]> No Title

By the way, even if you're just doing a quick calculation, I recommend not using a calculator. Enter the data into an Excel spreadsheet so that you can $\mathrm{a}\mathrm{d}\mathrm{d}\mathrm{/}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{/}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{/}$save calculations as needed. Sometimes I see an obviously invalid result and when I ask, ""How did you get that resulti) What numbers did you $\mathrm{u}\mathrm{s}{\mathrm{e}}^{\mathrm{i}\mathrm{\right)}}$“ the answer is ""I put the numbers into the calculator and this was the result I got.“ But how do you know you entered the numbers and formulas correcdyi) What if you need to re‐do the calculation for a slightly different set of numbersi)

Examples of the use of units and scrutiny

These examples, particularly the first one, also introduce the concept of ""back of the envelope“ estimates, a powerful engineering tool.

Example 1. Drag force and power requirements for an automobile

A car with good aerodynamics has a drag coefficient (C) of 0.2. The drag coefficient is defined as the ratio of the drag force (F) to the $\mathit{d}\mathrm{\setminus }\mathit{y}\mathit{n}\mathit{a}\mathrm{Reject}\mathit{p}\mathit{p}\mathit{c}\mathit{p}\mathit{r}\mathit{e}\mathrm{J}\mathrm{J}\mathrm{\text{'}}\mathit{u}\mathit{r}\mathit{e}$ of the flow $\mathrm{=}\mathrm{1}\mathrm{/}\mathrm{2}\mathrm{p}{\mathrm{v}}^{\mathrm{2}}$ (where $\mathrm{p}$ is the fluid density and $\mathrm{v}$ the fluid velocity far from the object) multiplied by the cross‐section area (A) of the object, i.e.

${\mathit{F}}_{\mathit{D}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{\mathit{C}}_{\mathit{D}}\mathit{\rho }{\mathrm{v}}^{\mathrm{2}}\mathit{A}$ (Equation 7)

The density of air at standard conditions is 1.18 $\mathrm{k}\mathrm{g}\mathrm{/}{\mathrm{m}}^{\mathrm{3}}\mathrm{.}$

(a) Estimate the power required to overcome the aerodynamic drag of such a car at 60 miles per hour.

Power $\mathrm{=}$ Force $\mathrm{x}$ velocity

$\mathrm{v}\mathrm{=}\mathrm{6}\mathrm{0}$ miles/hour $\mathrm{x}$ (5280 ft/mile) $\mathrm{x}\mathrm{\left(}\mathrm{m}\mathrm{/}\mathrm{3}\mathrm{.}\mathrm{2}\mathrm{8}\mathrm{f}\mathrm{t}\mathrm{\right)}\mathrm{x}\mathrm{\left(}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{/}\mathrm{6}\mathrm{0}\mathrm{min}\mathrm{\right)}\mathrm{x}\mathrm{\left(}\mathrm{min}\mathrm{/}\mathrm{6}\mathrm{0}\mathrm{s}\mathrm{\right)}\mathrm{=}\mathrm{2}\mathrm{6}\mathrm{.}\mathrm{8}\mathrm{m}\mathrm{/}\mathrm{s}$

Estimate cross‐section area of car as 2 $\mathrm{m}\mathrm{x}\mathrm{3}\mathrm{m}\mathrm{=}\mathrm{6}{\mathrm{m}}^{\mathrm{2}}$

${\mathrm{F}}_{\mathrm{D}}\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{5}\mathrm{x}\mathrm{0}\mathrm{.}\mathrm{2}\mathrm{x}\mathrm{1}\mathrm{.}\mathrm{1}\mathrm{8}\mathrm{k}\mathrm{g}\mathrm{/}{\mathrm{m}}^{\mathrm{3}}\mathrm{x}\mathrm{\left(}\mathrm{2}\mathrm{6}\mathrm{.}\mathrm{8}\mathrm{m}\mathrm{/}\mathrm{s}{\mathrm{\right)}}^{\mathrm{2}}\mathrm{x}\mathrm{6}{\mathrm{m}}^{\mathrm{2}}\mathrm{=}\mathrm{5}\mathrm{1}\mathrm{0}$ kg $\mathrm{m}\mathrm{/}{\mathrm{s}}^{\mathrm{2}}\mathrm{=}\mathrm{5}\mathrm{1}\mathrm{0}$ Newton

Power $\mathrm{=}{\mathrm{F}}_{\mathrm{D}}\mathrm{x}\mathrm{v}\mathrm{=}\mathrm{5}\mathrm{1}\mathrm{0}$ kg $\mathrm{m}\mathrm{/}{\mathrm{s}}^{\mathrm{2}}\mathrm{x}\mathrm{2}\mathrm{6}\mathrm{.}\mathrm{8}\mathrm{m}\mathrm{/}\mathrm{s}\mathrm{=}\mathrm{1}\mathrm{.}\mathrm{3}\mathrm{7}\mathrm{x}\mathrm{1}{\mathrm{0}}^{\mathrm{4}}$ kg ${\mathrm{m}}^{\mathrm{2}}\mathrm{/}{\mathrm{s}}^{\mathrm{3}}\mathrm{=}\mathrm{1}\mathrm{.}\mathrm{3}\mathrm{7}\mathrm{x}\mathrm{1}{\mathrm{0}}^{\mathrm{4}}\mathrm{W}\mathrm{=}\mathrm{1}\mathrm{8}\mathrm{.}\mathrm{3}$ horsepower, which is reasonable

(b) Estimate the gas mileage of such a car. The heating value of gasohne is 4.3 $\mathrm{x}\mathrm{1}{\mathrm{0}}^{\mathrm{7}}\mathrm{J}\mathrm{/}\mathrm{k}\mathrm{g}$ and its density is 750 $\mathrm{k}\mathrm{g}\mathrm{/}{\mathrm{m}}^{\mathrm{3}}\mathrm{.}$

Fuel mass flow required $\mathrm{=}$ power (Joules/s)/ heating value (Joules/kg)

$\mathrm{=}\mathrm{1}\mathrm{.}\mathrm{3}\mathrm{7}\mathrm{x}\mathrm{1}{\mathrm{0}}^{\mathrm{4}}$ $\mathrm{k}\mathrm{g}$ ${\mathrm{m}}^{\mathrm{2}}\mathrm{/}{\mathrm{s}}^{\mathrm{3}}\mathrm{/}\mathrm{4}\mathrm{.}\mathrm{3}\mathrm{x}\mathrm{1}{\mathrm{0}}^{\mathrm{7}}\mathrm{J}\mathrm{/}\mathrm{k}\mathrm{g}\mathrm{=}\mathrm{3}\mathrm{.}\mathrm{1}\mathrm{9}\mathrm{x}\mathrm{1}{\mathrm{0}}^{\mathrm{-}\mathrm{4}}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{s}$

Fuel volume flow required $\mathrm{=}$ mass flow / density

$\mathrm{=}\mathrm{3}\mathrm{.}\mathrm{1}\mathrm{9}\mathrm{x}\mathrm{1}{\mathrm{0}}^{\mathrm{-}\mathrm{4}}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{7}\mathrm{5}\mathrm{0}\mathrm{k}\mathrm{g}\mathrm{/}{\mathrm{m}}^{\mathrm{3}}\mathrm{=}\mathrm{4}\mathrm{.}\mathrm{2}\mathrm{5}\mathrm{x}\mathrm{1}{\mathrm{0}}^{\mathrm{-}\mathrm{7}}{\mathrm{m}}^{\mathrm{3}}\mathrm{/}\mathrm{s}\mathrm{x}$ $\mathrm{\left(}$3.281 $\mathrm{f}\mathrm{t}\mathrm{/}\mathrm{m}{\mathrm{\right)}}^{\mathrm{3}}\mathrm{x}\mathrm{7}\mathrm{.}\mathrm{4}\mathrm{8}$ gal/ft3 $\mathrm{=}\mathrm{1}\mathrm{.}\mathrm{1}\mathrm{2}\mathrm{x}\mathrm{1}{\mathrm{0}}^{\mathrm{-}\mathrm{4}}$ gal/sec

14