Head Loss in Innovyze H2OCalc

 

3.7 Head loss due to Transitions and Fittings (Local loss)

Whenever flow velocity changes direction or magnitude in a conduit (e.g., at fittings, bends, and other appurtenances) added turbulence is induced. The energy associated with that turbulence is eventually dissipated into heat that produces a minor head loss, or local (or form) loss. The local (minor) loss associated with a particular fitting can be evaluated by

                                                                                                      

where   V         =          mean velocity in the conduit (m/s, ft/s)

                K         =          loss coefficient for the particular fitting involved.

The table given below provides the loss coefficients (K) for various transitions and fittings.

 

Table 3-3: Typical Minor Loss Coefficients

Type of form loss			K
Expansion	Sudden	D1 < D2
	Gradual	D1/D2 = 0.8	0.03
		D1/D2 = 0.5	0.08
		D1/D2 = 0.2	0.13
Contraction	Sudden	D1 > D2
	Gradual	D2/D1 = 0.8	0.05
		D2/D1 = 0.5	0.065
		D2/D1 = 0.2	0.08
Pipe entrance	Square-edge		0.5
	Rounded		0.25
	Projecting		0.8
Pipe exit	Submerged pipe to still water		1.0
Tee	Flow through run		0.6
	Flow through side outlet		1.8
Orifice	(Pipe diameter  /orifice diameter)	D/d = 4	4.8
		D/d = 2	1.0
		D/d = 1.33	0.24
Venturi (long-tube)	(Pipe diameter  /throat diameter)	D/d = 3	1.1
		D/d = 2	0.5
		D/d = 1.33	0.2
Bend	90o miter bend with vanes		0.2
	90o miter bend without vanes		1.1
	45o miter bend		0.2
Type of form loss (continued)			K
Bend	45o smooth bend:      (bend radius  /pipe diameter)	r/D = 1	0.37
		r/D = 2	0.22
		r/D = 4	0.2
	90o smooth bend	r/D = 1	0.5
		r/D = 2	0.3
		r/D = 4	0.25
	Closed return bend		2.2
Sluice	Submerged port in wall		0.8
	As conduit contraction		0.5
	Without top submergence		0.2
Valve	Globe valve, fully open		10
	Angel valve, fully open		5.0
	Swing check valve, fully open		2.5
	Gate valve, fully open		0.2
	Gate valve, half open		5.6
	Butterfly valve, fully open		1.2
	Ball valve, fully open		0.1

       Source: Nicklow and Boulos (2005)

Water Hammer from Innovyze H20Calc

 

3.14 Water Hammer

Surge analysis is important to estimate the worst-case events in the Water Distribution Systems (WDS). Transient regimes in WDS are inevitable and will normally occur as a result of action at pump stations and control valves. Regions that are particularly susceptible to transients are high elevation areas, locations with either low or high static pressures, and regions far removed from overhead storage. They are generally characterized by fluctuating pressures and velocities and are critical precisely because pressure variations can be of high magnitude, possibly large enough to break or damage pipes or other equipment, or to greatly disrupt delivery conditions.

 

This section presents the calculation of potential surge using Joukowski equation, which is widely applied as a simplified surge analysis, and wave speed calculation. In the end, it provides the calculation of the inertia of pumps and motors, which are important for transients caused by pump failure.

Joukowski Expression

 

The pressure rise for instantaneous closure is directly proportional to the fluid velocity at cutoff and to the velocity of the predicted surge wave. Thus, the relationship used for analysis is simply the well-known Joukowski expression for sudden closures in frictionless pipes

where  

              =          surge pressure (m , ft)

            =          velocity change of water in the pipeline (m/s, ft/s)

            c          =          wave speed (m/s, ft/s)

            A          =          cross-sectional area (m², ft² h)

            g          =          gravitational acceleration (9.81 m/s², 32.17 ft/s²)

Wave Speed

 

The wave speed, c, is influenced by the elasticity of the pipe wall. For a pipe system with some degree of axial restraint a good approximation for the wave propagation speed is obtained using

where  E_f         =          elastic modulus of the fluid (for water, 2.19 GN/m², 0.05 Glb/ft²)

            ρ          =          density of the fluid (for water, 998 kg/m³, 1.94 slug/ft³)

            E_c         =          elastic modulus of the conduit (GN/m², Glb/ft²)

            D         =          pipe diameter (mm, inch)

            t           =          pipe thickness (mm, inch)

            K_R        =          coefficient of restraint for longitudinal pipe movement.

 

The constant K_R takes into account the type of support provided for the pipeline. Typically, three cases are recognized with K_R defined for each as follows (m is the Poisson’s ratio for the pipe material):

 

Case a: The pipeline is anchored at the upstream end only.

 

K_R =  1 -  m / 2

Case b: The pipeline is anchored against longitudinal movement.

 

K_R =  1 - m²

Case c: The pipeline has expansion joints throughout.

 

K_R =  1

The following table provides physical properties of common pipe materials.

 

Table 3-8: Physical Properties of Common Pipe Materials

Material	Young’s Modulus (Ec)		Poisson’s Ratio, μ
	GN/m2	Glb/ft2
Asbestos Cement	23 - 24	0.53 - 0.55	-
Cast Iron	80 - 170	1.8 - 3.9	0.25 - 0.27
Concrete	14 - 30	0.32 - 0.68	0.1 - 0.15
Reinforced Concrete	30 - 60	0.68 - 1.4	-
Ductile Iron	172	3.93	0.3
PVC	2.4 - 3.5	0.055 - 0.08	0.46
Steel	200 - 207	4.57 - 4.73	0.30

 

Inertia of Pumps and Motors

 

The combined inertia of pumps and motors driving them, including the connecting shafts and couplings, is required for transient analysis associated with the starting and stopping of pumps. The equations provided below are intended to be used as an initial guide to the inertia values that may be used as a reasonable first approximation, when more accurate data is not available. The total inertia for the pump/motor unit is the sum of both pump and motor inertias. The following inertia calculations are based on Thorley (2004).

Pump Inertias

 

From the linear regression analysis of 300 pump inertia data, two equations were developed for predicting the inertia I of pump impellers, including the entrained water and the shaft on which the impeller is mounted. The first equation represents the upper set of the data, and applies to single- and double-entry impellers, single and multistage, and horizontal and vertical, spindle machines.

                                                                                                   

where  I           =          pump inertia (kg m², lb ft²)

            C₁        =          coefficients (0.03768, 0.6674)

            P          =          power (kW, hp)

            N         =          pump speed (rev/min)

 

The second equation is for lower set of the data and represents relatively small, single-entry, radial flow impellers of lightweight design. This is applied to relatively small pumps of lightweight design.

                                                                                                      

where  C₂        =          coefficients (0.03407 in SI, 0.6244 in English)

Motor Inertias

 

Similar to pump inertia, linear regression of the motor inertia data yields the following equations.

                                                                                                       

where  C₃           =         coefficients (0.0043 in SI, 0.0648 in English).

Weirs from Innovyze H20Calc

 

3.11 Weir

Discharge in channels and small streams can be conveniently measured by using a weir. Weirs can be categorized in to two: sharp crested and broad crested.

Sharp-Crested Weir

 

A sharp-crested weir is a vertical plate placed in a channel that forces the liquid to flow through an opening to measure the flow rate. The type of the weir is characterized by the shape of opening.

Rectangular Sharp-Crested Weir

 

A vertical thin plate with a straight top edge is referred to as rectangular weir since the cross section of the flow over the weir is rectangular (see the following figure).

 

The discharge equation for a rectangular weir is given as

                                                                                                                            

where Q         =          discharge over the weir (m³/s, ft³/s)

            h           =          head (m, ft)

            L          =          weir length (m, ft)

C          =          weir coefficient

 typically given as 1.84 in SI, 3.33 in English.

Flow through the weir may not span the entire width of the channel (L) due to end contractions. Experiments have indicated that the reduction in length is approximately equal to 0.1nh, where n is the number of end contractions (e.g., could be 2 in the contracted rectangular weir), and h is head over the crest of the weir as defined above. Therefore, the formula for contracted weir (one with flow contraction due to end walls) is given as

                                                                                                     

Multiple-Step Sharp Crested Rectangular Weir

 

A multiple step weir is a rectangular weir with stepwise increase in length along the weir height. It helps to maintain low velocity across the weir during low flows and may be ecologically friendly as it allows fish freely pass across the weir.

   

The discharge equation for multi-step weirs is given as:

                                    

where  Q         =          discharge over weir (m³/s, ft³/s)

            h_i         =          head over the crest of the weir at step i (m, ft)

            L_i         =          length of the weir at step i (m, ft)

            C         =          the flow coefficient (1.86 in SI, 3.367 in English)

Cipolletti Sharp-Crested Weir

 

The Cipolletti (or trapezoidal) weir has side slopes of 4 vertical to 1 horizontal ratio as shown in the figure below. The discharge equation for a Cipolletti weir is given as

 

                                                                                                                            

where Q         =          discharge over weir (m³/s, ft³/s)

            h          =          head (m, ft)

            L          =          weir bottom length (m, ft)

            C         =          the flow coefficient (1.86 in SI, 3.367 in English)

 

 

Notice that L is measured along the bottom of the weir (called the crest), not along the water surface.

V-Notch Sharp-Crested Weir

 

With low flow rate, it is common to use a V-Notch weir (shown below).

 

 

The discharge equation for a V-Notch weir is given as

                                                                                                    

where Q         =          discharge over weir (m³/s, ft³/s)

            h          =          head (m, ft)

            θ          =          angle of notch (degree)

            C         =          the flow coefficient that typically range between 0.58 and 0.62.

 

The most commonly used value of the notch angle θ is 90^o; for this case (i.e., θ is 90^o), C is found to be around 0.585. 

Submerged Sharp-Crested Weir

 

The weir equations discussed above assume that the weir is free flowing. However, if the tailwater rises high enough, the weir will be submerged and the weir flow-carrying capacity will be reduced. Therefore, the discharge can be adjusted for submergence using the following equation:

                                                                                                  

where Q_s        =          discharge over a submerged weir (m³/s, ft³/s)

            Q         =          discharge computed using weir equations (m³/s, ft³/s)

            h_s         =          tailwater depth above the weir crest (m, ft)

            h          =          head upstream of the weir (m, ft)

            n          =          exponent, 1.5 for rectangular and Cipolletti weirs, 2.5 for a triangular weir.            

Broad-Crested Weir

If the weir is long in the direction of flow so that the flow leaves the weir in essentially a horizontal direction, the weir is a broad-crested weir.

 

The discharge equation for a broad crested weir is given as

                                                                                                                         

where Q         =          discharge over weir (m³/s, ft³/s)

            h          =          head (m, ft)

            L          =          crest length (m, ft)

            C         =          the flow coefficient that typically range between 2.4 and 3.087.

 

The flow coefficient C can be obtained from the following figure. Depending on the shape of the weir and head on the weir, the C value may range from 2.4 to 3.1.

 

Broad-Crested Weir Discharge Coefficients (Adapted from Normann et al., 1985)

Generic Weir

 

Any other type of weirs can be modeled as generic weir using the following equation.

                                                                                                                   

where Q         =          discharge over weir (m³/s, ft³/s)

            h          =          head above weir crest (m, ft)

            L          =          crest length (m, ft)

            C         =          weir coefficient

The weir coefficient value depends on the weir type, and is the function of the head above the weir crest.