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Journal of Food Engineering 48 (2001) 301±310 www.elsevier.com/locate/jfoodeng Modelling the pneumatic drying of food particles A.H. Pelegrina, G.H. Crapiste * PLAPIQUI (UNS±CONICET), Camino La Carrindanga, Km. 7, 8000 Bahõa Blanca, Argentina Received 4 May 2000; accepted 29 September 2000 Abstract...

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Modelling the pneumatic drying of food particles Guillermo Crapiste Journal of Food Engineering

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Journal of Food Engineering 48 (2001) 301±310 www.elsevier.com/locate/jfoodeng

Modelling the pneumatic drying of food particles A.H. Pelegrina, G.H. Crapiste * PLAPIQUI (UNS±CONICET), Camino La Carrindanga, Km. 7, 8000 Bahõa Blanca, Argentina Received 4 May 2000; accepted 29 September 2000

Abstract A one-dimensional model for pneumatic drying of food particles is presented. The model includes mass, momentum and energy transfer between the gas and the solid phases, taking into account variations of properties with humidity and temperature as well as solid shrinkage during drying. A plug-¯ow assumption is made for the dryer model and the non-spherical shape of particles is considered in drag and heat transfer coecients. The set of coupled non-linear ordinary di€erential equations is solved numerically for the velocity, moisture and temperature of particles and air along the dryer. The model is applied to simulate the drying of potato particles under di€erent conditions. Model predictions are used to illustrate the complex transport phenomena that occur during the process and the pro®les of temperature, moisture content and pressure developed through the dryer. E€ects of mass ¯owrates ratio, velocity and temperature of air, shape and size of particle on drying time and ®nal moisture content of the material are studied. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Pneumatic drying; Modelling; Simulation; Potato particles

1. Introduction Pneumatic or ¯ash drying of particulate materials is a very common operation in many industries. Pneumatic drying is widely used in the food industry to obtain high quality products, mainly granules and powders from vegetables and fruits. The process is characterised by cocurrent ¯ow, relatively small particle sizes, very short residence times, and relatively high air temperatures. Basically, the particles are continuously dried in a vertical duct while being conveyed by the heated airstream and then separated in a cyclone system. Mathematical modelling and simulation of a pneumatic dryer is a complex problem of coupled mass, momentum and energy transfer. Conditions and properties of the particles as well as air conditions vary considerably along the dryer in the direction of ¯ow. Several mathematical models have been presented to simulate the pneumatic drying process, most of them considering steady state one-dimensional ¯ow and uniform non-shrinking spherical particles (Thorpe, Wint, & Coggan, 1973; Matsumoto & Pei, 1984; Martin & Saleh, 1984; Saastamoinen, 1992). Adewumi and Arastoopour (1990) developed a two-dimensional ¯ow model for *

Corresponding author. Tel.: +54-291-4861600; fax: +54-2914861600. E-mail address: [email protected] (G.H. Crapiste).

vertical pneumatic conveying systems and studied the in¯uence of radial direction on velocity pro®les. Fyhr and Rasmuson (1997) presented a more complex model for a pneumatic dryer considering a distribution of particle sizes for steam drying of wood chips. More recently, Levy and Borde (1999) studied the drying of wet PVC particles in a large-scale pneumatic dryer and compared the results with predictions of a numerical simulation taking into account particle shrinkage. Application of comprehensive complex theories to model the drying of food particles is rather scarce (Crapiste & Rotstein, 1997) and only some simpli®ed models has been used to simulate pneumatic dryers for food products (Coggan, 1971; Matsumoto & Pei, 1984). In this work, the di€erential balances of momentum, mass and energy transfer for the gas and the solid phases are used to develop a one-dimensional ¯ow mathematical model for the pneumatic drying of foods. The model is improved by considering variations of air properties with humidity and temperature and of food properties with moisture content. Suitable equations are included to represent the heat and mass transfer between particles and the surrounding air as well as the water sorptional equilibrium of the product and its shrinkage during drying. Drag and convective heat transfer coecients for di€erent shapes are also taken into account. The model was applied to the drying of potato particles and was used to analyse the e€ect of di€erent variables as

0260-8774/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 0 - 8 7 7 4 ( 0 0 ) 0 0 1 7 0 - 9

302

A.H. Pelegrina, G.H. Crapiste / Journal of Food Engineering 48 (2001) 301±310

Nomenclature aw av A0p C Cd de D Dva f g h ky l M P q_ Q T U v V w_ W X y

water activity (±) particle surface area per unit particle volume (m 1 ) projected area of particle (m2 ) speci®c heat …J kg 1 C 1 † drag coecient (±) equivalent particle diameter (m) diameter of dryer duct (m) di€usion coecient of water vapour in air (m2 s 1 ) friction factor (±) acceleration due to gravity (m s 2 ) heat transfer coecient (W m 2 C 1 ) mass transfer coecient (m s 1 ) characteristic particle length (m) molecular weight (kg mol kg 1 ) pressure (Pa) heat ¯ow (W m 2 ) heat loss by unit length (W m 1 ) temperature (C) overall heat transfer coecient (W m 2 C 1 ) velocity (m s 1 ) volume (m3 ) drying rate (kg m 2 s 1 ) mass ¯ow (kg s 1 ) moisture content (kg kgdp1 ) mol fraction

solid to air mass ¯owrates ratio, velocity and temperature of air, and particle size and shape.

2. Balance equations The momentum, mass and energy di€erential balances for the continuous gas phase (moist air) and the dispersed solid phase (food particles) were considered in developing the mathematical model. The main following assumptions were made: · Steady-state ¯ow. · One-dimensional plug ¯ow. In most cases, variations of air and particle velocity in the radial direction can be neglected (Adewumi & Arastoopour, 1990; Fyhr & Rasmuson, 1997). · Interactions particle±particle in mass, momentum and heat transfer are negligible, a reasonable assumption in most practical situations (Fyhr & Rasmuson, 1997). · Momentum transfer due to friction forces can occur between the duct wall and both the continuous and the dispersed phases. · Heat transfer can occur between the continuous gas phase and the duct wall, but heat transfer between the particles and the wall is negligibly small. · The gas phase, a mixture of dry air and water vapour, has ideal behaviour. Its properties change as a function of temperature and humidity. · The solid is hygroscopic and can shrink during drying. Its properties vary as a function of moisture content.

Sc Pr Nu

absolute humidity (kg kgdg1 ) axial direction or height of the pneumatic dryer (m) Reynolds number based on the diameter of duct ˆ qg vg Dlg 1 (±) Reynolds number around a single particle ˆ …vg de qg lg 1 (±) Schmidt number ˆ lg qg 1 Dva1 (±) Prandtl number ˆ Cg lg kg 1 (±) Nusselt number ˆ hdkg 1 (±)

Greeks DHs o DHvap e k l q / w

heat of sorption …J kg 1 † latent heat of evaporation for water …J kg 1 † volume fraction of air thermal conductivity …W m 1 C 1 † viscosity …kg m 1 s 1 † density …kg m 3 † shape factor for heat and mass transfer (±) shape factor for momentum transfer (±)

Y z Re Rep

the vp †

Subscripts dg dry air dp dry product g gas phase p particle v water vapour w liquid water

· The solid feed is uniform in size and shape. Solid particles can be non-spherical, the shape a€ecting the drag and heat transfer coecients. · Internal resistance does not control mass and energy transfer between solid particles and air. Based on the above assumptions the process can be described by the following balance equations, which have been written in a suitable form for numerical solution:  Momentum balance Particle A0p qg dvp vp ˆ Cd …vg dz 2Vp qp

vp †jvg

vp j 

1

! qg g qp

1 fp v2 : 2D p

…1†

Air vg

dvg ˆ dz

1 dP g qg dz  Cd …vg

2 2 vf D g

vp †jvg

vp j

A0p …1 e† e 2Vp _ v …1 e† wa …vg e qg

vp †: …2†

The right-hand terms in these hydrodynamic equations account for all the forces that act on each phase: particle drag, gravitation and wall friction in Eq. (1), and pressure, gravitation, wall friction and particle drag in Eq. (2). In the gravity term, is used for upwards ¯ow and

A.H. Pelegrina, G.H. Crapiste / Journal of Food Engineering 48 (2001) 301±310

‡ for downwards ¯ow. The last term in Eq. (2) accounts for the momentum ¯ux due to mass transfer.  Mass balance Particle dX 1 d ˆ q vp …1 dz Wdp dz p Air

 e†A ˆ

_ ‡ X† av w…1 : qp vp

 av w…1 _ ‡ X † Wdp dY 1 d ˆ qg vg eA ˆ : dz Wdg dz qp v p Wdg

…3†

…4†

Particle dTp av ˆ q_ dz Cp qp vp h

Air

i

_ wDH s …Tp † :

dTg av …1 e† _ … q† ˆ e dz qg vg Cg _ v Cv …1 e† wa …Tg e qg vg Cg

…5†

Q Wdg Cg …1 ‡ Y† Tp †:

…6†

The terms in the right-hand side of Eq. (6) account for heat transfer between air and particles, heat losses through the duct wall and energy ¯ux due to mass transfer, respectively. The following relationship is derived from the mass balances and can be replaced in Eqs. (2) and (6) …1

e† e

ˆ

qg vg Wps …1 ‡ X † : qp vp Wgs …1 ‡ Y †

…7†

3. Single-particle models The model for pneumatic drying is completed by combination of the balance equations given above with the single-particle models that allow the evaluation of the source terms of mass and energy transport between _ respectively. At particles and air due to drying, w_ and q, moderate solid loading, the heat transfer to particles and the drying rate are assumed as though the particles were isolated from their neighbours. The heat transfer between particles and air is assumed to be governed only by convection due to a di€erence in temperature at the surface boundary layer, then: q_ ˆ h…Tg

Tp †:

can be neglected. Otherwise, a more complex one- or two-dimensional heat conduction model for the particle should be considered. In this paper, the drying rate …w† is assumed to be controlled mainly by convection at the boundary layer in the particle±air interface, and is de®ned as: w_ ˆ ky …Y 

…9†

Mv a …X ; Tp †Psat …Tp †  w : Mdg P aw …X ; Tp †Psat …Tp †

…10†

This is the simplest drying model for a single particle and, depending on the product and the particle size, its application may be restricted to suciently high water contents and relatively mild drying conditions. When the internal resistance for mass transfer is important, substantial moisture content variations within the product may arise and the drying process is partially controlled by moisture migration to the surface. In these cases more complex drying models have to be used, i.e., one- or two-dimensional di€usive equations, the characteristic drying curve or the receding front model (Crapiste & Rotstein, 1997).

4. Complimentary equations In this section, we present several complimentary equations that are required in order to solve the hydrodynamic-drying model represented by Eqs. (1)±(6) along with the single-particle model given by Eqs. (8) and (10). Basically these are equations for gas and solid properties, de®nitions of some geometrical factors, and empirical correlations for drag coecient, friction factors and convective heat and mass transfer coecients. 4.1. Gas properties Thermophysical and transport properties of wet air were calculated from equations for binary mixtures and correlation of experimental data for dry air and water vapour (Mayhew & Rogers, 1968; Raznjevic, 1976; Reid, Prausnitz, & Poling, 1977). The resulting equations, as a function of absolute humidity and temperature, can be summarised as: Density

…8†

This assumption applies in several practical situations of pneumatic drying where, because of the thermal conductivity of food products and the relatively small particle size, the Biot number for heat transfer is low and the internal resistance to heat conduction in the solid

Y †;

where Y  means the equilibrium absolute humidity at the surface, evaluated from sorption isotherm data for the product as Y ˆ

 Energy balance

303

qg ˆ

PMg : R…Tg ‡ 273:16†

Viscosity ydg ldg yv lv lg ˆ ; ‡ ydg ‡ yv /dgv yv ‡ ydg /vdg

…11†

…12†

304

/dgv

A.H. Pelegrina, G.H. Crapiste / Journal of Food Engineering 48 (2001) 301±310

h

1=2

1 ‡ …ldg =lv † …Mv =Mdg † ˆ  1=2 8…1 ‡ Mdg =Mv †

1=4

i2

Heat of sorption ;

/vdg ˆ

lv Mdg / ; ldg Mv dgv

lv ˆ

1:758  10

6

‡ 3:7  10 8 Tg ;

…14†

ldg ˆ 7:0808  10

6

‡ 3:88  10 8 Tg :

…15†

…13†

Speci®c heat Cg ˆ …Cdg ‡ Cv Y †=…1 ‡ Y †;

…16†

Cdg ˆ 974:740 ‡ 0:127Tg ;

…17†

Cv ˆ 10384:59

50:37Tg ‡

0:074Tg2 :

…18†

Thermal conductivity

2

…19†

3:55  10 4 Tg ‡ 6:11  10 7 Tg2 ; …20†

kdg ˆ 7:10  10

3

‡ 6:241  10 5 Tg :

…21†

Di€usion coecient 2:5116

Dva ˆ 0:175  10

6

…Tg ‡ 273:16† : …Tg ‡ 1166:44†

…22†

4.2. Solid properties

o DHvap ( 2502:54 2:3858Tp i0:5 ˆ h 2 7:3292106 15:956…Tp ‡ 273:16†

Shrinkage 8 0:0231 ‡ 0:2175X > < Vp 0:0231 ‡ 0:2175Xo ˆ 0:1866 ‡ 0:1333X Vpo > : 0:1866 ‡ 0:1333Xo

Potato was used as a model food to carry out the simulations. Thus, solid properties were calculated from proposed equations for potato (Lozano, Rotstein, & Urbicain, 1983; Crapiste & Rotstein, 1986; Ratti, 1994): Density 32:9X

259 exp‰ 3:446X Š:

…23†

Speci®c heat Cp ˆ

1610:88 4178:62X ‡ : 1‡X 1‡X

…24†

Water sorption equilibrium 

…26†

Tp 6 65:5; Tp P 65:5: …27†

1:89 6 X 6 4:5; …28† 0 6 X 6 1:89:

Two ``area to volume'' ratios appear in the previous balances. The relationship between projected area and volume of particle (A0p =Vp ) is used in the momentum balance. On the other hand, the relationship between surface area and volume of particle (av ˆ A=V ) is the exchange area involved in mass and heat transfer equations. When the particles are irregular in shape, an equivalent diameter (de ) is de®ned as the diameter of a sphere having the same volume than the particle. The sphericity shape factor of a particle (/) is then de®ned as the ratio of the surface area of this sphere to the real surface area of the particle. From this de®nition av can be written as: av ˆ

qp ˆ 1202



4.3. Geometrical factors

ydg kdg yv kv ‡ ; kg ˆ ydg ‡ yv /dgv yv ‡ ydg /vdg kv ˆ 7:028  10

 1:4798 X ‡ 25655:16 DHs ˆ 1‡X    X  exp 2:5396 ; 1‡X o DHvap

1 1 ln aw ˆ 55:544 Tp ‡ 273:16 358:8    X  exp 2:5396 : 1‡X



X 1‡X



1:4798

6 : /de

…29†

In a similar way, the ratio A0p =Vp can be expressed as: A0p 1:5 ˆ ; Vp wde

…30†

where w is the shape factor for momentum transfer de®ned as the ratio of the projected area of the equivalent sphere to the real projected area of the particle. The value of / and w depends on the particle shape. Values for some common geometric forms that can represent food particles, originally like grains and legumes or after cutting or grinding, are presented in Table 1. The shape factor w was estimated considering that particles align themselves with their maximum cross-section normal to the direction of relative motion. 4.4. Drag coecient

…25†

Data for the drag coecient (Cd ) were obtained from the literature (Pettyjohn & Christiansen, 1948; Clift,

305

A.H. Pelegrina, G.H. Crapiste / Journal of Food Engineering 48 (2001) 301±310 Table 1 Shape factors and constants of Eq. (30) for di€erent particle geometry Geometry

/

w

l=de

C1

C2

C3

Sphere Spheroid …a ˆ c ˆ 2b† Spheroid …a ˆ 2c ˆ 2b† Cylinder …h ˆ 3d† Cylinder …3h ˆ d† Parallelepiped …a ˆ b ˆ c† Parallelepiped …a ˆ b ˆ c=3†

1 0.913 0.929 0.779 0.756 0.806 0.719

1 0.630 0.794 0.714 0.630 1.209 0.838

1.571 1.565 1.247 1.904 1.680 1.612 1.118

)4.587 )3.546 )3.311 )1.287 )1.311 )2.602 )2.263

7.868 7.067 7.177 4.246 3.742 5.982 5.674

)0.1138 )0.1172 )0.1507 )0.1841 )0.1763 )0.1546 )0.1659

Grace, & Weber, 1978) and ®tted to the following equation: ln Cd ˆ C1 ‡ C2 ReCp 3 ;

Rep < 500:

…31†

The Rep number is de®ned in terms of the gas±particle slip velocity, the di€erence between air and solid velocities. The resulting parameters for the studied geometrical shapes are shown in Table 1. The equation proposed by Pettyjohn and Christiansen (1948) was used at higher Rep numbers Cd ˆ 5:31

4:88/

Rep > 500:

…32†

4.5. Friction factors The wall±air friction factor (f) was estimated from standard correlation for ¯ow in tubes (Bird, Stewart, & Lightfoot, 1960): 8 < 16 Re < 2100; f ˆ Re …33† : 0:25 0:079 Re 4000 < Re < 100 000:

4.6. Heat and mass transfer coecients Several correlations can be found in the literature to calculate the convective heat transfer coecient (h). The Ranz±Marshall equation (Bird et al., 1960) was used for spherical particles: Nul ˆ 2 ‡ 0:6 Pr1=3 Re1=2 :

…36†

For non-spherical particles h was estimated from Luikov (1980): 1=2

Nul ˆ 0:662 Pr1=3 Rel ;

…37†

where the dimensionless numbers Nu and Re are function of a characteristic length l which represents a ¯ow length over the particle and depends on its geometry. Values of l for di€erent shapes are presented in Table 1. The convective mass transfer coecient (ky ) was calculated from the psychrometric ratio:  0:56 h Sc ˆ Cg …1 ‡ Y† : …38† ky Pr 4.7. Heat losses

The friction factor for the wall±particle interaction (fp ) was calculated by using the following correlation proposed by Yang (1978):  B 1 e …1 e†vt fp ˆ A 3 e vg vp  A ˆ 0:0126; B ˆ 0:979 vg P 1:5vt ;  …34† A ˆ 0:041; B ˆ 1:021 vg < 1:5vt :

The heat loss through the duct wall (Q) was calculated by using the equation

The terminal velocity vt is given by: 8 0:71 0:153de1:14 g0:71 …qp qg † > > > 2 < Rep < 1000; < l0:143 q0:29 g g …35† vt ˆ > de2 …qp qg †g > > : Rep 6 0:1: 18lg

5. Results and discussion

This correlation was originally obtained for spherical particles. Their use is extended to other particle shapes in terms of the equivalent diameter because of the lack of additional experimental informa...