Applied Energy 86 (2009) 2244–2248

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Applied Energy

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Prediction of thermal conductivity of ethylene glycol–watersolutions by using artificial neural networks

Hüseyin Kurt a,*, Muhammet Kayfeci b

a Karabuk University, Technical Education Faculty, 78050 Karabuk, Turkeyb S.Demirel University, Graduate School of Natural and Applied Science, 32260 Isparta, Turkey

a r t i c l e i n f o

Article history:Received 13 November 2007Received in revised form 27 August 2008Accepted 12 December 2008Available online 4 February 2009

Keywords:Thermal conductivityEthylene glycol–water solutionsArtificial neural network

0306-2619/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.apenergy.2008.12.020

* Corresponding author. Tel.: +90 3704338200; faxE-mail address: huseyin_kurt@hotmail.com (H. Ku

a b s t r a c t

The objective of this study is to develop an artificial neural network (ANN) model to predict the thermalconductivity of ethylene glycol–water solutions based on experimentally measured variables. The ther-mal conductivity of solutions at different concentrations and various temperatures was measured usingthe cylindrical cell method that physical properties of the solution are being determined fills the annularspace between two concentric cylinders. During the experiment, heat flows in the radial direction out-wards through the test liquid filled in the annual gap to cooling water. In the steady state, conductioninside the cell was described by the Fourier equation in cylindrical coordinates, with boundary conditionscorresponding to heat transfer between the solution and cooling water. The performance of ANN wasevaluated by a regression analysis between the predicted and the experimental values. The ANN predic-tions yield R2 in the range of 0.9999 and MAPE in the range of 0.7984% for the test data set. The regressionanalysis indicated that the ANN model can successfully be used for the prediction of the thermal conduc-tivity of ethylene glycol–water solutions with a high degree of accuracy.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Heating or cooling fluids are important to many industrial sec-tors, including transportation, energy supply and production, andelectronics. The thermal conductivity of these fluids plays a vitalrole in the development of energy-efficient heat transfer equip-ment. Numerous methods have been proposed on the measure-ment of their thermal conductivity under different ranges oftemperature and pressure over the past several decades. Thesemeasured methods can be classified into two types, namely, stea-dy-state and unsteady-state methods [1–5]. The unsteady-statemethods are well developed, but require expensive equipment.Among various steady-state methods cylindrical method is one ofthe most commonly used techniques for the measurement of thethermal conductivity of liquids. This method is practical, inexpen-sive and its accuracy is comparable with other methods [1]. In thisstudy, the cylindrical method was used to measure the thermalconductivity of ethylene glycol–water solutions. Ethylene glycolbased water solutions are the most commonly used antifreeze fluidfor standard heating and cooling applications, where the tempera-ture in the heat transfer fluid can be below 0 �C. This is the situa-tion especially for cooling systems where the fluid operates withtemperatures below the freezing point of water. It is also common

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in heating applications that temporarily may not be operating insurroundings with freezing conditions such as solar collector appli-cation and water cooled engines. The physical properties of ethyl-ene glycol based water solutions vary significantly with thepercent of ethylene glycol and the temperature of the fluid. Theproperties differs so much from the pure water that a heat transfersystems with ethylene glycol should be calculated thoroughly forthe actual temperatures and solutions. Adding ethylene glycol tofluids can achieve the effect of lowering the freezing point and rais-ing the boiling point. However, the thermal conductivity of ethyl-ene glycol is lower than that of water; hence when added tofluid, it can affect the heat transfer performance. This problemcan be resolved by adding nanoparticles with high thermal con-ductivity to aqueous solutions of ethylene glycol. Jwo and Teng[6], Beck et al. [7], Liu et al. [8] investigated the thermal propertiesof ethylene glycol–water containing nanoparticles using the tran-sient hot-wire method. The results indicated that nanobrine withethylene glycol of volume fractions ranging from 10% to 90% hasbetter thermal properties than aqueous solution of ethylene glycol.

The objective of this study is to develop an ANN model to pre-dict the thermal conductivity of ethylene glycol–water solutionsbased on experimentally measured variables. The thermal conduc-tivity of liquids have been predicted by using this model and com-pared with experimentally measured values. The ANN has beenapplied successfully in various fields of modeling and predictionin many engineering systems, mathematics, medicine, economics,

Fig. 1. Cross-section of the equipment used for thermal conductivity measurement.

• Ti • To

r1

r2

r3

r4

Heater

Sample

Cooling water

Fig. 2. Thermal resistances related to equipment.

H. Kurt, M. Kayfeci / Applied Energy 86 (2009) 2244–2248 2245

meteorology and many others. It has become increasingly popularduring the last decade. The advantages of ANN compared to classi-cal methods are its high speed, simplicity and large capacity whichreduce engineering effort [9,10]. Some recent applications aremade in thermal engineering [9–25]. Considerable amount of re-search can be found about thermal engineering applications ofneural networks in literature such as; solar radiation estimation,thermal load prediction, cooling energy consumption, estimationof the heat transfer coefficient, heat transfer analysis of heat ex-changer and HVAC systems, energy consumption, fuel consump-tion, and engine emission analysis, etc.

2. Materials and method

2.1. Design of equipment

In order to measure the thermal conductivity of liquids such asthe ethylene glycol–water solutions, equipment has been designedand constructed. The thermal conductivity of solution at differentconcentrations and various temperatures was measured usingthe method, based on a cylindrical cell, where the solution whoseproperties are to be determined fills the annular space betweentwo concentric cylinders. The equipment is shown schematicallyin Fig. 1. It consists of a coaxial inner and outer cylinder. The innercylinder is made of copper and the outer cylinder is made of galva-nize. They have 26/28 and 42/48 mm of inner/outer diameters, and145 mm of length. An electrical heater was located into the innercylinder. It was made of a constantan wire (resistance: 160 X,maximum power: 302 W, voltage: 220 V), coiled around a ceramiccore and electrically insulated by settling in a glass tube. The innercylinder and the glass tube were separated by a 1.5 mm annulargap which was filled with test liquid. Cooling water flows with aflow rate of 2 l/min in the annular gap between inner and outercylinders. Front and back sides of the equipment was covered withinsulators to prevent axial heat transfer. Therefore, the axial heatloss from the front and back sides is considered to be small; andthus it is neglected in the measurement of thermal conductivity.During the experiment, heat flows in the radial direction outwardsthrough the test liquid filled in the annual gap to the cooling water.Two calibrated Fe–Constantan thermocouples were used to mea-sure the outer surface temperature of the glass tube (Ti) and the in-ner cylinder (To). The thermocouples were positioned in the middleof test section and connected to a multi channel digital read outwith an accuracy of 0.1 �C. The required measurements for calcula-tion of the thermal conductivity are the Ti and To temperatures, ad-justed voltage and current of the heater. The voltage and currentapplied to the heater is measured by using a digital multi-meterwith accuracy of 0.1 V and 0.05 A. The power input to the heaterwas controlled by means of a dimmer, which allowed the adjust-ment of the current with a stability of 0.1%.

2.2. Calculation of the thermal conductivity

In the steady state, conduction inside the cell was described bythe Fourier equation in cylindrical coordinates, with boundary con-ditions corresponding to heat transfer between the glass tube andouter surfaces of the inner cylinder kept at constant temperatures,and the thermal conductivity of liquid in the gap, as given by thefollowing equation:

k ¼ln r2

r1

� �

2pL DT_Qe�

lnr3r2

� �2pLkc

24

35ðW=mKÞ ð1Þ

where the heat input, Qe (W) was calculated from measurement ofthe current and voltage through the heater (Qe = IV), DT is temper-ature difference between Ti (�C) and To (�C), kc is thermal conductiv-ity of copper (W/mK), L is length of cylinders (m), r1 is outer radiusof the glass tube (m), and r2, r3 are inner and outer radius of the in-ner cylinder (m) as shown in Fig. 2.

The accuracy of the equipment was tested by determining ther-mal conductivity of the pure water at temperatures ranging from10 to 40 �C. The expected thermal conductivity values of the purewater were 0.575, 0.597, 0.612, and 0.628 W/mK at 10, 20, 30,

Input Layer

Hidden Layer

Output Layer

Bias Bias

Temperature

Concentration

Density

Thermal Conductivity

Fig. 3. The architecture of ANN used for this study.

Table 1The range of input and output parameters and normalization values.

Parameters Minimum Maximum Normalization value

Temperature, T (�C) 10 80 100Concentration, X (%) 0 100 120Density, q (kg/m3) 974.08 1138.25 1200Thermal conductivity, k (W/mK) 0.214 0.645 0.666

2246 H. Kurt, M. Kayfeci / Applied Energy 86 (2009) 2244–2248

and 40 �C, respectively. The calculated thermal conductivity valuesof the pure water according to Eq. (1) by using measured temper-ature difference, current, and voltage on the equipment were0.507, 0.552, 0.582, and 0.604 W/mK at 10, 20, 30, and 40 �C,respectively. Mean relative error between the expected and calcu-lated thermal conductivity values is 7.659%. After the accuracy ofthe device is tested, experiments are repeated using various ethyl-ene glycol–water solutions at the same conditions with the purewater experiments. The thermal conductivity was calculatedaccording to Eq. (1) by using measured temperature difference,current, and voltage.

3. Artificial neural network (ANN) principles

Networks with biases can represent relationships between in-puts and outputs more easily than networks without biases. Atransfer function, consisting generally of algebraic equations,may be linear or non-linear. The weighted sum of input compo-nents is calculated as [9–10]

NETj ¼Xn

i¼1

wijxi þ bi ð2Þ

where Netj is the weighted sum of the jth neuron for the input re-ceived from the preceding layer with n neurons, wij is the weightbetween the jth neuron and the ith neuron in the preceding layer,xi is the output of the ith neuron and bi is bias value in the precedinglayer. The output of the jth neuron, outj, is calculated using a sig-moid function as follows:

outj ¼ f ðNETjÞ ¼1

1þ expð�kNETjÞð3Þ

where k is a constant used to control the slope of the semi-linear re-gion. The sigmoid non-linearity activates in every layer except theinput layer.

The most basic and commonly used ANN is the multi-layer per-ception (MLP). This consists of at least three or more layers, an in-put layer, an output layer, and a number of hidden layers. Backpropagation (BP) algorithm, as one of the most famous trainingalgorithms for the MLP, is a gradient descent technique to mini-mize the error e through a particular training pattern in which itadjusts the weights by a small amount at a time [16,24].

To develop an ANN model, the network is processed throughtwo stages: training/learning stage and testing/validation stage.In the training stage, the network is trained to predict an outputbased on input data. In the testing stage, the network is tested tostop or continue training it, and it is used to predict an output. Itis also used to calculate different measures of error. The networktraining process is stopped when the testing error is within the tol-erance limits [10–13].

The BP algorithm is the most popular and extensively used algo-rithm. It consists of two phases: the feed forward pass and back-ward pass process. During the feed forward pass, the processingof information is propagated from the input layer to the outputlayer. In the backward pass, the difference between the output va-lue obtained from the network by feed forward process and desiredoutput is compared with the prescribed tolerance limits, and theerror in the output layer is computed. This obtained error is prop-agated backwards to the input layer in order to update the connec-tion [10,24].

3.1. Application of ANN on the experimental data

The objective of developed ANN model is to predict the thermalconductivity of ethylene glycol–water solutions based on the tem-perature, concentration and density. The available data set fromthe experimental study was divided into two sets; training and

testing sets. The experimental data set consists of 40 values. TheANN model was trained using 32 randomly selected data (account-ing for 80% of the total data) while the remaining eight data(accounting for 20%) was utilized for testing of the network perfor-mance. The ANN modeling program was implemented underMATLAB.

The BP learning algorithm has been used in feed-forward, whichincluded an input layer, a hidden layer and an output layer net-work. The number of neurons in the input layer is equal to thenumber of input parameters and the number of neurons in the out-put layer is output. Optimal number of the neurons in the hiddenlayer was determined by trying different networks. The numberof neurons was increased from 3 to 8 based on the trial and errormethod in the hidden layer. It is found that, best structure of theANN model has four neurons in the hidden layer. Therefore, devel-oped ANN architecture has a configuration of 3–4–1 neurons asshown in Fig. 3.

After development of the ANN architecture, experimentally ob-tained input and output data was normalized using a simple nor-malization method in the (0–1) range in order to improvetraining characteristics by using normalization values given in Ta-ble 1. Maximum and minimum values of the inputs and outputsare also given in Table 1. There are three input parameters in theinput layer, namely the temperature difference, concentrationand density of the ethylene glycol–water solutions. The output isthe thermal conductivity of solution. Then normalized data waspresented to the network as input data for training of the network.The chosen network architecture was 3–4–1 with a binary sigmoidtransfer function and the BP learning algorithm.

The connection weights were initialized randomly at the begin-ning of the training process. The network was trained until the er-ror, which is defined as the sum of the squares of differencesbetween desired output and network output, becomes acceptable.The sum of the squares error (SSE) was chosen as 0.0001 in thisstudy. If the SSE was greater than 0.0001, the network was runagain through all the input data until the SSE is within the requiredtolerance. To find out the optimum result, 200 iterations weredone.

The prediction performances of the networks were evaluatedusing root-mean-squared (RMS), mean absolute percentage error(MAPE), SSE and statistical coefficient of multiple determination(R2) values, which were calculated by the following expressions:

Table 2Weight and bias values between input and hidden layer.

Weights Number of hidden layer neurons (i)

1 2 3 4

w1i 2.31 7.0774 2.0369 1.1095w2i �20.6802 �3.0686 15.4676 �3.206w3i �1.0559 �6.0763 �6.0197 14.3968bi 2.011 2.4261 4.9572 9.4867

Table 3Statistical parameters of the ANN used for prediction of thermal conductivity.

Training Test

MAPE (%) 0.3383 0.7984RMS 0.0021 0.0046SSE 0.00001424 0.00001663R2 0.9999 0.9999Cov 0.0035 0.0078

H. Kurt, M. Kayfeci / Applied Energy 86 (2009) 2244–2248 2247

RMS ¼ ð1=pÞX

j

ðjtj � ojjÞ2 !1=2

ð4Þ

MAPE ð%Þ ¼ o� to

� �� 100 ð5Þ

SSE ¼X

j

ðoj � tjÞ2 ð6Þ

R2 ¼ 1�P

jðtj � ojÞ2PjðojÞ2

!ð7Þ

where t is the target (network output) value, o is the output (de-sired) value, p is the pattern.

The explicit formulation of the thermal conductivity is derivedby using the parameters, as inputs, weights and normalization fac-tors, the proposed ANN model. All necessary parameters are ob-tained from the trained ANN model, and the explicit expressionis formed from the weights of the trained ANN model. Each inputis multiplied by a connection weight. In the simplest case, productsand biases are simply summed, then transformed through a trans-fer function (logistic sigmoid) to generate a result, and finally out-puts are more easily obtained. Inputs and outputs are normalizedbefore the learning process of the ANN. To get an accurate resultfrom the formula proposed in this study, normalization valueshave to be considered as well. It should be noted that the proposedformulation is valid between the maximum and minimum valuesof the input parameters given in Table 1. In order to calculate thethermal conductivity in a functional form in terms of the temper-ature, concentration and density of ethylene glycol–water solution,is given as follows:

k ¼ 3=2

1þ e7:6323

1þe�F1� 2:3396

1þe�F2þ12:3381

1þe�F3� 9:8273

1þe�F4�3:4119

� � ð8Þ

where values of F1–F4 were obtained by employing these indepen-dent variables from Eqs. (9)–(12)

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8Measured

Pre

dict

ed

+ 5 %

-5 %

MAPE= 0.3383 % R2 = 0.9999

Fig. 4. Comparison of experimentally measured and ANN-predicted values ofthermal conductivity for the training data set.

F1 ¼ 2:31�T � 20:6802�X � 1:0559�qþ 2:011 ð9ÞF2 ¼ 7:0774�T � 3:0686�X � 6:0763�q� 2:4261 ð10ÞF3 ¼ 2:0369�T þ 15:4676�X � 6:0197�qþ 4:9572 ð11ÞF4 ¼ 1:1095�T � 3:206�X þ 14:3968�q� 9:4867 ð12Þ

where the constants are the connection weight values (wi,j) and lastterms are bias, which are constant in all equations and their valuesare given in Table 2. The three input parameters are temperature(�C), concentration (%), and density (kg/m3) of the ethylene gly-col–water solutions. In the ANN model, four hidden neurons areused, thus four pairs of equations is required, which represent thesummation and activation functions of each neuron of the hiddenlayer, respectively.

4. Results and discussion

In this study, the ANN model is developed for predicting thethermal conductivity of ethylene glycol–water solutions. The pre-dictions of trained ANN for temperature (T), concentration (X)and density (q) of the solutions as a function of the experimentalones are shown in Fig. 4 for the training data set. To evaluate theaccuracy of the ANN predictions, each graph is provided with astraight line indicating the perfect prediction. Figures also showanother prediction performance measurement that is within ±5%error band based on error analysis, respectively. All of the temper-ature prediction errors for the training data set are within the ±5%error band. The performance of the neural network prediction wasevaluated by a regression analysis between the predicted and theexperimental values. The ANN predictions yield the statistical coef-ficient of multiple determination (R2) in the range of 0.9999 andmean absolute percentage error (MAPE) in the range of 0.3383%for the training data set, as shown in the Table 3. The R2 and MAPEvalues are within an acceptable range. A comparison of the ANNpredicted and experimentally measured thermal conductivity val-ues for testing data set are shown in Fig. 5. Note that in Fig. 5, thecomparisons were made using only the values from the test dataset, which was not introduced to the network during the trainingprocess and was selected randomly from experimentally obtaineddata set. The ANN prediction and experimental values for thesetemperatures yield R2 of 0.9999 and MAPE of 0.7984%, as shownin Table 3. The regression coefficients obtained from testing ofthe ANN were perfect and within the acceptable limits in bothcases. As the correlation coefficient approaches to 1, the accuracyof the prediction improves. In the presented case, the correlationcoefficients range is very close to 1, which indicates excellentagreement between the experimental and the ANN predicted re-sults. Figures also show another prediction performance measure-ment that is within ±5% error band based on error analysis. All ofthe thermal conductivity prediction errors for the test data setare within the ±5% error band. Variation in the predicted thermalconductivity is shown in Fig. 6 as function of temperature and con-centration of the ethylene glycol–water solutions. From Fig. 6, itcan be seen that the thermal conductivity of solution was affectedby the temperature and concentration of the solution.

In our previous paper [26], regression analysis of calculatedthermal conductivity values by using classical statistical analysis

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8Measured

Pre

dict

ed

+ 5 %

-5 %

MAPE= 0.7984 % R2 = 0.9999

Fig. 5. Comparison of experimentally measured and ANN-predicted values ofthermal conductivity for the test data set.

2248 H. Kurt, M. Kayfeci / Applied Energy 86 (2009) 2244–2248

method were made and thermal conductivity was expressed as afunction of temperature and concentration, and temperature anddensity. The classical statistical analysis method yields the statisti-cal coefficient of multiple determination (R2) in the range of 0.9861and mean relative error (MRE) in the range of 2.6% for function oftemperature and concentration, and 0.9909 and 1.024% for func-tion of temperature and density. The regression analysis parame-ters obtained from the classical statistical analysis methodindicated that the ANN model can successfully be used for the pre-diction of the thermal conductivity of ethylene glycol–water solu-tions with a high degree of accuracy.

5. Conclusion

This paper presents an application of the ANN in the prediction ofthe thermal conductivity of ethylene glycol–water solutions basedon the temperature, concentration and density. In this study, toprove whether neural networks can be used for the prediction ofthermal conductivity of the liquids. From the presented results itis proved that ANNs can be used with satisfactory accuracy for theprediction of thermal conductivity of the ethylene glycol–watersolutions. This study helps application engineers determine thethermal conductivity of the ethylene glycol–water solutions easilywithout exhaustive experiments, thus saving both money and time.The ANN model based on a back propagation algorithm was devel-

10 20 30 40 50 60 70 80 0 20 40 80 100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Ther

mal

Con

duct

ivity

(W

/mK)

Temperature (ºC)Concentration (%)

Fig. 6. Predicted thermal conductivity as function of temperature and concentra-tion of the ethylene glycol–water solutions.

oped, which has a one hidden layer and 3–4–1 neuron configuration.The performance of the ANN prediction and experimental resultswas measured using the root-mean-squared (RMS), mean absolutepercentage error (MAPE), sum of the squares error (SSE), the statis-tical coefficient of multiple determination or correlation coefficients(R2) values. The developed ANN model showed a good regressionanalysis with the R2 in the range of 0.9999 and MAPE in the rangeof 0.7984% for the test data set. The ANN predicted the temperaturesof solar cooker within ±5% error. As the regression coefficients indi-cated that the ANN approach could be considered as an alternativeand practical technique to evaluate the thermal conductivity ofthe ethylene glycol–water solutions based on the temperature, con-centration and density with a high degree of accuracy.

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