Question

Combustion gases with \(k=1.33\) enter a converging nozzle at stagnation temperature and pressure of \(350^{\circ} \mathrm{C}\) and \(400 \mathrm{kPa},\) and are discharged into the atmospheric air at \(20^{\circ} \mathrm{C}\) and \(100 \mathrm{kPa}\). The lowest pressure that will occur within the nozzle is \((a) 13 \mathrm{kPa}\) \((b) 100 \mathrm{kPa}\) \((c) 216 \mathrm{kPa}\) \((d) 290 \mathrm{kPa}\) \((e) 315 \mathrm{kPa}\)

Short answer

Based on the given inlet and exit conditions and the specific heat ratio (k), the lowest pressure that occurs within the converging nozzle is the critical pressure (P*), which is found to be 216 kPa.

Step-by-step solution

Convert given temperatures to Kelvin

To work with absolute temperatures, we will convert the given temperatures from Celsius to Kelvin. T0 = 350°C + 273.15 = 623.15 K Te = 20°C + 273.15 = 293.15 K

Find the critical pressure (P*)

In isentropic flow through a nozzle, the pressure at which the maximum flow velocity is achieved is called the critical pressure (P*). We can determine this pressure using the following relation for isentropic flow: \(\frac{P^{*}}{P_{0}}=\left(\frac{2}{k+1}\right)^{k / (k-1)}\) Plugging in the values of k and P0, we get: P* = P0 * (2 / (k+1))^(k / (k-1)) P* = 400 kPa * (2 / (1.33+1))^(1.33 / (1.33-1)) P* ≈ 216 kPa

Compare critical pressure (P*) to the exit pressure (Pe)

Since we are asked to find the lowest pressure that occurs within the nozzle, we would need to compare the critical pressure (P*) to the exit pressure (Pe) to determine the lowest pressure. If the exit pressure is equal to or greater than the critical pressure, the pressure within the nozzle is never lower than the exit pressure. However, if the exit pressure is lower than the critical pressure, the lowest pressure will be the critical pressure. Pe = 100 kPa P* = 216 kPa Since Pe < P*, the lowest pressure that occurs within the nozzle is the critical pressure (P*). Thus, the correct answer is (c) 216 kPa.

Key concepts

Isentropic Flow

Isentropic flow pertains to a condition in thermodynamics where a fluid or gas flow happens in a manner that maintains constant entropy. In simpler terms, it implies a reversible and adiabatic flow with no heat transfer or friction, ideally preserving the total entropic value.

For gases moving through nozzles, such as in the combustion case from our exercise, isentropic flow can be useful for predicting changes in various properties like temperature, pressure, and density. The equation \( P^{*} = P_0 \left(\frac{2}{k+1}\right)^{\frac{k}{k-1}} \) given in the solution utilizes this concept to determine the gas's critical pressure, assuming that the process is isentropic.

One of the hallmarks of this flow dynamic is that the speed of sound in the medium, and hence the Mach number, play significant roles. Subsonic to supersonic transitions occur within these types of flow regimes, and the isentropic relations can predict conditions at which such changes happen.

Critical Pressure

Critical pressure, often symbolized as \( P^{*} \), is a key concept in the study of fluid dynamics within nozzles. It represents the pressure at which the flow speed reaches the speed of sound, known as Mach 1. Beyond this critical point, the flow can potentially transition from subsonic to supersonic speeds.

In our exercise, the critical pressure was calculated for gases within a nozzle using the isentropic flow relation. The formula used takes into consideration the heat capacity ratio (\( k \))—a property of the gas—which affects how the pressure drops throughout the nozzle.

Understanding the relationship between critical and exit pressures is vital. If the exit pressure is below the critical pressure, it suggests that the flow within the nozzle experiences a choked condition, where the mass flow rate is maximized and cannot increase further unless the stagnation conditions change. In our exercise, the critical pressure was found to be 216 kPa, which became the lowest pressure in the nozzle since it was lower than the atmospheric exit pressure.

Stagnation Temperature and Pressure

The concepts of stagnation temperature and pressure are fundamental in thermodynamics and fluid mechanics, particularly in the analysis of flow through nozzles and diffusers.

Stagnation temperature, referred to as \( T_0 \), is the temperature a flowing gas or fluid would attain if it were brought to a standstill isentropically. In other words, it's the maximum temperature the fluid would have if all its kinetic energy were converted into thermal energy. In the context of our exercise, the stagnation temperature was given as 350°C, which was then converted to Kelvin for calculations.

Similarly, stagnation pressure (\( P_0 \)) is the pressure a fluid would have if it were isentropically decelerated to zero velocity. It represents the maximum pressure achievable and is essential for calculating the critical pressure during isentropic flow through nozzles as seen with our 400 kPa initial condition.

These concepts help describe the total energy content of a flowing fluid, providing a useful simplification in calculating the flow properties when a fluid is slowed down or stopped without losing energy through heat transfer or frictional dissipation.