Question

The bulk density of the produce having porosity and true density of 40% and \(50 \text{ kg}\cdot\text{m}^{-3}\) will be

• A. \(20 \text{ kg}\cdot\text{m}^{-3}\)
• B. \(40 \text{ kg}\cdot\text{m}^{-3}\)
• C. \(80 \text{ kg}\cdot\text{m}^{-3}\)
• D. \(30 \text{ kg}\cdot\text{m}^{-3}\)

Hint:

To solve this mentally:
If the porosity (void space) is 40%, then the solid space fraction must be 60% (\(100% - 40% = 60%\)).
The bulk density is simply 60% of the true density:
\[ \rho_b = 50 \times 0.60 = 30 \text{ kg}\cdot\text{m}^{-3} \]

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In agricultural process engineering, the physical properties of grains and bulk agricultural products are characterized by different types of densities:
- True Density (\(\rho_t\)): The mass of the material divided by its actual solid volume (excluding any pore spaces between particles).
- Bulk Density (\(\rho_b\)): The mass of the material divided by the total volume it occupies (including the void spaces between particles).
- Porosity (\(\epsilon\)): The fraction of the total bulk volume that is occupied by void spaces (air).
Key Formula or Approach:
The relationship between porosity, bulk density, and true density is given by the formula:
\[ \epsilon = \left( 1 - \frac{\rho_b}{\rho_t} \right) \times 100% \] Or, in decimal form:
\[ \epsilon = 1 - \frac{\rho_b}{\rho_t} \]

Step 2: Detailed Explanation:
Let us perform the calculations using the given values:
- Porosity (\(\epsilon\)) = 40% = 0.40
- True Density (\(\rho_t\)) = \(50 \text{ kg}\cdot\text{m}^{-3}\)
1. Substitute these values into the decimal form of the formula:
\[ 0.40 = 1 - \frac{\rho_b}{50} \] 2. Rearrange the terms to isolate the fraction:
\[ \frac{\rho_b}{50} = 1 - 0.40 \] \[ \frac{\rho_b}{50} = 0.60 \] 3. Solve for the bulk density (\(\rho_b\)):
\[ \rho_b = 0.60 \times 50 \] \[ \rho_b = 30 \text{ kg}\cdot\text{m}^{-3} \] The bulk density of the produce is \(30 \text{ kg}\cdot\text{m}^{-3}\).

Step 3: Final Answer:
The correct option is (D).