• Support PF! Buy your school textbooks, materials and every day products Here!

Calculate the cost of the energy lost to heat per hour per meter

  • Thread starter NasuSama
  • Start date
  • #1
326
3

Homework Statement



A small city requires about ##17 \mbox{MW}## of power. Suppose that instead of using high-voltage lines to supply the power, the power is delivered at ##120 V##.

Assuming a two-wire line of 0.60cm -diameter copper wire, estimate the cost of the energy lost to heat per hour per meter. Assume the cost of electricity is about ##9.5\, \mbox{ cents}/\mbox{kWh} ##

Homework Equations



  • ##R = \dfrac{\rho \ell}{A}##
  • ##P = I^2R##
  • ##W = IV##

The Attempt at a Solution



Given that a small city requires that amount of watts ##W##

##W = IV##
##I = \dfrac{W}{V}##

By the resistance formula, rewrite it in terms of ohms per meter, which is

##R = \dfrac{\rho}{A}##

But we also know that the "face" of the wire is circle, so

##R = \dfrac{\rho}{\pi r^2}##

which is equivalent to

##R = \dfrac{\rho}{\dfrac{d^2\pi}{4}}##

where ##d## is the diameter of the wire. For the power rate, we have double the power rate of one wire, so

##P = 2I^2R##
##P = 2\left( \dfrac{W}{V}\right)^2 \left(\dfrac{\rho}{\dfrac{d^2\pi}{4}} \right)##

The units of that ##P## are now ##\mbox{W}/\mbox{s}##. By unit conversion, we obtain

##P = 7.2\left( \dfrac{W}{V}\right)^2 \left(\dfrac{\rho}{\dfrac{d^2\pi}{4}} \right)##

Given that the cost of electricity is around 9.5 cents per kilowatts hour,

##\text{Cost of lost energy}=7.2\cdot 9.5\left( \dfrac{W}{V}\right)^2 \left(\dfrac{\rho}{\dfrac{d^2\pi}{4}} \right)##

##\text{Cost of lost energy} = 68.4\left( \dfrac{W}{V}\right)^2 \left(\dfrac{\rho}{\dfrac{d^2\pi}{4}} \right)##

So converting cents into dollars, we finally obtain

##\text{Cost of lost energy} = .684\left( \dfrac{W}{V}\right)^2 \left(\dfrac{\rho}{\dfrac{d^2\pi}{4}} \right)##

By substitution, I got around $8156585 per hour per meter, but Mastering Physics marked it incorrect. I don't see where I went wrong.
 
Last edited:

Answers and Replies

  • #2
CWatters
Science Advisor
Homework Helper
Gold Member
10,529
2,295
For starters

r2 <> d/4
 
  • #3
CWatters
Science Advisor
Homework Helper
Gold Member
10,529
2,295
I don't understand the bit that says...

The units of that P are in W/s.
Power is measured in Watts not Watts per second.

Where does the 7.2 come from exactly?
 
  • #4
326
3
CWatters said:
Power is measured in Watts not Watts per second.
I mean ##P## is treated as Watts per second as a rate.

CWatters said:
Where does the 7.2 come from exactly?
I used the unit conversions to convert second into hours and Watts into kiloWatts
 
Last edited:
  • #5
326
3
Never mind. I found the solution. Anyway, thanks for the mistake catch!
 
  • #6
32
7
To all those checking out this thread to figure out the solution to this kind of question (like I was), the problem here was that NasuSama multiplied the power by 3600s (the time in an hour). This is unnecessary because the question was posed in KWh, where you're to find how much power is lost in 1 hour.
 

Related Threads on Calculate the cost of the energy lost to heat per hour per meter

Replies
1
Views
6K
  • Last Post
Replies
1
Views
13K
  • Last Post
Replies
1
Views
17K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
15
Views
4K
Replies
1
Views
4K
Replies
4
Views
8K
Replies
6
Views
30K
Replies
2
Views
1K
Top