Newton's Law of Cooling and Stefan's Law

[maverick280857]

Hello

I've been thinking about this for a while and having consulted quite a few resources on the internet, decided to post some issues related to heat transfer by forced convection/radiation here...

Stefan's Law states that the rate of heat transfer by radiation from the body to the surroundings when the temperature of the body is $$\theta$$ and that of the surroundings is $$\theta_{0}$$ is given by

$$\frac{dQ}{dt}_{net} = -\epsilon \sigma A (\theta^4 - \theta_{0}^4)$$

where $$\epsilon$$ and A are the emissivity and surface area of the body and $$\sigma$$ is the Stefan-Boltzmann constant.

APPROXIMATING Stefan's Law:

Set $$\theta = \theta_{0} + \Delta \theta$$

so that
$$\theta^4 - \theta_{0}^4 = \theta_{0}^4(1 + \frac{\Delta\theta}{\theta})^4 - \theta_{0}^4 \cong 4\theta_{0}^3 \Delta\theta$$

Substituting the approximate value of the difference of the fourth powers of the body and surrounding temperatures (obtained above) in Stefan's Law, we get

$$\frac{dQ}{dt} = -4 \epsilon \sigma A \theta_{0}^3 (\theta - \theta_{0})$$

(The binomial approximation used here is mathematically valid for $$\frac{\Delta\theta}{\theta_{0}} << 1$$.)

Newton's Law of Cooling (http://scienceworld.wolfram.com/physics/NewtonsLawofCooling.html) states that the rate of cooling (by forced convection) of a body is directly proportional to the temperature difference between the body and the surroundings:

$$\frac{d\theta}{dt} = -k(\theta - \theta_{0})$$

where k is a constant.

I have read in a book that Newton's Law of Cooling is a special case of the more general Stefan's Law. The book has shown how the fourth power difference is linearized to the simple temperature difference as follows:

$$\frac{dQ}{dt} = mC\frac{d\theta}{dt}$$ so

$$\frac{d\theta}{dt} = \frac{1}{mC}\frac{dQ}{dt} = \frac{-4 \epsilon \sigma A \theta_{0}^3}{mC} (\theta - \theta_{0})$$

Comparing this form with Newton's Law, we get

$$k = \frac{4 \epsilon \sigma A \theta_{0}^3}{mC}$$

According to the book therefore, this means that k depends on the emissivity as as well as the specific heat C.

I have the following two questions:

1. When in a practical situation, can I linearize the fourth power difference to the linear difference? What is the threshold difference above which this approximation is not valid? (I ask this because I have calculated the ratio of heat loss by stefan's law and Newton's law for a temperature difference of 10 degrees C and I found the ratios to be 0.12 and 2--the difference between them is large enough to suggest that this approximation is bad).

2. Strictly speaking (if one does ignore the mathematical illusion above...that linearizing a higher order differential equation represents a totally different law) does Newton's Law hold as a special case of Stefan's Law or are the two different? I think they should be different since the former refers to heat loss by forced convection and the latter by radiation (according to Wolfram Scienceworld).

I will probably add to this post a while later but for now, this is a sufficient input for discussion.

Thanks and cheers
Vivek

 

[Tide]

What base temperature did you use for your comparison?

 

[Clausius2]

Hello
.

I have read in a book that Newton's Law of Cooling is a special case of the more general Stefan's Law. The book has shown how the fourth power difference is linearized to the simple temperature difference as follows:

.......
Comparing this form with Newton's Law, we get

$$k = \frac{4 \epsilon \sigma A \theta_{0}^3}{mC}$$

According to the book therefore, this means that k depends on the emissivity as as well as the specific heat C.

What's that?.

I've never heard about that equation before!. Convection and Radiation are NOT relationed directly. They are two different mechanism of heat transfer. It's impossible to obtain a coefficient of convection $$k$$ so easily and as a function of such primary variables!. No confusion between radiation and convection is possible.

The Newton's law of cooling is a very simple law, a very simple formulation if we except a little detail: obtaining the coefficient $$k$$ is a very hard task. In order to obtain it, no consideration of radiation transfer is needed (it will be a misconception to need it, and I'm not going to change my opinion unless some thermal engineer comes here and says the contrary). In particular, $$k$$ depends on the external flow of cooling fluid that surrounds the solid walls. So that you have to solve the complete flow field to figure k out (there are correlations available too).

 

[Integral]

I am glad to hear you say that Clausius that is similar to my thoughts.

The best way to find k in Newtons law of cooling is experimentally for each special case. It is considered a bulk coefficient which will include heat transfer by all mechanisms. Of course as one would expect any experimentally determined coefficient will only be meaningful on a small range of temperatures.

 

[Clausius2]

I am glad to hear you say that Clausius that is similar to my thoughts.

The best way to find k in Newtons law of cooling is experimentally for each special case. It is considered a bulk coefficient which will include heat transfer by all mechanisms. Of course as one would expect any experimentally determined coefficient will only be meaningful on a small range of temperatures.

Yes. Have you ever employed those horrible correlations? (I'm not sure if "correlation" exists in english). A correlation is an expression (long and bored) involving the flow properties, conditions and dynamics trough the Numbers of the Fluid Mechanics. I remember some fatal times I spent calculating this in a few exams :zzz: .

 

[maverick280857]

I was just as surprised as you folks are when I read it the first time, because when I read Newton's Law as a problem it clearly mentioned forced convection as the mode of heat transfer (and besides, I couldn't doubt what the wolfram site says anyway). This is like saying that two equations are so similar that they are equivalent! I know this is incorrect but when I visited

http://www.math.tamu.edu/~francis.narcowich/m308h/m308h_prj1.html

I read the the line, "the time of death can be determined using Newton's law of cooling rather than the more accurate, but also more complicated, Stefan-Boltzmann law that Newton's law approximates". Please have a look and let me know what you think.

Despite the university site mentioning this, I still believe Newton's Law and Stefan's Law are two different things. As a matter of fact, the approximation which linearizes the temperature difference is strictly valid (as I pointed out in my first post) if the temperature difference to initial temperature ratio is small "enough".

I would also like to know what the practical/acceptable standard of this smallness is. I read that if the ratio is something like (5/60) it is okay to approximate the fourth power difference to a linear difference. But saying that the approximation is Newton's Law just because the equations look similar is wrong because after all, mathematical approximations won't change the mode of heat transfer! (I went ahead and even computed a few more such ratios, only to be startled by the difference--which is quite obvious without calculation too--in the answers obtained through the two routes).

I recently learned from a friend--who was taught this--that the k in Newton's Law is a sum of two constants...one for radiation and one for convection and that Newton's Law holds both for convection and radiation!

(I hope more people read this so that this apparent misconception is cleared).

Cheers
Vivek

 

[Clausius2]

I was just as surprised as you folks are when I read it the first time, because when I read Newton's Law as a problem it clearly mentioned forced convection as the mode of heat transfer (and besides, I couldn't doubt what the wolfram site says anyway). This is like saying that two equations are so similar that they are equivalent! I know this is incorrect but when I visited

http://www.math.tamu.edu/~francis.narcowich/m308h/m308h_prj1.html

I read the the line, "the time of death can be determined using Newton's law of cooling rather than the more accurate, but also more complicated, Stefan-Boltzmann law that Newton's law approximates". Please have a look and let me know what you think.

Despite the university site mentioning this, I still believe Newton's Law and Stefan's Law are two different things. As a matter of fact, the approximation which linearizes the temperature difference is strictly valid (as I pointed out in my first post) if the temperature difference to initial temperature ratio is small "enough".

I would also like to know what the practical/acceptable standard of this smallness is. I read that if the ratio is something like (5/60) it is okay to approximate the fourth power difference to a linear difference. But saying that the approximation is Newton's Law just because the equations look similar is wrong because after all, mathematical approximations won't change the mode of heat transfer! .

Ok, I see you are aware of such misconception. They are not equivalent at all, and are not relationed physically. They are relationed if you want mathematically, because the linearized Steffan-Boltzmann has a shape like the Newton's. But it is only its external shape.

Therefore, the link you've provided are telling you a cocks and bulls story, its a fatal misconception mixing both of them in spite of having the same mathematics. I assure you if Newton goes up again, surely he will kick off those who are written that.

I recently learned from a friend--who was taught this--that the k in Newton's Law is a sum of two constants...one for radiation and one for convection and that Newton's Law holds both for convection and radiation!

Tell your friend that's another misconception.

 

[maverick280857]

CONCLUSION: Newton's Law is for Forced Convection ONLY!

Yes I believe this is one of those instances when mathematics applied wrongly can lead to misleading results. You can linearize any differential equation of a higher order or employ an algebraic approximation under strict conditions but it can't change the mechanism--it can't change nature. Thats what I've been trying to convince everyone who has argued these problems with me so far.

Quite frankly, I think the tamu link (given in my earlier post) is a useless resource for students seeking clarifications in this regard. As a concrete example of this, there is a problem which approximates stefan's law and calls the approximated form as Newton's law: calculate the ratio of the rates of cooling of two spherical bodies of the same material when the temperature of the first body is 55 degrees C and that of the second is 60 degrees C (the temperature of the environment being 70 degrees in both cases). First off, the problem does not mention the mode of heat transfer--assuming it to be radiation, you would go ahead and apply stefan's law but your answer would be way different from that given by the author because he used the approximated stefan's law in the first place simply saying, "Since the temperature difference is small 'enough', we can use Newton's Law of cooling...". Clearly this is quite wrong conceptually.

I think it is high time people make a distinction between mathematics and its mechanical usage in physics. There is a bigger list of misconceptions in the minds of most students nowadays because of such errors on the part of a few more professional people.

Cheers
Vivek

 

[Clausius2]

Good clarification, Vivek.

But there is another little detail: The Newton's Law of cooling is not emplyed only with forced convection. Natural convection and Mixed convection effects can be included also in the convective coefficient h.

 

[maverick280857]

But there is another little detail: The Newton's Law of cooling is not emplyed only with forced convection. Natural convection and Mixed convection effects can be included also in the convective coefficient h.

Yes quite rightly so, but where is the h you are referring to Clausius? I mean are you referring to some other form of the law which is different from the one mentioned above?

Cheers
Vivek

 

[Clausius2]

Yes quite rightly so, but where is the h you are referring to Clausius? I mean are you referring to some other form of the law which is different from the one mentioned above?

Cheers
Vivek

Ooops! Sorry. In the majority of books the convective coefficient is called with letter h. But here h=k.