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## Main Question or Discussion Point

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 [tex]\theta[/tex] and that of the surroundings is [tex]\theta_{0}[/tex] is given by

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

where [tex]\epsilon[/tex] and A are the emissivity and surface area of the body and [tex]\sigma[/tex] is the Stefan-Boltzmann constant.

Set [tex]\theta = \theta_{0} + \Delta \theta[/tex]

so that

[tex]\theta^4 - \theta_{0}^4 = \theta_{0}^4(1 + \frac{\Delta\theta}{\theta})^4 - \theta_{0}^4 \cong 4\theta_{0}^3 \Delta\theta[/tex]

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

[tex]\frac{dQ}{dt} = -4 \epsilon \sigma A \theta_{0}^3 (\theta - \theta_{0})[/tex]

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

[tex]\frac{d\theta}{dt} = -k(\theta - \theta_{0})[/tex]

where k is a constant.

I have read in a book that

[tex]\frac{dQ}{dt} = mC\frac{d\theta}{dt}[/tex] so

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

Comparing this form with Newton's Law, we get

[tex]k = \frac{4 \epsilon \sigma A \theta_{0}^3}{mC}[/tex]

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

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 [tex]\theta[/tex] and that of the surroundings is [tex]\theta_{0}[/tex] is given by

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

where [tex]\epsilon[/tex] and A are the emissivity and surface area of the body and [tex]\sigma[/tex] is the Stefan-Boltzmann constant.

**APPROXIMATING Stefan's Law**:Set [tex]\theta = \theta_{0} + \Delta \theta[/tex]

so that

[tex]\theta^4 - \theta_{0}^4 = \theta_{0}^4(1 + \frac{\Delta\theta}{\theta})^4 - \theta_{0}^4 \cong 4\theta_{0}^3 \Delta\theta[/tex]

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

[tex]\frac{dQ}{dt} = -4 \epsilon \sigma A \theta_{0}^3 (\theta - \theta_{0})[/tex]

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

**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:[tex]\frac{d\theta}{dt} = -k(\theta - \theta_{0})[/tex]

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:[tex]\frac{dQ}{dt} = mC\frac{d\theta}{dt}[/tex] so

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

Comparing this form with Newton's Law, we get

[tex]k = \frac{4 \epsilon \sigma A \theta_{0}^3}{mC}[/tex]

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