An Orifice Flow Analysis on the Basis of Density and Viscosity Effects
of Fluids
SANTOSH KUMAR PANDA
Mechnical Engineering Department,
NIST Institute of Science and Technology,
Berhampur, Odisha, PIN- 761008,
INDIA
Abstract: - The orifice flow measurement produces errors due to highly turbulent and backflow in downstream which
have to be overcome. Determine the Δp, and coefficient of discharge (Cd) for flow through the orifice CFD analysis. The
input parameters vary with the variance of Reynolds number (Re), fluids with the deviation of density and viscosity, area
ratio (σ). The range of Re (20000- 100000), σ (0.2- 0.6), density (ρr), and viscosity (µr) ratio vary as per the fluids
considered. The various incompressible and compressible fluids are considered for the study of flow through the orifice
based on the difference in density and viscosity properties. The fluids considered for studies are Air, Ammonia (NH3),
Carbon dioxide (CO2), Hydrogen (H2), and Sulphur dioxide (SO2) as compressible fluid category, and water (H2O), liquid
ammonia (LNH3), liquid hydrogen (LH2), liquid oxygen (LO2), liquid R12 (Dichlorodifluoromethane) as incompressible
fluid category. Correlations are also proposed from the above numerical database to determine Cd as a function of Re, σ, ρr,
and µr for the orifice. The correlation provides a significant contribution to the viscous fluid flow measurement with the
flow through the orifice.
Key-Words: - Area ratio, Coefficient of discharge, Multi-fluid, Orifice, Pressure drop, Viscosity.
Received: April 12, 2023. Revised: September 18, 2023. Accepted: November 14, 2023. Published: December 31, 2023.
1 Introduction
Orifice flow meter is a popular flow measuring
equipment used for fluid flow measurement for a
wider range of applications. The repeatability of
measuring, dependability for a longer duration, a
wider range of flow limit, and simplicity of design
make the orifice more adaptable. In comparison to
the flow meter such as the nozzle and venturimeters
orifice gives better Δp to determine the Cd.
Bernoulli's principle is used to determine the fluid
energy along the flow through the orifice. Two
pressure taps are placed (one 2D upstream and the
other 0.5D distance downstream) to determine the
Δp with the help of pressure measuring devices.
The Δp of the flow changes because of the vena-
contractor (smallest cross-sectional area) of the
orifice. A literature-based study reported on flow
through an orifice to know the fundamental of the
orifice, [1]. Re and σ play an important role in the
prediction of Δp and Cd based on correlation for
different geometrical profile of the orifice. A
perforated plate orifice is used to develop
correlation, which is the function of Re and σ for
the prediction of Δp and Cd, [2], [3]. Square-edged
and concentric orifices used to develop correlation
as a function of plate thickness, σ and Re, [4]. The
pressure loss coefficient for a circular orifice is
determined based on a correlation, function of Re,
σ, and reduction ratio, [5]. A comparison study for
various profiles such as sloping-approach orifices,
sharp-edged and streamlined orifices evaluated Δp
and energy loss in a flood conduit to minimize the
cavitations effect, [6]. Discharge coefficient for
orifices and nozzles are evaluated with equations to
an orifice pipe flow, [7]. A series of correlations
expanded for Δp for a circular orifice on the
variation of non-dimensional orifice number,
thickness, and area ratio, [8]. Number of
compressible natural fluids (air, CO2, Ar, and He,
and mixture of Ar-He) are considered as working
fluids to determine Cd for various flow devices
such as orifice and nozzles, [9]. Air and water were
considered for a numerical study for an orifice flow
on the variation of Re, area, and space ratio to
predicted Δp, [10]. Several fluids applied for orifice
flow analysis for different fluids, [11], [12], [13],
[14], [15], [16], for various applications of the flow
through the orifice. High viscous fluids for small
size orifices used such as molasses, [17], viscous
fluids, [18], and high viscous fluids, [19], to
determine the flow coefficient of an orifice flow.
The orifice flow was used for a sharp-edge circular
orifice with R113 (Tri-chloro-tri-fluoro-methane) to
develop several correlations for different fluid
flows, [20], [21], [22]. Water-based flow analysis
through the orifice considers for estimation of flow
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characteristics with variation of input parameters,
[23], [24], [25], [26]. Multiple fluids are used in
multi-phase analysis such as air-water, [27], [28],
[29], [30], steam [31], wet gas [32], petroleum
products [33], for the orifice flow to analyze two-
phase studies.
Based on the literature study a flow analysis of
an orifice to predict velocity profile, discharge
coefficient (Cd), and pressure drop (Δp) for
different compressible and incompressible fluids.
Two no. of correlations will be estimated for the
fluids (Compressible and Incompressible) flow
through the orifice. The input parameters are varied
such as Reynolds number (Re), area ratio (σ), and
properties of fluids such as viscosity (µr) and
density (ρr) ratio to carry out the numerical studies.
2 Numerical Methodology
A geometrical model was prepared for the orifice
with pipe flow for the numerical analysis of
different fluids. Unstructured mesh is generated for
the model with the use of CFD based software. The
optimum solution to the numerical problem was
achieved with the generation of high-class and
aspect ratios for the grids. The mesh is discretized
with a finite-volume method (FVM). Wall effects
of the flow domain were improved by introducing
more no grids to enable grid-independent studies.
The circumference of the mesh volume improved
by creating O-grid mesh size for uniform
volumetric elements. Figure 1 presents a line sketch
of the orifice model for the present studies. The
dimension of the orifice is represented with
nomenclature presented the Figure 1. The ‘D’ is the
diameter of the pipe, ‘d’ is the diameter of the
orifice, and ‘l’ is the thickness of the orifice. The
Three different ‘D’ sizes are considered as 50.8 to
101.6 mm. The ‘d’ dimensions are selected to
maintain the σ. Figure 2(a) depicts the orifice
province mesh structure and mesh volume of the
orifice province. The Figure 2(b) is a enlarge view
of two different orifice mesh geometries near the
orifice zone shown for the present analysis.
Fig. 1: Line sketch of the orifice model
(a) Mesh
topology
Fig. 2: Geometry and computational mesh of a
present flow model
The fluids of compressible and incompressible
flow are considered for the numerical study based
on density and viscosity properties. The fluids are
taken into account based on categories of property
that differ from the based fluids. The base fluids for
compressible flow are air and water for
incompressible fluids. The fluids chosen for
incompressible fluids are LH2O, O2, LH2, R12, and
LNH3 as per the fluid property. Similar to
compressible fluids are air, NH3 gas, CO2, H2 gas,
and SO2. The fluids and their properties are
mentioned in Table 1.
Table 1. Property of the fluids
Name
ρ
(kg/m
3)
μ
(kg/m-
s)
Na
me
ρ
(kg/m
3)
μ x10-5
(kg/m-s)
Incompressible
Compressible
H2O
998.2
0.001
Air
1.225
1.79
LO2
1142
0.0002
NH
3
0.689
4
1.02
LH2
70.85
1.33e-
5
CO
2
1.787
8
1.70
R12
liquid
1130
0.0002
H2
0.081
9
0.841
NH3
liquid
610
0.0001
5
SO2
2.77
1.20
2.1 Governing Equation
The assumptions considered for the study are
steady flow, and Newtonian fluid with the
neglected compressibility factor.
Continuity
.0u
(2)
Momentum
2
..u u p u g


(3)
K.E
.( ) . tk
k
uk k P







(4)
Energy dissipation rate equation
2
12
.. tk
u C P C
kk








(5)
P= production term, μt= eddy viscosity and
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tk
C
,
12
0.09, 1.44, 1.44, 1, 1.3
k
C C C

2.2 Grid Independence Study
A suitable optimized grid size is required to get
accurate results for a numerical problem. The grid
independence test is the method to get the
optimized grid size for a geometric model. The
present study has considered five different models
as per the σ concern. The grid-independent study
for a model is reported in Table 2. The model size
is σ = 0.36, pipe diameter (D) = 101.6 mm
considered with Re = 60000 for grid-independent
study for different grids as reported in Table 2. The
grid shapes make an O-grid type and the wall
intensity increases by introducing more no grids for
accurate results. The mesh intensity increases near
the orifice section and upstream of the orifice along
the flow. The mesh element size decreases from
0.008 to 0.002 by increasing the number of mesh
sizes. The mesh sizes considered for the
independent study are 65.2, 103.5, 147.7, 315.7,885
million, among these 147.7 million mesh sizes give
better results for the Δp as mentioned in Table 2.
The % increase in the Δp calculated based on the
formula mentioned in eq. (6) The grids on the mesh
increase nearly the orifice to get a better flow
profile. The study found 147.7 million gives better
results as reported in Table 2.
Table 2. Mesh Independence study
Sl no
1
2
3
4
5
Mesh
Size
6524
9
10357
3
14771
0
31570
0
88503
3
Δp
(kPa)
3434.
6
3591.
2
3653.
2
3655.
4
3656
%
increas
e in Δp
---
4.55
1.69
0.06
0.01
new mesh previous mesh
previous mesh
p - p
% increase in p 100
p

(6)
2.3 Numerical Solution
The governing equations used for the present study
are mass, momentums, turbulent kinetic energy,
and dissipation rate of turbulent energy (standard
K-ε model) for the compressible and
incompressible fluids. The governing equations are
present in Eqs. (2)–(5). Steady-state simulation
carried out with the gravitational force acting
against gravity and pressure based solver used for
the numerical studies. Velocity inlet and pressure
outlet boundary conditions used for the inlet and
outlet to the orifice, and no-slip condition are used
for the wall to define the flow domain. The fluids
were selected as per the numerical studies required.
The turbulence intensity was selected with 5% at
the inlet with fully developed flow by assuming
hydraulic diameter. SIMPLE (Semi-implicit method
for pressure linked equation) algorithm used
pressure-velocity coupling. The grids are
discretized with the least square cell method.
Standard scheme for pressure terms, second order
upwind scheme has been used for momentum and
turbulent equation terms. The under-relaxation
factor (URF) used for pressure is 0.3, momentum
0.4 – 0.7, turbulent equation 0.8. The residual scale
is considered as required the converge criteria. The
solution is initialized before the simulation starts
and the output result is collected after convergence
of the problem.
3 Result Analysis
The computational envisaged result evaluated with
the existing analysis represented in (Roul and Das,
2012) of the flow-through orifice analysis for the
validation of the study. The above literature
considered H2O as a single phase fluid. The
parameters considered in the above article are an
orifice of 60 mm diameter, σ = 0.54 for different Re
= 30000 to 200000 to determine Δp. Similar
computational case studies were taken account for
the validation of the present analysis. Upto 5%
error was found in the present study on the
prediction of Δp.
The static pressures (St. Pr.) for different fluids
along the flow length of the orifice are shown in
Fig. 3(a) compressible fluids and Figure 4(a) for
incompressible fluids. The decreasing orders of the
St. Pt. along the compressible fluids are CO2,
NH3 gas, H2 gas, Air, and SO2. The graph
concludes that H2 has maximum St. Pt. compared
to other fluids. From the incompressible fluid point
of view H2O gives maximum St. Pt. because of the
high ρr, µr. The static Δp along the flow path of the
orifice for variation in Re is shown in Figure 3(b)
for compressible fluids and Figure 4(b) for
incompressible fluids.
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0.0 0.3 0.6 0.9 1.2 1.5
0
1000
St. Pr, Pa
L,m
Air
NH3
CO2
H2
SO2
(a) Variation of fluids
0.0 0.2 0.4 0.6 0.8 1.0
-4000
0
4000
8000
12000
St. Pr, Pa
L,m
Re = 20000
Re = 40000
Re = 60000
Re = 80000
Re = 100000
b) Variation with Re for air
Fig. 3: St. Pr. profiles for the compressible flow
0.0 0.3 0.6 0.9 1.2 1.5
-5000
0
5000
10000
15000
St. Pr, Pa
L,m
H2O
NH3
H2
O2
R12
(a) Variation of fluids
0.0 0.3 0.6 0.9 1.2 1.5
-10000
0
10000
20000
30000
40000
50000
St. Pr, Pa
L,m
Re = 20000
Re = 40000
Re = 60000
Re = 80000
Re = 100000
(b) Variation with Re for water
Fig. 4: St. Pr. profiles for the incompressible flow
Figure 5(a) and Figure 5 (b) demonstrate the Δp
along the orifice flow for compressible and
incompressible fluid on the variation of Re. The
graphs found H2 has maximum Δp for compressible
fluids and H2O gives maximum Δp compared to the
other incompressible fluids along the variation of
Re. The H2 has maximum because of better fluid
properties compared to other fluids. The results
were reported because of larger St. Pr. profile as
mentioned in the above figures. The Δp increases as
the Re increases because of an increase in the
velocity of fluids because of a directly proportional
relation.
20000 40000 60000 80000 100000
0
20000
40000
Δp,Pa
Re
Air
NH3
CO2
H2
SO2
(a) Compressible flow
20000 40000 60000 80000 100000
0
5000
10000
15000
20000
25000
30000
35000
Δ p, Pa
Re
H2O
LNH3
LH2
LO2
LR12
(b) Incompressible flow
Fig. 5: Δp on the variation of Re
The Δp varies with Re for different σ as shown
in Figure 6(a) and Figure 6(b) for compressible and
incompressible fluids for an orifice flow. The
smaller σ founds more Δp for compressible and
incompressible flows because of the narrow
passage. The Δp decreases as the σ value increases
with a wider orifice area to the fluid flows. Figure 6
also found that the Δp is more for incompressible
flow compared to compressible flow for a specified
σ value. Figure 7 reported the phenomena of Cd for
an orifice flow. The Cd for compressible fluids
shows that the values not vary much for different
fluids as values vary from 0.56 to 0.67 for the
variation of Re along the orifice flow. The Cd
values decrease as the Re increases as concluded
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because of smooth flow with the increase in the
velocity of flow.
20000 40000 60000 80000 100000
0
3000
6000
9000
12000
15000
Δp,Pa
Re
σ=0.36
σ=0.48
σ=0.6
(a) Compressible flow
20000 40000 60000 80000 100000
0
10000
20000
30000
40000
50000
60000
Δp, Pa
Re
σ=0.36
σ=0.48
σ=0.6
(b) Incompressible flow
Fig. 6: Δp on the variation of σ for orifice
20000 40000 60000 80000 100000
0.55
0.56
0.57
0.58
0.59
0.60
0.61
0.62
Cd
Re
Air
NH3
CO2
H2
SO2
(a) Compressible flow
20000 40000 60000 80000 100000
0.66
0.67
0.68
0.69
Cd
Re
H2O
LNH3
LH2
LO2
LR12
(b) Incompressible flow
Fig. 7: Cd on the variation of Re for orifice
3.1 Correlation
Two correlations are proposed for Cd for
compressible and incompressible fluid along the
orifice flow. The correlations are a function of Re,
σ, and the non-dimensional parameters of density
and viscosity. GARCH (Generalized
Autoregressive Conditional Heteroskedasticity) is a
MATLAB code that works on the principle of
multi-regression tools to get coefficients of the
correlations such as C1, C2, C3, ..., etc. The
correlation is represented in eq. (7-8) with
correlation coefficient to be predicted. GARCH is a
statistical tool used for the prediction financial
outcome for a particular period from the present
tendency which fits for the present numerical
studies for developing the correlation. The
correlation yields results with a 10% maximum
margin of error. The correlation is as follows
Cd= f(Re, σ, µr, ρr)
(7)
3 4 5
2
1.(Re) . . .
C C C
C
d r r
CC
(8)
Where
r
r
b
r
A
avd
,,,Re
The correlation functions Cd = f (Re, σ, µr, ρr)
and correlation coefficient (C1, C2, C3, C4, C5) for
compressible and incompressible fluids are
mentioned in Table 3. The correlation coefficient
predicted the factorial relation between the input
and output of the parameter for the analysis. The
numerical result fits with the correlation data of an
error within ±10% for both compressible and
incompressible fluids. The error ±10% mentioned
as a marginal trend line for the graphical
presentation. The correlation is a function of Re, σ,
and the property parameter µr, ρr which is a
function of fluids. The range for the correlations is
Re = 20000 to 100000, σ = 0.3 to 0.6, µr, and ρr as
the selection of the fluids.
Table 3. Correlation coefficients
Type of
fluids
C1
C2
C3
C4
C5
Compressi
ble
1.1
6
0.033
25
0.439
24
0.490
07
0.385
05
Incompres
sible
0.7
03
0.011
95
0.035
3
-
0.056
91
0.074
19
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4 Conclusions
Numerical studies for an orifice were carried out to
determine the velocity profile, Static pressure
profile (St. Pr.), Cd , and Δp for a fluid flow. The
flow fluids are considered as per the
compressibility and incompressibility nature based
on the properties of density and viscosity. Re and σ
are the other input parameters for the study. The
output parameters are compared with air for
compressible flow and water for incompressible
flow fluids. The study concludes that changing
viscosity and density affects the Δp, Cd , and other
flow profiles. The Cd value predicts 0.56 to 0.67
which is particular for different applications of
orifice flow. Based on the numerical prediction of
the results two different correlations developed for
compressible and incompressible fluids. The above
study considers of different categories of fluids and
properties such as the compressibility of fluids for
further analysis. The correlations developed based
on the µr, and ρr and predicted up to ±10%
agreement for the numerical prediction database.
The study will help to calibrate the mass flow
measurement and controlling device for different
fluids.
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WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.12
Santosh Kumar Panda
E-ISSN: 2224-3461
146
Volume 18, 2023