# Newton's Law of Cooling

#### John Ramelb

Joined Feb 2, 2014
26
dT/dt = k ( T - Tm )

John and Mary have just built a new house, and the builder informs them that the insulation in the house provides for a time constant of 5 hrs. Suppose that at midnight furnace fails w/ the outside temp. at a constant 10°F and the inside temp. at 70°F. a) Det. the initial - value of problem that describes the future temp. inside the house. b) Determine the temp. inside the house at 5:00 A.M

some one help me please. give me tip to solve the problem.
what the Times constant of 5 hrs means.?

#### MrChips

Joined Oct 2, 2009
24,141
Study how the voltage across a capacitor with capacitance C diminishes over time as the capacitor discharges through a resistor with resistance R.

Write down the equation for the voltage across the capacitor as a function of time.

#### WBahn

Joined Mar 31, 2012
26,398
Somewhere in all of this you should have been exposed to the concept of a time constant.

It's like a "half life" (do you understand what that is?).

The time constant characterized something that changes exponentially (in particular, a decaying exponential change from one level to another). Whatever the value of something was at time T, the value at time T+τ, where τ is the time constant, will be 1-(1/e) of the way toward it's final value. In rough terms, that works out to 2/3. So if something starts at, say, a distance of 19m and is changing exponentially toward a final distance of 100m with a time constant of 5 minutes, then it is changing by a total of 81m. After 5min it will be 2/3 of the way, which would be 19m+(2/3)*81m which is 73m, leaving it 27m to go. After another 5 minutes it will have moved 2/3 of this distance, or 18m closer to its final destination placing it at 91m. Another 5 minutes will take it to 97m and yet another 5 minutes will take it to 99m. As you can see, the closer it gets the slower its progress is. In theory it never gets there; in practice we usually consider it to have "arrived" at it's final state after about five time constants because it will be within about 0.5% of the amount of the change being made.

#### #12

Joined Nov 30, 2010
18,223
I've been finding K for buildings for 30 years and never knew it went back to Newton.

#### THE_RB

Joined Feb 11, 2008
5,438
Don't you know? He was sitting under a building and an air conditioner fell on his head. Hence the new law.

#### #12

Joined Nov 30, 2010
18,223
Oops! I knew I should have used lag bolts.

#### GopherT

Joined Nov 23, 2012
8,012
dT/dt = k ( T - Tm )

John and Mary have just built a new house, and the builder informs them that the insulation in the house provides for a time constant of 5 hrs. Suppose that at midnight furnace fails w/ the outside temp. at a constant 10°F and the inside temp. at 70°F. a) Det. the initial - value of problem that describes the future temp. inside the house. b) Determine the temp. inside the house at 5:00 A.M

some one help me please. give me tip to solve the problem.
what the Times constant of 5 hrs means.?

This problem can be solved if John and Mary were not present.

However, a sedentary adult human generates about 100 watts so that will need to be taken into account and more information about the size of the house and heat capacity of the air (relative humidity) must be known.

Another consideration is whether or not John has any idea how to fix the furnace. If he does, any heat added to the house during each furnace restart attempt will need to be considered.

If John has no idea how to fix the furnace, he will likely do what needs to be done to keep Mary warm until morning and then he and Mary can no longer be considered sedentary. Input energy from John and Mary will be (100 watts + X) for the seconds that John and Mary engage in additional 'activity'.

If John does not know how to keep the house warm, AND John does not know how to keep Mary warm, then Mary will leave and stay with her mother and john may as well be dead so you can go back and use the original assumptions of the problem.

------------

NOTE: IF John's Time Constant is longer than four hours, he must stop and call his doctor.

Last edited:
#12

#### MrChips

Joined Oct 2, 2009
24,141
Heating and cooling laws follow the exponential function.

If all John has on his mind is $$\int e^x$$ then the time constant is going be stretched out a bit longer.

#### WBahn

Joined Mar 31, 2012
26,398
And remember,

$$\int e^x \; = \; f(u^n)$$