Question

The filter medium resistance has dimensions of______

• A. ML$^{-1}$
• B. L$^{-1}$
• C. M$^{-1}$L$^{-1}$
• D. M$^{-1}$L

Hint:

Think of resistance physically: Just like specific cake resistance \( r \) has units of m/kg, the filter medium resistance \( R_m \) is defined per unit length of flow path and has units of \( \text{m}^{-1} \) (reciprocal meters), which directly yields \( L^{-1} \).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the physical dimensions of the "filter medium resistance" (\( R_m \)) used in cake filtration theory.
This belongs to the study of filtration in mechanical operations.

Step 2: Key Formula or Approach:
The fundamental filtration equation (derived from Darcy's Law) is:
\[ \frac{dt}{dV} = \frac{\mu}{A \cdot \Delta P} \cdot \left[ \frac{r \cdot c \cdot V}{A} + R_m \right] \]
where:
\( V \) is volume of filtrate (\( [L^3] \)),
\( t \) is filtration time (\( [T] \)),
\( \mu \) is filtrate viscosity (\( [M L^{-1} T^{-1}] \)),
\( A \) is filtration area (\( [L^2] \)),
\( \Delta P \) is pressure drop (\( [M L^{-1} T^{-2}] \)),
\( r \) is specific cake resistance,
\( c \) is mass of dry cake per unit volume of filtrate,
\( R_m \) is filter medium resistance.

Step 3: Detailed Explanation:

• Dimensional Consistency Method:
Inside the brackets, the terms \( \frac{r \cdot c \cdot V}{A} \) and \( R_m \) are added together.
By the principle of dimensional homogeneity, they must have the same physical dimensions.
\[ [R_m] = \left[ \frac{r \cdot c \cdot V}{A} \right] \]

• Let's evaluate the dimensions of the pressure drop equation directly:
The fluid velocity through the filter medium is:
\[ v = \frac{1}{A} \cdot \frac{dV}{dt} = \frac{\Delta P}{\mu \cdot R_m} \]
Rearrange to solve for \( R_m \):
\[ R_m = \frac{\Delta P}{\mu \cdot v} \]

• Substitute the dimensions of each variable into this expression:
\[ [\Delta P] = \text{Pressure} = [M L^{-1} T^{-2}] \]
\[ [\mu] = \text{Viscosity} = [M L^{-1} T^{-1}] \]
\[ [v] = \text{Velocity} = [L T^{-1}] \]

• Combine the dimensions:
\[ [R_m] = \frac{[M L^{-1} T^{-2}]}{[M L^{-1} T^{-1}] \cdot [L T^{-1}]} = \frac{[M L^{-1} T^{-2}]}{[M T^{-2}]} = [L^{-1}] \]


Step 4: Final Answer:
The dimensions of filter medium resistance are \( L^{-1} \) (reciprocal length).