Shell Momentum Balance
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10/8/2015 1 ChE 130 Prepared by: Engr. Sandra Enn Bahinting Shell Momentum Balance Average Velocity in Overall Mass Balance If the velocity is not constant but varies across the surface area, an average or bulk velocity is defined as For the case of incompressible flow through a cicular pipe of radius R, the velocity profile is parabolic for laminar flow as follows:
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momentum balancevelocity profilelaminar flowannulusaverage velocity its about the momentum balance of a given figure
Transcript of Shell Momentum Balance
10/8/2015
1
ChE 130
Prepared by:
Engr. Sandra Enn Bahinting
Shell Momentum Balance
Average Velocity in Overall Mass Balance
If the velocity is not constant but varies across the
surface area, an average or bulk velocity is
defined as
For the case of incompressible flow through a
cicular pipe of radius R, the velocity profile is
parabolic for laminar flow as follows:
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Derive an expression for the average or bulk
velocity to use in the overall mass-balance
equation.
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At steady state:
sum of forces acting on control volume =
rate of momentum out – rate of momentum into volume
pressure forces becomes
The drag force acting on the cylindrical surface at
the radius r is the shear stress times the are
2𝜋𝑟Δ𝑥 . Hence,
net rate of momentum efflux = rate of momentum
out – rate of momentum in
𝜏𝑟𝑥
In fully developed flow, the pressure gradient (Δp/Δx) is constant and
becomes (Δp/L).
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Fron Newton’s Law of viscosity,
Using the boundary condition at the wall, vx=0 at
r=R, the velocity distribution is
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Once the velocity profile has been established,
various derived quantities can be obtained:
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Example: Glycerine (CH2OH-CHOH-CH2OH) at 26.5C is flowing through a horizontal tube 1 ft long and with 0.1in inside diameter. For a pressure drop of 40 psi, the volume flow rate is 0.00398 ft3/min. The density of
glycerine at 26.5C is 1.261 g/cm3. From the flow data, find the viscosity of glycerine in cp and Pa-s.
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Shell Momentum Balance for Falling film
Momentum flux due to
molecular transport
Conservation of momentum at steady state:
Rearranging and letting ∆x 0
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For Newtonian fluid
Maximum velocity is at x = 0. Therefore,
The average velocity is then,
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The relationship between the average and
maximum velocity is,
Vzave = (2/3)Vzmax
The volumetric flow rate is obtained by multiplying
the average velocity by the cross-sectional area
In falling films, the mass flow rate per unit width of wall Γ in kg/m-s is defined as Γ=ρδvzave and a
Reynold’s number is defined as
Example:
An oil has a kinematic viscosity of 2 x 10-4 m2/s
and a density of 0.8 x 103 kg/m3. If we want to
have a falling film of thickness of 2.5 mm on a
vertical wall, what should the mass rate of flow
of the liquid be?
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Flow Through an Annulus Steady-state axial flow of
an incompressible liquid
in an annular region
between two coaxial cylinder of radii κR and
R. The fluid is flowing
upward.
Using momentum balance on a thin cylindrical
shell,
There will be a maximum in velocity at r=λR, where
the shear is zero.
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Using Newton’s Law of viscosity,
The momentum and velocity profile in an annulus
are;
The following relations can be obtained:
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Velocity Profiles in Pipes
When fluid is flowing in a circular pipe and the
velocities are measured at different distances
from the pipe wall to the center of the pipe, it
has been shown that in both laminar and
turbulent flow, the fluid in the center of the pipe
is moving faster than the fluid near the walls.
For viscous or laminar flow the velocity profile is
a true parabola.
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