![Further progress in lieu of the past research :
Governingequationsforunsteadynonlinearflowthroughorifice[2,3,4]
9
Continuity equation : 𝜵. 𝝆. 𝒖 𝒌 = 𝟎
Momentum Equation: 𝜵. 𝝆. 𝒖 𝒌
𝟐
= −𝜵𝒑 + 𝜵𝝉 𝒌 + 𝝆. 𝒈
Here, 𝝉 𝒌 = 𝟐𝝁 𝜵𝒖 𝒌 −
𝟐
𝟑
𝝁 𝒖 𝒌 = 𝑆ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠
Turbulent kinetic energy : 𝜵. 𝒖 𝒌. 𝝆. 𝒌 = 𝜵. 𝝁 +
𝒖 𝒌
𝝈 𝒌
𝜵. 𝒌 + 𝑮 𝒌
Where, 𝜌; 𝑘 = Density of fluid (kg/m^3) ; Kinetic energy.
𝛻 = Difference in a quantity
𝑢 𝑘 = Velocity component (m/s)
p= Pressure drop (N/m^2)
𝜇 = Kinematics viscosity (m^2 / s)
𝐺 𝑘 = Generation of turbulence kinetic energy due to the mean velocity gradients (kg.S^3 /m)
𝜎 𝑘= Turbulent Prandtl number for kinetic energy
8/21/2018](https://image.slidesharecdn.com/nonlinearflowthroughorificemeter-180821071441/85/A-study-on-Nonlinear-flow-through-an-orifice-meter-9-320.jpg)
![Further progress in lieu of the past research :
Governingequationsforunsteadynonlinearflowthroughorifice[2,3,4]
10
At the microscopic scale, the incompressible nonlinear flow is governed by the well known Navier-Stokes
equation: 𝝆
𝝏𝑼
𝝏𝒕
+ 𝑼. 𝜵𝑼 = 𝜵𝒑 + 𝝁𝜵 𝟐
𝑼 + 𝑭
For nonlinear flow, the expression for the pressure gradient can be derived from modified Darcy’s law and
Forccheimer’s law: 𝜵𝒑 =
𝝁
𝒌 𝟎 𝑨
𝒒 𝟏 + 𝑭 𝟎
Where,
𝜌
𝜕𝑈
𝜕𝑡
+ 𝑈. 𝛻𝑈 = Force components caused by rate of momentum change and convective acceleration
𝛻𝑝= Pressure gradient
F= Body force vector
𝜇 = Coefficient of friction, q= Rate of flow
𝐹0 = Forccheimer number=
𝑘0 𝜗
𝜇
,in which 𝜗 the velocity of flowing fluid.
𝑘0 = Permeability of orifice plate
A = Flow area
8/21/2018](https://image.slidesharecdn.com/nonlinearflowthroughorificemeter-180821071441/85/A-study-on-Nonlinear-flow-through-an-orifice-meter-10-320.jpg)
![Further progress in lieu of the past research :
Governingequationsforunsteadynonlinearflowthroughorifice[2,3,4]
11
Th. Discharge, 𝑸 𝒂 =
𝑨 𝟏 .𝑨 𝟐 𝟐𝒈𝑯
𝟏−
𝑨 𝟐
𝑨 𝟏
𝟐
× 𝟐 𝑷 𝟏 − 𝑷 𝟐 . 𝝆
Act. Discharge, 𝑸 𝒃 =
𝑪 𝒅 .𝑨 𝟐
𝟏−
𝑨 𝟐
𝑨 𝟏
𝟐
× 𝟐
𝝁
𝒌 𝟎 𝑨
𝒒 𝟏 + 𝑭 𝟎 . 𝝆
Where,
𝐶 𝑑 = Discharge coefficient = Q(actual)/Q(theoretical)
𝐴1 = Cross sectional area of pipe
𝐴2 = Cross sectional area of orifice
𝑃1, 𝑃2= Static Pressures, 𝑃1 − 𝑃2 = 𝛻𝑝
𝜌 = Fluid density
8/21/2018
𝑯 = 𝒉 ×
𝑺 𝟏
𝑺 𝟐
− 𝟏
Where,
H = head (in meters of fluid flowing through the
pipe)
h= differential manometer reading
S1= Specific gravity of manometric liquid
S2= Specific gravity of working liquid (i. e.
Water, S2=1)](https://image.slidesharecdn.com/nonlinearflowthroughorificemeter-180821071441/85/A-study-on-Nonlinear-flow-through-an-orifice-meter-11-320.jpg)
![Results & Discussion
12
Discharge(Qa) vs. Head (H) [1]
8/21/2018
• The discharge through the orifice is
directly proportional to the square root
of head of fluid flowing through the pipe.
• This graph shows that the variation of
discharge with respect to head of fluid of
large pipe.
LARGE PIPE](https://image.slidesharecdn.com/nonlinearflowthroughorificemeter-180821071441/85/A-study-on-Nonlinear-flow-through-an-orifice-meter-12-320.jpg)
![138/21/2018
Results & Discussion
• The discharge through the orifice is
directly proportional to the square root
of head of fluid flowing through the pipe.
• This graph shows that the variation of
discharge with respect to head of fluid of
small pipe.
Discharge(Qa) vs. Head (H) [1]
SMALL PIPE](https://image.slidesharecdn.com/nonlinearflowthroughorificemeter-180821071441/85/A-study-on-Nonlinear-flow-through-an-orifice-meter-13-320.jpg)
![148/21/2018
Results & Discussion
• The discharge through the orifice is
directly proportional to the square root
of head of fluid flowing through the pipe.
• The head is also directly proportional to
the differential manometric head.
• This graph shows that the variation of
discharge with respect to differential
manometric head of large pipe.
LARGE PIPE
Discharge(Qa) vs. Manomatric Head (h) [1]](https://image.slidesharecdn.com/nonlinearflowthroughorificemeter-180821071441/85/A-study-on-Nonlinear-flow-through-an-orifice-meter-14-320.jpg)
![158/21/2018
Results & Discussion
• The discharge through the orifice is
directly proportional to the square root
of head of fluid flowing through the pipe.
• The head is also directly proportional to
the differential manometric head.
• This graph shows that the variation of
discharge with respect to differential
manometric head of small pipe.
Discharge(Qa) vs. Manomatric Head (h) [1]
SMALL PIPE](https://image.slidesharecdn.com/nonlinearflowthroughorificemeter-180821071441/85/A-study-on-Nonlinear-flow-through-an-orifice-meter-15-320.jpg)
![168/21/2018
Results & Discussion
• The pressure gradient inside the orifice is
directly proportional to the Forccheimer
number of fluid flowing through the
orifice plate.
• So, the discharge is also directly
proportional to the Forccheimer number.
• This graph shows that the variation of
discharge with respect to Forccheimer
number of large pipe.
Discharge(Qa) vs. Forccheimer number (F0 ) [1]](https://image.slidesharecdn.com/nonlinearflowthroughorificemeter-180821071441/85/A-study-on-Nonlinear-flow-through-an-orifice-meter-16-320.jpg)
![178/21/2018
Results & Discussion
• The pressure gradient inside the orifice is
directly proportional to the Forccheimer
number of fluid flowing through the
orifice plate.
• So, the discharge is also directly
proportional to the Forccheimer number.
• This graph shows that the variation of
discharge with respect to Forccheimer
number of small pipe.
SMALL PIPE
Discharge(Qa) vs. Forccheimer number (F0 ) [1]](https://image.slidesharecdn.com/nonlinearflowthroughorificemeter-180821071441/85/A-study-on-Nonlinear-flow-through-an-orifice-meter-17-320.jpg)
![188/21/2018
Results & Discussion
• The pressure gradient is defined as the
difference between the pressures inside
orifice.
• The pressure gradient inside the orifice is
directly proportional to the Forccheimer
number of fluid flowing through the
orifice plate.
• This graph shows that the variation of
discharge with respect to pressure
gradient of large pipe.
Discharge(Qa) vs. Pressure Gradient ( 𝛻P) [1]
LARGE PIPE](https://image.slidesharecdn.com/nonlinearflowthroughorificemeter-180821071441/85/A-study-on-Nonlinear-flow-through-an-orifice-meter-18-320.jpg)
![198/21/2018
Results & Discussion
• The pressure gradient is defined as the
difference between the pressures inside
orifice.
• The pressure gradient inside the orifice is
directly proportional to the Forccheimer
number of fluid flowing through the
orifice plate.
• This graph shows that the variation of
discharge with respect to pressure
gradient of small pipe.
SMALL PIPE
Discharge(Qa) vs. Pressure Gradient ( 𝛻P) [1]](https://image.slidesharecdn.com/nonlinearflowthroughorificemeter-180821071441/85/A-study-on-Nonlinear-flow-through-an-orifice-meter-19-320.jpg)
![208/21/2018
Results & Discussion
• This graph shows that the variation of
theoretical discharge and actual
discharge with respect to the
Forccheimer number.
• In this review work, the actual discharge
is find out from the Navier-Stocks
equation with the help of standard
software.
Discharge(Qa & Qb ) vs. Forccheimer number (F0 ) [1]](https://image.slidesharecdn.com/nonlinearflowthroughorificemeter-180821071441/85/A-study-on-Nonlinear-flow-through-an-orifice-meter-20-320.jpg)
A study on Nonlinear flow through an orifice meter
- 1. A Study on Nonlinear Flow Through Orifice Meter Presented by Biplab Bhattacharjee PhD Scholar Enrolment No.:- 16EDPER001 Department of Production Engineering 1 Under the supervision of Dr. Prasun Chakraborti Dr. Kishan Choudhuri Associate Professor Assistant Professor Department of Mechanical Engineering Department of Production Engineering Course Work Seminar Topic: 8/21/2018
- 2. Content 2 • Introduction • Venturimeter vs. Orificemeter • Literature Survey • Working Principle • Formulation • Results & Discussion • Conclusion • References 8/21/2018
- 3. Introduction 3 • Orifice meter is a simple device used for the measuring the discharge through pipes. It also works on the same principle as that of venturi meter. • As such where the space is limited, the orifice meter may be used for the measurement of discharge through pipes . • Orifice Plate is the heart of the Orifice Meter. It restricts the flow and develops the Differential Pressure which is proportional to the square of the flow rate. The flow measuring accuracy entirely depends upon the quality of Orifice plate. • Some important reasons of nonlinear flow are as follows: The role of stretching in fluid mixing Clustering of particles suspended in liquid. Nonlinear waves on fluid interfaces. Multiple stability in granular flow. 8/21/2018
- 4. Applications of Orifice Meter • The concentric orifice plate is used to measure flow rates of pure fluids and has a wide applicability as it has been standardized. • The eccentric and segmental orifice plates are used to measure flow rates of fluids containing suspended materials such as solids, oil mixed with water and wet steam. Advantages of Orifice Meter Limitations of Orifice Meter • Pressure recovery at downstream is poor, that is, overall loss varies from 40% to 90% of the differential pressure. • In the upstream straightening vanes are a must to obtain laminar flow conditions. • Gets clogged when the suspended fluids flow. • The orifice plate gets corroded and due to this after sometime, inaccuracy occurs. Moreover the orifice plate has low physical strength. • The coefficient of discharge is low. 8/21/2018 4 • It is very cheap and easy method to measure flow rate. • It has predictable characteristics and occupies less space. • Can be used to measure flow rates in large pipes.
- 5. Venturimeter vs.Orificemeter 58/21/2018 Venturimeter: Venturi meter is a device used to measure the speed and flow rate of fluid through a pipe. Venturi tube flow meter is often used in applications where the fluid flow has higher turndown rates, lower pressure drops. Orificemeter: With an orifice plate, the fluid flow measured through the difference in pressure from the upstream side to the downstream side of a partially obstructed pipe.
- 6. Literature Survey Sl. No. Title of Paper with Author(s) Methodology Used Conclusion 1. Flow Through an Orifice From a Transverse Stream K. A. Andrews and R. H. Sabersky Journal of Fluids Engineering (1990) The approaching flow is symmetrical in respect to the orifice and the present investigation consists of a set of exploratory experiments to obtain discharge coefficients for this kind of orifice flows and, in particular, to ascertain the effect of the velocity in the pipe on these coefficients. • In the present series of experiments increases in discharge coefficients of as much as 30-40 percent have been measured. In some instances, slight decreases, of the order of 10 percent were noted. • Factors that were not systematically varied are the thickness of the boundary layer in relation to the width of the slot, the velocity gradient in the boundary layer, the effects of laminar or turbulent flow in the boundary layer, as well as the geometrical aspects of the orifice slots. 2. Acoustic Nonlinearity of an Orifice Uno Inoard and Hartmut Islng The Journal of the Acoustical Society of America (1967) The acoustic nonlinearity of an orifice in a plate has been investigated by measuring simultaneously the oscillatory flow velocity in the orifice and the acoustic-pressure fluctuations producing the flow. The relation between the pressure and velocity amplitudes, which is linear at sufficiently low pressures, is found to approach a square-law relation at large velocity amplitude. • The flow through the orifice meter is nonlinear flow. • The acoustic nonlinearity in terms of an orifice impedance is found out. 68/21/2018
- 7. Literature Survey Sl. No. Title of Paper with Author(s) Methodology Used Conclusion 3. Nonlinear Acoustic Properties Of An In-Duct Orifice Lin Zhou This project experimentally investigates the acoustic properties of an orifice with bias flow under medium and high sound level excitation. • The steady pressure difference over the orifice neither keeps constant nor increases so much as when the mean flow velocity keeps constant. • High level acoustic excitation causes the decreasing of the mean flow velocity and it makes resistance decrease in region I and II. 4. Analysis of flow through an orifice meter: CFD simulation Manish S. Shah et. al. Chemical Engineering Science (2012) Computational Fluid Dynamics (CFD) simulation has been used to predict the orifice flow with better accuracy. CFD simulations have been performed using OpenFOAM-1.6 solver and validated with the published experimental data. • Very good agreement between experimental data and CFD predictions viz. energy balance, flow pattern, pressure recovery, velocity profiles, pressure profiles and sensitivity analysis of turbulence model parameters, validates the CFD predictions in orifice flow. • It is also concluded that the CFD technique can be used as an alternative and cost effective tool towards replacement of experiments required for estimating discharge coefficient, empirically. 78/21/2018
- 8. Working: The orifice plate, being fixed at a section of the pipe, creates an obstruction to the flow by providing an opening in the form of an orifice to the flow passage. When an orifice plate is placed in a pipe carrying the fluid whose rate of flow is to be measured, the orifice plate causes a pressure drop which varies with the flow rate. This pressure drop is measured using a differential pressure sensor and when calibrated this pressure drop becomes a measure flow rate. Working Principle of Orifice meter 8/21/2018 8
- 9. Further progress in lieu of the past research : Governingequationsforunsteadynonlinearflowthroughorifice[2,3,4] 9 Continuity equation : 𝜵. 𝝆. 𝒖 𝒌 = 𝟎 Momentum Equation: 𝜵. 𝝆. 𝒖 𝒌 𝟐 = −𝜵𝒑 + 𝜵𝝉 𝒌 + 𝝆. 𝒈 Here, 𝝉 𝒌 = 𝟐𝝁 𝜵𝒖 𝒌 − 𝟐 𝟑 𝝁 𝒖 𝒌 = 𝑆ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 Turbulent kinetic energy : 𝜵. 𝒖 𝒌. 𝝆. 𝒌 = 𝜵. 𝝁 + 𝒖 𝒌 𝝈 𝒌 𝜵. 𝒌 + 𝑮 𝒌 Where, 𝜌; 𝑘 = Density of fluid (kg/m^3) ; Kinetic energy. 𝛻 = Difference in a quantity 𝑢 𝑘 = Velocity component (m/s) p= Pressure drop (N/m^2) 𝜇 = Kinematics viscosity (m^2 / s) 𝐺 𝑘 = Generation of turbulence kinetic energy due to the mean velocity gradients (kg.S^3 /m) 𝜎 𝑘= Turbulent Prandtl number for kinetic energy 8/21/2018
- 10. Further progress in lieu of the past research : Governingequationsforunsteadynonlinearflowthroughorifice[2,3,4] 10 At the microscopic scale, the incompressible nonlinear flow is governed by the well known Navier-Stokes equation: 𝝆 𝝏𝑼 𝝏𝒕 + 𝑼. 𝜵𝑼 = 𝜵𝒑 + 𝝁𝜵 𝟐 𝑼 + 𝑭 For nonlinear flow, the expression for the pressure gradient can be derived from modified Darcy’s law and Forccheimer’s law: 𝜵𝒑 = 𝝁 𝒌 𝟎 𝑨 𝒒 𝟏 + 𝑭 𝟎 Where, 𝜌 𝜕𝑈 𝜕𝑡 + 𝑈. 𝛻𝑈 = Force components caused by rate of momentum change and convective acceleration 𝛻𝑝= Pressure gradient F= Body force vector 𝜇 = Coefficient of friction, q= Rate of flow 𝐹0 = Forccheimer number= 𝑘0 𝜗 𝜇 ,in which 𝜗 the velocity of flowing fluid. 𝑘0 = Permeability of orifice plate A = Flow area 8/21/2018
- 11. Further progress in lieu of the past research : Governingequationsforunsteadynonlinearflowthroughorifice[2,3,4] 11 Th. Discharge, 𝑸 𝒂 = 𝑨 𝟏 .𝑨 𝟐 𝟐𝒈𝑯 𝟏− 𝑨 𝟐 𝑨 𝟏 𝟐 × 𝟐 𝑷 𝟏 − 𝑷 𝟐 . 𝝆 Act. Discharge, 𝑸 𝒃 = 𝑪 𝒅 .𝑨 𝟐 𝟏− 𝑨 𝟐 𝑨 𝟏 𝟐 × 𝟐 𝝁 𝒌 𝟎 𝑨 𝒒 𝟏 + 𝑭 𝟎 . 𝝆 Where, 𝐶 𝑑 = Discharge coefficient = Q(actual)/Q(theoretical) 𝐴1 = Cross sectional area of pipe 𝐴2 = Cross sectional area of orifice 𝑃1, 𝑃2= Static Pressures, 𝑃1 − 𝑃2 = 𝛻𝑝 𝜌 = Fluid density 8/21/2018 𝑯 = 𝒉 × 𝑺 𝟏 𝑺 𝟐 − 𝟏 Where, H = head (in meters of fluid flowing through the pipe) h= differential manometer reading S1= Specific gravity of manometric liquid S2= Specific gravity of working liquid (i. e. Water, S2=1)
- 12. Results & Discussion 12 Discharge(Qa) vs. Head (H) [1] 8/21/2018 • The discharge through the orifice is directly proportional to the square root of head of fluid flowing through the pipe. • This graph shows that the variation of discharge with respect to head of fluid of large pipe. LARGE PIPE
- 13. 138/21/2018 Results & Discussion • The discharge through the orifice is directly proportional to the square root of head of fluid flowing through the pipe. • This graph shows that the variation of discharge with respect to head of fluid of small pipe. Discharge(Qa) vs. Head (H) [1] SMALL PIPE
- 14. 148/21/2018 Results & Discussion • The discharge through the orifice is directly proportional to the square root of head of fluid flowing through the pipe. • The head is also directly proportional to the differential manometric head. • This graph shows that the variation of discharge with respect to differential manometric head of large pipe. LARGE PIPE Discharge(Qa) vs. Manomatric Head (h) [1]
- 15. 158/21/2018 Results & Discussion • The discharge through the orifice is directly proportional to the square root of head of fluid flowing through the pipe. • The head is also directly proportional to the differential manometric head. • This graph shows that the variation of discharge with respect to differential manometric head of small pipe. Discharge(Qa) vs. Manomatric Head (h) [1] SMALL PIPE
- 16. 168/21/2018 Results & Discussion • The pressure gradient inside the orifice is directly proportional to the Forccheimer number of fluid flowing through the orifice plate. • So, the discharge is also directly proportional to the Forccheimer number. • This graph shows that the variation of discharge with respect to Forccheimer number of large pipe. Discharge(Qa) vs. Forccheimer number (F0 ) [1]
- 17. 178/21/2018 Results & Discussion • The pressure gradient inside the orifice is directly proportional to the Forccheimer number of fluid flowing through the orifice plate. • So, the discharge is also directly proportional to the Forccheimer number. • This graph shows that the variation of discharge with respect to Forccheimer number of small pipe. SMALL PIPE Discharge(Qa) vs. Forccheimer number (F0 ) [1]
- 18. 188/21/2018 Results & Discussion • The pressure gradient is defined as the difference between the pressures inside orifice. • The pressure gradient inside the orifice is directly proportional to the Forccheimer number of fluid flowing through the orifice plate. • This graph shows that the variation of discharge with respect to pressure gradient of large pipe. Discharge(Qa) vs. Pressure Gradient ( 𝛻P) [1] LARGE PIPE
- 19. 198/21/2018 Results & Discussion • The pressure gradient is defined as the difference between the pressures inside orifice. • The pressure gradient inside the orifice is directly proportional to the Forccheimer number of fluid flowing through the orifice plate. • This graph shows that the variation of discharge with respect to pressure gradient of small pipe. SMALL PIPE Discharge(Qa) vs. Pressure Gradient ( 𝛻P) [1]
- 20. 208/21/2018 Results & Discussion • This graph shows that the variation of theoretical discharge and actual discharge with respect to the Forccheimer number. • In this review work, the actual discharge is find out from the Navier-Stocks equation with the help of standard software. Discharge(Qa & Qb ) vs. Forccheimer number (F0 ) [1]
- 21. Conclusion 21 • Within the limits of the experimental uncertainty and range of Reynolds number (feasible) investigated, the results obtained for the discharge coefficient through an orifice plate agree with the empirical relation as awarded by the earlier investigators. • The performance of the orifice meter with nonlinear flow can be somewhat evaluated by the linear relationship between flow rate and maximum pressure drop through an orifice. • The vena-contracta length depends on the surface roughness of the inner wall of the pipe and sharpness of the orifice plate at the edge. • The flow through the orifice is significantly affected by pressure gradient and velocity of fluid flow as observed under a modification attempted by me. • Forccheimer number is a very important parameter in case of nonlinear fluid flow rate (Qb) through the hole of the orifice plate owing to the factors like velocity of flowing fluid, coefficient of friction and permeability of the orifice plate. 8/21/2018
- 22. References 22 • K. A. Andrews and R. H. Sabersky (1990); “Flow Through an Orifice From a Transverse Stream”, Journal of Fluids Engineering • Uno Inoard and Hartmut Islng (1967), “Acoustic Nonlinearity of an Orifice”, The Journal of the Acoustical Society of America • Lin Zhou (2010), “Nonlinear Acoustic Properties Of An In-Duct Orifice”, International Conference on Engineering Trends 2010, North Korea • Manish S. Shah, Jyeshtharaj B. Joshi, Avtar S. Kalsi, C.S.R. Prasad, Daya S. Shukla, (2012) “Analysis of flow through an orifice meter: CFD simulation”, Chemical Engineering Science 8/21/2018