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Next: The Impulse Function in Up: The Impulse Function Previous: The Impulse Function   Index

One of the functions that is used in calculating the forces is the Impulse function. The Impulse function is denoted here as , but in the literature some denote this function as . To explain the motivation for using this definition consider the calculation of the net forces that acting on section shown in Figure (4.9). To calculate the net forces acting in the x-direction the momentum equation has to be applied

The net force is denoted here as . The mass conservation also can be applied to our control volume

Combining equation (4.104) with equation (4.105) and by utilizing the identity in equation (4.42) results in

Rearranging equation (4.106) and dividing it by results in

Examining equation (4.107) shows that the right hand side is only a function of Mach number and specific heat ratio, . Hence, if the right hand side is only a function of the Mach number and than the left hand side must be function of only the same parameters, and . Defining a function that depends only on the Mach number creates the convenience for calculating the net forces acting on any device. Thus, defining the Impulse function as

In the Impulse function when ( M=1 ) is denoted as

The ratio of the Impulse function is defined as

This ratio is different only in a coefficient from the ratio defined in equation (4.107) which makes the ratio a function of and the Mach number. Hence, the net force is

To demonstrate the usefulness of the this function consider a simple situation of the flow through a converging nozzle

Solution

The solution is obtained by getting the data for the Mach number. To obtained the Mach number, the ratio of is needed to be calculated. To obtain this ratio the denominator is needed to be obtained. Utilizing Fliegner's equation (4.51), provides the following

and

Isentropic Flow Input: PAR k = 1.4
M T/T0 ρ/ρ0 A/A* P/P0 PAR F/F*
0.273534 0.985256 0.963548 2.21206 0.949342 2.1 0.966656

With the area ratio of the area ratio of at point 1 can be calculated.

And utilizing again Potto-GDC provides

Isentropic Flow Input: A/A* k = 1.4
M T/T0 ρ/ρ0 A/A* P/P0 PAR F/F*
0.111636 0.997514 0.993796 5.2227 0.991325 5.1774 2.19489

The pressure at point 1 is

The net force is obtained by utilizing equation (4.111)

Next: The Impulse Function in Up: The Impulse Function Previous: The Impulse Function   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21