About Harlan Bengtson

Dr. Bengtson has B.S. and M.S. degrees from Iowa State University and a PhD from the University of Colorado, all in Chemical Engineering. He is now retired after 30 years in engineering education, in teaching and administrative positions. His areas of expertise are environmental engineering, hydrology, engineering science and renewable energy. He is a licensed Professional Engineer in Missouri. He did consulting work while in academia and had prior industrial experience. Dr. Bengtson has authored numerous publications, presentations and technical reports. He is currently active as a technical writer.

Critical Depth Open Channel Flow Spreadsheet

Posted on July 23, 2011 by Harlan Bengtson

Where to Find a Critical Depth Open Channel Flow Spreadsheet

To obtain a critical depth open channel flow spreadsheet for calculating critical depth and/or critical slope for open channel flow, click here to visit our spreadsheet store. Read on for information about the use of a critical depth open channel flow spreadsheet for critical depth and critical slope calculations.

The Froude Number and Critical, Subcritical and Supercritical Flow

Any particular example of open channel flow will be critical, subcritical, or supercritical flow. In general, supercritical flow is characterized by high liquid velocity and shallow flow, while subcritical flow is characterized by low liquid velocity and relatively deep flow. Critical flow is the dividing line flow condition between subcritical and supercritical flow.

The Froude number is a dimensionless number for open channel flow that provides information on whether a given flow is subcritical, supercritical or critical flow. The Froude number is defined to be: Fr = V/(gL)^1/2 , where V is the average velocity, g is the acceleration due to gravity, and L is a characteristic length for the particular type of open channel flow. For flow in a rectangular channel: Fr = V/(gy)^1/2 , where y is the depth of flow. For flow in an open channel with a shape other than rectangular: Fr = V/[g(A/B)]^1/2 , where A is the cross-sectional area of flow, and B is the surface width.

The value of the Froude number for a particular open channel flow situation gives the following information:

For Fr < 1, the flow is subcritical
For Fr = 1, the flow is critical
For Fr > 1, the flow is supercritical

Calculation of Critical Depth

It is sometimes necessary to know the critical depth for a particular open channel flow situation. This type of calculation can be done using the fact that Fr = 1 for critical flow. It is quite straightforward for flow in a rectangular channel and a bit more difficult, but still manageable for flow in a non-rectangular channel.

For flow in a rectangular channel (using subscript c for critical flow conditions), Fr = 1 becomes: V_c/(gy_c)^1/2 = 1. Substituting V_c = Q/A_c = Q/by_c and q = Q/b (where b = the width of the rectangular channel), and solving for y_c gives the following equation for critical depth: y_c = (q²/g)^1/3. Thus, the critical depth can be calculated for a specified flow rate and rectangular channel width.

For flow in a trapezoidal channel, Fr = 1 becomes: V_c/[g(A/B)_c]^1/2 = 1. Substituting the equation above for V_c together with A_c = y_c(b + zy_c) and B_c = b + zy_c² leads to the following equation, which can be solved by an iterative process to find the critical depth:

Calculation of Critical Slope

After the critical depth, y_c , has been determined, the critical slope, S_c , can be calculate using the Manning equation if the Manning roughness coefficient, n, is known. The Manning equation can be rearranged as follows for this calculation:

Note that R_hc , the critical hydraulic radius, is given by:

R_hc = A_c/P_c, where P_c = b + 2y_c(1 + z²)^1/2

Note that calculation of the critical slope is the same for a rectangular channel or a trapezoidal channel, after the critical depth has been determined. The Manning equation is a dimensional equation, in which the following units must be used: Q is in cfs, A_c is in ft², R_hc is in ft, and S_c and n are dimensionless.

Calculations in S.I. Units

The equations for calculation of critical depth are the same for either U.S. or S.I. units. All of the equations are dimensionally consistent, so it is just necessary to be sure that an internally consistent set of units is used. For calculation of the critical slope, the S.I. version of the Manning equation must be used, giving:

In this equation, the following units must be used: Q is in m³/s, A_c is in m², R_hc is in m, and S_c and n are dimensionless.

A Critical Depth Open Channel Flow Spreadsheet Screenshot

The critical depth open channel flow spreadsheet template shown below can be used to calculate the critical depth and critical slope for a rectangular channel with specified flow rate, bottom width, and Manning roughness coefficient. Why bother to make these calculations by hand? This Excel spreadsheet and others with similar calculations for a trapezoidal channel are available in either U.S. or S.I. units at a very low cost in our spreadsheet store.

References

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

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

3. Bengtson, Harlan H. Open Channel Flow II – Hydraulic Jumps and Supercritical and Nonuniform Flow – An online, continuing education course for PDH credit.

Natural Convection Heat Transfer Coefficient Calculator Spreadsheet

Posted on May 24, 2011 by Harlan Bengtson

Where to Find a Natural Convection Heat Transfer Coefficient Calculator Spreadsheet

For an Excel spreadsheet to use as a natural convection heat transfer coefficient calculator, click here to visit our spreadsheet store. Why search for heat transfer coefficient correlations or use online calculators, when you can buy a spreadsheet to use as a natural convection heat transfer coefficient calculator for five different configurations for only $14.95. Read on for information about natural convection heat transfer coefficients and Excel spreadsheets to obtain a value for them.

Convection heat transfer takes place between a solid surface and fluid that is at a different temperature and is in contact with the surface. If the fluid is flowing past the surface due to an external driving force like a fan or pump, then the heat transfer is called forced convection. When fluid motion is due to density differences within the fluid (caused by temperature variation), then the heat transfer is called natural convection or free convection.

Newton’s Law of Cooling for Natural Convection Heat Transfer Coefficient Calculator

Newton’s Law of Cooling [ Q = hA(T_s – T_f) ] is a simple expression used for the rate of convective heat transfer with either forced or natural convection. The parameters in Newton’s Law of Cooling are:

Q, the rate of forced convection heat transfer (Btu/hr – U.S. or W – S.I.)
T_s, the solid temperature (^oF – U.S. or ^oC – S.I.)
T_f, the fluid temperature (^oF – U.S. or ^oC – S.I.)
A, the area of the surface that is in contact with the fluid (ft² – U.S. or m² – S.I.)
h, the convective heat transfer coefficient (Btu/hr-ft²–^oF – U.S. or W/m²-K – S.I.)

Dimensionless Nusselt, Rayleigh, Grashof, and Prandtl Numbers

A natural convection heat transfer coefficient calculator typically makes estimations using correlations of dimensionless numbers, specifically correlations of Nusselt number (Nu) with Prandtl number (Pr), Grashof number (Gr), and/or Rayleigh number (Ra), where Ra = GrPr. The Nusselt, Grashof and Prandtl numbers are defined in the box at the left.

Following is a list of the parameters that appear in these dimensionless numbers, with units are given for both the U.S engineering system and S.I. system of units:

D, a characteristic length parameter (e.g. diameter for natural convection from a circular cylinder or a sphere or height of a vertical plate) (ft for U.S., m for S.I.)
ρ, the density of the fluid (slugs/ft³ for U.S., Kg/m³ for S.I.)
μ, the viscosity of the fluid (lb-sec/ft² for U.S., N-s/m² for S.I.)
k, the thermal conductivity of the fluid (Btu/hr-ft-^oF for U.S., W/m-K for S.I.)
C_p, the heat capacity of the fluid (Btu/lb-^oF for U.S., J/kg-K for S.I.)
g, the acceleration due to gravity (32.17 ft/sec² for U.S., 9.81 m/s² for S.I.)
β, the coefficient of volume expansion of the fluid ( ^oR for U.S., K for S.I.)
ΔT, the temperature difference between the solid surface and the fluid ( ^oF for U.S., ^oC or K for S.I.)

The following sections provide equations for estimating the heat transfer coefficient for several common natural convection configurations.

Natural Convection Heat Transfer Calculator for a Vertical Plane

The box at the right shows two correlations for convection heat transfer between a vertical plane and a fluid of different temperature in contact with it. The first can be used for all values of Rayleigh number and the second is only for laminar flow, indicated by Ra < 10⁹. The screenshot image below shows an example of an Excel spreadsheet to use as a natural convection heat transfer coefficient calculator for a vertical plate using the two equations shown here.

An Excel Spreadsheet as a Natural Convection Heat Transfer Calculator

For low cost, easy to use Excel spreadsheet packages to use as a natural convection heat transfer coefficient calculator for natural convection from a vertical plane, a horizontal plane, an inclined plane, a horizontal cylinder or a sphere in either U.S. or S.I. units (for only $16.95), click here to visit our spreadsheet store.

References

1. Incropera, F.P., DeWitt, D.P, Bergman, T.L., & Lavine, A.S., Fundamentals of Heat and Mass Transfer, 6th Ed., Hoboken, NJ, John Wiley & Sons, (2007).

2. Lienhard, J.H, IV and Lienhard, J.H. V, A Heat Transfer Textbook: A Free Electronic Textbook

3. Bengtson, Harlan H, Fundamentals of Heat Transfer, an online continuing education course for engineering PDH credit

4. Bengtson, Harlan H., Convection Heat Transfer Coefficient Estimation, an online continuing education course for PDH credit.

Sharp Crested Rectangular Weir Flow Calculator Spreadsheet

Posted on May 3, 2011 by Harlan Bengtson

Where to Find a Rectangular Weir Flow Calculator Spreadsheet

For a rectangular weir flow calculator spreadsheet, click here to visit our spreadsheet store. Obtain a convenient, easy to use spreadsheet to use as a rectangular weir flow calculator at a reasonable price. Read on for information about Excel spreadsheets that can be used as contracted rectangular weir open channel flow calculators.

The following section, which gives background on sharp crested rectangular weirs in general, also appears in the companion article, “Suppressed Rectangular Weir Calculations with an Excel Spreadsheet”

Background on Sharp Crested Rectangular Weirs in General

The picture at the left shows a rectangular weir measuring open channel flow rate in a natural channel. The diagram below right shows a longitudinal cross-section of a sharp crested weir, with some of the terminology and parameters often used for sharp crested weirs included on the diagram.

The weir crest is the top of the weir. For a rectangular weir it is the straight, levelbottom of the rectangular opening through which water flows over the weir. The term nappe is used for the sheet of water flowing over the weir. The equations for calculating flow rate over a weir in this article require free flow, which takes place when there is air under the nappe. The drawdown is shown in the diagram as the decrease in water level going over the weir due to the acceleration of the water. The head over the weir is shown as H in the diagram; the height of the weir crest is shown as P; and the open channel flow rate in the open channel (and over the weir) is shown as Q.

Image Credits: Rectangular, Sharp-Crested Weir: flowmeterdirectory.co.uk

Sharp Crested Weir Parameters: H. H. Bengtson, Ref #2

The Francis Equation for a Rectangular Weir Flow Calculator

A contracted rectangular weir is one for which the weir extends across only part of the channel, so that the length of the weir, L, is different from as the width of the channel. The picture at the left shows a contracteded rectangular weir being used to measure the flow of water in a triangular open channel. The diagram below right shows some of the key parameters used in contracted rectangular weir flow rate calculations. Specifically, the height of the weir crest, P, the head over the weir, H, the weir length, L, and the channel width, B, are shown on the diagram of a contracted rectangular weir in a rectangular channel. The U.S. Bureau of Reclamation, in their Water Measurement Manual (Ref #1 below), recommend the use of the Francis equation (shown below) for completely contracted rectangular weirs, subject to the condition that H/L < 0.33, B – L > 4 H_max, and P > 2H_max.

For U.S. units: Q = 3.33(L – 0.2H)H^3/2, where

Q is the water flow rate in ft³/sec,
L is the length of the weir in ft,
H is the head over the weir in ft,
B is the width of the channel in ft, and
H_max is the maximum expected head over the weir in ft.

For S.I. units: Q = 1.84(L – 0.2H)H^3/2, where

Q is the water flow rate in m³/sec,
L is the length of the weir in m, and
H is the head over the weir in m.
B is the width of the channel in m, and
H_max is the maximum expected head over the weir in m.

Image Credits: Contracted Rectangular Weir picture: Food and Agricultural Organization of the United Nations.

Contracted Rectangular Weir Diagram – Bengtson, Harlan H.

The Kindsvater-Carter Formula for a Rectangular Weir Flow Calculator

If any of the three required conditions given in the previous section are not met, then the more general Kindsvater- Carter Equation, shown below should be used.

U.S. units: Q = C_e(2/3)[(2g)^1/2](L + k_b)(H + 0.003)^3/2

S.I. units: Q = C_e(2/3)[(2g)^1/2](L + k_b)(H + 0.001)^3/2

C_e is a function of L/B and H/P, while k_b is a function of L/B. There are graphs, tables and equations available for obtaining values for C_e and k_b for specified values of L/B and H/P. The equations given below were prepared from information in Reference #3 at the end of the article.

C_e is dimensionless, so the equation for C_e is as a function of L/B and H/P is the same for both S.I. and U.S. units and is as follows:

C_e = α(H/P) + β, where β = 0.58382 + 0.016218(L/B), and

α = [-0.0015931 + 0.010283(L/B)]/[1 – 1.76542(L/B) + 0.870017(L/B)²]

The equation for k_b as a function of L/B has different constants for S.I. and U.S. units. The two versions of the equation for k_b are as follows:

U.S. units: for 0 < L/B < 0.35: k_b = 0.007539 + 0.001575(L/B) – (k_b is in ft)

for 0.35 < L/B < 1.0: k_b = -0.34806(L/B)⁴ + 0.63057(L/B)³ – 0.37457(L/B)² + 0.09246(L/B) – 0.000197 – (k_b is in ft)

S.I. units: for 0 < L/B < 0.35: k_b = 0.002298 + 0.00048(L/B) – (k_b is in m)

for 0.35 < L/B < 1.0: k_b = -0.10609(L/B)⁴ + 0.1922(L/B)³ – 0.11417(L/B)² + 0.028182(L/B) – 0.00006 – (k_b is in m)

Note that if H/L < 0.33, B – L > 4 H_max, and P > 2H_max, then the Francis Equation and the Kindsvater-Carter Equation will give nearly the same value for Q. As conditions diverge more and more from the requirements, the calculations from the two equations will diverge more and more. In these cases the value calculated by the Kindsvater-Carter formula should be used.

An Excel Spreadsheet as a Contracted Rectangular Weir Flow Calculator

The Excel spreadsheet template shown below can be used as a contracted rectangular weir flow calculator, using both the Francis equation and the Kindsvater-Carter equation. Only four input values are needed. They are the height of the weir crest above the channel invert, P; the width of the channel, B; the weir length L; and the measured head over the weir, H. With these four input values, the Excel formulas will calculate the parameters needed and check on whether the conditions required for use of the Francis equation are met. If the conditions are all met, then the value of Q calculated with the Francis equation can be used. If any of the conditions aren’t met, then the value of Q calculated with the Kindsvater-Carter formula should be chosen. This Excel spreadsheet and others for suppressed and contracted rectangular weir calculations are available in either U.S. or S.I. units at a very low cost in our spreadsheet store.

References

1. U.S. Dept. of the Interior, Bureau of Reclamation, 2001 revised, 1997, 3rd ed, Water Measurement Manual

2. Bengtson, H.H., Sharp Crested Weirs for Open Channel Flow Measurement, an Amazon Kindle ebook.

3. Bengtson, H.H., Open Channel Flow Measurement – Weirs and Flumes, An online continuing education course for PDH credit for Professional Engineers

4. Bengtson, H. H., Sharp-Crested Weirs for Open Channel Flow Measurement, An online continuing education course for PDH credit for Professional Engineers.

5. Merkley, Gary P., Weirs for Flow Measurement Open Course Ware, Utah State University.

Suppressed Rectangular Weir Calculations with Excel Spreadsheets

Posted on May 2, 2011 by Harlan Bengtson

Introduction to Suppressed Rectangular Weir Calculations

For an Excel spreadsheet to make suppressed rectangular weir flow calculations, click here to visit our spreadsheet store. Read on for information about Excel spreadsheets that can be used as suppressed rectangular weir open channel flow calculators.

As shown in the diagrams and pictures below, the rectangular refers the the shape of the water cross-section as it goes over a sharp crested rectangular weir, which consists of a plate placed in an open channel so that the water is forced to flow through the rectangular open in the weir plate. It can be used for open channel flow rate measurement, by measuring the height of water above the weir crest (the straight, level top of the weir opening), which can then be used to calculate the water flow rate over the weir.

Background on Sharp Crested Rectangular Weir Calculations in General

The weir crest is the top of the weir. For a rectangular weir it is the straight, level bottom of the rectangular opening through which water flows over the weir. The term nappe is used for the sheet of water flowing over the weir. The equations for calculating flow rate over a weir in this article require free flow, which takes place when there is air under the nappe. The drawdown is shown in the diagram as the decrease in water level going over the weir due to the acceleration of the water. The head over the weir is shown as H in the diagram; the height of the weir crest is shown as P; and the open channel flow rate in the open channel (and over the weir) is shown as Q.

Image Credits: Rectangular, Sharp-Crested Weir: flowmeterdirectory.co.uk

Sharp Crested Weir Parameters: H. H. Bengtson, Ref #2

The Francis Equation for Suppressed Rectangular Weir Calculations

A suppressed rectangular weir is one for which the weir extends across the entire channel, so that the length of the weir, L, is the same as the width of the channel, B. The picture at the left shows a suppressed rectangular weir being used to measure the flow of water in an open channel. The diagram below right shows some of the key parameters used in suppressed rectangular weir flow rate calculations. Specifically, the height of the weir crest, P, the head over the weir, H, and the weir length, L (equal to channel width, B) are shown on the diagram. The U.S. Bureau of Reclamation, in their Water Measurement Manual (Ref #1 below), recommend the use of the Francis equation (shown below) for suppressed rectangular weirs, subject to the condition that H/P < 0.33 and H/B < 0.33:

For U.S. units: Q = 3.33 B H^3/2, where

Q is the water flow rate in ft³/sec,
B is the length of the weir (and the channel width) in ft, and
H is the head over the weir in ft.

For S.I. units: Q = 1.84 B H^3/2, where

Q is the water flow rate in m³/sec,
B is the length of the weir (and the channel width) in m, and
H is the head over the weir in m.

The same condition for H/P and H/B apply.

Image Credits: Suppressed Rectangular Weir Picture – U.S. Dept. of the Interior, Bureau of Reclamation, 2001 revised, 1997 third edition, Water Measurement Manual.

Suppressed Rectangular Weir Diagram – Bengtson, Harlan H.

The Kindsvater-Carter Formula for Suppressed Rectangular Weir Calculations

If either of the requirements in the previous section (H/P < 0.33 and H/B < 0.33) are not met the the more general Kindsvater- Carter Equation, shown below should be used.

U.S. units: Q = [0.075(H/P) + 0.602](2/3)[(2g)^1/2](L – 0.003)(H + 0.003)^3/2

S.I. units: Q = [0.075(H/P) + 0.602](2/3)[(2g)^1/2](L – 0.001)(H + 0.001)^3/2

Note that if H/P < 0.33 and H/B < 0.33, then the Francis Equation and the Kindsvater-Carter Equation will give nearly the same value for Q. As H/P and/or H/B increase more and more above the 0.33 limit the calculations from the two equations will diverge more and more. In these cases the value calculated by the Kindsvater-Carter formula should be used.

An Excel Spreadsheet for Suppressed Rectangular Weir Calculations

The Excel spreadsheet template shown below can be used for suppressed rectangular weir calculations, to calculate the water flow rate over a suppressed rectangular weir, using both the Francis equation and the Kindsvater-Carter equation. Only three input values are needed. They are the height of the weir crest above the channel invert, P; the width of the channel, B (which equals the weir length L); and the measured head over the weir, H. With these three input values, the Excel formulas will calculate H/P and H/B. If both of these are less than 0.33, then the value of Q calculated with the Francis equation can be used. If either of the conditions aren’t met, then the value of Q calculated with the Kindsvater-Carter formula should be chosen. This Excel spreadsheet and others for suppressed and contracted rectangular weir calculations are available in either U.S. or S.I. units at a very low cost in our spreadsheet store.

References

1. U.S. Dept. of the Interior, Bureau of Reclamation, 2001 revised, 1997, 3rd ed, Water Measurement Manual

2. Bengtson, H.H., Sharp Crested Weirs for Open Channel Flow Measurement, an Amazon Kindle ebook.

3. Bengtson, H.H., Open Channel Flow Measurement – Weirs and Flumes, An online continuing education course for PDH credit for Professional Engineers

4. Bengtson, H. H., Sharp-Crested Weirs for Open Channel Flow Measurement, An online continuing education course for PDH credit for Professional Engineers.

5. Bengtson, H.H., “A Sharp Crested Rectangular Weir Equations Spreadsheet,” an online blog article.

6. Merkley, Gary P., Weirs for Flow Measurement Open Course Ware, Utah State University.

ISO 5167 Orifice Coefficient Calculation Spreadsheet

Posted on April 19, 2011 by Harlan Bengtson

Introduction to an ISO 5167 Orifice Coefficient Calculation Spreadsheet

For an ISO 5167 orifice coefficient calculation spreadsheet, click here to visit our spreadsheet store. Read on for information about Excel spreadsheets that can be used to make these ISO 5167 orifice meter calculations.

When ISO 5167 came out in 1991, it included three standard configurations for the pressure taps in an orifice flow meter and equations to calculate the orifice discharge coefficient for a specified ratio of orifice diameter to pipe diameter for any of those three standard pressure tap configurations. This provided greater flexibility for orifice meters, because orifice plates with different orifice diameters could be used in a given orifice meter, while still allowing accurate determination of the orifice discharge coefficient.

Background on ISO 5167 Orifice Coefficient Calculation Spreadsheet

An orifice meter is a simple device for measuring pipe flow rate through the use of a circular plate with a hole in the center (the orifice plate), held in place between pipe flanges, as shown in the diagram at the left. The fluid pressure decreases downstream of the orifice plate due to the accelerated flow. The pressure difference shown in the diagram as P₁ – P₂ can be measured and used to calculate the flow rate passing through the meter (and thus the pipe flow rate) using the equation shown at the right. This equation allows calculation of pipe flow rate, Q, for measured pressure difference, P₁ – P₂, and known density of the fluid, ρ, the ratio of orifice diameter to pipe diameter, β, the cross-sectional area of the orifice, A_o, and the orifice discharge coefficient, C_d.

For more details about the orifice, flow nozzle, and venturi meter, see the article, “Excel Spreadsheets for Orifice and Venturi Flow Meter Calculations.”

ISO 5167 Standard Pressure Tap Locations

Prior to ISO 5167 coming out in 1991, the downstream pressure tap of an orifice meter was typically located at the vena contracta (the minimum jet area downstream of the orifice plate) as shown in the diagram above. The correlations in place for determining the orifice discharge coefficient were for the downstream pressure tap at the vena contracta. Unfortunately, the distance of the vena contracta fro the orifice plate changes with orifice diameter, so changing to an orifice plate with a different hole diameter required moving the downstream pressure tap in order to be able to accurately estimate the orifice discharge coefficient.

The three standard pressure tap configurations identified for orifice flow meters, known as corner taps, flange taps, and D – D/2 taps, are shown in the diagram at the left. As shown in the diagram, the distance of the pressure taps from the orifice plate is given as a fixed distance, or as a function of the pipe diameter, independent of the orifice diameter, so the orifice discharge coefficient can be calculated for several orifice diameters in a given orifice meter.

Equations for ISO 5167 Orifice Coefficient Calculation Spreadsheet

Included in ISO 5167 is an equation allowing calculation of the orifice discharge coefficient, C_d, for known values of β (d/D), Reynolds number, Re, and L₁ & L₂, where L₁ is the distance of the upstream pressure tap from the orifice plate and L₂ is the distance of the downstream pressure tap from the orifice plate. For corner taps: L₁ = L₂ = 0; for flange taps: L₁ = L₂ = 1″ ; and for D-D/2 taps: L₁ = D & L₂ = D/2. The ISO 5167 equation for the orifice discharge coefficient is:

C_d – 0.5959 + 0.0312 β^2.1 – 0.1840 β⁸ + 0.0029 β^2.5(10⁶/Re)^0.75 + 0.0900(L₁/D)[β⁴/(1 – β⁴)] – 0.0337(L₂/D)β³

This equation is usable to find the orifice discharge coefficient for an orifice flow meter with any of the three standard pressure tap configurations, but not for any other arbitrary values of L₁ and L₂. The introduction of these standard pressure tap configurations and the equation for C_d, allows a given orifice flow meter to conveniently use different size orifice openings and cover a wide flow measurement range.

An iterative (trial and error) calculation is needed to get a value for C_d, because the upstream velocity needed for Re isn’t known until C_d is known. The ISO 5167 orifice coefficient calculation spreadsheet template shown in the screenshot at the right will calculate the orifice discharge coefficient based on the indicated input information. The spreadsheet uses an iterative calculation procedure. It is necessary to assume a value for Re to start the process and replace that value with the calculated Re as any times as necessary until the two Re values are the same. This ISO 5167 orifice calculation spreadsheet is available in either U.S. or S.I. units at a very low cost in our spreadsheet store.

References:

1. U.S. Dept. of the Interior, Bureau of Reclamation, 2001 revised, 1997 third edition, Water Measurement Manual.

2. International Organization of Standards – ISO 5167-1:2003 Measurement of fluid flow by means of pressure differential devices, Part 1: Orifice plates, nozzles, and Venturi tubes inserted in circular cross-section conduits running full. Reference number: ISO 5167-1:2003.

3. Bengtson, Harlan H., Flow Measurement in Pipes and Ducts, An online continuing education course.

4. Bengtson, Harlan H., “Orifice and Venturi Pipe Flow Meters: for Liquid Flow or Gas Flow,” an Amazon Kindle e-book.

Air Density Calculator Excel Spreadsheet

Posted on April 18, 2011 by Harlan Bengtson

Where to Find an Air Density Calculator Excel Spreadsheet

To find an air density calculator Excel spreadsheet to use as an air density calculator, click here to visit our spreadsheet store. Why use an online calculator or look in tables, when you can get an air density calculator excel spreadsheet to use as an air density calculator here? Read on for information about Excel spreadsheets that can be used to calculate the density of air (and other gases) at different pressures and temperatures with the ideal gas law.

Gas density background for an Air Density Calculator Excel Spreadsheet

Pressure and temperature have significant effects on the density of gases, so some means of determining the density of air and other gases at specified temperatures and pressures is needed for a variety of fluid mechanics applications. Fortunately, the ideal gas law provides a means of doing this for many gases over ranges of temperature and pressure that are of interest.

The Ideal Gas Law for use in an Air Density Calculator Excel Spreadsheet

A common form for the ideal gas law equation is PV = nRT, giving the relationship among T, the absolute temperature of the gas; P, its absolute pressure; V, the volume occupied by n moles of the gas; and R, the ideal gas law constant.

The density of the gas can be introduced into this equation, through the fact that molecular weight (MW) has units of mass/mole, so that n = m/MW. This leads to the ideal gas law written as: PV = (m/MW)RT. Solving this equation for m/V (which is equal to the gas density, ρ) gives the following equation for gas density as a function of its MW, pressure and temperature: ρ = (MW)P/RT.

A commonly used set of U.S. units for this equation is as follows:

ρ = density of the gas in slugs/ft³,

MW = molecular weight of the gas in slugs/slugmole (or kg/kgmole, etc.) (NOTE: MW of air = 29),

P = absolute gas pressure in psia (NOTE: Absolute pressure equals pressure measured by a guage plus atmospheric pressure.),

T = absolute temperature of the gas in ^oR (NOTE: ^oR = ^oF + 459.67)

R = ideal gas constant in psia-ft³/slugmole-^oR.

For conditions under which air can be treated as an ideal gas (see the next section), the ideal gas law in this form can be used to calculate the density of air at different pressures and temperatures.

The air density calculator excel spreadsheet template shown in the screenshot below will calculate the density of a gas for specified molecular weight, pressure and temperature. This Excel spreadsheet is available at a very reasonable price in our spreadsheet store and can be used with either U.S. or S.I. units. These spreadsheets also contain tables of critical temperature and critical pressure for several common gases.

But When Can I Use the Ideal Gas Law to Calculate the Density of Air?

A good question indeed, because air and other gases for which you may need a density value are real gases, not ideal gases. It is fortunate, however, that many real gases behave almost exactly like an ideal gas over a wide range of temperatures and pressures. The ideal gas law works best for high temperatures (relative to the critical temperature of the gas) and low pressures (relative to the critical pressure of the gas). See table at the left for values of critical temperature and critical pressure for several common gases. For many practical, real situations, the ideal gas law gives quite accurate values for the density of air (and many other gases) at different pressures and temperatures.

S.I. Units for the Ideal Gas Law

The ideal gas law is a dimensionally consistent equation, so it can be used with any consistent set of units. For SI units the ideal gas law parameters are as follows:

ρ = density in kg/m³,

P = absolute gas pressure in pascals (N/m²),

T = absolute temperature in ^oK (NOTE: ^oK = ^oC + 273.15)

R = ideal gas constant in Joules/kgmole-K

References:

1. Bengtson, Harlan H., Flow Measurement in Pipes and Ducts, An online PDH course for Professional Engineers

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

3. Applied Thermodynamics ebook, http://www.taftan.com/thermodynamics/

4. Bengtson, Harlan H., “Gas Property Calculator Spreadsheet,” an Amazon Kindle e-book.

Parshall Flume Discharge Calculation – Open Channel Flow Measurement with Excel

Posted on April 14, 2011 by Harlan Bengtson

Where to find a Parshall Flume Discharge Calculation Spreadsheet

For a Parshall flume discharge calculation Excel spreadsheet to make open channel flow measurement calculations, click here to visit our spreadsheet store. Obtain a convenient, easy to use Parshall flume discharge calculation spreadsheet at a reasonable price. Read on for information about Excel spreadsheets that can be used for Parshall flume/open channel flow measurement calculations.

Parshall flumes are used for a variety of open channel flow measurement. They are especially good for flows containing suspended solids, as for example the flow in wastewater treatment. As seen in the picture at the right, the plan view of a Parshall flume is similar to that of a venturi flume, with a converging section, a throat, and a diverging section. A Parshall flume, however, also has prescribed variations in the channel bottom slope as shown in the diagram in the next section. Flow rate through a Parshall flume can be calculated based on a measured head, using equations that will be discussed in a later section. A Parshall flume must be constructed with prescribed dimensions as shown in the next section.

Image Credit: City of Batavia, Illinois

Flume Configuration and Dimensions for Parshall Flume Discharge Calculations

The diagram at the left shows the general configuration of a Parshall flume with a plan and elevation view. The width of the throat is typically used to specify the size of a Parshall flume. The table at the right below, shows the standard dimensions for Parshall flumes with throat widths ranging from 1 ft to 8 ft. Similar information is available for throat widths down to 1 inch and up to 50 ft.

Such a range of sizes covers a very wide range of flow rates. A 1 inch flume will carry a flow of 0.03 cfs at 0.2 ft of head, while a 50 ft Parshall flume will carry 3,000 cfs at a head of 5.7 ft. For the range of throat widths in the table, the other dimensions in the diagram are constant at the following values:

E = 3′-0″, F = 2′-0″, G = 3′-0″,

K = 3 inches, N = 9 inches,

X = 2 inches, Y = 3′

Free Flow and Submerged Flow in Parshall Flume Discharge Calculation

For “free flow” through a Parshall flume, the flow rate through the throat of the flume is unaffected by the downstream conditions. For free flow, a hydraulic jump will be visible in the throat of the Parshall flume. For flow situations where downstream conditions cause the flow to back up into the throat, the hydraulic jump isn’t visible, and the flow is said to be “submerged flow” rather than “free flow.”

The ratio between head measurements at the two locations, H_a and H_b, as shown in the diagram at the left above, can be used as a quantitative criterion to differentiate between free flow and submerged flow. The values of H_b/H_a for free flow and for submerged flow, for several ranges of throat width from 1″ to 8′ are as follows:

For 1” < W < 3” : free flow for H_b/H_a < 0.5; submerged flow for H_b/H_a > 0.5

For 6” < W < 9” : free flow for H_b/H_a < 0.6; submerged flow for H_b/H_a > 0.6

For 1’ < W < 8’ : free flow for H_b/H_a < 0.7; submerged flow for H_b/H_a > 0.7

For 8’ < W < 50’ : free flow for H_b/H_a < 0.8; submerged flow for H_b/H_a > 0.8

Excel Formulas for Free Flow Parshall Flume Discharge Calculation

The free flow equation for Parshall flume discharge calculation is Q_free = C H_aⁿ, where

Q_free = the open channel flow rate through the Parshall flume under free flow conditions, cfs for U.S. units or m³/s for S.I.
H_a = the head measured at the correct point in the converging section of the Parshall flume as described in the previous section, ft for U.S. units or m for S.I. units
C and n are constants for a given Parshall flume throat width, W.

The tables below give the constants C and n in the equations for free flow Parshall flume discharge calculation for both U.S. units and for S.I. units.

The screenshot at the right shows a Parshall flume discharge calculation spreadsheet that will calculate flow rate through the Parshall flume under free flow conditions in S.I. units for a selected throat width and a specified value for the measured head. This Excel spreadsheet and one for submerged flow calculation are available in either U.S. or S.I. units at a very low cost in our spreadsheet store.

Excel Formulas for Submerged Flow Parshall Flume Discharge Calculation

The submerged flow equations for Parshall flume discharge calculation, as used by the Excel formulas in the spreadsheet below, are summarized for U.S. units and for S.I. units in the diagrams below:

The primary submerged flow equation Parshall flume discharge calculation is: Q_subm = Q_free – Q_corr, where

Q_subm = the flow rate through the Parshall flume for a submerged flow condition, in cfs for U.S. units or m³/s for S.I. units

Q_free = the flow rate calculated with the equation, Q_free = C H_aⁿ, as described in the previous section, in cfs for U.S. units or m³/s for S.I. units

Q_corr is a flow correction factor calculated from the equations shown above for the correct throat width, W, in cfs for U.S. units or m³/s for S.I. units

The screenshot of an Excel spreadsheet template shown at the left will carry out submerged flow Parshall flume discharge calculation in U.S. units for a selected throat width and a specified value for the measured heads, H_a and H_b. This Excel spreadsheet and one for free flow calculation are available in either U.S. or S.I. units at a very low cost in our spreadsheet store.

References

1. U.S. EPA, Recommended Practice for the Use of Parshall Flume and Palmer Bowlus Flumes in Wastewater Treatment plants, EPA600/2-84-180, 1984

2. Wahl, Tony L., Equations for Computing Submerged Flow in Parshall Flumes, Bureau of Reclamation, Denver, Colorado, USA

3. U.S. Dept. of the Interior, Bureau of Reclamation, Water Measurement Manual, 2001 revised, 1997 third edition

Watershed Time of Concentration Calculation with an Excel Spreadsheet

Posted on April 11, 2011 by Harlan Bengtson

Where to find Excel Spreadsheets for Watershed Time of Concentration

To obtain an Excel spreadsheet for watershed time of concentration calculations, click here to visit our spreadsheet store. Obtain a convenient, easy to use spreadsheet for watershed time of concentration calculation at a reasonable price. Read on for information about Excel spreadsheets that can be used for watershed time of concentration calculations.

The time of concentration for a watershed is the time for rainfall that lands on the farthest point of the watershed to reach the outlet. The main reason for interest in the watershed time of concentration is for its use as the storm duration in finding the design rainfall intensity to use in Rational Method calculation of peak storm water runoff rate.

The reason that the watershed time of concentration is used as design storm duration is because it gives the largest peak storm water runoff rate for a given return period. This can be reasoned out as follows: If the storm duration is less than the time of concentration, then the storm will end before runoff from the entire watershed reaches the outlet. Thus flow from the entire watershed will never all be contributing to the outflow. If the storm duration is greater than the time of concentration, then the storm will continue longer than it takes for the entire watershed to contribute to the outflow, but the storm intensity will be less for a storm of longer duration than one of short duration for a given return period. Thus the maximum peak storm water runoff rate for a specified return period on a given watershed will be for a storm with duration equal to the time of concentration of that watershed.

We can now move on to a discussion of how to calculate values for the time of concentration of a given watershed.

Methods for Estimating Watershed Time of Concentration

There are several empirical equations that have been developed for calculating travel time/time of concentration for different types and conditions of watersheds. Some examples are the Kerby equation, the Izzard equation, the Manning Kinematic equation, the Bransby Williams equation, the National Resources Conservation Service (NCRS) method, and the Manning equation. The following three equations will be discussed in this article: 1) the Manning Kinematic equation for use with overland sheet flow, 2) the NRCS method for shallow concentrated flow, and 3) the Manning equation for channel flow. These three methods are recommended by the U.S. Soil Conservation Service (SCS) in ref #1 at the end of this article. The Iowa Stormwater Management Manual (ref #2) also recommends these three methods. Typically overland sheet flow will occur in the upper portion of the watershed, followed by shallow concentrated flow, with channel flow for the final portion of watershed before the outlet.

Calculations with the Manning Kinematic Equation

The boxes at the right show the Manning Kinematic equation for U.S. and for S.I. units. The parameters in the Manning Kinematic equation and their units are as follows:

t₁ = overland sheet flow runoff travel time, min (NOTE: many places show the constant being 0.007 for U.S. units giving the time in hours. The equations in the boxes both give travel time in minutes.)
n = Manning roughness coefficient, dimensionless*
L = length of flow path, ft (S.I. – m)
P = 2 year, 24 hr rainfall depth, in (S.I. – m)
S = ground slope, ft/ft (S.I. m/m)

*See table of n values below.

The screenshot of an Excel spreadsheet template shown below will calculate overland sheet flow travel time with U.S. units using the Manning kinematic equation, based on the input values entered for the other parameters listed above. A tables with values of the Manning roughness coefficient for various overland flow conditions is also given below. This Excel spreadsheet and others for time of concentration calculations are available in either U.S. or S.I. units at a very low cost in our spreadsheet store.

Watershed Time of Concentration Calculations with the NRCS Method

The Manning Kinematic equation is recommended for travel length of no greater than 300 ft in ref #1 and for no greater than 100 ft in ref #2. Both of these references recommend use of the NCRS method for the shallow concentrated flow that normally develops within 100 to 300 ft into the watershed. The NCRS method calculates the velocity of the shallow concentrated flow first, based on the slope and the type of surface. The travel time is then calculated as travel length divided by velocity of flow. The equations used for the NRCS method are:

t₂ = L/(60V) ( for either U.S. or S.I. units )
V = 16.1345 S^0.5 for U.S. units ( V = 4.9178 S^0.5 for S.I. units) for an unpaved surface
V = 20.3282 S^0.5 for U.S. units ( V = 6.1960 S^0.5 for S.I. units) for a paved surface

An explanation of each of the parameters used in these equations follows:

L is the length of the flow path in ft for U.S. or m for S.I. units
V is the velocity of flow in ft/sec for U.S. or m/s for S.I. units
S is the slope of the flow path, which is dimensionless for either U.S. or S.I. units
t₂ is the travel time for shallow concentrated flow in minutes (for either U.S. or S.I. units)

The screenshot of an Excel spreadsheet template shown at the left will calculate shallow concentrated flow travel time with S.I. units using the NRCS method, based on the input values indicated. This Excel spreadsheet and others for time of concentration calculations are available in either U.S. or S.I. units at a very low cost at www.engineeringexceltemplates.com or in our spreadsheet store.

Calculation of Travel Time with the Manning Equation

The Manning equation is used for quite a variety of open channel flow calculations. It is recommended in ref#1 and ref #2 for any channel flow portion of the watershed runoff path. The following equations are used for Manning equation calculations:

The Manning equation in U.S. units: Q = (1.49/n)A(R^2/3)(S^1/2)
The Manning equation in S.I. units: Q = (1.0/n)A(R^2/3)(S^1/2)
R = A/P
V = Q/A
t₃ = L/(60V)

An explanation of the parameters in these equations and their U.S. and S.I. units follows:

Q = channel flow rate in cfs for U.S. units or m³/s for S.I. units
V = average velocity of flow in ft/sec for U.S. units or m/s for S.I. units
R = hydraulic radius of the channel (= A/P) in ft for U.S. units or m for S.I. units
A = channel cross-sectional area in ft² for U.S. units or m² for S.I. units
P = wetted perimeter of channel in ft for U.S. units or m for S.I. units
S = channel bottom slope, which is dimensioness for either set of units
n = Manning roughness coefficient for channel
L = length of flow path in ft for U.S. units or m for S.I. units
t₃ = travel time for channel flow in min for either set of units

The screenshot of an Excel spreadsheet template shown at the right will calculate channel flow travel time with U.S. units using the NRCS method, based on the input values indicated. This Excel spreadsheet and others for time of concentration calculations are available in either U.S. or S.I. units at a very low cost at www.engineeringexceltemplates.com or in our spreadsheet store.

The overall time of concentration can now be calculated as the sum of t₁, t₂ and t₃.

References:

1. U.S. Soil Conservation Service, Technical Note – Hydrology No N4, June 17, 1986.

2. Iowa Stormwater Management Manual, Section on Time of Concentration.

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

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

5. Bengtson, Harlan H., “Spreadsheets for Rational Method Hydrological Calculations,” an Amazon Kindle e-book.

V Notch Weir Calculator Excel Spreadsheet

Posted on April 2, 2011 by Harlan Bengtson

Where to Find a V Notch Weir Calculator Excel Spreadsheet

To obtain a V notch weir calculator Excel spreadsheet, click here to visit our spreadsheet store. Why use online calculators or hand calculations when you can buy a V-notch weir calculator excel spreadsheet for only $11.95. Read on for information about Excel spreadsheets that can be used as v-notch weir open channel flow calculators.

As you can see in the diagrams and picture below, the name, v notch weir, is a good description of the device, simply a v shaped notch in a plate placed in an open channel so that the water is forced to flow through the v notch. It can be used to measure the open channel flow rate, because the height of water above the point of the v notch can be correlated with flow rate over the weir. The v-notch weir works well for measuring low flow rates, because the flow area decreases rapidly as the head over the v notch gets small.

Background for Sharp Crested Weirs

The v notch weir is only one of several possible types of sharp crested weirs. The image at the left shows a picture of a v-notch weir. Acknowledgement of Image Source: RS Hydro www.rshydro.co.uk The diagram below right shows a longitudinal cross-section of a sharp crested weir with several commonly used parameters identified on the diagram. The weir crest is the term used for the top of the weir. In the case of a v notch weir, the crest is the point of the v-shaped notch. The term nappe refers to the sheet of water flowing over the weir. The equations to be discussed in this article for calculating flow over a v-notch weir require free flow over the weir. This means that there must be air under the nappe, as shown in the diagram. The drawdown is the decrease in water level going over the weir caused by the acceleration of the water. The measurement, H, shown in the diagram is referred to as the head over the weir. P in the diagram is the height of the weir crest, and the open channel flow rate (also the flow rate over the weir) is shown as Q.

Picture Credit: U.S. Forest Service

A V Notch Weir Calculator Excel Spreadsheet for a 90 Degree Notch Angle

The equation shown below is recommended by the U.S. Dept. of the Interior, Bureau of Reclamation in their Water Measurement Manual (ref #1 below) for calculations with a fully contracted, 90^o, v notch, sharp crested weir with free flow conditions and 0.2 ft < H < 1.25 ft.

In U. S. units: Q = 2.49H^2.48, where Q is discharge in cfs and H is head over the weir in ft.

In S.I. units: Q = 1.36H^2.48, where Q is discharge in m³/s and H is head over the weir in m.

The conditions for the v notch weir to be fully contracted are:

H/P < 0.4, H/B < 0.2, P > 1.5 ft (0.45 m), B > 3 ft (0.9 m)

The diagram above shows the parameters H, P, θ and B for a v notch weir as used for open channel flow rate measurement in a v notch weir calculator excel spreadsheet.

Screenshot of a V Notch Weir Calculator Excel Spreadsheet

The screenshot below shows a v notch weir calculator excel spreadsheet for making 90^o, v-notch weir calculations in U.S. units. Based on specified values for H, P, & S, along with H_max, the maximum expected head over the weir, the spreadsheet checks on whether the required conditions for fully contracted flow are met and then calculates the flow rate, Q. This Excel spreadsheet and others for v notch weir calculations are available in either U.S. or S.I. units at a very low cost (only $11.95) in our spreadsheet store.

References:

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

2. Bengtson, Harlan H., Open Channel Flow III – Sharp Crested Weirs, an online continuing education course for PDH credit, http://www.online-pdh.com/engcourses/course/view.php?id=87

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

Partially Full Pipe Flow Calculator with Excel Spreadsheets

Posted on March 29, 2011 by Harlan Bengtson

Where to Find Partially Full Pipe Flow Calculator Spreadsheets

To obtain Excel spreadsheets for partially full pipe flow calculations, click here to visit our spreadsheet store for partially full pipe flow calculator spreadsheets. Read on for information about Excel spreadsheets that can be used as a partially full pipe flow calculator.

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(R^2/3)(S^1/2) for (U.S. units) or Q = (1.0/n)A(R^2/3)(S^1/2) for (S.I. units)] applies if the flow is uniform flow For background on the Manning equation and 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, however, 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/Q_full and V/V_full 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. Q_fulland V_full 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 Calculations?

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(R^2/3)(S^1/2)], 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 #1 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/n_full, in terms of y/D for y < D/2 are as follows

n/n_full = 1 + (y/D)(1/3) for 0 < y/D < 0.03
n/n_full = 1.1 + (y/D – 0.03)(12/7) for 0.03 < y/D < 0.1
n/n_full = 1.22 + (y/D – 0.1)(0.6) for 0.1 < y/D < 0.2
n/n_full = 1.29 for 0.2 < y/D < 0.3
n/n_full = 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, n_full; 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.

Excel Spreadsheet/Partially Full Pipe Flow Calculator for Pipe More than Half Full

The parameters used in partially full pipe flow calculations with the pipe more 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 more than pipe radius, as shown below.

h = 2r – y
θ = 2 arccos[ (r – h)/r ]
A = πr² – K = πr² – r²(θ – sinθ)/2
P = 2πr – S = 2πr – rθ

The equation used for n/n_full for 0.5 < y//D < 1 is: n/n_full = 1.25 – [(y/D – 0.5)/2]

An Excel spreadsheet like the one shown above for less than half full flow, and others for partially full pipe flow calculations, are available in either U.S. or S.I. units at a very low cost at www.engineeringexceltemplates.com.

References

1. Bengtson, Harlan H., Uniform Open Channel Flow and The Manning Equation, an online, continuing education course for PDH credit.

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

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

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

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

6. Bengtson, H.H., “Manning Equation Partially Filled Circular Pipes,” An online blog article

7. Bengtson, H.H., “Partially Full Pipe Flow Calculations with Spreadsheets“, available as an Amazon Kindle e-book and as a paperback.