Question

Complete the following exercises dealing with slopes: (a) At the triple point of water, evaluate the ratio of the slope of the vaporization line to the slope of the sublimation line. Use steam table data to obtain a numerical value for the ratio. (b) Consider the superheated vapor region of a temperature-entropy diagram. Show that the slope of a constant specific volume line is greater than the slope of a constant pressure line through the same state. (c) An enthalpy-entropy diagram (Mollier diagram) is often used in analyzing steam turbines. Obtain an expression for the slope of a constant-pressure line on such a diagram in terms of \(p-v-T\) data only. (d) A pressure-enthalpy diagram is often used in the refrigeration industry. Obtain an expression for the slope of an isentropic line on such a diagram in terms of \(p-v-T\) data only.

Short answer

Part a: Use Clausius-Clapeyron equation to calculate and ratio steam table values. Parts b, c, and d: Derive slope equations from p-v-T data using thermodynamic properties and relationships.

Step-by-step solution

- Understanding the Ratio of Slopes (Part a)

To find the ratio of the slope of the vaporization line to the slope of the sublimation line at the triple point, we need data from steam tables for the triple point of water.

- Gather Data for Triple Point (Part a)

The triple point of water occurs at approximately 0.01°C and 611.657 Pa. The slopes can be derived from the Clausius-Clapeyron equation: \[\frac{dP}{dT} = \frac{L}{T(V_g - V_l)}\] for vaporization and \[\frac{dP}{dT} = \frac{L}{T(V_s - V_i)}\] for sublimation, where \(L\) is the latent heat, \(V_g\), \(V_l\), \(V_s\), \(V_i\) are specific volumes of gas, liquid, solid, and ice.

- Use Clausius-Clapeyron Equation (Part a)

Using steam table values at the triple point:For vaporization, find the latent heat \(L_{vl}\) and volumes \(V_g\) and \(V_l\).For sublimation, find the latent heat \(L_{si}\) and volumes \(V_s\) and \(V_i\).Then calculate \(\frac{dP}{dT}\) for both phases.

- Calculate the Ratio (Part a)

Substitute the values into the Clausius-Clapeyron equation for both slopes and calculate their ratio: \[\frac{(dP/dT)_{vaporization}}{(dP/dT)_{sublimation}}\].

- Analyze Superheated Vapor (Part b)

For a superheated vapor region, use the fact that on a T-s diagram, isochoric lines (constant specific volume) have a steeper slope than isobaric lines (constant pressure). Show mathematically that \[\left.\frac{dT}{dS}\right|_v > \left.\frac{dT}{dS}\right|_p\].

- Establish Expression for Isentropic Line (Part b, c)

For an isentropic line on a temperature-entropy diagram, use the Maxwell relations and the specific volume and entropy definitions to show that \[\left. \frac{dT}{dS} \right|_v > \left.\frac{dT}{dS} \right|_p\].

- Compute Slope on Mollier Diagram (Part c)

Derive the slope of a constant-pressure line in a Mollier (enthalpy-entropy) diagram by setting up the relationship between temperature, pressure, and specific volume, resulting in the equation: \[ \left( \frac{\partial h}{\partial S} \right)_p \].

- Slope of an Isentropic Line (Part d)

For a pressure-enthalpy diagram, use the p-v-T relationship to derive: \[ \left. \frac{dv}{dH} \right|_S = \frac{1}{v(T)} (dP/dT)_{S} \]

Key concepts

Clausius-Clapeyron equation

The Clausius-Clapeyron equation is an essential relation in thermodynamics. It provides a way to describe the phase change between different states of matter, such as from liquid to vapor, or solid to vapor. The equation is given as: \[\frac{dP}{dT} = \frac{L}{T(V_{2} - V_{1})}\] Here, \( P \) is the pressure, \( T \) is the temperature, \( L \) is the latent heat of the phase change, and \( V_{2} \) and \( V_{1} \) are the specific volumes of the two phases involved. For example, in the case of water vaporizing to steam, \( V_{2} \) would be the specific volume of steam, and \( V_{1} \) would be the specific volume of liquid water. This equation helps us calculate the slope of the phase boundaries in a pressure-temperature diagram, which is critical in thermodynamics for understanding how substances behave under different pressures and temperatures.

Superheated vapor region

In thermodynamics, the superheated vapor region refers to the state of a substance where it is heated above its boiling point, without changing its phase from gas to liquid. This region is important for engineering applications like steam turbines. On a temperature-entropy (T-s) diagram, the behavior of the superheated vapor can be analyzed. In the T-s diagram:

• Isochoric lines (constant specific volume) often have steeper slopes.
• Isobaric lines (constant pressure) have shallower slopes compared to isochoric lines.

It follows mathematically by showing that the slope of the isochoric line, \(\left.\frac{dT}{dS}\right|_v\), is greater than that of the isobaric line, \(\left.\frac{dT}{dS}\right|_p\). Understanding this region helps in designing efficient thermal systems, ensuring that the vapor does not condense back into the liquid.

Enthalpy-entropy diagram

An enthalpy-entropy diagram, also known as a Mollier diagram, is a graphical representation used in thermodynamics to visualize changes in enthalpy (\( h \)) and entropy (\( s \)) during thermodynamic processes. It is especially useful in analyzing the behavior of steam turbines and refrigeration cycles. To understand the slopes of different lines in this diagram:

• A constant-pressure line (isobaric) on an enthalpy-entropy diagram shows how enthalpy changes with entropy at a constant pressure.
• The slope of an isobaric line can be expressed as \( \left( \frac{\partial h}{\partial s} \right)_p \).

This slope can be derived in terms of specific volume (\( v \)), temperature (\( T \)), and pressure (\( P \)). The resulting equation signifies how the enthalpy and entropy change concurrently under constant pressure, an essential factor in optimizing steam turbines and other thermal systems.

Pressure-enthalpy diagram

In the refrigeration industry, a pressure-enthalpy (P-h) diagram is frequently used to represent refrigeration cycles and analyze their performance. This diagram plots the pressure of a refrigerant against its enthalpy. Key observations for a P-h diagram include:

• Isentropic processes (constant entropy) appear as vertical lines, showing no change in entropy with varying enthalpy.
• To find the slope of an isentropic line, use thermodynamic relationships and the p-v-T data, represented by: \(\left. \frac{dv}{dH} \right|_s = \frac{1}{v(T)} \left(\frac{dP}{dT}\right)_{s}\).

Understanding this diagram aids in designing more efficient refrigeration systems by clearly depicting the state changes and energy transformations during the refrigeration cycle.

Specific volume

Specific volume is a fundamental thermodynamic property defined as the volume occupied by a unit mass of a substance, typically denoted as \( v \) (m³/kg). It is an inverse of density and provides key insights into the substance's behavior under different conditions. Specific volume is crucial across various thermodynamic processes:

• During phase changes, the difference in specific volumes between phases determines the slopes of phase boundaries in phase diagrams, as seen in the Clausius-Clapeyron equation.
• In the superheated region, knowing the specific volume helps analyze properties on T-s and P-h diagrams.

Recognizing how specific volume changes with temperature and pressure is essential for accurately predicting the behavior of gases and refining the performance of thermodynamic systems.