The density (ρ) of a substance can be defined as its mass (m) per unit volume (V). The specific volume (v_{g}) is the volume per unit mass and is therefore the inverse of density. In fact, the term ‘specific’ is generally used to denote a property of a unit mass of a substance (see Equation 2.1.2).
The SI units of density (ρ ) are kg/m³, conversely, the units of specific volume (v_{g}) are m³/kg.
Another term used as a measure of density is specific gravity. It is a ratio of the density of a substance (ρ_{s}_{}_{}_{}) and the density of pure water (ρ_{w}_{}_{}_{}) at standard temperature and pressure (STP). This reference condition is usually defined as being at atmospheric pressure and 0°C. Sometimes it is said to be at 20°C or 25°C and is referred to as normal temperature and pressure (NTP).
The density of water at these conditions is approximately 1 000 kg/m³. Therefore substances with a density greater than this value will have a specific gravity greater than 1, whereas substances with a density less than this will have a specific gravity of less than 1.
Since specific gravity is a ratio of two densities, it is a dimensionless variable and has no units. Therefore in this case the term specific does not indicate it is a property of a unit mass of a substance. Specific gravity is also sometimes known as the relative density of a substance.
Heat, work and energy
Energy is sometimes described as the ability to do work. The transfer of energy by means of mechanical motion is called work. The SI unit for work and energy is the joule, defined as 1 N m.
The amount of mechanical work carried out can be determined by an equation derived from Newtonian mechanics:
Work = Force x Displacement
It can also be described as the product of the applied pressure and the displaced volume:
Work = Applied pressure x Displaced volume
Example 2.1.1
An applied pressure of 1 Pa (or 1 N/m²) displaces a volume of 1 m³. How much work has been done?
Work done = 1 N/m² x 1 m³ = 1 N m (or 1 J)
The benefits of using SI units, as in the above example, is that the units in the equation actually cancel out to give the units of the product.
The experimental observations of J. P. Joule established that there is an equivalence between mechanical energy (or work) and heat. He found that the same amount of energy was required to produce the same temperature rise in a specific mass of water, regardless of whether the energy was supplied as heat or work.
The total energy of a system is composed of the internal, potential and kinetic energy. The temperature of a substance is directly related to its internal energy (ug). The internal energy is associated with the motion, interaction and bonding of the molecules within a substance. The external energy of a substance is associated with its velocity and location, and is the sum of its potential and kinetic energy.
The transfer of energy as a result of the difference in temperature alone is referred to as heat flow. The watt, which is the SI unit of power, can be defined as 1 J/s of heat flow.
Other units used to quantify heat energy are the British Thermal Unit (Btu: the amount of heat to raise 1 lb of water by 1°F) and the kilocalorie (the amount of heat to raise 1 kg of water by 1°C). Conversion factors are readily available from numerous sources.
Specific enthalpy
This is the term given to the total energy, due to both pressure and temperature, of a fluid (such as water or steam) at any given time and condition. More specifically it is the sum of the internal energy and the work done by an applied pressure (as in Example 2.1.1).
The basic unit of measurement is the joule (J). Since one joule represents a very small amount of energy, it is usual to use kilojoules (kJ = 1 000 joules).
The specific enthalpy is a measure of the total energy of a unit mass, and its units are usually kJ/kg.
Specific heat capacity
The enthalpy of a fluid is a function of its temperature and pressure. The temperature dependence of the enthalpy can be found by measuring the rise in temperature caused by the flow of heat at constant pressure. The constant-pressure heat capacity cP, is a measure of the change in enthalpy at a particular temperature.
Similarly, the internal energy is a function of temperature and specific volume. The constant-volume heat capacity cv, is a measure of the change in internal energy at a particular temperature and constant volume.
Because the specific volumes of solids and liquids are generally smaller, then unless the pressure is extremely high, the work done by an applied pressure can be neglected. Therefore, if the enthalpy can be represented by the internal energy component alone, the constant-volume and constant-pressure heat capacities can be said to be equal.
Therefore, for solids and liquids: c
_{P}_{}_{}_{} c
_{v}
Another simplification for solids and liquids assumes that they are incompressible, so that their volume is only a function of temperature. This implies that for incompressible fluids the enthalpy and the heat capacity are also only functions of temperature.
The specific heat capacity represents the amount of energy required to raise 1 kg by 1°C, and can be thought of as the ability of a substance to absorb heat. Therefore the SI units of specific heat capacity are kJ/kg K (kJ/kg °C). Water has a large specific heat capacity (4.19 kJ/kg °C) compared with many fluids, which is why both water and steam are considered to be good carriers of heat.
The amount of heat energy required to raise the temperature of a substance can be determined from Equation 2.1.4.
Example 2.1.2
Consider a quantity of water with a volume of 2 litres, raised from a temperature of 20°C to 70°C.
At atmospheric pressure, the density of water is approximately 1 000 kg/m³. As there are 1 000 litres in 1 m³, then the density can be expressed as 1 kg per litre (1 kg/l). Therefore the mass of the water is 2 kg.
The specific heat capacity for water can be taken as 4.19 kJ/kg °C over low ranges of temperature.
Therefore: Q = 2 kg x 4.19 kJ/kg °C x (70 - 20)°C = 419 kJ
If the water was then cooled to its original temperature of 20°C, it would also release this amount of energy in the cooling application.
Entropy (S)
Entropy is a measure of the degree of disorder within a system. The greater the degree of disorder, the higher the entropy. The SI units of entropy are kJ/kg K (kJ/kg °C).
In a solid, the molecules of a substance arrange themselves in an orderly structure. As the substance changes from a solid to a liquid, or from a liquid to a gas, the arrangement of the molecules becomes more disordered as they begin to move more freely. For any given substance the entropy in the gas phase is greater than that of the liquid phase, and the entropy in the liquid phase is more than in the solid phase.
One characteristic of all natural or spontaneous processes is that they proceed towards a state of equilibrium. This can be seen in the second law of thermodynamics, which states that heat cannot pass from a colder to a warmer body.
A change in the entropy of a system is caused by a change in its heat content, where the change of entropy is equal to the heat change divided by the average absolute temperature, Equation 2.1.5.
When unit mass calculations are made, the symbols for entropy and enthalpy are written in lower case, Equation 2.1.6.
To look at this in further detail, consider the following examples:
Example 2.1.3
A process raises 1 kg of water from 0 to 100°C (273 to 373 K) under atmospheric conditions.
Specific enthalpy at 0°C (h_{f}_{}_{}_{}) = 0 kJ/kg (from steam tables)
Specific enthalpy of water at 100°C (h_{f}_{}_{}_{}) = 419 kJ/kg (from steam tables)
Calculate the change in specific entropy
Since this is a change in specific entropy of water, the symbol ‘s’ in Equation 2.1.6 takes the suffix ‘f’ to become s_{f}.
Example 2.1.4
A process changes 1 kg of water at 100°C (373 K) to saturated steam at 100°C (373 K) under atmospheric conditions.
Calculate the change in specific entropy of evaporation
Since this is the entropy involved in the change of state, the symbol ‘s’ in Equation 2.1.6 takes the suffix ‘_{fg}’ to become s_{fg}_{}_{}_{}.
Specific enthalpy of evaporation
of steam at 100°C (373 K) (h_{fg}_{}_{}_{}) = 2 258 kJ/kg (from steam tables)
Specific enthalpy of evaporation
of water at 100°C (373 K) (h_{fg}_{}_{}_{}) = 0 kJ/kg (from steam tables)
Calculate:
The total change in specific entropy from water at 0°C to saturated steam at 100°C is the sum of the change in specific entropy for the water, plus the change of specific entropy for the steam, and takes the suffix ‘g’ to become the total change in specific entropy s_{g}_{}_{}_{}.
Therefore
Example 2.1.5
A process superheats 1 kg of saturated steam at atmospheric pressure to 150°C (423 K). Determine the change in entropy.
As the entropy of saturated water is measured from a datum of 0.01°C, the entropy of water at 0°C can, for practical purposes, be taken as zero. The total change in specific entropy in this example is based on an initial water temperature of 0°C, and therefore the final result happens to be very much the same as the specific entropy of steam that would be observed in steam tables at the final condition of steam at atmospheric pressure and 150°C.