Calculating theoretical heating rate of sand

[Bacat]

I have a small steel cylinder centered inside a larger steel cylinder. The remaining space is filled with 60% silica sand and 40% air. I am heating the larger cylinder in a furnace at a known heating rate and I want to calculate the theoretical rate of heating of the internal cylinder. This is not homework- it's motivated by some reactor design research I am doing.

Some facts about the system:

Outer cylinder: volume = 6095.9 cc
Inner cylinder: volume = 395.3 cc
Walls of both cylinders are 1 mm thick

Calculated mass and volume of silica sand: 9063.9 g in 3420.36 cc
Calculated mass and volume of air: 2.75 g in 2280.24 cc

Heat capacities:
Steel: 500 J/kg C
Sand: 733 J/kg C
Air: 1040 J/Kg C

Thermal conductivities:
Steel: 19 W/m C
Sand: 1.3 W/m C
Air: 0.024 W/m C

Furnace: Begins at 30C and heats at 5C per minute until it reaches 500C. The real question is how long does it take for the internal cylinder to reach 500C?

I'm not sure which set of equations to use for solving this system. I've had basic thermodynamics courses and PDEs but don't remember how to calculate heat in a 3D body.

Should I be using Heat Conduction in a Cylinder to solve this or is there a better way? I think I need to work with the thermal conductivity instead of the specific heat capacity because I am interested in the amount of time it takes for the heat to conduct through the sand. Is that right?

Q = 2 \pi k \ell r_m \frac{T_1-T_2}{r_2-r_1}

where r_m = \frac{r_2-r_1}{\ln r_2 - \ln r_1}

 

[Chestermiller]

It depends on the precise geometry (not shown), and involves solution of the transient heat conduction equation involving both thermal conductivity and heat capacity.