NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Chapter 3 Jet Flow Theory
§ 1 Theory of Isothermal Free Jet
Free jet,Jet without the limitation of solid boundary
1,General Properties
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
1) turning section
OH dS 2
14?
2) beginning region and basic region
3) core with velocity of beginning one
4) turbulent boundary layer - the part between
inner boundary (v=vo) and outer boundary (v=0)
5) polar point,cross point of outer boundary
surface
6) Similarity
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
In basic region,the distribution of non-dimensional
velocity is complete similar
i,e,at corresponding points
21
21
?
?
?
?
?
?
?
?
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
?
?
?
?
?
?
m
x
m
x
cc
v
v
v
v
y
y
y
y
Where vm - velocity at x axis
yc - distance of points with
velocity (1/2)vm with x axis.
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
)()( byfyyfvv
cm
x ??
(3-1)
b - half of jet width
In beginning region,the velocity distribution in
boundary layer is similar at different sections
)()(
0 b
yf
y
yf
v
v
b
cx ?
?
??
Where,
00 9.01.0
0
vvb
c
yy
yyy
???
???
(3-2)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
7) Attenuation of vm
)(,67.0
,67.0
000
0
0
r
as
f
v
v
r
as
vv
r
as
x
m
??
??
(3-3)
2,Turbulent properties
1) in core region,y↑,x↑,nt↑
2) in basic region,ntmax > 3 nto
3) vx >> vy ≈vz,v’x ≈ v’y ≈ v’z
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
In core region,
)(
'
'
)(
'
'
2
1
b
y
f
v
v
b
y
f
v
v
ym
y
xm
x
?
?
(3-4)
4) mixing length
ltx/b = 0.11
lty/b = 0.10
(3-5)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
3,Outer boundary and iso-velocity line
In basic region
dy
dv
l
dt
db
dy
dv
lv
vvvv
dy
dv
v
dt
db
x
t
x
tx
xyxy
x
y
??
?
???
?
'
)''('',0
'
22
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
From the similarity of velocity
m
tt
mx
v
dt
db
c o n s t
b
l
b
l
b
v
dy
dv
??
???
?
?
?
?
?
??
?
?
?
?
?
?
.,,
21
Also,
xc o n s tb
c o n s t
dx
db
dx
db
v
dt
db
v
dt
dx
dt
dx
dx
db
dt
db
m
m
??
?
?
?
?
?
(3-6)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
For
.
.
c o n st
v
v
c o n st
b
y
m
x ?
??
?
?
?
?
?
(3-7)
i,e,
.
.
c o n st
v
v
c o n st
x
y
m
x ?
? (3-8)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Also for beginning region
.
.
c o n st
x
y
c o n st
b
y
?
?
.
.
0
0
c o n st
v
v
c o n st
v
v
x
x
?
? (3-9)
(3-10)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
§ 2,Semi.- Theory and Semi-Experiencial
Formular for Free Jet
1,Momentum Transfer of Isothermal Free Jet
In free jet,
)/(
.
0
2
0
2
smkgm o m e n t u mJ
c o n s tdfvJ
dy
dp
F
x
??
??
?
? ?
(3-11)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
1) Free jet with circular section
? ? ???
?
b b
xx c o n s ty d yvy d yvJ
y d ydf
0 0
22,22
2
????
?
(3-12)
Initial Momentum at exit of nozzle
20200 RvJ ???
When the fluid is non-compressible
? ???
?
?
???
?b x
R
dy
R
y
v
v
0
00
2
0
12
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Define,
? ? ???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
?
?
?
?
?
?
???
??
1
0
1
0
22
0
2
000
2
0
00
1)(2
,
????
?
d
v
v
R
b
v
v
d
R
b
R
b
v
v
v
v
v
v
v
v
v
v
b
y
m
xmm
m
x
m
m
xx
From experiment,
? ???
?
?
?
?
?
?
??
1
0
2
22/3
0 6 4 6.0
)1(
??
?
d
v
v
v
v
m
x
m
x
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
mv
v
R
b 0
0
3.3??
(3-13)
At transition section,vm = v0
3.3
0
???
?
?
???
?
trsR
b (3-14)
The quantity of gas passes any section
? ???????????????????? b
m
xm
x dv
v
R
b
v
vvRdyyvG
0
1
0
00
0
2
022 ????
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
m
m
x
mm
v
v
G
G
d
v
v
v
v
v
v
R
b
vRG
0
0
1
0
2
0
2
0
2
0
0
2
00
13.2
0 9 8 5.0
8.103.3
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ??
??
(3-15)
At transition section,vm = v0
13.2
0
???
?
?
???
?
trsG
G (3-16)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Define average velocity of jet at a section
mav
mm
av
av
vv
v
v
G
G
v
v
R
b
b
R
G
G
v
v
bfRf
f
G
v
f
G
v
2.0
13.2,3.3
,
,
0
0
0
0
2
0
00
22
00
0
0
0
??
??
?
?
?
?
?
?
??
???
?
?
? ??
(3-17)
Which indicates the section average velocity equals
20% of vm.
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
For calculating b,G,vav,vm = f (x) is needed
define
ax
b
ax
y
b ?? ??,
a - experience constant
? ??
??
?
f
ax
y
f
x
y
f
v
v
d
v
v
R
ax
v
v
m
x
m
xm
b
??
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
'
5.0
0
22
0
2
0
From theory of jet,
536.0
2
0
???
?
?
???
?? ??? d
v
vb
m
x
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
0
0
96.0
R
axv
v m
?? (3-17)
At transition section,vm = v0
Distance of transition from polar point xn
a
Rx
n
096.0?
(3-18)
Depth of polar point s0
0
0
as
R
ax
b ??
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
From jet theory,b/ax = ?b = 3.4
29.0
0
0 ??
R
as (3-19)
The distance of a section to exit of nozzle is s
000
0
0
29.0 RasRasRasRax ???? (3-20)
29.0
96.0
0
0 ?
??
R
asv
v m (3-21)
a = 0.07 ～ 0.08
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
These relations are valid in the basic region of free
jet with circular section.
Other two correlations for velocity distribution
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
??
x
y
v
v
x
y
v
v
m
x
m
x
1 7 0.0
c o s15.0
96e x p
2
?
(3-22)
(3-23)
Semi-angle of jet au = 4.85°
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
2) Plane Parallel Free Jet
y
x
b
b0
s0 s
x
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
With unit thickness 1
?? b x dyvJ 0 2?
From momentum conservation,
? ??b x c o n s tbvdyv0 0202,??
(3-24)
Assume gas is non-compressible
0
0
2.1
b
axv
v m
?
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
From jet theory,?b = b/ax = 2.4
2
0
0
46.3 ??
?
?
???
??
mv
v
b
b (3-25)
On transition section,vm = v0
46.3
0
?bb
(3-26)
Distance of transition section to polar point xn
44.1
0
?bax n
or
a
bx
n
044.1?
(3-27)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Assume diffusion angle is same in basic region and
beginning region,
41.0
2.1
41.0
41.0
4.2
0
0
000
0
0
0
0
0
0
?
??
????
??
??
?
b
asv
v
b
as
b
as
b
as
b
ax
b
as
ax
b
as
b
ax
b
m
b
??
(3-28)
(3-29)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
a - experiment constant,≈ 0.1 - 0.11
Gas quantity pass any section
?
?
??
?
y
m
x
y
x
v
v
dyv
bvG
G
dyvG
0
0
000
0
42.1
1
2
(3-30)
At transition section,vm = v0
mav
av
av
t r s
vv
b
b
G
G
f
f
G
G
v
v
b
b
f
f
f
G
v
G
G
41.0
,
42.1
0
0
0
0
00
0
?
?????
??
?
?
?
?
?
?
?
?
?
? (3-31)
(3-32)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
From experiment data of Van der Hegge Zijnen,
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
x
y
v
v
x
y
v
v
m
x
m
x
1 9 2.0
c o s15.0
75e x p
2
?or
(3-33)
2,Heat Transfer of Non-Isothermal Free Jet
From Taylor’s turbulence theory
m
x
m v
v
T
T ?
?
? (3-34)
Where ?T = T - Th
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
When the entropy of the jet is calculated on temperature
difference,
? ??m c o n s tT d m0, (3-35)
For circular section jet,
? ???b x RvTy d yTv0 20002 ??
Change to
? ???????????????? ax
b
m
x
m
mm
ax
yd
ax
y
v
v
T
T
R
ax
v
v
T
T
0
2
000
5.0)(
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Define ? = y/ax,?b = b/ax = 3.4 (from jet theory)
29.0
7.0
29.0
52.0
00
2/3
0
0 ?
?
??
?
?
??
?
?
??
?
?
??
?
?
?
?
?
?
? R
as
d
v
v
R
asT
T
b
m
x
m
?
??
(3-36)
(a = 0.07 - 0.08)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
For plane parallel jet,
41.0
04.1
0
0 ?
?
?
?
b
asT
T m (a = 0.1 - 0.11) (3-37)
It can be proved that the diffusion of combined
components is completely same with temperature,
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Chapter 3 Jet Flow Theory
§ 1 Theory of Isothermal Free Jet
Free jet,Jet without the limitation of solid boundary
1,General Properties
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
1) turning section
OH dS 2
14?
2) beginning region and basic region
3) core with velocity of beginning one
4) turbulent boundary layer - the part between
inner boundary (v=vo) and outer boundary (v=0)
5) polar point,cross point of outer boundary
surface
6) Similarity
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
In basic region,the distribution of non-dimensional
velocity is complete similar
i,e,at corresponding points
21
21
?
?
?
?
?
?
?
?
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
?
?
?
?
?
?
m
x
m
x
cc
v
v
v
v
y
y
y
y
Where vm - velocity at x axis
yc - distance of points with
velocity (1/2)vm with x axis.
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
)()( byfyyfvv
cm
x ??
(3-1)
b - half of jet width
In beginning region,the velocity distribution in
boundary layer is similar at different sections
)()(
0 b
yf
y
yf
v
v
b
cx ?
?
??
Where,
00 9.01.0
0
vvb
c
yy
yyy
???
???
(3-2)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
7) Attenuation of vm
)(,67.0
,67.0
000
0
0
r
as
f
v
v
r
as
vv
r
as
x
m
??
??
(3-3)
2,Turbulent properties
1) in core region,y↑,x↑,nt↑
2) in basic region,ntmax > 3 nto
3) vx >> vy ≈vz,v’x ≈ v’y ≈ v’z
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
In core region,
)(
'
'
)(
'
'
2
1
b
y
f
v
v
b
y
f
v
v
ym
y
xm
x
?
?
(3-4)
4) mixing length
ltx/b = 0.11
lty/b = 0.10
(3-5)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
3,Outer boundary and iso-velocity line
In basic region
dy
dv
l
dt
db
dy
dv
lv
vvvv
dy
dv
v
dt
db
x
t
x
tx
xyxy
x
y
??
?
???
?
'
)''('',0
'
22
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
From the similarity of velocity
m
tt
mx
v
dt
db
c o n s t
b
l
b
l
b
v
dy
dv
??
???
?
?
?
?
?
??
?
?
?
?
?
?
.,,
21
Also,
xc o n s tb
c o n s t
dx
db
dx
db
v
dt
db
v
dt
dx
dt
dx
dx
db
dt
db
m
m
??
?
?
?
?
?
(3-6)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
For
.
.
c o n st
v
v
c o n st
b
y
m
x ?
??
?
?
?
?
?
(3-7)
i,e,
.
.
c o n st
v
v
c o n st
x
y
m
x ?
? (3-8)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Also for beginning region
.
.
c o n st
x
y
c o n st
b
y
?
?
.
.
0
0
c o n st
v
v
c o n st
v
v
x
x
?
? (3-9)
(3-10)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
§ 2,Semi.- Theory and Semi-Experiencial
Formular for Free Jet
1,Momentum Transfer of Isothermal Free Jet
In free jet,
)/(
.
0
2
0
2
smkgm o m e n t u mJ
c o n s tdfvJ
dy
dp
F
x
??
??
?
? ?
(3-11)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
1) Free jet with circular section
? ? ???
?
b b
xx c o n s ty d yvy d yvJ
y d ydf
0 0
22,22
2
????
?
(3-12)
Initial Momentum at exit of nozzle
20200 RvJ ???
When the fluid is non-compressible
? ???
?
?
???
?b x
R
dy
R
y
v
v
0
00
2
0
12
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Define,
? ? ???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
?
?
?
?
?
?
???
??
1
0
1
0
22
0
2
000
2
0
00
1)(2
,
????
?
d
v
v
R
b
v
v
d
R
b
R
b
v
v
v
v
v
v
v
v
v
v
b
y
m
xmm
m
x
m
m
xx
From experiment,
? ???
?
?
?
?
?
?
??
1
0
2
22/3
0 6 4 6.0
)1(
??
?
d
v
v
v
v
m
x
m
x
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
mv
v
R
b 0
0
3.3??
(3-13)
At transition section,vm = v0
3.3
0
???
?
?
???
?
trsR
b (3-14)
The quantity of gas passes any section
? ???????????????????? b
m
xm
x dv
v
R
b
v
vvRdyyvG
0
1
0
00
0
2
022 ????
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
m
m
x
mm
v
v
G
G
d
v
v
v
v
v
v
R
b
vRG
0
0
1
0
2
0
2
0
2
0
0
2
00
13.2
0 9 8 5.0
8.103.3
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ??
??
(3-15)
At transition section,vm = v0
13.2
0
???
?
?
???
?
trsG
G (3-16)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Define average velocity of jet at a section
mav
mm
av
av
vv
v
v
G
G
v
v
R
b
b
R
G
G
v
v
bfRf
f
G
v
f
G
v
2.0
13.2,3.3
,
,
0
0
0
0
2
0
00
22
00
0
0
0
??
??
?
?
?
?
?
?
??
???
?
?
? ??
(3-17)
Which indicates the section average velocity equals
20% of vm.
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
For calculating b,G,vav,vm = f (x) is needed
define
ax
b
ax
y
b ?? ??,
a - experience constant
? ??
??
?
f
ax
y
f
x
y
f
v
v
d
v
v
R
ax
v
v
m
x
m
xm
b
??
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
'
5.0
0
22
0
2
0
From theory of jet,
536.0
2
0
???
?
?
???
?? ??? d
v
vb
m
x
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
0
0
96.0
R
axv
v m
?? (3-17)
At transition section,vm = v0
Distance of transition from polar point xn
a
Rx
n
096.0?
(3-18)
Depth of polar point s0
0
0
as
R
ax
b ??
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
From jet theory,b/ax = ?b = 3.4
29.0
0
0 ??
R
as (3-19)
The distance of a section to exit of nozzle is s
000
0
0
29.0 RasRasRasRax ???? (3-20)
29.0
96.0
0
0 ?
??
R
asv
v m (3-21)
a = 0.07 ～ 0.08
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
These relations are valid in the basic region of free
jet with circular section.
Other two correlations for velocity distribution
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
??
x
y
v
v
x
y
v
v
m
x
m
x
1 7 0.0
c o s15.0
96e x p
2
?
(3-22)
(3-23)
Semi-angle of jet au = 4.85°
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
2) Plane Parallel Free Jet
y
x
b
b0
s0 s
x
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
With unit thickness 1
?? b x dyvJ 0 2?
From momentum conservation,
? ??b x c o n s tbvdyv0 0202,??
(3-24)
Assume gas is non-compressible
0
0
2.1
b
axv
v m
?
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
From jet theory,?b = b/ax = 2.4
2
0
0
46.3 ??
?
?
???
??
mv
v
b
b (3-25)
On transition section,vm = v0
46.3
0
?bb
(3-26)
Distance of transition section to polar point xn
44.1
0
?bax n
or
a
bx
n
044.1?
(3-27)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Assume diffusion angle is same in basic region and
beginning region,
41.0
2.1
41.0
41.0
4.2
0
0
000
0
0
0
0
0
0
?
??
????
??
??
?
b
asv
v
b
as
b
as
b
as
b
ax
b
as
ax
b
as
b
ax
b
m
b
??
(3-28)
(3-29)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
a - experiment constant,≈ 0.1 - 0.11
Gas quantity pass any section
?
?
??
?
y
m
x
y
x
v
v
dyv
bvG
G
dyvG
0
0
000
0
42.1
1
2
(3-30)
At transition section,vm = v0
mav
av
av
t r s
vv
b
b
G
G
f
f
G
G
v
v
b
b
f
f
f
G
v
G
G
41.0
,
42.1
0
0
0
0
00
0
?
?????
??
?
?
?
?
?
?
?
?
?
? (3-31)
(3-32)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
From experiment data of Van der Hegge Zijnen,
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
x
y
v
v
x
y
v
v
m
x
m
x
1 9 2.0
c o s15.0
75e x p
2
?or
(3-33)
2,Heat Transfer of Non-Isothermal Free Jet
From Taylor’s turbulence theory
m
x
m v
v
T
T ?
?
? (3-34)
Where ?T = T - Th
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
When the entropy of the jet is calculated on temperature
difference,
? ??m c o n s tT d m0, (3-35)
For circular section jet,
? ???b x RvTy d yTv0 20002 ??
Change to
? ???????????????? ax
b
m
x
m
mm
ax
yd
ax
y
v
v
T
T
R
ax
v
v
T
T
0
2
000
5.0)(
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Define ? = y/ax,?b = b/ax = 3.4 (from jet theory)
29.0
7.0
29.0
52.0
00
2/3
0
0 ?
?
??
?
?
??
?
?
??
?
?
??
?
?
?
?
?
?
? R
as
d
v
v
R
asT
T
b
m
x
m
?
??
(3-36)
(a = 0.07 - 0.08)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
For plane parallel jet,
41.0
04.1
0
0 ?
?
?
?
b
asT
T m (a = 0.1 - 0.11) (3-37)
It can be proved that the diffusion of combined
components is completely same with temperature,
