What-if scenario: How would the answers change if the velocity were 500 m/s? 

Time Saver: To reproduce the visual  solution, copy and paste this TEST-code on the I/O panel of the appropriate daemon, click the Load button followed by the Super-Calculate button.
 

Step 1: If the problem involves an working fluid other than a perfect gas, use the generic approach outlined in the Approach section.
 
 
 
 
 

Step 2: A Stagnation state has the same j and s as the given state.
 
 

The gas dynamics daemon only handles perfect gas as the working fluid. Treating this problem as a generic thermodynamic problem, answering the questions described in the Approach  section. This leads you to the following daemon page: HOME. Daemons. Systems. Open. SteadyState. Generic. SingleFlow. PhaseChange .

Let  State-0 and State-1 represent the stagnation and the given states.

State-1: Enter  Vel1 (250 m/s), x1 (95%),  T1 (250^oC), and  Calculate. The stagnation enthalpy, j, is calculated as 2747 kJ/kg. and p1 is 3973 kPa.

State-0: Enter  Vel0 (0 m/s), j0 ('=j1'), s0 ('=s1') and   Calculate. The stagnation properties are calculated as: T0=260^oC , p0=4678 kPa , x0=97% and rho0=24.348 kg/m3 . 
 

For the what-if study, select the ..Specific.GasDynamics daemon.  Calculate.  The stagnation properties are calculated as part of the state: T_t1=266^oC , p_t1=4512 kPa. The answers can also be obtained by using the Table Panel. Because Mach1 is calculated as 0.4414, enter M_i=0.4414 in the Table Panel to obtain p/p_t=0.88, T/T_t=0.969, A/Astar=1.477.

Time Saver: To reproduce the visual  solution, copy and paste this TEST-code on the I/O panel of the appropriate daemon, click the Load button followed by the Super-Calculate button.
 

Step 1: This problem is very similar to the earlier eXAMPLE. Use the same SingleFlow H2O daemon.
 
 
 
 

Step 2: Assume an arbitrary mass flow rate of 1 kg/s
 
 
 

Step 3: Check your answer against a different mass flow rate
 

The change in area is independent of the mass flow rate. Let us assume a mass flow rate of of 1 kg/s through the duct, obtain the answer and then see if the answer depends on the choice of the mass flow rate. We use the same daemon as in the previous eXAMPLE.

State-1: Enter  Vel1 (250 m/s), x1 (95%),  T1 (250^oC), mdot1 (1 kg/s) and Calculate. A1, is calculated as 1.91 cm^2 . 

State-2: Enter  mdot2 ('=mdot1'), j2 ('=j1'), s2 ('=s1'), Vel2 (500 m/s) and  Calculate to obtain  A2=1.509 cm^2 . This is a reduction of 21%. 

Now change the mass flow rate mdot1 to 2 kg/s and Calculate and Super-Calculate. The new areas are: A1=3.828 cm^2. and A2=3.018 cm^2., also a 21% reduction.
 

What-if scenario: How would the answer change if the gas dynamics tables were used for H2O?

Solution

This problem is very similar to the previous problem. We use the same daemon as in the previous eXAMPLE.

We will use different guesses for Vel2 until the throat area, indicated by convergence and divergence on two sides, is found.
 

Time Saver: To reproduce the visual  solution, copy and paste this TEST-code on the I/O panel of the appropriate daemon, click the Load button followed by the Super-Calculate button.
 

Step 2: Calculate State-1
 

Step 3: Assume a throat velocity and Calculate, State-2, the throat State.
 

Step 4: Iterate on the throat velocity until the minimum area is found
 
 

State-1: Enter  Vel1 (250 m/s), x1 (95%),  T1 (250^oC), mdot1 (1 kg/s) and Calculate. A1, is calculated as 1.91 cm^2

State-2: Enter  mdot2 ('=mdot1'), j2 ('=j1'), s2 ('=s1') and the first guess for Vel2 as 400 m/s.   Calculate to obtain  A2=1.511 cm^2. Now change Vel2 to 500 m/s and Calculate A2=1.509 cm^2 . Likewise for Vel2= 600 m/s  A2=1.663 cm^2 . A plot of A2 vs. Vel2 shows that the area decreases and then increases again as Vel2 increases.

Now that we have approcimately located the neck of the nozzle, we can refine the solution  as follows. Vel2= 550 m/s  A2=1.57 cm^2 . Vel2= 450 m/s  A2=1.49 cm^2 . Vel2= 475 m/s  A2=1.491 cm^2 . Vel2= 425 m/s  A2=1.495 cm^2 . The best answer can be shown to be 462 m/s.

Note that although the throat area depends on the mass flow rate, the sonic velocity can be shown, in a similar manner, to be independent of the flow rate.

For the what-if study, launch the Gas Dynamics daemon and evaluate State-1 and State-2 as described in the TEST-Codes above. For State-2, Mach2=1, T_t2=T_t1 (energy balance), p_t1=p_t2 (entropy balance), and Astar2=Astar1 (mass balance). The throat area is calculated as A2=Astar1= 1.646 cm^2. Yet another way is to go to the Table Panel after State-1 is evaluated and find A/Astar for M_i=M1=0.4414 producing  A/Astar=1.4812. Therefore, A2=Astar=A1/1.4812=2.433/1.4812=1.646 cm^2 . 

What-if scenario: How woulde the answer change if the cruising speed were increased to 350 m/s?

Once the baseline case is established, you can perform parametric studies by changing one or more variables and clicking the Calculate and Super-Calculate buttons.
 

Step 2: Calculate the States taking advantage of isentropic flow equations
 
 

Step 3: Analyze the Device
 

Step 4: Calculate and Super-Calculate 
the desired results
 
 
 
 

Step 5: For the what-if study, change Vel1, Calculate and Super-Calculate.
 

Answering the questions described in the Approach  section leads you to the appropriate daemon page:  HOME. Daemons. Systems. Open. SteadyState. Specific. GasDynamics . (..Specific.PowerCycles is also a possibility).

Let  State-1 represent the i-State and State-2 the e-State for the diffuser, and State-2 represent the i-State and State-3 the e-State for the compressor. Instead of basing the analysis on unit mass, base it on a unit mass flow rate of  1 kg/s.

State-1: Enter mdot1 (1 kg/s), p1 (54 kPa, it is not always quite accurate to assume the inlet pressure to be equal to the ambient pressure), T1(256 K), Vel1 (275 m/s) and   Calculate. Leave z1 at its default value of zero (you could change it to 32,000 fl, but it does not change between the inlet and exit). The stagnation pressure is calculated as 87 kPa . 

State-2: Enter Vel2 (=0), T_t2 ('=T_t1') (or j2=j1),  p_t2 ('=p_t1') (or s2=s1), and  mdot2 ('=mdot1'). Calculate. 

State-3: Enter Vel3 (=0),  p_t3 (700 kPa), mdot3 ('=mdot1'), s3('=s2'), and Calculate.  The stagnation temperature is calculated as 532 K. 

On the Analysis panel, load State-2 as the i-State and State-3 as the e-State of the compressor. Enter the known device variable Qdot (=0),  and T_B as 256 K (its value does not affect the calculations as the heat transfer is assumed to be zero).  A Calculate and Super-Calculate produce Wdot_ext=-240 kW. Notice that   Sdot_gen (is calculated as part of the solution to be zero, as expected for an isentropic (adiabatic and reversible) flow. 

For the what-if study, go to States panel,  choose State-1, change Vel1  to its new value (350 m/s) and Calculate . Get back to the Analysis panel and do a   Super-Calculate . All the answers are updated. The new value for Wdot_ext=-216 kW. (the power requirement dropped because of the higher stagnation pressure of the incoming air.)

In this problem, even if you assume the velocity to be non-zero at the inelt and exit of the compressor, the answers do not change. Also note that, State-2 is not necessary, as it becomes internal to the system, if one constructs a control volume around the diffuser and the compressor bundled together. 

What-if scenario:   How would the answers change if the gas were argon instead?

Solution

Answering the questions described in the Approach  section leads you to the appropriate daemon page:  HOME. Daemons. Systems. Open. SteadyState. Specific. GasDynamics . 
 

Step 2: Calculate State-1

State-1: Enter  Vel1 (250 m/s), p1 (95%)  and T1 (250^oC), and Calculate.

State-2: Enter  Mach2 ('=mdot1'), A2, T_t2 ('=T_t1'), p_t2 ('=p_t1') and  Calculate to obtain  mdot2=0.32 kg/s . 

Now change the working fluid to Ar and Super-Calculate. The new mass flow rate is calculated as 0.44 kg/s.