More Examples for Moving Shocks

Figure 5.24: The results for Example (5.12)

Figure 5.23: Figure for example (5.12)

Solution

This problem is known as the suddenly open valve problem in which the shock choking phenomenon occurs. The time it takes for the shock to travel from the valve depends of the pressure ratio ${P_y \over P_x} = 30$

$\rule[-0.1in]{0.pt}{0.3 in}\mathbf{M_x}$	$\mathbf{M_y}$	$\mathbf{{M_x}^{'}}$	$\mathbf{{M_y}^{'}}$	$\mathbf{{T_y} \over {T_x} }$	$\mathbf{{P_y} \over {P_x}}$	$\mathbf{{P_0}_y \over {P_0}_x }$
5.0850	0.41404	0.0	1.668	5.967	30.00	0.057811

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The direct calculation will be by using the ``upstream'' Mach number, $M_x = M_{sx} = 5.0850$ . Therefore, the time is

$$t = {distance \over M_{sx} \sqrt{kRT_x}} = { 3 \over 5.0850 sqrt{1.4 \times 287 \times 300}} \sim 0.0017 [sec]$$

The mass flow rate after reaching the exit under these assumptions remains constant until the uncooled material reaches the exit. The time it takes the material from the valve to reach the exit is

$$t = {distance \over {M_y}^{'} \sqrt{kRT_y}} = { 3 \over 1.668 sqrt{1.4 \times 287 \times 300 \times 5.967}} \sim 0.0021 [sec]$$

Figure: The results for Example (5.12)

During that difference of time the material is get heated instead of cold to a high temperature. The suggestion of the engineer to increase the pressure will decrease the time but will increase the temperature at the exit during this critical time period. Thus, this suggestion contradicts the purpose of required the manufacturing needs.

To increase the pipe diameter will not change the temperature and therefore will not change the effects of heating. It can only increase the rate after the initial heating spike.

A possible solution is to have the valve very close to the pipe exit. Thus, the heating time is reduced significantly. There is also the possibility of steps increase in which every step heat released will not be enough to over heat the device. The last possible requirement programmable value and very fast which its valve probably exceed the moving shock the valve downstream. The plot of the mass flow rate and the velocity are given in Figure (5.24).

Solution

To be developed.