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A novel strategy for predicting the performance of open vertical refrigerated display cabinets based on the MTF model and ASVM algorithm
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 4 1 3 e1 4 2 4
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A novel strategy for predicting the performance of open vertical refrigerated display cabinets based on the MTF model and ASVM algorithm Zhikun Cao a,*, Bo Gu a,**, Graham Mills b, Hua Han a a
Institute of Refrigeration and Cryogenics, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dong Chuan Road, Min Hang, Shanghai 200240, PR China b Environ Australia Pty Ltd, 100 Pacific Highway, North Sydney, NSW, 2060, Australia
article info
abstract
Article history:
This paper presents a novel strategy for predicting the performance of open vertical refrig-
Received 19 November 2009
erated display cabinets which is based on a modified two-fluid (MTF) model and an adaptive
Received in revised form
support vector machine (ASVM) algorithm. A MTF model (physical model) was built for open
5 February 2010
vertical refrigerated display cabinets, and then an ASVM algorithm (machine learning
Accepted 5 April 2010
algorithm) was built. To verify the quantity of air leakage from the cabinet during operation,
Available online 9 April 2010
an important performance factor of display cabinets, an MTF model was built. After the training and validation data sets were constructed from the output of the MTF model, the
Keywords:
problem was solved using an ASVM algorithm. The defrosting water quantity and total
Display cabinet
energy consumption / total display area (TEC/TDA), achieved from the experiments by using
Open cabinet
the predicted combination of the controlled parameters, were found to be reduced by 39.2%
Prediction
and 19.3%, respectively, from the experimental results of the original display cabinet.
Modelling
ª 2010 Elsevier Ltd and IIR. All rights reserved.
Flow Air Leakage Simulation
Strate´gie innovante pour pre´voir la performance des meubles de vente frigorifiques fonde´e sur un mode`le a` deux fluides modifie´s et un algorithme Mots cle´s : Meuble de vente ; Meuble ouvert ; Pre´vision ; Mode´lisation ; E´coulement ; Air ; Fuite ; Simulation
* Corresponding author. Tel.:þ86 34 20 62 60; fax: þ86 21 34 20 68 14. ** Corresponding author. E-mail addresses: [email protected] (Z. Cao), [email protected] (B. Gu). 0140-7007/$ e see front matter ª 2010 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2010.04.006
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Nomenclature actual values ai ASVM adaptive support vector machine C penalty coefficient C1, C2, Cm empirical constants in the turbulent energy equations Cm1, Cm2 empirical constants in the mass transfer rate equations empirical constants in the momentum and heat Cf, Ct transfer equations specific heat at constant pressure (J kg1 K1) Cp d dimension D mesh diameter (mm) DEC direct energy consumption (kWh day1) ERM empirical risk minimization f fluid friction force per unit volume (N m3 s1) h Prandtl mixing length (m) ambient relative humidity (%) ha i the enthalpy value (kJ kg1) H surface heat transfer (J m3 s1) J production rate of turbulent energy (kgm1 s1) k turbulent kinetic energy (m2 s2) m fluid mass transfer rate per unit volume (kg m3 s1) the defrosting water quantity (kg) Mc n number of the dataset predicted values pi Q calorific value (W) R reversal counting RBF radial basis function REC refrigeration electrical energy consumption (kWh day1) MTF modified two-fluid the effective air supply area (m2) Se the total air supply area of orifice plates (m2) St
1.
Introduction
Refrigerated display cabinets play an important role in the cold food chain as they are the final stage and face the consumer directly at the time of purchase. Due to their high degree of exposure to the ambient environment, refrigerated display cabinets can consume an enormous amount of energy. According to statistics published by the Electric Power Research Institute, refrigeration contributes about 60% of the total energy consumed by a supermarket (Evans et al., 2007). Open vertical display cabinets, despite being high consumers of energy are a common type of cabinet which makes it possible to display large amounts of food in a small area. Therefore, the open vertical cabinet is probably one of the most important components in the supermarket system even though cabinets with permanent glass doors can reduce energy consumption by 50% (Adams, 1985). Moreover, the quantity of air leakage is the key factor affecting the performance of display cabinets. Many studies show that because the only barrier between the refrigerated load and the ambient air
SRM T TDA TEC ! U V
structured risk minimization temperature ( C) total display area (m2) total energy consumption (kWh day1) velocity vector (m s1) fluid volume (m3)
Greek symbols g secondary relaxation factor d volume fraction mass fraction dm e loss coefficient h scale factor m dynamic viscosity (kg m1 s1) effective dynamic viscosity (kg m1 s1) me turbulent dynamic viscosity (kg m1 s1) mt q angle of air supply outlet ( ) l convergence coefficient s standard deviation r density (kg m3) c free parameter (the bandwidth of RBF kernel) u coefficient sk, se, sT, sr turbulent Prandtl number relaxation factors xi ; xi z dissipation rate of turbulent energy (m2 s3) Subscripts 1 fluid 1 (the turbulent air) 2 fluid 2 (the non-turbulent air) a ambient cabinet inlet cin cabinet outlet cout eo evaporator outlet i, j serial numbers r air return s air supply se saturated evaporating
is one or more forced-air curtains, there may be a significant entrainment of ambient wet hot air. Such operating conditions also lead to poor food temperature control, the formation of dew or frost on the surface of the evaporator and food in the cabinets as well as significantly increased energy consumption (Getu and Bansal, 2007; Navaz et al., 2005; D’Agaro et al., 2006). Although many studies have been conducted on the performance of display cabinets (Howell, 1993; Brolls, 1986; Axell and Fahle´n, 1995, 1998, 2003; Billiard and Gautherin, 1993; Chandrasekharan et al., 2006; Cortella, 2002), most of them only emphasize a few aspects by performing experiments or computational fluid dynamics simulations. As a result, few studies have presented systematic strategies for performance prediction by investigating the relationship between multiple parameters. This paper is an attempt to fill this gap. In particular, we investigate strategies for the predicting the performance of open vertical refrigerated display cabinets using the MTF model and ASVM algorithm. The two-fluid (TF) model, proposed by Spalding (Spalding, 1984), can be summarized as follows:
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 4 1 3 e1 4 2 4
(1) Turbulent flow is considered as comprehensive motion for the interaction between each fluid; (2) There is a respective volume fraction for the two fluids, i.e., the two fluids exist respectively with certain probability; (3) The two fluids, whose motions respectively follow their controlled differential equations, entrain each other as a continuum; (4) The dividing characteristics of the two fluids are the concentration, temperature, flow direction or volume. Recently, the TF model has become a good method for modeling fluid dynamics applications, such as free jets (Ilegbusi and Spalding, 1987a), flat plate flows (Ilegbusi and Spalding, 1989), conduit flows (Ilegbusi and Spalding, 1987a), near-wall flows (Ilegbusi and Spalding, 1987b), Couette flows (Spalding, 1987), stratified flows (Shen et al., 2003), tundish problems (Sheng and Jonsson, 2000) and air entrainment (Yu et al., 2008; Cao et al., 2010a). In these cases, the predicted results are in good agreement with the experimental results. The Support vector machine (SVM) is a very promising technique developed by Vapnik and his group at AT&T Bell Labs (Vapnik, 1995; Cortes and Vapnik, 1995; Boser et al., 1992; Burges, 1996). This new machine learning algorithm can be seen as an alternative training technique for Polynomial, Radial Basis Function and Multi-Layer Perceptron classifiers. Unlike traditional artificial neural algorithms (Yigit and Ertunc, 2006), which use the empirical risk minimization (ERM) principle, SVM algorithm uses an approximate implementation of the structured risk minimization (SRM) induction principle which demands fewer experimental samples to obtain a good level of performance reducing experimental time and cost (Vapnik and Chervonenkis, 1991). Training a SVM is equivalent to solve a quadratic programming problem with linear and box constraints in a number of variables equal to the number of data points (Vapnik, 1998). Moreover, SVM is able to learn good separating hyperplanes in high dimensional feature space (Cristianini and Shawe, 2000). Rather than estimating parameter coefficients from variables contained in the data matrix, SVM estimates parameter coefficients by using the observations. It is therefore possible for a SVM’s feature space to be equivalent to the number of observations used in training the SVM. Reducing the number of estimated coefficients is referred to as sparseness and is often achieved by optimization techniques. It is by controlling the hyperplane margin measures through generalization theory that SVM rarely overfit the data and achieves sparseness. Therefore, SVM is created by applying optimization theory, usually with Lagrange multipliers or KarusheKuhneTucker conditions, to the margins. Moreover, the introduction of Vapnik’s e-insensitive loss function and kernel function has extended SVM to solve problems involving nonlinear regression estimation and noise characterization. Based on the statistical learning theory, the SVM which is a novel method with good theoretical properties, has been developed and widely used recently in the HVAC and refrigeration field (Liang and Du, 2007; Esen et al., 2008; Ren et al., 2008; Cao et al., 2009, 2010b; Li et al., 2009). To predict the performance of the refrigerated display cabinets, A MTF model (physical model) was built for open
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vertical refrigerated display cabinets, and then an ASVM algorithm (machine learning algorithm) was built.
2.
MTF model and ASVM algorithm
2.1.
TF model
As shown in Fig. 1, the cooling air supplied by the air curtain and rear grilles is defined as fluid 1 (the turbulent fluid), and the ambient air is defined as fluid 2 (the non-turbulent fluid). Based on the initial two-fluid model (Ilegbusi, 1994), the TF model of a display cabinet without phase diffusion is described as follows. Energy equations: v v m m vT1 v m vd1 ! d1 t þ þ T1 e ðd1 r1 U 1b T1 Þ ¼ sT Pr vxb sT vxb vxb vxb vxb H12 v v m vd2 H12 ð1Þ ! þvT2 m þm vT2 d2 Cp þ T2 e T2 m ðd2 r2 U 2b T2 Þ ¼ sT vxb Cp vxb vxb Pr vxb vxb (2) Turbulent kinetic energy and dissipation rate of the turbulent energy equations: v v m vk v m vd1 ! d1 t þ m þ k e þ d1 ðJ r1 zÞ ðd1 r1 U 1a kÞ ¼ sk vxb vxb vxb vxb sr vxb (3) v v m vz v m vd1 ! d1 t þ m þ e e ðd1 r1 U 1a zÞ ¼ se vxb vxb vxa vxa sr vxa z þ d1 ðC1 J C2 r1 zÞ k
ð4Þ
where J, the rate of production of turbulent energy, is given by
J ¼ mt
! ! ! v U 1a v U 1b v U 1a þ vxb vxa vxb
(5)
Momentum equations: ! ! v vp v v U 1a v U 1b ! ! d 1 me ðd1 r1 U 1a U 1b Þ ¼ d1 þ þ vxb vxa vxb vxa vxb v ! me vd1 U 1a þ d1 ðr1 r2 Þga þ sr vxb vxb ! þ U 2a m þ f12 ! ! v vp v v U 2a v U 2b ! ! d2 m ðd2 r2 U 2a U 2b Þ ¼ d2 þ þ vxb vxa vxb vxa vxb v ! me vd2 ! U 2a U 2a m f12 þ sr vxb vxb
ð6Þ
(7)
Continuity equations: v v me vd1 ! þm ðd1 r1 U 1b Þ ¼ vxb vxb sr vxb v v m vd2 ! m ðd2 r2 U 2b Þ ¼ vxb vxb sr vxb
(8)
(9)
The constraint of the volume fraction: d 1 þ d2 ¼ 1
(10)
where the subscript 1 or 2 denotes fluid 1 or fluid 2; the ! subscripts a and b denote the coordinate system (Fig. 1); U denotes a velocity vector; Cp denotes specific heat at
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Model 1, Eq. (16) Model 2, Eq. (17) MTF model (ρ / ρ =2), Eq. (18)
0.5 0.4
MTF model (ρ / ρ =1.5), Eq. (18)
0.3
Model 1 neglected m from fluid 1 to fluid 2
m/ ρ U -U /h
0.2 0.1 0.0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-0.1
0.9
1.0
δ
-0.2 -0.3 -0.4 -0.5
Model 2 over predicted m from fluid 1 to fluid 2
Fig. 2 e Results of different mass transfer rate equations.
2.2.
Fig. 1 e Open vertical refrigerated display cabinet.
constant pressure; d is the volume fraction of the fluid, given by d ¼ Se/St (Yu et al., 2008); Se denotes the effective air supply area; St denotes the total air supply area of the orifice plates. The turbulent dynamic viscosity : mt ¼ Cm rk2 =z
(11)
The effective dynamic viscosity : me ¼ mt þ m
(12)
The surface heat transfer : ! ! H12 ¼ Ct Cp r1 d1 d2 ðT2 T1 Þj U 2b U 1b j=h
ð13Þ
The inter-fluid friction force per unit volume : ! ! ! ! f12 ¼ Cf r1 d1 d2 ð U 2b U 1b Þj U 2b U 1b j=h
ð14Þ
The Prandtl mixing length : h ¼ cm3=4 k3=2 =z
(15)
Equations for the mass transfer rate between the two fluids (Malin and Spalding, 1984): ! ! m ¼ Cm1 r1 d1 d2 j U 2b U 1b j=h (16) and (Sheng and Jonsson, 2000), ! ! m ¼ Cm2 r1 d1 d2 ðd2 0:5Þj U 2b U 1b j=h
Since the TF model evaluates the air leakage quantity from a refrigerated display cabinet from the defrosting water quantity, in order to increase the accuracy of the results it is necessary to improve the two mass transfer rate equations in the original TF model given by Eqs. (16) and (17). From a physical viewpoint, the non-turbulent fluid (fluid 2) is entrained by the turbulent fluid (fluid 1) while a small quantity of the turbulent fluid is entrained by the nonturbulent fluid. Fig. 2 shows the relationship between the ! ! dimensionless number m=ðr1 j U 2b U 1b j=hÞ and different values of d1 calculated from a number of different models. Since m is always positive in Eq. (16), model 1 cannot be used to investigate the entrainment of fluid 2 by fluid 1. Moreover, because of the added control factor (d2 ¼ 0.5) in Eq. (17), only an equivalent fluid 2 can be entrained by fluid 1. Therefore, both model 1 and model 2 have limitations. Yu et al. (2008) obtained better results using a modified TF model for open vertical display cabinet based on the weighted volume fraction factor. However, practical operation of display cabinets usually involves fluid 2 being hot wet air whose density is lower than that of fluid 1. As a result, the influence of air density cannot be neglected in the mass transfer rate equations. In this study, the TF model was modified using a weighted mass fraction factor in developing the MTF model as described below: (See Appendix 1 for detailed derivation):
(17)
The empirical constants employed in the TF model are listed in Table 1 (Shen et al., 2003; Sheng and Jonsson, 2000).
Table 1 e The values of the constants employed in the TF model. Cm1
Cm2
Cm
sk
se
sT
sr
C1
C2
Cf
Ct
0.1
10.0
0.09
1.0
1.3
1.0
0.95
1.44
1.92
0.05
0.05
MTF model
m¼
! ! Cm1 r1 d1 d2 j U 2b U 1b j 1 h d1 þ d2 ðr2 =r1 Þ ! ! Cm2 r1 d1 d2 ðd2 0:5Þj U 2b U 1b j 1 þ d1 ðr1 =r2 Þ þ d2 h
(18)
As shown in Fig. 2 (For convenient comparison, it is assumed that r1/r2 ¼ 2, r1/r2 ¼ 1.5) the MTF model shows that in the mutual entrainment of the two fluids the entrainment rate of fluid 1 by fluid 2 is less than that of fluid 2 by fluid 1, and that the entrainment rate changes with density which corresponds to expectations from practical operation.
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2.3.
SVM algorithm
where,
For regression problems, SVM ensures the model can be applied generally by fitting an observation sample set using functionf ðxÞ ¼ uT Fðxi Þ þ b. Specially, F(x) is the nonlinear map from the input space to the higher-dimensional feature space, with f(x) representing the resulting linear function in that feature space. Here, the y-SVM regression (Scho¨lkopf et al., 2000; Wang et al., 2005; Chen et al., 2006) which yields simple form and good performance is employed in this paper. In addition, the SVM maps the observed sample data from the nonlinear problem into a higher-dimensional feature space. Using Mercer’s theorem, when the process of solving the optimized dual problem in the higher-dimensional feature space only referred to the inner-product operation, there was one function, Kðx; xi Þ ¼ 4ðxÞ$4ðxi Þ, which just equaled the inner product of the nonlinear transform in this higher-dimensional feature space. By using the inner product directly derived by K (x,xi), the calculation can be greatly simplified without the use of a complex nonlinear transform (Na and Webb, 2004). K(x,xi) therefore becomes the kernel function by satisfying these conditions. Using this kernel function, the nonlinear problem in the lower dimensional space is transformed into a linear problem in the higher-dimensional feature space. Here, the Gaussian radial basis function (RBF) kernel function is employed as it yields better prediction performance (Lin and Lin, 2004). ! kx xi k2 (19) Kðx; xi Þ ¼ exp c2 After solving the convex quadratic optimization problem the result then becomes (see Appendix 2 for detailed derivation): f ðxÞ ¼
l X
ai ai K xi ; xj þ b
(20)
i¼1
The learning machine constructed by Eq. (20) is simply the y-SVM regression where the support vectors are the vectors which make the coefficients in Eq. (20) non-zero. These support vectors make up the training sample with coefficients which are non-zero. By ignoring the fitting errors which are less than e, the e-insensitive loss function possesses some anti-noise ability. This can be shown as follows: Lðy; f ðxÞÞ ¼ L y f ðxÞje
(21)
jy f ðxÞje ¼
2.4.
0 jy f ðxÞj e jy f ðxÞj e otherwise
(22)
The adaptation of ASVM algorithm
Using the loss coefficient e-insensitive loss function from Eq. (21), the SVM regression is constructed. In most circumstances e is a value which does not change when the sample is determined. However, when the level of interference is changed, or the alteratione ¼ jy f ðxÞj exceeds a particular value, the antinoise ability of the insensitive loss function (Eq. (21)), decreases or even disappears because e loses its ability to judge as the threshold. To avoid this and enhance the adaptive ability of the SVM eis adjusted dynamically in this research. As shown in Fig. 3, reversal counts are used as a basic control index in the univariate statistical control chart, where E and s are the expectation and the standard deviation of the noise inherent in a process variable. When the process variable first exceeds the threshold as illustrated in Fig. 3, the reversal counting starts (R ¼ 1), and increments (R ¼ R þ 1) every time the process variable exceeds the threshold in either direction. Usually, the maximum tolerable number of reversals in both directions is chosen to be 4, with the maximum tolerable being 2 in the same direction (Salsbury, 1999). Due to this, the dynamic control equations of e can be described as follows: eiþ1 ¼ ð1 þ hÞei ; when R ¼ 1; 3ðmaximum tolerable number of reversals in the yþ direction ; or R ¼ 2; 4 maximum tolerable numberof reversals in the y direction ; or R ¼ 1; 2; 3; 4
(23)
where h is the scale factor whose value can chosen as follows: h ¼ ðei ei Þ=emax
(24)
Due to Eq. (24) being a unilateral contact function, convergence is ensured through the introduction of Eq. (25). eiþ1 ¼ ð1 gÞei ; when ei < gEi1 ; eiþ1 < gEi
(25)
where g is the secondary relaxation factor 0 < g < 1 which can effect the final degree of convergence. In addition, when the value of e ¼ jy f ðxÞj is in the interval excluding the constraint conditions of Eqs. (23) and (25) e then becomes: eiþ1 ¼ ei
2.5.
Fig. 3 e Illustration of reversal counts.
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(26)
Controlled parameters determination
Before the application of the strategy, the controlled parameters corresponding to the performance of the refrigerated display cabinet should be confirmed. The choice of data to be collected should ensure that the algorithm is easily applied to the development of commercial display cabinet products.
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For refrigerant cycle: REC ¼ Qtot =COP
(27)
where Qtot is the total heat extraction, kWh/day Qtot ¼
N max X
Fn $Dt
(28)
n¼1
where Fn is the transient heat extraction, kW; Dt is the time; h; Fn ¼ 0 kW during stopping and defrost time. For a remote compression-type refrigerating system, COP is calculated from the following equation (in Section 5.3.6.3.3 of EN ISO 23953-2, 2005): COP ¼
H$Tse Tc Tse
(29)
where Tc is the condensing temperature, K; H is the coefficient; H ¼ 0.34, when Tc ¼ 308.15. The heat extraction rate in kilowatts is defined as Fn ¼ qm $ðicout icin Þ
(30)
where qm is the mass flow rate of refrigerant; icin, icout is the enthalpy value of refrigerant at cabinet inlet and outlet, respectively. For each individual measuring instant, where n indicates the measuring sample (Fn ¼ 0 kW during stopping and defrost time). The refrigeration electrical energy consumption for a cabinet intended for a remote compression-type refrigerating system, REC is calculated from the following equation: REC ¼ Qtot $
Tc Tse Tc Tse ¼ 24 tdeft $F24deft $ H$Tse H$Tse
(31)
where tdeft is the defrosting time, h. Therefore, the saturated vapour temperature Tse is chosen, which represents the refrigerating coefficient of performance COP; the refrigerant temperature of evaporator outlet Teo is chosen, which represents the cooling capacity of display cabinet (the total heat extraction Qtot) with the refrigerant temperature of display cabinet inlet Tin and the refrigerant flow qm (saturated liquid is assumed). Because the destination is predict the performance of display cabinets, the parameters Tse and Teo which affects TEC/TDA of display cabinets should be involved. For the air curtain cycle: The heat extraction: Q ¼ Gðia ic Þ
(32)
where G is the infiltration air mass through air curtain; ia is the
enthalpy value of ambient air; ic is the enthalpy value of air curtain. Therefore, the ambient temperature Ta, the relative humidity ha (saturation is assumed), the air supply temperature Ts and the air return temperature Tr which represent the enthalpy difference between the ambient air and air curtain are chosen. Also, the angle of air supply outlet q and the mesh diameter of the honey comb structured air supply outlet D which represent G are chosen.
2.6.
ASVM algorithm architecture
The architecture of the ASVM algorithm, applied to the refrigerated display cabinet system, is shown in Fig. 4. The vectors input to the ASVM model are the attributes which influence the quantity of air leakage. These include evaporator outlet superheat (Teo Tse), the difference between the ambient air temperature and supply air temperature (Ta Ts), the difference between the ambient air temperature and the return air temperature (Ta Tr), the angle of air supply outlet q, the diameter of the honey comb structured air supply outlet mesh D, and the ambient relative humidity ha. The output vector is the defrosting water quantity Mc. Using the Gaussian RBF kernel function the ASVM model can achieve multilayer perception including hidden layers.
2.7.
ASVM algorithm parameter determination
Results obtained by Cristianini and Shawe (2000), Kwok (2000), and Van et al. (2001) in their work on SVM evidence frameworks show that convergence precision can be improved through reasonable selection of AVSM parameters. The model convergence precision l can be presented as follows. l ¼ jðR RMSEiþ1 Þ ðR RMSEi Þj
where the relative root mean square error (RMSE) is shown as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pn i¼1 ai pi =ai R RMSE ¼ 100% (34) n where n is the number of the dataset, and ai and pi are the actual values and predicted values, respectively. The values of l vary with different g values in the loss function. The results obtained from the training data show that when 0.05 < g < 0.2, there is a high degree of convergence precision. In this study, g was selected to be 0.13. For multivariate d-dimensional problems, the RBF width parameter c is set so that cd ˛ð0:1; 0:5Þ. Cherkassky and Ma (2004) showed that such values yield good SVM performance for a range of various regression data sets. Here, d ¼ 6 and c ¼ 0.85. Moreover, Cherkassky and Ma (2004) also showed the penalty coefficient C obtained from the training data then becomes: C ¼ max y þ 3sy j; y 3sy
Fig. 4 e Algorithm architecture of ASVM of refrigerated display cabinets.
(33)
(35)
wherey and sy are the mean and the standard deviation of the training data y values. Scho¨lkopf et al. (1998) showed that the y-SVM algorithm can automatically select e by the priori chosen y. Wheny˛½0; 1,
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 4 1 3 e1 4 2 4
Dataset calculated from MTF model
Scaling
Testing set
Initializing ε
Training set
New ε Training ASVM algorithm
Accuracy control
ε adaptive adjustment
Reversal counts judgment criteria
N
Y Trained ASVM with optimized ε
ASVM output
Fig. 5 e Flow chart of MTF-ASVM prediction strategy for refrigerated display cabinets. the training error displays an increasing trend as y values increase. The evalue corresponding to y ¼ 0.2 was selected as the initial e value.
2.8. Flow chart of the prediction strategy based on MTF-ASVM As shown in Fig. 5, the defrosting water quantity for the display cabinet was calculated using an MTF model to save a significant amount of experimental time and cost. After scaling the input control parameters corresponding to different defrosting water quantities, the defrosting water quantity and other controlled parameters were set as the output vector and the input vectors of the training and validation datasets of the ASVM algorithm. Using the adaptive e, each team of input vectors produces one output from the trained ASVM algorithm. In addition, as machine learning is based on statistical learning theory the ASVM algorithm has a significant calculation advantage over the MTF model which based on a physical model. This saves further time and cost. Finally, the input parameters corresponding to the minimum defrosting water quantity that meets requirements are used as the predicted parameters which realize the objective of reducing the quantity of air leakage from vertical refrigerated display cabinets.
3.
Methodology
3.1.
Experimental facilities/display cabinet
The object of this investigation is a remote open vertical refrigerated display unit with a single chilled air curtain (Fig. 1). The dimensional features are: length 2.5 m, height 2 m,
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depth 1.1 m with a display opening height of 1.65 m. This display cabinet system consists of three loops: a refrigerant loop, a condensing air loop and an evaporating air curtain loop. The refrigerant loop is a single-stage vapour compression plant, using R22 as the working fluid, the fundamental components of which are as follows: a reciprocating open type compressor, a fin-tube evaporator, a thermostatic expansion valve, and a fin-tube condenser. As shown in Fig. 6, the experiments were carried out in a test chamber with length 8 m, height 3.5 m and depth 6 m. In particular, the air in the test chamber moved parallel to the longitudinal axis of the cabinet at uniform velocity between the range of 0.1 and 0.2 ms1. There is the special HVAC system (including the humidifier) for the test chamber. Referring to the EN ISO 23953-2, the test chamber should supply the air moving parallel to the longitudinal axis of the cabinet (with the velocity of 0.1 and 0.2 ms1) which makes the test chamber uniformly achieve the required ambient air. T-type thermocouples (coppereconstantan) with an accuracy of 0.1 C and piezoelectric pressure gauges with an accuracy of 0.1% of the full-scale range were used to measure the temperature and pressure, respectively. The T-type thermocouples were mounted in the display cabinet which was loaded with test packages, some of which are M-packages. All these measurements were collected by an HMAM 25203 lab data acquisition system at intervals of 60 s for a period of 12 h. Moreover, the defrosting water quantity is weighted by the compute a charge scale (CPS CC220) with an accuracy of 0.015% of the full scale range and a range of 0e100 kg.
3.2.
Experimental design
Referring to the EN ISO 23953-2 (2005) standard, the air leakage quantity was tested by the defrosting water quantity. The factors which significantly influence the defrosting water quantity are summarized as follows: air supply temperature Ts, air return temperature Tr, ambient temperature Ta, saturated evaporating temperature Tse, evaporator outlet temperature Teo, angle of air supply outlet q, the mesh diameter of the honey comb structured air supply outlet D, and ambient relative humidity ha. Many authors follow the approach of Rigot (Rigot, 1990) to define an “infiltration ratio” which describes the amount of air leakage and is based on a simple energy balance from the knowledge of Ts, Ta and Tr. However, the parameters Ts, Ta and Tr which could be directly measured, were selected here in order to simplify the calculation. Experience from the process of developing medium temperature display cabinets (Getu and Bansal, 2006; Navaz et al., 2005; Chen and Yuan, 2004) and the actual situation of the experimental display cabinet, the range of design parameters values mentioned above are as follows. 3 ðTeo Tse Þ 10; 14 ðTa Ts Þ 31; 4 ðTa Tr Þ 22; ð36Þ 0 q 10; 3 D 8; 0:4 ha 0:8 Referring to the M1 standard (the temperature inside cabinet is between 1 C and 5 C) (EN ISO 23953-2, 2005), 60 groups of control parameters were randomly selected as the experimental operating conditions for the air leakage experiments with an ambient temperature of 25 C in the test
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Fig. 6 e Experimental facility of refrigerated display cabinet test.
4.
Results and discussion
4.1.
MTF model e results and discussion
This section summarizes the results obtained using the MTF model to simulate the performance of a display cabinet. The objective of this work was to verify the effectiveness of the MTF model in predicting the performance of these cabinets. In order to do this, the results obtained using the MTF model were compared with the results from Yu’s model (Yu et al., 2008). As shown in Fig. 7, it can clearly be seen that the predictive accuracy of the MTF model is better than that of the Yu’s model. Generally, with a 95% confidence interval, the discrepancy between the defrosting water quantity predicted by the model and experimental results is less than 8%. By comparison, the error band of Yu’s model is 15%. In addition, compared with 12 h of experimental test time required for each measurement period, the MTF model only needs 0.5 h to measure the same conditions, saving experimental time and cost in producing higher accuracy results.
selected as the training dataset and the validation dataset for the ASVM algorithm. However, as it is very important to cover the entire expected operational range of the display cabinet in training the ASVM model it was ensured that the training dataset used to develop the model covered the maximum and minimum values of the model input and output variables. Using 60 validation datasets Fig. 8(a) shows the high degree of output regressive accuracy produced by the ASVM algorithm. With a 95% confidence interval, the predicted discrepancy of the ASVM algorithm was less than 5% for the defrosting water quantity. In addition, the ASVM algorithm has a convergence speed which is an order of magnitude higher than that of the MTF model using the same computer (PIV 2.66 G CPU, 1 G memory). By adding noise to the dataset, the robustness of the ASVM algorithm was investigated. Since the ASVM approach is not sensitive to a particular noise distribution (Cristianini and Shawe, 2000), Gaussian noise was selected. In the following implementation, the same values for the free parameters (C, c, and initial e) were used when developing the prediction model using the noisy dataset.
20
18
ASVM algorithm e results and discussion
In total 120 defrosting water quantity and controlled parameter datasets obtained from the MTF model were randomly
+15%
17
16
+8%
15
-15% -8%
14
4.2.
MTF model Yu's model
19
Predicted Mc kg
chamber (because the ambient temperature is kept at 25 C, thus leading to supply temperature in the range from 6 to 11 C, and return temperature in the range 3e21 C). After testing was undertaken for a period of defrosting, the upper night curtain (Fig. 6) was pulled down to isolate the air inside the display cabinet from the external environment. Then the defrosting controller was set (the defrosting period and the defrosting time were set to 4 h and 35 min, respectively), and the defrosting water quantity was weighted after the defrosting period. Moreover, each measurement was lasted for 3 defrosting periods. After that, the accuracy of the MTF model was validated by comparing of the experimental defrosting water quantity and that predicated by the model.
14
15
16
17
18
19
Experimental Mc kg
Fig. 7 e Prediction accuracy of MTF model and weighted model based on volume fraction (Yu’ model).
20
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19
Predicted Mc kg
b 20
20
19
R-RMSE=3.93%
18
18
17
17
Predicted Mc kg
a
16
+5% 15
-5%
14 13
15 14 13 12
11
11
11
12
13
14
15
16
17
18
19
+10%
16
12
10 10
ξ~N 0, 0.082 R-RMSE=6.34%
-10%
10 10
20
11
12
13
19
ξ~N 0, 0.102
19
18
R-RMSE=9.97%
18
17
18
19
20
17
18
19
20
ξ~N 0, 0.122
R-RMSE=11.77%
17
+18%
Predicted Wc kg
+15%
16
16 15
15 14
14
13
13
-15% 12
12
11
11
10 10
16
d 20
20
17
Predicted Mc kg
15
Experimental Mc kg
Experimental Mc kg
c
14
11
12
13
14
15
16
17
18
19
-18%
10 10
20
11
12
13
Experimental Mc kg
14
15
16
Experimental Mc kg
Fig. 8 e Effect of Gaussian noise with different level on the prediction performance of ASVM: (a) Without noise; (b) x w N (0, 0.082); (c) x w N (0, 0.102); (d) x w N (0, 0.122).
4.3.
As shown in Fig. 8(b)e(d), the Gaussian noise targets (x) with standard deviations of s (s ¼ 0.08, s ¼ 0.10, s ¼ 0.12) were added to each data point of the validation dataset output. From Fig. 8 it can be seen that the ASVM algorithm developed in this work is robust against Gaussian noise. For a noise level of s ¼ 0.08, with a 95% confidence interval, the results predicted by the ASVM show only a slight decrease (within 10% error band) in accuracy when compared to the results obtained in the absence of noise, with the other statistical metrics such as R-RMSE also indicating good agreement of the predicted results. Even for higher noise levels (s ¼ 0.10, s ¼ 0.12), the predicted performance is still good (15% and 18%).
Method validation
By inputting a group of parameters (e.g. q, ha, D, etc in the set of inequalities (36)), the corresponding value of defrosting water quantity could be obtained from the trained ASVM. If all the values of Mc, each one corresponding to a group of the controlled parameters from within the value ranges in inequality (36), are calculated from the trained ASVM, the minimum Mc can be obtained from the results pool, and also come the corresponding parameters to be employed as an improved condition for design or further experimental validation. However, the least interval of each controlled parameter in the set of inequalities (36) should be set according to the measuring accuracy of temperature and humidity (e.g.
Table 2 e Compare of defrosting water quantity, TEC/TDA and controlled parameters. Parameters Original cabinet Current cabinet
(Teo Tse) (K)
(Ta Ts) (K)
(Ta Tr) (K)
q ( )
D (mm)
ha (%)
Mc (kg)
TEC/TDA (kWh m2 day1)
7 5
28 27
17 20
0 7
3 6
60 40
18.75 11.40
11.89 9.97
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referring to EN ISO 23953-2, the required measuring accuracy of ambient temperature is 1 C; the required measuring accuracy of ambient relative humidity is 5%), and also the minimum machining dimension of q and D, etc. Otherwise, there is no practical significance of the results of Mc. Here, the least interval of (Teo Tse), (Ta Ts), (Ta Tr), q, D and ha are 0.1 (the measuring accuracy of Teo and Tse are both 0.1 C), 1.0, 1.0, 1.0, 1.0 and 5, respectively. After the group of parameters corresponding to the minimum defrosting water quantity was used to improve a display cabinet, a verification test of the defrosting water quantity and energy consumption was carried out using the M1 standard under an ambient temperature of 25 C. Referring to the EN ISO 23953-2 (2005) standard, the performance of the display cabinet was evaluated using the daily energy consumption per m2 display area which was determined as follows: TEC=TDA ¼ ðREC þ DECÞ=TDA
(37)
where TEC is the total energy consumption in kilowatt hours per day, TDA is the total display area, REC is the refrigeration electrical energy consumption in kilowatt hours per day, and DEC is the direct energy consumption in kilowatt hours per day (including fan, lighting and heaters, etc.). As shown in Table 2, the experimental air leakage quantity from the current display cabinet is better than the original display cabinet. When the following control parameters were used (air supply temperature 2 C and air return temperature 5 C), the defrosting water quantity and the TEC/TDA for the current display cabinet were observed to reach 11.40 kg and 9.97 kWh m2 day1, which is a decrease of 39.2% and 19.3%, respectively. Moreover, the defrosting water quantity and the TEC/TDA reached 15.88 kg and 10.78 kWh m2 day1 when the relative humidity was set to 60 % on the basis of current display cabinet. Because the training sets of ASVM are obtained from the MTF model, the relative error of the prediction strategy between the predicted (Mc ¼ 9.61 kg) and measured (Mc ¼ 11.4 kg) defrosting water quantity is 15.7% owing to the accumulative error both of the MTF (w8%) and ASVM (w5%). Though the high accuracy could be achieved if ASVM is trained by the measured data, it would consume a large amount of experimental time and resources.
This study also revealed that the prediction strategy using the MTF model and ASVM algorithm has a high degree of accuracy and robustness against noise. It was found that these models can be used by industry to design effective and energy-saving prediction strategies for the manufacture display cabinets.
Acknowledgments The study was supported by Natural Science Foundation of China (Grant No. 50876059).
Appendix 1 Volume fraction: d1 ¼
V1 V2 ; d2 ¼ V1 þ V2 V1 þ V2
(38)
Mass fraction: dm1 ¼
r1 V1 V1 r2 V2 V2 ¼ ; dm2 ¼ ¼ r1 V1 þ r2 V2 V1 þ r2 V2 r1 V1 þ r2 V2 r1 V1 þ V2 r1 r2 (39)
By Eq. (39) dividing by Eq. (38), Eq. (40) is obtained: d2 1þ 1 þ VV21 dm1 V1 þ V2 d1 ¼ ¼ ¼ ; r d1 V1 þ 2 V2 1 þ r2 V2 1 þ r2 d2 r1 r1 d1 r1 V1 (40) V1 d1 þ1 þ1 dm2 V1 þ V2 V2 d ¼r ¼ ¼ 2 1 d2 V1 þ V2 r1 V1 þ 1 r1 d1 þ 1 r2 r2 d2 r2 V2 According to Eq. (40), the mass fraction can be given as follows: d1 þ d2 1 dm1 ¼ ¼ ; d1 þ d2 ðr2 =r1 Þ d1 þ d2 ðr2 =r1 Þ (41) d1 þ d2 1 dm2 ¼ ¼ d1 ðr1 =r2 Þ þ d2 d1 ðr1 =r2 Þ þ d2
Appendix 2 5.
Conclusions
An effective strategy has been developed to predict the performance of open vertical refrigerated display cabinets. In order to deal with the nonlinear relationship between control parameters, the MTF model and ASVM algorithm are introduced to establish the objective function for the defrosting water quantity. The use of the MTF model and AVSM algorithm was found to be an effective method to avoid the large amounts of time and cost required to solve these prediction problem through the use of conventional design methods. Verification and evaluation of the prediction strategy was then carried out using experimental data. The results obtained by the experiments show that the defrosting water quantity and energy consumption of the predicted model were, respectively, decreased by 39.2% and 19.3%.
For the observation sample set of xi ˛Rn ; yi ˛R; i ¼ 1; 2; .; l; there are some derived properties of an SVM which are expanded upon below. One of these can be presented as the following optimization problem: !# l 1 1X 2 x þ xi ; and kuk þ C ye þ min l i¼1 i x;xi ˛Rl ;e;b˛R 2 ððu xi Þ þ bÞ yi e þ xi ; yi ððu xi Þ þ bÞ e þ xi ; xi ; xi 0; e 0 (42) Pl
i¼1 ðxi þ xi Þ
where ye þ 1=l is the data approximation error (ERM), 1=2kuk2 is the regularization term (confidence interval) y 0 is a constant, C is the penalty coefficient, xi and xi are both the relaxation factors, and e is the loss coefficient (which is a pre-set parameter near zero (qeski, 2002)) which controls the
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 4 1 3 e1 4 2 4
width of the e-intensive zone used to fit the training data. By controlling the values of the ERM and confidence interval, the above optimization was carried out using the SRM principle to minimize the risk in the given function set with a nested structure. Such an optimization defines a compromise between the accuracy of a given data set and the complexity of the approximating function using the SRM principle. The penalty coefficient C represents equilibrium between the two terms. The Lagrange function of Eq. (42) is as follows: l l X 1 CX xi þ xi be hi xi þ hi xi L ¼ kuk2 þ Cye þ 2 l i¼1 i¼1 l X ai xi þ yi uT Fðxi Þ b þ e
i¼1 l X
ai xi þ uT Fðxi Þ þ b yi þ e
ð43Þ
i¼1
From optimization theory, the partial derivatives of the Lagrange function for u; b; e; xi ; xi are equal to zero and the condition for an optimal solution can be shown to be as follows: 8 > P P > > u ¼ li¼1 ai ai Fðxi Þ; li¼1 ai ai ¼ 0 > > < P (44) Cy li¼1 ai ai b ¼ 0 > > > C C > > : ai hi ¼ 0; ai hi ¼ 0 l l Here, the Gaussian RBF kernel function is employed as it yields better prediction performance. Kðx; xi Þ ¼ exp
kx xi k2 c2
! (45)
Using the principal of duality and this kernel function, the optimized dual solution has the following form:
P P min Wða; a Þ ¼ li¼1 lj¼1 ai ai aj aj K xi ; xj a;a ˛Rl
Pl i¼1
P li¼1 ai ai yi ; and ai ai ¼ 0; 0 ai ; ai Cl; i; j ¼ 1; 2; ,,,; l:
ð46Þ
After solving the convex quadratic optimization problem the result then becomes: f ðxÞ ¼
l X
ai ai K xi ; xj þ b
(47)
i¼1
When am and an , the vectors of a and a*, are chosen in the open interval (0, C/l ), b can be computed by taking into account Eq. (42).
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journal homepage: www.elsevier.com/locate/ijrefrig
A novel strategy for predicting the performance of open vertical refrigerated display cabinets based on the MTF model and ASVM algorithm Zhikun Cao a,*, Bo Gu a,**, Graham Mills b, Hua Han a a
Institute of Refrigeration and Cryogenics, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dong Chuan Road, Min Hang, Shanghai 200240, PR China b Environ Australia Pty Ltd, 100 Pacific Highway, North Sydney, NSW, 2060, Australia
article info
abstract
Article history:
This paper presents a novel strategy for predicting the performance of open vertical refrig-
Received 19 November 2009
erated display cabinets which is based on a modified two-fluid (MTF) model and an adaptive
Received in revised form
support vector machine (ASVM) algorithm. A MTF model (physical model) was built for open
5 February 2010
vertical refrigerated display cabinets, and then an ASVM algorithm (machine learning
Accepted 5 April 2010
algorithm) was built. To verify the quantity of air leakage from the cabinet during operation,
Available online 9 April 2010
an important performance factor of display cabinets, an MTF model was built. After the training and validation data sets were constructed from the output of the MTF model, the
Keywords:
problem was solved using an ASVM algorithm. The defrosting water quantity and total
Display cabinet
energy consumption / total display area (TEC/TDA), achieved from the experiments by using
Open cabinet
the predicted combination of the controlled parameters, were found to be reduced by 39.2%
Prediction
and 19.3%, respectively, from the experimental results of the original display cabinet.
Modelling
ª 2010 Elsevier Ltd and IIR. All rights reserved.
Flow Air Leakage Simulation
Strate´gie innovante pour pre´voir la performance des meubles de vente frigorifiques fonde´e sur un mode`le a` deux fluides modifie´s et un algorithme Mots cle´s : Meuble de vente ; Meuble ouvert ; Pre´vision ; Mode´lisation ; E´coulement ; Air ; Fuite ; Simulation
* Corresponding author. Tel.:þ86 34 20 62 60; fax: þ86 21 34 20 68 14. ** Corresponding author. E-mail addresses: [email protected] (Z. Cao), [email protected] (B. Gu). 0140-7007/$ e see front matter ª 2010 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2010.04.006
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i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 4 1 3 e1 4 2 4
Nomenclature actual values ai ASVM adaptive support vector machine C penalty coefficient C1, C2, Cm empirical constants in the turbulent energy equations Cm1, Cm2 empirical constants in the mass transfer rate equations empirical constants in the momentum and heat Cf, Ct transfer equations specific heat at constant pressure (J kg1 K1) Cp d dimension D mesh diameter (mm) DEC direct energy consumption (kWh day1) ERM empirical risk minimization f fluid friction force per unit volume (N m3 s1) h Prandtl mixing length (m) ambient relative humidity (%) ha i the enthalpy value (kJ kg1) H surface heat transfer (J m3 s1) J production rate of turbulent energy (kgm1 s1) k turbulent kinetic energy (m2 s2) m fluid mass transfer rate per unit volume (kg m3 s1) the defrosting water quantity (kg) Mc n number of the dataset predicted values pi Q calorific value (W) R reversal counting RBF radial basis function REC refrigeration electrical energy consumption (kWh day1) MTF modified two-fluid the effective air supply area (m2) Se the total air supply area of orifice plates (m2) St
1.
Introduction
Refrigerated display cabinets play an important role in the cold food chain as they are the final stage and face the consumer directly at the time of purchase. Due to their high degree of exposure to the ambient environment, refrigerated display cabinets can consume an enormous amount of energy. According to statistics published by the Electric Power Research Institute, refrigeration contributes about 60% of the total energy consumed by a supermarket (Evans et al., 2007). Open vertical display cabinets, despite being high consumers of energy are a common type of cabinet which makes it possible to display large amounts of food in a small area. Therefore, the open vertical cabinet is probably one of the most important components in the supermarket system even though cabinets with permanent glass doors can reduce energy consumption by 50% (Adams, 1985). Moreover, the quantity of air leakage is the key factor affecting the performance of display cabinets. Many studies show that because the only barrier between the refrigerated load and the ambient air
SRM T TDA TEC ! U V
structured risk minimization temperature ( C) total display area (m2) total energy consumption (kWh day1) velocity vector (m s1) fluid volume (m3)
Greek symbols g secondary relaxation factor d volume fraction mass fraction dm e loss coefficient h scale factor m dynamic viscosity (kg m1 s1) effective dynamic viscosity (kg m1 s1) me turbulent dynamic viscosity (kg m1 s1) mt q angle of air supply outlet ( ) l convergence coefficient s standard deviation r density (kg m3) c free parameter (the bandwidth of RBF kernel) u coefficient sk, se, sT, sr turbulent Prandtl number relaxation factors xi ; xi z dissipation rate of turbulent energy (m2 s3) Subscripts 1 fluid 1 (the turbulent air) 2 fluid 2 (the non-turbulent air) a ambient cabinet inlet cin cabinet outlet cout eo evaporator outlet i, j serial numbers r air return s air supply se saturated evaporating
is one or more forced-air curtains, there may be a significant entrainment of ambient wet hot air. Such operating conditions also lead to poor food temperature control, the formation of dew or frost on the surface of the evaporator and food in the cabinets as well as significantly increased energy consumption (Getu and Bansal, 2007; Navaz et al., 2005; D’Agaro et al., 2006). Although many studies have been conducted on the performance of display cabinets (Howell, 1993; Brolls, 1986; Axell and Fahle´n, 1995, 1998, 2003; Billiard and Gautherin, 1993; Chandrasekharan et al., 2006; Cortella, 2002), most of them only emphasize a few aspects by performing experiments or computational fluid dynamics simulations. As a result, few studies have presented systematic strategies for performance prediction by investigating the relationship between multiple parameters. This paper is an attempt to fill this gap. In particular, we investigate strategies for the predicting the performance of open vertical refrigerated display cabinets using the MTF model and ASVM algorithm. The two-fluid (TF) model, proposed by Spalding (Spalding, 1984), can be summarized as follows:
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 4 1 3 e1 4 2 4
(1) Turbulent flow is considered as comprehensive motion for the interaction between each fluid; (2) There is a respective volume fraction for the two fluids, i.e., the two fluids exist respectively with certain probability; (3) The two fluids, whose motions respectively follow their controlled differential equations, entrain each other as a continuum; (4) The dividing characteristics of the two fluids are the concentration, temperature, flow direction or volume. Recently, the TF model has become a good method for modeling fluid dynamics applications, such as free jets (Ilegbusi and Spalding, 1987a), flat plate flows (Ilegbusi and Spalding, 1989), conduit flows (Ilegbusi and Spalding, 1987a), near-wall flows (Ilegbusi and Spalding, 1987b), Couette flows (Spalding, 1987), stratified flows (Shen et al., 2003), tundish problems (Sheng and Jonsson, 2000) and air entrainment (Yu et al., 2008; Cao et al., 2010a). In these cases, the predicted results are in good agreement with the experimental results. The Support vector machine (SVM) is a very promising technique developed by Vapnik and his group at AT&T Bell Labs (Vapnik, 1995; Cortes and Vapnik, 1995; Boser et al., 1992; Burges, 1996). This new machine learning algorithm can be seen as an alternative training technique for Polynomial, Radial Basis Function and Multi-Layer Perceptron classifiers. Unlike traditional artificial neural algorithms (Yigit and Ertunc, 2006), which use the empirical risk minimization (ERM) principle, SVM algorithm uses an approximate implementation of the structured risk minimization (SRM) induction principle which demands fewer experimental samples to obtain a good level of performance reducing experimental time and cost (Vapnik and Chervonenkis, 1991). Training a SVM is equivalent to solve a quadratic programming problem with linear and box constraints in a number of variables equal to the number of data points (Vapnik, 1998). Moreover, SVM is able to learn good separating hyperplanes in high dimensional feature space (Cristianini and Shawe, 2000). Rather than estimating parameter coefficients from variables contained in the data matrix, SVM estimates parameter coefficients by using the observations. It is therefore possible for a SVM’s feature space to be equivalent to the number of observations used in training the SVM. Reducing the number of estimated coefficients is referred to as sparseness and is often achieved by optimization techniques. It is by controlling the hyperplane margin measures through generalization theory that SVM rarely overfit the data and achieves sparseness. Therefore, SVM is created by applying optimization theory, usually with Lagrange multipliers or KarusheKuhneTucker conditions, to the margins. Moreover, the introduction of Vapnik’s e-insensitive loss function and kernel function has extended SVM to solve problems involving nonlinear regression estimation and noise characterization. Based on the statistical learning theory, the SVM which is a novel method with good theoretical properties, has been developed and widely used recently in the HVAC and refrigeration field (Liang and Du, 2007; Esen et al., 2008; Ren et al., 2008; Cao et al., 2009, 2010b; Li et al., 2009). To predict the performance of the refrigerated display cabinets, A MTF model (physical model) was built for open
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vertical refrigerated display cabinets, and then an ASVM algorithm (machine learning algorithm) was built.
2.
MTF model and ASVM algorithm
2.1.
TF model
As shown in Fig. 1, the cooling air supplied by the air curtain and rear grilles is defined as fluid 1 (the turbulent fluid), and the ambient air is defined as fluid 2 (the non-turbulent fluid). Based on the initial two-fluid model (Ilegbusi, 1994), the TF model of a display cabinet without phase diffusion is described as follows. Energy equations: v v m m vT1 v m vd1 ! d1 t þ þ T1 e ðd1 r1 U 1b T1 Þ ¼ sT Pr vxb sT vxb vxb vxb vxb H12 v v m vd2 H12 ð1Þ ! þvT2 m þm vT2 d2 Cp þ T2 e T2 m ðd2 r2 U 2b T2 Þ ¼ sT vxb Cp vxb vxb Pr vxb vxb (2) Turbulent kinetic energy and dissipation rate of the turbulent energy equations: v v m vk v m vd1 ! d1 t þ m þ k e þ d1 ðJ r1 zÞ ðd1 r1 U 1a kÞ ¼ sk vxb vxb vxb vxb sr vxb (3) v v m vz v m vd1 ! d1 t þ m þ e e ðd1 r1 U 1a zÞ ¼ se vxb vxb vxa vxa sr vxa z þ d1 ðC1 J C2 r1 zÞ k
ð4Þ
where J, the rate of production of turbulent energy, is given by
J ¼ mt
! ! ! v U 1a v U 1b v U 1a þ vxb vxa vxb
(5)
Momentum equations: ! ! v vp v v U 1a v U 1b ! ! d 1 me ðd1 r1 U 1a U 1b Þ ¼ d1 þ þ vxb vxa vxb vxa vxb v ! me vd1 U 1a þ d1 ðr1 r2 Þga þ sr vxb vxb ! þ U 2a m þ f12 ! ! v vp v v U 2a v U 2b ! ! d2 m ðd2 r2 U 2a U 2b Þ ¼ d2 þ þ vxb vxa vxb vxa vxb v ! me vd2 ! U 2a U 2a m f12 þ sr vxb vxb
ð6Þ
(7)
Continuity equations: v v me vd1 ! þm ðd1 r1 U 1b Þ ¼ vxb vxb sr vxb v v m vd2 ! m ðd2 r2 U 2b Þ ¼ vxb vxb sr vxb
(8)
(9)
The constraint of the volume fraction: d 1 þ d2 ¼ 1
(10)
where the subscript 1 or 2 denotes fluid 1 or fluid 2; the ! subscripts a and b denote the coordinate system (Fig. 1); U denotes a velocity vector; Cp denotes specific heat at
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i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 4 1 3 e1 4 2 4
Model 1, Eq. (16) Model 2, Eq. (17) MTF model (ρ / ρ =2), Eq. (18)
0.5 0.4
MTF model (ρ / ρ =1.5), Eq. (18)
0.3
Model 1 neglected m from fluid 1 to fluid 2
m/ ρ U -U /h
0.2 0.1 0.0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-0.1
0.9
1.0
δ
-0.2 -0.3 -0.4 -0.5
Model 2 over predicted m from fluid 1 to fluid 2
Fig. 2 e Results of different mass transfer rate equations.
2.2.
Fig. 1 e Open vertical refrigerated display cabinet.
constant pressure; d is the volume fraction of the fluid, given by d ¼ Se/St (Yu et al., 2008); Se denotes the effective air supply area; St denotes the total air supply area of the orifice plates. The turbulent dynamic viscosity : mt ¼ Cm rk2 =z
(11)
The effective dynamic viscosity : me ¼ mt þ m
(12)
The surface heat transfer : ! ! H12 ¼ Ct Cp r1 d1 d2 ðT2 T1 Þj U 2b U 1b j=h
ð13Þ
The inter-fluid friction force per unit volume : ! ! ! ! f12 ¼ Cf r1 d1 d2 ð U 2b U 1b Þj U 2b U 1b j=h
ð14Þ
The Prandtl mixing length : h ¼ cm3=4 k3=2 =z
(15)
Equations for the mass transfer rate between the two fluids (Malin and Spalding, 1984): ! ! m ¼ Cm1 r1 d1 d2 j U 2b U 1b j=h (16) and (Sheng and Jonsson, 2000), ! ! m ¼ Cm2 r1 d1 d2 ðd2 0:5Þj U 2b U 1b j=h
Since the TF model evaluates the air leakage quantity from a refrigerated display cabinet from the defrosting water quantity, in order to increase the accuracy of the results it is necessary to improve the two mass transfer rate equations in the original TF model given by Eqs. (16) and (17). From a physical viewpoint, the non-turbulent fluid (fluid 2) is entrained by the turbulent fluid (fluid 1) while a small quantity of the turbulent fluid is entrained by the nonturbulent fluid. Fig. 2 shows the relationship between the ! ! dimensionless number m=ðr1 j U 2b U 1b j=hÞ and different values of d1 calculated from a number of different models. Since m is always positive in Eq. (16), model 1 cannot be used to investigate the entrainment of fluid 2 by fluid 1. Moreover, because of the added control factor (d2 ¼ 0.5) in Eq. (17), only an equivalent fluid 2 can be entrained by fluid 1. Therefore, both model 1 and model 2 have limitations. Yu et al. (2008) obtained better results using a modified TF model for open vertical display cabinet based on the weighted volume fraction factor. However, practical operation of display cabinets usually involves fluid 2 being hot wet air whose density is lower than that of fluid 1. As a result, the influence of air density cannot be neglected in the mass transfer rate equations. In this study, the TF model was modified using a weighted mass fraction factor in developing the MTF model as described below: (See Appendix 1 for detailed derivation):
(17)
The empirical constants employed in the TF model are listed in Table 1 (Shen et al., 2003; Sheng and Jonsson, 2000).
Table 1 e The values of the constants employed in the TF model. Cm1
Cm2
Cm
sk
se
sT
sr
C1
C2
Cf
Ct
0.1
10.0
0.09
1.0
1.3
1.0
0.95
1.44
1.92
0.05
0.05
MTF model
m¼
! ! Cm1 r1 d1 d2 j U 2b U 1b j 1 h d1 þ d2 ðr2 =r1 Þ ! ! Cm2 r1 d1 d2 ðd2 0:5Þj U 2b U 1b j 1 þ d1 ðr1 =r2 Þ þ d2 h
(18)
As shown in Fig. 2 (For convenient comparison, it is assumed that r1/r2 ¼ 2, r1/r2 ¼ 1.5) the MTF model shows that in the mutual entrainment of the two fluids the entrainment rate of fluid 1 by fluid 2 is less than that of fluid 2 by fluid 1, and that the entrainment rate changes with density which corresponds to expectations from practical operation.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 4 1 3 e1 4 2 4
2.3.
SVM algorithm
where,
For regression problems, SVM ensures the model can be applied generally by fitting an observation sample set using functionf ðxÞ ¼ uT Fðxi Þ þ b. Specially, F(x) is the nonlinear map from the input space to the higher-dimensional feature space, with f(x) representing the resulting linear function in that feature space. Here, the y-SVM regression (Scho¨lkopf et al., 2000; Wang et al., 2005; Chen et al., 2006) which yields simple form and good performance is employed in this paper. In addition, the SVM maps the observed sample data from the nonlinear problem into a higher-dimensional feature space. Using Mercer’s theorem, when the process of solving the optimized dual problem in the higher-dimensional feature space only referred to the inner-product operation, there was one function, Kðx; xi Þ ¼ 4ðxÞ$4ðxi Þ, which just equaled the inner product of the nonlinear transform in this higher-dimensional feature space. By using the inner product directly derived by K (x,xi), the calculation can be greatly simplified without the use of a complex nonlinear transform (Na and Webb, 2004). K(x,xi) therefore becomes the kernel function by satisfying these conditions. Using this kernel function, the nonlinear problem in the lower dimensional space is transformed into a linear problem in the higher-dimensional feature space. Here, the Gaussian radial basis function (RBF) kernel function is employed as it yields better prediction performance (Lin and Lin, 2004). ! kx xi k2 (19) Kðx; xi Þ ¼ exp c2 After solving the convex quadratic optimization problem the result then becomes (see Appendix 2 for detailed derivation): f ðxÞ ¼
l X
ai ai K xi ; xj þ b
(20)
i¼1
The learning machine constructed by Eq. (20) is simply the y-SVM regression where the support vectors are the vectors which make the coefficients in Eq. (20) non-zero. These support vectors make up the training sample with coefficients which are non-zero. By ignoring the fitting errors which are less than e, the e-insensitive loss function possesses some anti-noise ability. This can be shown as follows: Lðy; f ðxÞÞ ¼ L y f ðxÞje
(21)
jy f ðxÞje ¼
2.4.
0 jy f ðxÞj e jy f ðxÞj e otherwise
(22)
The adaptation of ASVM algorithm
Using the loss coefficient e-insensitive loss function from Eq. (21), the SVM regression is constructed. In most circumstances e is a value which does not change when the sample is determined. However, when the level of interference is changed, or the alteratione ¼ jy f ðxÞj exceeds a particular value, the antinoise ability of the insensitive loss function (Eq. (21)), decreases or even disappears because e loses its ability to judge as the threshold. To avoid this and enhance the adaptive ability of the SVM eis adjusted dynamically in this research. As shown in Fig. 3, reversal counts are used as a basic control index in the univariate statistical control chart, where E and s are the expectation and the standard deviation of the noise inherent in a process variable. When the process variable first exceeds the threshold as illustrated in Fig. 3, the reversal counting starts (R ¼ 1), and increments (R ¼ R þ 1) every time the process variable exceeds the threshold in either direction. Usually, the maximum tolerable number of reversals in both directions is chosen to be 4, with the maximum tolerable being 2 in the same direction (Salsbury, 1999). Due to this, the dynamic control equations of e can be described as follows: eiþ1 ¼ ð1 þ hÞei ; when R ¼ 1; 3ðmaximum tolerable number of reversals in the yþ direction ; or R ¼ 2; 4 maximum tolerable numberof reversals in the y direction ; or R ¼ 1; 2; 3; 4
(23)
where h is the scale factor whose value can chosen as follows: h ¼ ðei ei Þ=emax
(24)
Due to Eq. (24) being a unilateral contact function, convergence is ensured through the introduction of Eq. (25). eiþ1 ¼ ð1 gÞei ; when ei < gEi1 ; eiþ1 < gEi
(25)
where g is the secondary relaxation factor 0 < g < 1 which can effect the final degree of convergence. In addition, when the value of e ¼ jy f ðxÞj is in the interval excluding the constraint conditions of Eqs. (23) and (25) e then becomes: eiþ1 ¼ ei
2.5.
Fig. 3 e Illustration of reversal counts.
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(26)
Controlled parameters determination
Before the application of the strategy, the controlled parameters corresponding to the performance of the refrigerated display cabinet should be confirmed. The choice of data to be collected should ensure that the algorithm is easily applied to the development of commercial display cabinet products.
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i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 4 1 3 e1 4 2 4
For refrigerant cycle: REC ¼ Qtot =COP
(27)
where Qtot is the total heat extraction, kWh/day Qtot ¼
N max X
Fn $Dt
(28)
n¼1
where Fn is the transient heat extraction, kW; Dt is the time; h; Fn ¼ 0 kW during stopping and defrost time. For a remote compression-type refrigerating system, COP is calculated from the following equation (in Section 5.3.6.3.3 of EN ISO 23953-2, 2005): COP ¼
H$Tse Tc Tse
(29)
where Tc is the condensing temperature, K; H is the coefficient; H ¼ 0.34, when Tc ¼ 308.15. The heat extraction rate in kilowatts is defined as Fn ¼ qm $ðicout icin Þ
(30)
where qm is the mass flow rate of refrigerant; icin, icout is the enthalpy value of refrigerant at cabinet inlet and outlet, respectively. For each individual measuring instant, where n indicates the measuring sample (Fn ¼ 0 kW during stopping and defrost time). The refrigeration electrical energy consumption for a cabinet intended for a remote compression-type refrigerating system, REC is calculated from the following equation: REC ¼ Qtot $
Tc Tse Tc Tse ¼ 24 tdeft $F24deft $ H$Tse H$Tse
(31)
where tdeft is the defrosting time, h. Therefore, the saturated vapour temperature Tse is chosen, which represents the refrigerating coefficient of performance COP; the refrigerant temperature of evaporator outlet Teo is chosen, which represents the cooling capacity of display cabinet (the total heat extraction Qtot) with the refrigerant temperature of display cabinet inlet Tin and the refrigerant flow qm (saturated liquid is assumed). Because the destination is predict the performance of display cabinets, the parameters Tse and Teo which affects TEC/TDA of display cabinets should be involved. For the air curtain cycle: The heat extraction: Q ¼ Gðia ic Þ
(32)
where G is the infiltration air mass through air curtain; ia is the
enthalpy value of ambient air; ic is the enthalpy value of air curtain. Therefore, the ambient temperature Ta, the relative humidity ha (saturation is assumed), the air supply temperature Ts and the air return temperature Tr which represent the enthalpy difference between the ambient air and air curtain are chosen. Also, the angle of air supply outlet q and the mesh diameter of the honey comb structured air supply outlet D which represent G are chosen.
2.6.
ASVM algorithm architecture
The architecture of the ASVM algorithm, applied to the refrigerated display cabinet system, is shown in Fig. 4. The vectors input to the ASVM model are the attributes which influence the quantity of air leakage. These include evaporator outlet superheat (Teo Tse), the difference between the ambient air temperature and supply air temperature (Ta Ts), the difference between the ambient air temperature and the return air temperature (Ta Tr), the angle of air supply outlet q, the diameter of the honey comb structured air supply outlet mesh D, and the ambient relative humidity ha. The output vector is the defrosting water quantity Mc. Using the Gaussian RBF kernel function the ASVM model can achieve multilayer perception including hidden layers.
2.7.
ASVM algorithm parameter determination
Results obtained by Cristianini and Shawe (2000), Kwok (2000), and Van et al. (2001) in their work on SVM evidence frameworks show that convergence precision can be improved through reasonable selection of AVSM parameters. The model convergence precision l can be presented as follows. l ¼ jðR RMSEiþ1 Þ ðR RMSEi Þj
where the relative root mean square error (RMSE) is shown as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pn i¼1 ai pi =ai R RMSE ¼ 100% (34) n where n is the number of the dataset, and ai and pi are the actual values and predicted values, respectively. The values of l vary with different g values in the loss function. The results obtained from the training data show that when 0.05 < g < 0.2, there is a high degree of convergence precision. In this study, g was selected to be 0.13. For multivariate d-dimensional problems, the RBF width parameter c is set so that cd ˛ð0:1; 0:5Þ. Cherkassky and Ma (2004) showed that such values yield good SVM performance for a range of various regression data sets. Here, d ¼ 6 and c ¼ 0.85. Moreover, Cherkassky and Ma (2004) also showed the penalty coefficient C obtained from the training data then becomes: C ¼ max y þ 3sy j; y 3sy
Fig. 4 e Algorithm architecture of ASVM of refrigerated display cabinets.
(33)
(35)
wherey and sy are the mean and the standard deviation of the training data y values. Scho¨lkopf et al. (1998) showed that the y-SVM algorithm can automatically select e by the priori chosen y. Wheny˛½0; 1,
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 4 1 3 e1 4 2 4
Dataset calculated from MTF model
Scaling
Testing set
Initializing ε
Training set
New ε Training ASVM algorithm
Accuracy control
ε adaptive adjustment
Reversal counts judgment criteria
N
Y Trained ASVM with optimized ε
ASVM output
Fig. 5 e Flow chart of MTF-ASVM prediction strategy for refrigerated display cabinets. the training error displays an increasing trend as y values increase. The evalue corresponding to y ¼ 0.2 was selected as the initial e value.
2.8. Flow chart of the prediction strategy based on MTF-ASVM As shown in Fig. 5, the defrosting water quantity for the display cabinet was calculated using an MTF model to save a significant amount of experimental time and cost. After scaling the input control parameters corresponding to different defrosting water quantities, the defrosting water quantity and other controlled parameters were set as the output vector and the input vectors of the training and validation datasets of the ASVM algorithm. Using the adaptive e, each team of input vectors produces one output from the trained ASVM algorithm. In addition, as machine learning is based on statistical learning theory the ASVM algorithm has a significant calculation advantage over the MTF model which based on a physical model. This saves further time and cost. Finally, the input parameters corresponding to the minimum defrosting water quantity that meets requirements are used as the predicted parameters which realize the objective of reducing the quantity of air leakage from vertical refrigerated display cabinets.
3.
Methodology
3.1.
Experimental facilities/display cabinet
The object of this investigation is a remote open vertical refrigerated display unit with a single chilled air curtain (Fig. 1). The dimensional features are: length 2.5 m, height 2 m,
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depth 1.1 m with a display opening height of 1.65 m. This display cabinet system consists of three loops: a refrigerant loop, a condensing air loop and an evaporating air curtain loop. The refrigerant loop is a single-stage vapour compression plant, using R22 as the working fluid, the fundamental components of which are as follows: a reciprocating open type compressor, a fin-tube evaporator, a thermostatic expansion valve, and a fin-tube condenser. As shown in Fig. 6, the experiments were carried out in a test chamber with length 8 m, height 3.5 m and depth 6 m. In particular, the air in the test chamber moved parallel to the longitudinal axis of the cabinet at uniform velocity between the range of 0.1 and 0.2 ms1. There is the special HVAC system (including the humidifier) for the test chamber. Referring to the EN ISO 23953-2, the test chamber should supply the air moving parallel to the longitudinal axis of the cabinet (with the velocity of 0.1 and 0.2 ms1) which makes the test chamber uniformly achieve the required ambient air. T-type thermocouples (coppereconstantan) with an accuracy of 0.1 C and piezoelectric pressure gauges with an accuracy of 0.1% of the full-scale range were used to measure the temperature and pressure, respectively. The T-type thermocouples were mounted in the display cabinet which was loaded with test packages, some of which are M-packages. All these measurements were collected by an HMAM 25203 lab data acquisition system at intervals of 60 s for a period of 12 h. Moreover, the defrosting water quantity is weighted by the compute a charge scale (CPS CC220) with an accuracy of 0.015% of the full scale range and a range of 0e100 kg.
3.2.
Experimental design
Referring to the EN ISO 23953-2 (2005) standard, the air leakage quantity was tested by the defrosting water quantity. The factors which significantly influence the defrosting water quantity are summarized as follows: air supply temperature Ts, air return temperature Tr, ambient temperature Ta, saturated evaporating temperature Tse, evaporator outlet temperature Teo, angle of air supply outlet q, the mesh diameter of the honey comb structured air supply outlet D, and ambient relative humidity ha. Many authors follow the approach of Rigot (Rigot, 1990) to define an “infiltration ratio” which describes the amount of air leakage and is based on a simple energy balance from the knowledge of Ts, Ta and Tr. However, the parameters Ts, Ta and Tr which could be directly measured, were selected here in order to simplify the calculation. Experience from the process of developing medium temperature display cabinets (Getu and Bansal, 2006; Navaz et al., 2005; Chen and Yuan, 2004) and the actual situation of the experimental display cabinet, the range of design parameters values mentioned above are as follows. 3 ðTeo Tse Þ 10; 14 ðTa Ts Þ 31; 4 ðTa Tr Þ 22; ð36Þ 0 q 10; 3 D 8; 0:4 ha 0:8 Referring to the M1 standard (the temperature inside cabinet is between 1 C and 5 C) (EN ISO 23953-2, 2005), 60 groups of control parameters were randomly selected as the experimental operating conditions for the air leakage experiments with an ambient temperature of 25 C in the test
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i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 4 1 3 e1 4 2 4
Fig. 6 e Experimental facility of refrigerated display cabinet test.
4.
Results and discussion
4.1.
MTF model e results and discussion
This section summarizes the results obtained using the MTF model to simulate the performance of a display cabinet. The objective of this work was to verify the effectiveness of the MTF model in predicting the performance of these cabinets. In order to do this, the results obtained using the MTF model were compared with the results from Yu’s model (Yu et al., 2008). As shown in Fig. 7, it can clearly be seen that the predictive accuracy of the MTF model is better than that of the Yu’s model. Generally, with a 95% confidence interval, the discrepancy between the defrosting water quantity predicted by the model and experimental results is less than 8%. By comparison, the error band of Yu’s model is 15%. In addition, compared with 12 h of experimental test time required for each measurement period, the MTF model only needs 0.5 h to measure the same conditions, saving experimental time and cost in producing higher accuracy results.
selected as the training dataset and the validation dataset for the ASVM algorithm. However, as it is very important to cover the entire expected operational range of the display cabinet in training the ASVM model it was ensured that the training dataset used to develop the model covered the maximum and minimum values of the model input and output variables. Using 60 validation datasets Fig. 8(a) shows the high degree of output regressive accuracy produced by the ASVM algorithm. With a 95% confidence interval, the predicted discrepancy of the ASVM algorithm was less than 5% for the defrosting water quantity. In addition, the ASVM algorithm has a convergence speed which is an order of magnitude higher than that of the MTF model using the same computer (PIV 2.66 G CPU, 1 G memory). By adding noise to the dataset, the robustness of the ASVM algorithm was investigated. Since the ASVM approach is not sensitive to a particular noise distribution (Cristianini and Shawe, 2000), Gaussian noise was selected. In the following implementation, the same values for the free parameters (C, c, and initial e) were used when developing the prediction model using the noisy dataset.
20
18
ASVM algorithm e results and discussion
In total 120 defrosting water quantity and controlled parameter datasets obtained from the MTF model were randomly
+15%
17
16
+8%
15
-15% -8%
14
4.2.
MTF model Yu's model
19
Predicted Mc kg
chamber (because the ambient temperature is kept at 25 C, thus leading to supply temperature in the range from 6 to 11 C, and return temperature in the range 3e21 C). After testing was undertaken for a period of defrosting, the upper night curtain (Fig. 6) was pulled down to isolate the air inside the display cabinet from the external environment. Then the defrosting controller was set (the defrosting period and the defrosting time were set to 4 h and 35 min, respectively), and the defrosting water quantity was weighted after the defrosting period. Moreover, each measurement was lasted for 3 defrosting periods. After that, the accuracy of the MTF model was validated by comparing of the experimental defrosting water quantity and that predicated by the model.
14
15
16
17
18
19
Experimental Mc kg
Fig. 7 e Prediction accuracy of MTF model and weighted model based on volume fraction (Yu’ model).
20
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i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 4 1 3 e1 4 2 4
19
Predicted Mc kg
b 20
20
19
R-RMSE=3.93%
18
18
17
17
Predicted Mc kg
a
16
+5% 15
-5%
14 13
15 14 13 12
11
11
11
12
13
14
15
16
17
18
19
+10%
16
12
10 10
ξ~N 0, 0.082 R-RMSE=6.34%
-10%
10 10
20
11
12
13
19
ξ~N 0, 0.102
19
18
R-RMSE=9.97%
18
17
18
19
20
17
18
19
20
ξ~N 0, 0.122
R-RMSE=11.77%
17
+18%
Predicted Wc kg
+15%
16
16 15
15 14
14
13
13
-15% 12
12
11
11
10 10
16
d 20
20
17
Predicted Mc kg
15
Experimental Mc kg
Experimental Mc kg
c
14
11
12
13
14
15
16
17
18
19
-18%
10 10
20
11
12
13
Experimental Mc kg
14
15
16
Experimental Mc kg
Fig. 8 e Effect of Gaussian noise with different level on the prediction performance of ASVM: (a) Without noise; (b) x w N (0, 0.082); (c) x w N (0, 0.102); (d) x w N (0, 0.122).
4.3.
As shown in Fig. 8(b)e(d), the Gaussian noise targets (x) with standard deviations of s (s ¼ 0.08, s ¼ 0.10, s ¼ 0.12) were added to each data point of the validation dataset output. From Fig. 8 it can be seen that the ASVM algorithm developed in this work is robust against Gaussian noise. For a noise level of s ¼ 0.08, with a 95% confidence interval, the results predicted by the ASVM show only a slight decrease (within 10% error band) in accuracy when compared to the results obtained in the absence of noise, with the other statistical metrics such as R-RMSE also indicating good agreement of the predicted results. Even for higher noise levels (s ¼ 0.10, s ¼ 0.12), the predicted performance is still good (15% and 18%).
Method validation
By inputting a group of parameters (e.g. q, ha, D, etc in the set of inequalities (36)), the corresponding value of defrosting water quantity could be obtained from the trained ASVM. If all the values of Mc, each one corresponding to a group of the controlled parameters from within the value ranges in inequality (36), are calculated from the trained ASVM, the minimum Mc can be obtained from the results pool, and also come the corresponding parameters to be employed as an improved condition for design or further experimental validation. However, the least interval of each controlled parameter in the set of inequalities (36) should be set according to the measuring accuracy of temperature and humidity (e.g.
Table 2 e Compare of defrosting water quantity, TEC/TDA and controlled parameters. Parameters Original cabinet Current cabinet
(Teo Tse) (K)
(Ta Ts) (K)
(Ta Tr) (K)
q ( )
D (mm)
ha (%)
Mc (kg)
TEC/TDA (kWh m2 day1)
7 5
28 27
17 20
0 7
3 6
60 40
18.75 11.40
11.89 9.97
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referring to EN ISO 23953-2, the required measuring accuracy of ambient temperature is 1 C; the required measuring accuracy of ambient relative humidity is 5%), and also the minimum machining dimension of q and D, etc. Otherwise, there is no practical significance of the results of Mc. Here, the least interval of (Teo Tse), (Ta Ts), (Ta Tr), q, D and ha are 0.1 (the measuring accuracy of Teo and Tse are both 0.1 C), 1.0, 1.0, 1.0, 1.0 and 5, respectively. After the group of parameters corresponding to the minimum defrosting water quantity was used to improve a display cabinet, a verification test of the defrosting water quantity and energy consumption was carried out using the M1 standard under an ambient temperature of 25 C. Referring to the EN ISO 23953-2 (2005) standard, the performance of the display cabinet was evaluated using the daily energy consumption per m2 display area which was determined as follows: TEC=TDA ¼ ðREC þ DECÞ=TDA
(37)
where TEC is the total energy consumption in kilowatt hours per day, TDA is the total display area, REC is the refrigeration electrical energy consumption in kilowatt hours per day, and DEC is the direct energy consumption in kilowatt hours per day (including fan, lighting and heaters, etc.). As shown in Table 2, the experimental air leakage quantity from the current display cabinet is better than the original display cabinet. When the following control parameters were used (air supply temperature 2 C and air return temperature 5 C), the defrosting water quantity and the TEC/TDA for the current display cabinet were observed to reach 11.40 kg and 9.97 kWh m2 day1, which is a decrease of 39.2% and 19.3%, respectively. Moreover, the defrosting water quantity and the TEC/TDA reached 15.88 kg and 10.78 kWh m2 day1 when the relative humidity was set to 60 % on the basis of current display cabinet. Because the training sets of ASVM are obtained from the MTF model, the relative error of the prediction strategy between the predicted (Mc ¼ 9.61 kg) and measured (Mc ¼ 11.4 kg) defrosting water quantity is 15.7% owing to the accumulative error both of the MTF (w8%) and ASVM (w5%). Though the high accuracy could be achieved if ASVM is trained by the measured data, it would consume a large amount of experimental time and resources.
This study also revealed that the prediction strategy using the MTF model and ASVM algorithm has a high degree of accuracy and robustness against noise. It was found that these models can be used by industry to design effective and energy-saving prediction strategies for the manufacture display cabinets.
Acknowledgments The study was supported by Natural Science Foundation of China (Grant No. 50876059).
Appendix 1 Volume fraction: d1 ¼
V1 V2 ; d2 ¼ V1 þ V2 V1 þ V2
(38)
Mass fraction: dm1 ¼
r1 V1 V1 r2 V2 V2 ¼ ; dm2 ¼ ¼ r1 V1 þ r2 V2 V1 þ r2 V2 r1 V1 þ r2 V2 r1 V1 þ V2 r1 r2 (39)
By Eq. (39) dividing by Eq. (38), Eq. (40) is obtained: d2 1þ 1 þ VV21 dm1 V1 þ V2 d1 ¼ ¼ ¼ ; r d1 V1 þ 2 V2 1 þ r2 V2 1 þ r2 d2 r1 r1 d1 r1 V1 (40) V1 d1 þ1 þ1 dm2 V1 þ V2 V2 d ¼r ¼ ¼ 2 1 d2 V1 þ V2 r1 V1 þ 1 r1 d1 þ 1 r2 r2 d2 r2 V2 According to Eq. (40), the mass fraction can be given as follows: d1 þ d2 1 dm1 ¼ ¼ ; d1 þ d2 ðr2 =r1 Þ d1 þ d2 ðr2 =r1 Þ (41) d1 þ d2 1 dm2 ¼ ¼ d1 ðr1 =r2 Þ þ d2 d1 ðr1 =r2 Þ þ d2
Appendix 2 5.
Conclusions
An effective strategy has been developed to predict the performance of open vertical refrigerated display cabinets. In order to deal with the nonlinear relationship between control parameters, the MTF model and ASVM algorithm are introduced to establish the objective function for the defrosting water quantity. The use of the MTF model and AVSM algorithm was found to be an effective method to avoid the large amounts of time and cost required to solve these prediction problem through the use of conventional design methods. Verification and evaluation of the prediction strategy was then carried out using experimental data. The results obtained by the experiments show that the defrosting water quantity and energy consumption of the predicted model were, respectively, decreased by 39.2% and 19.3%.
For the observation sample set of xi ˛Rn ; yi ˛R; i ¼ 1; 2; .; l; there are some derived properties of an SVM which are expanded upon below. One of these can be presented as the following optimization problem: !# l 1 1X 2 x þ xi ; and kuk þ C ye þ min l i¼1 i x;xi ˛Rl ;e;b˛R 2 ððu xi Þ þ bÞ yi e þ xi ; yi ððu xi Þ þ bÞ e þ xi ; xi ; xi 0; e 0 (42) Pl
i¼1 ðxi þ xi Þ
where ye þ 1=l is the data approximation error (ERM), 1=2kuk2 is the regularization term (confidence interval) y 0 is a constant, C is the penalty coefficient, xi and xi are both the relaxation factors, and e is the loss coefficient (which is a pre-set parameter near zero (qeski, 2002)) which controls the
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 4 1 3 e1 4 2 4
width of the e-intensive zone used to fit the training data. By controlling the values of the ERM and confidence interval, the above optimization was carried out using the SRM principle to minimize the risk in the given function set with a nested structure. Such an optimization defines a compromise between the accuracy of a given data set and the complexity of the approximating function using the SRM principle. The penalty coefficient C represents equilibrium between the two terms. The Lagrange function of Eq. (42) is as follows: l l X 1 CX xi þ xi be hi xi þ hi xi L ¼ kuk2 þ Cye þ 2 l i¼1 i¼1 l X ai xi þ yi uT Fðxi Þ b þ e
i¼1 l X
ai xi þ uT Fðxi Þ þ b yi þ e
ð43Þ
i¼1
From optimization theory, the partial derivatives of the Lagrange function for u; b; e; xi ; xi are equal to zero and the condition for an optimal solution can be shown to be as follows: 8 > P P > > u ¼ li¼1 ai ai Fðxi Þ; li¼1 ai ai ¼ 0 > > < P (44) Cy li¼1 ai ai b ¼ 0 > > > C C > > : ai hi ¼ 0; ai hi ¼ 0 l l Here, the Gaussian RBF kernel function is employed as it yields better prediction performance. Kðx; xi Þ ¼ exp
kx xi k2 c2
! (45)
Using the principal of duality and this kernel function, the optimized dual solution has the following form:
P P min Wða; a Þ ¼ li¼1 lj¼1 ai ai aj aj K xi ; xj a;a ˛Rl
Pl i¼1
P li¼1 ai ai yi ; and ai ai ¼ 0; 0 ai ; ai Cl; i; j ¼ 1; 2; ,,,; l:
ð46Þ
After solving the convex quadratic optimization problem the result then becomes: f ðxÞ ¼
l X
ai ai K xi ; xj þ b
(47)
i¼1
When am and an , the vectors of a and a*, are chosen in the open interval (0, C/l ), b can be computed by taking into account Eq. (42).
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