The Thermal Conductivity Of Bakery Products Biology Essay

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In this work, an artificial neural network approach was used to model and predict the thermal conductivity of bakery products as a function of product moisture content, temperature and apparent density. Some of the bakery products considered in the work include bread, bread dough, cake, tortilla chip, whole wheat dough, etc.

Data on thermal conductivity of bakery products were obtained from the literature for a wide range of product moisture contents, temperatures and apparent densities resulting from baking conditions.

In developing the ANN model, several configurations were evaluated. The optimal ANN model was found to be a network with eight neurons in each of the two hidden layers. The optimal model was capable of predicting the thermal conductivity values of various bakery products for a wide range of conditions with a mean relative error (MRE) of 4.878 X 10-2 %, a mean absolute error (MAE) of 0.54 % and a standard error (SE) of 0.15 %. The simplest ANN model, which had one hidden layer and ten neurons, predicted thermal conductivity values with a mean relative error (MRE) of 3.388 X 10-2 %, a mean absolute error (MAE) of 0.34 % and a standard error (SE) of 0. 11 %.

Since the simplest ANN model had the least values of all three errors (MRE, MAE and SE) when compared with other configurations, including the optimal ANN model, it is, however, regarded as the best ANN model and is, thus, recommended.

KEYWORDS: thermal conductivity of bakery products, back-propagation, mean absolute error, mean relative error, standard error.

INTRODUCTION

The changes that occur in the baking process of a bakery product include physical, chemical and biochemical changes. These changes include expansion of volume, evaporation of water, formation of a porous structure, denaturation of protein, gelatinization of starch, formation of crust and browning reaction. During baking, heat is transferred mainly by convection from the heating media, and by radiation from oven walls to the product surface and then by conduction to the geometric center. At the same time, moisture diffuses outward to the product surface [1, 2]. The temperature and moisture distribution within the porous product can be predicted using diffusion equations of heat and water. A knowledge of the product properties, including thermal conductivity as a function of processing conditions is needed in order to predict the temperature and water distribution in the product during baking [2, 3].

In predicting thermal properties of a material at desired conditions, several modeling approaches have been proposed and none of them was found suitable for use over a wide range of foods [4]. According to Murakami et al., the most promising approach is based on chemical composition, temperature and physical characteristics. The series model of specific heat, density and thermal diffusivity has been successfully applied to many food materials including porous materials such as baking products. However, for the prediction of thermal conductivity of porous food, there is still some theoretical argument for the use of the structural models [4]. Murakami and Okos (1989) evaluated nine different structural models with specific types of porous foods and found that parallel and perpendicular models showed 12 - 97% and 18 - 61% standard error. Among the models, Keey's model was found to be the best prediction model for porous grains and powders. The model produced standard errors of <28% for full fat dry milk and <10% for other food materials. In addition, all structural model neglect interactions between components, phase transition and distillation heat transfer, which may be significant in the baking process [1]. Hence, most thermal conductivity models reported are usually empirical rather than theoretical.

Linko and Zhu also stated in [5] that of all the various modeling approaches of predicting the thermal conductivity of wide range of foods, including bakery products, the neural network-based models have proven to be excellent. Amongst the major benefits of using ANN are excellent management of uncertainties, noisy data and non-linear relationships. Neural network modeling has generated increasing acceptance and is an interesting method in the estimation, prediction and control of bioprocesses [5].

Commonly, neural networks are adjusted, or trained, so that a particular input leads to a specific target output. Such a situation is shown in fig. 1.0. There, the network is adjusted, based on a comparison of the output and the target, until the network output matches the target. Typically, many such input/target pairs are needed to train a network.

NN including connections (weights) between neurons

Target

Compare

Input

Output

Adjust weights

Fig 1.0: adjusting / training of NN

Neural networks have been trained to perform complex functions in various fields including pattern recognition, identification, classification, speech, vision and control systems. Today, neural networks can be trained to solve problems that are difficult for conventional computers of human beings.

Table 1: Thermal conductivity of dough and bakery products, according to Baik et al. 1999

Product

Temperature (0C)

Moisture content (%)

Apparent density (kg/m3)

Thermal conductivity (W/m K)

Technique

Reference

Bread

22

28 - 36

190 - 210

0.091 - 0.104

Line heat source

Bakshi and Yoon (1984)

Bread dough

-43.5 - 21

43.5 - 46.1

1100

0.500 - 0.920

Transient hot strip

Lind (1988)

French bread

22

42.0

161.4

0.0989

Line heat source

Sweat (1985)

Yellow cake batter

22

35.5 - 41.5

300 - 694

0.121 - 0.223

Line heat source

Sweat (1973)

Cup cake batter

20 - 104

25 - 37

272 - 815

0.068 - 0.290

Line heat source

Baik, Sablani, Marcotte, and Castaigne (1999)

Tortilla chip

25

1.4 - 35.6

520 - 880

0.09 - 0.23

Line heat source

Moreira et al (1995)

Whole wheat dough

35 - 62.5

42.4 - 46.0

1035 - 1093

0.221 - 0341

Modified guarded hot plate

Gupta (1993)

Baked chapatti

35 - 62.5

38 - 48.7

1050 - 1200

0.142 - 0.343

2.0 SIMULATION / TRAINING OF ANN

Several ANN models were simulated / trained using the thermal conductivity data set. The feed forward network structure with input output and hidden layers was used for this research. The generalized network structure is shown in fig 2 and fig 3. A commercial software package, MATLAB version 7.0.4.365 (R14) Service Pack2 with embedded neural network toolbox, was employed for this work.

Fig 2: Generalized Multilayer Neural Network

Fig 3: Generalized Multilayer Neural Network obtained with MATLAB.

For this work, the input layer consisted of three (3) neurons which corresponded to product moisture content, temperature and apparent density, while the output layer had one neuron representing the thermal conductivity. The number of hidden layers and neurons within each hidden layer can be varied based on the complexity of the problem and data set. Moreover, the number of hidden layers was varied from 1 to 2. The neurons within each of these layers were varied from 2 to 16 with increments of two. This resulted in a total of 16 networks.

The optimal configuration was based upon minimizing the difference between the neural network predicted values and the desired outputs. The data sets of 52 cases obtained from other literature [1] were divided into two sets. The first set consisted of 36 (70%) cases for training / testing and 16 (30%) cases for validation (simulation), chosen randomly from the set of 52 cases.

The back-propagation algorithm was utilized in model training. A hyperbolic-tangent transfer function was also used in all cases. Properly trained backpropagation networks tend to give reasonable answers when presented with inputs that they have never seen. Typically, a new input leads to an output similar to the correct output for input vectors used in training that are similar to the new input being presented. This generalization properly makes it possible to train a network on a representative set of input / target pairs and get good results without training the network on all possible input / output pairs.

The back propagation algorithm uses the supervised training technique where the network weights and biases are initialized randomly at the beginning of the training phase. For a given set of inputs to the network, the response to each neuron in the output layer is calculated and compared with the corresponding desired output response. The errors associated with desired output response are adjusted in the way that reduces these errors in each neuron from the output to the input layer.

In order to avoid the potential problem of over-training or memorization while employing the back propagation algorithm, the option of saving the best result is saved during the selected number of training cycles of 2,000.

2.1 SELECTION OF OPTIMAL CONFIGURATION

Upon using the following criteria: mean relative error (MRE), mean absolute error (MAE) and standard error (MAE) and standard error (SE), the performances of the various ANN configurations were compared. These are defined as follows:

n

MAE = 1 / n ∑ | KD - KP| (4.1)

i = 1

n

MRE = 1 / n ∑ | KD - KP| [%] (4.2)

i = 1 KD

n

SE = - ∑ | KD - Kp| (4.3)

i = 1

n - 1

where n is the number of data points, KD and KP are the desired and predicted values of thermal conductivity respectively. The optimum configuration of the network was chosen by selecting the lower value from the different configuration of the network.

3.0 RESULTS AND DISCUSSION

Once a given ANN configuration was trained using the input data, its performance was evaluated using the same data set. The analysis was repeated several times. The ANN configuration (out of 16) that minimized the three error measures: (i) mean relative error; MRE, (ii) mean absolute error; MAR and (iii) standard error; SE, was selected as the optimum. The error measures associated with different ANN configurations for prediction of thermal conductivity are presented in Table 2. The optimal ANN configuration included two (2) hidden layers with eight (8) neurons in each layer. The MAE, MRE and SE for this optimal configuration were 0.0054W/mK, 4.8776 X 10-4 W/mK (4.8776 X 10-2 %) and 0.0015 W/mK respectively.

Table 2: Error parameters in the prediction of thermal conductivity with different neural network configurations.

No of hidden layers

No of neurons in each hidden layer

MRE ( %)

MAE (W/mK)

SE (W/mK)

1

2

0.17

0.0199

0.0046

1

4

4.2

0.2212

0.0970

1

6

3.5077 X 10-2

0.0038

0.0011

1

8

0.18

0.0191

0.0050

1

10

3.388 X 10-2

0.0034

0.0011

1

12

0.26

0.0297

0.0073

1

14

9.0897 X 10-2

0.0149

0.0041

1

16

0.12

0.0115

0.0038

2

2

0.24

0.0298

0.0075

2

4

0.41

0.0641

0.0125

2

6

0.37

0.0606

0.0121

2

8

4.8776 X 10-2

0.0054

0.0015

2

10

0.29

0.0441

0.0106

2

12

0.22

0.0382

0.0095

2

14

0.17

0.0166

0.0047

2

16

0.22

0.0323

0.0072

To reveal the credibility of prediction (with the training data set) from the optimal ANN, predicted values of thermal conductivity, k are plotted against the desired values of thermal conductivity, k as shown in fig 4. The results demonstrate a very good agreement between the predicted and the desired values of thermal conductivity. Similarly, that of the simplest ANN model is shown in fig 5. These two figures (figs. 4 and 5) showed straight line curves, demonstrating correlation between the predicted and targeted thermal conductivities when the points are joined together (best fit-points).

Fig 4 : Correlation of targeted versus neural network values of thermal conductivity for

two hidden layers with eight neurons each (Training data set).

Fig 5: Correlation of targeted versus neural network values of thermal conductivity for

one hidden layers with ten neurons (Training data set).

The simplest ANN model with one hidden layer and ten neurons predicted thermal conductivity with a 0.34% (0.0034 W/mK) MAE, 3.386 X 10-4 W/mK MRE and 0.0011 W/mK SE being the smallest of all values of the error measures. Considering the inherent variation in the input data set, the simplest ANN configuration can be considered.

The performance of the optimal neural networkl was validated using a second data set consisting of 16 cases not previously used in the training. The optimal neural network predicted thermal conductivity values with an MRE of 0.0217 W/mK, MAE of 0.1544 W/mK and SE of 0.0156 W/mK for two hidden layers with eight neurons. The graph between the predicted and the desired values of thermal conductivity for the optimal ANN model is shown in fig.6. There seemed to be no correlation between the predicted and the targeted thermal conductivities because of the few number of cases (16) used for the validating data set, as compared to that used for the training set (36).

Fig 6: Correlation of targeted versus neural network values of thermal conductivity for

two hidden layers with eight neurons each (Validating data set).

This data set was also used with the simplest ANN configuration of one hidden layer consisting of ten neurons. The model predicted thermal conductivity with a 4.73% (0.0473 W/mK) MRE, 25.6% (0.2559 W/mK) MAE and 11.6% (0.1163 W/mK) SE. The graph between the predicted and the targeted values of thermal conductivity for the simplest ANN model is shown in fig 7. Similar reasons could also be given for the non-correlation between the predicted and targeted thermal conductivities for the validating data set, i.e. fewer number of cases (16) was used.

Fig 7:Graph of targeted versus neural network values of thermal conductivity for

one hidden layer with ten neurons (Validating data set).

3.1 DISCUSSION OF RESULTS

In this work, thermal conductivity of the bakery products was modeled, by simulation, as a function of product moisture content, temperature and apparent density. As a result, both the predicted and targeted / desired thermal conductivities are plotted separately against each of the three variables that the thermal conductivity depends on: moisture content (%), temperature (0C) and apparent density (kg/m3). The curves were obtained for (i) the optimal ANN configuration; two hidden layers with eight neurons (see figs. 8 to 10), and (ii) the simplest ANN configuration; one hidden layer with ten neurons (see figs. 11 and 13).

In all these diagrams, black triangles (∆) were used to represent the predicted thermal conductivity while purple stars (*) were used to represent the targeted / desired thermal conductivity. Figs. 8 to 13 showed a considerable and good agreement between the predicted and desired / targeted values of thermal conductivities as the points considered (i.e. triangles and stars) intersected at some locations on the curve and nearly intersected at remaining locations.

Since most of the points (triangles) of the predicted values of thermal conductivity intersected with most of the points (stars) of the thermal conductivity, it can be concluded that the predicted thermal conductivity is a good, efficient and credible prediction in comparison to the desired / targeted thermal conductivity (Figs 8 to 13) for both the optimal and simplest ANN models.

Fig 8: Predicted and targeted thermal conductivities versus temperature (two hidden

layers with eight neurons)

Fig 9: Predicted and targeted thermal conductivities versus moisture content (two hidden

layers with eight neurons)

Fig 10: Predicted and targeted thermal conductivities versus apparent density (two hidden

layers with eight neurons)

Fig 11: Predicted and targeted thermal conductivities versus temperature (one

hidden layer with ten neurons)

Fig 12: Predicted and targeted thermal conductivities versus moisture content (one

hidden layer with ten neurons)

Fig 13: Predicted and targeted thermal conductivities versus apparent density (one hidden

layer with ten neurons)

Finally, there is a great disagreement or deviation between the predicted values of thermal conductivity and the targeted values of thermal conductivity (say, when plotted against density) for an ANN model that has one hidden layer with four neurons, as shown in Fig.14. Similarly, when the predicted and targeted values of thermal conductivity are plotted against moisture content for an ANN model that has two (2) hidden layers with twelve (12) neurons, there was no agreement, as shown in Fig.15. These ANN models are said to be inefficient models because of the non-intersection of most of their points (i.e.triangles and stars).

Fig 14: Predicted and targeted thermal conductivities versus density (one hidden

layer with four neurons)

Fig 15: Predicted and targeted thermal conductivities versus moisture content (two hidden

layers with twelve neurons)

Tables 3 and 4 show the error parameters in the prediction of thermal conductivity for one and two hidden layers with 2 to 16 neurons in each hidden layer, respectively. The corresponding charts of

Table 3 and 4 are shown in figs 16 and 17 respectively. Figure 16 clearly shows that the three errors considered: MRE, MAE and SE, are least for neural network having two hidden layers with eight neurons in each hidden layer when compared with other ANN configurations (i.e. MRE = 4.8776X10-2 %, MAE = 0.0054 W/mK, SE = 0.0015 W/mK). This is, therefore, regarded as the optimal ANN configuration.

Table 3: Error prediction for one hidden layer with their corresponding neurons

No. of hidden Layers

No. of Neurons in each hidden Layers

MRE (%)

X 10- 2

MAE (W/mK)

X 10- 3

SE (W/mK)

X 10- 4

1

2

17

19.9

46

1

4

420

221.2

970

1

6

3.5

3.8

11

1

8

18

19.1

50

1

10

3.4

3.4

11

1

12

26

29.7

73

1

14

9.1

14.9

41

1

16

12

11.5

38

Table 4: Error prediction for two hidden layers with their corresponding neurons

No. of hidden Layers

No. of Neurons in each hidden Layers

MRE (%)

X 10- 2

MAE (W/mK)

X 10- 3

SE (W/mK)

X 10- 4

2

2

24

29.80

75

2

4

41

64.10

125

2

6

37

60.60

121

2

8

4.88

5.40

15

2

10

29

44.10

106

2

12

22

38.20

95

2

14

17

16.60

47

2

16

22

32.30

72

Fig 16: Chart showing the error values for Fig 17: Chart showing the error values for

two hidden layers one hidden layer

4.0 CONCLUSION

In this work, an ANN model was developed for calculating the thermal conductivity of a variety of bakery products under a wide range of conditions of moisture content, temperature and apparent density. The optimal model consisted of two hidden layers with eight neurons in each hidden layer, and was able to produce thermal conductivity values with an MAE (mean absolute error) of 54 X 10-4W/mK, MRE (mean relative error) of 40878 X10-4W/mK and an SE (siandard error) of 15 X 10-4W/mK (Table 3).

However, the simplest ANN model has one hidden layer with ten neurons. This also showed a good prediction with an MRE of about 3.388 X 10-4W/mK, MAE of 34 X 10-4W/mK and SE of

11 X10-4W/mK (Table 4).

From these values, it can be deduced and concluded that the simplest ANN model (with one hidden layer and ten neurons), when compared with the optimal ANN model (with two hidden layers and eight neurons in each hidden layer), has (i) smaller mean relative error, (ii) smaller mean absolute error and (iii) lesser standard error. Therefore, this model is recommended.

The results obtained from the curves (Figs 8 to 13) for both the optimal and simplest ANN models demonstrated a considerable and good agreement between the predicted and desired / targeted values of thermal conductivities as nearly all their points (circles and crosses) intersected one another. Hence, the predicted thermal conductivity is a good prediction.

5.0 RECOMMENDATION

Of all the ANN configurations created, tested and simulated, the simplest ANN model with one hidden layer and ten neurons produced the thermal conductivity with the least values of the three errors considered, namely: mean relative error, mean absolute error and standard error. Upon this fact, the model is consequently recommended.

The developed model can be used to estimate thermal conductivity of different bakery products for a wide range of conditions and can easily be incorporated aiding in the numerical analysis of heat and moisture transfer during baking.

The scope of ANN is increasing daily and, thus, there is need for more experts in this field.

Nevertheless, professionals such as engineers, doctors, mathematicians, physicists and the likes, should be incorporated into this encompassing field. To make it more appealing to the less matthematical ambitious people,simple paradigms should be developed.

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