The Polytropic Analysis of Centrifugal
JOHN IVI. SGHULTZ
Senior Engineer, York Division,
B o r g - W o r n e r Corporation,
York, Pa. Mem. A S M E
The real-gas equations of polytropic analysis are derived in terms of
compressibility
functions X and Y which supplement the familiar compressibility factor, Z. A polytropic head factor, f, is introduced to adjust test results for deviations from
perfect-gas
behavior.
Functions X and Y are generalized and plotted for gases in corresponding
states.
Introduction
LI HE thermodynamic design and test evaluation of
centrifugal compressors is frequently based upon a polytropic
analysis employing perfect-gas relations. In many instances
real-gas relations would be more accurate, but these are virtually
unknown. The purpose of this paper is to derive the real-gas
equations of polytropic analysis and to show their application to
centrifugal compressor testing and design.
T o do so we must supplement the familiar compressibility
factor, Z, by two additional functions, X and Y. Like Z, these
compressibility functions can be generalized and plotted for gases
in corresponding states. Thus another purpose of this paper is to
publish generalized charts of the compressibility functions X
and Y.
We shall also discuss isentropic analysis, relate it to polytropic
analysis, and show how the two can be combined to advantage.
In this connection we shall define a polytropic head factor, / , to
adjust polytropic head measurements for test gas deviations from
perfect-gas behavior.
T o conclude our study we shall consider a numerical example
employing the real-gas equations of polytropic analysis. A comparison of the results with those obtained by perfect-gas relations
will reveal the inaccuracy of the latter. Many similar instances
are regularly encountered in centrifugal compressor applications.
0, 1, 2, and 3 locate these states on a Pressure-Volume diagram
of the gas, Fig. 1. Let a smooth path, p, be drawn between
points 0 and 3, passing through or between points 1 and 2.
Curve p may be regarded as the path of a reversible process
whose energy balance is
QP + Wp = lh -
Ho
(1)
where Qp is reversible heat input, TFP is net reversible mechanical
energy input, and H is enthalpy from the test data.
For our adiabatic test compressor the energy balance is
IF =
H
HA
„+
IV
(2)
2<7
where W is shaft-work input to the test gas and v is gas velocity.
For convenience we shall rewrite (2)
TF
ARE
= H3 -
(2a)
Ho
Origin of Polytropic Analysis
Imagine we have tested an adiabatic (uncooled), three-stage,
centrifugal compressor without side-flow, and have determined the
thermodynamic states of the test gas at the compressor inlet (state
0) and at the outlet of each stage (states 1, 2, and 3). Let points
C o n t r i b u t e d b y the P o w e r Test C o d e s C o m m i t t e e and presented
at the W i n t e r A n n u a l M e e t i n g , N e w Y o r k , N . Y . , N o v e m b e r 2 7 D e c e m b e r 2, 1960, of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. M a n u s c r i p t received at A S M E Headquarters, S e p t e m b e r
19, 1960. Paper N o . 6 0 — W A - 2 9 6 .
SPECIFIC VOLUME
Fig. 1 P r e s s u r e — V c l u m e
centrifugal compressor test
diagram
of
a
Nomenclatureacoustic velocity, fps
cP
= specific heat at constant pressure, ft-lb/lb deg R
= (dH/dT)r
Cy
= specific heat at constant volume,
cP
ft-lb/lb deg R
CV = (bE/bT)y
e = polytropic efficiency
dP
e = V
dH
E
isentropic efficiency
internal energy, ft-lb/lb
E
H-PV
es
= polytropic head factor
0 = standard gravitational acceleration
0 = 32.174 ft./sec 2
f
H = enthalpy, ft-lb/lb
J rp — Joule-Thomson coefficient,
deg R ft 2 /lb
J f ~ (dT/dP) „
k = specific heat ratio
k = Cp/ Cy
L
= compressibility function
L
=
T_ / dP'
P
\i>T
Journal of Engineering for Power
m
—
m
=
M
M
n
=
n
=
P
P
Pc
Pr
=
=
=
=
=
=
polytropic temperature exponent
P dT ,
along p
T dP
impeller Mach number
u/a
polytropic volume exponent
V dP ,
- - —
along p
path of constant efficiency e
absolute pressure, psf
absolute critical pressure, psf
reduced pressure
(Continued On next page)
JANUARY
1 96 2 / 6 9
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where A K E is the kinetic energy increase through the compressor.
Combining (1) and (2a) we obtain
Qp + wp
= W -
(3)
ARE
showing that path p requires both heat and mechanical energy to
keep abreast of mechanical energy alone in the test compressor.
Only if the compressor were reversible would every test point, and
p, coincide with the isentrope So where Qp = 0.
Evidently the relative magnitude of Qp is an index of irreversibility or inefficiency. Conversely, the relative magnitude of 1VP
is an index of reversibility or efficiencj'. Let us define an efficienej'
TF„
Hz
W„
H0
W
-
A
(4)
KE
where e is compressor efficiency with respect to path p.
To evaluate e we note that
r3
VdP
W,
-I
Pa
r
1
VdP
H 3 —
Ho
V
TdS
(7)
where T is absolute temperature and S is entropy, the equation
of an isentrope (dS = 0) is
1 = V
dP
(S)
dH
which coincides with (0) for
Q„ = 0
W„ = W -
MiE
(9)
e = 1
We complete our review without finding any discrepancy between (0) and our original discussion. In fact, we conclude that
(6) is uniquely suited to our purpose. The next step is to discover
whether (6) can be rearranged to permit direct integration of (5)
in (4a).
(o)
where absolute pressure P and specific volume T" arc related by p.
Thus (4) may be written
J P0
dll = VdP +
Derivation of Equations
For any homogeneous gas we may regard H as a function of P
and T and write
(4a)
and could be evaluated by graphic integration.
The result would be somewhat arbitrary, however, because p was
not precisely defined. A path equation would eliminate this uncertainty and is in fact essential if our analysis is to have much
utility. Let us redefine p therefore by the path equation
dH
dH
dP
dP
dT
dp
]
dH
dP )r
( dT ) „ ( dH
(10)
)F
The Joule-Thomson coefficient, JT, and the specific heat at constant pressure, CP, are defined
dP
J;
where e is that constant for which p passes through points 0 and 3.
B y integrating (6) we obtain (4a), hence constant e in (6) corresponds to efficiency e in (4a), and p may be considered the path
of constant efficiency between 0 and 3.
Reviewing our previous discussion we note that (6) eliminates
points 1 and 2 from any role in determining p. This generalizes
our analysis to compressors of any number of stages and even to
individual stages themselves. Consideration of the latter topic
reveals a unique property of (6).
Let points 0 and 1, 1 and 2, 2 and 3 in Fig. 1 be joined bj- paths
Vh
Pi defined by efficiencies ei, ei, e3 in (6). Should these stage
efficiencies all be equal (6) requires that p for the over-all compressor coincide with P l , p 2 , Pz for the individual stages, and that e for
the over-all compressor equal ei, e2, ei.
Continuing our review, we note that p should coincide with the
isentrope So for Qp = 0. From the general thermodynamic relation
dH
+
dT_
^
(11)
dH \
vr )r
.
so that (10) becomes
dH
dP
=
Cr \
dT
-
dp
Jt
(10a)
Among the general thermodynamic relations for any homogeneous
substance we find
Jf - V
\ dT
Cp
(12)
A general equation of state for any gas is
PV
= ZRT
(13)
-Nomenclaturep,
P,
Qp
R
^
=
p/p
= absolute total pressure, psf
= reversible heat input along p
= individual gas constant,
ft-lb/lb deg R
1545.4
molecular weight
entropy, ft-lb/lb deg R
absolute temperature, deg R
459.69 + deg F
absolute critical temperature,
deg R
T, = reduced temperature
S
T
T
Tc
=
=
=
=
70 / JANUARY
1962
Tr = T/Tc
T, = absolute total temperature,
deg R
u = impeller rim speed, fps
v = gas velocitj', fps
V = specific volume, f t 3 / l b
IF = shaft-work input to gas, ft-lb/lb
IFp = poly tropic head, ft-lb/lb
1FS = isentropie head, ft-lb/lb
A' = compressibility function
1' = compressibility function
p_ / ay
Y
~ V
V dP , T
compressibility factor
Z
PV
Z
RT
AKE
=
AICE
M
kinetic energy increase, ft-lb/lb
2g
= polytropic head coefficient
isentropie head coefficient
Note: =
Multiply Btu by 778.26 to obtain
ft-lb.
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where Z is the compressibility factor and R the individual gas
constant. From (13) we obtain
t)F
^
Z
r
dP
L 1 ~ -Z \ &P
1 +
(12a)
\ bT JP
P
where m
T
dT
dP
<1H
cIP
bZ
P
VbT
Y1F
dT
(
—
dP
P
Z
P_ rfT
ZR. r i
T
Cp L e
dP
—
T / bZ
(106)
)
\bTjr
L
z
jr
\
Z
+
'
V <>T )p_
(6 o)
C p — (jy — 1
bP \
/ bV_
bT )v
\ bT
I
bE
c.
where E is internal energy.
bP \
_ P^
T
T
(bZ\
T
bZ
Z
\ bT J v
1 +
(20a)
/ bZ
1 +f
\bT
dT
P
dZ
Z
dP
_
~
1
P
dT
P
dV
T
dP
V
dP
C)
P dV
is (6a) or (6b) and — —j^ is (6c) or (6rf).
T
/ bZ
Z
V bT
(15 a)
At this point let us pause to review our progress. We have discovered several rearrangements of (6), each determining path p
of constant efficiency e. These may be summarized conveniently
by
By inserting (15a) into (6a) our path equation may be written
P
dT
T
dP
P_ dV
T
dP
(
1 + -
k
T
)
/ bZ
Z
\bT
Z
+
1 +
/,,
(dJ'l
T f bZ
lyr
P_ dZ^
n - 1
~Z dP
n
(13)
Cy
dT
+
dP
n
where
Cp
ZR
For any homogeneous gas we may regard V as a function of P
and T and write
dV_
bT
P
dP
Cp
(H
(\ ei + ,J ) . ( L{M
1 + X)
bP
bT
bP /T \ bT Jy \bV
,P
(20)
From (13) we obtain
F
,bPjT
P
P
1
~ Z
( bZ
\bP / T
(21)
By inserting (14) and (21) into (19) our path equation may be
written
Journal of Engineering for Power
(6 0)
1
(19)
Y - m(l +
X)
I t-- " I"'
and
dF
(6/)
dP
V
(6b)
where
k
m
(17)
so that (15) becomes
1 +
\bT
Finally, by differentiating (13) our path equation may be written
P
where —
From (13) we obtain
bT )v~
JL JL
F dP
(16)
bT , y
1
^ bP )T_
r -i + T
Z \ZT\_
_e
r
( H
2
/
bZ
r +
i \ bT ) J
(15)
In (15) Cv is specific heat at constant volume and is defined
1 +¥
By inserting (6b) and (20a) into (6c) our path equation may be
written
Another general thermodynamic relation for any homogeneous
substance is
JL JL
(6c)
is (6a) or (6b).
By inserting (10b) and (13) into (6) our path equation may be
written
Cy = ZR
T
*
In (20) we may combine (14), (17), and (21) to obtain
and (10a) becomes
CP -
p_ / bz
(14)
\ bT / P,
so that (12) becomes
CP
dv
V
+
T / bZ
1 +
bT
p
and
\
/ bZ >
T (bV\
V bT ,)p~
v
Y = 1 - —
Z
L = 1 +
T
I
/p
-
1
bZ
P_ / bV
bP
V
\dP
( bZ
T
( bP
bT
P
\<>T/v
JANUARY
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(22)
1962
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In terms of compressibility functions (22) we may write (12a),
(15a), and (20a)
X
JT —
cp
X)
(1 + AQ
L =
Y
(1 + A )
ZR
( 1
X
\
(1 +
«,.—>• co
(20b)
where subscript V denotes constant specific volume.
(6c), (6/), and (22)
Y
Y -
m( 1 +
mT = 0
nT =
7
ms
7 + *
Y
1
ke
(1 + A')
— e
(25)
k - 1
(6 h)
X)
From
where subscript T denotes constant temperature.
In a perfect gas X = 0, Y — Z = 1, and our equations reduce to
A") 2
1
(24)
(15b)
From (20b) we see that one of our compressibility functions can
be eliminated. We chose to retain X and 5' so that
k - 1
From (Ge), (6/), and
5"
(12b)
Cy = ZRL( 1 +
-
thej' are not paths of constant efficiency.
(22)
=
k - 1
k
1
(6/1
its' = A-
and
mI{' = » ! r ' = 0
(1 +
C p — Cy — ZR
A)2
rin' = » ? ' = 1
(15 c)
Y
Recalling that p is an isentrope when e = 1, let us examine this
special case:
ZR ,
„
(k »>s = —
(1 + A ) = '
ns = Z
Y -
(1 +
k
ms( 1 + X)
A)
(60
Y
where subscript S denotes constant entropj'. The result is confirmed by the general thermodynamic relation for any homogeneous substance
dV
)
dP J a =
1
(
k \dP
-
(23)
)
J7
which, by (6/) and (22) may be written
P_ /
_
1
V
~
~ iis
[ dP )s
ZR
n„
(k
\
=
_
-
1
Y -
l\
~ K ~ )
m „ ( l + A)
=
(2'a)
JT'
= 0
(12c)
Cy' = /?
(15 d)
where superscript ' denotes a perfect gas. It is interesting to
compare the previous real-gas equations with these perfect-gas
relations. The former are entirely rigorous and hold for any
homogeneous gas whatever.
Some of our results have been published before. Edmister and
McGarry [I] 1 derived the equivalent of ms = {ZR/CP){ 1 + X)
in 1949 and published generalized charts of {Z/T)X and ZR( 1 +
A") for gases in corresponding states. In 1951, Edmister [2]
ZR
derived the equivalent of m„ = —— X and theorized that an
CP
equation of the form m =
k
(23a)
Another special case is the path of constant enthalpy for which
" L "
= 1
Cp' -
Y
1
mv'
XY
I — + X I might represent the
CP \e
J
general case. Of course the equivalents of our perfect-gas relations are well known, having been published in many textbooks.
Returning to (6/) we see that none of our rearrangements of (6)
permits direct integration of (5) because m and n are variables.
We suspect, however, that they are relatively constant compared to P, V, and T. If this be the case, we can integrate (6/)
as if m and n were constants and obtain
( T T A T 2
(1 + A )
~
Y ( l +
—
= constant
PV"
= constant
(6j)
n-1
where subscript H denotes constant enthalpy.
A seemingly trivial case is that of constant pressure for which
e = 0:
(6A-)
Subscript P denotes constant pressure.
Two other paths are also described bj- (6/) although in general
72
/ JANUARY
1962
P "
Z
(6m)
= constant
The three paths defined by (6m) are all approximations of path p,
becoming identical with p as m and n become constant.
Our analysis is called "polytropic" because the polytropic process is commonly defined by the path equation PV" = constant.
This path is called a "polytrope." One of our equations (6m)
1
Numbers in brackets designate References at end of paper.
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is a polytrope, hence we call e "polytropic efficiency," Qp "reversible polytropic heat input," and W p "net reversible polytropic work input" or "polytropic head." A less common but
more fundamental definition of the polytropic process is the path
equation V(clP/dH) = constant, for which p is a polytrope by (6).
This definition suits our analysis better, but in either case our
equations remain the same
Test Evaluation
For reasons not pertinent to our discussion it is appropriate to
relate compressor head to compressor speed by a polytropic head
coefficient, p, which is defined
gwp
(26)
2.U2
where g is standard gravitational acceleration and u is impeller
rim speed. The summation is taken over the number of impellers
within the compressor. For our three-stage test compressor
Let us rearrange (Gm) to belter advantage:
p =
(26a)
?ti2 + ur - f us-
er, if all the impellers are of the same diametei,
sir,
* =
Z,o_
Z
(On)
(0-
p_
Po
and
log ( T / T o )
=
log ( P / P „ )
m
log
(P/Po)
(60)
log ( V J V )
Individual stage coefficients
p.*,
can be determined from
individual stage heads and speeds. Should these coefficients all
be equal, and should the stage efficiencies also be equal, (5), (6),
and (26) require that p for the over-all compressor equal p. 1, p.2,
for the individual stages. This is another unique property of (6).
Perhaps our detailed discussion of polytropic analysis has obscured the simplicity of the actual test procedure which may be
summarized:
1 Determine P, V and H at the compressor inlet and outlet,
states 0 and 3.
2 Compute
log ( P 3 / P c )
=
_ n 1'1
1 =
log (ZQ/Z)
"
Now we can integrate (5), at least approximately, by (6/), (6/1),
(60), and (13):
W„ =
log (F0/F3)
log ( P / P „ )
n
(266)
3u2
n
IF.
n -
1
(P.Vz
-
P0F0)
(27)
rF
c1
Z
VdP = R I
— dT
Jpc
Jt0 ">
H3 -
H0
W„
ill2 + U2'2 + W32
N
—
\ J
n—1
rp \ rnn
-
ZMJ 0
L\
The accuracy of W p in (27) depends upon the constancy of n
along p. Later we shall discover means to minimize this dependence.
1
Isentropic Analysis
TO
n—l
w.
n - 1
_P
ZqRTQ
-
1
Po
(5a)
w,
(PV
-
P0V0)
There is another thermodynamic analysis of centrifugal compressors called "adiabatic" or "isentropic." This differs from
polytropic analysis by substituting isentropic head, IF.;, for polytropic head, TF,,. The isentropic head of a compressor is the net
mechanical energy input required by a reversible adiabatic compressor having the same inlet state and outlet pressure. The
path of a reversible adiabatic compression process is an isentrope.
For our test compressor, Fig. 1, this path is S0 and the isentropic head is
TF.s =
(
,
R
R
I
)
M
~
R
Hi -
Ho I
Z O T O )
(28)
P , = P3
S 4 = S„
WD
which from (5a) and (60) can be approximated by
V
,og
w
=
WP
P»F 0 In ( P / P 0 )
when
/
(In
=
P_\
(PV
+
P„F„
log (P:,/Pc)
log ( F 0 / F 4 )
(28a)
Po
n = 1
and evaluate e for our test compressor by (4a), (5a), and (60).
Another useful test result is the polytropic head coefficient.
Journal of Engineering for Power
IFs s
{PsVi
~
PoVS)
The accuracy of TVs in (28a) depends upon the constancy of n s
along So.
JANUARY
1 96 2
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/ 73
The isentropie efficiency is
IF, = /
Hi — H o
cs =
Ws
Ho
H,
W
-
A KE
/PF -
log
log
(29)
P 0 F„'
PF
PoFo
and the isentropie head coefficient is
gWs
V-s
(AH, -
Ho)
Ul' + »22 + M32
(30)
Isentropie stage efficiencies and head coefficients may be evaluated from individual stage heads and speeds.
Unlike their polytropic counterparts, however, equal isentropie
stage efficiencies and equal isentropie stage head coefficients
must exceed es and p.s for the over-all compressor. The disparity increases as the pressure ratio or the number of stages
increases and varies from one gas to another. This difficulty
may be traced to the general thermodynamic relation
A
\ ds
= T
(31)
IF, = JPoVo hi ( P / P 0 )
when
Hs — Ho
>>s
ns
ns
-
(PVS
1
P„Fo)
-
log ( P / P . )
log ( F o / F . , )
Our test procedure becomes:
1 Determine P, V, and H at the compressor inlet and outlet,
states 0 and 3. Determine V and II at the isentropie outlet, state
4.
2 Compute
requiring that pressure lines diverge on a Mollier chart. The rest
follows from (28), (29), and (30). As a consequence, the extension of centrifugal compressor test results to other compressors or
other gases is accomplished better by polytropic than by isentropie analysis.
log
=
"
(P3/P0)
log ( F „ / F 3 )
ns
log
(P3/P0)
log
(Vo/Vi)
Polytropic Head Factor
Hi — Ho
Despite its shortcomings isentropie analysis does have one advantage: Ws in (28) is exact, whereas IF, in (5a) and (27) is
approximate. In a similar manner TFS in (28a) is approximate,
and the similarity suggests that the ratio of (28) to (28a) must
nearly equal the ratio of (5) to (5a). Accordingly, we define a
polytropic head factor
Hi — Ho
We had better incorporate / into (5a) and our test procedure
summary. The former becomes
w„
=
/
\n -
1
P 0 F„
71—1
«
-
ft/
-
PoVo)
(27a)
(PSF 3 -
IF
P„F„)
1F„
H 3 — Ho
PoVo)
which is the ratio of (28) to (28a). Multiplying the approximate
TF„ of (5a) or (27) by / we approach the exact Wp of (5) independent of the constancy of n along p. A closer approach might
be obtained if / were determined along Si or
but So is more
convenient and should be sufficient. I11 many cases / is so near
unity as to be superfluous.
p\
(P,Vi
(32)
(.P3V< -
n = 1
1
71-1
M =
ffJFp
i'r +
+ «32
Equations (27a) are limited to tests of uncooled centrifugal c o m pressors without side-flow. In (56) and (27a) W p is nearly independent of the const ancj r of n along p.
Compressor Design
Relations similar to (27a) are emplo3Ted to design centrifugal
compressors when detailed thermodynamic tables or charts of
the design gas are unavailable. The problem is to determine TF,
u, the number of stages, F 0 , V3, and T3 from Po, To, and P 3 .
Gathering (4), (56), (6h), (6n), (13), and (26) we obtain
rp \ 17111
=
./'
/i. - 1
ZollTo
P_
=
/
' \n - 1
ft.
L
\ V
ZR
m = - . . - I — + A' ) =
CP
Y -
ZT
=
/
(PF -
1962
m( 1 +
X)
(1 +
X)
-(h:
/'r,l"u)
mZT
74 / JANUARY
(1 + X ) 2
1
- ZoTo) |
(56)
(33)
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W = ^
e
+
Similar to (5c) is a version of (6) containing (6/), (6/i), and (13):
MCE
dH
M
V0 =
(1
(6 p)
+eX)
For anj' homogeneous gas (6p) must hold along a path of constant efficiency. An approximate integration of (6p) is
ZoRTo
Po
H Fa =
CP
dT
gw,
Ho ^
CP(T
Vo
{Pz/PoY'"
T, = T0(P,/Po)">
For (33) a mean value of either CP or k is estimated from whatever thermodynamic data are available. Mean values of A', Y,
and Z are obtained from generalized compressibility charts for
gases in corresponding states, as is Zo at the compressor inlet.
For best accuracy a trial-and-error solution is required; i.e.,
the discharge temperature must be assumed in order to estimate
mean values of X, Y, and Z. The selection of e and p is based
upon test data and experience. The usual assumption for / is
unity. Frequently A KE is negligible. The separation of « from
the number of stages in Z v 2 involves considerations not pertinent
to our analysis.
P±
i\Po
ir =
e
.
P
Po
('•Hr)
(33b)
+ 1
Po
( t
= Po
- 1
bW.
+ 1
dS
ZR ( 1
p
\
( T -
(6r)
)
1
for any homogeneous gas along a path of constant efficiency
approximate integration of (6r), using (6m), is
ZoRTo
Zp -
Po
.log (Zo/Z)
r„{pjp„y
HI-
AVo
ZoiPz/Po)1'
This results from a version of (5a) in which we integrate (5) approximately by (6/), (dh), (6n), and (13) assuming CP, X, and into be relatively constant compared to P and T:
f T
[)'„ = R
IV = H
II',. = e(H
dT
£
(
i t :
t - )
CpfJ -
P \
In
An
Z
((is)
(Zo
Po
B y trial-and-error the correct combination of Sz and Z3 is found
to satisfy (6s). Then V3, T3, and Hz are read directly from the
tables or charts and IF and u are computed by (2a), (4), and (26).
In summary
,S ^ So
Z
Z
—
- dT
m
"
Jr.
1
R
-
i
log
1
)
P>
—
i 0
log (Zo/Z3)
Ho + A KE
-
(33 c)
-
Ho)
p(5c)
To)
Helacions similar to (6r) and (6s) are
(I
H'„
) w>
C pTo
dP
" + A KE
•i\ =
' •'
x
+
Individual stage values of CP or k, X, Y, Z, e, and n are employed for (33b).
When detailed thermodynamic tables or charts are available
the designer can obtain F 0 and So directlj' and can determine V3,
T3, and Hz from S 3 . Knowledge of Sz stems from (6), (7), and
(13) which produce
H
FII =
(6?)
IF
(33«)
,
To)
eX)
where CP and X are mean values along p.
Having determined v by (33) or (33a), a stage-by-stage analysis
is required to determine stage outlet values of P, V, and T from
stage inlet values P ( „ Vo, and To. For this purpose (33) and
(33a) are modified by
An alternate form of (33) eliminates Y but requires Z3:
<VA,
-
(1 +
ry/'„
+
A
Journal of Engineering for Power
1
S -
dS
_
C_P / 1 -
dT
~
T
S„
; CP
e
\l+eA
1 -
(60
e
1 + eX
111
To J
JANUARY
1 96 2 / 7 5
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which follow from (6/), (6/t), and (6c). The differential equation
(6<) must hold for any homogeneous gas along a path of constant
efficiency. The approximate integration (6f) contains mean
values of C P and X .
B y taking the ratio of (61) to (69) we find
1
/ & -
So \
r T - T
\H,
Ho)
.In
-
3
0
(27 b)
(T3/TA)_
which is another test evaluation equation like (27) and (27a).
Although (276) was derived from real-gas equations its accurac} 7
is suspect because it is also a perfect-gas relation.
Probably
better is
e =
'log (Z0/Z3)
Z3T3 — ZqTQ
1—
H 3 — //<
Zo — Z:i
log
{Zz'l\)
"s
Vs =
i^s
e
ix
W,
"S
1
-
1
1]
e
M
TF„
m, s [(P/P„)'» -
1]
ZR
ms = —
(1 +
X)
+
A
(1 +
X)
CP =
Y
AT ) P
—
V \
AV
AT/P
Both (34) and (34a) reduce to the perfect-gas equation
(P/Po)""' -
=
(P/Po)'"
(K -
L\
ms
^
»i, 5 (i + X)
R
_ k -
CP
~
1
k
(346)
for X = 0, y = Z = 1. In many cases the complications of (34),
or even the simpler (34a), can be avoided by using (346) as a first
approximation to estimate Fa' by (35).
With TV in (27a) an e'
is obtained corresponding to es'.
Then
r
PV
(1 +
X)
(1 +
es_
es
Ms
e'
e
M
AT
1
Y
1
~ 1
ms'
Y
CPT
m(l +
(34a)
A H\
es
k
Y -
m[(P/Po)"'s -
71-1
R
Y -
Ws
Po
U ) _ U )
PV
Vs
m = >ns
derived from (4), (5a), (6s), and (13).
Another design technique with thermodynamic tables or charts
is to determine V3 and T3 from H3 which is found by converting
from polytropic to isentropic analysis. The conversion is accomplished by combining (4), (56), (6/1), (61), (11), (13), (15c),
(16), (18), (22), (26), (29), and (30) to obtain
e_s =
eA
(27c)
(Z0r0)
ns — 1
P \ "s
-
namic tables or charts of the design gas. As before, e and fj. are
based upon test data and experience. Note that CP, Cv, k, X,
and Y computed for ms and ns may differ from those computed
for m and n.
Often the mean values of CP, X, and Z required for ms do not
differ appreciably from those required for m. In that case (34)
can be simplified by (5c) to become
(34)
X)
I T " " ! " '
(34c)
Usually es/e by (34c) is sufficiently accurate for practical purposes.
Having determined es and
by (34), (34a), or (34c) the designer combines (28), (29), and (30) for his isentropic analysis:
Ws
= Hi -
P
= P3
H
S 4 = So
T
IF =
( AF \
ZM2
Ws
A ICE
(35)
gJVs
=
Ms
V
c
_
\AP
)T
_
(AH -
A Tjv
V
HS = Ho +
AT
FAP^
)v
= CP
k =
PF(I +
xy
TY
Cp
Cv
For (34) the values of P, F, T, and H necessary to compute mean
values of CP, CV, k, X, and Y are obtained from the thermody76 / J A N U A R Y
196 2
es
^
Values of P, V, T, H, and S for (35) are obtained from the thermodynamic tables or charts.
When a similar compressor has been tested with the design gas
es and /J.s can be inferred directly from the test data and no conversion from e and ^ is necessary. Equations (33), (33a), (336),
(33c), (34), (34a), (346), and (35) are limited to the design of uncooled centrifugal compressors without side-flow.
An important application of polytropic analysis is the evaluation of equivalent performance tests where a centrifugal compres-
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sor designed for one gas is shop-tested with another. B y duplicating certain design parameters the test conditions become equivalent to the design conditions and test values of e and p. from (27a)
can be used in (33), (33a), or (33c) to predict field performance.
Usually isentropic analysis is inaccurate for this purpose because
of difficulties resulting from (31).
One of the design parameters of centrifugal compressors is the
acoustic velocity of the gas. For an infinitestimal disturbance
this can be shown to be
SP'
{- g
a = V \ where a is acoustic velocity.
(36) we find
(36)
= ^
^
(36a)
The designer relates acoustic velocity to compressor speed by a
speed coefficient or impeller Mach number, M , defined
M :
(37)
At the compressor inlet .1/ becomes
»i
il/„ =
(37a)
«o
By inserting (56), ( 6 « ) , (26), and (36a) into (37a) we obtain
n—1
1 +1
-
i).w
Pi
( p ^
n
u- 1
=
=
(38)
for the first stage, where ns is evaluated at the inlet and m and n
are mean values. Similar relations apply to the other stages and
to the over-all compressor.
Sometimes it is necessary to subdivide individual stages and
consider impeller and diffuser performance separately. Referring
again to Fig. 1, imagine we have a test point locating the thermodynamic state of the gas between an impeller and its diffuser.
Let constant-efficiency paths, pt and p,h representing the impeller
and diffuser, connect this state with the stage inlet and outlet
states. The polytropic head, 1FP, or \VpJ, of the impeller or diffuse!' is the net reversible mechanical energy input along p; or pd.
(26 c)
It follows from (26), (26c), and (39) that
Pi
IF,,
AH;
e,i;
IF, -
A ICE,
IF.
1F„„
IF,, -
AHj
IF
AICE,
Wpd
c,i
(4c)
- A ICEd
It follows from (4) and (4c) that
+
e,
" V
_
ed
"I,
Journal of Engineering for Power
(89)
c
where IFP and e are stage head and efficiency.
(40)
ed
Total Pressure and Temperature
Throughout our entire discussion P and T have represented
"static" pressure and temperature as distinguished from "int a c t " or " t o t a l " values. By total pressure and temperature, P t
and T„ we mean the stagnation values resulting from an isentropic deceleration of a flowing gas. The distinction is onlj'
important in compressor tests where v2/'2g is appreciable compared to }V P ; e.g., in the testing of impellers or diffusers.
The difficulty is that a temperature-sensing device in a gas
stream "feels" 2', rather than T and registers some intermediate
temperature depending upon its particular characteristics. Assuming that these characteristics are known we can surmount the
problem by installing a pressure probe to measure P,. Then by
combining (4), (5c), (6i), and (6n) with e = 1 we can write
V2 ^
ZRT
2g
ms
T
P
(
'
~\P
ms
CP{T,
T
(1 +
-
T)
X)
'LL
ZR
(41)
k -
Y
1
k
(1 +
X)
For small ratios P,/P, (41) simplifies to
v2
2g
I
(P, -
P)V
P, -
P>
(41a)
r
Side-Flow and Cooling
AKEd
In (46) the shaft-work input, IF,, to the impeller is also the
shaft-work input, IF, to the entire stage. The shaft-work input
to the diffuser, IF,,, is zero. Thus (46) becomes
W -
Pi
where p and e are the stage head coefficient and efficiency. Equations (4c), (26c), (39), and (40) are used with our previous relations
to separately consider impeller and diffuser performance.
T. (4 6)
|
e<
Impeller and diffuser efficiencies, e, and ed, are defined by (4):
IF„,
"
P'
gw„
. gWpi
Pel = '
By inserting (13) and (23a) into
a = V^gPV
p^idn
(39) also relates individual stage heads and efficiencies to those of
the over-all compressor.
Impeller and diffuser head coefficients jit, and pd are defined by
(26):
Thus far none of our analysis has included compressors having
either side-flow or cooling. The treatment of side-flow requires
that the compressor be subdivided into units of constant mass
flow rate, with due consideration given to the mixing effects when
the side-flow is inward. The homogeneity of such a mixture entering the next stage may be questionable but is usually assumed
for practical purposes.
A similar procedure is employed for interstage cooling since the
compressor can be subdivided into uncooled units and the intercooler treated separately. Most cases of liquid injection cooling
also fall into this category. Even diaphragm cooling can be
handled this way if most of the cooling takes place between the
diffuser outlet and the inlet of the succeeding stage.
If a cooled compressor is tested for over-all values of e and p.
these are only significant for the tested ratio of gas cooling to
An equation like
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mechanical energy input. Both the magnitude of this ratio and
its distribution within the compressor are controlling factors.
*
Other Applications
Y = 1
Polytropic analysis is not limited to centrifugal compressors
alone but can be applied to any machinery handling compressible
fluids. This includes turbines as well as compressors, provided
that e is replaced by the reciprocal of turbine efficiency.
Nor is our analysis limited exclusively to machinery. The
determination of orifice and nozzle flow rates can be accomplished
by (4) or (29) using e = es = 1, with (56) or (5c), (6f), (6ft),
and an empirical flow coefficient. The results are similar to
(41) with the addition of an approach velocity factor and a flow
coefficient.
Another application is the P, V, T relation of a throttling process, which is represented by (6j) and (6ft). The P, V, T relations
of constant volume and constant temperature processes are represented by (6ft) and (24) or (25).
At first glance a constant pressure process appears to be
indeterminate by (6k) and (6n). However, we can rearrange
(6h) to supplement (6/i) by
mPn P
= 1 +
A
(On)
Wr/Pr
-
(22 a)
T\
/
dZ
Z
\ bPr / Tr
and determine X and Y from the slopes of constant Pr and Tr
lines representing Z. The usual plot of Z contains lines of constant Tr versus Pr from which Y can be determined directly. A
cross-plot to obtain lines of constant PT versus Tr is necessary to
determine X .
In this manner X and Y were obtained from the Nelson-Obert
charts throughout the region 0 ^ Pr ^ 3 and 0.6 ^ Tr ^ 5.
These graphic determinations were plotted along with values
calculated by finite differences in (22) for a common refrigerant
gas, dichlorodifluoromethane (Refrigerant-12) [5], which reproduces the Nelson-Obert charts within 3 per cent. The preliminary
plots were checked and adjusted by a second set of calculations,
this time employing (15c) for Refrigerant-12.
The final generalized charts of X and Y are shown in Figs. 2
and 3. Examination of these charts reveals the possibility of
large deviations from the perfect-gas values X = 0, Y = 1. For
pressures and temperatures outside the charted region X can
become negative.
Numerical Example
and write (6ft)
T
\ I +.r
(6»)
where subscript P denotes constant pressure.
Generalized Compressibility Functions
In connection with (33) we mentioned the necessity of obtaining
A", 1", and Z from generalized compressibility charts for gases in
corresponding states. Many such charts have been published
since 1931 relating Z to Pr and Tr. B y (13) Z is defined
PV
RT
(13 a)
Pr
T,
Pc
(42)
JL
In (42) P, and Tc are absolute critical pressure and temperature.
Gases having the same values of P r and T r are said to be in
corresponding states and often have nearly the same value of Z.
The Nelson-Obert charts [3, 4], published in 1953, correlated the
^-values of twenty-six different gases within 2'/> per cent throughout the region 0 ^ Pr ^ 10 and 0.6 ^ Tr ^ 15, except near
PT = Tr = 1. These gases included air, argon, benzene, carbondioxide, ethane, iso-butane, methane, neon, oxygen, nitrogen,
propane, and propylene. However, it was also reported that five
gases, ammonia, helium, hydrogen, methyl-fluoride, and steam
correlated less accurate^, so that Pt and Tr are not universal
compressibility parameters. Sometimes better results are obtained if pseudo values of Pe and Tc are employed in (42). Nelson
and Obert found this to be the case with helium and hydrogen.
For gases whose Z-values can be correlated accurately by P r and
Tn using either actual or pseudo values of Pc and Tc. it is also
possible to correlate X and 1". With (42) we may write (22)
196 2
Perhaps it would be instructive to consider the polytropic
analysis of an actual compression process involving a real gas.
Imagine we have tested a centrifugal compressor with Refrigerant-12 and obtained the following test data:
Po =
10 psia
Pi =
130 psia
T0 =
-10 F
T3 =
210 F
In a table of thermodynamic properties [5] we find:
V0 = 3.8861 ft 3 /lb
H, =
106.520 B t u / l b
Ho = 76.880 B t u / l b
V, = 0.37297 ft 3 /lb
V3 = 0.41676 ft. 3 /lb
Hi =
97.8-39 B t u / l b
B y (27a) we obtain:
while reduced pressure, Pn and reduced temperature, Tr, are
defined
78 / JANUARY
dZ_
=
n =
1.1488
Wp =
ns =
1.0944
e =
f =
1.015
17,285 ft-lb/lb
0.749
The deviation of f from 1.000 shows that ns varies along SoLet us explore this variation by tabulating n s and the related
properties along So. We can also tabulate these properties along
p, substituting n for ns with e = 0.749. In addition to the end
points our table should include the approximate mid-points of So
and p where TFl5 and IF, are each about half their final totals.
The results are shown in Table 1.
Table 1 shows a variation in n s along So of 7.5 per cent, hence
it is not surprising that TFS calculated with a constant mean ns
= 1.0944 is in error by 1.5 per cent. The variation in n along p
is 6.9 per cent. The similarity of this to n s reinforces our decision to adjust TFP by / = 1.015 determined from n s and WsTable 1 also shows an appreciable deviation of n s from its perfect-gas value, k. Were we to use a constant mean k = 1.173 to
predict TFS, Vi, and A T s by perfect-gas relations (61) in (5a) and
(6n) our errors would be 7, 17, and 22 per cent, respectively.
By using the real-gas equations (6i) in (5b) and (6ft) with a constant mean Ar = 0.265, Y = 1.076, a n d / = 1.015 the corresponding errors would be 1 / 2 , 1, and l /s per cent.
The foregoing mean values of k, X, and Y were estimated by
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Table 1
P (psia)
V (ft 3 /lb)
T (deg F)
H (Btu/lb)
S (Btu/lb deg R )
Cp (Btu/lb deg R ) . . .
Cy (Btu/lb deg R ) . . .
k
X
Y
X
ms
ns
m
-Along S 0
Middle
39
1. 1460
74..11
87..419
0..18171
0. 15343
0..13160
1. 1659
0. 220
1. 057
0. 943
0. 123
1. 103
Inlet
10
3. 8861
-10
76. 880
0. 18171
0. 13603
0. 11716
1. 1611
0. 101
1. 038
0. 974
0. 131
1. 119
Outlet
130
0..37297
159 .96
97 839
0 .18171
0 . 17462
0..14558
1. 1995
0. 519
1 .152
0. 882
0. 126
1. 041
numerical averages with double weight given the mid-point, as
k =
1.1611 + 2 X 1.1659 + 1.1995
The accuracy of our results could be improved slightly by refining these constant mean values, but perfect accuracy is impossible since (6m) cannot reproduce (8) perfectly.
Now imagine it were necessary to design a centrifugal compressor for the application we have just discussed, but without
benefit of detailed thermodynamic data. Suppose our only information were:
10 psia
R =
12.779 ft-lb/lb deg R.
Vo = 3.8861 f t y i b
Pc =
596.9 psia
T0 =
-10 F
T„ =
233.6 F
P3 =
130 psia
e =
0.749
k =
1.160
The solution of this problem requires that we assume T3 and
verify our assumption by (33). To make a long story short let us
assume the correct T3 = 210 F and observe the results. T o
estimate mean values of X and Y we need a tabulation like
Table 1 and therefore we must choose an approximate mid-point
of p. A reasonable choice would be at the square root of the
over-all pressure ratio and half the assumed temperature rise;
i.e., at P = 36 psia and T = 100 F. The results are shown in
Table 2 with A" and 7 from Figs. 2 and 3.
Table 2
Inlet
0.0168
0.649
0.11
1.04
Pr
Tr
.V
Y
Middle
0.0603
0.807
0.17
1.05
Outlet
0.2178
0.966
0.36
1.10
From Table 2 we compute the mean X = 0.20 and the mean
1" = 1.06. Assuming / = 1.000 we find by (33)
m =
0.1559
V3 = 0.4141 ft 3 /lb
n =
1.1455
T3 = 211.1 F
Wp
=
16,967 ft-lb/lb
Comparing these results with the correct data from our test
discussion we find the errors in TFP, V3, and A71 to be 2, 1, and
Vs per cent, respectively. Had we used perfect-gas relations the
results would have been
Journal of Engineering for Power
Middle
40
1.1862
102
91 .678
0.18909
0.15518
0.13436
1.1550
0.187
1.056
0.952
Outlet
130
0.41676
210
106.520
0.19518
0.17294
0.14806
1.1680
0.355
1.106
0.912
0. 171
1. 177
0.153
1 .113
0.146
1.101
m' =
0.1842
TV = 0.4794 f t ' / l b
n' =
1.2257
T3'
= 26t.5 F
W p ' = 18,347 ft-lb/lb
k = 1.173
Po =
Inlet
10
3 .8861
-10
76..880
0 .18171
0 .13603
0 .11716
1. 1611
0. 101
1, 038
0. 974
and our errors would have been 6, 15, and 23 per cent.
Now suppose it were necessary to design a compressor for this
same application but with complete thermodynamic data
available. Using k = 1.167 and e = 0.749 in (34b) we would
obtain es' = 0.701. By (35) we would find TV = 0.41801 ft 3 /lb
and by (27a) e' = 0.744. Then
~
e
= 0.942 and es = 0.706 in (34c)
From the data of Table 1 we find the correct es = 0.707.
Alternatively we might have used (33c) to find S3 = 0.19502
B t u / l b deg R and T3 = 209.4 F. In Table 1 the correct values
are 0.19518 and 210.0. This is only 0.3 deg F less accurate than
the preceding isentropic solution and far less complicated.
T o consider three different aspects of the same compression
process we used three different ^--values. The first, 1.173, was
the mean value along So. The second, 1.160, was the mean
value along p. The third, 1.167, was the mean value between
So and p. Each was the most appropriate for its particular
application. No single value would have produced comparable
accuracies in all three instances.
Conclusions and Recommendations
The real-gas equations of polytropic analysis indicate, and our
numerical example confirms, that accuracies within a few per
cent require more thermodynamic data than exist for some applications. Compressor users should recognize such applications
for what they are and not expect the impossible. Furthermore,
when thermodj'namic tables or charts are prepared it would be
most convenient if CP or k, X, Y, and Z were included.
Generalized compressibility data are helpful when specific data
are lacking. Since compressibility functions X and Y have been
defined and generalized here for the first time future investigators may be able to improve some regions in Figs. 2 and 3.
Experience may disclose a need for extending these figures beyond
Pr = 3 and Tr = 5. Investigations to improve or extend Figs.
2 and 3 are recommended.
Most of the real-gas equations have also been derived and
published here for the first time.
Future rearrangements
and derivations maj' enhance their utility. In this connection a
derivation to generalize the polytropic head factor, / , would be
quite useful.
Finally, the A S M E Power Test Code for Centrifugal, Mixed
Flow, and Axial Flow Compressors and Exhausters (PTC10-1949)
should be rewritten to include polytropic analysis and equivalent
JANUARY
1962 / 79
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x NoiiONinj
80
/
JANUARY
1962
Ainiaiss3ddiAioo
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A NOIlONfld
Journal of Engineering for Power
AllliaiSS3dd^OO
J A N U A R Y 1 96 2 / 8 1
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performance testing. The present code is based upon isentropic
analysis and therefore cannot provide for equivalent tests.
Acknowledgments
In the graphic determinations of A" and Y for Figs. 2 and 3 the
author was assisted by Miss Mary H. Butler, Mr. Kenneth E.
Dodrer, and Mr. George D. Ferree, all of the York Division,
Borg-Warner Corporation. For the calculations involving R e frigerant-12 the author was furnished detailed specific heat data
b}' Mr. Robert C. McHarness of the " F r e o n " Products Division,
E. I. duPont de Nemours & Company, Inc.
82 / J A N U A R Y
1962
References
1 W. C. Edmister and R. J. McGarry, "Gas Compressor Design," Chemical Engineering Progress, vol. 45, 1949, pp. 421-434.
2 W. C. Edmister, "Compressor and Expander Design," Chemical
Engineering Progress, vol. 47, 1951, pp. 191-198.
3 L. C. Nelson and E. F. Obert, "Generalized Properties of Gases,"
TRANS. ASME, vol. 76, 1954, pp. 1057-1066.
4 L. C. Nelson and E. F. Obert, "Generalized Compressibility
Charts," Chemical Engineering, vol. 61, 1954, pp. 203-208.
5 "Thermodynamic Properties of Freon-12 Refrigerant (Diclilorodifluoromethane)," E. I. duPont de Nemours & Company, Inc.,
Copyright_1955_and 1956.
Transactions of the A S M E
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