Derivation of the Energy Conservation

In the field of engineering thermodynamics, the Law of Conservation of Energy—commonly known as the First Law of Thermod

In the field of engineering thermodynamics, the Law of Conservation of Energy—commonly known as the First Law of Thermodynamics—serves as the fundamental cornerstone. It dictates a simple yet profound reality: energy can neither be created nor destroyed; it can only be transformed from one form to another or transferred between a system and its surroundings.

To apply this principle to real-world engineering problems, we must transition from abstract philosophy to mathematical modeling. This requires defining the nature of the system under study. In this article, we will focus on the Closed System (also referred to as a Control Mass), a system characterized by the fact that while energy may cross its boundaries in the form of heat or work, the mass within the system remains constant.
Before deriving the conservation equation, we must categorize the various forms of energy present within a closed system. The total energy of a system, denoted as $E$, is the sum of its microscopic and macroscopic components.

Internal Energy ($U$)

Internal energy is a state function representing the energy stored at the molecular level. It encompasses the kinetic energy of molecules (translation, rotation, and vibration) as well as the potential energy arising from intermolecular forces.

Kinetic Energy ($KE$)

Kinetic energy refers to the energy the system possesses due to its macroscopic motion through space. For a system with mass $m$ moving at a velocity $v$, it is expressed as:
$$KE = \frac{1}{2}mv^2$$

Potential Energy ($PE$)

Potential energy is the energy the system possesses due to its position within a force field, such as a gravitational field. For a mass $m$ at a height $h$ relative to a reference plane, it is expressed as:
$$PE = mgh$$

Summing these components, the Total Energy of the system is:
$$E = U + KE + PE$$

Mechanisms of Energy Transfer

Energy crosses the boundary of a closed system via two distinct mechanisms:

  • Heat ($Q$): Energy transfer driven solely by a temperature difference between the system and its surroundings.
  • Work ($W$): Any energy transfer that is not heat. This includes mechanical work (such as moving a piston), electrical work, or shaft work.

2. The Derivation of the Energy Conservation Equation

The derivation follows a logical progression: the change in the system's total energy must equal the net energy exchanged with the environment.

Step 1: The Energy Balance Principle

Based on the principle of conservation, we establish the fundamental balance:
$$\Delta E_{system} = \text{Energy In} - \text{Energy Out}$$

In a closed system, the only modes of transfer are $Q$ and $W$.

Step 2: Establishing Sign Conventions

In thermodynamics, the direction of energy flow is critical. To maintain mathematical consistency, engineers follow a standard sign convention:

  • Heat ($Q$): Heat entering the system is considered positive ($Q > 0$), while heat leaving the system is negative ($Q < 0$).
  • Work ($W$): Work done by the system on its surroundings is considered positive ($W > 0$), whereas work done on the system by the surroundings is negative ($W < 0$).

Under these conventions, the net energy transferred to the system is represented by $Q - W$.

Step 3: Mathematical Formulation

Combining the energy balance with the total energy definition, we arrive at:
$$\Delta (U + KE + PE) = Q - W$$

In many practical engineering applications—such as analyzing gases within a stationary piston-cylinder assembly—the changes in kinetic and potential energy are often negligible ($\Delta KE \approx 0$ and $\Delta PE \approx 0$). In such cases, the equation simplifies to the most widely used form of the First Law for closed systems:
$$\Delta U = Q - W$$

3. Expansion: Boundary Work

In closed systems involving compressible fluids, the most frequent form of work encountered is Boundary Work ($W_b$). This occurs when the movement of the system's boundaries (e.g., a piston moving upward) results in a change in volume.

If a system undergoes a process from volume $V_1$ to $V_2$ under varying pressure $P$, the boundary work is calculated as the integral of pressure with respect to volume:
$$W_b = \int_{V_1}^{V_2} P , dV$$

Substituting this into our energy equation provides a highly practical tool for analyzing expansion and compression processes:
$$U_2 - U_1 = Q - \int_{V_1}^{V_2} P , dV$$

4. Practical Case Study

To illustrate the application of these equations, consider the following engineering scenario:

Problem Statement:
A $2\text{ kg}$ mass of an ideal gas is contained within a piston-cylinder device. The gas undergoes a constant-pressure expansion from an initial volume of $0.01\text{ m}^3$ to a final volume of $0.03\text{ m}^3$. During this process, the system absorbs $500\text{ J}$ of heat. Neglecting changes in kinetic and potential energy, determine the change in the system's internal energy ($\Delta U$).

Solution:

  1. Identify Known Variables:

    • Pressure ($P$): $200\text{ kPa} = 200,000\text{ Pa}$
    • Initial Volume ($V_1$): $0.01\text{ m}^3$
    • Final Volume ($V_2$): $0.03\text{ m}^3$
    • Heat Transfer ($Q$): $+500\text{ J}$ (positive because heat is absorbed)

  2. Calculate Boundary Work ($W$):
    Since the pressure is constant, the integral simplifies to:
    $$W = P(V_2 - V_1)$$
    $$W = 200,000\text{ Pa} \times (0.03\text{ m}^3 - 0.01\text{ m}^3)$$
    $$W = 200,000 \times 0.02 = 4000\text{ J}$$

  3. Apply the First Law:
    $$\Delta U = Q - W$$
    $$\Delta U = 500\text{ J} - 4000\text{ J}$$
    $$\Delta U = -3500\text{ J}$$

Conclusion:
The internal energy of the system decreased by $3500\text{ J}$. This indicates that the energy required to perform the work on the surroundings exceeded the energy supplied by the heat input, necessitating a drawdown of the system's internal energy.

Summary

The derivation $\Delta (U + KE + PE) = Q - W$ provides a rigorous framework for quantifying energy transformations. By decomposing total energy into its constituent parts and carefully applying sign conventions to heat and work, engineers can accurately predict the behavior of complex thermal systems, from internal combustion engines to refrigeration cycles. Mastery of this derivation and the nuances of boundary work is essential for any advanced study in thermal sciences.