A Spreadsheet for Choked Air Flow Calculation

Where to Find a Spreadsheet for Choked Air Flow Calculation

To obtain a spreadsheet for choked air flow calculations click here to visit our spreadsheet store.  Intended for use in making fanno flow calculations with adiabatic compressible flow of air, you can buy a convenient spreadsheet for choked air flow calculations for a very reasonable price.  This spreadsheet will also calculate the frictional pressure drop, and is available in either U.S. or S.I. units.  Read on for information about a choked air flow calculation spreadsheet for compressible pipe flow.

Conditions for Choked Air Flow Calculation with a Fanno Flow Excel Spreadsheet

For flow of air in a pipe with the frictional pressure drop less than 20% of the inlet air pressure, satisfactory results can be obtained using incompressible flow calculations with the Darcy-Weisbach equation.  For background information on that type of calculation, 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 discussed here using a Fanno Flow Excel spreadsheet.

With adiabatic compressible flow conditions, the maximum air flow rate that can pass through a pipe is that which occurs when the exit Mach number is 1.0.  This is referred to as “choked flow.”

Equations for Choked Air Flow Calculations with a Fanno Flow Excel Spreadsheet

Shown below are the primary equations used in the fanno flow excel spreadsheet being discussed here for compressible flow of air in a pipe.  These equations are for compressible pipe flow with negligible heat transfer with the surroundings, but including effects of friction in the flow.

Fanno Flow Excel Spreadsheet EquationsThe parameters in these fanno flow equations are as follows:

  • f = Moody friction factor
  • L = pipe length
  • D = pipe diameter
  • k = adiabatic constant for the flowing air
  • M = mach number
  • T = air temperature
  • P = air pressure
  • ρ = 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 Choked Air Flow Calculation Spreadsheet Using Fanno Flow Equations

The screenshot below shows part of a fanno flow excel spreadsheet for making compressible air flow calculations.  These calculations are for air flow in a pipe, in S.I. units. This Excel spreadsheet and others for compressible pipe flow calculations in a fanno flow excel spreadsheet are available in either U.S. or S.I. units at a very reasonable cost  in our spreadsheet store.Choked Air Flow Calculation Spreadsheet

Reference:

Bengtson, Harlan H.,  “Compressible Flow Pressure Drop Calculator Spreadsheet,” an online blog article

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Trickling Filter Design Calculations Spreadsheet

Where to Find a Trickling Filter Design Calculations Spreadsheet using the NRC Formula

To obtain a trickling filter design calculations spreadsheet using the NRC formula, click here to visit our spreadsheet store.  For use in making trickling filter design calculations with the NRC equation, you can buy a convenient trickling filter design calculations spreadsheet for a very reasonable price.  This spreadsheet will use the NRC equations to calculate the trickling filter diameter needed for either a single stage or a two stage system and will calculate the estimated treatment efficiency for a specified trickling filter system, in either U.S. or S.I. units.  Read on for information about a trickling filter design calculation spreadsheet that uses the NRC formula.

Trickling Filter Design Calculations Background

The trickling filter process is widely used for biological wastewater treatment all over the world.  It is an attached growth biological wastewater treatment process.  The aerobic microorganisms grow on a solid medium and the wastewater to be treated flows through the bed.  A flow diagram for a typical single stage trickling filter wastewater treatment plant is shown in the diagram below.  The diagram shows the typical flow pattern through the primary clarifier, trickling filter and secondary clarifier.

trickling filter design calculations spreadsheet flow diagram

Example Trickling Filter Design Calculations Spreadsheet

A trickling filter design calculations spreadsheet  is partially shown in the image below.  It can be used to calculate the trickling filter diameter needed for either a single stage or a two stage trickling filter system using the  NRC formula.  The spreadsheet will also calculate an estimated treatment efficiency for a specified single stage or two stage trickling filter system. This Excel spreadsheet, as well as others for wastewater treatment calculations, is available in either U.S. or S.I. units for a very reasonable price in our spreadsheet store.

Trickling Filter Design Calculations Spreadsheet Screenshot

References for Further Information:

1. Metcalf & Eddy, Inc, (revised by Tchobanoglous, G, Burton, F.L., Stensel, H.D., Wastewater Engineering Treatment and Reuse, 4th Edition, New York, NY, 2003.

2. Vesilind, P.A. and Morgan, S.M., Introduction to Environmental Engineering, 2nd Edition, Belmont, CA, Brooks/Cole, 2004.

3. Bengtson, H. H., “Trickling Filter NRC Equation Spreadsheet“, a blog article at www.EngineeringExcelTemplates.com

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A Spreadsheet to Calculate Oxygen Requirement Activated Sludge Process

Where to Find a Spreadsheet to Calculate Oxygen Requirement Activated Sludge Process

To obtain a spreadsheet to calculate oxygen requirement activated sludge processclick here to visit our spreadsheet store.  For use in calculating oxygen requirements and blower specifications, you can buy a convenient spreadsheet to calculate oxygen requirement activated sludge for a very reasonable price.  This spreadsheet will calculate the oxygen requirement and blower specifications for BOD removal or for BOD removal and nitrification for specified wastewater flow and characteristics and diffuser characteristics, in either U.S. or S.I. units.  Read on for information about Excel spreadsheets that can be used to calculate oxygen requirement activated sludge process parameters.

Oxygen Requirement Activated Sludge Background

The theoretical oxygen requirement for BOD removal can be approximated as ranging from  0.90 lb O2/lb BOD removed at an SRT of 5 days to 1.3 lb O2/lb BOD removed at an SRT of 20 days.  This leads to the equation:

O2 requirement in lb/day  =  0.90  +  [(SRT - 5)/(20 - 5)](1.3 – 0.9)(lb/day BOD removed)

A design increase factor (safety factor) is typically multiplied times the calculated oxygen requirement to get the design oxygen transfer rate.  The design oxygen transfer efficiency and the air density at design conditions can then be used to calculate the design air flow rate.  The blower horsepower can then be calculated from the following equation:

Blower hp  =  (Qair)(ΔP)/(229*η)

Where:

  • Qair is the air flow rate in cfm to be delivered by the blower
  • ΔP is the pressure rise across the blower in psi  = outlet pressure – inlet pressure
  • η is the blower efficiency

If oxygen is to be provided for nitrification as well as BOD removal, then additional calculations are needed to estimate the oxygen requirement for nitrification.

Example Spreadsheet to Calculate Oxygen Requirement Activated Sludge Process

A spreadsheet to Calculate oxygen requirement activated sludge process is partially shown in the image below.  It can be used to calculate the oxygen requirement and blower specifications for an activated sludge wastewater treatment system.  This Excel spreadsheet, as well as others for wastewater treatment calculations, is available in either U.S. or S.I. units for a very reasonable price in our spreadsheet store.

A spreadsheet to calculate oxygen requirement activated sludge process

Reference:  Bengtson, Harlan H., “Activated Sludge Oxygen Requirement,”  an online blog article.

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Sequencing Batch Reactor Design Calculation Spreadsheet

Where to Find a Sequencing Batch Reactor Design Calculation Spreadsheet

To obtain a Sequencing Batch Reactor Design Calculation excel spreadsheet, click here to visit our spreadsheet store.  For use in wastewater treatment design calculations, you can buy a convenient sequencing batch reactor design calculation excel spreadsheet for a very reasonable price?  This spreadsheet will make calculations for BOD removal, nitrification, denitrification, and biological phosphorus removal for given information about wastewater flow rate and characteristics, in either U.S. or S.I. units.  Read on for information about Excel spreadsheets that can be used for SBR design calculations.

Sequencing Batch Reactor Design Calculation Background

A typical activated sludge wastewater treatment process operates as a continuous flow process, with incoming wastewater flow coming into a primary clarifier and treated effluent continuously coming off from the secondary clarifier.  A sequencing batch reactor wastewater treatment system, on the other hand, operates as a batch system.  Two or more tanks are required.  While one tank is receiving influent wastewater (the “fill” part of the cycle), another tank is undergoing aeration (the “react” part of the cycle), settling (the “settle” part of the cycle) and decanting of treated effluent (the “decant” part of the cycle).  This is illustrated in the diagram below.

Sequencing Batch Reactor Design Calculation Cycle Diagram

Sequencing Batch Reactor Design Calculation Applications

A sequencing batch reactor wastewater treatment system has a great deal of flexibility.  It can be used for traditional BOD removal and nitrification using the four cycle components shown above.  In that case there may be aeration for at least part of the fill cycle.  If denitrification is to be accomplished also, then there should be no aeration during the fill cycle.  If the SBR wastewater treatment system is to be designed for biological phosphorus removal as well, then an anaerobic react period is needed after the fill portion of the cycle, and an anoxic react is needed after the aerobic react part of the cycle, as shown in the diagram below.

Sequencing Batch Reactor Design Calculation Diagram for phosphorus removal

Example Sequencing Batch Reactor Design Calculation Excel Spreadsheet

The Sequencing Batch Reactor Design Calculation excel spreadsheet partially shown in the image below can be used to make a variety of design calculations for an SBR wastewater treatment system.  Based on input information about the wastewater flow rate and characteristics, as well as the treatment objectives, the spreadsheet leads the user through calculations for deciding on times for each part of the SBR cycle, tank number and size, and checks on the adequacy of the design.  This Excel spreadsheet, as well as others for wastewater treatment calculations, is available in either U.S. or S.I. units for a very reasonable price in our spreadsheet store.

Sequencing Batch Reactor Design Calculations Spreadsheet Screenshot

Reference:

Bengtson, Harlan H.,  “SBR Wastewater Treatment Plant Design Spreadsheet,” an online blog article.

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Wastewater Neutralization Calculations Spreadsheets

Where to Find a Wastewater Neutralization Calculations Spreadsheet

To obtain a Wastewater Neutralization Calculations excel spreadsheet, click here to visit our spreadsheet store.  For use in calculating dosage of acid or caustic to neutralize wastewater, you can buy a convenient wastewater neutralization calculations excel spreadsheet for a very reasonable price?  This spreadsheet will calculate the dosage and flow rate of acid or caustic needed for a given pH change in a given flow of wastewater in either U.S. or S.I. units.  Read on for information about Excel spreadsheets that can be used for wastewater neutralization calculations.

Wastewater Neutralization Calculations Background

The chemical typically used to adjust wastewater pH upward when it is too low is caustic soda (sodium hydroxide – NaOH).  On the other hand, if the wastewater pH is too high, sulfuric acid (H2SO4) is commonly used to bring the pH down.  Sodium hydroxide is a strong base, so it ionizes virtually completely to Na+ and OH- ions in a water solution.  Similarly, sulfuric acid is a strong acid, so it ionizes virtually completely to H+ and SO4= ions in a water solution.

Equations for Wastewater Neutralization Calculations

The equations relating pH to hydrogen ion concentration, [H+], and relating pOH to hydroxide ion concentration, [OH-], are needed in order to make wastewater neutralization calculations.  These equations can each be written in two ways.  The definition of pH is:  pH  =  – log[H+], which can also be written as:  [H+] = 10-pH.  Also, the definition of pOH is:  pOH  =  – log[OH-], which can be written as:  [OH-]  =  10-pOH.

Example Wastewater Neutralization Calculations Excel Spreadsheet

The Wastewater Neutralization Calculations excel spreadsheet partially shown in the image below can be used to calculate the acid dosage or the caustic dosage needed for a specified pH change in a specified flow rate of wastewater, as discussed above.  Based on input information about the caustic soda or sulfuric acid to be used, the spreadsheet calculates the daily flow needed and the daily cost of the chemical.  This Excel spreadsheet, as well as others for wastewater treatment calculations, is available in either U.S. or S.I. units for a very reasonable price in our spreadsheet store.

Wastewater Neutralization Calculations Spreadsheet

Reference

Bengtson, Harlan H., “Wastewater Neutralization with an Excel Spreadsheet,”  an informational online blog article.

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Design Storm Hyetograph Generation Spreadsheet

Where to Find a Design Storm Hyetograph Generation Spreadsheet

To obtain a Design Storm Hyetograph Generation excel spreadsheet, click here to visit our spreadsheet store.  For use in stormwater management calculation, you can buy a convenient Design Storm Hyetograph generation excel spreadsheet for a very reasonable price?  This spreadsheet will generate a design storm hyetograph using the Chicago storm model, a triangular storm model or the “alternating blocks” procedure in either U.S. or S.I. units.  Read on for information about Excel spreadsheets that can be used for design storm hyetograph generation.

Models for Design Storm Hyetograph Generation

Several different hyetograph models can be used for design storm hyetograph generation, including the Chicago storm, triangular, or rectangular (constant intensity design storm) models or the “alternating blocks” procedure for constructing a design storm hyetograph.  An initial step typically needed is the generation of an equation for storm intensity as a function of storm duration at the design location, for the design recurrence interval.

The Chicago Storm Hyetograph

Chicago design storm hyetograph equationFor example, the Chicago storm hyetograph model uses the equation at the left for the portion of the hyetograph before the peak storm intensity.  A slightly different equation is used for the portion of the design Chicago Design Storm Hyetograph Examplestorm hyetograph that is after the peak storm intensity.  The resulting hyetograph has the general shape shown in the diagram at the right.  A user specified parameter is r, which is the fraction of the hyetograph that is before the point of peak storm intensity.  The triangular hyetograph model is similar in shape, but the lines before and after the peak storm intensity are straight instead of curved.

Example Design Storm Hyetograph Generation Excel Spreadsheet

The Design Storm hyetograph generation excel spreadsheet partially shown in the image below can be used to generate a triangular or Chicago storm hyetograph as discussed above.  The portion shown is for generating an equation for storm intensity as a function of storm duration.  This Excel spreadsheet, as well as others for stormwater management calculations, is available in either U.S. or S.I. units for a very reasonable price in our spreadsheet store.

Design Storm Hyetograph Generation Spreadsheet Screenshot

References

1. American Iron and Steel Institute, Modern Sewer Design, 4th Edition, 1999.

2. Bengtson, Harlan H., “Chicago Storm Hyetograph Generation Spreadsheet,”  an online informational blog article.

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Air Viscosity Temperature Calculator Spreadsheet

Where to Find an Air Viscosity Temperature Calculator Spreadsheet

To obtain an Air Viscosity Temperature Calculator excel spreadsheet, click here to visit our spreadsheet store.  Why use online calculators or tables to find the viscosity of air at a specified pressure and temperature when you can buy a convenient air viscosity temperature calculator excel spreadsheet for only $4.95?  This spreadsheet will calculate the viscosity of air at specified pressure and temperature in either U.S. or S.I. units.  Read on for information about Excel spreadsheets that can be used as an Air viscosity temperature calculator for specified pressure and temperature.

Air Viscosity Temperature Calculator Spreadsheet Applications

An  Air Viscosity Temperature calculator excel spreadsheet  can be used for any situation where a value of air viscosity is needed at a specified pressure and temperature.  This could include calculations for air flow in a pipe, drag force or drag coefficient calculations for flow of an object through air, and any other calculation requiring the Reynolds number for air flow or flow through air.  For example, see the related article, Fanno Flow Excel Spreadsheet for Air Flow in a Pipe.

Equations for an Air Viscosity Temperature Calculator Spreadsheet

Equations are available for an air viscosity temperature calculator to calculate the viscosity of air at specified temperature and pressure.  The spreadsheet shown in  the diagram below calculates air density using an equation for air viscosity as a function of temperature ratio, Tr , and density ratio, ρr  , where in U.S. units:  Tr   =  T/238.5 with T in degrees R  and ρr    =  ρ/0.6096 with ρ in slugs/ft3.  Since the air density is needed for this calculation, the spreadsheet also calculates the density of air at the specified air temperature and pressure.  The complete equations are included in the spreadsheet discussed above and shown in the screenshot below.

Example Air Viscosity Temperature Calculator Excel Spreadsheet

The Air Viscosity Temperature calculator excel spreadsheet shown in the image below can be used to calculate the viscosity of air at given temperature and pressure as discussed above.  This Excel spreadsheet and others for fluid properties calculations, in either U.S. or S.I. units are available for very reasonable prices in our spreadsheet store.

Air Viscosity Temperature Calculator Spreadsheet

References

1. Bengtson, Harlan H, “Air Viscosity Calculator Pressure Temperature Spreadsheet,”  An online informational blog article.

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Manning Equation Open Channel Flow Calculator Excel Spreadsheets

Where to Find a Manning Equation Open Channel Flow Calculator Spreadsheet

To obtain a Manning equation open channel flow calculator excel spreadsheet, click here to visit our spreadsheet store.  Why use online calculators or make open channel flow/Manning Equation calculations by hand when you can buy a variety of Manning equation open channel flow calculator excel spreadsheets or spreadsheet packages for prices ranging from $6.95 to $27.95?  Read on for information about Excel spreadsheets that can be used as a Manning equation open channel flow calculator.

picture for a Manning equation open channel flow calculatorAn excel spreadsheet can conveniently be used as a Manning equation open channel flow calculator.  The Manning equation can be used for water flow rate calculations in either natural or man made open channels.  Uniform open channel flow calculations with the Manning equation use the channel slope, hydraulic radius,  flow depth, flow rate, and Manning roughness coefficient.   Image Credit: geograph.org.uk

Uniform Flow for a Manning Equation Open Channel Flow Excel Spreadsheet

Diagram for a Manning Equation Open Channel Flow Calculator SpreadsheetOpen channel flow may be either uniform flow or nonuniform flow, as illustrated in the diagram at the left.  For uniform flow in an open channel, there is always a constant volumetric flow of liquid through a reach of channel with a constant bottom slope, surface roughness, and hydraulic radius (that is constant channel size and shape).  For the constant channel conditions described, the water will flow at a constant depth (usually called the normal depth) for the  particular volumetric flow rate and channel conditions. The diagram above shows a stretch of uniform open channel flow, followed by a change in bottom slope that causes non-uniform flow, followed by another reach of uniform open channel flow.  The Manning Equation, which will be discussed in the next section, can be used only for uniform open channel flow.

Equation and Parameters for a Manning Equation Open Channel Flow Calculator Excel Spreadsheet

The Manning Equation is:

Q = (1.49/n)A(R2/3)(S1/2) for the U.S. units shown below, and it is:

Q = (1.0/n)A(R2/3)(S1/2) for the S.I. units shown below.

  • Q is the volumetric water flow rate in the reach of channel (ft3/sec for U.S.) (m3/s for S.I.)
  • A is the cross-sectional area of flow  (ft2for U.S.) (m2for S.I.)
  • P is the wetted perimeter of the flow  (ft for U.S.)  (m for S.I.)
  • R is the hydraulic radius, which equalsA/P(ft for U.S.) (m for S.I.)
  • S is the bottom slope of the channel, (dimensionless or ft/ft -U.S. & m/m – S.I.)
  • n is the empirical Manning roughness coefficient, which is dimensionless

The equation V = Q/A, a definition for average flow velocity, can be used to express the Manning Equation in terms of average flow velocity,V, instead of flow rate,Q, as follows:

V = (1.49/n)(R2/3)(S1/2) for U.S. units with V expressed in ft/sec.

Or V = (1.0/n)(R2/3)(S1/2) for S.I. units with V expressed in m/s.

It should be noted that the Manning Equation is an empirical equation.  The U.S. units must be just as shown above for use in the equation with the constant 1.49 and the S.I. units must be just as shown above for use in the equation with the constant 1.0.

The Manning Roughness Coefficient for a Manning Equation Open Channel Flow Calculator Excel Spreadsheet

Manning Equation Open Channel Flow Calculator Manning Roughness CoefficientsAll calculations with the Manning equation (except for experimental determination of n) require a value for the Manning roughness coefficient, n, for the channel surface.  This coefficient, n, is an experimentally determined constant that depends upon the nature of the channel and its surface.  Smoother surfaces have generally lower Manning roughness coefficient values and rougher surfaces have higher values. Many handbooks, textbooks and online sources have tables that give values of n for different natural and man made channel types and surfaces. The table at the right gives values of the Manning roughness coefficient for several common open channel flow surfaces for use in a Manning equation open channel flow calculator excel spreadsheet.

Example Manning Equation Open Channel Flow Excel Spreadsheet

The Manning equation open channel flow calculator excel spreadsheet shown in the image below can be used to calculate flow rate and average velocity in a rectangular open channel with specified channel width, bottom slope, & Manning roughness, along with the flow rate through the channel.  This Excel spreadsheet and others for Manning equation open channel flow calculations for rectangular, trapezoidal or triangular channels, in either U.S. or S.I. units are available for very reasonable prices in our spreadsheet store.

Manning Equation Open Channel Flow Calculator Excel Spreadsheet

References

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

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

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

4.  Bengtson, Harlan H., “Manning Equation Open Channel Flow Excel Spreadsheets,”  an online blog article, 2012.

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Backwater Curve Calculations Spreadsheet

Where to Find a Backwater Curve Calculations Spreadsheet

To obtain a backwater curve calculations spreadsheet to calculate surface profiles for non uniform open channel flow, click here to visit our spreadsheet store.  Obtain a convenient, easy to use backwater curve calculations spreadsheet at a reasonable price.  Read on for information about the use of an Excel spreadsheet for non uniform flow open channel surface profile step wise calculations.

Background on Non Uniform and Uniform Open Channel Flow

uniform nonuniform flow diagram - backwater curve calculations spreadsheetsThe diagram at the right illustrates uniform and nonuniform open channel flow.  Uniform flow in an open channel consists of a constant volumetric flow of liquid through a reach of channel with a constant bottom slope, surface roughness, and hydraulic radius (that is constant channel size and shape).  For those constant channel conditions, the water will flow at a constant depth, called the normal depth, for the  particular channel conditions and volumetric flow rate. The diagram shows a reach of uniform open channel flow, followed by a change in bottom slope that causes non-uniform flow, ending with another reach of uniform open channel flow.  This article is about means of calculating the surface profile (depth vs distance down the channel) for a reach of non uniform flow.


Classifications of Non Uniform Open Channel Flow for a Backwater Curve Calculations Spreadsheet

backwater curve calculations spreadsheet - uniform surface profile typesClassifications of Non Uniform Open Channel Flow (Mild or Steep Channel Slope)

The diagram above shows the three possible non uniform flow patterns for a mild slope (channel slope less than the critical slope) and the three for a steep slope (channel slope greater than the critical slope).  The three mild slope classifications are M1, M2, and M3.  The “M” indicates mild slope and the number shows the relationship among depth of flow, y, critical depth, yc, and normal depth, yo , as shown in the diagram.  Similarly the three steep slope classifications are S1, S2, and S3, with the numbers having the same meaning.  The diagram shows a typical physical situation that will give rise to each of these six types of non uniform open channel flow.

The Energy Equation for a Backwater Curve Calculations Spreadsheet

The energy equation (the first law of thermodynamics applied to a flowing fluid), which has many applications in fluid mechanics, can be used for non uniform open channel flow surface profile stepwise calculations.  The diagram below shows the parameters that will be used at each end of a reach of channel with non uniform flow.

Backwater curve calculations spreadsheet - non uniform flow parametersA Reach of Open Channel with Non Uniform Flow

The energy equation written across a reach of channel is illustrated graphically in the diagram above.  The sum of the three items on the upstream end of the channel reach must equal the sum of the three items on the downstream end of the channel reach, giving the equation:

Where the parameters in the equation are as follows:

  • y1 =  the upstream depth of flow in ft (m for S.I. units)
  • y2 =  the downstream depth of flow in ft (m for S.I. units)
  • V1 =  the upstream average velocity in ft/sec (m/s for S.I. units)
  • V2 =  the downstream average velocity in ft/sec (m/s for S.I. units)
  • g  =  the acceleration due to gravity  =  32.17 ft/sec2 (9.81 m/s2 for S.I. units)
  • ΔL  =  the horizontal length of the channel reach in ft (m for S.I. units)
  • So =  the bottom slope of the channel, which is dimensionless
  • Sf =  the slope of the energy grade line (thus head loss is hL = SfΔL)

For specified flow rate, Q, channel bottom slope, So , Manning roughness coefficient, n, and channel width for a rectangular channel, the energy equation can be used to calculate the length, ΔL, for transition from a known upstream depth, y1 , to a selected downstream depth, y2 .  This process can be repeated as many times as necessary to determine the total distance to a specified downstream depth.

The energy equation can be rearranged to give the following equation for ΔL:

The Manning equation is typically used to calculate the slope of the energy grade line, Sf .  Although the Manning equation only applies for uniform flow, the use of mean cross-sectional area and mean hydraulic radius with a relatively small step for the calculation gives a good approximation.  The equation for Sf is as follows:

Sf =  {Qn/[1.49Am(Rhm2/3)]]}2, where  Am is the mean area and Rhmis the mean hydraulic radius between sections 1 and 2.  For S.I. units, the 1.49 constant in this equation becomes 1.00.

Screenshot of a Backwater Curve Calculations Spreadsheet

Consider a 20 ft wide rectangular channel with bottom slope equal to 0.0003, carrying 1006 cfs.  The normal depth for this flow is 10 ft.   An M1 backwater curve is generated due to a downstream obstruction.  Calculate the channel length for the transition from a depth of 12 ft to a depth of 12.5 ft in this backwater curve.

Solution: The spreadsheet shown in the screenshot below shows the solution.  It actually has the entire M1 curve from a depth of 10 ft to a depth of 16 ft.  It shows DL for the transition from 12 ft depth to 12.5 ft depth to be 3853 ft.

The Excel spreadsheet template shown above can be used to calculate an M1 surface profile for a rectangular channel with specified flow rate, bottom width, bottom slope, and Manning roughness coefficient.  Why bother to make these calculations by hand?  This backwater curve calculations spreadsheet and others with similar calculations for a trapezoidal channel, and for any of the six mild or steep nonuniform flow surface profiles 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, 4th 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 FlowAn online, continuing education course for PDH credit.

4.  Bengtson, Harlan H., “Non Uniform Flow in Open Channels“, an online blog article

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C-Channel Bending Stress and Torsional Stress Spreadsheet

Where to Find a  C-Channel Bending Stress and Torsional Stress Spreadsheet

For a C-channel bending stress and torsional stress spreadsheetclick here to visit our spreadsheet store.  Obtain a convenient, easy to use C-channel bending stress and torsional stress spreadsheet at a reasonable price. Read on for information about calculating combined bending and torsional warping stresses in light-gage steel C-channel bending members.

Background for Calculations with Combined Bending and Torsional Warping Stresses.

The usual calculation of stresses in bending members applies to loads that act along one of the principal axes passing through both the centroid and the shear center. For symmetrical sections, the centroid and the shear center coincide and this requirement is easily accomplished.  If this is the case, the beam will deflect in the plane of these applied loads without any rotational displacements. Many light-gage structural bending members are shaped into configurations that have cross sections where the shear center and the Diagram for C-channel bending stress and torsional stress spreadsheetcentroid do not coincide.  In these cases, if the loads do not pass through the shear center, the beam will rotate as well as displace along the direction of the applied loads.  C-shaped channel sections fall into this category.  A realistic analysis of these shapes must calculate both the usual bending stresses and torsional stresses.  The torsional warping stresses act in the same direction as the bending stresses.  The bending stresses vary from top to bottom of the section with the maximum stresses equally distributed across the top and bottom flanges .The torsional moments will cause sideways bending of the flanges with maximum warping stresses occurring at the ends of the flanges. This is shown in the figure at the left above.  The maximum combined stresses therefore may occur at either of the points A. B or C. a

Calculations for a  C-Channel Bending Stress and Torsional Stress Spreadsheet

The current edition of the AISI Specification for the Design of Cold-Formed Steel Members includes a requirement that combined bending and warping stresses be calculated in order to determine the maximum combined stress.  From these combined stresses a reduction factor is calculated.  This factor reduces the moment capacity established by bending alone.  This reduction factor assures that the maximum combined stresses will not cause premature failure of the beam.

The calculation of these maximum combined stresses is time consuming.  The spreadsheet available at www.EngineeringExcelSpreadsheets.com does these calculations for C-shaped members subject to the common load case of a uniformly distributed load.  Three possibilities are considered:

  1. Top flange of the member braced against torsion only at its end supports.
  2. Top flange of the member braced at its ends and mid-span.
  3. Top flange braced at its ends and third-points.

For a C-shape, the centroid of the cross-section is on the same side of the web as the flanges.  The shear center is on the opposite side of the web from the flanges and hence the centroid.

Vertical loads applied to the top flange of the channel will not pass through the shear center.  The stresses in this case may be calculated by analyzing the member for bending alone then combining these with the torsional warping stresses calculated independently.  The torsional warping stresses are in line with the member similar to the bending stresses. They can be combined algebraically.

loading diagram for c-channel bending stress and torsional stress spreadsheet

Equations for a  C-Channel Bending Stress and Torsional Stress Spreadsheet

Torsional analysis requires calculation of the angle of rotation and its second derivative along the length of the beam.  Derivation of this approach may be found in the references. Reference 1 provides formulas for the angle of rotation for 12 loading and boundary support conditions.  In this spreadsheet, we assume simple torsional support for the boundary conditions at the end supports of the beam. This means the cross-section can warp freely at the ends of the member and the warping normal stresses are zero. If the member is part of a continuous beam, the user may input the beam end moments from separate analysis. In Reference 1, Case 4 applies to a uniformly distributed applied torsional moment along the span.  Case 3 applies to a concentrated applied torsion moment applied anywhere along the length of the beam.  These are the basis of this spreadsheet.

The formulas for the angle of rotation, Ɵ, are as follows:

Case 3 Concentrated torsional moment applied at a point αL from the left support with pinned end boundary condition:

Ɵ = TL/GJ [(1.0 – α)(z/L) + ((sinh αL/a)/(tanh L/a) – (cosh αL)) (a/L)sinh z/a)]

Case 4 Uniformly distributed torsional moment along length of beam with pinned end boundary condition:

Ɵ = (t a2/GJ )[(L2/a2)(z/L – z2/L2) + cosh z/a – (tanh L/2a) (sinh z/a – 1.0)]

The terms in these formulas and their units are described in the spreadsheet.

The spreadsheet consists of four sheets.  Each is for one of the loadings shown above.  Only the first sheet requires input data.  The user must place this required data in the amber colored cells.  All other cells are locked to protect the integrity of the spreadsheet. The final results for all possible bracing conditions are summarized on the last sheet.  The user selects the proper solution for the actual bracing condition.  This c-channel bending stress and torsional stress spreadsheet is available at www.EngineeringExcelTemplates.com.

References:

1. Heins, C.P. and P.A. Seaburg, “Torsional Analysis of Rolled Steel Sections”, Bethlehem Steel Design File, 1963.

2. Galambos, T.V., “Structural Members and Frames”, Prentice Hall, 1968.

3. Yu, Wei-Wen, “Cold-Formed Steel Design”, John Wiley & Sons, 1991.

4. Seaburg, P.A., “C Channel Beam Design Spreadsheet“, an online blog article.

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