Fuzzy logic to predict the heat transfer in an air cooler equipped with different tube inserts

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International Journal of Thermal Sciences 53 (2012) 141e147

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International Journal of Thermal Sciences

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Fuzzy logic to predict the heat transfer in an air cooler equipped with differenttube inserts

Alimohammad Karami a,*, Ehsan Rezaei a, Mohsen Shahhosseni b, Masood Aghakhani c

aDepartment of Mechanical Engineering, Kermanshah University of technology, Kermanshah, IranbDepartment of Chemical Engineering, Kermanshah University of technology, Kermanshah, IrancDepartment of Mechanical Engineering, Razi University, Kermanshah, Iran

a r t i c l e i n f o

Article history:Received 6 March 2011Received in revised form21 October 2011Accepted 24 October 2011Available online 22 November 2011

Keywords:Air cooled heat exchangerFuzzy logicMamdani fuzzy inferenceHeat transferTube inserts

* Corresponding author. Tel: þ98 21 88957750; faxE-mail addresses: [email protected]

yahoo.com (E. Rezaei), [email protected] (Mrazi.ac.ir (M. Aghakhani).

1290-0729/$ e see front matter � 2011 Elsevier Masdoi:10.1016/j.ijthermalsci.2011.10.016

a b s t r a c t

This paper highlights the application of fuzzy logic to predict the heat transfer in an air cooled heatexchanger equipped with three types of tube inserts including; butterfly, classic and jagged twisted tape.In the study which was carried out in the range of Reynolds number, the maximum heat transfer wasobtained by the butterfly insert with an inclined angle of 90�. The results have also revealed that the heattransfer rates obtained by employing the classic and jagged inserts, reduced by increasing twist ratio.Moreover, it has been also shown that, fuzzy logic is a powerful technique used for predicting the heattransfer due to its low error rate. The average error of fuzzy technique as compared with experimentaldata was found to be 0.68% for this study.

� 2011 Elsevier Masson SAS. All rights reserved.

1. Introduction

1.1. Heat transfer enhancement in heat exchangers

It is commonly known that the heat transfer rate of heatexchangers especially for single-phase flows can be improved uponthrough many enhancement techniques. In general, heat transferenhancement (HTE) techniques can be divided into two categories:(1) active techniques which need an external power source and (2)passive techniques which do not need an external power source.Some examples of passive HTE methods include: insertion oftwisted stripes and tapes [1,2], insertion of coil wire and helicalwire coil [3,4] and mounting of turbulent decaying swirl flowdevices [5,6]. Despite the high pressure drop caused by an insert inembedded tubes, the use of tube inserts in heat exchangers hasreceived lots of attention during the last two decades [2,7]. Theincrease in turbulence intensity and swirling flowmay be the mainreasons for HTE induced by tube inserts. An experimental studywas carried out on heat transfer in a round tube equipped with

: þ98 21 88956207.m (A. Karami), Ehsan.rezaei@. Shahhosseni), aghakhani@

son SAS. All rights reserved.

propeller type swirl generators by Eiamsa-ard et al. [8]. The effectsof the blade angle, pitch ratio and number of blades on the Nusseltnumber, pressure loss were also studied. In another work, Eiamsa-ard et al. [9] examined the heat transfer enhancement inside tubesusing louvered strip tube inserts. Their experiments were carriedout with louvered strip inserts with forward and backwardarrangements accompanied by various inclined angles. Kongkait-paiboon et al. [10] investigated the heat transfer of a circular tubefitted with perforated conical-rings. Chang et al. [11,12] studied theheat transfer enhancement in a tube fitted with serrated twistedtapes and broken twisted tapes. Saha [13] studied the heat transferand pressure loss behaviors in rectangular and square ducts withcombined internal axial corrugations and twisted tapes with andwithout oblique teeth. Many studies have been carried out onporous media insert as a kind of satisfying inserts for enhancingheat transfer in the tube flow. Shabanian et al. [14] studied heattransfer enhancement in an air cooler equipped with different tubeinserts. They showed that using different tube inserts (butterfly,jagged and classic twisted tape inserts), increased the heat transferfrom the air cooler. Also, they included that using the butterflyinsert with an inclined angle of 90�, maximum heat transfer wasobtained. Moreover, results revealed that the thermal performancefactor decreased with the increase in Reynolds number, due to themore significant role of inserts in increasing the turbulence inten-sity at lower velocities. In the present work, the effect of twist ratio

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Fig. 1. A schematic representation of a triangular membership function.

A. Karami et al. / International Journal of Thermal Sciences 53 (2012) 141e147142

of classic and jagged twisted tape inserts and inclined angle ofbutterfly inserts on heat transfer in an air cooled heat exchangerwere modeled based on experimental data collected by Shabanianet al. [14].

Fig. 2. (a)Schematic diagram of the experimental

1.2. Fuzzy logic (FL)

Fuzzy logic (FL) is a method introduced for the first time in 1965by Zadeh [15] and has been used to model the experiments.In research works, modeling of experiments can be helpful in orderto reduce the cost of experimentation and at the same time topredict the results of which are not possible to be performed due tosome restrictions.

Fuzzy logic starts with the concept of a fuzzy set. A fuzzy set isa set without a crisp, clearly defined boundary. It can containelements with only a partial degree of membership. In other words,a fuzzy set is an extension of a classical set. If X is the universe ofdiscourse and its elements are denoted by x, then a fuzzy set A in Xis defined as a set of ordered pairs.

A ¼ fx;mAðxÞjx˛Xg (1)

Where, mA(x) is called the membership function (or MF) of x in A.Themembership functionmaps each element of X to amembership

rig and (b)The used tools in the experiment.

Fig. 3. Comparison of experimental results of Shabanian et al. [14] for the Nusseltnumber for the plain tube with the results of Gnielinski [21].

Table 1Selected structure of the proposed fuzzy model.

Type of fuzzy inference system (FIS) Mamdani

Inputs/output 4/1Input membership function types TriangularOutput membership function types TriangularNumber of input membership functions 7/3/3/4Number of output membership function 100Rules weight 1Number of fuzzy rules 252And method MinImplication method MinAggregation method MaxDefuzzification method Centroid

A. Karami et al. / International Journal of Thermal Sciences 53 (2012) 141e147 143

value between 0 and 1. In other words, a membership function (MF)is a curve that defines how each point in the input space is mappedto amembership value (or degree of membership) between 0 and 1.The input space is sometimes referred to as the universe ofdiscourse, a fancy name for a simple concept. The membershipfunctions are of many types such as trapezoidal, generalized bell,Gaussian, bell, sigmoidal, pi shaped, z- shaped s-shaped and etc.More information about the membership functions and theirproperties can be found in [16]. The most commonly simple usedmembership function is triangular membership function [16,17]defined as

y ¼ trimfðx; parametersÞ (2)

y ¼ trimfðx; ½ a b c �Þ (3)

The triangular curve is a function of a vector x, and depends onthree scalar parameters a, b, and c, as given by

mAðxÞ ¼

8>>>>>>>>>>><>>>>>>>>>>>:

0;

x� aibi � ai

;

ci � xci � bi

;

0;

x � ai

ai � x � bi

bi � x � ci

ci � x

9>>>>>>>>>>>=>>>>>>>>>>>;

(4)

or, more compactly, by

f ðx; ai; bi; ciÞ ¼ max�min

�x� aibi � ai

;ci � xci � bi

�;0

�(5)

As it can be observed in Fig. 1, the parameters a and b locate the“feet” of the triangle and the parameter c locates the peak.

To start with the fuzzy logic, following stepsmust be considered:

1.2.1. Choosing a fuzzy inference system (FIS)The first step is to choose a fuzzy inference system (FIS)which its

duty is to map an input space to an output space. Two types of fuzzyinference systems have been widely employed in various applica-tions such as automatic control, data classification, decision analysis,expert systems, and computer vision: Mamdani fuzzy models [18]and Sugeno fuzzy models [19], which is similar to the Mamdanimodel in many respects. The first two parts of the fuzzy inferenceprocess, including the fuzzification of the inputs and applying thefuzzy operator, are exactly the same. The main difference betweenMamdani and Sugeno inference systems is that the Sugeno outputmembership functions are either linear or constant [17].

1.2.1.1. Fuzzifying inputs. The second step is to take the input valuesand fuzzifying them through fuzzification operators. A fuzzificationoperator has the duty of transforming crisp numerical value,limited to the universe of discourse of the input variable into fuzzysets through the generation of membership values (always theinterval between 0 and 1) for a fuzzy variable using membershipfunctions.

1.2.1.2. Applying fuzzy operators. I case we have more than oneinput, the use of fuzzy operators will be inevitable. Because, initialparts of each fuzzy rulemust be connected to each other through thenecessary fuzzy operators in order to determine the output valuesbetween 0 and 1. The first operation is AND operation being equiv-alent tomin (A, B), where A and B aremembership values limited tothe range (0, 1). AND operation is of two types, min (minimum) andprod (product) AND method. Using the same reasoning, we can

replace the OR operation with the max function, so that A OR Bbecomes equivalent tomax (A, B). OR operation is also of two types,max (maximum), and probor (probabilistic OR) method [17]. Theprobabilistic OR method is calculated according to the equation

proborðA;BÞ ¼ Aþ B� AB (6)

For example, consider a fuzzy rule which is consist of two initialparts as given below:if input1 is mf6 and input2 is mf13, then outputis mf48.

The twodifferent parts of the above rule (input1 ismf6 and input2is mf13) yielded the fuzzy membership values 0.8 and 0.5 respec-tively. The fuzzyORoperator simply selects themaximumof the twovalues, 0.8, and the fuzzy operation for this rule is complete. If weused the probabilistic OR method, the result would be 0.9. Finally,the operation NOT A becomes equivalent to the operation 1-A.

1.2.1.3. Applying implication method. After fuzzifying the input andapplying any necessary fuzzy operators, that result must be appliedto the consequent. This process is called Implication. A consequentis a fuzzy set represented by a membership function, which assignsan entire fuzzy set to the output. The consequent is reshaped usinga single number associated with the initial part of the fuzzy rule.The input for the implication process is a single number given bythe initial part of the fuzzy rule, and the output is a fuzzy set.Implication is implemented for each rule. There are two methodsfor applying the implication process, used by the ANDmethod: min(minimum), which truncates the output fuzzy set, and prod(product), which scales the output fuzzy set [17].

1.2.1.4. Aggregating all outputs. After applying implication Method,the aggregation process takes place. Aggregation is a process whichthe fuzzy sets that represent the outputs of each rule are combinedinto a single fuzzy set. Aggregation only occurs once for each output

Fig. 6. Type of inserts membership functions.Fig. 4. Mamdani system used.


variable, just prior to the final step, defuzzification. The output ofthe aggregation process is one fuzzy set for each output variable.Aggregation methods are of three types, maximum (max), proba-bilistic OR (probor), and sum method [17].

1.2.1.5. Defuzzification. The final step is defuzzification. The processof converting a Fuzzy set into a single value is called defuzzification.Defuzzification methods are of three types, centroid, bisector,middle of maximum, largest of maximum, and smallest ofmaximum. The most commonly used method of defuzzification iscentroid method which returns the center of area under the curve.More information on the defuzzification methods is availableelsewhere [17].

2. Experimental apparatus

A schematic viewof the experimental rig [14] is shown in Fig. 2a.The rig consists of two fans and a set of copper tubes. The set oftubes has three sections including a calming section, bent tube andoutlet section. The fluid enters the calming section which hasa length of 2 m to eliminate the entrance effect. The temperatureand pressure are measured at the end of this section at the inlet ofbent tube section. Then, the fluid passes through nine bends in the6.5 m length of bent tube and reaches the outlet section. Thepressure and the temperature are measured at the outlet section.The 50 W fans with 1400 rpm rotation speed are placed at a 20 cmdistance beneath the bent tube and entire assembly is enclosed ina 60 � 100� 50 cm cubic channel. Hot water from a 100 L reservoirequipped with heaters enters the bent tube after passing throughrotameter with 58 �C temperature. Water volumetric flow ratevaries from 100 L/hr to 400 L/hr which correspond to Reynoldsnumbers from 4021 to 16118. The tube inlet and outlet waterpressure and temperature are measured through two pressuretransmitters and a copper-constantan thermocouple. Moreover, inorder to determine the average Nusselt number, the temperaturesat 20 different positions on the outer surface of the tube aremeasured. All twenty temperature sensing probes are connected toa data logger set. In the experiments [14], three various tube insertsare placed in the bent tube. Fig. 2b shows the bent tube, fan andtube inserts used in the experiment. The tube applied here has

Fig. 5. Reynolds number membership functions.

a 17 mm of inner diameter and a 1 mm thickness. The classic andjagged twisted tape inserts have 15 mmwidth and 1 mm thicknesswith four twist ratios as 1.76, 2.35, 2.94 and 3.53. The butterflyinserts are made from an aluminum sheet with a 0.5 mm thicknessand consist of a holding rod with a 1.9 mm diameter. These insertsare used at three inclined angles of 45�, 90� and 135� between thebutterfly piece and the rod with a 6cm pitch length. As can be seenin Fig. 2b, the edge of jagged insert is cut and warped upward todisturb the fluid motion near the tube wall. In the butterflyarrangement, pieces are placed on the rod to increase the flowturbulence intensity in the tube. Also the pieces on the rod aretwisted slightly in order to reduce the blocking effect.

3. Data reduction

In order to express the experimental results in a more efficientway, the measured data are reduced using the following procedure:

The heat transfer rate resulted from the hot fluid in the tubes isexpressed as [14]:

Q ¼ mCpðTo � TiÞ (7)

On the other hand, the heat transfer rate to the air surroundedthe tube is approximated by:

Q ¼ hAðTw � TbÞ (8)

where,

Tb ¼ ðTo þ TiÞ=2 and Tw ¼�X

Tw�.

20 (9)

Tw is the local wall temperature and is measured at the outerwall surface of the tubes. The relations used in calculation of theaverage heat transfer coefficient and the average Nusselt numberare as follows:

h ¼ mCpðTo � TiÞ=AðTw � TbÞ (10)

Nu ¼ hDh=K (11)

Fig. 7. Inclined angles of butterfly inserts membership functions.

Table 2Part of rules involved in fuzzy model.

No Rules No Rules

1 If (Re is mf1) and (I is mf1) and(q is mf2) then (Nu is mf30)

10 If (Re is mf3) and (I is mf2) and(R is mf3) then (Nu is mf14)

2 If (Re is mf4) and (I is mf1) and(q is mf2) then (Nu is mf 70)





13 If (Re is mf2) and (I is mf3) and(R is mf2) then (Nu is mf 17)











Fig. 8. Twist ratios of the jagged and classic twisted tape inserts membershipfunctions.


In addition, the Reynolds number is obtained according to thefollowing equation:

Re ¼ UDh=n (12)

In the present work, the uncertainties of experimentalmeasurements are determined based on ANSI/ASME [14,20]. Themaximum uncertainties for Nu and Re are estimated at 7% and 5.2%,respectively.

Table 3Comparison of average Nusselt numbers, between fuzzy logic and experiments [14].

No Exp Fuzzy logicpredictions

Error% No Exp Fuzzy logicpredictions

Error%

1 94.280 95.0000 0.75 40 99.0000 100.000 1.002 125.714 126.000 0.22 41 114.104 113.000 0.963 152.857 152.000 0.56 42 127.140 126.000 0.894 177.142 178.000 0.48 43 37.8910 37.9000 0.025 200.000 199.000 0.50 44 50.7500 50.9000 0.296 224.285 225.000 0.31 45 65.8460 66.5000 0.997 244.286 243.000 0.52 46 79.2650 79.5000 0.298 114.44 113.000 1.25 47 91.0060 89.9000 1.209 156.020 155.000 0.65 48 104.400 105.000 0.5710 186.660 186.000 0.35 49 116.160 116.000 0.1311 218.050 217.000 0.48 50 59.0440 58.7000 0.5812 245.000 246.000 0.40 51 81.9600 82.0000 0.0413 271.100 272.000 0.33 52 102.161 103.000 0.8214 295.000 295.000 0.00 53 120.060 121.000 0.7815 94.2840 95.0000 0.75 54 137.297 137.000 0.2016 125.011 126.000 0.22 55 153.350 152.000 0.8817 147.140 147.000 0.09 56 171.350 170.000 0.7818 169.721 170.000 0.16 57 54.5730 53.5000 1.9619 191.421 191.000 0.22 58 78.9190 79.5000 0.7320 214.281 215.000 0.29 59 97.8380 97.7000 0.1421 232.861 233.000 0.05 60 116.216 116.000 0.1822 49.0730 48.3000 1.57 61 133.243 134.000 0.5623 69.1300 69.1000 0.14 62 147.838 147.000 0.5624 85.4150 84.7000 0.83 63 163.780 165.000 0.7425 101.830 103.000 1.80 64 51.7460 50.9000 1.60

4. Results and discussion

In order to predict the effects of Reynolds number and differenttypes of inserts, on heat transfer in an air cooled heat exchangera fuzzy logic model was used. The plan for the experimentationwasto conduct the experiments without the inserts whereby the Nus-selt numbers were obtained and then compared with the resultsobtained with well-known correlations under similar condition asshown in Fig. 3. Indicating good agreement with the Gnielinski’sNusselt number [21] with tolerances of�5.9%. Three different typesof inserts namely butterfly, the jagged and classic twisted tapeinserts were used in this study, and their effects on heat transferfrom the air cooled heat exchanger were investigated. The aim ofcurrent study was to determine the effect of two main factors, suchas Reynolds number, and the types of inserts i.e. butterfly inserts,having inclined angle and, classic and jagged twisted tape insertswith their corresponding twist ratio, on heat transfer from the aircooled heat exchanger using fuzzy logic. The structure of the fuzzymodel used in this study is shown in Table 1. In order to performfuzzy logic, input and output variables with their correspondinglevels were determined. Moreover, the following levels for differentinput variables were considered. Reynolds number (Re) in sevenlevels ranging from 4021 to 16118, type of inserts (I) in three levels(i.e three types of inserts as mentioned before), twist ratio (R) ofclassic and jagged twisted tape inserts in four levels from 1.76 to3.53, inclined angle (q) of butterfly inserts in three levels from 45�

to 135�, and finally average Nusselt number, as output variablewereconsidered for experimentation. After data reduction, the values ofaverage Nusselt number for seventy seven different tests were

Fig. 9. Average Nusselt number membership functions.

determined. The fuzzy inference system, Mamdani, used in thisstudy is shown in Fig. 4. Symmetric triangular membership func-tions for output and input variables were defined. To build up theReynolds membership functions, seven symmetric triangles wherethe peak of each triangle is the main values in experiments wereused [14]. The membership functions of inclined angle and twistratio is established using three and four triangle, respectively,

26 115.942 116.000 0.05 65 73.8550 74.3000 0.6027 133.120 134.000 0.66 66 91.4280 92.5000 1.1728 149.153 150.000 0.56 67 108.435 108.000 0.4029 44.1530 43.1000 2.36 68 124.308 124.000 0.2430 62.4920 61.3000 1.90 69 140.180 139.000 0.8431 79.6880 79.5000 0.23 70 156.054 157.000 0.6032 93.4660 92.6000 0.92 71 49.6300 48.3000 2.6033 108.338 108.000 0.31 72 68.6400 69.1000 0.6734 121.757 121.000 0.62 73 85.4150 84.7000 0.8335 135.730 137.000 0.93 74 101.830 103.000 1.1436 42.1080 43.1000 2.30 75 115.607 116.000 0.3437 57.4140 58.7000 2.20 76 131.120 131.000 0.0938 72.7200 71.7000 1.40 77 144.680 144.000 0.4739 86.8900 87.3000 0.47

Fig. 10. Experimental results of Shabanian et al. [14] versus predicted results fromfuzzy logic.


similar to those of Reynolds number. For the types of inserts, threetriangles were used in which the peak of each triangle was 0, 1, 2.The digits 0, 1, 2 indicate the butterfly, classic and jagged inserts,respectively. To establish the output membership functions, onehundred triangular symmetric membership functions were used inthe range of zero to 99, where 0,1,.,99 indicate locations of trianglepeaks. Subsequently, the range of zero to 99 was marked by

Fig. 11. Average Nusselt number of an air cooler with (a) The jagged twisted tapeinserts, versus Reynolds number and twist ratio, (b) The classic twisted tape inserts,versus Reynolds number and twist ratio, (c) The butterfly inserts, versus Reynoldsnumber and inclined angle.

37.89e295 respectively. Having this sort of replacement, wetherefore, obtained one hundred triangular membership functionsfor the average Nusselt number. Figs. 5 to 8, show membershipfunctions for input variables including Reynolds number, types ofinserts, twist ratio of classic and jagged twisted tape inserts andinclined angle of butterfly inserts. In addition the average Nusseltnumber membership functions are shown in Fig. 9. Some parts ofseventy seven rules, which were chosen for the fuzzy model, areshown in Table 2. The values of average Nusselt number and errorsof the fuzzy model developed using experimental results of Sha-banian et al. [14] are shown in Table 3. As it can be seen in this tableand Fig. 10, the average error is 0.68% indicating good agreementbetween experimental data with the one predicted by fuzzy model.According to Fig. 11, increasing the Reynolds number increases theaverage Nusselt number for different types of inserts. Fig. 11 a andb show the effects of twist ratio of jagged and classic twisted tapeinserts on average Nusselt number for different Reynolds numbers.It can be understood from these figures that, in both classic andjagged inserts, the heat transfer increases from 13% to 22% and15%e21% respectively by decreasing the twist ratio from 3.53 to1.76 [14]. This behavior can be understood by the fact that, the swirlintensity increases by decreasing the twist ratio and therefore,increases the Nusselt number. As it can be seen from Fig. 11c, thetube fitted with 90� inclined butterfly insert has higher Nusseltnumber as compared with other two types of butterfly inserts. Ascan be seen in this figure, heat transfer increases from 23% to 26%by the convergence of the inclined angle to 90�. This result may beexplained by the generation of stronger turbulence intensity andmore rapid mixing of flow created by this insert.

5. Conclusions

The heat transfer from an air cooler equipped with three typesof inserts namely the butterfly, classic and jagged twisted tapeinserts, placed in a bent tube of the air cooled heat exchanger, weremodeled using fuzzy logic. This model was used on the basis ofobtaining relationship between four input variables namely Ray-leigh number, types of inserts, inclined angles of butterfly inserts,twist ratios of the classic and jagged twisted tape inserts, andaverage Nusselt number.

The following results were obtained from the study:

1. Average Nusselt number increased with the increase in Rey-nolds number.

2. Average Nusselt number had a decreasing trend with respect totwist ratio in classic and jagged twisted tape inserts.

3. Maximum heat transfer was obtained by employing butterflyinsert with inclination angle of 90� in Reynolds number of16118 and in addition, minimum heat transfer rate was ob-tained by employing classic twisted tape insert with twist ratioof 3.53 in Reynolds number of 4021.

4. Fuzzy logic is a reliable technique due to its high accuracy.

Nomenclature

A heat transfer area (m2)Q heat transfer rate (W)Cp specific heat capacity (kJ/kg K)D Inner diameter of the smooth tube (m)Dh hydraulic diameter (m)h heat transfer coefficient (W/m2,K)I type of insertK thermal conductivity (W/m,K)m mass flow rate (kg/s)


Nu Nusselt numberPi axial distance of twist pitch (m)P static pressure (Pa)R twist ratio Y ¼ Pi/DRe Reynolds numberT temperature (K)u mean velocity (m s�1)

Greek lettersn kinematic viscosity (m2/s)q inclined angle

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Fuzzy logic to predict the heat transfer in an air cooler equipped with different tube inserts

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