DISPOSICIONES ADICIONALES Disposición adicional primera. Zona de servicio

Table 4.1 lists the names of the main contributors, and their contributions, to pipe flow theories in chronological order.

The Colebrook–White transition formula represents the culmination of all the previous work and can be applied to any fluid in any pipe operating under turbulent flow conditions. The later contributions of Moody, Ackers and Barr are mainly concerned with the practical application of the Colebrook–White equation.

There are three major concepts described in the table. These are 1. The distinction between laminar and turbulent flow

2. The distinction between rough and smooth pipes

3. The distinction between artificially roughened pipes and commercial pipes

To understand these concepts, the best starting point is the contribution of Reynolds, followed by the laminar flow equations, before proceeding to the more complex turbulent flow equations.

taBle 4.1 Chronological Development of Pipe Flow Theories

Date Name Contribution

1839–1841 Hagen and Poiseuille Laminar flow equation 1850 Darcy and Weisbach Turbulent flow equation

1884 Reynolds Distinction between laminar and turbulent flow – Reynolds number

1902 Hazen–Williams formula Empirical formula for flow of water in pipes 1913 Blasius Friction factor equation for smooth pipes

1914 Stanton and Pannell Experimental values of the friction factor for smooth pipes

1930 Nikuradse Experimental values of the friction factor for artificially rough pipes

1930s Prandtl and von Kármán Equations for rough and smooth friction factors 1937–1939 Colebrook and White Experimental values of the friction factor for

commercial pipes and the transition formula 1944 Moody The Moody diagram for commercial pipes

1958 Ackers The Hydraulics Research Station Charts and Tables for the design of pipes and channels

4.2.1 laminar and turbulent Flow

Reynolds’ experiments demonstrated that there were two kinds of flow – laminar and turbulent – as described in Chapter 3. He found that transition from laminar to turbulent flow occurred at a criti- cal velocity for a given pipe and fluid. Expressing his results in terms of the dimensionless parameter Re = ρDV/μ, he found that for Re less than about 2000 the flow was always laminar and that for Re greater than about 4000 the flow was always turbulent. For Re between 2000 and 4000, he found that the flow could be either laminar or turbulent and termed this the transition region.

In a further set of experiments, he found that for laminar flow the frictional head loss in a pipe was proportional to the velocity and that for turbulent flow the head loss was proportional to the square of the velocity.

These two results had been previously determined by Hagen and Poiseuille (hf ∝ V) and

Darcy and Weisbach (h_f∝ V2_{), but it was Reynolds who put these equations in the context of}

laminar and turbulent flow.

4.3 FundaMental ConCePts oF PIPe FloW

4.3.1 Momentum equation

Before proceeding to derive the laminar and turbulent flow equations, it is instructive to con- sider the momentum (or dynamic) equation of flow and the influence of the boundary layer.

Referring to Figure 4.2, showing an elemental annulus of fluid, thickness δr, length δl, in a pipe of radius R, the forces acting are the pressure forces, the shear forces and the weight of the fluid. The pressure forces act in the directions shown on the upstream and downstream sections of the annulus, which can be considered as a control volume (refer to Chapter 2). The shear forces are due

τ2πrδl p2πrδr _{ρg2πrδrδl} δr δl r R z Q θ

(

)

(

)

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Flow in Pipes and Closed Conduits

to the velocity gradient across the pipe, the velocity being zero at the pipe wall and a maximum at the centre. The sum of the forces acting is equal to the rate of change of momentum. In this case the rate of change of momentum is zero, since the flow is steady and uniform. Hence, allowing for the pressure force to vary with distance along the pipe and the shear force to vary with the radius then

p r r p p

l l r r r l r r r r l

2π δ −_ +d_d δ_2π δ +τ π δ2 −_τ+d_dτ δ π δ δ_2 ( + ) ++ρ π δ δg2 r l r sinθ=0 Setting sin θ = −dz/dl and dividing by 2πr δr δl gives

−d − − − = d d d d d p l r r g z l τ τ ρ 0

(ignoring second-order terms), or

−d_dp_l*− −_rτ d_drτ=0

where p*(= p + ρgz) is the piezometric pressure measured from the datum z = 0. As

1 1 r r r r r r r r d d d d d d ( )τ = _ τ τ+ _ = τ τ+ then −d − = d d d p l r r r * 1 _{( )τ 0} Rearranging, d d d d r r r p l ( )τ = − * Integrating both sides with respect to r,

τr = −d_dp_l*r2 +constant 2

At the centreline r = 0, and therefore constant = 0. Hence, τ = −d d p l r * 2 (4.1)

Equation 4.1 is the momentum equation for steady uniform flow in a pipe. It is equally appli- cable to laminar or turbulent flow, and relates the shear stress τ at radius r to the rate of head

loss with distance along the pipe. If an expression for the shear force can be found in terms of the velocity at radius r, then the momentum equation may be used to relate the velocity (and hence discharge) to head loss.

In the case of laminar flow, this is a simple matter. However, for the case of turbulent flow it is more complicated, as will be seen in the following sections.

4.3.2 development of Boundary layers

Figure 4.3a shows the development of laminar flow in a pipe. At entry to the pipe, a laminar boundary layer begins to grow. However, the growth of the boundary layer is halted when it reaches the pipe centreline, and thereafter the flow consists entirely of a boundary layer of thickness r. The resulting velocity distribution is as shown in Figure 4.3a.

For the case of turbulent flow shown in Figure 4.3b, the growth of the boundary layer is not suppressed until it becomes a turbulent boundary layer with the accompanying laminar sub- layer. The resulting velocity profile therefore differs considerably from the laminar case. The existence of the laminar sublayer is of prime importance in explaining the difference between smooth and rough pipes.

Expressions relating shear stress to velocity have been developed in Chapter 3, and these will be used in explaining the pipe flow equations in the following sections.

Boundary layer (a) Velocity profile u Laminar sub-layer

Turbulent boundary layer Laminar boundary layer

(b)

Velocity profile

u

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Flow in Pipes and Closed Conduits