Hydrogen sulfide is a contaminant in many gas streams like biogas, syngas, flue gases etc. Hydrogen sulfide is toxic even at concentrations of parts per million and can cause corrosion of

equipment that these gases encounter. Halothiobacillus sp are chemolithoautotrophic microorganisms that play an important role in global carbon and sulfur cycles. They derive energy

from oxidizing hydrogen sulfide $($H$2$S$)$ to elemental sulfur $($S$)$ under aerobic conditions. These microorganisms are used as biocatalysts in the commercial ThiopaqO&G process for

desulfurization of natural gas, refinery gas, synthesis gas and biogas. ThiopaqO&G is marketed by Paqell B$.$V$.$ a joint venture company between Shell Oil and Paques B$.$V$.$ of Netherlands. In the process of removing hydrogen sulfide, it recovers the sulfur in the form of elemental sulfur, a valuable resource. ThiopaqO&G is a replacement for the traditional high temperature catalytic Claus process.

Consider the treatment of biogas, made up of $60\%($by volume$)$ methane, $38\%($by volume$)$ carbon dioxide and $2\%($by volume$)$ hydrogen sulfide, produced by an anaerobic digester in a continuously fed stirred tank bioreactor in which Halothiobacillus sp microbes are cultivated to convert hydrogen sulfide to elemental sulfur.

The process works as follows: Biogas is mixed with dry air and

sparged into the bioreactor. The bioreactor is operated at $38$ deg C and is not at steady state. A continuous stream of nutrient solution $($ammonium chloride$)$ is added, and the product stream is withdrawn. Off gases are vented continuously to maintain a constant headspace total pressure of $1$

atm. A schematic diagram of this arrangement is shown in the figure below.

a$)$ Construct unsteady state mass balance equations to describe the following liquid phase and gas

phase concentrations in the bioreactor.

Liquid phase concentrations of

$1.$ Halothiobacillus sp

$2.$ Total inorganic sulfide $($sum of dissolved hydrogen sulfide, bisulfide ion and sulfide ion$)$

$3.$ Elemental sulfur

$4.$ Total ammonia nitrogen $($free ammonia nitrogen and ammonium ion$)$

$5.$ Dissolved oxygen

$6.$ Total inorganic carbon $($sum of dissolved carbon dioxide $($carbonic acid$),$ bicarbonate ion

and carbonate ion$)$

Gas phase partial pressures for

$7.$ Carbon dioxide

$8.$ Methane

$9.$ Oxygen

$10.$ Nitrogen

$11.$ Hydrogen sulfide

Include

$$ equilibrium expressions for bisulfide, sulfide, bicarbonate, carbonate and pH$.$

$$ total off gas flow rate

Assume that

$$ The gas bubbles within the liquid and the head space are all well mixed and is at $1$ atm

pressure with total volume VG

$$ The mixed biogas and air stream is introduced at $1$ atm pressure and $25$ deg C$.$

$$ The stoichiometry for conversion is hydrogen sulfide to elemental sulfur and microbial

biomass is known.

b$)$ Explain all notations with units of measurement used in your equations.

c$)$ Explain all additional assumptions made to write the equations.

d$)$ List all specifications required to solve the eleven unsteady state mass balance equations.

e$)$ Ensure that you have sufficient equations and specifications required to uniquely solve the set

of ordinary differential equations. Explain.

f$)$ DO NOT SOLVE THESE EQUATIONS