Question- Newton's Law of Cooling

[sankalpmittal]

Homework Statement

A hot body placed in the surrounding of temperature θ_o obeys Newton's law of cooling dθ/dt = -k(θ-θ_o), where k is any constant. Its temperature is θ₁ at t=0. t is time. The specific heat capacity of body is "s" and its mass is "m". Find the time starting from t=0 in which it will lose 90% of the maximum heat.

Homework Equations

http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/diffeqs/cool.html

The Attempt at a Solution

I solved it via integration and got the correct answer as well. As far as I know, it can be solved without integration:

The maximum heat body can lose has to be ms(θ₁-θ_o). If only 90 percent of this maximum heat is lost, we have, Heat lost = 9ms(θ₁-θ_o)/10. As ms is constant we have, Temperature difference = 9(θ₁-θ_o)/10

Initial temperature of body at time t=0 : θ₁
Final temperature of body when it cools as per condition : θ₁ - 9(θ₁-θ_o)/10 = (θ₁ + 9θ_o)/10

Average temperature of body = (Initial temperature + Finial temperature)/2 = (11θ₁ + 9θ_o)/20

In magnitude only, we can write law as, Δθ/Δt= k(θ-θ_o)

Now change in temperature Δθ = Final temperature of body when it cools as per condition - Initial temperature of body at time t=0 = 9(θ₁-θ_o)/10

Now Δt = t-0=t
θ-θ_o = Average temperature of body - θ_o = 11(θ₁-θ_o)/20

Now putting all these values in Δθ/Δt= k(θ-θ_o), we get

9/t = 11k/2 => t = 18/11k

But the answer is t= ln(10)/k = 2.303/k.

Where did I go wrong ?

Please help !

Thanks in advance...

 

[Mute]

Homework Statement

A hot body placed in the surrounding of temperature θ_o obeys Newton's law of cooling dθ/dt = -k(θ-θ_o), where k is any constant. Its temperature is θ₁ at t=0. t is time. The specific heat capacity of body is "s" and its mass is "m". Find the time starting from t=0 in which it will lose 90% of the maximum heat.

Homework Equations

http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/diffeqs/cool.html

The Attempt at a Solution

I solved it via integration and got the correct answer as well. As far as I know, it can be solved without integration:

The maximum heat body can lose has to be ms(θ₁-θ_o). If only 90 percent of this maximum heat is lost, we have, Heat lost = 9ms(θ₁-θ_o)/10. As ms is constant we have, Temperature difference = 9(θ₁-θ_o)/10

Initial temperature of body at time t=0 : θ₁
Final temperature of body when it cools as per condition : θ₁ - 9(θ₁-θ_o)/10 = (θ₁ + 9θ_o)/10

Average temperature of body = (Initial temperature + Finial temperature)/2 = (11θ₁ + 9θ_o)/20

In magnitude only, we can write law as, Δθ/Δt= k(θ-θ_o)

Now change in temperature Δθ = Final temperature of body when it cools as per condition - Initial temperature of body at time t=0 = 9(θ₁-θ_o)/10

Now Δt = t-0=t
θ-θ_o = Average temperature of body - θ_o = 11(θ₁-θ_o)/20

Now putting all these values in Δθ/Δt= k(θ-θ_o), we get

9/t = 11k/2 => t = 18/11k

But the answer is t= ln(10)/k = 2.303/k.

Where did I go wrong ?

Please help !

Thanks in advance...

Your problem is that the equation Δθ/Δt= k(θ-θ_o) is only valid when Δθ and Δt are very small, but they're not small here. You're basically treating the problem as though the heat flow is constant over the total duration of the problem, but you know that's not true: as the temperature difference between the body and the environment decreases, the rate of heat flow also decreases. That's why you need to solve the problem by integration.

 

[sankalpmittal]

Your problem is that the equation Δθ/Δt= k(θ-θ_o) is only valid when Δθ and Δt are very small, but they're not small here. You're basically treating the problem as though the heat flow is constant over the total duration of the problem, but you know that's not true: as the temperature difference between the body and the environment decreases, the rate of heat flow also decreases. That's why you need to solve the problem by integration.

But my textbook and reference books say that Newton's law of cooling in only valid when temperature difference between body and surrounding is not very large that is small. So why cannot I use the method in "Attempt at solution" here ?

In other words the books say that Newton's law of cooling itself is true for only small temperature difference. Question says that the body obeys Newton;'s law of cooling. So temperature difference between body and surrounding is also small. Am I correct ?

 

[Mute]

But my textbook and reference books say that Newton's law of cooling in only valid when temperature difference between body and surrounding is not very large that is small. So why cannot I use the method in "Attempt at solution" here ?

In other words the books say that Newton's law of cooling itself is true for only small temperature difference. Question says that the body obeys Newton;'s law of cooling. So temperature difference between body and surrounding is also small. Am I correct ?

Small compared to what? You need to be clear on what "small" and "large" are in reference to when using approximation techniques.

In this case, using your "attempt at solution" method requires that Δθ = θ(t + Δt) -θ(t) is small in the sense that Δθ = θ(t + Δt) -θ(t) ≈ θ'(t)Δt is a valid approximation, which requires that Δt is small compared to the timescale 1/k in the problem. Your method finds that the time between the initial and final condition is ~ 1.64/k, which is of course not small compared to the typical 1/k timescale. Hence, the conditions under which you used your approximation are violated, as indicated by your solution.

Note that these conditions of "smallness" are different from the conditions under which even the differential form of Newton's law, θ'(t) = k(θ-θo), holds. There, the "small" temperature difference is likely referring to the difference between the body and the environment. Probably, the "smallness" is determined by requiring that the heat-transfer coefficient between the two bodies is approximately constant (temperature independent). This temperature-scale could be much larger than the Δθ = θ(t + Δt) -θ(t), and so has no bearing on whether you can approximate Δθ = θ(t + Δt) -θ(t) ≈ θ'(t)Δt (except to say that the whole law is invalid if the heat-transfer coefficient is not constant).

 

[sankalpmittal]

Small compared to what? You need to be clear on what "small" and "large" are in reference to when using approximation techniques.

In this case, using your "attempt at solution" method requires that Δθ = θ(t + Δt) -θ(t) is small in the sense that Δθ = θ(t + Δt) -θ(t) ≈ θ'(t)Δt is a valid approximation, which requires that Δt is small compared to the timescale 1/k in the problem... Snip

Great Explanation Mute ! I get the concept now ! Thanks a lot ! :)