## Conditions Required for Compressible Pipe Flow Calculator Spreadsheet

With air flow in a pipe having a frictional pressure drop less than 20% of the inlet air pressure, the Darcy-Weisbach equation will give useable results with incompressible flow equations.  For background information on incompressible pipe flow calculations, see the article, “Pipe Flow-Friction Factor Calculations with Excel Spreadsheets.”  If the frictional pressure drop is more than 20% of the incoming air pressure, then Fanno Flow equations should be used, as in the compressible pipe flow calculator spreadsheet being discussed in this article.

## Fanno Flow Equations for a Compressible Pipe Flow Calculator Spreadsheet

The equations shown below are those used in the compressible pipe flow calculator spreadsheet being discussed here.  These equations are for compressible pipe flow with heat transfer with the surroundings being neglected, but effects of friction in the flow being included.

The parameters in these fanno flow equations are as follows:

• f is the Moody friction factor
• L is the pipe length
• D is the pipe diameter
• k is the adiabatic constant for the flowing air
• M is the mach number
• T is the air temperature
• P is the air pressure
• ρ is the air density

Parameters with an asterisk refer to conditions at the “choke point” in the pipe, where the Mach number is 1.

## Screenshot for a Compressible Pipe Flow Calculator Spreadsheet

The screenshot below shows part of a compressible pipe flow calculator spreadsheet.  This spreadsheet makes fanno flow calculations  for compressible air flow in a pipe, in S.I. units. This compressible pipe flow calculator spreadsheet and others for fanno flow, compressible pipe flow calculations are available in either U.S. or S.I. units at a very reasonable cost  in our spreadsheet store.

## Where to Find Storm Sewer Hydraulic Design Spreadsheets

One part of storm sewer hydraulic design is determination of the design pipe diameter and sewer slope for the storm sewer pipe between adjacent manholes.  Although storm sewers are circular pipes, the storm water typically flows under gravity, rather than as pressure flow, so the Manning equation for open channel flow can be used for the calculations.  A storm sewer hydraulic design spreadsheet typically makes hydraulic calculations for full pipe flow.  For full pipe flow, the hydraulic radius becomes: R = A/P = (πD2/4)/(πD) = D/4.

## The Manning Equation in a Storm Sewer Hydraulic Design Spreadsheet

The general form of the Manning equation in terms of velocity is: V = (1.49/n)(R2/3)(S1/2) for U.S. units and  V = (1.0/n)(R2/3)(S1/2) for S.I. units.  As noted above, R = D/4 for full pipe flow, so the Manning equation in U.S. units becomes  V = (1.49/n)[(D/4)2/3](S1/2) -U.S. units or V = (1.0/n)[(D/4)2/3](S1/2) – S.I units, for full pipe, gravity flow in a storm sewer pipe.  The parameters in the equations are as follows:

• V is the flow velocity in the pipe (ft/sec – U.S. and m/s – S.I.).
• n is the Manning roughness coefficient, an empirical, dimensionless constant.
• D is the pipe diameter (ft -U.S. and m – S.I.).
• S is the pipe slope, which is dimensionless.

The volumetric flow rate is related to the other parameters through the equation Q = VA or, for a circular pipe flowing full:  Q = (πD2/4)V, where Q will be in cfs for U.S. units or m3/s for S.I. units.

## Calculation of Diameter and Slope with a Storm Sewer Hydraulic Design Spreadsheet

The required diameter and slope for the length of storm sewer between two manholes can be calculated with a storm hydraulic sewer design spreadsheet using the equations presented in the last section (Mannings equation and Q = VA) together with the typical design criteria that 1) the full pipe flow rate that the pipe can carry must be at least equal to the design peak storm water runoff rate to the inlet for that section of storm sewer and 2) the full pipe velocity must be equal to or greater than a specified minimum velocity.  The diagram above shows a sectional view of a storm sewer pipe between two manholes and the parameters being discussed here. The calculation procedure is illustrated by the example in the next section.

## Example Storm Sewer Hydraulic Design Calculations

Problem Statement: For a section of storm sewer between two manholes, the design flowrate is: Qdes = 6.4 cfs. The required minimum full pipe storm water velocity is: V min= 3 ft/sec.  The Manning roughness coefficient (concrete pipe) is: n = 0.011.  Find a standard pipe diameter and sewer slope that will meet the two criteria: Qfull > Qdes and Vfull > Vmin for this section of storm sewer pipe.

Problem Solution: First the pipe diameter needed for a full pipe velocity of 3 ft/sec at design flow rate will be calculated using the equation: Q = VA.   Then the Manning equation will be used to calculate the sewer slope needed to give full pipe velocity equal to 3 ft/sec with the next larger standard pipe size.

Step 1:  The equation, Q = VA becomes: Qfull = Vfull(πD2/4). Substituting known values for Qfull and Vfull, the equation becomes: 6.4 = 3(πD2/4).  Solving for D gives: D = 1.65 ft = 19.8 in.  From the list of standard storm sewer pipe sizes in the next section it can be seen that the next standard size larger than 19.8 inches is 21 “, so that will be used for the diameter.

The Manning equation will then be used to calculate the slope for D = 21 in. = 1.75 ft, and V = 3 ft/sec. The Manning equation is: V = (1.49/n)[(D/4)2/3](S1/2).  Substituting values for V, D, and n gives:  3 = (1.49/0.011)[(1.75/4)2/3](S1/2).  Solving this equation for S gives: S = 0.00148.

Thus, the solution is: D = 21″, S = 0.00148. These values of D and S will give Qfull > 6.4 cfs, because Qfull = 6.4 cfs for Vfull = 3 ft/sec and D = 19.8″. With D = 21 ” and V = 3 ft/sec, Qfull must be greater than 6.4 cfs. The equation Q = (πD2/4)V can be used to check this.

Standard Pipe Sizes

Standard U.S. pipe sizes in inches for most types of pipe used as storm sewers:                          4, 6, 8, 10, 12, 14, 16, 18, 21, 24, 27, 30, 33, 36, 39, 42, 48, 54, 60

Standard S.I. pipe sizes in mm for most types of pipe used as storm sewers:                           100, 150, 200, 250, 300, 350, 400, 450, 500, 600, 650, 700, 750, 800, 850, 900, 950, 1000, 1050

## Use of Excel Spreadsheets for Storm Sewer Design Calculations

For information on making storm sewer calculations with Excel spreadsheets, see the related article: “Excel Spreadsheets for Storm Sewer Hydraulic Design.”  For low cost, easy to use spreadsheets for several types of storm water calculations, including storm sewer hydraulic design, click here to visit our spreadsheet store.

References

1. Bengtson, Harlan H., Hydraulic Design of Storm Sewers, Including the Use of Excel, an online, continuing education course for PDH credit.

2. McCuen, Richard H., Hydrologic Analysis and Design, 2nd Ed, Upper Saddle River, NJ, Prentice Hall, 1998.

3. Steele, E.W. and McGhee, T.J., Water Supply and Sewerage, New York, NY, McGraw-Hill Book Co, 1979.

## Where to find a Storm Sewer Design Spreadsheet

The storm sewer design spreadsheet discussed in this article uses Excel formulas with the rational method to find design storm water runoff rate and the Manning equation to find pipe diameter and slope.

The hydraulic portion of stormwater sewer design proceeds in the form of calculations between each pair of manholes in the storm sewer line. The first part of the spreadsheet is essentially a rational method design spreadsheet used to determine the design stormwater runoff flow rate for each section of storm sewer being designed. The next part of the spreadsheet is used to calculate the pipe diameter and slope for each section of storm sewer with the Manning Equation. Finally, the pipe invert elevation at each manhole is calculated in the last part of the spreadsheet.  Each part of the storm sewer design spreadsheet will be discussed briefly in the next several sections, followed by presentation and discussion of an Excel spreadsheet template to make the calculations.

## Peak Storm Water Runoff Rate for Storm Sewer Design Spreadsheet

The rational method equation (Q = CiA for U.S. units and Q = 0.0028 CiA for S.I. units) is widely used to calculate the design stormwater runoff rate to use for a variety of storm water projects, including storm water sewer design.  The parameters in the rational method equations are:

• Q, the design storm water runoff rate (cfs – U.S. and m3/s – S.I.)
• C, the runoff coefficient, which is an estimate of the fraction of rainfall that becomes surface runoff (dimensionless)
• i, the design rainfall intensity (in/hr – U.S. and mm/hr – S.I.)
• A, the runoff area that drains to the section of sewer pipe being designed (acres – U.S. and ha – S.I.)

The storm sewer design spreadsheet being discussed here will assume that the manhole locations have already been determined, as shown in the diagram at the right.  A street map like this would be used to determine the area draining to each of the manhole inlets for the length of storm sewer being designed.

## Criteria Used in Storm Sewer Design Spreadsheet

Following are the criteria typically used to calculate the design pipe diameter and sewer slope for a length of sewer pipe:

1. The pipe must be sized to carry the design peak stormwater runoff rate.
2. The velocity in the sewer pipe must be greater than or equal to the design minimum velocity (usually 3 ft/s).

The use of these design criteria, together with the Manning equation

[ Q = (1.49/n)(A)(R2/3)(S1/2) ]  and Q = VA, to calculate the pipe diameter and slope is discussed and illustrated with an example in the article, “Storm Sewer Hydraulic Calculations with the Manning Equation.”  The procedure is also illustrated in the spreadsheet template presented later in this article.

## Invert Elevations at Manholes in the Storm Sewer Design Spreadsheet

The sewer pipe invert elevation (or depth) at the uppermost manhole is determined by the minimum required depth of cover above the sewer pipe to protect it from freezing. This required minimum cover is usually specified by a state or local agency.  For subsequent manholes, the required minimum cover, the required pipe slope, and the ground surface elevations from a street/manhole map like that shown in a previous section above, are used to calculate the pipe invert elevations.  Calculation of the invert elevations at manholes with a storm sewer design spreadsheet is presented in the next section.

## Putting it together in a Storm Sewer Design Spreadsheet

The storm sewer design spreadsheet template shown in the two images below contains design calculations for a storm sewer line along one of the streets on the manhole layout map shown above in the second section of this article.  The spreadsheet makes the calculations described above.  The various parts of the spreadsheet will now be discussed briefly with reference to the column numbers given on the spreadsheet.

Columns 1, 2, and 3 contain information from a scale street/manhole map, such as the one shown earlier in this article. Column 4 is the calculated cumulative area draining to downstream sections of storm sewer pipe. The uppermost part of the sewer line in this example is the manhole at 8th Street and Maple Avenue.  An estimate of the runoff coefficient is given in column 5.  Column 6 shows the inlet time from the farthest point in the drainage area. For the uppermost section of sewer pipe, the inlet time is equal to the time of concentration.  For the other sections of sewer pipe in the line being designed, the time of concentration is the inlet time to the first inlet plus the pipe flow time to the inlet of the pipe section currently being designed.  This is calculated in column 7.

Column 8 is the calculated design rainfall intensity. The portion of the Excel template shown at the right below has the Excel formulas for derivation of an equation for storm intensity vs storm duration for a given return period, using linear regression of storm duration, δ, vs the inverse of storm intensity, 1/i.   This requires at least some values of i vs δ, from  I-D-F data for the location of interest.  This linear regression makes use of the fact that the relationship between i and δ is typically of the form i = a/(δ + b), where a and b are constants. Column 9 is the calculation of peak storm water runoff rate (the design flow rate) with the rational method equation:  Q = CiA.

Columns 10 through 15 use of the Manning Equation and Q = VA to determine the minimum standard pipe diameter and sewer slope needed, as well as to make a check on Vfull and Qfull when the pipe is receiving the design stormwater runoff flow rate.  This set of calculations is discussed in some detail in the article: Storm Sewer Hydraulic Calculations with the Manning Equation.”

Columns 16 and 17 are used to calculate the pipe flow time to be used for the time of concentration calculation in column 7. Columns 18 and 19 give ground surface elevations taken from the manhole layout map. Columns 20 and 21 calculate the pipe invert elevations. The invert elevation of the uppermost end of the pipe is taken to be the surface elevation minus the minimum cover (taken to be 5′ ) plus the pipe diameter. The invert elevation at the lower end of the pipe section is calculated using the sewer slope that was previously determined. Columns 22 and 23 are a check on the depth of cover at each manhole, and column 24 is a listing of the final design pipe slope.

References

1. Bengtson, Harlan H., Hydraulic Design of Storm Sewers, Including the Use of Excel, an online, continuing education course for PDH credit.

2. McCuen, Richard H., Hydrologic Analysis and Design, 2nd Ed, Upper Saddle River, NJ, 1998.

3. Knox County Tennessee, Stormwater Management Manual, section on the Rational Method

## Where to Find Storm Sewer Design Spreadsheets

One part of storm sewer hydraulic design is determination of the design diameter and slope for the storm sewer pipe between adjacent manholes.  Although storm sewers are circular pipes, the storm water typically flows under gravity, rather than as pressure flow, so the Manning equation for open channel flow can be used for the calculations.  A storm sewer design spreadsheet typically makes hydraulic calculations for full pipe flow.  For full pipe flow, the hydraulic radius becomes: R = A/P = (πD2/4)/(πD) = D/4.

## The Manning Equation in a Storm Sewer Design Spreadsheet

The general form of the Manning equation in terms of velocity is: V = (1.49/n)(R2/3)(S1/2) for U.S. units and  V = (1.0/n)(R2/3)(S1/2) for S.I. units.  As noted above, R = D/4 for full pipe flow, so the Manning equation in U.S. units becomes  V = (1.49/n)[(D/4)2/3](S1/2) -U.S. units or V = (1.0/n)[(D/4)2/3](S1/2) – S.I units, for full pipe, gravity flow in a storm sewer pipe.  The parameters in the equations are as follows:

• V is the flow velocity in the pipe (ft/sec – U.S. and m/s – S.I.).
• n is the Manning roughness coefficient, an empirical, dimensionless constant.
• D is the pipe diameter (ft -U.S. and m – S.I.).
• S is the pipe slope, which is dimensionless.

The volumetric flow rate is related to the other parameters through the equation Q = VA or, for a circular pipe flowing full:  Q = (πD2/4)V, where Q will be in cfs for U.S. units or m3/s for S.I. units.

## Calculation of Diameter and Slope with a Storm Sewer Design Spreadsheet

The required diameter and slope for the length of storm sewer between two manholes can be calculated with a storm sewer design spreadsheet using the equations presented in the last section (Mannings equation and Q = VA) together with the typical design criteria that 1) the full pipe flow rate that the pipe can carry must be at least equal to the design peak storm water runoff rate to the inlet for that section of storm sewer and 2) the full pipe velocity must be equal to or greater than a specified minimum velocity.  The diagram above shows a sectional view of a storm sewer pipe between two manholes and the parameters being discussed here. The calculation procedure is illustrated by the example in the next section.

## Example Storm Sewer Design Spreadsheet Calculations

Problem Statement: For a section of storm sewer between two manholes, the design flowrate is: Qdes = 6.4 cfs. The required minimum full pipe storm water velocity is: V min= 3 ft/sec.  The pipe roughness coefficient (concrete pipe) is: n = 0.011.  Find a standard pipe diameter and pipe slope that will meet the two criteria: Qfull > Qdes and Vfull > Vminfor this section of storm sewer pipe.

Problem Solution: First the pipe diameter needed for a full pipe velocity of 3 ft/sec at design flow rate will be calculated using the equation: Q = VA.   Then the Manning equation will be used to calculate the pipe slope needed to give full pipe velocity equal to 3 ft/sec with the next larger standard pipe size.

Step 1:  The equation, Q = VA becomes: Qfull = Vfull(πD2/4). Substituting known values for Qfull and Vfull, the equation becomes: 6.4 = 3(πD2/4).  Solving for D gives: D = 1.65 ft = 19.8 in.  From the list of standard storm sewer pipe sizes in the next section it can be seen that the next standard size larger than 19.8 inches is 21 “, so that will be used for the diameter.

The Manning equation will then be used to calculate the slope for D = 21 in. = 1.75 ft, and V = 3 ft/sec. The Manning equation is: V = (1.49/n)[(D/4)2/3](S1/2).  Substituting values for V, D, and n gives:  3 = (1.49/0.011)[(1.75/4)2/3](S1/2).  Solving this equation for S gives: S = 0.00148.

Thus, the solution is: D = 21″, S = 0.00148. These values of D and S will give Qfull > 6.4 cfs, because Qfull = 6.4 cfs for Vfull = 3 ft/sec and D = 19.8″. With D = 21 ” and V = 3 ft/sec, Qfull must be greater than 6.4 cfs. The equation Q = (πD2/4)V can be used to check this.

Standard Pipe Sizes

Standard U.S. pipe sizes in inches for most types of pipe used as storm sewers:                        4, 6, 8, 10, 12, 14, 16, 18, 21, 24, 27, 30, 33, 36, 39, 42, 48, 54, 60

Standard S.I. pipe sizes in mm for most types of pipe used as storm sewers:                    100, 150, 200, 250, 300, 350, 400, 450, 500, 600, 650, 700, 750, 800, 850, 900, 950, 1000, 1050

## Use of Excel Spreadsheets for Storm Sewer Design Calculations

For information on making storm sewer calculations with Excel spreadsheets, see the related article: “Excel Spreadsheets for Storm Sewer Hydraulic Design.”  For low cost, easy to use spreadsheets for several types of storm water calculations, including storm sewer hydraulic design, click here to visit our spreadsheet store.

References

1. Bengtson, Harlan H., Hydraulic Design of Storm Sewers, Including the Use of Excel, an online, continuing education course for PDH credit.

2. McCuen, Richard H., Hydrologic Analysis and Design, 2nd Ed, Upper Saddle River, NJ, Prentice Hall, 1998.

3. Steele, E.W. and McGhee, T.J., Water Supply and Sewerage, New York, NY, McGraw-Hill Book Co, 1979.

## Inlet Control and Outlet Control for a Pipe Culvert Design Spreadsheet

One of the general conditions for pipe culvert design calculations is inlet control, in which the flow rate through the culvert is controlled at the inlet end of the culvert by the culvert diameter and other inlet conditions.  The other general condition is outlet control, in which the flow rate is controlled by the outlet conditions and the entire length of the culvert.

## Pipe Culvert Inlet Control Design Spreadsheet Calculations

An equation that relates culvert parameters for inlet control conditions in a pipe culvert design spreadsheet is:

where:

• HW = headwater depth above inlet invert (ft – U.S. or m – S.I.)
• D = inside height of the culvert (ft – U.S. or m – S.I.)
• Q = discharge (cfs – U.S. or m3/s – S.I.)
• A = cross-sectional area of culvert (ft2 – U.S. or m2 – S.I.)
• S = culvert slope (dimensionless)
• K1 = 1.0 for U.S. units or 1.811 for S.I. units
• Ks = slope constant = -0.5 for a non-mitered or + 0.7 for a mitered inlet
• Y and c are constants dependent on the type of culvert and type of inlet.

## Pipe Culvert Outlet Control Design Calculations

An equation that relates culvert parameters for outlet control conditions in a pipe culvert design spreadsheet is:

Where:

• hL = the head loss in the culvert barrel for full pipe flow (ft – U.S. or m – S.I.)
• Ku = 29 for U.S. units or 19.63 for S.I. units
• n = Manning roughness coefficient for the culvert material
• L = length of the culvert barrel (ft – U.S. or m – S.I.)
• R = hydraulic radius of the full culvert barrel = A/P (ft – U.S. or m – S.I.)
• A = cross-sectional area of the culvert barrel (ft2 – U.S. or m2 – S.I.)
• P = perimeter of the culvert barrel, ft or m
• V = velocity in the culvert barrel, ft/sec or m/s
• Ke = loss coefficient for pipe entrance

## A spreadsheet screenshot for pipe culvert design calculations

The Excel spreadsheet screenshot below shows part of a spreadsheet for circular culvert design calculations based on inlet control.   Based on the indicated input values, the spreadsheet will calculate the minimum required pipe culvert diameter and the headwater depth for the next larger standard culvert diameter.

Reference

1.  Hydraulic Design of Highway Culverts,Third Edition,  Publication No. FHWA-HIF-12-026, U.S. DOT/Federal Highway Administration, April, 2012.

## Introduction to Partially Full Pipe Flow Calculations

The Manning equation can be used for flow in a pipe that is partially full, because the flow will be due to gravity rather than pressure.  the Manning equation [Q = (1.49/n)A(R2/3)(S1/2) for (U.S. units) or Q = (1.0/n)A(R2/3)(S1/2) for (S.I. units)] applies if the flow is uniform flow.  For background on the Manning equation, open channel flow and the conditions for uniform flow, see the article, “Manning Equation/Open Channel Flow Calculations with Excel Spreadsheets.

Direct use of the Manning equation as a partially full pipe flow calculator, isn’t easy.  This is because of the rather complicated set of equations for the area of flow and wetted perimeter for partially full pipe flow.  There is no simple equation for hydraulic radius as a function of flow depth and pipe diameter.  As a result graphs of Q/Qfull and V/Vfull vs y/D, like the one shown at the left are commonly used for partially full pipe flow calculations.  The parameters, Q and V in this graph are flow rate an velocity at a flow depth of y in a pipe of diameter D.  Qfull and Vfull can be conveniently calculated using the Manning equation, because the hydraulic radius for a circular pipe flowing full is simply D/4.

With the use of Excel formulas in an Excel spreadsheet, however, the rather inconvenient equations for area and wetted perimeter in partially full pipe flow become much easier to work with.  The calculations are complicated a bit by the need to consider the Manning roughness coefficient to be variable with depth of flow as discussed in the next section.

## Is the Manning Roughness Coefficient Variable for Partially Full Pipe Flow?

Using the geometric/trigonometric equations discussed in the next couple of sections, it is relatively easy to calculate the cross-sectional area, wetted perimeter, and hydraulic radius for partially full pipe flow  with any specified pipe diameter and depth of flow.  If the pipe slope and Manning roughness coefficient are known, then it should be easy to calculate flow rate and velocity for the given depth of flow using the Manning Equation                             [Q = (1.49/n)A(R2/3)(S1/2)], right?   No, wrong!  As long ago as the middle of the twentieth century, it had been observed that measured flow rates in partially full pipe flow aren’t the same as those calculated as just described.  In a 1946 journal article (ref #3 below), T. R. Camp presented a method for improving the agreement between measured and calculated values for partially full pipe flow.  The method developed by Camp consisted of using a variation in Manning roughness coefficient with depth of flow as shown in the graph above.

Although this variation in Manning roughness due to depth of flow doesn’t make sense intuitively, it does work.  It is well to keep in mind that the Manning equation is an empirical equation, derived by correlating experimental results, rather than being theoretically derived.  The Manning equation was developed for flow in open channels with rectangular, trapezoidal, and similar cross-sections.  It works very well for those applications using a constant value for the Manning roughness coefficient, n.  Better agreement with experimental measurements is obtained for partially full pipe flow, however, by using the variation in Manning roughness coefficient developed by Camp and shown in the diagram above.

The graph developed by Camp and shown above appears in several publications of the American Society of Civil Engineers, the Water Pollution Control Federation, and the Water Environment Federation from 1969 through 1992, as well as in many environmental engineering textbooks (see reference list at the end of this article).  You should beware, however that there are several online calculators and websites with equations for making partially full pipe flow calculations using the Manning equation with constant Manning roughness coefficient, n.  The equations and Excel spreadsheets presented and discussed in this article use the variation in n that was developed by T.R. Camp.

## Excel Spreadsheet/Partially Full Pipe Flow Calculator for Pipe Less than Half Full

The parameters used in partially full pipe flow calculations with the pipe less than half full are shown in the diagram at the right.  K is the circular segment area; S is the circular segment arc length; h is the circular segment height; r is the radius of the pipe; and θ is the central angle.

The equations below are those used, together with the Manning equation and Q = VA, in the partially full pipe flow calculator (Excel spreadsheet) for flow depth less than pipe radius, as shown below.

• h = y
• θ = 2 arccos[ (r - h)/r ]
• A = K = r2(θ – sinθ)/2
• P = S = rθ

The equations to calculate n/nfull, in terms of y/D for y < D/2 are as follows

• n/nfull = 1 + (y/D)(1/3) for 0 < y/D < 0.03
• n/nfull = 1.1 + (y/D – 0.03)(12/7) for 0.03 < y/D < 0.1
• n/nfull = 1.22 + (y/D – 0.1)(0.6) for 0.1 < y/D < 0.2
• n/nfull = 1.29 for 0.2 < y/D < 0.3
• n/nfull = 1.29 – (y/D – 0.3)(0.2) for 0.3 < y/D < 0.5

The Excel template shown below can be used as a partially full pipe flow calculator to calculate the pipe flow rate, Q, and velocity, V, for specified values of pipe diameter, D, flow depth, y, Manning roughness for full pipe flow, nfull; and bottom slope, S, for cases where the depth of flow is less than the pipe radius.  This Excel spreadsheet and others for partially full pipe flow calculations are available in either U.S. or S.I. units at a very low cost in our spreadsheet store.

## Other Partially Full Pipe Flow Calculations

Spreadsheets for a variety of other partially full pipe flow calculations, including calculations for more than half full pipe flow, normal depth, and required diameter, are available at very low cost in our spreadsheet store.

References

1. Bengtson, Harlan H., Partially Full Pipe Flow Calculations with Spreadsheets, an e-book available at Amazon.com.

2. Bengtson, Harlan H.,  Spreadsheet Use For Partially Full Pipe Flow Calculations, an online, continuing education course for PDH credit.

3. Camp, T.R., “Design of Sewers to Facilitate Flow,” Sewage Works Journal, 18 (3), 1946

4. Chow, V. T., Open Channel Hydraulics, New York: McGraw-Hill, 1959.

5. Steel, E.W. & McGhee, T.J., Water Supply and Sewerage, 5th Ed., New York, McGraw-Hill Book Company, 1979

6.  ASCE, 1969. Design and Construction of Sanitary and Storm Sewers, NY

## Venturi Meter Equation

Venturi meters function by sending pipe flow through a constricted area (the venturi throat), as shown in the diagram at the right.  Due to the increased fluid velocity passing through the constriction, there will be a decreased pressure at that location.   The pipe flow rate can then be calculated from the measured pressure difference between the undisturbed pipe flow and the flow through the constriction.

A general equation for calculating flow rate through a venturi meter is shown at the left.  The parameters in the equation and their units are as shown below:

• Q is the flow rate through the pipe and through the meter  (cfs – U.S. or m3/s – S.I.)
• C is the discharge coefficient, which is dimensionless
• A2 is the constricted area perpendicular to flow  (ft2 – U.S. or m2 – S.I.)
• P1 is the undisturbed upstream pressure in the pipe  (lb/ft2 – U.S. or N/m2 – S.I.)
• P2 is the pressure in the pipe at the constricted area, Ao (lb/ft2 – U.S. or N/m2 – S.I.)
• β = d/D = (diam. at A2/pipe diam.), which is dimensionless
• ρ is the fluid density (slugs/ft3 – U.S. or kg/m3 – S.I.)

## Venturi Meter ISO 5167 Calculations

ISO 5167-4: 2003 provides discharge coefficient values for three venturi meter variations, subject to the venturi meeting a set of specifications and guidelines given in the publication.  The three venturi meter variations are i) “as cast” convergent section, ii) “machined” convergent section, and iii) “rough welded sheet iron” convergent section.  For each of these three variations, ISO 5167-4:2003 specifies a range for pipe diameter, diameter ratio (d/D), and Reynolds number in the pipe.

Excel Spreadsheets for Venturi Meter ISO 5167 Flow Rate Calculation

The image below shows part of an Excel spreadsheet that can be used for ISO 5167 venturi meter calculations.  For this spreadsheet and other low cost, easy to use spreadsheets for gas flow or liquid flow venturi meter calculations in S.I. or U.S. units, click here to visit our spreadsheet store.

References

1. Bengtson, H. H., Flow Measurement in Pipes and Ducts, an online continuing education course.

2. Munson, B. R., Young, D. F., & Okiishi, T. H., Fundamentals of Fluid Mechanics, 4th Ed., New York: John Wiley and Sons, Inc, 2002.

3. U.S. Dept. of the Interior, Bureau of Reclamation, 2001 revised, 1997 third edition, Water Measurement Manual, available for on-line use or download at: http://www.usbr.gov/pmts/hydraulics_lab/pubs/wmm/index.htm

4. International Organization of Standards -Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full. Part 4, Reference number: ISO 5167-4:2003

## Where to Find Spreadsheets to Calculate Water Flow Rates for Pipe Sizes

For Excel spreadsheets make pipe flow pressure drop calculations or to calculate water flow rates for pipe sizes with the Hazen Williams equation, click here to visit our spreadsheet store.  Read on for information about the Hazen Williams equation and its use to calculate water flow rates for pipe sizes and different lengths.

Excel spreadsheets are very convenient to calculate water flow rates for pipe sizes with the Hazen Williams equation.  Both U.S. and S.I. units will be used in the Hazen Williams spreadsheets and calculations discussed in this article.

## Limitations of the Hazen Williams Equation to Calculate Water Flow Rates for Pipe Sizes

The Hazen Williams equation is intended for turbulent water flow rates in pipes at normal ambient temperatures.  The turbulent flow requirement isn’t typically a problem, because most practical transport of water in pipes is in the turbulent flow regime.  The Hazen Williams formula works best for water temperature that isn’t too far above or below 60oF.  For calculation of flow rates for pipes with a fluid other than water or for water at a temperature that is far above or far below 60oF, the Darcy Weisbach equation is an alternative to the Hazen Williams equation.  For more information about this alternative, see the post, “Friction Factor/Pipe Flow Calculations with Excel Spreadsheets.”

## Forms of the Hazen Williams Equation to Calculate Water Flow Rates for Pipe Sizes

The Hazen Williams equation is sometimes expressed as an equation for velocity in the pipe and sometimes as an equation for pipe flow rate.  It also can be expressed in terms of the pipe head loss or the frictional pressure drop.  Finally, the Hazen Williams equation can be written to use either U.S. or S.I. units.

As an equation for Velocity, the traditional form of the Hazen Williams equation, is:

in U.S. units: V = 1.318 C R0.633 S0.54, where:

• V is the water flow velocity, ft/sec
• C is the Hazen Williams coefficient, dimensionless (depends on pipe material and age)
• R is the hydraulic radius, ft (R = cross-sectional area of flow/wetted perimeter)
• S is the slope of the energy grade line, dimensionless (S = head loss/pipe length = hL/L)

in S.I. units: V = 0.85 C R0.633 S0.54, where the parameters are as defined above with V in m/s and R in meters.

As an equation for water flow rate in a circular pipe, the hydraulic radius is R = A/P = (πD2/4)/(πD)  = D/4 and  Q = VA = V(πD2/4).  Substituting these equations into those for velocity give the following form of the Hazen Williams equation:

in U.S. units: Q = 193.7 C D2.63 S0.54, where:

• Q is the water flow rate in the pipe, gal/min (gpm)
• D is the pipe diameter, ft
• C and S are the same as defined above

in S.I. units: Q = 0.278 C D2.63 S0.54, where the parameters are as defined above with Q in m3/s and D in meters.

In terms of frictional pressure drop, ΔP instead of frictional head loss, hL, the Hazen Williams equation is:

in U.S. units: Q = 0.442 C D2.63 (ΔP/L)0.54, where

• Q is the water flow rate in the pipe, gpm,
• D is the pipe diameter, inches
• L is the pipe length, ft
• ΔP is the pressure difference across pipe length L, psi

In S.I. units: Q = (3.763 x 10-6) C D2.63 (ΔP/L)0.54, where

• Q is the water flow rate in the pipe, m3/hr,
• D is the pipe diameter, mm
• L is the pipe length, m
• ΔP is the pressure difference across pipe length L,  kN/m2

# The Hazen Williams Coefficient – C

The Hazen Williams equation can be used to calculate water flow rates for pipe sizes, only if values of the Hazen Williams coefficient, C, can be obtained for the pipe materials in use. Values of C can be found on internet sites and in handbooks & textbooks. The table at the left shows C values for some commonly used pipe materials.

Source: Toro Ag Irrigation

# A Table of Values of Flow Rates for Pipe Sizes and Lengths

The tables below were prepared using the equations: Q = 0.442 C D2.63 (ΔP/L)0.54(U.S.) and  Q = 0.278 C D2.63 (ΔP/L)0.54 (S.I.) with units as given above, to calculate the water flow rates for PVC pipe with diameters from 1/2 inch to 6 inches (1 mm to 30 mm) and length from 5 ft to 100 ft (12 m to 150 m), all for a pressure difference of 20 psi (140 kn/m2) across the particular length of pipe. The Hazen Williams coefficient was taken to be 150 per the table in the previous section.

# Excel Spreadsheets to Calculate Water Flow Rates for Pipe Sizes

The table shown above, showing water flow rates for pipe sizes, can be calculated with an Excel spreadsheet like the one shown below.  It has Excel formulas entered to calculate water flow rates for pipe sizes using the Hazen Williams equation.  This spreadsheet allows for entering the Hazen Williams coefficient for the proper pipe material, the pressure drop, and the pipe diameter(s) and length(s) of interest.  The Excel formulas then calculate water flow rates for the entered pipe sizes and lengths.  The spreadsheet shown below uses S.I. units.  This spreadsheet and a similar one using U.S. units is available at a very reasonable price from our spreadsheet store.  Those spreadsheets are also set up to calculate pipe diameter, length, or pressure drop if the other parameters are known.

References

1. Bengtson, H., Fundamentals of Fluid Flow, An online continuing education course for PDH credit.

2. Munson, B. R., Young, D. F., & Okiishi, T. H., Fundamentals of Fluid Mechanics, 4th Ed., New York: John Wiley and Sons, Inc, 2002.

3. Liou, C.P., “Limitations and Proper Use of the Hazen-Williams Equation,” Journal of Hydraulic Engineering, Vol. 124, No. 9, Sept. 1998, pp. 951-954.

## Where to Get Spreadsheets for Frictional Pressure Drop in Pipe Calculations

Excel spreadsheets are very convenient for Darcy Weisbach equation/pipe flow calculations, such as frictional pressure drop in pipe flow or use as a friction factor calculator. This is true, at least in part, because some of the calculations require iterative solutions.  The Darcy Weisbach equation is applicable to pressure flow in pipes, rather than gravity flow (as in sewer pipes), which is handled by open channel flow equations like the Manning equation.  The Darcy Weisbach equation provides the relationship among the following parameters: pipe diameter and length, pipe flow rate, and  frictional pressure drop or head loss.  Any one of these can be calculated if the others are known along with the density and viscosity of the fluid.

## A Friction Factor Calculator and the Darcy Weisbach Equation

The Darcy Weisbach equation  is ΔPf = ρf(L/D)(V2/2), with the parameters in the equation as follows: ΔPf is the frictional pressure drop in pipe for flow of a fluid at average velocity, V, through a pipe of length, L, and diameter, D.  The Reynolds number for the flow (Re) and the relative roughness of the pipe (e/D) are needed to get a value for the friction factor, f.  The Moody Diagram at the right shows the nature of the dependence of the friction factor, f,  on Re and e/D.

Equations for f as a function of Re and e/D would be more convenient than a graph like the Moody Diagram for use with Excel spreadsheets making  pipe flow calculations with the Darcy Weisbach equation.  Such equations are shown in the box at the left, giving the relationships between Moody friction factor and Re & e/D for four different portions of the Moody diagram.  The four portions of the Moody diagram are:

• laminar flow (Re < 2100 – the straight line at the left side of the Moody
• smooth pipe turbulent flow (the dark curve labeled “smooth pipe” in the Moody diagram – f is a function of Re only in this region)
• completely turbulent region (the portion of the diagram above and to the right of the dashed line labeled “complete turbulence” – f is a function of e/D only in this region)
• transition region (the portion of the diagram between the “smooth pipe” solid line and the “complete turbulence” dashed line – f is a function of both Re and e/D in this region and this is not an explicit equation for f)

The table above right gives pipe roughness values for several common pipe materials.  These can be used to calculate the pipe roughness ratio, e/D.

## A Spreadsheet for Frictional Pressure Drop in Pipe Flow Calculations

For a low cost Moody friction factor calculator download, that will calculate f for Reynolds number above 2100, see: www.engineeringexceltemplates.com.  Also find other spreadsheets for frictional pressure drop in pipe flow calculation,

The Excel spreadsheet screenshot below shows a friction factor calculator spreadsheet available  at our spreadsheet store in either U.S. or S.I. units at a very low cost (only \$6.95).

References

1.  Munson, B. R., Young, D. F., & Okiishi, T. H., Fundamentals of Fluid Mechanics, 4th Ed., New York: John Wiley and Sons, Inc, 2002.

2. Darcy Weisbach equation history – http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm

3. Source for pipe roughness values – http://www.efunda.com/formulae/fluids/roughness.cfm

4. Bengtson, H.H., Pipe Flow/Friction Factor Calculations with Excel, an online continuing education course for Professional Engineers.

5. Bengtson, H.H., Advantages of Spreadsheets for Pipe Flow/Friction Factor Calculations, an Amazon.com e-book.