# Power used to heat water and efficiency math problem

• courtneywetts
In summary, the household uses 100 gallons of hot water per day and the water temperature must be raised from 10°C to 50°C. To find the average power needed to heat the water, the mass of 100 gallons of water (378 kg) must be multiplied by the specific heat of water (4.184 J/(g*K)) and the temperature change (40 K). This results in an average power of 732 W. For part B, the solar panel's efficiency of 55% must be taken into account, and the amount of solar power needed to produce 732 W must be calculated using the table provided. The table gives the average monthly insolation in Albuquerque, NM for June, and the necessary power can

## Homework Statement

A household uses 100 gallons of hot water per day (1 gallon of water has a mass of 3.78 kg) and the water temperature must be raised from 10°C to 50°C

a) find the average power needed to heat the water.

b)Then assume you have solar collectors with 55% average efficiency at converting sunlight to thermal energy when tilted at your location's latitude. Using the table find the collector area needed to supply this power in Albuquerque in June.

From looking at the text, the tilted altitude is 296.

Q=mcΔt

## The Attempt at a Solution

For part A I think I need to find the mass of 100 gallons of water? This would be 3780 kg. (make sure I am right on this)

I would then use the formula I stated above. So

Q= (3780) (4.184) (40) -is this correct for water?

Then the answer would be 632620.8

I am not sure if this is right.

For part B I have no idea what formula I should be using

Your mass is 3780 kg per day, you need this in kg per second to get the units for power. So you have to divide your answer by 1 day converted to seconds.

For part b, they are telling you that 55% of the total energy is what you calculated above.

Also what table are they referring to?

rock.freak667 said:
Your mass is 3780 kg per day, you need this in kg per second to get the units for power. So you have to divide your answer by 1 day converted to seconds.

For part b, they are telling you that 55% of the total energy is what you calculated above.

Also what table are they referring to?

So to get it for a day I would need to know how many seconds are in a day? This would be 86400.
That would then mean 3780 divided by 86400?

The table is average monthly insolation (W/m^2) and temperature for selected US cities

In part B how would I find the area?

courtneywetts said:
So to get it for a day I would need to know how many seconds are in a day? This would be 86400.
That would then mean 3780 divided by 86400?

The table is average monthly insolation (W/m^2) and temperature for selected US cities

In part B how would I find the area?

Yes that is correct.

The units of the insulation is W/m^2 and your energy is in W, so if you divide them, what units would you get?

rock.freak667 said:
Yes that is correct.

The units of the insulation is W/m^2 and your energy is in W, so if you divide them, what units would you get?

Would you end up with just m^2?

I am confused with what I am trying to find in this problem still though

You would need to actually put the table, I assumed they were giving your the heat transferred per unit area.

courtneywetts said:

## Homework Statement

I would then use the formula I stated above. So

Q= (3780) (4.184) (40) -is this correct for water?

Then the answer would be 632620.8

I am not sure if this is right.

For part B I have no idea what formula I should be using

The specific heat of water of about 4.2 is in J/(g*K). Per gram, not per kilogram.
So you are off by a factor of 1000, to start with.

Last edited:
rock.freak667 said:
You would need to actually put the table, I assumed they were giving your the heat transferred per unit area.

What the table gives is

Albuquerque, NM for June:

Horizontal- 338 W/m^2
Titled at latitude- 296 W/m^2
Average Temp (°C)- 23.4

I am not sure what to do with this information to solve part B

1. First find the correct value of the power.
2. Then calculate how much power is necessary to collect from the sun if the conversion efficiency is 55%. In other words, how much power is necessary so that 55% of it is the answer in part (1).
3. And third, you know that you 296 W for each m^2. How many m^2 you need to get the power in part (2).

nasu said:
1. First find the correct value of the power.
2. Then calculate how much power is necessary to collect from the sun if the conversion efficiency is 55%. In other words, how much power is necessary so that 55% of it is the answer in part (1).
3. And third, you know that you 296 W for each m^2. How many m^2 you need to get the power in part (2).

How would I go about calculating the power?

Just the way you did. Find the energy and divide by time (one day in seconds, to get watts).
I meant to use consistent units so you get the correct value.
Either mass in grams and specific heat in J/(g*K) or mass in kg and specific heat in J/(kg*K).
In the first case the value is 4.180, in the second 4180.

nasu said:
Just the way you did. Find the energy and divide by time (one day in seconds, to get watts).
I meant to use consistent units so you get the correct value.
Either mass in grams and specific heat in J/(g*K) or mass in kg and specific heat in J/(kg*K).
In the first case the value is 4.180, in the second 4180.

I am confused on how to find the energy.

I got 7322 watts from part A.

I am not sure if this is right or relevant to part B? What numbers should I be using?

Now I think you are 10 times higher. Did you use the mass as 3780 kg, as you wrote in the first post?
100 gallons is about 378 kg. Not thousands.
So the average power required is about 732 W.
Now, the panel only converts 55% of the solar power.
How much solar power is required so that 55% of it is 732 W?
This is the first step for part B.

nasu said:
Now I think you are 10 times higher. Did you use the mass as 3780 kg, as you wrote in the first post?
100 gallons is about 378 kg. Not thousands.
So the average power required is about 732 W.
Now, the panel only converts 55% of the solar power.
How much solar power is required so that 55% of it is 732 W?
This is the first step for part B.

I did (.04375)(4184)(40) and got 7322

but if the answer is 732 would I do (.55) (732)?

Am I looking for power input or output?

What is .04375? Edit. Never mind. It's mass of water per second, right?
It's just that you converted the gallons wrong. So that's why you get this higher values. I told you this already.

No, you need to find of what number 732 is 55%.
That number is obviously larger than 732, not smaller.
Do you understand what 55% of something means?

nasu said:
What is .04375? I got this for the mass?

Sorry, I realized what you mean while you were posting yourself. Please see my edit above. (post 15)

How would I convert the gallons correctly?

So then would I be doing 732/.55 to get 1330?

Sorry I am really confused

1 gallon is about 3.78 liters. And 1 liter of water is about 1 kg.
So 1 gallon of water has a mass of about 3.78 kg.
Then 100 gallons have 378 kg. And not 3780 kg, as you used in your calculations.

Yes, the solar power collected by the panels should be about 1330 W.
Now there is just one more step. One square meter of panel collects 296 W.
How many m^2 you need to collect 1330 W.

nasu said:
1 gallon is about 3.78 liters. And 1 liter of water is about 1 kg.
So 1 gallon of water has a mass of about 3.78 kg.
Then 100 gallons have 378 kg. And not 3780 kg, as you used in your calculations.

Yes, the solar power collected by the panels should be about 1330 W.
Now there is just one more step. One square meter of panel collects 296 W.
How many m^2 you need to collect 1330 W.

So I am trying to find out how many m^2 are in 1330 W?

Would I do 1330/296 to get 4.5 m^2?

courtneywetts said:
So I am trying to find out how many m^2 are in 1330 W?

Would I do 1330/296 to get 4.5 m^2?

Even though your first sentence does not make sense (there are no m^2 in 1330 or any number of watts) , the second one looks OK.

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