91Ó°ÊÓ

Liquid water at 60 bar and \(250^{\circ} \mathrm{C}\) passes through an adiabatic expansion valve, emerging at a pressure \(P_{\mathrm{f}}\) and temperature \(T_{\mathrm{f}} .\) If \(P_{\mathrm{f}}\) is low enough, some of the liquid evaporates. (a) If \(P_{\mathrm{f}}=1.0\) bar, determine the temperature of the final mixture \(\left(T_{\mathrm{f}}\right)\) and the fraction of the liquid feed that evaporates \(\left(y_{\mathrm{v}}\right)\) by writing an energy balance about the valve and neglecting \(\Delta \dot{E}_{\mathrm{k}}\) (b) If you took \(\Delta \dot{E}_{\mathrm{k}}\) into account in Part (a), how would the calculated outlet temperature compare with the value you determined? What about the calculated value of \(y_{\mathrm{v}} ?\) Explain. (c) What is the value of \(P_{\mathrm{f}}\) above which no evaporation would occur? (d) Sketch the shapes of plots of \(T_{\mathrm{f}}\) versus \(P_{\mathrm{f}}\) and \(y_{\mathrm{v}}\) versus \(P_{\mathrm{f}}\) for 1 bar \(\leq P_{\mathrm{f}} \leq 60\) bar. Briefly explain your reasoning.

Step by step solution

01

Apply The Energy Balance Principle

Since it is given that the part (a) of the problem neglects the kinetic energy change \(\Delta \dot{E}_{\mathrm{k}}\), start with applying the first law of thermodynamics for an open system considering the initial and final states: \( \Delta \dot{E_{\mathrm{u}}} + \Delta \dot{E_{\mathrm{p}}} + \Delta \dot{E_{\mathrm{k}}} = 0 \). This simplifies to \( \Delta h = 0 \) for the isenthalpic process since \( \Delta \dot{E_{\mathrm{k}}} \) is neglected and the system is adiabatic.

02

Find The Final Temperature

Now, knowing the initial properties of water and using the steam tables, we can find the total enthalpy at the inlet. We can equate this to the enthalpy of the mixture at the outlet to find the final temperature \(T_{\mathrm{f}}\). For this, it's also important to note that enthalpy is a function of both temperature and pressure.

03

Calculate The Evaporated Fraction

For calculating the evaporated fraction \(y_{\mathrm{v}}\), we can use the specific enthalpies of the liquid and vapor phases obtained from the steam tables and set up a mass balance around the valve. The fraction can be obtained by dividing the mass of the vapor by the total mass.

04

Analyse The Effect Of Kinetic Energy

In part (b), we need to discuss the possible effect of taking into account \(\Delta \dot{E_{\mathrm{k}}\). As the valve is a restriction to the flow, we can expect an increase in velocity, hence, an increase in kinetic energy after expansion. This would mean a decrease in internal energy and a lower \(T_{\mathrm{f}}\) and \(y_{\mathrm{v}}\).

05

Find The Critical Pressure

To find the value of \(P_{\mathrm{f}}\) at which no evaporation would occur, we need to find the pressure at which inlet enthalpy corresponds to a saturated liquid. Above this pressure, water remains in a liquid phase due to the higher boiling point.

06

Interpret The Plots

To sketch the plots of \(T_{\mathrm{f}}\) versus \(P_{\mathrm{f}}\) and \(y_{\mathrm{v}}\) versus \(P_{\mathrm{f}}\), it's important to understand that \(T_{\mathrm{f}}\) decreases with an increase in \(P_{\mathrm{f}}\), while \(y_{\mathrm{v}}\) increases with the decrease in \(P_{\mathrm{f}}\). The values of \(P_{\mathrm{f}}\) above which no evaporation occurs will be the points where \(T_{\mathrm{f}}\) remains constant and \(y_{\mathrm{v}}\) becomes zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Adiabatic expansion

Adiabatic expansion occurs when a fluid such as liquid water, passes through a valve or some other restriction without any heat exchange with its surroundings. This means that during the expansion process, the system neither gains nor loses heat.

In thermodynamics, this is classified under the isenthalpic process, where the enthalpy before and after expansion remains constant (\( \Delta h = 0 \)). Here, as water expands from 60 bar to 1 bar pressure, the absence of external heat exchange results in some of the initial internal energy being converted into work done in expansion instead.

This can lead to a temperature drop in the fluid, since as pressure decreases, the capacity for kinetic motion and thus thermal energy also may change. This particular exercise focuses on how expansion might lead to partial evaporation due to pressure decrease, potentially changing some of the water from liquid to vapor phase.

Energy balance

Energy balance in thermodynamics is a crucial principle that applies to any system undergoing change. In our exercise, a simplified energy balance is applied, which serves to assess what's happening inside the adiabatic expansion valve.

Given that the exercise deals with an adiabatic process without kinetic energy consideration (\( \Delta \dot{E}_{k} \) is neglected), the energy equation boils down to maintaining constant enthalpy (\( \Delta h = 0 \)). This means that whatever energy is residing in the liquid water at the initial state must equal the sum of energies in the resulting mix of phases after passing through the valve.

This equilibrium allows us to predict the temperature and degree of vaporization in the output state. Such an analysis shows how one can deduce both the temperature change and the evaporation fraction using principles of energy conservation effectively.

Phase change

Phase change is a fascinating thermodynamic process, especially when substances like water change from liquid to vapor. In this exercise, the phase change occurs during adiabatic expansion when water at high pressure passes through a valve, dropping significantly in pressure.

At a lower pressure of 1 bar, as seen in the exercise, some of the liquid water in this condition evaporates. This transition happens because the water reaches its boiling point and changes into vapor, a process that requires a certain amount of energy known as the latent heat of vaporization. This energy requirement can cause the water temperature to drop, leading to partial evaporation.

The fraction of liquid that evaporates (\( y_{v} \)) can be calculated through mass balance, where the amount of initial mass lost as vapor is taken from the initial mass, indicative of the phase change extent.

Steam tables

Steam tables are an essential tool in thermodynamics used widely to find properties of water and steam under different conditions of pressure and temperature. They become particularly useful when dealing with the energy calculations in phase changes.

In this problem, the steam tables help identify the initial enthalpy of water at 60 bar and 250°C and help find the enthalpies at the lower pressure and mixed state.

By referring to steam tables, you can find values such as specific enthalpy and entropy, which permit accurate determination of temperature (\( T_{f} \)) and evaporation fraction (\( y_{v} \)) at the final state after expansion. These tables thus play a crucial role in the energy balance calculations and help accurately gauge phase transitions with changing pressure.