Continuous PBR Simulator — Documentation (EN)

1. Overview

This single-page application (SPA) simulates the operation of a photobioreactor (PBR) in continuous mode at steady state, plotting volumetric productivity $P_b$ and biomass concentration $C_b$ against dilution rate $D$.

In a continuous culture, the reactor is fed and purged at a constant volumetric flow rate $Q$. The dilution rate is defined as:

$$D = \frac{Q}{V}$$

where $V$ is the photobioreactor volume. At steady state, $D$ equals the net specific growth rate of the culture — which is why the chart's X axis, labelled $\mu$ (h⁻¹), effectively represents the value of $D$ at which the system operates stably for each biomass concentration.

As in the batch simulator, two limiting hypotheses about how cells experience light are compared:

Hypothesis	Description	Assumed mixing
Light integration (Iav)	Every cell sees the spatially averaged irradiance $I_{av}$	Perfect / instantaneous
Local integration (Local)	The rate is the spatial average of local rates $\mu(I(x))$	Poor / slow

The model enforces that the local rate can never exceed the Iav rate ($\mu_{Local} \le \mu_{Iav}$), as a reasonable physical constraint.

2. Input Parameters

2.1 Light Distribution

Parameter	Symbol	Units	Default	Description
Incident irradiance	$I_o$	µmol/m²·s	1000	Surface photon flux density
Absorption coefficient	$k_a$	m²/g	0.10	Biomass-specific absorption
Path length / Radius	$L$ or $R$	m	0.10	Reactor thickness or radius
Incidence angle	$\theta$	°	0	Direct light angle vs. normal
Lower diffuse angle	$\alpha_1$	°	-45	Lower bound of diffuse cone
Upper diffuse angle	$\alpha_2$	°	45	Upper bound of diffuse cone
Max. $C_b$ range	—	g/m³	4000	Biomass range explored

Geometry selector: Flat/Direct Beer-Lambert 1-D. Flat/Diffuse angular cone integration. Cylindrical 2-D numerical (30×30 grid).

2.2 Kinetic Model

Parameter	Symbol	Units	Default	Notes
Max. specific growth rate	$\mu_{max}$	h⁻¹	0.080	All models
Half-saturation irradiance	$I_k$	µmol/m²·s	150	Molina, Bannister, Van Oorshot
Shape exponent (Molina)	$n$	—	2.0	Molina only
Shape exponent (Bannister)	$n_b$	—	2.0	Bannister only
Saturation parameter	$\alpha_{sat}$	µmol/m²·s	300	Camacho-Rubio only
Inhibition coefficient	$\kappa$	—	0.10	Camacho-Rubio only
Maintenance coefficient	$m$	h⁻¹	0.005	All models

3. Mathematical Model

3.1 Dilution rate

$$D = \frac{Q}{V}$$

3.2 Steady-state balance

$$D = \mu(I) - m$$

3.3 Volumetric productivity

$$P_b = \mu \cdot C_b$$

3.4 Compensation point

$$\mu(I) = m \quad (D=0)$$

Solved numerically via bisection (25 iterations).

3.5 Local rate integration

$$\mu_{av} = \frac{1}{V}\iiint_V \mu\!\left(I(\mathbf{x})\right)\, dV - m$$

3.6 Average irradiance (general form)

$$I_{av} = \frac{1}{V}\iiint_V I(\mathbf{x})\, dV$$

3.7 Light attenuation — flat panel, direct

$$I(x) = I_o \, e^{-k_a C_b x / \cos\theta}$$ $$I_{av} = \frac{I_o}{k_a C_b L/\cos\theta}\left(1-e^{-k_a C_b L/\cos\theta}\right)$$

3.8 Light attenuation — flat panel, diffuse

$$I(x) = \frac{I_o}{\alpha_2-\alpha_1}\int_{\alpha_1}^{\alpha_2} e^{-k_a C_b x/\cos\alpha}\, d\alpha$$

3.9 Light attenuation — cylindrical, direct

$$I(x,y) = I_o \, e^{-k_a C_b (y_B-y)/\cos\theta}, \qquad y_B=\sqrt{R^2-x^2}$$

3.10 Kinetic models

Molina et al.$$\mu = \mu_{max} \frac{I^n}{I_k^n + I^n}$$

Bannister$$\mu = \frac{\mu_{max}\, I}{\left(I_k^{n_b} + I^{n_b}\right)^{1/n_b}}$$

Van Oorshot$$\mu = \mu_{max}\left(1 - e^{-I/I_k}\right)$$

Camacho-Rubio$$\mu = (1+\kappa)\frac{\mu_{max}\,I}{2\alpha_{sat}}\left[\left(1+\kappa+\frac{\alpha_{sat}}{I}\right) - \sqrt{\left(1-\kappa-\frac{\alpha_{sat}}{I}\right)^2+4\kappa}\right]$$

4. Outputs

4.1 Chart

Trace	Axis	Style	Colour
$C_b$ — Iav / Local	Right (g/m³)	Solid	Green / Orange
$P_b$ — Iav / Local	Left (g/m³·h)	Dashed	Green / Orange

X axis: $\mu$ (h⁻¹), equivalent to $D$ at steady state.

4.2 Optimal operation & 4.3 Compensation

Optimal operation reports $\mu$ (=$D$), $C_b$ and $P_b$ at the point maximising productivity, for both models. Compensation reports $C_b$ and $I_{av}$ at $D=0$.

4.4 CSV export

Exports 201 rows: Cb, mu_Iav, mu_av, Pb_Iav, Pb_av, sweeping $C_b$ from 0 to the configured maximum.

5. Interface Features

Language selector: ES / EN / FR.
Colour theme: 5 themes with intensity slider (white → theme colour) via 🎨 button.
Documentation: direct-access buttons in ES/EN.

6. Technical Notes

All computation is client-side JavaScript; no server required.
Cylindrical: 30×30 grid masked to circular section.
Diffuse flat panel: 40×25 nodes. Direct flat panel: 100 nodes (trapezoidal rule).
Compensation point: bisection, 25 iterations, range 0–30000 g/m³.