CellAveragePBE

A cell-average population-balance solver for yeast fermentation, written in Julia. The package couples a size-structured cell population — a mesoscopic integro-PDE for the number density N(x, t) over cell mass x — to a well-mixed bioreactor chemistry model (glucose, ethanol, oxygen, CO₂). The fully coupled system reproduces the sustained metabolic oscillations characteristic of the Henson-group yeast model (Zhu et al. 2000; Zhang et al. 2002).

An independent Julia implementation of the published model described under References — written from the literature, not ported from an existing codebase.

What it demonstrates

• Cell Average Technique (Kumar et al. 2006) for the division/breakage integro-PDE — non-integer daughter masses are redistributed onto neighboring grid nodes with consistent number and mass placement.
• Upwind finite-volume advection for cell growth up the mass axis.
• Coupled stiff ODE integration — n population cells + 8 chemistry ODEs (gas–liquid mass transfer, substrate uptake integrated over the population), solved with a BDF method.
• Idiomatic, documented, tested Julia — parameters in one immutable struct, small composable functions, a 26-test suite.

The model

∂N(x,t)/∂t + ∂[g·N(x,t)]/∂x  =  B_div(x,t)  −  [ S(x) + D ]·N(x,t)

• g = K_net(...) — mass-specific growth rate (Monod kinetics on the local chemistry)
• B_div — cell-average division birth from the breakage kernel b(x, y) and selection rate S(x)
• D — dilution rate

coupled to 8 chemistry ODEs for glucose, ethanol, their effective (sensed) concentrations, and dissolved/gas-phase O₂ and CO₂.

Layout

src/
  parameters.jl   all physical/numerical constants in one keyword struct
  grid.jl         mass grid + cell-average interpolation weights
  kernels.jl      division-mass thresholds, breakage, selection, ethanol-formation
  kinetics.jl     Monod growth kinetics (concentration-clamped for stability)
  cellaverage.jl  Cell Average Technique birth term
  chemistry.jl    the 8 coupled bioreactor ODEs
  model.jl        RHS assembly, initial conditions, solve drivers
test/runtests.jl  26 tests (grid, kernels, kinetics, quadrature, RHS)
notebook/         population_balance.ipynb + rendered PDF

Running it

using Pkg; Pkg.activate("."); Pkg.instantiate()

using CellAveragePBE

# Coupled bioreactor — sustained metabolic oscillations
out = solve_pbe(n = 60, tspan = (0.0, 25.0), p = PBEParameters(D = 0.3), N_scale = 1/500)
N, chem = split_state(out.sol.u[end], out.grid.n)

# Population balance alone, in a fixed chemical environment
outp = solve_population(n = 80, tspan = (0.0, 4.0))

Run the tests with julia --project=. -e 'using Pkg; Pkg.test()' (or include("test/runtests.jl")).

The notebook/ directory contains a narrated walk-through (population_balance.ipynb) and its rendered PDF, covering the kernels, the growth-plus-division mechanism, and the coupled oscillations.

References

The numerics and the yeast model are both taken from the literature; this package is an independent Julia implementation of them. BibTeX entries for everything below are in docs/references.bib.

Population balance discretization

• J. Kumar, M. Peglow, G. Warnecke, S. Heinrich and L. Mörl, "Improved accuracy and convergence of discretized population balance for aggregation: the cell average technique," Chemical Engineering Science 61(10), 3327–3342, 2006. — the cell average technique itself, i.e. the scheme in src/cellaverage.jl.
• J. Kumar, M. Peglow, G. Warnecke and S. Heinrich, "An efficient numerical technique for solving population balance equation involving aggregation, breakage, growth and nucleation," Powder Technology, 2008. — cell average extended to combined growth and breakage, which is the combination solved here.
• S. Kumar and D. Ramkrishna, "On the solution of population balance equations by discretization — I. A fixed pivot technique," Chemical Engineering Science 51(8), 1311–1332, 1996. — the fixed pivot technique: the predecessor the cell average method improves on, not the method implemented here. Note this is a different Kumar (S., with Ramkrishna) from the cell-average papers (J. Kumar) — the two are easy to conflate.
• D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic Press, 2000. — general reference for the population balance itself.

Yeast cell population balance model

• G.-Y. Zhu, A. Zamamiri, M. A. Henson and M. A. Hjortsø, "Model predictive control of continuous yeast bioreactors using cell population balance models," Chemical Engineering Science 55, 6155–6167, 2000.
• Y. Zhang, A. M. Zamamiri, M. A. Henson and M. A. Hjortsø, "Cell population models for bifurcation analysis and nonlinear control of continuous yeast bioreactors," Journal of Process Control 12, 721–734, 2002.
• M. A. Henson, "Dynamic modeling of microbial cell populations," Current Opinion in Biotechnology, 2003. — review covering this model class.

The division kernels, the piecewise-linear transition/division masses, the three-mode Monod kinetics and the effective (sensed) concentrations all follow the Henson-group model; the kinetic and yield parameters in src/parameters.jl are that model's published values rather than fitted ones.