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Chapter 5: Problem 3

When a liquid or a gas occupies a volume, it may be assumed to fill the volume completely. On the other hand, when solid particles occupy a volume, there are always spaces (voids) among the particles. The porosity or void fraction of a bed of particles is the ratio (void volume)/(total bed volume). The bulk density of the solids is the ratio (mass of solids)/(total bed volume), and the absolute density of the solids has the usual definition (mass of solids)/(volume of solids). Suppose \(600.0 \mathrm{g}\) of a crushed ore is placed in a graduated cylinder, filling it to the \(184 \mathrm{cm}^{3}\) level. One hundred \(\mathrm{cm}^{3}\) of water is then added to the cylinder, whereupon the water level is observed to be at the \(233.5 \mathrm{cm}^{3}\) mark. Calculate the porosity of the dry particle bed, the bulk density of the ore in this bed, and the absolute density of the ore.

Short Answer

Expert verified
The porosity of the dry particle bed is approximately 73.1%, the bulk density of the ore in this bed is approximately 3.26 g/cm\u00b3 and the absolute density of the ore is approximately 12.12 g/cm\u00b3.

Step by step solution

01

Calculate the bulk density

The bulk density of the solids is given by the formula: \( Bulk Density = \frac{Mass of Solids}{Total Bed Volume} \). Substituting the given values: \( Bulk Density = \frac{600.0 g}{184 cm^{3}} = 3.26 g/cm^{3} \).
02

Calculate the volume of solids

The volume of solids can be determined from the volume of the liquid displaced when the solid is introduced into it. Here, the volume of the solids is equal to the rise in the water level when the ore was added. This is given by the formula: \( Volume of Solids = Final Water Level - Initial Water Level \). Substitute the provided values to find: \( Volume of Solids = 233.5 cm^{3} - 184 cm^{3} = 49.5 cm^{3} \).
03

Calculate the absolute density

The absolute density of the solids is given by the formula: \( Absolute Density = \frac{Mass of Solids}{Volume of Solids} \). Substituting the determined values: \( Absolute Density = \frac{600.0 g}{49.5 cm^{3}} = 12.12 g/cm^{3} \).
04

Calculate the porosity or void fraction

The porosity or void fraction of a bed of particles is given by the formula: \( Porosity = \frac{Void Volume}{Total Bed Volume} \). First, calculate the void volume. Void Volume = Total bed Volume - Volume of Solids = 184 cm^{3} - 49.5 cm^{3} = 134.5 cm^{3}. Then, using the formula, we have: \( Porosity = \frac{134.5 cm^{3}}{184 cm^{3}} = 0.731 or 73.1% \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bulk Density
Bulk density is an important concept that helps us understand the overall density of a mixture, including both the solid particles and the spaces (voids) between them. It refers to the mass of the particles divided by the total volume they occupy, which includes the pores within the material as well as the spaces between individual grains.

To calculate bulk density, use the formula:
  • Bulk Density = \( \frac{\text{Mass of Solids}}{\text{Total Bed Volume}} \)
This formula accounts for the entire volume occupied by the particles in their container and is measured in grams per cubic centimeter (g/cm鲁).

For example, if you have 600 grams of ore occupying a 184 cm鲁 volume, the bulk density is \( \frac{600.0 \, g}{184 \, cm^{3}} = 3.26 \, g/cm^{3} \).

An understanding of bulk density is crucial for applications in industries such as agriculture, geology, and construction, where material properties need to be carefully considered.
Absolute Density
Absolute density provides a measure of how dense the particles themselves are, excluding any voids or spaces between them. It is calculated by dividing the mass of the solids by their actual volume, not including the space between particles.

The formula to find absolute density is:
  • Absolute Density = \( \frac{\text{Mass of Solids}}{\text{Volume of Solids}} \)
This property's typical unit is also grams per cubic centimeter (g/cm鲁).

Continuing with our example, if the volume of solids is determined by how much they displace the water (49.5 cm鲁 in this case), the absolute density will be \( \frac{600.0 \, g}{49.5 \, cm^{3}} = 12.12 \, g/cm^{3} \).

Absolute density is especially significant when considering material strength and compaction processes, as it reflects the true density of the material matter.
Void Fraction
The void fraction, often referred to as porosity, is the measure of the empty spaces in a material compared to its total volume. This ratio indicates how much of the volume is occupied by void space rather than the material itself.

To calculate the void fraction, you use the following relationship:
  • Void Fraction (Porosity) = \( \frac{\text{Void Volume}}{\text{Total Bed Volume}} \)
Void volume can be found by subtracting the volume of solids from the total volume of the bed.

For instance, with a total bed volume of 184 cm鲁 and a volume of solids calculated as 49.5 cm鲁, the void volume becomes 134.5 cm鲁. Thus, the porosity is \( \frac{134.5 \, cm^{3}}{184 \, cm^{3}} = 0.731 \) or 73.1%.

Understanding void fraction is vital in fields like civil engineering, hydrogeology, and material science, where the extent to which materials can absorb or allow passage of fluids is crucial.
Volume of Solids
Volume of solids is the actual space occupied by the material particles, with no inclusion of the air spaces or voids between them. Determining this is essential for assessing the absolute density of the material.

It is established by observing how much space the solid itself takes up, which is achieved through displacement methods when the solid is immersed in a fluid.

Using the displacement principle, if the initial water level is measured at 184 cm鲁 and rises to 233.5 cm鲁 after placing the solid in it, the volume occupied by the solid particles can be computed as:
  • Volume of Solids = Final Water Level - Initial Water Level
Therefore, the volume of solids would be \( 233.5 \, cm^3 - 184 \, cm^3 = 49.5 \, cm^3 \).

This concept is particularly valuable in mineral processing and building materials testing, where accurate volume and density are essential for quality control.

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Most popular questions from this chapter

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