This document discusses the relationships between flow rate, pressure drop, and shear stress for laminar flow in pipes. It provides equations to calculate flow rate from shear stress and pressure drop data. The key relationships are: 1) Pressure drop is directly proportional to flow rate for laminar flow. 2) Shear stress at the wall is related to pressure drop by an equation involving pipe diameter and length. 3) Shear stress decreases linearly from the wall to the center of the pipe in laminar flow. 4) The flow rate can be calculated from experimentally measured shear stress and pressure drop data using integration methods like Simpson's rule.

1. Calculation of Flow Rate and
Pressure Drop Relationship for
Laminar Flow using τ-γ Data

2. Contents
 Flow Rate
 Pressure Drop
 Laminar Flow In Pipes
 Relation between pressure drop and flow rate
 Determine the relationship b/t shear stress at wall &
pressure drop for steady flow
 How does Shear stress vary with radial location for the
flow?
 Shear Stress In Pipes
 Calculation of Flow Rate and Pressure Drop Relationship
for Laminar Flow using τ-γ Data

3. Flow Rate
• The volumetric flow rate (also known as volume flow rate, rate
of fluid flow or volume velocity) is the volume of fluid which
passes per unit time; usually represented by the symbol Q
(sometimes V̇ ). The SI unit is m3/s (cubic metres per second).
• Another unit used is sccm (standard cubic centimeters per
minute).
• In US units and imperial units, volumetric flow rate is often
expressed as ft3/s or gal/min.

4. Formula
timet
volumeV
tVQ


 /

5. • The volume of a portion of the fluid in a pipe can be
written as V=Ad, where A is the cross sectional area of
the fluid and dddd is the width of that portion of fluid
• But the term d/t is just the length of the volume of fluid
divided by the time it took the fluid to flow through its
length, which is just the speed of the fluid. So we can
replace d/t with v in the previous equation and get
• Q=Av
t
d
A
t
Ad
t
V
Q 

7. Pressure Drop
• Pressure drop is defined as the difference in total pressure between two
points of a fluid carrying network.
• Pressure drop occurs when frictional forces, caused by the resistance to
flow, act on a fluid as it flows through the tube.
• The main determinants of resistance to fluid flow are fluid velocity through
the pipe and fluid viscosity.
• Pressure drop increases proportional to the frictional shear forces within
the piping network. A piping network containing a high relative roughness
rating as well as many pipe fittings and joints, tube convergence,
divergence, turns, surface roughness and other physical properties will
affect the pressure drop.
• High flow velocities and / or high fluid viscosities result in a larger
pressure drop across a section of pipe or a valve or elbow. Low velocity
will result in lower or no pressure drop.

8. • Pressure drop = Pressure Loss + Head Loss +
Frictional losses
• The equation for pressure loss is:
• Where,
• ρ= fluid density in kg per cubic meter
• L = Length of pipe (m)
• D = Pipe diameter (m)
• V = Mean flow velocity (m/s)
• fD = Darcy Friction Factor

9. Laminar flow
• Also known as streamline
flow
• Occurs when the fluid flows
in parallel layers, with no
disruption between the
layers
• 3 Conditions
• fluid moves slowly
• viscosity is relatively high
• flow channel is relatively
small.

10. Relation between pressure drop and flow rate:
• Under laminar flow
conditions, pressure drop
is proportional to
volumetric flow rate. At
double the flow rate, there
is double the pressure
drop.

11. Determine the relationship b/t shear stress at wall
& pressure drop for steady flow
Fig 1:
An element of fluid extendingg over the
whole of the pipe cross section
• The fig 1 shows the flow
with a suitable element of
fluid, extending over the
whole cross secton of the
pipe. For the conditions
specified, the fluids
momentum remains
constant so the net force
actiong on the fluid is
zero.

12. Continue
• Three forces act on the element in the x-direction:
• The pressure P1 pushes the fluid in the direction of flow,
the pressure P2 pushes against the flow and the frictional
drag between the fluid and the pipe wall acts against the
flow.
• The upstream pressure P1 acts over the cross-sectional
area of the element, so that the force acting on the elemt
in the direction of flow is given by
force acting in flow direcion=πr2
i P1 ....1
• Where the radius of the element is the same as the inside
as the inside radius ri of the pipe.

13. Continue
• The downstreampressure P2 acts on the element against the flow, as
does the drag of the pipe wall on the fluid. The shear stress at the wall is
called the wall shear stress and is denoted by τw . This shear stress acts
over the area of the element in contact with the wall. The force acting
against the flow is therefore given by
force acting against flow=πri
2 P2 + 2πri Lτw .....2
• Where L is the length of the element.
• The net force being zero requires that
πri
2 (P1 - P2 ) - 2πri Lτw =0 ......3
• The wall shear stress τw is therefore related to the pressure drop ▲P by
.......4

14. Continue
• The pressure drop is entirely caused by fluid friction.
• The above equation can be written as:
......5
• Where ▲Pf is the frictional component of the pressure
drop and di is the inside diameter of the pipe.

15. How does Shear stress vary with radial location for
the flow?
The variation of shear stress τrx with radial coordinate r
can be determined by making a force balance similiar to
above derivation.
In this case, the force balance, equivalent to equation
3,can be written as
πr2 (P1 - P2 ) - 2πr Lτrx =0 .......6
The shear stress τrx at distance r from the centre-line is
therefore given by
......7

16. Shear Stress In Pipes
• Consider steady, fully developed flow in a straight pipe of lenght L and
internal diameter as shown above in eq:(6), a force balance on a
cylindrical element of the fluid can be written as
πr2 ▲Pf - 2πr Lτrx =0 ........9
• Where ▲Pf is the frictional component of the pressure drop over the pipe
length. in the case of fully developed flow in a horizontal pipe ▲Pf is the
only component of the pressure drop.
• Rearranging equation (9), the shear stress is given by:
.....10

17. Continue
• A special case of equation (10) is the shear stress τw at
the wall
.........11

18. Calculation of Flow Rate and Pressure Drop
Relationship for Laminar Flow using τ-γ Data
• When the data are in the form of shear stress-shear
rate values,the flow rate can be calculated by using
Rabinowitsch-Mooney equation
......12
• Where di is diameter of pipe to be used and τw the wall
shear stress.
• The value of τw is given by equation (11) which is
.......13

19. Continue…
• In relation (12), the value of share stress and shear rate can be
calculated by Simpson’s rule whose formula is given as,
..........14
• By substituting the answer of Eq {13} and {14} in Eq {12}
• We will get the value of volumetric average velocity and
• We have also relation:Q=Av
• Putting the value of this velocity we can calculate the
flow rate-pressure drop relation for the laminar flow .

20. References:
• Fluid flow for Chemical Engineers By F.A Holland &
R.BRAGG
• http://www.alicat.com/
Numerical Related to this Relation.
• Example no 3.2 (from Fluid flow for Chemical Engineers
By F.A Holland & R.BRAGG)