, 0 ≤ r ≤ R ∂X 0, t M = 0, t > 0 ∂r NO:

− D

∂X R, t M = βSX R, t − X T U, t > 0 ∂r NOI

(12) (13)

(14)

179

The Robin-type boundary condition (Equation 14) was proposed in order to study the effect

180

of the external resistance to moisture transfer at the interface gas-solid, in which β is an apparent

181

mass transfer coefficient, on the assumption that the rate of moisture loss is directly proportional to

182

the excess moisture content above equilibrium [10]. This parameter can also be defined as a

183

partitional constant that links the water concentration difference to the vapor concentration

184

difference at the boundary [41-42]. Applying the initial and boundary conditions, Equation (11) can

185

be written in terms of the Biot number for mass transfer (BiX ) as [33]:

r ∞ sin ]λZ R_ X−X T 2R BiX D = Y 4 exp J−λ4Z < 4 t>K X: − X T r R Sλ + BiX BiX − 1 U sin λZ ZO3 Z

186

Substituting Equation (15) in Equation (10), it gives [43]:

(15)

9

X−X T Bi4X D = 6Y 4 4 exp J−λ4Z < 4 t>K X: − X T R λ Sλ + BiX BiX − 1 U ZO3 Z Z ∞

187 188 189

(16)

Besides the isothermal approach, Equation (16) is valid only for the following assumptions: isotropic material, negligible shrinkage and uniform initial moisture content. The eigenvalues, λ n , are obtained from the following transcendental equation: λZ ctg λZ + BiX − 1 = 0

(17)

190

In preliminary studies, it was verified that only the first two terms of the series provided

191

enough accuracy to predict average moisture content as a function of the time. Solving Equation

192

(16) for n > 2 did not improve the fitting, and the coefficient of determination remained unaltered.

193

Thus, Equation (16) could be simplified to: X = 6c

λ34 Sλ34

Bi4X

+ BiX BiX − 1 U +

exp <−λ34

D t> R4

Bi4X

λ44 Sλ44 + BiX

D exp <−λ44 4 t>d X: − X T # + X R BiX − 1 U

(18) T

194

In order to develop the coupled heat and mass transfer model and to analyze the influence of

195

the heat transfer on moisture transport by diffusion, Equation (18) is derived with respect to time to

196

obtain the drying rate: ∂X = − X: − X T #6Bi4X ∂t

D λ34 D t 1 ∙ c< 4 > exp *− + A E R R4 λ34 + BiX BiX − 1

197

Considering that D

(19)

D λ44 D t 1 + < 4 > exp *− + A Ed 4 R R4 λ4 + BiX BiX − 1 depends only on temperature for the simultaneous heat and mass

198

transfer model [44], the effect of the heat transfer on the drying kinetics can be described according

199

to the Arrhenius relationship:

10

D By substitution of D

200

= D: exp A−

Ea E R T + 273.15

in Equation (19) by the definition presented in Equation (20), one can

201

obtain an expression for the drying rate with a new set of parameters:

202

fg fh

∙c

= − X: − X T #6Bi4X ∙ D: Ea λ34 t Ea 1 exp exp D: exp A− A− E *− E+ A 4 E 4 4 R R T + 273.15 R R T + 273.15 λ3 + BiX BiX − 1

D: Ea λ44 t Ea 1 + 4 exp A− E exp *− 4 D: exp A− E+ A 4 Ed R R T + 273.15 R R T + 273.15 λ4 + BiX BiX − 1 203

(20)

(21)

Taken a = 3/R for a spherical particle, Equation (6) can be rearranged to give [37]: 3 dT h ]R_ T − T# dX ∆H = +* + dt ρc dt c

(22)

204

Hence, the final set of differential equations of the coupled heat and mass transfer model

205

proposed is constituted by Equations (21) and (22). The values of BiX , D: and Ea were obtained in

206

preliminary study by fitting Equation (18) together with Equation (20) to experimental drying

207

kinetics data. For all the operating conditions tested, BiX was equal to 2.5, indicating that drying of

208

acai berry wastes is internally controlled by moisture diffusion. The values of D: and Ea were

209

1.39×10‒4 m2·min‒1 and 26.88 kJ·mol‒1 for Tg = 42 °C, respectively, and 2.94×10‒4 m2·min‒1 and

210

25.05 kJ·mol‒1 for Tg = 57 °C, respectively. As expected, the values of the activation energy are

211

very close. The values of λ3 and λ4 for BiX = 2.5 were 2.1746 and 5.0037, respectively, obtained in

212

the work of Carslaw and Jaeger [45]. By those parameter values, the system of differential

213

equations (Equations 21 and 22) was solved simultaneously in a Matlab environment using

214

ode23tb. The initial condition applied to solve Equation (21) was X = X: at t = t0. The initial

215

condition for Equation (22) was T = T0 at t = t0.

216 217

3. Material and methods

218

3.1. Biomass

11 219

The spherical-shape biomass studied in this work consisted of acai berry solid wastes

220

(sphericity greater than 0.9), which were collected from pulp extraction step of the processing plant

221

of acai berry fruit (Euterpe oleracea Mart.) localized in Brazil. The residue consisted mainly of the

222

kernel of the fruit, covered with a rough layer of fibers and some pulp residues, as shown in Figure

223

1. Small samples of the solid waste were packaged in polyethylene vessels, which were then kept in

224

a refrigerator at a temperature of about -6 °C. Prior to drying experiments, the samples were

225

withdrawn from the refrigerator and placed in a suspended sieve for 12 h at room temperature

226

(25 °C) to remove superficial water. This procedure gave initial moisture content with an average

227

value of 0.49 kg H2O ·kg−1 dry material. No pre-treatment was applied to the samples to perform the

228

drying experiments.

229

•

Figure 1

230

Individual particles of acai berry waste had an average diameter of 13.40 ± 0.8 mm, specific

231

mass of 1250 ± 11 kg·m−3 and specific heat (dry) of 1850 ± 125 J·kg−1·°C−1. The mean particle

232

diameter was determined with the aid of a digital caliper. The specific mass of the particles was

233

determined by liquid picnometry using water at temperature of 25 °C, while the specific heat of the

234

dry material was obtained by calorimetry using cold water at 4 °C and solids at temperature of

235

25 °C.

236

The thermogravimetric analysis (TGA) of the dried biomass in an environment of O2 is

237

shown in Figure 2. The heating rate was equal to 10 °C/min. The analysis shows that there is a

238

thermal resistance up to 200 °C. At higher temperatures, there is a significant and fast degradation

239

of the residue of about 50%. This suggests that for applications up to 200 °C, the residue can be

240

used for industrial purposes and indicates the stability of the material for drying up to this

241

temperature. Higher temperatures are also interesting for thermochemical conversion. Referring to

242

drying applications, inlet air temperature with a value of 200 °C is the maximum one to be

243

employed without significant thermal degradation of the material. However, even temperature

244

values below 200 °C must be employed with care, since a significant emission of pollutants

12 245

(volatile organic compounds) begin to appear at low temperature [46]. Even though acai residues

246

drying could be performed up to 200 °C without significant thermal degradation, use of low to

247

moderate temperature permits an efficient and environmentally friendly drying [46]. All physical and thermal properties of the solid waste presented were obtained from a

248 249 250

representative sample by the quartering method. •

Figure 2

251 252

3.2. Drying unit and experimental methodology

253

The experiments were performed in a laboratory-scale tunnel dryer, depicted in Figure 3,

254

which consisted of a drying tunnel 0.1 m of diameter and 1.2 m long, thermally isolated by glass

255

wool insulation and thermal conductivity of about 0.035 W·m−1·K−1, a fan, an air heater constituted

256

of a circular heating element connected to a PID controller, which kept the air temperature within

257

±0.5 °C of the set point. The monitoring of the air temperature inside the drying tunnel was

258

performed based on a temperature sensor, Pt-100, linked to the automatic temperature controller.

259

The relative humidity of the air was detected and monitored by a thermo-hygrometer (Politerm,

260

accuracy ±2.0% and resolution 0.01% and 0.01 °C), inserted in the middle position of the cross

261

section of the drying tunnel. The air velocity was measured by a thermo-anemometer (Highmed,

262

accuracy ±3.0% and resolution 0.01 m/s and 0.1 °C), placed in a point above the samples in order to

263

establish a well-defined velocity of the drying medium.

264

•

Figure 3

265

The experimental unit was developed in order to provide a convection environment with a

266

well-defined air flow direction and reliable data with the aid of both online temperature data-

267

logging data and a controlling system. Fixed bed dryers, like the tunnel dryer developed in this

268

work, are the simplest type of laboratory-scale dryer and commonly utilized in drying experiments

269

because of the greater surface area exposed to the air stream with moderate conditions, along with

270

its simplest conception and operation. In terms of the scientific point of view, this type of dryer is

13 271

very suitable for obtaining drying kinetics in laboratory-scale, since it is very simple to maintain

272

constant the operational conditions of the air, which allows obtaining reliable drying kinetics data

273

that are the basis for energy, modeling, optimization and scale-up studies. Moreover, it is

274

appropriate for particle-level drying. In this drying unit, the samples were subjected to a single-

275

phase air flow under specific conditions of velocity and temperature. As the particle-level

276

experiments were employed in this work, particles of acai berry wastes were inserted in a metallic

277

wire, which was fixed in an insulation block made of polystyrene foam-like material 0.001 m of

278

thick with thermal conductivity of about 0.025 W·m−1·K−1 (Figure 3). This system was inserted

279

1.0 m from the fan, inside the drying tunnel, and it was easily removed for periodic weighting of the

280

sample.

281

The experimental methodology used to obtain drying kinetics data was based on the

282

gravimetric method, which is a standard technique for the absolute determination of moisture of the

283

sample in a laboratory-scale [10]. Initially, the drying unit was adjusted to the desired operational

284

condition of velocity and temperature of the air. The drying unit took approximately twenty minutes

285

to achieve the thermal equilibrium. In such condition, the system constituted of the sample, the wire

286

and the insulated block, was inserted into the tunnel to start the drying kinetics experiments, in

287

which the sample of initially known mass was submitted to the drying process and weighed in pre-

288

established time intervals on an analytical scale (Marte, accuracy ±0.0001 g, resolution 0.0001 g

289

and maximum capacity 2 kg). This procedure, in which was assumed constant dry matter in the

290

system, was repeated until there was no significant mass variation of the sample, indicating that the

291

“practical equilibrium” was attained. Each weighting took about 12 seconds. The initial and final

292

sample weights were about from 3 to 4 g and from 2.1 to 2.8 g, respectively. The dry mass was

293

obtained by keeping the sample in an oven at 105±3 °C for 24 hours. After this period, the sample

294

was kept for 30 minutes in a desiccator to allow it to equilibrate with room temperature, and then

295

took to the analytical scale. This procedure allowed obtaining the drying kinetics of the acai solid

14 296

waste. From the observed values of sample mass with time, the average moisture content in dry

297

basis could be determined by the following fashion, according to the gravimetric method: i= X

m m: − mF J1 + < >K − 1 m: mF

(23)

298

To obtain the temperature of the solid as a function of the time, the same experimental

299

procedure was used to attain the thermal equilibrium of the drying system for the desired operating

300

condition. After this period, the thermocouple was inserted in the center of the particle and then the

301

acquisition system was initiated to take the initial temperature of the sample, rapidly placed into the

302

drying tunnel. The temperature was recorded at every 30 seconds. The data acquisition was stopped

303

when the temperature of the solid remained constant. The temperature of the samples during the

304

experiments was measured separately from the weight measurements, but in the same drying

305

conditions. A pre-calibrated K-type thermocouple (Chromel-alumel, 1.61 mm of diameter, ± 0.5 °C

306

of accuracy) was inserted at the center of the particle to measure its temperature along the time. The

307

thermocouple was connected to the data acquisition system, which consisted of a temperature data

308

logger (Akso, AK176) linked to a computer, where the observed data were recorded. The

309

temperature of the sample was recorded at every 30 seconds. Software Akso Data Logger Linker,

310

version 1.0, was used to save experimental data into a computer text file and then imported into a

311

software package like spreadsheet editors.

312

A total of eight experiments were carried out, in which particles of acai berry waste were

313

dried in the tunnel dryer at dry bulb temperatures of 42 °C and 57 °C and wet bulb temperatures of

314

24 °C and 31 °C, respectively. Low-temperature drying holds a large potential for biomass drying,

315

as it is possible to minimize the emissions of volatile organic compounds to ambient air [46]. The

316

range of the inlet temperature of the gas was chosen in order to analyze the effect of this variable in

317

in the heating rate of the solids, taking in account the experimental deviations. Such wide range is

318

interesting when analyzing the coupling between heat and mass transfer and the approach to the

319

isothermal condition.

15 320

The employed air velocity for those drying conditions was equal to 1.8 m·s−1, which is in the

321

practical range of tunnel dryers reported in literature. In the drying tunnel, this velocity value gave

322

an average air mass flux of 2.13 kg·s−1·m−2. All the operational conditions tested were chosen

323

according to the limitations of the equipment and because such conditions allow the analysis of

324

external and internal controlling mechanisms.

325

The total drying time ranged from 2000 (57 °C) up to 3900 minutes (42 °C). The initial

326

temperature of the samples was about 28 °C and depended on the room temperature, as well as the

327

relative humidity of the ambient air, which depended on the laboratory conditions. Each experiment

328

was repeated once to estimate the experimental errors associated with moisture and temperature

329

measurements. The average standard deviation and variance for moisture content were equal to

330

0.0827 g−1 water·g−1dry material and 6.84×10−3 g−1 water·g−1dry materia and for temperature were

331

1.0792 °C and 1.1647 °C, respectively. The uncertainties of the temperature measurements can be

332

attributed to the complex flow pattern across the single particle. On the other hand, the diameter of

333

the thermocouple is smaller than that of the particle, diminishing the experimental errors associated

334

with temperature measures. Referring to the experimental errors of the gravimetric method, it is not

335

always possible to obtain representative measures at the end of the drying, where the experimental

336

error increases due to the low moisture content and small size of the samples.

337

The drying methodology employed in this work was chosen because the conclusions

338

obtained at particle level can be applicable to understand the drying behavior of a thin-layer of an

339

industrial dryer (known as deep bed), which is divided into several layers of very small thickness.

340

The tunnel dryer in laboratory scale can be considered an “ideal” one, as the results obtained may

341

serve to comprehend other drying technologies at industrial scale, such as rotary dryer, solar dryer,

342

fluidized bed dryer, among others. Furthermore, the isolated particle method permits the analysis in

343

a simple way of the intrinsic behavior of the drying kinetics. This is difficult to attain in some

344

systems where a layer of particles is subjected to a perpendicular air flow that is affected by the

345

porosity distribution, which is a function of the shape and size of the particles. With respect to the

16 346

mathematical modelling, the heat transfer coefficients can be estimated in a more reliable way

347

through the empirical correlations for an isolated sphere. The main limitation of the single particle

348

drying kinetics is the accuracy in the determination of the sample mass, mainly at the end of the

349

experiments, where the experimental error increases when the moisture content of the particle is

350

close to the dynamic equilibrium condition.

351 352

4. Results and discussions

353

The coupled mass and energy transfer model (non-isothermal) represented by Equations (21)

354

and (22) gave adequate predictions for the observed values of temperature and moisture content of

355

the solid as a function of the time, as shown by Figures 4 and 5, mainly for the highest value of the

356

temperature of the drying medium studied. For the lowest value of the inlet air temperature, as

357

shown in Figure 5, predicted values of moisture content somewhat overestimates observed ones at

358

the middle-end period of drying, but the dynamic equilibrium predicted by the model is close to the

359

practical equilibrium determined experimentally. These results indicate that the proposed model can

360

be utilized as an alternative for parameters estimation in this condition, if an optimization method is

361

applied, and that it is suitable for simulation of operational conditions not studied in this work. To

362

some extent, the prediction error could be attributed to the estimated properties that depend on the

363

internal structure, moisture content and the nature of the solid, such as effective transport

364

parameters.

365

•

Figure 4

366

•

Figure 5

367

As presented in Figure 4, increasing the inlet temperature of the gas led to higher heating

368

rates. The fast increasing of the temperature of the particle with time until the stationary state being

369

reached was satisfactory predicted by the mathematical model proposed in this work, indicating an

370

intense heat exchange between solid and gas phases across the boundary layer. This is in

371

accordance with the results shown in Figure 5, as the drying rate was also influenced by the inlet

17 372

temperature of the gas. As more thermal energy is supplied to the system, lower is the total drying

373

time, which may lead to less energy consumption. According to Barati and Esfahani [47], higher

374

initial condition is associated with lower energy needed to increase the temperature, leading to less

375

energy consumption due to experiencing less temperature variation. In drying of biomass, the goal

376

is to use the lower amount of energy possible for a higher moisture evaporation rate for the required

377

final properties of the dried fuel in terms of water content. It is also important to point out that

378

neither the temperature curves nor the moisture content ones denoted the presence of the critical

379

temperature period or the constant drying rate period, indicating that the drying of acai berry wastes

380

is limited by the moisture migration inside the particle. These results are in accordance with

381

Equation (19), which is based on the mass transfer by diffusion.

382

The rapid approach of the particle temperature to the drying air asymptote, as presented in

383

Figure 4, is an indicative that the flat temperature profile inside the particle can be considered [23].

384

Besides, the fact that the time to reach the stationary condition for the heat transfer is lower than

385

that for moisture removal reinforces that assumption. In the view of temperature history of crop

386

materials during drying, the fast approach to the stationary state was also observed in convective

387

drying of maize kernels [48]. According to these discussions, the isothermal approach initially

388

applied to mathematical modeling of convective drying of acai berry solid waste is a valid

389

simplification, which is mainly strengthened by the good prediction of the macroscopic energy

390

balance. In other words, this means that the thermal energy supplied to the dryer is balanced by the

391

latent heat of vaporization of water. The analysis of the isothermal condition is important to be

392

checked, since the main underlying problem in convective dryers is the need to supply the latent

393

heat of evaporation to ensure the maximum energy efficiency possible [49]. Another statement that

394

can be made is that a microscopic energy balance is not necessary to describe the drying behavior.

395

Referring to the coupling between heat and mass transfer in drying, it has been stated by the

396

theories previously presented that in drying of capillary-porous materials there are significative

397

temperature gradients in the material and the coupling effect of these gradients, together with

18 398

moisture evaporation, influences the drying kinetics [25-27]. According to these theories, if the

399

coupling effect is not taken into account, the exclusion of this effect potentially contributed in a

400

large part of the error of the models. Hence, if the complexity of the drying problem arises,

401

Equations (1) and (2) or (3) and (4) are necessary to describe the drying behavior. However, as a

402

simple model is more attractive, like those presented in Equations (21) and (22), it is necessary to

403

investigate in what extent is the coupling between heat and mass transfer. This may be

404

accomplished, as suggested by Gely and Giner [50], by comparing predicted data of moisture

405

content as a function of the time by the isothermal model (Equation 18) and by the non-isothermal

406

model (Equation 21), taking the influence of the instantaneous solid temperature on drying kinetics

407

by Equation (20). Such comparison is presented in Figure 6.

408

•

Figure 6

409

It is possible to note that the drying curves predicted by the non-isothermal is similar to

410

those predicted by the analytical solution of the diffusive model for the isothermal drying

411

assumption. The small difference between predicted data by both models suggests the approach to

412

the isothermal condition, as the resistance to heat transfer does not play a significative hole in

413

drying of acai berry solid wastes.

414

According to the results presented in Figure 6, the analytical solution of the diffusive model

415

can be solely used to properly describe the drying kinetics of the problem studied in this work. As it

416

is possible to verify in Table 1, the analysis of variance (ANOVA), performed with Microsoft

417

Excel (Microsoft Office Professional Plus 2019), shows that the difference between predicted

418

curves is not significative for a confidence interval of 95%, since p > 0.05 and F < Fcritical. Tukey’s

419

statistical test, performed with software Past (Hammer and Harper, version 2.17c, 2013), revealed

420

non-significant differences between the curves presented in Figure 6 at a confidence level of 95%.

421

•

Table 1

422

Few studies presented by literature use the derived equation of the analytical solution of the

423

diffusive model considering the Robin-type boundary condition at the surface of the particle to

19 424

predict drying kinetics data and relating the simulation results with the analysis of the isothermal

425

condition approach in drying, along with a non-isothermal model. The methodology of analysis

426

proposed is the main contribution of this work along with the analysis of the mathematical

427

modelling of drying kinetics at particle scale. However, the discussions drawn considered only a

428

spherical particle and uniform temperature distribution inside the solid material, as the heat transfer

429

was governed by external control. Thus, for larger particles of agricultural residues, the isothermal

430

condition must be analyzed with care as internal temperature gradients may play a key hole in

431

drying process. On the other hand, the methodology proposed can be also extended to other fruit

432

stones and kernels of crop products with size similar to acai berry wastes, such as jambolão (3.11

433

mm Sauter diameter) [51], almonds (12.87 mm geometric mean diameter) [52], jatropha (11.57 mm

434

geometric mean diameter) [53], olive stone (3 mm average diameter) [54] and apricots (9.138 mm

435

geometric mean diameter) [55].

436 437 438

5. Conclusions •

From the characterization experiments, it was obtained the following thermophysical

439

properties of acai berry waste (fruit stone): diameter of 13.40 mm, specific mass of

440

1250 kg·m−3, specific heat of 1850 J·kg−1·°C− and sphericity of 0.91;

441

•

The non-isothermal model provided accurate predicted data, despite previous studies

442

presented in literature that proposed more complex models to describe drying

443

behavior of particulate materials in terms of the coupling between heat and mass

444

transfer phenomena;

445

•

Statistical analysis and simulation results showed that there was not a significant

446

difference between the drying curves predicted by both non-isothermal and

447

isothermal (analytical solution of the diffusive model) models, which demonstrates

448

that a macroscopic energy balance is adequate to predict temperature history of the

449

solid as a function of the time, and that the analytical solution of the diffusive model

20 450

can be solely used to describe the drying kinetics of convective drying of acai berry

451

wastes;

452

•

Predicted results by both diffusive and non-isothermal models indicate that there is

453

not a strong coupling between heat and mass transfer, leading to the assumption of

454

the isothermal drying condition and moisture diffusion control;

455

•

The main transport parameters of the non-isothermal model were the heat transfer

456

coefficient, estimated from the empirical correlation of Whitaker for an isolated

457

sphere (47.78 W·m‒2·°C‒1, for T = 42 °C, and 42.55 W·m‒2·°C‒1 for T = 57 °C) and

458

the pre-exponential factor and activation energy (1.39×10‒4 m2·min‒1 and 26.88

459

kJ·mol‒1 for Tg = 42 °C, respectively, and 2.94×10‒4 m2·min‒1 and 25.05 kJ·mol‒1 for

460

Tg = 57 °C, respectively), estimated from the Arrhenius correlation, used to link heat

461

and mass balances from the effective moisture diffusivity. The optimum value of

462

mass transfer biot number was equal to 2.5, indicating that the moisture transfer by

463

diffusion controls the drying dynamics;

464

•

As the particle size of the biomass plays a key hole in coupling between heat and

465

mass transfer, and therefore it has an important impact in drying, the present study

466

can be also extended to other fruit stones with size similar or smaller than the acai

467

residues (Dp < 13 mm), such as jambolão, almonds, jatropha, olive stone and

468

apricots, in which the internal temperature gradients can be neglected.

469

•

The findings obtained in this work may be useful for optimization studies, which will

470

permit the balance between operation costs and the energy consumption of drying in

471

order to obtain the most suitable situation to be implemented in pre-thermal

472

treatment process of the biomass studied in this work.

473 474 475

21 476

Acknowledgement The authors express their gratitude to Fundação de Amparo à Pesquisa do Estado de Minas

477 478

Gerais (FAPEMIG), Brasil, for funding provided to Project APQ-03095-18.

479 480

Nomenclature av Bim cps cpds D0 Deff Dp Ea h hb hs hv hw keff kg m m0 md Pr r R ReD Rg t T Tg T0 !" w X Xk X0 Xeq β λ δ ∆H

Specific particle surface Biot number for mass transfer Specific heat of the solid Specific heat of the dry solid Pre-exponential factor of the Arrhenius relationship Diffusivity tensor of bound water Effective moisture diffusivity Particle diameter Diffusivity tensor of water vapor Activation energy Dimensionless diffusivity tensor Heat transfer coefficient Enthalpy of bound water Enthalpy of solid phase Enthalpy of water vapor Enthalpy of liquid phase Effective thermal conductivity Thermal conductivity of the gas Mass of the solid at time t Initial mass of the solid at time t = 0 Mass of dry solid Prandtl number Radial coordinate Radius of the particle Reynolds number Ideal gas constant Time Temperature Temperature of the gas Initial temperature of the solid at time t = 0 Velocity of the liquid phase Vapor mass fraction Moisture content at time t Moisture content at time t (average) Initial moisture content at time t = 0 Equilibrium moisture content

m²/m³ J/kg·°C J/kg·°C m²/min m²/min m²/min m m²/min J/mol W/m²·°C J/kg J/kg J/kg J/kg W/m·°C W/m·°C kg kg kg m J/mol·K min °C °C °C m/min kg/kg kg/kg kg/kg kg/kg

Greek symbols Apparent coefficient of mass transfer Roots of the transcendental equation Thermogradient coefficient Latent heat of evaporation

m/min kg/°C J/kg

22

ε ε ε µ µs ρg ρs ρv ρw ϕ

Gas volume fraction Solid volume fraction Liquid volume fraction Gas viscosity Gas viscosity evaluated at the surface temperature Specific mass of the gas phase Specific mass of the solid phase Specific mass of the water vapor Specific mass of the liquid phase Phase conversion factor

kg/m·s kg/m·s kg/m³ kg/m³ kg/m³ kg/m³ -

481

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Table Captions Table 1: Analysis of variance (ANOVA) for the predicted results of both isothermal and nonisothermal model. Figure Captions Figure 1: Particles of moist acai berry waste: (a) fruit stones and (b) fruit stone cut in half.

615 616

Figure 2: Thermogravimetric analysis (TGA) curves showing the mass loss profile of acai berry wastes.

617

Figure 3: Schematic sketch of the drying unit.

618 619

Figure 4: Observed temperature data of the single particle as a function of the time predicted by the non-isothermal model.

620 621

Figure 5: Observed moisture content data as a function of the time predicted by the non-isothermal model.

622 623

Figure 6: Predicted data of moisture content as a function of the time by the isothermal (diffusive model) and the non-isothermal model (coupled heat and mass transfer).

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Table 1: Analysis of variance (ANOVA) for the predicted results of both isothermal and non-isothermal model.

Tg [°C]

F

Fcritical

p-Value

42

0,0383

3,9361

0,8452

57

0,0111

3,9381

0,9160

(a)

(b)

•

Mathematical modelling considering the coupling between heat and mass transfer.

•

Comparison between observed and predicted data for different inlet air conditions.

•

Diffusive model can be solely used to predict experimental moisture content data.

•

The isothermal drying condition approach can help to study other biomass drying.

•

Estimated parameters values are in the range of biological materials drying.

Declaration of interests x The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: