- #1
maverick280857
- 1,789
- 5
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.
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
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