---
title: "Machine temperature after one hour of heating"
subject: "Electric Machine Design"
chapter: 2
key: "semester_5/electric_machine_design:machine-temperature-after-hour"
source: https://www.bctengineer.com/q/semester_5/electric_machine_design/machine-temperature-after-hour
site: BCT Engineering
---

# Machine temperature after one hour of heating

Electric Machine Design · Chapter 2: Heating and cooling of electric machine

- Asked in: 2076 Asoj, 8 marks
- Video: [Module_1_Part_5_ Heating of Electrical Machines(Problems)](https://www.youtube.com/watch?v=6U22xE42xpE) — Jyothi G K
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/electric_machine_design/machine-temperature-after-hour
- Whole chapter: https://www.bctengineer.com/md/semester_5/electric_machine_design/chapter-2

## Question

The initial temperature of a machine is 40°C. Calculate the temperature of the machine after 1 hour if its final steady temperature rise is 80°C and the heating time constant is 2 hours. The ambient temperature is 30°C.

## Short answer

With initial rise 10 °C and final rise 80 °C, after 1 h the rise is θ = 80(1−e⁻⁰·⁵) + 10 e⁻⁰·⁵ ≈ 37.54 °C, so machine temperature ≈ 67.5 °C.

## Answer

When a machine starts with an initial rise $\theta_i$ above ambient, its rise after time $t$ is

$$
\theta = \theta_m\left(1-e^{-t/\tau}\right) + \theta_i\,e^{-t/\tau}
$$

where $\theta_m$ is the final steady rise and $\tau$ the heating time constant. This follows from the heat balance: heat produced = heat stored + heat dissipated, $P\,dt = G h\,d\theta + S\lambda\theta\,dt$, whose solution is the expression above with $\tau = Gh/(S\lambda)$.

#### Data
- Ambient temperature = 30 °C
- Initial temperature = 40 °C, so initial rise $\theta_i = 40 - 30 = 10\ ^\circ$C
- Final steady rise $\theta_m = 80\ ^\circ$C
- $\tau = 2$ h, $t = 1$ h

#### Calculation

$$
\begin{aligned}
e^{-t/\tau} &= e^{-1/2} = 0.6065\\
\theta &= 80(1-0.6065) + 10(0.6065)\\
&= 31.48 + 6.07 = 37.54\ ^\circ\text{C}
\end{aligned}
$$

Temperature of the machine $= 30 + 37.54 = 67.54\ ^\circ$C.

```text
 rise
 80 |- - - - - - - - - - - - - - -  final
    |              ___....-----
    |        _.--''
 37.5|- - -.'  (t = 1 h)
    |   /
 10 |_.'
    +-------------------------> t
    0      1 h
```

**Answer: temperature after 1 hour ≈ 67.5 °C (rise of 37.5 °C above ambient).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. With ambient 30 °C and initial temperature 40 °C, the initial rise θi is:
   A. 40 °C
   B. 70 °C
   C. 30 °C
   D. 10 °C
   Answer: D. Rise is measured above ambient: 40 - 30 = 10 °C. (Key point: Initial rise)

2. For a machine that starts with initial rise θi, the rise is:
   A. θm(1 - e^(-t/τ)) + θi e^(-t/τ)
   B. θm e^(-t/τ)
   C. θi(1 - e^(-t/τ))
   D. θm(1 - e^(-t/τ)) - θi
   Answer: A. The solution has a heating part from zero plus the decaying initial rise. (Key point: Formula)

3. After 1 h the actual temperature is about:
   A. 37.5 °C
   B. 67.5 °C
   C. 80 °C
   D. 110 °C
   Answer: B. Rise = 80(0.3935) + 10(0.6065) = 37.54 °C, so temperature = 30 + 37.54 = 67.5 °C. (Key point: Result)

## Related in Electric Machine Design

- [Heating constant and 1-hour rating (40 kVA)](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Aoil-temperature-40kva-rating) — Electric Machine Design, chapter 2, asked 2×
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