A Dimensionless Model for Estimating the Dehumidification

4thInternationalConferenceOnBuildingEnergy,Environment

ADimensionlessModelforEstimatingtheDehumidificationEffectivenessofLiquidDesiccantDehumidifiers:ApplicationofLeastSquareSupport

VectorMachineA. Zendehboudi, X. Li*, P. Song

Department of Building ScienceSchool of Architecture, Tsinghua University, Beijing 100084, China

SUMMARYThis study introduces a novel dimensionless model based on Least Square Support Vector Machine (LSSVM) that is practical and powerful for estimating the dehumidification effectiveness of liquid desiccant dehumidifiers. This model takes into consideration the number of transfer unit, thermal capacity ratio of air to desiccant, and difference between air and desiccant inlet parameters as the inputs. The performance of the aforementioned model for the estimation of the dehumidification effectiveness is further assessed against a theoretical model in the open literature. It was found that the LSSVM model is more in the agreement with the actual data and further improvement is achievable compared to the theoretical model, with the coefficient of determination and mean square error values 0.998 and 5.586E-05, respectively. All the results indicate that the suggested dimensionless model is promising for accurate and reliable outcomes, which could be practical for other dehumidifiers with different geometries.

INTRODUCTIONHeating, ventilation, and air-conditioning (HVAC) systems utilize around 50% of the total energy used in buildings (Pérez-Lombard et al. 2008). The statistics indicated that the energy consumption for dehumidification process is responsible for 20-40% of the total energy utilized in HVAC systems (Huangand Zhang 2013). As a matter of fact, next-generation HVACtechnologies have emerged as interesting alternatives to theconventional vapor-compression system for dehumidificationand cooling. Obviously, from the previous literature, it wasfound that the desiccant cooling systems are one of the newtechnologies, which are able to control separately the latentand sensible loads, resulting in reducing energy consumption,and keeping safe the environment by removing condensedwater and conventional refrigerant from the mechanism ofcooling systems (Abdul-Wahab et al. 2004).Desiccants can take on a liquid or solid form. The liquiddesiccants are more preferable because of various benefits,such as need of lower regeneration temperature, pressuredrop, etc (Zurigat et al. 2004). Triethylene glycol aqueoussolution is regarded as the earliest desiccant in such systems,which then replaced by lithium chloride (LiCl) aqueoussolution due to the benefits of the latter one (Liu et al. 2011).Dehumidifier is one of the most important components in theliquid desiccant systems, and a great deal of research hasfocused on the component to evaluate and model itsperformance. In the past years, different theoretical andempirical models for predicting the performance of thedesiccant systems have been developed by differentresearchers, such as Gandhidasan (1994), Martin andGoswami (2000), Liu et al. (2006), Liu et al. (2007), Moon etal. (2009), and Park et al. (2016).

Nowadays, researchers pay more attention to the application of artificial neural algorithms, which are the new computational techniques for problem solving and determining optimal values. These approaches have been implemented for the prediction of the operating parameters of liquid desiccant systems to present a reliable and easy-to-use model for optimizing the design and operating the device. These models are a solution of nonlinear problems when the relation between input-output pairs are not clear. Different attempts have been made in the literature to describe the applicability of such models to liquid desiccant systems performance prediction, such as Zeidan et al. (2010), Gandhidasan and Mohandes (2011), Mohammad et al. (2013a), and Mohammad et al. (2013b). In Table 1, the summery of studies conducted on liquid desiccant systems by means of soft computing approaches is summarized. It is clear from the literature that the previous works on the prediction of dehumidifiers using artificial intelligence techniques have mainly presented the models which are related to specific physical geometry of the component. Support Vector Machines (SVMs) are relatively a new machine learning algorithm, which have attracted ample attentions in solving different complex functions owing to the merits such as the easy implementation, robustness, and high efficiency (Zendehboudi 2016). Owning to the proven proficiency of SVMs in images retrieval (Tao et al. 2006), fault diagnosis (Tian et al. 2015), and text detection (Kim et al. 2001), this article aims to apply a modified version of SVM, i.e., Least Square Support Vector Machine (LSSVM), todevelop a practical and powerful dimensionless model forestimating the dehumidification effectiveness of liquiddesiccant dehumidifiers. This model takes into considerationthe number of transfer unit, thermal capacity ratio of air todesiccant, and difference between air and desiccant inletparameters as the inputs. The performance of theaforementioned model for the estimation of thedehumidification effectiveness is further assessed against atheoretical model in the open literature.

METHODOLOGYLeast square support vector machine model

SVM theory has been first introduced by Vapkin (1998). This approach uses kernel functions to map the input variables to high-dimensional feature spaces and find a hyperplane via a nonlinear mapping. LSSVM was introduced by Suykens and Vandewalle (1999), which is considered as the modified version of SVM, along with two main advantages: (1) the inequality constraints are transferred into equality constraints; (2) a set of linear equation problems is solved instead of aconvex quadratic programming (Zendehboudi 2016).

ISBN: 978-0-646-98213-7 COBEE2018-Paper059 page 162


Table 1. Summary of studies on liquid desiccant dehumidifiers using soft computing. Author Material Method Input Output

Zeidan et al. (2010) CaCL2 ANN

1. Air and desiccant mass flow rates2. Air and desiccant temperature3. Air specific humidity4. Desiccant inlet concentration

1. Air and desiccant temperatures2. Desiccant vapor3. Mass of evaporated water

Gandhidasan and Mohandes (2011) LiCL ANN

1. Air humidity2. Air and desiccant flow rates3. Air and desiccant temperatures4. Desiccant concentration5. Cooling water temperature6. Dimensionless temperature ratio

1. Water condensation rate2. Desiccant concentration3. Desiccant temperature

Mohammad et al. (2013a) TEG ANN

1. Air and desiccant temperatures2. Air and desiccant flow rates3. Desiccant concentration4. Air humidity

1. Water condensation rate2. Dehumidifier effectiveness

Mohammad et al. (2013b) LiCL ANN

1. Air and desiccant flow rates2. Air humidity3. Air and desiccant temperatures

1. Moisture removal rate2. Dehumidification effectiveness

Current study LiCL GA-LSSVM

1. Number of transfer unit2. Thermal capacity ratio of air todesiccant3. Difference between air and desiccantinlet parameters

1. Dehumidification effectiveness

In below, a summarized explanation of LSSVM theory is presented. Suppose a regression problem over data set Z = {χi, gi│i=1,2, 3, …, n} where χi ϵ Rq, q is the dimensional input vector, gi ϵ R is the corresponding target value, and n refers to the training data size. The support vector regression model that fit the relationship between input-output is as follow:

(1)

In Eq. (1), ω is the weight vector, b is the bias, and q (c) is representative of a mapping function, which maps χ into higher dimensional feature space. The regression equation is transformed to an optimization problem by minimizing the cost function, which can be formulated as in (2), with the constraint of Eq. (3):

(2)

(3)

In Eqs. (2) and (3), ¡ represents for the regularization or penalty parameter that balances the model’s complexity and the training error, and ei is the regression error during training process. To solve the optimization problem, we should define Lagrangian function as:

(4)

Where ai refers to the Lagrange multipliers. Based upon Karush-Kuhn-Tucker conditions, we can get:

(5)

Where α and b are determined by solving Eq. (5). As a result, LSSVM model which is able to predict the corresponding output value of the new input vectors is represented as:

(6)

In Eq. (6), Ƙ(c, ci) is the kernel function. There are different kernel functions, which are used in LSSVM model. Among which, radial basis functions (RBF) is well-known and has been repeatedly applied in various studies due to advantages such as simplicity, notable regression capability, and reliability (Zendehboudi 2016). Therefore, in this study, RBF was selected as the kernel function, as shown in Eq. (7).

K χ, χi( ) = exp (− χ− χi

2/ σ2 ) (7)

The approximation ability of LSSVM is seriously affected by kernel parameter (σ2) and regularization parameter (ϒ). GA, which has already proven its ability in determining the LSSVM parameters, was executed for the purpose of determining the optimal magnitudes of these two parameters so that leads to higher accuracy. The best values of σ2 and ϒ were indicated as 8.608540161647952 and 3.981258374460578E+03 for the dehumidification effectiveness.

( ) . T bg w q c= +

( ) T 2

1

1ω, e ω . ω , , 2 2

n

ii

Min eb ew =

¡= + åõ

( ) Ts.t. ω . b , i 1 , 2, 3, , ni ieg q c= + + = …

( ) ( ) T

1( , , , ) ω, e ω . b

n

i i i ii

b e ew a a q c g=

é ù= - ë -+ + ûåõ

( )

( )

1

1

. 0

0

0 .

0 . 0

0

n

i ii

n

ii

i ii

Ti i i

i

b

ee

eb

w a q cw

a

a

gw q ca

=

=

¶= ® =

¶¶

ìïïï

=ïïíïïïï + +

= ®¶¶

= ®

-ïî

= ¡¶¶

= ® =¶

å

å

1( ) ( , )

n

i ii

Z K bg c a c c=

= = +å



Data collection and construction of datasets

To construct and develop a reliable predictive mode, the experimental data samples were extracted from the report of Park et al. (2016). According to the analytical solution results of Liu et al. (2007), the dehumidification effectiveness is as a function of m*, К, and NTUm, which is shown as Eqs. (8-10). Therefore, the dimensionless input and output parameters could be obtained, as shown in Table 2.

,

,

a p e

s p s

m cm

m c* = (8)

,

, ,

a in

a in e in

w wk

w w

*-=

- (9)

mm

a

ANTU

ma

= (10)

where m* is the thermal capacity ratio of air to desiccant, ma and ms are for respective the mass flow rate of air and desiccant, Cp,e and Cp,s are the specific heat of air in equilibrium with desiccant and desiccant, К is the dimensionless humidity ratio difference between air and desiccant inlet parameters, ωa,in and ωe,in are the humidity ratio of inlet air and air in equilibrium with desiccant, ω* is the humidity ratio of intersection point of inlet desiccant in concentration line and inlet air isenthalpic line, NTUm is the number of mass transfer unit, αm is the mass transfer coefficient, and A is the heat and mass transfer area.

RESULTSANDDISCUSSIONIn this investigation, MATLAB 2015b has been used to construct the model network architecture. To develop and check the validity of the constructed model, the dataset was randomly divided to two sets of training and testing data, which are 70% and 30% of the total data, respectively. The testing data were not introduced to the network during the training process. After the training process, the validity and over fitting of the network was checked with the testing data. Likewise, after developing the model, to examine the suggested model’s accuracy and predictability, Liu et al. (2007) model was developed considering the same data set and input variables. This model has an expression as follows:

ˆ ˆ(1 )

ˆ ˆ(1 m) (1 m)

ˆ1 (1 e ) (1 e ).ˆ ˆ

NTU m NTU mNTU

NTU NTU

e mm e m e

e k- - -

- -

- - - -= +

- -(11)

To visualize the performance of the developed models in a better way, regression plots between the predicted and actual values, scatter plots of error distribution as a function of the actual data, and contrast between the actual and predicted versus the number of data points are presented. Additionally, two statistical error tests, the coefficient of determination (R2) and the root mean squared error (RMSE), were utilized in this study. The statistical error tests were used to demonstrate the deviation between the estimated data samples using the introduced models and the observed data samples. It is worth to mention that the method consists of the lowest values of RMSE as well as the highest values of R2 is chosen as the best-fit technique in this contribution. These parameters can

Table 2. The dimensionless experimental data.

Number m* К NTUm εm

1 0.12 0.86 1.21 0.71

2 0.21 0.85 1.10 0.68

3 0.18 0.79 0.83 0.59

4 0.16 0.90 1.22 0.71

5 0.13 0.97 2.21 0.9

6 0.16 0.88 1.04 0.66

7 0.16 0.94 1.41 0.77

8 0.20 0.86 0.94 0.63

9 0.09 0.98 1.98 0.88

10 0.11 0.95 1.41 0.79

11 0.10 0.87 1.14 0.71

12 0.15 0.94 1.57 0.8

13 0.07 0.97 1.97 0.87

14 0.17 0.76 0.63 0.48

15 0.24 0.80 0.78 0.56

16 0.21 0.68 0.32 0.29

17 0.21 0.67 0.42 0.35

18 0.18 0.87 1.03 0.66

19 0.30 0.79 0.77 0.55

20 0.33 0.84 1.03 0.66

21 0.07 0.71 0.49 0.41

22 0.05 0.73 0.53 0.43

23 0.07 0.74 0.54 0.43

24 0.09 0.79 0.69 0.52

25 0.07 0.71 0.42 0.35

26 0.17 0.73 0.49 0.4

27 0.53 0.91 1.26 0.73

28 0.29 0.95 1.50 0.79

29 0.22 0.93 1.33 0.75

30 0.25 0.93 1.57 0.8

31 0.14 0.95 1.65 0.82

32 0.11 0.96 1.66 0.82

33 0.29 0.89 1.18 0.71

34 0.14 0.97 1.95 0.88

35 0.18 0.80 0.89 0.65

be calculated through Eqs. (12) and (13).

n2

i i2 i 1

n2

ii 1

(actual - predicted )R 1-

(actual - average(actual))

=

=

=å

å(12)



n2 0.5

i ii 1

1RMSE ( (actual - predicted ) )n =

= å (13)

The values of R2 and RMSE of these two different predictive approaches are presented in Table 3. Indeed, it is noticeable that a favorable agreement exists between the estimations using the GA-LSSVM and the actual data. The results show that Liu et al. (2007) model cannot be practical to precisely model the dehumidification effectiveness.

Table 3. Statistical parameters of the developed models.

Parameter Model R2 MSE

e GA-LSSVM 0.9980 5.586E-05

Liu et al. (2007) 0.7961 0.0058

(a)

(b)

Figure 1. Regression plot between the predicted and actual dehumidification effectiveness: (a) GA-LSSVM; (b) Liu et al.

(2007) model.

To visualize further the predictive abilities of the proposed models, Figure 1 presents the regression plots between the

predicted results and the related actual data for the dehumidification effectiveness. In these plots, the vertical axis represents the estimated values and the horizontal axis is the actual values. This figure indicates the degree of agreement between the actual data and predicted values using both the GA-LSSVM and the model of Liu et al. (2007). By sketching the diagonal line, it is observed that the predicted and actual values are in a great consistency using the suggested GA-LSSVM model because the distribution of the predicted points around the unit slop line is lower, proving the ability of the model to reproduce the actual data samples with high accuracy. Results are found to be less scattered for low dehumidification effectiveness values than for high ones using Liu et al. (2007) model.

(a)

(b)

Figure 2. Scatter plot of the relative deviations as a function of the actual samples: (a) GA-LSSVM; (b) Liu et al. (2007)

model.

Figure 2 represents the relative deviation for both models. As it is clear from this figure, the relative deviations of the model of Liu et al. (2007) are higher than those of the developed GA-LSSVM presented in the current study. The predicted data samples using the developed GA-LSSVM fall within an error band of -4.63% and +3.07%. This fact highlights that the predicted values by the GA-optimized approach are more reliable and accurate. The results generated by the correlation



of Liu et al. (2007) was presented to have a better understanding about the suggested model in the current study. It is observed that this model generally underpredicts the experimental data samples as the data points are below the zero line. Comparison of the relative deviations of these two approaches for the prediction of this problem highlighted that the GA-LSSVM method is a good solution by presenting accuracy improvements in comparison with the another one. Figure 3 shows the contrast plot where the predicted and actual data samples are plotted as a function of the number of data samples. This evaluation shows truthfully of a model and provides the tendency of a predictive method in terms of overestimation and underestimation. This figure shows that, for all data samples, there is a high overlap between the estimated and the actual trends and the developed GA-LSSVM outcomes excellently follow the trend of actual samples. Once again, these result shows high integrity of the suggested model in prediction of the dehumidification effectiveness based on the given data and proves that the actual samples are precisely modeled using the designed network.

(a)

(b)

Figure 3. Contrast plot of the actual and predicted data versus the number of samples: (a) GA-LSSVM; (b) Liu et al.

(2007) model.

CONCLUSIONSA LSSVM approach was developed to offer a novel dimensionless model for the accurate estimation of the dehumidification effectiveness of liquid desiccant dehumidifiers by considering the number of transfer unit, thermal capacity ratio of air to desiccant, and difference between air and desiccant inlet parameters as the input vectors. Genetic Algorithm was implemented to optimize the tuning parameters of the LSSVM method. Meanwhile, to visualize the robustness and reliability of the suggested novel model, a well-known reported model in the literature was developed by utilizing the same data set and inputs. Based on a comparative analysis via various graphical and statistical error analyses, it was concluded that the performance of the GA-LSSVM highly outperforms the another one and presents predictions with reasonable errors. The maximum generalization error of the developed LSSVM, according to all data samples, was within -4.63% and +3.07% error band with an MSE=5.586E-05 and R2=0.9980. The suggested model is highly valuable to industries, which could be used for other dehumidifiers with different geometries in the reported dimensionless ranges to determine and monitor the outcomes with a favorable precision.

ACKNOWLEDGEMENTThis study was supported by the Innovation Research Groups of the National Natural Science Foundation of China (Grant No. 51521005).

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A Dimensionless Model for Estimating the Dehumidification