Question

Imagine that you have a box that is 15 centimeters (6 inches) square. You also have a never-ending supply of rectangular sticks measuring \(15 \times 3 \times 3\) centimeters \((6 \times 1.2 \times 1.2\) inches). How could you arrange the sticks in the box to model a system with very low porosity \((\sim 0\) percent) and high porosity ( \(\sim 40\) percent)?

Short answer

Use 25 sticks tightly for low porosity, or leave gaps to use 15 sticks for high porosity ( 40% void space).

Step-by-step solution

Understanding Porosity

Porosity is defined as the volume of voids divided by the total volume of the system. In this case, we need to arrange the sticks to achieve two different porosity levels, one close to 0 percent and one close to 40 percent.

Arrange Sticks for Low Porosity

To achieve very low porosity, nearly 0 percent, arrange the sticks tightly without any gaps. This means placing the 15 cm dimension along the sides of the box and the 3 cm side along the other direction, filling the box completely with no empty spaces. Since each stick is designed to perfectly fit within the 15 cm square in this configuration, stacking sticks face-to-face will fill the entire volume without voids.

Calculate Sticks Needed for Low Porosity

The box has a volume of 15 cm x 15 cm x 15 cm = 3375 cubic centimeters. Each stick has a volume of 15 cm x 3 cm x 3 cm = 135 cubic centimeters. Therefore, the number of sticks needed to fill the box is \( \frac{3375}{135} = 25 \) sticks.

Arrange Sticks for High Porosity

To achieve approximately 40 percent porosity, arrange the sticks in a way that creates voids. You may stack the sticks such that there are regular gaps between them, for example, alternating the direction of the sticks every few layers or arranging them at an angle when permissible, ensuring about 40 percent of the total box volume remains void.

Calculate Required Void Volume

For 40% porosity, 40% of the box's volume should be voids, which amounts to 0.4 x 3375 = 1350 cubic centimeters of space, leaving 2025 cubic centimeters for the sticks. \( \frac{2025}{135} = 15 \) sticks will fit in this arrangement, creating voids.

Key concepts

Void Volume

Void volume refers to the empty spaces within a particular volume, like a box in this context. When arranging objects, like rectangular sticks, within a defined space, the void volume is the part that remains unoccupied.

To calculate porosity, which expresses the fraction of void volume relative to the total volume, use this formula:

• Porosity (in percentage) = \( \left( \frac{\text{Void Volume}}{\text{Total Volume}} \right) \times 100 \)

For instance, if the box has a total volume of 3375 cubic centimeters, and 40% porosity is required, you must leave 1350 cubic centimeters as voids. Understanding the concept of void volume is essential when optimizing for different porosity levels, as it directly impacts how densely or loosely materials, like the sticks, are packed.

Rectangular Sticks Arrangement

To arrange the sticks effectively, consider their dimensions and how they fit within the box. The sticks each measure 15 cm, 3 cm, and 3 cm. To minimize porosity, arrange the sticks so no gaps are present. This often means laying them flat and snug together within the box dimensions.

For low porosity (near 0%), align the long side of the sticks parallel to a box side. Place the 15 cm side of each stick along the 15 cm side of the box. This way, the sticks occupy the entire box volume without leaving spaces. Completely filling the box means carefully stacking the sticks face-to-face and side-by-side until no empty space remains. When targeting higher porosity ( 40%), introduce gaps intentionally. You can stagger the sticks or arrange them at angles to create voids, ensuring they occupy only part of the box volume. This approach increases the void space between each layer, effectively achieving the desired porosity. Always visualize the arrangement options to find the best configuration that meets your porosity goals.

Volume Calculation

Calculating the volumes involved is crucial in determining the number of sticks that fit for various porosity levels. The volume of the entire box is found by multiplying its dimensions: 15 cm x 15 cm x 15 cm = 3375 cubic centimeters.

Each stick comes with its own volume, calculated using its dimensions: 15 cm x 3 cm x 3 cm = 135 cubic centimeters. Knowing these values allows us to determine how many sticks fill the box, both entirely and partially.

• For low porosity: Divide the box volume by the stick volume: \( \frac{3375}{135} = 25 \) sticks occupy the full box with no void spaces.
• For high porosity: First, determine the non-void volume (the part sticks will fill), then divide: \( \frac{2025}{135} = 15 \) sticks, corresponding to leaving 40% of the box as empty space.

By understanding these volume calculations, it becomes easier to manage how the sticks are arranged to meet specific porosity requirements, whether near-zero or significantly greater.