Question 31.1: The design of aeration systems for aerobic-fermentation proc......
The design of aeration systems for aerobic-fermentation processes is based on gas–liquid mass-transfer. Microorganisms grow in a liquid suspension and feed on dissolved nutrients such as glucose and mineral salts. Aerobic microorganisms in liquid suspension also require dissolved oxygen for growth. If oxygen is not supplied at a rate sufficient to support cell growth, the cells will die.
In the present process, Aerobacter aerogenes is being cultivated within a continuous flow fermenter of 3.0 m³ liquid volume (V) and tank diameter (d_{T}) of 1.5 m. Fresh nutrient medium containing a trace amount of dissolved O_{2} at concentration 0.010\ gmole\ O_{2}/m^{3} enters the fermenter at a flow rate of 1.8\,\mathrm{m^{3}}/\mathrm{h}. At steady-state conditions, the aerobic fermenter operates at a cell concentration (c_{X})\mathrm{~of~}5.0\ k\mathrm{g/m^{3}~} of liquid culture. The cell concentration is determined by the specific growth rate of the organism and the nutrient composition of the liquid medium, details of which will not be presented here. The liquid cell suspension consumes oxygen proportional to the cell concentration according to the rate equation
R_{A}=-q_{o}c_{X}
where q_{o} is the specific oxygen consumption rate of the cells, equal to 20\operatorname{gmole}\operatorname{O}_{2}/kg cells h, which is assumed to be constant. Determine the K_{L}a value necessary to ensure that the dissolved oxygen concentration in the liquid culture (c_{A}) is maintained at 0.050\,\mathrm{gmole}/m^{3},. Also, determine the power input into a 3.0 m³ fermenter if the gas flow rate into the fermenter is 1.0 m³ air/min at the process conditions of 298 K and 1.0 atm. Assume that the bubbles are noncoalescing. At 298 K, Henry’s law constant for dissolution of O_{2} in the liquid nutrient medium is 0.826 m³ atm/gmole.
The required K_{L}a is backed out from a material balance on dissolved oxygen (species A) within the well-mixed liquid phase of the fermenter. Recall equation (31-2):
\dot{V}_{o}(c_{A o}-c_{A})+K_{L}a\cdot V\bigl(c_{A}^{*}-c_{A}\bigr)+R_{A}\,V=0 (31-2)
\dot{V}_{o}(c_{A o}-c_{A})+K_{L}a V\bigl(c_{A}^{*}-c_{A}\bigr)+R_{A}\,V=0
Inserting R_{A}=-q_{o}c_{X} and solving for the required K_{L}a yields
K_{L}a={\frac{q_{o}c x-{\frac{\dot{V}_{o}}{V}}(c_{A o}-c_{A})}{c_{A}^{*}-c_{A}}} (31-4)
The saturation concentration of dissolved oxygen is determined by Henry’s law:
c_{A}^{*}={\frac{p_{A}}{H}}={\frac{0.21\mathrm{~atm}}{0.826{\frac{{\mathrm{m}}^{3}\cdot{\mathrm{atm}}}{{\mathrm{gmole}}}}}}=0.254{\frac{\mathrm{gmole}\,\mathrm{O}_{2}}{\mathrm{m^{3}}}}
The partial pressure of oxygen (p_{A}) is presumed constant, as the rate of O_{2} transferred to the sparingly soluble liquid is very small in comparison to the molar flow rate of O_{2} in the aeration gas. Finally,
K_{L}a = \frac{\left({\frac{20\ {\mathrm{gmole}}\,O_{2}}{{\mathrm{kg}}\,\mathrm{cells}\cdot h}}{\frac{5.0\,{\mathrm{kg}}\,\mathrm{cells}}{\mathrm{m}^{3}}} – \frac{1.8\,\mathrm{m^{3}/h}}{3.0\,\mathrm{m^{3}}}(0.010-0.050)\frac{\mathrm{gmole}\,\mathrm{O}_{2}}{\mathrm{m^{3}}}\right)\frac{1\ \mathrm{h}}{3600\ s} }{(0.254-0.050){\frac{\mathrm{gmobte}\,\mathrm{O}_{2}}{\mathrm{m}^{3}}}} = 0.136\ s^{-1}
The power input to the aerated tank is backed out from the correlation
(k_{L}a)_{O_{2}}=2\times10^{-3}\left(\frac{P_{g}}{V}\right)^{0.7}\left(u_{g s}\right)^{0.2} (30-33)
where k_{L}a has units of s^{-1},P_{g}/V has units of W/m³, and u_{g s} has units of m/s. The superficial velocity of the gas through the empty tank is
u_{g s}={\frac{4Q_{g}}{\pi d_{T}^{2}}}={\frac{(4)\left({\frac{1.0~m^{3}}{\mathrm{min}}}{\frac{1\,\mathrm{min}}{60\,s}}\right)}{\pi(1.5\,\mathrm{m})^{2}}}=0.0094{\frac{\mathrm{m}}{\mathrm{s}}}
If the gas is sparingly soluble in the liquid, the interphasemass-transfer process is liquid-phase controlling so that K_{L}a \cong k_{L}a. Therefore,
0.136=2.0 x 10^{-3}\left({\frac{P_{g}}{V}}\right)^{0.7}(0.0094)^{0.2}
or
{\frac{P_{g}}{V}}=1572{\frac{\mathrm{W}}{\mathrm{m^{3}}}}
The total required power input (P_{g}) for the 3.0 m³ aerated fermenter is 4716 W.