Write a computer program (or use a spreadsheet) to implement the solution procedure of Exercise

Write a computer program (or use a spreadsheet) to implement...

Key Concepts

Stability and Convergence Analysis

Stability and convergence are key criteria in numerical methods, ensuring that the solution approximates the true behavior of the system as the discretization parameters are refined. Stability analysis determines whether errors will dampen out over time, while convergence ensures that as mesh sizes and time steps approach zero, the numerical solution approaches the exact solution of the PDE.

Temporal Discretization (Time Step)

Time step refers to the interval at which the system state is updated in the simulation. Shorter time steps can capture the transient behaviors more accurately and enhance the stability of the solution, but similarly may result in higher computational demands. Understanding the influence of the time step length is essential for ensuring that the temporal evolution of the system is modeled accurately.

Spatial Discretization (Mesh Size)

Mesh size refers to the resolution of the spatial grid used in the numerical method. A smaller mesh size leads to a more detailed approximation of the spatial domain, which can improve the accuracy of the simulation but may also increase computational cost. The choice of mesh size is crucial in balancing precision and efficiency in numerical analysis.

Numerical Solution of Partial Differential Equations

This concept involves discretizing continuous equations, such as heat conduction equations, and solving them approximately by replacing derivatives with difference equations. It is central to modeling physical phenomena numerically, and allows one to implement and simulate the behavior of the system using computer programs or spreadsheets.

Transcript

00:01 In this problem, we're asked to write a program to solve the general case of problem 27 in the chapter, in which the states, velocities, and area are input variables, and we want to use a constant specific heat and find the diffuser, exit area, temperature, and pressure.

00:20 All right, so i didn't write, well, i did write a program, but i'm going to kind of walk through how you might do this in a real analysis.

00:32 Of a more formal analysis.

00:35 So what happened, what i did here is one of the things i did is just, you know, i have all of the equations that we need here.

00:45 So we have conservation of energy, isontropic expansion, ideal gas law for the exit and for the inlet, and conservation of mass, and again the mass flow rates in terms of, terms of the cross -sectional area, the velocity, and the inlet, i mean in the specific gravity.

01:11 So what we can see here is if we actually count up here, we have 12 different parameters that we could play with, which makes things kind of difficult.

01:21 You know, they told us to, you know, some of the ones to use, the states, the velocities and our input, and the state's velocities and area are input variables.

01:33 Well, that's, quite a lot.

01:37 So what i did is i did something that's called non -dimensionalization.

01:43 So it's just redefining variables.

01:47 And so that things are relative to one another.

01:50 And so what i said is that let's call this te hat.

01:54 The hats mean that they are dimensionless.

01:58 And let's call that, that's the exit temperature divided by the inlet temperature.

02:02 So it's the exit inlet temperature ratio.

02:06 The inlet and exit velocities, we can nonditionalize those using the square root of the specific heat at constant pressure times the inlet temperature.

02:24 We can nondimensionalize the exit pressure using the inlet pressure.

02:29 So this is the pressure ratio from the inlet to the exit or the exit to the inlet.

02:35 The specific volumes we can nondimensionalize using the inlet pressure, the inlet temperature, and the ideal gas constant.

02:45 And then the inlet exit area, we can nondimensionalize with the inlet area.

02:52 So what happens is if we, well, we would, in here, in this case, we would, you know, solve these, solve for t .e., solve for these, solve for these, solve for these, and plug them all into here.

03:04 And what happens is a lot of things drop out.

03:08 And this isn't getting these variables as kind of a trial and error, not really a trial and error thing, but you know, you do this one and you say, okay, now let's see, i need to make, i need to make, i want the tis to drop out.

03:22 So let's see here.

03:23 If i change these vs to something like this, then all the tis drop out of this equation.

03:31 And the c .p drops out of this equation.

03:34 And so in the end, what this comes down to, and again, we have this unit conversion because this is cut kilojoules in here.

03:42 So we had a unit version of a thousand, and then we multiplied through by the two, and that's why we get the 2 ,000.

03:48 So conservation of energy becomes this equation here, where, again, notice that ti is gone.

03:57 And so is cp.

03:59 And then the isontropic expansion equation, if we notice that, you know, this.

04:06 This just becomes t -hat and this just divide through here.

04:11 So we just get, we have no pi and we have no t -i in this equation anymore.

04:18 And then this equation here just becomes this.

04:25 And we can see that, in fact, we have no r in here anymore.

04:33 This one actually just says that the inlet specific volume in non -dimensional terms is just one, just because that's the way we defined it.

04:44 So this is, you know, and then these two equations, or this equation here, at least this part of it here, we wind up with this expression here, and again, this is just one.

04:58 So we can have, you see, we don't have any, we don't have the inlet temperature, we don't have the inlet pressure, we don't have cp, we don't have r, and we don't have ai anymore, the inlet area.

05:15 So we've cut down the parameters a lot.

05:19 We still have k, because k is actually itself a nondimensional parameter.

05:26 So what we have left with is k, t -e -hat, v -e -hat, v -i -hat, p -e -hat, little v -e -e -hat, and a -e -e -hat.

05:38 And that gives us seven parameters.

05:41 Instead of 12.

05:44 So by making this change of variables, we got these equations into a kind of more simpler compact form.

05:53 And we can then analyze these equations...

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