δe_cv = q - w + ∑_in 𝑚̇(ℎ + v^2/2 + gz) - ∑_out 𝑚̇(ℎ + v^2/2 + gz)

Definition

This equation represents the energy balance for a control volume, illustrating how the change in internal energy (δe_cv) is affected by heat transfer (q), work done (w), and the energy contributions from mass flow into and out of the system. It connects to fundamental principles by emphasizing how energy can neither be created nor destroyed, but only transformed or transferred, aligning with the first law of thermodynamics.

5 Must Know Facts For Your Next Test

1. The term δe_cv represents the change in internal energy within a control volume, essential for analyzing thermodynamic systems.
2. Heat transfer (q) can be positive or negative, indicating whether the system is gaining or losing energy.
3. Work done (w) on the system is considered positive, while work done by the system is negative, affecting the overall energy balance.
4. The summation terms account for the energy associated with mass entering and exiting the control volume, including enthalpy (h), kinetic energy (v^2/2), and potential energy (gz).
5. This equation is a direct application of the first law of thermodynamics and illustrates how mass transfer impacts energy conservation in fluid systems.

Review Questions

• How does the equation δe_cv = q - w + ∑_in 𝑚̇(ℎ + v^2/2 + gz) - ∑_out 𝑚̇(ℎ + v^2/2 + gz) illustrate the conservation of energy?
  • The equation demonstrates conservation of energy by showing that any change in internal energy within a control volume (δe_cv) is balanced by heat added to the system (q), work done on or by the system (w), and the net energy associated with mass flow into and out of the control volume. It captures how energy enters and exits through mass transfer and highlights that all forms of energy—thermal, kinetic, and potential—must be accounted for to maintain this balance.
• Analyze how variations in mass flow rate (𝑚̇) influence the overall energy balance represented in this equation.
  • Variations in mass flow rate directly impact both summation terms in the equation, affecting how much energy enters or leaves the control volume. A higher mass flow rate increases the total enthalpy and kinetic/potential energies contributed by incoming mass, leading to a larger positive contribution to δe_cv. Conversely, if more mass exits than enters, this could lead to a decrease in internal energy. Therefore, controlling mass flow rates is crucial for maintaining desired energy states in thermodynamic systems.
• Evaluate the implications of heat transfer (q) being zero in practical applications of this equation on a control volume.
  • If heat transfer (q) is zero in practical applications, it implies that there is no thermal interaction with the surroundings. In this scenario, the change in internal energy (δe_cv) would solely depend on work done on or by the system and the net mass flow effects. This situation might occur in adiabatic processes where insulation prevents heat exchange. The overall effect would simplify analysis since only work and mass transfer would contribute to changes in energy, emphasizing how crucial these factors are when designing systems like turbines or compressors where minimizing heat loss is essential.

Related terms

First Law of Thermodynamics: A principle stating that energy cannot be created or destroyed, only transformed from one form to another, often expressed in terms of heat and work.

Control Volume: A defined region in space used to analyze the flow of mass and energy, where the boundary can be fixed or moving.

Mass Flow Rate (𝑚̇): The amount of mass passing through a surface per unit time, crucial for understanding energy transfer in systems with fluid motion.