Transformer exceeding its rated maximum temperature

[clembo]

Homework Statement

An oil-filled transformer has a c.m.r. of 500 kVA which allows it to run continuously in an ambient of 35°C with an oil temperature rise of 50°C. The transformer has a time constant of 1.5. hours and the ratio of copper losses to iron losses at c.m.r. is 1.5:1.

After switch-on, the transformer supplies a load of 600 kVA for one hour and the load then rises to 750 kVA. Calculate the period of time that the transformer can supply this load without exceeding its rated maximum temperature. Sketch the heating curves for the two loads.[/B]

Homework Equations

The temperature rise of a transformer is given by the equation
[/B]

-t/τ​

θ = θm ( 1 - e )

where

θ is the temperature rise at any time t,
θm is the final steady temperature,
τ is the heating time constant (determined by the heat capacity and efficiency of cooling).

The Attempt at a Solution

From the above equation I understand the temperature rise, final steady temperature and time constant. But what does anyone know what the "e" signifies. Any help on this question would be very appreciated. I can't find much information on this topic online.

Many thanks

Andy[/B]

 

[scottdave]

I believe e in this case is Euler's Number, which is approximately 2.718281828

See this. https://www.mathsisfun.com/algebra/exponential-growth.html

 

[scottdave]

Another thing - it is not really clear in your notation, but it should be $$θ = θm(1 - e^{-t/τ})$$
to indicate exponential "decay" upward toward θm.

 

[clembo]

Another thing - it is not really clear in your notation, but it should be $$θ = θm(1 - e^{-t/τ})$$
to indicate exponential "decay" upward toward θm.

Thank you Scottdave that is very helpful.

 

[clembo]

So far I have:

The max temperature my oil filled transformer can handle without damage or ageing occurring is 85 deg.

With an ambient temperature of 35 deg and a load of 500 kVA the transformer will run continuously quite happy.

My transformer time constant is 1.5
Ratio of copper losses to iron losses at CMR is 1.5:1

For a 500 kVA transformer I can work out using:

θ=θm(1−e−t/τ)

= 50(1-e-t/1.5)

=24.33 deg

This would mean that my transformer is running at 59.33 deg

I am struggling to work out where to go from here. Is it just a % of this figure as 600 kVA is 20% overloaded?

 

[scottdave]

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