EFFECTS OF FLUID DYNAMIC COMPRESSIBILITY AND FLOW INERTIA ON MASS FLOW RATE AND THERMAL FLUX ACROSS AN INSULATED PIPE: SIMULINK APPROACH

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com
 EFFECTS OF FLUID DYNAMIC COMPRESSIBILITY AND FLOW INERTIA ON
 MASS FLOW RATE AND THERMAL FLUX ACROSS AN INSULATED PIPE:
 SIMULINK APPROACH
 Nkwor Chimezie Agbafor*1, Onyenobi Chinwendu Samuel*2,
 Oriaku Justice Chibuzor*3, Mr. Ewurum Tennison. I*4
 *1,2,3,4Department Of Mechanical Engineering, Federal Polytechnic Nekede, Owerri.
 ABSTRACT
The study, effects of fluid dynamic compressibility and flow inertia on mass flow rate and thermal flux across an
insulated pipe was successfully achieved through simulink approach. Block models gotten from simscap-
thermal and hydraulics were used to represent all the elements of pipe flow model. The insulated pipe element
was modeled to retain hydraulic diameter of 0.1128 m within a length of 5 m. The shape factor and internal
surface roughness were chosen to be 64 and 1.5e-5m, respectively. Under fluid dynamic compressibility and
inertia influences, initial temperature and pressure were set to 298k or 25℃ and 1 atm respectively. Turbulent
regime Nusselt number correlation coefficients were 0.023, 0.8, 0.33, 0, and 0. In addition, the mass flow rate
from the reservoir has initial value of 0.6 kg/s, with the signal blocks adjusted to 0.1, 100, 1 for chirp signal and
6, 0, 5 for ramp signal respectively. Ode23t solver was used to run the simulation for 10seconds. System model
mass flow rate, thermal flux and temperature difference were found to be 0.058kg/s, -275J/s and 100K or -
173℃ within 10seconds of simulation. These results portrayed the effects of dynamic compressibility and
inertia on mass flow rate and thermal flux. Furthermore, simulation was run without dynamic compressibility
and inertia influences and values of mass flow rate, thermal flux and temperature difference were found to be
0.6 kg/s( same with initial value), 5.0344× 104 J/s and -19k or 254℃ within 10seconds of simulation. Results
indicated that fluid dynamic compressibility and flow inertia increases conductive heat transfer and reduces
mass flow rate through an insulated pipe element. The study also added MATLAB codes for component
operational equation models and simulation graphs. The researchers made the following recommendations:
The influence of fluid dynamic compressibility and flow inertia must be compromised if maximum mass flow
rate and heat leakage are of paramount, this research can also be done in future using different design models
and other advanced software for generalization.
Keywords: Simulation, Fluid Dynamics, Compressibility, Flow Inertia, Simulink, Simscap, Insulated Pipe, Block
Models.
 I. INTRODUCTION
Fluid flows are encountered in engineering systems, metrology, aeronautics, combustion, hydrodynamics, etc.
The study of fluid dynamics is required for designing, analyzing and optimization of engineering devices such as
aircraft, piping systems and heat exchangers (Anderson, 2009).
Rajput (2011) stated that kinematics involves merely the description of the motion of fluids in terms of space-
time relationship. Kinetics covers the action of forces in producing or changing motion of fluids. Fluid dynamics,
in its simplest term studies fluids in motion, which involves the consideration of both the kinematics and
kinetics. He also added that the basic equations that govern fluid dynamics are continuity equation, energy
equation, and momentum equation.
A compressible flow is that flow in which the density of the fluid changes during flow as can be seen in all real
fluids where their densities change with change in pressure or temperature. Compressibility affects the drag co-
efficient of bodies by formation of shock waves, which lowers discharge, increases stagnation pressure and
decreases fluid flow in convergent-divergent sections (Rajput, 2008).
Simulink is a MATLAB-based graphical programming environment for modeling, simulating and analyzing
multi-domain dynamical systems. The primary interface consists of graphical block diagrams or tools and a
customizable set of block libraries. There are no doubts that fluid dynamic compressibility and flow inertia to a
greater extent influences mass flow rate and thermal heat loss through a pipe element. It may lower operational
performance service life of mechanized fluid systems. Hence, researchers aimed at studying effects of fluid
www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
 [421]

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com
dynamic compressibility and flow inertia on mass flow rate and thermal flux across an insulated pipe using
SIMULINK approach.
 II. METHODOLOGY
An effect of fluid dynamic compressibility and flow inertia on mass flow rate and thermal flux across an
insulated pipe was carried out using SIMULINK in MATLAB. Simulink in the matlab command window contains
block models that were used to represent all the elements of pipe flow model. The block models gotten from
simscap- thermal and hydraulics, includes:
1. Clock Block Properties
2. Conductive Heat Transfer Block Properties
3. Demux Block Properties
4. Ideal Heat Flow Source Block Properties
5. Ideal Temperature Sensor Block Properties
6. Mass Flow Rate Source (TL) Block Properties
7. Mass Flow Rate & Thermal Flux Sensor (TL) Block Properties
8. Mux Block Properties
9. PS-Simulink Converter Block Properties
10. Pipe (TL) Block Properties
11. Product Block Properties
12. Ramp Block Properties
13. Reservoir (TL) Block Properties
14. Simulink-PS Converter Block Properties
15. Solver Configuration Block Properties
16. Thermal Liquid Settings (TL) Block Properties
17. Thermal Reference Block Properties
18. Transfer Fcn Direct Form II Block Properties
19. XY scope. Block Properties
20. chirp Block Properties
21. Block Type Count
Block ports were connected as shown in fig 1.0 below. The insulated pipe element as shown in table 1.6 was
modeled to retain hydraulic diameter of 0.1128 m within a length of 5 m. the shape factor and internal surface
roughness were chosen to be 64 and 1.5e-5m, respectively. Under fluid dynamic compressibility and inertia
influences, initial temperature and pressure were set to 298k or 25℃ and 1 atm respectively. Turbulent regime
Nusselt number correlation coefficients were 0.023, 0.8, 0.33, 0, and 0.
Ode23t solver was chosen due to its tight tolerance in fixed step solving. Simulation was allowed to run for
10seconds, see table 1.0 below.
 III. RESULTS AND PRESENTATIONS
 Table 1: Simulation Parameter
 Simulation Parameter Value

 Solver ode23t

 RelTol 1e-3

 Refine 1

 MaxOrder 5

 ZeroCross on
www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
 [422]

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com
 Simulation Parameter Value
 Simulation time 10seconds
 Table 2: PS-Simulink Converter Block Properties
 Sub Class Left Port Right Pseudo Affine
 Name Physical Domain Unit
 Name Type Port Type Periodic Conversion

 PS-Simulink
 network_engine_domain ps_output input output off K off
 Converter

 PS-Simulink
 network_engine_domain ps_output input output off kg/s off
 Converter1

 PS-Simulink
 network_engine_domain ps_output input output off J/s off
 Converter2
 Table 3: Transfer Fcn Direct Form II
 Name Num Coef Vec Den Coef Vec Vinit Rnd Meth Do Satur

 Transfer Fcn Direct Form II [0.2 0.3 0.2] [-0.9 0.6] 0.0 Floor off
 Table 4: XY Scope Block Properties
 Name Xmin Xmax Ymin Ymax St

 XY Graph -1 200 -1 10 -1
 Table 5: Signal Block Properties
 Name F1 T F2

 Chirp Signal 0.1 100 1
 Initial
 Ramp Signal Slope =6 Start time =0
 output=5
 Table 6: Block Type Count
 BlockType Count Block Names

 Scope 4 Scope, Scope1, Scope2, Scope4

 PS-Simulink PS-Simulink Converter, PS-Simulink Converter1, PS-Simulink
 3
Converter (m) Converter2

 Reservoir (TL) (m) 2 Reservoir (TL), Reservoir (TL)1

Conductive Heat
 2 Conductive Heat Transfer, Conductive Heat Transfer1
 Transfer (m)

 chirp (m) 1 Chirp Signal

 XY scope. (m) 1 XY Graph

 Transfer Fcn Direct Form II
 1 Transfer Fcn Direct Form II
 (m)

 Thermal Reference (m) 1 Thermal Reference

www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
 [423]

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com
 BlockType Count Block Names

 Thermal Liquid
 1 Thermal Liquid Settings (TL)
Settings (TL) (m)

 Solver
 1 Solver Configuration
Configuration (m)

 Simulink-PS
 1 Simulink-PS Converter
Converter (m)

 Ramp (m) 1 Ramp

 Product 1 Product

 Pipe (TL) (m) 1 Pipe (TL)

 Mux 1 Mux

 Mass Flow Rate &

Thermal Flux Sensor 1 Mass Flow Rate & Thermal Flux Sensor (TL)

 (TL) (m)

Mass Flow Rate
 1 Mass Flow Rate Source (TL)
Source (TL) (m)

Ideal Temperature
 1 Ideal Temperature Sensor
 Sensor (m)

Ideal Heat Flow
 1 Ideal Heat Flow Source
 Source (m)

 Demux 1 Demux

 Clock 1 Clock
 Table 7: Hydraulic Pipe Model
 Pipe length 5m
 Hydraulic diameter 0.1128m
 Cross area 0.01 m2
 Shape factor 64
 Internal surface roughness 1.5e-5m
 Laminar flow upper margin 2e+3
 Turbulent flow lower margin 4e+3
 Laminar regime nusselt number correlation
 [ 1.86 0.33 0.33 0.33 0.14 ]
 coefficients
 turbulent regime nusselt number correlation [ 0.023 0.8 0.33 0 0 ]

www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
 [424]

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com
 coefficients
 Fluid dynamic compressibility on
 Fluid inertia on
 Initial temperature 298k
 Initial pressure 1 atm
 Initial mass flow rate from A to B 0.6 kg/S
 Solver Ode23t
Pipe (TL) block models pressure and temperature dynamics in a pipe due to frictional effects and conductive
heat transfer with the wall. Based on the selected effects, fluid dynamic compressibility, and flow inertia can
also be accounted for. Including flow inertia means that the water hammer effect can be simulated.
Gravitational effects are not taken into account.
Viscous friction is evaluated using the Darcy-Weisbach law. The heat exchange coefficient with the wall is
evaluated using Nusselt number correlations.
Mass flow rate block represents a mechanical energy source that is powerful enough to maintain a constant
mass flow rate regardless of the pressure differential across the ports. The source adds no loss-related heat to
the flow.
Block connections A and B correspond to the thermal liquid inlet and outlet ports, respectively. A positive mass
flow rate results in fluid flowing from port A to port B.
 IV. DESIGN ANALYSIS
The flow inertia force possessed by fluid mass in motion can be estimated with the equation below:
 = × 
 
 = × . . × …… (1.0) ( Rajput, 2008) or
 
 = × 
 = × ….(1.1)
 ℎ = , = c , = ,
 = , = 
This can as well be established in terms of fluid resistance as shown below according to Poiseuille’s equation of
flow below
 
 =
 ℎ 
 2 − 1
 = ….. (1.2)
 
Basic equations of compressible fluid flow require the basic principles of conservation of mass, energy,
momentum and basic thermodynamics laws.
In case of one dimensional flow, mass per second = = ….(1.3)
 ℎ = , = 
Differentiating the above equation, gives continuity equation in differential expression.
 
 + + = 0 …(1.4)
 
 = (ℎ ) × × 
 0.6 / = 1000 × × 0.01
 0.6
 = = . m/s
 1000×0.01
The momentum equation for compressible flow in x-corodinate is given as:
 ∑ = ( )2 − ( )1 …(1.5)

www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
 [425]

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com
Energy equation is also given as below as pressure changes with density of fluid:
 2
 ∫ + + = ….(1.6)
 2

Heat loss by conduction through the pipe element is given as below:
 2 ( 1 − 2 )
 = … . . (1.7)
 ℎ / 
Where = ℎ , / 
 1 2 = , = ℎ
For an insulated pipe element, we have:
 2 ( 1 − )
 = 2 
 3
 …..(1.8)
 r1 2 ….
 +
 1 2

 = , = and insulated material.
When fluid flows through a pipe, it experiences some resistance to its motion, due to which its velocity and
energy (head) are reduced. The loss of head or energy is given in Darcy-Weisbach formula below:
 4 2
 ℎ = … (1.9)
 ×2 

 ℎ = ℎ , = y 
 = ℎ , = , = ℎ .
The universal velocity distribution equation, according to Prandtl’s hypothesis for both smooth and rough pipe
is as below:
 
 = + 2.5 ( ) …..(10.0)
 
 0
 = ℎ , = √ , = r , 
 
 = .
Karman-Prandtl equation for rough pipe velocity distribution is as shown;
 
 = 5.75 10 ( ⁄ ) + 8.5 …..(10.1)
 
 ℎ = ℎ ℎ ℎ = × 30
Navier –Stokes equations of motion for general analysis of a dynamic viscous flow is given below;
 1 2 2 2 
 − . = − [ + + ] …(10.2)
 2 2 2
 1 2 2 2 
 − . = − [ + + ] …(10.3)
 2 2 2
 1 2 2 2 
 − . = − [ + + ] …(10.4)
 2 2 2

Where represents body force or gravity force per unit mass of fluid.

www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
 [426]

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com

 Fig 1: Details for MODEL_PIPE_FLOW [User: Ewurum Tennison : 01-Feb-2023 00:58:34]
 MASS FLOW RATE, kg/s

 SIMULATION TIME(s)

 Fig 2: Mass Flow Rate Graph

www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
 [427]

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com
 TEMPERATURE DIFFERENCE, K

 SIMULATION TIME(s)
 Fig 3: Temperature Difference Graph
 THERMAL FLUX, J/s

 SIMULATION TIME(s)
 Fig 4: Thermal Flux Graph

www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
 [428]

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com
 MASS FLOW RATE, kg/s

 SIMULATION TIME(s)

 Fig 5: Mass Flow Rate without Dynamic Compressibility and Inertia
 THERMAL FLUX, J/s

 SIMULATION TIME(s)

 Fig 6: Thermal Flux with no Dynamic Compressibility and Inertia.

www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
 [429]

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com

 TEMPERATURE DIFFERENCE, K

 SIMULATION TIME(s)

 Fig 7: Temperature Difference with no Dynamic Compressibility and Inertia
 SIGNAL, J/s

 SIMULATION TIME(s)

 Fig 8: Ramp Signal Input Graph

www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
 [430]

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com

 SIGNAL, J/s

 SIMULATION TIME(s)

 Fig 9: Sine wave Signal Input Graph
MATLAB Codes for Component Mass_Flow_Source
component mass_flow_source < foundation.thermal_liquid.two_port_source
% Mass Flow Rate Source (TL)
variables(Access = protected)
 power = { 0, 'J/s' }; % Mechanical power generated by the source
end
parameters
 commanded_mass_flow = { 0.6, 'kg/s' }; % Mass flow rate
 length = { 1e-1, 'm'}; % Characteristic longitudinal length
 area = { 1e-2, 'm^2'}; % Pipe cross-sectional area
end
function setup
 % Parameter range checking
 if length

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com
 Gth = if k*area/(length/2) B.Q;
end
equations
 T == A.T - B.T;
 Q == S;
end
end
MATLAB Codes for Component Conductive_ Heat__Transfer

www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
 [432]

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com
component conduction < foundation.thermal.branch
% Conductive Heat Transfer
parameters
 area = { 1e-4, 'm^2' }; % Area
 thickness = { 0.1, 'm' }; % Thickness
 th_cond = { 401, 'W/(m*K)' }; % Thermal conductivity
end
function setup
 % Parameter range checking
 if area

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com
 % Pressures and temperatures at fluid boundaries
 p_A = A.p;
 T_A = A.T;
 p_B = B.p;
 T_B = B.T;
 in
 % No pressure or temperature difference across the sensor
 T_A == T_B;
 p_A == p_B;
 % Measured through variables
 M == mdot;
 H == Phi;
 end
end
end
MATLAB Codes for Component Ideal Temperature Sensor
component temperature
% Ideal Temperature Sensor :0.8
% The block represents an ideal temperature sensor, that is, a device
% that determines the temperature differential measured between two points
% without drawing any heat. The temperature differential, T, is returned
% at the physical signal port T. Connections A and B are conserving thermal
% ports.
%
% The sensor is oriented from A to B and the measured temperature is
% determined as T = T_A - T_B.
nodes
 A = foundation.thermal.thermal; % A:left
 B = foundation.thermal.thermal; % B:right
end
outputs
 T = { 0, 'K' }; % T:right
end
variables(Access=private)
 Qs = { 0, 'J/s' };
 Ts = { 0, 'K' };
end
branches
 Qs : A.Q -> B.Q;
end
equations
 Ts == A.T - B.T;
 Qs == 0;
 Ts == T;

www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
 [434]

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com
end
end
MATLAB Codes for Component Temperature Reservoir
component temperature_reservoir < foundation.thermal_liquid.one_port_source
% Reservoir (TL)
% Thermal liquid port A is the reservoir inlet and is therefore at
% reservoir pressure.
parameters
 pressure_spec = { 1 , '1' }; % Reservoir pressure specification
 % 1 - Atmospheric pressure
 % 2 - Specified pressure
 reservoir_pressure = { 1.01325, 'bar'}; % Reservoir pressure
 reservoir_temperature = { 293.15, 'K' }; % Reservoir temperature
 area = { 1e-2, 'm^2' }; % Inlet pipe cross-sectional area
end
parameters (Access = protected)
 p_reservoir = { 1.01325, 'bar'};
end
function setup
 % Parameter range checking
 if pressure_spec == 2
 if reservoir_pressure < A.p_min
 pm_error('simscape:GreaterThanOrEqual', 'Reservoir pressure', 'minimum valid pressure' )
 end
 if reservoir_pressure > A.p_max
 pm_error('simscape:LessThanOrEqual', 'Reservoir pressure', 'maximum valid pressure' )
 end
 end
 if reservoir_temperature < A.T_min
 pm_error( 'simscape:GreaterThanOrEqual', 'Reservoir temperature', 'minimum valid temperature' )
 end
 if reservoir_temperature > A.T_max
 pm_error( 'simscape:LessThanOrEqual', 'Reservoir temperature', 'maximum valid temperature' )
 end
 if area

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com

equations
 let
 % Thermal conductance
 length = sqrt(area/(4*pi)); % characteristic length of pipe connection
 Gth = if k*area/length

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
 ( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:02/February-2023 Impact Factor- 6.752 www.irjmets.com
1) The influence of fluid dynamic compressibility and flow inertia must be compromised if maximum mass
flow rate and heat leakage are of paramount.
2) This research can also be done in future using different design models and other advanced software for
generalization.
 ACKNOWLEDGMENTS
We thank the Department of Mechanical Engineering in Federal Polytechnic Nekede, Owerri, Imo State, Nigeria,
for given us the opportunity to use their facility and library during the research period.
Funding
This research article was sponsored by the authors.
Disclosure of conflict of interest
This research article is original and the corresponding author hereby confirms that all of the other authors have
read and approved the manuscript with no ethical issues and with declaration of no conflict of interest.
 VIII. REFERENCES
[1] Anderson, J.D. (2009). Computational Fluid Dynamics. National Air and Space Museum, Smithsonian
 Institution, Washington, DC. Springer-Verlag Berlin Heidelberg.
[2] Nataraj, M. & Singh, R.S. (2013). Analyzing pump impeller for performance evaluation using RSM and
 CFD. www. Researchgate.com
[3] Rajput, R.K. (2008). Fluid Mechanics and Hydraulic Machines. New Dehi: Khanna Publishers.
[4] Rajput, R.K. (2011). Fluid Mechanics and Hydraulic Machines. New Dehi: Khanna Publishers.

www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
 [437]