Revista Mexicana de ngeniería uímica · T10 T12 T4 T6 T1 X3 T7 T5 Pure water P2 P1 X1 X2 X4 T3 Fig. 1. Schematic diagram of absorption heat transformer integrated to water puriﬁcation

Revista Mexicana de Ingeniería Química

CONTENIDO

Volumen 8, número 3, 2009 / Volume 8, number 3, 2009

213 Derivation and application of the Stefan-Maxwell equations

(Desarrollo y aplicación de las ecuaciones de Stefan-Maxwell)

Stephen Whitaker

Biotecnología / Biotechnology

245 Modelado de la biodegradación en biorreactores de lodos de hidrocarburos totales del petróleo

intemperizados en suelos y sedimentos

(Biodegradation modeling of sludge bioreactors of total petroleum hydrocarbons weathering in soil

and sediments)

S.A. Medina-Moreno, S. Huerta-Ochoa, C.A. Lucho-Constantino, L. Aguilera-Vázquez, A. Jiménez-

González y M. Gutiérrez-Rojas

259 Crecimiento, sobrevivencia y adaptación de Bifidobacterium infantis a condiciones ácidas

(Growth, survival and adaptation of Bifidobacterium infantis to acidic conditions)

L. Mayorga-Reyes, P. Bustamante-Camilo, A. Gutiérrez-Nava, E. Barranco-Florido y A. Azaola-

Espinosa

265 Statistical approach to optimization of ethanol fermentation by Saccharomyces cerevisiae in the

presence of Valfor® zeolite NaA

(Optimización estadística de la fermentación etanólica de Saccharomyces cerevisiae en presencia de

zeolita Valfor® zeolite NaA)

G. Inei-Shizukawa, H. A. Velasco-Bedrán, G. F. Gutiérrez-López and H. Hernández-Sánchez

Ingeniería de procesos / Process engineering

271 Localización de una planta industrial: Revisión crítica y adecuación de los criterios empleados en

esta decisión

(Plant site selection: Critical review and adequation criteria used in this decision)

J.R. Medina, R.L. Romero y G.A. Pérez

Vol. 13, No. 3 (2014) 907-917

COEFFICIENT OF PERFORMANCE PREDICTION BY A POLYNOMIAL MODELOF ABSORPTION HEAT TRANSFORMER

PREDICCION DEL COEFICIENTE DE DESEMPENO POR UN MODELOPOLINOMIAL PARA UN TRANSFORMADOR TERMICO

B.A. Escobedo-Trujillo1, F.A. Alaffita-Hernandez1, D. Colorado2∗ and J. Siqueiros3

1Facultad de Ingenierıa, Universidad Veracruzana.Campus Coatzacoalcos, Av. Universidad km 7.5 Col. SantaIsabel, C.P. 96535, Coatzacoalcos, Veracruz, Mexico

2Centro de Investigacion en Recursos Energeticos y Sustentables. Universidad Veracruzana. Av. Universidad km7.5 Col. Santa Isabel, C.P. 96535, Coatzacoalcos, Veracruz, Mexico

3Secretarıa de Innovacion, Ciencia y Tecnologıa, Av. Atlacomulco no. 13, esq. Calle de la ronda, Col.Acapantzingo, Cuernavaca, Morelos, C.P. 62440

Received March 16, 2014; Accepted June 25, 2014

AbstractA polynomial model is developed to predict the coefficient of performance of a water purification processintegrated to an absorption heat transformer. The range of the coefficient of performance operations was from0.21 to 0.39. This model used: inlet temperature in the generator which comes from the absorber, outlettemperature in the absorber that comes from the generator, inlet temperature in the absorber that comes fromthe generator, water-lithium bromide solution inlet concentration in the generator that comes from the absorberand the pressure in the absorber and generator. A polynomial model is presented in order to obtain coefficientof performance prediction with a determination coefficient of 0.9919. Level surfaces of the coefficient ofperformance against the inlet variables for the polynomial model and residual analysis were presented withthe aim of validating the model. This work has the purpose of providing faster and simpler solutions insteadof the complex equations used for the analysis of the heat transformer in order to obtain accurate coefficientof performance prediction. The operation variable with the greater contribution of determination coefficient ispresented.

Keywords: lithium bromide solution, residual analysis, Gaussian distribution, water purification.

ResumenUn modelo polinomial es desarrollado para predecir el coeficiente de desempeno para un sistema depurificacion de agua integrado a un transformador termico. El rango de operacion del coeficiente de desempenofue desde 0.21 a 0.39. El modelo usa: temperatura de entrada en el generador el cual proviene del absorbedor,temperatura de salida en el absorbedor el cual proviene del generador, temperatura de entrada en el absorbedorel cual proviene del generador, concentracion de entrada de la solucion de bromuro de litio en el generadorproveniente del absorbedor y la presion en el absorbedor y generador. Un modelo polinomial es presentado conel objetivo de predecir el coeficiente de desempeno con un coeficiente de determinacion de 0.9919. Superficiesde nivel del coeficiente de desempeno contra las variables de entrada del modelo polinomial y el analisisresidual son presentados con el objetivo de validar el modelo. Este trabajo tiene el objetivo de proveer unasolucion rapida y simple en lugar de ecuaciones complejas usadas en el analisis del transformador termico conel objetivo de obtener apropiadas predicciones del coeficiente de desempeno. La variable de operacion con lamayor contribucion en el coeficiente de determinacion es presentada.

Palabras clave: solucion de bromuro de litio, analisis residual, distribucion Gaussiana, purificacion de agua.

∗Corresponding author. E-mail: [email protected] (921) 203 65 16

Publicado por la Academia Mexicana de Investigacion y Docencia en Ingenierıa Quımica A.C. 907

Escobedo-Trujillo et al./ Revista Mexicana de Ingenierıa Quımica Vol. 13, No. 3 (2014) 907-917

1 Introduction

In linear multiple regression analysis, the goal is topredict, knowing the measurements collected fromthe data. In fact, the polynomial fitting is anattractive technique used for estimating the dependentvariable in a system. The polynomial model has beenused in several fields as: biomedical, environmental,socioeconomic studies Perez et al. (2009), compositedynamic envelopes thermal performance forecastingLazaros et al. (2013) and electronic applicationsHossain et al. (2011). Castilla et al. (2013) proposedthe use of approximated models with the aim ofreducing the computational cost required to computethe index, allowing its use in real-time control ofsystems and decreasing the size of the network sensor.The Authors compared an artificial neural networkversus a polynomial model. Although, in general, theresults obtained with the ANN model are better. Thepolynomial model can be derived in an easier way.

Absorption heat transformers which operate ina cycle opposite to that of absorption heat pumpscan be used to reduce the CO2 discharge and toreuse large amounts of industrial waste heat Horuzet al. (2009). Physical, theoretical and empiricalmodels have been satisfactorily tested in orderto model heat transformers and their components.Sencan et al. (2007) presented a comparison ofdifferent methods for modelling heat transformerspowered by a solar pond. The coefficient ofmultiple determination R2 value by: linear regression(0.26), pace regression (0.76), sequential minimaloptimization (0.77), M5 model tree (0.52), M5’rules (0.51), decision table (0.25) and neural networkmethods (0.99), for the estimation of coefficient ofperformance. The comparison and methodology isbased on experimental results by Rivera and Romero(1998). Siqueiros and Romero (2007) showed thatproposed water purification coupled with a systemabsorption heat transformer is capable of increasingthe original value of performance for more than120% by recycling part of the energy from a waterpurification system by thermodynamic simulation.Hernandez et al. (2008) and Hernandez et al.(2009) developed a forecasting model for a waterpurification process integrated in an absorption heattransformer by direct and inverse artificial neuralnetwork with 16 variables in the inlet layer to obtainon-line prediction of coefficient of performance basedon experimental results by Morales (2005). Sozenet al. (2007) analyzed the first and second lawsof thermodynamics with the aim of evaluating an

absorption heat transformer coupled to a solar pondoperating with the aqua/ammonia, the maximumtemperature of the useful heat produced was ' 150oC. Colorado et al. (2009) and (2011) describeda physical-empirical model for a vertical helicalevaporator and condenser under the heat transformeroperation conditions. Juarez et al. (2009) presented adynamic model to describe heat and mass transfer of ahorizontal pipe absorber applied to a heat transformerused to purify water. Rivera et al. (2003) evaluated thetheoretical integration between single and double stageabsorption heat transformer with a distillation columnof butane and pentane, the numerical results showedthat it is possible to reduce the energy consumption inthe re-boiler up to 43% by using a single stage heattransformer.

A heat transformer experimental rig has beenstudied by different authors. Rivera et al. (2011)applied the first and second law of thermodynamicsto analyze the performance of an experimental heattransformer used for water purification. Resultsshowed that decreasing the absorber temperature canlead to a better performance with low flow ratiosand a greater amount of purified water. Sekarand Saravanan (2011) described an absorption heattransformer working with a water-lithium bromidesolution coupled with a seawater distillation system.A maximum coefficient of performance of 0.38 anda maximum temperature lift of 20 oC have beenreached under operation conditions, the quality of thewater has been accepted as drinking water accordingto the Bureau of Indian Standard. Flores (2013)proposed a correlation for the heat transfer coefficientbased on energy balance and Wilson plot method withexperimental information of a double tube condenserunder operation conditions of a heat transformer.

Actually, the integration of heat transformers withindustry processes have demonstrated great interest.Moreover, engineers need more efficient tools to dataprocessing and integration process assessment. Asto the Author’s knowledge, most of the theoreticalinquiries regarding the absorption heat transformermodelling were carried out with complex empirical,semi empirical and thermodynamic assessment withsatisfactory results. The aim of the present workdiscusses two main ideas; firstly, present a newmethodology which provides faster and simplersolutions instead of the complex equations used forthe analysis of the heat transformer in order toobtain accurate coefficient of performance predictionwith a polynomial model and secondly, to determinethe operation variable with the largest contribution

908 www.rmiq.org


to the determination coefficient of prediction. Thepolynomial model presented in this work wascompared with an artificial neural network reported inthe literature.

2 Experimental dataFigure 1 shows a schematic diagram of the absorptionheat transformer integrated to a water purificationprocess. The absorber gives us a useful heat quantityQAB, produced by the heat transformer from theevaporator, condenser and generator Siqueiros et al.(2007).

Experimental database provided by Morales et al.(2005) consists on different coefficient of performancevalues (COP), obtained from a portable water

purification process coupled to an absorption heattransformer with energy recycling. An experimentaldata set was obtained at different initial concentrationsof LiBr in LiBr + H2O mixture, different temperaturesin the absorber, the generator, the evaporator, and thecondenser as well as different pressures in the absorberand the generator. Transitory and steady states weretaken into account for each initial concentration ofthe mixture. After 2 hours from start-up, data wascollected for 4 hours. Experiments were carried outat eight different initial conditions with at least tworeplicates. The arrangement was 8 + 2 with 4 hoursof information acquisition. Thus, a database of 6786samples was obtained. This data was sufficient forthe polynomial model. A summary of the operatingparameters is shown in Table 1.

Condenser Generator

AbsorberEvaporator

PumpPumpPump

Inpure

water

Waste

heatAuxiliar

condenserPatm

PAB

PGE

T CO TGE ,T EV TAB

PumpPump

T8

T9

T10 T12

T4

T6 T1X3

T7 T5

Pure

water

P2

P1

X1 X2

X4

T3

Fig. 1. Schematic diagram of absorption heat transformer integrated to water purification process with energyrecycling.

www.rmiq.org 909


Table 1. Range of experimental operating conditions used to obtain thecoefficient of performance values

Temperatures (oC) Operation range Instrumentation label (see Fig. 1)Tin GE−AB 76.29-91.53 T1Tin EV−AB 74.56-89.93 T2Tout AB−GE 84.31-98.27 T3Tin AB−GE 74.99-92.58 T4Tout GE−CO 76.29-91.53 T5Tout GE−AB 77.03-83.89 T6

Tin CO 40.37-65.03 T7Tout CO 26.77-33.79 T8Tin EV 28.52-85.33 T9

Tout EV−AB 74.56-89.93 T10Concentrations (%)

Xin AB 51.66-55.36 X1Xout AB 50.75-54.36 X2Xin GE 50.75-54.36 X3Xout GE 53.16-56.07 X4

Pressure (in Hg absolute)PAB 7.00-11.50 P1PGE 19.00-21.10 P2

Thermodynamic properties of the LiBr + H2Omixture were estimated with Alefeld correlationscited by Torres-Merino (1997). The inlet andoutlet-temperatures of each component (AB, GE,CO and EV) were obtained experimentally. At thesame time, the pressure of two components (ABand GE) was registered with a temperature-pressureacquisition system (thermocouple conditioner andAgilent equipment with commercial software). Inletand outlet-concentrations in the AB and GE wereestablished by a refractometer (refraction index). Inthis process, LiBr + H2O mixture was used as theworking fluid in the absorber and generator, whereasonly H2O was used in the evaporator and condenser.A total of 16 variables (10 levels of temperature, 4of concentration and 2 of pressure) were used andregistered. Only the experimental database of anabsorption heat transformer was used in this study.

2.1 Experimental uncertainty analysis

In the experimental rig, the quantities measureddirectly were: mass flow rate, pressure and bulktemperature. The rotameter, for the working fluid,had an accuracy of ±3% on the measurement. Themanovacuometer uncertainty was estimated to be less

than 0.5% of full scale, the scale of manovacuometeris from 0.033 bar to 2.07 bar. The measurementsof thermocouples, T-type copper-constantan, had anaccuracy of ±0.5 oC.

3 Methodology

In this section, the objective is to describe amethodology to build up a polynomial model thatrelates the coefficient of performance with theprevious operation variables (see Table 1).

In general, relationship between the variables ofinterest, coefficient of performance and a number ofassociated (or inlet) variables T1, . . . ,T10, X1, . . . ,X4,P1 and P2 which are unknown but can beapproximated by a polynomial model of the form:

COP = p(T1, . . . ,T10,X1, . . . ,X4,P1,P2) + ε, (1)

where p is an unknown polynomial function and ε isthe random error. It is assumed that ε is a randomvariable with Gaussian distribution.

To determine the polynomial model described in

910 www.rmiq.org


Eq. (1), define the following predictor variable set:

A :={Y1, . . . ,Y16, YiY j, YiY jYk, YiY jYkYn, YiY jYk

YnYm|i = 1, . . . ,16; j = i, . . . ,16;k = j, . . . ,16;n = k, . . . ,16;m = n, . . . ,16}

where Y1 = T1, . . . ,Y10 = T10, Y11 = X1, . . . ,Y14 = X4,Y15 = P1 and Y16 = P2. As can be seen, it hasto be checked with all combinations of variables upto polynomials of degree five, because polynomialswith a degree greater than five only give a benefit of0.0001 in the determination coefficient (R2), a simplepolynomial is required. The cardinality of set Aequals to 51084,since that there are 16, 136, 2176,9996 and 38760, terms of the form Yi, YiY j,YiY jYk,YiY jYkYn, YiY jYkYnYm, respectively. Therefore, thereare 251084 − 1 subsets of A without the empty set.

Linear multiple regression between A subsets andexperimental coefficient of performance (COP) is usedto find the simplest polynomial that fits the data. Asyou can notice, in the above paragraph, there are toomany subsets of A, to discard some, the followingcriterion is used: if the coefficient of determinationof Yi and Yn

i differed by less than 0.0001 then, Yni is

discarded (n = 1,2, . . . ,5). Under this criterion, it iseasy to find the degrees of the monomials associatedto the polynomial model in Eq. (1) (see Table 2.)

Table 2 shows that Y12 and Y13, Y1 and Y5, Y2and Y10 have the same coefficient of determination,therefore, it can dispense with one of the two variables.Therefore the number of combinations that it needsto test is: 213 − 1 = 8191 (the products YiY j,YiY jYk,YiY jYkYn, YiY jYkYnYm for i , j , k , n , m wereeliminated due to that provided few benefits to thecoefficient of determination).

Then, the coefficient of determination is calculated(R2) between each combination of variables and thecoefficient of performance (COP) and the one with thehighest R2 coefficient is chosen.

4 Results

Remember that it seeks the simplest possiblepolynomial, therefore those with less variablepolynomials are prefered. After trying manycombinations:

COP ≈ 1.9123× 104 + 0.0247(Tin GE−AB)−0.0259(Tout AB−GE) + 0.0027(Tin AB−GE)−12.1769(Xin GE) + 0.1170(Xin GE)2

+1.0797× 103(PAB)− 240.0364(PAB)2

+26.4762(PAB)3 − 1.4490(PAB)4

+0.0315(PAB)5 − 3.0444× 103(PGE)+148.9803(PGE)2 − 2.4298(PGE)3.

(2)

The fit of the polynomial model in Eq. (2) wasexpressed by the determination coefficient R2 whichwas found to be 0.9919, indicating that 99.19% of thevariability in the coefficient of performance could beexplained by this polynomial model. Coefficient ofperformance can be seen as a linear combination offactors (see Table 3) plus an independent coefficient(1.9223×104).

The polynomial (2) is not only the one with thehighest determination coefficient, but also one of theshortest in terms of variables related.

Size of the confidence intervals (0.0004) (seeTable 3) and coefficient of determination R2 = 0.9919shows that the monomials T1,T3,T4,X3,X32,P1,P12,P13,P14,P15,P2,P22,P23 have a linearrelationship with the coefficient of performance (eachtreated as distinct and independent terms), to betterappreciate this multivariable linearity see Figure 2.All the previously mentioned, supports the idea thatthe model is indeed valid.

Table 2. Monomials associated to ourpolynomial model.

Variable Degree R2

Y1 T1 1 4.2078× 10−6

Y2 T2 1 0.0934Y3 T3 1 0.0058Y4 T4 2 0.1977Y5 T5 1 4.2078× 10−6

Y6 T6 2 0.1714Y7 T7 1 0.0550Y8 T8 1 0.0090Y9 T9 1 0.0096Y10 T10 1 0.0934Y11 X1 2 0.3121Y12 X2 2 0.3139Y13 X3 2 0.3139Y14 X4 2 0.1183Y15 P1 5 0.5488Y16 P2 3 0.1238

www.rmiq.org 911


Table 3. Each coefficient are shown with their correspondingconfidence interval with a confidence level of 99%.

Factor Coefficient Confidence IntervalT1 0.0247 [0.0245, 0.0249]T3 -0.0259 [-0.0261, -0.0257]T4 0.0027 [0.0025, 0.0029]X3 -12.1769 [-12.2857, -12.0680]X32 0.1170 [0.1159, 0.1180]P1 1.0797 ×103 [1.0659×103, 1.0936×103]P12 -240.0364 [-243.1142, -236.9585]P13 26.4762 [26.1369, 26.8155]P14 -1.4490 [-1.4676, -1.4304]P15 0.0315 [0.0311, 0.0319]P2 -3.0444 ×103 [-3.1268×103, -2.9620×103]P22 148.9803 [144.9588, 153.0017]P23 -2.4298 [-2.4952, -2.3644]

0.15 0.2 0.25 0.3 0.35 0.4 0.450.2

0.25

0.3

0.35

0.4

Predicted Coefficient of Performance

Exp

erim

enta

l C

oe

ffic

ient

of P

erf

orm

ance

Fig. 2. Experimental versus predicted coefficient of performance for database.

On the other hand, remember that the variablesinvolved in the polynomial (see Eq. (2)) are: Y1 =

T1,Y3 = T3,Y4 = T4,Y13 = X3,Y15 = P1 andY16 = P2, then to give a geometric idea of thebehaviour of the polynomial predictor p, experimentaland predicted coefficients of performance are plotted,however it is noteworthy that the polynomial has sixdifferent variables therefore, level surfaces are plottedfor each variable, the two-dimensionals and three-dimensionals are later shown in Figures 3, 4 and 5.As can be seen, there is good agreement betweenpredicted values and experimental data points.

4.1 Residual analysis

In this subsection, a residual analysis to verify thegoodness of the polynomial fit in Eq. (2) is presentedby Montgomery (2001) and (2010) and by Devore(2004).

In section 4, it is supposed that in Eq.(1), ε hada Gaussian distribution. Under assumption that ε hasa zero mean, the expected value (E) can be taken onboth sides of the equation (1) as follows:

912 www.rmiq.org


75 80 85 90 95

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Tin GE−AB

Coeffi

cien

tofPerform

ance

Predicted COP

Experimental COP

80 85 90 95 100

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Tout AB−GE

Predicted COP

Experimental COP

70 75 80 85 90 95

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Tin AB−GE

Predicted COP

Experimental COP

50 51 52 53 54 55

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Xin GE

Coeffi

cien

tofPerform

ance

Predicted COP

Experimental COP

7 8 9 10 11 12

0.2

0.25

0.3

0.35

0.4

0.45

0.5

PAB

Predicted COP

Experimental COP

20 20.5 21 21.5

0.2

0.25

0.3

0.35

0.4

0.45

0.5

PGE

Predicted COP

Experimental COP

Fig. 3. Level surfaces. Comparison of experimental and predicted coefficient of performance for each variablesignificant in the polynomial model.

8085

90

8590

95

0.2

0.25

0.3

0.35

Tin GEABTout ABGE

COP

Experimental COP

Predicted COP

8085

9080

90

0.2

0.25

0.3

0.35

Tin GEABTin ABGE80

8590

52

540.2

0.25

0.3

0.35

Tin GEABXin GE

8085

90

8

10

0.2

0.25

0.3

0.35

Tin GEABPAB

COP

8085

90

20

20.5

210.2

0.25

0.3

0.35

Tin GEABPGE85

9095

80

90

0.2

0.25

0.3

0.35

Tout ABGETin ABGE

8590

9552

540.2

0.25

0.3

0.35

Tout ABGEXin GE

COP

8590

958

10

0.2

0.25

0.3

0.35

Tout ABGEPAB

Fig. 4. Level surfaces. Comparison of experimental and predicted coefficient of performance for each two variablessignificants in the polynomial model.

www.rmiq.org 913


8590

95

20

20.5

210.2

0.25

0.3

0.35

Tout ABGEPGE

COP

Experimental COP

Predicted COP

7580

8590

52

540.2

0.25

0.3

0.35

Tin ABGEXin GE75

8085

90

8

10

0.2

0.25

0.3

0.35

Tin ABGEPAB

7580

8590

20

20.5

210.2

0.25

0.3

0.35

Tin ABGEPGE

COP

5152

5354

8

10

0.2

0.25

0.3

0.35

Xin GEPAB51

5253

54

20

20.5

210.2

0.25

0.3

0.35

Xin GEPGE

78

910

11

20

20.5

210.2

0.25

0.3

0.35

PABPGE

COP

Fig. 5. Level surfaces. Comparison of experimental and predicted coefficient of performance for each two variablessignificants in the polynomial model.

E(COP) =E(p(T1, . . . ,T10,X1, . . . ,X4,P1,P2) + ε)E(p(T1, . . . ,T10,X1, . . . ,X4,P1,P2)) + E(ε)

COP =p(T1, . . . ,T10,X1, . . . ,X4,P1,P2).

Note that from the last equation, the experimentalcoefficient of performance and the polynomial p(·)would be equal in average. In order to know if εhas a Gaussian distribution with a zero mean and aconstant variance, In Figure 6 a standardized residualshistogram by Montgomery (2001) was obtained as canbe seen, the image is bell-shaped except for the threepeaks to the left of the zero indicating that it does nothave a perfect symmetry, to ensure that the residualshave a zero mean and constant variance. They wereplotted versus coefficient of performance (COP) (seeFigure 7) along with a pair of horizontal lines 3 and-3. There are points that come out of these lines,such points are called outliers, in the graph there are81 outliers, i.e. 98.8% of the standardized residualsare within -3 to 3. Finally, it is crucial to knowwhether these abnormal residues affect the polynomialsignificantly, for this reason the 81 outliers wereremoved and came back to do the regression withoutthese points and obtained a determination coefficientof R2 = 0.9937. The difference is 0.0018, thisresult infers that outliers do not represent a significantamount, so the hypothesis, that the standardized

residuals follow a Gaussian distribution with a zeromean and constant variance, is accepted.

Finally, to confirm that residues are independent,figures 7 and 8 are plotted, wherein, in fig. 8 theconfidence intervals (gray lines) standardized for eachresidue (points in the middle) and outliers (black linesat the ends) can be observed. Both images show thatthe average of residues is zero (for the distribution toboth sides of the axis x), the variance is constant (in theform of a gray band at the residues) and the residuesare independent. This ensures the goodness of thepolynomial model in Eq. (2) (see Chapter 4 and 13from Devore (2004) for details).

−10 −8 −6 −4 −2 0 2 4 6 8 100

50

100

150

200

250

300

350

400

450

500

StandarizedResiduals

Fig. 6. Histogram of the standarized residuals.

914 www.rmiq.org


0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4−10

−8

−6

−4

−2

0

2

4

6

8

10

Predicted Coefficient of Performance (COP)

Sta

nd

arize

d R

esid

ua

ls

Fig. 7. Predicted coefficient of performance versusstandarized residuals.

1000 2000 3000 4000 5000 6000

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

Residual Case Order Plot

Residuals

Case Number

Fig. 8. Confidence intervals on the residuals withconfidence level of the 95%.

Once the model is validated, it can assured thatthere is a linear relationship between the coefficient ofperformance and the monomials T1,T3,T4, X3,X32,P1,P12,P13, P14,P15, P2,P22,P23. For all theabove, there is a polynomial relationship betweenthe coefficient of performance and the variablesT1,T3,T4, X3,P1 and P2. It is important to note thatthese variables are related to the absorption-desorptionprocess.

On the other hand, the variables that have animpact on the coefficient of performance in thepolynomial model are T1,T3,T4,X3,P1 and P2.Table 4 shows the contribution of each term of thepolynomial model on coefficient of determinationR2. The variable with the biggest contribution to the

Table 4. The contribution of each term of thepolynomial model on determination coefficient R2

Factor R2

T1 4.2078× 10−6

T3 0.0058T4 0.1738X3 0.1577

X32 0.1604X3 + X32 0.3139

P1 0.0869P12 0.0916P13 0.0948P14 0.0965P15 0.0968

P1 + P12 + P13 + P14 + P15 0.5488P2 7.4473× 10−5

P22 6.2589× 10−5

P23 5.1838× 10−5

P2 + P22 + P23 0.1238

coefficient of determination of the coefficient ofperformance predicted is P1 and the contribution is≈ 0.5488. Another important contribution is X3 witha ≈ 0.3139. Therefore, the absorber pressure is themain operation variable from the point of view of themethodology, followed by the water-lithium bromidesolution inlet concentration in the generator that comesfrom the absorber.

The coefficients of the polynomial model wereestimated from the experimental data applying themethod of least squares.

4.2 Comparison between a polynomialmodel and an Artificial NeuralNetwork(ANN) model

The polynomial model has been carefully verifiedusing, whenever possible, an artificial neural modelreported in the open literature. Figure 9 showsa further comparison of the developed polynomialwith an artificial neural model by Hernandezet al. (2009). The comparison is basedon a simulated coefficient of performance versusexperimental results. The determination coefficientsof 0.9919 and 0.9981 were calculated for polynomialand artificial neural network models respectively.The greatest discrepancies between polynomial modeland artificial neural network were observed for acoefficient of performance from 0.2 to 0.26.

www.rmiq.org 915


0.2 0.25 0.3 0.35 0.4

0.2

0.3

0.4

0.5

ANN

Polynomial

Fig. 9. Approaches to the Coeficient ofPerformance by an Artificial Neural Network (ANN)and multivariable polynomial (Polynomial).

It can be seen that the coefficient of performanceprediction could be calculated with an artificial neuralnetwork or the polynomial methodology discussed inthis work, with great confidence. It’s important to notethat the artificial neural network by Hernandez et al.(2009) performs COP prediction with 51 coefficients(weights and bias) and the polynomial model onlywith 14 coefficients. Therefore, both models canbe used, at a time, to approximate the coefficient ofperformance.

Conclusion

A polynomial model with six inlet operation variableswas successfully developed to predict coefficient ofperformance in a water purification process integratedto a heat transformer. The results obtained with thefit showed high accuracy with the experimental dataR2 > 0.99. Very high confidence level of the 95% foreach coefficient of the polynomial model was verified.

This methodology can be used to simplify thecomplex analysis of the heat transformer with highconfidence. The operation variable with the largestcontribution to the determination coefficient is theabsorber pressure P1. Consequently, greater efforts inon-line estimation, control pressure developments andequipment in a heat transformer are recommended.This methodology can be used in order to predict theperformance of absorption systems. The polynomialmodel could be used to assess in the heat transformertechnology integration in other kinds of systems,for instance, renewable energy systems or chemicalindustry.

AcknowledgmentWe are grateful to the anonymous reviewer and theeditor-in-chief for useful comments on an earlierversion of this manuscript.

NomenclatureCOP coefficient of performance [−]P pressure [inHg]Q heat flow [W]R2 coefficient of determination [−]T temperature [oC]X concentration [% w

w ]E expected value [−]Greek lettersσ2 standard deviation [−]SubscriptAB absorberCO condenserEV evaporatorGE generator

ReferencesCastilla M, Alvarez JD, Ortega MG, Arahal

MR. (2013) Neural network and polynomialapproximated thermal comfort models forHVAC systems. Building and Environment,59:107-115.

Colorado D, Santoyo-Castelazo E, Hernandez JA,Garcıa-Valladares O, Siqueiros J, Juarez D.(2009) Heat transfer of a helical double-pipe vertical evaporator: Theoretical analysisand experimental validation. Applied Energy,86:1144-1153.

Colorado D, Hernandez JA, Garcıa-Valladares O,Huicochea A, Siqueiros J. (2011) Numericalsimulation and experimental validation of ahelical double-pipe vertical condenser. AppliedEnergy, 88:2136?2145.

Devore J. (2004) Probability and statistics forengineering and the sciences. ThomsonLearning, Inc. 6th. edition, Canada.

Flores O, Velazquez V, Meza M, Horacio H,Juarez D, Hernandez JA. (2013) Estimationof the condensation heat transfer coefficientfor steam water at low pressure in a coileddouble tube condenser integrated to a heat

916 www.rmiq.org


transformer. Revista Mexicana de IngenierıaQuımica, 12:303-313.

Hernandez JA, Juarez-Romero D, Morales LI,Siqueiros J. (2008) COP prediction for theintegration of a water purification process ina heat transformer: with and without energyrecycling. Desalination, 219:66-88.

Hernandez JA, Bassam A, Siqueiros J, Juarez-Romero D. (2009) Optimum operatingconditions for a water purification processintegrated to a heat transformer with energyrecycling using neural network inverse.Renewable Energy, 34:1084-1091.

Horuz I, Kurt B. (2009) Single stage and doubleabsorption heat transformers in an industrialapplication. International Journal of EnergyResearch,33:787-798.

Hossain Khan A, Muntasir T, Zahidur Rahman ASM,Kumar Acharjee UK, Abu layek A. (2011)Multiple polynomial regression for modelinga MOSFET in saturation to validate theearly voltage. IEEE symposium on industrialelectronics and applications, 261-266.

Juarez D, Shah N, Pliego-Solorzano F, Hernandez JA,Siqueiros J, Huicochea A. (2009) Heat and masstransfer in a horizontal pipe absorber for a heattransformer. Desalination and water treatment,10:238?244.

Mavromatidis LE, Bykalyuk A, Lequay H.(2013) Development of polynomial regressionmodels for composite dynamic envelopesthermal performance forecasting. AppliedEnergy,104:379?391.

Montgomery D., Peck E, Geoffrey V. (2001)Introduction to linear regression analysis. Wileyseries in Probability and Statistics. 3th edition,USA.

Montgomery D., Runger G. (2010) Probabilidad yestadıstica aplicadas a la ingenierıa. LIMUSA.2do. edition, USA.

Morales-Gomez LI. (2005) Estudio experimentalsobre un sistema portatil de purificacion de agua

integrado a un transformador termico, M. S.Thesis. Mexico. CIICAp-UAEM.

Perez-Gonzalez A, Vilar-Fernandez JM, Gonzalez-Manteiga W. (2009) Asymptotic properties oflocal polynomial regression with missing dataand correlated errors. Annals of the Institute ofStatistical Mathematics, 61:85-109.

Rivera W, Romero RJ. (1998) Thermodynamicdesign data for absorption heat transformers.part seven: operating on an aqueous ternaryhydroxide. Applied Thermal Engineering, 18(3-4):147?56.

Rivera W, Cerezo J, Rivero R, Cervantes J, BestR. (2003) Single stage and double absorptionheat transformers used to recover energy ina distillation column of butane and pentane.International Journal of Energy Research,27:1279-1292.

Rivera W, Huicochea A, Martınez H, Siqueiros J,Juarez D, Cadenas E. (2011) Exergy analysisof an experimental heat transformer for waterpurification. Energy, 36:320-327.

Sekar S, Saravanan R. (2011) Experimentalstudies on absorption heat transformer coupleddistillation system. Desalination, 274:292-301.

Sencan A, Kizilkan O, Bezir NC, KalogirouS. (2007) Different methods for modelingabsorption heat transformer powered by solarpond. Energy Conversion and Management,48:724?735.

Siqueiros J, Romero RJ. (2007) Increase of COP forheat transformer in water purification systems.Part I ? Increasing heat source temperature.Applied Thermal Engineering, 27:1043?1053.

Sozen A, Serdar-Yucesu H. (2007) Performanceimprovement of absorption heat transformer.Renewable Energy, 32:267-284.

Torres-Merino J. (1997) Contacteurs gaz-liquidepour pompes a chaleur a absorption multi-etagees, Ph. D. thesis France. Inst NatlPolytechnique de Lorraine.

www.rmiq.org 917

Documents

Revista Mexicana de ngeniería uímica · T10 T12 T4 T6 T1 X3 T7 T5 Pure water P2 P1 X1 X2 X4 T3 Fig. 1. Schematic diagram of absorption heat transformer integrated to water puriﬁcation