Many of the new "compostable" food containers made from biodegradable plastics $($such as PLA$)$

require high temperatures and high moisture content to properly decompose. I have designed a

new composter, the Skip$-$o$-$Poster that is a plastic $($HDPE$)$ drum with a four impeller shaft down

the center for easy and rapid mixing of the compost. The HDPE drum is $1\mathrm{.}0m$ tall with a $0\mathrm{.}5m$

diameter. It is completely insulated on the outside of the barrel, except for the cover, which is a

$2\mathrm{.}5cm$ thick HDPE sheet.

I need to do a heat transfer calculation so that I can claim that the average temperature of the compost

in the Skip$-$o$-$Poster is at least $70C.$ That is the approximate temperature needed for composting

biodegradable plastics such as PLA.

a$)$ Sketch and clearly label the problem. Clearly state all given information and assumptions.

b$)$ What is the average compost temperature at steady state when the Skip$-$o$-$Poster is $\frac{3}{4}$ filled with compost?

Can I make my claim of an average temperature of $70C?$

Hint: The average temperature in the compost can be taken as the average of the temperature at

the top and bottom of the compost $($and NOT the container walls$)\frac{T_{\text{t}\text{o}\text{p}}+T_{\text{b}\text{o}\text{t}}}{2}.$

c$)$ If the compost bed were FULL $($instead of $\frac{3}{4}$ full$),$ would the average temperature go up$,$ down, or not

change. Explain with reference to the calculations made in the problem $($use equations$).$

A simple statement of up$,$ down, no change will get no points.

Show all work and units, and clearly state All ASSUMPTIONS.

Useful Information $($start here!$)$:

heat transfer related to the mixing shaft may be neglected

calculations should be made for winter conditions $T_{\text{o}\text{u}\text{t}}=5C$ and $h_o=5\mathrm{.}0\frac{W}{m^2}-K$

compost decomposition gives a constant net heat generation $=85\frac{W}{m^3}$ of compost.

when the compost bed is not full, assume the air layer above it is stagnant and moist.

thermal conductivities $(k)$ : Compost $=1\mathrm{.}0\frac{W}{m}-K$; Air $($moist$)=0\mathrm{.}38\frac{W}{m}-K$;HDPE$=0\mathrm{.}5\frac{W}{m}-K$