CHEE456_Design_Project

Biotrickling Filter Mass Balance

Gas Phase

Odorous compounds initially enter the biotrickling filter in the gas phase. As they travel through the column, these components will be in contact with the trickling liquid phase, where mass transfer occurs due to the concentration gradient between the two phases.

The following assumptions are made:

The trickling liquid uniformly covers the biofilm. Its contact area can thus be expressed using the bed void fraction.

The void fraction is assumed to be constant throughout the column.

Plug flow is assumed, so mass transport is predominantly convective.

The superficial velocity is constant throughout the column.

Neglecting liquid phase and interfacial transfer with biofilm

Using these assumptions, a microscopic model is developed over a differential volume of the column. The vertical axis is represented by the variable 𝑧.

$$ \epsilon \frac{\delta C_g}{\delta t} = -V_0 \frac{\delta C_g}{\delta z} - \alpha R_g$$

At steady-state:

$$ \frac{\delta C_g}{\delta z} = -\alpha\frac{R_g}{V_0} $$

$$ Rg = D\frac{\delta S}{\delta x} |{x=0} $$

$$ \frac{\delta C_g}{\delta z} = -\alpha\frac{1}{V0}D\frac{\delta S}{\delta x} |{x=0} $$

Biofilm

The contaminants are degraded as they travel through the biofilm. Their degradation is modeled using Monod kinetics.

$$ \frac{\delta S}{\delta t} = D \frac{\delta^2S}{\delta x^2} - Rb$$

At steady-state:

$$ D \frac{\delta^2S}{\delta x^2} = Rb$$

$$ Rb = -\frac{1}{Y{X/S}}\frac{dX}{dt} - \frac{1}{Y_{P/X}}\frac{dP}{dt}$$

$$ Rb = X(-\frac{\mu{net}}{Y_{X/S}} - \frac{qp}{Y{P/X}})$$

$$ D \frac{\delta^2S}{\delta x^2} = -X(\frac{\mu{net}}{Y{X/S}} + \frac{qp}{Y{P/X}}) $$

Boundary conditions

The following boundary conditions are used:

$$ Cg(z=0) = C{g0} $$

$$ S(x=0) = \frac{C_g(z)}{m} $$

$$ \frac{\delta S}{\delta x}|_{x=\delta} = 0$$

Together:

$$ D\frac{dS}{dx}(x=0) = \frac{X}{Y{X/S}} \frac{\mu{max} S}{K+S} \delta $$

$$ \frac{\delta C_g}{\delta z} = -\alpha\frac{1}{V0} \frac{X}{Y{X/S}}\frac{\mu_{max} S}{K+S}\delta $$

$$ \frac{\delta C_g}{\delta z} = -\alpha\frac{1}{V0} \frac{X}{Y{X/S}}\frac{\mu_{max} \frac{C_g}{m}}{K+ \frac{C_g}{m}}\delta $$

Biotrickling Filter Energy Balance

Energy stored in the substrates is balanced by the heat stored by biomass formation and metabolic heat generation.

$$\Delta H_s \Delta S^T = \Delta Hc \Delta X^T + Q{met}$$

$$\frac{\Delta Hs}{Y{X/S}} = \Delta Hc + \frac{Q{met}}{\Delta X^T}$$

$$\frac{\Delta Hs}{Y{X/S}} = \Delta H_c + \frac{1}{y_H}$$

Where $y_H$ is the amount of biomass produced per unit of heat generated.

To obtain the cooling rate for the bioreactor, the following equation is used:

$$ q{cool} = \frac{Q{met}}{\Delta t} = \frac{\mu_{net} X^T}{y_H}$$