Use of Temperature Front Parameters to Compute the Drying Front

J. agric. Engng Res. (1996) 65, 313 – 323

Use of Temperature Front Parameters to Compute the Drying Front Sz. Szo¨ ke; V. Wertz; E. Persoons Unite´ de Ge´ nie rural, Faculte´ des Sciences agronomiques, Universite´ catholique de Louvain, Place Croix du Sud 2 bte 2, B-1348 Louvain-la-Neuve, Belgium (Receiy ed 5 September 1994; accepted in rey ised form 9 July 1996)

Small bins were used to conduct cooling and drying experiments on wheat, barley and peas. Four conceptual models are described. They consider different ways in which water may leave the grain. A new equation has been developed to take the hysteresis phenomenon into account. The temperature front occurring at the beginning of the drying process is modelled. An optimisation procedure is used to adjust the parameters for the heat and mass transfer coefficients to this temperature front using experimental data. The best adjusted parameters are then used to compute the drying front. The four models give similar results and the general quality of the models with all three products are similar. Information extracted from the exchanges taking place in the temperature front are not sufficient to compute the drying front accurately. On the contrary, it is found that parameters obtained by optimising the models using results from drying experiments and the drying front give better predictions. ÷ 1996 Silsoe Research Institute

1. Introduction As shown in Fig. 1 , deep bed drying may be divided into two successive phenomena. Usually, a fast move-

a b » wa wg r Ω cp dτ dz E

Notation Convection coefficient, W (kg d.m.)21 K21 Mass transfer coefficient, (kg water) (kg d.m.)21 s21 Porosity of grain Relative humidity of air Grain moisture content dry basis Density, kg m23 Bin cross sectional area, m2 Specific heat at constant pressure, J kg21 K21 Time step, s Height step in bin, m Energy, J

0021-8634 / 96 / 120313 1 11 $25.00 / 0

313

h J k K L m n P qm r r0 τ T

ya x z

Enthalpy, J kg21 Quality criterion Index Water source, kg water s21 (kg d.m.)21 Energy flow, W (kg d.m.)21 Mass of water, kg Number of samples Pressure, Pa Air flow rate, kg s21 Vaporization heat of water, J kg21 Vaporization heat of water at 08C, J kg21 Time, s Celsius temperature, 8C Air velocity (Bin supposed empty), m / s Air moisture content, kg water / kg dry air Height in bin, m

Subscripts a of air des for desorption c corrected dm dry matter e estimated, predicted eq in equilibrium g of grain m mass, measured p of grain wall v vapour w liquid water z at vertical location z τ at time τ 1 entering the layer 2 leaving the layer 3 in void air at time τ 4 in void air at time τ 1 dτ 5 lost by grain 6 in the grain at time τ 7 in the grain at time τ 1 dτ 9 given to the grain 10 for desorption ÷ 1996 Silsoe Research Institute

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¨ KE ET AL . Sz. SZO

Temperature front Drying front

Air temperature, °C

20 18 16 14 12 10

11

12

13

14

Time (June, 1992)

Fig. 1. Temperature front and drying front. Initial grain temperature: ,188C. Drying air temperature: 18 – 198C. Distance from bin bottom: d , 0?2 m; s , 0?6 m; j , 1?0 m; h , 1?4 m

ment of the temperature profile through the grain with time, called the temperature front, decreases grain temperature without changing moisture content (m.c.). It is followed by a slow movement of the same profile, called the drying front, that extracts water from the product and slowly increases its temperature. The object of this paper is to examine if a good forecast of drying front movement can be obtained by fitting four test models to the temperature front to obtain convection and mass transfer coefficients and then to apply these model parameters to calculations of the drying front. If this is possible, a prior knowledge of grain characteristics, such as convection and mass transfer coefficients, may not be necessary.

2. The model Four different models were developed and tested in nine drying experiments. Considering a single grain, some general aspects of water loss may be expressed as follows. The outside cell layers of a grain protect it against desiccation. Most of the water losses occur through the micropyle. Models considering a seed as a homogenous and isotropic volume are conceptually false. A seed can be seen as a small reservoir reacting against water exchanges to guarantee its survival and eventually its germination. Differences in the physical and chemical properties of a grain between species but also in a single species may be very important. The energy needed for water evaporation at the grain surface may have different sources, conduction from inside the grain kernel or convection from the air. Only small changes are observed in the bulk density during drying. In fact, the water content

decreases in the same range as the volume, the first balancing the effects of the second. Neglecting the desorption enthalpy is a dangerous assumption if drying occurs under about 14% moisture content (w.b.). Two different desorption isotherms are necessary to compute this enthalpy. The theoretical approach of Meunier and Wauters1 is used in this paper. The development of a mass transfer relation based on microscopic observations is not yet feasible. The structural complexity of a grain, the great variability between species and the difficulty of measuring any physical properties at a very small scale except by nuclear magnetic resonance (NMR) (Song and Litchfeld,2 Song et al.3) does not allow this approach at present.

2.1 . The water balance A thin layer of grain is considered and air is blown from the bottom to the top of the layer. The air void between the grains is the control volume. Water comes in with air at the bottom (m1) and leaves with it at the top (m2). The grain loses water (m5). This increases the water content of the inlet air from m3 to m4. During the same time, the water content of the product changes from m6 to m7. During the time dτ , water entering in the incoming air is given by m 1 5 q m dτ x(z,τ), (1) where x(z,τ) describes the air moisture content at vertical location z at time τ . Water leaving in the outgoing air is m 2 5 qm dτ x(z1dz ,τ )

(2)

Water in the control volume changes from m 3 5 » r aΩ dz x(z,r)

(3)

m 4 5 » r aΩ dz x(z,τ1dτ )

(4)

to where » represents the grain porosity; r a the air density, kg / m3; Ω the bin cross-section area, m2; dz the layer thickness, m; and qm the flow rate, kg dry air s. The amount of water lost by the grain is given by m 5 5 KΩ dz r dm dτ

(5)

where K represents water mass leaving the product, kg water kg d.m. s and rdm represents the amount of dry matter per unit volume of product. It is important to mention that K is a function, not a constant. This function K is not introduced in Eqn 5 at the moment,

USE OF TEMPERATURE FRONT PARAMETERS

because many different proposals are possible for it, but K will be detailed in Section 2?4. K is related to the mass of dry matter rather than the volume because grain volume can change by up to 25% during drying. Alternatively K could have been related to the grain surface area. This would create problems as grain surface area is difficult to measure and it also changes in an almost unpredictable manner during drying. The stability of the porosity of wheat is clearly expressed by the results shown in Table 1. These values were measured with an air pycnometer. The small bulk density (686 – 692 kg / m3) is easily explained by the small size of the sample used in a pycnometer. In bins of 400 kg, values in the range of 680 – 750 kg / m3 were measured. The grain water content decreases from the value m 6 5 r dmΩ dz §g(z,τ)

315

2.2 . The energy balance Energy (E1) enters the layer with the incoming air and energy (E2) leaves the volume with the outgoing air. Convection gives heat to the grain (E9) so its energy increases from E6 to E7. Water is lost by the product (E5) and leaves the layer as vapour. The air void enthalpy changes from E3 to E4. During the time period dτ , a heat balance shows that incoming energy is E 1 5 qm dτ (c pa Ta(z,τ) 1 xz,τ)(r0 1 c pv Ta(z,τ)))

(14)

where qm is an air flow, kg of dry air s. Writing the energy lost at the top gives E2 q m dτ [c pa Ta(z1dz ,τ ) 1 x(z1dz ,τ )(r0 1 c pv Ta(z1dz ,τ ))] (15)

(6)

The energy of the air void changes from the value

to the value m 7 5 r dmΩ dz wg(z,τ1dτ ) ,

(7)

where wg represents the grain moisture content (d.b.) The water balance of the air void can be written as m 1 2 m 2 1 m 5 5 m 4 2 m3

(8)

or

E 3 5 » r aΩ dz [c pa Ta(z,τ ) 1 x(z,τ )(r0 1 c pv Ta(z,τ ))] to

E 4 5 » r aΩ dz [c pa Ta(z,τ 1dz ) 1 x(z,τ 1dz )(r0 1 c pv Ta(z,τ 1dr ))] (17) From convection, the air gives energy to the grain

uqm dτ x uzz1dz 1 KΩ dz r dm dτ 5 u» r aΩ dz x uττ 1dz (9)

After simplifying and taking the limit, the following model is found q m ­x » r a ­x 2 (10) 1K5 Ωr dm ­z r dm ­τ For the product, the water balance is expressed by the equation m5 5 m6 2 m7 (11) where each term can be replaced by its corresponding expression to give KΩ dz r dm dτ 5 ur dmΩ dz w guττ 1dτ

(12)

and after simplification K5 2

­w g ­τ

(13)

Grain m.c. w.b.

Real density , kg / m 3

0?171 0?115 0?094

1351 1361 1369

E 9 5 LΩ dz r dm dτ

Bulk density , kg / m 3 Porosity 686 694 692

0?492 0?490 0?495

(18)

L is expressed as W kg d.m. Similarly as for K , L is not a constant but a function which will be detailed in Section 2.3. Many possibilities exist to compute these values, as will be shown later. The grain energy changes from E6 5 Ω dz r dm(c pdm 1 wg(z,τ )c pw)Tg(z,τ )

(19)

to E 7 5 Ω dz r dm(cpdm 1 wg(z,τ 1dz )c pw)Tg(z,τ 1dτ ) (20) where wg is the grain moisture content, dry basis. The heat needed to increase the energy of adsorbed water from its value to that of free water is taken from either the air or grain. Its value is given by E 10 5 (KΩ dz r dm dτ ) Dh des

Table 1 Effects of grain moisture content on porosity

(16)

(21)

where Dhdes is the amount of energy to free one kg of adsorbed water (at a liquid state). It is clear that the enthalpy of the adsorbed water is smaller than the enthalpy of free water at the same temperature. This value is easily computed from the desorption isotherms. A number of different physical models are possible to explain loss of moisture from grain. Four of these are considered below.

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2.2.1. Model 1 , y apour at the air temperature The temperature profile of a single grain could be as shown in Fig 2 . Water moves within the grain as a vapour. The energy needed for the desorption and evaporation comes from the grain and the grain loses vapour at the air temperature. Vapour enthalpy is given by E 5 5 (KΩ dz r dm dτ )(r0 1 c pv Ta) (22) 2.2 .2 . Model 2 , y apour at the grain temperature The product is generally at a lower temperature than the surrounding air (Fig. 3 ). The lost vapour does not reach the air temperature, but only that of the grain. The difference from the equation above is small because these two temperatures are very close throughout the drying process. Vapour enthalpy is given by E 5 5 (KΩ dz r dm dτ )(r0 1 c pv Tg) (23) 2.2 .3. Model 3 , liquid water at the air temperature Water is transferred as a liquid in the cell walls and cytoplasm. Vapour state is only reached at the surface, after contact with the air. The temperature profile is given in Fig. 4 . The grain energy is decreased by the value of liquid water enthalpy E 5 5 (KΩ dz r dm dτ )(c pw Ta)

(24)

2.2 .4. Model 4 , liquid water at the grain temperature As for Model 2, the grain temperature is lower than that of the air (Fig. 5 ). The air must give the lost water energy to increase its temperature and vapourise it. The heat lost by the product is the liquid water enthalpy as described by

Distance from the centre of the grain kernel

Fig. 3. Temperature profile for model 2 . Grain loses y apour at temperature Tg

wall moisture content. Both temperature and humidity gradients coexist inside the grain. We could therefore imagine a temperature situation as described in Fig. 6 . where Tp is the wall temperature. This possibility will not be discussed further in this paper because the values of Ta and Tp cannot be measured separately.

2.3 . Heat transfer Two possibilities concerning the heat transfer are possible. If only the drying period is considered, an isenthalpic transfer is a realistic assumption. In other terms, the energy given by the air to the grain is used completely for evaporating water E1 5 E2

(26)

In other cases, the heat transfer must be defined, for example, by (27) L 5 a (Ta 2 Tg)

Reality is far more complex than the above descriptions. During the drying process, the endosperm moisture content decreases later and slower than the

where a is a convection coefficient, in W kg d.m. K . This coefficient may also be a function of wa, as shown later. This equation is separated from the main equation set because it represents just one possibility among others. An energy balance is always true, a

Fig. 2. Temperature profile for model 1 . Grain loses y apour at temperature Ta

Fig. 4. Temperature profile for model 3 . Grain loses water at temperature Ta

E 5 5 (KΩ dz r dm dτ )(c pw Tg)

(25)

317

USE OF TEMPERATURE FRONT PARAMETERS

function L, K may be defined in many different ways. K is expressed here to make a distinction between fundamental physical laws (heat and mass balances) and intuitive laws. It is necessary now to define the relation between the b parameter and the air and product properties. Again, a number of assumptions are possible as shown later in this paper.

Fig. 5. Temperature profile for model 4 . Grain loses water at temperature Tg

2.5 . Dey elopment of model 4

convection equation is just a model describing what we think is the most important phenomenon taking place in this case.

For all four models, the mathematical resolution is similar. As an example, model 4 is considered below. Liquid water leaves the grain at the grain temperature, therefore E 1 2 E 2 1 E 5 2 E9 2 E 10 5 E 4 2 E 3 ,

2.4. Mass transfer Many different equations connecting the mass transfer parameter K with some physical property of the air or of the grain have been described in the literature. They consider, for example, the difference in the vapour pressure of the grain and the vapour pressure of the same product in equilibrium with the drying air, the grain temperature, the seed size, its bulk density, its moisture content, the difference between the grain moisture content and that at equilibrium. This last proposal is realistic because the water moves within the grain in a liquid state as explained by Irudayaraj et al .4 An extended Darcy law is probably not far from the truth. It is proposed that K 5 b (w g 2 w geq)

(29)

in another form uqm dτ [c pa Ta 1 x (r0 1 c pv Ta)]uzz1dz

1 Ω dz r dm dτ [K (cpw Tg 2 Dh des) 2 L 5 u» r aΩ dz [c pa Ta 1 x (r0 1 c pv Ta)]uττ1dτ .

(30)

If the moisture content of the air is considered as constant during a time period, we have 2

S

D

qm ­Ta ­x c pa 1 r0 1 [K (c pw Tg 2 Dh des) 2 L] Ωr dm ­z ­z 5

S

D

»r a Ta ­x cpa 1 r0 r dm ­τ ­τ

(31)

(28)

where K is still a water source, expressed in kg d.m. s. b represents the facility of water movement inside the grain and is expressed in s21. As explained for

where cpa is the specific heat of dry, not humid air. The energy balance of the product is E7 2 E 6 5 E 9 2 E 5

(32)

or uΩ dz r dm(c pdm 1 w g cpw)Tguττ 1dτ

5 Ω dz r dm dτ [L 2 K (c pw Tg)] (33) This relationship may be simplified into the following form ­Tg (34) 5 L 2 K (c pw Tg) ­τ Note that this equation is only true if the term cpg 5 (c pdm 1 w g c pw) is constant between the time τ (cpdm 1 w g c pw)

Fig. 6 . Real temperature profile

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¨ KE ET AL . Sz. SZO

and τ 1 dτ . In other words, the specific heat of grain is supposed constant during a computation time.

2.5.1. Integration Six relations are defined as follows 2

x(z,τ ) 5

q m ­x » r a ­x 1K5 , Ωr dm ­z r dm ­z

S

wg(z ,τ ) 5 (36)

D

(cpdm 1 wg cpw)

S

D

»r a ­Ta ­x c pa 1 r0 , r dm ­τ ­τ

­Tg 5 L 2 K (cpwTg), ­τ

(39)

K 5 b (w g 2 w geq) 5 b Dw g.

(40)

If K and L are replaced by Eqns (39) and (40)

E

» r a ­x q m ­x 1 5 b (w g 2 wgeq), r dm ­τ Ωr dm ­z ­wg 5 2b (wg 2 w geq), ­τ 2

S

qm ­Ta ­x c pa 1 r0 Ωr dm ­z ­z

D

(41)

1 (b (f g 2 w geq)(c pw Tg 2 Dh des) 2 a (Ta 2 Tg)) 5

S

D

»r a ­Ta ­x c pa 1 r0 , r dm ­τ ­τ

(cpdm 1 w g c pw)

wg(z,τ 21) 1 b Dτ wgeq(z,τ ) 1 1 b Dτ

Sr»r Dz 1 Ωqr a

m

dm

L 5 a (Ta 2 Tg),

­Tg ­τ

5 a (Ta 2 Tg) 2 b (wg 2 w geq)(cpw Tg).

2.5 .2. Resolution with a finite difference scheme We have a differential equation system concerning the variables Ta , Tg , x and wg. Integrating the equation

,

(42)

(43)

qm Dτ [r0(x(z,τ ) 2 z(z21,τ )) 2 c pa Ta(z21,τ )] Ωr dm cps

(38)

qm Dτ x(z21,τ ) Ωr dm

Ta(z,τ ) 5 Dτ Dz [b (wg(z,τ ) 2 wgeq(z,τ )(cpw Tg(z,τ ) 2 Dhdes(z,τ )) »r a 1 a Tg(z,τ )] 2 Dz [r0(x(z,τ ) 2 x(z,τ 21)) 2 c pa Ta(z,τ 21)] r dm 2

(37)

»r a x(z,τ 21) Dz r dm

»r a qm Dz 1 Dτ r dm Ωr dm

(35)

qm ­Ta ­x c pa 1 r0 1 [K (c pw Tg 2 Dh des) 2 L] Ωr dm ­z ­z

5

b (wg(z,τ ) 2 wgeq(z,τ ) Dτ Dz 1 1

­w g K5 2 , ­τ 2

set (41) gives

dm

D

Dτ 1 a Dτ Dz (44)

Tg(z,τ ) 5 (cpdm 1 w g c pw)Tg(z,τ 21) 1 a Dτ Ta(z,τ ) . (cpdm 1 w g c pw) 1 a Dτ 1 b Dτ (wg(z,τ ) 2 wgeq(z,τ )c pw (45)

2.6 . Related equations We have seen that the four essential properties needed to define a drying process are the air temperature Ta, its moisture content x , the grain temperature Tg and its moisture content wg. The dynamic behaviour of the model is governed by Eqns (42) to (45) wich give the values of Ta, x , Tg and wg. However, other variables are needed to allow computation of a drying front. These variables are linked to the four previous variables through the following relations. Eqns (46), (48) and (49) were obtained by curve fitting of data given by Houberechts5. Equation (47) is a purely theoretical law. The vapour pressure (Pv) and the heat of vaporization (r ) for water, above 08C are Pv 5 610?71 1 45?06T 1 1?32T 2 1 0?0325T 3 1 1?4036 1024 T 4 1 4?132 1026 T 5 2 4?908 1029 T 6 , r 5 2501?9 2 2?3982T 1 0?0010814T 2 ,

(46)

the absolute moisture content of air xa 5

0?6221w a Pv , P 2 w a Py

(47)

USE OF TEMPERATURE FRONT PARAMETERS

the vapour enthalpy hv 5 2501?39 1 1?83022T 1 0?000248067T 2 2 1?07504 1025 T 3 ,

(48)

and the air enthalpy h a 5 x a h v 1 1?00189T 1 4?77 1025 T 2 2 0?000378648. (49) The specific heat of grain depends on its moisture content and the specific heat of dry matter, supposed equivalent to 1?65 kJ / kg K21. Last but not least, the desorption isotherm as given by Iglesias and Chirife6 (for wheat here)

S

wg 1 2ln(1 2 w a) 5 (1 2 w g) 100 0?0019

D

1/2?3636

.

(50)

Note that in Eqns (42) to (45), values are needed for two parameters a and b . In a model fitting phase, these parameters are adjusted so as to make the model correspond to actual measurements (see Section 4.2 below). In a second step, namely the prediction phase, these values are used to compute the drying front.

The experimental apparatus is shown in Fig. 7 . The bins consist of vertical columns (G) 0?61 m in diameter and 1?5 m high, wrapped with 20 mm industrial insulating foam. They are equipped with fully perforated floors (F) and are placed on a weighing scale. Air can be blown through only one at a time. There are sampling ports every 0?2 m up the column. Temperature sensors are fixed every 0?2 m along a cable at the centre of the bin. These correspond with the sampling

Heating unit

Fog separator

Evaporator Aspiration t°

A

ϕ

Grain

t°

B

C t°

t°

Heating regulation

Cooling system regulation

G F

Air flow manually controlled

Sample ports Bellow

D

t° ϕ Scale

E

ports. The temperature sensor cable will move downward as the grain shrinks, so corrections must be made in the model before comparing the measured and computed results to take this effect into account. Outside air passes through a Freon evaporator (A) and a fog separator (B) to remove any liquid water in suspension. Saturation is reached and temperature is measured. A heating unit (C) of 6 kW and an 11 kW centrifugal fan (D) are used. A diaphragm (E) and two pressure sensors measure the air flow. The cooling and heating units are electronically controlled. Air velocities of about 0?1 m / s are used. It is possible to calculate the exact air characteristics knowing only the temperatures. However, there are three humidity sensors to measure the outside air, the heated air and at the exit air from the column. The grain was artificially rewetted before the experiment, for at least 24 h. A concrete mixer was used to homogenize the grain before each experiment. Samples of grain were taken every 3 h for moisture content measurements. Temperatures were recorded automatically each minute during the temperature front and every 5 min during the drying front. The experiments were conducted during the summers of 1992 and 1993.

4. Results and discussion

3. Materials and methods

ϕ t°

319

Diaphragm

Fan

Fig. 7. General arrangement of the drying apparatus

As expected, the best parameters found when considering wheat, barley and peas are different. The important differences in their physical properties easily explain this result. Nevertheless, the problems encountered and the general quality of the models with all three products are similar. For clarity, only numerical values for wheat are given below.

4.1. Models fitted to the temperature front All models were fitted to the temperature front, which occurs approximately during the first 2 h. The best parameters were then used to compute the drying front, which needed about 4 d to pass through the whole bin. Results of models 1 and 2 are not distinguishable. The same remark is true for models 3 and 4. The convection coefficient a is far greater in model 2 than in model 4. This is quite normal as heat given by the air to the grain in order to evaporate water, is included in that coefficient in the first case. For all experiments, results are slightly better with models 3 and 4 than with the other two models. The numerical results presented here are for model 4.

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4.2. The temperature front Relations between the convection coefficient a , the mass transfer coefficient b and the air and grain properties were examined using the model described. An optimisation procedure was applied to nine experiments (five on wheat, three on barley and one on peas) involving a temperature front. The best result was obtained with the following relationships

a 5 91?94 1023 2 74?125 1023w a, 25

b 5 1?296 10

25

1 21?124 10 w g

Air temperature, °C

28 26 24 22 20 18 16 0900

0930

1000

1030

Time, h

(51) (52)

The convection term, a depends on the air humidity. The mass transfer parameter b depends on the grain moisture content. This relationship must be treated with care because the grain moisture content does not change enough during the temperature front to ensure its validity. The optimized parameters obtained from all experiments drying the same grain were used to predict the temperature front for one experiment. The results are shown in Fig. 8 . These parameters may be used for cooling processes exceeding thirty degrees when dry grain is concerned. The decrease of accuracy with measurement height is explained by the increasing computation time needed to reach higher positions in the bin. As layer properties are calculated from the properties of the layer below and that of the same layer earlier in time, the error is a cumulative process. Therefore, the higher the layer or the later the considered period, the greater will be the inaccuracy. For very wet wheat (wg 5 0?2), the results were found to be less accurate (Fig. 9 ). The temperature front moves very fast and high temperature and humidity gradient may be observed. The model, as with every model, is a simplification of the real

Fig. 9. Modelling of the temperature front and comparison with experimental results for wet wheat grain. Calculated results are all shown by full lines. d , 0?2 m; s , 0?6 m; j , 1?0 m; h , 1?4 m

exchanges occurring in the grain and no better result was found with this kind of model, whatever the relations that described the water and heat exchanges.

4.3 . The drying front The parameters found for the temperature front were tried on the drying process (Fig. 10 ). The simulation shows that these parameters are not suitable for predicting the temperature evolution during drying. Conceptually, it is certain that the grain characteristics are dependent, among others, on the desorption isotherms. This information is not obtainable from a single temperature front, during which the grain moisture content does not fall more than 0?5%. This simulation is worse by far than was expected. It is of course possible to optimize the parameters during the drying front itself. The initial

50 Air temperature, °C

Air temperature, °C

50 40 30 20

45 40 35 30 25 20

1600

1700

1800

1900

Time, h

Fig. 8. Modelling of the temperature front and comparison with experimental results for dry wheat grain. Calculated results are all shown by full lines. d , 0?2 m; s , 0?6 m; j , 1?0 m; h , 1?4 m

1200

1800

0000

0600

1200

1800

0000

Time, h

Fig. 10. The drying front computed using the temperature front parameters Drying of wheat at 0.16 initial m.c. (w.b.). Calculated results are all shown by full lines. d , 0?2 m; s , 0?6 m; j , 1?0 m; h , 1?4 m

321

USE OF TEMPERATURE FRONT PARAMETERS

4.4 . Improy ements 4.4 .1. The hysteresis A sample of dry grain being rewetted does not reach the same moisture content as a sample of wet grain being dried, both in contact with the same air. The difference in moisture content between these two samples may be called the hysteresis amplitude. This amplitude can be obtained through experimental measurements, Fig. 13 , but in our model it is considered as an adjustable parameter. In computations, the hysteresis was handled in quite a simple way: if the difference in moisture content of a grain and the same product in equilibrium (wg) is smaller than the half amplitude of hysteresis, nothing occurs. A modification is made to Eqn (40) to consider the corrected moisture content difference instead of the real moisture content difference. This is shown in Fig. 14 . The angular points on the curve induce convergence problems in the optimisation algorithm. In order to avoid these problems, a smoothed version of this curve is obtained as follows Dwgc 5

2uDw gu arctan (K6 Dw g) π

(53)

The advantages of this equation are that only one parameter is needed (K6), no singularities appear, the hysteresis can vary from zero (a straight line) to a flat curve, as shown in Fig. 15 .

0·14 0·12 0·10 0·08 0·06 1200

1800

0000

1200

0600

1800

0000

Time, h

Fig. 12. Model of grain moisture content during the drying front in wheat. Calculated results are all shown by full lines. d , 0?4 m; j , 0?8 m; s , 1?2 m

The best result was obtained with the following relationship 2uDw gu arctan (106 Dw g) Dwgc 5 . (54) π 4.4.2 . The quality criteria Many different criteria exist to evaluate the quality of a result. During an experiment, the grain temperature is measured at seven different levels in the bin approximately 3000 times. During a prediction, all these points are computed and the mean square of the difference between measured and computed temperatures is used. This criterion does not take the results of moisture content into account. With all models described in this paper, consideration of only the temperature error gives interesting results. The results of moisture content always show the same inaccuracy: the computed drying front moves too fast. In other terms, the grain moisture content computed during a prediction decreases faster than that observed in the experimental data. In a second step, another criterion has been used to adjust the parameters, which takes both temperature

45 40 35 30 25 20 1200

1800

0000

0600

1200

1800

0000

Time, h

Fig. 11. Model of grain temperature during the drying front itself Drying of wheat at 0.16 initial m.c . (w.b.). Calculated results are all shown by full lines. d , 0?2 m; s , 0?6 m; j , 1?0 m; h , 1?4 m

Grain moisture content w.b.

Air temperature, °C

50

0·16 Air temperature, °C

conditions are defined precisely when the temperature front leaves the bin (Fig. 1 ). Model 4 was optimized using the same data just during the drying front and gave the results presented in Figs 11 and 12 .

0.22 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0·0

Desorption

Adsorption Hysteresis amplitude

0·1

0·2

0·3

0·4

0·5

0·6

0·7

Air moisture content

Fig. 13. Hysteresis amplitude

0·8

0·9

1·0

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¨ KE ET AL . Sz. SZO

Corrected moisture content difference

0·10

0·05

Angles

0·00

–0·05

–0·10 –0·10

–0·05 0·00 0·05 Moisture content difference

0·10

Fig. 14 . Hysteresis and angles (hysteresis amplitude 0?02)

and moisture content into account. An arbitrary weight must be given to the moisture content mean square error to give it the same importance as the error in temperature. The error in temperature, added to 60 000 times the error in m.c. is chosen. This value of 60 000 corresponds to the ratio of the square of the mean temperature of grain during drying (408C) to the square of its mean m.c. (0?16). The same importance is then given approximately to both errors. These values are only given as an example and must be chosen in accordance with the experiment temperature and moisture content. Accordingly, the quality criterion used here is

O

1 n J5 [(Te(k) 2 Tm(k))2 1 60 000(wge 2 w gm)2]. (55) n k51 where Te(k) and wge are the predicted temperature and moisture content at time τ , Tm(k) and wgm are the

Corrected moisture content difference

0·10

0·05

measured temperature and moisture values and n is the number of samples. This last definition of the quality criterion was used to give the results shown in Table 2 under the heading ‘model 4’. As an example, if the difference between the predicted temperature and the measured temperature at each prediction time and at each position in the bin is 1?5 K, the quality criterion has a value of 1?52 5 2?25. In the last line of Table 2, the parameters are fitted to all five experiments on wheat. The quality criterion is hence worse than when these parameters are fitted to each experiment separately. For all experiments on wheat, the quality criterion which takes account of both temperature and moisture content is lower than 8?56. As an example, Figs 11 and 12 show temperature predictions during the drying front for a model for which the quality criterion has a value of 4?6. 4.4 .3. The air flow rate It was discovered that a correction to the air flow rate greatly improved the models. The errors (for wheat) obtained with and without a corrected flow rate are given in Table 2. The air flow rate is multiplied by a parameter. Fitting this parameter gives a value between 0?52 and 0?92, depending on the experiment. The reason for this is presently unknown and further research is needed to elucidate this surprising result.

5. Conclusions This paper has investigated the accuracy of predictions for the evolution of temperature and moisture content with time in the process of drying grain. This process can be divided in two successive steps which have been called the temperature front and the drying front. The conceptual models which are used to describe these two phenomena are such that if their parameters are tuned on the first step (the temperaTable 2 Improvement with a corrected air flow rate (for wheat)

0·00

–0·05

–0·10 –0·10

–0·05 0·00 0·05 Moisture content difference

0·10

Fig. 15. Hysteresis without angles K6 y alue: —— , 10; ??????? , 20; ----- , 40; – – – , 100

Experiment

Model 4

Model 4 and corrected flow rate

1 2 3 4 5 1–5

11?17 4?44 9?05 2?60 8?45 8?56

4?48 1?36 2?39 2?27 1?82 6?38

Error J (Eqn 55) found in different experiments. A smaller value implies a better result.

USE OF TEMPERATURE FRONT PARAMETERS

ture front), the models do not give accurate predictions for the second step (the drying front). On the contrary, the work has shown that the parameters should be adjusted using the results of several drying experiments for a given product on the basis of the drying front. The work has presented conceptual models which do not need assumptions about the shape of the grain. Instead, some parameters have to be tuned using the results of specific experiments. Further modifications have been made to these conceptual models to take account of hysteresis between absorption and desorption and to provide adjustment of the air flow rate, which have improved the prediction significantly. For tuning of the models, a quality criterion was used that takes account of the errors that occur in the prediction of both temperature and moisture content.

323

References 1

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