1.1 Density and specific gravity

Definition of density

Density describes how much mass is packed into a given volume of a substance. It tells you how closely packed the particles (atoms, molecules, or ions) are within a material, and it plays a central role in understanding fluid behavior across nearly every application in this course.

Mass per unit volume

Density is defined as the ratio of an object's mass to its volume:

$\rho = \frac{m}{V}$

where $\rho$ is density, $m$ is mass, and $V$ is volume. The greater the mass per unit volume, the higher the density. Lead, for instance, has a much higher density than wood because its atoms are more massive and more tightly packed into the same space.

Common units

• SI unit: kilogram per cubic meter ($\text{kg/m}^3$)
• Also commonly used: gram per cubic centimeter ($\text{g/cm}^3$) and pound per cubic foot ($\text{lb/ft}^3$)
• A handy reference: water has a density of $1000 \text{ kg/m}^3$, which equals $1 \text{ g/cm}^3$, at standard temperature and pressure
• Converting between units is straightforward with the right factors (e.g., $1 \text{ g/cm}^3 = 1000 \text{ kg/m}^3$)

Factors affecting density

Density isn't a fixed number for a given substance. It shifts with changes in temperature and pressure, and accounting for these variations matters when you need accurate predictions in fluid dynamics problems.

Temperature effects

For most substances, density decreases as temperature increases. Higher temperature means particles gain kinetic energy and vibrate more, spreading apart and occupying more space. The thermal expansion coefficient quantifies this relationship.

Water is a good example: its density drops from $1000 \text{ kg/m}^3$ at $4°\text{C}$ to about $998 \text{ kg/m}^3$ at $20°\text{C}$. That change seems small, but it matters in precision engineering and large-scale systems.

Pressure effects

Density increases with increasing pressure. Pressure forces particles closer together, reducing the volume they occupy. How much the density changes depends on the substance's compressibility.

This effect is especially noticeable in gases. Air density increases with depth in the atmosphere because the weight of the air above compresses the air below. For liquids, the effect is much smaller since liquids are nearly incompressible.

Density of liquids

Liquids have a fixed volume but take the shape of their container. Their density directly influences phenomena like buoyancy and hydrostatic pressure, which you'll use throughout this course.

Water density

Water serves as the standard reference liquid in fluid dynamics. At $4°\text{C}$, it reaches its maximum density of approximately $1000 \text{ kg/m}^3$. This is actually an unusual property: most substances are densest in their solid phase, but water is densest as a liquid just above freezing.

Dissolved substances change water's density. Seawater, for example, has an average density of about $1025 \text{ kg/m}^3$ because of dissolved salts.

Other common liquids

Different liquids vary widely in density based on their molecular structure and composition. Some approximate values:

• Ethanol: $789 \text{ kg/m}^3$
• Olive oil: $920 \text{ kg/m}^3$
• Mercury: $13{,}600 \text{ kg/m}^3$

These differences have practical consequences. In an oil-water separator, oil (less dense) naturally rises above water (more dense), allowing the two liquids to separate into distinct layers without any energy input.

Density of gases

Gases have much lower densities than liquids or solids because their particles are spread far apart. Unlike liquids, gas density is highly sensitive to both temperature and pressure.

Ideal gas law

The ideal gas law relates pressure, volume, temperature, and the amount of gas:

$PV = nRT$

where $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is absolute temperature (in Kelvin).

You can rearrange this to get a density form. Since $n = m/M$ (mass divided by molar mass) and $\rho = m/V$:

$\rho = \frac{PM}{RT}$

This tells you gas density is proportional to pressure and inversely proportional to temperature. At standard conditions ($0°\text{C}$ and $1 \text{ atm}$), air has a density of approximately $1.29 \text{ kg/m}^3$.

The ideal gas law assumes gas particles have negligible volume and don't interact with each other, which works well at moderate temperatures and low pressures.

Real gas behavior

Real gases deviate from ideal behavior, especially at high pressures or low temperatures where intermolecular forces and finite particle volume become significant. Equations of state like the van der Waals equation account for these effects and give more accurate density predictions.

Carbon dioxide ($CO_2$) at high pressures is a classic example: its measured density deviates significantly from ideal gas law predictions because of intermolecular attractions between molecules.

Measurement techniques

Accurate density measurements are essential for characterizing fluids and validating theoretical predictions. The methods fall into two categories.

Direct measurement methods

Direct methods involve measuring mass and volume separately, then calculating density:

1. Measure the mass of the sample using a balance or scale
2. Measure the volume using a graduated cylinder, pycnometer, or by liquid displacement
3. Calculate density: $\rho = m/V$

A pycnometer is a small glass flask with a precise, known volume. You weigh it empty, fill it with your liquid, and weigh it again. The mass difference divided by the known volume gives you a very accurate density.

Indirect measurement methods

Indirect methods infer density from other physical properties:

• Hydrometer: A calibrated float that sinks to different depths depending on the liquid's density. The denser the liquid, the higher the float rides.
• Oscillating U-tube densimeter: Measures how the vibration frequency of a fluid-filled tube changes with density. Higher density lowers the frequency.
• Coriolis flow meter: Measures density of a flowing fluid based on the Coriolis effect on vibrating tubes.

Hydrometers are common in practical settings like breweries (measuring sugar content) and auto shops (checking battery electrolyte concentration).

Definition of specific gravity

Specific gravity (SG) compares the density of a substance to a reference substance. It gives you a quick, unitless way to express how dense something is relative to a familiar standard.

Ratio of densities

Specific gravity is defined as:

$SG = \frac{\rho_{\text{substance}}}{\rho_{\text{reference}}}$

• For liquids, the reference is water at $4°\text{C}$ ($1000 \text{ kg/m}^3$)
• For gases, the reference is air at standard conditions

If a liquid has a density of $800 \text{ kg/m}^3$, its specific gravity is $800/1000 = 0.8$. That immediately tells you it's lighter than water and will float on it.

Dimensionless quantity

Because specific gravity is a ratio of two densities with the same units, the units cancel out, making it dimensionless. This is one of its main advantages: you can compare substances without worrying about unit systems.

• $SG > 1$: the substance is denser than the reference (it sinks in the reference fluid)
• $SG < 1$: the substance is less dense than the reference (it floats)

Glycerin, with a specific gravity of about 1.26, is 1.26 times denser than water.

Specific gravity of liquids

Specific gravity is widely used in industrial and scientific settings to compare liquid densities in a standardized way.

Water as reference

Water at $4°\text{C}$ is the standard reference, with $SG = 1$. Since water's density is $1000 \text{ kg/m}^3$, the specific gravity of any liquid numerically equals its density in $\text{g/cm}^3$. This makes conversions very convenient.

Ethanol has a specific gravity of about 0.79, meaning it's less dense than water. Mercury has a specific gravity of 13.6, meaning it's 13.6 times denser.

Hydrometers

Hydrometers are the go-to instrument for measuring liquid specific gravity. They consist of a weighted glass float with a calibrated stem.

How they work: you lower the hydrometer into the liquid, and it sinks until the buoyant force equals its weight. In a denser liquid, it doesn't sink as far, so the scale reading is higher. In a less dense liquid, it sinks deeper.

Common applications include:

• Brewing: measuring sugar concentration in wort
• Petroleum: characterizing crude oil and fuel grades
• Battery maintenance: checking the electrolyte specific gravity to assess charge level in lead-acid batteries

Specific gravity of gases

For gases, specific gravity compares a gas's density to that of air, which is useful in gas mixing, storage, and safety applications (e.g., knowing whether a leaked gas will rise or settle).

Air as reference

Air at standard conditions ($0°\text{C}$, $1 \text{ atm}$) is the reference, with $SG = 1$. Helium has a specific gravity of about 0.14, which is why helium balloons float: helium is far less dense than the surrounding air.

Ideal gas approximation

For gases behaving close to ideal, you can estimate specific gravity using molecular weights alone:

$SG = \frac{MW_{\text{gas}}}{MW_{\text{air}}}$

where $MW_{\text{air}} \approx 29 \text{ g/mol}$. This works because at the same temperature and pressure, equal volumes of ideal gases contain the same number of molecules (Avogadro's principle), so density scales directly with molecular weight.

Methane ($CH_4$), with a molecular weight of $16 \text{ g/mol}$, has $SG \approx 16/29 \approx 0.55$. This means methane is lighter than air and will rise if released, which is important for ventilation design and safety.

Relationship between density and specific gravity

These two properties are closely linked, and you'll often need to convert between them in fluid dynamics problems.

Conversion factors

Converting is straightforward:

$\rho_{\text{substance}} = SG \times \rho_{\text{reference}}$

And in reverse:

$SG = \frac{\rho_{\text{substance}}}{\rho_{\text{reference}}}$

If a liquid has $SG = 0.9$ and the reference is water at $1000 \text{ kg/m}^3$, then the liquid's density is $0.9 \times 1000 = 900 \text{ kg/m}^3$.

Dimensionless analysis

The dimensionless nature of specific gravity makes it useful in scaling and comparing fluid systems. Dimensionless numbers like the Reynolds number and Froude number incorporate density (or density ratios) to characterize flow behavior, and using specific gravity simplifies comparisons across different fluids.

Archimedes' principle is a direct application: the buoyant force on a submerged object equals the weight of the displaced fluid, which depends on the fluid's density. Comparing specific gravities of the object and fluid tells you immediately whether the object floats or sinks.

Applications in fluid dynamics

Density and specific gravity show up constantly in fluid dynamics. Here are the three most important applications you'll encounter in this unit.

Buoyancy calculations

Buoyancy is the upward force a fluid exerts on an immersed object. It depends on the density difference between the object and the fluid:

• If $\rho_{\text{object}} < \rho_{\text{fluid}}$, the object floats
• If $\rho_{\text{object}} > \rho_{\text{fluid}}$, the object sinks

Oil-water separators exploit this principle: oil droplets ($SG \approx 0.8\text{–}0.9$) rise through water ($SG = 1$) and collect at the surface.

Hydrostatic pressure

Hydrostatic pressure is the pressure a fluid at rest exerts due to its own weight:

$P = \rho g h$

where $P$ is pressure, $\rho$ is fluid density, $g$ is gravitational acceleration ($9.81 \text{ m/s}^2$), and $h$ is the depth below the fluid surface.

This means denser fluids produce higher pressures at the same depth. Mercury ($\rho = 13{,}600 \text{ kg/m}^3$) generates 13.6 times more pressure per meter of depth than water, which is why mercury barometers can be much shorter than water barometers.

Flow behavior predictions

Density influences how fluids flow in several ways:

• The Reynolds number ($Re = \frac{\rho v L}{\mu}$) depends on density and determines whether flow is laminar or turbulent
• Density differences between fluid regions drive buoyancy-driven flows like natural convection in heat transfer
• In pipe flow, pressure drop is proportional to fluid density, which directly affects pumping power requirements

Understanding density's role in these calculations is foundational for the rest of the course.