A THEORETICAL
AND EXPERIIw1ENTAL
INVESTIGATION
OF so~m ASPECTS
OF THE HILSCH TUBE
by
WILLIAM
L. SLAGER
Submitted in Partial Fulfillment
of the Requirements for the
Degree of Bache1o~ of Science
at the
MASSACHUSETTS
INSTITUTE
OF TECHNOLOGY
JUne, 1959
)
.~
Signature of Author
f
)
..
\
,
t
Dept. of Mechaftica~ 1~l)G~A1,~rlngJl1
MF1-Y25,1959
'I
Certified by
~
Accepted by
.Chairman,
(/
Thesis S~)rrvisor
\
.~
Depar~~~al
committee
on theses
32 Hereford Street
Boston 15, ~,lassachusetts
~fay
25,1959
Professor Alvin Sloane
Secretary of the Faculty
Massachusetts Institute of Technology
Cambridge 39, Massachusetts
Dear Sir,
In accordance with the requirements of the
Massachusetts Institute of Technology, I hearby
submit my thesis entitled "A Theoritical and
Experimental Investigation of Some Aspects of the
Hilseh Tube", in partial fulfillment of the
requirements for the degree of Bachelor of Science~
in Mechanical Engineering.
Sincerely yours,
t.
William L. Slager
Acknowledgement
The major part of this thesis was carried out
under the guidance of Professor J. L. Smith,
of the
M. I. T. Mechanical Engineering Department.
His knowledge and previous experienc~gained
in working with cyclone seperators and allied equipment using the vortex principle, were invaluable and
continually served to stimulate my interest.
I would also like to thanlcMr. C. B. Dolber and
especially Mr. C. W. Christiansen, who helped me
immensely in the construction of the experimental
equipment.
Abstract
A theor'.eticalanalysis of the internal operation
of the Hilsch tube or vor~ex refrigerator was carried
out.
The analysis assumed that a major factor
influencing the performance of the device "'lasthe
frictional effects of wall shear forces inside the
refrigerator.
The type of solution obtained by
assuming an incompressible working fluid has been
visually observed in previous experiments where this
assumption \trasvalid.
A method of obtaining a solution when a
compressible fluid is used is outlined.
A vortex chamber attachment having as its purpose
the pre-acceleration of the working fluid before
entrance into the Hilsch tube was analyzed and a theoretical design criteriOn was obtained.
The attachment was tested experimentally and was
foUnd to deliver the same flow rate of refrigerated
fluid
~OO F colder then a similar vortex tube with no
such device, while maintaining the initial inlet
pressure.
~he effect of placing various circular flow reI
stricting weirs at the hot air exit of the refrigerator
was observed experimentally to have, an apparently
Table of Contents
............................ page 1
page 9
Purpose of Thesis ............................
Theoretical Analyses. . .. . . . . . . . .. . . .. .. ... .... page 10
page'll
Part I .........................
Introduction •.••••
Discussion of Results of Part I.
page 24
... ... . . .... .. . . . . . . .. . .. . . . . ... page
.
. page
Part III.
page
Experimental Work ............................
Discussion of Experimental Results .. .. .. page
. page
Bibliography •••• . . . .. . . .
. ..... .. .. . . .. .. . ... page
Appendix •....•••
Part lIe •
'
32
37
45
49
55
57
--1--
Ilntroduction
Historical
The early history of the Hilsch tube may be best
summarized by the following excerpt from Scientific
American.
«(1 )'
"Shortly after the end of World War II...
":lordcame
to the U.S. that t~e Germans had developed a remarkably simple device with which one could reach temperatures as low as the freezing point of mercury. The
device 1-laSsaid to consist only of an air compressor
and three pipes.
The details of construction were not
available~but it was reported that the device had in
effect realized Maxwell's demon, a fanciful means of
seperating heat from cold without work."
n •••
When physicists heard that the Germans had
developed a device which could achieve low temperatures
they were intrigued though obviously skeptical.
One
physicist Robert M. Milton, investigated the matter
first hand for the U.S. Navy."
"Milton discovered that the device was most
ingenious, though not quite as miraculous as had been
rumored. It consisted of a T-shaped assembly of pipe
joined by a novel fitting (Fig.l).
\then compressed
air is admitted to the leg of the T, hot air comes out
- 2 -
of one arm of the T and cold air out of the other arm I
Obviously"however~work must be done to compress the
air."
"Theorigln
of the device is obscure. The prin-
ciple is said to have been discovered by a Frenchman
who left some early experimental models in the path of
the German Army when France was occupied.
These were
turned over to a German physicist named Rudolf Hilsch
who was working on low temperature devices for the
German war effort.
Hilsch made improvements on the
Frenchman's design but found that it was no more efficient then conventional methods of refrigeration in
achieving fairly low temperatures, subsequently the
device became knovm as the Hilsch tube. t1
Hilsch's Investigation
After the war Hi).sch published a paper,"The Use of
the Expansion of Gases in a Centrifugal Field as a
in "Vlhichhe presented the results
Cooling Process t1
(~)
of his experimentation with the device(Fig. 1).
He
found that by introducing compressed air into the pipe
.
0
containing his ingenious spiral mechanism at 68 F , he
could obtain by suitably varying the significent parameters, a maximum cold air temperature of approximately
o
-60 F and a maximum
o '
.
\;ho"f,
air temperature of approxi-
mately 390 F although not simUltaneously.
- 3
..... hot.. pspe with
at left c,nd
~stopcock
Fig •. I_
-4 -
The parameters which influenced the.tempe.ratures
of.the gas flows he found t9 be; 1) the temperatureJ1
pressure~and flow rate of the compressed air into the
pipe; 2) the ratio of the diameters of the hot pipe,
cold air diaphragm and compressed air inlet nozzle;
3) the ratio of distribution of flow in both dlrectlon~
which he varied by setting a valve on the end of the
hot pipe and; 5) the external pressure outside of the
device.
The length of the cold pipe had no effect in
the tube's performance.
He also observed that.by increasing the size of
the hot.pipe, inlet nozzle and diaphragm in geometric
proportion, lower values of cold temperatures were
'obtained at the ~ame inlet pressure and the thermodynamic efficiency of t~e device increased.
In spite of the impressive performance of the
Hllsch tUbe, it's efficiency was found to be low.
The
best efficiency Hllsch could obtain with the larger
size pipe was 20%.
Various Theories About the Hilsch Tube
\ Hilsch successfully recorded the external data
which could be obtained from the devic~ but.presented
only a qualitative picture of the phenomenon occuring
inside the tube.
No one has yet adeQ.uately analyzed
the internal operation of this device mathematically,
- 5 -
and there seems to be some.disagreement about the.signi).
ficant aerodynamic .mechanism "Thich is responsible for
the temperature seperation~
In the usual experimental setup (FiS. 2), the air.J
which is introduced by a nozzle or nozzles tangentially
placed on the tube wall, spirals do~m the hot tube. The
flow through the cold air diaphragm is regulated by
changing the back pressure in the hot pipe 'Hith a valve
attached at it's end.
Experimentally it is found that the motion of the
air near the tube walls resembles that of an irrotational
or free vortex and the motion of the air in the center
of the pipe is a rotational or forced vortex.
A typical tangential velocity distribution obtained in the Hilsch tube is shown in Fig. 2.1illustrating
r
the nature of the transition from a free to a forced
vortex within the tube.
According to one theory, as the tangential velocity
of the outer irrotational flow increases with decreasing
radius, viscous effects become important and the shear
forces between the air particles act to reduce the velocity ,of the inner layers of air such that near the
centerline of the pipe there is a core of air rotating
as a solid body.
During this process the. inner layers
surrender part of ~heir kinetic energy to the layers
near" the \'lalland subseQuently some of this cold core
- 6 --
T'j
.o'st-rl butlon
f.c-rce.J
\10"'.
1'\
~
'
-
+~
~
f' ccd _ Va
.~----I
.........
,/
/
..........
-----------.---.-----------.--...
----------.-----------l
I
I
I
l
.._
z
I
Z+AZ
- 7 -
is forced out the cold air diaphrgm by pressure forces.
(3)
Another theory contends that near the wall the
,
flow
outer layers of the irrotational~are
slowed by boundary-
laye~ stresses to a Rmall velocity.
The intermediate
layers '-1i
th a higher rotational veloci ty transfer some
of their energy to the outer layers:in an'attempt' to
speed them up.
As the flow progresses dOlvn the tube,
the overall tangential velocity profiles decrease due
'to the accumulated frictional "effect. Aga.in the
overall result is to create a core of cold air near the
center of' the pipe and a hot one near the wall.
(4)
Another mechanism of heat transfer theorized to be
present v1ithin the tube again involves the bo~darylayer.
Molecules of air entering the boundary layer on
the wall are slowed to a small velocity.
Because of
their lack of angular momentum these particles are unstable with respect to the rest of the centrifugal
field.
They tend to pile up into a packet,which is
eventually swept away from the wall and moves to a more
stable position near the center of the pipe, where the
angUlar momentum is also small.
The net result of the
process is the accumalatlon of low enerGY molecules in
the center of the pipe and high energy ones near the',-,.
wall •.
, • (4)
'.'
- 8 -
Difficulty of a-Direct Solution
It is quite possible that all three of the previously mentioned theories :figure into an adequate solution of'the Hi1sch tUbe.
s
There seems to be agreement
-
that in the phenomenon vi"cous forces', \'lal1shear- stresses
and compressibility effects are not neglible and occur
in at least two dimensions.
The only mathematical tool which can handle these
effects simu~taneous1y is the Navier-Stokes equations
coupled with the energy equations for a Newtonian fluid.
However, these equations become, complex in the extreme
when all the above effects are considered, and a~ present only special cases of these equations have been
solved by "neglecting one or more of the above effects.
Even if a solution were discovered it might still
only be approximate because of the high Reynold's
numbers and correspondingly turbulent flow within the
tube.
\
- 9 -
Purpose of Thesis
One of the objects of this thesis was to study
the second of the aforementioned theories.
5y con-
sidering wall shear forces as a major factor in the
performance of the ~ilsch tube, it was hoped that the
results obtained would clarify some aspects of the
internal operation of this device.
Another aim was to tryout
chamber pre-accelerating
experimentally a vortex
device which seemed promising
,.;henpx'evious data vias considered.
This device v[ould
increase the inlet air tangential velocity in a manner
'similar to a converging-diverging
nozzle, but had the
advantage of introducing the flow evenly at the perimeter of the hot pipe while
J
.using the same inlet pres-
sure as a tube with no such device.
It was hoped that
such a device might increase the efficiency of the
Hilsch tube.
-10
.Theoretical Analysis
List of symbols not otherwise defined in text
f - pipe friction factor - dimensio~ess
F - force - lbf
H - Bernoulli constant or 'total head' - ft.
k - dimensionless ratio of specific heats of air - 1.40
Greek symbols
€ - ratio of cold fluid flow to total flow
~ - fluid density - lbm/
r - wall
ft3
2
shear stress - lbf/ In
Subscripts
o - applies to wall of hot pipe (Fig.
3)
I - applies to outer radius of vortex pre-accelerator
(Fig. 8)
c - pertains to boundary between rotational and
- 11 -
and irrotational flow (Fig. 3)
h - hot fluid
r - radial direction
s - stagnation conditions
.w - inside radius of weir
z - axial direction
Q (':* _
tangential direction
refers to points where the Mach number is one.
- 12 -
Part I
Derivation of the Flow Pattern of the Hot Fluid
This study presents the theoretical results obtained assuming that an important factor in the operation
of the Hilsch tube is the frictional effects of wall
shear stresses.
It is limited to an incompressible fluid
and deals specifically with the hot fluid inside the pipe.
The following assumptions were made:
#1
The flow pattern was broken up into two regions,
an outer irrotational layer and an inner rotational
coreJseperated by a boundary along which the pressure in the fluid was assumed to be constant.
furthur
It was
assumed that along the boundary the radial
velocity ''laszer.o~ so that no fluid crossed the boundary.
#2
The fluid dens! ty ''lastaken to be constant.
#3
neslec ted..!
All shear stresses in the fluid '\-lere
except those at the wall vlhlch ,..,ere taken to be
2
1=
#4
f~Vg
• (5)
8
The tansential.velocity
was assumed to be much
greater then either the radial or axial velocities~
#5
This led to the assumption that the pressure
gradient in the radial direction 1s given by
d'P-
~
2
.Jlg~'
= ~ ;:-.
- 13--
#6
The form of the tangential velocity distribution
was taken as
V
9
to be determined.
#7
= C;Z)
where C(z) was a function
At any given cross section of the hot pipe~the
axial velocity in the outer core was assumed to have
a constant value given by the continuity equation.
#8
The fluid flow was assumed to be steady and the
pipe walla adiabatic_
Figure 3 shows the nature of some of these assump-
tiona and illustrates an element of the flow within the
hot pipe.
With the above assumptions the continuity, linear
and angular momentum equations may now be applied by
utilizing the integral method, and yield a solution for
the flow pattern in the outer layers of hot fluid.
The continuity equation is
1a)
~~
(V-n)
Jcontrol
c1A
(6)
•
surface,
Taking a small cross section ,of the flow at z
haVing a Width ofAz
(Fig. 3)Jthen at z the mass flow
into the element is
rc
1 q ) Wi = / ~ V ( - 211'r dr)
n r
Z
o
At
Z
,;+ AZ the mass flay;out of the element is
c
Jrr
o
Then
d['rc'j
(-21f1'\ dr-) + -
~ V
Z
-
dz
~
a Vz (-211'r dr)
r'
0
- 14 -
dr.,.
dr)
li
out'- win:: 0=, -dzr.fC;~"~Vz;(-21Yr
'
o
,Upon integration,
,
wh
le)
=~
sirieeV,z, .is constant
2
"fe
obtain
'-'2
Vz . (11'ro -1r~c) •
Differentiating equation lc, we obtain another
equation which will be useful later,
2
.2 dVz
o (Tir -1Tr )-::
~
0
c ,dz
'
dre
2fir VzC
dz
J
inserting equation Ie into the left hand side of the
above equation arid simplifing , we have finally,
ld)"
dVz
2'trcwh
'dre
=
~ (ftr2 _ flr2)2 dz
dz
o
c
The angular momentum equation is for steady f~ow,
-
The torque aetine;'on the element in Fig. 3 is
l'Fxr :: 21rr~zt~xro)::
~211'r;~z~
•
At z the integral
/~(V9~r)(ij.n)
~n
'
dA = ]\vgrvZ
ro
(-21T'r dr)
and is direct~d in the positive z direction.
At z
+4Z
- 15 -
Inserting these expressions into the anGular momen.
tum equation 2a and substituting in assumption #6 for
Ve'
2b)
1'ie
arrive at
2 ft r~ '/. == -
~
(~c ~ C( z) Vz
dz'.or o
.
o
E-211 r dr).]
Removing C(z) from beneath the integral sign but
not outside the brackets, the remaining 'integral in eq.
2b is eQual to ths flow rate • Using eq. Ib we find
2c
r
211
dC(z)
r; ~ = ":"'
\'lh---
dz
Since the Reynolds number in a typical Hilsch tube
has'a large value of approximately 5 106 or greater, we
0
"rillmake the additional assumption that the friction
factor f maybe
taken as constant.
Then
'.
.
3a)
~ =
f
_~VQ
8
f'~ C (z) 2
2
:::
8r 02'
Inserting this expression into the angular momentum equation (20)
dC(z)
3b)
dz
=
we have
11f~
C(z)2
4 "Th
Integrating the above equation and choosing the
Vinlet nozzle
initial condition at z
0 , C (z ) = Co
.r
=
=
o
we obtain
30) .
C(z)
- 16 -
Applying the linear momentum equation in the z
direction, 1fIhichis
:z F
~.
4a)
, Z
Then at z
....,.
rcFZ = /
p (- 217'
r:
dr ) "
ro
acting in the positive z direction.
At
z .f.AZ
-F
= / rc -P (-211 r dr)
z--+
0Z
r0
+ -d
dz
ifC
r Pr0
.
(-21l"r
dr) ~~ z
acting in the negative z direction.
The pressure forces exerted along the core boundary
in the positive z direction are
~
Fc
--.
=
211 rcAz
211rc6z
-211r4zP
c
drc
since
Pc .sin
0<.
~
Pc tan 0<.
drc
~
c dz
is negative. -Therefore
dz
~-[r
dz
r
P
o
At.z the momemtum integral is
and at z +.6
Z
the integral is
(-21Yr dr)l az
'j
•
~ 17 -
J~
V(V:;)
out
c1A
Z
. _Ins~rting the above expressions into equation 4a, we
get cancelling 6 z t s.
+
~[.l~v~
QZ
ro
(-21J'r drl} ..
. Removing one of the Vz's from the integral but not
the brackets on the left hand side of the equation, the
remaining integra1 is equal to the f1o,'/'
rate. (eq.- 1b)
Then
4c)
~Jc
r drl]
P (-2ft
=
dVz
-
l'lh -
d
-
2 II r P
drc
-
dz
c c dz
dz L ro
Using assumption #5, we can now find the pressure
distribution. Given
d
5a)
--
d
2
... Vg
P
r
= ~-
C ( Z )2
= ~-----....3
r
r
by integrating this equation we have
~ C( z)
2
~ = - 2 r2
.
+
A(z).
The constant A(z) of the partial r integration being'
a funtion of z.
be determined.
-distribution 1s
rc' P = PCJA(z) may
The final expression for the pressure
Since at all zJr=
- 18 -
2
5b')
P" __..p ~
. 'c-'
0
'
[-=- _ 2-] .
( z)
2
.2
rc
r
2,
Reinserting this expression into equation 4c and
remembering that r'c = rc(z) and is held constant during
the r integration,we get
-
d
{(,'
P
nz.
'""c
1
," ~ C,( z ) 2
-2
+
c
2 r
2
-
(11ro -ffr
-wh -
dVz
2
)
C
- 211' r
dz
P
C
drc
c dz
.. '.
Differentiating this expression and inserting eq. ld
dVz
eliminating --- , we obtain
dz .
+ 1J' ~
c (z )
2
dr c
]
'Q'
dz
+ 2 11~ 0 ( z)
In":'
dC(z)
C
ro
]
=
0
and defining the
Inserting eq. 3b to eliminate
dz
substitution
6)'-
Az
. 11 f ~ Co
=
z
1+
4 't"h
so that
00
drc
-dz
= A
C(z)
z
it f ~ Co
drc
4 vlh
dAz
"'Te get
2
dr c [ 2 11 r c wh
dA
~ (1Tr2 _11];2) 2
.z
.
0
c
2
2
' 11~ Co r 0
r3i\.2
c
z
-
+
11'~ _~~
rcA~
j
=
- 19 -
=
\
[ 1T ~ ;;
/\ z.
_ 11~ + 21i~ In
rc
I' c]
•
ro
1
Rewriting2this
7)
dA
equ~ti~n in a diff~ren~ form we have,
1f_~_C_o_ 1)'~coro]
(211rc v1h
3
3
A
+
-2
2
2
A
z
z
( -.r,-.
r
~ (n'r -11 r )
=
~
2r~(~r~
CeO
dz
Co
\
-11 r;)
c
r
Accordingly
ro
differential
substituting
(7 )
1
U
=~
i\~
di\z .
= - ---
so that
drc
z
2
dU
drc
we have
]u
2
l1t(ro
2 [~_
r
c
c~ [~ eftI';
r3
c
~11r~)
rc
.
+ 2f1~ In rc
ro
The general form of this equation
If
..
8b)
1
dQ
U=-'Q,
drc
.
c ]
as a Bernoulli
\~11ch may be recognized
equation.
+ 211~In-
~2.
c
1
is
•
- 20 -
where A is an arbitrary constant.
Fortunately in eq.7b the numerator is the exact
differential of the denominator, so that eq. 8b. is
satsified ''!hen
8d ) ~
~ (11'r2 -1Tr;)
0 2
[
=
.
+
rc
r
211~ In -2
1
•
.ro
Inserting this expression into eq. 80 which conviently oancells the denominator of h(rc)
and integrating,
vie get
9a)
U
=
Equation 9a may be re\'Tri
tten in dimensionless form by
ro
writing the equation in terms of the groups
and
)'Z
•
The final solution of tp.e flo,vpat t?rn is
then found to be,
10)
'A
.8 [
2
~.YQ
(1 -
2
r )
r~ c
=
1
( 1 - ~)
ro
2
r
In(r~ ) ]
•
:+
A
- 21 -
Determination of A
The. solution for the flo,-;pattern , eq. 10, is
however not as yet complete since neither of the boundary conditions
z
=
0 ,
rc
=
ro
z =00, rc
or
=0
yield a finite value for the constant A.
One method of determining the constant is to apply
Bernoulli's equation at the beginning of the pipe, where
an average value of the head is taken.
This condition could be concievably realized if
the flow were introduced evenly through a peripheral
sf.:L-t in the wall at the pipe entrance. (See bottom of
de , then in this region
If the slit is ~ z ,-;i
Fig. 8)
there is no wall friction and thus the head can be
taken as constant over this small A z.
This implies that all variations in z are approximatelyzero
over an interval of axial distance, which
in the limit can be taken to approach zero.
Bernoulli's equation for an incompressible fluid
and a horizontal pipe is
lla)
f
T
V
+ ~+
2
2
2
z
V
--
2
=
gH •
(5)
Denoting the initial boundary core.radius at
beginning of the pipe by ri , then at z
from eq. 3c., P
= Pc
+
~c~
2
[-=-ri - -=-]
r
2
=
0
Vg
=
modifying
r
- 22 -
eq. 5b and Vz =..
Wh
2
2
~ (1Yro -rrri)
is obtained.~rom eq.lc.
Inserting the "above expressions into eq. l1a , we have
after simplification
lIb)
c; "
Pc
T
= gH .•
r}+
+ 2
Since we have assumed that for a short distance
near t~e pipe entrance, all variations in z are neglible
dH
.
and H
constant , we may set --o.
drc
Performing the differentiation and simplifing we
=
=
can solve for wh as
.,
"
\
12a)
~'~~-''''t'".,
''lh
=
.... ,... ,.,......
ro C 0
1T~
J2
3
1~ft1i
ffi~
•
This can be written in dimensionless form by
collecting a A r
l2b)
group.
The aqua tion then become's,
a
I
A =
ro
Combining eq. 10 and 12b.atz
= r1
= 0 , where rc
we may now determine A algebracially as
- 23 -
[-
r 4
r2
ro
r
--1 In~
13) A =
+
0
0
(1-
r2
i
2
r
0
r2
.-..1 - 1]
r2,
f
Equations 12b and 13 in combination with equation
10, is the'complete solution of the initial problem,
r
and gives curves of the boundary radius rc or --£
r
The 0
ve~sus distance down the pipe z or :Xz•
geometrical and fluid constants of the solution are
contained in the yariable constant /\r
o
•
Assuming a knowledge of the physical constants in
'the system'~r
is calculated.
This allows the initial
o
core radius ratio,'to be determined from eq. 12b. This
value is then used to determine A in eq. 13 , which
may be inserted in eq. 10 to give the b~undary curve.
In Table. 1 in the. appendix , values of the above
parameters necessary for obtaining any desired curves of
r
versus z for different flow inlet condition are
c
computed and listed.
- 24 -
Discussion of the Results.of Part I
A typical solution of the above equations.reveals that there are two possible modes of .hot ai'r
flow down the pipe.
An example of the solutions
obtainable is shown 'in Fig. £a.
One solution which curves away from the wall
toward,the centerline of the pipe corresponds to an
overall decrease in the tangential and axial velocity,
as the flow proceeds down the pipe.
A second solution whi~h.curves toward the wall,
.away from the pipe centerline, corresponds to a flow
,
,
situation where the tangential velocity, while having
exactly the same value along the wall as the previous
case , now has a much reduced value at the larger core
radius of the second case.
This means that much of
the energy contained in the tangential velocity in the
'first instance is now used to increase 'the axial
velocity along the wall.
The limiting value of core radius in the second
case corresponds to the condition where all of the
whirl energy of the flow has been converted into a
maximum axial velocity while still allowing passage
of all the fluid as required by th~ continuity equation.
Since no consideration of frictional shear 'stresses
in the axial direction was taken into account in the
above solution, at some point down the pipe this layer
.. .'
,!.
•:~..'
~
- 25 -
- 26 -
would probably experience a hydraulic jump
when the
(16)
critical Reynold.s number for air '\AlaS
reached.
'
Both types of solutions, especially the second one
which seems curious, were observed visually by my thesis
advisor Prof. J. L. Smith during his doctoral Investigation of a cyclone seperator.
The model he used to
(4)
approximate this seperator was very similar to the one
outlined above.
The flow conditions were such that the
approximation of incompressible flow was.J very nearl.y
true.
Comparison of Model with Data of Reed
The following comparison will show the nature of
of the'velocity, pressure and streamlineprofl1es
as
derived theoretically and their deficiencies when
compared with actual data.
A. Reed's Masters thesis
This data is taken from G.
performed at M.l.T. in
diameter.
tangentially mounted inlet nozzle of
8
.One of his runs.is reproduced in Fig. 5 •
For this particular run,
Fa
=
95 psia
Ts = 560 R ~ s = .460 Ibm! ft3
0
= .• 452
- 27 -
• 532. A *P s
=
w
wh
.
.
ITs
=
=
1bm/ sec.
.235,
(6)
=
w(l .- €)
Taking the boundary line of constant pressure to
be 14 psig , then ~
may be found by assuming: an
avg.
adiabatic pipe.
L
~
= .196 Ibm/
=~s(Pc\k
Pal
avg.
ft3
Since
~* =
0
= 466
(6)
.833Ts
R
.and
V
~
= 49 ~
(6)
= 1060 ft/sec •
Then
A
=
11~ C
.
r
0
0
4 wh
ro
This determines
=
11.30
r1
.
in eq. 12b and consequently
r
ri
eq. 13 .' This parR~eters were computed to be
in
A
~
=
and A = - 15.76.
We may use a point on Reed's 14 psig curve to
determine a value of f by matching this point'on the
curvet
A. z
=
sirice
11 f,< Co
1+,---- z
4 '\'/h
=
1 + 132 f z
•
.953
- 28 -
1
From Reedls data the psig line begins at z
rc
rc = .6l0u and
= .594 •
ro
-
=
-ft.
3
At this point we then have using eq. 10
;',
Solving this equation gives a value f
=
.292 , which
.is 10 times larger than a good average value of f =' .03.
Knowing A, eq. 10 may now be used to calculate Az
(17)
Aro
versus z •
Other lines of constant pressure may be found by
using eq~ 5b.
P - Pc - -;-~_.-:-~-.(1
_(::)2J
knowing A versus rc •
z
The computations i~dicated above were carried out
and are presented in Fig. 6b.
As'.a furthur comparison
additional curves were calculated for a friction factor
one tenth of the value gotten above.
When the
t\'lO
graphs (Fig 5 and Fig. 6b) are com-
pared it will be noted that there is very little variation in pressure i11 the theoretlcal case as compared to
the actual example, although there is a general resemblance
in the shape .of the constant pressure curves.
29
- 30 -
t~7 ••••
~••).
I:::.
..
::::
~.
.::: : :: ::: .
- 31 -
vfuat seems logically to be the main fault with the
theory is that it assumes an incompressible fluid., This
is of course not accurate when dealing with pressures
measured in atmospheres.
The next part of the analysis
deals with the above fault.
- 32-
Part II
M~thod of Obtaining a Compressible, Solution
The solution obtained in Part I can be modified to
incorporate compressibility.
One immediate object of
~uch a solution would be to obtain the tangential velocity
distribution.
By then applying the steady flow energy
equation to the Hilsch tube the average cold core
temperature could be determined.
At the present time
however the following analysis seems to indicate that
to obtain this distribution would require at least a
complicated series solution and possibly numerical
integration.
The previous assumptions are all retained except
that:
#2'
The density'is allowed to vary adiabatically as
1
~
(;)k_ .
=~c
c
#3'
The definition of shear stress must be slightly
modified to account for variation of density along
the wall
r
f~
0=
.
2
oVg
8
From the previous assumption
#5 together with
assumption #2' leads to expressions for the pressure
- 33 -
and density in the fluid which can be readily found by
integration.
The constant of integration is eliminated
as \'Taspreviously done in eq. 5b, using the assu.mp-.
tion of a constant boundary pressure and density.
The final results are
16a) P =
Po{I
-k 1;:0 C(Z)2[ :~ - :2J}~
+(\
2
C( z ) [_1
(k - 1) ~
16b) 0 = Pol {
+0
,""
2 kP
.
r
c
2
_ :2]}
k ~ 1
c
These may be writeen
160)
.,
P = Po ~(Z)
. L..
16d)
~.=
[ M ( z)
C
~'i
_S_(Z)~k-l
- 2
r
.
.
\'There
=
S(z)
\l\: . -
1)
S(z)
C ( Z ) 2~ ...
~a~d
2 k Pc
M(z) = 1 + ~
rc
The continuity equation is now written as
17a )
~
[
d~
Z;C({ ( r, z) V z (- 211 r dr)]
= a
~o
or a1~~rnatlvely
r
17b)
wh =
fO ~ (r, z)
ro
r
Vz(-21l'r dr) = Vz..( ~ (r, z)( -211 rdr)
- 34 -
The angular
slightly
modifying
2 'it r
18a)
momentum
.
Removing
2
0
"0
C (z)
the brackets,
equation
is as previously
eq. 2b
= - -
d [ rc
f~ (r,' ~) ,e ( z)
ro
dz
from the integral
the remaining
]
,V~ ( -2 fT r dr •
~
in eq. 18'? but not fl"om
inte~ral
is equal
to wh' so
,that
2
l8b) 2 f( r0
Inserting
r
dp(z)
0
= -
11' f.
=
dz
4
The axial momentum
19a)-d
frc
!P(-2rf
C(
•
into eq. 18b) we get finally
z )2
~(z,ro)
equation
is as previously
j' *"rc "'.
, f~.cr,z)v;,(-211
d~ ,ro ~.
- 21fr
Removing
V
z
from beneath
and substituting
integral
C
P
, eqo4b
r dr)
]
drc
-
c dz
the integral
on the right
side
wh from eq. 17b for the resulting
we have
r]c P(r,z)(-2fl'r
d;l ro,j
19b) d
the following
c c dzdrc .
drJ = _ w .dYz - 211'r p
h
dz
The last term of the right
using
,'
wh
r dr , =. -
ro'.
dz
.dz',
#2'
assumption
de ( z )
18e)
'\ih
integral
side may be eliminated
theorum
by
- 35 -
r;c
d
r 0 g ( r, z) dr =
d g ( r; z) tlr + g ( r ,z) ~r 0 . _ g( r ,z) dr 0
dZ{:O
{O QZ
0
dz
0
dz
20)
(8)
The last term on the right side -is zero since ro is 'constant.
Applying this to eq. 19b we find
Jrr
0 ()
P ( r, z )
.az .
o
.
dr c
+ P(r, ,z-)i-211' r )0,
0 dz
(-2# r dr)
=
. dr
- 211r
Since P(r
,z)
=
P
c
c
r~ a'P(r, z) ,
,21)
fr
c
P ---.9 •
dz
0
,-this simplifies to
(-21l'r
dVz
dr)
= - ''1 ---
h dz
o 0 z.
dV
'In order to remove ---Z from eq. 21 we oan use the
dz
continuitY,equation (eQ.17a). First removing V from
z
under the integral in l7a and carrying through the z
differentiation
we have
o = dV zr;o l( (r, z )( -211' r dr ~ + V Z~
dz tro
.
~
dz
Applying the integral re1ation,eq.20
(.1ro ~ (
r, z) (,~21J' r _
dr )]
-
to the second
integral in the above equation there results
,
o =, - d V z~r
fC ~(r,z)(-211
. dz
ro
'_
r dr)
~ + V [ fr co~ ( r, z ) (-211r
Z
r0 a z
+ Vz ~(rc,z)(-21Trc)
Inser:ting,~(rc~z) = ~c and eq. 17b and solving for
dV
'
--L we get
dz
dr)
]
- 36 -
dV
z
22) -=
dz
-
The term 211rc ~ c being; present since it has been
previously assumed that no fluid crosses the boundary
from the rotational to the irrotational region.
Inserting eq. 22 into eq. 21, we have the final result,'
23)[
rc 'aP(r,z)
ro d
Z
2 [ rc d~(r.z)
(-211'r dr) = V Z
f
ro
d
Z
.
(-217 r dr) - 21frc~c]
The expressions for P and ~ eq. l6c and 16d may
navy be differentiated with respect to z and inserted in
the integrals in eq. 23.
Integrating these over rand
substituting for Vz from eq. l7b as well as eliminating
from the result: by using eq. l8c , there will
dz
result a function of the form
dC(z)
This expression coupled with eq. l8c which is of the
form
_ h2
rl
J
de (z)
dz
J
rc ' C(Z)J
=
0
should be adequate for a solution.
- 37-
Part III
Introduction
In looking over the data of various other experimentors who have studied the Hilsch tube (Ref. 9 to 14),
it will be noticed that although the pressure at the
nozzle exit
is high, the pressure along the wall 90
or 180 degrees around the pipe in the nozzle exit plane
has dropped a considerable amount.
A pressure tap at
the wall will read 35 to 80 per-cent of the nozzle throat
pressure.
The higher pressure Hi1sch tubes having the
greatest losses.
This is apparently due not only to expansion of
the entering gas inward toward the centerline, but
axially do\Vllthe pipe as well.
It was felt that if this pressure loss or part
of it could be utilized to increase the velocity of
the air entering the chamber above a Mach number of
one, then better performance might be expected.
In Fig 7 some previous data is plotted and shows
roughly the nature of the pressure loss.
Taking an
example; if the air has an initial stagnation pressure
of 190 psia , then the throat ,pressure in a converging
nozzle ",auldbe 100 psia.
By introducing this sas
directly into the pipe we would suffer 65% losses
and the wall pressure would be 35 'psla.
... 38 -
.:
;
.1
! ..
u:_
,
,Q.~
't'
"trj
:LL':
()
rf)
,0
.....
"
.1
t>
0
+
~
"
J ...
I'
i
,
I
":1:'
"
"
N
l'1
.
,
0
I
0
0
0
r-.
'lil;'I:
,;I
;;:L.;;;
•
I:::
.'.
:l
..
I
I
i i i
i
ii;;
1...
: :
.
I
i
J';,
l
••
"'_
••
1:
:
,
••.••.•
..
_
j.;::
It: 1 i i llJ LL..tuh,~il,LLdilli1.l..i:1.:.lHh~
- 39 -
In ord~r.to utilize some of this pressure loss we
could for example introduce the gas through a vortex
chamber.attachment shown at the bottom of Fig.8
and
allow the flow to accelerate as it moved inwardly around
the chamber to a higher tangential velocity and lower
pressure.
By dropping the pressure of the gas leaving
the acceleration chamber to 50 psia, the gas entering the
pipe would now only suffer 30% losses, and we would still
have a wall pressure of 35 psia while gaining 50 psia
worth of increase in the tangential velocity.
This pressure loss seems to be a probable explanation for the observed fact that higher pressure devices,
although reaching lower cold temperatures, are less
efficient.
This idea of using a vortex chamber to pre-accelerate
the flow was tried out experimentally and is discussed
in the section on experimental work.
Analysis of a Vortex Pre-Accelerator
The following analysis was carried out in order to
get an idea of a design criterian for the chamber.
It was assumed that in the chamber there was a free
or irrotational vortex, no friction, and symetry in the
tangential and-axial directions.
The density and pres-
sure were allowed to vary adiabatically.
Writing the continUity and Euler's equations in
- 40 -
polar coordinates
and introducing the a.boveassumptions,
(6)
the equations simplify to:
Continuity,
(.!.rr
24a) ~
+
d Vr\ +
arl
V
-a ~
=
O.
rdr
Tangential Direction,
<)
25a)
V
Vg
-
r"dr
+ -
Vr
r
V"
~
=
0
1
ap
Radial Direction,
26)
V
'dVr
-
-
rar
Vg
2
-.-,
r
=
~ or
The continuity equation may be rewritten as
d
-
~r
(~V
r
r) = 0
and gives upon integration
24b)'
w: =
211 r~ Vr A-Z
The equation in the tangential dipection ma~ be
immediately integrated to give the result
25b)
where
C1
=
Vin1et nozzle
r1
By this substituting this expression for Vg and the
adiabatic gas law
- 41 -
1
27) ~ =~I(":"\K
.
P1)
into eq. 26 we have
= _
-=(PI\
~,
I
P'
kO
P -;
~ r
Multiplying through bya rand
Vr2
1
k P -k-
C2
-+-=
1
2
-.,!=1..
1
2
2r
integrating, we find
-
..
p-.
k
+ B
~ I (k-l )
•
The initial conditions
When P = PI'
are used to determine B.
The equation becomes after simplification
.P
28a) -
PI
=
l
Using
k
k-l
1 -+
('2'
2kRTl
2
VI - V
r
- ~
Ci)~k-rr~
eq. 27 as well as the perfect gas law, to
obtain the relatibn between the pressure and density
ratios~ ahd substituting in the expression for Vr from
eq. 24b , we get an expression for the density ratio as
28b):1 [1 :::T1 (vi =
+
:~ - 41T;:2AZ2~2~
K~I
The equation for the temperature is found in a
similar fashion to be
28c)
-
k-l (2V
2kRTI
1
- 42 -
Eq. 28b may be modified by substitutiong in the
continui ty equation
WI
= ~ I VI
and the
Al
defini tioD
of the Mach number
29 )
(6)
to give after simplification
(~Xl
f:11
+ k:l
Multiplying
Mi
[(1 - :~) - ~ ~ 4fT~~ 4z2]
the above equation
bY(~~T2 and re,olritine;
in terms of the temperature ratio,
T by T
, r by r
o k+l
T
30)
it
. then replacing
the final result arrived at will be
0
kIT
(-T:) - - ~ T: )
2
k-l [
ri ~+ It-l
2(
---MI
k 1
~li ~ - ---2
1 + --;-
.ro
2
A~ J2=
21Tr06
An additional useful result may be gotten from
the definition of the Mach number (eq.29) and the
=
tangential velocity equation (25b), Co
Vg ro
o
=
Vg rl
1
/kRTo
=
/kRT1
-!kRT
o
giving the result
_r~go
31) -
MI
It
=
is
r1.rTl
-1-;ro
also
To
hal1d~to know that
(rl)
ro max
the radial Mach number is one.
At this point
occurs
vrhen
0
1 +
32)(
k-l
1'"
for
1
r2
7~1i 1 - -r21
0
=
:~)
(
)
,
k + 1
max
2
ro
k-l
2
-MI
2
In.figure 8 equation 30 is solved and plotted with
T
Po
r
the aid of eqs. 31, 32, e.nd33 as --2 and versus - l
PI
ro
TI
for an inlet air Mach number of 1; k was taken as 1.4.•
-
Plot
'I:t
11; ... ----
of
11-4
-
MI-!
e~.3Q
~
.a
'.4
......
-----.,.----
.'0 +--...;~--...._~~~-+-_+_---....__I_---_hl_---____:lI4_----~
."0 .....---i---+---f'-----f-i&+~trYrI_--~~_+_----__t-----~
.50 +--+----A---.....,~~f---:l~__:~~_\_~r_-_+----_t-----~M
3O-+-_~-..,.i__--,I'o--+----_+----..&Ti~_+_Jtr_-_+_-----_;
.-z.o~-~-M~=:L
'.00
z.oo
- 45 -
Experimental Work
Object
one of the objects of the experiments ~~dertaken,.
was to test, by comparitive data, the idea of adding a
vortex chamber pre-accelerating attachment, in an attempt
to improve the efficiency of the Hilsch tube.
A second object of the experimentation was to
examine the effect of placing various size flow restricting circular weirs at the end of the hot pipe in
an attempt to increase the performance of the refrigerator.
This second idea originally arose from erroneous
theoretical considerations which indicated that by
placing a weir of the proper size on the pipe, it might
be possible to decrease the length to diameter ratio from
an experimentally observed optimum of 50 to a lesser
value while still maintaining the same level of performance.
The analysis was wrons but since an effect,
even though apparently negative, was observed; data
and discussion of the idea is included in this thesis
.for future reference.
Procedure
A simple experimental setup was constructed from
partially available equiptment in order to test the above
- 46 -
. 3"
ideas.
The vortex chamber used had a
l"
'inside dia4
meter and was fed by two --- diameter converging
8
nozzles/placed symetrically.
Thermocouples were placed directly in the hot and
2
from both ends of the hot pipe. A.l'1eirand needle
valve device was built and installed on the end of the
hot pip.e and an orifice flow meter was placed on the
cold end of the pipe. Additional details of construction are shown in Figs. 13 and 14 in the appendix.
The total flow was estimated at a known Ps and Ts
by completely closing up the hot end and blowing all
the air through the flow meter on the cold end, after
which it was only necessary to measure the cold flow.
The flow measurement data is listed in Table 2 in the
appendix.
Using a hot pipe of length to diameter ratio of
. 3"
33 '\'/i
th an inside dj ameter of , three weir to pipe
r
214
radius ratios ~
of 1,---,--- were tried out '\'/i th a
3"
r
3'
2
7"
.--- cold flow diRphragm in runs 1, 2, and 3 and a --
16
32
cold diaphragm in runs 4, 5, and 6.
The plots of this
data are sho\'ffi
in Figs. 9 and 10 •
3"
pipe was then removed and a .546" inside
The 4
diameter pipe with the same length to diameter ratio,
- 47 -
was inserted
in its place.
A cold diaphragm
geometri-
cally similar to that used in runs 1, 2, and 30f
diameter
.136" \vas then inserted.
This reduced pipe radius now allowed
the flow to
-.:tt
J
pre-accelerate
entering
in the -
.
4
diameter
vortex
chamber before
the hot pipe.
For this vortex
chamber
the pipe radius
ratio of
.750
zero.
r
For this ---l ratio, the pressure
P
be --2
p
=
press~re
p
=
1
.50 •
ro
With an initial
in the air inlet nozzle
.528 Ps
=
90 psia,
80
ratio is seen to
Ps of 170 psla the
throat is then
that theoretically
Po should
(6)
be 45 psia.
Thl-s value is 8 psia higher
then the data of Fig. 7
indicates.
With this smaller pipe, weir to pi~e radius
2
1
ratios of 1, -,
and
"lere again tried out in
3
2
runs 7, 8, and 9.
The plot of this data is sho~~ in Fig. 11 •
A compari son of runs 1 and 7 , sho\'lingthe effect
of preaccelerating
Fig. 12 •
the flow with no weir is plotted
in
- 48 -
Table 3 in the appendix
calculations
•
performed •
lists all data taken and
- 49 -
Discussion
of Experimental
Results
The effect of placing
the hot pipe without
various
weirs at the end of
the pre-accelerator
attached
to
the pipe, is seen in Figs. 9 and 10 to be observable
but apparertly
detrimental,
sizes were used.
although
only three weir'
weir sizes ,
Using intermediate
r
r
especially
the ratios of --~ = 1 and -
between
give improved
might concievably
L
by increasing
vl
2
=-
ro
3
1"0
or equal performance
in this case from 33 to
the ~ratio
D
the experimental
optimum.
The effect of pre-accelerating
the flow before
it
f'ft:
entered
tube.
the~did improve
The temperature
the performance
of the Hilsch
of the cold air was fOQ~d to be
200F colder for the same flow rates,
then a geometri-
cally similar tube with no pre-acceleration,
as may be
seen in Fig. 12 •
With both types of tubes it was noted that with
the smaller
size weirs, the flow resistance
great, as to raise the pressure
value.
This set
be passed
restricted
a
minimum
inside the pipe to a hiBh
These smaller weirs
half the value-found
optimum.
thus
by forcing. the ratio of cold air
flow to total flow above the value
For this setup
so
amount of flo1"!vl111chcould
out the cold end.
performance
became
e was
for optimum
approximately
10%, a figure about
.by other experimentors
This was attributed
operation.
to be an
to the lack of a sharp
- 50 -
edged cold orifice.
In its place, a relatively
( I inch) piece of threaded
hole drilled
through
size
it was used.
This idea was verified
orifice
bar with the proper
long
in the device.
by placing'a
sharp edged cold
For best performance
found to increase 'to 16% , as is indicated
tabled in the appendix.
E "Jas then
in run #10,
- 51 -
- 52 ~
i
l'
i
I
'
-~--i
.Z4J-
~
I
i
I
l-'~-':
0-.
!
-
j---,-
-i
I
I
________
.L
j
.
,-
- 53-
;;::'.' :0;::1;\i:,~.;i; .•..
; .•S:-;:;.;..:j.> • I.':,. :ili ...
b, ':~:': ~i1i,:[';I;:;:~[i~~!
[;~a;:~,~', rt~ ':!~~.~.;!<'--I~ i; ; I,,,: iii[!i:: Ii'
:;::~1:~~I~:~JT::;Hdj
:;r+~(
f: :J~.:r:-1:()V~<:~~~n~s :::::::::::::::;1::;:
;:;; :lIi ::::fH::
..
:'.',
..
. . ..
...
'.
~;~ ~!~~~:.~
- 54 -
:_.~-:---~.,o.
i
: . :.
-r
I
I
!'
';"
I
:__.._~._--j--!
..
I
I
I
.
,
-
I
!
:
i
_
i
I'
,
I"
; ....
-SO.'
"
--.- ..-. 1--
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Bibliography
1.
Scientific American, Nov. 1958 , P. 145.
2.
Hilsch, R. ,"The Use of the Expansion of Gases in
A Centrifugal Field as a Cooling Praces s" , Revie,'l
of Scientific Instruments, Vol. 18 , No.2,
Feb •
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3.
Lay, J. E., "An Experimental and Analytical
Study of Vortex-Flow Temperature Se~ration
Superposition of Spiral and Axial Flow",
by
ASlI1E
paper 58 - A - 90 •
4.
Smith, J.L., "An Experimental and ~nalytic Stu~
of the Vortex in Cyclone Seperators" Phd Thesis
Course II M.I.T., Jan.1959 •
5.
Hunsaker, J.C. and Rightmire, B.G., l1Ensineering
Applications of Fluid M~chanics", Mc Gral'l Hill
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6.
Shapiro, A.H., "The Dynamics and Thermodynamics
of CO!!m~essible Fluid FIo\-,",Vol.l , Roland Press
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7.
Martin
vi. T.,
and Reissner E., "Elementary
.
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Hildebrand, F.B.,"Advanced Calculus for
Engineers", Prentice-Hall Inc,1949.
- 56 -
9.
Reed G.A., "vortex
thesis
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R.J.Jr.
,andSolniclr,R.L.,
B.S.Thesis
of Centrifugal
Haddox,R. , Hunter,
B.S. thesis
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Nickerson,
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t
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Rouse,H. and Howe,J.vl.,
"Basic Mechanics of Fluids",
John Wiley and Sons, Inc.
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Refriser.ation"
Course II ~I.I.T.
Course II,
Wilson, vl.A. , :'Foster,E.
Engineerinp;",
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W.R.
Hunter, J :vl., and Bruce H. Mayer , "Centrifugal
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J.W. and Plunkett,
Study of Centrifugal
Refrigeration"
14.
Refriger?-tion",.
Course II M.I.T.,1947.
"Experimental
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1947.
M.N. and Sweeny A.N.,"Experimenta1
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"~xperimental
of Vortex Refrigeration",B.S.thesis
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Fattah,
M. 'S.
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Tube Refrigeration",
Gied't,W.H.,"Principles
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of Engineerin6
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1957, P.I07.
Heat Transfer",
- 57 -
Appendix
Details
of Construction of Apparatus
Table 2
..................................
................................
Table 3
..................................
Table 1
page 58
page 60
page 62
page 63
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