Problem 29 - An Introduction to Thermal Physics

Question

Liquid HFC-134a at its boiling point at 12 bars pressure is throttled to 1 bar pressure. What is the final temperature? What fraction of the liquid vaporizes? $$\begin{array}{rrrrrr}\begin{array}{r}P \\\\(\text { bar })\end{array} & \begin{array}{r}T \\\\\left(^{\circ} \mathrm{C}\right)\end{array} & \begin{array}{r}H_{\text {liquid }} \\\\(\mathrm{kJ})\end{array} & \begin{array}{r} H_{\text {gas }} \\\\(\mathrm{kJ})\end{array} & \begin{array}{r}S_{\text {liquid }} \\\\(\mathrm{kJ} / \mathrm{K}) \end{array} & \begin{array}{r}S_{\text {gas }} \\\\(\mathrm{kJ} / \mathrm{K})\end{array} \\\\\hline 1.0 & -26.4 & 16 & 231 & 0.068 & 0.940 \\\1.4 & -18.8 & 26 & 236 & 0.106 & 0.932 \\\2.0 & -10.1 & 37 & 241 & 0.148 & 0.925 \\ 4.0 & 8.9 & 62 & 252 & 0.240 & 0.915 \\\6.0 & 21.6 & 79 & 259 & 0.300 & 0.910 \\\8.0 & 31.3 & 93 & 264 & 0.346 & 0.907 \\\10.0 & 39.4 & 105 & 268 & 0.384 & 0.904 \\\12.0 & 46.3 & 116 & 271 & 0.416 & 0.902 \end{array}$$ Properties of the refrigerant HFC-134a under saturated conditions (at its boiling point for each pressure). All values are for 1 kg of fluid, and are measured relative to an arbitrarily chosen reference state, the saturated liquid at \(-40^{\circ} \mathrm{C} .\) Excerpted from Moran and Shapiro (1995). Properties of superheated (gaseous) refrigerant HFC-134a. All values are for \(1 \mathrm{kg}\) of fluid, and are measured relative to the same reference state as in Table 4.3. Excerpted from Moran and Shapiro (1995).

Step-by-Step Solution

Answer

The final temperature is -26.4°C, and 46.5% of the liquid vaporizes.

Step 1: Understand the Throttling Process

Throttling is an isenthalpic process, meaning the enthalpy before and after the throttling remains constant. Therefore, the enthalpy at 12 bars for the liquid state is the same as the final enthalpy at 1 bar.

Step 2: Identify Initial Conditions

At 12 bars, the liquid HFC-134a is at its boiling point with an enthalpy, denoted as \(H_{liquid, 12 \, ext{bar}} = 116 \text{ kJ/kg}\). This value will remain constant after throttling.

Step 3: Locate Enthalpy at 1 bar

The enthalpy just before throttling, \(116 \text{ kJ/kg}\), is used to find the state of the fluid after throttling. At 1 bar, check the enthalpy table for values close around 116 kJ/kg. The liquid at 1 bar has \(H_{liquid, 1 \, ext{bar}} = 16 \text{ kJ/kg}\) and the gas has \(H_{gas, 1 \, ext{bar}} = 231 \text{ kJ/kg}\).

Step 4: Apply Quality Factor Formula

In the two-phase region, the enthalpy is given by \(H = x H_{gas} + (1-x) H_{liquid}\), where \(x\) is the quality or mass fraction of vapor. Solve for \(x\) using:\[116 = x (231) + (1-x) (16)\]

Step 5: Solve Quality Equation

Re-arrange and solve the equation from Step 4 to find \(x\):\[116 = 231x + 16 - 16x\]Solving, we find:\[x = \frac{116 - 16}{231 - 16} = \frac{100}{215} \approx 0.465\]

Step 6: Determine Final Temperature

Since the fluid is two-phase after throttling, the temperature corresponds to the saturated temperature at 1 bar, which is \(-26.4^{\circ} C\).

Key Concepts

Throttling Process

In thermodynamics, the throttling process is an essential concept often used in refrigeration and air conditioning systems. It is characterized by an adiabatic process where a fluid passes through a restriction, such as a valve or porous plug, without exchanging heat with its surroundings. During throttling, the enthalpy of the fluid remains constant. This means that the energy in the form of enthalpy before and after the process is unchanged, even though other properties like pressure and temperature may vary.
This isenthalpic property makes throttling valuable for processes that involve flow rate regulation without additional energy input or output. Throttling devices are typically used to lower the pressure of a fluid while controlling its flow rate. Although there is no work done or heat transfer, some kinetic energy is converted efficiently into work, enabling fluid regulation and cooling in practical applications.

Enthalpy

Enthalpy is a key concept in thermodynamics, particularly when analyzing energy changes during various processes like the throttling in this exercise. It is a measure of the total energy content of a system, taking into account internal energy and the energy required for the system to expand against its surroundings—captured by the equation: \[ H = U + PV \] where \( H \) is enthalpy, \( U \) is internal energy, \( P \) is pressure, and \( V \) is volume.
During throttling, the enthalpy remains constant, highlighting how no heat is exchanged, and work is not produced, even as the fluid undergoes pressure changes. By examining the enthalpy values before and after throttling, one can determine other parameters, such as the phase or temperature of the fluid after throttling.

Saturation Temperature

Saturation temperature is the temperature at which a liquid boils and turns into vapor at a given pressure, or conversely, where vapor condenses into liquid. For a given substance, this temperature changes with pressure. In a saturated state, both liquid and vapor phases coexist in equilibrium.
The saturation temperature is critical in processes like throttling because it helps determine the conditions at which phase change—boiling or condensation—can occur. Knowing the saturation temperature allows you, for example, to determine the temperature after throttling at a new lower pressure. In this exercise, the fluid reaches a two-phase state post-throttling, where the saturation temperature of \(-26.4 ^{\circ} C\) at 1 bar indicates the fluid's boiling point at reduced pressure.

Vapor-Liquid Equilibrium

Vapor-liquid equilibrium (VLE) is a critical concept when examining systems undergoing phase changes, such as in refrigeration cycles. VLE occurs when the rates of evaporation and condensation are equal, allowing both liquid and vapor phases to exist simultaneously in equilibrium. This equilibrium depends on specific pressure and temperature conditions.
In the context of throttling, VLE plays a significant role because the process often brings a fluid to a state where it exists as a mixture of both liquid and vapor—known as the two-phase region. Understanding VLE allows us to determine the properties of mixtures, such as those found after a throttling process, by using metrics like the quality factor \( x \), which represents the fraction of the mass that is vapor. In this exercise, calculating the quality \( x \) helps assess how much of the liquid has vaporized during the throttling to 1 bar pressure.