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HEAT TRANSFER which indicates that in fully developed flow in a tube, the viscous and pressure forces balance each other. Dividing by 2 drdx and rearranging,
r Px (r )x Px dx (r )r dr dx dr 0 (835) Taking the limit as dr, dx → 0 gives Substituting d(r )
dr dP
dx r 0 (836) (d /dr) and rearranging gives the desired equation,
d
d
r dr r dr dP
dx (837) The quantity d /dr is negative in tube flow, and the negative sign is included
to obtain positive values for . (Or, d /dr
d /dy since y R r.) The
left side of this equation is a function of r and the right side is a function of x.
The equality must hold for any value of r and x, and an equality of the form
f(r) g(x) can happen only if both f(r) and g(x) are equal to constants. Thus
we conclude that dP/dx constant. This can be verified by writing a force
balance on a volume element of radius R and thickness dx (a slice of the tube),
which gives dP/dx
2 s/R. Here s is constant since the viscosity and
the velocity profile are constants in the fully developed region. Therefore,
dP/dx constant.
Equation 8–37 can be solved by rearranging and integrating it twice to give
dP
dx 1
4 (r) C1lnr C2 (838) The velocity profile (r) is obtained by applying the boundary conditions
/ r 0 at r 0 (because of symmetry about the centerline) and
0 at
r R (the noslip condition at the tube surface). We get
R2 dP
1
4 dx (r) r2
R2 (839) Therefore, the velocity profile in fully developed laminar flow in a tube is
parabolic with a maximum at the centerline and minimum at the tube surface.
Also, the axial velocity is positive for any r, and thus the axial pressure gradient dP/dx must be negative (i.e., pressure must decrease in the flow direction because of viscous effects).
The mean velocity is determined from its definition by substituting
Eq. 8–39 into Eq. 8–2, and performing the integration. It gives
m 2
R2 R rdr
0 2
R2 R
0 R2 dP
1
4 dx r2
rdr
R2 R2 dP
8 dx (840) Combining the last two equations, the velocity profile is obtained to be
(r) 2 m 1 r2
R2 (841) cen58933_ch08.qxd 9/4/2002 11:29 AM Page 433 433
CHAPTER 8 This is a convenient form for the velocity profile since m can be determined
easily from the flow rate information.
The maximum velocity occurs at the centerline, and is determined from
Eq. 8–39 by substituting r 0,
2 max (842) m Therefore, the mean velocity is onehalf of the maximum velocity. Pressure Drop
A quantity of interest in the analysis of tube flow is the pressure drop P since
it is directly related to the power requirements of the fan or pump to maintain
flow. We note that dP/dx constant, and integrate it from x 0 where the
pressure is P1 to x L where the pressure is P2. We get
P1 P2 dP
dx P
L L (843) Note that in fluid mechanics, the pressure drop P is a positive quantity, and
is defined as P P1 P2. Substituting Eq. 8–43 into the m expression in
Eq. 8–40, the pressure drop can be expressed as
Laminar flow: 8L
R2 P 32 L
D2 m m (844) In practice, it is found convenient to express the pressure drop for all types of
internal flows (laminar or turbulent flows, circular or noncircular tubes,
smooth or rough surfaces) as (Fig. 8–18) ∆P 2 P f m
L
D2 (845) D m L where the dimensionless quantity f is the friction factor (also called the
Darcy friction factor after French engineer Henry Darcy, 1803–1858, who
first studied experimentally the effects of roughness on tube resistance). It
should not be confused with the friction coefficient Cf (also called the Fanning
2
f/4.
friction factor), which is defined as Cf
s(
m /2)
Equation 8–45 gives the pressure drop for a flow section of length L provided that (1) the flow section is horizontal so that there are no hydrostatic or
gravity effects, (2) the flow section does not involve any work devices such as
a pump or a turbine since they change the fluid pressure, and (3) the cross sectional area of the flow section is constant and thus the mean flow velocity is
constant.
Setting Eqs. 8–44 and 8–45 equal to each other and solving for f gives the
friction factor for the fully developed laminar flow in a circular tube to be
Circular tube, laminar: f 64
D m 64
Re (846) This equation shows that in laminar flow, the friction factor is a function of
the Reynolds number only and is independent of the roughness of the tube ρm
Pressure drop: ∆P = f L
D2 2 FIGURE 8–18
The relation for pressure
drop is one of the most general
relations in fluid mechanics, and it is
valid for laminar or turbulent flows,
circular or noncircular pipes, and
smooth or rough surfaces. cen58933_ch08.qxd 9/4/2002 11:29 AM Page 434 434
HEAT TRANSFER surface. Once the pressure drop is available, the required pumping power is
determined from
·
Wpump ·
VP (847) ·
where V is the volume flow rate of flow, which is expressed as
·
V D4 P
128 L (848) This equation is known as the Poiseuille’s Law, and this flow is called the
Hagen–Poiseuille flow in honor of the works of G. Hagen (1797–1839) and
J. Poiseuille (1799–1869) on the subject. Note from Eq. 8–48 tha...
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This note was uploaded on 01/28/2010 for the course HEAT ENG taught by Professor Ghaz during the Spring '10 term at University of Guelph.
 Spring '10
 Ghaz

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