# Relationship between the height of the filter cake and the applied solution

Situation
When a solid is applied to a suction filter for filtration, a cake is formed which is washed with the help of washing water. The filtrate flowing through contains the parts of the solids to be separated, the cake contains the product and should be obtained as pure as possible.

Theoretically: Using the Carman-Kozeny-Equation with constant volume flow one can calculate the specific resistance of the cake ($$\alpha$$) and the resistance of filter media ($$\beta_M$$):

$$\dot V = \frac{dV}{dt} = \frac{\Delta p(t) \cdot A_F}{\alpha \cdot \mu \cdot w \cdot (\frac{\dot V \cdot t}{A_F}) + \mu \cdot \beta_M} \text{ with } \dot V = \frac{dV}{dt} = const.$$

Resulting in a regression line for $$\Delta p(t)$$ from which $$\alpha$$ and $$\beta_M$$ can be calculated as follows:

$$\alpha = \frac{k \cdot A_F^2}{\dot V \cdot \mu \cdot w} \quad \quad \quad \quad \beta_M = \frac{A_F \cdot d}{\mu \cdot \dot V}$$

where k and d are the components of the regression line $$\Delta p(t) = k \cdot t + d$$.

Having those values, one can use the viscosity $$\mu$$, the solid content $$w$$, the porosity $$\epsilon$$ and cake height $$h_c$$ to calculate for volume flow and time dependency for different pairs (time-flow).

$$A_F \cdot h_c \cdot (1-\epsilon) \cdot \rho_{solid} = V_F \cdot c$$

The filtration time increases quadratically with the height of the filter cake. Therefore, the less solid is used, the less filtration time is required (displacement washing). Although the height of cake ($$h_c$$) is often neglected, but must have a bigger influence than one might see.

When considering the design of a process, I asked myself whether it makes a difference if one works with different volumes (of solids):

Question
Assuming that 600kg of solids are to be filtered, is a filtration of 600kg faster than a double application of filter cakes of 300kg each? (Or 200kg+400kg) (Neglected time used for pumping from different vessels, etc.)

Legend:
$$\dot V$$ ... volumen flow [L/h]
$$V_F$$ ... volume of filtrate [L]
$$t$$ ... time [s]
$$h_c$$ ... height of cake [m]
$$A_F$$ ... area of filter [m²]
$$\rho_{solid}$$ ... density of the solid [g/L]
$$c$$ ... concentration of suspension
$$\alpha, \beta_M, \mu, \epsilon, w$$ are written within the text

• Some one with an idea? Aug 12 at 12:07
• Anyone with input? Aug 16 at 11:58