Chapter 7
FINITE ELEMENT MODELING AND ANALYSIS OF
LPGDS NOZZLE
This chapter contains model for the flow through the supersonic nozzle in LPGDS
process. The model was developed using Finite Element Method and solved by ANSYS
FLUENT 12.0 software to predict the particle velocity, pressure and temperature at
the exit of the nozzle.
7.1
MODELLING THE FLOW OF MIXTURE IN THE LPGDS
PROCESS USING FINITE ELEMENT MODELING
The mathematical representation of the flow through nozzle in the LPGDS process
involves the continuity, momentum, and constitutive equations (Maev and
Leshchynsky, 2008; Rajaratnam, 1976). The following assumptions are made to
simplify the analysis.
7.1.1 Assumptions
The gas flow model uses isentropic relationships and linear nozzle geometry. The
assumptions for the calculation are as follows:
i.
The gas obeys the ideal gas law.
ii.
There is no friction impeding the gas flow.
iii.
The gas flow is adiabatic, i.e., no heat exchange occurs with the surroundings.
iv.
Steady-state conditions exist.
v.
Expansion of the gas occurs in a uniform manner without shock or discontinuities.
vi.
Flow through the nozzle is one-dimensional.
vii.
Particles do not influence gas conditions.
viii.
Collision between particles is neglected.
ix.
The effect of the presence of particles does not affect the space charge.
127
x.
The formation of thin coating does not affect thermo fluid field locally in the
impinging region.
The equations are written for a fluid region Ωf using the Cartesian coordinate system
xi in an Eulerian reference frame, with the subscript (or index), i = 1,2,3 or 1 and 2 for
two-dimensional problems, where all the variables may be functions of the position x
= x 1 , x2 .
7.1.2 The Governing Equations for flow through the nozzle in
LPGDS process
The following governing equations are required for finite element modeling (FEM)
for the flow in the LPGDS process (Vlcek, 2003):
7.1.2.1 Continuity equation (conservation of mass):
According to the law of conservation of mass, the total time rate of change of mass in
a fixed region is zero. The conservation equations used for turbulent flow are obtained
from those of laminar flow using a time averaging procedure commonly known as
Reynolds averaging (Rajaratnam, 1976). It can be stated as
(7.1)
Where ρg is the density of the gas, and vj is the velocity vector in the jth direction.
7.1.2.2 Momentum equation (conservation of momentum):
The law of conservation of linear momentum states that the total time rate of change
of linear momentum is equal to the sum of the external forces acting on the region.
Appling this law for the LPGDS process gives:
+
)=-
+
+ ρggi
(7.2)
where τij is the stress and gi is the gravitational acceleration.
128
7.1.2.3 Constitutive equation:
It is necessary to specify the relation between the state of stress and the history of the
flow in order to obtain a closed set of equations. Such a relation is known as a
constitutive equation. In the case of a Newtonian fluid, the state of stress depends only
on the instantaneous local rate of strain; the extra stress is proportional to the rate of
strain tensor.
The stress is given by
τij =[(μ + μt)[
+
] - μt
. δij ]
(7.3)
where μ is the molecular viscosity and δij = 1 for i = j; otherwise δij = 0.
μt is the turbulent viscosity given by μt = ρg C μ
where C = 0.09 is a constant, kt is the kinetic energy of turbulence, and ε is the
dissipation of kinetic energy of turbulence.
This constitutive equation predicts, in particular, that the stress responds
instantaneously to any deformation and has no memory; the stress vanishes as soon as
the driving motion has ceased.
ρ g.
=0
(7.4)
From equation (7.2) and (7.3), we get:
ρg(
+
-
[(μ + μt)[
]-
μt
. δij]
– ρg. gi =0
On solving we get:
ρ g [Vj.
]+
-
[(μ + μt) [
+
] ] - ρg gi =0
(7.5)
7.1.3 Mixed finite element model
7.1.3.1 Weak form
The starting point for the development of the finite element models of equations (7.4),
(7.5) to (7.3) is their weak form, where ωi is the weight function, which will be
equated, in the Rayleigh–Ritz–Galerkin finite element models, to the interpolation
function used for (p, u, t) respectively (Reddy and Gartling, 1994; Huang et al., 1999).
129
The weighted integral statements of two equations over a typical element Ωe are given
by:
. ω1.dx = 0
(7.6)
The weak form for =n (7.5) is written as:
ω [ρ g (Vj
ρ g Vj
)+
-
)+
ρg gi.dx =
[(μ + μt) [
+
]] - ρggi ].dx =
(μ + μt)
.p.dx -
ω2.ds
(7.7)
μ + μt)
.dx
.dx -
ω2.ds
(7.8)
On constitutive equation =n (7.3):
[τij - [(μ + μt) [
τij.dx -
+
]+
[(μ + μt) [
μt
. δij].dx =0
[(μ + μt) [
].dx-
(7.9)
].dx+
.δij.dx =0
.
μt
(7.10)
7.1.3.2 Finite element model
For the Rayleigh–Ritz–Galerkin finite element models, the choice of weight functions
is restricted to spaces of approximation functions used for the velocity fields,
pressure, and extra stresses (Reddy and Gartling, 1994; Huang et al., 1999). Suppose
that the dependent variables (ui, pi, τij) are approximated by expressions of the form.
Vi= Σα1 Vj=αT Vj
(7.11a)
Vj= Σα1 Vi =αT Vi
(7.11b)
Pi =Σγm pm =γT p
(7.11c)
τij= ΣΨn τijn=ΨT τij
(7.11d)
The weight functions (ω1, ω2, ω3) have the following correspondence:
ω1 ≈ χ m
(7.12a)
ω2≈ α1
(7.12b)
ω3 ≈Ψn
(7.12c)
Substitution of equations (7.11) and (7.12) in equation (7.6), (7.8), (7.10) results in the
following finite element equations:
130
(a) Continuity equation:
(7.13)
(b) Momentum:
+
α
(μ + μt)
α
-
.dx] Vi -
gi .dx] =
α
(μ + μt)
.dx] Vj -
.ds
(7.14)
(c) Constitutive:
ΨT τij.dx +
.
.μt.
[(μ + μt) [
].dx]Vj -
[(μ + μt) [
α
].dx].Vi
.δij.dx].Vj=0
(7.15)
The discrete system is given by the following matrix equations, here the momentum
and continuity equations are written as a single system:
+
=
(7.16)
and for the constitutive equation:
-
+
=
(7.17)
where the sum on the repeated indices is implied. The coefficient matrix shown in the
equations (7.16) and (7.17) are defined by:
K11 = (2S11+S22) (μ + μt)
K22 = (S11+2S22) (μ + μt)
K21 = [K12] T
=
.dx
Q
=
Fi =
N=
(μ + μt)
.ds +
.dx
gi .dx]
ΨT .dx
131
L1=
[(μ + μt) [
D1=
.
μt
].dx
.δij.dx
The equation (7.16) and (7.17) becomes
C (u) + KU=F
(7.18)
N -LU +DU = 0
(7.19)
where the vector of unknowns is UT = {Vi, Vj, P}. Equations (7.18) and (7.19)
represent the finite element equation for flow through nozzle in the LPGDS process in
the steady state case. Equation (7.18) is recognized as the standard form for a
Newtonian problem used to calculate the velocity and pressure value (Reddy and
Gartling, 1994; Huang et al., 1999). Equation (7.19) is the finite element analogue of
the implicit constitutive equation and is seen as an equation analogue of the extra
stress. A stability condition for the resulting matrix problem requires that the extra
stresses be approximated to the same order as the velocity components. The above
equations (7.18) and (7.19) are solved after substitution of the boundary conditions.
7.2
MODELING PROCEDURE FOR LPGDS NOZZLE
The procedure for modeling of nozzle for LPGDS is done by using the ANSYS 12.0
[Module CFD Fluent] workbench software as described in following steps:
STEP 1: Determination of Problem Domain:
This is the first step, in which the geometry of the problem is developed. The 2-D
model for the problem is taken in the present research work. The analysis type is 2-D
PLANER. In this work, six different Nozzles under investigation named as N1, N2,
N3, N4, N5, and N6 were considered. The various dimensions for these nozzles as
shown in Table 7.1.
ABCDEF are the different coordinates, shown in 2-D view of the nozzle in Figure
7.1. The positions of the coordinates in the reference plane are shown in Table 7.2.
132
TABLE 7.1: Different Nozzle Configurations (All Dimensions in mm)
CONFIGURATION (mm)
Inlet diameter di
Throat diameter dt
Outlet diameter de
Convergent length Lu
Divergent length Ld
Injector diameter Id
Research References No:
N1
8
4
6
50
125
6
41
N2
16
3
6
30
140
4
28
N3
8
2
6
28
72
6
5
N4
20
4
4.6
17
70
0.9
34
N5
22
2
6
30
180
2
34
N6
8
2
6
10
40
6
23
Figure 7.1: 2-D view of Nozzle for the present work
TABLE 7.2: Coordinates For Nozzles in Modeling
Nozzles
N1
N2
N3
N4
N5
N6
(-50,-4)
(-50,4)
(0,2)
(125,3)
(125,-3)
(0,-2)
(-30,-8)
(-30,8)
(0,1.5)
(140,3)
(140,-3)
(0,-1.5)
(-28,-4)
(-28,4)
(0,1)
(72,3)
(72,-3)
(0,-1)
(-17,-10)
(-17,10)
(0,2)
(70,2.3)
(70,-2.3)
(0,-2)
(-30,-11)
(-30,11)
(0,1)
(180,3)
(180,-3)
(0,-1)
(-10,-4)
(-10,4)
(0,1)
(40,3)
(40,-3)
(0,-1)
Coordinates
A
B
C
D
E
F
STEP 2: Meshing and Boundary Conditions:
The total surface is meshed using ANSYS meshing. The mapped free meshing
contains the quadrilateral surface element shown in Figure 7.2. After the meshing is
done, the boundary conditions are defined as given. In the meshing the surface name
is created as inlet- outlet. Nozzle inlet is taken as pressure inlet and powder feeder is
taken as the mass flow inlet. The outer surface of the nozzle is taken as the wall and
the inside surface is for fluid flow. The number of nodes and elements for different
nozzle configurations has been defined in Table 7.3.
133
Figure 7.2: Nozzle after meshing and boundary Conditions (domain and finite element mesh)
TABLE 7.3: Number of nodes and elements for different nozzles
Nozzles
NODES
ELEMENTS
N1
2914
1328
N2
738
308
N3
864
376
N4
1498
674
N5
1114
492
N6
1148
497
Boundary conditions:
The boundary conditions described for flow through the nozzle in the LPGDS process
(Figure 7.2) are as given below:
1. At the inlet 1 of the nozzle :
P=Pi , T= Ti
2. At the inlet 2 of the nozzle :
M=Mi with gravity feeder.
3. The outlet of nozzle:
P=Po, V=Vo, T=To
4. At the nozzle surface:
Vx1 =0
Nozzle configurations:
Six different types of nozzles are under investigation in this research work. In all
these configurations the powder is injected axially just starting the divergent section
134
of the nozzle. The inlet operating conditions for the all nozzle configurations are same
to measure the output conditions for different nozzle configurations.
The inlet pressure (Pi) at the nozzle is 1 MPa, and the mass flow rate (Mi) is 0.05 gs-1.
The nozzle surface is assumed to be the wall surface having no slip. The powder is
fed through the injector by mean of gravity feeder. The pressure Po, velocity Vo and
temperature To is to be measured at the outlet of the nozzle.
7.3
ANALYSIS PROCEDURE
The Analysis is carried out in the ANSYS 12.0 [Module CFD FLUENT]. The steps
that are followed are given below which include all the conditions and the boundaries
values for the problem statement.
STEP 1: Checking of Mesh and Scaling
The Fluent solver is opened where 2DDP is selected. The meshed file then undergoes
a checking where the number of nodes for different nozzles are displayed. After this
grid check is done following which smoothing and swapping of grid is done.
Following this the scaling is done. Scale is scaled to mm. Grid created was changed to
mm. After this defining of various parameters are done.
STEP 2: Solvers and Material Selection
The solver is defined first. Solver is taken as pressure based and formulation as
implicit, space as 2D PLANER and time as steady. Velocity formulation as absolute
and gradient options as Green-Gauss Cell based are taken. Energy equation is taken
into consideration. The viscous medium is also taken. The analysis is carried using the
k-epsilon RNG model. In the multiphase, the EULERIAN multiphase model is used
[Fluent User’s guide].
The air is used as carrier gas .The properties of air are mentioned in Table 7.4.
Table 7.4: Various properties of air
Property
Density, ρ ( kg/m3)
specific heat capacity, Cp ( J/kg K)
Thermal conductivity, h (W/m K)
Viscosity,µ (kg/m s)
Value
1.138
1040.67
0.0242
1.663 x 10-5
135
The powder material used is copper. The properties for copper are given in Table 7.5.
Table 7.5: Various properties of copper material
Property
Density, ρ ( kg/m3)
Heat capacity, C ( J.kg-1 .K-1)
Thermal conductivity, K (W6'.K-1)
Elastic modulus, E (GPa)
Poisson ratio, µ
Value
8 960
383
386
124
0.34
STEP3: Boundary Conditions
 Nozzle inlet:
The Nozzle inlet is taken as the pressure inlet. Input the following values as:
Pressure (Pi) = 1 MPa
Temperature (Ti) = 623 K
 Powder feeder inlet:
The powder feeder inlet is taken as the mass flow inlet. The powder material is at
room temperature. Input the following values as:
Mass flow rate (Mi) = 0.05 g/s.
 Outlet :
The Nozzle was set as pressure outlet.
 Upper and lower boundary condition:
On the wall, u = v = w =0; T = TR on the nozzle surface, no-slip boundary conditions
for the velocities are used. The temperature is constant and equals room temperature
(TR).
STEP 4: Controls Set Up
The solutions controls are set as listed below in Table 7.6. The under relaxation factor
was set as given.
Table 7.6: Control set-up
Pressure
Density
Body forces
Momentum
Volume fraction
Turbulence kinetic energy
Turbulence dissipation energy
Energy
0.3
1
1
0.7
0.5
0.8
0.8
1
136
Pressure Velocity Coupling was taken as SIMPLE
Discretization Equation selected is as given
Volume fraction- First Order Upwind
Turbulence kinetic energy- First Order Upwind
Turbulence dissipation energy- First Order Upwind
Momentum-First Order Upwind
Energy- Second Order Upwind
STEP 5: Initialization
Solution initialization is done. Initial values of velocity are taken as zero along x, y
and z direction. Temperature is taken as 623 K. Residual monitorization is done and
convergence criteria are set up. The convergence criteria of various parameters are
listed below.
Continuity- 0.001
X-Velocity- 0.001
Y-Velocity- 0.001
Z-Velocity- 0.001
Energy- 1e-06
The number of iterations is then set up and iterations starts. The iteration continues till
the convergence is reached.
STEP 6: Plotting of contours and graphs
After the solution of the problem, next step is to plot the contours of velocity, pressure
and temperature on the surface and the graphs for same. The different shades of the
colors in the contour represent the different value of velocity, pressure and
temperature at the different positions.
The graph is plotted along the line. The beginning of the line from the starting of
nozzle, convergent section till the end, divergent section. The line is on the axis on
which y =0. The Y-axis of the graph is velocity, pressure and temperature
respectively, and the X-axis of the graph is the total length of the nozzle.
137
7.4
RESULTS AND DISCUSSION
The simulation procedure involves the determination of velocity, temperature and
pressure through the nozzle. The velocity, temperature and pressure distribution were
analyzed by the finite element method using the ANSYS 12.0 having module
FLUENT 12.0. The resultant pressure, temperature and the velocity of flow during the
process cycle were determined.
At the first instant, a model has been built using ANSYS 12.0 software considering
the pressure inlet for the air and the mass flow inlet for the injector. The mesh file for
the nozzle is also created in ANSYS 12.0 and defining the boundary conditions. Then
CFD software FLUENT (ANSYS 12.0) was used for the solving process. Since in the
finite element analysis, the results are available only on the nodes, the geometry of the
problem is meshed such that the nodes can be found on the locations where the values
of the degree of freedom (flow velocity, temperature and resultant pressure) are
needed. A finer meshing is required in the regions where the changes in gradients of
the degree of freedom are higher. For fulfilling all these requirements, the
quadrilateral meshing is used for meshing the geometry. In the solution procedure, the
Eulerian multiphase model is used, with the viscous K-ε model.
7.4.1 for nozzle N1
7.4.1.1 Velocity distribution
The patterns of velocity distribution obtained using FLUENT (ANSYS 12.0) ( Figure
7.3) indicates the flow with maximum velocity near the throat section of the nozzle
and the velocity decreases non-linearly, and maintain the critical velocity at the outlet
of nozzle. Different colors (or shades) indicate different values of the velocity. Figure
7.3(a) show the distribution of the velocity in the nozzle. The velocity at the inlet of
nozzle is low as compared to throat section and has maximum value at throat section.
Figure 7.3(b) also shows the graphical representation of the velocity through the
nozzle. The velocity at the outlet of the nozzle is about 945ms-1.
138
7.4.1.2 Pressure distribution
Figure 7.4(a) shows the resultant pressure contours through the nozzle along the total
length of the nozzle. The resultant pressure at the entrance of the nozzle is maximum
and decreases along the length towards the exit of the nozzle. Different colors (or
shades) of contour indicate different values of the resultant pressure. This pressure
variation can be believed to be the cause of splat formation on the work piece when
processed by LPGDS. The resultant pressure near about the powder feeder is below
the atmospheric pressure. The pressure at the outlet of the nozzle N1 is about 1.5 bars.
7.4.1.3 Temperature distribution
Figure 7.5(a) shows the temperature distribution through the LPGDS nozzle. The
contours of the temperature are shown through the nozzle. It shows that the temp at
inlet is maximum and when the powder material mixes with the carrier gas, the
powder is at room temperature. The critical temperature is attained at the outside of
nozzle. The measured temperature at the outlet of the nozzle is 555K.
7.4.2 for nozzle N2
7.4.2.1 Velocity distribution
Figure 7.6 (a) shows the velocity distribution through the nozzle N2 having
configuration in the Table 7.1. It shows that the velocity near about the throat is
maximum and the final velocity attained is just near about 700 ms-1.
7.4.2.2 Pressure distribution
Figure 7.7 shows the distributions of the pressure through the nozzle. The different
colors and shades show the different values of pressure into the nozzle. The pressure
near about the injector is less than atmospheric pressure. The pressure at the outlet of
the nozzle is about 1.25 bars.
7.4.2.3 Temperature distribution
Figure 7.8 shows the distribution of temperature through the nozzle along the length.
The temperature at the inlet of nozzle is about 623K and the powder material at the
139
room temperature. Figure 7.8(b) shows the graphical representation of the temperature
distribution. The outlet temperature for the nozzle is 548 K.
140
(a)
(b)
Figure 7.3: Velocity distribution through the nozzle (contours of velocity) N1
141
(a)
(b)
Figure 7.4: Pressure distribution through the nozzle N1
142
(a)
(b)
Figure 7.5: Temperature distribution through the nozzle N1
143
(a)
(b)
Figure 7.6: Velocity distribution through nozzle N2
144
(a)
(b)
Figure 7.7: Pressure distribution through the nozzle N2
145
(a)
(b)
Figure 7.8: Temperature distribution through nozzle N2
146
7.4.3 for nozzle N3
7.4.3.1 Velocity distribution
Figure 7.9(a) shows the velocity distribution into the nozzle N3 as discussed in the
Table 7.1. The different colors of the nozzle show the different values of the velocity
in to the different parts of the nozzle. Figure 7.9(b) also shows the graphical view of
the velocity through the nozzle N3. The velocity calculated at the outlet of the nozzle
is about 550 ms-1.
7.4.3.2 Pressure distribution
Figure 7.10 shows the pressure distribution through the nozzle. The pressure near
about the injector is less than 1 bar. It means the powder material is being enabled to
suck into the nozzle. The pressure at the outlet of the nozzle is about 1.3 bars.
7.4.3.3 Temperature distribution
Figure 7.11 shows the temperature distribution through the nozzle at the different
sections of the nozzle. Figure 7.11(b) also shows the graphical view of the
temperature through the nozzle. It shows the higher temperature at the inlet and the
critical temperature at the outlet of the nozzle. The temperature at the outlet of the
nozzle is about 520 K.
7.4.4 for nozzle N4
7.4.4.1 Velocity distribution
Figure 7.12 shows the velocity distribution through the nozzle. It shows that the flow
attains very high velocity at the throat of the nozzle and is low at the outlet. The graph
for velocity vs. position is also shown below in Figure 7.12(b). The velocity measured
at the outlet of the nozzle is about 510 ms-1.
7.4.4.2 Pressure distribution
In Figure 7.13, the pressure distribution through the nozzle N4 is shown. The
dimensions of the nozzle N4 is discussed in Table 7.1. The pressure near about the
147
injector is below the atmospheric pressure. The pressure measured at the outlet of
nozzle is 1.2 bars.
7.4.4.3 Temperature distribution
Figure 7.14 shows that the inlet temperature is the temperature of the carrier gas. The
powder material is mixed at room temperature with the air and the outlet temperature
is less than the inlet temperature. The temperature at the outlet of the nozzle N4 is 530
K.
148
(a)
(b)
Figure 7.9: Velocity distribution through the nozzle (contours of velocity) N3
149
(a)
(b)
Figure 7.10: Pressure distribution through the nozzle N3
150
(a)
(b)
Figure 7.11: Temperature distribution through nozzle N3
151
(a)
(b)
Figure 7.12: Velocity distribution through the nozzle (contours of velocity) N4
152
(a)
(b)
Figure 7.13: Pressure distribution through the nozzle N4
153
(a)
(b)
Figure 7.14: Temperature distribution through nozzle N4
154
7.4.5 for nozzle N5
7.4.5.1 Velocity distribution
Figure 7.15 shows the velocity distribution through the nozzle. It shows that the flow
attains very high velocity at the outlet of the nozzle which is due to small injector
diameter. The graph for velocity vs. position is also shown below in Figure 7.15 (b).
The velocity at the outlet of the nozzle is 1100 ms-1.
7.4.5.2 Pressure distribution
Figure 7.16 shows the pressure distribution through the nozzle N5. The dimensions of
the nozzle N5 is discussed in Table 7.1. The pressure near about the injector is below
the atmospheric pressure. The pressure at the outlet of the nozzle is 1.1 bars.
7.4.5.3 Temperature distribution
Figure 7.17 shows the distribution of the temperature into the nozzle. The different
colors show the different values of the temperature through the nozzle. Temperature
at the outlet of the nozzle is 540 K.
7.4.6 for Nozzle N6
7.4.6.1 Velocity distribution
Figure 7.18(a) shows the velocity distribution through the nozzle. The different colors
show the different values of velocity at different positions. The graph for velocity vs.
position is also shown below in Figure 7.18(b). The velocity at the outlet of the nozzle
is 540 ms-1.
7.4.6.2 Pressure distribution
Figure 7.19 shows the pressure distribution through the nozzle N6. The dimensions of
the nozzle N6 is discussed in Table 7.1. The pressure near about the injector is below
the atmospheric pressure. The pressure at the outlet of the nozzle is 1.30 bars.
155
7.4.6.3 Temperature distribution
Figure 7.20 shows the distribution of the temperature into the nozzle. The different
colors show the different values of the temperature through the nozzle. Temperature
at the outlet of the nozzle is 530 K.
Table 7.7 presents the comparison of outlet velocity, pressure and temperature for the
nozzle configurations under study. From the values, it had been found that nozzle N4
and N6 are optimum according to the study because there is a proper mixing of
powder particles with the carrier gas as a result of which the momentum and heat
transfer from the gas to the particle takes place. In the nozzle N3 there is not proper
mixing of the powder material due to shorter length of the divergent section.
Moreover the pressure at the section of powder injection is below atmospheric which
shows that this pressure gradient helps the powder enter the nozzle from the powder
feeder under the action of gravity resulting in higher velocity of the powder particles
at the exit of the nozzle.
TABLE 7.7: Comparison of results for chosen nozzles
Inlet conditions at
Inlet conditions at
Outlet conditions
Boundary 1
Boundary 2
predicted at
Boundary 3
Carrier
gas
Pi
(bar)
Ti
(K)
Nozzles
N1
Air
10
623
N2
Air
10
N3
Air
N4
Powder
material
Size
(µm)
Mi
(gs-1)
Ti2
(K)
Vo
(ms-1)
Po
(bar)
To
(K)
Cu
50
0.05
300
945
1.5
555
623
Cu
50
0.05
300
680
1.25
548
10
623
Cu
50
0.05
300
550
1.25
520
Air
10
623
Cu
50
0.05
300
510
1.20
530
N5
Air
10
623
Cu
50
0.05
300
1100
1.10
540
N6
Air
10
623
Cu
50
0.05
300
540
1.30
530
156
(a)
(b)
Figure 7.15: Velocity distribution through the nozzle (contours of velocity) N5
157
(a)
(b)
Figure 7.16: Pressure distribution through the nozzle N5
158
(a)
(b)
Figure 7.17: Temperature distribution through nozzle N5
159
(a)
(b)
Figure 7.18: Velocity distribution through the nozzle (contours of velocity) N6
160
(a)
(b)
Figure 7.19: Pressure distribution through the nozzle (contours of velocity) N6
161
(a)
(b)
Figure 7.20: Temperature distribution through the nozzle (contours of velocity) N6
162