I am trying to develop an equation to determine the cooling rate of a steel tube in air. I'm using Fourier's Law, Stefan-Boltzman Law, Newton's Law as well as the specific heat capacity equation. The tube transfers heat by radiation and convection to the air and it transfers heat through conduction to the steel cooling bed it is laying on. The equation I have determined from these is $$T_{tube, final} = T_{tube, intial}- \frac{(\dot Q_{radiation} + \dot Q_{convection} + \dot Q_{conduction})\times\Delta t}{\rho VC_{p}}$$ where, $$\dot Q_{radiation}=\epsilon\sigma A(T_{tube, initial}^4-T_{air}^4),$$ $$\dot Q_{convection}=hA(T_{tube, initial}-T_{air}),$$ $$\dot Q_{conduction}=kA\frac{T_{tube, intial}-T_{cooling bed}}{dx}.$$ The thermodynamic properties of the steel are: $$C_{p}=416\frac{J}{kgK}$$ $$\rho=7667 \frac{kg}{m^3}$$ $$k=24.2 \frac{W}{mK}$$ $$\epsilon=0.86$$ My assumptions making this equation are:

  • Air temperature and cooling bed temperature is at a constant 300 K.
  • Temperature dependent properties are constant.
  • The heat transfer coefficient is 5 W/(m^2K).
  • The temperature throughout the tube is constant.

I know the tube starts at about 1300 K and drops to about 1150 K in about 40 seconds, when I use this equation though I get really low final temperatures, sometimes even negative ones. Can you help point me in the right direction?

  • $\begingroup$ Correction: The temperature is consistent throughout the tube not constant $\endgroup$
    – Lahey
    Jan 30, 2019 at 6:01
  • $\begingroup$ There seems to be a typo in the units of your heat transfer coefficient. I think it would also be helpful to specify the radius of the steel component (both inner and outer, if the component is hollow). $\endgroup$ Jan 30, 2019 at 7:25
  • $\begingroup$ The tube is 145 mm OD with and ID is 129 mm and a length of 29 m. $\endgroup$
    – Lahey
    Jan 30, 2019 at 15:29
  • $\begingroup$ The cooling bed temperature is not going to be constant with a very hot tube laying on it. Convection will be severely constrained because air cannot get under the tube. I suggest you use a different mathematical model, where there is an overall heat transfer coefficient that can be determined from the boundary conditions at $t=0$ and $t=40s$. $\endgroup$ Jan 30, 2019 at 16:30

1 Answer 1


Calculating from physics for such situations is always a bit dicey. I think two assumptions are quite far from reality. The temperature of the tube is certainly not constant. It is in contact with a cooling bed which is at 300K. So the part of the tube in contact must be at 300K too. So there is really a very large gradient. Secondly the assumption of a constant value for temperature dependent properties is not valid for such a large variation. These could very easily give you results off by orders of magnitude from experimental results

  • $\begingroup$ Thank you for your comment but this does not give a suggestion or help in any way. $\endgroup$
    – Lahey
    Jan 30, 2019 at 6:09
  • $\begingroup$ @Lahey you asked for someone to help point you in the right direction, and this answer has done that by pointing out that you may need to revise your assumptions of constant temperature in the tube and constant thermodynamic properties. $\endgroup$ Jan 30, 2019 at 7:28
  • $\begingroup$ These properties do not change significantly with temperature and I do not have the data or the means to get it. The temperature of the tube also is assumed to be consistent throughout because I also have no means of getting that data. $\endgroup$
    – Lahey
    Jan 30, 2019 at 15:31

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