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Journal of Food Engineering 86 (2008) 422–432 www.elsevier.com/locate/jfoodeng

Data-based mechanistic modelling of three-dimensional temperature distribution in ventilated rooms filled with biological material Vo Tan Thanh a,b, Erik Vranken a, Daniel Berckmans a,* a

Division of Monitoring Modelling Management of Bioresponses (M3-BIORES), Catholic University of Leuven, Belgium b Department of Food Technology, Cantho University, Viet Nam Received 18 June 2007; received in revised form 19 October 2007; accepted 23 October 2007 Available online 4 November 2007

Abstract In this paper, a data-based mechanistic modelling approach was developed to real time monitoring and online adaptive control of three-dimensional temperature distribution in both an individual biological product and in a given movement. Boxes of potatoes were used as biological material. During the experiments, step inputs on an air inlet temperature were applied while airspace and potatoes temperature were recorded. The simplified refined instrument variable algorithm was used as a model parameter identification tool to obtain the best model order and parameters. By model compacting, several physically meaningful parameters were found to present the temperature distribution between the products and the airspace. A third  order transfer function from the dynamic response of airspace from inlet air temperature with a high coefficient of determination R2T > 0:9 and a low standard error (SE < 0.01) explained the heat exchange in a system. Two physically meaningful parameters were found from the model parameters. A local volumetric fresh air concentration b1 was defined to the temperature distribution in the airspace, and a local ‘cooling rate’ a2 was presented for the product temperature distribution. The values of b1 and a2 existing in the model could be used to design a control system for real time monitoring and online adaptive control of three-dimensional temperature distribution in the process room. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Modelling; Temperature; Porous media; Storage process

1. Introduction Heating (i.e. drying, cooking and sterilisation) and cooling (i.e. chilling and cold storage) are common thermal processes in the food industry. These thermal processing techniques are widely used to improve quality and safety of food products and to extend shelf life of the products. The principle of many thermal processes of solid foods is based on heat exchanges between products and the flow around.

*

Corresponding author. E-mail address: [email protected] (D. Berckmans).

0260-8774/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2007.10.022

Conductive heat transfer through solid food products is normally modelled by Fourier’s equation of heat conduction under the boundary conditions of the governing equations. Most heat transfer models can only be solved analytically for simple cases. Numerical methods are useful for estimating the thermal behaviour of foods under complex but realistic conditions such as variation in initial temperature, non-linear and non-isotropic thermal properties, irregular-shaped bodies and time dependent boundary conditions (Puri and Anantheswaran, 1993). In solving the models, the finite difference and finite element methods are widely used (Zhou et al., 1995; Erdogdu et al., 1999; Jia et al., 2000a,b,c,d, 2001; Ahmad et al., 2001; Wang and Sun, 2002a,b,c). In recent years, the finite volume method

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Nomenclature heat capacity of air supply, J kg1 °C1 heat capacity air in the WMZ2, J kg1 °C1 heat capacity of air in the WMZ1, J kg1 °C1 heat capacity of potatoes, J kg1 °C1 model parameter model parameter model parameter model parameter heat transfer coefficient between WMZ1 and WMZ2, J s1 m2 °C1 k2 heat transfer coefficient between WMZ2 and environment, J s1 m2 °C1 km heat transfer coefficient between WMZ1 and potato, J s1 m2 °C1 m mass of one potato, kg Qc potato heat production, J s1 S1 surface area between WMZ1 and WMZ2, m2 S2 surface area between WMZ2 and environment, m2 Sm surface of one potato, m2 t0(t  s) supply air temperature at small temperature perturbations about steady state, °C T0 supply air temperature at the steady state condition, °C T0(t  s) supply air temperature, °C tbuff(t) buffer temperature at small temperature perturbations about steady state, °C T buff buffer temperature at the steady state condition, °C Tbuff(t) buffer zone temperature (WMZ2), °C Cp,0 Cp,buff Cp,i Cp,m K1 K2 K3 Km k1

was the main computational scheme used in commercial computational fluid dynamics (CFD) software packages. Computational fluid dynamics is a simulation tool, which uses powerful computer and applied mathematics to model fluid flow situations for the prediction of heat, mass and momentum transfer and optimal design in industrial processes (Xia and Sun, 2002), for analysing complex flow behaviour (Scott and Richardson, 1997), temperature distribution in storage room (Ville and Smith, 1996; Chua et al., 2002; Verboven et al., 2004; Chao and Wan, 2004), in the food industry (Wang and Sun, 2003), etc. Despite of CFD has already showed to be a valuable tool to design and optimise in many areas, CFD models suffer from their complex nature and can not be used in applications where reduced order model is mandatory, e.g. in controller algorithms (Zerihun Desta et al., 2004). The term data based mechanistic modelling (DBM) was first used by Young (2001) and this approach to modelling has been used since then in many different areas of the environment, engineering, science and social science (see, e.g. Young, 1998; Young, 2006). In DBM models, the model structure is first identified using objective methods of

ti(t)

Ti Ti(t) Tlab tm(t) Tm Tm(t) V Vbuff Vc Vm Vi a1 a2 b1 c0 cbuff ci cm n e

temperature in the well mixed zone i (WMZ1) at small temperature perturbations about steady state, °C temperature in the WMZ1 at the steady state condition, °C temperature in the well mixed zone i (WMZ1), °C temperature of environment, °C potato temperature at small temperature perturbations about steady state, °C potato temperature under steady state condition, °C potato temperature, °C volume of the WMZ1, m3 volume of the buffer zone WMZ2, m3 part of the ventilation rate entering the WMZ1, m3 s1 volume of one potato, m3 volume of air in the WMZ1, m3 model parameter cooling rate, s1 local volumetric concentration of fresh air rate, s1 density of air supply, kg m3 density of air in the buffer zone (WMZ2), kg m3 density of air in the WMZ1, kg m3 potato density, kg m3 ratio of volume and surface of potato, m porosity of media

time-series analysis based on a given, general class of time-series model (here linear, continuous-time transfer functions or the equivalent ordinary differential equations). But the resulting model is only considered fully acceptable if, in addition to explaining the data well, it also provides a description that has relevance to the physical reality of the system under study. In the present paper, this DBM approach is applied to the problem of modelling imperfect mixing in the forced ventilation of buildings, based on previous research concerned with the modelling and control of glasshouse and environmental systems (e.g. Young et al., 2000). Note that in this previous reference and in the present related study, the mechanistic interpretation is aided by the use of differential equation models, since the equations of heat and mass transfer are normally formulated in these terms. Objectives of this study focused on (1) modelling to predict airspace temperature in a room filled with biological products; (2) developing a data based mechanistic model for controlling of three-dimensional temperature distribution in air ventilated rooms filled with biological products; (3) finding an algorithm for real time monitor and online

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adaptive control for three-dimensional temperature in both an individual biological product and a regional movement air.

3

dimensional measuring grid covering a large part of the room, of which sensor numbers 1–36 are used in this research. The temperature sensors are located in two vertical planes: a ‘front sensor plane’ (0.375 m from the front wall) and a ‘rear sensor plane’ (0.375 m from the back wall). Thermocouples are further located in the slotted air inlet, in the exhaust outlet, in the buffer zone and in the laboratory hall. The accuracy of the thermocouples is 0.1 °C and the time constant is less than 3 s. An intelligent measurement and data collection unit with programmable measurement frequency is used for the data acquisition with a sampling rate of 60 s for 36 thermocouple channels. A more detailed description of the laboratory test room is given in literature (Berckmans et al., 1992). On the floor of the test installation, 12 large tubes with a diameter of 90 mm were installed, which were connected to the air outlet. These tubes were perforated with 24 holes, each with a diameter of 10 mm. The test installation was filled up with 144 boxes, each box (0.4  0.6  0.3 m) contained approximate 4.17 kg of potatoes (total 480 kg of potatoes) representing the physical presence of material in thermal processing, drying and storage processes. The 36 thermocouples (type T) were inserted inside 36 potatoes to measure the potato temperature (Fig. 2). The position of these 36 potatoes was closed to thermocouples that measure temperature in the airspace.

2. Materials and methods 2.1. Laboratory test room The laboratory test room used in this study is represented in Fig. 1. It is a transparent mechanically ventilated room with a length of 3 m, a height of 2 m and a width of 1.5 m. It has a slot inlet in the left sidewall just beneath the ceiling and an asymmetrically positioned, circular air outlet in the right sidewall just above the floor. The air volume in the test room is 9 m3. An envelope chamber of length 4 m, width 2.5 m and height 3 m is built around the test room to reduce disturbing effects of varying laboratory conditions (fluctuating temperature, opening doors, etc.). The volume of this surrounding buffer zone is 21 m3. A mechanical ventilation system enables an accurate control of the ventilation rate in the range 70–420 m3 h1 and this with an accuracy of 6 m3 h1. A heat exchanger unit is provided in the supply air duct to regulate the temperature of the inflowing air. To measure the spatio-temporal temperature distribution in the test chamber, 36 calibrated type T thermocouples are located in a three-

Envelope chamber

3m

3m

2.5 m Thermocouples 2m Inlet Front sensor plane

Rear sensor plane

1.55 m Outlet

Duct

1.5 m

Holes

4m

34

28

22

16

10

4

31

25

19

13

7

1

35

29

23

17

11

5

32

26

20

14

8

2

36

30

24

18

12

6

33

27

21

15

9

3

Numbered sensors in the front plane

Numbered sensors in the rear plane

Fig. 1. Test chamber.

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Fig. 2. Boxes of potatoes and inserted sensor in potato.

2.2. Experiments

Times-series data

Dynamic experiments were conducted in an air ventilated room filled with the boxes of potatoes, the ventilation rates were varied between 160 and 320 m3 h1 (16.6–37.6 volumes change per hour). In each experiment at a specific ventilation rate, after 2 h of steady state, the inlet temperature was adjusted from 6 to 20 °C and from 20 to 6 °C. After each step, the temperatures in air space and potatoes were monitored for 10 h with interval of 60 s per sample (Fig. 3).

1 Identification experiments

Identification of minimally parameterised model

Physical knowledge

Physical interpretation of the identified model

Databased phase

2 No

Mechanistic phase

Model physically meaningful?

2.3. Data based mechanistic modelling approach

Yes

Data based mechanistic (DBM) modelling approach was applied to estimate the temperature distributions in a room filled with biological products from the inlet temperature. This approach is illustrated in Fig. 4. The most parametrically efficient differential equation model structure is

Fig. 4. Data based mechanistic modelling approach (Young, 2002).

b Temperature (°C)

a Temperature (°C)

End

20

6

2

Time (h)

12

20

6

2

Time (h)

12

Fig. 3. Step up air inlet (a) and step down air inlet (b) of dynamic experiments; (—) inlet temperature setting; (–  –) measured response.

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first identified statistically from the experimental time-series data in an inductive manner. After this initial blackbox modelling stage, the model is interpreted in a physically meaningful parameter, mechanistic manner based on the nature of the system under study and the physical laws that are most likely to control its behaviour (Young, 2006). 2.4. Data based phase The time-series data obtained from dynamic experiments were used in the ‘Data based phase’ by means of mathematical identification techniques. Although other techniques are available, the simplified refined instrumental variable (SRIV) approach to identifying continuous-time transfer function models was employed as a method for model identification, since it not only yields consistent estimates of the parameters but also exhibits close to optimum performance in the model order reduction context (Young, 1984). Comparison  of the fitted model based on coefficient of determination R2T , Young critical identification (YIC) and standard error was used to select the best fitted model. In the present example, the continuous-time transfer function model obtained from this SRIV identification phase of the DBM analysis has a third order structure. This forms the basis for subsequent mechanistic modelling, which involves the definition of a mechanistic model that has the same third order structure, as discussed in the following section. 2.5. Data based mechanistic model in room filled with biological products Research on ventilated rooms (De Moor and Berckmans, 1993) has demonstrated that the empty test chamber that is used in this study is an imperfectly mixed airspace with considerable spatio-temporal gradients of temperature. Within such an imperfectly mixed airspace it is always possible to define a number of well mixed zones (WMZs) around temperature sensors in which there exists a good mixing at acceptably low-temperature gradient (Berckmans and Goedseels, 1986; Berckmans et al., 1992). Assuming such a well mixed zone (WMZ1) with a volume V in which n potatoes are located and Vi is a volume of airspace. The volume of V has porosity e, and the surface of heat exchange between WMZ1 and the buffer zone (WMZ2) is S1. It is assumed that there is perfectly mixed air in WMZ2 and the volume is Vbuff and surface of heat exchange between WMZ2 and environment is S2. A uniform temperature in each potato was assumed, water loss is neglected and the respiration heat from potatoes was by constant during the experiments. The heat transfer expresses in this system are shown in Fig. 5. The heat exchange for energy flow between air inlet and WMZ1, with no time delay between WMZ1 with potatoes and WMZ2 can be formulated as

5

dT i ðtÞV i ci C p;i ¼ V c T 0 ðt  sÞc0 C p;0  V c T i ðtÞci C p;i þ Qc dt þ nk m S m ðT m ðtÞ  T i ðtÞÞ þ k 1 S 1 ðT buff ðtÞ  T i ðtÞÞ

ð1Þ

Analogue, the heat balance in WMZ2 (inside buffer zone) dT buff ðtÞV buff cbuff C p;buff ¼ k 1 S 1 ðT i ðtÞ  T buff ðtÞÞ dt þ k 2 S 2 ðT lab  T buff ðtÞÞ

ð2Þ

Assuming density and specific heat of air inlet, air in WMZ1 and in WMZ2 are equal (c0 = c1 = cbuff = c; Cp,0 = Cp,i = Cp,buff = Cp), and m, Vm, Sm are mass, volume, and surface of one potato, the ratio of volume of potato and surface denotes as n ¼ VS mm The air space Vi in relate to V and e in WMZ1 can be expressed as:     1e 1e Vi V Vi ¼ V i ¼ nV m ¼ nnS m or nS m ¼ e e n ð3Þ Eq. (1) can be written as dT i ðtÞ V c Vc Qc nS m k m ¼ T 0 ðt  sÞ  T i ðtÞ þ þ ðT m ðtÞ dt Vi Vi V i cC p V i cC p k1S1  T i ðtÞÞ þ ðT buff ðtÞ  T i ðtÞÞ ð4Þ V i cC p Replacing Eq. (3) in Eq. (4), results in Eq. (5) dT i ðtÞ V c Vc Qc ¼ T 0 ðt  sÞ  T i ðtÞ þ dt Vi Vi V i cC p   1  e km þ ðT m ðtÞ  T i ðtÞÞ e ncC p k1S1 ðT buff ðtÞ  T i ðtÞÞ þ V i cC p

ð5Þ

The heat transfer from WMZ1 to the potato is dT m ðtÞ ¼ S m k m ðT i ðtÞ  T m ðtÞÞ or dt dT m ðtÞ Smkm ¼ ðT i ðtÞ  T m ðtÞÞ dt mC p;m

mC p;m

ð6Þ

With m = Vmcm and n ¼ VS mm , Eq. (6) can be minimized to Eq. (7) dT m ðtÞ km ¼ ðT i ðtÞ  T m ðtÞÞ dt cm C p;m n

ð7Þ

The set of three Equations (2), (5) and (7) present the heat transfer in an air ventilated potato room. These three equations can be minimized to dT i ðtÞ Qc ¼ b1 T 0 ðt  sÞ  b1 T i ðtÞ þ þ K m ðT m ðtÞ dt V i cC p  T i ðtÞÞ þ K 1 ðT buff ðtÞ  T i ðtÞÞ dT buff ðtÞ ¼ K 3 ðT i ðtÞ  T buff ðtÞÞ þ K 2 ðT lab  T buff ðtÞÞ dt

ð8Þ ð9Þ

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Tlab

WMZ2 Tbuff (t), Vbuff , γbuff , Cp,buff, S2 k1S1(Tbuff (t)-Ti(t))

Air enter the WMZ1 T0(t-τ), Vc, γ 0, Cp,0 WMZ1 VcT0 (t-τ ) γ 0Cp,0

VcTi(t) γ iCp,i

Vi, Ti(t), ε, S1,γi, Cp,i Air exits the WMZ1 Vc, Ti(t), Vi, γ i, Cp,i Qc

kmSm(Ti(t)-Tm(t)) Potato Tm(t), γm, Cp,m, Sm, km, ξ

Fig. 5. Scheme of heat transfer in a selected well mixed zone containing several potatoes: Cp,0, Cp,i, Cp,buff, Cp,m, heat capacity of air supply, air in the WMZ1, air in the WMZ2, and potato; k1, k2, km, heat transfer coefficient between WMZ1 and WMZ2, buffer zone and environment, and airspace to potato; n, number of potatoes in the WMZ1; T0(t  s), Ti(t), Tbuff(t), Tlab, Tm(t), temperature of air supply, the WMZ1, the WMZ2, environment, and potato; t, time; Vc, part of the ventilation rate entering the WMZ1; V, Vi, Vbuff, volume of the WMZ1, air in the WMZ1, the WMZ2; Qc, potato heat production; c0, ci, cbuff, cm, density of air supply, air in the WMZ1, air in the WMZ2, and potato; S1, S2, Sm, surface area between WMZ1 and WMZ2, WMZ2 and environment, and surface of one potato; s, time delay; n, ratio of volume and surface of potato; e, porosity of media.

dT m ðtÞ ¼ a2 ðT i ðtÞ  T m ðtÞÞ ð10Þ dt k1 S1 k2 S2 k1 S 1 Vc with  b1 ¼ V i ; K 1km¼ V i cCp ; K 2 ¼ V buff cCp ; K 3 ¼ V buff cCp ; K m ¼ 1e k m ; a2 ¼ cm Cp;m n. e ncC p Under the steady state condition Equations (8)–(10) will be Qc þ T m ðK m Þ þ T buff K 1 ¼ 0 b1 T 0  T i ðb1 þ K m þ K 1 Þ þ V i cC p ð11Þ T i ðK 3 Þ  T buff ðK 3 þ K 2 Þ þ T lab K 2 ¼ 0

ð12Þ

T i a2  a2 T m ¼ 0

ð13Þ

tm ðtÞ ¼

ð19Þ

where a1 = b1 + Km + K1. The set of three Equations (17)–(19) is presented in Fig. 6. This figure shows that the temperature of the airspace ti(t) and potatoes tm(t) can be controlled by adjusting the supply air temperature t0(t  s) when the estimated parameters b1, a1, a2, K1, K2, K3, Km are known. And the system of Equations (17)–(19) can be minimized to ti ðtÞ ¼

With a very small temperature perturbation around steady state (t0(t  s), tbuff(t), ti(t) and tm(t)), and after subtracting Eqs. (8)–(10) from Eqs. (11)–(13) results in dti ðtÞ ¼ b1 t0 ðt  sÞ  ðb1 þ K m þ K 1 Þti ðtÞ þ K m tm ðtÞ þ K 1 tbuff ðtÞ dt ð14Þ dtbuff ðtÞ ¼ K 3 ðti ðtÞ  tbuff ðtÞÞ  K 2 tbuff ðtÞÞ ð15Þ dt dtm ðtÞ ¼ a2 ðti ðtÞ  tm ðtÞÞ ð16Þ dt Laplace transformation on Equations (14)–(16) result in the following equations: b1 K1 Km t0 ðt  sÞ þ tbuff ðtÞ þ tm ðtÞ ð17Þ ti ðtÞ ¼ s þ a1 s þ a1 s þ a1 K3 tbuff ðtÞ ¼ ti ðtÞ ð18Þ s þ ðK 2 þ K 3 Þ

a2 ti ðtÞ s þ a2

b0 s 2 þ b1 s þ b2 t0 ðt  sÞ s3 þ a1 s2 þ a2 s þ a3

tm(t)

ð20Þ

α2 s + α2

Km

t0(t-τ)

β1

+

1 s + α1

ti(t)

K1

tbuff (t)

K3 s + ( K 2 + K3 )

Fig. 6. Third order block diagram of system.

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Eq. (20) is a third order of transfer function with

3. Results and discussion

b0 ¼ b 1 b1 ¼ b1 ða2 þ K 2 þ K 3 Þ

ð21Þ ð22Þ

b2 ¼ b1 a2 ðK 2 þ K 3 Þ

ð23Þ

a0 ¼ 1 a1 ¼ a1 þ a2 þ K 2 þ K 3 a2 ¼ K 2 a2 þ K 2 a2 þ a2 a1 þ a1 K 2 þ a1 K 3  K 1 K 3  K m a2 a3 ¼ a1 K 2 a2 þ a1 K 3 a 2 þ K 1 K 3 a2  K m a2 K 2  K m a2 K 3 Eqs. (21)–(23) can be minimized to a22 b1  b1 a2 þ b2 ¼ 0 or a22 b0  b1 a2 þ b2 ¼ 0

7

3.1. Temperature distribution in airspace and potatoes during experiment Temperature distribution in airspace and potatoes are presented in Figs. 7 and 8, after 5 h in the steady state condition with constant ventilation rate 200 m3 h1 (23.5 volumes change per hour). These figures have shown the non-uniformity of temperature in each case. 3.2. Uniformity of air and potato temperature in experiment

ð24Þ

With a2 value obtained from a root of Eq. (24). From a2 ¼ cm Ckmp;m n, with cm, Cp, m and n constant, a heat transfer coefficient (km) is proportional to a2. So, a2 is a value presenting the ability of heat transfer from well mixed zone to biological products. a2 value (s1) is the same structure with ‘cooling rate’ term in Newton’s law of heating/cooling processes. In cooling storage process, cooling rate is an important parameter to predict product temperature, cooling time and air velocity distribution. In the other way, Km parameter is related to heat transfer coefficient km and the relationship between a2 and Km is    1  e C p;m cm ð25Þ Km ¼ a2 e Cp c Consequently, the parameters from the fitted model (third order transfer function) are basic to identify two important values which presented for temperature distribution in airspace (b1) and products (a2).

The temperature uniformity index Itemp is a useful index to quantify the spatial homogeneity of temperature in a ventilated airspace. Itemp is defined as the volumetric part (in %) of airspace with the temperature between the limiting value Tavg  DT and Tavg + DT, where Tavg is the average of air temperature in the room, and DT is the acceptable temperature gradient, Pn voli ð26Þ I temp ¼ i¼1 VOL where voli, volume of well mixed zone, m3 (voli = 0 if temperature is not in Tavg ± DT). The temperature uniformity index in airspace is related to ventilation rate and acceptable temperature in Table 1, and the same method to calculate in case of potato bulk is in Table 2. Higher acceptable temperature gradient and higher ventilation rate resulted in a higher uniformity index in both cases. Maximum Itemp reached 67.67% for both airspace and potatoes. The uniformity of potato temperature is

Fig. 7. Temperature distribution in airspace with ventilation rate 200 m3 h1.

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Fig. 8. Temperature distribution in potato bulk with ventilation rate 200 m3 h1.

Table 1 Temperature uniformity index as a function of acceptable temperature gradient and ventilation rates in the airspace Ventilation rate (m3 h1)

Acceptable temperature gradient (°C)

160

200

240

280

300

320

0.20 0.40 0.60 0.80 1.00

2.78 16.67 33.33 44.44 66.67

2.78 8.33 22.22 36.11 52.78

5.56 19.44 27.78 38.89 63.89

11.11 19.44 25.00 33.33 50.00

13.89 13.89 27.78 36.11 58.33

5.56 19.44 22.22 36.11 61.11

Table 3 Estimated parameters from the third order of transfer function at position (6) and (30) with ventilation rate of 200 m3/h as an example of calculating method Sensor position

[n, m, s]

Parameter estimates

6

[1, 0, 60]

a1 = 0.0090 b0 = 0.0031 a1 = 0.0108 a2 = 0.0001 b0 = 0.0036 b1 = 0.00004 a1 = 0.7568 a2 = 0.0067 a3 = 0.00002 b0 = 0.0740 b1 = 0.0003 b2 = 0.00001

[2, 1, 60]

Table 2 Temperature uniformity index as a function of acceptable temperature gradient and ventilation rates in potato bulk Acceptable temperature gradient (°C) 0.20 0.40 0.60 0.80 1.00

[3, 2, 60]

Ventilation rate (m3 h1) 160

200

240

280

300

320

16.67 27.78 36.11 41.67 55.56

8.33 25.00 36.11 38.89 66.67

5.56 22.22 36.11 47.22 66.67

5.56 16.67 27.78 38.89 55.56

8.33 19.44 25.00 33.33 41.67

8.33 19.44 30.56 36.11 52.78

30

[1, 0, 60] [2, 1, 60]

[3, 2, 60]

not only depends on the uniformity index of air space but also air flow pattern in the room. 3.3. Data based phase In data based phase, the set of time-series data from experiment were used. Fig. 9 shows the inlet, airspace and potato temperature at positions (6) and (30) at ventila-

a1 = 0.0115 b0 = 0.0060 a1 = 0.0134 a2 = 0.0001 b0 = 0.0070 b1 = 0.0001 a1 = 0.2586 a2 = 0.0021 a3 = 0.00001 b0 = 0.0451 b1 = 0.0002 b2 = 0.00001

Relative standard error (%) 2.67 23.2 72.5 19.6 22.4 18.8 10.3 23.8 3.4 8.2 63.3 7.0 2.6 10.3 82.1 38.0 27.1 20.0 17.0 141.5 8.1 12.0 39.2 11.0

R2T

YIC values

SE

0.924

9.23

0.059

0.954

8.10

0.027

0.997

9.74

0.002

0.927

9.34

0.146

0.950

8.24

0.074

0.997

9.73

0.005

R2T , coefficient of determination (Young et al., 1996); YIC, Young identification criterion (Young et al., 1996); SE, standard error of determination; n, m and s, denominator, numerator and time delay; and a1, a2, a3, b0, b1, b2, parameters in the first and second order of transfer function.

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tion rate 200 m3 h1 as illustration for calculation. In this figure the airspace temperature reached the inlet temperature at the end, while potato temperature changed slowly in the experiment. The continuous-time SRIV algorithm in the CAPTAIN Toolbox for Matlab (Taylor et al., 2007) was used to identify and estimate the parameters in the third order transfer

9

function at 36 positions in the room at the ventilation rates changes from 160 to 320 m3 h1. Position 6 and 30 at ventilation rate 200 m3 h1 were selected as examples in this paper. A fully functional version of the CAPTAIN Toolbox can be downloaded from http://www.es.lancs.ac.uk/ cres/captain/). The full results of the SRIV identification analysis are given in Table 3.

Fig. 9. Dynamic temperature data at position 6 (a) and 30 (b) with ventilation rate 200 m3/h: (. . . . . .) inlet temperature, °C; (—) airspace temperature, °C; ( ) potatoes temperature, °C.

Fig. 10. Partial contour of parameter b1 (s1) at front plane with ventilation rate 200 m3 h1.

Please cite this article in press as: Thanh, V. T. et al., Data-based mechanistic modelling of three-dimensional ..., Journal of Food Engineering (2007), doi:10.1016/j.jfoodeng.2007.10.022

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From Table 3, the third order transfer function gave the best fit based on the coefficient of determination (>99%), very low of YIC values (