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