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Steady Incompressible Flow in Pressure Conduits

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Laminar and Turbulent Flow in Pipes Laminar Paths of Particles don t obstruct each other Viscous forces are dominant Velocity of fluid particles only changes in ... – PowerPoint PPT presentation

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Title: 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.
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