Bioreaction Engineering, Downstream Processing and Process Control

Downstream separation techniques

Downstream separation techniques exploit one physical or chemical property that differs between the product and the impurities. Choosing a method starts with identifying which property separates the target most clearly. The main pairings are the following.

• Density — particles settle in a gravitational or centrifugal field, so centrifugation depends on the density difference between the particle and the medium.
• Diffusivity — small solutes pass through a semipermeable membrane by diffusion down a concentration gradient, as in dialysis.
• Solubility (partitioning) — a solute distributes between two immiscible phases according to its solubility, quantified by the partition coefficient K = C_{\text{upper}}/C_{\text{lower}}, as in solvent extraction.
• Size — molecules are separated by hydrodynamic size or mass, as in ultrafiltration, size-exclusion chromatography and ultracentrifugation, because sedimentation depends mainly on mass and shape. Ion exchange is not a size-based method because it separates by net charge.

Chromatography media are named after the interaction they exploit. Protein A is a bacterial protein that binds the Fc region of immunoglobulins, so it is an affinity ligand. Sephadex is a cross-linked dextran gel used for size-exclusion separation. Phenyl-Sepharose carries hydrophobic phenyl groups for hydrophobic interaction chromatography, and DEAE-cellulose carries positively charged groups that bind negatively charged proteins, which makes it an anion exchanger.

In adsorption chromatography on a polar solid such as silica, uncharged solutes are held reversibly by van der Waals forces, namely dipole, induced-dipole and dispersion interactions, rather than by covalent or ionic bonds.

In aqueous two-phase extraction, a protein partitions between two aqueous polymer phases. The fraction recovered in the upper phase is E = \dfrac{KR}{KR+1}, where R is the ratio of upper to lower phase volumes. A large product of KR therefore gives near-complete recovery.

In gel filtration, large molecules are excluded from the pores and elute first, while small molecules enter the pores and elute last, so the elution order is the reverse of the size order.

If a recombinant protein is secreted into the medium in soluble form, cell disruption and solubilisation of inclusion bodies are unnecessary. Cells are first removed from the broth, after which ultrafiltration concentrates the protein and liquid chromatography achieves high purity.

Chemostat steady-state kinetics

A chemostat is a continuous stirred-tank culture in which sterile medium enters and culture leaves at the same volumetric flow rate F, so the dilution rate is D = F/V. At steady state, the specific growth rate equals the dilution rate, \mu = D, because cells leave the reactor at the rate they grow. With Monod kinetics, \mu = \mu_{max}S/(K_s+S), so the residual substrate is S = K_sD/(\mu_{max}-D) and the biomass is X = Y_{X/S}(S_0 - S).

For a product, the steady-state balance is 0 = q_pX - DP, which gives P = q_pX/D, where q_p is the specific product formation rate.

At steady state the biomass formation rate is DX, so the oxygen uptake rate is DX/Y_{X/O_2}, and the oxygen transfer rate must match it.

Washout occurs when D exceeds the maximum growth rate available at the feed substrate concentration, \mu_{max}S_0/(K_s+S_0). Biomass then falls to zero. Since no substrate is consumed, the exit substrate concentration equals the feed concentration.

The minimum feed flow to avoid washout is F_{crit} = D_{crit}V, where D_{crit} is the washout dilution rate.

In a CSTR with cell recycle and no purge, biomass leaves only in the effluent, so the mean cell retention time is \theta_c = VX/(QX_e). This can be much longer than the hydraulic residence time V/Q.

Biomass elemental composition

The biomass is written as an empirical formula CH_aO_bN_c normalised to one carbon atom. Its coefficients are fixed by elemental balances on C, H, O and N, using the consumed substrate, nitrogen source, oxygen and the products. Balances are written per mole of biomass, so when a balance is done per mole of product, the coefficients must be divided by the stoichiometric factor of biomass.

For an empirical formula determined from elemental analysis, ash is removed first because it is inorganic. Masses are converted to moles per 100 g of ash-free dry matter, then each value is divided by the moles of nitrogen to normalise to one nitrogen atom.

Once the biomass formula is known, the oxygen requirement follows from the oxygen balance, for example per mole of glucose consumed. Oxygen consumed equals the oxygen in the products minus the oxygen in the substrate and nitrogen source.

The respiratory quotient is the molar ratio of CO_2 evolved to O_2 consumed. Hydrocarbon substrates are more reduced than sugars, so their RQ is lower.

The heat released by a reaction can be estimated from heats of combustion, because every compound is ultimately burned to CO_2, H_2O and N_2, while O_2 has zero heat of combustion. The heat of reaction is the sum over reactants minus the sum over products. Multiplying by the specific growth rate gives the specific rate of heat production.

Nutrient requirements follow from the elemental formula. For C_5H_7O_2N, the nitrogen mass fraction is 14/113, and a phosphorus requirement stated as a fraction of nitrogen on a weight basis is converted directly from the nitrogen needed for a given dry weight.

Feedback control instrumentation

A feedback loop contains a sensor and transmitter that measure the controlled variable, a controller that compares the measured value y_m with the set point r to form the error e = r - y_m, a final control element such as a valve, and the process. The controlled variable is the quantity held at its set point, the manipulated variable is the input the valve adjusts to achieve this, and a load flow that the controller does not set acts as a disturbance. In a tank level loop, the inlet flow is the manipulated variable, while the outlet flow is a disturbance.

A proportional controller gives an output proportional to the error, u = K_c e. The integral term gives an output proportional to the integral of the error with respect to time, \dfrac{K_c}{\tau_I}\int e\,dt, not to the derivative of the error, which is the role of derivative action.

For a first-order process under proportional-only control, the closed loop leaves a steady-state offset. Adding integral action (PI control) drives the offset to zero.

The proportional band is the percentage change in input that produces a full-scale change in output, and it is related to the gain by PB = 100/K_p. A 25% band therefore corresponds to a gain of 4.

A process with transfer function G(s) = K_p/(\tau_p s+1) has one first-order lag and is classed as a first-order process.

Oxygen mass transfer coefficient

Oxygen transfer from air bubbles to the broth is described by \text{OTR} = k_La\,(C^* - C_L), where k_L is the liquid-side film coefficient, a is the gas-liquid interfacial area per unit volume, C^* is the saturation concentration and C_L is the dissolved oxygen concentration. The product k_La is the volumetric mass transfer coefficient.

In the steady-state method, the oxygen uptake rate equals the oxygen transfer rate, so k_La is obtained from the measured uptake rate divided by the driving force. Units must be converted consistently, since an area in m^2/m^3 equals m^{-1}, and 1\ \text{m}^{-1} = 0.01\ \text{cm}^{-1}.

When oxygen transfer is limiting, C_L approaches zero and the maximum biomass concentration is X_{max} = k_La\,C^*/q_{O_2}, where q_{O_2} is the specific oxygen uptake rate.

Raising the air flow rate increases gas holdup and interfacial area, and smaller bubbles give more area for the same holdup, since a = 6\varphi/d_{32}. Both increase k_La. Pure oxygen raises the driving force (C^*-C_L) but does not change k_La, and lower agitation reduces dispersion and so lowers a.

First-order process response

A first-order process obeys \tau_p\dfrac{dy}{dt} + y = K_p f(t), where K_p is the gain and \tau_p is the time constant. With zero initial condition, the Laplace transform gives the transfer function G(s) = \dfrac{K_p}{\tau_p s + 1}, so the time constant multiplies s in the denominator and the gain stays in the numerator.

For a step input of magnitude M, the response is y(t) = MK_p\left(1 - e^{-t/\tau_p}\right). The input magnitude multiplies the whole response, so the steady-state value is MK_p.

An instrument with first-order response reaches a fraction f of its final value at t = -\tau\ln(1-f). For 95%, t = \tau\ln 20 \approx 3\tau.

Fermenter scale-up criteria

Scale-up assumes geometric similarity, so all linear dimensions, including the impeller diameter D, scale together. Under constant power per unit volume, P/V \propto N^3D^2 is held fixed, which gives N \propto D^{-2/3}. A tenfold increase in diameter therefore reduces the agitator speed by a factor of 10^{2/3}\approx 4.6.

In turbulent agitation, constant shear corresponds to a constant tip speed, ND = constant, because shear stress scales with \rho(ND)^2. Under geometric similarity with the same aspect ratio, D \propto V^{1/3}, so a volume increase of 1000 times, from 20 L to 20,000 L, corresponds to a tenfold diameter increase and a tenfold lower impeller speed.

Degree of reduction

The degree of reduction counts the electron equivalents available in an organic compound, assigning +4 to carbon, +1 to hydrogen, -2 to oxygen and -3 to nitrogen when it is present as ammonia. For a compound C_cH_hO_o, the total is \gamma = 4c + h - 2o, and the value per carbon-mole is \gamma_C = \gamma/c.

Compounds with the same \gamma_C carry the same electron equivalents per carbon, which matters for matching substrates and products in yield calculations. Glucose, lactic acid and acetic acid all have \gamma_C = 4, while formic acid has \gamma_C = 2 because its single carbon is highly oxidised.

For biomass of the form CH_{1.8}O_{0.5}N_{0.2}, with nitrogen counted as -3, the degree of reduction per carbon-mole is 4 + 1.8 - 1 - 0.6 = 4.2. This value is used to balance electrons between substrate, biomass and product.

Fed-batch fermentation

A fed-batch culture is a cultivation mode in which substrate is added to the bioreactor during the run while no broth is withdrawn, so the working volume increases with time. Compared with batch operation, it keeps the substrate concentration low, which avoids overflow metabolism, catabolite repression and excessive oxygen demand, and it allows the cells to remain in the growth phase longer. Compared with continuous culture, no cell-containing broth leaves the vessel, so biomass accumulates.

• Substrate feed rate — the volumetric feed rate F multiplied by the feed concentration s_f gives the substrate mass supplied per unit time, F s_f. Example: a feed of 2 L/h at 30 g/L supplies 60 g of substrate per hour.
• Quasi-steady state — the assumption that the culture changes slowly enough that the specific growth rate equals the dilution rate D = F/V, where V is the instantaneous culture volume.
• Maximum biomass yield — when all fed substrate is converted to biomass at Y^M_{XS}, the biomass increment equals Y^M_{XS} times the substrate fed, and the total biomass is the initial biomass plus this increment.