Question

Steam is throttled from 2.5 MPa and \(400^{\circ} \mathrm{C}\) to 1.2 MPa. Estimate the temperature change of the steam during this process and the average Joule-Thomson coefficient.

Short answer

Short Answer: To find the final temperature, look up the initial enthalpy in steam tables and find the temperature at the final pressure with the same enthalpy, due to the isenthalpic throttling process. Calculate the temperature change by subtracting the initial temperature from the final temperature. To find the average Joule-Thomson coefficient, divide the temperature change by the pressure change.

Step-by-step solution

Determine the initial enthalpy using steam tables

Refer to a steam table and look up the values for the initial pressure and temperature (2.5 MPa, \(400^{\circ} \mathrm{C}\)). Find the specific enthalpy at this state, \(h_1\).

Find the final state of the steam

Since throttling is an isenthalpic process, the final enthalpy, \(h_2\), will be equal to \(h_1\). Look up the final pressure (1.2 MPa) in the steam table and find the temperature \(T_2\) that corresponds to the same specific enthalpy \(h_2 = h_1\).

Calculate the temperature change

Subtract the initial temperature \(T_1\) from the final temperature \(T_2\) to find the temperature change during the throttling process: \(\Delta T = T_2 - T_1\).

Calculate the average Joule-Thomson coefficient

The Joule-Thomson coefficient \(\mu_{JT}\) is defined as: $$\mu_{JT} = \frac{\partial T}{\partial P}\Bigg| _{H}$$ Since we know the temperature change and the pressure change, we can find the average Joule-Thomson coefficient as: $$\mu_{JT_{avg}} = \frac{\Delta T}{\Delta P}$$ Note that \(\Delta P = P_2 - P_1\).

Key concepts

Throttling Process

The throttling process is a common thermodynamic procedure often used in refrigeration and air conditioning systems. It occurs when a fluid is forced to pass through a valve or a porous plug, which results in a pressure drop without performing external work.

During a throttling process, there is no heat exchange with the surroundings, which means it is adiabatic, and the total enthalpy of the fluid remains constant. As the fluid's pressure decreases, the temperature may either increase or decrease depending on the fluid's properties, specifically the Joule-Thomson coefficient.

Real-World Applications

Throttling is not just a theoretical concept but is used in various applications such as the expansion valve in a refrigeration cycle, where the refrigerant is throttled to a lower pressure before entering the evaporator.

Isenthalpic Process

An isenthalpic process is one in which the enthalpy of the fluid remains constant throughout. This is precisely what happens during a throttling process. Since there’s no heat transfer and no work done on or by the system, the enthalpy before and after the process remains unchanged.

Understanding this property is crucial in thermodynamics for analyzing the behavior of fluids when they undergo pressure changes without changing their total internal energy content.

Application in Calculations

In the context of our exercise, knowing that throttling is an isenthalpic process allows us to quickly ascertain that the final enthalpy after throttling will equal the initial enthalpy, making steam tables a valuable tool for finding the corresponding temperatures.

Steam Tables

Steam tables are essential reference materials in thermodynamics that list the properties of water and steam at different pressures and temperatures. These tables are used to determine various thermodynamic properties, such as enthalpy, entropy, specific volume, and temperature.

For example, when a problem involves steam or water as the working fluid, a student or engineer can consult these tables to find the properties associated with the current state of the substance. They are an essential tool in solving problems related to the Joule-Thomson effect, the design of steam turbines, or refrigeration cycles.

Enthalpy

Enthalpy is a measure of the total heat content of a thermodynamic system and is often used in the analysis of energy transfer. It is denoted by the letter 'H' and includes both the internal energy of the system and the energy associated with the pressure and volume of the system.

When a substance is allowed to expand or contract at constant enthalpy, such as in our throttling exercise, its temperature change can be linked to the pressure drop using the Joule-Thomson coefficient. This concept is important because it governs how heat and work are interrelated in processes that involve phase changes or the mixing of substances.