DI system uniformity

A basic relation used in hydraulic design of a pipeline system is the one describing the dependence of discharge Q (say in m³/s) on head loss h_f(m) caused by friction between the flow of fluid and the pipe wall. This section discusses two of the most commonly used head-loss relations: the Darcy-Weisbach and Hazen-Williams equations.

The Darcy-Weisbach equation is used to describe the head loss resulting from flow in pipes in a wide variety of applications. It has the advantage of incorporating a dimen-sionless friction factor that describes the effects of material roughness on the surface of the inside pipe wall and the flow regime on retarding the flow. The Darcy-Weisbach equa-tion can be written as

h_f,_DW f DL 2 V g

  0.0826 2 Q D

2

 Lf5 (2.16)

where h_f,_DW= head loss caused by friction (m), f = dimensionless friction factor, L = pipe length (m), D = pipe diameter (m), V = Q/A = mean flow velocity (m/s), Q = dis-charge (m³/s), A = cross-sectional area of the pipe (m²), and g = acceleration caused by gravity (m/s²).

For noncircular pressure conduits, D is replaced by 4R, where R is the hydraulic radius.

The hydraulic radius is defined as the cross-sectional area divided by the wetted perime-ter or, R = A/P.

Note that the head loss is directly proportional to the length of the conduit and the fric-tion factor. Obviously, the rougher a pipe is and the longer the fluid must travel, the greater the energy loss. The equation also relates the pipe diameter inversely to the head loss. As the pipe diameter increases, the effects of shear stress at the pipe walls are felt by less of the fluid, indicating that wider pipes may be advantageous if excavation and construction costs are not prohibitive. Note in particular that the dependence of the discharge Q on the pipe diameter D is highly nonlinear; this fact has great significance to pipeline designs because head losses can be reduce dramatically by using a large-diameter pipe, whereas an inappropriately small pipe can restrict flow significantly, rather like a partially closed valve.

For laminar flow, the friction factor is linearly dependent on the Re with the simple relationship f = 64/Re. For turbulent flow, the friction factor is a function of both the Re and the pipes relative roughness. The relative roughness is the ratio of equivalent uniform sand grain size and the pipe diameter (e/D), as based on the work of Nikuradse (1933), who

experimentally measured the resistance to flow posed by various pipes with uniform sand grains glued onto the inside walls. Although the commercial pipes have some degree of spa-tial variance in the characteristics of their roughness, they may have the same resistance char-acteristics as do pipes with a uniform distribution of sand grains of size e. Thus, if the veloc-ity of the fluid is known, and hence Re, and the relative roughness is known, the friction fac-tor f can be determined by using the Moody diagram or the Colebrook-White equation.

Jeppson (1976) presented a summary of friction loss equations that can be used instead of the Moody diagram to calculate the friction factor for the Darcy-Weisbach equation.

These equations are applicable for Re greater than 4000 and are categorized according to the type of turbulent flow: (1) turbulent smooth, (2) transition between turbulent smooth and wholly rough, and (3) turbulent rough.

For turbulent smooth flow, the friction factor is a function of Re:

兹 1 苶f

  2log (Re兹f苶) (2.17)

For the transition between turbulent smooth and wholly rough flow, the friction factor is a function of both Re and the relative roughness e/D. This friction factor relation is often summarized in the Colebrook White equation:

兹

When the flow is wholly turbulent (large Re and e/D), the Darcy-Weisbach friction fac-tor becomes independent of Re and is a function only of the relative roughness:

兹 1 苶f

  1.14 2log (e/D) (2.19)

In general, Eq. (2.16) is valid for all turbulent flow regimens in a pipe,, where as Eq.

(2.22) is merely an approximation that is valid for the hydraulic rough flow. In a smooth-pipe flow, the viscous sublayer completely submerges the effect of e on the flow. In this case, the friction factor f is a function of Re and is independent of the relative roughness e/D. In rough-pipe flow, the viscous sublayer is so thin that flow is dominated by the roughness of the pipe wall and f is a function only of e/D and is independent of Re. In the transition, f is a function of both e/D and Re.

The implicit nature of f in Eq. (2.18) is inconvenient in design practice. However, this difficulty can be easily overcome with the help of the Moody diagram or with one of many available explicit approximations. The Moody diagram plots Re on the abscissa, the resis-tance coefficient on one ordinate and f on the other, with e/D acting as a parameter for a family of curves. If e/D is known, then one can follow the relative roughness isocurve across the graph until it intercepts the correct Re. At the corresponding point on the opposite ordinate, the appropriate friction factor is found; e/D for various commercial pipe materials and diameters is provided by several manufacturers and is determined experimentally.

A more popular current alternative to graphical procedures is to use an explicit mathematical form of the friction-factor relation to approximate the implicit Colebrook-white equation. Bhave (1991) included a nice summary of this topic. The popular net-work-analysis program EPANET and several other codes use the equation of Swanee and Jain (1976), which has the form

(2.20)

To circumvent considerations of roughness estimates and Reynolds number depen-dencies, more direct relations are often used. Probably the most widely used of these empirical head-loss relation is the Hazen-Williams equation, which can be written as

Q CuCD^2.63S^0.54 (2.21)

where C_u unit coefficient (Cu 0.314 for English units, 0.278 for metric units), Q  discharge in pipes, gallons/s or m³/s, L length of pipe, ft or m, d  internal diameter of pipe, inches or mm, C Hazen-Williams roughness coefficient, and S = the slope of the energy line and equals h_f/L.

The Hazen-Williams coefficient C is assumed constant and independent of the dis-charge (i.e., R e). Its values range from 140 for smooth straight pipe to 90 or 80 for old, unlined, tuberculated pipe. Values near 100 are typical for average conditions. Values of the unit coefficient for various combinations of units are summarized in Table 2.2.

In Standard International (SI) units, the Hazen-Williams relation can be rewritten for head loss as

h_{f ,HW} 10.654^

Q C





0.

1 54

D 1

4.87 L (2.22)

where h_f,HWis the Hazen-Williams head loss. In fact, the Hazen-Williams equation is not the only empirical loss relation in common use. Another loss relation, the Manning equa-tion, has found its major application in open channel flow computations. As with the other expressions, it incorporates a parameter to describe the roughness of the conduit known as Manning’s n.

Among the most important and surprisingly difficult hydraulic parameter is the diam-eter of the pipe. As has been mentioned, the exponent of diamdiam-eter in head-loss equations is large, thus indicating high sensitivity to its numerical value. For this reason, engineers

EGL

H₁ H₂ H₃

(a)

(b)

Q₁ Q₁

Q₁

EGL

H₁ H₂

Q₂

FIGURE 2.4 Flow in series and parallel pipes.

and analysts must be careful to obtain actual pipe diameters often from manufacturers; the use of nominal diameters is not recommended. Yet another complication may arise, how-ever. The diameter of a pipe often changes with time, typically as a result of chemical depositions on the pipe wall. For old pipes, this reduction in diameter is accounted for indirectly by using an increased value of pipe resistance. Although this approach may be reasonable under some circumstances, it may be a problem under others, especially for unsteady conditions. When ever possible, accurate diameters are recommended for all hydraulic calculations. However, some combinations of pipes (e.g., pipes in series or par-allel; Fig. 2.4) can actualy be represented by a single equivalent diamenter of pipe.