Next: Supersonic Branch Up: Isothermal Flow Previous: The Entrance Limitation of Index

Comparison with Incompressible Flow

The Mach number of the flow in some instances is relatively small. In these cases, one should expect that the isothermal flow should have similar characteristics as incompressible flow. For incompressible flow, the pressure loss is expressed as follows

$\displaystyle P_{1} - P_{2} = \frac{4fL}{D} { U ^{2} \over 2}$

(8.39)

Now note that for incompressible flow $U_{1} = U_{2}= U$ and $\frac{4fL}{D}$ represent the ratio of the traditional $h_{12}$ . To obtain a similar expression for isothermal flow, a relationship between $M_{2}$ and $M_{1}$ and pressures has to be derived. From equation (8.39) one can obtained that

$\displaystyle M_{2} = M_{1} {P_{1} \over P_{2}}$

(8.40)

Substituting this expression into (8.40) yields

$\displaystyle \frac{4fL}{D} = {1 \over k{M_{1}}^{2}} \left( 1 - \left( {P_{2} \over P_{1}} \right)^{2} \right) - \ln \left( {P_{2} \over P_{1}} \right)^{2}$

(8.41)

Because

is always positive there is only one solution to the above equation even though M2.

Expanding the solution for small pressure ratio drop, $P_{1} - P_{2} /P_{1}$ , by some mathematics.

denote

$\displaystyle \chi = {P_{1} -P_{2} \over P _{1} }$

(8.42)

Now equation (8.41) can be transformed into

$\displaystyle \frac{4fL}{D} = { 1 \over k {M_{1}}^{2}} \left( 1 - \left( { {P_{... ...}} \right)^{2} \right) -\ln \left( { 1 \over { P_{2} \over P_{1} }} \right)^{2}$

(8.43)

$\displaystyle \frac{4fL}{D} = { 1 \over k {M_{1}}^{2}} \left( 1 - \left( { 1 -\chi } \right)^{2} \right) -\ln \left( { 1 \over 1 -\chi } \right)^{2}$

(8.44)

$\displaystyle \frac{4fL}{D} = { 1 \over k {M_{1}}^{2}} \left( 2\chi - \chi^{2} \right) -\ln \left( { 1 \over 1 -\chi } \right)^{2}$

(8.45)

now we have to expand into a series around $\chi=0$ and remember that

$\displaystyle f(x) = f(0) + f'(0) x + f''(0) {x^2 \over 2} + 0\left(x^{3}\right)$

(8.46)

and for example the first derivative of

$\displaystyle \left. {d \over d\chi } \ln \left( { 1 \over 1 -\chi } \right)^{2} \right\vert _{\chi= 0} =$

$\displaystyle \left. \left( 1 - \chi \right)^{2} \times {\left[ (-2) (1-\chi)^{-3} \right] } {(-1)} \right\vert _{\chi=0} = 2$

(8.47)

similarly it can be shown that $f''(\chi=0)=1$ equation (8.45) now can be approximated as

$\displaystyle \frac{4fL}{D} = {1 \over k{M_{1}}^{2}} (2\chi - \chi^{2} ) - \left( 2 \chi - \chi^{2}\right) + f \left( \chi^{3} \right)$

(8.48)

rearranging equation (8.48) yields

$\displaystyle \frac{4fL}{D} = {\chi \over k{M_{1}}^{2}} \left[ (2 - \chi ) - k{M_{1}}^{2} \left( 2 - \chi \right) \right] + f \left( \chi^{3} \right)$

(8.49)

and further rearrangement yields

$\displaystyle \frac{4fL}{D} = {\chi \over k{M_{1}}^{2}} \left[ 2(1 - k{M_{1}}^{2} ) - \left( 1 + k{M_{1}}^{2} \right) \chi \right] + f \left( \chi^{3} \right)$

(8.50)

in cases that $\chi$ is small

$\displaystyle \frac{4fL}{D} \approx {\chi \over k{M_{1}}^{2}} \left[ 2(1 - k{M_{1}}^{2} ) - \left( 1 + k{M_{1}}^{2} \right) \chi \right]$

(8.51)

The pressure difference can be plotted as a function of the $M_{1}$ for given value of $\frac{4fL}{D}$ . Equation (8.51) can be solved explicitly to produce a solution for

$\displaystyle \chi = { 1 -k{M_{1}}^{2} \over 1 +k{M_{1}}^{2}} - \sqrt{{ 1 -k{M_... ...} \over 1 +k{M_{1}}^{2}} - {k{M_{1}}^{2} \over 1 +k{M_{1}}^{2} } \frac{4fL}{D}}$

(8.52)

A few observations can be made about equation (8.52).

Next: Supersonic Branch Up: Isothermal Flow Previous: The Entrance Limitation of Index

Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21

If you want the whole book or parts in pdf or other formats, then click here. You also can get the best and the largest gas dynamics tables in the world.

About Potto Project

Potto Project has been created by Dr. Genick Bar-Meir and friends to build free software and textbooks for college students.

Potto Project is under open content licenses, which means that you will always have the freedom to use it, make copies of it, and improve it. You are encouraged to make use of these freedoms and share the textbooks and program with your family and friends!