Steady Incompressible Flow in Pressure Conduits

1
Steady Incompressible Flow in Pressure Conduits
2
Laminar and Turbulent Flow in Pipes

• Laminar
• Paths of Particles dont obstruct each other
• Viscous forces are dominant
• Velocity of fluid particles only changes in
  magnitude
• Lateral component of velocity is zero
• Turbulent
• Paths do intersect each other
• Inertial forces are dominant
• Velocity of fluid particles change in magnitude
  and direction
• Lateral components do exist.

3
Laminar and Turbulent Flow in Pipes

• If we measure the head loss in a given length of
  uniform pipe at different velocities , we will
  find that, as long as the velocity is low enough
  to secure laminar flow, the head loss, due to
  friction, is directly proportional to the
  velocity, as shown in Fig. 8.1. But with
  increasing velocity, at some point B, where
  visual observation of dye injected in a
  transparent tube would show that the flow changes
  from laminar to turbulent, there will be an
  abrupt increase in the rate at which the head
  loss varies. If we plot the logarithms of these
  two variables on linear scale or in other words,
  if we plot the values directly on log-log paper,
  we will find that, after passing a certain
  transition region (BCA in Fig. 8.1), the lines
  will have slopes ranging from about 1.75 to 2.

4
Laminar and Turbulent Flow in Pipes

• Thus we see that for laminar flow the drop in
  energy due to friction varies as V, while for
  turbulent flow the friction varies as Vn, where n
  ranges from about 1.75 to 2. The lower value of
  1.75 for turbulent flow occurs for pipes with
  very smooth walls as the wall roughness
  increases, the value of n increases up to its
  maximum value of 2.
• If we gradually reduce the velocity from a high
  value, the points will not return along line BC.
  Instead, the points will lie along curve CA. We
  call point B the higher critical point, and A the
  lower critical point.
• However, velocity is not the only factor that
  determines whether the flow is laminar or
  turbulent. The criterion is Reynolds number.

5
Laminar and Turbulent Flow in Pipes
6
Reynolds Number

• Ratio of inertia forces to viscous forces is
  called Reynolds number.
• Where we can use any consistent system of
  units, because R is a dimensionless number.

7
Significance of Reynolds Number

• To investigate the development of Laminar and
  Turbulent flow
• Investigate Critical Reynolds Number
• Develop a relationship between head loss (hL) and
  velocity.

8
Critical Reynolds Number

• The upper critical Reynolds number, corresponding
  to point B of Fig. 8.1, is really indeterminate
  and depends on the care taken to prevent any
  initial disturbance from effecting the flow. Its
  value is normally about 4000, but experimenters
  have maintained laminar flow in circular pipes up
  to values of R as high as 50,000. However, in
  such cases this type of flow is inherently
  unstable, and the least disturbance will
  transform it instantly into turbulent flow. On
  the other hand, it is practically impossible for
  turbulent flow in a straight pipe to persist at
  values of R much below 2000, because any
  turbulence that occurs is damped out by viscous
  friction. This lower value is thus much more
  definite than the higher one, and is the real
  dividing point the two types of flow. So we
  define this lower value as the true critical
  Reynolds number.

9
Critical Reynolds Number

• It will be higher in a converging pipe and lower
  in a diverging pipe than in a straight pipe.
  Also, it will be less for flow in a curved pipe
  than in a straight one, and even for a straight
  uniform pipe it may be as low as 1000, where
  there is excessive roughness. However, for normal
  cases of flow in straight pipes of uniform
  diameter and usual roughness, we can take the
  critical value as
• Rcrit
  2000

10
Problem

• Q In a refinery oil (? 1.8 x 10-5 m2/s) flows
  through a 100-mm diameter pipe at 0.50 L/s. Is
  the flow laminar or turbulent?
• Solution
• Q 0.50 Liter/s 0.0005 m3/s
• D 100 mm 0.1 m
• Q AV (pD2/4)V, V (4Q)/(pD2)
• V (4 x 0.0005)/(3.14 x 0.1 x0.1) 0.0637
  m/s
• R (DV)/? (0.1 x 0.0637)/(1.8 x 10-5)
  354
• Since R lt Rcrit2000, the flow is
  laminar.

11
Hydraulic Radius

• For conduits having non-circular cross sections,
  we need to use some value other than the diameter
  for the linear dimension in the Reynolds number.
  The characteristic dimension we use is the
  hydraulic radius, defined as
• Rh A/P
• Where A is the cross sectional area of the
  flowing fluid, and P is the wetted perimeter,
  that portion of the perimeter of the cross
  section where the fluid contacts the solid
  boundary, and therefore where friction resistance
  is exerted on the flowing fluid. For a circular
  pipe flowing full,
• Full-pipe flow Rh (p r2)/(2pr)
  r/2 D/4

12
Friction Head Loss in Conduits

• This discussion applies to either laminar or
  turbulent flow and to any shape of cross section.
• Consider steady flow in a conduit of uniform
  cross section A, not necessarily circular as
  shown Fig. below. The pressures at sections 1 and
  2 are p1 and p2, respectively. The distance
  between the sections is L.

13
Friction Head Loss in Conduits

• For equilibrium in steady flow, the summation of
  forces acting on any fluid element must be equal
  to zero (i.e., SFma0). Thus, in the direction
  of flow,
• p1A p2A ?LAsina t0(PL) 0 (1)
• where we define t0, the average shear stress
  (average shear force per unit area) at the
  conduit wall.
• Nothing that sina (z2 z1)/L and dividing
  each term in eq. (1) by ?A gives,
• p1A/(?A) p2A/(?A) ?LA(z2 - z1)/(?AL)
  t0(PL)/(?A)
• p1/? p2/? z2 z1 t0(PL)/(?A) .
  (2)
• From the left hand sketch of the Fig., we can
  see that the head loss due to friction at the
  wetted perimeter is
• hf (z1 p1/?) (z2 p2/?) .. (3)

14
Friction Head Loss in Conduits

• The eq.(3) equation indicates that hf depends
  only on the values of z and p on the centerline,
  and so it is the same regardless of the size of
  the cross-sectional area A. Substituting hf from
  eq.(3) and replacing A/P by Rh in eq.(2), we get,
• hf t0L/(Rh?)
  (4)
• This equation is applicable to any shape of
  uniform cross section, regardless of whether the
  flow is laminar or turbulent.

15
Friction Head Loss in Conduits

• For a smooth-walled conduit, where we can
  neglect wall roughness, we might assume that the
  average fluid shear stress t0 at the wall is
  some function of ?, µ, V and some characteristic
  linear dimension, which we will here take as the
  hydraulic radius Rh. Thus
• t0 f(?, µ, V, Rh)
• Using the pi theorem of dimensional analysis
  to better determine the form of this
  relationship, we choose ?, Rh and V as primary
  variables, so that
• ?1 µ ?a1 Rhb1 Vc1
• ?2 t0 ?a2 Rhb2 Vc2
• With the dimensions of the variables being
  ML-1T-1 for µ, ML-1T-2 for t0, ML-3 for ?,
  L for Rh, and LT-1 for V,

16
Friction Head Loss in Conduits

• the dimensions for ?1 are
• ?1 µ ?a1 Rhb1 Vc1
• M0L0T0 (ML-1T-1) (ML-3)a1 (L)b1 (LT-1)c1
• For M 0 1 a1
• For L 0 -1 3a1 b1 c1
• For T 0 -1 c1
• The solution of these simultaneous equations
  is
• a1 b1 c1 -1, from which
• ?1 µ ?-1 Rh-1 V-1
• ?1 µ /(? Rh V) R-1
• where (? Rh V)/µ is a Reynolds number with Rh
  as the characteristic length.

17
Friction Head Loss in Conduits

• the dimensions for ?2 are
• ?2 t0 ?a2 Rhb2 Vc2
• M0L0T0 (ML-1T-2) (ML-3)a2 (L)b2 (LT-1)c2
• For M 0 1 a2
• For L 0 -1 3a2 b2 c2
• For T 0 -2 c2
• The solution of these simultaneous equations
  is
• a2 -1, c2 -2, b2 0, from which
• ?2 t0 ?-1 V-2
• ?2 t0 /(?V2)
• We can write ?2 ?( ?1-1), which results in
• t0 ? V2 ? (R)
• .

18
Friction Head Loss in Conduits

• Setting the dimensionless term ? (R) ½ Cf ,
  this yields
• t0 Cf ? V2/2
• Where Cf average friction-drag coefficient
  for total surface (dimensionless)
• Inserting this value of t0 and ? ?g, in
  eq. (4), which is
• hf t0L/(Rh?) , we get
• hf Cf (L/Rh)(V2/2g) (5)
• which can apply to any shape of smooth-walled
  cross section. From this equation , we may easily
  obtain an expression for the slope of the energy
  line,
• S hf / L Cf /Rh (V2/2g) (6)
• which we also know as the energy gradient.

19
Friction in Circular Conduits

• Head loss due to friction, hf Cf
  (L/Rh)(V2/2g)
• Energy gradient, S hf / L Cf /Rh (V2/2g)
• For a circular pipe flowing full, Rh D/4,
  and
• f 4Cf , where f is friction factor (also
  some times called the Darcy friction factor) is
  dimensionless and some function of Reynolds
  number.
• Substituting values of Rh and Cf into above
  equations, we obtain (for both smooth-walled and
  rough-walled conduits) the well known equation
  for pipe-friction head loss,
• Circular pipe flowing full (laminar or
  turbulent flow)
• hf f (L/D) (V2/2g) . (7)
• and hf /L S f /D (V2/2g) .. (8)
  

20
Friction in Circular Conduits

• For a circular pipe flowing full, by
  substituting Rh r0/2, where r0 is the radius of
  the pipe in the eq. (4), we get
• hf t0L/(Rh?) 2t0L/(r0?)
• where the local shear stress at the wall, t0,
  is equal to the average shear stress t0 because
  of symmetry.

21
Friction in Circular Conduits

• The shear stress is zero at the center of the
  pipe and increases linearly with the radius to a
  maximum value t0 at the wall as shown in Fig.
  8.3. This is true regardless of whether the flow
  is laminar or turbulent.
• From eq.(4) hf t0L/(Rh?), we have
• t0 hf (Rh?)/L, substituting eq.(7) and Rh
  D/4 into this, we obtain
• t0 f (L/D)(V2/2g)(D/4)(?/L)
• t0 (f /4) ? (V2/2g) or t0 (f /4) ?
  (V2/2) where ??g
• With this equation, we can compute t0 for
  flow in a circular pipe for any experimentally
  determined value of f.

22
Friction in Circular Conduits

• For laminar flow under pressure in a circular
  pipe,
• We may use the pipe-friction equation (7) with
  this value of f as given by the above equation.

23
Problem

• Q Stream with a specific weight of 0.32 lb/ft3
  is flowing with a velocity of 94 ft/s through a
  circular pipe with f 0.0171. What is the shear
  stress at the pipe wall?
• Solution
• ? 0.32 lb/ft3
• V 94 ft/s
• f 0.0171
• g 32.2 ft/s2
• t0 ?
• t0 (f /4) ? (V2/2g)
• t0 (0.0171/4)(0.32)(94x94)/(2x32.2)
• t0 0.187 lb/ft2

24
Problem

• Q Stream with a specific weight of 38 N/m3 is
  flowing with a velocity of 35 m/s through a
  circular pipe with f 0.0154. What is the shear
  stress at the pipe wall?
• Solution
• ? 38 N/m3
• V 35 m/s
• f 0.0154
• g 9.81 m/s2
• t0 ?
• t0 (f /4) ? (V2/2g)
• t0 (0.0154/4)(38)(35x35)/(2x9.81)
• t0 9.13 N/m2

25
Problem

• Q Oil of viscosity 0.00038 m2/s flows in a 100mm
  diameter pipe at a rate of 0.64 L/s. Find the
  head loss per unit length.
• Solution
• ? 0.00038 m2/s
• D 100 mm 0.1 m
• Q 0.64 L/s 0.00064 m3/s
• g 9.81 m/s2
• hf /L ?
• Q AV (pD2/4)V, V (4Q)/(pD2)
• V (4 x 0.00064)/(3.14 x 0.1 x 0.1)
  0.0815 m/s
• R (DV)/? (0.1 x 0.0815)/(0.00038)
  21.45

26
Problem

• f 64 / R 64/21.45 2.983
• hf /L S f /D (V2/2g)
• hf /L (2.983/0.1)(0.0815x0.0815)/(2x9.81)
• hf /L 0.010 m/m

27
Friction in Non-Circular Conduits

• Most closed conduits we use in engineering
  practice are of circular cross section however
  we do occasionally use rectangular ducts and
  cross sections of other geometry. We can modify
  many of the equations for application to non
  circular sections by using the concept of
  hydraulic radius.
• For a circular pipe flowing full, that
• Rh A/P (p D2/4)/(pD) D/4
• D 4 Rh
• This provides us with an equivalent diameter,
  which we can substitute into eq. (7) to yield
• hf f (L/4Rh)(V2/2g)

28
Friction in Non-Circular Conduits

• and when substitute into equation of Reynolds
  number, we get
• R (DV?)/µ (4RhV?)/µ (4RhV)/?
• This approach gives reasonably accurate
  results for turbulent flow, but the results are
  poor for laminar flow, because in such flow
  viscous action causes frictional phenomena
  throughout the body of the fluid, while in
  turbulent flow the frictional effect occurs
  largely in the region close to the wall i.e., it
  depends on the wetted perimeter.

29
Entrance Conditions in Laminar Flow

• In the case of a pipe leading from a
  reservoir, if the entrance is rounded so as to
  avoid any initial disturbance of the entering
  stream, all particles will start to flow with the
  same velocity, except for a very thin film in
  contact with the wall. Particles in contact with
  the wall have zero velocity and with the slight
  exception, the velocity is uniform across the
  diameter.

30
Entrance Conditions in Laminar Flow

• As the fluid progresses along the pipe,
  friction origination from the wall slows down the
  streamlines in the vicinity of the wall, but
  since Q is constant for successive sections, the
  velocity in the center must accelerate, until the
  final velocity profile is a parabola as shown in
  Fig. 8.3. Theoretically, this requires an
  infinite distance, but both theory and
  observation have established that the maximum
  velocity in the center of the pipe wall reach 99
  of its ultimate value in a distance
• Le 0.058 RD
• We call this distance the entrance length. For
  a critical value of R 2000, the entrance length
  Le equals 116 pipe diameters. In other cases of
  laminar flow with Reynolds number less than 2000,
  the distance Le will be correspondingly less in
  accordance with the above equation.

31
Entrance Conditions in Laminar Flow

• Within the entrance length the flow is
  unestablished that is the velocity profile is
  changing. In this region, we can visualize the
  flow as consisting of a central inviscid core in
  which there are no frictional effects, i.e., the
  flow is uniform, and an outer, annular zone
  extending from the core to the pipe wall. This
  outer zone increases in thickness as it moves
  along the wall, and is known as the boundary
  layer. Viscosity in the boundary layer acts to
  transmit the effect of boundary shear inwardly
  into the flow. At section AB the boundary layer
  has grown until it occupies the entire cross
  section of the pipe. At this point, for laminar
  flow, the velocity profile is a perfect parabola.
  Beyond section AB, for the same straight pipe the
  velocity profile does not change, and the flow is
  known as (laminar) established flow or (laminar)
  fully developed flow.

32
Entrance Conditions in Laminar Flow

• The flow will continue as fully developed so
  long as no change occurs to the straight pipe
  surface. When a change occurs, such as at a bend
  or other pipe fitting, the velocity profile will
  deform and will require some more flow length to
  return to established flow. Usually such fittings
  are so far apart that fully developed flow is
  common but when they are close enough it is
  possible that established flow never occurs.