Multiphase transfer model for intermittent microwave-convective drying of food: Considering shrinkage and pore evolution

Accepted Manuscript

Multiphase transfer model for intermittent microwave-convective drying of food: Considering shrinkage and pore evolution Mohammad U.H. Joardder , C. Kumar , M.A. Karim PII: DOI: Reference:

S0301-9322(16)30266-X 10.1016/j.ijmultiphaseflow.2017.03.018 IJMF 2591

To appear in:

International Journal of Multiphase Flow

Received date: Revised date: Accepted date:

5 May 2016 16 January 2017 19 March 2017

Please cite this article as: Mohammad U.H. Joardder , C. Kumar , M.A. Karim , Multiphase transfer model for intermittent microwave-convective drying of food: Considering shrinkage and pore evolution, International Journal of Multiphase Flow (2017), doi: 10.1016/j.ijmultiphaseflow.2017.03.018

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.

ACCEPTED MANUSCRIPT

Highlights: 

Multiphase modelling of IMCD provides a better understanding of the physics



Consideration of shrinkage in model provide more realistic results



Intermittency of microwave result in uniform moisture and tempurature

AC

CE

PT

ED

M

AN US

CR IP T

distribution

ACCEPTED MANUSCRIPT Multiphase transfer model for intermittent microwave-convective drying of food: Considering shrinkage and pore evolution Mohammad U. H. Joarddera,b*, C. Kumara, and M. A. Karima a

Chemistry, Physics and Mechanical Engineering, Queensland University of Technology, Brisbane, Australia b Rajshahi University of Engineering and Technology, Rajshahi, Bangladesh

CR IP T

Abstract

Intermittent microwave convective (IMCD) drying is an advanced drying technology that improves both energy efficiency and food quality during drying. Although many experimental studies on IMCD have been conducted, there is no complete multiphase

AN US

porous media model describing the physics of the process available in the literature. A multiphase porous media model considering liquid water, gases and the solid matrix of food during drying can provide in-depth understanding of IMCD process. Currently there is no IMCD model that have taken shrinkage and pore evolution

M

during drying into consideration. In this study, first a multiphase porous media model

ED

with shrinkage (IMCD2) has been developed for IMCD. Then the model has been compared with IMCD model without shrinkage (IMCD1). Simulated temperature,

PT

moisture content, density, porosity from IMCD2 are then validated against experimental data. The profile of vapour pressures and evaporation during IMCD are

CE

also presented and discussed.

AC

Keywords: Intermittent Microwave Convective Drying, Mathematical Model, COMSOL Multiphysics, Multiphase Porous media

1

Introduction Intermittent drying is a comparatively improved drying method for plant-based

food materials. Intermittency in drying can be attained by supplying the drying energy by varying the drying temperature, airflow, humidity, or operating pressure

ACCEPTED MANUSCRIPT [1]. It can also be achieved by changing the mode of energy input such as microwave, convection, radiation, and conduction. Supply of

same amount of

energy during entire drying process causes severe quality degradation [2] and wastage of energy. Case hardening or other heat damages can be occurred on the surface of the sample in case of constant external energy application. Intermittent

CR IP T

application of secondary energy reduces the chances of overheating as temperature and moisture are redistributed during tempering periods. Consequently, surface damage and overall quality degradation can be minimized.

AN US

Due to uneven power distribution of microwave (MW), continuous supply of MW energy in food material causes uneven heating [3]. Intermittent use of MW can overcome this overheating problem[4]. This intermittent MW maintains uniform distribution of moisture and temperature within the food materials during drying and

M

therefore improves the process efficiency and product quality [5]. Many researchers

ED

incorporated intermittent use of microwave energy with different drying techniques such as vacuum drying [6, 7] and freeze drying [8], and found superior quality

PT

attributes of dried samples.

CE

Soysal et al. [9] compared the energy efficiency of IMCD and convective drying and found that the IMCD was 4.7–11.2 times more energy efficient process

AC

compared to convective drying. In another study Soysal et al. [5] found that IMCD treated red pepper showed better sensory attributes, colour and texture compared to convective drying and continuous microwave convective drying. Researchers have attempted to develop models for continuous microwave convective drying without considering intermittency. Those are only either empirical models [10, 11], or single-phase diffusion based models [12, 13]. As these models

ACCEPTED MANUSCRIPT did not take into intermittency of microwave power account, IMCD process could not be described with these models. There are a some empirical models for IMCD available in the literature [14]. However, these empirical models are only applicable for specific experimental conditions [15, 16] and therefore can not be generalised. Apart from empirical

CR IP T

models, there are some diffusion based theoretical MW heating models that considered intermittency of microwave power [17-20]. These intermittent MW heating models only considered heat transfer in the food materials and neglected the None of these intermittent heating models investigated the

AN US

mass transfer.

temperature distribution and redistribution during the tempering period due to intermittent use microwave power, which is critical for overcoming overheating problem in IMCD. Therefore, it can be concluded that currently there is no

M

theoretical model that can describe multiphase heat and mass transfer mechanism during IMCD. Moreover, no IMCD drying model has considered shrinkage during

ED

drying. As shrinkage is indispensable phenomena during drying, a realistic model

PT

must take shrinkage into consideration. Level of shrinkage is directly related to the pore formation and evolution during drying. Therefore, proper prediction of porosity

CE

is essential in order to develop a comprehensive drying model appropriate for porous

AC

media.

In this study, two multiphase porous media IMCD models have been

developed; one considering shrinkage and the other without considering shrinkage. The model that without consideration of shrinkage is termed as as IMCD1 and the model that takes shrinkage into consideration is referred as IMCD2. The simulated moisture and temperature profiles were compared with experimental results and thus the models were validated. The temperature distribution and redistribution during

ACCEPTED MANUSCRIPT tempering period due to the intermittent use of microwave application was also investigated. Finally, different transport mechanisms such as pressure driven, diffusion and binary diffusion driven and capillary driven flows during IMCD were investigated.

Mathematical model

CR IP T

2

In this section, the equations for multiphase porous media transport are developed in order to describe the heat, mass and momentum transfers for IMCD. Transport mechanisms involved in drying and assumptions and input parameters

AN US

considered for the model have also been presented.

The models developed in this study considered transport of liquid water, vapour and air inside the sample. The mass conservation equations contain convection,

M

diffusion and evaporation of water and vapour and the energy equations include the conduction, convection and microwave heat generation term.

Momentum

ED

conservation is developed from Darcy’s equations. Evaporation is considered as

PT

distributed throughout the domain and a non-equilibrium evaporation formulation is used for evaporation-condensation phenomena. Problem description and assumptions

CE

2.1

AC

A schematic of the sample together with transport mechanisms is presented in

Figure 1. Heat and mass transfer take place at all boundaries except the symmetry boundary (r=0). The apple slice is considered as a porous media where the pores are filled with three transportable phases, namely liquid water, air and water vapour as shown in Figure 1.

ACCEPTED MANUSCRIPT Please insert Figure 1 about here

It is assumed that all three phases are continuous and local thermal equilibrium exists, which means that the temperatures in all three phases are equal.

CR IP T

Liquid water transport occurred due to convective flow resulting from gas pressure gradient, capillary flow and evaporation. Vapour and air transport take place from gas pressure gradients and binary diffusion.

The driving force of deformation is associated with a change in moisture

AN US

content and the gradients in gas pressure. Structural deformation and porosity of the material affect multiphase transport while heat and mass transfer change mechanical properties of the sample that result in pore formation and evolution. Eventually, the

M

drying problem experiences two-way coupled with material deformation, and heat

ED

and mass transport as illustrated in Figure 2. Please insert Figure 2 about here

PT

To develop the multiphase IMCD model of an apple sample, the following

AC

CE

assumptions have been taken into consideration: 

Non-equilibrium condition prevails during evaporation.



Evaporation takes place through the entire domain.



All phases maintain local thermal equilibrium.

The governing equations for multiphase transport phenomena in porous media, and Arbitrary–Lagrangian– Eulerian (ALE) for moving mesh and solid mechanics for large deformations were solved using Finite Element software COMSOL Multiphysics 4.4 (COMSOL, Burlington, MA). Mass, energy and momentum

ACCEPTED MANUSCRIPT conservation for various species were solved using transport of dilute species, heat transfer in fluid modules, Darcy’s Law modules, respectively. ALE framework and Non-linear Solid Mechanics module were used for moving mesh and solid momentum balance respectively.

2.2.1

Governing equations

CR IP T

2.2

Mass balance equations

The instantaneous representative elementary volume (REV) for the sample is presented in Figure 2. The REV ( V ) is the summation of the volume of gas,

AN US

water, and solid phase. Therefore, V  Vg  Vw  Vs

(1)

Where, Vw is the volume of water (m3), V g is the volume of gas (m3), and

M

Vs is the volume of solid (m3). The apparent porosity,  , is defined as the volume



Vg  Vw

PT

ED

fraction occupied by water and gas. Thus,

V

(2)

.

CE

Water saturation ( S w ) and gas saturation ( S g ) are defined as the fraction of

AC

pore volume occupied by the individual phase,

Sw 

and S g 

Vw Vw  , Vw  Vg V Vg Vw  Vg



Vg

V

(3)

 1  Sw ,

(41)

The mass concentrations of water ( cw ), vapour ( c v ), and air ( c a ) in kg/m3 are calculated by,

ACCEPTED MANUSCRIPT cw   wS w , cv 

(2)

pv S g , RT

and ca 

(3)

pa S g , RT

(4)

CR IP T

Where, R is the universal gas constant (J/mol/K),  w is the density of water (kg/m3), p v is the partial pressure of vapour (Pa), T is the temperature of product (K) and p a is the partial pressure of air (Pa).

AN US

Mass conservation of liquid water

The mass conservation equation for the liquid water is expressed by, (8)

M

*  S w  w     nw    I , t

*  Where, nw is water flux (kg/m2s), and I  Revap is the evaporation rate of liquid

ED

water to vapour (kg/m3s).

PT

The total flux of the liquid water is due to the gradient of liquid pressure,

CE

pw  P  pc , as given Darcy’s Law [21]:

AC

k w k r ,w k w k r ,w k w k r ,w  nw    w p w    w P   w pc .

w

w

(9)

w

Where, P is the total gas pressure (Pa), p c is the capillary pressure (Pa), k w is

the intrinsic permeability of water (m2), k r , w is the relative permeability of water, and

 w is the viscosity of water (Pa.s). Capillary pressure in the pressure that tries to hold the water within the pore. Most of the plant-based food materials are classified as capillary-porous materials having pore diameter less than 10-7 m [22]. The capillary

ACCEPTED MANUSCRIPT pressure during drying mainly depends on moisture content and temperature gradient although. There is also very small effect of pore diameter on capillary pressure. As very little experimental data of capillary diffusivity relating moisture content and temperature is available in literature, there is no study that dealt with the effect of pore size on capillary pressure.

CR IP T

considering that the capillary pressure depends upon concentration ( cw ) and temperature (T) for a particular material [22], further breakdown of Equation 9 can be presented as,

AN US

k w k r ,w k w k r ,w pc k w k r ,w pc  nw    w P   w c w   w T . w  w c w  w T

(10)

The capillary diffusivity due to the temperature gradient is known as a Soret effect and is insignificant compared to the capillary diffusivity due to concentration

M

gradients [22]. Therefore, the capillary diffusivity due to temperature will be

ED

neglected in this work. Substituting the above into Equation 8 can be written as,

(11)

PT

k k k k    S w  w        w w r ,w P   w w r ,w pc    Revap t w  w T  

CE

Mass conservation of water vapour

AC

The conservation of water vapour can be written as follows:  S g  gv     nv   Revap , t

(12)

Where, v is the mass fraction of vapour,  g is the density of the gas (kg/m3) and

 nv is the vapour mass flux (kg/m2s).

 As gas is a binary mixture, nv can be written as [23],

ACCEPTED MANUSCRIPT k g kr ,g  nv    vv P  S g  g Deff , g v ,

v

(13)

where, k r , g is the relative permeability of gas (m2), k g is the intrinsic permeability of gas (m2), Deff , g is the binary diffusivity of vapour and air (m2/s) , and

CR IP T

 g is the viscosity of gas (Pa.s) . Mass fraction of air

After calculating the mass fraction of vapour, v from water vapour mass

AN US

conservation equation, the mass fraction of air,  a , can be calculated from

 a  1  v .

2.2.2

(14)

Continuity equation for pressure

M

The gas pressure, P, can be determined by the total mass balance for the gas phase, as shown in eqn (15).

ED

 gS g     ng   Revap , t

(15)

AC

CE

PT

 Where, the gas flux, ng , is given by,

k k  ng    g g r , g P .

i

(16)

Here,  g is the density of gas phase, given by,

g 

PM g RT

,

M g is the molecular weight of gas (kg/mol).

(17)

ACCEPTED MANUSCRIPT 2.2.3

Energy equation Assuming thermal equilibrium of all the phases, the energy balance equation

that considers conduction, convection, radiation heat transfer, energy sources and sinks due to evaporation and condensation can be written as:   T    ng hg  nw hw   .(keff T )  h fg Revap  Qm f (t ) t

(18)

CR IP T

 eff c p eff

where,  eff is the effective density (kg/m3), T is the temperature (K) of each phase, hg is the enthalpy of gas (J), k eff is the effective thermal conductivity (W/m/K),

AN US

hw is the enthalpy of water (J), h fg is the latent heat of evaporation (J/kg), c p eff is the effective specific heat (J/kg/K), and f(t) is an intermittency function of microwave source (Qmic).

Thermos-physical properties

ED

2.3.1

Input parameters and properties

M

2.3

The thermo-physical properties of the REV were obtained by the volume-

AC

CE

PT

weighted average of different phases as calculated by the following equations,

 eff   S g  g  S w  w   1    s

(19)

c p eff   S g c pg  S w c pw   1   c ps

(20)

and keff   S g kth, g  S w kth,w   1   kth,s

(21)

where  s is the solid density (kg/m3). k th , g , k th , w , and k th , s are the thermal

conductivities of gas, water, and solid, (W/m/K) respectively and c pg , c pw , and c ps are the specific heat capacity of gas, water, and solid (J/kg/K), respectively.

ACCEPTED MANUSCRIPT 2.3.2

Evaporation rate A non-equilibrium formulation as described in Ni et al. [24] was used to

calculate the evaporation rate: Revap  K evap

Mv  pv,eq  pv . RT

(22)

CR IP T

where, pv ,eq is the equilibrium vapour pressure (Pa), M v is the molecular weight of vapour (kg/mol), p v is the vapour pressure (Pa), and Kevap is material and process dependent the evaporation constant (s-1).

AN US

Equation (22) for calculating evaporation rate has been used in literature for transport with phase change in other hygroscopic materials

[25, 26] and it is

consistent with statistical rate theory based analysis for evaporation from pure water

M

surface [27].

It is worthy to mention that K denotes evaporation rate constant and defined as

ED

transition of a molecule from liquid water to vapour. For different plant based food materials, K is used in the range of 1000-100000 s-1 [28]. However, is K=100000 s-1

PT

value is used, convergence in the numerical scheme and the computation time

CE

becomes a huge concern. In addition, it is found that there is very negligable difference in the results for K=1000 s-1 and higher values. Taken all these into

AC

consideration, K=1000 s-1 has been used in this study. The equilibrium vapour pressure of apple, as presented in Equation 23,can be

obtained from the sorption isotherm [29].



pv,eq  Pv,sat exp  0.182M 0.696  0.232 e 43.949M M

0.0411

ln Pv,sat



(23)

ACCEPTED MANUSCRIPT where, M is the moisture content (dry basis), Pv , sat is the saturated vapour pressure of water (Pa) which can be related to S w through the following equation:

M db 

S w  w . 1    s

(24)

CR IP T

The saturated vapour pressure of water, Pv , sat , is a function of temperature and is calculated using the regression equation developed by Vega-Mercado et al. [30]:  5800.2206 / T  1.3915  0.0486T  0.4176 x10 4 T 2  Pv ,sat  exp  . 7 3  0.01445 x10 T  6.656 ln T  

(25)

AN US

The vapour pressure, p v , is obtained from partial pressure relations given by,

pv   v P ,

(26)

M

Where,  v is the mole fraction of vapour and P is the total pressure (Pa).

ED

The mole fraction of vapour,  v , can be calculated from the mass fractions and

v 

v M a , v M a   a M v

(27)

CE

PT

molecular weight of vapour and air as shown in equation (27):

Where, M v is the molecular mass of vapour (kg/mol), M a is the molecular

AC

mass of air (kg/mol). 2.3.3

Solid deformation Plant-based food materials are deformable under the condition of drying due to

simultaneous heat and mass transfer [31, 32]. From the multiphase transport model, it is easy to calculate the related forces that causes deformations during drying. The material properties are also the dominant factors for deformations.

ACCEPTED MANUSCRIPT Mechanical properties during drying are not available in the literature. If we consider a very basic realistic model, a Neo-Hookean constitutive model, as presented in Equation 28, can be used in order to determine material deformation. Ws 

1 1  ( I1  3)   ln J el   ( ln J el ) 2 2

(28)

CR IP T

In the equation, I1 is the first invariant of the right-Cauchy-Green tensor, whereas, µ is the shear modulus of the material and λ is the Lame’s constant which



E 2(1   )

AN US

can be calculated using the following equations:  

Where, E is the Elastic modulus

and 

m

E (1   )(1  2 )

and

is the Poisson ratio.

Eventually, the constitutive model depends on E and  . The value of these two

M

mechanical characteristics significantly changes throughout the drying process. There is no literature that used constitutive model with variable mechanical

ED

characteristics during IMCD. Gulati et al [33] applied a moisture dependent elastic

PT

modulus for potato sample in their mechanistic model for multiphase modelling during convective drying. The reference work of Gulati et al [31] was the work of

CE

Yang and Sakai [34] where they determined variable Elastic modulus for different moisture contents. They found an increasing E with decreasing moisture content.

AC

However, Yang and Sakai [34] conducted their experiments only at room temperature. Therefore, this variable E cannot be taken into consideration as E strongly depends on temperature and values determined in room temperature cannot be the same as E at higher temperatures. Materials remain rubbery with various degree of softness at higher temperatures during drying. Moreover, the value of E

ACCEPTED MANUSCRIPT should show a decreasing trend with a decrease of moisture content [35] until the sample reaches the glass phase when the sample starts to show brittleness.

Gulati et al. [33] considered that mechanical properties are correlated only with

CR IP T

moisture contents. However, the mechanical properties of processed plant-based food materials also depend upon cell wall strength, inter-cellular adhesion, ease of fluid flow and cell wall porosity [36]. Furthermore, processing temperature has a significant effect on the mechanical properties. Most of the fruits and vegetables

AN US

show decreasing firmness of tissue with the increase in drying temperature [37]. In this study, the volumetric deformation due to moisture migration will be calculated by using a semi-empirical shrinkage velocity approach as described below

2 Dw  T  Tg  L  T0  Tg 0 

(29)

ED

vs 

M

[38]:

PT

Dw is effective water diffusivity and this can be obtained from the total flux of the liquid water at any particular time of drying. The concept of shrinkage velocity

CE

has also been found in other literature [39-41].

AC

2.3.4

Porosity calculation

The instantaneous bulk volume from the above shrinkage velocity approach of

the material during drying can be calculated by the following equation:

V   r0  vst  L0  vst  2

(30)

where V is the instantaneous volume of water that changes with drying time t, r and r0 are the instantaneous and initial radius of the sample, L and L0 are the

ACCEPTED MANUSCRIPT instantaneous and initial thickness of the sample. Over the drying duration, the instantaneous radius and the thickness of the sample can be determined with the application of shrinkage velocity approach. Assuming that the mass of solid phase remains constant, the instantaneous

sV (1   )  sV0 (1  0 ) (31)

 1

AN US

after simplification

CR IP T

total porosity can be determined by the following relationships:

1  0 V V0

(32)

M

Therefore, gas porosity can be expressed as

2.3.5

(33)

ED

    Sg Initial conditions

PT

The initial conditions for equations 8, 9, 15, and 18 are given by[42], (34)

wv (t 0)  0.0262 ,

(35)

P(t 0)  Pamb ,

(36)

and T(t 0)  303K ,

(37)

AC

CE

cw(t 0)   wS w0

respectively. 2.3.6

Boundary conditions

Figure 3, shows the boundary conditions considered in this study.

ACCEPTED MANUSCRIPT

Please insert Figure 3 about here

 Total vapour flux, nv ,total , from a surface with only gas phase can be written as,

 p -p   nv ,total  hmv v v air RT

CR IP T

(38)

 Where, nv ,total is the total vapour flux at the surface (kg/m2s), hmv is the mass

AN US

transfer coefficient (m/s), and pv ,air is the vapour pressure of ambient air (Pa).

In a multiphase problem, the total vapour flux from the surface is the combination of evaporation from liquid water and existing vapour on the surface.

 p -p   nw  hmvS w v v air RT

(39)

 p -p   nv  hmvS g v v air RT

(40)

PT

ED

(39) and (40) respectively.

M

The boundary conditions for water and vapour phase can be written by equations

CE

In food drying, the pressure at the boundary is equal to the ambient pressure,

AC

Pamb . Hence, the pressure boundary condition can be expressed as, P  Pamb .

(41)

Energy is transferred by convective heat transfer whereas heat is lost due to

evaporation at the sample surface as shown by equation (42). q surf  hT (T  Tair )  hmvS w

 pv -pv air h RT

fg

.

(42)

ACCEPTED MANUSCRIPT Here, hT is the heat transfer coefficient (W/m2/K) and Tair is the drying air temperature (K). 2.3.7

Values of Input parameters The input parameters for the model are listed in Table 1. Some of the

CR IP T

parameters that are not listed in Table 1 are derived and discussed in the subsequent sections.

2.3.8

AN US

Please enter Table 1 about here

Permeability

Permeability is an important propety for describing the water transport due to the pressure gradient in porous media. The value of the permeability determines the

M

extent of pressure generation inside the material. The smaller the permeability, the

ED

lower the moisture transport rate and the higher the internal pressure, and vice versa. According to Darcy’s law for laminar flow, pressure drop and flow rate show

PT

a linear relationship in a porous material [43]. The slope of this line is related to

AC

CE

permeability as shown in the equation below.

k

Q L A P

(43)

Where, k is the permeability, Q is the volumetric flow rate, � is the fluid

viscosity, A is the sample cross sectional area, and

is the pressure gradient.

The permeability k of a sample is the product of intrinsic permeability, k i , of the material and relative permeability, ki ,r , of the fluid in that material [21]. Therefore, it can be written,

ACCEPTED MANUSCRIPT k  ki ki ,r .

(44)

The intrinsic permeability, ki, represents the permeability of a liquid or gas in the fully saturated state. The relative permeability of a phase is a dimensionless measure of the effective permeability of that phase. It is the ratio of the effective

CR IP T

permeability of that phase to the absolute permeability. The intrinsic permeability depends on the pore structure of the material and the relationship can be expressed as [44]:

3 1   2

0.39    0.77

AN US

k w  5.578  10 12

The gas intrinsic permeability kg was found to be 7.4 10 and 6.5 10

13

(45)

12

 1.2 1012 m2

 2.4 10 13 m 2 at a moisture level of 36.0% (db) of 60.0% (db) 12

m 2 as the value of kg that comes from

M

respectively. In this study, we used 4.0 10

ED

the average value of gas intrinsic permeability at the above mentioned moisture level [44].

PT

Relative permeabilities are generally expressed as functions of liquid saturation. Feng et al. [44], reported relative permiabilities for liquid and gas as

AC

CE

follows:

2.3.9

k r ,w  S w

3

and k r , g  1.01e 10.86S

(46)

w

(47)

Viscosity of water and gas

Viscosities of water [45] and gas [46] as a function of temperature are given by,

ACCEPTED MANUSCRIPT

 w   we

1540   19.143  T  

(48)

 T  and  g  0.017 10 3    273 

0.65

(49)

2.3.10 Effective gas diffusivity

CR IP T

The effective gas diffusivity can be calculated as a function of gas saturation and porosity according to the Bruggeman correction [47] as given by the following equation:

Deff , g  Dva S g  .

(50)

AN US

4/3

Here, binary diffusivity, Dva , can be written as,

(51)

1.81

M

Dva  2.3 10

5

and

ED

Where was considered as

P0  T    P  T0 

,

. In this study effective gas diffusivity

[48].

PT

2.3.11 Capillary diffusivity of liquid water

CE

Liquid in food material can flow from the higher concentration to the lower concentration of water due to the difference in capillary action. This is referred to as

AC

―unsaturated‖ flow and is extremely important in drying. Capillary force is the main driving force for liquid water in drying [47]. The capillary diffusivity, Dc , is a function of capillary pressure is considered as the proportional to

p c . S w

It can be found in the literature that the capillary pressure increases significantly at lower saturation levels and when it reaches irreducible saturation the

ACCEPTED MANUSCRIPT value becomes infinity. Therefore, that situation is neglected to avoid numerical instability. The underlying physics is that as S w approaches 1, more water becomes free and the resistance of the solid matrix to the flow of free water becomes almost zero. Therefore, Dc is very large at high moisture content. Therefore, the concentration gradient is small [47] and as an outcome, the capillary diffusivity can

CR IP T

be very close to effective moisture diffusivity for very wet material when vapour diffusion is insignificant.

Considering that the highest value corresponds to the highest saturation of

AN US

water, a relationship between capillary diffusivity and moisture as given by Ni [47] and shown in equation (52) is used in this study.

Dc  108 exp  6.88  8M wb  .

(52)

M

In this study, a similar function was developed for apple by analysing the

ED

values of different effective diffusivities presented in the literature [11, 39, 49, 50]. 2.3.12 Microwave power absorption

PT

Lambert’s Law has been widely used for developing microwave heating models in literature [12, 13, 51-55]. In this study, Lambert’s Law has been used to

CE

calculate the microwave energy absorption inside the food samples. This law

AC

considers exponential attenuation of microwave absorption within the product, given by,

Pmic  P0 exp 2 h z  .

Here,

(53)

the incident power at the surface (W), α is the attenuation constant, h

is the thickness of the material, and (h-z) is the distance from top surface (towards centre).

ACCEPTED MANUSCRIPT The attenuation constant, α is given by,

    '' 2  1      1    '        2   

(54)

CR IP T

   2  '     

Where,  is the wavelength of microwave in free space (   12.24cm at 2450MHz and air temperature 200C) and ε' and ε" are the dielectric constant and the dielectric loss, respectively.

AN US

The volumetric heat generation, Qmic (W/m3) is then calculated by; Qmic 

Pmic V

(55)

ED

2.3.13 Dielectric constant

M

Where, V is the volume of the sample (m3).

The dielectric constant, ε' and dielectric loss, ε", are the most important

PT

parameters that control the microwave power absorption of the materials. Here we use the data of Martín-Esparza et al. [56] in a quadratic regression analysis in which

CE

the intercept of the  ' and   versus M wb graph was set to 0.1 in order to avoid

AC

numerical singularity in  ' and   when M wb is zero. The resulting quadratic expression are found to be,

 '  36.638M wb 2  30.289M wb  0.1

(56)

   13.543M wb 2  26.8150.M wb  0.1.

(57)

ACCEPTED MANUSCRIPT 2.3.14 Heat and mass transfer coefficients

The heat transfer coefficients are calculated from well-established correlations of Nusselt number for laminar and turbulent flow over flat plates as shown in

CR IP T

equations (58) and (59) [16, 39, 57].

hT L  0.664 Re 0.5 Pr 0.33 (Turbulent) k

(58)

Nu 

hT L  0.0296 Re 0.5 Pr 0.33 (laminar) k

(59)

AN US

Nu 

Where L is characteristics length (m), Re is the Reynolds number, and Pr is the Prandtl number.

M

Mass transfer coefficients were calculated by the Sherwood number (Sh) and

Sh 

hmL  0.332 Re 0.5 Sc 0.33 (Turbulent) Dva

(60)

PT

ED

the Schmidt number (Sc) as shown in equations (60) and (61):

hmL  0.0296 Re 0.8 Sc 0.33 . (laminar) Dva

(61)

CE

Sh 

AC

Here Dva is the binary diffusivity of vapour and air (m2/s).The values of Re, Sc

and Pr, were calculated from following equations:

and

,

(62)

,

(63)

,

(64)

ACCEPTED MANUSCRIPT

Here,

is density of air (

drying air velocity (m/s),

),

is dynamic viscosity of air(

is thermal conductivity of air (

),

is

).

In this study, the heat and mass transfer coefficients calculated using the above equations were found to be hT =16.746 W/(m2K) and hm =0.017904 m/s respectively.

CR IP T

After gathering all the required input parameters and initial conditions for the governing equations and boundary conditions, the multiphase porous media IMCD models were simulated in COMSOL Multiphysics. The simulated results were then

3

Materials and methods

AN US

validated by experimental data as discussed in the following sections.

In this section the experimental procedures, sample preparation and data

IMCD drying

ED

3.1

M

acquisition method are discussed.

The intermittency in MW energy was achieved by placing a sample in a

PT

microwave oven for 20s followed by convection drying of the same sample for 80s (tempering period) in the convection dryer. The experiments were conducted with a

CE

Panasonic Microwave Oven (Model NNST663W) having inverter technology with

AC

internal cavity dimension of 352mm (W) x230mm (H) x347mm (D). The inverter technology enables accurate and continuous power supply at lower power settings [58]. The microwave oven is able to supply 10 accurate power levels with a maximum of 1100W at 2.45GHz frequency. The apple slices were placed in the centre of the microwave cavity, in order to achieve an even absorption of microwave energy. The moisture loss was recorded at 10 mins intervals at the end of each tempering period with a digital balance

ACCEPTED MANUSCRIPT (specification: 0.001g accuracy). Drying tests were performed based on the American Society of Agricultural and Biological Engineers (ASABE S448.1) Standard. All experiments were done in triplicate, and the standard deviation was calculated. 3.2

Sample preparation

CR IP T

Fresh Granny Smith apples were used for the intermittent microwave drying experiments were obtained from local supermarkets. The samples were stored at 5±10C to keep them as fresh as possible before they were used in the experiments.

AN US

The samples were cut to a thickness of 10 mm and a diameter of approximately 40 mm.

Electronic Moisture Analyser (KERN MLS_A Version 3.1) was used to measure the moisture content of the samples. Ten samples were used for the moisture

M

content measurement. The initial moisture content of the fresh apple was found to be

3.2.1

ED

85±0.75% wb.

Uncertainty analysis

PT

Uncertainty analysis of the experiments was done following the procedure

CE

undertaken by Kumar et al.[59]. If the result R of an experiment is calculated from a set of independent variables so that, R  R X 1, X 2 , X 3..........., X N  . Then the overall

AC

uncertainty can be calculated using the following expression 1/ 2

2     N  R  R    .X i   X     i 1  i 

and the relative uncertainty can be expressed as follows:

(65)

ACCEPTED MANUSCRIPT (66)

1/ 2

2     N  1 R  e    . .X i   R  i 1  R X i    

R

Uncertainty analysis of temperature

CR IP T

The temperature was directly obtained from the calibrated thermal image camera and the accuracy was within the ASHRAE recommended range, which is ±0.50C. Therefore, the uncertainty of the temperature would be

(67)

AN US

T  Tmeasured  0.5 Uncertainty analysis of moisture content

The dry basis moisture content, ratio of the weight of moisture, Wm to that of

M

bone dry weight, Wd , of the sample was calculated from the following equation. Wm W  Wd  Wd Wd

(68)

ED

M

M M W W .Wd and .W  .Wd   2 W Wd Wd Wd

M M



W W  Wd



W .Wd W  Wd .Wd

CE

PT

Therefore, M 

AC

Now the relative uncertainty associated with the measurement of moisture

content of sample can be expressed as: 1/ 2

2 2   W   W .Wd         em   W  Wd   W  Wd .Wd     

(69)

The present work considered W  10 g and Wd  1g for the apple sample to be dried in the drying chamber. As these two values are obtained using the same load

ACCEPTED MANUSCRIPT cell, and as per manufacturer’s specification, the percentage error of load cell is

 0.1% , therefore, W  Wd  0.0001 . Substituting all the values in Eq. (24), the relative uncertainty for moisture content, em , is found to be  1.1% . 3.3

Glass transition temperature

CR IP T

In order to measure heat flow and temperature associated phase transitions in any materials, a thermal analysis technique named differential scanning calorimeter (DSC) is used. The measured parameters are expressed as a function of temperature and time. Firstly, about 10 mg of Granny Smith samples of different moisture

AN US

contents were placed into aluminium pans (20 µl) of DSC. Samples were weighted after hermetically sealed with lids. The mass of each sample was checked prior to sample preparation and found to be within ± 0.1mg weight differences from the

M

empty reference pan.

ED

Please insert Figure 4 about here In this study, DSC Q100 (TA Instruments, USA), as shown in Figure 4 was

PT

used. N2, 50 ml/min was deployed as purge gas. After cooling the sample at -650C, thermos-analytical curves were obtained by heating the sample to 1000C. The

CE

thermos-analytical curves were investigated using TA software for measuring glass

AC

transition temperature. 3.4

Power absorption ratio The tests were conducted at three power levels 100W, 200W and 300W, with a

water sample. The volume of water sample was taken as the same volume of apple to obtain accurate power absorption. Water was heated for 60s, and thermal images were taken by the thermal imaging camera (FLIR i7) before and after heating. The water was properly agitated to measure the average rise of temperature. The absorbed

ACCEPTED MANUSCRIPT power, P0 , can be calculated by Equation (70) for various load volume and applied microwave power. P0  mwC pw

(70)

Thermal imaging

CR IP T

3.5

T , t

A thermal image camera allows monitoring temperature distribution in the heated material. An FLIR i7 thermal camera that provided 10000-pixel images was used for this. It allowed temperature accuracy within ± 2°C or 2% of reading to meet

AN US

the standard. It is capable of tracing the optimised temperature range from -20° to 250°C. The FLIR i7 thermal imaging camera was also used to measure the temperature distribution on the surface. Accurate measurement of temperature by thermal imaging cameras depends on the emissivity values. The emissivity value of

M

apple was found in the range between 0.94 and 0.97 [60] and set in the camera before

3.6

ED

taking images.

Particle density determination

PT

Particle density was determined with the gas (helium) pycnometer

CE

(Quantachrome pentatype 5200e, USA), as shown in Figure 5. Helium is used to measure the skeleton (particle/solid/true) density due to its small molecular size,

AC

which makes it possible to access the smallest pores of up to 3.5X10-10 m. Therefore, all of the pores, even the closed intercellular pores, can be accessed, as the cell walls contain numerous pores that are larger than 3.5 X10-10m. Ten measuring cycles were carried out using this instrument to calculate the mean value for particle density of each sample. Therefore, particle volume, Vp can be determined from the known mass and particle density of the sample.

ACCEPTED MANUSCRIPT

Please insert Figure 5 about here The bulk density, ρb, was calculated from the bulk volume and the weight of the sample. Bulk volume was measured by the glass bead displacement method. The

CR IP T

density of the same sample was measured both before and after coating. The sample was first covered with conventional correcting fluid to fill up the open pores as there are many open pores that are large enough to allow the glass beads access (Lab Glass, 57 µm) into the pores. The density of glass beads (glass bead packing density)

AN US

was determined from the weight of glass beads required to fill the known volume vial. Details of determination of shrinkage and porosity can be found in Joardder,

3.7

Numerical analysis

M

Kumar, Brown and Karim [61].

ED

The simulation was performed using a finite element based engineering

PT

simulation software COMSOL Multiphysics 4.4.,. The software facilitated all steps in the modelling process, including defining

CE

geometry, meshing, specifying physics, solving, and then visualising the results. COMSOL Multiphysics can handle the variable properties, which are a function of

AC

the independent variables. Therefore, this software was very useful in drying simulation where material properties changed with temperature and moisture content. The simulation methodology and implementation strategy followed in IMCD multiphase is shown in Figure 6. Please insert Figure 6 about here

ACCEPTED MANUSCRIPT A combination of a rectangular function and an analytic function in COMSOL Multiphysics was used to develop a microwave intermittency function as shown in Figure 7. Then, it was multiplied by the heat generation term in the energy equations to implement intermittency of the microwave heat source.

CR IP T

Please insert Figure 7 about here

Since heat and mass transport phenomena ocur at the transport boundaries, a finer mesh (maximum element size 0.01mm) was chosen at those boundaries to

AN US

capture this phenomenon more accurately.

Moreover, triangular mesh with a homogenous element size was used for better accuracy within the rectangular domain. Figure 8 shows the mesh chosen for the

M

model.

ED

Please insert Figure 8 about here

PT

To ensure that the results are grid-independent, several grid sensitivity tests were conducted. Once the simulation solution satisfies these tests, the solution is

CE

considered independent of mesh. In order to observe the mesh independency,

AC

variation of average temperature with a number of cells is plotted, as shown in Figure 8.

From Figure 9, it is clear that up to 30000 cells there is a significant variation

found in the selected parameter (average volumetric temperature). It can also be seen from that acceptable difference exists for cell number more than 60000. This manifest that we have achieved a solution value that is not dependent of the mesh resolution. Therefore, for the further study we can use 60000 cells.

ACCEPTED MANUSCRIPT Please insert Figure 9 about here

The maximum time step was set to 1s to prevent the solver from taking too large time steps that can result in convergence problems. In addition to this, linear

CR IP T

element order for all dependent variables was used to get the robust performance of solving linear elements.

Moving mesh module of COMSOL Multiphysics, Arbitrary–Lagrangian– Eulerian (ALE) approach has been used to account structural deformation happening

AN US

in food sample. This approach can provide structural shrinkage along with domain mesh movement. To solve the circular dependency of the variables, the variables were treated as dependent variables. The simulation was repeated using updated

M

sample size and materials properties after each cycle. The total number of degrees of freedom to be solved for was 805526. The simulations were run on a 12 core, 2.0

ED

GHz Windows workstation with 16 GB memory for a total run time of 10 hours. The simulations were performed using a Windows 7 computer with Intel Core i7 CPU,

CE

PT

3.4GHz processor and 16 GB of RAM.

4

Results and discussion

AC

In this section, profiles of moisture, temperature, pressure, fluxes and

evaporation rate are presented and discussed. Validation of the models developed is also performed by comparing moisture content and temperature gathered from experiments.

ACCEPTED MANUSCRIPT 4.1

Average moisture content The evolution of average moisture content obtained from IMCD1 and IMCD2

models and experiments are compared in Figure 10. It can be seen from the figure that both models show agreeable trends as found in experiments. However, IMCD2 shows better match having R2 values 0.98

compared to IMCD1 which shows

CR IP T

relatively low value of R2 with 0.96.

AN US

Please insert Figure 10 about here

This difference in result is due to higher temperature developed in IMCD1 model resulting higher moisture migration rate. It was found that moisture content

after 4500 s of drying.

Distribution and evolution of water and vapour

ED

4.2

M

(dry basis) of apple slice dropped from its initial value of 6.14 kg/kg to 1.58 kg/kg

PT

The predicted distribution of water saturation and vapour saturation along the half thickness of the material at different times is shown in Figure 11 and 12,

CE

respectively. The result obtained from both the models show that over the course of drying the water saturation near the surface was lower than that in the centre region.

AC

Moreover, due to an adequate supply of drying air, all of the water at the surface migrated through convective flow instantly from the surface. This nature of moisture distributions within the sample, i.e. that the core contained higher moisture content compared to the surface, seem to be consistent with the findings of Chemkhi et al. (2009). Figure 11 it is clearly demonstrated that concentration of water decreased throughout the drying period for both models.

ACCEPTED MANUSCRIPT

Please insert Figure 11 about here Please insert Figure 12 about here Unlike the moisture distributions, vapour saturation was found to increase

CR IP T

with drying time within the sample (Figure 12). However, the vapour saturation at and near the surface was lower than the centre, because the vapour coming from the surface was immediately removed by the drying air through convection. In addition to this, it is clearly apparent from Figure 12 that concentration of vapour of any point

AN US

of the sample is decreased with time for both the models.

Please insert Figure 13 about here

M

A higher vapour fraction is found in these IMCD drying compared to

ED

convective drying due to higher vapour generation in IMCD. It is also observed that average vapour fraction fluctuates with the intermittency of microwave application.

PT

However, vapour fraction is relatively higher (up to 0.45) in IMCD1 compare to IMCD2, as shown in Figure 13. A possible explanation for this might be that

CE

unchanged volume consideration causes more volumetric heating resulting higher

AC

vapour generation during IMCD1. 4.3

Temperature evolution and distribution Figure 14 shows average surface temperature evolution in the material in

IMCD1 and IMCD2. Fluctuations of temperature in both models are found as expected because temperature increases during microwave on-time and drops during the tempering periods. Due to sudden exposure of the sample to higher air temperature and volumetric heating, the surface temperature rose sharply at the

ACCEPTED MANUSCRIPT beginning of the drying process. For IMCD1 model, the temperature continued to rise but for IMCD2 temperature rise stabilised and fluctuated within the range of 6065 0C. The inconsistency may be due to the fact that higher absorption of microwave energy was resulted as the volume of the sample was assumed constant in IMCD1

CR IP T

(as absorption power is proportional to the sample volume). The trend of temperature obtained for IMCD2 is consistent with that of Sunjka et al.[62] who found almost stable temperature for Cranbarry during microwave convective drying after 15 min.

AN US

of drying. It proves that consideration of shrinkage and porosity evolution in IMCD2 provides a realistic change of thermos-physical properties of the sample. Please insert Figure 14 about here To compare the temperature distribution within the apple sample during

M

IMCD1 and IMCD2, simulation results are shown in Figure 15. From the figure, it is

ED

apparent that the surface temperature was always lower than the central temperature over the course of drying. This higher interior temperature could be due to

PT

evaporation cooling and higher absorption of MW power towards the core due to

CE

higher moisture content.

AC

4.4

Please insert Figure 15 about here

Vapour pressure, equilibrium vapour pressure, and saturated pressure The comparison between vapour pressure, equilibrium vapour pressure and

saturation vapour pressure at the surface during IMCD1 and IMCD2 are presented in Figure 16. A proper understanding the nature of these three pressures is essential for proper understanding of drying kinetics. Please insert Figure 16 about here

ACCEPTED MANUSCRIPT It can be seen that the saturated vapour pressure varies with temperature. The saturated vapour pressure obtained from simulation is compared with literature [63] and found to be consistent with available data. The equilibrium vapour pressure was calculated from the sorption isotherm of apple and, was found to be lower than the saturation vapour pressure as expected for IMCD. All the pressures show some

CR IP T

fluctuations due to temperature fluctuations resulting from intermittency of microwave energy source.

At the initial stage of drying, vapour pressure was found to be higher than

AN US

equilibrium vapour pressure which resulted in higher evaporation at this time. After this period, values of vapour pressure and equilibrium vapour pressures in IMCD 2 nearly overlapped; however, a constant discrepancy among these two pressures was found for IMCD1. Gas pressure

M

4.5

presented.

ED

In Figure 17, spatial gradient of the gas pressure ( Pg  P  Pamb ) within the sample is

CE

PT

Please insert Figure 17 about here

AC

Figure 17 indicates that the gas pressure is maximum at the core region and

progressively drops towards the surfaces. Due to lower gas porosity, the migration of gas from the interior is restricted while the sample contains higher moisture content and this causes a rise in total gas pressure. In microwave heating, this type of gas pressure distribution is in agreement with the available literature [50, 64]. The regions having zero gas pressure maintain an atmospheric pressure. Increase of gas

ACCEPTED MANUSCRIPT porosity results in the increase of gas migration from the centre and eventually a lower gas pressure gradient can be observed. 4.6

Evaporation rate As the IMCD models considered non-equilibrium approaches, calculation and

visualisation of evaporation within the sample are of great interest. Figure 18

CR IP T

illustrates the spatial distribution of the evaporation rate over the course of IMCD. From the figure, it is quite clear that a higher evaporation rate occurs near the surface and maintains a non- equilibrium condition.

AN US

Please insert Figure 18 about here

It is clear from Figure 18 is that the evaporation maintains a zone, which starts

M

not exactly from surface, rather beneath the surface and decreasing to zero evaporation towards the core (as demonstrated in Figure 19). This result may be

ED

explained by the fact that the liquid water saturation becomes lower near the surface of the sample and contributes no difference between the vapour pressure and

PT

equilibrium vapour pressure. Eventually, the evaporation front moves inside of the

CE

material as drying progress. Another interesting pattern emerged from the graph in Figure 18 is that the

AC

peak of evaporation moves towards the centre. IMCD1 shows the random nature of the evaporation zone throughout the drying time; whereas, IMCD2 provides a welldistributed evaporation zone. In other words, consideration of pore formation and evolution makes difference between IMCD1 and IMCD2 evaporation rate profiles as porosity plays important role in determining the transport phenomena during drying.

ACCEPTED MANUSCRIPT 4.7

Liquid water and vapour fluxes The major advantage of a multiphase porous media model is that the relative

contribution of liquid water and vapour fluxes due to diffusion (due to a concentration gradient) and convection (due to gas pressure gradient) can be illustrated. Liquid water fluxes

CR IP T

4.7.1

Water fluxes due to capillary pressure (diffusive flow) and gas pressure gradients (convection flow) are plotted in Figure 19 and Figure 20, respectively.

AN US

Water capillary flux

It can be seen from Figure 19 that the capillary flux is higher at about 0.4–0.8 mm beneath the surface and this peak is moving towards the core with drying time in

M

both models. It is worthy to note here that capillary flux ( Dc cw ) is directly proportional to moisture gradient ( cw ) of the sample. Moreover, the gradient is

ED

higher initially near the surface and the peak of gradient moves towards the centre

PT

with time. These factors may explain the nature of the spatial distribution of water capillary flux within the sample.

CE

In addition, with time the pick capillary flux reduces due to moisture migration

as shown in Figure 19. The capillary flux in IMCD1 does not represent the typical

AC

trend concentration gradient. Please insert Figure 19 abut here

Water convective flux The water flux due to the gas pressure gradient (Figure 20) showed a similar pattern of water diffusion flux distribution, although with lower magnitude than

ACCEPTED MANUSCRIPT capillary flux for IMCD2. Therefore, it is clear from this result that diffusive water flux dominates over the convective flux throughout the food processing time. Please insert Figure 20 about here

CR IP T

The convective water flux increased from zero (at the centre) to a peak at approximately 0.5 mm beneath the surface. This could be due to the higher-pressure gradient near the surface resulting in higher convective flow. However, in convection drying, the gradient of pressure needed a closer inspection, because there may not be

AN US

enough pressure development inside the sample.

However, IMCD1 showed higher convective flux than capillary flux with increasing trend while drying progress, which cannot be justified. This rather

M

contradictory result may be due to the fact that higher temperature developed in the sample for IMCD1. It is also apparent that the convective flux is higher than

ED

capillary flux in the case of IMCD1, which also not acceptable as capillary flux water

PT

must be higher than convective flux during low-temperature drying. The pattern of liquid fluxes show that liquid water concentration prediction

CE

differs in IMCD1 and IMCD2 is mainly due to convective flux. As convective flux is directly affected by available porosity of the sample, erroneous prediction of

AC

convective flux may occur if shrinkage is not considered. 4.7.2

Vapour fluxes Figure 21 and 22 show the spatial distribution of the diffusive and convective

fluxes of vapour, respectively. Please insert Figure 21 about here

ACCEPTED MANUSCRIPT Figure 21 shows that vapour flux due to binary diffusion occurring near the surface with zero flux in the centre of the sample. The findings of this study in this regard is consistent with that found by Ousegui et al. (2010) who found a similar pattern of capillary vapour flux. The spatial distribution of convective flux during IMCD is presented in Figure 22. It can be observed that due to controlled application

CR IP T

of microwave the temperature could not reach much higher to causes huge evaporation.

Please insert Figure 22 about here

AN US

Therefore, gas pressure developed within the sample is minimal and within the range of 1-10 Pa. From the above figure, it is clear that convective vapour flux is occurring near the surface with zero flux at centres depending on the gas pressure gradient at any time. However, IMCD1 does not capture this trend due to unrealistic

M

temperature development at the end of drying. Similar to water concentration, vapour

ED

concentration prediction was not found correct in case of IMCD1 again due to this impractical temperature development.

PT

In brief, the vapour and water fluxes caused by all sources showed that the

CE

fluxes were minimal at the centre and gradually increase until a peak is achieved. From the pick, these start decreasing towards the surface and show zero fluxes

AC

except vapour diffusive flux. 4.8

Gas porosity Figure 23 shows the porosity evolution of the sample against moisture content.

In the figure, there is a clear trend of increasing porosity with a decrease of moisture content. This increase of porosity is the direct consequence of the increase in gas saturation during drying. Moreover, porosity prediction from IMCD2 agrees closely

ACCEPTED MANUSCRIPT with experimental value whereas IMCD1 failed to predict a reasonable value of porosity over the course of drying.

CR IP T

Please insert Figure 23 about here

The proper prediction of porosity is essential for correct prediction of heat and mass transfer as porosity affects thermo-physical properties of the sample. The effect of shrinkage and porosity evolution on thermal conductivity, effective heat

4.8.1

Thermal conductivity

AN US

capacity and effective density is discussed below:

Thermal conductivity is defined as the ability of a material to conduct heat. The

M

structure of the foodstuff, its composition, heterogeneity of food and processing

ED

conditions influence the thermal conductivity of foods [65, 66]. It is clear from Figure 24 that the effective density decreases sharply in IMCD1

PT

resulting in higher gas porosity. Dry porous solids are very poor heat conductors as air occupies the pores resulting in low thermal conductivity. This is the same case for

AC

CE

IMCD1.

4.8.2

Please insert Figure 24 about here

Effective specific heat capacity Like effective thermal conductivity, effective specific heat capacity depends on

the pore characteristics and distribution of different phases, such as air, water, ice, and solids. Please insert Figure 25 about here

ACCEPTED MANUSCRIPT It is clearly apparent from Figure 25 that effective specific diffusivity highly depends on the porosity of the sample. Therefore, neglecting shrinkage during multiphase model should not be justified. 4.8.3

Effective density Effective density, generally, depends on shrinkage and pore evolution over the

CR IP T

course of drying.

Please insert Figure 26 about here

Figure 26 shows the effective density of the apple sample at different moisture

AN US

content. The graph clearly shows that effective density is significantly affected by the distribution of multiphase in the sample. The effective density found in IMCD2 is consistent with experimental results.

Conclusion

M

5

ED

Two non-equilibrium multiphase porous media IMCD models have been developed. Incorporation of shrinkage and pore evolution proved a significant

PT

advancement relative to existing approaches. A number of parameters of the model

CE

were validated by experimental results and demonstrated that a good agreement existed. One of the significant findings to emerge from this study is that

AC

consideration of shrinkage and pore evolution during IMCD can explain the physics of heat and mass transfer more accurately. In addition to this, the model can manifest the characteristics of different parameters such as evaporation rate, capillary diffusion, effective thermal conductivity, gas pressure successfully, which is not possible by using simpler models. In addition to this, the results of above-mentioned parameters can be utilised to answer particular inquiries in practical situations where real time observation of these parameters is not possible. As this model is a physics

ACCEPTED MANUSCRIPT based model, it provides a deeper insight of heat and mass transfer over the course of drying. Therefre, this model can be utilised for predicting energy requirement,

AC

CE

PT

ED

M

AN US

CR IP T

quality enhancement and process optimisation.

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

6 References 1. Kumar, C., M.A. Karim, and M.U.H. Joardder, Intermittent drying of food products: A critical review. Journal of Food Engineering, 2014. 121(0): p. 48-57. 2. Zeki, B., Chapter 22 - Dehydration, in Food Process Engineering and Technology. 2009, Academic Press: San Diego. p. 459-510. 3. Joardder, M.U.H., C. Kumar, and M.A. Karim, Effect of moisture and temperature distribution on dried food microstucture and porosity. Proceedings of From Model Foods to Food Models: The DREAM Project International Conference, 2013. 4. Kumar, C., M.U.H. Joardder, T.W. Farrell, G.J. Millar, and M.A. Karim, A Mathematical Model for Intermittent Microwave Convective (IMCD) Drying of Food Materials. Drying Technology, 2015: p. null-null. 5. Soysal, Y., Z. Ayhan, O. Eştürk, and M.F. Arıkan, Intermittent microwave–convective drying of red pepper: Drying kinetics, physical (colour and texture) and sensory quality. Biosystems Engineering, 2009a. 103(4): p. 455-463. 6. Mothibe , K.J., Wang, C.Y. ,Mujumdar A.S., Zhang , M.;, Microwave-Assisted PulseSpouted Vacuum Drying of Apple Cubes. Drying Technology, 2014. 32(15). 7. Jiang, H., M. Zhang, A.S. Mujumdar, and R.-X. Lim, Comparison of drying characteristic and uniformity of banana cubes dried by pulse-spouted microwave vacuum drying, freeze drying and microwave freeze drying. Journal of the Science of Food and Agriculture, 2014. 94(9): p. 1827-1834. 8. Wang, Y., M. Zhang, A.S. Mujumdar, and K. Mothibe, Microwave-Assisted PulseSpouted Bed Freeze-Drying of Stem Lettuce Slices—Effect on Product Quality. Food and Bioprocess Technology, 2013. 6(12): p. 3530-3543. 9. Soysal, Y., Arslan, M. and Keskin, M.;, Intermittent Microwave-convective Air Drying of Oregano. Food Science and Technology International, 2009. 15(4): p. 397-406. 10. Bhattacharya, M., P. Srivastav, and H. Mishra, Thin-layer modeling of convective and microwave-convective drying of oyster mushroom (Pleurotus ostreatus). Journal of Food Science and Technology, 2013: p. 1-10. 11. Esturk, O., Intermittent and Continuous Microwave-Convective Air-Drying Characteristics of Sage (Salvia officinalis) Leaves. Food and Bioprocess Technology, 2012. 5(5): p. 1664-1673. 12. Arballo, J.R., L.A. Campanone, and R.H. Mascheroni, Modeling of Microwave Drying of Fruits. Part II: Effect of Osmotic Pretreatment on the Microwave Dehydration Process. Drying Technology, 2012. 30(4): p. 404-415. 13. Hemis, M., R. Choudhary, and D.G. Watson, A coupled mathematical model for simultaneous microwave and convective drying of wheat seeds. Biosystems Engineering, 2012. 112(3): p. 202-209. 14. Esturk, O. and Y. Soysal, Drying Properties and Quality Parameters of Dill Dried with Intermittent and Continuous Microwave-convective Air Treatments. Tarim Bilimleri Dergisi-Journal of Agricultural Sciences, 2010. 16(1): p. 26-36. 15. Kumar, C., A. Karim, M.U.H. Joardder, and G.J. Miller. Modeling Heat and Mass Transfer Process during Convection Drying of Fruit. in The 4th International Conference on Computational Methods. 2012a. Gold Coast, Australia. 16. Perussello, C.A., C. Kumar, F. de Castilhos, and M.A. Karim, Heat and mass transfer modeling of the osmo-convective drying of yacon roots (Smallanthus sonchifolius). Applied Thermal Engineering, 2014. 63(1): p. 23-32. 17. Yang, H.W. and S. Gunasekaran, Comparison of temperature distribution in model food cylinders based on Maxwell's equations and Lambert's law during pulsed microwave heating. Journal of Food Engineering, 2004. 64(4): p. 445-453.

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

18. Gunasekaran, S. and H.-W. Yang, Effect of experimental parameters on temperature distribution during continuous and pulsed microwave heating. Journal of Food Engineering, 2007. 78(4): p. 1452-1456. 19. Gunasekaran, S. and H.-W. Yang, Optimization of pulsed microwave heating. Journal of Food Engineering, 2007. 78(4): p. 1457-1462. 20. Gunasekaran, S., Pulsed microwave-vacuum drying of food materials. Drying Technology, 1999. 17(3): p. 395-412. 21. Bear, J., Dynamics of fluids in porous media. 1972: American Elsevier Pub. Co. 22. Datta, A.K., Porous media approaches to studying simultaneous heat and mass transfer in food processes. I: Problem formulations. Journal of Food Engineering, 2007. 80(1): p. 8095. 23. Bird, R.B., E.W. Stewart, and N.E. Lightfoot, Transport Phenomena. 2007: John Wiley & Sons. 24. Ni, H., A.k. Datta, and K.e. Torrance, Moisture transport in intensive microwave heating of biomaterials: a multiphase porous media model. International Journal of Heat and Mass Transfer, 1999. 42(8): p. 1501-1512. 25. Le, C.V., N.G. Ly, and R. Postle, Heat and mass transfer in the condensing flow of steam through an absorbing fibrous medium. International Journal of Heat and Mass Transfer, 1995. 38(1): p. 81-89. 26. Scarpa, F. and G. Milano, The Role of Adsorption and Phase Change Phenomena in the Thermophysical Characterization of Moist Porous Materials. International Journal of Thermophysics, 2002. 23(4): p. 1033-1046. 27. Fang, G. and C.A. Ward, Temperature measured close to the interface of an evaporating liquid. Physical Review E, 1999. 59(1): p. 417-428. 28. Halder, A., A. Dhall, and A.K. Datta, Modeling transport in porous media with phase change: Applications to food processing. Journal of Heat Transfer, 2011. 133(3). 29. Ratti, C., G.H. Crapiste, and E. Rotstein, A New Water Sorption Equilibrium Expression for Solid Foods based on Thermodynamic Considerations. Journal of Food Science, 1989. 54(3): p. 738-742. 30. Vega-Mercado, H., M. Marcela Góngora-Nieto, and G.V. Barbosa-Cánovas, Advances in dehydration of foods. Journal of Food Engineering, 2001. 49(4): p. 271-289. 31. Joardder, M.U.H., C. Kumar, R.J. Brown, and M.A. Karim, Effect of cell wall properties on porosity and shrinkage of dried apple. International Journal of Food Properties, 2015. In Press. 32. Joardder, M.U.H., C. Kumar, and M.A. Karim, Food structure: Its formation and relationships with other properties. Critical Reviews in Food Science and Nutrition 2015. 33. Gulati, T. and A.K. Datta, Mechanistic understanding of case-hardening and texture development during drying of food materials. Journal of Food Engineering, 2015. 166: p. 119-138. 34. Yang, H., Sakai, N., and Watanabe, A.;, Drying model with non-isotropic shrinkage deformation undergoing simultaneous heat and mass transfer. Drying technology, 2001. 19(7): p. 1441–1460. 35. Krokida, M.K. and C. Philippopoulos, Rehydration of Dehydrated Foods. Drying Technology: An International Journal, 2005. 23(4): p. 799-830. 36. Ormerod, A.P., J.D. Ralfs, R. Jackson, J. Milne, and M.J. Gidley, The influence of tissue porosity on the material properties of model plant tissues. Journal of Materials Science, 2004. 39(2): p. 529-538. 37. Bourne, M.C., Effect of Temperature on Firmness of Raw Fruits and Vegetables. Journal of Food Science, 1982. 47(2): p. 440-444.

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

38. Joardder, M.U.H., A. Karim, R.J. Brown, and C. Kumar, Porosity: Establishing the Relationship between Drying Parameters and Dried Food Quality. 2016, Switzerland: Springer. 39. Golestani, R., A. Raisi, and A. Aroujalian, Mathematical Modeling on Air Drying of Apples Considering Shrinkage and Variable Diffusion Coefficient. Drying Technology, 2013. 31(1): p. 40-51. 40. Aprajeeta, J., R. Gopirajah, and C. Anandharamakrishnan, Shrinkage and porosity effects on heat and mass transfer during potato drying. Journal of Food Engineering, 2015. 144(0): p. 119-128. 41. Karim, M.A. and M.N.A. Hawlader, Mathematical modelling and experimental investigation of tropical fruits drying. International Journal of Heat and Mass Transfer, 2005. 48(23-24): p. 4914-4925. 42. Kumar, C., M.U.H. Joardder, T.W. Farrell, and M.A. Karim, Multiphase porous media model for intermittent microwave convective drying (IMCD) of food. International Journal of Thermal Sciences, 2016. 104: p. 304-314. 43. Scheidegger, A.E., The physics of flow through porous media, 3rd edition. University of Toronto press. Toronto, 1974. 44. Feng, H., J. Tang, O.A. Plumb, and R.P. Cavalieri, Intrinsic and relative permeability for flow of humid air in unsaturated apple tissues. Journal of Food Engineering, 2004. 62(2): p. 185-192. 45. Truscott, S., A heterogenous three-dimensional computational model for wood drying. 2004, Queensland University of Technology. 46. Gulati, T. and A.K. Datta, Enabling computer-aided food process engineering: Property estimation equations for transport phenomena-based models. Journal of Food Engineering, 2013. 116(2): p. 483-504. 47. Ni, H., Multiphase moisture transport in porous media under intensive microwave heating. 1997, Cornell University, Ithaca, New York, United States: Ithaca, New York, United States. 48. Datta, A.K., Porous media approaches to studying simultaneous heat and mass transfer in food processes. II: Property data and representative results. Journal of Food Engineering, 2007. 80(1): p. 96-110. 49. Feng, H., J. Tang, and S. John Dixon-Warren, Determination of Moisture Diffusivity of Red Delicious Apple Tissues by Thermogravimetric Analysis. Drying Technology, 2000. 18(6): p. 1183-1199. 50. Feng, H., J. Tang, R.P. Cavalieri, and O.A. Plumb, Heat and mass transport in microwave drying of porous materials in a spouted bed. AIChE Journal, 2001. 47(7): p. 1499-1512. 51. Mihoubi, D. and A. Bellagi, Drying-Induced Stresses during Convective and Combined Microwave and Convective Drying of Saturated Porous Media. Drying Technology, 2009. 27(7-8): p. 851-856. 52. Zhou, L., V.M. Puri, R.C. Anantheswaran, and G. Yeh, Finite element modeling of heat and mass transfer in food materials during microwave heating — Model development and validation. Journal of Food Engineering, 1995. 25(4): p. 509-529. 53. Salagnac, P., P. Glouannec, and D. Lecharpentier, Numerical modeling of heat and mass transfer in porous medium during combined hot air, infrared and microwaves drying. International Journal of Heat and Mass Transfer, 2004. 47(19-20): p. 4479-4489. 54. Abbasi Souraki, B. and D. Mowla, Experimental and theoretical investigation of drying behaviour of garlic in an inert medium fluidized bed assisted by microwave. Journal of Food Engineering, 2008. 88(4): p. 438-449.

ACCEPTED MANUSCRIPT

List of Figure

Figure 1: Schematic showing 3D sample, 2D axisymmetric domain and Representative Elementary Volume (REV) with the transport mechanism of different phases

AC

1.1

CE

PT

ED

M

AN US

CR IP T

55. Khraisheh, M.A.M., T.J.R. Cooper, and T.R.A. Magee, Microwave and air drying I. Fundamental considerations and assumptions for the simplified thermal calculations of volumetric power absorption. Journal of Food Engineering, 1997. 33(1–2): p. 207-219. 56. Martín-Esparza, M.E., N. Martínez-Navarrete, A. Chiralt, and P. Fito, Dielectric behavior of apple (var. Granny Smith) at different moisture contents. Journal of Food Engineering, 2006. 77(1): p. 51-56. 57. Montanuci, F.D., C.A. Perussello, L.M. de Matos Jorge, and R.M.M. Jorge, Experimental analysis and finite element simulation of the hydration process of barley grains. Journal of Food Engineering, 2014. 131: p. 44-49. 58. Panasonic. Linear outpur for even cooking. 2013 [cited 2013 12 September ]; Available from: http://shop.panasonic.com/learn/microwaves/microwaveinverter/?sc_mc=vanity_inverter_01172012. 59. Kumar, C., G.J. Millar, and M.A. Karim, Effective Diffusivity and Evaporative Cooling in Convective Drying of Food Material. Drying Technology, 2014. 33(2): p. 227-237. 60. Hellebrand, H.J., H. Beuche, and M. Linke, Determination of thermal emissivity and surface temperature distribution of horticultural products, in Sixth International Symposium on Fruit, Nut and Vegetable Production Engineering. 2001: Potsdam, Germany. 61. Joardder, M.U.H., C. Kumar, R.J. Brown, and M.A. Karim, A micro-level investigation of the solid displacement method for porosity determination of dried food. Journal of Food Engineering, 2015. 166(0): p. 156-164. 62. Sunjka, P.S., T.J. Rennie, C. Beaudry, and G.S.V. Raghavan, Microwave-Convective and Microwave-Vacuum Drying of Cranberries: A Comparative Study. Drying Technology, 2004. 22(5): p. 1217-1231. 63. Çengel, Y.A. and M.A. Boles, Thermodynamics: an engineering approach. 2006: McGraw-Hill Higher Education. 64. Wei, C.K., H.T. Davis, E.A. Davis, and J. Gordon, Heat and mass transfer in water-laden sandstone: Microwave heating. AIChE Journal, 1985. 31(5): p. 842-848. 65. Sablani, S.S. and M.S. Rahman, Using neural networks to predict thermal conductivity of food as a function of moisture content, temperature and apparent porosity. Food Research International, 2003. 36(6): p. 617-623. 66. Hamdami, N., J.Y. Monteau, and A. Le Bali, Effective thermal conductive evolution as a function of temperature and humidity during freezing of high porosity model. Trnasictions of the institution of chemical engineering, 2003. 81: p. 1123-1128.

Figure2: Flow chart showing coupling between multiphase transport and solid mechanics Figure3: Boundary condition of multiphase IMCD model Figure 4: Determination of Glass transition temperature using DSC Figure 5: Particle density determination of sample using He pycnometer Bulk density determination Figure 6: Simulation strategy in COMSOL Multiphysics

ACCEPTED MANUSCRIPT Figure 7: Intermittency function of microwave power Figure 8: Mesh for the simulation of IMCD drying Figure 9: Mesh independence check of the simulation Figure 10: Comparison between predicted and experimental values of average moisture content during IMCD with and without shrinkage Figure 11: Spatial distribution of water saturation with times for IMCD1 and IMCD2

CR IP T

Figure 12: Spatial distribution of vapour with different time for IMCD1 and IMCD2 Figure 13: Average vapour mass fraction in apple sample during IMCD drying Figure 14: Top surface average temperature obtained for IMCD1 and IMCD2

Figure 15: Temperature distribution within apple sample during IMCD1 and IMCD2

AN US

Figure 16: Average vapour pressure, equilibrium vapour pressure and saturation pressure at surface

Figure 17: Gas pressure gradient across the half thickness the sample in different times during IMCD1 and IMCD2

M

Figure 18: Spatial distribution of evaporation rate at different drying times during IMCD1 and IMCD2

ED

Figure 19: Spatial distribution water flux due to capillary diffusion at different drying times during IMCD1 and IMCD2

PT

Figure 20: Spatial distribution water flux due to gas pressure gradient at different drying times during IMCD1 and IMCD2

CE

Figure 21: Spatial distribution vapour flux due to binary diffusion at different drying times during IMCD1 and IMCD2

AC

Figure 22: Spatial distribution of vapour flux due to gas pressure at different drying times during IMCD1 and IMCD2 Figure 23: Change of porosity with moisture removal Figure 24: Effective thermal conductivity with moisture content during IMCD Figure 25: Effective specific heat capacity with moisture content during IMCD Figure 26: Effective density with moisture content during IMCD

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 1: Schematic showing 3D sample, 2D axisymmetric domain and and Representative Elementary Volume (REV) with the transport mechanism of different phases.

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

Figure2: Flow chart showing coupling between multiphase transport and solid mechanics

Figure 3: Boundary condition of multiphase IMCD model

CR IP T

ACCEPTED MANUSCRIPT

CE

PT

ED

M

AN US

Figure 4: Determination of Glass transition temperature using DSC

AC

Figure 5: Particle density determination of sample using He pycnometer Bulk density determination

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

Figure 6: Simulation strategy in COMSOL Multiphysics

Figure 7: Intermittency function of microwave power

CR IP T

ACCEPTED MANUSCRIPT

AN US

Figure 8: Mesh for the simulation of IMCD drying

M

61.5

ED

61

PT

60.5

60

AC

CE

Average volumetric tempurature ( 0C)

62

59.5

59 0

20000

40000

60000 80000 Number of cells

Figure 9: Mesh independence check of the simulation

100000

120000

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

Figure 10: Comparison between predicted and experimental values of average moisture content during IMCD with and without shrinkage

Figure 11: Spatial distribution of water saturation with times for IMCD1 and IMCD2.Color legend represents water saturation. Different sizes of sample of IMCD2 refers sample deformation with time.

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

Figure12: Spatial distribution of vapour with different time for IMCD1 and IMCD2. Color legend represents vapour saturation. Different sizes of sample of IMCD2 refers sample deformation with time.

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 14: Top surface average temperature obtained for IMCD1 and IMCD2

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

Figure 15: Temperature distribution within apple sample during IMCD1 and IMCD2. Color legend represents sample temperature at oC. Different sizes of sample of IMCD2 refers sample deformation with time.

ACCEPTED MANUSCRIPT Figure 16: Average vapour pressure, equilibrium vapour pressure and saturation

AN US

CR IP T

pressure at surface

AC

CE

PT

ED

M

Figure 17: Gas pressure gradient across the half thickness the sample in different times during IMCD1 and IMCD2. Color legend represents gas pressure gradient in Pa. Different sizes of sample of IMCD2 refers sample deformation with time.

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

Figure 18: Spatial distribution of evaporation rate at different drying times during IMCD1 and IMCD2. Color legend represents evaporation rate in kg/m3s.

ACCEPTED MANUSCRIPT

AN US

CR IP T

Figure 19: Spatial distribution water flux due to capillary diffusion at different drying times during IMCD1 and IMCD2

AC

CE

PT

ED

M

Figure 20: Spatial distribution water flux due to gas pressure gradient at different drying times during IMCD1 and IMCD2. Color legend represents water flux in kg/m2s

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

Figure 21: Spatial distribution vapour flux due to binary diffusion at different drying times during IMCD1 and IMCD2. Color legend represents water flux in kg/m2s

AN US

CR IP T

ACCEPTED MANUSCRIPT

ED

M

Figure 22: Spatial distribution of vapour flux due to gas pressure at different drying times during IMCD1 and IMCD2. Color legend represents vapour flux in kg/m2s

1

0.7

Without shrinkage With shrinkage Experimental

CE

Porosity

0.8

PT

0.9

0.6 0.5

AC

0.4 0.3 0.2

0.1 0 2.5

3

3.5

4

4.5

5

5.5

Moisture content (kg/kg of solid) Figure 23: Change of porosity with moisture removal

6

6.5

ACCEPTED MANUSCRIPT

0.7 0.6 0.5

CR IP T

Effective thermal conductivity (W/m.K)

0.8

Without shrinkage

0.4

With shrinkage

0.3 0.2 2.5

3

3.5

4

4.5

5

5.5

6

6.5

AN US

Moisture content (kg/kg solid)

M ED

Without shrinkage With shrinkage

PT

4000 3800 3600 3400 3200 3000 2800 2600 2400 2200 2000

CE

Effective specific heat capasity (J/kg.K)

Figure 24: Effective thermal conductivity with moisture content during IMCD

2.5

3

3.5

4

4.5

5

5.5

Moisture content (kg/kg solid)

6

6.5

AC

Figure 25: Effective specific heat capacity with moisture content during IMCD

ACCEPTED MANUSCRIPT

750 650 Without shrinkage

550

With shrinkage Experimental

450

CR IP T

Effective density (kg/m3)

850

350 2.5

3

3.5

4 4.5 5 5.5 Moisture content ( kg water/kg solid)

6

AN US

Figure 26: Effective density with moisture content during IMCD

List of Table

Table 1: Input properties for the model Value

M

Parameter

This work

10 mm

This work

Equivalent porosity, initial,  0

0.922

[37, 38].

Water saturation, initial, S w0

0.794

[37, 38].

Initial saturation of vapour, S v 0

0.15

[37, 38].

Gas saturation, initial,

0.19

[37, 38].

Initial temperature, T0

303K

Vapour mass fraction, wv

0.026

Calculated

Evaporation constant, Kevap

1000

This work

Drying air temperature, Tair

333K

This work

Universal gas constant, Rg

8.314 J mol-1K-1

[39]

AC

CE

PT

Sample thickness, Ths

40 mm

ED

Sample diameter, Dias

Reference

6.5

ACCEPTED MANUSCRIPT Value

Reference

Molecular weight of water,

18.016 g mol-1

[39]

Molecular weight of vapour,

18.016 g mol-1

[39]

Molecular weight of gas (air),

28.966 g mol-1

[39]

Latent heat of evaporation,

2.26e6 J kg-1

[39]

Ambient pressure,

101325 Pa

3734 J kg-1K-1

Measured

Water,

4183 J kg-1K-1

[40]

Vapour,

1900 J kg-1K-1

[40]

Air,

1005.68 J kg-1K-1

[40]

Apple solid, k th,s

0.46 W m-1K-1

[41]

Gas, k th, g

0.026 W m-1K-1

[42]

Water, k th,w

0.59  9.8 10 4 T

[42]

1419 kg m-3

This study

AN US

Apple solid,

M

Specific heats

CR IP T

Parameter

ED

Thermal conductivity

Apple solid,

AC

Air,  v

CE

Vapour,  v

PT

Density

Water,  w

Ideal gas law, kg m-3 Ideal gas law, kg m-3 1000, kg m-3