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A colleague's nephew has emailed this science question from an assignment (A level physics?) as my colleague thinks I'm brilliant at all academic things. I can't remember any of it .

Question:



So far I've thought:

Total resistance will be (10,000m x 0.03 x 10^-6 ohms/m)

Ohm's Law(s): P = I x V and V = I x R, so knowing V: voltage (415 or 32k), P:required power (200kw) we can get figures for current for both voltages, in two ways, i.e.

If P = I x V then I = P/V, ditto if V = I x R then I = V/R. Presumably if you put your figures for V and P in, you'll get different I: currents for the lower voltage, thus proving it's impractical?

But I don't know what what the relevance of the cross-sectional area or it being single phase AC are.

I also can't remember what the SI units are in each case: m, ohms, amps, but is the power SI unit watts?

About £6000

I just had a eureka moment in the bath.

V = I x R (or I = V/R) tells us the (I) current figure that is the minimum necessary to overcome the given resistance, correct?

And in P = I x V (or I = P/V) the (I) current figure is that actually supplied by each of the suggested voltages.

So the I value from the second equation needs to be higher than that produced by the first equation, in other words there needs to be at least as much current as we get from V = I x R to overcome the resistance.

In other words, if you put figures in, I bet with 415V the current figure in Amps will be not high enough, but with 32 kilovolts it will. Hence a 415v system will be "not practical".

Still don't know why we need to know the X-sectional area of the cable or that it's AC tho'

I know where the question is going, but there is something wrong with this:

That set of numbers seems wrong to me, and doesn't fit with the spirit of the question. I calculate the resistance of 100mm2 of alu to be 100x their figure.
edited to add: no, the question isn't clear.


Yes, I think you're Right JTB. If I'm right with where I think the question is going, the X-area is fairly redundant.
The AC doesn't really matter, the RMS issues sort themselves out for this question.

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Views expressed are personal opinions and are not necessarily shared by the Safe Speed campaign

I think the cross-sectional area has nothing to do with the power loss calculation, which is a simple matter of applying Ohm's Law.
However, it probably has a lot to do with how hot the cable gets - but that's getting a bit beyond my comfort zone.

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Only when ideology, prejudice and dogma are set aside does the truth emerge - Kepler

The latter part implies the resistance of the cable per meter (so apparently making the X-area redundant), but I now recognise that figure as the "specific resistance" (rho).
That really wasn't clear!

Anyway

R = rho * L/A = 0.03e-6 * 1 / (0.1 * 0.1) = 3e-6 ohms per meter.
For 10km = 0.03 Ohms

For 415V, the current must be 200kW/415V = 482A. Vdrop for two runs of cable (to close the circuit) = 29V
That means only 386V at the outlet.
The cable would dissipate/waste 14KW - that's a lot of power loss!

For 32KV, the current must be 200kW/415V = 6.25A. Vdrop for two runs of cable (to close the circuit) = 0.375V
The cable would waste 2.3W - not at all significant.

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Or

R = rho * L/A = 0.03e-6 * 1 / (0.1 * 0.1) = 3e-4 ohms per meter.
For 10km = 3 Ohms

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I think I have to disagree with you Toltec.

Not only have you multiplied the 'mantissa' (?) by 100, you also adjusted the 'exponent' by that factor too.

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Yes, I'd forgotten completely about that - it's been a very long time

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Only when ideology, prejudice and dogma are set aside does the truth emerge - Kepler

IIRC, the West Coast Main Line uses aluminium TUBE rather than cable to increase surface area, and reduce resistance and weight.

Is that any help?

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Time to take responsibility for our actions.. and don't be afraid of speaking out!

This is a very good point, one I hadn't considered. I suspect the questioner didn't consider that either, so I'll let myself off

Yes - in the real world, it is a bit silly having an AC power cable as large as 100mm2, the "skin effect" would render much of the cross-sectional area useless.
Realistically, one could triple (finger in air estimate) the DC resistance of such a cable, as well as the voltage drops and power losses.

All this is assuming "100mm2" means '(100 millimetres) squared' as opposed to '100 square millimetres'. If the latter is the correct interpretation then my earlier maths is waaaaay out, and the 200kw wouldn't be merely impractical - it would actually be outright impossible!

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I'm not so sure whether to give the question setter 100 lines or you, young Steven!

100mm²

I can't help think that he (the question setter) didn't think this question through. A similar thing happened to me in a question which involved the shutter speed and frame rate in a movie projector.

I gave the right answer - which I got wrong, because the question failed to take into account factors such as the pausing of each frame!

_________________
Time to take responsibility for our actions.. and don't be afraid of speaking out!

I have given this some thought - and the best answer would be for them to buy a big diesel powered industrial generator, and generate on site - supplemented by a 125 meter wind turbine if planning conditions allow, for periods when the plant is on shut down or maintenance!

This of course would all come to nought if any NIMBYs called the planning decision in for a public enquiry, as the legal fees would outweigh the benefits!

They could of course be in a position to supply back to the grid - thus sidestepping the environmental levy!

_________________
Time to take responsibility for our actions.. and don't be afraid of speaking out!

Bit late on this since I had an early night last night but my two pennorth is that even if the 100mm^2 means what it says - a cable 1.27cm in diameter - than the proposal is ludicrous at 415 and quite workable at 32kV. With a cable that size at 50Hz I think skin effect is negligible.

If the X-section are is 100mm^2 -(10mm * 10mm) that is 10e-4 m^2
So R = pL/A = .03 10E-6 * 10e4 /10e-4 = .03 10e (-6 + 4 + 4) = .03 e2 = 3 ohms
Because it is single phase the loop resistance is twice that, 6 ohms.

200kW at 415volts is almost 500Amps and ohms law shows voltage drop of 3000volts. Greater than the sourse voltage!
At 32 kv you are looking at just over 6amps for a 40volt drop. Quite manageable and rather in line with what I remember from my time at the CEGB.

The units for resistivity are ohm metre not ohms per metre so the question is very misleading.

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When I see an adult on a bicycle, I do not despair for the future of the human race. H.G. Wells
When I see a youth in a motor car I do d.c.brown

Thanks all


What does pL stand for in that equation? Just so I'm clear?

p should be Greek rho, the symbol for resistivity - aka specific resistance - which is measured in ohms * meters (not ohms/meter as in the question)
L is the length of the cable in metres
A is the cross sectional area of the cable in square metres

_________________
When I see an adult on a bicycle, I do not despair for the future of the human race. H.G. Wells
When I see a youth in a motor car I do d.c.brown

Yes, I would agree with DCB's analysis. The issue is voltage drop at the high current required at the lower voltage level. It's easy to forget that there are two cable lengths - line and return. Skin effect is negligible at 50Hz.

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Malcolm W.
The views expressed in this post are personal opinions and do not represent the views of Safespeed.

In copper the "skin depth" at 50 hz is just under a centimeter. That means that there is no significant skin effect for cables up to 2cm diameter. However if there is any harmonic component then the skin depth for that will be less.
Skin effect is taken into account in designing large busbars and switchgear, and equipment designed for 50Hz working has to derated if it is used at 400Hz

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When I see an adult on a bicycle, I do not despair for the future of the human race. H.G. Wells
When I see a youth in a motor car I do d.c.brown

DCB is quite correct. Switchgear and bussbars tend to be made from solid copper parts. Litz construction can be used on large cables to mitigate skin effect.

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Malcolm W.
The views expressed in this post are personal opinions and do not represent the views of Safespeed.

Both it seems!


(aimed at myself, not you DCB)
What is the recognised written short-hand notation for '(100 millimetres) squared' and '100 square millimetres' ?

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Views expressed are personal opinions and are not necessarily shared by the Safe Speed campaign