[Physics] Transient radiation–heating a slab


Hey guys I really need help on this problem.

A ceramic slab of dimentions 5cm x 10 cm x .25 cm has to be heated to $177\,^{\circ}{\rm C}$. The ceramic slab travels on a conveyor belt traveling at $.1 \frac{m}{s}$. The slab is initially at a temperature of $18\,^{\circ}{\rm C}$ when it enters a radiant heater on the conveyor belt. The heater surrounds the whole slab and emits heat as a black body at $1000\,^{\circ}{\rm C}$. Since the heat surrounds the slab, all the energy that leaves the heater is absorbed by the slab. The slab has an emissivity of .85. Find the amount of time it takes to heat the slab to $177\,^{\circ}{\rm C}$. The slab has properties $c_{p}=1500\frac{J}{Kg K}$ and $\rho=900\frac{kg}{m^{3}}$ and $k=1 \frac{W}{mK}$

I started off by creating an energy balance on the slab as it's in the heater. I assumed convection that is caused by the conveyor belt traveling at such a slow speed to be insignificant to all calculations.
$$\dot{E}_{in}- \dot{E}_{out}+\dot{E}_{g}=\dot{E}_{st}$$
$$A\alpha\sigma T^{4}_{heater}-A\epsilon\sigma T^{4}=\rho c_{p}V\frac{dT}{dt}$$
$$A\sigma(T^{4}_{heater}-\epsilon T^{4})=\rho c_{p}V \frac{dT}{dt}$$

I then plugged in the values and got:
$$44.185-1.428×10^{-11} T^{4}=\frac{dT}{dt}$$

I realized that this comes out to be a VERY nasty integral so I tried to use matlab and numerically integrate it using Euler's method. My graph for that came out to be linear which I knew was not right and I came here straight away to figure out what I was doing wrong

Best Answer

I haven't checked your numbers, but I too get an equation that looks like:

$$\frac{dT}{dt} = a - bT^4$$

I used your figures and graphed dT/dt in Excel over the range 18 to 177C, and it does indeed remain almost constant over this range. That's because the $1.428x10^{-11} T^4$ term is small compared to 44.185. Physically this means the temperature difference between the slab and the oven is approximately constant.

However I think there's an error in your numbers. If you set T = 1000C (i.e. 1273K) shouldn't dT/dt come out as zero? With your numbers I get when T = 1273 dT/dt = 6.68.

Related Question