Quasi-one-dimensional flow A=A(x), p=p(x), r=r(x), T=T(x), M=M(x) See the new Atlas 5 RD-180 Rocket Motor Analysis |
Nozzle 3.7
New Version (3.7.0.5) INSTRUCTION MANUAL A DeLaval Nozzle Analysis Program for Microsoft Windows By AeroRocket Purchase | MAIN PAGE | PRODUCTS | CONSULTING | MISSION | RESUME | Copyright © 1999-2014 John Cipolla/AeroRocket |
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Nozzle Instructions |
2-D Plume Instructions |
Turbulent Free Jet Instructions |
Nozzle 3.7 is a one-dimensional and two-dimensional, compressible flow computer program for the analysis of converging-diverging
nozzles. Nozzle models inviscid, adiabatic and hence isentropic flow of a
calorically perfect gas through variable-area ducts. Nozzle internal flow may be entirely subsonic, entirely
supersonic or a combination of subsonic and supersonic including
shock waves in the diverging part of the nozzle. Shock waves are
clearly identified as vertical red lines on all plots. The cross-sectional
shape in the axial direction of the nozzle is specified by selecting
from five standard nozzle types or by defining nozzle geometry
using the Free-Form nozzle geometry method. Nozzle plots color
contours of pressure ratio, temperature ratio, density ratio,
and Mach number and has a slider bar that displays real-time values
of all nozzle flow properties. New in this version is the ability
to determine shock-angle, jet-angle (plume-angle) and Mach number
for axisymmetric and two-dimensional nozzles in the region near
the lip for underexpanded and overexpanded flow. The plume analysis capability
has been greatly enhanced in the new version of
Nozzle 3.7.0.4.
NOZZLE
INSTRUCTIONS Back To Top
Summary of Features
1) Specify nozzle geometry as either Parabolic, Conical, Bell,
Imported, or Free-Form. Free-Form nozzle shapes may use up to
30 points to define nozzle geometry.
2) Standard and Import nozzle shapes may have up to 1000 axial
points defining the cross-sectional area distribution of the nozzle.
3) Select either the classical isentropic and normal-shock relations
method or the MacCormack backward-predictor forward-corrector
finite difference method to determine characteristics of nozzle
internal flow.
4) Locate internal shock waves quickly using the slider bar that
displays nozzle property verses axial location in real time.
5) Determine Mach number (V/c), pressure ratio (P/P0), density
ratio (R/R0) and temperature ratio (T/T0) at each axial location
in the nozzle.
6) Determine shock wave location, Mach number before the shock
wave, Mach number after the shock wave and nozzle area at the
shock wave location.
7) Specify fluid properties for a large number of inert gases,
liquid fluid rocket propellants and solid fuel rocket propellants
or specify your own.
8) Specify the units of analysis as MKS (meter-newton-seconds),
CGS (centimeter-newton-seconds), IPS (inch-pound-seconds) and
FPS (foot-pound-seconds).
9) Enlarge all plots for easy data reduction and output all data
and plots to a color printer.
10) Print Detailed Results to a printer and Save Data File As for
displaying X, R coordinates and flow properties in CSV format for spreadsheet
display and review.
11) Fast solution - most analyses completed in less than 15 seconds.
12) Generate color contour plots for Mach number (Mn), Pressure
ratio (P/P0), Temperature ratio (T/T0), and density ratio (R/R0).
13) Determine
exterior flow properties in the nozzle-lip region for underexpanded
nozzles and overexpanded nozzles.
14) Added a hybrid rocket
motor propellant having the following fuel and oxidizer to the list of
combustion gases: 85% Nitrous Oxide, 15% HTPB.
15)
Made the SSME example
(shown below) the start-up data for Nozzle program analyses. Data easily cleared
for new data entries.
16)
Two-dimensional plume analysis using the method of characteristics for underexpanded (Patm
< Pexit)
flow.
17) Nozzle_Examples.zip in the Nozzle directory includes 34 nozzle
examples used for validation purposes.
18)
Design Conditions routine for those who wish to quickly design
subsonic/supersonic wind tunnels or efficient every-day nozzles
19) Added Turbulent Circular and Turbulent 2-D Free Jet analysis
capability based on theory from Viscous Fluid Flow by Frank M. White
20) NEW!
Display Conical nozzle geometry in Computer Aided Design (CAD) formatted
LINES and CIRCLES for generating imported shapes using the
Imported shape option command. Accessed by clicking File then CAD
Input For Conical Shapes then SHOW NOZZLE CAD. Use these LINE
and CIRCLE values in any CAD program to generate the text file required
to generate a Nozzle 3.7 import geometry file.
21) NEW!
For overexpanded
nozzles, the value for pressure ratio (Pa/P3), Mach number (M3) and oblique
shock diameter at point-2 are inserted into the 2-D Plume Analysis
and
the Method of Characteristics results are automatically displayed starting at
the end of the external oblique shock wave pattern as illustrated in Figure-16.
Previously, the plume analysis did not compute the expansion wave pattern for
overexpanded nozzles. For underexpanded nozzles the value for pressure ratio (Pa/Pe),
exit Mach number (Me) and nozzle exit diameter (De) are inserted and plume
results automatically displayed from nozzle exit to several diameters downstream
as in previous versions of Nozzle.
Inert Gases |
|||||
Dry Air | Hydrogen | Helium | Water Vapor | Argon | Carbon Dioxide |
Carbon Monoxide | Nitrogen | Oxygen | Nitrogen Monoxide | Nitrous Oxide | Chlorine |
Methane | |||||
Liquid Fuel Propellant Gases |
|||||
Oxygen, 75% Ethyl Alcohol(1.43) | Oxygen, Hydrazine(.09) | Oxygen, Hydrogen(4.02) | |||
Oxygen, RP-1(2.56) | Oxygen, UDMH(1.65) | Fluorine, Hydrazine(2.3) | |||
Fluorine, Hydrogen(7.60) | Nitrogen Tetroxide, Hydrazine(1.34) | Nitrogen Tetroxide, 50% UDMH, 50% Hydrazine(2.0) | |||
Nitric Acid, RP-1(4.8) | Nitric Acid, 50% UDMH, 50% Hydrazine(2.20) | ||||
Solid Fuel Propellant Gases |
|||||
Ammonium Nitrate, 11% Binder, 4-20% Mg | Ammonium Perchlorate, 18% Binder, 4-20% Al | Ammonium Perchlorate, 12% Binder, 4-20% Al | |||
Hybrid Rocket Motor Propellant Gases |
|||||
85% Nitrous Oxide, 15% HTPB | |||||
User-Defined Gases |
|||||
Specify custom or user-defined gases by inserting Ratio of specific heats (Cp/Cv) and Gas constant (R=Cp-Cv) in the nozzle data entry section. |
A) BACKGROUND THEORY - NOZZLE INTERNAL FLOW
As the exit back pressure,
Pe is reduced below Po, flow through the nozzle begins. If Pe
is only slightly less than Po, the flow throughout the nozzle
is subsonic and the pressure profile along the axis would be like
curve A in Figure 1. Reducing Pe increases the mass flow rate
through the nozzle. As the flow rate increases, the pressure at
the throat decreases until it reaches the critical pressure as
indicated by curve B (PCRIT-1). The exit pressure Pe which
exactly corresponds to sonic conditions at the throat can be easily
determined from isentropic flow relations. The flow is subsonic
everywhere in the nozzle except at the throat, and mass flow is
the maximum possible for the given nozzle and the reservoir conditions.
Suppose the exit pressure is now reduced to a value corresponding
to curve F (PCRIT-3) in Figure 1 where no shocks are present
in the nozzle. The exit pressure at F is such that the entire
expansion is isentropic and the flow is supersonic in the diverging
portion of the nozzle. The value for pressure is simply obtained
from the isentropic relationships for Mach number, pressure, temperature
and density and represents an optimal nozzle design. The pressure
within the nozzle exit cannot be reduced further and when the
external pressure is reduced to G the fluid leaving the nozzle
changes its pressure through a complicated flow pattern outside
the nozzle. Thus, the curves B and F represent the two limiting
cases of exit pressure for isentropic flow in such a nozzle. For
exit pressures below that at B, a shock wave forms within the
diverging part of the nozzle, changing the flow from supersonic
to subsonic and compressing the gas exactly enough to match the
nozzle exit conditions. Because of the entropy rise across the
shock, the overall flow through the nozzle is not isentropic,
although the flow on either side of the shock can still be considered
isentropic. The lower limit for this kind of flow pattern is given
by a shock occurring exactly at the exit of the nozzle as indicated
by curve D (PCRIT-2). The flow conditions for exit pressures
between curves B and D may be computed with aid of the isentropic
relationships and normal shock analysis. At still lower exit pressures
the flow adjusts itself through a series of two-dimensional or
three-dimensional shock waves and the average exhaust velocity
is generally still supersonic.
The designer must chose an appropriate condition from the previous
possibilities for his particular application. When the flow leaves
the nozzle at supersonic speeds and its pressure exactly equals
the surroundings (curve F) , the nozzle is called correctly
expanded (PCRIT-3). If the exit area of the nozzle
is less than the correctly expanded value for a given back pressure,
the nozzle is underexpanded and the fluid leaving the nozzle
has a pressure greater than the surroundings (curve G). On the
other hand, if the exit area of the nozzle is too large, shock
waves form within or just outside the nozzle and the flow is called
overexpanded. The particular mode of operation of any nozzle
can be quickly checked by first establishing the limiting pressure
curves B and D and comparing them with the specified exit pressure.
REFERENCE: Pages 299 and 300 Fluid Flow, Sabersky and Acosta.
Figure 1 and Figure 2: Flow conditions
depending on Pressure Ratio (Pe/Pc)
-------------------------- SHAPES Nozzle 3.7 CAN MODEL -------------------------- | |
Figure 3: Nozzle 3.7 determines isentropic internal flow properties for axisymmetric nozzles where the cross-sectional area distribution is directly specified by throat diameter, exit diameter and nozzle program axial shape selections. Exit area, Ae for the circular cross-section case is, Ae=p/4*De^2. Using this relationship the exit diameter, De for the circular cross-section case is, De=sqr(4/p*Ae), where Ae is nozzle exit area. A similar relationship determines throat diameter for the circular nozzle case to be, Dt=sqr(4/p*At). In practice De and Dt are specified directly without having to apply these simple relationships. Compressible fluid flow properties only vary along the x-axis of the one-dimensional nozzle. |
Figure 4: Nozzle 3.7 determines isentropic internal flow properties for rectangular nozzles where the cross-sectional area distribution is specified by the equivalent circular area defined by throat diameter, exit diameter and nozzle program axial shape selections. Exit area, Ae for the rectangular cross-section case is, Ae=Wz*Hy=p/4*De^2. Using this relationship the equivalent exit diameter, De for the rectangular cross-section case is, De=sqr(4/p*Wz*Hy), where Wz is nozzle exit width in the z-direction and Hy is nozzle exit height in the y-direction. A similar relationship determines throat diameter for the rectangular nozzle case to be, Dt=sqr(4/p*Wz*Hy), where Wz is nozzle throat width in the z-direction and Hy is nozzle throat height in the y-direction. Compressible fluid flow properties only vary along the x-axis of the one-dimensional nozzle. |
B)
STEP-BY-STEP NOZZLE ANALYSIS EXAMPLE (REFER TO FIG 5 and FIG
6)
BASIC DIMENSIONS ARE FOR THE SPACE SHUTTLE MAIN ENGINE (SSME)
1) Using the Units
pull-down menu check Length(inch), Pressure(lb/in^2) to
specify the units of the analysis.
2) Select Bell nozzle in the Nozzle Shapes section.
The data entries for a Bell nozzle having a bell shape will appear.
Please see notes below.
3) Enter an entrance temperature of 5400 degrees R.
4) Enter an entrance pressure of 3000 psia.
5) Enter an Atmospheric pressure of .0017 psia to simulate vacuum
conditions in space. Please see Note-5 for other options, for
example the optimal design condition where no shocks are present.
6) Using the Gases pull-down menu select OXYGEN, HYDROGEN
as the working fluid in the nozzle. By selecting OXYGEN, HYDROGEN
the value for the ratio of specific heats (g) and the Gas Constant (Rgas) are
automatically specified in the appropriate spaces in thi s case
having units, in^2 *sec^2/R.
7) Enter total nozzle length as 127 inches (The converging section
is 6 inches long and the diverging section is 121 inches long).
8) Enter throat diameter as 10.3 inches.
9) Enter the throat location from the origin as 6 inches.
10) Enter the upstream throat radius as 7.725 inches (1.5 * Rthroat).
11) Enter the downstream throat radius as 1.967 inches (0.382
* Rthroat) .
12) Enter the entry angle of parabolic section as 32 degrees.
13) Enter the exit diameter as 90.7 inches.
14) Enter the total number of grids along the nozzle axis as 1000
points. A maximum of 1000 nozzle X,Y coordinates may be defined.
15) In the Solve Flow section select the Classical gasdynamics
method option.
16) Click the SOLVE NOZZLE FLOW command button to determine
nozzle flow characteristics using the method specified in step
(16).
SPECIAL NOTE ON UNITS: Nozzle 3.7 design Units should be specified
before dimensional data and fluid properties data are entered. Please see the
nozzle design example in the section labeled, STEP-BY-STEP NOZZLE ANALYSIS
EXAMPLE. In this step-by-step example of the SSME, Units are specified at
the beginning of each Project not in the middle or end of each Project. If
however, the user decides to “short circuit” the normal input procedure here is
what happens if Units are changed "after" entering nozzle dimensional and
fluid properties data. First, Entrance temperature and Entrance pressure are
reinitialized to 5400 degrees Rankin and 1000 psig or their values in the
specified units. The value for Atmospheric (back) pressure remains unchanged
from the value entered by the user because it is not considered a basic property
for the isentropic variations in the nozzle from chamber to nozzle exit. Then,
Ratio of specific heats and Gas Constant are converted to their proper values in
the specified units. The remaining nozzle dimensional values are unchanged
because Nozzle 3.7 assumes the user knows which units are required. To properly
operate Nozzle 3.7 the user needs to know that changing Units is
considered a basic operational change that reinitializes the nozzle fluid
dynamic analysis similar to choosing a new nozzle shape.
Figure 5: Nozzle Dimension Locations
NOZZLE RESULTS (REFER
TO FIGURE 7)
17) Use the slider-bar
to see real-time results for Nozzle radius [Y(X)], Nozzle cross-sectional
area [A(X)], Mach number [Mn], Pressure ratio [P/Po], Temperature
ratio [T/To] and Density rio [R/Ro].
18) After selecting the desired plot variable option-button, enlarge
the plot by clicking the ENLARGE PLOT command button. The
plots can be printed from the enlarged plot screen.
19) Nozzle results may be sent directly to a printer in text form
by clicking File and then Print Detailed Results.
20) Click the Show results option to display the Results
section.
The results of the analysis are:
a) Mach number (Mn) at exit is 5.182
b) Pressure ratio (P/P0) at exit is .0007
c) Temperature ratio at exit is .2227
d) Thrust produced is 466,151.266 pounds.
e) Mass flow rate through nozzle is 1024.329 pounds per second.
f) Thrust coefficient (CF) is 1.865
SSME ACTUAL MEASUREMENT
Mass flow rate: 1035 pounds per second (1.0% difference)
Vacuum thrust: 470,000 pounds (0.80% difference)
Note-1: The Bell nozzle shape uses a parabolic curve approximation
from the throat to the nozzle exit. For an approximate G.V.R.
Rao Bell nozzle configuration the contour immediately upstream
of the throat is a circular arc with radius 1.5*Rthroat. The divergent
part of the nozzle immediately downstream of the throat is made
up of a circular section with a radius of 0.382*Rthroat and then
a parabola to the exit of the nozzle.
Note-2: If Free-Form Shape
is selected in step (2) the Imported and Graphical Shapes
entry box appears. Enter all required data and then bring up the
Free-Form screen by double-clicking on the DEFINE FREE-FORM
NOZZLE SHAPE icon. Generate the nozzle shape by dragging up
to 30 points into position on the screen and then return to the
main screen.
Note-3: In step (2) Import a list of X,Y nozzle coordinates by
clicking File and then Import Nozzle
Shape. The data must be in the following format: First line:
[Total number of nozzle X,Y coordinates]. Second and subsequent
lines: [Point number], [X nozzle location], [Y nozzle location]
and have a .TXT file delimiter. A maximum of 1000 nozzle
X,Y coordinates may be defined. Use the AeroGRID option of
AeroRocketCAD to generate the
fluid boundary of a nozzle by moving several
points into place and by specifying the number of intermediate grids. See the
simple nozzle example in the AeroGRID section.
Note-4: In step (15) the selection of the MacCormack
finite difference method will allow Nozzle to use the forward-predictor
backward-corrector finite difference CFD method to compute nozzle
flow. This option computes curve F (PCRIT-3) which is the
optimum design condition when no shocks are present in the nozzle
(isentropic) and the flow is entirely supersonic in the diverging
part of the nozzle. For optimum nozzle expansion the nozzle exit
pressure, P2 is equal to the external pressure, Patm. Rocket nozzles
are normally designed using the PCRIT-3 flow expansion
condition for optimal performance. This method is only accurate
if the residuals are reduced to at least 1.0E-6.
In practice the number of nozzle points is usually less than 50,
the CFL should be about 0.80, the starting Mach number should
be around 0.001 and finally the total number of iterations should
be at least 750 and sometimes up to 2000.
Note-5: To compute an optimal nozzle design when no shocks
are present and If the Classical gasdynamics method is
selected insert 0.0 for the Atmospheric (back) Pressure.
SOLVE the flow and the value for PCRIT-3 and therefore
atmospheric pressure is automatically determined and displayed
in the Results section and reflected in the input data
section. To compute the case where the flow is sonic (M=1) at
the throat and subsonic everywhere else (PCRIT-1) insert
a value for Exit Pressure just slightly smaller than the
Entrance Pressure. SOLVE the flow and an estimate
for PCRIT-1 appears in the status bar at the bottom of
the screen. Insert this estimate for PCRIT-1 into the value
for Exit Pressure and SOLVE again. The new value
for the Exit Pressure in the Results section is
the new value for PCRIT-1.
Note-6: Please remember to "Click" back using
the Return icon. Using the [X] box will kill the results
and delete the modifications or may hang the application.
Figure 6: Input data for ideal expansion, no shock in nozzle.
Figure 7: Results for ideal expansion, no shock in nozzle.
Figure 8: Mach number contour plot for ideal expansion, no shocks
in nozzle.
Figure 9: Results where back pressure is 100 psig causing a shock
in the diverging part of the nozzle. This figure is not part of
the SSME example.
This
part of the description is to illustrate the Free-Form
screen and is not part of the SSME nozzle example.
Figure 10: Free-Form screen for generating nozzle geometry and
is not part of the SSME nozzle example.
OVEREXPANDED (Pa > Pe)
AND UNDEREXPANDED (Pa < Pe) EXTERNAL NOZZLE FLOW
Nozzle uses two-dimensional oblique shock and Prandtl-Meyer
expansion theory to predict shock-angles (b, b2),
jet-angle (q) and Mach number for underexpanded and overexpanded
flow. If
the nozzle is axisymmetric the solution is valid in the immediate vicinity of the nozzle-exit.
Further than a half diameter from the nozzle-exit, axisymmetric expansions and
shocks are not accurately defined by two-dimensional oblique shock and Prandtl-Meyer
theory. For this reason the external nozzle analysis is accurate for two-dimensional
flow at large distances from the nozzle exit and a good approximation for
axisymmetric flow within a half diameter from the nozzle exit. A nozzle is underexpanded when Pa/Pc <
Pe/Pc and is characterized as a nozzle that experiences a series
of external Prandtl-Meyer expansion waves starting from the lip
of the nozzle. Similarly, a nozzle is overexpanded when Pa/Pc
> Pe/Pc and is characterized as a nozzle that experiences a
series of oblique shocks starting from the lip
of the nozzle. In this analysis, Pa/Pc is the ratio of the atmospheric
(Pa) or back pressure to the chamber pressure (Pc) and Pe/Pc is
the ratio of the nozzle exit pressure (Pe) to the chamber pressure
(Pc). Variables with subscript (c) are related to the entrance
of the nozzle and variables with subscript (jet) for underexpanded flow and (2),
(3) for overexpanded flow are related to
the exterior region of the nozzle after the nozzle-exit. Finally, variables
with subscript (a) are related to the atmospheric pressure or
back pressure of the environment.
For underexpanded flow, Nozzle determines flow properties using the
Pa/Pc and Pe/Pc pressure ratio criteria. Then,
Nozzle uses Me the exit Mach number to determine the Prandtl-Meyer
function in region (e) of the flow. Mjet is computed using
the isentropic expansion equation by assuming Pjet = Pa. Then,
using Mjet the Prandtl-Meyer function is determined in region
(jet) of the flow. Finally, the outer boundary or jet-angle is
determined using the relationship, Theta(q) = n(Mjet) - n(Me). Theta
(q) is defined as the angle the jet boundary makes
with the horizontal axis of the nozzle-lip.
For overexpanded flow,
Nozzle determines flow properties using the Pa/Pc and Pe/Pc
pressure ratio criteria. Then, Nozzle
computes the pressure ratio across the initial and reflected oblique shocks to
compute Mach number in region 2 and 3 and oblique shock angles (b, b2) using two-dimensional gas dynamics. Figure-11 and
Figure-12 define the variables used for overexpanded flow and underexpanded
flow, respectively. Figure-13 illustrates the SSME nozzle having
an atmospheric pressure of 14.7 psia and the resulting overexpanded
external flow with shock and jet boundary. Finally, Figure-14
illustrates the SSME nozzle having an atmospheric pressure of
2.0692 psia with a slightly underexpanded external flow and a
jet boundary set at almost 0.0 degrees. This condition can be
understood to mean optimal expansion when no shocks are present
and the nozzle is exhausting directly into the atmosphere.
Figure11: Overexpanded nozzle (Pa/Pc > Pe/P Figure 12: Underexpanded nozzle (Pa/Pc < Pe/Pc) |
Figure
13: SSME overexpanded nozzle where Patm/Pc > Pe/Pc and external
lip shocks and reflected shocks occur.
Figure
14: SSME properly expanded nozzle (Slightly Underexpanded) where Patm/Pc
< Pe/Pc.
2-D PLUME
ANALYSIS BY THE METHOD OF CHARACTERISTICS
Back To Top
OVEREXPANDED NOZZLES
FROM POINT-2 AND UNDEREXPANDED
NOZZLES FROM POINT-1
For underexpanded (Patm
< Pexit) nozzles the external flow must adjust itself
through a series of expansion and compression waves as the flow proceeds
downstream. Starting at nozzle-exit the flow goes through a Prandtl-Meyer expansion wave to adjust
the flow to
ambient pressure (Pa). Then, to maintain the constant-pressure boundary condition on
the outer boundary of the plume, the expansion wave reflects off the outer boundary as a compression wave. This process is repeated through several cycles
of expansion waves and compression waves that reflect off the boundary of the jet's plume.
The routine in NOZZLE that models the external jet plume assumes
the exit flow is supersonic, two-dimensional and underexpanded. The user may run
the plume analysis at any time by clicking FILE and then clicking 2-D
Plume Analysis. The plume analysis
is used by first inserting the ratio of ambient pressure to exit pressure
(Pa/Pe), exit-plane Mach number, ratio of specific heats and nozzle exit
diameter.
If results for the flow on the main screen are for an underexpanded
nozzle, the value for pressure ratio (Pa/Pe), exit Mach number (Me) and nozzle
exit diameter (De) are inserted and plume results automatically displayed from
nozzle exit to several diameters downstream.
However, If results for the flow on the main screen are for an overexpanded
nozzle, the value for pressure ratio (Pa/P3), Mach number (M3) and oblique shock
diameter at point-2 are inserted into the plume analysis and results automatically displayed
starting at the end of the external oblique shock wave pattern as illustrated
below. The user must
first click SOLVE and click Show Contours before performing a
2-D Plume Analysis. Otherwise, the user can override the automatic input and manually insert
another value for Pa/Pe. The other 2-D plume input variables, Nozzle exit plane
Mach number (Me), Specific heat ratio (Gamma) and Nozzle exit diameter (De) may
be entered manually and the external 2-D plume computed for underexpanded flow. The user may step though the flow field by clicking
the LOCATION button to display the physical location in the plume and the
properties at that location. Finally, the screen may be enlarged and
screen results printed by the click of a button. In these illustrations, "e"
represents the correctly expanded value at the exit-plane of the
converging-diverging nozzle and "a" represents the atmospheric or back pressure
of the environment which is 14.7 psig at sealevel.
The following
steps generate solutions for overexpanded
and underexpanded flows.
STEP-1: Perform fluid flow analysis on main screen and click Show
Contours then click Plot external pattern and contours
to generate the oblique shock wave geometry for overexpanded flow or
the expansion wave geometry for underexpanded flow.
Please refer to Figure-15, below
Figure-15: Test case from Gas Dynamics: Theory and Applications,
Example-3 on page 97 to 99 for an overexpanded nozzle.
Example-3 predicts that a nozzle defined by M_{1} = 2.44, Pb/Pc = 0.1
and A_{e}/A_{t} = 2.5 that
b
= 29.9 deg,
q
= -7 deg, and M_{2} = 2.14
Also, this example displays the CAD definitions for a conical nozzle
design composed of two LINEs and two CIRCLEs.
STEP-2: Click File then click 2-D Plume Analysis to
generate underexpanded nozzle properties from nozzle-exit to several
diameters downstream. OR generate overexpanded nozzle properties
from point-2 on the main screen plot of the oblique shock
wave region to several diameters downstream. Please refer to Figure-16, below.
Figure
16:Two-Dimensional plume analysis for underexpanded flow where Patm
< Pexit. Nozzle 3.7 starts the 2-D plume
Method of Characteristics analysis starting at Point-2 of the overexpanded
reflected oblique shock wave in Region-2.
PROPERLY DESIGNED NOZZLES THAT
EXHAUST DIRECTLY
INTO THE ATMOSPHERE
WITH NO SHOCKS FOR DESIGNING
WIND TUNNELS AND MORE EFFICIENT NOZZLES
For some users the relative complexity of standard Nozzle computer
program features are to complex to apply for routine nozzle design. The new
Design Conditions routine is for those who need nozzle designs where the
diverging nozzle flow is entirely subsonic or supersonic, including the exit jet
where Pexit = Patm. Results from Design Conditions is suitable for
designing subsonic or supersonic wind tunnels or for designing efficient nozzles
where no shocks are present in the diverging part of the nozzle or in the jet
exhaust. Please see Figure-1 and Figure-2 where design condition
refers to flows that leave the nozzle at
supersonic velocity and whose exit pressure equals the surroundings (curve F).
The nozzle is called correctly
expanded (PCRIT-3) for a supersonic design condition nozzle. Simply
specify exit Mach number (Me) or Pressure Ratio (Pc/Pe) for either subsonic
(PCRIT-1) or supersonic (PCRIT-3) exit flow as depicted by Curve-B or
Curve-F in Figure-1 and Figure-2.
For Curve-B and Curve-F the
area ratio (Ae/At) exactly equals the critical ratio (Ae/Astar) for subsonic and
supersonic correctly expanded flow. The throat velocity becomes sonic (M = 1),
mass flux reaches a maximum and the exit pressure (Pexit) exactly equals the
atmospheric pressure (Patm) or in other words Pexit = Patm. The applicable
nozzle equations required for design condition nozzles is displayed in the
Basic Equations menu. For a complete understanding of the technical aspects
of nozzle design please refer to Fluid Mechanics by Frank M. White, pages
513 to 547 in chapter, Compressible Flow. Quickly plot color contours and
flow properties verses axial location for Mach number (Mn), Pressure Ratio
(Pc/P), Temperature Ratio (Tc/T) and density Ratio (Rc/R). Please be aware these
ratios are the inverse of the flow properties computed in the main nozzle
analysis.
Figure
17: Main screen for the Design Conditions routine showing properly
designed nozzle results.
Figure
18: Main screen for the Design Conditions routine showing color contours
of Mn verses axial location.
Figure
19 Main screen for the Design Conditions routine showing Mn verses
axial location.
Figure
20: Main screen for the Design Conditions routine showing page one
of three pages of nozzle equations.
TURBULENT
CIRCULAR AND 2-D FREE JET ANALYSIS Back To Top
Steady-State
Viscous Boundary Layer Analysis of a Free Jet issuing from an orifice: The
sketch on the left shows a typical subsonic streamline pattern for circular and
two-dimensional, turbulent free jets. For free jets at some distance down stream of the
beginning of the jet or wake, the viscous boundary layer approximations apply and the
velocity profiles become nearly similar in shape when normalized by local
velocity and jet width. Similarity holds well for jets and wakes to determine
the velocity profile along the axis of the jet. Please see Viscous Fluid Flow
by Frank M. White, starting on page 505 for a complete derivation of the
boundary layer approximation for the circular and two-dimensional free jet flow
used in this analysis.
The turbulent free jet geometry is completely defined by specifying Nozzle-exit
radius (b1), Jet Computational length (Lmax) and Jet starting (centerline)
velocity (Umax). Results include velocity profile plots at each of five
predetermined axial locations as a percentage of Lmax, velocity contour plots
having a maximum of 256 plot levels and velocity vector plots. In addition, this
routine includes the ability to save the five-station velocity profile results
in CSV format for spreadsheet applications. Finally, all Free Jet analysis
screens may be sent to the printer.
Figure 21: Typical free jet streamline pattern with definitions.
Figure 22: Turbulent Circular Free Jet velocity profiles at each of five
predetermined axial locations as a percentage of Lmax.
Figure 23: Turbulent Circular Free Jet velocity contour plot having a maximum of
256 plot levels. Line illustrates jet boundary.
Figure
24: Turbulent Circular Free Jet velocity vector plot. Line illustrates jet
boundary.
ATLAS RD-180 ROCKET
MOTOR ANALYSIS
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The
RD-180 rocket engine is a Russian designed and built dual-combustion chamber,
dual-nozzle rocket engine used to provide first-stage power for the US built Atlas 5 launch vehicle. The two
combustion chambers of the RD-180 share a single turbopump fueled by a mixture
of RP-1 (kerosene) and LOX (Liquid oxygen) that uses an extremely efficient,
high-pressure staged combustion cycle. The RD-180 rocket engine operates on an
oxidizer to fuel mixture ratio (O/F) of 2.72 and like its predecessor the
RD-170, employs an oxygen-rich pre-burner. The thermodynamics of the staged
combustion cycle allows the efficient oxygen-rich pre-burner to provide greater
than usual thrust to weight operation. However, to achieve greater efficiency the rocket motor must
tolerate high pressure and high temperature gaseous oxygen that must be cycled
through the engine.
The following Nozzle 3.7 input variables are used to model the RD-180
rocket motor. Chamber pressure for the RD-180 rocket motor is 26.7 MPa, nozzle
exit diameter is 1.4 m, nozzle area ratio (Ae/At) is 36.87 which establishes the throat
diameter to be 0.2306 meters, atmospheric pressure is 0.10135293 MPa. Entrance
temperature, Ratio of specific heats and Gas constant are defined by specifying
RP-1 and LOX as propellant and oxidizer for propulsion. The remaining input
dimensions are used to define an approximate Bell nozzle geometry for the rocket
motor. Nozzle 3.7 determines thrust for a single nozzle of the RD-180
which represents half of the total thrust generated. For the two nozzle RD-180
rocket motor total thrust is 3.79 MN at sea level and 4.1 MN in vacuum. Results
for this analysis are tabulated in Table-1 below which shows that
Nozzle 3.7 is capable of predicting thrust for the RD-180 rocket motor
within 1.5% of measured results.
Atlas 5-400 launch vehicle (left) and the RD-180
rocket motor (right).
Figure 25: Input data for sea level
RD-180 Nozzle 3.7 analysis used to populate the results listed in Table-1,
below.
Figure 26:
RD-180 analysis displaying Isp, CF, V_{exhaust} and external shock
pattern compared to engine test at sea
level.
The inserted exhaust plume of an RD-180 rocket motor is not part of Nozzle
3.7
output but the color contour plot is actual output.
Table-1, RD-180 Analysis Results |
Sea Level (Patm = 101,353 N/m^2) | Vacuum (Patm = 5 N/m^2 ) | ||
Thrust, lbf (MN) | Isp, sec | Thrust, lbf (MN) | Isp, sec | |
Nozzle 3.7.0.x | 851,421 (3.79) | 302 | 921,567 (4.15) | 327 |
RD-180 Test | 860,568 (3.83) | 311 | 933,400 (4.10) | 338 |
Difference | 1.1% | 2.9% | 1.3% | 3.3% |
Nozzle Minimum System Requirements
(1) Screen resolution: 800 X 600
(2) System: Windows 98, XP, Vista, Windows 7 (32 bit and 64 bit), NT or Mac with emulation
(3) Processor Speed: Pentium 3 or 4
(4) Memory: 64 MB RAM
(5)
English (United States) Language
(6)
256 colors
Please note this web page requires your
browser to have
Symbol fonts to properly display Greek letters (a,
m, p,
∂
and w)
ADDITIONAL REQUIREMENT: Input data for all AeroRocket programs must use a period (.)
and not a comma (,) and the computer must be set to the English (United States)
language. For example, gas constant should be
written as Rgas = 355.4 (J / kg*K = m^2 / sec^2*K)
and not Rgas = 355,4. The English (United States)
language is set in the
Control Panel by clicking Date, Time, Language and
Regional Options then Regional and Language Options
and finally by selecting English (United States). If periods are not used in all inputs
and outputs the
results will not be correct.
NOZZLE REVISIONS
Nozzle 2.7 and Nozzle
2.8 Features
1) Nozzle outputs nozzle shapes in X,Y format. First, the user
must run the program or click Plot Shape to generate the
points describing the nozzle. The user may output X,Y nozzle coordinates
and all axially varying nozzle parameters using the Save Data
File As command. The data file created using Save Data
File As has the .CSV extension to distinguish it from
the imported shape file that has the .TXT extension.
2) Nozzle shows up on the Status Bar. The program may be
minimized, maximized or terminated using the window controls.
3) Nozzle can model ultra-small nozzle shapes. Very small nozzles
use scientific notation while larger (Greater than .001 diameter)
nozzles use standard output format.
4) Mass flow rate in kg/sec or lbm/sec added to the output.
Nozzle 2.9 Features
1) Fixed a few minor problems involving display of very small
nozzle dimensions and output results.
Nozzle 3.0 Features
1) Fixed error in the computation of thrust and mass flow rate.
2) Fixed a few spelling errors.
Nozzle 3.1 Features
1) Fixed error in the computation of thrust and mass flow rate.
Nozzle 3.2 Error
Fix
1) Fixed an error in Nozzle that manifested itself when analyzing
nozzles with area ratios (Ae/At) greater than 60. Specifically,
for larger area ratios and for pressure ratios (Pe/Pc) sufficient
for a shock to form in the diverging portion of the nozzle, Nozzle
would incorrectly determine that the flow was sonic at the throat
and subsonic everywhere else in the nozzle. The problem was that
the constant PCRIT was miss-dimensioned as a string variable when
it should have been defined as a single precision variable. This
was an intermittent precision problem because sometimes the results
were correct and sometimes incorrect depending on the value of
the pressure ratio and area ratio.
Nozzle 3.3 Features
and Error Fixes
1) Added
color contour plots for Mach number (Mn), Pressure ratio (P/P0),
Temperature ratio (T/T0), and density ratio (R/R0).
2) Fixed a few FREE-FORM screen nozzle geometry errors. Nozzle
would occasionally fail to analyze some FREE-FORM nozzle geometries
when the ratio of specific heats were less than 1.4.
3) Cleaned up a few presentation errors and enhanced results display.
Nozzle 3.4 Features
1) Added
the ability to specify the upstream radius and downstream radius
on either side of the throat for Conical and Bell nozzle shapes.
Nozzle 3.5 Features
(12/14/03)
1) Added
the ability to interrupt the nozzle analysis or to Stop the nozzle
analysis.
2) Improved the initial slope of the parabolic portion of the
Bell nozzle shape.
Nozzle 3.6.2 Features and Error Fix (06/02/04)
1) Added
the ability to determine underexpanded and overexpanded external
flow in the vicinity of the nozzle-lip region.
2) Fixed confusion concerning Nozzle exit (back) pressure and
Atmospheric pressure (Patm). These two quantities should always
be identical, but confusion about these entries caused thrust
to be computed incorrectly. Now, the user enters only the Atmospheric
(back) pressure. Previously, this entry did not accept pressures
less than the optimal design condition (Pdesign) where no shocks
are present in the nozzle and the flow exhausts directly into
the atmosphere. However, to allow for underexpanded and overexpanded
nozzles this constraint needed to be lifted. Now, a small non-zero
value may be specified for the atmospheric pressure (Patm) corresponding
to near-vacuum conditions.
Nozzle 3.6.3 Error Fix (02/22/05)
1) Under certain
conditions when the maximum velocity exceeded Mach 7 in the diverging part of the
nozzle, Nozzle would erroneously insert a shock wave. This condition has been
fixed by increasing the upper limit of the maximum exit velocity (Me) to Mach 20
which increases the upper limit of the area ratio (Ae/At) to over 15,000.
2) Under certain conditions when the user decided to Cancel a nozzle geometry
Import, Nozzle would repeatedly provide an error message. The user would
have to perform a CTR-ALT-DEL to exit the program.
Nozzle 3.6.4 Features and Error Fix (07/25/05)
1) When displaying
external flow contour plots for overexpanded nozzles the value of Mjet
was inadvertently displaying the normal component of Mach number across the
oblique shock emanating from the nozzle lip. Instead, the total Mach number in
the jet region behind the oblique shock wave should have been displayed.
2) Added a hybrid rocket motor propellant having the following fuel and oxidizer
to the list of combustion gases: 85% Nitrous Oxide, 15% HTPB.
Nozzle 3.6.5, 3.6.6 Feature (10/31/05)
1) Added a plume analysis for supersonic, two-dimensional and underexpanded
nozzles.
Nozzle 3.6.7 Features and Error Fix (11/26/06)
1) Included Nozzle_Examples.zip in the Nozzle directory which includes 34
nozzle examples used for validation purposes.
2) The gas Nitrogen Dioxide in the Gases pull-down menu should be labeled
Nitrous Oxide (N2O). (Fixed)
3) Program terminated if attempting to read a misspelled or non-existent file
using the Open Project command. (Fixed)
Nozzle 3.7.0.1 Features (12/12/08)
1) Added a
Design Conditions routine for those who wish to quickly design subsonic or
supersonic wind tunnels or more efficient every-day nozzles when no shocks are
present in the diverging part of the nozzle or in the exhaust jet. Design
Conditions quickly plots color contours and flow properties verses axial
location for Mach number (Mn), Pressure Ratio (Pc/P), Temperature Ratio (Tc/T)
and density Ratio (Rc/R). No changes were made to the main nozzle analysis.
Nozzle 3.7.0.2 Features (09/14/09)
1) Added Turbulent
Circular and Turbulent 2-D Free Jet analysis capability to Nozzle 3.7
based on the theory presented in Viscous Fluid Flow by Frank M. White,
starting on page 505.
2) For Nozzle 3.7, fixed all input data text boxes for 32 bit and 64 bit
Windows Vista. When operating earlier versions of Nozzle 3.7 in Windows Vista the input data
text boxes failed to show their borders making it difficult to separate each
input data field from adjacent input data fields.
Nozzle 3.7.0.3 Features (01/12/10)
1) In the
Design Conditions routine increased the number of input digits from 3 to 6
digits after the decimal point. Nothing else has been modified.
Nozzle 3.7.0.4 Features (08/21/10)
1) This version displays
Conical nozzle geometry in Computer Aided Design (CAD) formatted LINES
and CIRCLES for generating imported shapes using the Imported shape
option command. Accessed by clicking File then CAD Input For Conical
Shapes then SHOW NOZZLE CAD.
2) For overexpanded
nozzles, the value for pressure ratio (Pa/P3), Mach number (M3) and oblique
shock diameter at point-2 are inserted into the 2-D Plume Analysis and
the Method of Characteristics results are automatically displayed starting at
the end of the external oblique shock wave pattern as illustrated in Figure-16.
Previously, the plume analysis did not compute the expansion wave pattern for
overexpanded nozzles. For underexpanded nozzles the value for pressure ratio (Pa/Pe),
exit Mach number (Me) and nozzle exit diameter (De) are inserted and plume
results automatically displayed from nozzle exit to several diameters downstream
as in previous versions of Nozzle. Fixed a minor error in the computation of jet
Mach number for overexpanded nozzles.
Nozzle 3.7.0.5 Features (11/28/11)
NEW!
1) This version displays
specific impulse (Isp) in place of Thrust Coefficient (CF) in the
Results section. Also, Exhaust Velocity (V_{exhaust}) is
also displayed as illustrated in Figure-27 in length units (meters/sec in this
case) selected by the user.
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For
more information about
Nozzle 3.7
please
contact AeroRocket at
aerocfd@aerorocket.com.
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