Summary 1: Storing Energy and Systems

What is an Energy Balance?

The energy balance:

  • is just like a mass balance.
  • is based on the 1st Law of Thermodynamics.
  • accounts for energy stored in a system.
  • accounts for energy transferred in/out of the system.
  • is based on a single equation.

Why do we care about energy balances?

  • To understand and predict system behavior.
  • To design and optimise processes.
  • To maximise efficiency.
  • To ensure safety.

Basic Principles

Conservation of Energy = First Law of Thermodynamics

  • Energy cannot be created or destroyed, only transformed.
  • All forms of energy are equal in value.
  • Not all energy is useful (2nd Law of Thermodynamics).

Storing energy

Energy is stored at two levels:

Macroscopic

  • Kinetic energy
  • Potential energy

Microscopic

  • Internal energy

Differences vs Absolute Values

Absolute energy is not important, only differences in energy are important.

  • does the energy change over time?
  • does the energy change when the system undergoes some change?
    • the temperature changes
    • a phase change occurs
    • a reaction occurs
    • a solution is formed

System and Surroundings

For a mass balance, we track mass entering / leaving the system and accumulation within the system.

Figure 1: A simple mass balance diagram

System and Surroundings

For energy balance, we do the same thing.

Figure 2: A simple energy balance diagram

System and Surroundings

Defining the system boundary is critical for both mass and energy balances.

Key Terms

System — a specified region of space or matter that is under consideration. Surroundings — everything outside the system boundary.

States

The state of a system is an important thermodynamic concept.

  • The “state” of a system describes everything about it.
  • We fully define the “state” if we know or choose enough “state properties”.
  • The Gibbs phase rule can tell us how much of these state properties we need to know.
    • We can’t over specify!!! Sometimes called the degrees of freedom.

State Properties

State properties are either intensive or extensive.

  • Extensive properties
    • do depend on the size of the system.
    • e.g. mass, volume, energy.
  • Intensive properties
    • do not depend on the size of the system.
    • e.g. temperature, pressure, density.

State Property examples

State properties include:

  • density (or specific volume or molar volume), \(\rho\) (or \(v\))
  • pressure, \(P\)
  • temperature, \(T\)
  • energy
  • entropy, \(S\)
  • composition, \(x_i\) (mole fraction of component \(i\))

Gibbs Phase Rule

Gibbs phase rule:

\[F = C - \pi + 2\]

where:

  • F = degrees of freedom (number of independent intensive properties)
  • C = number of components
  • \(\pi\) = number of phases

Example 1.1: Gibbs Phase Rule

Determine how many intensive properties need to be defined or known to fully define:

  1. Liquid water
  2. A mixture of liquid water and ice
  3. A mixture of liquid ethanol and water

State changes

State changes can be caused by:

  • Temperature changes
  • Pressure changes
  • Phase changes
  • Composition changes (reaction or solution formation)

State changes and energy

Important

The path between the initial and final states does not alter the change in energy between the states. Only the initial and final states matter.

The energy associated with a state change is always the same1, regardless of the path taken.

We can choose a path that is convenient for our calculations.

State change - path independence

Figure 3: The energy change associated with a state change is independent of the path

Example 1.2: Hydroelectric power

A hydroelectric power station generates electricity by allowing water to fall from a height. The water is stored in a reservoir at a height of 100 m above the turbine. The flow rate of water through the turbine is 10 m³/s. The density of water is 1000 kg/m³. Calculate the power generated by the turbine.

Problem 1.1: Pumping Water

Cooling water is pumped to the top of a 50 m cooling tower. The water enters the cooling tower through a nozzle at high velocity. The flow rate of water is 2 L/s, and the cross-sectional area of nozzle at the exit is 0.005 m². Calculate the power required by the pump.