How Long Does It Take to Heat a Room Using Newton's Law of Cooling?

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Homework Help Overview

The discussion revolves around calculating the time required to heat a large room using an air cooled condensing unit, applying principles from Newton's Law of Cooling. The room's volume is approximately 27,195 cubic feet, and the unit rejects 102,600 BTU/hr into the space, with an initial ambient temperature of 95 degrees Fahrenheit.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the relationship between heat transfer, specific heat, and temperature change. There are attempts to apply formulas involving specific heat and density to calculate the energy required to heat the air. Questions arise regarding the specific heat and density of air, as well as the validity of the volume conversion from cubic feet to cubic meters.

Discussion Status

Some participants have provided calculations and expressed their findings, while others have raised questions about the assumptions made, particularly regarding unit conversions and the mass of air being heated. There is an ongoing exploration of different interpretations of the problem and the related equations.

Contextual Notes

There is a noted discrepancy in the volume conversion from cubic feet to cubic meters, which may affect the calculations. Participants are also considering the implications of using different units and the assumptions inherent in the problem setup.

esw6838
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I have a problem where I have an air cooled condensing unit located in a large room within a building. The volume of the room is approximately 27,195 cubic feet. The air cooled condensing unit rejects 102,600 btu/hr to the space. Assuming the ambient air temperature of the space is 95 degrees fahrenheit, how can I calculate how long will it take to heat the volume of air 1 degree fahrenheit? Any help is appreciated. Thanks.
 
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https://www.physicsforums.com/showthread.php?t=94379

What is the specific heat of air? What is the density of air? How is specific heat related to temperature change? What have you done to try to solve this problem?
 
If I am correct:

Q = Cp(T2-T1) and
P = QVp/time

Cp = specific heat and
p = density
P = power
V =Volume
T2 = final temp
T1 = initial temp
 
esw6838 said:
If I am correct:

Q = Cp(T2-T1) and
P = QVp/time

Cp = specific heat and
p = density
P = power
V =Volume
T2 = final temp
T1 = initial temp

The mass (Vρ) belongs in your first equation; Q depends on how much air is being heated. It does not belong in the second equation since it is already included in Q.
 
I first solve for Q. After I know Q, I can solve for time since I know the power is 102,600 btu per hour. I used SI units and came up with the following:

Cp = 1,006 J per KgK
p = 1.15 Kg per cubic meter
V = 481 cubic meters
T1 = 308.15K (95F)
T2 = 310.9K (100F)
P = 30,060W (102,600 btu per hour

I wind up with time = 50 sec. I thought it would take longer. Interesting.
 
esw6838 said:
I first solve for Q. After I know Q, I can solve for time since I know the power is 102,600 btu per hour. I used SI units and came up with the following:

Cp = 1,006 J per KgK
p = 1.15 Kg per cubic meter
V = 481 cubic meters
T1 = 308.15K (95F)
T2 = 310.9K (100F)
P = 30,060W (102,600 btu per hour

I wind up with time = 50 sec. I thought it would take longer. Interesting.

I did not check all your conversions, but the volume looks a bit small to me. 1 meter is less than 3⅓ feet, and (3⅓ feet)³ is around 37. 27,195 ft³ should be around 740 m³
 

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