Question

Combustion gases with a specific heat ratio of 1.3 enter an adiabatic nozzle steadily at \(800^{\circ} \mathrm{C}\) and \(800 \mathrm{kPa}\) with a low velocity, and exit at a pressure of 85 kPa. The lowest possible temperature of combustion gases at the nozzle exit is \((a) 43^{\circ} \mathrm{C}\) \((b) 237^{\circ} \mathrm{C}\) \((c) 367^{\circ} \mathrm{C}\) \((d) 477^{\circ} \mathrm{C}\) \((e) 640^{\circ} \mathrm{C}\)

Short answer

Answer: (b) 237°C

Step-by-step solution

Adiabatic equation and parameters

The adiabatic equation is represented by the formula (assuming ideal gas behavior): \(p_1^{(\gamma - 1)/\gamma} \cdot T_1^{1 - 1/\gamma} = p_2^{(\gamma - 1)/\gamma} \cdot T_2^{1 - 1/\gamma}\), where: \(p_1\) = initial pressure (800 kPa) \(T_1\) = initial temperature (800 + 273.15 K) \(T_2\) = final temperature \(p_2\) = final pressure (85 kPa) \(\gamma\) = specific heat ratio (1.3)

Rearrange the adiabatic equation for final temperature

In order to solve for \(T_2\), we need to rearrange the adiabatic equation to isolate \(T_2\). The rearranged equation is given by: \(T_2 = \frac{p_2^{(\gamma - 1)/\gamma} \cdot T_1^{1 - 1/\gamma}}{p_1^{(\gamma - 1)/\gamma}}\) Now, we can plug in the given values to determine the final temperature.

Calculate the final temperature

Plugging in the given values into the rearranged adiabatic equation, we obtain: \(T_2 = \frac{(85 \times 10^3)^{(1.3 - 1)/1.3} \cdot (800 + 273.15)^{1 - 1/1.3}}{(800 \times 10^3)^{(1.3 - 1)/1.3}}\) After calculating this expression, we find that the temperature \(T_2\) is approximately 510.15 Kelvin.

Convert the final temperature to Celsius

Since the given answer choices are in Celsius, we need to convert the final temperature to Celsius: \(T_2^{\circ} \mathrm{C} = T_2 - 273.15\) \(T_2^{\circ} \mathrm{C} = 510.15 - 273.15 = 237^{\circ} \mathrm{C}\) The lowest possible temperature of combustion gases at the nozzle exit is \((b) 237^{\circ} \mathrm{C}\).

Key concepts

Specific Heat Ratio

The specific heat ratio, often denoted by the Greek letter \(\gamma\), is a dimensionless number that represents the ratio of the specific heat capacity at constant pressure (\(C_p\)) to the specific heat capacity at constant volume (\(C_v\)). It is a crucial parameter in the study of thermodynamics, particularly when dealing with gases, as it affects how the temperature of a gas changes as it expands or compresses.

For ideal gases, \(\gamma\) helps determine the nature of several thermodynamic processes. In an adiabatic process, for instance, where there is no heat exchange with the surroundings, the pressure and volume of the gas change in a way that is described by the specific heat ratio. Since the value of \(\gamma\) varies depending on the gas, knowing this ratio is essential for accurately predicting the behavior of a gas during processes such as expansion or compression in a nozzle, as seen in the exercise.

For monatomic gases, such as helium, \(\gamma\) typically has a value around 1.67, while for diatomic gases like nitrogen or oxygen, it's around 1.4. The specific heat ratio provided in the original exercise, 1.3, suggests we're likely dealing with a polyatomic gas, where the molecules have more complex structures and thus, different energetic properties.

Adiabatic Process

An adiabatic process is a thermodynamic process in which a system (usually a gas) does not exchange heat with its surroundings. This means that all the energy in the system is conserved, and any work done by the system will cause a change in its internal energy, resulting in a change in temperature.

During an adiabatic expansion, a gas does work on its surroundings and cools down, while in an adiabatic compression, the surroundings do work on the gas, causing it to heat up. This type of process is described by the adiabatic equation shown in the original exercise:\[p_1^{(\gamma - 1)/\gamma} \cdot T_1^{1 - 1/\gamma} = p_2^{(\gamma - 1)/\gamma} \cdot T_2^{1 - 1/\gamma}\]
Here, \(p\) stands for pressure, \(T\) for temperature, and subscripts 1 and 2 refer to the initial and final states of the gas, respectively. The absence of heat exchange makes the adiabatic process especially relevant in situations where the process happens too quickly for heat to be transferred, such as in the expansion or compression of gases in engines or nozzles.

Ideal Gas Behavior

The concept of ideal gas behavior is a simplified model that helps in understanding and predicting how gases respond to changes in pressure, volume, and temperature. According to the ideal gas law, the relationship between these variables is given by the equation \(PV = nRT\), where:\

• \(P\) is the pressure of the gas,
• \(V\) is the volume it occupies,
• \(n\) is the amount of substance in moles,
• \(R\) is the ideal gas constant, and
• \(T\) is the absolute temperature of the gas.
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Furthermore, an ideal gas is assumed to consist of perfectly elastic particles that are point-like, meaning they have no volume and do not attract or repel each other.

In practice, real gases only behave like an ideal gas under a certain range of temperatures and pressures, and deviations become notable under extreme conditions. However, for many engineering applications, such as the nozzle problem in our exercise, assuming ideal gas behavior simplifies calculations and still provides sufficiently accurate results.