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Question 6.AE.7: An enclosed fan-cooled induction motor has a thermal time co......

An enclosed fan-cooled induction motor has a thermal time constant of τ_θ=3 h and a steady-state rated temperature of θ_f=120 °C at an ambient temperature of θ_{amb}=40 °C. The motor has at time t=0 \_a temperature of θ=θ_{amb}, and the motor is fully loaded at t=0_+. Calculate the time t_{95\%} when the fully loaded motor reaches 95% of its final temperature θ_f=120 °C.

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With Eq. 6-39 one obtains
\theta=\theta_f-\left(\theta_f-\theta_o\right) \cdot e^{\left(-t / \tau_\theta\right)}              (6-39)

\begin{gathered} 0.95 \cdot \theta_f=\theta_f-\left(\theta_f-\theta_{a m b}\right) e^{\left(\frac{-t_{95 \%}}{\tau_\theta}\right)} \\ 0.95 \cdot 120=(120-40) e^{\left(\frac{-t_{95 \%}}{\tau_\theta}\right)} \end{gathered}

or t_{95 \%}=2.59 \cdot \tau_\theta=7.77 \mathrm{~h}.

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