Anaerobic digestion of biomass is well known to produce biogas, which is mostly

methane and carbon dioxide, but the protein content hydrolyses to primary amino

acids. Leucine $C_6{H}_{13}N{O}_2$ is one of the most common amino acids, so we will presume

that leucine is the sole product of the initial hydrolysis of proteins fed to an anaerobic

digestion process. The hydrolysis reaction is represented as:

$C_6{H}_{13}N{O}_2+a{H}_{2}ObC{H}_{4(g)}+cC{O}_{2(g)}+dN{H}_{3(aq)}$

$3$a$.$ Balance the stoichiometric equation above. Hint: Consider the $N-$balance

first.

An anaerobic digester with a volume of $100000L$ is fed leucine $($molecular weight $131$

$\frac{g}{m}ol)$ at the rate of $131k\frac{g}{h}r$ and water at ten times the stoichiometric ratio. This

excess water makes the hydrolysis reaction effectively first order with respect to

leucine concentration. The outlet mass flow rate is identical to the inlet flow rate. The

tank initially contains only the bioculture for anaerobic digestion and water. The

bioculture is retained in the tank by selectively permeable membranes on the outlet,

so that only water, ammonia and leucine can leave in the liquid outlet. Methane and

carbon dioxide gas evolve from the headspace.

$3$b$.$ Draw a schematic of the system labelling all streams. List your assumptions to

be able to solve the problem.

$3$c$.$ Suppose there is no anaerobic digestion metabolism. Treat the liquid outflow as

having the same volume as the aqueous inflow. Water has molecular weight $18\frac{g}{m}ol$

and density $1\frac{g}{m}L$ at the operating conditions. Write a leucine material balance for

the no metabolism, and solve it for an arbitrary time after the feed stream starts.

What will the concentration $c_{{?}_{?}}(l)($in moles per litre$)$ of leucine be after $30$ minutes?

For information, suppose you have a differential equation in time $\frac{dy}{dt}=B-A{y}_{{?}_{?}}.$

The substitution $u={?}_{\frac{?}{\text{b}\text{a}\text{r}}(A)}^B-y>y={?}_{\frac{?}{\text{b}\text{a}\text{r}}(A)}^B-u$ has solution $u=u_w^{\text{'}}$exp$(-at).$

$3d.$ Now suppose that the anaerobic digestion is active. Ammonia attacks the cell

membranes of the anaerobic bacterial consortia, inhibiting its metabolism and

therefore its production. This inhibition effect is modelled by a rate law for the

disappearance of leucine $r$ :

$r=k_l{c}_l-k_a{c}_a$

Where $c_a$ is the molar concentration of ammonia. If $k_l=1\mathrm{.}0mi{n}^{-1},$ and $k_a=0\mathrm{.}1mi{n}^{-1},$

what is the rate reaction at the initial condition in the digester?

$3$e$.$ Write a material balance on the system that takes into account the inhibition on

the fermenter. If there is no accumulation in the digester, calculate the value of the

steady concentration of leucine, $c_{l{0}_{{?}_{?}}}.$

$3$f$.$ With the metabolism and inhibition of $3d$ active, solve the transient material

balance for the concentration $c_l$ of leucine at any time $t$ after the initial condition.

What is the concentration of leucine $30$ minutes after the digestion starts?