Voltage, electricity and transformers question?

[Arkane1337]

Homework Statement

An electric power plant generates electricity @ 22 kV
and 100 A. The voltage is stepped up to 240 kV before
being transmitted to a city 15 km away over a copper
wire that has a radius of 2 mm. Energy is lost in heating
the wire during transmission. If the transformer is 95%
efficient, and the price of electricity is 15¢/kWh, How
much money is saved per day by stepping the voltage up
instead of transmitting it at the original voltage?

Homework Equations

P1 = Ip^2 R
P2 = Is^2 R

Specific resistance of copper = pi d^2 R /4 L , where d = 4x10-3 m (?)
Sp.resistance of copper = 1.7x120^-4 ohm-m (?)

The Attempt at a Solution

No definitive idea of how to solve it unfortunately :/

 

[tiny-tim]

Welcome to PF!

Hi Arkane1337! Welcome to PF!

(try using the X² tag just above the Reply box )

Show us how far you've got, and where you're stuck, and then we'll know how to help!

 

[Arkane1337]

So far I have a few variables mapped out, and apparently, the first equation and 'part' figured out:

I = 100A, V_p = 22kV, r =2mm,

Equationing:
Part 1
V_sI_s = V_pI_p(.95)

I_s = [(22)(100)(.95)] / [240kV]

= 8.708A (secondary transformer output current apparently)

Part2
Power Loss with/out step ups: (?)
P₁ = I_p²*R
P₂ = I_s²*R

The specific resistance of copper is also 1.678 * 10^-6 ohm-cm apparently

I'm not sure if what I have so far is right or truly relevant, nor what I should do next. A friend of mine claims that he found the answer to be $14,497,500 saved, but I'm not sure if that is correct. (Haven't gotten a chance to take a look at his actual work either.)

 

[epenguin]

You've got V (RMS I presume) and I of the generator. That can tell you the total power, which is consumed - turned into heat - between the city, the wires and the transformer when there is one.

5% of that is power is consumed by the transformer. So you can work that out straightaway.

For the power consumed in the lines it would be more convenient to use V which you already have than I which you never need to know here. From the V's that you are given you can already work out the ratio of what you waste in the wires in the two situations. You can already see that bit is very in favour of upping the voltage.

For the actual amount I think though you do have to work out the resistance of the wire.

Expect the low voltage one to waste more than 5% otherwise it's pointless.

I notice that they haven't included in the problem any step-down transformers at the other end of the wire. There will be several of them and at least two steps down before the current arrives in the houses. A lot of transformers, presumably consuming more energy than the generator one? Never thought of it before. So when they say money saved does the generator companies costs end when the electricity arrives at the city?

But you are not asked that.

Also I thought the big transformers at the generator end did quite a bit better than 95%, very efficient.

The academic excercise is some use for some principles, but an engineer along here to explain some nitty gritties of power transmission would be additionally educative.

 

[paranormal]

I would approach this problem by first understanding the basic principles of electricity and transformers. Voltage is a measure of the potential difference between two points in an electrical circuit, and is often referred to as the "pressure" that drives the flow of electrons. Electricity is the flow of electric charge, typically carried by electrons, through a conductive material. Transformers are devices that are used to change the voltage of an alternating current (AC) electrical circuit.

In this scenario, the power plant is generating electricity at a voltage of 22 kV and a current of 100 A. The voltage is then stepped up to 240 kV before being transmitted to a city 15 km away. This is done through the use of a transformer, which increases the voltage while decreasing the current. This is because high voltage is more efficient for long-distance transmission, as it reduces the amount of energy lost due to resistance in the transmission wires.

In order to calculate the money saved per day by stepping the voltage up instead of transmitting it at the original voltage, we need to consider the efficiency of the transformer and the cost of electricity. The efficiency of the transformer is given as 95%, meaning that 95% of the energy put into the transformer is successfully transmitted to the city. The remaining 5% is lost as heat, which is why the wire heats up during transmission.

To calculate the energy lost during transmission, we can use the formula P = I^2R, where P is power (in watts), I is current (in amperes), and R is resistance (in ohms). We know the current (100 A) and the length of the wire (15 km), but we need to calculate the resistance. The specific resistance of copper is given as 1.7x10^-8 ohm-m, but we need to convert this to ohms per meter (ohms/m) by multiplying it by the cross-sectional area (pi x r^2) and dividing by the length (15 km or 15,000 m). This gives us a resistance of approximately 0.007 ohms/m.

Using this value for resistance, we can calculate the power lost during transmission at the original voltage (22 kV) and the stepped up voltage (240 kV). This gives us 154,000 watts and 1,680,000 watts, respectively. Multiplying these values by the cost of electricity (15 cents per kWh) and the efficiency