Modelling of heat transfer coefficients during condensation inside an enhanced inclined tube

Abstract

In this study, experiments were conducted for the flow of R-134a condensing in an enhanced inclined tube at a saturation condensing temperature of 40 °C. The enhanced tube had a helix angle of 14° with a mean internal diameter of 8.71 mm. The mass velocities were varied from 200 to 600 kg m−2 s−1, while the inclination angles were varied from − 90° to + 90°. It was found that the inclination angle had a considerable effect on the flow patterns and the thermal performance. It was also found that the maximum heat transfer coefficients were obtained at tube inclinations of between − 15° and − 5° (downward flows). By using the experimental data and artificial neural networks (ANN), a model was proposed to predict the heat transfer coefficients during condensation inside the enhanced inclined tube. By using four statistical criteria, the performance of the proposed model was examined against experimental data, and it was found that ANN was a useful tool for the prediction of the heat transfer coefficients based on the effective parameters of vapour quality, mass velocity and inclination angle.

Full text

Vol.:(0123456789)
1 3
Journal of Thermal Analysis and Calorimetry
https://doi.org/10.1007/s10973-020-09930-2
Modelling ofheat transfer coecients duringcondensation
insideanenhanced inclined tube
D.R.E.Ewim1 · A.O.Adelaja2,3,5· E.J.Onyiriuka4· J.P.Meyer2· Z.Huan1
Received: 19 January 2020 / Accepted: 9 June 2020
© Akadémiai Kiadó, Budapest, Hungary 2020
Abstract
In this study, experiments were conducted for the flow of R-134a condensing in an enhanced inclined tube at a saturation
condensing temperature of 40°C. The enhanced tube had a helix angle of 14° with a mean internal diameter of 8.71mm.
The mass velocities were varied from 200 to 600kgm−2s−1, while the inclination angles were varied from − 90° to + 90°. It
was found that the inclination angle had a considerable effect on the flow patterns and the thermal performance. It was also
found that the maximum heat transfer coefficients were obtained at tube inclinations of between − 15° and − 5° (downward
flows). By using the experimental data and artificial neural networks (ANN), a model was proposed to predict the heat transfer
coefficients during condensation inside the enhanced inclined tube. By using four statistical criteria, the performance of the
proposed model was examined against experimental data, and it was found that ANN was a useful tool for the prediction of
the heat transfer coefficients based on the effective parameters of vapour quality, mass velocity and inclination angle.
Keywords Condensation· Heat transfer coefficient· Smooth tube· Enhanced tube· ANN
List of symbols
Acs
Test section cross-sectional area (m2)
Ai
Internal surface area (m2)
e Fin height (m)
G Mass velocity (kgm−2s−1)
h Heat transfer coefficient (Wm−2K−1)
Lt Heat transfer length of test section (m)
Q.
Heat transfer rate (W)
T Temperature (K)
Tsat Saturation temperature (K)
̄
Tw,i
Average wall inner temperature (K)
x Vapour mass fraction (−)
Greek symbols
𝛽
Inclination angle (°)
Subscripts
a Actual
pred Predicted
r, in Inlet refrigerant temperature
r, out Outlet refrigerant temperature
w, test Water side of the test section
Abbreviations
EF Enhancement factor
MAE Mean absolute error
MP Membership function
MRE Mean relative error
RMSE Root-mean-square error
MSE Mean squared error
MAPE Mean absolute percentage error
R Correlation coefficient
R2 Coefficient of determination
RMSE Root-mean-square error
* D. R. E. Ewim
ewimdr@tut.ac.za
* A. O. Adelaja
aadelaja@unilag.edu.ng
* J. P. Meyer
josua.meyer@up.ac.za
Z. Huan
huanz@tut.ac.za
1 Department ofMechanical Engineering, Mechatronics
andIndustrial Design, Tshwane University ofTechnology,
Private Bag X860, Pretoria0001, SouthAfrica
2 Department ofMechanical andAeronautical
Engineering, University ofPretoria, Private Bag X20,
Hatfield,Pretoria0028, SouthAfrica
3 Department ofMechanical Engineering, University ofLagos,
Akoka,Yaba, LagosState101017, Nigeria
4 Department ofMechanical Engineering, University ofBenin,
PMB 1154, BeninCity, EdoState, Nigeria
5 Unilag-LG Air Conditioning Academy, University ofLagos,
Akoka,Yaba, LagosState101017, Nigeria

D.R.E.Ewim et al.
1 3
Introduction
Condensation is a critical heat transfer process because it
finds applications in refrigeration, air conditioning, power
plants, automotive processes, chemical processes and so
forth. Hence, a thorough understanding of this rather
complicated two-phase process is essential for the proper
design and optimisation of condensers. One of the meth-
ods of increasing the heat transfer coefficient is to make
the size of a condenser larger. However, from a practical
viewpoint, this is not advisable due to capital and main-
tenance costs. This drawback possibly necessitated the
invention of enhanced tubes, which have been widely used
to increase the thermal performance of condensers, lead-
ing to the design of compact heat exchangers and savings
in terms of material cost. Enhanced tubes are characterised
by the presence of fins, which can be axial, helical, cross-
grooved or herringbone.
A lot of researchers [1–17] have studied enhanced tube
surfaces effects on flow patterns, pressure differences and
heat transfer. In those studies, it was found that enhanced
tubes generally improved thermal performance but with
corresponding pressure drop increase. Another method of
increasing the heat transfer coefficient is by experimenting
with the positioning of the condenser (in other words, its
inclination). However, there are limited studies [18–22] on
the effects of inclination on the heat transfer characteristics
of smooth and enhanced tubes. In these limited studies,
when the vapour qualities and mass velocities are below
average, there is a notable increase in the effect of tube
inclination. Furthermore, it was found that the flow pattern
and the arrangement of the liquid phase were altered by
the orientation of the tube.
In general, studies can be experimental, theoretical,
computational or analytical. The challenge with experi-
mental studies is that they are usually capital-intensive
to carry out because of the nature of the equipment and
instrumentation needed. Furthermore, they are time-con-
suming and challenging. On the other hand, computational
fluid dynamics (CFD) work is still developing, and limited
study has yet coupled the effect of microfin and inclination
on heat transfer coefficients in tubes.
Soft computing methods such as an artificial neural net-
work (ANN) have been touted as an optimisation tool for
the future. These methods are gaining significant ground in
a variety of engineering applications such as making deci-
sion, recognising pattern, making computers, processing
information, systems control, manipulating mathematical
expressions and robotics. ANN is a soft computing tool
that has been used to successfully model, optimise and
predict heat transfer coefficients and other thermal perfor-
mance parameters in a variety of heat transfer applications
[23–50]. The advantage of ANN is that it is faster than
other tools, and it allows understanding of intricacies of
thermal systems that otherwise would have been impos-
sible to characterise with straightforward analytical meth-
ods. However, the challenge is that no ANN study has
coupled inclination for condensation inside enhanced
inclined tubes. A brief review of some literature on the
applicability of ANN for heat transfer literature will be
presented below.
Hu etal. [51] developed an (ANN) model of a heat
exchanger operating on existing mechanical ventilation and
air-conditioning (MVAC) system in order to predict its per-
formance. They tested two approaches and considered all the
components of the MVAC system. They obtained both static
and dynamic models for a heat exchange system mounted in
an air handler unit (AHU) by using the neural network tech-
nique. The AHU is the most essential module of the MVAC
system. The obtained model was shown to be very good.
Romero-Mendez etal. [23] studied the rate of convection
of a refrigerant flowing inside a small tube by usingartifi-
cial neural networks (ANNs). In their experimental study,
they used the inverse Rankine refrigeration cycle to estimate
heat transfer data for R-134a refrigerant mini-tubeevapora-
torset-up with constant heat flux. Their large data were
split into 75:25 ratio of which 75% was employed for train-
ing the ANN, while 25% was used for prediction purposes.
They trained several neural network configurations, and they
selected the most accurate to predict the thermal charac-
teristics of the flow. It was observed that ANN was a good
predictive tool for finding out the convective heat transfer
rates in a mini-tubeevaporator.
Heat transfer characteristics were investigated during
the evaporation of R-134a inside a vertical smooth and five
pieces of the corrugated tube by Balcilar etal. [33]. They
used their experimental data to numerically determine the
best artificial intelligence method for predictive purposes.
Heat flux, mass velocity, two-phase friction factor, vapour
quality, evaporating temperature, two-phase multiplier,
temperature difference Reynolds number, Weber number,
Froude number, Bond number, depth of corrugation and
helix angle for the tested corrugated tubes were the ANN
input. Also, the ANN outputs were the measured pressure
drop and measured condensation heat transfer coefficient.
They modelled the evaporation heat transfer characteristics
of R-134a by numerous ANN methods such as radial basis
function networks (RBFNs) and multi-layer perceptron
(MLP). They observed that the MLP and RBFN methods
were in good agreement with experimental data. Further-
more, the reliance of the ANNs output from input values was
studied, and they developed new correlation for ANN-based
heat transfer coefficient from their study.
An artificial neural network (ANN) model was devel-
oped for predicting friction factor in smooth and microfin

Modelling ofheat transfer coefficients duringcondensation insideanenhanced inclined tube
1 3
tubes under heating, cooling and isothermal conditions by
Cebi etal. [52]. Data taken from a vertically positioned heat
exchanger experimental set-up were used to train the ANN.
The ANN had the following configurations tested: a radial
basis function networks, multi-layered feed-forward neural
network with backpropagation algorithm, and hybrid PSO-
neural network algorithm. The ANN had the following as
its inputs: experimental condition depending on isothermal,
heating, or cooling conditions, the ratio of cross-sectional
flow area to hydraulic diameter and mass flow rate, while
the output of the ANN system was the friction factor. They
observed accurate prediction of the friction factors by the
ANN system regardless of the tube type. A dependency
study led to the discovery that the tube geometry was found
to be the strongest parameter that affected the network.
Azizi and Ahmadloo [29] employed ANN to predict the
experimental heat transfer coefficient data of Meyer etal.
[21]. Their network was trained by using a total of 440
experimental data points extracted from the literature. The
inputs to their MLP ANN model were: the inclination angle,
saturation condensing temperature, mass velocity and mean
vapour quality, while the output or target variable was the
heat transfer coefficient. They tested many MLP networks
and found that the optimum structure was eighteen neurons
which made it probable to determine with high accuracy for
the heat transfer coefficient.
A robust prediction technique was developed by Zende-
hboudi and Li [26] to predict the pressure difference dur-
ing condensation in inclined smooth tubes. They evaluated
several universal intelligent models: genetic algorithm-
least square support vector machine (GA-LSSVM), hybrid
approach-adaptive neuro-fuzzy inference system (Hybrid-
ANFIS), particle swarm optimisation-artificial neural net-
work (PSO-ANN) and genetic algorithm-power law com-
mittee with intelligent systems (GA-PLCIS), for correctly
determining the frictional pressure drop (ΔPfric), and meas-
ured pressure drop (ΔP). In accordance with their compara-
tive results, they stated that the GA-LSSVM, Hybrid-ANFIS
and GA-PLCIS models gave correct predictions for the con-
densation process. They also observed that the GA-PLCIS
models, by combining the merits of the single developed
models, indicated the best performance. Based on the
results, they recommended the GA-PLCIS as an easy-to-use
and also applicable model for predicting with high accuracy
the performance of the system even in challenging situations
such as low qualities and low mass velocities.
Azizi etal. [53] predicted the void fraction for gas–liquid
flow in inclined pipes using an artificial neural network. The
input parameters were the superficial Reynolds number of
liquid and gas and the inclination angles of the pipe. They
used 301 experimental data points from the work of Ghajar
and Bhagwat’s [54] work to develop the ANN model. It
was found that the ANN predictions showed a very good
agreement with the experimental void fraction data. Fur-
thermore, the accuracy of their proposed ANN model was
compared with the predictions of 17 void fraction corre-
lations available in the technical literature. For all cases,
it was found that their trained ANN model showed better
performance than the studied correlations.
The HTC and various flow distribution of oil–water two-
phase flow in a horizontal and slightly upward inclined
(+ 4° and + 7°) tube was studied by Boostani etal. [55].
They discovered that the inclination angle and flow pattern
highly influenced oil–water heat transfer coefficients. They
also developed an artificial neural network (ANN) model
for forecasting the HTC. The input variables to their model
were: superficial water Reynolds number, the superficial oil
Reynolds number, inclination angles and flow pattern, while
the output variables were the HTCs. They therefore obtained
an optimal ANN model that had a good prediction for every
position and flow patterns.
Noori Rahim Abadi etal. [56] proposed an adaptive
neuro-fuzzy inference system (ANFIS) to optimize and
predict the pressure difference and heat transfer coefficients
during the condensation of R-134a in smooth inclined tubes.
They examined the performance of three different ANFIS
structure identification methods. For the training, they used
the experimental data of [18, 21, 57]. They also compared
their model to the numerical simulations of [58–60]. It was
found that the numerical simulations performed better than
the ANN model but commented that the error of the ANFIS
methods was within the acceptable uncertainties of the
experimental data. They concluded that the ANFIS model
was useful for quick and dependable results.
Therefore, it can be concluded that no ANN study has
coupled the effect of tube enhancement and inclination
angles to describe the heat transfer coefficients during con-
densation. This paper, therefore, studies the applicability of
ANN to model for the effect of inclination and enhancement,
vapour quality and mass velocities on the measured heat
transfer coefficients.
Experimental
Figure1 is a well-known set-up earlier used for condensation
studies [8, 18, 20–22, 57, 61–76]. The experimental bench
was made up of a vapour compression refrigerant system
with water cycles associated with each heat exchanger. The
inclination angle of the test section (β) changed from − 90°
to 90°, with the reference as 0° (horizontal flow).
The inner tube of the test section was 1.49m long, with
a helix angle of 14°. Furthermore, the inner and outer diam-
eters were 8.92mm and 9.55 mm, respectively. The fin
height, pitch, roughness and circumferential number were
0.21mm, 0.445mm, 0.0235 and 60, respectively. The pre-,

D.R.E.Ewim et al.
1 3
test, post- and bypass condensers as well as the evaporator,
and all the water and refrigerant lines were insulated.
One hundred and sixty-ninecombinations of refriger-
ant mass velocities between 200 and 600kgm−2s−1, mean
vapour qualities between 0.1 and 0.9 and a saturation con-
densing temperature of 40°C were considered. Measure-
ments were conducted with heat transfer rates between 230
and 270W. These rates were applied to control the span of
the vapour quality within the test section (Fig.2).
The data reduction in the heat transfer coefficients was
exhaustively discussed in our previous studies and will not
be repeated in this study. However, it is worth noting that the
heat transfer coefficients were calculated as:
Furthermore, the enhancement factor (EF), which is the
heat transfer coefficients obtained for the microfin tube
(1)
𝛼
=
̇
Q
w,test
A
i
ΔT
divided by the heat transfer coefficients obtained for a simi-
lar smooth tube, was calculated as:
Neural andneuro‑fuzzy networks
The idea of artificial neural networks (ANNs) was built on
the paradigm that an intelligent machine may only be a real-
ity if the brains architecture is adequately replicated. ANN
is the very centre of deep learning where they are used to
classify billions of images and suggest videos/items to bil-
lions of web users etc. ANN outperforms other machine
learning techniques when very large and complex datasets
are involved [78].
A perceptron is a neural network composed of a single
layer of linear threshold unit (LTU), in which all the inputs
are connected to each neuron. Input neurons are connec-
tions using pass through neurons (i.e. these neurons output
whatever signals they get), and a bias neuron is an extra
neuron added that outputs one all the time. A perceptron can
be called a multi-output classifier since it can classify into
three different binary classes. Perceptrons were not able to
solve trivial problems like the exclusive OR (XOR) classifi-
cation problem. Researchers had to drop the perceptron idea
until it was found that its limitation can be eliminated by
stacking multiple perceptrons. The resulting ANN is called
a multiple-layer perceptron (MLP) which can solve the XOR
problem. An MLP is made up of one input layer, one or more
hidden layers which are LTUs and one last LTU which is the
output layer, and all layer excluding the output layer includes
a bias neuron and is fully connected to the next layer. When
an ANN has two or more hidden layers, it is called a deep
neural network (DNN). An MLP is a class of feed-forward
artificial neural network. All the nodes except the input
nodes use a nonlinear activation function. It is a supervised
learning algorithm using the backpropagation technique for
(2)
EF
=
𝛼
microfin
𝛼smooth
.
= High pressure R134a
= Low pressure R134a
= Water
sight
glass
sight
glass
Post-condenser Pre-condenser
g
β
Expansion
valves Bypass condenser
Compressor
Evaporator
Suction accumulator
DPT
Test condenser
Fig. 1 Experimental set-up and test section
360°
N
d
o
d
i
d
m
Helix angle d
o
A
A
SECTION A-A
Fig. 2 Representations of the enhanced tube (Adapted from Thome [77])

Modelling ofheat transfer coefficients duringcondensation insideanenhanced inclined tube
1 3
training. It is different from a linear perceptron in the fact
that it has multiple layers and uses nonlinear activation [78].
ANN can be categorised into feed-forward (no loop is
formed by network connection) and feedback networks (has
loop(s)). MLPs (a type of feed-forward network) are com-
monly used. The input layer neurons only act to pass the
input signals to neurons in the hidden layer. Every hidden
layer’s neurons sum up its input signals x after weighing
them with the power of the corresponding connections from
the input layers and calculates its output with respect to the
sum. The output is calculated likewise. A gradient decent
method in the backpropagation algorithm is the most widely
used in MLP training method. The radial basis function
(RBF) has of three layers: the input layer consists of neurons
possessing a linear function that to pass the input signals to
the hidden layer, but weighting is not applied to the linkage
between the input and hidden layers. The RBF is performed
by the hidden neurons. The design and building of ANN
models involve: data acquisition, cleaning of data, creating
the network, training the network and test the performance
of the resultant model [79].
Results ofexperimental studies
The general trend found in the experiments is shown in
Figs.3 and 4. In Fig.3, the effect of inclination angle on
heat transfer coefficient in the microfin tube for different
vapour qualities for mass fluxes of (a) 200kgm−2s−1 (b)
300kgm−2s−1 and (c) 400kgm−2s−1 is shown. In Fig.4,
the effect of inclination angle on heat transfer coefficient in
the microfin tube for different mass fluxes for mean vapour
qualities of (a) 0.25, (b) 0.50 and (c) 0.75 is shown. In
Fig.5, the effect of inclination angle on the heat transfer
enhancement factor for different vapour qualities for mass
fluxes of (a) 200kgm−2s−1 (b) 300kgm−2s−1 and (c)
400 kgm−2s−1 is presented. In general, it was found that an
increase in mean vapour quality and mass fluxes translated
in increased heat transfer coefficients and pressure drops
for both the microfin tube. For the microfin tube, higher
heat transfer coefficients were obtained during the down-
ward flow orientation (i.e. from − 90° to 0°), compared with
upward flow orientations.
The inclination angle which resulted in the maximum heat
transfer coefficients was between − 15° (downward flow) and
− 5° (downward flow) for the microfin tube. When compared
with the heat transfer coefficients obtained with a smooth
tube having a similar diameter at the same operating condi-
tions from the works of [21, 57], heat transfer enhancement
factors of between 0.8 and 2.4 were obtained depending on
the inclination angle, mean vapour quality and mass flux.
Figures3 and 4 also indicate a reversal in the trend for
upward and downward flow. For instance, there is an increase
(a)
(b)
(c)
Inclination angle, β/°
Inclination angle, β/°
Inclination angle, β/°
Heat transfer coefficient,
α/W m
–2
K
–1
-90 -60 -30 0 30 60 90
1500
2000
2500
3000
3500
4000
4500
5000
x
m
= 0.25
x
m
= 0.5
x
m
= 0.75
-90-60-30 0 30 60 90
1000
2000
3000
4000
5000
6000
7000
8000
x
m
= 0.1
x
m
= 0.25
x
m
= 0.5
x
m
= 0.75
x
m
= 0.9
-90 -60 -30 0 30 60 90
2000
3000
4000
5000
6000
7000
8000
9000
10000
x
m
= 0.25
x
m
= 0.5
x
m
= 0.75
x
m
= 0.9
Heat transfer coefficient, α/W m
–2
K
–1
Heat transfer coefficient, α/W m
–2
K
–1
Fig. 3 Effect of inclination angle on heat transfer coefficient in
the microfin tube for different vapour qualities for mass fluxes of
a 200kg m−2s−1, b 300kg m−2s−1 and c 400kg m−2s−1

D.R.E.Ewim et al.
1 3
in the heat transfer coefficients from an inclination angle
(β = − 90°) to the highest value at inclination angle between
− 5° and − 15° (downward flow), after which there is a reversal
in the trend of the heat transfer coefficients for upward flow.
The effect of inclination during the downward flow can be seen
most clearly for mean vapour quality between 10% and 50%,
while it is practically insignificant for higher vapour qualities.
At an inclination angle range of +30° and +60° (upward flow),
the minimum heat transfer coefficients were found. This may
be attributed to the increase in liquid holdup which necessi-
tates a higher hence thermal resistance in the flow for all the
vapour qualities (except for a few cases, i.e. vapour quality of
10% for mass flux of 200kgm−2s−1, vapour qualities of 10%
and 50% for mass flux of 300kgm−2s−1 and vapour qualities
of 0.25 and 0.75 for mass flux of 400kgm−2s−1).
Results ofANN predictions
In this study, an ANN model was trained using 156 examples
for predicting heat transfer coefficient with 70 (Training):15
(Validation):15 (Test) splits. Several networks with single
hidden layers were created to predict the heat transfer coef-
ficient. The number of neurons was varied from 1 - 18 with
the same random number generation seed to ensure consist-
ency and reproduction of results. A seed number of 1 was
therefore used. The mean square error (MSE), mean abso-
lute percentage error (MAPE), correlation coefficient (R),
coefficient of determination (R2) and the root-mean-square
error (RMSE) were used for performance evaluation. These
performance indices were calculated for both the training
and the test datasets. It is worthy of note what dataset was
shuffled. Shuffling the data ensures that each example point
creates an “independent” change on the model, avoiding bias
by the example points before them, which will lead to poor
performance when the model is used on unseen data.
Equations(3) to (7) show the expressions for calculat-
ing MSE, MAPE, R, R2 and RMSE. In the equations,
Ypred
represents the predicted value,
̄
Ypred
represents the mean of
the predicted values,
Yexp
represents the experimental values
(that is the target variable), and
̄
Yexp
represents the mean of
the experimental values.
(3)
MSE
=1
n
n
∑
m=1
(
Ypred,m −Yexp,m
)2
(4)
MAPE
=
[
1
n
n
∑
m
=
1|||||
Yexp,m −Ypred,m
Yexp,m
|||||]
×
100
(5)
R
=
∑n
m=1(Ypred,m −
̄
Ypred)(Yexp,m −
̄
Yexp )
�∑
n
m=1(Ypred,m −
̄
Ypred)2(Yexp,m −
̄
Yexp )
2
-90-60-30030 60 90
1500
2000
2500
3000
3500
4000
G = 200 kg m
2
s
–1
G = 300 kg m
2
s
–1
G = 400 kg m
2
s
–1
-90-60-30 0 30 60 90
2000
3000
4000
5000
6000
7000
8000
-90 -60 -30 0 30 60 90
2500
3000
3500
4000
4500
5000
5500
6000
6500
7000
7500
8000
Heat transfer coefficient, α/W m–2 K–1
Inclination angle, β/°
Inclination angle, β/°
Inclination angle, β/°
(a)
(c)
Heat transfer coefficient, α/W m–2 K–1
Heat transfer coefficient,
α/W m –2 K–1
(b)
G = 200 kg m
2
s
–1
G = 300 kg m
2
s
–1
G = 400 kg m
2
s
–1
G = 600 kg m
2
s
–1
G = 200 kg m
2
s
–1
G = 300 kg m
2
s
–1
G = 400 kg m
2
s
–1
Fig. 4 Effect of inclination angle on heat transfer coefficient in the
microfin tube for different mass fluxes for mean vapour qualities of
a 25%, b 50% and c 75%

Modelling ofheat transfer coefficients duringcondensation insideanenhanced inclined tube
1 3
According to the testing performance of the networks in
Table1, the network having seven neurons (Fig.5) in the
hidden layer for predicting the heat transfer coefficient was
found to be the optimum network with the best performance.
This network had an MSE of 22114, a MAPE of 0.1504%,
R2 of 0.8954 and RMSE of 579.0337.
The MSE with Epoch is plotted in Fig.6, and it can be
seen the best validation performance was at Epoch 9 and it
took a total of 15 Epochs to train the neural network; at the
15th Epoch, the MSE is lower for train and higher for valida-
tion, while the MSE for test and best is very close.
The error histogram with 20 bins is shown in Fig.7 and
instances are plotted on the y-axis and the errors on the
(6)
R
2=1−

n
m=1

Yexp,m −Ypred,m
2

n
m=1
Y
exp,m
−
̄
Y
exp 
2
(7)
RMSE
=




1
n
n

m=1

Ypred,m −Yexp,m

2
Fig. 5 A neural network with
seven neurons
Table 1 Performance of various ANNs for predicting heat transfer coefficient with one hidden layer
S/NMSE train MSE test RMSE train RMSE test R2 train R2 test MAPE train/% MAPE test/%
1 3.78E+05 2.92E+05 614.995 665.3367 0.8822 0.8659 1.5381 2.0402
2 2.20E+05 1.69E+05 468.4987 506.5801 0.9316 0.9223 1.9246 1.6996
3 2.79E+05 2.17E+05 528.3642 573.5888 0.913 0.9003 1.2233 1.4914
4 1.67E+05 3.77E+05 407.0388 755.558 0.9484 0.8271 0.8012 0.7895
5 1.35E+05 4.59E+05 366.486 834.4564 0.9582 0.7891 0.1814 0.4105
6 1.61E+05 3.02E+05 401.5539 676.5012 0.9498 0.8614 0.2953 1.0801
7 1.19E+04 2.21E+05 268.105 579.0337 0.9776 0.8954 0.1439 0.1504
8 1.19E+05 5.41E+05 345.1562 905.921 0.9629 0.7514 0.0641 0.1733
9 1.16E+05 5.3472E+05 340.127 900.39 0.764 0.7544 1.1252 5.1725
10 1.78E+05 2.74E+05 421.6121 644.761 0.9446 0.8741 0.2637 1.7603
11 8.56E+04 4.52E+05 696.8291 327.6762 0.5488 0.7925 1.8639 4.9372
12 1.01E+05 2.97E+05 317.9987 671.2352 0.9685 0.8635 0.4718 0.0218
13 2.82E+03 3.24E+05 304.6631 701.0231 0.9711 0.8511 0.7149 2.66
14 1.36E+05 4.68E+05 368.1497 842.0766 0.9578 0.7852 0.2864 0.3039
15 1.64E+05 3.29E+05 405.3459 707.1771 0.9488 0.8465 0.5876 2.102
16 2.18E+05 8.79E+05 466.9905 1.15E+03 0.9321 0.5965 2.1974 5.4349
17 2.68E+05 5.62E+05 517.4747 922.9988 0.9166 0.7419 4.73 4.6709
18 1.73E+05 5.88E+05 415.4508 944.5694 0.9462 0.7287 1.2714 3.559
Best validation performance is 97598.5999 at epoch 9
15 Epochs
0
103
104
105
106
Mean squared error/mse
107
108
51
01
5
Train
Validation
Test
Best
Fig. 6 Plot of mean squared error against Epochs

D.R.E.Ewim et al.
1 3
x-axis. It can be observed that error ranges from − 774.9 to
786.5 in a bell shape.
Figure8 shows the regression plot for training, validation,
test and all dataset. It can be observed from the plot that the
R-value of the training dataset was 0.98998; for the valida-
tion set, it was 0.98871; for the test set, it was found to be
0.98226, and for all the set, it was found to be 0.98877. From
these, we may deduce that the ANN model fits the dataset
satisfactorily. The predictions of the ANN model are plotted
in Figs.9–14.
Figure9 shows the variation of heat transfer coeffi-
cient with respect to inclination angles for a mass flux of
200kgm−2s−1 for various vapour qualities. First, for mass
flux of 200kgm−2s−1, we can generally see that the ANN
model closely predicted the experimental results. How-
ever, for vapour quality of 0.75 and 0.50, from − 30 to − 10
degrees and from − 90 to − 60 degrees, respectively, the
ANN model has higher deviations. This may be attributed to
the fact that the ANN model had difficulties learning about
these regions due to the complexity of the two-phase flow.
20
Error histogram with 20 bins
Training
Validation
Test
Zero error
15
10
5
0
− 774.
9
− 692.
7
− 610.
5
− 528.
3
− 446.
2
− 364
− 281.
8
− 199.
6
− 117.
5
− 35.27
46.91
Errors = Targets − Outputs
Instances
129.1
211.3
293.4
375.6
457.8
540
622.2
704.3
786.5
Fig. 7 Plot of error histogram with 20 bins
Fig. 8 Regression plot for
training, validation, test and all
dataset 8000
8000
7000
6000
6000
5000
4000
4000
Data
Fit
Y = T
Data
Fit
Y = T
Data
Fit
Y = T
Data
Training: R = 0.98998
Validation: R = 0.98871
All: R = 0.98877
Test: R = 0.98226
Fit
Y = T
3000
2000
2000 8000600040002000
Heat transfer coefficient/predicted
8000
7000
6000
5000
4000
3000
2000
Heat transfer coefficient/predicted
8000
7000
6000
5000
4000
3000
2000
Heat transfer coefficient/predicted
Heat transfer coefficient/Experimental Heat transfer coefficient/Experimental
8000600040002000 8000600040002000
Heat transfer coefficient/experimental
Heat transfer coefficient/experimental
8000
7000
6000
5000
4000
3000
2000
Heat transfer coefficient/predicted

Modelling ofheat transfer coefficients duringcondensation insideanenhanced inclined tube
1 3
Figure10 shows the variation of the heat transfer coef-
ficient with respect to inclination angles for a mass flux of
300kgm−2s−1 for various vapour qualities. From Fig.10, it
may be observed that for a mass flux of 300kgm−2s−1, the
ANN model closely followed the experimental (target) vari-
able and had more errors at low vapour qualities of 0.1 espe-
cially for inclination angles of − 90 to 0 degrees, while for
other vapour qualities, these inclinations were not difficult
to learn by the ANN model. Furthermore, for inclinations of
− 30 to 0, 60 to 90 degrees and vapour qualities of 0.9, the
ANN model also had difficulty in learning.
Figure11 shows the variation of heat transfer coeffi-
cient with respect to inclination angles for a mass flux of
400kgm−2s−1 for various vapour qualities. From Fig.9, for
a mass flux of 400kgm−2s−1, the problematic zones for the
ANN model were the vapour qualities of 0.50, 0.75 and 0.9,
with inclination angles of 0 to 90 degrees, 0 to 10, − 90, and
90 degrees, and 60 to 90 degrees. These zones involve com-
plex interactions which the ANN model was unable to learn.
The reason for this behaviour could be related to the lack of
training data for the mentioned range of inclination angles.
Figure12 shows the plot of heat transfer coefficients for
25% vapour quality varying the mass fluxes with several
inclination angles. It may be seen that the ANN model had
no difficulties learning the heat transfer coefficients (the tar-
get variable from the experiment). The interactions for this
case were easier for the proposed ANN model to learn.
Figure13 shows the variation of the heat transfer coef-
ficient for different inclinations and vapour qualities of 50%
under different mass fluxes. It was found that the ANN
model had difficulties to predict the heat transfer coeffi-
cient in the following zones 200kgm−2s−1, − 90 to − 60
degrees, 10 degrees; 300kgm−2s−1, 30 to 60 degree; and
400kgm−2s−1, 0 to 60 degrees. This is due to complex
interactions in these regions.
Figure14 shows the variation of heat transfer coef-
ficient for different inclinations and vapour qualities of
75% under different mass fluxes. We can observe from
Fig.12 that the ANN model had difficulties in predicting
the heat transfer coefficient for the following regions: for
7000
6000
5000
4000
3000
2000
1000
− 100 − 50 0
Inclination angle (
β
)/°
50 100
Heat transfer coefficient
(α)/W m−2 k−1
Fig. 9 Effect of inclination angle on the heat transfer coefficient in
the microfin tube for different vapour qualities for the mass flux of
200kgm−2s−1
7000
6000
5000
4000
3000
2000
8000
Heat transfer coefficient
(α)/W m
−2
k
−1
− 100 − 50 0
Inclination angle (
β
)/°
50 100
Fig. 10 Effect of inclination angle on the heat transfer coefficient in
the microfin tube for different vapour qualities for the mass flux of
300kgm−2s−1
− 100 − 50 0
Inclination angle (β)/°
50 100
7000
6000
5000
4000
3000
2000
8000
9000
10000
Heat transfer coefficient
(α)/W m−2 k−1
Fig. 11 Effect of inclination angle on the heat transfer coefficient in
the microfin tube for different vapour qualities for the mass flux of
400kgm−2s−1

D.R.E.Ewim et al.
1 3
a mass flux of 200kgm−2s−1, − 60 to − 10 degrees and
30–90 degrees; for a mass flux of 300kgm−2s−1, − 10 to
0 and 90 degrees; for a mass flux of 400kgm−2s−1, 0 to
60 degrees. Once again, the relationship in these zones
made it difficult for the ANN model to learn the target
variables at these conditions.
Conclusions
Experiments were carried out during the condensation of
R-134a in an enhanced inclined tube at mass fluxes between
200 and 600kgm−2s−1. The mean vapour qualities ranged
from 0.1 to 0.9, while the saturation temperature was main-
tained at 40°C.
Based on the experimental results, an ANN model was
proposed for the prediction of the condensation heat transfer
coefficients. Following from that, the following conclusions
were made:
1. The maximum condensation heat transfer coefficients
for the enhanced tube was at inclination angles between
− 15° and − 5°. Using these angles, more efficient heat
exchangers can now be designed.
2. The ANN model was able to satisfactorily predict the
heat transfer coefficients
3. ANN can be a very handy tool for heat transfer predic-
tion especially where a large pool of experimental data
is available.
Acknowledgements The funding obtained from Tshwane University
of Technology, NRF, TESP, Stellenbosch University/the University of
Pretoria, SANERI/SANEDI, CSIR, TUT, EEDSM Hub and NAC is
acknowledged and duly appreciated.
7000
6000
5000
4000
3000
2000
1000
Heat transfer coefficient
(α)/W m−2 k−1
− 100 − 50 0
Inclination angle (
β
)/°
50 100
Fig. 12 Effect of inclination angle on the heat transfer coefficient in
the microfin tube for different mass fluxes for mean vapour quality of
25%
7000
6500
6000
5500
5000
4500
4000
3500
3000
2500
2000
7500
Heat transfer coefficient
(α)/W m−2 k−1
− 100 − 50 0
Inclination angle (
β
)/°
50 100
Fig. 13 Effect of inclination angle on the heat transfer coefficient in
the microfin tube for different mass fluxes for mean vapour quality of
50%
7000
6500
6000
5500
5000
4500
4000
3500
3000
2500
2000
7500
8000
Heat transfer coefficient
(α)/W m
−2
k
−1
− 100 − 50 0
Inclination angle (
β
)/°
50 100
Fig. 14 Effect of inclination angle on the heat transfer coefficient in
the microfin tube for different mass fluxes for mean vapour quality of
75%

Modelling ofheat transfer coefficients duringcondensation insideanenhanced inclined tube
1 3
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