• Negative Pressure Head

© 2014 LMNO Engineering,Make selections:Research, and Software, Ltd.Density, ρ:Flowrate, Q:Fluid Velocity, V:Downstream Velocity, V 2:ft/sPressure Difference, P 2-P 1:Elevation Difference, Z 2-Z 1:Pipe Diameter, D:Downstream Diameter, D 2:inchPipe Area, A:Downstream Area, A 2:ft2Units in Bernoulli Applications calculator:cm=centimeter, ft=foot, g=gram, gal=U.S. Gallon, in=inch, kg=kilogram, lb=pound, m=meter, mbar=millibar,min=minute, mm=millimeter, N=Newton, s=second.Bernoulli EquationTopics:IntroductionThe Bernoulli equation is named in honor of Daniel Bernoulli (1700-1782). Manyphenomena regarding the flow of liquids and gases can be analyzed by simply using theBernoulli equation. However, due to its simplicity, the Bernoulli equation may notprovide an accurate enough answer for many situations, but it is a good place tostart. It can certainly provide a first estimate of parameter values.Modifications to the Bernoulli equation to incorporate viscous losses, compressibility,and unsteady behavior can be found in other (more complex) calculations on this websiteand in the links shown at the right. When viscous effects are incorporated, the resultingequation is called the 'energy equation'.The Bernoulli equation assumes that your fluid and device meet four criteria:1.

Fluid is incompressible, 2. Fluid is inviscid, 3. Flow is steady, 4.Flow is along a streamline.The Bernoulli equation is used to analyze fluid flow along a streamline from a location1 to a location 2. Most liquids meet the incompressible assumption and many gasescan even be treated as incompressible if their density varies only slightly from 1 to 2.The steady flow requirement is usually not too hard to achieve for situationstypically analyzed by the Bernoulli equation.

Steady flow means that the flowrate(i.e. Discharge) does not vary with time. The inviscid fluid requirement impliesthat the fluid has no viscosity. All fluids have viscosity; however, viscous effectsare minimized if travel distances are small.To aid in applying the Bernoulli equation to your situation, we have included manybuilt-in applications of the Bernoulli equation.

They are described below. Foradditional information about the Bernoulli equation and applications, please see the at the bottom of this page.ApplicationsPitot TubeA pitot tube is used to measure velocity based on a differential pressure measurement.The Bernoulli equation models the physical situation very well. In theBernoulli equation, Z 2=Z 1 and V 2=0 for a pitottube. A pitot tube can also give an estimate of the flowrate through a pipe or ductif the pitot tube is located where the average velocity occurs (average velocity timespipe area gives flowrate). Oftentimes, pitot tubes are negligently installed in thecenter of a pipe. This would give the velocity at the center of the pipe, which isusually the maximum velocity in the pipe, and could be twice the average velocity.Dam (or weir)Using the Bernoulli equation to determine flowrate over a dam assumes that the velocityupstream of the dam is negligible (V 1=0) and that the nappe is exposed toatmospheric pressure above and below.

Experiments have shown that the Bernoulliequation alone does not adequately predict the flow, so empirical constants have beendetermined which allow better agreement between equations and real flows. To obtainbetter accuracy than the Bernoulli equation alone provides, use our weir calculations (, ).Sluice gateA sluice gate is often used to regulate open channel flows, and the Bernoulli equationdoes an adequate job of modeling the situation. A may or may notoccur downstream of a sluice gate. Be sure that Z 2 is not measured downstream of a hydraulic jump.The Bernoulli equation cannot be usedacross hydraulic jumps since energy is dissipated. Usually for sluice gates Z 1Z 2,so the Bernoulli equation can be simplified to Q = Z 2 W (2 g Z 1) 1/2(Munson et al., 1998) - which is the equation used in our calculation.Circular hole in tank (or pipe connected to tank)Non-circular hole in tank (or duct, i.e. Non-circular conduit, connected to tank)Diagrams showing some situations which can be modeled with these twoselections:The Bernoulli equation does not account for viscous effects of the holes in tanks orfriction due to flow along pipes, thus the flowrate predicted by our Bernoulli equationcalculator will be larger than the actual flow.

V 1 is automatically setto 0.0 - implying that location 1 is a device that has a large flow area so that thevelocity at location 1 (e.g. In a tank) is negligible compared to the velocity of fluidleaving the tank.

For comprehensive calculations which include viscous effects, trythe following calculations:,.Circular pipe diameter change, Non-circular duct area changeVenturi flow meter (C=0.98), Nozzle flow meter (C=0.96),and Orificeflow meter (C=0.6)This selection is useful for determining the change in static pressure in a pipe due toa diameter change, determining flowrate, or designing a flow meter. Locations 1 and2 should be as close together as possible; otherwise, viscous effects due to pipe frictionwill impact the pressure. However, flow meters normally have specified locations forthe pressure taps.If you select 'Solve for D, W, or A', the diameter and/or or area at location2 (D 2 and/or A 2) will be computed.

For all but the flowmeters, if you instead need to compute the diameter (or area) at location 1 (D 1,A 1), you can 'fool' the calculation by reversing the signs on yourpressure and elevation differences and enter the diameter (area) at location 2 as D 1(or A 1). Then, the D 2 (or A 2) computed willactually be at location 1. You cannot do this for the flow meters since they requirethat D 1 is greater than D 2.The venturi flowmeter analysis is based on the Bernoulli equation except for anempirical coefficient of discharge, C. V 2 in the equation at the top ofthis page is known as the theoretical throat velocity. In our calculation, thevelocity that is output for V 2 is the actual throat velocity, CV 2.Flowrate is computed as Q=CV 2A 2 (Munson et al.

1998) and A 2=πD 2 2/4. For simplicity, our Bernoulli venturi calculation usesa fixed value of C=0.98. However, it is well known that C is not fixed at 0.98 butvaries as a function of Reynolds number and the material from which the meter isconstructed. Also, most relationships for C are only valid for certain ranges of D 1,D 2 and D 2/D 1. For a more rigorous and accurateventuri calculation (yet one that has limits on some of the variables), please visit ourcomprehensive.For nozzle and orifice flow meters, Z 2-Z 1 is fixed at 0.0 sincethese meters are typically installed in horizontal pipes (or the elevation differencebetween locations 1 and 2 is negligible).

Nozzle and orifice meters tend to have agreater impact on the flow (greater energy loss) than venturi meters reflected bygenerally lower C values. The C value is incorporated into the Bernoulli equation asdescribed above in the above paragraph for venturi meters. The C values of 0.96 and0.6 are typical values that can be used for nozzles and orifices but will producesubstantial error for certain Reynolds numbers and geometries since C actually is afunction of pressure tap locations, Reynolds number, diameter ratio, and pipe diameter.For more rigorous and accurate equations and computations (yet ones that havelimits on some of the variables), use our comprehensive nozzle and orifice calculations (, ).Variables(dimensions shown in. F=Force units, L=Length units, M=Mass units, T=Time units, where F=M-L/T 2)Subscripts:1 indicates upstream location2 indicates downstream location.A = Cross-sectional area (i.e. Area normal to flow direction) L 2.If pipe is circular, then A= π D 2/4D = Diameter L. For flow meters, D 2 is the flow meter's throatdiameter.g = Acceleration due to gravity = 9.8066 m/s 2 = 32.174 ft/s 2.The calculation converts all input variables to SI, performs computations, then convertsoutput variables to specified units.H = Water depth above top of dam L.P = Pressure F/L 2.Q = Flowrate (i.e.

Discharge) L 3/T.W = Channel width L.Z = Elevation L.p = Mass density of fluid (Greek letter'rho') M/L 3. Densities built-into the calculation are atstandard atmospheric pressure which is 1 atmosphere (atm). Note that 1 atm=101,325N/m 2=1.01325 bar=760 mm Hg (0 oC)=2116 lb/ft 2=14.7 psi=30.01 inch Hg(60 oF)=29.92 inch Hg (32 oF).Error Messages given by calculationMessages are shown if you solve for a variable that is not used for theapplication. For example, you cannot solve for pressure difference across a sluicegate since the pressure difference is, by default, 0.0 since both the upstream anddownstream water surfaces are exposed to atmospheric pressure.Messages are also shown if a variable is entered as negative when it must be positive,such as a diameter. Additionally, a message will be shown if entered values resultin a physically infeasible (impossible) situation - such as flow moving upward in acontracting pipe and you entered a positive value for the pressure difference, P 2-P 1.The pressure difference would have to be negative in order to have upward flow in acontracting pipe (it may need to be quite a bit negative if Z 2-Z 1 islarge).ReferencesCitedMunson, B. Young, and T.

Fundamentals of FluidMechanics. John Wiley and Sons, Inc.

3ed.Useful ResourcesGerhart, P. Gross, and J. Fundamentals of FluidMechanics.

Addison-Wesley Publishing Co. 2ed.Potter, M. Mechanics of Fluids.Prentice-Hall, Inc.Roberson, J. Engineering Fluid Mechanics.Houghton Mifflin Co.Streeter, V. Wylie, and K. Fluid Mechanics.WCB/McGraw-Hill.

Negative Pressure Head

8ed.White, F. Fluid Mechanics. McGraw-Hill, Inc.© 2000-2014 LMNO Engineering, Research, and Software, Ltd. (AllRights Reserved)Please contact us for consulting or other questions.LMNO Engineering, Research, and Software, Ltd.7860 Angel Ridge Rd. Athens, Ohio 45701 USA Phone: (740) 592-1890To:More accurate calculations for specific applications:Tank discharge calculations:Pipe flow:Relief valve:Others.