# Geothermal heat extraction math problem

courtneywetts

## Homework Statement

The actual efficiency of a geothermal power plant using a 200 C geothermal resource is only about 7%. Take this value as typical of California's Geysers geothermal field, which produces 900 MW of electric power.

a) Find the actual rate of geothermal heat extraction at Geysers

b) it's estimated that 35EJ of energy could be extracted from the Geysers before the temperature dropped from 250 C to 150 C minimum for a vapor-dominated geothermal plant. Given your answer in (a) how long will the Geysers geothermal resource last before the temperature reaches 150 C

## Homework Equations

I am not sure what equation to use

## The Attempt at a Solution

Starting at part A I know I have to find the rate of geothermal heat extraction but do not know the formula to do so.

Does this have something to do with change in temperature? It's the only thing I can think to do but am not sure if this is right

Staff Emeritus
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You are given the amount of electric power produced by the plant along with the efficiency. How many Joules per second are required to give this power? (Hint: 1 watt = 1 Joule / second)

Gnaw on this first before tackling part b.

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How many Joules per second are required to give this power? (Hint: 1 watt = 1 Joule / second)
I don't see the relevance of that. The question can be answered in MW, surely.
Courtneywetts, there's nothing special about heat extraction for part a. You have power input, a conversion system that's 7% efficient, and power output of 900MW. So how much power is input?
Part b doesn't have much to do with temperature either. We can replace the references to temperatures with references to states and still make a sensible question: the resource will change from state A to state B when 35EJ have been extracted; we know from part a the rate of energy extraction; how long will it take to change from state A to state B?

courtneywetts
I don't see the relevance of that. The question can be answered in MW, surely.
Courtneywetts, there's nothing special about heat extraction for part a. You have power input, a conversion system that's 7% efficient, and power output of 900MW. So how much power is input?
Part b doesn't have much to do with temperature either. We can replace the references to temperatures with references to states and still make a sensible question: the resource will change from state A to state B when 35EJ have been extracted; we know from part a the rate of energy extraction; how long will it take to change from state A to state B?

What formula would I be using for part A? Is there a specific formula to use? I am not sure how to find how much power is in input.

Staff Emeritus
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I don't see the relevance of that. The question can be answered in MW, surely.
Courtneywetts, there's nothing special about heat extraction for part a. You have power input, a conversion system that's 7% efficient, and power output of 900MW. So how much power is input?
Part b doesn't have much to do with temperature either. We can replace the references to temperatures with references to states and still make a sensible question: the resource will change from state A to state B when 35EJ have been extracted; we know from part a the rate of energy extraction; how long will it take to change from state A to state B?

Part a of the problem asks for the rate of energy extraction. I thought that it was important for the user to understand that power is equivalent to the rate of energy production from the source, especially in light of part b.

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What formula would I be using for part A? Is there a specific formula to use? I am not sure how to find how much power is in input.
If the power input is P and the power output is P', what would the efficiency be?

courtneywetts
If the power input is P and the power output is P', what would the efficiency be?

I am confused what this means. If the power plant is 7% efficient and 900 MW then am I trying to find the power input? If so how do I do this?

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What's the definition of efficiency?

courtneywetts
What's the definition of efficiency?

Achieving maximum productivity with minimal effort.

Is this what I am looking for?

Staff Emeritus
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Does that statement apply to your problem, or is it a personal ideal?

Have you looked at your textbook for a definition of efficiency (the thermal kind, not the personal actuation kind)?

You can always Google the term, if that helps.

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Achieving maximum productivity with minimal effort.

Is this what I am looking for?
The efficiency of a process is the fraction of energy (or other resource) which is makes it through into the output. If P in and P' out the efficiency is P'/P.
In thermodynamics the concept of efficiency can get a little strange. Since there is a theoretical limit on conversion of heat to work, based on the temperatures involved, the process will have a theoretical maximum efficiency as well as an actual eficiency, so in asessing the process it is ressonable to think in terms of the ratio of those two efficiencies rather than the actual efficiency.
Conversely, a heat pump can produce more heat out than work in, so this normal definition of efficiency can result in values over 100%.

courtneywetts
The efficiency of a process is the fraction of energy (or other resource) which is makes it through into the output. If P in and P' out the efficiency is P'/P.
In thermodynamics the concept of efficiency can get a little strange. Since there is a theoretical limit on conversion of heat to work, based on the temperatures involved, the process will have a theoretical maximum efficiency as well as an actual eficiency, so in asessing the process it is ressonable to think in terms of the ratio of those two efficiencies rather than the actual efficiency.
Conversely, a heat pump can produce more heat out than work in, so this normal definition of efficiency can result in values over 100%.

Then in the problem the efficiency is 7% and the output is 900 MW. Am I right on this? So I would then use these numbers to find the rate of heat extraction?

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Then in the problem the efficiency is 7% and the output is 900 MW. Am I right on this? So I would then use these numbers to find the rate of heat extraction?
Yes, just using the Eff = Pout/Pin formula.

courtneywetts
Yes, just using the Eff = Pout/Pin formula.

Okay so then would this be .07 * 900= 63?

Staff Emeritus
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No, 900 MW is what comes out of this process.

courtneywetts
No, 900 MW is what comes out of this process.

So then 900 MW is the pout

How do I arrange this equation.

I am using .07 right?

courtneywetts
I think I figured it out

7% * (Total Wattage) = 900 MW
Total Wattage = 900 MW / (7/100) = 12,857.14 MW = 12.9 GW

Is this right?

Homework Helper
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I think I figured it out

7% * (Total Wattage) = 900 MW
Total Wattage = 900 MW / (7/100) = 12,857.14 MW = 12.9 GW

Is this right?
Yes.

courtneywetts
Yes.

I am confused how to do part B.

What formula would I be using?

Where am I using this 12.9 GW?