Initially the design for the insulation called for 2-4" of insulation on only the outside of the pressure vessel. Steady-state analysis with 4" of insulation proved to be the most effective at keeping the surface temp below 120F and also minimizing heat loss. This was based on mineral wool insulation which typically comes in 2" thick boards. See figures 1 and 2 below, 4" insulation corresponds to an ro of 0.413 meters (this is the outer radius of the vessel, so 12.375" of that is the pipe radius). As can be seen, there is little reduction in either heat flux or outer temperature at this radius, and as the insulation typically comes in 2" thick pieces, 4" is an easily constructed amount.
Figure 1: Surface temperature as a function of outer radius, 400F internal steady-state temperature.
Figure 2: Heat flux as a function of outer radius, 400F internal steady-state temperature.
Based on this, an FEA analysis was run using 4" of insulation on the outside of the tank and a constant internal temperature of 400F. As the true profiles of the end caps were not known, they were assumed to be half-spheres, and in fact the true caps have less surface area than that, so the values seen here are slightly, though not excessively, high for heat flux. Convective conditions were assumed to be 10 W/m2-K on the outside and 30 W/m2-K on the inside. This relatively high internal condition is due to the fact that there will be a blower system in the vessel circulating the gas. In fact, variations in this internal condition were conducted for both steady-state and transient conditions with little change in the heat flux or surface temperatures. As can be seen in figure 3 below, the surface temperature at steady-state will average between 80 and 110F, which is safe to touch without the risk of burns, and this agrees with the manual calculations above.
Figure 3: Steady-state surface temperature of entire pressure vessel.
Figure 4: Steady-state heat flux on pressure vessel.
Figure 4 shows the heat flux on the pressure vessel at the steady-state conditions discussed before. The units are in lbf/in-s, so the outer average value of 0.43 lbf/in-s is equivalent to 473W when the entire outer surface of the vessel is accounted for (32.75" outer diameter). This is not a particularly worrisome value, and is easily obtained (a standard home toaster oven has between 1-2 kW worth of heating elements). This is the power needed to feed the heaters at steady-state to maintain the internal temperature.
For the heat-up stage, first the power needed to heat up the volume of Nitrogen gas was computed, using the relationship Q=mCpdt, where m is the total material mass, Cp is the specific heat of the material, and dt is the time elapsed. For the desired heat up of 15F/min, this comes out to roughly 23 minutes total, yielding a total power input of 1kW. Again, this is very obtainable. When conducting an FEA analysis on the pressure vessel, however, figure 5 resulted, indicating a maximum heat flux that equated to almost 67kW! (The maximum heat flux will be on the inner surface, which was used as the area for the conversion, 24" diameter). Also, as seen in figure 6, steady state is not reached for at least 25 hours! As a note, the simulation to obtain this profile was based on a constant internal temperature of 400F, convecting to the inner surface of the vessel via the fan circulation. Since the time to reach steady state is so long, this assumption of starting right off with the full temperature has practically no effect on the maximum flux seen, and so this simplification was made as a consideration to computer processing time.
Figure 5: Heat flux profile for externally insulated vessel during heat-up.
Figure 6: Transient heat-up temperature profile for externally insulated pressure vessel.
This massive power requirement is due to the fact that the pressure vessel needs to come up to temperature before steady-state is reached, and with well over 1000 pounds of steel, this does not occur quickly or cheaply (power-wise). After some research, it was found that most autoclaves actually use internal insulation to combat this problem. Since the internal diameter was already limited, the decision was made to try only 2" of insulation inside the vessel, with another 2" outside the vessel. The results are in figures 7 and 8 below. The same procedure that was used for the earlier simulation was used here as well.
Figure 7: Heat flux profile during heating of internally-insulated pressure vessel.
Figure 8: Temperature profile during heating of internally-insulated pressure vessel.
Now the maximum flux was only about 8.6kW (occurring early on in the operation), which is quite manageable. Steady-state is still not reached for over a day, but this is not necessary to the autoclave operation. Taking the 8.6kW for heating the vessel, and combining it with the 1kW to heat the fluid in the desired time period yields a little under 10kW, and so the recommendation was for 12kW of heating element power to allow for the parts being autoclaved to be accommodated as well (not knowing the amount of parts being fired at any given time). Just as a check, steady-state analysis of this new insulation configuration was conducted, finding little change in conditions from the externally-insulated case. See figures 9 and 10 below.
Figure 9: Steady-state heat flux for internally-insulated pressure vessel. 385W.
Figure 10: Steady-state temperature for internally-insulated pressure vessel. 80-110F.
For reference, the physical properties used are listed below.
Pipe:
Cp (specific heat capacity) = 11778.92895 in2/s2-F (440 J/kg-K)[1]
α (thermal expansion coefficient) = 6.5*10-6 1/F[1]
E (elastic modulus) = 29*106 lbf/in2[2]
ρ (density) = .284 lbf-s2/in4[2]
k (thermal conductivity) = 7.1203 lbf/s-F (57 W/m-K) [1]
Insulation:
Cp = 22411.296 in2/s2-F [1]
α = .65*10-6 1/F (assumption, has no effect on temperatures/fluxes)
E = 29*103 lbf/in2 (assumption, again, no effect on temperatures/fluxes)
ρ = .0095737 lbf-s2/in4 (265 kg/m3) [1]
k = .006246 lbf/s-F (.05 W/m-K) [1]
Nitrogen:
Cp = 1056 J/kg-K (at 500K, worst-case scenario)[3]
ρ = 12.486 kg/m3 (based on pressure, worst-case scenario)[3]
Convective Conditions:
outside: h = .03172 lbf/in-s-F (10 W/m2-K) - assumption based on mostly still air[1]
outside environment temperature: T = 68F
inside: h = .09516 lbf/in-s-F (30 W/m2-K) - assumption based on rapid air, variations in this value had insignificant effect on fluxes and temperatures[1]
inside environment temperature: T = 400F
References:
[1] Incropera, DeWitt, Bergman, Lavine: Fundamentals of Heat and Mass Transfer, 6th Edition.
[2] Beer, Johnson, DeWolf: Mechanics of Materials, 4th Edition.
[3] Moran, Shapiro: Fundamentals of Engineering Thermodynamics, 5th Edition.