Surface Area per Unit Volume of a Circular Polyester Fiber

To calculate the surface area per unit volume of a circular polyester fiber, we need to consider the fineness and density of the fiber.

1. Understanding the Given Values

- Fineness: The fineness of a fiber is a measure of its linear density and is typically expressed in denier. Denier is defined as the weight in grams of 9,000 meters of the fiber. In this case, the fineness is given as 1.5 denier.

- Density: The density of a substance is its mass per unit volume. In this case, the density of the polyester fiber is given as 1.38 g/cm3.

2. Calculating the Mass of the Fiber

To calculate the mass of the fiber, we can use the formula:

Mass = Fineness / 9,000

In this case, the fineness is given as 1.5 denier. Converting denier to grams:

1 denier = 1 gram / 9,000

So, the mass of the fiber can be calculated as:

Mass = 1.5 g / (1 g / 9,000) = 1.5 * 9,000 = 13,500 g

3. Calculating the Volume of the Fiber

The volume of the fiber can be calculated using the formula:

Volume = Mass / Density

In this case, the mass of the fiber is 13,500 g and the density is 1.38 g/cm3. Converting grams to cubic millimeters (mm3):

1 cm3 = 1,000,000 mm3

So, the volume of the fiber can be calculated as:

Volume = 13,500 g / (1.38 g/cm3 * 1,000,000 mm3/cm3) = 9.78 mm3

4. Calculating the Surface Area of the Fiber

The surface area of the fiber can be calculated using the formula for the surface area of a cylinder:

Surface Area = 2πr2 + 2πrh

Since the fiber is circular, the radius (r) is half of the diameter. The diameter can be calculated using the formula:

Diameter = (Fineness / π)1/2

In this case, the fineness is given as 1.5 denier. Converting denier to millimeters (mm):

1 denier = 1 mm

So, the diameter of the fiber can be calculated as:

Diameter = (1.5 mm / π)1/2 = 1.22 mm

The radius can be calculated as half of the diameter:

Radius = 1.22 mm / 2 = 0.61 mm

The height (h) of the fiber is the length of the fiber, which is not given in the question. Therefore, the height cannot be determined.

5. Calculating the Surface Area per Unit Volume

To calculate the surface area per unit volume, divide the surface area by the volume:

Surface Area per Unit Volume = Surface Area / Volume

Since we cannot determine the height of the fiber, we cannot calculate the surface area. Therefore, the surface area per unit volume cannot be determined.

Conclusion

The surface area per unit volume