(Solved): \#3. Solid-state fermentation (SSF) is an aerobic bioprocess that uses solid materials as substrate ...

\#3. Solid-state fermentation (SSF) is an aerobic bioprocess that uses solid materials as substrates for microbial production of enzymes, antibiotics, and biopesticides. Overheating is a significant problem with SSF due to microbial heat generation. A cylindrical SSF bioreactor has a radius of $$0.035\mathrm{m}$$ and a height of $$0.35\mathrm{m}$$, as shown to the right (top and side views). The fungus Aspergillus niger is being grown on soybean meal in the SSF bioreactor for production of industrial enzymes. The microbial heat generation during the process is $$6600\mathrm{W}/{\mathrm{m}}^3$$. Assume that heat transfer in the substrate only occurs by conduction, with a constant thermal conductivity of $$0.1\mathrm{W}/\mathrm{m}?\mathrm{K}$$. In order to prevent overheating, cool water at $${\mathrm{T}}_{?}$$ flows past the entire outer surface of the bioreactor with a convection heat transfer coefficient of $$100\mathrm{W}/{\mathrm{m}}^2?\mathrm{K}$$. The thin metal bioreactor wall separating the substrate from the water is being ignored in this simplified analysis. a) Write out the appropriate steady state governing equation for conduction in the substrate in the SSF bioreactor. Simplify as much as possible. Justify modeling the bioreactor as a long cylinder. b) Write out the boundary conditions needed to solve the heat conduction equation. c) Solve the differential equation to find an expression for temperature as a function of radial position in bioreactor $$(T(r))$$. (note that $$C_2$$ will be written in terms of $$T_{?}$$, which is currently unknown) d) The maximum temperature in the SSF bioreactor will occur at the center $$(r=0)$$ and should not exceed $$40^{?}\mathrm{C}$$. Using your expression from part (c), set the center temperature to $$40^{?}\mathrm{C}$$ and calculate the water temperature $$\left(T_{?}\right)$$ needed to provide adequate cooling to the bioreactor.