### 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

*\[ \Delta P = {8 \over \pi^2} \rho fL { Q^2 \over d^5}
\]*

where *\(\Delta P\)* is the pressure drop, *\(\rho\)*
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

*\[ \Delta P = A_1 \rho fL {Q^2 \over d^5}
\]*

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