Refrigeration units can be rated in “tons.” A 1-ton air condit...

Question

Refrigeration units can be rated in “tons.” A 1-ton air conditioning system can remove sufficient energy to freeze 1 ton (2000 pounds = 909 kg) of 0°C water into 0°C ice in one 24-h day. Assume the hot part of a day averages 35°C and the interior of a house is maintained at 22°C by the continuous operation of a 6-ton air conditioning system for 6 hours a day. How much does this cooling cost the homeowner per day, and per month?Assume the work done by the refrigeration unit is powered by electricity that costs $0.13 per kWh and that the unit’s coefficient of performance is only 18% of an ideal refrigerator. 1 kWh = 3.60 x 10⁶ J .

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem. A commercial building uses an air conditioning system with a capacity rating of five tons. It can remove sufficient heat to convert five tons, which is also equal to 4545 kg of water at 0.0 °C into ice. Within 24 hours. The ac runs for 10 hours every weekday when the external temperature averages around 38 °C while maintaining an internal temperature of 20 °C. If the electricity cost is 0.12 cents per kilowatt and efficiency is just 15% compared with an ideal refrigerator. Find out what the daily and monthly costs are for cooling. This particular building, the latent heat of fusion for water is 333,000 joules per kilogram. Awesome. So it appears that our final answer or I should say answers that we're trying to solve for our end B is, we're trying to figure out what the daily and monthly costs are for cooling this particular building. So that's what we're ultimately trying to solve for. We're trying to solve for the daily and monthly costs it takes in order to cool this particular building. So with that in mind, let's read off our multiple choice answers to see which one might be our final answer pair. And let's note that our first answer is for the daily cost in American dollars. And our second answer is the monthly cost also in American dollars. Ok. So, or actually doesn't specify us D but it's in dollars for the unit of measurement for money price. So A is 8.6 and 170. B is 8.7 and 170 C is 8.6 and 180 D is 8.7 and 180. Ok. So first off, we need to figure out what the energy removed by the five ton air conditioning system is. So, in order to help us solve for that, we need to quickly recall that the latent heat effusion or water is 333,000 joules per kilogram. So we can therefore using this value, we can recall and use the equation to help us solve for Q, the heat. So, oh, sorry, not Q, the heat, it's Q the energy. So we're using Q which is the energy removed by the five ton system in one day. And we're calling this Q is equal to 4545 kg because this is the amount of water and five tons in kilogram units multiplied by our latent heat effusion which is 333,000 and its units are jewels per kilogram. So when we plug that into our calculator, we will find that Q is equal to 1.513485 multiplied by 10 to the power of nine and its units are Jews per day. Awesome. So now we need to figure out what the coefficient of performance is, which we're gonna denote the coefficient of performance as cop. So let us note and recall that the ideal cop which we're gonna denote as COP ideal for a refrigerator working between 20 °C, which is equivalent to 293 Kelvin and 38 °C, which is equivalent to 311. Kelvin would be equal to when we write this equation out COP ideal is equal to the cold temperature or the cold reservoir temperature divided by the hot temperature, which we're gonna denote as th and cold reservoir STC. So th minus TC. So COP ideal is equal to TC divided by th minus TC, which is equal to plugging in our known value. So we're gonna use the Calvin units instead of degrees Celsius. So our TC would be equal to 293. Kelvin divided by 311. Kelvin minus 293. Kelvin, which will mean that our cop ideal is equal to when we round to two decimal places. 16.28. Awesome. So now we need to note and recall once again that we're told that the actual cop is 15% of the ideal cop. So therefore, we could write at cop is equal to and let's write 15% as a decimal. So it's 0.15 multiplied by cop idea, which is equal to when we plug in our known variables, 0.15 multiplied by 16.28 which will mean that our co is approximately equal to from roll 2.44 when we round to two decimal places. So now at this stage, we need to solve for the total work done. And let's denote this value as capital W. So let us recall and use the equation that will help us offer for the work done per day by this five ton system which states that the work is equal to Q divided by cop, which is equal to 1.513485 multiplied by 10 to the power of nine Jews per day. And what we're doing right now is we're plugging in our value for Q and now we're gonna plug in our value for the cop which we determined to be 2.44. So when we plug that into our calculator. We will get 6.203 multiplied by 10 to the power of eight jewels. Awesome. Awesome. And this is, don't forget that it's jewels per day is our unit snapped. So now we need to figure out what our work. Now, now we need to convert our work done to kilowatts because all her is we're trying to figure out the cost. And right now our electricity, electricity cost is 0.12 which is 12 cents per kilowatt. So we need to convert jewels per day and we're trying to convert our work done value to get it into kilowatt units. So with that in mind, we need to note that the work per hour is and let's call this work per hour. So um we're gonna call this W subscript hour is equal to 6.204 multiplied by 10 to the power of eight and its units are jewels per day and this is divided by 24 hours. These are 24 hours in one day which to keep things simple, we're just gonna do a lowercase H to denote our. So when we plug this into our calculator, we will get 2.58 five and it's gonna be multiplied by 10 to the power of seven and its new units will be jewels hour and this is when we round to three decimal places. So now we need to consider that it's operating for 10 hours. So we need to say that W subscript 10 hours. So this is the work done per 10 hours of work is equal to 10 hours, multiplied by two point. So multiplied by 2.585 multiplied by 10 to the power of seven jewels per hour is equal to. So this is our work for 10 hours is equivalently equal to two. So when you say two point 585, multiplied by, when we plug this into our calculator, we will get that two point that this is equal to 2.585 multiplied by 10 to the power of eight jewels. So now we need to convert jewels to kilowatts like we were talking about a moment ago. So for w 10 hours, this will be equal to 2.5 85 multiplied by 10 to the power of eight jules divided by 3.60 multiplied by 10 to the power of six jewels per kilowatt. Awesome. Ok. Killer. What? Which is equal to let me plug that into our calculator. 71.81 kg watts when we run to two decimal places. So now we can officially start to solve for the cost per day. So let's call, so the cost per day. So cost dot per day. So we're gonna call this C dot P dot D. So cost per day. So CPD is equal to 71.8 one kilowatts and we need to multiply this by conversion factor which says that 12 cents. So 0.12 per kilowatt is equal to me round to one decimal place eight $0.6. And that's our first answer. Hooray. We did it. So now we could solve for the cost monthly cost. So we need to recall and note that the number of weekdays in a month, assuming that there's four weeks in a month, we could say that the number of weekdays. So number of weekdays, NWD. So the number of weekdays is equal to five days per and we're gonna say per one and I'll do a lowercase W per week. So per one week, multiplied by four weeks, which will mean that that is equal to so four weeks, which is equal to 20 days total. Awesome. You as you can see the week units cancel out. So now we need to say that the monthly cost, we need to take 20 days and we need to multiply it by our cost per day, which I went ahead and we rounded to one decimal place. But in your calculator, you'll get eight point 6172, which we just rounded to the one decal place. But let's use this big old long number to make sure that our final answer is more accurate. So 8.6172 and this is the cost per day. And when we multiply that out in our calculator and we round to the nearest pole number. We will get $170 and that is our final answer. All right, we did it. So looking at our multiple choice answers, the correct answer has to be the letter A. The daily cost is equal to $8.6 and the monthly cost is equal to $170. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Key Concepts

Coefficient of Performance (COP)

Energy Consumption Calculation

Cost of Electricity

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