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• In course of flow through a pipe, a fluid may be withdrawn from the side tappings along the length of the pipe as shown in Fig. 37.1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:

Fig. 37.1 Flow through pipes with side tappings

• 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 over a small length dx of the pipe at any section can be written as

(37.1)

where, 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, 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 q per 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

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

• Hence,

(37.2)

Substituting 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)

• 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.