Boundary layer concepts:

In this case the boundary layer develops all over the circumference. The initial development of the boundary layer is similar to that over the flat plate. At some distance from the entrance, the boundary layers merge and further changes in velocity distribution becomes impossible. The velocity profile beyond this point remains unchanged. The distance upto this point is known as entry length. It is about 0.04 Re × *D. *The flow beyond is said to be fully developed. The velocity profiles in the entry region and fully developed region are shown in Fig. 7.3.1*a*. The laminar or turbulent nature of the flow was first investigated by Osborn Reynolds in honour of

**Figure: **Boundary layer development (pipe flow)

Whom the dimensionless ratio of inertia to viscous forces is named. The flow was observed to be laminar till a Reynolds number value of about 2300. The Reynolds number is calculated on the basis of diameter (*ud*/*v*). In pipe flow it is not a function of length. As long as the diameter is constant, the Reynolds number depends on the velocity for a given flow. Hence the value of velocity determines the nature of flow in pipes for a given fluid. The value of the flow Reynolds number is decided by the diameter and the velocity and hence it is decided at the entry itself.

The development of boundary layer in the turbulent range is shown in Fig. 7.3.1*b*. In this case, there is a very short length in which the flow is laminar. This length, *x, *can be calculated using the relation *ux*/*v *= 2000. After this length the flow in the boundary layer turns turbulent. A very thin laminar sublayer near the wall in which the velocity gradient is linear is present all through. After some length the boundary layers merge and the flow becomes fully developed. The entry length in turbulent flow is about 10 to 60 times the diameter.

The velocity profile in the fully developed flow remains constant and is generally more flat compared to laminar flow in which it is parabolic.

__Laminar flow through cicular conduits:__

In laminar region the flow is smooth and regular. The fluid layers do not mix macroscopically (more than a molecule at a time). If a dye is injected into the flow, the dye will travel along a straight line. Laminar flow will be maintained till the value of Reynolds number is less than of the critical value (2300 in conduits and 5 × 10^{5} in flow over plates). In this region the viscous forces are able to damp out any disturbance.

The friction factor, *f *for pipe flow defined as4 ts /r*u*^{2}/2*g*o) is obtainable as *f *= 64/Re where ts is the wall shear stress, *u *is the average velocity and Re is the Reynolds number. In the case of flow through pipes, the average velocity is used to calculate Reynolds number. The dye path is shown in Fig. 7.4.1.

**Figure 7.4.1 **Reynolds Experiment

In turbulent flow there is considerable mixing between layers. A dye injected into the flow will quickly mix with the fluid. Most of the air and water flow in conduits will be turbulent. Turbulence leads to higher frictional losses leading to higher pressure drop. The friction factor is given by the following empirical relations.

These expressions apply for smooth pipes. In rough pipes, the flow may turn turbulent below the critical Reynolds number itself. The friction factor in rough pipe of diameter *D, *with a roughness height of e, is given by

**3.5 Hydraulic Gradient and Energy Gradient.**

The hydraulic grade line, or the hydraulic gradient, in open flow is the water surface, and in pipe flow it connects the elevations to which the water would rise in piezometer tubes along the pipe. The energy gradient is at a distance equal to the velocity head above the hydraulic gradient. In both open and pipe flow the fall of the energy gradient for a given length of channel or pipe represents the loss of energy by friction. When considered together, the hydraulic gradient and the energy gradient reflect not only the loss of energy by friction, but also the conversions between potential and kinetic energy.

In the majority of cases the end objective of hydraulic computations relating to flow in open channels is to determine the curve of the water surface. These problems involve three general relationships between the hydraulic gradient and the energy gradient. For uniform flow the hydraulic gradient and the energy gradient are parallel and the hydraulic gradient becomes an adequate basis for the determination of friction loss, since no conversion between kinetic and potential energy is involved. In accelerated flow the hydraulic gradient is steeper than the energy gradient, and in retarded flow the energy gradient is steeper than the hydraulic gradient. An adequate analysis of flow under these conditions cannot be made without consideration of both the energy gradient and the hydraulic gradient.

**DARCY–WEISBACH EQUATION FOR CALCULATING PRESSURE DROP:**

In the design of piping systems the choice falls between the selection of diameter and the pressure drop. The selection of a larger diameter leads to higher initial cost. But the pressure drop is lower in such a case which leads to lower operating cost. So in the process of design of piping systems it becomes necessary to investigate the pressure drop for various diameters of pipe for a given flow rate. Another factor which affects the pressure drop is the pipe roughness. It is easily seen that the pressure drop will depend directly upon the length and inversely upon the diameter. The velocity will also be a factor and in this case the pressure drop will depend in the square of the velocity (refer Bernoulli equation).

Hence we can say that

The proportionality constant is found to depend on other factors. In the process of such determination Darcy defined or friction factor *f *as

This quantity is dimensionless which may be checked. Extensive investigations have been made to determine the factors influencing the friction factor. It is established that in laminar flow *f *depends only on the Reynolds number and it is given by

In the turbulent region the friction factor is found to depend on Reynolds number for smooth pipes and both on Reynolds number and roughness for rough pipes. Some empirical equations are given in section 7.4 and also under discussions on turbulent flow. The value of friction factor with Reynolds number with roughness as parameter is available in Moody diagram, given in the appendix. Using the definition of Darcy friction factor and conditions of equilibrium, expression for pressure drop in pipes is derived in this section. Consider an elemental length *L *in the pipe. The pressures at sections 1 and 2 are *P*1 and *P*2.

The other force involved on the element is the wall shear t_{0}.

Net pressure force in the element is (*P*1 –*P*2)

Net shear force in the element is t_{0} P*DL*

Force balance for equilibrium yields

From the definition friction factor

Substituting and letting (*P*1 –*P*2) to be Δ*P*.

This reduces to

This equation known as Darcy-Weisbach equation and is generally applicable in most of the pipe flow problems. As mentioned earlier, the value of *f *is to be obtained either from equations or from Moody diagram. The diameter for circular tubes will be the hydraulic diameter *D**h* defined earlier in the text. It is found desirable to express the pressure drop as head of the flowing fluid.

In this case as

The velocity term can be replaced in terms of volume flow and the equation obtained is found useful in designs as *Q *is generally specified in designs.

Substituting in (7.8.6), we get

It is found that

Another coefficient of friction *C**f *is defined as *C**f *= *f */4

In this case

Now a days as Darcy-Weisbach equation are more popularly used as value of *f *is easily available.

__Moody diagram____ __

The Moody chart or Moody diagram is a graph in non-dimensional form that relates the Darcy-Weisbach friction factor, Reynolds number and relative roughness for fully developed flow in a circular pipe. It can be used for working out pressure drop or flow rate down such a pipe.

Pressure drop can then be evaluated as:

where is the density of the fluid, is the average velocity in the pipe, is the friction factor from the Moody chart, is the length of the pipe and is the pipe diameter.

The basic chart plots Darcy–Weisbach friction factor against Reynolds number for a variety of relative roughnesses and flow regimes. The relative roughness being the ratio of the mean height of roughness of the pipe to the pipe diameter or .

The Moody chart can be divided into two regimes of flow: laminar and turbulent. For the laminar flow regime, the Darcy–Weisbach friction factor was determined analytically by Poiseuille and is used. In this regime roughness has no discernible effect. For the turbulent flow regime, the relationship between the friction factor and the Reynolds number is more complex and is governed by the Colebrook equation which is implicit in :

In 1944, Lewis Ferry Moody plotted the Darcy–Weisbach friction factor into what is now known as the Moody chart.

The Fanning friction factor is 1/4 the Darcy–Weisbach one and the equation for pressure drop has a compensating factor of four.

__Minor loses in pipe flow:__

Additional frictional losses occur at pipe entry, valves and fittings, sudden decrease or increase in flow area or where direction of flow changes. The frictional losses other than pipe friction are called minor losses. In a pipe system design, it is necessary to take into account all such losses. These losses are generally expressed as *h** _{f}*=

*Cu*/2

_{m}^{2}*g*where

*C*is constant, the value of which will depend on the situation and is called the loss coefficient. The expression is applicable both for laminar and turbulent flows.

**(***i***) Loss of head at entrance: **At the entrance from the reservoir into the pipe, losses take place due to the turbulence created downstream of the entrance. Three types of entrances are known.

**(a) Bell mouthed: **This is a smooth entrance and turbulence is suppressed to a great extent and *C *= 0.04 for this situation.

**(b) Square edged entrance: **Though it is desirable to provide a bell mouthed entrance it will not be always practicable. Square edged entrance is used more popularly. The loss coefficient, *C *= 0.5 in this case.

**(c) Reentrant inlet: **The pipe may sometimes protrude from the wall into the liquid.Such an arrangement is called reentrant inlet. The loss coefficient in this case is about 0.8.

**Figure: **Types of entrance

**(***ii***) Loss of head at submerged discharge: **When a pipe with submerged outlet discharges into a liquid which is still (not moving) whole of the dynamic head *u*^{2}/2*g *will be lost. The loss coefficient is 1.0. The discharge from reaction turbines into the tail race water is an example. The loss is reduced by providing a diverging pipe to reduce the exit velocity.

**(***iii***) Sudden contraction: **When the pipe section is suddenly reduced, loss coefficient depends on the diameter ratio. The value is 0.33 for *D*_{2}/*D*_{1} = 0.5. The values are generally available in a tabular statement connecting *D*2/*D*1 and loss coefficient. Gradual contraction will reduce the loss. For gradual contraction it varies with the angle of the transition section from 0.05 to 0.08 for angles of 10° to 60°.

**(***iv***) Sudden expansion: **Here the sudden expansion creates pockets of eddying turbulence leading to losses. The loss of head *h**f *is given by

where *u*1 and *u*2 are the velocities in the smaller and larger sections. Gradual expansion will reduce the losses.

**(***v***) Valves and fittings : **Losses in flow through valves and fittings is expressed in terms of an equivalent length of straight pipe.

For gate valves *L *= 8*D*, and for globe valves it is 340 *D*. For 90° bends it is about 30 *D*.

__NETWORKS OF PIPES:__

Complex connections of pipes are used in city water supply as well as in industrial systems. Some of these are discussed in the para.

**Pipes in Series—Electrical Analogy**

Series flow problem can also be solved by use of resistance network. Consider equation as given below

For given pipe specification the equation can be simplified as

**Note: **The dimension for *R *is *s*2/*m*5. For flow in series *Q *is the same through all pipes. This leads to the relation

The *R *values for the pipe can be calculated. As the total head is also known *Q *can be evaluated. The length *L *should include minor losses in terms of equivalent lengths. The circuit is shown in Fig.

**Figure **Equivalent circuit for series flow

**Example 7.12**. A reservoir at a level with respect to datum of 16 m supplies water to a ground level reservoir at a level of 4 m. Due to constraints pipes of different diameters were to be used. Determine the flow rate

The resistance values are calculated using simplified equation

__Pipes in Parallel__

Such a system is shown below

**Case **(*i*) The head drop between locations 1 and 2 are specified: The total flow can be determined using

As *h**f *and all other details except flow rates *Q*1, *Q*2 and *Q*3 are specified, these flow rates can be determined.

Total flow

The process can be extended to any number of connections

**Case **(*ii*) Total flow and pipe details specified. One of the methods uses the following steps:

1. Assume by proper judgement the flow rate in pipe 1 as *Q*1.

2. Determine the frictional loss.

3. Using the value find *Q*2 and *Q*3.

4. Divide the total *Q *in the proportion *Q*1 : *Q*2 : *Q*3 to obtain the actual flow rates.

**Case **(*iii*) Electrical analogy is illustrated or in problem Ex. 7.13 and Ex. 7.14.

**Example 7.13 **The details of a **parallel pipe system **for water flow are given below.

1. If the frictional drop between the junctions is 15 m of water, determine the total flow rate

2. If the total flow rate is 0.66 m3/s, determine the individual flow and the friction drop.

The system is shown in Fig. Ex. 7.13.

**Case **(*i*) Let the flows be *Q*1, *Q*2 and *Q*3. Total flow *Q *= *Q*1 + *Q*2 + *Q*3, using equation 7.11.15

The flow rates are calculated individually with *h**f *= 15 m and totalled.

**Case **(*ii*) Total flow is 0.66 m3/s . Already for 15 m head individual flows are available. Adopting method 2 the total flow is divided in the ratio of *Q*1 : *Q*2 : *Q*3 as calculated above.

Calculation for frictional loss.