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Units for force on pipe bend calculation:
cfm=cubic feet per minute, cfs=cubic feet per second, cm=centimeter, g=gram, ft=foot,
gal=gallon (US), gpd=gallons (US) per day, gph=gallons (US) per hour, gpm=gallons (US) per
minute, Hg=mercury, hr=hour, H_{2}O=water, in=inch, kg=kilogram, lb=pound, m=meter, mbar=millibar,
MGD=Million gallons (US)/day, min=minute, mm=millimeter, N=Newton, Pa=Pascal, psi=pound
per square inch, s=second.
Discussion
In order to properly size thrust blocks or other devices to hold a pipe in
place, the momentum equation is used to compute the necessary resistive force to hold the
pipe stationary. This is also called thrust restraint design. Forces in a pipe bend in the horizontal plane are caused by the fluid's
momentum and pressure. If the pipe undergoes a bend in the vertical plane, where the
entrance to the bend is above the exit (or viceversa), then the weight of the liquid and
pipe material within the bend will contribute to the force. Since computing the volume of
fluid and pipe material within a bend requires considerably more input, we kept our
calculation relatively simple by keeping it in the horizontal plane. Our calculation is
also valid for incompressible gases, but  due to a gas's low density  the force required
to hold a gas pipe in place is typically small compared to the force required to hold a
liquid pipe in place unless pressures are high. The forces F_{x} and F_{y}
computed by the calculation are the x and y components of the total force F.
The four diagrams below further explain the bend angle b and the
direction of the computed force for various situations. The pipe bends are in the
horizontal plane. Plain arrow (>) indicates flow direction. Bold arrow (>) indicates resistive force that must be applied to
keep the bend in place.
Equations for force on pipe bend calculation
The equations used in our calculation can be found in nearly any college level fluid
mechanics textbook (e.g. Munson et al., 1998) or fluid mechanics reference handbook. The
force equations are based on linear momentum conservation. P_{2} is computed using
the Bernoulli Equation, which assumes negligible friction loss around the bend.
Subscript 1 is upstream of bend; Subscript 2 is downstream of bend;
Subscript x is xcomponent of force; Subscript y is ycomponent of force. For a pipe bend
in the horizontal plane where friction effects around the bend are negligible:
F_{x} = P_{1}A_{1}  P_{2}A_{2}
cos(b)  d Q [V_{1} + V_{2} cos(b)]
F_{y} = P_{2} A_{2} sin(b) + d V_{2} Q
sin(b) F = (F_{x}^{2} + F_{y}^{2})^{1/2}
Q=VA A=π D^{2} / 4
P_{2} = P_{1} + d (V_{1}^{2}  V_{2}^{2})
/ 2
Variables
To use the equations above, a consistent set of units must be used. The variables below
show the SI (System International) units for each variable. Our calculation allows a
variety of other units with the conversions made internally within the program.
A=Pipe flow area (m^{2})
b=Pipe angle shown in the figure above (valid range is 0 to 2π radians or 0 to 360^{o})
d=Liquid (or gas) mass density (kg/m^{3})
D=Pipe diameter (m)
F=Resistive (reaction) force (N)
P=Gage pressure (N/m^{2}, relative to atmospheric pressure)
Note: P_{1} and P_{2} cannot physically be below 0.0 absolute pressure
(101,325 N/m^{2} gage). Further, if P_{1} or P_{2} approach the
liquid's vapor pressure, flashing to vapor may occur and the calculation will not be
accurate. Vapor pressure is not checked by the program, but a message will appear if P_{1}
or P_{2} is less than 101,325 N/m^{2} gage.
Q=Flowrate (m^{3}/s)
V=Velocity (m/s)
π=Greek letter pi, 3.1415926....
Messages given by calculation
"Need Density>0", "Need Q>0", "Need D_{1}>0",
"Need D_{2}>0", "Need pipe angle b>0", "Need
P_{1}>0 absolute", "Need 0 < b < 6.28
radian", "Need 0 < b < 360 degrees". These are
initial checks of input data.
"Need P_{2}>0 absolute". Message will appear after
computing downstream pressure. Physically, pressure cannot be less than 0.0 absolute,
which is a complete vacuum. The calculation method does not account for liquid flashing to
vapor if the pressure drops below the liquid's vapor pressure.
Reference
Munson, Bruce R., Donald F. Young, and Theodore H. Okiishi. 1998. Fundamentals of Fluid
Mechanics. John Wiley and Sons, Inc. 3ed.
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LMNO Engineering, Research, and Software, Ltd.
7860 Angel Ridge Rd. Athens, Ohio 45701 USA Phone and
fax: (740) 5921890
LMNO@LMNOeng.com http://www.LMNOeng.com
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