Design & analysis of laminar flow meter

DESIGN & ANALYSIS OF
LAMINAR FLOW METER
Paper made
By
Abhijit Roy
Pursuing M.Tech in Material Science & Engineering.
B.Tech in Mechanical Engineering.
Address- Bhattapukur near Suktara club chowmuhani
Agartala, Tripura
Pin- 799003
Contact-
Email- abhijitroy9619@gmail.com
Mobile- 8794150689
September, 2019

i
“You can’t teach people everything they need to know. The
best you can do is position them where they can find what
they need to know when they need to know it.”
-Seymour Papert (MIT Mathematician)

ii
Dedication
To my Teachers and Parents

Abstract
Various types of discharge measuring instruments are currently used to measure discharge of
liquids through conduits. These are venture meter, orifice meter, rotameter, pitot tubes etc.
We are presenting a new more accurate flow measuring device with more accuracy, called
laminar flow meter.
Laminar flow elements differ from other metering devices in that they are specifically
designed to operate in laminar flow regime. Pipe flows are generally considered laminar if
Reynolds number is less than 2000. The simplest form of laminar flow element is merely a
length of small diameter tubing. Hagen Poiseuille viscous flow relation gives for
incompressible fluids a linear relation between volume flow rate and pressure drop in the
tubing. We use this principle to calculate the discharge through the pipe by measuring the
differential head using a manometer. Extremely small flows can be measured in this way. A
three feet length of 0.004 inch diameter tubing measuring pressure difference with a 2 inch
water inclined manometer gives a threshold sensitivity of about 0.000175 𝑖𝑛3/h. when
hydrogen is flowing. Another highlight of laminar flow elements is that, it has the advantages
accruing from a linear (rather than square root) relation between flow rate and pressure drop:
these are principally a large accurate range of as much as 100:1 (compared with 3:1 or 4:1 for
square root devices), accurate measurement of average flow rates in pulsating flow, and ease
of integrating Δp signals to compute total flow. The laminar elements also can measure
reversed flows with no difficulty. They are usually less sensitive to upstream and downstream
flow disturbances than the other flow measuring devices.
Laminar flow meter has some disadvantages too. It includes clogging from dirty
fluids, high cost, large size, and pressure loss(all the measured Δp is lost).
.
iii

List of Tables
Tables Page no
Table 7.1: Density of liquids. 13
Table 8.3.2: Test details 17
iv

v
List of Figures
Figures Page no
Fig.2.1: Venturimeter 2
Fig.2.2: Orifice Plate 3
Fig.3.1: Velocity Distribution 6
Fig.4.1: Entrance Length 9
Fig.7.1: U tube Manometer 12
Fig 8.1 : Laminar Flow Meter 14
Fig.8.2: Laminar flow Meter Model 18

List of symbols, abbreviations
ρ – density
µ - viscosity
Re – Reynold’s number
P – Pressure
Q – Discharge
D – Diameter
L – Length
V – Velocity
g – Acceleration due to gravity

Contents
Page no
Quotes i
Dedication ii
Abstract iii
List of tables iv
List of figures v
Notations and Abbreviations vi

Contents
Chapters Page No
1. Introduction 1
2. Pressure based flow measuring devices 2
(a) Venturimeter
(b) Orifice plate 3
(c) Dall tube 5
(d) Pitot tube
3. Hagen Poiseuille Equation 6
4. Entrance length in pipe flow 9
5. Reynolds Number 10
6. Laminar Flow 11
7. Liquid Column Gauge 12
8. Laminar flow Meter 14
(a) Design 15
(b) Construction 16
Test Results 17
(c) Laminar Flow Meter Model 18
9. Conclusion 19
10.Future scope & advantages 20
References 21

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Chapter-1
Introduction
Measuring the rate volume of fluid flow through a duct is one of the most important
engineering task with a lot applications in different fields. Depending on the situation of use
Fluid flow rate may be very high or very low. A common flow measuring device or a
common principle of operation will not suffice for all situations.
We are designing a simple flow measuring device called Laminar flow meter to be
used where volume flow rate of fluid is extremely low and fluid is in very low pressure.
Commonly used flow measuring devices such as venture meter, orifice meter, pitot tube,
rotameter etc will cannot be good enough to measure such small volume flow rate. More over
most of these are square root devices (volume flow rate is proportional to square root of
differential pressure head) and are not very accurate enough. But Laminar flow meter is
based on Hagen Poiseulle principle and there is a linear relation between volume flow rate
and differential head.
Since we do not have such an accurate and special purpose flow measuring
device in our labs we decided to build laminar flow meter and to study the principles and
characteristics of the device.

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Chapter-2
Pressure based flow measuring device
( a) Venturimeter
The Venturi effect is a jet effect; as with a funnel the velocity of the fluid increases as the
cross sectional area decreases, with the static pressure correspondingly decreasing. According
to the laws governing fluid dynamics, a fluid's velocity must increase as it passes through a
constriction to satisfy the principle of continuity, while its pressure must decrease to satisfy
the principle of conservation of mechanical energy. Thus any gain in kinetic energy a fluid
may accrue due to its increased velocity through a constriction is negated by a drop in
pressure. An equation for the drop in pressure due to the Venturi effect may be derived from
a combination of Bernoulli's principle and the continuity equation.
The limiting case of the Venturi effect is when a fluid reaches the state of choked flow, where
the fluid velocity approaches the local speed of sound. In choked flow the mass flow rate will
not increase with a further decrease in the downstream pressure environment.
However, mass flow rate for a compressible fluid can increase with increased upstream
pressure, which will increase the density of the fluid through the constriction (though the
velocity will remain constant). This is the principle of operation of a de Laval nozzle.
Increasing source temperature will also increase the local sonic velocity, thus allowing for
increased mass flow rate.
Fig 2.1- Venturimeter

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Referring to the diagram above, using Bernoulli's equation in the special case of
incompressible flows (such as the flow of water or other liquid, or low speed flow of gas), the
theoretical pressure drop at the constriction is given by:
𝑝 𝑝 =
𝜌
(𝑣2 − 𝑣2)......................... eq.(2.1)
1 − 2 2 2 1
where is the density of the fluid, is the (slower) fluid velocity where the pipe is
wider, is the (faster) fluid velocity where the pipe is narrower (as seen in the figure). This
assumes the flowing fluid (or other substance) is not significantly compressible - even though
pressure varies, the density is assumed to remain approximately constant.
(b) Orifice plate
An orifice plate is a device used for measuring flow rate. Either a volumetric or mass flow
rate may be determined, depending on the calculation associated with the orifice plate. It uses
the same principle as a Venturi nozzle, namely Bernoulli's principle which states that there is
a relationship between the pressure of the fluid and the velocity of the fluid. When the
velocity increases, the pressure decreases and vice versa.

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Fig 2.2 - Orifice plate
Q = Cd A2
1
1−β4 √2(P1 − P2)/ρ....................................eq.(2.2)
An orifice plate is a thin plate with a hole in the middle. It is usually placed in a pipe in which
fluid flows. When the fluid reaches the orifice plate, the fluid is forced to converge to go
through the small hole; the point of maximum convergence actually occurs shortly
downstream of the physical orifice, at the so-called vena contracta point (see drawing to the
right). As it does so, the velocity and the pressure changes. Beyond the vena contracta, the
fluid expands and the velocity and pressure change once again. By measuring the difference
in fluid pressure between the normal pipe section and at the vena contracta, the volumetric
and mass flow rates can be obtained from Bernoulli's equation.
√

5 @a.roy
Other pressure based flow measuring devices
(c) Dall tube
The Dall tube is a shortened version of a Venturi meter, with a lower pressure drop than an
orifice plate. As with these flow meters the flow rate in a Dall tube is determined by
measuring the pressure drop caused by restriction in the conduit. The pressure differential is
typically measured using diaphragm pressure transducers with digital readout. Since these
meters have significantly lower permanent pressure losses than orifice meters, Dall tubes are
widely used for measuring the flow rate of large pipeworks.
(d) Pitot tube
A pitot tube is a pressure measurement instrument used to measure fluid flow velocity. The
pitot tube was invented by the French engineer Henri Pitot in the early 18th century and was
modified to its modern form in the mid-19th century by French scientist Henry Darcy. It is
widely used to determine the airspeed of an aircraft and to measure air and gas velocities in
industrial applications. The pitot tube is used to measure the local velocity at a given point in
the flow stream and not the average velocity in the pipe or conduit.
Multi-hole pressure probes
Multi-hole pressure probes (also called impact probes) extend the theory of pitot tube to more
than one dimension. A typical impact probe consists of three or more holes (depending on the
type of probe) on the measuring tip arranged in a specific pattern. More holes allow the
instrument to measure the direction of the flow velocity in addition to its magnitude (after
appropriate calibration). Three holes arranged in a line allow the pressure probes to measure
the velocity vector in two dimensions. Introduction of more holes, e.g. five holes arranged in
a "plus" formation, allow measurement of the three-dimensional velocity vector.

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Chapter-3
Hagen- poiseuille equation
In fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille
law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in a
fluid flowing through a long cylindrical pipe. It can be successfully applied to blood flow
in capillaries and veins, to air flow in lung alveoli, for the flow through a drinking straw or
through a hypodermic needle. It was experimentally derived independently by Gotthilf
Heinrich Ludwig Hagen in 1839 and Jean Louis Marie Poiseuille in 1838, and published by
Poiseuille in 1840 and 1846.
The assumptions of the equation are that the fluid is viscous and incompressible; the flow is
laminar through a pipe of constant circular cross-section that is substantially longer than its
diameter; and there is no acceleration of fluid in the pipe. For velocities and pipe diameters
above a threshold, actual fluid flow is not laminar but turbulent, leading to larger pressure
drops than calculated by the Hagen–Poiseuille equation.
∆𝑃 =
128𝜇𝐿𝑄.......................................................
eq.(3.1)
𝜋𝑑4
Fig 3.1 – Velocity Distribution

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Considering the equilibrium of force in a fluid element in a pipe flow, the value of shear
stress is obtained as,
𝜏 = −
𝜕𝑝 𝑟
𝜕𝑥 2
…………………………(i)
Equation of shear stress is,
𝜏 = 𝜇
𝑑𝑢...............................................
(ii)
𝑑𝑦
But y is measured from the pipe wall. Hence
y = R – r and dy = - dr……(iii)
Hence
τ = - µ 𝑑𝑢 ..................................(iv)
𝑑𝑟
Substituting for τ in (i), τ = µ
d𝑢
𝑑𝑦 we obtain,
𝑑𝑢
=
1 𝜕𝑝
r .....................................(v)
𝑑𝑟 2µ 𝜕𝑥
Integrating and removing the arbitrary constants from the boundary condition r =R when
u = 0 we get the equation for velocity distribution,
1 𝜕𝑝
u =
2µ 𝜕𝑥
[ 𝑅2 − 𝑟2] ...................... (vi)

8 @a.roy
Thus we have finally the following parabolic velocity profile:
𝑢 = −
1 𝜕𝑝
(𝑅2
− 𝑟2
)…................... (vii)
𝑧 4𝜇 𝜕𝑧
The maximum velocity occurs at the pipe centerline ( ):
𝜇 =
𝑅2
(−
𝜕𝑝
)
𝑧 𝑚𝑎𝑥 4𝜇 𝜕𝑧
……………………..(viii)
The average velocity can be obtained by integrating over the pipe cross section:
𝑢 =
1
𝑅
𝑢
∙ 2𝜋𝑟𝑑𝑟 = 0.5 𝑢 ……………………..(ix)
𝑧 𝑎𝑣𝑔 𝜋𝑅2 ∫0 𝑧 𝑧 𝑚𝑎𝑥
The Hagen–Poiseuille equation relates the pressure drop ∆𝑝 across a circular pipe of
length L to the average flow velocity in the pipe 𝑢 𝑧 𝑎𝑣𝑔 and other parameters. Assuming that
the pressure decreases linearly across the length of the pipe, we have,
−
𝜕𝑝
=
∆𝑝
(constant).................................. (x)
𝜕𝑧 𝐿
Substituting this and the expression for into the expression for 𝑢 𝑧 𝑎𝑣𝑔 , and noting
that the pipe diameter 𝐷 = 2𝑅, weget:
𝑢 𝑧 𝑎𝑣𝑔 =
𝐷2
32𝜇
∆𝑃......................................................
(xi)
𝐿
Rearrangement of this gives the Hagen–Poiseuille equation:
∆𝑃 =
32𝜇𝐿𝑢 𝑧 𝑎𝑣𝑔
… … … … … … … . . (𝑥𝑖𝑖)
𝐷2

9 @a.roy
Chapter-4
Entrance length in pipe flow
Fig 4.1- Entrance Lenght
• The fluid adjacent to the wall sticks to the wall due to friction effects. This is the no-slip
condition and occurs for all liquids.
• This boundary layer grows until it reaches all parts of the pipe.
• Inside the inviscid core, viscosity effects are not important.
• The entrance region for laminar flow is given by
𝐿
= 0.05 Re.........................(eq 4.1)
𝐷
• Past here the flow is fully developed.

10 @a.roy
Chapter- 5
Reynold’s Number
In fluid mechanics, the Reynolds number (Re) is a dimensionless number that gives a
measure of the ratio of inertial forces to viscous forces and consequently quantifies the
relative importance of these two types of forces for given flow conditions.
The concept was introduced by George Gabriel Stokes in 1851, but the Reynolds number is
named after Osborne Reynolds (1842–1912), who popularized its use in 1883.
Reynolds numbers frequently arise when performing dimensional analysis of fluid dynamics
problems, and as such can be used to determine dynamic similitude between different
experimental cases.
They are also used to characterize different flow regimes, such as laminar or turbulent flow:
laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is
characterized by smooth, constant fluid motion; turbulent flow occurs at high Reynolds
numbers and is dominated by inertial forces, which tend to produce
chaotic eddies, vortices and other flow instabilities.
For flow in a pipe or tube, the Reynolds number is generally defined as:
𝑅𝑒 =
𝜌𝑉𝐷 𝐻
𝜇
𝑉𝐷 𝐻
=
𝑣
𝑄𝐷 𝐻
=
𝑣𝐴
… … … … … … … … … … … … . 𝑒𝑞. (5.1)
where:
 is the hydraulic diameter of the pipe; its characteristic travelled length, , (m).
 is the volumetric flow rate (m3
/s).
 is the pipe cross-sectional area (m²).
 is the mean velocity of the fluid (SI units: m/s).
 is the dynamic viscosity of the fluid (Pa·s or N·s/m² or kg/(m·s)).
 is the kinematic viscosity ( (m²/s).
 is the density of the fluid (kg/m³).

11 @a.roy
Chapter-6
Laminar Flow
Laminar flow (or streamline flow) occurs when a fluid flows in parallel layers, with no
disruption between the layers. At low velocities the fluid tends to flow without lateral mixing,
and adjacent layers slide past one another like playing cards. There are no cross currents
perpendicular to the direction of flow, nor eddies or swirls of fluids. In laminar flow the
motion of the particles of fluid is very orderly with all particles moving in straight lines
parallel to the pipe walls. In fluid dynamics, laminar flow is a flow regime characterized by
high momentum diffusion and low momentum convection.
When a fluid is flowing through a closed channel such as a pipe or between two flat plates,
either of two types of flow may occur depending on the velocity of the fluid: laminar flow
or turbulent flow. Laminar flow tends to occur at lower velocities, below the onset of
turbulent flow. Turbulent flow is a less orderly flow regime that is characterised by eddies or
small packets of fluid particles which result in lateral mixing. In nonscientific terms laminar
flow is "smooth", while turbulent flow is "rough".
The type of flow occurring in a fluid in a channel is important in fluid dynamics problems.
The dimensionless Reynolds number is an important parameter in the equations that describe
whether flow conditions lead to laminar or turbulent flow. In the case of flow through a
straight pipe with a circular cross-section, at a Reynolds number below the critical value of
approximately 2040[4]
fluid motion will ultimately be laminar, whereas at larger Reynolds
number the flow can be turbulent. The Reynolds number delimiting laminar and turbulent
flow depends on the particular flow geometry, and moreover, the transition from laminar
flow to turbulence can be sensitive to disturbance levels and imperfections present in a given
configuration.

12 @a.roy
Chapter-7
Liquid Column Gauge
By using Bernoulli's principle and the derived pressure head equation, liquids can be used
for instrumentation where gravity is present. Liquid column gauges consist of a vertical
column of liquid in a tube that has ends which are exposed to different pressures. The column
will rise or fall until its weight (a force applied due to gravity) is in equilibrium with the
pressure differential between the two ends of the tube (a force applied due to fluid pressure).
A very simple version is a U-shaped tube half-full of liquid, one side of which is connected to
the region of interest while the reference pressure (which might be the atmospheric
pressure or a vacuum) is applied to the other. The difference in liquid level represents the
applied pressure. The pressure exerted by a column of fluid of height h and density ρ is given
by the hydrostatic pressure equation, P = hgρ. Therefore the pressure difference between the
applied pressure Pa and the reference pressure P0 in a U-tube manometer can be found by
solving Pa − P0 = hgρ. In other words, the pressure on either end of the liquid (shown in blue
in the figure to the right) must be balanced (since the liquid is static) and so Pa = P0 + hgρ.
Fig 7.1- U- Tube Manometer

13 @a.roy
Although any fluid can be used, mercury is preferred for its high density (13.534 g/cm3
) and
low vapour pressure. For low pressure differences well above the vapour pressure of
water, water is commonly used (and "inches of water" or "Water Column" is a common
pressure unit). Liquid-column pressure gauges are independent of the type of fluid being
measured and have a highly linear calibration. They have poor dynamic response because the
fluid in the column may react slowly to a pressure change.
Chart of liquids and their densities
Table 1
Liquid Density(kg per 𝑚3)
petrol 720
kerosene 817.15
diesel 832
mercury 13594

14 @a.roy
Chapter-8
Laminar Flow Meter
Fig 8.1- Laminar Flow Meter

15 @a.roy
(a) Design
We decided to design our Laminar flow meter in such a way that it can be connected to a 1.5
inch diameter pipe which is commonly used for fluid flows. Now to create laminar flow
through small diameter tube, we planned to insert many small tubes (external diameter 0.6
mm and internal diameter 0.4 mm ) into the pipe. If they are inserted to the same pipe it’s
cross sectional area for fluid flow could be reduced considerably for two reasons, thickness of
material of pipe and voids between tubes. Hence we calculated number of tubes required to
maintain the cross sectional area. So cross sectional area of pipe was divided by internal cross
sectional area of small tube to obtain the total number tubes required.
Number of tubes required = 91
Now a larger diameter pipe is required to contain 91 tubes in them. We found that (
considering the availability in the market also) a 2.5 inch diameter pipe can contain 91 tubes
in them.To avoid loss of head due to sudden contraction and loss of head due to sudden
expansion we decided to use two reducers on both ends of larger diameter pipe.
Then we calculated the largest discharge possible in the device, maintaining laminar flow in
small tubes.We found corresponding head difference generated in the pipe for a length of 30
cm. Since the head difference was not high enough to use a mercury manometer we decided
to use an inverted u tube manometer and a liquid lighter than water.
Then we studied the effect of entrance effect on instrument. We found that largest entrance
length for maximum flow in our device. It was 10 cm. Hence we decided to use a pipe of 40
cm pipe considering entrance length too. Since there was a possibility of water trapping
inside the larger pipe we planned to connect an air vent in the bigger pipe. To make space to
connect air vent length of bigger pipe was finally set to 50 cm.
To ensure laminar flow in tubes we set the upper limit of reynolds number to 500.

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(b) Construction
A small hole was cut in 2.5 inch diameter pipe at exact positions where manometer were to
be fixed to tap pressure difference. Then 91 small tubes were inserted into the pipe and glued
well. Voids between tubes were filled with glues. To avoid experimental errors pressure was
tapped from three tubes and all were connected to common manometer. The cut hole in PVC
pipe was well sealed. Inverted u tube manometer was filled with low density fluid and was
connected to pressure tapping tubes. Reducers were connected at both ends small diameter
pipes were connected to it. Valves were connected at both end to control fluid flow and to
seal water inside. Manometer was glued to board and board was fixed to the larger pipe.

17 @a.roy
(c) Test Details
Fluids Height(mm) Pressure Drop(N/𝑚2) Volume flow
rate(𝐿/𝑠)
Water 10 97.97 0.01
Diesel 5 49.98 0.0629
Kerosene 7 68.58 0.173

18 @a.roy
(d) Laminar flow Meter Model:
Fig 8.2- Laminar flow meter model & its element diagram

19 @a.roy
Chapter- 9
Conclusion
For a given Laminar flow, the Reynolds number has to be less than 2100, but as per our
results the number is a lot greater than it. The following error can be explained due as a result
of human error due to hand adjustments on the outer surface flow and the manometer reading
fluctuations due to the periodic change in diameter. Manometer readings play a vital role in
conducting this experiment successfully, because of the fluctuations that were encountered
we were unable to achieve the desired results for this experiment. As a group we are of an
opinion as such, if the following experiment were to be repeated many times with different
set of manometer and different fluid than accurate and precise results can be achieved. In this
experiment, we enlightened ourselves with ta deeper understanding of the Bernoulli’s
equation along with the core principles of fluid dynamics.

20 @a.roy
Chapter- 10
Future scope of this Project
Laminar flow meter is used when pressure drop is flow and gives high accuracy in flow rate.
Hence, in future we can used this device in small industry, chemical industry and medical
sciences to measure small volume flow rate. The volumetric rate can also be converted to a
mass flow rate using density connection at a given temperature and pressure.
Advantages
 Can be used to measure low flow rates.
 Ability to measure the flow of high viscous liquid.
 Linear relationship between flow rate & pressure drop.
 Low noise.

21 @a.roy
References
 Fluid mechanics, R K Benzal
 Batchelor, G. (2000). Introduction to Fluid Mechanics.
 Noakes, Cath & Sleigh, Andrew (January 2009). "Real Fluids". An Introduction to
Fluid Mechanics.
 Rogers, D.F. (1992). Laminar flow analysis

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