Carbon dioxide enters an adiabatic nozzle at 8 MPa and 450 K with a low velocity and leaves at 2

Carbon dioxide enters an adiabatic nozzle at 8 MPa and 450 K...

Carbon dioxide enters an adiabatic nozzle at $8 \mathrm{MPa}$ and $450 \mathrm{K}$ with a low velocity and leaves at $2 \mathrm{MPa}$ and $350 \mathrm{K}$ Using the generalized enthalpy departure chart, determine the exit velocity of the carbon dioxide.

Key Concepts

Adiabatic Process

An adiabatic process is one in which there is no heat transfer between the system and its surroundings. In the context of nozzle flows, this means that the energy exchange is solely through work interactions (like acceleration) and changes in the fluid's internal energy, simplifying the energy balance for the flow.

Steady-Flow Energy Equation

The steady-flow energy equation relates the energy content of a fluid entering and exiting a device such as a nozzle. For an adiabatic nozzle, this equation shows that the drop in enthalpy of the fluid is converted into an increase in kinetic energy, which is used to determine the exit velocity.

Generalized Enthalpy Departure

Generalized enthalpy departure refers to the correction applied to the ideal gas enthalpy to account for real gas behavior. Departure charts are used for substances like carbon dioxide under high-pressure conditions, providing a way to adjust the thermodynamic properties based on deviations from ideality.

Conversion of Enthalpy Change to Kinetic Energy

In nozzle flow, the decrease in enthalpy, calculated using the departure chart for real fluids, is directly related to the increase in the fluid's kinetic energy. This conversion is fundamental for determining the exit velocity, particularly when the inlet velocity is negligible.

Transcript

00:01 For this problem on the topic of thermodynamic property relations, we are told that carbon dioxide enters an adiabatic nozzle at a given state with a low velocity and leaves at a different state.

00:12 We want to use the generalized enthalpy departure charts to determine the exit velocity of the carbon dioxide.

00:19 So we assume that steady operating conditions exist, kinetic and potential energy changes are negligible, and the nozzle's adiabatic and so heat transfers also negligible.

00:30 Now we have to first write the steady balance, steady flow energy balance equation for this process.

00:38 And the rate of change of energy in minus the rate of change of energy out of the nozzle.

00:46 So e.

00:47 Dot in minus e.

00:48 Dot out is equal to the change in the rate of energy flow.

00:54 That's delta e.

00:55 Dot for the system.

00:57 But we know that the system is at a steady state, so this change in rate of flow of energy is zero.

01:05 This implies that the rate at which energy is entering the nozzle e.

01:11 Dot in is equal to the rate at which energy exits the nozzle, e.

01:16 Dot out.

01:19 And we can write this in terms of the enthalpy h1 into the system plus the change in kinetic energy, which is v1 squared over 2, where v1 is the inlet velocity, and this is equal to the final enthalpy, enthalpy at the outlet of the nozzle h2, plus the exit velocity v2, alt squared over 2, which is the exit kinetic energy.

01:52 So we can rearrange this equation, and we can find the exit velocity of the nozzle, assuming that the inlet velocity is very low.

02:00 So v2 is equal to the square root of 2 into h1 minus h2.

02:08 So that's our simplified energy balance equation.

02:13 Now we need to find the enthalpy departures of co2 at these specified states from the generalized enthalpy departure chart.

02:20 So we'll first find the reduced temperature of the inlet state, tr1...

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