Next: Flow with pressure losses
Up: Mass Flow Rate (Number)
Previous: Mass Flow Rate (Number)
Index
To explain the material better some instructors
invented problems, which have mostly academic proposes,
(see for example, Shapiro (problem 4.5)).
While these problems have a limit applicability in reality, they have
substantial academic value and therefore presented here.
The situation where the mass flow rate per area given with one
of the stagnation properties and one of the static properties,
e.g.
and
or
T_{0} and
present difficulty for the
calculations.
The use of the regular isentropic Table is not possible
because there isn't variable represent this kind problems.
For this kind of problems a new Table was constructed and
present here^{4.5}.
The case of
T_{0} and
This case considered to be simplest case and will
first presented here.
Using energy equation (4.9) and
substituting for Mach number
results in
Rearranging equation
(
4.56) result in
And further Rearranging equation
(
4.57) transformed
it into
Equation (
4.58) is
quadratic equation for density,
ρ when all other variables
are known.
It is convenient to change it into
The only physical solution is when the density is positive and
thus the only solution is
For almost incompressible flow the density is reduced and the
familiar form of perfect gas model is seen since stagnation
temperature is approaching the static temperature for very small Mach
number (
).
In other words, the terms for the group over the underbrace
approaches zero when the flow rate (Mach number) is
very small.
It is convenient to denote a new dimensionless density as
With this new definition equation (
4.60)
is transformed into
The dimensionless density now is related to a dimensionless group
that is a function of Fn number and Mach number only!
Thus, this dimensionless group is function of Mach number only.
Thus,
Hence, the dimensionless density is
Again notice that the right hand side of equation
(
4.65) is only function of
Mach number (well, also the specific heat,
).
And the values of
were tabulated in
Table (
4.2) and Fn
is tabulated in the next Table (
4.1).
Thus, the problems is reduced to finding tabulated values.
The case of P_{0} and T
A similar problem can be described for the case of stagnation
pressure,
, and static temperature,
.
First, it is shown that the dimensionless group
is a function of Mach number
only (well, again the specific heat ratio,
also).
It can be noticed that
Thus equation (
4.66) became
The right hand side is tabulated in the ``regular'' isentropic
Table such (
4.2).
This example shows how a dimensional analysis is used to solve
a problems without actually solving any equations.
The actual solution of the equation is left as exercise (this example
under construction).
What is the legitimacy of this method?
The explanation simply based the previous experience in which
for a given ratio of area or pressure ratio (etcetera) determines the
Mach number.
Based on the same arguments, if it was shown that a group
of parameters depends only Mach number than the Mach is
determined by this group.
The method of solution for given these parameters is by calculating the
and then using the table to find the corresponding
Mach number.
The case of
&rho_{0} and T or P
The last case sometimes referred to as
the ``naughty professor's question'' case dealt here is when
the stagnation density given with the static temperature/pressure.
First, the dimensionless approach is used later analytical method
is discussed (under construction).
The last case dealt here is of the stagnation density with static
pressure and the following is dimensionless group
It was hidden in the derivations/explanations of the above
analysis didn't explicitly state under what conditions these
analysis is correct.
Unfortunately, not all the analysis valid for the same conditions
and is as the regular ``isentropic'' Table,
(4.2).
The heat/temperature part is valid for enough
adiabatic condition while the pressure condition requires also
isentropic process.
All the above conditions/situations require to have
the perfect gas model as the equation of state.
For example the first ``naughty professor'' question is sufficient
that process is adiabatic only (
T_{0},
, mass flow rate per area.).
Fliegner's number a function of Mach number
M 













0.00E+00 
1.400E06 
1.000 
0.0 
0.0 
0.0 
0.0 
0.050001 
0.070106 
1.000 
0.00747 
2.62E05 
0.00352 
0.00351 
0.10000 
0.14084 
1.000 
0.029920 
0.000424 
0.014268 
0.014197 
0.20000 
0.28677 
1.001 
0.12039 
0.00707 
0.060404 
0.059212 
0.21000 
0.30185 
1.001 
0.13284 
0.00865 
0.067111 
0.065654 
0.22000 
0.31703 
1.001 
0.14592 
0.010476 
0.074254 
0.072487 
0.23000 
0.33233 
1.002 
0.15963 
0.012593 
0.081847 
0.079722 
0.24000 
0.34775 
1.002 
0.17397 
0.015027 
0.089910 
0.087372 
0.25000 
0.36329 
1.003 
0.18896 
0.017813 
0.098460 
0.095449 
0.26000 
0.37896 
1.003 
0.20458 
0.020986 
0.10752 
0.10397 
0.27000 
0.39478 
1.003 
0.22085 
0.024585 
0.11710 
0.11294 
0.28000 
0.41073 
1.004 
0.23777 
0.028651 
0.12724 
0.12239 
0.29000 
0.42683 
1.005 
0.25535 
0.033229 
0.13796 
0.13232 
0.30000 
0.44309 
1.005 
0.27358 
0.038365 
0.14927 
0.14276 
0.31000 
0.45951 
1.006 
0.29247 
0.044110 
0.16121 
0.15372 
0.32000 
0.47609 
1.007 
0.31203 
0.050518 
0.17381 
0.16522 
0.33000 
0.49285 
1.008 
0.33226 
0.057647 
0.18709 
0.17728 
0.34000 
0.50978 
1.009 
0.35316 
0.065557 
0.20109 
0.18992 
0.35000 
0.52690 
1.011 
0.37474 
0.074314 
0.21584 
0.20316 
0.36000 
0.54422 
1.012 
0.39701 
0.083989 
0.23137 
0.21703 
0.37000 
0.56172 
1.013 
0.41997 
0.094654 
0.24773 
0.23155 
0.38000 
0.57944 
1.015 
0.44363 
0.10639 
0.26495 
0.24674 
0.39000 
0.59736 
1.017 
0.46798 
0.11928 
0.28307 
0.26264 
0.40000 
0.61550 
1.019 
0.49305 
0.13342 
0.30214 
0.27926 
0.41000 
0.63386 
1.021 
0.51882 
0.14889 
0.32220 
0.29663 
0.42000 
0.65246 
1.023 
0.54531 
0.16581 
0.34330 
0.31480 
0.43000 
0.67129 
1.026 
0.57253 
0.18428 
0.36550 
0.33378 
0.44000 
0.69036 
1.028 
0.60047 
0.20442 
0.38884 
0.35361 
0.45000 
0.70969 
1.031 
0.62915 
0.22634 
0.41338 
0.37432 
0.46000 
0.72927 
1.035 
0.65857 
0.25018 
0.43919 
0.39596 
0.47000 
0.74912 
1.038 
0.68875 
0.27608 
0.46633 
0.41855 
0.48000 
0.76924 
1.042 
0.71967 
0.30418 
0.49485 
0.44215 
0.49000 
0.78965 
1.046 
0.75136 
0.33465 
0.52485 
0.46677 
0.50000 
0.81034 
1.050 
0.78382 
0.36764 
0.55637 
0.49249 
0.51000 
0.83132 
1.055 
0.81706 
0.40333 
0.58952 
0.51932 
0.52000 
0.85261 
1.060 
0.85107 
0.44192 
0.62436 
0.54733 
0.53000 
0.87421 
1.065 
0.88588 
0.48360 
0.66098 
0.57656 
0.54000 
0.89613 
1.071 
0.92149 
0.52858 
0.69948 
0.60706 
0.55000 
0.91838 
1.077 
0.95791 
0.57709 
0.73995 
0.63889 
0.56000 
0.94096 
1.083 
0.99514 
0.62936 
0.78250 
0.67210 
0.57000 
0.96389 
1.090 
1.033 
0.68565 
0.82722 
0.70675 
0.58000 
0.98717 
1.097 
1.072 
0.74624 
0.87424 
0.74290 
0.59000 
1.011 
1.105 
1.112 
0.81139 
0.92366 
0.78062 
0.60000 
1.035 
1.113 
1.152 
0.88142 
0.97562 
0.81996 
0.61000 
1.059 
1.122 
1.194 
0.95665 
1.030 
0.86101 
0.62000 
1.084 
1.131 
1.236 
1.037 
1.088 
0.90382 
0.63000 
1.109 
1.141 
1.279 
1.124 
1.148 
0.94848 
0.64000 
1.135 
1.151 
1.323 
1.217 
1.212 
0.99507 
0.65000 
1.161 
1.162 
1.368 
1.317 
1.278 
1.044 
0.66000 
1.187 
1.173 
1.414 
1.423 
1.349 
1.094 
0.67000 
1.214 
1.185 
1.461 
1.538 
1.422 
1.147 
0.68000 
1.241 
1.198 
1.508 
1.660 
1.500 
1.202 
0.69000 
1.269 
1.211 
1.557 
1.791 
1.582 
1.260 
0.70000 
1.297 
1.225 
1.607 
1.931 
1.667 
1.320 
0.71000 
1.326 
1.240 
1.657 
2.081 
1.758 
1.382 
0.72000 
1.355 
1.255 
1.708 
2.241 
1.853 
1.448 
0.73000 
1.385 
1.271 
1.761 
2.412 
1.953 
1.516 
0.74000 
1.415 
1.288 
1.814 
2.595 
2.058 
1.587 
0.75000 
1.446 
1.305 
1.869 
2.790 
2.168 
1.661 
0.76000 
1.477 
1.324 
1.924 
2.998 
2.284 
1.738 
0.77000 
1.509 
1.343 
1.980 
3.220 
2.407 
1.819 
0.78000 
1.541 
1.362 
2.038 
3.457 
2.536 
1.903 
0.79000 
1.574 
1.383 
2.096 
3.709 
2.671 
1.991 
0.80000 
1.607 
1.405 
2.156 
3.979 
2.813 
2.082 
0.81000 
1.642 
1.427 
2.216 
4.266 
2.963 
2.177 
0.82000 
1.676 
1.450 
2.278 
4.571 
3.121 
2.277 
0.83000 
1.712 
1.474 
2.340 
4.897 
3.287 
2.381 
0.84000 
1.747 
1.500 
2.404 
5.244 
3.462 
2.489 
0.85000 
1.784 
1.526 
2.469 
5.613 
3.646 
2.602 
0.86000 
1.821 
1.553 
2.535 
6.006 
3.840 
2.720 
0.87000 
1.859 
1.581 
2.602 
6.424 
4.043 
2.842 
0.88000 
1.898 
1.610 
2.670 
6.869 
4.258 
2.971 
0.89000 
1.937 
1.640 
2.740 
7.342 
4.484 
3.104 
0.90000 
1.977 
1.671 
2.810 
7.846 
4.721 
3.244 
0.91000 
2.018 
1.703 
2.882 
8.381 
4.972 
3.389 
0.92000 
2.059 
1.736 
2.955 
8.949 
5.235 
3.541 
0.93000 
2.101 
1.771 
3.029 
9.554 
5.513 
3.699 
0.94000 
2.144 
1.806 
3.105 
10.20 
5.805 
3.865 
0.95000 
2.188 
1.843 
3.181 
10.88 
6.112 
4.037 
0.96000 
2.233 
1.881 
3.259 
11.60 
6.436 
4.217 
0.97000 
2.278 
1.920 
3.338 
12.37 
6.777 
4.404 
0.98000 
2.324 
1.961 
3.419 
13.19 
7.136 
4.600 
0.99000 
2.371 
2.003 
3.500 
14.06 
7.515 
4.804 
1.000 
2.419 
2.046 
3.583 
14.98 
7.913 
5.016 
<>
Solution
The first thing that need to be done is to find the mass flow per area
and it is
It can be noticed that the total temperature is
and
the static pressure is 1.5[Bar].
The solution is based on section equations
(4.60) through
(4.65).
It is fortunate that PottoGDC exist and it can be just plug into it
and it provide that
Isentropic Flow 
Input: mDot P T0 
k = 1.3 
M 
T/T0 
ρ/ρ0 
A/A* 
P/P0 
PAR 
F/F* 
0.171236 
0.995621 
0.985478 
3.47565 
0.981162 
3.41018 
1.53921 
The velocity can be calculated as
The stagnation pressure is

Next: Flow with pressure losses
Up: Mass Flow Rate (Number)
Previous: Mass Flow Rate (Number)
Index
Created by:Genick BarMeir, Ph.D.
On:
20071121