Plastic kayaks are manufactured using a process called rotomolding, which is diagrammed in Fig. 9.19. Molten plastic is injected into a mold, which is then spun on the long axis of the kayak until the plastic cools, resulting in a hollow one-piece craft. Suppose that the induction motors used to

Question

Plastic kayaks are manufactured using a process called rotomolding, which is diagrammed in Fig. 9.19. Molten plastic is injected into a mold, which is then spun on the long axis of the kayak until the plastic cools, resulting in a hollow one-piece craft. Suppose that the induction motors used to spin the molds consume 50 kW at a pf of 0.8 lagging from a [latex]220 \underline{/ 0^{\circ}}-\mathrm{V} \mathrm{rms}[/latex], 60-Hz line. We wish to raise the pf to 0.95 lagging by placing a bank of capacitors in parallel with the load.

Accepted Answer

The circuit diagram for this problem is shown in Fig. 9.20 . [latex]P_{L}=50 \mathrm{~kW}[/latex] and since [latex]\cos ^{-1} 0.8=36.87^{\circ}, heta_{ ext {old }}=36.87^{\circ}[/latex]. Therefore,

[latex]Q_{ ext {old }}=P_{ ext {old }} an heta_{ ext {old }}=(50)\left(10^{3} ight)(0.75)=37.5 \mathrm{kvar}[/latex]

Hence,

[latex]\mathbf{S}_{ ext {old }}=P_{ ext {old }}+j Q_{ ext {old }}=50,000+j 37,500[/latex]

and

[latex]\mathbf{S}_{ ext {cap }}=0+j Q_{ ext {cap }}[/latex]

Since the required power factor is 0.95

[latex]\begin{aligned} heta_{ ext {new }} &=\cos ^{-1}\left(\mathrm{pf}_{ ext {new }} ight)=\cos ^{-1}(0.95) \\&=18.19^{\circ}\end{aligned}[/latex]

Then

[latex]\begin{aligned}Q_{ ext {new }} &=P_{ ext {old }} an heta_{ ext {new }} \\&=50,000 an \left(18.19^{\circ} ight) \\&=16,430 \mathrm{var}\end{aligned}[/latex]

Hence

[latex]\begin{aligned}Q_{ ext {new }}-Q_{ ext {old }} &=Q_{ ext {cap }}=-\omega C V^{2} \mathrm{rms} \\16,430-37,500 &=-\omega C V^{2} \mathrm{rms}\end{aligned}[/latex]

Solving the equation for C yields

[latex]\begin{aligned}C &=\frac{21,070}{(377)(220)^{2}} \\&=1155 \mu \mathrm{F}\end{aligned}[/latex]

By using a capacitor of this magnitude in parallel with the industrial load, we create, from the utility's perspective, a load pf of 0.95 lagging. However, the parameters of the actual load remain unchanged. Under these conditions, the current supplied by the utility to the kayak manufacturer is less and therefore they can use smaller conductors for the same amount of power. Or, if the conductor size is fixed, the line losses will be less since these losses are a function of the square of the current.

Question 9.14: Plastic kayaks are manufactured using a process called rotom...

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