Flow Through Pipes With Side Tappings | Fluid Mechanics for Mechanical Engineering PDF Download

Flow Through Pipes With Side Tappings

• In course of flow through a pipe, a fluid may be withdrawn from the side tappings along the length of the pipe as shown in Fig. 37.1 
• If the side tappings are very closely spaced, the loss of head over a given length of pipe can be obtained as follows:

• The rate of flow through the pipe, under this situation, decreases in the direction of flow due to side tappings. Therefore, the average flow velocity at any section of the pipe is not constant. 
• The frictional head loss dh_f over a small length dx of the pipe at any section can be written as

                                                        (37.1)       

here, V_x is the average flow velocity at that section.

• If the side tappings are very close together, Eq. (37.1) can be integrated to determine the loss of head due to friction over a given length L of the pipe, provided,  V_x can be replaced in terms of the length of the pipe.
• Let us consider, for this purpose, a Section 1-1 at the upstream just after which the side tappings are provided. If the tappings are uniformly and closely spaced, so that the fluid is removed at a uniform rate qper unit length of the pipe, then the volume flow rate Q_x at a distance x from the inlet Section 1-1 can be written as

     Q_X = Q₀ - q_x

where, Q₀ is the volume flow rate at Sec.1-1.

• Hence,

                                   (37.2)

Substituting V_x  from Eq. (37.2) into Eq. (37.1), we have,

                                             (37.3)                           

Therefore, the loss of head due to friction over a length L is given by

                      (37.4a)    

Substituting V_x from Eq. (37.2) into Eq. (37.1), we have,     

• Here, the friction factor f has been assumed to be constant over the length L of the pipe. If the entire flow at Sec.1-1 is drained off over the length L , then,
  
  

Equation (37.4a), under this situation, becomes

                                    (37.4b)

where, V₀  is the average velocity of flow at the inlet Section 1-1.
Equation (37.4b) indicates that the loss of head due to friction over a length L of a pipe, where the entire flow is drained off uniformly from the side tappings, becomes one third of that in a pipe of same length and diameter, but without side tappings.      

Losses In Pipe Bends

• Bends are provided in pipes to change the direction of flow through it. An additional loss of head, apart from that due to fluid friction, takes place in the course of flow through pipe bend.
• The fluid takes a curved path while flowing through a pipe bend as shown in Fig. 37.2.  
  
  
  
  
  

  Whenever a fluid flows in a curved path, there must be a force acting radially inwards on the fluid to provide the inward acceleration, known as centripetal acceleration

  This results in an increase in pressure near the outer wall of the bend, starting at some point A (Fig. 37.2) and rising to a maximum at some point B . There is also a reduction of pressure near the inner wall giving a minimum pressure at C and a subsequent rise from C to D . Therefore between A and B and between C and D the fluid experiences an adverse pressure gradient (the pressure increases in the direction of flow).
  Fluid particles in this region, because of their close proximity to the wall, have low velocities and cannot overcome the adverse pressure gradient and this leads to a separation of flow from the boundary and consequent losses of energy in generating local eddies. Losses also take place due to a secondary flow in the radial plane of the pipe because of a change in pressure in the radial depth of the pipe. 
  This flow, in conjunction with the main flow, produces a typical spiral motion of the fluid which persists even for a downstream distance of fifty times the pipe diameter from the central plane of the bend. This spiral motion of the fluid increases the local flow velocity and the velocity gradient at the pipe wall, and therefore results in a greater frictional loss of head than that which occurs for the same rate of flow in a straight pipe of the same length and diameter.
  The additional loss of head (apart from that due to usual friction) in flow through pipe bends is known as bend loss and is usually expressed as a fraction of the velocity head as    , where V is the average velocity of flow through the pipe. The value of K depends on the total length of the bend and the ratio of radius of curvature of the bend and pipe diameter R/D. The radius of curvature R is usually taken as the radius of curvature of the centre line of the bend. The factor K varies slightly with Reynolds number Re in the typical range of Re encountered in practice, but increases with surface roughness.

Losses In Pipe Fittings

• An additional loss of head takes place in the course of flow through pipe fittings like valves, couplings and so on. In-general, more restricted the passage is, greater is the loss of head. 

• For turbulent flow, the losses are proportional to the square of the average flow velocity and are usually expressed by  , where V is the average velocity of flow. The value of K depends on the exact shape of the flow passages. Typical values of K are
  

• Since the eddies generated by fittings persist for some distance downstream, the total loss of head caused by two fittings close together is not necessarily the same as the sum of the losses which,each alone would cause.
  These losses are sometimes expressed in terms of an equivalent length of an unobstructed straight pipe in which an equal loss would occur for the same average flow velocity. That is

                                                     (37.5)

where, L_e represents the equivalent length which is usually expressed in terms of the pipe diameter as given by Eq. (37.5). Thus  L_e /d depends upon the friction factor f, and therefore on the Reynolds number and roughness of the pipe.

Power Transmission By A Pipeline

• In certain occasions, hydraulic power is transmitted by conveying fluid through a pipeline. For example, water from a reservoir at a high altitude is often conveyed by a pipeline to an impulse hydraulic turbine in an hydroelectric power station. The hydrostatic head of water is thus transmitted by a pipeline. Let us analyse the efficiency of power transmission under this situation.

The potential head of water in the reservoir = H ( the difference in the water level in the reservoir and the turbine center)                                                            

The head available at the pipe exit (or at the turbine entry) = H_E = H = h_f

Where h_f is the loss of head in the pipeline due to friction.

• Assuming that the friction coefficient and other loss coefficients are constant, we can write

h_f  = RQ²

Where Q is the volume flow rate and R is the hydraulic resistance of the pipeline. Therefore, the power available P at the exit of the pipeline becomes 

For P to be maximum, for a given head  H,dp/dQshould be zero. This gives

                                                (37.6)

 is always negative which shows that P has only a maximum value (not a minimum) with Q.

• From Eq. (37.6), we can say that maximum power is obtained when one third of the head available at the source (reservoir) is lost due to friction in the flow. 
• The efficiency of power transmission n_p is defined as

1. 
   
   1. The efficiency n_p equals to unity for the trivial case of Q = 0. 
   2. For flow to commence and hence n_p is a monotonically decreasing function of Q from a maximum value of unity to zero. 

   3. The zero value of n_p corresponds to the situation given by     when the head H available at the reservoir is totally lost to overcome friction in the flow through the pipe.

• The efficiency of transmission at the condition of maximum power delivered is obtained by substituting RQ²from Eq. (37.6) in Eq. (37.7) as
  
  
  
  Therefore the maximum power transmission efficiency through a pipeline is 67%.