Question

Steam is accelerated as it flows through an actual adiabatic nozzle. The entropy of the steam at the nozzle exit will be (greater than, equal to, less than) the entropy at the nozzle inlet.

Short answer

Answer: The entropy of the steam at the nozzle exit is greater than the entropy at the nozzle inlet.

Step-by-step solution

Identify the process

The given process is adiabatic, meaning there is no heat transfer between the steam and the surroundings.

Define the concept of entropy

Entropy is a thermodynamic property that measures the level of disorder in a system. In general, for any process: - If the process is reversible, the entropy remains constant (ΔS = 0). - If the process is irreversible, the entropy increases (ΔS > 0).

Determine entropy change in an adiabatic process

As there is no heat transfer in an adiabatic process, the process can either be reversible or irreversible. In a reversible adiabatic process (also called isentropic), the entropy remains constant (ΔS = 0). In an irreversible adiabatic process, the entropy increases (ΔS > 0).

Apply the concept of entropy change to the given problem

The question describes an "actual" adiabatic nozzle. In the real world, most processes are irreversible due to factors like friction or turbulence. Hence, we will consider the process to be irreversible adiabatic.

Conclude the relation between the entropy at the nozzle inlet and exit

Since the given adiabatic process is irreversible, the entropy of the steam at the nozzle exit will be greater than the entropy at the nozzle inlet (ΔS > 0).

Key concepts

Entropy in Thermodynamics

Entropy in thermodynamics is a fundamental concept that represents the degree of disorder or randomness in a system. It serves as a measure of how energy is dispersed or spread out at a specific temperature. Whenever a process occurs that changes the state of a system, entropy provides insight into the feasibility of the process and the direction it will naturally proceed.

In accordance with the Second Law of Thermodynamics, the total entropy of an isolated system can never decrease over time. It can either remain constant for a perfectly reversible process, which is an idealized concept and very rare in practice, or it increases for an irreversible process, which is the common scenario in real-world applications.

When considering an adiabatic process—where there is no heat exchange with the surroundings—the entropy change (\( \triangle S \)) is determined entirely by the internal changes within the system. If such a process is reversible (idealized), \( \triangle S = 0 \); the entropy is conserved. However, real adiabatic processes are typically irreversible, leading to an increase in entropy (\( \triangle S > 0 \)). Factors like friction, turbulence, or heat generation within the system contribute to this increase, signaling a natural progression towards disorder.

Reversible and Irreversible Processes

In thermodynamics, processes are categorized as either reversible or irreversible based on their ability to return to the initial state without leaving any change in the system and the surroundings. A reversible process is an idealized concept where the system changes state in such a way that the process can be reversed without any net energy or entropy change. These processes are carried out infinitely slowly, allowing the system to remain in equilibrium throughout the entire process.

On the other hand, irreversible processes are the norm in nature. They occur spontaneously and cannot be reversed without affecting the surroundings or the universe as a whole. Irreversible processes include natural occurrences like mixing of substances, chemical reactions, heat transfer with a finite temperature difference, and phenomena involving friction and turbulence.

It's essential to recognize that all real processes are irreversible to some extent, contributing to the increase in the universe's overall entropy. In the context of nozzle flow, even if the nozzle is designed to be adiabatic, the actual flow within it will incur irreversible losses due to factors such as viscosity and the resulting friction effects.

Nozzle Flow Thermodynamics

In the realm of thermodynamics, nozzle flow is a practical application where the principles of energy conversion are utilized to accelerate a fluid. In a nozzle, the fluid's potential energy and internal energy (primarily as enthalpy) are partly converted into kinetic energy, causing the fluid to speed up. Nozzle flow is governed by the principles of fluid mechanics and thermodynamics, including conservation laws like those of mass, momentum, and energy.

For an adiabatic nozzle, we assume no heat is transferred to or from the fluid as it accelerates, meaning that the process is focused on the conversion of thermal energy to kinetic energy. During this process, if the flow were reversible and frictionless, we would have a constant entropy scenario, termed isentropic flow. However, in real conditions, the flow is affected by irreversibilities such as viscous dissipation (frictional effects) and turbulence, which lead to an inevitable increase in entropy.

The consequence of these irreversibilities becomes evident in factors like a drop in pressure, temperature, and a potential loss of efficiency in the energy conversion process. Nevertheless, understanding nozzle flow in thermodynamics is crucial in designing more effective engines, turbines, and various other systems where the efficient acceleration of fluids is sought.