# How to develop a model for cooling rate of steel tube in air?

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?

• Correction: The temperature is consistent throughout the tube not constant – Lahey Jan 30 '19 at 6:01
• 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). – user1476176 Jan 30 '19 at 7:25
• The tube is 145 mm OD with and ID is 129 mm and a length of 29 m. – Lahey Jan 30 '19 at 15:29
• 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 $t=40s$. – David White Jan 30 '19 at 16:30