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Frosting weight and refrigerating capacity prediction of fin evaporator based on random finite element method and ridgelet neural network
Accepted Manuscript
Frosting Weight and Refrigerating Capacity Prediction of Fin Evaporator Based on Random Finite Element Method and Ridgelet Neural Network Bin Zhao , Yi Ren , Diankui Gao , Chengjiang Shi , Lizhi Xu , Yuanyuan Zhang PII: DOI: Reference:
S0140-7007(18)30510-3 https://doi.org/10.1016/j.ijrefrig.2018.11.046 JIJR 4217
To appear in:
International Journal of Refrigeration
Received date: Revised date: Accepted date:
28 July 2018 25 November 2018 30 November 2018
Please cite this article as: Bin Zhao , Yi Ren , Diankui Gao , Chengjiang Shi , Lizhi Xu , Yuanyuan Zhang , Frosting Weight and Refrigerating Capacity Prediction of Fin Evaporator Based on Random Finite Element Method and Ridgelet Neural Network, International Journal of Refrigeration (2018), doi: https://doi.org/10.1016/j.ijrefrig.2018.11.046
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Highlights A prediction model of frost volume and refrigerating capacity of fin evaporator is established.
Ridgelet neural network is constructed through using ridgelet function as excitation function.
Improved genetic algorithm is used to training ridgelet neural network. Samples of finned evaporator are obtained based on random finite element method. Prediction precision and efficiency of frost volume and refrigerating capacity are improved.
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Frosting Weight and Refrigerating Capacity Prediction of Fin Evaporator Based on Random Finite Element Method and Ridgelet Neural Network
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Bin Zhao, Yi Ren, Diankui Gao, Chengjiang Shi, Lizhi Xu, Yuanyuan Zhang (School of Mechanical Engineering, Liaoning Shihua University, Fushun, Liaoning, 113001, China) Abstract: In order to confirm the proper defrosting measurement, the Frosting weight and refrigerating capacity of fin evaporator should be predicted correctly, therefore the new prediction model is proposed by combining the ridgelet neural network and random finite element method. Firstly, the ridgelet neural network is constructed through using ridgelet function as excitation function of hidden layer, and the basic structure of the ridgelet neural network is designed. Secondly, the training algorithm of ridgelet neural network is put forward based on improved genetic algorithm, the intervals of weights and thresholds can be changed dynamically for the improved genetic algorithm. Thirdly, the frosting random finite element model of fin evaporator is constructed, the frosting parameters of fin evaporator are considered as random variables, and the simulation results under different working condition can be used as testing samples and training samples. Finally, the Frosting weight and refrigerating capacity of fin evaporator are predicted based on new prediction model, simulation analysis shows that the new prediction model has highest prediction precision and efficiency.
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Key words: Frosting weight; Refrigerating capacity; fin evaporator; random finite element method; ridgelet neural network Nomenclatures
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scale factor
CE
scale factor of ridgelet function, total pressure of wet air, Pa translation factor location of ridgelet function coefficient
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a a' B b b' C
c fr
thermal conductivity of frost layer, W /( m K )
d1
import moisture content of fined evaporator, kg / kgdry air
d2
export moisture content of fined evaporator, kg / kgdry air
deq
equivalent diameter of finned tube, m
E
prediction error of ridgelet neural network
E (n)
sum of error energy
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E [T j ,low ]
upper bounds of meaning value of E [T j ]
E [T j ,upper ] low bounds of meaning value of E [T j ] sectional area of fin evaporator, m
Fa gˆ ( x)
2
Fourier transform of g (x)
ia ,i
enthalpy of import air, J / kg
ia ,o
enthalpy of export air, ia ,o , J / kg
~ { }
fin length along airflow direction, m
ma mass flow, kg /(m2 s)
heat conducting matrix between frost layer and fin of taking low bound of {~} and
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N
population size
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m and n coefficient
N0
~
fin thermal conductivity matrix of taking upper bound of {~} and low bound of { }
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fin thermal conductivity matrix of taking low bound of {~} and upper bound of
K
K
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h denotes the heat transfer coefficient, W /(m2 K )
~
N
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upper bound of { }
heat conducting matrix between frost layer and fin of taking upper bound of {~} and
~
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low bound of { }
n0
outer normal direction of boundary surface
Ps saturated pressure of water vapor under dry bulb temperature of wet air, Pa pc
cross probability
pm
mutation probability
Qf
3 internal heat strength, kJ / m
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3 heat of conducting from frost layer to fin per unit volume, W / m
Q
qm, mass flow rate of vapor that improve the thickness of frost layer, kg/s
r
parameter space
S d 1
d 1 dimensional space
Tw
surface temperature of fin, K
Ts
outer surface temperature of frost layer, K
~ j th component of T
Tj
T0 T p 1 Tp
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initial temperature, K
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heat flux density, kJ /(m2 s)
qw
node temperature vector in p 1 th step
derivative of node temperature vector to time e in p th step initial time,
t U u
time step, circumference of fin, m direction of ridgelet function
f
2 kinematic viscosity of air, m /s
vw
wind velocity, m / s
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t0
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matrixes Z and Y level set
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f t thermal conductivity of fin and tube, W /(m K ) f
thermal conductivity of air, W /( m K )
fr
thermal conductivity of frost layer, W /( m K )
~ ~ and random parameter vector
fr
thickness of frost layer, m
ij
connection weight
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ij*
optimal weight
relative humidity admissible function
ˆ
Fourier transform of function
Euler parameter boundary surface of fin surface matrix
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The basic function of refrigerators is to keep food fresh. The quality of food is decided by air temperature and air distribution of freezer and refrigerator, and the unsuitable air temperature and wind speed may affect the preservation of food, therefore it is necessary to ensure the stability of refrigeration performance for refrigerator [1]. Finned evaporator has characteristics of light quality, compact structure and cheap price, which has been widely applied in refrigerator. Finned evaporator frosting is a regular problem in the refrigerating system. When outer surface temperature of finned evaporator is lower than the freezing temperature, the heat and mass transfer between the wet air and cold surface of the evaporator. Frosting will be produced when the wet air passes through the surface of the evaporator. And the frost layer is the porous medium made up of bubbles and ice crystals, and the thermal conductivity of the frost layer is not high, therefore the heat exchange between air and surface of evaporator reduces, and the performance of the evaporator decreases accordingly. The water vapor condenses on the outer surface of evaporator, on the one hand, it can form thermal resistance, on the other hand, the air passage becomes narrow to improve the pressure drop, therefore the evaporator can not work successfully. The frosting of finned evaporator has become been concerned in recent years. Ali Bahadır Olcay et al. programmed an in-house-code to confirm frost ratio on evaporator surface by an image processing technique, and measured the evaporator capacities and pressure drops across the evaporator to compare with the theoretical findings [2]. Marco A.S.Timmermann et al. studied the thermal performance of peripheral-finned tube evaporator under frosting, and proposed an experimental and theoretical evaluation of frost formation on the moist air side of a peripheral finned-tube heat evaporator [3]. Diogo L. da Silva and Christian J. L. Hermes assessed the performance of fan-supplied tube-fin evaporators to identify the main parameters which affect the cooling capacity under frosting conditions and defrost cycle performance [4]. Gustavo G.Heidinger et al. carried out experiment based on ISO 23953 in an open multideck display cabinet with dual air curtain to evaluate the effect of the evaporator fins arrangement pattern on the overall thermal performance, and studied the increase on the operation time before defrost and minimization of the latent thermal load due water condensation and freezing between the evaporator fins [5]. Heat and mass transfer characteristics of frosting process of finned evaporator have been studied based on experiment and theoretical model. The frosting make can affect the air volume and refrigerating capacity of refrigeration system, and can lead to change of temperature and flow fields of cooling space, the energy consumption of refrigerating system increases accordingly. To ensure normal working of refrigerating system, the defrosting should be carried out for the finned
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evaporator, and the extra energy is consumed in defrosting process, and the refrigeration effect of the evaporator will be affected. Therefore it is necessary to predict frost volume and refrigerating capacity, and then the proper defrosting measurements can be made, and energy saving and material saving of the whole refrigerating system can be achieved to improve the economic benefits of refrigerator manufacturing enterprise. The artificial neural network has good prediction effect in nonlinear problem, and the frost volume and energy consumption have strong nonlinear characteristics, however artificial neural network has some disadvantages, for example, the network structure has difficult to be determined scientifically, the training efficiency of prediction is relatively low, and the global optimal solution is difficult to be found out. In order to make up for these shortcomings of artificial neural network, the wavelet transform can be combined with the traditional neural network to construct the wavelet neural network, the transfer function in hidden layer is replaced by wavelet function. Bin Shi et al. proposed an improved wavelet neural network to model the complicated crude distillation unit, in which new parametric updating rules were adopted to correctly obtain the performance of crude oil distillation unit [6]. Lili Huang and JunWang proposed a new hybrid neural network through combining discrete wavelet transform and stochastic recurrent wavelet neural network, and performed empirical experiments in the prediction of four energy market prices, and the effectiveness of proposed prediction model was presented through contrastive results of the different predictive models [7]. Madasthu Santhosh et al. proposed ensemble empirical mode decomposition algorithm to decompose the original wind speed data, and the developed adaptive wavelet neural network for predict the wind time series data [8]. Younes Solgi and Soheil Ganjefar proposed a variable structure controller based on fuzzy wavelet neural network, and introduced a complex nonlinear time delay system to evaluate it [9]. Weslly Puchalsky et al. evaluated wavelet neural network performance combined with five optimization techniques to obtain the best time series forecasting, and predicted the soybean sack price and product demands from a good company based on wavelet neural network [10]. The structure of prediction model is changed, and the training speed of prediction can be improved greatly without affecting the prediction precision, and defect of the neural network liable to fall into local optimal point can be eliminated. The important factor of restricting the development of neural network is the construction and selection of the sample set required of training neural network. Obtaining data trough experiments will cost a lot of manpower and material resources, in addition, the experiment has many uncertainties. In order to obtain the satisfactory prediction effect of wavelet neural network, the appropriate training samples should be obtained. The sample size can not be too small, otherwise, the training effect can not be achieved. The sample size cannot be excessively many, and the prediction meaning will lose. In addition, the selected samples should has sufficient representativeness and certain randomness. The random finite element method can be used to construct the sample collection, which combines the random method and finite element method. M. A. Hariri-Ardebili et al. carried out seismic analysis of gravity dams based on random finite element method, and proposed the results of a study that considers the spatial distribution of random variables in the context of random field theory [11]. A. Johari and A. Gholampour carried out reliability analysis of unsaturated slope based on conditional random finite element method, simulated conditional random fields by consideration possible fluctuations of measured soil properties, and estimated and implemented suctions in a finite element analysis to predict the
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unsaturated soil behavior [12]. The random variables can be calculated repeatedly, and the statistical analysis is carried out for obtained results to get the frost volume and refrigerating capacity of fin evaporator under different conditions. And the obtained samples based on random finite element method can be used to train and test the wavelet neural network. 1 Basic model of ridgelet neural network The wavelet neural network applies the continuous wavelet function to replace the excitation function of feed forward neural network [13]. The wavelet neural network inherits the advantages of wavelet transform and feed forward neural network, which has good self-adaptive ability, self-learning ability and strong modeling ability. The structure of wavelet neural network is shown in figure 1. The wavelet neural network has three layers, which are input layer, hidden layer and output layer.
Figure 1 Structural diagram of wavelet neural network
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The input sample in input layer is defined by x [ x1, x2 ,, xm ] , and the mother wavelet T
function is defined by g ( x) [ g1 ( x), g2 ( x),, gh ( x)] , which satisfies the following
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T
admissible condition [14]:
| gˆ ( x) |
2
dx
(1)
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R
where gˆ ( x) is the Fourier transform of g (x) . The wavelet basic function can be formed through translation and expansion of g (x) , which is expressed by [15]
g a , b ( x)
1 xb g( ) a a
(2)
where a denotes the scale factor, b denotes the translation factor. Changing value of parameter b of wavelet transform only affects location of window in time axis, Changing value of parameter a can change location of window on frequency axis and shape of window. If the
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Fourier transform ˆ of function
: R d R satisfies the following allowable condition
| ˆ ( ) |2 | |d d
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[16]:
then
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window becomes narrow, therefore the time domain resolution of wavelet transform decreases, and the frequency domain resolution of wavelet transform increases. The main advantages of wavelet transform are reflected in analyzing one dimensional piecewise smooth or bounded variation function. When the wavelet transform is generalized to two-dimensional or higher dimensions, the separable wavelet spanned by one dimensional wavelet functions has limited direction, which can not describe the high dimensional functions with linear or surface singularity. To overcome the disadvantages of the wavelet transform, the ridgelet transform is put forward which can shows the singularity in all directions. The ridgelet transform is a multiscale method, which is obtained through adding a directional vector to wavelet basis function. The ridgelet not only has local time-frequency resolution ability like wavelet transform, but also has strong direction selection and identification ability. Therefore the ridgelet function is used as the excitation function to construct the ridgelet neural network.
(3)
is called admissible function, the ridgelet function r generated from is the 1 ux b' ( ) a' a
(4)
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r ( x)
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ridgelet function, which is expressed by
where r denotes the parameter space, r [a' , u, b' ] , a' denotes the scale factor of ridgelet
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function, u denotes the direction of ridgelet function, b' denotes the location of ridgelet d 1 d 1 function. a' , b' R , u S , S denotes the d 1 dimensional space, || u || 1 .
For the ridgelet neural network, the excitation function is expressed by
ux a ' ) , i 1,2,, h b'
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gi ( x) i (
(5)
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2 Training algorithm of ridgelet neural network The feed forward neural network algorithm concludes gradient descent method and genetic algorithm, the gradient descent method is a kind of iteration training algorithm, which has many disadvantages, for example, the algorithm is easy to fall into local minimum point, the learning rate is difficult to be selected, and the convergence speed is relative low. The genetic algorithm is a intelligent optimal method generated based on natural selection and genetics, which can optimize the initial weight and threshold of artificial neural network. However, the optimizing the artificial neural network based on genetic algorithm firstly needs to give the interval ranges of weight and threshold. Because the interval ranges of weight and threshold is difficult to be evaluated, if the optimal weight and threshold is not within the interval ranges, the genetic algorithm has difficulty to find out the optimal solution. In addition, the number of individuals in
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Using a connection weight
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children generation is equal to that in parent generation, therefore the probability of producing excellent individuals is relative low, therefore the genetic algorithm has low convergence speed and is easy to fall into local optimization [17-19]. In order to improve the global search capability of genetic algorithm, the improved multiple children generation genetic algorithm is put forward to adaptively adjust the ranges of weight and threshold. The parent population composed of weight and threshold produce multiple children population, and the number of individuals in generated multiple children population is more than that in parent population generation. Then probability of producing individuals with higher fitness degree is improved, and the searching space of weights and thresholds of ridgelet neural network can be adjusted according to fitness degree value change of current multiple children generation
ij as an example, the adaptive adjustment process is given. It
assume that the initial range of weight
ij is [min , max ] , the initial weight population with
size of pop is generated randomly in range, and the individuals are ij 1 , ij 2 ,..., ijpop , the 0
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prediction. If the weight ij corresponding to minimum prediction error of ridgelet neural *
network is larger than
max , at this moment the optimal weight ij* exceeds the interval
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[min , max ] , the optimal weight can not be obtained based on traditional genetic algorithm. the individuals in population are gradually close to
max after several iterations, the distance
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0 between the worst individual ij 1 in population and min is getting bigger and bigger, and the
following operation is carried out [20]:
(6)
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ij01 - m i n
m i n m' i n
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(7)
m a x m' a x
(8)
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And then the new search interval [min , max ] is formed, one time interval movement is '
'
' ' completed. If the optimal weight ij is in [min , max ] , the optimal weight value can be *
' ' obtained based on the genetic algorithm. If the optimal weight ij is not in [min , max ] , the *
interval can be moved based on expressions (6)-(8). The improved genetic optimal algorithm procedure of ridgelet neural network is listed as follows: Step 1: Parameter initialization is carried out. The population size N0 , cross probability pc
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and mutation probability pm , and the initial interval of weight and threshold are defined by
[min , max ] , the prediction error of ridgelet neural network is defined by E . Step 2: The initial population with size of N0 is randomly generated in interval [min ,max ] . Step 3: The parent population is formed.
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Step 4: The sum of error energy E (n) is calculated, and the individual fitness degree of population is calculated based on linear fitness degree function according to E (n) .
Step 5: The population is sorted in descending order according to individual fitness degree, the top m individuals with higher fitness degree are selected, which are saved as "elite individual".
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Step 6: A pair of individuals in parent generation is selected based on roulette selection method. Step 7: Multiple individuals of children generations are produced based on cross operation. Two cross points are produced randomly on each pair of selected parent chromosomes to enter into cross operation. Changing part after the intersection of each pair of parent generation chromosomes can form the new chromosomes, finally 2 pc N0 new individuals can be produced. The number of chromosomes that does not enter into cross operation is (1 pc ) N0 , and the total number of individuals in multiple children generation is calculated by [21]
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Nl 2 pc N0 (1 pc ) N0 (1 pc ) N0
(9)
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Step 8: The new population is generated based on internal competition of the population. The mutated Nl individuals and saved m elite individuals of parent generation carry out population competition together, and the fitness degree of each individual is calculated, the
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( Nl m N0 ) individuals with lower fitness degree are eliminated.
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Step 9: It judges whether E (n) satisfies the condition E ( n) E , if it is satisfied, then weight and threshold are output, if it is not satisfied, the interval is moved based on expressions (6)-(8), and the new interval [min , max ] will be obtained, the population of the children '
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'
generation is used as parent generation, the optimal weight and threshold are searched in ' ' [min , max ] , the above process is carried out until the termination condition is reached.
3 Frosting model of finned evaporator based on random finite element model The structure of finned evaporator is shown in figure 2. Heat and mass transfer model of finned evaporator under the action of a certain thickness of frost layer is constructed, in order to improve the computing efficiency, four assumptions are given without affecting the computation accuracy, and the following assumption is made [22-24]: Assumption 1: The frosting process is quasi-static process, therefore the frosting process is
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steady in a time step. The thickness of frost layer is equal everywhere, and it is less than height of fin. The thickness of frost layer on fin surface is homogeneous. Assumption 2: The physical parameters of air and frost layer are constant. The effect of environment parameter on the physical parameters of air and frost layer is relative small, which can be ignored. Assumption 3: Heat transfer of fin goes along the fin, and the heat transfer direction of frost layer be perpendicular to surface of fin. The heat transfer in other direction can be ignored. Assumption 4: The top of fin is adiabatic.
Figure 2 Structural diagram of finned evaporator The heat of conducting from frost layer to fin is used as internal heat source. Because the frosting process is considered as a quasi steady state process, the steady heat conduction equation
fr (Ts Tw ) U fr A
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Q
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d 2Tw fr 2 Q 0 dx
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is constructed in a time step. And the microelement dx is taken along the fin, and the heat transfer differential model is expressed by [25] (10)
(11)
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where Tw denotes the surface temperature of fin, K ; Ts denotes the outer surface temperature
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of frost layer, K ; Q denote the heat of conducting from frost layer to fin per unit volume,
W / m 3 ; c fr denotes the thermal conductivity of frost layer, W /(m K ) ; fr denotes the
thermal conductivity of frost layer, W /( m K ) ; U denotes the circumference of fin, m ; A 2
denotes the sectional area, m ;
fr denotes the thickness of frost layer, m .
The change in the thickness of frost layer in a time step is calculated by
ft
qm , t A fr
(12)
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where qm, denotes mass flow rate of vapor that improve the thickness of frost layer, kg/s . The heat transfer coefficient on the air side is calculate by the following formula:
a 1.1 where
f d eq
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L m ) deq
(13)
f denotes the thermal conductivity of air, W /(m K ) ; deq denotes the equivalent
0.24 Ref 1000
n 0.45 0.006
m 0.28
(15)
0.08 Ref 1000
L L L 0.000425 ( ) 2 3 10 6 ( )3 d eq d eq d eq
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vw d eq
f
(16)
(17)
(18)
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where
(14)
L d eq
0.518 0.02315
Ref
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C (1.36
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diameter of finned tube, m ; L denotes the fin length along airflow direction, m ; Coefficients C , m and n are calculated by the following expressions:
f denotes the kinematic viscosity of air, m 2 /s ; vw denotes the wind velocity, m / s .
The moisture content of wet air is calculated by (19)
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Ps d 0.6 2 2 B Ps
where Ps is the saturated pressure of water vapor under dry bulb temperature of wet air, Pa ;
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is the relative humidity, B is the total pressure of wet air, Pa . The frosting weight is calculated by [26]
M f ma (d1 d 2 ) Fa d 0
(20)
2 where ma denotes the mass flow, kg /(m s) ; d 1 denotes the import moisture content of
fined evaporator, kg / kgdry air , d 2 denotes the export moisture content of fined evaporator,
kg / kgdry air , Fa is the sectional area of fin evaporator, m2 .
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The refrigerating capacity is calculated by [27]
Qa ma (ia,i ia,o ) Fa
(21)
where ia , i denotes the enthalpy of import air, J / kg ; ia ,o is the enthalpy of export air, ia ,o ,
J / kg . The three dimensional heat conduction model of fin and tube is expressed by
T 2T 2T 2T f t ( 2 2 2 ) Q f t x y z
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where
(22)
f t denotes the thermal conductivity of fin and tube, W /(m K ) ; Q f denotes the
3 internal heat strength, kJ / m .
- f t
T n0
h(T T f )
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The boundary condition of fin surface is expressed by
(23)
where denotes the boundary surface of fin surface; n0 denotes the outer normal direction of
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boundary surface; h denotes the heat transfer coefficient, W /(m2 K ) . The boundary condition of internal surface of the tube is expressed by
T n0
qw
(24)
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- f t
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where q w denotes the heat flux density, kJ /(m2 s) . The initial condition is listed as follows: (25)
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T0 T ( x, y, z , t ) t
0
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where t0 is initial time, T0 is the initial temperature, K . In order to separate the uncertain parameters, the temperature field finite element computing system is expressed by
[
Z ( c) 1 t t p ( f t , h)]T p1 Y (h, T f , qw ) Z ( c) T 0 t t
(26)
where denotes the Euler parameter, t denotes the time step, Z , and Y denotes the matrixes, T
p 1
denotes the node temperature vector in
derivative of node temperature vector to time e in p th step.
p 1 th step, T p denotes the
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Let
K [
Z ( c) ( f t , h)] t
N Y (h, T f , qw ) Z ( c)
(27)
1 t t p T t
(28)
The following equation can be obtained:
KT p1 N 0
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(29)
The boundary condition and affecting parameters of frosting process characteristics of randomness, the node temperature response are also random, therefore K , T
p1
and N are
random. The random finite element model of fin evaporator frosting is deduced, which is expressed by
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~~ ~ KTw N 0
(30)
The level set is carried out for expression (15), and the random interval equation can be obtained as follows [28]:
[ Kl o w, Ku p p e]r[Tl o w, Tup p e]r [ Nl o w, Nu p p e]r 0
(31)
The solution of expression (31) is defined by
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[Tl o w, Tu p p e]r [[T1,l o w, T1,u p p e],r[T2,l o w, T2,u p p e],r, [Tn,l o w, Tn,u p p e]]rT
(32)
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The upper and lower bounds of every random interval in (32) is defined by (33)
E [T j ,u p p e] r m a xE{(T j ) | KT N 0, K [ Kl o w, Ku p p e]r , N [ Nl o w, Nu p p e]r}
(34)
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E [T j ,l o w] m i n E { (T j ) | KT N 0, K [ Kl o w, Ku p p e]r , N [ Nl o w, Nu p p e]r}
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where E [T j ,low ] and E [T j ,upper ] upper and low bounds of meaning value of E [T j ] , where
~ T j is j th component of T obtained through solving random equilibrium equation
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~~ ~ KT N 0 . The solution of equation (15) can be denoted by [29]
~ T [T ] [Tl o w, Tu p p e]r ( 0 ,1)
(35)
The node temperature is function of every random parameter, the random parameter vectors are defined by
~ {~1, ~2 ,, ~m} ~
~ ~
~
{ 1 , 2 ,, m }
(36) (37)
~ where ~ is directly proportional to node temperature, is inversely proportional to node
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temperature. The
~ level set is carried out for ~ and , and the following expression cans be obtained
{ low , upper } {[ 1,low , 1,upper ],[ 2,low , 2,upper ],,[ m,low , m,upper ]} { low , upper } {[ 1,low , 1,upper ],[ 2,low , 2,upper ],,[ m,low , m,upper ]}
(38) (39)
The stiffness matrix and load matrix are calculated based on expressions (38) and (39), which are expressed by [30]
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K [ K ( low , , upper , )]
(40)
K [ K ( upper , , low , )]
(41)
N [ N ( low , , upper , )]
(42)
where K
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N [ N ( upper , , low , )]
(43)
is fin thermal conductivity matrix of taking low bound of {~} and upper bound of
~ { } , K is fin thermal conductivity matrix of taking upper bound of {~} and low bound of
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~ { } , N is heat conducting matrix between frost layer and fin of taking low bound of {~} ~
and upper bound of { } , N is heat conducting matrix between frost layer and fin of taking
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~
upper bound of {~} and low bound of { } .
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The equilibrium equation is obtained as follows: (44)
K T N 0
(45)
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K T N 0
~ , and the Taylor series expansion is The random vector with average value is defined by
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carried out for K , T , Q , K , T , Q , and the corresponding expressions are obtained as follows:
m
K K 0 K i i
(46)
i 1 m
K K 0 K i i
(47)
i 1
m
N - N 0- Ni i i 1
(48)
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m
N N 0 N i i
(49)
i 1
m
T T0- Ti i
(50)
i 1
m
T T0 Ti i
(51)
i 1
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Putting expressions (46)-(51) to expressions (44) and (45) obtains the following equation set:
K 0T0 N 0 0 K 0 Ti Qi K iT0
(37)
K 0T0 N 0 0 K 0 Ti Qi K iT0
(38)
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Based on the above equations, the solution of (31) can be obtained.
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Input random parameter to construct boundary and initial conditions
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Calculate the random finite element model
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Obtain the temperature distribution of fin evaporator
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Calculate the frosting weight and refrigerating capacity of fin evaporator
Random finite element method
Obtain training and testing samples
Set parameters of rigdelet neural network
Train ridgelet neural network
Test ridgelet neural network
Output prediction results of frosting volume and refrigerating capacity
Ridgelet neural network Figure 3 Prediction procedure of Frosting weight and refrigerating capacity of fin evaporator based on random finite element method and ridgelet neural network 4 Prediction simulation analysis of Frosting weight and refrigerating capacity of fin evaporator
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The prediction simulation analysis of Frosting weight and refrigerating capacity is carried out based on random finite element method and Ridgelet neural network. A new type fin evaporator is used as the researching objective, and its structure is shown in figure 4. The fin pitches are 7.5/15mm. The fin pitch is distance between adjacent fins, as seen from figure 4, the fin pitch of the top three rows of fins is 7.5m, the fin pitch of bottom row of fin is 15mm, the fin areas are 0.245m2.
Figure 4 Structural diagram of new type fin evaporator The parameter testing is carried out for the new type fin evaporator, the fridge evaporator cover is changed to transparent material for observation, and the a video camera is set inside cabinet for
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recording. The environmental temperature is 311 K , the relative humidity is 70%，the temperature of cold storage room is 275 K , the temperature of cold room is 257 K . The testing device is shown in figure 5.
Figure 5 Testing device of fin evaporator Frosting situations of different location of fin evaporator at moment with most Frosting weight
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are shown in figure 6.
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(a) Location 1
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(b) Location 2 Figure 6 Frosting situations of different location of fin evaporator at moment with most Frosting weight To verify the effectiveness of the prediction model, ten sets of data are obtained based on test, the initial and boundary conditions are listed as follows: the initial temperature of fin surface is
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240K, and the heat transfer coefficient is 226 W / m K . The testing results are listed in table 1. 2
The obtained testing data is used as testing samples to compare with the simulation results.
Number
Table 1 Testing data of fin evaporator (Evaporation temperature=299 K )
Time/
Inlet
Relative
Inlet air
Refrigerant
Frosting
Refrigerating
min
wind
humidity of
temperature
flow/kg/h
weight/g
capacity/W
speed/
imported
/K
m/s
air/%
1
2
0.7
35%
257
2.0
34.5
112.3
2
4
0.9
40%
259
2.1
42.3
106.6
3
6
1.0
45%
260
2.3
50.6
99.5
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4
8
1.1
50%
261
2.4
57.4
92.7
5
10
1.2
55%
263
2.6
69.3
87.5
6
12
1.3
60%
265
2.7
75.8
81.2
7
14
1.4
65%
266
2.8
84.8
78.3
8
16
1.6
70%
267
3.0
96.2
70.3
9
18
1.8
75%
269
3.2
105.3
63.1
10
20
2.0
80%
271
3.4
114.8
57.9
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The random finite element method is used to generate 100 samples, the top 90 samples are used as training samples, the other 10 samples are used as testing samples, the initial conditions of 10 testing samples are same as that of test. The parameters of multiple children genetic algorithm are set as follows: the population scale
N0 is 75, the initial interval of weights and threshold is [-15,15], the required precision is 1.5×
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10-5, the cross probability pc is 0.75, the mutated probability pm is 0.15, retention ratio of elite individuals is 15%. The ridgelet neural network concludes three layers, the input layer concludes inlet wind speed ( x1 ), relative humidity of imported air ( x 2 ), inlet air temperature ( x3 ) refrigerant flow ( x 4 ) and
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evaporation temperature ( x5 ). The number of hidden layer nodes are confirmed based on gradually increasing method, which is 6. The output layers concludes Frosting weight ( y1 ) and
The
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refrigerant capacity ( y 2 ).
frosting parameters
of
fin
evaporator
are
described
by
L R parameter,
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c ( 401,3,5) LR , fr (0.3,0.02,0.04) LR , fr (0.0005,0.00002,0.00001) LR ,
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ma (6 104 ,3 105 ,2 105 ) , Fa (0.245,0.03 ,0.003)LR ,
(0.75,0.002,0.002 ) LR .
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The randomness of every variable is described by
c ~ N (0, c2 ) , ~ N (0, 2 ) , fr
fr
~ N (0, 2 ) , m ~ N (0, m2 ) , F ~ N (0, F2 ) , ~ N (0, 2 ) . fr
fr
a
a
a
a
The =0.85 level set is carried out for every frosting parameter. In order to verify the effectiveness of ridgelet neural network optimized by improved genetic algorithm (IGA-RNN), the BP neural network optimized by genetic algorithm (GA-BPNN) and the B-spline wavelet neural network optimized by genetic algorithm (GA-BSWNN) is also applied to prediction the Frosting weight and refrigerating capacity of fin evaporator. The iteration times are 450. And the prediction performance indexes of the three models are listed in table 2. Table 2 Prediction performance indexes of the three models
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Performance index
Prediction model GA-BSWNN
GA-BPNN
Average absolute relative error/% (at 100s)
10.32
11.67
12.64
Maximum absolute relative error/% (at 100s)
31.54
45.37
66.92
Root mean square error/% (at 100s)
188.62
249.65
255.37
30s
1.43×10-2
3.17×10-2
6.29×10-2
40s
2.94×10-3
1.46×10-2
4.38×10-2
50s
1.47×10-3
7.45×10-3
8.59×10-3
60s
1.43×10-4
2.79×10-4
4.28×10-4
70s
1.16×10-4
5.32×10-4
6.83×10-4
80s
6.32×10-5
9.63×10-5
1.07×10-4
90s
4.97×10-5
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IGA-RNN
6.28×10-5
8.25×10-5
As seen from table 2, the ridgelet neural network optimized by improved genetic algorithm (IGA-RNN) has least average absolute relative error, minimum absolute relative error, root mean
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square error and E (n) . The BP neural network optimized by genetic algorithm (GA-BPNN) has biggest average absolute relative error, maximum absolute relative error, root mean square error
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and E (n) . The prediction performance indexes of the B-spline wavelet neural network optimized
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by genetic algorithm (GA-BSWNN) in between that of the above two prediction models. Results show that the improved genetic algorithm has best global searching ability, and has lowest risk of falling into local optimization. The ridgelet neural network optimized by improved genetic algorithm (IGA-RNN) can give full play to the advantages of the ridgelet neural network and improved genetic algorithm, therefore can effectively improve the prediction precision. The ridgelet neural network optimized by improved genetic algorithm (IGA-RNN) has also highest convergence speed among the three prediction models. The IGA-RNN, GA-BSWNN and GA-BPNN are used to carry out precision comparing analysis of Frosting weight and refrigerating capacity of fin evaporator for the ten testing samples, and the final prediction results are listed in table 3. Table 3 Prediction results comparison among the three prediction models Number
Frosting weight/g
Time/ min
IGA-RNN
GA-BSWN
GA-BPNN
N
Refrigerating capacity/W Real
IGA-RNN
GA-BSWNN
GA-BPNN
value
Real value
1
2
35.1
36.4
38.3
34.5
113.5
114.6
116.2
112.3
2
4
43.6
44.9
46.2
42.3
107.4
108.7
120.2
106.6
3
6
51.3
52.5
53.7
50.6
100.2
102.3
104.6
99.5
4
8
58.2
59.8
61.4
57.4
93.4
95.6
97.8
92.7
5
10
70.4
72.3
74.6
69.3
88.7
89.3
101.6
87.5
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12
76.2
77.6
79.2
75.8
82.5
84.2
86.1
81.2
7
14
85.3
86.8
88.4
84.8
79.3
80.8
82.4
78.3
8
16
96.9
97.7
99.1
96.2
71.7
73.2
75.8
70.3
9
18
106.4
107.7
109.2
105.3
64.6
65.8
67.5
63.1
10
20
115.3
117.0
118.9
114.8
58.3
59.9
61.4
57.9
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As seen from table 3, the prediction values of frosting weight and refrigerating capacity based on ridgelet neural network optimized by improved genetic algorithm (IGA-RNN) are closest to the real among the three prediction models, and the prediction values of frosting weight and refrigerating capacity based on BP neural network optimized by genetic algorithm (GA-BPNN) has maximum deviation from the real deviation. The prediction results based on B-spline wavelet neural network optimized by genetic algorithm (GA-BSWNN) are between in that of the above two prediction models. As seen from table 3, the error of IGA-RNN ranges from 1% to 3%, the error of GA-BSWNN ranges from 2% to 6%, and the error of GA-BPNN ranges from 3% to 11%. Therefore the IGA-RNN has highest prediction precision. Therefore the ridgelet neural network optimized by improved genetic algorithm has highest prediction precision, the main reason for this results concludes two aspects, on one hand, the ridgelet neural network has arbitrary approximation ability, on the other hand, the improved genetic algorithm improve the learning and generalization abilities of the ridgelet neural network to improve the correctness of prediction. 5 Conclusions The Frosting weight and refrigerating capacity are important parameters of describing the frosting of fin evaporator, therefore the prediction model based on ridgelet neural network optimized by improved genetic algorithm is constructed to predict the two parameter. The ridgelet neural network is constructed through using ridgelet function as excitation function of the hidden layer, and the ridgelet neural network has strong adaptability, fault tolerance and robustness depend on the advantages of ridgelet function. The improved genetic algorithm is used to optimize the ridgelet neural network, the improved genetic algorithm can dynamically regulate the intervals of weights and thresholds, the searching interval can be adaptively changed based on fitness degree of individuals in current multiple children generation to improve the prediction precision and efficiency of ridgelet neural network. The effect of parameters randomness on frosting temperature field of fin evaporator is considered, the random finite element model of frosting of fin evaporator is constructed to obtain the frost volume and refrigerating capacity of fin evaporator combing with frosting model of fin evaporator. The simulation results under different condition from random finite element model are used as the training samples and testing samples. In addition the testing device is designed to obtain the comparing samples using working conditions of testing samples to verify the correctness of simulation results. The simulation analysis results show that the ridgelet neural network has best prediction performance, the prediction precision and efficiency are all improved. Therefore the prediction model based on ridgelet neural network and improved genetic algorithm can effectively predict the frosting weight and refrigerating capacity. The prediction effect has the following main reasons: 1) The IGA-RNN has best prediction precision based on advantages of IGA and RNN. 2) The random finite element method introduces the random parameters into the heat transfer analysis of fin evaporator, and the fuzzy random parameters are used to describe the uncertainty of temperature field. Therefore the computation precision of the random finite element can be improved accordingly. The frosting
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situation of fin evaporator can be predicted correctly, and the proper defrosting measurements can be made to ensure stable operation of fin evaporator. The following contents should be studied in further, the optimal algorithm of ridgelet neural network should be improved for improving the prediction precision and efficiency. The other frosting parameters prediction model can be constructed based on this idea. The randomness of more frosting parameters can be considered to construct the perfect prediction.
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Frosting Weight and Refrigerating Capacity Prediction of Fin Evaporator Based on Random Finite Element Method and Ridgelet Neural Network Bin Zhao , Yi Ren , Diankui Gao , Chengjiang Shi , Lizhi Xu , Yuanyuan Zhang PII: DOI: Reference:
S0140-7007(18)30510-3 https://doi.org/10.1016/j.ijrefrig.2018.11.046 JIJR 4217
To appear in:
International Journal of Refrigeration
Received date: Revised date: Accepted date:
28 July 2018 25 November 2018 30 November 2018
Please cite this article as: Bin Zhao , Yi Ren , Diankui Gao , Chengjiang Shi , Lizhi Xu , Yuanyuan Zhang , Frosting Weight and Refrigerating Capacity Prediction of Fin Evaporator Based on Random Finite Element Method and Ridgelet Neural Network, International Journal of Refrigeration (2018), doi: https://doi.org/10.1016/j.ijrefrig.2018.11.046
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Highlights A prediction model of frost volume and refrigerating capacity of fin evaporator is established.
Ridgelet neural network is constructed through using ridgelet function as excitation function.
Improved genetic algorithm is used to training ridgelet neural network. Samples of finned evaporator are obtained based on random finite element method. Prediction precision and efficiency of frost volume and refrigerating capacity are improved.
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Frosting Weight and Refrigerating Capacity Prediction of Fin Evaporator Based on Random Finite Element Method and Ridgelet Neural Network
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Bin Zhao, Yi Ren, Diankui Gao, Chengjiang Shi, Lizhi Xu, Yuanyuan Zhang (School of Mechanical Engineering, Liaoning Shihua University, Fushun, Liaoning, 113001, China) Abstract: In order to confirm the proper defrosting measurement, the Frosting weight and refrigerating capacity of fin evaporator should be predicted correctly, therefore the new prediction model is proposed by combining the ridgelet neural network and random finite element method. Firstly, the ridgelet neural network is constructed through using ridgelet function as excitation function of hidden layer, and the basic structure of the ridgelet neural network is designed. Secondly, the training algorithm of ridgelet neural network is put forward based on improved genetic algorithm, the intervals of weights and thresholds can be changed dynamically for the improved genetic algorithm. Thirdly, the frosting random finite element model of fin evaporator is constructed, the frosting parameters of fin evaporator are considered as random variables, and the simulation results under different working condition can be used as testing samples and training samples. Finally, the Frosting weight and refrigerating capacity of fin evaporator are predicted based on new prediction model, simulation analysis shows that the new prediction model has highest prediction precision and efficiency.
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Key words: Frosting weight; Refrigerating capacity; fin evaporator; random finite element method; ridgelet neural network Nomenclatures
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scale factor
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scale factor of ridgelet function, total pressure of wet air, Pa translation factor location of ridgelet function coefficient
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a a' B b b' C
c fr
thermal conductivity of frost layer, W /( m K )
d1
import moisture content of fined evaporator, kg / kgdry air
d2
export moisture content of fined evaporator, kg / kgdry air
deq
equivalent diameter of finned tube, m
E
prediction error of ridgelet neural network
E (n)
sum of error energy
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E [T j ,low ]
upper bounds of meaning value of E [T j ]
E [T j ,upper ] low bounds of meaning value of E [T j ] sectional area of fin evaporator, m
Fa gˆ ( x)
2
Fourier transform of g (x)
ia ,i
enthalpy of import air, J / kg
ia ,o
enthalpy of export air, ia ,o , J / kg
~ { }
fin length along airflow direction, m
ma mass flow, kg /(m2 s)
heat conducting matrix between frost layer and fin of taking low bound of {~} and
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N
population size
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m and n coefficient
N0
~
fin thermal conductivity matrix of taking upper bound of {~} and low bound of { }
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fin thermal conductivity matrix of taking low bound of {~} and upper bound of
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h denotes the heat transfer coefficient, W /(m2 K )
~
N
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upper bound of { }
heat conducting matrix between frost layer and fin of taking upper bound of {~} and
~
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low bound of { }
n0
outer normal direction of boundary surface
Ps saturated pressure of water vapor under dry bulb temperature of wet air, Pa pc
cross probability
pm
mutation probability
Qf
3 internal heat strength, kJ / m
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3 heat of conducting from frost layer to fin per unit volume, W / m
Q
qm, mass flow rate of vapor that improve the thickness of frost layer, kg/s
r
parameter space
S d 1
d 1 dimensional space
Tw
surface temperature of fin, K
Ts
outer surface temperature of frost layer, K
~ j th component of T
Tj
T0 T p 1 Tp
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initial temperature, K
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heat flux density, kJ /(m2 s)
qw
node temperature vector in p 1 th step
derivative of node temperature vector to time e in p th step initial time,
t U u
time step, circumference of fin, m direction of ridgelet function
f
2 kinematic viscosity of air, m /s
vw
wind velocity, m / s
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t0
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matrixes Z and Y level set
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f t thermal conductivity of fin and tube, W /(m K ) f
thermal conductivity of air, W /( m K )
fr
thermal conductivity of frost layer, W /( m K )
~ ~ and random parameter vector
fr
thickness of frost layer, m
ij
connection weight
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ij*
optimal weight
relative humidity admissible function
ˆ
Fourier transform of function
Euler parameter boundary surface of fin surface matrix
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The basic function of refrigerators is to keep food fresh. The quality of food is decided by air temperature and air distribution of freezer and refrigerator, and the unsuitable air temperature and wind speed may affect the preservation of food, therefore it is necessary to ensure the stability of refrigeration performance for refrigerator [1]. Finned evaporator has characteristics of light quality, compact structure and cheap price, which has been widely applied in refrigerator. Finned evaporator frosting is a regular problem in the refrigerating system. When outer surface temperature of finned evaporator is lower than the freezing temperature, the heat and mass transfer between the wet air and cold surface of the evaporator. Frosting will be produced when the wet air passes through the surface of the evaporator. And the frost layer is the porous medium made up of bubbles and ice crystals, and the thermal conductivity of the frost layer is not high, therefore the heat exchange between air and surface of evaporator reduces, and the performance of the evaporator decreases accordingly. The water vapor condenses on the outer surface of evaporator, on the one hand, it can form thermal resistance, on the other hand, the air passage becomes narrow to improve the pressure drop, therefore the evaporator can not work successfully. The frosting of finned evaporator has become been concerned in recent years. Ali Bahadır Olcay et al. programmed an in-house-code to confirm frost ratio on evaporator surface by an image processing technique, and measured the evaporator capacities and pressure drops across the evaporator to compare with the theoretical findings [2]. Marco A.S.Timmermann et al. studied the thermal performance of peripheral-finned tube evaporator under frosting, and proposed an experimental and theoretical evaluation of frost formation on the moist air side of a peripheral finned-tube heat evaporator [3]. Diogo L. da Silva and Christian J. L. Hermes assessed the performance of fan-supplied tube-fin evaporators to identify the main parameters which affect the cooling capacity under frosting conditions and defrost cycle performance [4]. Gustavo G.Heidinger et al. carried out experiment based on ISO 23953 in an open multideck display cabinet with dual air curtain to evaluate the effect of the evaporator fins arrangement pattern on the overall thermal performance, and studied the increase on the operation time before defrost and minimization of the latent thermal load due water condensation and freezing between the evaporator fins [5]. Heat and mass transfer characteristics of frosting process of finned evaporator have been studied based on experiment and theoretical model. The frosting make can affect the air volume and refrigerating capacity of refrigeration system, and can lead to change of temperature and flow fields of cooling space, the energy consumption of refrigerating system increases accordingly. To ensure normal working of refrigerating system, the defrosting should be carried out for the finned
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evaporator, and the extra energy is consumed in defrosting process, and the refrigeration effect of the evaporator will be affected. Therefore it is necessary to predict frost volume and refrigerating capacity, and then the proper defrosting measurements can be made, and energy saving and material saving of the whole refrigerating system can be achieved to improve the economic benefits of refrigerator manufacturing enterprise. The artificial neural network has good prediction effect in nonlinear problem, and the frost volume and energy consumption have strong nonlinear characteristics, however artificial neural network has some disadvantages, for example, the network structure has difficult to be determined scientifically, the training efficiency of prediction is relatively low, and the global optimal solution is difficult to be found out. In order to make up for these shortcomings of artificial neural network, the wavelet transform can be combined with the traditional neural network to construct the wavelet neural network, the transfer function in hidden layer is replaced by wavelet function. Bin Shi et al. proposed an improved wavelet neural network to model the complicated crude distillation unit, in which new parametric updating rules were adopted to correctly obtain the performance of crude oil distillation unit [6]. Lili Huang and JunWang proposed a new hybrid neural network through combining discrete wavelet transform and stochastic recurrent wavelet neural network, and performed empirical experiments in the prediction of four energy market prices, and the effectiveness of proposed prediction model was presented through contrastive results of the different predictive models [7]. Madasthu Santhosh et al. proposed ensemble empirical mode decomposition algorithm to decompose the original wind speed data, and the developed adaptive wavelet neural network for predict the wind time series data [8]. Younes Solgi and Soheil Ganjefar proposed a variable structure controller based on fuzzy wavelet neural network, and introduced a complex nonlinear time delay system to evaluate it [9]. Weslly Puchalsky et al. evaluated wavelet neural network performance combined with five optimization techniques to obtain the best time series forecasting, and predicted the soybean sack price and product demands from a good company based on wavelet neural network [10]. The structure of prediction model is changed, and the training speed of prediction can be improved greatly without affecting the prediction precision, and defect of the neural network liable to fall into local optimal point can be eliminated. The important factor of restricting the development of neural network is the construction and selection of the sample set required of training neural network. Obtaining data trough experiments will cost a lot of manpower and material resources, in addition, the experiment has many uncertainties. In order to obtain the satisfactory prediction effect of wavelet neural network, the appropriate training samples should be obtained. The sample size can not be too small, otherwise, the training effect can not be achieved. The sample size cannot be excessively many, and the prediction meaning will lose. In addition, the selected samples should has sufficient representativeness and certain randomness. The random finite element method can be used to construct the sample collection, which combines the random method and finite element method. M. A. Hariri-Ardebili et al. carried out seismic analysis of gravity dams based on random finite element method, and proposed the results of a study that considers the spatial distribution of random variables in the context of random field theory [11]. A. Johari and A. Gholampour carried out reliability analysis of unsaturated slope based on conditional random finite element method, simulated conditional random fields by consideration possible fluctuations of measured soil properties, and estimated and implemented suctions in a finite element analysis to predict the
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unsaturated soil behavior [12]. The random variables can be calculated repeatedly, and the statistical analysis is carried out for obtained results to get the frost volume and refrigerating capacity of fin evaporator under different conditions. And the obtained samples based on random finite element method can be used to train and test the wavelet neural network. 1 Basic model of ridgelet neural network The wavelet neural network applies the continuous wavelet function to replace the excitation function of feed forward neural network [13]. The wavelet neural network inherits the advantages of wavelet transform and feed forward neural network, which has good self-adaptive ability, self-learning ability and strong modeling ability. The structure of wavelet neural network is shown in figure 1. The wavelet neural network has three layers, which are input layer, hidden layer and output layer.
Figure 1 Structural diagram of wavelet neural network
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The input sample in input layer is defined by x [ x1, x2 ,, xm ] , and the mother wavelet T
function is defined by g ( x) [ g1 ( x), g2 ( x),, gh ( x)] , which satisfies the following
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T
admissible condition [14]:
| gˆ ( x) |
2
dx
(1)
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R
where gˆ ( x) is the Fourier transform of g (x) . The wavelet basic function can be formed through translation and expansion of g (x) , which is expressed by [15]
g a , b ( x)
1 xb g( ) a a
(2)
where a denotes the scale factor, b denotes the translation factor. Changing value of parameter b of wavelet transform only affects location of window in time axis, Changing value of parameter a can change location of window on frequency axis and shape of window. If the
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Fourier transform ˆ of function
: R d R satisfies the following allowable condition
| ˆ ( ) |2 | |d d
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[16]:
then
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window becomes narrow, therefore the time domain resolution of wavelet transform decreases, and the frequency domain resolution of wavelet transform increases. The main advantages of wavelet transform are reflected in analyzing one dimensional piecewise smooth or bounded variation function. When the wavelet transform is generalized to two-dimensional or higher dimensions, the separable wavelet spanned by one dimensional wavelet functions has limited direction, which can not describe the high dimensional functions with linear or surface singularity. To overcome the disadvantages of the wavelet transform, the ridgelet transform is put forward which can shows the singularity in all directions. The ridgelet transform is a multiscale method, which is obtained through adding a directional vector to wavelet basis function. The ridgelet not only has local time-frequency resolution ability like wavelet transform, but also has strong direction selection and identification ability. Therefore the ridgelet function is used as the excitation function to construct the ridgelet neural network.
(3)
is called admissible function, the ridgelet function r generated from is the 1 ux b' ( ) a' a
(4)
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r ( x)
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ridgelet function, which is expressed by
where r denotes the parameter space, r [a' , u, b' ] , a' denotes the scale factor of ridgelet
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function, u denotes the direction of ridgelet function, b' denotes the location of ridgelet d 1 d 1 function. a' , b' R , u S , S denotes the d 1 dimensional space, || u || 1 .
For the ridgelet neural network, the excitation function is expressed by
ux a ' ) , i 1,2,, h b'
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gi ( x) i (
(5)
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2 Training algorithm of ridgelet neural network The feed forward neural network algorithm concludes gradient descent method and genetic algorithm, the gradient descent method is a kind of iteration training algorithm, which has many disadvantages, for example, the algorithm is easy to fall into local minimum point, the learning rate is difficult to be selected, and the convergence speed is relative low. The genetic algorithm is a intelligent optimal method generated based on natural selection and genetics, which can optimize the initial weight and threshold of artificial neural network. However, the optimizing the artificial neural network based on genetic algorithm firstly needs to give the interval ranges of weight and threshold. Because the interval ranges of weight and threshold is difficult to be evaluated, if the optimal weight and threshold is not within the interval ranges, the genetic algorithm has difficulty to find out the optimal solution. In addition, the number of individuals in
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Using a connection weight
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children generation is equal to that in parent generation, therefore the probability of producing excellent individuals is relative low, therefore the genetic algorithm has low convergence speed and is easy to fall into local optimization [17-19]. In order to improve the global search capability of genetic algorithm, the improved multiple children generation genetic algorithm is put forward to adaptively adjust the ranges of weight and threshold. The parent population composed of weight and threshold produce multiple children population, and the number of individuals in generated multiple children population is more than that in parent population generation. Then probability of producing individuals with higher fitness degree is improved, and the searching space of weights and thresholds of ridgelet neural network can be adjusted according to fitness degree value change of current multiple children generation
ij as an example, the adaptive adjustment process is given. It
assume that the initial range of weight
ij is [min , max ] , the initial weight population with
size of pop is generated randomly in range, and the individuals are ij 1 , ij 2 ,..., ijpop , the 0
0
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0
prediction. If the weight ij corresponding to minimum prediction error of ridgelet neural *
network is larger than
max , at this moment the optimal weight ij* exceeds the interval
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[min , max ] , the optimal weight can not be obtained based on traditional genetic algorithm. the individuals in population are gradually close to
max after several iterations, the distance
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0 between the worst individual ij 1 in population and min is getting bigger and bigger, and the
following operation is carried out [20]:
(6)
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ij01 - m i n
m i n m' i n
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(7)
m a x m' a x
(8)
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And then the new search interval [min , max ] is formed, one time interval movement is '
'
' ' completed. If the optimal weight ij is in [min , max ] , the optimal weight value can be *
' ' obtained based on the genetic algorithm. If the optimal weight ij is not in [min , max ] , the *
interval can be moved based on expressions (6)-(8). The improved genetic optimal algorithm procedure of ridgelet neural network is listed as follows: Step 1: Parameter initialization is carried out. The population size N0 , cross probability pc
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and mutation probability pm , and the initial interval of weight and threshold are defined by
[min , max ] , the prediction error of ridgelet neural network is defined by E . Step 2: The initial population with size of N0 is randomly generated in interval [min ,max ] . Step 3: The parent population is formed.
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Step 4: The sum of error energy E (n) is calculated, and the individual fitness degree of population is calculated based on linear fitness degree function according to E (n) .
Step 5: The population is sorted in descending order according to individual fitness degree, the top m individuals with higher fitness degree are selected, which are saved as "elite individual".
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Step 6: A pair of individuals in parent generation is selected based on roulette selection method. Step 7: Multiple individuals of children generations are produced based on cross operation. Two cross points are produced randomly on each pair of selected parent chromosomes to enter into cross operation. Changing part after the intersection of each pair of parent generation chromosomes can form the new chromosomes, finally 2 pc N0 new individuals can be produced. The number of chromosomes that does not enter into cross operation is (1 pc ) N0 , and the total number of individuals in multiple children generation is calculated by [21]
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Nl 2 pc N0 (1 pc ) N0 (1 pc ) N0
(9)
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Step 8: The new population is generated based on internal competition of the population. The mutated Nl individuals and saved m elite individuals of parent generation carry out population competition together, and the fitness degree of each individual is calculated, the
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( Nl m N0 ) individuals with lower fitness degree are eliminated.
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Step 9: It judges whether E (n) satisfies the condition E ( n) E , if it is satisfied, then weight and threshold are output, if it is not satisfied, the interval is moved based on expressions (6)-(8), and the new interval [min , max ] will be obtained, the population of the children '
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'
generation is used as parent generation, the optimal weight and threshold are searched in ' ' [min , max ] , the above process is carried out until the termination condition is reached.
3 Frosting model of finned evaporator based on random finite element model The structure of finned evaporator is shown in figure 2. Heat and mass transfer model of finned evaporator under the action of a certain thickness of frost layer is constructed, in order to improve the computing efficiency, four assumptions are given without affecting the computation accuracy, and the following assumption is made [22-24]: Assumption 1: The frosting process is quasi-static process, therefore the frosting process is
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steady in a time step. The thickness of frost layer is equal everywhere, and it is less than height of fin. The thickness of frost layer on fin surface is homogeneous. Assumption 2: The physical parameters of air and frost layer are constant. The effect of environment parameter on the physical parameters of air and frost layer is relative small, which can be ignored. Assumption 3: Heat transfer of fin goes along the fin, and the heat transfer direction of frost layer be perpendicular to surface of fin. The heat transfer in other direction can be ignored. Assumption 4: The top of fin is adiabatic.
Figure 2 Structural diagram of finned evaporator The heat of conducting from frost layer to fin is used as internal heat source. Because the frosting process is considered as a quasi steady state process, the steady heat conduction equation
fr (Ts Tw ) U fr A
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Q
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d 2Tw fr 2 Q 0 dx
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is constructed in a time step. And the microelement dx is taken along the fin, and the heat transfer differential model is expressed by [25] (10)
(11)
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where Tw denotes the surface temperature of fin, K ; Ts denotes the outer surface temperature
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of frost layer, K ; Q denote the heat of conducting from frost layer to fin per unit volume,
W / m 3 ; c fr denotes the thermal conductivity of frost layer, W /(m K ) ; fr denotes the
thermal conductivity of frost layer, W /( m K ) ; U denotes the circumference of fin, m ; A 2
denotes the sectional area, m ;
fr denotes the thickness of frost layer, m .
The change in the thickness of frost layer in a time step is calculated by
ft
qm , t A fr
(12)
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where qm, denotes mass flow rate of vapor that improve the thickness of frost layer, kg/s . The heat transfer coefficient on the air side is calculate by the following formula:
a 1.1 where
f d eq
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L m ) deq
(13)
f denotes the thermal conductivity of air, W /(m K ) ; deq denotes the equivalent
0.24 Ref 1000
n 0.45 0.006
m 0.28
(15)
0.08 Ref 1000
L L L 0.000425 ( ) 2 3 10 6 ( )3 d eq d eq d eq
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vw d eq
f
(16)
(17)
(18)
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where
(14)
L d eq
0.518 0.02315
Ref
)
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C (1.36
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diameter of finned tube, m ; L denotes the fin length along airflow direction, m ; Coefficients C , m and n are calculated by the following expressions:
f denotes the kinematic viscosity of air, m 2 /s ; vw denotes the wind velocity, m / s .
The moisture content of wet air is calculated by (19)
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Ps d 0.6 2 2 B Ps
where Ps is the saturated pressure of water vapor under dry bulb temperature of wet air, Pa ;
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is the relative humidity, B is the total pressure of wet air, Pa . The frosting weight is calculated by [26]
M f ma (d1 d 2 ) Fa d 0
(20)
2 where ma denotes the mass flow, kg /(m s) ; d 1 denotes the import moisture content of
fined evaporator, kg / kgdry air , d 2 denotes the export moisture content of fined evaporator,
kg / kgdry air , Fa is the sectional area of fin evaporator, m2 .
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The refrigerating capacity is calculated by [27]
Qa ma (ia,i ia,o ) Fa
(21)
where ia , i denotes the enthalpy of import air, J / kg ; ia ,o is the enthalpy of export air, ia ,o ,
J / kg . The three dimensional heat conduction model of fin and tube is expressed by
T 2T 2T 2T f t ( 2 2 2 ) Q f t x y z
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where
(22)
f t denotes the thermal conductivity of fin and tube, W /(m K ) ; Q f denotes the
3 internal heat strength, kJ / m .
- f t
T n0
h(T T f )
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The boundary condition of fin surface is expressed by
(23)
where denotes the boundary surface of fin surface; n0 denotes the outer normal direction of
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boundary surface; h denotes the heat transfer coefficient, W /(m2 K ) . The boundary condition of internal surface of the tube is expressed by
T n0
qw
(24)
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- f t
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where q w denotes the heat flux density, kJ /(m2 s) . The initial condition is listed as follows: (25)
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T0 T ( x, y, z , t ) t
0
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where t0 is initial time, T0 is the initial temperature, K . In order to separate the uncertain parameters, the temperature field finite element computing system is expressed by
[
Z ( c) 1 t t p ( f t , h)]T p1 Y (h, T f , qw ) Z ( c) T 0 t t
(26)
where denotes the Euler parameter, t denotes the time step, Z , and Y denotes the matrixes, T
p 1
denotes the node temperature vector in
derivative of node temperature vector to time e in p th step.
p 1 th step, T p denotes the
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Let
K [
Z ( c) ( f t , h)] t
N Y (h, T f , qw ) Z ( c)
(27)
1 t t p T t
(28)
The following equation can be obtained:
KT p1 N 0
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(29)
The boundary condition and affecting parameters of frosting process characteristics of randomness, the node temperature response are also random, therefore K , T
p1
and N are
random. The random finite element model of fin evaporator frosting is deduced, which is expressed by
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~~ ~ KTw N 0
(30)
The level set is carried out for expression (15), and the random interval equation can be obtained as follows [28]:
[ Kl o w, Ku p p e]r[Tl o w, Tup p e]r [ Nl o w, Nu p p e]r 0
(31)
The solution of expression (31) is defined by
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[Tl o w, Tu p p e]r [[T1,l o w, T1,u p p e],r[T2,l o w, T2,u p p e],r, [Tn,l o w, Tn,u p p e]]rT
(32)
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The upper and lower bounds of every random interval in (32) is defined by (33)
E [T j ,u p p e] r m a xE{(T j ) | KT N 0, K [ Kl o w, Ku p p e]r , N [ Nl o w, Nu p p e]r}
(34)
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E [T j ,l o w] m i n E { (T j ) | KT N 0, K [ Kl o w, Ku p p e]r , N [ Nl o w, Nu p p e]r}
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where E [T j ,low ] and E [T j ,upper ] upper and low bounds of meaning value of E [T j ] , where
~ T j is j th component of T obtained through solving random equilibrium equation
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~~ ~ KT N 0 . The solution of equation (15) can be denoted by [29]
~ T [T ] [Tl o w, Tu p p e]r ( 0 ,1)
(35)
The node temperature is function of every random parameter, the random parameter vectors are defined by
~ {~1, ~2 ,, ~m} ~
~ ~
~
{ 1 , 2 ,, m }
(36) (37)
~ where ~ is directly proportional to node temperature, is inversely proportional to node
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temperature. The
~ level set is carried out for ~ and , and the following expression cans be obtained
{ low , upper } {[ 1,low , 1,upper ],[ 2,low , 2,upper ],,[ m,low , m,upper ]} { low , upper } {[ 1,low , 1,upper ],[ 2,low , 2,upper ],,[ m,low , m,upper ]}
(38) (39)
The stiffness matrix and load matrix are calculated based on expressions (38) and (39), which are expressed by [30]
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K [ K ( low , , upper , )]
(40)
K [ K ( upper , , low , )]
(41)
N [ N ( low , , upper , )]
(42)
where K
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N [ N ( upper , , low , )]
(43)
is fin thermal conductivity matrix of taking low bound of {~} and upper bound of
~ { } , K is fin thermal conductivity matrix of taking upper bound of {~} and low bound of
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~ { } , N is heat conducting matrix between frost layer and fin of taking low bound of {~} ~
and upper bound of { } , N is heat conducting matrix between frost layer and fin of taking
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~
upper bound of {~} and low bound of { } .
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The equilibrium equation is obtained as follows: (44)
K T N 0
(45)
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K T N 0
~ , and the Taylor series expansion is The random vector with average value is defined by
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carried out for K , T , Q , K , T , Q , and the corresponding expressions are obtained as follows:
m
K K 0 K i i
(46)
i 1 m
K K 0 K i i
(47)
i 1
m
N - N 0- Ni i i 1
(48)
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m
N N 0 N i i
(49)
i 1
m
T T0- Ti i
(50)
i 1
m
T T0 Ti i
(51)
i 1
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Putting expressions (46)-(51) to expressions (44) and (45) obtains the following equation set:
K 0T0 N 0 0 K 0 Ti Qi K iT0
(37)
K 0T0 N 0 0 K 0 Ti Qi K iT0
(38)
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Based on the above equations, the solution of (31) can be obtained.
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Input random parameter to construct boundary and initial conditions
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Calculate the random finite element model
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Obtain the temperature distribution of fin evaporator
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Calculate the frosting weight and refrigerating capacity of fin evaporator
Random finite element method
Obtain training and testing samples
Set parameters of rigdelet neural network
Train ridgelet neural network
Test ridgelet neural network
Output prediction results of frosting volume and refrigerating capacity
Ridgelet neural network Figure 3 Prediction procedure of Frosting weight and refrigerating capacity of fin evaporator based on random finite element method and ridgelet neural network 4 Prediction simulation analysis of Frosting weight and refrigerating capacity of fin evaporator
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The prediction simulation analysis of Frosting weight and refrigerating capacity is carried out based on random finite element method and Ridgelet neural network. A new type fin evaporator is used as the researching objective, and its structure is shown in figure 4. The fin pitches are 7.5/15mm. The fin pitch is distance between adjacent fins, as seen from figure 4, the fin pitch of the top three rows of fins is 7.5m, the fin pitch of bottom row of fin is 15mm, the fin areas are 0.245m2.
Figure 4 Structural diagram of new type fin evaporator The parameter testing is carried out for the new type fin evaporator, the fridge evaporator cover is changed to transparent material for observation, and the a video camera is set inside cabinet for
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recording. The environmental temperature is 311 K , the relative humidity is 70%，the temperature of cold storage room is 275 K , the temperature of cold room is 257 K . The testing device is shown in figure 5.
Figure 5 Testing device of fin evaporator Frosting situations of different location of fin evaporator at moment with most Frosting weight
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are shown in figure 6.
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(a) Location 1
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(b) Location 2 Figure 6 Frosting situations of different location of fin evaporator at moment with most Frosting weight To verify the effectiveness of the prediction model, ten sets of data are obtained based on test, the initial and boundary conditions are listed as follows: the initial temperature of fin surface is
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240K, and the heat transfer coefficient is 226 W / m K . The testing results are listed in table 1. 2
The obtained testing data is used as testing samples to compare with the simulation results.
Number
Table 1 Testing data of fin evaporator (Evaporation temperature=299 K )
Time/
Inlet
Relative
Inlet air
Refrigerant
Frosting
Refrigerating
min
wind
humidity of
temperature
flow/kg/h
weight/g
capacity/W
speed/
imported
/K
m/s
air/%
1
2
0.7
35%
257
2.0
34.5
112.3
2
4
0.9
40%
259
2.1
42.3
106.6
3
6
1.0
45%
260
2.3
50.6
99.5
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4
8
1.1
50%
261
2.4
57.4
92.7
5
10
1.2
55%
263
2.6
69.3
87.5
6
12
1.3
60%
265
2.7
75.8
81.2
7
14
1.4
65%
266
2.8
84.8
78.3
8
16
1.6
70%
267
3.0
96.2
70.3
9
18
1.8
75%
269
3.2
105.3
63.1
10
20
2.0
80%
271
3.4
114.8
57.9
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The random finite element method is used to generate 100 samples, the top 90 samples are used as training samples, the other 10 samples are used as testing samples, the initial conditions of 10 testing samples are same as that of test. The parameters of multiple children genetic algorithm are set as follows: the population scale
N0 is 75, the initial interval of weights and threshold is [-15,15], the required precision is 1.5×
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10-5, the cross probability pc is 0.75, the mutated probability pm is 0.15, retention ratio of elite individuals is 15%. The ridgelet neural network concludes three layers, the input layer concludes inlet wind speed ( x1 ), relative humidity of imported air ( x 2 ), inlet air temperature ( x3 ) refrigerant flow ( x 4 ) and
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evaporation temperature ( x5 ). The number of hidden layer nodes are confirmed based on gradually increasing method, which is 6. The output layers concludes Frosting weight ( y1 ) and
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refrigerant capacity ( y 2 ).
frosting parameters
of
fin
evaporator
are
described
by
L R parameter,
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c ( 401,3,5) LR , fr (0.3,0.02,0.04) LR , fr (0.0005,0.00002,0.00001) LR ,
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ma (6 104 ,3 105 ,2 105 ) , Fa (0.245,0.03 ,0.003)LR ,
(0.75,0.002,0.002 ) LR .
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The randomness of every variable is described by
c ~ N (0, c2 ) , ~ N (0, 2 ) , fr
fr
~ N (0, 2 ) , m ~ N (0, m2 ) , F ~ N (0, F2 ) , ~ N (0, 2 ) . fr
fr
a
a
a
a
The =0.85 level set is carried out for every frosting parameter. In order to verify the effectiveness of ridgelet neural network optimized by improved genetic algorithm (IGA-RNN), the BP neural network optimized by genetic algorithm (GA-BPNN) and the B-spline wavelet neural network optimized by genetic algorithm (GA-BSWNN) is also applied to prediction the Frosting weight and refrigerating capacity of fin evaporator. The iteration times are 450. And the prediction performance indexes of the three models are listed in table 2. Table 2 Prediction performance indexes of the three models
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Performance index
Prediction model GA-BSWNN
GA-BPNN
Average absolute relative error/% (at 100s)
10.32
11.67
12.64
Maximum absolute relative error/% (at 100s)
31.54
45.37
66.92
Root mean square error/% (at 100s)
188.62
249.65
255.37
30s
1.43×10-2
3.17×10-2
6.29×10-2
40s
2.94×10-3
1.46×10-2
4.38×10-2
50s
1.47×10-3
7.45×10-3
8.59×10-3
60s
1.43×10-4
2.79×10-4
4.28×10-4
70s
1.16×10-4
5.32×10-4
6.83×10-4
80s
6.32×10-5
9.63×10-5
1.07×10-4
90s
4.97×10-5
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E (n)
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IGA-RNN
6.28×10-5
8.25×10-5
As seen from table 2, the ridgelet neural network optimized by improved genetic algorithm (IGA-RNN) has least average absolute relative error, minimum absolute relative error, root mean
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square error and E (n) . The BP neural network optimized by genetic algorithm (GA-BPNN) has biggest average absolute relative error, maximum absolute relative error, root mean square error
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and E (n) . The prediction performance indexes of the B-spline wavelet neural network optimized
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by genetic algorithm (GA-BSWNN) in between that of the above two prediction models. Results show that the improved genetic algorithm has best global searching ability, and has lowest risk of falling into local optimization. The ridgelet neural network optimized by improved genetic algorithm (IGA-RNN) can give full play to the advantages of the ridgelet neural network and improved genetic algorithm, therefore can effectively improve the prediction precision. The ridgelet neural network optimized by improved genetic algorithm (IGA-RNN) has also highest convergence speed among the three prediction models. The IGA-RNN, GA-BSWNN and GA-BPNN are used to carry out precision comparing analysis of Frosting weight and refrigerating capacity of fin evaporator for the ten testing samples, and the final prediction results are listed in table 3. Table 3 Prediction results comparison among the three prediction models Number
Frosting weight/g
Time/ min
IGA-RNN
GA-BSWN
GA-BPNN
N
Refrigerating capacity/W Real
IGA-RNN
GA-BSWNN
GA-BPNN
value
Real value
1
2
35.1
36.4
38.3
34.5
113.5
114.6
116.2
112.3
2
4
43.6
44.9
46.2
42.3
107.4
108.7
120.2
106.6
3
6
51.3
52.5
53.7
50.6
100.2
102.3
104.6
99.5
4
8
58.2
59.8
61.4
57.4
93.4
95.6
97.8
92.7
5
10
70.4
72.3
74.6
69.3
88.7
89.3
101.6
87.5
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6
12
76.2
77.6
79.2
75.8
82.5
84.2
86.1
81.2
7
14
85.3
86.8
88.4
84.8
79.3
80.8
82.4
78.3
8
16
96.9
97.7
99.1
96.2
71.7
73.2
75.8
70.3
9
18
106.4
107.7
109.2
105.3
64.6
65.8
67.5
63.1
10
20
115.3
117.0
118.9
114.8
58.3
59.9
61.4
57.9
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As seen from table 3, the prediction values of frosting weight and refrigerating capacity based on ridgelet neural network optimized by improved genetic algorithm (IGA-RNN) are closest to the real among the three prediction models, and the prediction values of frosting weight and refrigerating capacity based on BP neural network optimized by genetic algorithm (GA-BPNN) has maximum deviation from the real deviation. The prediction results based on B-spline wavelet neural network optimized by genetic algorithm (GA-BSWNN) are between in that of the above two prediction models. As seen from table 3, the error of IGA-RNN ranges from 1% to 3%, the error of GA-BSWNN ranges from 2% to 6%, and the error of GA-BPNN ranges from 3% to 11%. Therefore the IGA-RNN has highest prediction precision. Therefore the ridgelet neural network optimized by improved genetic algorithm has highest prediction precision, the main reason for this results concludes two aspects, on one hand, the ridgelet neural network has arbitrary approximation ability, on the other hand, the improved genetic algorithm improve the learning and generalization abilities of the ridgelet neural network to improve the correctness of prediction. 5 Conclusions The Frosting weight and refrigerating capacity are important parameters of describing the frosting of fin evaporator, therefore the prediction model based on ridgelet neural network optimized by improved genetic algorithm is constructed to predict the two parameter. The ridgelet neural network is constructed through using ridgelet function as excitation function of the hidden layer, and the ridgelet neural network has strong adaptability, fault tolerance and robustness depend on the advantages of ridgelet function. The improved genetic algorithm is used to optimize the ridgelet neural network, the improved genetic algorithm can dynamically regulate the intervals of weights and thresholds, the searching interval can be adaptively changed based on fitness degree of individuals in current multiple children generation to improve the prediction precision and efficiency of ridgelet neural network. The effect of parameters randomness on frosting temperature field of fin evaporator is considered, the random finite element model of frosting of fin evaporator is constructed to obtain the frost volume and refrigerating capacity of fin evaporator combing with frosting model of fin evaporator. The simulation results under different condition from random finite element model are used as the training samples and testing samples. In addition the testing device is designed to obtain the comparing samples using working conditions of testing samples to verify the correctness of simulation results. The simulation analysis results show that the ridgelet neural network has best prediction performance, the prediction precision and efficiency are all improved. Therefore the prediction model based on ridgelet neural network and improved genetic algorithm can effectively predict the frosting weight and refrigerating capacity. The prediction effect has the following main reasons: 1) The IGA-RNN has best prediction precision based on advantages of IGA and RNN. 2) The random finite element method introduces the random parameters into the heat transfer analysis of fin evaporator, and the fuzzy random parameters are used to describe the uncertainty of temperature field. Therefore the computation precision of the random finite element can be improved accordingly. The frosting
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situation of fin evaporator can be predicted correctly, and the proper defrosting measurements can be made to ensure stable operation of fin evaporator. The following contents should be studied in further, the optimal algorithm of ridgelet neural network should be improved for improving the prediction precision and efficiency. The other frosting parameters prediction model can be constructed based on this idea. The randomness of more frosting parameters can be considered to construct the perfect prediction.
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