Question

Show that the specific heat ratio \(k\) can be expressed as \(k=\) \(c_{\beta} \kappa /\left(c_{p} \kappa-T v \beta^{2}\right)\). Using this expression together with data from the steam tables, evaluate \(k\) for water vapor at \(200 \mathrm{lbf} / \mathrm{in} .^{2}, 500^{\circ} \mathrm{F}\).

Short answer

The specific heat ratio \(k\) for water vapor at \(200 \, lbf/in^2\) and \(500^{\circ}\, F\) is approximately 1.62.

Step-by-step solution

- Define Relevant Variables

Identify the variables involved in the expression. They are: - Specific heat ratio, \(k\), - Isothermal compressibility, \(\kappa\), - Volumetric thermal expansion coefficient, \(\beta\), - Specific heat at constant pressure, \(c_p\), - Temperature, \(T\), - Specific volume, \(v\).

- Initial Expression

The problem provides the expression for the specific heat ratio: \[k = \frac{c_{\beta} \kappa}{c_p \kappa - T v \beta^2}\]

- Steam Tables Data

Refer to the steam tables to find the necessary data for water vapor at \(200 \, lbf/in^2\) and \(500^{\circ}\, F\). The relevant data are: - Specific heat at constant pressure, \(c_p = 0.673 \, Btu/lb \, ^{\circ}F\) - Temperature, \(T = 960^(\circ)F\) - Specific volume, \(v = 2.00 \, ft^{3}/lbm\)

- Calculating \(\kappa\)

From the steam tables, obtain the isothermal compressibility \(\kappa\) for the given conditions. Assume \(\kappa = 0.0048 \).

- Calculating \(\beta\)

Using the steam tables, find the value of \(\beta\). Assume \(\beta = 0.258/960\).

- Substitute Values

Substitute the values obtained into the expression for \(k\): \[ k = \frac{c_{\beta} \kappa}{c_p \kappa - T v \beta^2}\] Plugging in the values: \[ k = \frac{(0.258/960)(0.0048)}{0.673(0.0048) - (960)(2)(0.258/960)^2}\]

- Simplify Expression

Simplify the expression to obtain the final value of \(k\): \[ k = \frac{(0.0002695)(0.0048)}{(0.673 \times 0.0048) - (960 \times 2 \times 0.0000283)} \= \frac{1.2936 \times 10^{-6}}{0.0032304 - 0.0542}.\] Solving the denominator and finalizing the result: \[k \approx 1.62.\]

Key concepts

thermodynamics

Thermodynamics is a branch of physics that deals with heat, work, and the forms of energy involved in physical processes. One of the key concepts in thermodynamics is the specific heat ratio, denoted by the symbol \( k \). This ratio helps in understanding the behavior of gases under different thermodynamic processes.
Specific heat refers to the amount of heat per unit mass required to raise the temperature by one degree Celsius. There are two types of specific heats: \( c_p \) (specific heat at constant pressure) and \( c_v \) (specific heat at constant volume). The specific heat ratio \( k \) is defined as \( k = \frac{c_p}{c_v} \).

In thermodynamics, we often use steam tables to find properties like specific heat, temperature, and pressure for different substances in different states. These tables are essential tools for engineers and scientists working with thermodynamic systems.

isothermal compressibility

Isothermal compressibility \( \kappa \) is a measure of the relative volume change of a fluid or solid in response to a change in pressure at a constant temperature. The formula for isothermal compressibility is:
\[ \kappa = -\frac{1}{V} \left(\frac{\partial V}{\partial P}\right)_T \]
Here, \( V \) is the volume, and \( P \) is the pressure.
Isothermal compressibility is important because it helps to understand how a substance behaves under varying pressures. For instance, if a substance has a high compressibility, it means that its volume changes significantly with small changes in pressure. This property is particularly significant in fluid dynamics and material science.
When evaluating water vapor at specific conditions, we refer to steam tables to find the corresponding value of \( \kappa \) for precise calculations, like in the provided solution where \( \kappa \) for water vapor is assumed as 0.0048 at the specified conditions.

volumetric thermal expansion coefficient

The volumetric thermal expansion coefficient \( \beta \) represents the change in volume of a material per unit change in temperature. It's mathematically expressed as:
\[ \beta = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P \]
Here, \( V \) is the initial volume, and \( T \) is the temperature. The coefficient \( \beta \) indicates how much a material expands or contracts when heated or cooled.
An essential application of \( \beta \) is in understanding material behaviors under temperature fluctuations. For example, materials with a high \( \beta \) will expand more with a small increase in temperature. This is especially important in fields like engineering and construction, where temperature changes can affect material stability and integrity.
In thermodynamics exercises, we often use steam tables to find the precise value of \( \beta \) for specific substances. For example, in our specific heat ratio evaluation, \( \beta \) for water vapor was calculated as \( 0.258/960 \) based on steam table data.

steam tables data

Steam tables are an essential resource in thermodynamics, providing vital data about the properties of water and steam at various temperatures and pressures. These tables contain information such as specific volume, specific heat, entropy, and enthalpy of water and steam.

Steam tables are particularly useful when working with problems involving phase changes and thermodynamic cycles. They allow accurate calculations by providing the necessary data under different states.
In the provided solution, steam tables were used to gather the necessary data for water vapor at \( 200 \, lbf/in^2 \) and \( 500^\circ F \). From these tables, values for specific heat \( c_p \), temperature \( T \), specific volume \( v \), isothermal compressibility \( \kappa \), and the volumetric thermal expansion coefficient \( \beta \) were obtained.
For students and professionals, steam tables are indispensable tools for making precise and accurate thermodynamic calculations, ensuring the reliability of results in real-world applications.