*Corr. Author’s Address: Hijjawi Faculty for Engineering Technology Yarmouk University, Irbid, Jordan, Amjad.Alsakarneh@yu.edu.jo 401
Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 401-408 Received for review: 2023-04-07
© 2023 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2023-05-22
DOI:10.5545/sv-jme.2023.597 Original Scientific Paper Accepted for publication: 2023-05-29
Fuzzy and Matlab/Simulink Modelling  
of the Air Compression Refrigeration Cycle
Alsakarneh, A. – Lina Momani, L. – Tabaza, T.
Amjad Alsakarneh
1,*
 – Lina Momani
2
 – Taha Tabaza
3
1 
Yarmouk University, Hijjawi Faculty for Engineering Technology, Department of Mechanical Engineering, Jordan 
2 
Higher Colleges of Technology, Fujairah Men’s College, Engineering Division, UAE 
3 
AL-Zaytoonah University of Jordan, Department of Mechanical Engineering, Jordan
The coefficient of performance (COP) for a gas refrigeration cycle was estimated using Matlab/Simulink and fuzzy logic. A Matlab/Simulink 
model of the gas refrigeration cycle was developed, and the output was compared to theoretical data. Additionally, fuzzy logic was used to 
estimate the COP for arbitrary low- and high-pressure levels. Simulation results were used to develop a multi-input-multi-output (MIMO) fuzzy 
Takagi-Sugeno-Kang (TSK)-based model. Both the Matlab/Simulink and the MIMO fuzzy model were found to be very well correlated with 
theoretical results, with an error of less than 2 %. These results demonstrate the effectiveness of using fuzzy logic to analyse gas refrigeration 
cycles and suggest that this approach can be extended to analyse other thermodynamic cycles.
Keywords: coefficient of performance, Matlab/Simulink, Takagi-Sugeno-Kang, refrigeration cycles
Highlights
• 	 The coefficient of performance (COP) for a gas refrigeration cycle was assessed using a modern intelligent tool, specifically 
fuzzy logic.
• 	 To verify the results obtained by fuzzy logic, a simulation study was conducted over a wide range of low and high pressures in 
the cycle.
• 	 The fuzzy logic approach was found to produce results very similar to the simulated COP, with a root mean square error of less 
than 2 %.
• 	 Fuzzy logic was found to be a fast approach for estimating the COP with limited knowledge of gas refrigeration cycles.
0  INTRODUCTION
Thermodynamics has a wide range of applications, 
including refrigeration, which involves the transfer 
of heat from a lower temperature region to a higher 
temperature one [1] and [2]. The air compression 
refrigeration cycle is a well-known example of this 
process, utilizing air as a working fluid instead of 
chlorofluorocarbon (CFC). This type of refrigeration 
is commonly used in airplane air conditioning systems, 
where pressurized air from the engine compressors 
is utilized. In recent years, researchers have shown 
more interest in this type of refrigeration due to its 
environmental advantages, as it has a positive impact 
on the ozoneosphere. Additionally, the working fluid 
for this refrigeration cycle is readily available at no 
cost, making it economically attractive. However, due 
to the low efficiencies of the compressor and expander 
[3] and [4], the coefficient of performance (COP) of 
this refrigeration cycle is relatively low.
The closed-air refrigeration cycle is typically 
comprised of four components: the air cooler, 
refrigerator, compressor, expander, and motor. Fig. 
1 displays the four components of the refrigeration 
cycle, where the refrigerator holds air at a low 
pressure and the air cooler holds air at a high pressure. 
Many studies, such as [5] and [6], have investigated 
the simulation and thermodynamic analysis of this 
cycle. Gigiel et al. [7] compared the air cycle and the 
CFC conventional freezing cycle. Further research 
regarding the optimization of the air refrigeration 
cycle using finite time thermodynamics and entropy 
generation minimization has been discussed in [8] and 
[9].
Fig. 1.  Air refrigeration cycle (dashed and solid lines stand for high 
and low pressures, respectively)
Refrigeration system designers aim to maximize 
efficiency while minimizing energy consumption 
and production costs. Several variables have an 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 401-408
402 Alsakarneh, A. – Lina Momani, L. – Tabaza, T.
effective role in determining which combination 
of properties can be used to increase the coefficient 
of performance (COP) of the system. Kizilkan [10] 
conducted a performance analysis of a variable-
speed refrigeration system using artificial neural 
networks. The objective of the study and the learning 
procedure was to identify the optimal set of weights 
that would produce the correct output for any input. 
The output of the network was compared to a desired 
response to generate an error, and the performance 
of the network was evaluated in terms of the desired 
signal and the convergence criterion. Fuzzy logic and 
artificial neural networks (ANNs) have been widely 
used in the literature to model complex mechanical 
systems [11] and [12]. Sahin [13], for example, used 
ANNs and adaptive neuro-fuzzy (ANFIS) to analyse 
the performance of a single-stage vapour compression 
refrigeration system with an internal heat exchanger 
using refrigerants R134a, R404a, and R407c. A 
comprehensive review of the applications of artificial 
neural networks for refrigeration cycles was presented 
in [14].
This research work proposes an alternative 
method for estimating the coefficient of performance 
(COP) using fuzzy logic as a tool, eliminating the 
need for analytical modelling of the refrigeration 
cycle. The researchers developed two models to 
estimate the COP of gas refrigeration cycles: the first 
model utilized a Matlab/Simulink model and direct 
mathematical relations and models, while the second 
model employed fuzzy logic. Both models utilized 
the low and high pressures of the cycle as inputs and 
the COP as an output. These results can be applied to 
modelling and simulating other refrigeration cycles, 
specifically, and thermodynamic cycles, generally.
1  METHODS
In this section, the gas refrigeration cycle model 
development using Matlab/Simulink is described. 
Matlab/Simulink has been extensively used by 
researchers for modelling and simulating mechanical 
systems. Milecki and Rybarczyk [15], for instance, 
employed Matlab/Simulink to simulate the behaviour 
of an electrohydraulic proportional valve. The 
development process starts with the calibration of the 
thermodynamic properties of air, as described below:
 h = f (P
r
), (1)
 h = f (T), (2)
where h is the enthalpy (kJ/kg), P
r
 is the relative 
pressure, and T is the temperature in Kelvin.
To obtain the enthalpy fitted model, a power 
function in the form of (h = a x
b
 + c) is suggested. By 
utilizing the curve-fitting toolbox (CFT) in Matlab, 
the following results are obtained:
 h(P
r
)  = 333.70 × P
r
0.2539
 – 65.14, (3)
 h(T)  = 0.391 × T
1.136
 + 43.91. (4)
To evaluate the accuracy of this model, the root 
mean square error (RMSE) is utilized, which is less 
than 1 % for the enthalpy for both Eqs. (3) and (4). 
Figs. 2a and b depict the original data, obtained 
from the thermodynamics tables, and the fitted data 
generated through this model.
In addition to the enthalpy data, the relative 
pressure as a function of temperature and temperature 
as a function of relative pressure are also required. 
These relationships are generated using the curve 
fitting toolbox, and the resulting equations are as 
follows:
 PT T
r
 

1025 10 5 586
11 4 333 .
., (5)
 TP P
rr
   395 125 4
0 221 .
.. (6)
To assess the performance of the developed 
model, the root mean square error (RMSE) is used as 
an evaluation metric. The RMSE is determined to be 
less than 1.5 % for both Eqs. (5) and (6), indicating 
good agreement between the fitted data and the 
original data obtained from the thermodynamics 
tables. This is further supported by Figs. 2c and d, 
which show the fitted data closely overlapping with 
the original data.
Using the fitted functions described earlier, the 
researchers developed a complete Simulink model of 
the gas refrigeration cycle. This model takes the low 
and high pressures of the cycle as input and produces 
the COP as output. A block diagram of the Simulink 
model is depicted in Fig. 3.
In the compressor subsystem, it is assumed that 
the air enters the compressor with a temperature of 285 
K and a pressure of one atmospheric pressure. Using 
Eqs. (4) and (5), the enthalpy and relative pressure 
can be calculated. The thermodynamic properties of 
the air at the exit can be determined by applying the 
conservation of mass and energy principles.
P
P
P
P
rr 2
2
1
1
= .
The exit enthalpy and temperature can be 
determined using the computed value of the P
r
 from 
Eqs. (3) and (6). 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 401-408
403 Fuzzy and Matlab/Simulink Modelling of the Air Compression Refrigeration Cycle 
In the Turbine subsystem, the air enters the 
turbine with the third state, which is the exit of the 
condenser and exits the turbine with the fourth state. 
The enthalpy and relative pressure of the third state 
obtained from the condenser subsystem are used to 
calculate the enthalpy and relative pressure of the 
fourth state, using Eqs. (4) and (5), respectively.
a)           b) 
c)           d) 
Fig. 2.  Displays a comparison between the simulated data obtained from the developed models and the data obtained from thermodynamics 
tables; a) the enthalpy as a function of relative pressure, b) the enthalpy as a function of temperature,  
c) the relative pressure as a function of temperature, and d) the temperature as a function of relative pressure
Fig. 3.  The gas refrigeration cycles model

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 401-408
404 Alsakarneh, A. – Lina Momani, L. – Tabaza, T.
 P
P
P
P
rr 4
4
3
3
= .
Using the relative pressure, the enthalpy and 
temperature can be determined using Eqs. (3) and (6). 
In the evaporator model, the absorbed energy in 
the condenser (q
L
) can be calculated as the difference 
between the enthalpies of the first and fourth states, 
which is given by the following equation:
 qh h
L

14
. (7)
2  EXPERIMENTAL
The components of a fuzzy expert system include 
fuzzification, a knowledge base, a fuzzy inference 
engine, and defuzzification. A summary of how these 
components operate is provided below:
1. Fuzzification: To perform fuzzification in a fuzzy 
expert system, input values are transformed into 
fuzzy values according to a specific membership 
function. In this case, the proposed method uses a 
Gaussian function for input membership.
2. Knowledge Base: In the knowledge base 
component of a fuzzy expert system, rules are 
created based on the expertise and experience 
of the system's designer. Typically, each rule is 
given equal weight, though some rules may be 
given more weight based on expert judgement. 
This component establishes the connections and 
regulations between input and output values. The 
proposed method for this system uses nine rules 
that encompass all possible combinations of the 
input membership functions.
3. Fuzzy inference engine: The fuzzy inference 
engine component of a fuzzy expert system 
utilizes expert knowledge-based rules to make 
inferences from the available information. One 
popular method of inference is Takagi-Sugeno-
Kang (TSK) inference. In TSK inference, the 
existing rules are combined using the Max-Min 
operation.
4. Defuzzification: Defuzzification is the process 
of converting fuzzy information into a specific 
value. One common technique for defuzzification 
is the weighted average method, which is simple 
and widely used. The method of defuzzification 
that uses weighted averages is also known as the 
Sugeno defuzzification method. It is suitable for 
fuzzy sets with symmetrical output membership 
functions and produces results that are similar to 
the centroid of area (COA) method.
The proposed method involves the development 
of a fuzzy logic model that is based on the input-output 
data of the COP obtained from the mathematical 
models and relations of the gas refrigeration cycle. 
The data set that is generated is then used to structure 
the fuzzy model, including the input and output 
membership functions, as well as the if-then rules. 
We attempted to optimize the membership functions 
used for the input variables by trying two and four 
functions. However, the accuracy of the two-function 
approach was only about 95 %, while the four-
function approach was very close to the three-function 
approach, with an error of about 1 %. Therefore, 
we decided to adopt the three-function approach, 
as it provided a better correlation compared to the 
two-function approach and faster computation time 
compared to the four-function approach. 
The refrigeration cycle includes four sub-systems: 
the evaporator, compressor, turbine, and condenser. 
However, the proposed fuzzy model only considers the 
input variables, which are the low and high pressures, 
and the output variable, which is the COP. For the low 
and high pressures, three membership functions are 
assumed, using Gaussian functions as follows:
 LPIMF ei
i
xA
B
i
i


 
2
2
123 ,, ,, where (8)
a)           b) 
Fig. 4.  The input membership functions before tuning of a) low pressure, and b) high pressure

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 401-408
405 Fuzzy and Matlab/Simulink Modelling of the Air Compression Refrigeration Cycle 
 HPIMF ei
i
xC
D
i
i


 
2
2
123 ,, ,, where (9)
where L
i
 and H
i
 are the i
th
 membership function of 
low and high pressures, respectively, A
i
, B
i
, C
i
 and 
D
i
 are the design parameters of i
th
 input membership 
functions. Fig. 4 shows the input and output 
membership functions.
For the if-then rules, Table 1 lists the if-then rules 
together with the output membership functions.
Where COP were calculated as follows:
 COPa LbHc
i
iiii i
   


1
9
. (10)
The authors previously presented a method to 
estimate the coefficients a, b, and c of the TSK model 
using the available experimental or simulation data. 
This was achieved by generating an optimized TSK 
model through the application of a nonlinear least 
square method, as described in reference [16].
3  NUMERICAL EXAMPLES RESULTS
After performing simulations using the Matlab/
Simulink model with various input pressures, a dataset 
was generated with the COP as the output. The range 
of low pressures considered was from 100 kPa to 200 
kPa, while the range of high pressures was from 1.0 
MPa to 3.0 MPa. The MATLAB code used the COP 
data generated from the Simulink model and applied 
the least square method to estimate the optimal values 
of the membership function coefficients.
The accuracy of the results is demonstrated by 
the RMSE value of 0.01953, which means that the 
expected COP error based on the model is less than 
2 % on each occasion. Table 2 provides all the If-
Then rules and output membership functions (OMF). 
Increasing the order of the OMF could be a potential 
area for future work, but it may require high-speed 
computers.
Fig. 5a depicts the relationship between the input 
variables of low and high pressures and the output 
variable of COP, as predicted by the developed model. 
The accuracy of the model is evident from the plot, 
with the error being negligible. Furthermore, Fig. 5b 
displays the residuals of the developed surface with 
a maximum error of less than 0.02, indicating the 
accuracy of the model's predictions.
a)           b) 
Fig. 5.  The input membership functions after tuning of a) low pressure; and b) high pressure
Table 1.  The if-then rules and output membership functions
No. If low pressure is
and
If high pressure is
then
The COP is
1. LPIMF
1
HPIMF
1
a
1
×L + b
1
×H + c
1
2. LPIMF
2
HPIMF
1
a
2
×L + b
2
×H + c
2
3. LPIMF
3
HPIMF
1
a
3
×L + b
3
×H + c
3
4. LPIMF
1
HPIMF
2
a
4
×L + b
4
×H + c
4
5. LPIMF
2
HPIMF
2
a
5
×L + b
5
×H + c
5
6. LPIMF
3
HPIMF
2
a
6
×L + b
6
×H + c
6
7. LPIMF
1
HPIMF
3
a
7
×L + b
7
×H + c
7
8. LPIMF
2
HPIMF
3
a
8
×L + b
8
×H + c
8
9. LPIMF
3
HPIMF
3
a
9
×L + b
9
×H + c
9

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 401-408
406 Alsakarneh, A. – Lina Momani, L. – Tabaza, T.
Table 2.  The if-then rules and output membership functions
No. If low pressure is
and
If high pressure is
then
The COP is
1. LPIMF
1
HPIMF
1
0.09796×L + 0.08642×H – 0.02512
2. LPIMF
2
HPIMF
1
–0.7031×L + 0.3672×H + 0.2381
3. LPIMF
3
HPIMF
1
0.1053×L + 0.3493×H – 0.2462
4. LPIMF
1
HPIMF
2
0.06101×L – 0.6759×H + 0.2887
5. LPIMF
2
HPIMF
2
–0.4113×L + 0.2595×H – 0.3855
6. LPIMF
3
HPIMF
2
–0.09852×L + 0.63×H – 0.1404
7. LPIMF
1
HPIMF
3
–0.1456×L – 0.03702×H – 0.3712
8. LPIMF
2
HPIMF
3
0.1469×L + 0.1681×H – 0.3202
9. LPIMF
3
HPIMF
3
0.6347×L + 0.4321×H + 0.4061
a) 
b) 
Fig. 6.  a) the normalized COP as a function of the normalized input pressures, and  
b) the residual between the calculated COP and the predicted COP using fuzzy logic

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 401-408
407 Fuzzy and Matlab/Simulink Modelling of the Air Compression Refrigeration Cycle 
4  COMPARISONS BETWEEN SIMULINK  
AND FUZZY MODELING RESULTS
Although the proposed approach does not inherently 
require a dynamic model, the Matlab/Simulink 
is employed to facilitate the utilization of the 
developed model for future research endeavours, 
e.g., implementing control strategies for refrigeration 
cycles and to make the developed model easily 
accessible to other researchers, enabling them to 
further investigate and contribute to the field of 
refrigeration systems.
Table 3 presents a comparison between the 
methods used by Matlab/Simulink and Fuzzy logic 
modelling to estimate the COP of refrigeration cycles. 
Table 3.  A simple comparison of the Matlab/Simulink and Fuzzy logic modelling
Matlab/Simulink Fuzzy logic
Mathematical modelling is employed to express the equations governing 
the gas refrigeration cycle through block diagrams.
The model is built using the raw data available from the gas refrigeration 
cycle.
To estimate the COP , complete mathematical models and equations for all 
sub-domains of a refrigeration system are necessary.
The detailed interaction between sub-systems of a refrigeration system is 
not necessary for fuzzy modelling. Instead of requiring expertise, a data 
set can be used to develop the Fuzzy model of the COP .
Classical mathematical programming models rely on well-defined 
coefficients for all refrigeration cycle components. However, in some 
cases, model parameters can only be roughly estimated. Additionally, to 
obtain the COP , all tables must be used.
After developing the fuzzy model of the gas refrigeration cycle, a tuning 
process is typically carried out to optimize it. Once optimized, the fuzzy 
model can estimate the COP independently without the need for tables or 
physical components.
With its graphical user interface (GUI) environment, Matlab/Simulink 
provides an easier way to simulate gas refrigeration cycles, as well as 
other engineering systems.
To use fuzzy logic in refrigeration systems, an expert in the field is 
needed to provide an initial guess of the input/output membership 
functions and to construct the If-Then rules.
5  CONCLUSIONS
In this research, the COP of gas refrigeration cycles 
was estimated using both Matlab/Simulink and fuzzy 
logic approaches. Matlab/Simulink used mathematical 
equations and relations to estimate the COP, while 
fuzzy logic used available data sets for the same 
purpose. The performance of the fuzzy logic approach 
was evaluated across a wide range of low and high 
pressures in the cycle. The results showed that fuzzy 
logic provided a quick way to estimate the COP with 
less knowledge of gas refrigeration cycles. The fuzzy 
logic found to be comparable in accuracy to Matlab/
Simulink and mathematical modelling with a RMSE 
of less than 2 %, highlighting the effectiveness of 
fuzzy logic as a tool to model refrigeration cycles, in 
particular, and thermodynamics cycles in general.
6  ACKNOWLEDGEMENTS
This paper's research was supported by Yarmouk 
University, Jordan and Al-Zaytoonah University of 
Jordan, Jordan.
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