Bernoulli Equation Calculator with Applications 
Pitot tube, dam, sluice gate, tank discharge, pipe expansion, orifice, nozzle, venturi. Bernoulli equation provides a first estimate of flow, pressure, elevation, or diameter. Assumptions: No viscous effects, fluid is incompressible, flow is steady. 
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Topics: Introduction Applications Variables Error Messages References
Introduction
The Bernoulli equation is named
in honor of Daniel Bernoulli (17001782). Many phenomena regarding the
flow of liquids and gases can be analyzed by simply using the Bernoulli
equation. However, due to its simplicity, the Bernoulli equation may not
provide an accurate enough answer for many situations, but it is a good place to
start. 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 website and in the links shown above. When
viscous effects are incorporated, the resulting equation 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 location 1 to a location 2. Most liquids meet the incompressible assumption and many gases can 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 situations typically analyzed by the Bernoulli equation. Steady flow means that the flowrate (i.e. discharge) does not vary with time. The inviscid fluid requirement implies that the fluid has no viscosity. All fluids have viscosity; however, viscous effects are minimized if travel distances are small. Flow along a 100 km river has significant viscous effects (friction between the channel material and the flowing fluid), but viscous effects along just a 10 m reach of that channel where a sluice gate occurs would be minimal. Sluice gates are typically analyzed with the Bernoulli equation (Munson et al., 1998).
To aid in applying the Bernoulli equation to your situation, we have included
many builtin applications of the Bernoulli equation. They are described
below. For additional information about the Bernoulli equation and
applications, please see the references at
the bottom of this page.
Pitot Tube
A pitot tube is used to measure velocity based on a differential pressure measurement. The Bernoulli equation models the physical situation very well. In the Bernoulli equation, Z_{2}=Z_{1} and V_{2}=0 for a pitot tube. A pitot tube can also give an estimate of the flowrate through a pipe or duct if the pitot tube is located where the average velocity occurs (average velocity times pipe area gives flowrate). Oftentimes, pitot tubes are negligently installed in the center of a pipe. This would give the velocity at the center of the pipe, which is usually 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 velocity upstream of the dam is negligible (V_{1}=0) and that the nappe is exposed to atmospheric pressure above and below. Experiments have shown that the Bernoulli equation alone does not adequately predict the flow, so empirical constants have been determined which allow better agreement between equations and real flows. To obtain better accuracy than the Bernoulli equation alone provides, use our weir calculations (Rectangular weir, Vnotch weir, Cipoletti weir).
Sluice gate
A sluice gate is often used to regulate open channel flows, and the Bernoulli
equation does an adequate job of modeling the situation. A hydraulic jump
may or may not occur downstream of a sluice gate. Be sure that
Z_{2} is not measured downstream of a hydraulic jump. The
Bernoulli equation cannot be used across hydraulic jumps since energy is
dissipated. Usually for sluice gates Z_{1}>>Z_{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)
Noncircular
hole in tank (or duct, i.e. noncircular conduit, connected to
tank)
Diagrams showing some situations which can be modeled with these two selections:
The Bernoulli equation does not account for viscous effects of the holes in
tanks or friction due to flow along pipes, thus the flowrate predicted by our
Bernoulli equation calculator will be larger than the actual flow.
V_{1} is automatically set to 0.0  implying that location 1 is a device
that has a large flow area so that the velocity at location 1 (e.g. in a tank)
is negligible compared to the velocity of fluid leaving the tank. For
comprehensive calculations which include viscous effects, try the following
calculations: Discharge from
a tank, Circular liquid
or gas pipes using DarcyWeisbach losses, Circular water pipes using
HazenWilliams losses.
Circular pipe diameter change, Noncircular duct area
change
Venturi flow meter (C=0.98), Nozzle flow meter
(C=0.96), and Orifice flow meter (C=0.6)
This selection is useful for determining the change in static pressure in a pipe due to a diameter change, determining flowrate, or designing a flow meter. Locations 1 and 2 should be as close together as possible; otherwise, viscous effects due to pipe friction will impact the pressure. However, flow meters normally have specified locations for the pressure taps.
If you select "Solve for D, W, or A", the diameter and/or or area at location 2 (D_{2} and/or A_{2}) will be computed. For all but the flow meters, 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 your pressure 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 will actually be at location 1. You cannot do this for the flow meters since they require that D_{1} is greater than D_{2}.
The venturi flowmeter analysis is based on the Bernoulli equation except for an empirical coefficient of discharge, C. V_{2} in the equation at the top of this page is known as the theoretical throat velocity. In our calculation, the velocity that is output for V_{2} is the actual throat velocity, CV_{2}. Flowrate is computed as Q=CV_{2}A_{2} (Munson et al. 1998) and A_{2}=¶ D_{2}^{2}/4. For simplicity, our Bernoulli venturi calculation uses a fixed value of C=0.98. However, it is well known that C is not fixed at 0.98 but varies as a function of Reynolds number and the material from which the meter is constructed. 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 accurate venturi calculation (yet one that has limits on some of the variables), please visit our comprehensive venturi flowmeter calculation.
For nozzle and orifice flow meters, Z_{2}Z_{1} is fixed at
0.0 since these meters are typically installed in horizontal pipes (or the
elevation difference between locations 1 and 2 is negligible). Nozzle and
orifice meters tend to have a greater impact on the flow (greater energy loss)
than venturi meters reflected by generally lower C values. The C value is
incorporated into the Bernoulli equation as described above in the above
paragraph for venturi meters. The C values of 0.96 and 0.6 are typical
values that can be used for nozzles and orifices but will produce substantial
error for certain Reynolds numbers and geometries since C actually is a function
of pressure tap locations, Reynolds number, diameter ratio, and pipe diameter.
For more rigorous and accurate equations and computations (yet ones that
have limits on some of the variables), use our comprehensive nozzle and orifice
calculations (liquid flow thru
nozzle, liquid flow thru
orifice, gas flow thru
orifice).
Variables
(dimensions shown in [])
Subscripts:
1 indicates upstream location
2
indicates downstream location.
A = Crosssectional area (i.e. area normal to flow direction)
[L^{2}]. If pipe is circular, then A= ¶
D^{2}/4
D = Diameter [L]. For flow meters,
D_{2} is the flow meter's throat diameter.
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
converts output 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 builtinto the calculation are at
standard atmospheric pressure which is 1 atmosphere (atm). Note that 1
atm=101,325 Pa=1.01325 bar=760 mm Hg (0^{ o}C)=2116
lb/ft^{2}=14.7 psi=30.01 inch Hg (60^{ o}F)=29.92 inch Hg
(32^{ o}F).
Error Messages given by
calculation
Messages are shown if you solve for a variable
that is not used for the application. For example, you cannot solve for
pressure difference across a sluice gate since the pressure difference is, by
default, 0.0 since both the upstream and downstream 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 result in a physically infeasible (impossible) situation  such
as flow moving upward in a contracting 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 a
contracting pipe (it may need to be quite a bit negative if
Z_{2}Z_{1} is large).
References
Cited
Munson, B. R., D. F.
Young, and T. H. Okiishi. 1998. Fundamentals of Fluid Mechanics.
John Wiley and Sons, Inc. 3ed.
Useful Resources
Gerhart, P. M, R. J. Gross, and J. I. Hochstein.
1992. Fundamentals of Fluid Mechanics. AddisonWesley Publishing
Co. 2ed.
Potter, M. C. and D. C. Wiggert. 1991. Mechanics of Fluids. PrenticeHall, Inc.
Roberson, J. A. and C. T. Crowe. 1990. Engineering Fluid Mechanics. Houghton Mifflin Co.
Streeter, V. L., E. B. Wylie, and K. W. Bedford. 1998. Fluid Mechanics. WCB/McGrawHill. 8ed.
White, F. M. 1979. Fluid Mechanics. McGrawHill,
Inc.
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(All Rights Reserved)
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