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Water Velocity: Its Impact on the Accuracy of Hydraulic Calculations

September 1996

By: Roland Huggins, P.E.

Does excessive water velocity cause inaccuracies that are sufficient enough to invalidate hydraulic calculations based on the Hazen-Williams equation? I wouldn't be surprised if you've never had a problem with this issue since it is often ignored by the industry but the NFPA sprinkler Handbook does identify it as a concern. While the handbook also identifies that NFPA 13 does not stipulate a maximum limit on velocities, it does state: "Twenty feet per second (ft/s) maximum velocities may result in more conservative designs." At velocities beyond this, the Hazen-Williams formula is not as conservative as other methods for certain pipe types, such as Darcy-Weisbach.Even though NFPA does not specify a value, other agencies, such as Factory Mutual and Wausau, both use 20 ft/s and some governmental agencies, such as Naval Facilities, use 32 ft/s.The primary objective of this evaluation is to quantify the actual impact water velocity has on the accuracy of the calculations, thus providing a basis for those AHJs who intend to implement a maximum water velocity. I've also been curious for some time about the actual impact velocity has on the accuracy of hydraulic calculations.It should be noted that this evaluation does not address the impact of velocity on the use of total pressure versus normal pressure in hydraulic calculations.

Background
This evaluation is based on comparing calculated friction losses.As we all know, as water flows through a pipe, there is a loss of energy due to friction. This loss occurs because the water particles rub against the pipe as well as each other.Technically, this loss is not constant, but changes with the water velocity. There are three primary phases of water velocity that affect friction loss. These phases are laminar, transition, and turbulent flows. In the laminar and transition phases the friction loss factor changes drastically with water velocity. These phases occur at very low flow rates, with laminar flow occurring in a 1 inch steel pipe at rates less than 1 gpm and transition flow at less than 2 gpm. Fortunately, the flows encountered in the sprinkler industry are only in the turbulent phase, where the impact on the friction loss factor is relatively linear.In estimating friction losses, some equations account for this change, while others do not.

It is worth noting that even though we talk about the calculated loss, which implies an absolute accuracy, the calculation is only a mathematical estimation of the real losses.

The friction loss has been calculated by the Hazen-Williams equation and then compared to that obtained by the Darcy-Weisbach equation for a range of flow rates and pipe sizes.

These equations are:

Darcy-Weisbach: hL = f x v2

                                                                2 x D x g

hL: friction loss (ft)

f:  friction factor

v:  water velocity (ft/s)

D: pipe diameter (ft)

G: gravitational constant (ft/s2)

 

Hazen-Williams:  p = 4.52 x Q1.85

                                    C1.85 x D1.85

p: friction loss (psi)

Q: flow rate (gpm)

C: friction factor

D: pipe diameter (in)

 

The Darcy-Weisbach equation was selected as the benchmark since it is commonly accepted as providing a more accurate estimate of energy loss due to fluid flow, hereafter referred to as friction loss. This position is supported by the Sprinkler Handbook as indicated in the introduction of this article. The reason it is viewed as more accurate is that it is based on scientific principles (the continuity equation) and accounts for changes such as water velocity and viscosity; whereas, the Hazen-Williams equation, being empirically based from the observations of many pipeline flows, treats the effect of these variables as a constant.

One of the main problems with the Darcy-Weisbach equation is that it is much more difficult to apply. For instance, the friction factor is not a constant for the same pipe material but changes with the water velocity as well as the pipe diameter. The typical method for determining the friction factor is the use of the Moody diagram (see Figure 1). As shown by the diagram, as the Reynolds number increases (due to an increase in water velocity), the friction factor decreases. A severe restriction on its use is that there is little or no information readily available on the roughness of pipe material except for new piping. Hazen-Williams uses the same friction factor (the "C" value) based solely on pipe material, regardless of pipe size or water velocity. Though more difficult to apply, the Darcy-Williams equation can be used for any fluid, at any temperature, whereas the Hazen-Williams equation is applicable to only water and is most accurate when the water temperature is near 60° F.

Results

The results from the two equations were compared over a flow range of 10 feet per second (ft/s) to 40 ft/s. Friction factors were based on the relative roughness for new pipe which is determined by dividing the specific roughness by the pipe diameter. The assigned specific roughness, as indicated by a pipe manufacturer, was 0.00015 ft and the Hazen-Williams "C" factor was 140. As indicated in the Sprinkler handbook new steel pipe actually has a "C" value of 140 but 120 is used in standard hydraulic calculations to account for the reduced "C" value of aged pipe. In order to evaluate the impact of the roughness of the material, both steel (schedule 40) and copper (type M) were evaluated.Even though copper is called tubing, for simplicity it will hereafter be referred to as pipe.The specific roughness assigned to the copper was 0.000005 ft and the "C" factor was 150. The pipe diameter was also varied from 1 inch to 4 inches. As final parameter needing identification is that the water viscosity was determined based on a temperature of 60° F.

See Figure 2 for the results of the friction loss calculations.Within this figure the columns showing the friction loss are identified as Loss/ft Ð HW for Hazen-Williams and DW for Darcy-Weisbach. The indicated value for Darcy-Weisbach was converted from ft to psi by multiplying it by 0.434. Since the Darcy-Weisbach is being used as the benchmark, the percentage that Hazen-Williams varies from it is identified as Difference HW/DW, which is actually (HW-DW)/DW. The percentages indicated were calculated using the whole number, not the rounded-off values shown in the figure. A negative value indicates Hazen-Williams has estimated a lower friction loss.

Discussion

In discussing the results, references to deviation generally relates to the difference by percentage of the friction loss calculated by the Hazen-Williams equation when compared to that by Darcy-Weisbach. It is also worth mentioning that a seemingly large percentage deviation is not always significant, particularly when evaluating small numbers. For instance, 0.003 psi deviates from .0025 psi by 20% but there is only a real difference of 0.0005 psi. Since the roughness of the material completely changes the pattern of the results, steel and copper are addressed separately.

In looking at the results for steel pipe, the only thing that is clearly indicated is that an increase in velocity always increases the deviation. Unfortunately, (from perspective of identifying a maximum value), as the velocity increases, the amount of deviation generally decreases. For example, in a 1 inch pipe there is a 2.45 increase in percentage between 10 ft/s and 15 ft/s but between 35 ft/s and 40 ft/s the increase is only 0.89. The most prominent issue is that at 10 ft/s 1-inch pipe deviates 11.2% and 4 inch deviates 2.7% while at 40 ft/s 1- inch deviates 21.9% and 4 inch 14.8%.These are large percentages but presented as a real numbers, 40 ft/s in a 1-inch pipe cerates a deviation of 0.6 psi but in the 4-inch pipe it is only 0.08 psi.

There are two items of interest that I would also like to address. We've identified that, up to a point, the smaller the pipe size, the greater the deviation.   It appears though, that for pipe sizes starting at 3 inches, there is a consistent deviation.  This eliminates, beyond this point, pipe diameter as an influential parameter.  It is also worth noting that steel pipe has a negative deviation indicating a lower calculated friction loss, which is less conservative.

As already mentioned the roughness of the material is important since smoother material experiences less deviation. For example, at 40 ft/s 1 inch steel deviates 21.9%, whereas copper is only 4.8% and 4 inch steel is 14.8% with copper at 7.5%. Allow me to revert to steel pipe momentarily as it relates to pipe roughness.  Even though this evaluation did not review existing pipe with lower "C" values, it does support the position that Hazen-Williams becomes less accurate for lower "C" values.The SFPE Handbook of Fire Protection Engineering, Second Edition, indicates that lower "C" values below 100 consists of a 5% reduction when the velocity exceeds 3 ft/s, with an additional 5 % reduction each time the velocity doubles.

Copper does not produce a significant deviation but it does produce an interesting change of pattern in the results. The rougher steel pipe had a negative deviation, whereas copper has a positive deviation (except for 1-inch pipe at 10 ft/s which has a deviation of less than 1%). This indication of additional friction loss actually provides a conservative estimate. Another shift is that the larger the diameter, the greater the deviation. Again, one should greater the deviation in percentage by the change in actual values. In this case, at 40 ft/s 4 inch copper deviates 7.5% but the actual difference in pressure is only 0.03 psi.

Conclusions

There is not a single velocity that stands out as a logical maximum, when applied to the different pipe sizes and materials, which would allow us to say "stop here, do not exceed." I wouldn't even call the variances large, except for 1-inch steel pipe where we see up to a 22% variance.Even though this is a large variance, in real numbers it only represents a difference of 0.6 psi. Considering the lack of accuracy of the water supply data, which is typically no better than plus or minus 10 psi (anyone who has performed a water flow test and seen how much the pressure gauge bounces, knows what I mean), is 0.6 psi significant? I wouldn't think so, particularly when the high deviation is typically accompanied by a high friction loss (discussed further below). The important thing is the use of a consistent methodology throughout the industry. I don't want to imply that allowing a design to go over by 0.6 psi is acceptable, since this would interfere with the consistency of the methodology. The theoretical, calculated difference of 0.6 psi is what I'm referring to.

It doesn't seem appropriate to have a single maximum velocity applied to all pipe size when only the smaller diameter pipe is adversely affected. After all, what do we gain by having a velocity restriction of 40 ft/s in a 4-inch steel pipe? Sure the deviation is 14.8% (sounds big) but the actual pressure difference is only 0.08 psi (yes, in the hundredths). Thus, we would be restricting the velocity of all pipe sizes when only the smallest diameter pipes are moderately affected. Or for the sake of theoretical purity, should we restrict the deviation to a specified percentage? This also does not seem appropriate since large percentages often do not reflect a meaningful real value. >Another method could be assigning a maximum real value allowed for deviation such as 0.2 psi.  The problem with this approach is that there will be a different maximum velocity for each pipe size as well as for each type pipe material. Would there be sufficient improvement in sprinkler performance to warrant the increase in confusion? I wouldn't think so.

There is a naturally occurring restriction which is called friction loss. In small diameter pipe where a high velocity creates the greatest variance, the friction loss is also the greatest. Few sprinkler designs can tolerate a friction loss of 2.2 psi/ft (1 inch steel at 40 ft/s) which creates a 22 psi loss between sprinklers at a 10 ft. spacing (and that's with no equivalent lengths for fittings). From the perspective of accuracy of the friction loss equations, this seems the best method.

There is another drastic method for obtaining a perceived increase in accuracy, and that is to start using the Darcy-Weisbach instead of the Hazen-Williams equation. There are several items which indicated this approach is undesirable particularly for standard sprinkler systems. The first item is that even though the Darcy-Weisbach equation is more accurate for specific conditions, many of these conditions for a sprinkler system will change over time.This offsets the accuracy since an average must now be assumed. A second item is the extensive amount of information accrued on the Hazen-Williams equation, particularly for existing piping. A final item is the successful performance of current systems, which demonstrates an acceptable degree of accuracy. Thus, there is no reason to change to a different equation for performing hydraulic calculations for sprinkler systems.

In closing, there is not sufficient deviation to warrant a maximum water velocity of 20 ft/s simply for the sake of a more conservative design nor does a high velocity create sufficient inaccuracies to invalidate hydraulic calculations based on the Hazen-Williams equation.

Figure 2    Result of the friction loss calculations.

4" Schedule 40 Steel Pipe (4.026 inch)

 

 

 

 

Velocity

Q

C

Loss/ft

Fric Factor

Loss/ft

Difference

(ft/s)

(gpm)

HW

HW

DW

DW

HW/DW

10

397

140

0.035

0.018

0.036

-2.68%

15

595

140

0.075

0.175

0.079

-5.80%

20

794

140

0.127

0.174

0.14

-9.26%

25

992

140

0.192

0.173

0.217

-11.74%

30

1190

140

0.269

0.171

0.309

-13.12%

32

1270

140

0.303

0.171

0.352

-13.96%

35

1389

140

0.357

0.169

0.416

-14.10%

40

1587

140

0.457

0.167

0.537

-14.79%

 

 

 

 

 

 

 

3" Schedule 40 Steel Pipe (3.068 inch)

 

 

 

 

Velocity

Q

C

Loss/ft

Fric Factor

Loss/ft

Difference

(ft/s)

(gpm)

HW

HW

DW

DW

HW/DW

10

230

140

0.048

0.0188

0.05

-2.41%

15

346

140

0.102

0.0185

0.11

-6.68%

20

461

140

0.174

0.0183

0.193

-9.65%

25

576

140

0.263

0.018

0.297

-11.16%

30

691

140

0.369

0.0178

0.422

-12.59%

32

737

140

0.416

0.0178

0.48

-13.43%

35

806

140

0.491

0.0177

0.572

-14.10%

40

922

140

0.628

0.0175

0.738

-14.85%

 

 

 

 

 

 

 

2" Schedule 40 Steel Pipe (2.067 inch)

 

 

 

 

Velocity

Q

C

Loss/ft

Fric Factor

Loss/ft

Difference

(ft/s)

(gpm)

HW

HW

DW

DW

HW/DW

10

105

140

0.077

0.021

0.082

-6.57%

15

157

140

0.163

0.0202

0.178

-8.60%

20

209

140

0.277

0.0199

0.311

-11.14%

25

261

140

0.418

0.0196

0.479

-12.75%

30

314

140

0.586

0.0195

0.687

-14.67%

32

335

140

0.66

0.0194

0.777

-15.05%

35

366

140

0.779

0.0193

0.925

-15.75%

40

418

140

0.998

0.0192

1.202

-17.00%

 

 

 

 

 

 

 

1" Schedule 40 Steel Pipe (1.049 inch)

 

 

 

 

Velocity

Q

C

Loss/ft

Fric Factor

Loss/ft

Difference

(ft/s)

(gpm)

HW

HW

DW

DW

HW/DW

10

27

140

0.17

0.0248

0.191

-11.22%

15

40

140

0.359

0.024

0.416

-13.67%

20

54

140

0.612

0.0238

0.734

-16.62%

25

67

140

0.925

0.0235

1.132

-18.34%

30

81

140

1.296

0.0232

1.61

-19.51%

32

86

140

1.46

0.0232

1.831

-20.29%

35

94

140

1.723

0.0231

2.181

-21.01%

40

108

140

2.206

0.0229

2.825

-21.90%

 

 

 

 

 

 

 

4" Type M Copper Tubing (3.935 inch)

 

 

 

 

Velocity

Q

C

Loss/ft

Fric Factor

Loss/ft

Difference

(ft/s)

(gpm)

HW

HW

DW

DW

HW/DW

10

379

150

0.032

0.015

0.031

3.19%

15

569

150

0.067

0.014

0.065

4.04%

20

758

150

0.115

0.0134

0.11

4.11%

25

948

150

0.173

1.0128

0.164

5.40%

30

1137

150

0.243

0.0124

0.229

5.87%

32

1213

150

0.274

0.0123

0.259

5.70%

35

1327

150

0.323

0.0121

0.305

6.01%

40

1516

150

0.413

0.0117

0.385

7.46%

 

 

 

 

 

 

 

1" Type M Copper Tubing (1.055 inch)

 

 

 

 

Velocity

Q

C

Loss/ft

Fric Factor

Loss/ft

Difference

(ft/s)

(gpm)

HW

HW

DW

DW

HW/DW

10

27

150

0.148

0.0195

0.149

-0.71%

15

41

150

0.314

0.018

0.31

1.22%

20

54

150

0.535

0.017

0.521

2.64%

25

68

150

0.808

0.0165

0.79

2.27%

30

82

150

1.133

0.0158

1.09

3.92%

32

87

150

1.276

0.0157

1.232

3.58%

35

95

150

1.507

0.0153

1.437

4.87%

40

109

150

1.929

0.015

1.84

4.84%

 

image

 

 

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