Article pp.589-622 du Vol.23 n°5-6 (2003)

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REVUE BIBLIOGRAPHIQUE REVIEW

Grain drying: A review

S.R. Parde¹ , D.S. Jayas^1*, and N.D.G. White²

SUMMARY

Grain drying, as a research field, now relies extensively on the use of mathematical modelling and computer simulation to describe the grain drying processes. Several mathematical models have been developed by researchers to explain the important phenomena associated with drying, namely, equilibrium moisture content, thin-layer and deep-bed drying. These models have limitations in that they are based on several assumptions. This paper reviews the various models evolved around thin-layer and deep-bed drying of grains.

List of Symbols

A = particle surface area, m² C_pa = specific heat of air, kJkg^–1 K^–1 C_pg = specific heat of grain, kJkg^–1 K^– C_pl = specific heat of liquid, kJkg^–1 K^–

C_pw= specific heat of water vapour, kJkg^–1 K^–1 D = diffusivity of vapour in porous body, m² h^–1 G_a = mass flow rate of air, kgh^–1 m^–2

G_g = mass flow rate of grain, kgh^–1 m^–2 H = humidity ratio, kg H₂O/kg_{dry air}

h’ = convective heat transfer coefficient, kJh^–1 m^–2 K^–1 h’_D = convective mass transfer coefficient, kJh^–1 m^–2 K^–1 h_fg = heat of evaporation, kJ/kg

k = drying constant, h^–1

k_ii = phenomenological constant

1. Department of Biosystems Engineering, University of Manitoba, Winnipeg, MB, Canada R3T 2N2.

2. Cereal Research Centre, Agriculture and Agri-Food Canada, 195 Dafoe Road, Winnipeg, MB, Canada R3T 2M9.

* Correspondance : jayas@Ms.UManitoba.CA.

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m = net rate of drying, kg H₂Om^–3s^–1) m_w = mass flow rate of water, kg/h M_w = molecular weight of water = 18.01

M = average moisture content of grain kernel, % dry basis M₀ = initial moisture content, % dry basis

M_e = equilibrium moisture content, % dry basis M_sur= moisture content at the surface, % dry basis P_s = saturated vapour pressure, kg/m²

P_vwb= saturated vapour pressure at wet bulb temperature, kg/m² P_v_α = vapour pressure of air, kg/m²

q = rate of heat transfer, kJh^–1 m^–2 Q = travel rate of drying zone, kgm^–2 s^–1 RH = equilibrium relative humidity, decimal r = radial distance, m

R = radius of a sphere, m R₀ = Universal gas constant R_v = gas constant for water vapour t = time, h

T_a = temperature, °C

T_abs= temperature absolute, K T_g = grain temperature, °C T_g0 = initial grain temperature, °C T_wb = wet bulb temperature, °C T^ = temperature of the air stream, °C T_a0 = initial air temperature, °C

V = volume, m³

v_a = velocity of air, m h^–1 v_g = velocity of grain, m h^–1 V_m = volume in a monolayer, m³ z = thickness of a bed of grain, m ρa = density of drying air, kg_{dry air}^m³ ρd = density of dry grain, kg^m³ ε = porosity of the bed, decimal

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1 – INTRODUCTION

Annual loss of grain, on a global basis, from harvesting to consumption is estimated at 10-25% (HALL, 1980). The magnitude of these losses varies from country to country. These losses are significantly higher in developing countries because of warm climates which cause deterioration of stored grains, and also because of lack of knowledge and proper facilities for drying and storage. The post-harvest loss is proportional to the production of grain and increases with an increase in the production. The reduction in post-harvest losses depends on proper threshing, cleaning, drying and storage of the crops. A reduction in loss of crop at one stage may have a far reaching effect on the overall reduction of the loss. For example, over-drying of paddy will increase the storage life but will also increase the breakage percentage of rice during milling (ADAMS, 1977). This suggest that drying plays a major role in increasing the efficiency of food production systems.

Drying is the removal of moisture from grain by the application of heat and is done to maintain the quality of grain during storage. This prevents the growth of bacteria and fungi and the development of insects and mites (BALA, 1997). Heat is normally supplied during drying by heating air either artificially or by natural means, and the vapour pressure or concentration gradient thus created causes the movement of moisture from inside the kernel to the surface. The speed and efficiency of drying depend on the temperature and humidity of the drying air.

Agricultural dryers are often categorized according to whether the air temperature is low (up to 5°C above ambient temperature), medium (40-250°C) or high (up to 1000°C) (PARRY, 1985). High temperature drying can speed up the drying process, but it often results in the decline in germination of grain. To have a better understanding of the drying process, the basic concept of equilibrium moisture content is very important. The objective of this paper was to review grain drying theories and the models associated with them.

2 – EQUILIBRIUM MOISTURE CONTENT AND ISOTHERMS

Data relating equilibrium moisture content (EMC) and equilibrium relative humidity (ERH) are necessary to design handling, storing and drying systems for hygroscopic materials. Equilibrium moisture content determines the mini- mum moisture content to which grain can be dried under a given set of drying conditions. It can be defined as the limiting moisture content approached by a material after it has been exposed to a particular environment for an infinitely long period of time (OLESEN, 1987). The relative humidity of the air surrounding a cereal grain in equilibrium with its environment is called the equilibrium relative humidity. The curves obtained by plotting EMC against ERH or water activity (a_w) for fixed temperatures are known as isotherms. The sorption and desorption isotherms are the plot of EMC versus ERH at a given temperature for a material subjected to wetting and drying environments (figure 1). The difference

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in these two phenomena at a given temperature for a material is hysteresis (OLESEN, 1987). A number of theoretical, semi-theoretical and empirical models have been proposed for calculating the moisture equilibria of cereals.

Figure 1

Typical sorption and desorption isotherms for grain.

2.1 Equilibrium moisture content models

BABBIT (1949) was among the first to present EMC data for wheat. HENDER- SON (1952) presented an empirical relationship to describe the sorption/desorption, or both isotherms as:

where a and b are material constants determined from the experimental data.

Other symbols are defined in the List of Symbols. To incorporate the effect of temperature, the relationship was further expressed as:

where K and N are constants determined from the experimental data.

Henderson’s equation, in the form described above, has been found inadequate for cereal grains (BROOKER et al., 1974). THOMSON et al. (1968) modified the Hender- son equation by adding another constant. The Modified-Henderson equation is:

where A, B and C are constants.

For calculating EMC values of rough rice, ZURITZ and SINGH (1985) presented the following equation:

where c₀ = 3.88368E9, c₁ = –3.52486, c₂ = –1.1205E-2, c₃ = 1.30047.

Desorption Adsorption

Relative humidity (%)

Moisture content (%)

(

aMe^b

)

RH= −

− exp

1 (1)

(

KTabsMe^N

)

(2) RH= −

− exp

1

( )

(3)

(

Â ^Tâbs ^C ^Mâ^B

)

RH= − +

− exp

1

( )

^RH ^c^T^abs^c

(

^c^T^abs^c^M^e

)

(4) n =− 0 1exp 2 3

l

(5)

STROHMAN and YOEGER (1967) proposed the following equation to represent the isotherms of corn at various moisture contents:

where a, b, c and d are experimental constants. This equation is valid over the whole range of moisture content, relative humidity and temperature.

CHUNG and PFOST (1967), based upon the assumption that change in free energy for sorption is related to moisture content, developed a following equation to describe the relationship between EMC and ERH:

where m and n are temperature-dependent constants.

PFOST et al. (1976) modified eq. (6) to obtain a better fit and presented a modified equation of the form:

where m, n and q are constants. The above equation fits grain EMC data over the 20-90% relative humidity range and it was adopted as the American Society of Agricultural Engineers (ASAE) standard D245.4 (ASAE, 1991).

BRUNAEUR et al. (1938) presented an equation, called the BET equation for multilayer molecular adsorption. The model considered internal surfaces of a grain kernel as an array of specific adsorption sites capable of adsorbing a number of water molecules. The BET equation is represented as:

where c is a product constant related to the adsorption of the water vapour.

The BET model is satisfactory in the relative humidity range below 50%.

HARKINS and JURA (1944) presented an EMC equation based on the potential theory that the total work required to adsorb (or desorb) a molecule can be considered to be equal to the sum of the work required to overcome the field strength in bringing a vapour molecule to just above the surface, plus the work of condensation. The equation is of the form:

where d and e are product constants dependent on the grain temperature.

The Harkins-Jura equation predicted grain moisture equilibrium isotherms satis- factorily at relative humidity values above 30%.

SMITH (1947) assumed the existence of two principal classes of sorbed moisture in polymers:

1) bound moisture held on the adsorbent by intermolecular forces in excess of forces responsible for condensation; and

2) normally condensed moisture.

( )

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[

a bMe nPs c dMe

]

RH=exp exp( )l + exp

( ) ( ) ( )

abs ^e (6) T nM

R RH m

n = − exp −

0

l

( ) ( ) (

e

)

(7)

abs

q nM T RH m

n −

+

= − exp

l

( ) ( )

^RH (8) c V c c V RH V

RH

m

m ⎟⎟⎠⎞

⎜⎜⎝⎛ − +

− =

1 1

1

( )

V2 (9) d e RH

n = −

l

(6)

He also accepted the multilayer concept of the BET equation for the condensed moisture and presented an EMC equation:

where f and g are experimentally determined constants dependent on the temperature.

The Smith equation fits the experimental EMC curves well for 50% < RH <

95% (BECKER and SALLANS, 1956).

HALSEY (1948) developed an equation to describe the condensation of multi- layers. The equation is:

Because the use of the R₀T_absterm does not eliminate the temperature dependence of constants A and C, IGLESIAS and CHIRIFE (1976b) simplified it to the form:

IGLESIAS and CHIRIFE (1976a) also analysed the parameter A of the Halsey equation (eq. 11) and proposed a new Modified-Halsey equation of the form:

OSWIN (1946) presented an equation which is a mathematical series expansion for a sigmoid-shaped curve and is expressed as:

where K and N are constants.

CHEN (1988, cited by CHEN and MOREY, 1989) found the parameter K to be a linear function of temperature. The Oswin equation was modified as:

A consensus has developed recently among food scientists and engineers of the superiority of the Guggenheim-Anderson-de Boer (GAB) equation for the prediction of the water activity of food products (VAN DEN BERG, 1984). The equation is derived from a physical adsorption model related to the BET theory.

The GAB equation reads:

)

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(

^RH

n g f

V= − l 1−

⎟⎟⎠ (11)

⎜⎜⎝ ⎞

= ⎛ − e^−c abs

T M R RH A

0

exp

(

^A^M^e^c

)

(12) RH=exp − ⁻

( )

RH

(

A B Tabs

)

Me^C (13) n =−exp +

. .

⁻

l

(14)

N

e RH

K RH

M ⎟

⎠

⎜ ⎞

⎝

⎛

= − 1

(15)

1

−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ⎟⎟⎠⎞ +

⎜⎜⎝⎛ +

=

C

e abs

M BT RH A

(

^B ^RH

)(

^A ^B^B^C^RH^RH^B ^C ^RH

)

(16)

M_e −

.

−

.

+

. .

=

.

1 1

(7)

Equation (16) can be used to accurately describe the majority of food isotherms up to about 90% RH (VAN DEN BERG, 1986).

JAYAS and MAZZA (1993), while evaluating three parameter equations for determining the EMC of oats, modified eq. (16) to incorporate the effect of temperature on EMC data. The original GAB eq. (16) was modified to:

2.2 Comparing models and discussion

PIXTON and HOWE (1983) examined the suitability of equations (eqs. 3, 7-10, 12 and 13) for cereals and other products. They observed that CHUNG and PFOST equation (eq. 7) gave an excellent fit to the observed data relating moisture content and equilibrium relative humidity. The Henderson equation (eq. 3) was suitable for flour. They maintained that only the Chung and Pfost transfor- mation could be used for cereals.

The Modified-Henderson equation (eq. 3) and the Modified-Chung-Pfost equation (eq. 7) were adopted by the ASAE (1991) as standard equations for describing EMC/ERH relationships of various agricultural grains. CHEN and MOREY (1989) compared four models, Modified Henderson, Modified Chung- Pfost, Modified Halsey and Modified Oswin (eqs. 3, 7, 13 and 15) for 36 sets of isotherm data. They found that the Modified-Henderson and Chung-Pfost equations were suitable for most starchy grains and high fibre materials. The Modi- fied-Halsey equation was a good model for one sorption isotherm of corncobs.

The Modified-Oswin equation served as an excellent model for popcorn, corncobs and some varieties of corn and wheat. For the high protein and oil products, Modified-Halsey and Modified-Oswin equations served as good models, while Modified-Henderson and Chung-Pfost showed a poor fit. Also the Modi- fied-Oswin equation showed the smallest mean relative percentage deviation and standard error. They, therefore, recommended that the ASAE Standard D245.4 needed to be partially revised with inclusion of Modified-Oswin and Modified-Halsey equations for certain products. For starchy grains, they found that no model was suitable for sorption isotherms for all data sets. They, therefore, concluded that no ERH model could be considered as the “Universal EMC/ERH equation” for cereal grains and other seeds.

LOMAURO et al. (1985a, 1985b) compared several sorption isotherm equa- tions for different grains, fruits, vegetables and meats. They concluded that the GAB equation gave a good fit for over 75 percent of the food products studied, and described the food isotherms better than the two parameter equations of Chung-Pfost. They further stated that the GAB equation has a theoretical basis and it therefore supplies fundamental information on the thermodynamic properties (e.g., heat of vapourization, pore structure) of a food product in contrast to the Modified-Chung-Pfost relationship which is empirical in nature.

CHEN and JAYAS (1998) evaluated the GAB equation (eq. 16) to describe the sorption data for several agricultural products. They found that the equation did not give a good fit for starchy products, because for most products, the para-

( )

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(

^B ^RH

)

^A

[

^B^B ^C^RH^T ^B^RH

(

^C ^T

)

^RH

]

M_e

•

+ •

− •

•

= •

/ 1

1

/

(8)

meters in the equation did not exhibit the Arrhenius-type relationship with temperature. In another work by JAYAS and MAZZA (1993), EMC data of oats were determined and evaluated using Modified-Henderson, Chung-Pfost, Halsey, Oswin and Modified-GAB equations (eqs. 3, 7, 13, 15 and 17). They found that the Modified-Chung-Pfost equation was the best and the Modified-GAB equation (eq. 17) was the second best in describing EMC data of oats.

The models cited above relate to what is termed the “static EMC”. The concept of a “dynamic EMC”, relevant to short-term, high temperature drying has been suggested by some researchers. It has been found that under such conditions a surface moisture content greater by several percent, at low humi- dities may be observed. SIMMONDS et al. (1953) introduced the concept of a dynamic EMC to justify the logarithmic drying law, in which it is assumed that dM/dt = – k(M – M_e). The dynamic EMC is that function giving a straight line when lllln(M – M_e) is plotted against time. Although BAKKER-ARKEMA and HALL

(1965) rejected the idea of a dynamic EMC citing an investigation of the drying of alfalfa wafers, CHEN and JAYAS (1998) recently investigated the application of dynamic EMC to the drying of corn and rice. They found that the dynamic EMC values were different from static EMCs in the high and low RH ranges.

The thin-layer drying equations for corn kernels and rice were better fitted using dynamic EMC values. Further research on this topic was recommended.

Basically, which equation is better for the design depends on which relationship represents the experimental data best within the range of moisture contents and temperatures encountered in the dryer under study. Pure empirical equations may provide better fittings and more accurate prediction (BROO- KER et al., 1974), nevertheless, theoretical or semi-theoretical equations are capable of giving insight into the physical principles of moisture sorption processes.

Current strategies for the management of grain stores is moving away from the prophylactic use of pesticides and towards a more integrated approach.

This involves a combination of measures including cooling grain with ambient air, monitoring for insects and the selective use of pesticides. In this strategy for a more physically oriented control of pests, the EMC/ERH relationship plays a vital role. A balance of sensible and latent heat shows that the evaporation of 1% moisture from grain requires sufficient heat to reduce the temperature by more than 10°C. Also, a 10°C fall in temperature at constant moisture content reduces the ERH typically by 3 to 4% due to temperature effects on the sorption isotherms. The integrated approach synthesises such beneficial effects into an overall strategy to maintain grain quality.

3 – THEORIES OF DRYING

Grain is dried using ambient or high temperature air in various grain drying systems, commonly referred to as dryers. Drying of grain requires considerable energy, between 1000 to 2000 kJ/kg of water (BAKKER-ARKEMA, 1986). Optimal design of grain dryers can result in considerable savings in energy and opera-

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ting costs. Optimal design by trial and error is time consuming and costly, but can be accomplished quickly and inexpensively with the use of simulation modelling. The optimization can be successful if sufficiently accurate dryer simulation models are developed which can predict the performance of a dryer for evaluating new dryer designs, drying efficiency, rates and energy use.

Grain drying simulation consists of a set of heat and mass transfer equations which describes: (a) the exchange of heat and moisture between grain and air;

(b) the adsorption and desorption rates of heat and moisture transfer; (c) the equilibrium relations between grain and air; and (d) the psychrometric properties of moist air. While equations in group (a) are based on the governing laws of energy and mass conservation, and equations in group (d) on the thermodynamic relations for mixtures of dry air and water vapour, equations in groups (b) and (c) are material-dependent and are based on experimental results. Grain drying is usually accomplished either in a thin-layer or in a deep-bed.

3.1 Thin-layer drying

JAYAS et al. (1991) have described the term “thin-layer” as:

1) A single grain kernel freely suspended in drying air or one layer of grain kernels.

2) A polylayer of many grain thicknesses if the temperature and relative humidity of the drying air is in the same thermodynamic state at the time of drying.

This suggests that drying a single kernel is analogous to drying grains in a thin-layer using any drying method and the velocity, temperature and relative humidity of the drying air may change the thickness of a thin-layer. Grains, when dried as a single particle under constant external conditions, may exhibit a constant-rate moisture loss during the initial drying period if the initial moisture content is high followed by a falling rate during the drying period (figure 2).

LEWIS (1921) and SHERWOOD (1929) were the first to carry out investigations on drying theory. Sherwood classified drying by the following two points:

1) Water evaporates at the solid surface and there is very little resistance to internal diffusion of liquid as compared to the removal of vapour from the surface (constant rate drying period).

2) Water evaporates at the solid surface and from the interior of the solid and resistance to internal diffusion of liquid is great as compared with the resistance to the removal of vapour from the surface (falling rate drying period).

BROOKER et al. (1974) explained the constant rate period of drying in biological products using the heat and mass transfer equations.

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Figure 2

Drying during the constant and falling rate periods.

When a stream of unsaturated air is passed over a thermometer covered with a wetted wick, evaporation takes place. Heat is transferred from the air to the wick until a temperature equilibrium is reached. At this point the convective heat transfer is equal to the latent heat needed to evaporate the liquid from the wick. The rate of heat transfer from the air at T_α to the wick at T_wb, neglecting radiation to the atmosphere and conduction along the thermometer stem, from a wick area A is:

The mass transfer from the wick to the air is due to the concentration difference or partial vapour pressure difference between the boundary layer of the wick and the air stream. The rate of evaporation is:

The rate of energy required for evaporation is:

At equilibrium conditions, the energy quantities in eqs. (18) and (20) are equal. Combining eqs. (18) and (20) results in the following expression for the moisture loss relationship for biological products during the constant-rate drying period:

During the falling-rate drying period the surface of a drying particle is not covered with a thin-layer of water because the internal resistance to moisture transport is greater than the external resistance. As the moisture content of a product falls below the critical point (this is when the drying rate of the product

M_critical

Time, t Mequilibrium

Drying rate, dM/ft

Moisture content, M

(

T Twb

)

(18) kA

h

q= ' _α−

(

vwb va

)

(19)

abs v

w D

w P P

T R

AM

m =h' −

(

vwb va

)

(20)

abs v

w D fg fg

w P P

T R

AM h h h

m

q= = ' −

( ) (

a wb

)

(21)

fg va vwb abs v

D T T

h A P h

T P R

A h dt

dM = ' − = ' −

(11)

changes from a constant rate to a falling rate), the driving potential of the drying process decreases along with the drying rate. Also, a moisture content gradient appears within the drying product and the product temperature rises above the wet-bulb temperature.

Cereal grains are mostly dried during the falling rate period. Prediction of the drying rate of a biological product during the falling rate period requires analysis of external transfer mechanisms (convective heat and mass transfer) and transfer mechanisms within the product (heat and mass diffusion). Many theories have been put forward for predicting drying behaviour of cereal grains in the falling-rate period. However, semi-theoretical and empirical relationships have proved useful to the dryer designers.

3.1.1 Drying equations

Thin-layer drying equations are useful in grain drying simulations. BROOKER

et al. (1974) have described the transfer of moisture in cereal grains by:

1) liquid movement due to surface forces (capillary flow);

2) liquid movement due to moisture concentration difference (liquid diffusion);

3) liquid movement due to diffusion of moisture on pore surfaces (surface diffusion);

4) vapour movement due to moisture concentration differences (vapour diffusion);

5) vapour movement due to temperature differences (thermal diffusion); and 6) water and vapour movement due to total pressure differences (hydrody-

namic flow).

LUIKOV (1966, 1980) developed the following equations to describe the drying of capillary porous products based on physical mechanisms mentioned above.

The coupling results from the combined effects of the moisture, temperature and total pressure gradients on the moisture, energy and total mass transfer.

The artificial drying of cereal grains occurs under circumstances when moisture flow due to a total pressure gradient is not significant in the temperature ranges employed during drying. The pressure terms in the Luikov system of equations can therefore be dropped (BROOKER et al., 1974). The modified system of equations describing cereal grain drying becomes:

P (22) k T k M t k

M

13 2 12 2 11

2 +∇ +∇

∇

⎟ =

⎠

⎜ ⎞

⎝

⎛

∂

P (23) k T k M t k

T

23 2 22 2 21

2 +∇ +∇

∇

⎟=

⎠

⎜ ⎞

⎝

⎛

∂

P (24) k T k M t k

P

33 2 32 2 31

2 +∇ +∇

∇

⎟ =

⎠

⎜ ⎞

⎝

⎛

∂

(12)

HUSSAIN et al. (1973) applied Luikov’s equations to a number of products. In drying of rough rice they found that the prediction agreed well with the experimental data, but the coupling effects of temperature and moisture were limited to certain cereal grains. Therefore, eqs. (25) and (26) were modified to:

Equations (27) and (28) describe the drying and thermal behaviour of cereal grains very well. However, since thermal diffusion is negligible in cereal grains, temperature gradients do not have to be considered. Luikov’s equations are therefore simplified to:

Since the moisture flow within a grain kernel takes place by diffusion, the transfer coefficient k₁₁ is called D, the diffusion coefficient.

For a constant value of D, eq. (29) can be written as:

where c is zero for planar symmetry, unity for a cylindrical body and two for a sphere.

CRANK (1979) described the solutions to diffusion eq. (30) in cartesian, sphe- rical and cylindrical coordinates. The following initial and boundary conditions were considered in solving eq. (30):

The solution of eq. (30) for the moisture content distribution along the sphere radius is:

T (25) k M t k

M

12 2 11

2 +∇

∇

⎟=

⎠

⎜ ⎞

⎝

⎛

∂

T (26) k M t k

T

22 2 21

2 +∇

∇

⎟=

⎠

⎜ ⎞

⎝

⎛

∂

M (27) t k

M

11

∇2

⎟ =

⎠

⎜ ⎞

⎝

⎛

∂

T (28) t k

T

22

∇2

⎟=

⎠

⎜ ⎞

⎝

⎛

∂

M (29) t k

M

11

∇2

⎟ =

⎠

⎜ ⎞

⎝

⎛

∂

⎟⎟⎠ (30)

⎜⎜⎝ ⎞

⎛

∂ + ∂

∂

= ∂

∂

r r

M c r D M t M

2 2

( )

r,0 M0 (31)

M =

( )

Rt Me (32) M , =

(33)

( ) ( )

( )

− _⎜⎜⎝^{⎛ −} _⎟⎟⎠^⎞

=

− =

−

∞ +

∑

= _n^R ⁿ_R ^D^t

r n r

R

t r M MR

M M t r M

n

e e

2 2 1 2

1 0

exp sin

2 1

, ,

π π π

(13)

Integrating eq. (33) over the volume of a sphere and then dividing by the total volume of sphere gives the average MR for the sphere.

Eq. (34) is frequently used to predict the drying of grain (PABIS and HENDER- SON 1962; BAKKER-ARKEMA and HALL, 1965; WATSON and BHARGAVA, 1974;

OSBORN et al., 1988).

Several researchers (HUSTRULID, 1963; BECKER and SALLANS, 1956) showed that by neglecting higher terms for large values of time, eq. (34) can be reduced to:

HENDERSON and PABIS (1961) showed that the drying rate constant k is related to an effective diffusion coefficient D and equivalent grain radius R by:

Thus eq. (35) becomes:

HENDERSON and PABIS (1961) further expressed the diffusion coefficient, D, in eq. (34) as a function of temperature in an Arrhenius-type relationship:

where C₁ and C₂ are constants depending on the particular grain.

Although the theoretical equations described in literature by BERGER and PEI

(1973), LUIKOV (1966, 1975), MIKHAILOV (1973), WHITNEY and PORTERFIELD (1968) and YOUNG (1969) can explain the complex phenomenon of moisture movement inside capillary-porous bodies such as grains and for predicting the moisture profile within a kernel (BRUCE, 1985; FORTES and OKOS, 1981a), but they are inconvenient for using in deep-bed simulation models (PARRY, 1985). Several researchers, therefore, have preferred to use simple semi-empirical equations for modelling thin-layers of cereals and oilseeds. BAKKER-ARKEMA (1984) has stated: “for several major small grains such as wheat and barley, the empirical drying rate equations need updating, and diffusion-type drying equations need to be developed for most grains.”

3.1.2 Semi-empirical and empirical equations

A semi-empirical grain drying model is based on the diffusion theory, assuming that the resistance to water diffusion in a kernel occurs in the outer layer of the kernel. Thus, using Newton’s law of cooling and assuming a similarity between the cooling and drying of a solid body, drying rate can be expressed as:

⎥⎦ (34)

⎢⎣ ⎤

=

∑

^∞₌ _n ⎡−ⁿ _R^D ^t MR

n

2 2 2 1

2

2 1 exp

6 π

π

⎥⎦ (35)

⎢⎣ ⎤

⎡−

= t

R MR 6₂exp ² D₂

π π

(36)

2 2

DR k= π

[ ]

^kt (37) MR= 6 exp −

π2

⎟⎠ (38)

⎜⎝ ⎞

⎛−

=

Tabs

C C

D ₁exp ²

(

M Me

)

(39) dt k

dM=− −

(14)

Upon integration, eq. (39) becomes:

where k is a constant.

Several investigators (SIMMONDS et al., 1953; KACHRU et al., 1971; WATSON

and BHARGAVA, 1974) have widely used eq. (40) to describe drying of wheat, rice and shelled corn. This equation however, cannot describe drying rate accurately throughout the drying period. It models the grain drying process during the latter part of drying. To improve the closeness of predictions with the experimental data during early periods of drying, HENDERSON (1974) proposed the following equation:

where a, b, k₁ and k₂ are calculated from the experimental data. Equation (41) was further modified by NOOMHORM and VERMA (1986) for drying of rough rice because this grain could not be treated as a homogeneous material. They proposed:

PAGE (1949, cited by JAYAS et al., 1991) modified eq. (40) by adding an expo- nent n to time t to improve the fit to thin-layer data for shelled corn and since then the modified equation has been used widely for characterizing thin-layer drying of cereals, oilseeds, ear corn and clover (WHITE et al., 1973; JAYAS and SOKHANSANJ, 1986, 1989; SINGH (JAYAS) et al., 1983; PABIS and HENDERSON, 1962; BRUCE, 1985). Page’s equation is:

where k and n are moisture transfer rate parameters and are dependent on the product being dried. Equation (43) considers moisture transfer due to liquid diffusion with resistance to the moisture transfer occurring in a thin outer layer of the kernel. It also shows that the rate of moisture transfer is dependent on the time elapsed. The moisture transfer rate is expressed as a function of time which apparently also accounts for vapour flux due to temperature gradients, irregular shape and anisotropy of kernels, shrinkage and expansion of kernels, thereby giving the equation a good predictability. FORTES and OKOS (1981a, 1981b) have stated that for low temperature drying, other factors affecting the moisture transfer rate may be product-dependent, because liquid diffusion due to a concentration gradient is the dominant moisture transport mechanism. The parameter n in eq. (43), therefore, can be assumed to be a product-dependent constant. This assumption makes it easy to compare the effects of independent variables such as temperature and relative humidity on the moisture transfer rates by direct comparison of parameter k of eq. (43), which otherwise is not possible because of the random adjustments in the parameters to give the best fitting curve (JAYAS et al., 1988).

MISRA and BROOKER (1980) developed a single thin-layer drying and rewet- ting equation using eq. (43). They concluded that the drying temperature and

( )

_M ^MR

( )

^kt (40) M

M t M

e

e = = −

−

− exp

0

( )

^t ^a

(

^k ^t

) (

^b ^k^t

)

(41) MR = exp − ₁ + − ₂

( )

^t ^a

(

^k^t

)

^b

(

^k ^t

)

(42) MR = exp− ₁ + exp − ₂

( )

^ktⁿ (43) MR=exp−

(15)

the air velocity significantly affected k, and the relative humidity of air and corn’s initial temperature affected n. BRUCE (1985) differentiated eq. (43) with respect to time as:

where drying rates depended on the time and moisture content. He found eq. (43) superior to eq. (40) since it depended on moisture content only.

3.1.3 Comparing drying equations

Several researchers have compared various forms of drying equations and have selected the one which best incorporated the drying data for a particular grain. MISRA and BROOKER (1980) compared the thin-layer drying data of corn compiled from eight sources. They fit eq. (43) to the data, with k as a function of temperature and air velocity, and n as function of initial moisture content and humidity. SYAREIF et al. (1984) fit eqs. (40), (42) and (43) to the drying data of sunflower seed in the range of 27 to 93°C. They found that the Page equation (eq. 43) had the best overall fit.

BRUCE (1985) compared eqs. (40) and (44) using experimental drying data for barley. He found that eq. (44) gave a better overall drying curve over the whole drying period. It did not however, predict the initial drying rate of barley.

3.1.4 Discussion on thin-layer drying equations

Not one of the theoretical or semitheoretical equations presented represents the drying process of cereal grains accurately over the full moisture content range from M_o to M_e. To obtain an acceptable agreement between experimental drying rates and those calculated with one of the drying equations, an arbi- trary value is often inserted into the MR term of the equations.

There appear to be two reasons why the drying equations based on diffusion theory do not represent the drying behaviour of cereal grains:

1) the improper choice of boundary conditions in solving the equations; and 2) the incorrect assumption that D and k are independent of moisture con-

tent.

For example, first the boundary conditions in eq. (32) indicate that the grain surface moisture content reaches the EMC instantaneously. This assumption is more of a simplification. It is more realistic to solve the diffusion equation with a convective type boundary condition:

Also, in the development of the diffusion type of drying equations it was assumed that the diffusion coefficient, D, is constant during the isothermal drying process and not dependent on the grain moisture content. If the drying takes place over a significant moisture content range, this assumption can lead to serious errors in the calculated moisture contents, especially in larger grains such as corn.

(

e

)

(44)

n M M

dt nkt

dM =− ⁻¹ −

[

^sur ^e

]

(45)

D r R

M M r h

D M = −

∂

=

'

(16)

Taking this limitation into consideration, several researchers (HUSTRULID, 1963; STEFFE and SINGH, 1982) have expressed diffusion coefficient as a function of grain moisture content and temperature. BRUCE (1985) has stated that a diffusion coefficient expressed as a function of temperature does not represent grain drying phenomena adequately.

Empirical drying equations often give the best results in predicting drying behaviour of cereal grains. The equations can be used within the temperature, relative humidity, airflow velocity and moisture content range for which they were derived.

3.2 Deep-bed grain drying

Deep-bed drying refers to the heterogeneous drying of grain in a deep layer (more than 20 cm deep) where drying is faster at the inlet end of drying air than at the exhaust end. It could be visualized as the drying of successive thin-layers of grains arranged one above the other. The rate of moisture removal is maxi- mum for the bottom layer and decreases exponentially for subsequent layers.

Grain during deep-bed drying is either stationary or moving, and is exposed to gradients of temperature and moisture content and is dried by convection (PABIS et al., 1998). The conditions of grain and air change with the position of the grain and time during deep-bed drying.

The drying of cereal grains in a deep-bed is analysed using thin-layer equations so that deep-bed drying can be mathematically simulated (BROOKER et al., 1974). In thin-layer drying, however, there are no gradients of temperature or moisture content and deep-bed drying, therefore, cannot be described by just one equation as can thin-layer drying.

Theories on deep-bed drying and the mathematical models based on these theories have been put forward by many researchers. Computer simulation techniques based on these models provide a better way of understanding the complex mechanisms and processes of drying of grain in a deep-bed. These models have been developed under the following assumptions (HUKILL 1954;

BARRE et al., 1971; BLOOM and SHOVE, 1971):

1) volume shrinkage is negligible during the drying process;

2) temperature gradients within the individual particles are negligible;

3) heat and mass transfer by conduction between kernels of grain are negligible;

4) bins walls are adiabatic, with negligible heat capacity;

5) accurate thin-layer drying equation and moisture equilibrium isotherm are known;

6) heat capacities of moist air and of grain are constant during short time periods;

7) airflow and grain flow are the plug type; and

8) and are negligible compared to and , where z is the deep-bed thickness in the direction of airflow.

t T

∂

t H

∂

z T

∂

z H

∂

(17)

These eight assumptions indicate that the grain-drying models represent adiabatic desorption from uniformly packed beds. Convective energy transfer controls heat transfer. Mass transfer from the bed particles to the inert gas is controlled by the desorption rate as described by a thin-layer drying equation.

Equilibrium exists between the water vapour in the air and on the kernel surfaces because desorption is considered to occur instantaneously. The interparti- cle void fraction is constant (e.g., m³ of air per m³ of bed), as are the bed height and the specific external surface area of the bed.

Deep-bed drying models are classified as (MOREY et al., 1978):

1) logarithmic models;

2) non-equilibrium models; and 3) equilibrium models.

Logarithmic models are simple and are applied in all grain drying theories.

Non-equilibrium models, which consider deep-bed as a series of thin-layers, include a set of partial differential equations to describe deep-bed drying, while the equilibrium models are applied in the stationary bed drying.

3.2.1 Logarithmic models

They are based on direct relationship between the rate of drying of a layer of grain in a deep-bed, , and the air temperature gradient, , across this layer. They involve simple heat balance equations and these facilitate rapid computation of drying times, moisture removal and drying zone properties.

HUKILL (1954) was the first to develop a logarithmic model for modelling grain drying processes in a deep-bed. Assuming that the (time) rate of drying at some depth z, after time t, is proportional to the (spatial) rate of decrease in air temperature at (z, t), he obtained:

This is equivalent to assuming that the (sensible) heat energy lost by the air provides solely the latent heat of vapourization and neglects sensible heating of the grain. He obtained a solution to eq. (46) of the form:

where x and τ are dimensionless depth and time variables, respectively. He, however, found that this model underestimated the time required to dry grain to a specified moisture content.

BAUGHMANet al. (1971) modified this equation and established a relationship between temperature and moisture gradients through the bed. He expressed it as:

where Q is the travel rate of the drying zone.

t M

∂

z T_a

∂

(46) t

h M z C T

G_a _pa ^a _d _fg

∂

= ∂

∂

∂ ρ

(47) 1

2 2

2

−

= _x+ ^x_τ MR

(48) z

Qh M z C T

G_a _pa ^a _fg

∂

− ∂

∂ =

∂

(18)

The solution of eqs. (46) and (48) with a thin-layer drying equation, e.g.:

yields the logarithmic grain drying model:

YOUNG and DICKENS (1975) used Hukill’s model to estimate the costs of grain drying in fixed bed and cross-flow systems and SABBAH et al. (1979) used such a model to simulate solar drying of grain.

Logarithmic models are useful because of their simplicity and hence compu- tational economy. They are, at best, only acceptable in low temperature, low airflow rate dryer simulations (PARRY, 1985).

3.2.2 Non-equilibrium models

Non-equilibrium models of grain drying assume that there is no heat and mass equilibrium between the drying air and grain throughout the bed. Based on this assumption a system of partial differential equations is derived from the laws of heat and mass transfer, the mathematical theory of drying of single solid bodies, and the general grain drying theory to represent the grain drying models. These models are the theoretical models for the grain drying systems provided there exists a temperature difference between the drying air and grain, and the temperature of drying air is assumed to be higher by at least 5°C than the ambient temperature. A number of such models explain the grain drying simulations in a deep-bed.

3.2.3 Model of BAKKER-ARKEMA et al. (1967)

BAKKER-ARKEMA et al. (1967) developed a set of partial differential equations to describe the heat and mass transfer in a stationary bed of biological materials. They considered the drying process as a non-stationary process with moisture content and temperature dependent on position, z, and drying time, t, i.e., M = M (z, t) and T_g = T_g (z, t). The temperature gradients within the kernel were assumed to be negligible so temperature of a grain kernel became the average temperature.

⎟ (49)

⎟

⎠

⎞

⎜⎜

⎝

⎟⎟⎠ ⎛

⎜⎜⎝ ⎞

⎛

−

= −

fg a pa e wb a

h G C M M

T T

0 0

(

M Me

)

(50) t k

M =− −

∂

(51)

0 = + −1

−

Y D

D

e e

e M

M M M

( )

(52)

(

a wb

)

pa a

e d

fg

T T C G

M M D kh

−

= −

0

ρ 0

kt (53) Y=

(19)

When modelling the grain drying process in a deep-bed, the following unk- nowns were identified:

1) M = M (z, t), the average moisture content of grain kernel;

2) T_g = T_g (z, t), the average temperature of a grain kernel;

3) H = H (z, t), the humidity ratio of drying air in a bed; and 4) T_a = T_a (z, t), the temperature of drying air in a bed.

Based on this, the grain drying process in a stationary bed was described using four partial differential equations. They were: mass and energy balances of grain and drying air and a drying rate equation of a thin-layer of grain.

1) Mass balance equation:

2) Drying rate equation:

3) Balance on the enthalpy of air:

4) Balance on the enthalpy of grain:

3.2.4 Spencer model

SPENCER (1969) presented a stationary deep-bed drying model using following assumptions:

1) the thermal properties of dry grain, moisture, and air are constant;

2) there is negligible heat transfer by conduction within the deep-bed;

3) the effect of condensation within the bed is negligible; and 4) the problem is one dimensional.

The four partial differential equations he gave were:

1) Air temperature equation:

2) Grain temperature equation:

(54) t

M z

G_a H _d

∂

− ∂

∂ =

∂ ρ

( )

_M ^MR

(

^k ^t

)

(55) M

M t M

e

e = = − •

−

− exp

0

(

^a ^g

)

(56)

pw a pa a

a T T

H C G C G

A h z

T −

+

= −

∂

∂ '

( ) ( )

(57)

z G H M C C

T T C h M C C

T T A h t T

a pl d pg d

g a pw fg pl d pg d

g a g

∂

∂ +

− + +

+

= −

∂

ρ ρ

'

(

^g ^a

)

(58)

a a a a

pa h T T

t T z

G T

C ⎟ = −

⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂

∂ ερ '

( ) ( )

(59)

(

pg pl

) ( )

d d fg g g a

M C C

t h M

T T h t T

ρ ε

−

+ ∂

− ∂ +

= −

∂

1 1 '

(20)

3) Mass balance equation:

4) Thin-layer drying equation

3.2.5 Sharp model

SHARP (1982) described drying in a stationary bed. He expressed the specific heat of moist air as:

And the specific heat of moist grain as:

By conducting energy and mass balances, SHARP (1982) presented the following partial differential equations to describe drying of grain in a stationary bed:

He further suggested that the terms containing and in the above equations be neglected.

3.2.6 Laws and Parry model

LAWS and PARRY (1983) presented a general mathematical model for grain drying in the five basic types of grain drying systems: stationary-bed, deep-bed, cross-flow, concurrent-flow and counter-flow. They considered moist air and moist grain as a binary mixture; assumed one-dimensional mass flow rates and

( )

(60) t M t

H z

G_a H _a _d

∂

− ∂

∂ = + ∂

∂

∂ ερ ρ 1 ε

(

^M^,^M ^,^T^,^t^...

)

(61) t f

M

= e

∂

( )

^H ^C ^C ^H (62) C = _pa+ _pw

( )

^M ^C ^C ^H (63) C = _pg+ _pl

(64)

(

^C ⁺^C ^H

)

^∂_∂^T_z ⁼

G_a _pa _pw ^a

(

^T ^T

)

^M_t ^h

(

^T ^T

) (

^C ^C ^H

)

^T_t

C_pw _a _g _a _g _a _pa _pw ^a

d ∂

+ ∂

−

∂ −

− ∂ ρ ε

ρ '

(65) t

H t

M z

G_a H _d _a

∂

− ∂

∂

− ∂

∂ =

∂ ρ ερ

(

^C^pg ^C^pw^M

)

^T_t^g ^h

(

^T^a ^T^g

)

^h^fg ^d ^M_t (66)

d ∂

+ ∂

−

∂ =

+ ∂ ρ

ρ '

(

M Me

)

(67) t k

M =− −

∂

t T_a

∂

t H

∂

(21)

constant densities for air and grain and neglected conductive and radiative heating effects. The partial differential model was presented as:

where z is the direction of airflow, y is the direction of grain flow, the vector {u} is {u} = [H, M, T_a, T_g]^T, and [A], [B] and {b} are expressed as:

where E_w(T_a) = e_w + C_pwT_a and E_pl(T_g) = e_pl + C_plT_g are the specific internal energies of vapour in the air and liquid (water) in the grain, respectively. The base energies of vapour in the air and water in the grain, e_w and e_pl, are constant. The symbols m and Ψ are functions of H, M, T_a, T_g, v_a and v_g and represent the net rate of mass transfer (drying) and energy transferred from the grain to the air, respectively. Here the net rate of drying is expressed as:

and the net rate of energy transferred from the grain to the air by:

where E_w(T_w) is the internal energy of the net water vapour produced.

Since in a stationary deep-bed, velocity of grain is zero, eq. (68) reduces to:

[ ]

^A ^u_z

[ ]

^B ^u_y

{}

^b (68) t

u =

⎭⎬

⎫

⎩⎨

⎧

∂ + ∂

⎭⎬

⎫

⎩⎨

⎧

∂ + ∂

⎭⎬

⎫

⎩⎨

⎧

∂

[ ]

(69)

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

=

0 0 0 0

0 0 0

0 0 0 0

0 0 0

a a

v v A

[ ]

(70)

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

=

g g

v B v

0 0 0

0 0 0 0

0 0 0

0 0 0 0

{ } [ ( ) ]

(71)

( )

[ ]

[ ( ) ]

( )

[ ]

^⎪^⎪^⎪

⎪⎪

⎭

⎪⎪

⎪

⎬

⎫

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

+ +

+

−

=

⎪⎪

⎭

⎪⎪⎬

⎫

⎪⎪

⎩

⎪⎪⎨

⎧

=

M C C

T mE

H C C

T mE m m

b b b b b

pl pg d

g pl

pw pa a

a w d

a

ρ ψ ερ ρ ψ ερ

4 3 2 1

(

e

)

(72)

dk M M

m= ρ −

(

^T^a ^T^g

)

^mE^w

^{( )}

^T^w (73)

h − −

=

Ψ '

{ } [ ] { } {}

^b (74) z A u t

u =

∂ + ∂

∂

(22)

Writing this in the form of differential equations (LAWS and PARRY, 1983):

with the following initial conditions:

3.2.7 Cross-flow models

In a cross-flow dryer, air flows in the z-direction (vertically downward) and grain in the y-direction (horizontally) (figure 3). By replacing the term ∂t by in eqs. (54) and (57) and introducing G_g = ρdv_g, BROOKER et al. (1974) presented a cross-flow model of deep-bed drying with the following partial differential equations:

(75) b1

dt dH=

(76) b3

dt dT_a

=

(77) b2

dt dM =

(78) b4

dt dT_g

=

( )

^z ^M

( )

^z ^z ^L (79) M ,0 = ₀ for 0≤ ≤

( )

^z ^T

( )

^z ^z ^L (80) T_g ,0 = _g₀ for 0≤ ≤

( )

⁰^,t =H0

( )

t ^fort>⁰ (81) H

( )

⁰^,t =T0

( )

t ^fort>⁰ (82)

T_a _a

vg

∂y

(

^a ^g

)

(83)

pw a pa a

a T T

H C G C G

A h z

T −

+

= −

∂

∂ '

( ) ( )

(84)

z G H M C G C G

T T C h M C G C G

T T A h y T

a pl g pg g

g a pw fg pl g pg g

a

g g

∂

∂ +

− + +

+

= −

∂

∂ '

(85) y

G M z G_a H _g

∂

− ∂

∂ =

∂

(

^M^,^M ^,^T^,^t^...

)

(86) t f

M

= e

∂

(23)

Figure 3

Schematic of a cross-flow dryer.

LAWS and PARRY (1983) presented a cross-flow model by reducing eq. (68) to:

where {u} = {u(z, y)} and the properties of air and grain entering the dryer are time independent.

The system of equations they presented were:

If the depth of a dryer is L and the width is W, the boundary conditions for this model are:

Grain flow

Air flow

[ ]

^A ^∂_∂

^{{ }}

_z^u ⁺

[ ]

^B ^∂_∂

{ } {}

_y^u ⁼ ^b (87)

1 (88) dz b v_a dH=

3 (89) dz b v_adT^a =

2 (90) dy b

v_gdM =

4 (91) dy b

v_gdT^g =

( )

^z ^M ^z ^W (92) M ,0 = ₀ for 0≤ ≤

( )

^z ^T ^z ^W (93) T_g ,0 = _g₀ for 0≤ ≤

( )

^y ^H ^y ^L (94) H 0, = ₀ for 0≤ ≤

( )

^y ^T ^y ^L (95) T_a 0, = _a₀ for 0≤ ≤

(24)

3.2.8 Concurrent-flow models

In a concurrent flow dryer (figure 4), both air and grain flow in the same direction z, but the velocity of the air, v_a, differs from the velocity of the grain, v_g. The grain drying model for a stationary deep-bed (eqs. 54, 55, 56 and 57) can be converted to represent steady-state drying in a concurrent-flow dryer (BROOKER et al., 1974) as:

Figure 4

Schematic of a concurrent flow-dryer.

Since z and y coincide in a concurrent-flow dryer, the general model of LAWS

and PARRY (1983) (eq. 68) can be written as:

where {u} = {u(z, t} and C = A + B. Here b₂ is proportional to b₁ and therefore:

(

^a ^g

)

(96)

pw a pa a

a T T

H C G C G

A h dz

dT −

+

= − '

( ) ( )

(97)

dz G dH M C G C G

T T C h M C G C G

T T A h dz dT

a pl g pg g

g a pw fg pl g pg g

g a g

+

−

− + +

= ' −

(98) dz

GdM dz G_a dH=−

(

^M^,^M ^,^T^,^t^...

)

(99) dt f

dM

= e

Air flow

Grain flow

(100)

{ } [ ] { } {}

^b

z C u t

u =

∂ + ∂

∂

(101) b2

dz v_gdM =

(102) b3

dz v_a dT^a =

(103) b4

dz v_g dt=

(104) constant

= + v M H

v_a _d _g

a ρ

ερ