Complicated Unit Expressions (Units: A Unit Conversion Program and Scientific Calculator)

5.6 Complicated Unit Expressions

The units program is especially helpful in ensuring accuracy and dimensional consistency when converting lengthy unit expressions. For example, one form of the Darcy–Weisbach fluid-flow equation is

$Δ P = \frac{8}{π^{2}} ρ f L \frac{Q^{2}}{d^{5}}$

where $Δ P$ is the pressure drop, $ρ$ is the mass density, $f$ is the (dimensionless) friction factor, $L$ is the length of the pipe, $Q$ is the volumetric flow rate, and $d$ is the pipe diameter. You might want to have the equation in the form

$Δ P = A_{1} ρ f L \frac{Q^{2}}{d^{5}}$

that accepted the user’s normal units; for typical units used in the US, the required conversion could be something like

You have: (8/pi^2)(lbm/ft^3)ft(ft^3/s)^2(1/in^5)
You want: psi
        * 43.533969
        / 0.022970568

The parentheses allow individual terms in the expression to be entered naturally, as they might be read from the formula. Alternatively, the multiplication could be done with the ‘*’ rather than a space; then parentheses are needed only around ‘ft^3/s’ because of its exponent:

You have: 8/pi^2 * lbm/ft^3 * ft * (ft^3/s)^2 /in^5
You want: psi
        * 43.533969
        / 0.022970568

Without parentheses, and using spaces for multiplication, the previous conversion would need to be entered as

You have: 8 lb ft ft^3 ft^3 / pi^2 ft^3 s^2 in^5
You want: psi
        * 43.533969
        / 0.022970568