Chapter 1 Units in Fluid Mechanics
Before beginning with problem solving methods it helps to recall some important quantities in fluid mechanics and their associated units. While the world has generally moved forward into standardizing the use of the SI unit system, the U.S. stubbornly holds onto the antiquated US (sometimes called the British Gravitational, BG) system. This means practicing engineers must be familiar with both systems, and be able to convert between the two systems.
These important quantities are shown in Table 1.1.
| Quantity | Symbol | Dimensions | US (or BG) Units | SI Units | US to SI multiply by |
|---|---|---|---|---|---|
| Length | L | \(L\) | \(ft\) | \(m\) | 0.3048 |
| Acceleration | a | \(LT^{-2}\) | \(ft/s^2\) | \(m/s^{2}\) | 0.3048 |
| Mass | m | \(M\) | \(slug\) | \(kg\) | 14.59 |
| Force | F | \(F\) | \(lb\) | \(N\) | 4.448 |
| Density | \(\rho\) | \(ML^{-3}\) | \(slug/ft^3\) | \(kg/m^3\) | 515.4 |
| Energy/Work | FL | \({ft}\cdot{lb}\) | \({N}\cdot{m}=joule (J)\) | 1.356 | |
| Flowrate | Q | \(L^{3}/T\) | \(ft^{3}/s\)=cfs | \(m^{3}/s\) | 0.02832 |
| Kinematic viscocity | \(\nu\) | \(L^{2}/T\) | \(ft^{2}/s\) | \(m^{2}/s\) | 0.0929 |
| Power | \(FLT^{-1}\) | \({ft}\cdot{lb/s}\) | \({N}\cdot{m/s}=watt (W)\) | 1.356 | |
| Pressure | p | \(FL^{-2}\) | \(lb/in^2=psi\) | \(N/m^2=Pa\) | 6895 |
| Specific Weight | \(\gamma\) | \(FL^{-3}\) | \(lb/ft^3\) | \(N/m^3\) | 157.1 |
| Velocity | V | \(LT^{-1}\) | \(ft/s\) | \(m/s\) | 0.3048 |
| (Dynamic) Viscocity | \(\mu\) | \(FTL^{-2}\) | \({lb}\cdot{s/ft^2}\) | \({N}\cdot{s/m^2}={Pa}\cdot{s}\) | 47.88 |
| Volume | \(\forall\) | \(L^3\) | \(ft^3\) | \(m^3\) | 0.02832 |
There are many other units that must be accommodated. For example, one may encounter the poise to describe (dynamic) viscosity (\(1~Pa*s = 10~poise\)), or the stoke for kinematic viscocity (\(1~m^2/s=10^4~stokes\)). Many hydraulic systems use gallons per minute (gpm) as a unit of flow (\(1~ft^3/s=448.8~gpm\)), and larger water systems often use millions of gallons per day (mgd) (\(1~mgd = 1.547~ft^3/s\)). For volume, the SI system often uses liters (\(l\)) instead of \(m^3\) (\(1~m^3=1000~l\)).
One regular conversion that needs to occur is the translation between mass (m) and weight (W), where \(W=mg\), where \(g\) is gravitational acceleration on the earth’s surface: \(g=9.81~m/s^2=32.2~ft/s^2\). When working with forces (such as with momentum problems or hydrostatic forces) be sure to work with weights/forces, not mass.
It is straightforward to use conversion factors in the table to manipulate values between the systems, multiplying by the factor shown to go from US to SI units, or dividing to do the \[{1*10^{-6}~m^2/s}*\frac{1 ~ft^2/s}{0.0929~m^2/s}=1.076*10^{-5} ~ft^2/s\]
Another example converts between two quantities in the US system: 100 gallons per minute to cfs: \[{100 ~gpm}*\frac{1 ~cfs}{448.8 ~gpm}=0.223 ~cfs\] The units package in R can do these conversions and more, and also checks that conversions are permissible (producing an error if incompatible units are used).
units::units_options(set_units_mode = "symbols")
Q_gpm <- units::set_units(100, gallon/min)
Q_gpm
#> 100 [gallon/min]
Q_cfs <- units::set_units(Q_gpm, ft^3/s)
Q_cfs
#> 0.2228009 [ft^3/s]Repeating the unit conversion of viscosity using the units package:
Example 1.1 Convert kinematic viscosity from SI to Eng units.
nu <- units::set_units(1e-6, m^2/s)
nu
#> 1e-06 [m^2/s]
units::set_units(nu, ft^2/s)
#> 1.076391e-05 [ft^2/s]The units package also produces correct units during operations. For example, multiplying mass by g should produce weight.
Example 1.2 Using the units package to produce correct units during mathematical operations.
#If you travel at 88 ft/sec for 1 hour, how many km would you travel?
v <- units::set_units(88, ft/s)
t <- units::set_units(1, hour)
d <- v*t
d
#> 316800 [ft]
units::set_units(d, km)
#> 96.56064 [km]
#What is the weight of a 4 slug mass, in pounds and Newtons?
m <- units::set_units(4, slug)
g <- units::set_units(32.2, ft/s^2)
w <- m*g
#Notice the units are technically correct, but have not been simplified in this case
w
#> 128.8 [ft*slug/s^2]
#These can be set manually to verify that lbf (pound-force) is a valid equivalent
units::set_units(w, lbf)
#> 128.8 [lbf]
units::set_units(w, N)
#> 572.9308 [N]