- #1
maverick280857
- 1,774
- 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 \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
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
:zzz: .
