model-based dimensionless neural networks for fin-and-tube condenser performance evaluation

Accepted Manuscript

Model-based dimensionless neural networks for fin-and-tube condenser performanceevaluation

Liang Yang, Ze-Yu Li, Liang-Liang Shao, Chun-Lu Zhang

PII: S0140-7007(14)00020-6

DOI: 10.1016/j.ijrefrig.2014.01.006

Reference: JIJR 2719

To appear in: International Journal of Refrigeration

Received Date: 6 December 2013

Revised Date: 10 January 2014

Accepted Date: 16 January 2014

Please cite this article as: Yang, L., Li, Z.-Y., Shao, L.-L., Zhang, C.-L., Model-based dimensionlessneural networks for fin-and-tube condenser performance evaluation, International Journal ofRefrigeration (2014), doi: 10.1016/j.ijrefrig.2014.01.006.

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http://dx.doi.org/10.1016/j.ijrefrig.2014.01.006

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ACCEPTED MANUSCRIPTDimensionless neural networks of condenser

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Model-based dimensionless neural networks for fin-and-tube condenser

performance evaluation

Liang Yang, Ze-Yu Li, Liang-Liang Shao*, Chun-Lu Zhang*

School of Mechanical Engineering, Tongji University, Shanghai 201804, China

ABSTRACT

The paper presents a dimensionless neural network modeling method for the fin-and-tube refrigerant-to-air

condensers which are widely used in air-cooled refrigeration and heat pump systems. The model-based

dimensional analysis method is applied to develop the dimensionless Pi-groups for the condenser performance.

The three-layer perceptron neural network is served as the performance model using the dimensionless Pi-groups

as its inputs and outputs. Compared with a well-validated tube-by-tube first-principle model, the standard

deviations of trained dimensionless neural networks are 0.66%, 4.83% and 0.11% for the heating capacity, the

refrigerant pressure drop and the air pressure drop, respectively. The accuracy is also consistent with the

previously developed dimensional neural networks. Furthermore, independent model validation using different

refrigerants shows that the dimensionless models have good potential in predicting the condenser performance if

the Pi-groups were in the range of training data.

Keywords: Condenser; Model; Neural network; Dimensional analysis

* Corresponding author. Tel.: +86-136-71825-133 (C.-L. Zhang).

E-mail address: [email protected] (L.-L. Shao); [email protected] (C.-L. Zhang)

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Nomenclature

A

Ac

Ao

b

area, m2

minimum flow area, m2

total surface area, m2

bias of neuron

cp

D

Dc

f

fn

specific heat at constant pressure, J kg-1 K-1

tube inner diameter, m

fin collar outside diameter, m

friction factor

functional equation

g

G

transfer function

mass flux, kg s-1 m-2

h enthalpy, J kg-1; output of the hidden layer

I, J, K

L

neuron number of the input, hidden and output layers, respectively

condenser total tube length, m

o transfer function in output layer

Q capacity, W

m mass flow rate, kg s-1

N

Nu

p

pcrit

Pr

Re

Rw

total number of data samples

Nusselt number

refrigerant pressure, Pa

refrigerant critical pressure, Pa

Prandtl number

Reynolds number

thermal resistance of tube wall and fins, K m2 W-1

T temperature,°C, K

u

U

velocity, m s-1; connection weight between input and hidden layers of neural network

overall heat transfer coefficient, W m-2 K-1

V volume flow rate, m3 s-1

w connection weight between hidden and output layers of neural network

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x vapor quality; input of neural network; variable

y output of neural network

∆p pressure drop, Pa

∆Tm mean time difference, K

Greek symbols

α heat transfer coefficient, W m-2 K-1

αg void fraction

λ thermal conductivity, W m-1 K -1

µ viscosity, Pa s

ρ density, kg m-3

σ surface tension, Pa m; ratio of the minimum flow area to frontal area

Subscripts

a air

db

f

fo

g

go

dry-bulb

refrigerant liquid

refrigerant liquid only

refrigerant vapor

refrigerant vapor only

in

NN

inlet

neural network

out outlet

r refrigerant

s

tp

wb

saturated refrigerant

two-phase

wet-bulb

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1 Introduction

Fin-and-tube heat exchangers are widely applied as the refrigerant-to-air condensers in refrigeration and

heat pump systems. How to predict the condenser performance has drawn a lot of attention since it’s put in use.

As the fin-and-tube condenser involves complex heat transfer processes and a large number of geometric

variables, very complex model and method are required to design this type of condensers (Domanski and Yashar,

2007; Jiang et al., 2006; Liu et al., 2004). However, owing to the fairly low robustness and time-consuming

simulations, this type of condenser design models are not recommended for direct use in the complex system

modeling, such as the multi-split air-conditioning systems.

Researchers have developed many simple semi-empirical or empirical models for fast and robust

simulations of heat exchanger performance in different systems, particularly complex systems. Among these

models, neural network (NN) theory is a fast developing branch because of its good generality and accuracy in

modeling multi-input multi-output nonlinear objects. For different purposes, NNs were used for prediction of

heat transfer coefficients (Jambunathan et al., 1996; Sablani et al., 2005; Wang et al., 2006; Zdaniuk et al., 2007),

prediction of heat exchanger performance (Akbari et al., 2012; Díaz et al., 1999; Hayati et al., 2009; Islamoglu,

2003; Jiang et al., 2012; Pacheco-Vega et al., 2001a; Pacheco-Vega et al., 2001b; Peng and Ling, 2009; Tan et al.,

2009; Wu et al., 2008; Xie et al., 2007; Zhao et al., 2010; Zhao and Zhang, 2010), optimization of heat

exchangers (Peng and Ling, 2008; Zdaniuk et al., 2011) and control of heat exchangers (Dı́az et al., 2001; Gang

and Wang, 2013; Vasičkaninová et al., 2011). For more information, there are two review papers. Yang (2008)

reviewed NN applications in thermal science and engineering. Recently Mohanraj et al. (2012) gave an overview

of NN applications in refrigeration, air-conditioning and heat pump systems.

From the NN applications in heat exchanger performance evaluation published to-date, we can find the

following main issues to be solved. Firstly, most NN models of heat exchangers performance were dimensional,

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which limits the generality of NNs. Secondly, most researchers were only concerned about the heat transfer

rate of heat exchanger and missed other important performance parameters such as pressure drops. Lastly,

the over-fitting risk was raised by training a relatively large NN with limited testing data.

This paper thus proposes a dimensionless NN model of fin-and-tube condenser performance. We look

forward to generality improvement from dimensional to dimensionless model. In addition to the heating

capacity, both air side and refrigerant side pressure drops are taken into account so that the NN model can

be well fit to the system modeling. A well-validated tube-by-tube first-principle condenser model is

employed as the training and testing data generator so that we can have sufficient data to cover the

envelope of dimensionless PI-groups and minimize the over-fitting risk.

2 Dimensional analysis of fin-and-tube condenser

For a given fin-and-tube condenser with certain working fluid, we can clearly identify the operating

parameters as the inputs and outputs of the condenser performance model (Zhao and Zhang, 2010).

However, this type of dimensional model would be very limited in use. Any changes on the working fluid

or the condenser configuration will lead to unpredictable results.

A general approach to better generality is to develop dimensionless Pi-groups for this problem using

the dimensional analysis method (Lienhard IV and Lienhard V, 2012). Furthermore, the dimensional

analysis method can be more effective in identifying dimensionless Pi-groups with a physics-based model.

Therefore, we do the dimensional analysis with a general fin-and-tube condenser model including

calculations of heating capacity, refrigerant pressure drop and air pressure drop.

2.1 Heating capacity

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The energy conservation equations for the condenser are

)()( outa,ina,ap,aouta,ina,a TTcmhhmQ −=−= (1)

)( inr,outr,r hhmQ −= (2)

The heat transfer rate equation of condenser is

),,,,(fn ins,outr,inr,outa,ina,rmr TTTTTUATUAQ =∆= (3)

where, fn(⋅) represents some function of the variables in the parentheses. The overall heat transfer coefficient U

is based on the refrigerant heat transfer area of condenser. Namely,

aa

rw

r

11

A

AR

U αα++= (4)

In addition, the refrigerant property relation

),( rsrr TThh = (5)

Generally speaking, the heating capacity Q and the leaving temperatures Ta,out, Tr,out are the unknowns to be

solved by the governing equations (1) ~ (3). When the heating capacity is solved, the remaining two can be

figured out simultaneously. Therefore, we can write the dimensional functional equation for the heating capacity

Q in terms of equations (1) ~ (5).

{ { {

−=

−−−−44 344 21321

K

inr,ina,ins,

KW

r

Kkg sW

ap,

s kg

ra

W

,,,,,fn1111

TTTUAcmmQ (6)

Since U and Ar always show up together as a product, we treat UAr as one independent variable. To feature the

heat transfer temperature difference, we also take Ts,in – Ta,in as the representative. Thus there are seven variables

in three dimensions (W, K, kg s-1). In terms of the Buckingham Pi-theorem (Lienhard IV and Lienhard V, 2012),

we look for 7 – 3 = 4 Pi-groups. They would have to be

)( ina,ins,ap,a

1 TTcm

Q

−=Π (7)

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ina,ins,

in2 TT

T

−=Π (8)

a

r3 m

m=Π (9)

ap,a

r4 cm

UA=Π (10)

The functional equation (5) is therefore reduced to a dimensionless one with less variables.

( )4321 ,,fn ΠΠΠ=Π (11)

According to equation (4), we know that U is a function of αr and αa for a given condenser. Usually, the

heat transfer coefficients αr and αa are related to many other variables in a very complex way. Therefore, the

Pi-group Π4 cannot be simply calculated unless αr and αa are figured out. It requires further dimensional analysis

on αr and αa.

2.1.1 Air-side heat transfer coefficient

To evaluate the air-side heat transfer coefficient αa, we introduce the Colburn j factor which is commonly

used to feature the air-side heat transfer of fin-and-tube heat exchangers (Wang and Chi, 2000; Wang et al.,

2000a).

32

aap,maxa

a Prcu

jρ

α= (12)

where umax = ua/σ. The term, σ, is the ratio of the minimum flow area to frontal area.

The Colburn j factor can be empirically correlated as the following form (Wang et al., 2000a).

2a1ReCCj = (13)

where C1 and C2 depend on the physical dimensions of the heat exchanger.

In terms of equations (12) and (13), we can directly have the dimensionless functional equation of the

air-side heat transfer coefficient.

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( )aaap,maxa

a Pr,Refn=cuρ

α (14)

It can be further rearranged as

( )aaa Pr,RefnNu = (15)

Therefore the Pi-groups for air-side heat transfer are

a

caaa4 Nu

λα D==Π (16)

a

caaa5 Re

µρ Du==Π (17)

a

ap,aa6 Pr

λµ c

==Π (18)

where Dc is the fin collar outside diameter.

2.1.2 refrigerant side heat transfer coefficient

In a condenser, the condensation heat transfer is predominant on refrigerant side. To simplify the problem,

we only consider refrigerant in-tube condensation heat transfer. The Shah correlation (Shah, 1979) is employed

since it has been widely validated with different refrigerants.

−+−==38.0

crit

04.076.08.0

ff

rr )(

)1(8.3)1(NuNu

pp

xxx

D

λα

(19)

where

0.4f

0.8ff PrRe023.0Nu= (20)

Equation (19) is for the local heat transfer coefficient. To get the average heat transfer coefficient, we have

∫=1

0 rr d)( xxαα (21)

Therefore, the dimensionless functional equation for refrigerant-side heat transfer would have to be

=

critffr ,Pr,RefnNu

p

p (22)

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But after trial-and-error, the pressure ratio is found having marginal impact on the condenser performance.

Consequently, the Pi-groups for refrigerant-side heat transfer are

r

rr4r Nu

λα D==Π (23)

f

fff7 Re

µρ Du==Π (24)

f

fp,ff8 Pr

λµ c

==Π (25)

2.1.3 Overall heat transfer coefficient of condenser

For a given condenser, the overall heat transfer coefficient can be expressed as a function of the heat

transfer coefficients on both sides. Namely,

),(fn ar αα=U (26)

or,

),(fn,fn 4r4aa

ca

f

r

f

ΠΠ=

=

λα

λα

λDDUD

(27)

Combining equations (10) and (27) to eliminate the overall heat transfer coefficient U, we have a new

countable Π4,

ap,a

f4 cm

Lπλ=Π (28)

Eventually, we obtain the Pi-groups and the dimensionless functional equation for the heating capacity of

condenser.

),,,,,,(fn 87654321 ΠΠΠΠΠΠΠ=Π (29)

where Π4 is defined by equation (28).

2.2 Refrigerant pressure drop

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In most cases, the gravity effect is negligible and two-phase region is the main portion of condenser.

Therefore, we concentrate on the two-phase pressure drop (Rohsenow et al., 1998) in a condenser.

onacceleratifrictionTP ppp ∆+∆=∆ (30)

where the acceleration pressure drop is

+−

−=∆gg

2

gf

22

onaccelerati )1(

)1(

d

d

αραρxx

z

pGp (31)

32fg

g )]()1([1

1

ρρα

xx−+= (32)

z is the direction of refrigerant flow.

Two-phase frictional pressure drop can be calculated by the Friedel correlation (Friedel, 1979).

f035.0045.0friction WeFr

24.3p

FHEp ∆

+=∆ (33)

where

f

2

fof 2ρG

D

Lfp =∆ (34)

+−=

fog

gof22)1(f

fxxE

ρρ

(35)

224.078.0 )1( xxF −= (36)

7.0

f

g

19.0

g

f

91.0

g

f 1

−

=

µµ

µµ

ρρ

H (37)

2tp

2

FrρgD

G= (38)

tp

2

Weσρ

DG= (39)

1

fgtp

1−

−+=ρρ

ρ xx (40)

Friction factor ffo is typically a function of the corresponding Reynolds number.

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In terms of equations (30)~(40), we can write the functional equation for refrigerant-side pressure drop.

{ { { { {

=∆Pa

inm Pa

s Pa

gf

m kg

gf

ms m kgPa

r ,,,,,,,,fn3-

1-2-

pDLGp σµµρρ321321

(41)

There are ten variables in four dimensions (kg, m, s, Pa). So we look for 10 – 4 = 6 Pi-groups. They would have

to be

in

r9 p

p∆=Π (42)

Dp

G

inf

f10 ρ

µ=Π (43)

f

g11 ρ

ρ=Π (44)

f

g12 µ

µ=Π (45)

in

13 Dp

σ=Π (46)

D

L=Π14 (47)

Since the condenser geometry variations, particularly the numerous options in refrigerant circuitry

(Domanski and Yashar, 2007) are too complex to be expressed as Pi groups, we are not able to deal with it in a

general way for the lumped NN model and have to keep it fixed. At this point, the dimensionless functional

equation for refrigerant pressure drop of condenser is

),,,(fn 131211109 ΠΠΠΠ=Π (48)

2.3 Air pressure drop

The relation between air-side friction factor and pressure drop is as follows (Wang and Chi, 2000; Wang et

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al., 2000a).

−+−

∆= 1)1(

2

outa,

ina,22a

aina,

ina,

ma,

o

ca ρ

ρσ

ρρρ

G

p

A

Af (49)

where σρρ amaxc uuG == is the mass flux of the air based on the minimum flow area.

The friction factor can be expressed as (Wang et al., 2000b)

4Re3CCf = (50)

where C3 and C4 depend on the physical dimensions of the heat exchanger.

According to equations (49) and (50), at certain air pressure, we can write the functional equation for air

pressure drop. Note that we don’t take the air side geometry variations, particularly the fin details into account

because there is no generic condenser level physics-based model or correlations can precisely describe the

geometry variations of air side and we have already ignored the geometry variations of refrigerant side.

{ { { { {

=∆−− Pa

ina,

m

c

s Pa

a

s m

a

m kg

outa,ina,

Pa

a ,,,,,13

pDufnp µρρ43421

(51)

There are seven variables in four dimensions (kg, m, s, Pa). So we look for 7 – 4 = 3 Pi-groups. They would have

to be

ina,

a15 p

p∆=Π (52)

ina,

outa,16 ρ

ρ=Π (53)

ina,c

aa17 pD

uµ=Π (54)

Therefore, the dimensionless functional equation for air pressure drop of condenser is

),(fn 171615 ΠΠ=Π (55)

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3 Condenser performance data bank

Sufficient data are very important for NN training and testing to mitigate the over-fitting risk.

Meanwhile, in order to fairly compare the present dimensionless NNs and the previous dimensional ones

(Zhao and Zhang, 2010), the same condenser (as shown in Fig. 1) and the performance data bank generated

by the same well-validated tube-by-tube first-principle model is used in this study. In the data bank, the

primary working fluid is R410A and there are 2074 sets of data for R410A in NNs training and testing. To

verify the generality of the present dimensionless NNs for different refrigerants, additional 36 sets of data

using R134a and R22 as working fluids are added for NN testing. The range of mass flow rates and

temperatures of R134a and R22 data is within the range of R410A data.

The ranges of Pi-groups for the heating capacity, refrigerant pressure drop and air pressure drop are

given in Tables 1 ~ 3, respectively. All the thermal properties of refrigerants are calculated using REFPROP

9.0 (Lemmon et al., 2010).

4 Dimensionless neural networks of condenser performance

4.1 Neural network

Nowadays neural networks are widely applied in nonlinear function approximation. Among hundreds

of types of NNs, the multi-layer perceptron (MLP) network is the most popular neural network in

engineering application, and a three-layer perceptron network is capable of approximating any function

with a finite number of discontinuities. Therefore, three-layer perceptron network was used in the previous

work (Zhao and Zhang, 2010) and is still used in the present work for apple-to-apple comparison.

The three-layer perceptron network consists of one input layer, one hidden layer and one output layer,

as it is shown in Fig.2. The number of neurons in the three layers is I, J, and K, respectively. The

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relationship of network inputs and outputs can be mathematically expressed as below.

),,1(1

,1,hidden JjbxughI

ijiijj L=

+= ∑=

(56)

),,1(,21

,out Kkbhwgy k

J

jjjkk L=

+= ∑

= (57)

where, uj,i is the linked weight between the ith neuron of input layer and the jth neuron of hidden layer. b1,j is the

bias of the jth neuron of hidden layer. wk,j is the linked weight between the jth neuron of hidden layer and the kth

neuron of output layer. b2,k is the bias of the kth neuron of output layer. g(x) represents the transfer function of

neurons.

Nonlinear differentiable transfer functions are commonly employed in the hidden layer of NN, and pure

linear transfer functions are usually employed in the output layer. In terms of the best practices we got from the

previous study (Zhao and Zhang, 2010), the log-sigmoid transfer function is used in the hidden layer. Namely,

xe

xg −+=

1

1)(hidden (58)

xxg =)(out (59)

The neural network toolbox of MATLAB version 2011b (MATLAB, 2011) is applied in this study. To be

fair, the dimensional NN shall be re-trained by the same version of MATLAB. In terms of the experience from

the previous work (Zhao and Zhang, 2010), the Bayesian-Regulation algorithm (trainbr) is used for NN training.

The functional equations (29), (48) and (55) will be modeled by the neural networks hereinafter.

4.2 Neural network training and testing

4.2.1 Heating capacity

To predict the heating capacity, the Pi-groups Π2 ~ Π8 are used as the inputs and Π1 is served as the output

of the dimensionless NN. As usual, we split the total R410A condenser performance data into two parts. The

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training samples are used for NN training and the testing samples are used to validate the prediction capability of

the trained NN. Table 4 shows the overall standard deviations at different percentages of training samples out

of total R410A data. At certain percentage, the standard deviation decreases with the number of hidden

neurons at the beginning. When the standard deviation reaches a minimum value, it rebounds with more

hidden neurons. The more the hidden neurons, the higher the over-fitting risk. On the other hand, at certain

number of hidden neurons, more samples out of the total R410A data for NN training will mitigate the

over-fitting risk. But a large number of training samples will lead to low applicability. As a trade-off, we

choose 5 hidden neurons and randomly pick 30% R410A data for NN training. The process is repeated

several times to ensure the repeatibility of training and testing accuracies.

As for the dimensional NN, we randomly pick 30% R410A data for NN training as well. The inputs

and output of NN are the same as the previous work (Zhao and Zhang, 2010). The overall statistical results are

shown in Table 5. In all tables of this section, the average deviation (A.D.) and the standard deviation (S.D.) are

defined as follows.

%1001

A.D. NN ×−

= ∑N y

yy

N (60)

%100)A.D.(1

1S.D. NN ×−

−−

= ∑N y

yy

N (61)

To balance accuracy and complexity of the dimensional NN, we choose 4 hidden neurons from Table 5.

Meanwhile, overall 0.66% S.D. is close to that of the dimensionless NN from Table 4. Since the training data are

sufficient, it’s not necessary to use the cross-validation method which will be more helpful if training data is

insufficient.

The comparison of dimensional and dimensionless NNs for the heating capacity is given in Table 6. The

accuracy of both NNs is very close. The dimensionless NN has more inputs and hidden neurons because it took

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into account the fluid properties so as to predict the condenser performance with different refrigerants.

To validate the predictability of trained NNs in different refrigerants, we use R134a and R22 data for

independent testing. Table 7 tells the results. The prediction accuracy of the dimensional NN looks not bad for

both refrigerants since all the dimensional NN inputs (mass flow rates and temperatures) are independent of

other fluid properties and the range of R22 and R134a data is inside the envelope of R410A training data. By

comparison, the prediction accuracy of the dimensionless NN for R134a and R22 is quite different. For R134a,

the accuracy is much worse than that of the dimensional NN. For R22, however, the accuracy is much nicer than

that of the dimensional one. The root cause can be found from Table 1. For R134a, the range of Π6 is completely

outside of the training envelope (R410A data). For R22, all Pi-groups are inside the training envelope. Therefore,

the dimensionless NN would give satisfactory predictions in different refrigerants only if the new data were

inside the envelope of training data. This condition shall be emphasized as long as the NN method is applied.

4.2.2 Refrigerant pressure drop

In terms of equation (48), the Pi-groups Π10~ Π13 are used as the inputs and Π9 is the output of the

dimensionless NN for refrigerant pressure drop. The trial-and-error NN training process is similar to previous

one and is therefore omitted. Table 8 gives a summary of the re-trained dimensional NN and the trained

dimensionless NN, where we can come to the following conclusions.

Even though we took into account the fluid properties, the dimensionless NN has one less input and one

more hidden neuron than the dimensional one. Therefore, the architecture complexity of both NNs is very close.

The overall deviations of R410A data are close to each other as well.

According to Table 2, both R134a and R22 data are outside the training envelope (R410A data). Therefore,

the prediction on R134a and R22, as expected, has larger deviations. By comparison, the dimensionless NN is

superior to the dimensional one.

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4.2.3 Air pressure drop

In terms of equation (55), the Pi-groups Π16 and Π17 are the inputs and Π15 is the output of the

dimensionless NN for air pressure drop. The trial-and-error NN training process is omitted because it is similar

as before. Table 9 gives a summary of the re-trained dimensional NN and the trained dimensionless NN. At the

same size of network, the dimensionless NN have much better accuracy over the dimensional one. In addition,

the fluid properties, as expected, have little influence on the air pressure drop.

To validate the above trained NNs, they are also compared with experimental data of the current specific

condenser. Figure 3 shows the schematic of the testing facility. Measurements, as shown in Figure 4, include the

air volume flow rate, the air entering and leaving dry-bulb temperatures, the air pressure drop through the

condenser, the refrigerant mass flow rate, the refrigerant entering and leaving temperatures and pressures. The

heating capacity of the condenser is obtained by the refrigerant mass flow rate multiplying the enthalpy

difference. The refrigerant enthalpy is a function of the refrigerant pressure and temperature. In the experiments,

measurement uncertainties of refrigerant pressure, air pressure drop, temperature, and refrigerant mass flow rate

are ±0.25%, ±0.1 mmH2O, ±0.1 K, and ±2%, respectively. In turns, the uncertainties of the heating capacity, the

refrigerant pressure drop, and the air pressure drop are about ±2%, ±10 kPa and ±0.1 mmH2O, respectively.

Figure 4 shows the comparison between the trained NNs and the lab test data of the heating capacity and

the refrigerant pressure drop. Good predictions are made for both the heating capacity and refrigerant pressure

drop. In addition, the air flow rate kept almost constant in all tests because the fan speed is fixed. The NN

prediction on the air pressure drop is about 5%.

Note that the present work focused on the method of developing dimensionless NNs for fin-and-tube

condenser performance evaluation. The condenser used in this work is simply a case for demonstration. All

weights and biases of the trained NNs are only meaningful for the specific condenser. Therefore, weights and

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biases of NNs are not detailed here. In theory, the NN is a nonlinear data regression model, which means its

accuracy largely depends on the training data set. The readers are suggested to use the method more than the

specific trained NNs when doing the similar work.

5 Conclusions

In the present work, we developed dimensionless NNs for the fin-an-tube condenser performance

evaluation using the model-based dimensional analysis method. We took into account the fluid properties in the

dimensionless Pi-groups and used the Pi-groups as the inputs and outputs of NNs. The dimensionless NNs have

equivalent accuracy as the dimensional ones. Compared with a well-validated tube-by-tube first-principle model,

the standard deviations of trained dimensionless neural networks are 0.66%, 4.83% and 0.11% for the heating

capacity, the refrigerant pressure drop and the air pressure drop, respectively. In predicting the condenser

performance with different refrigerants, the dimensionless NNs have promising prediction capability over the

dimensional ones.

This is a new attempt to the hybrid modeling of condenser performance using NNs and physics-based

models. There is still large room for improvement. Further investigations will involve accuracy improvement

outside the training envelope, better alternatives to NN (e.g. ANFIS - Adaptive Neuro Fuzzy Inference System,

SVM - Support Vector Machine), and so on.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 51206123), the

Innovation Program of Shanghai Municipal Education Commission (Grant No. 11ZZ30), and the China

Postdoctoral Science Foundation (Grant No. 2013M541539).

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Fig. 1

Tube length: 890mm

Tube diameter: 7.4mm

36 tubes per row;6 straight through

counter flow circuits;18 tubes / circuit

Air flow

Refrigerant

Tube pitch: 21.9mm

Row pitch:12.7mm

Fin space: 1.41mm

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� � �

…hj …hJ h1

Input layer

Hidden layer

Output layer

Bias

…yk

x1 …xi …xI

y1 …yK

o1 …oK …ok

Fig. 2

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Fig. 3

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Fig. 4

0

20

40

60

80

1000 20 40 60 80 100

0

5

10

15

0 5 10 15

Pre

dict

ed r

efr

ige

rant

pre

ssur

e dr

op (

kPa

)

Measured refrigerant pressure drop (kPa)

Pre

dict

ed

hea

ting

capa

city

(kW

)

Measured heating capacity (kW)

heating capacityrefrigerant pressure drop

10%

-10%

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Figure captions

Fig. 1 Schematic of the fin-and-tube condenser

Fig. 2 Architecture of three-layer perceptron neural network

Fig. 3 Schematic of the condenser performance testing facility

Fig. 4 Comparison of trained neural networks and experimental data

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Table 1 Range of Pi-groups for heating capacity

R410A R134a R22

Π1 0.724 ~ 1.247 0.805 ~ 0.982 0.809 ~ 0.923

Π2 2.667 ~ 14.17 4.25 ~ 8.5 4.25 ~ 8.5

Π3 0.023 ~ 0.205 0.039 ~ 0.103 0.040 ~ 0.106

Π4 0.233 ~ 0.446 0.259 ~ 0.419 0.268 ~ 0.432

Π5 35675 ~ 503155 39393 ~ 130088 45411 ~ 153970

Π6 2.291 ~ 2.899 3.15 ~ 3.164 2.427 ~ 2.485

Π7 138.5 ~ 234.4 150.7 ~ 226.1 150.7 ~ 226.1

Π8 0.745 ~ 0.764 0.745 ~ 0.745 0.745 ~ 0.745

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Table 2 Range of Pi-groups for refrigerant pressure drop

R410A R134a R22

Π9 0.0009 ~ 0.076 0.0155 ~ 0.100 0.0084 ~ 0.059

Π10 2.8E-9 ~ 1.6E-8 1.4E-8 ~ 2.9E-8 8.52E-9 ~ 1.90E-8

Π11 0.105 ~ 0.234 0.051 ~ 0.071 0.068 ~ 0.093

Π12 0.156 ~ 0.268 0.086 ~ 0.102 0.110 ~ 0.130

Π13 3.8E-8 ~ 2.0E-7 4.1E-7 ~ 6.8E-7 2.7E-7 ~ 4.5E-7

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Table 3 Range of Pi-groups for air pressure drop

R410A R134a R22

Π15 2E-4 ~ 4E-4 2E-4 ~ 4E-4 2E-4 ~ 4E-4

Π16 0.896 ~ 0.986 0.942 ~ 0.975 0.942 ~ 0.975

Π17 8.4E-9 ~ 1.31E-8 8.5E-9 ~ 1.28E-8 8.5E-9 ~ 1.28E-8

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Table 4 Standard deviations at different percentages of training samples out of total R410A data

No. of hidden neurons Percentage of training samples out of total data

10% 30% 50% 75%

1 6.67 6.64 6.55 6.62

2 2.02 2.03 2.04 2.00

3 1.05 1.01 1.02 1.01

4 0.89 0.86 0.86 0.85

5 0.86 0.63 0.64 0.63

6 0.67 0.59 0.59 0.57

7 0.57 0.46 0.44 0.47

8 0.55 0.38 0.43 0.38

9 0.48 0.33 0.33 0.36

10 0.45 0.27 0.25 0.29

11 0.49 0.23 0.21 0.22

12 0.50 0.23 0.22 0.17

13 0.57 0.26 0.22 0.16

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Table 5 Deviations of dimensional NN using all R410A data

No. of hidden neurons A.D.(%) S.D.(%)

1 0.29 4.92

2 0.07 1.26

3 -0.02 0.84

4 0.006 0.66

5 -0.02 0.53

6 0.02 0.39

7 0.001 0.37

8 0.004 0.26

9 0.013 0.21

10 -0.007 0.23

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Table 6 Comparison of dimensional and dimensionless NNs for heating capacity

Dimensional NN Dimensionless NN

NN architecture 5-4-1 7-5-1

No. of weights and biases 29 46

A.D. (%) 0.006 0.02

S.D. (%) 0.66 0.63

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Table 7 Comparison of dimensional and dimensionless NNs with different refrigerants


R134a A.D.% 4.37 12.41

S.D.% 0.75 13.22

R22 A.D.% 5.73 0.05

S.D.% 0.47 1.81

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Table 8 Summary of refrigerant pressure drop NNs




R410A A.D. (%) 0.13 0.28

S.D. (%) 4.26 4.83

R134a A.D. (%) 47.2 11.8

S.D. (%) 3.9 11.5

R22 A.D. (%) 32.2 1.54

S.D. (%) 4.1 7.4

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Table 9 Summary of air pressure drop NNs




R410A A.D. (%) 0.09 0.006

S.D. (%) 1.91 0.11

R134a A.D. (%) 1.0 0.006

S.D. (%) 1.17 0.07

R22 A.D. (%) 0.99 0.02

S.D. (%) 1.20 0.09

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ACCEPTED MANUSCRIPT

Highlights: > We developed dimensionless neural networks for fin-and-tube condensers. > Dimensionless Pi-groups were derived from model-based dimensional analysis method. > Three-layer perceptron neural network was served as the performance model. > Neural networks well predicted the condenser performance with different refrigerants.

model-based dimensionless neural networks for fin-and-tube condenser performance evaluation

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