Energy Transfer by Work

Work, $W$, is energy transfer associated with a force acting through a distance.

Sign convention:

• If the system does work, $W(+)$
• If work is done to the system, $W(-)$

$W$ is in units of $[\text{kJ}]$, and $w=\frac{W}{m}\left[\frac{\text{kJ}}{\text{kg}}\right]$. This is a process, not a property; it’s only quantified across a boundary.

Modes of work transfer:

• Work of compression/expansion
• Electrical work
• Shaft work
• Spring work
• Magnetic work
• Electrical polarization work
• etc.

Power is the rate at which work is done:

$$\dot{W}=\frac{dW}{dt}\left[\frac{\text{kJ}}{s}\right]=[\text{kW}]$$

Mechanical Work

Mechanical work is given by a force applied over a distance:

$$W_{1-2}={\int }_1^{2}Fdx[\text{kJ}]$$

For example, a spring is governed by $F=kx$, where $x$ is displacement from an undisturbed position. Then, we have:

$$W_{1-2}={\int }_1^{2}kxdx=\frac{1}{2}k(x_2^2-x_1^2)$$

Shaft Work

Shaft work is given by:

$$\begin{array}{rl}{W}_{\text{sh}}& =\left(\frac{T}{r}\right)(2\pi r)n\\ & =2\pi nT[\text{kJ}]\end{array}$$

• Here, $T/r$ is the force (recall that torque is given by $T=Fr$),
• $2\pi r$ is the circumference
• $n$ is the number of rotations

Shaft power is then given by:

$${\dot{W}}_{\text{sh}}=2\pi \dot{n}T$$

• $\dot{n}$ is revolutions/time, such as RPM.

Electrical Work

Electrical work is given by:

$$\begin{array}{rl}{W}_e& ={\int }_1^{2}VIdt[\text{kJ}]\\ & =VI\Delta t\end{array}$$

Electrical power is then given by ${\dot{W}}_e=VI[\text{kJ}]$.

A system does work on the surroundings if the sole effect could be equivalent to raising a weight; if a motor and pulley are 100% efficient, sole effect would be raising a weight. Therefore, electrical work = work.