CHE315 Mechanical Energy Balance. CHE315 2.9 SHELL MOMENTUM BALANCE INSIDE A PIPE Objective: To...
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Transcript of CHE315 Mechanical Energy Balance. CHE315 2.9 SHELL MOMENTUM BALANCE INSIDE A PIPE Objective: To...
CHE315 Mechanical Energy Balance
02
1 2
1
1221
22 FW
dpzzgvv s
p
p
avav
CHE315
2.9 SHELL MOMENTUM BALANCE INSIDE A PIPE
Objective:
To Derive a Velocity Profile
(eqn.)for Newtonian Fluids
flowing inside a pipe
Mechanical Energy Balance
CHE315
2.9 SHELL MOMENTUM BALANCE INSIDE A PIPE
x
r
P|x P|x+x R
xr
Incompressible Newtonian fluid
Let us consider the
following
simplifications:
One dimensional flow Laminar flow
Fully developed flow (no entrance effect and velocity is independent from x)
Mechanical Energy Balance
CHE315
2.9 SHELL MOMENTUM BALANCE INSIDE A PIPE
Fully developed flow (no entrance effect and velocity is independent from x)
Mechanical Energy Balance
CHE315
Net momentum efflux = rate of momentum out - rate of
momentum in
The pressure forces
=
xxxxxxrrprrppApA
22
rrxrrrx xrxr
22
x
r
P|x P|x+x R
xr
Mechanical Energy Balance
CHE315
So: xxxrrxrrrx rrprrpxrxr
2222
x
ppr
r
rrxxxrrxrrrx
r
dx
dp
dr
rd rx
Mechanical Energy Balance
r
L
p
dr
rd rx
CHE315
r
L
p
dr
rd rx
rdr
L
prd rx
Cr
L
pr rx
2
2
r
Cr
L
prx
2
If the momentum flux cannot be infinite
at r = 0, Then C must be zero:
r
L
ppr
L
p Lrx 22
0
Mechanical Energy Balance
CHE315
The shear stress
profileNewtonian
fluids:
dr
dvxrx
Equating the two equations: r
L
pp
dr
dv Lxrx 2
0
Integrating and
using the
boundary
condition vx
(r=R) = 0:
220 1
4 R
rR
L
ppv Lx
r
L
ppr
L
p Lrx 22
0
Mechanical Energy Balance
CHE315
This result means that the
momentum flux varies
linearly with the radius
r
L
ppr
L
p Lrx 22
0
vx = 0 vmax
Parabolic velocity profile
Momentum flux profile
rx = 0rx max
x
r
This result means that the
velocity distribution is
parabolic
220 1
4 R
rR
L
ppv Lx
Mechanical Energy Balance
CHE315
Using the definition of the average velocity:
We obtain in this case,
the Hagen-Poiseuille
equation:
A
av vdAA
v1
2020
328D
L
ppR
L
ppv LL
avx
20max 4
RL
ppv Lx
2maxx
avx
vv
Mechanical Energy Balance
CHE315
20
8R
L
ppv L
avx
20max 4
RL
ppv Lx
2maxx
avx
vv
Mechanical Energy Balance
220 1
4 R
rR
L
ppv Lx
CHE315 Mechanical Energy Balance
Quiz #2: Sec. 2.7
Next Class (Monday)
26/11/1432H