Application of experimental state data and spectroscopic term values shows that the thermodynamic and transport properties of reacting pH2-oH2 mixtures are considerably different than those of chemically frozen pH2 at temperatures below 300 R. Calculated H-S data also show that radiation-induced pH2-oH2 equilibration at constant enthalpy produces a temperature drop of at least 28 R, corresponding to an ideal shaft work loss of 15% or more for a turbine operating downstream from the point of conversion. Aside from differences in thermodynamic and transport properties, frozen pH2-oH2 mixtures may differ from pure pH2 on a purely hydrodynamical basis

Bullock, R. E.

Carter, H. G.

English

NASA Technical Reports Server

THERMODYNAMIC AND TRANSPORT PROPERTIES OF FROZEN AND
REACTING PH2-oH 2 MIXTURES
H. G. Carter and R. E. Bullock
General Dynamics Corporation
Fort Worth, Texas
Abstract
Application of experimental state data and spectroscopic
term values shows that the thermodynamic and transport proper-
ties of reacting PH2-oH 2 mixtures are considerably different
than those of chemically frozen pH 2 at temperatures below 300°R.
Such differences will be significant from the standpoint of
nuclear rocket design if it turns out that para- to ortho-
hydrogen conversion can be catalyzed by radiation in the cold
dense phase. Calculated H-S data also show that radiation-induced
PH2-oH 2 equilibration at constant enthalpy can produce a tem-
perature drop o_ at Least g8°R, corresponding to an ideal shaft
work loss of 15% or more for a turbine operating downstream from
the point of conversion. Aside from differences in thermodynamic
and transport properties, frozen PH2-oH 2 mixtlJres may differ
from pure pH 2 On a purely hydrodynamical basis.
Significance of Possible Para-
Orthohydro_en Conversion in
Nuclear Rockets
Hydrogen is generally a mixture of two
distinct N 2 components. A molecule of the
component called "orthohydrogen" may assume
quantized rotational energies that are not
accessible to the component called "para-
hydrogen," and vice versa. The resulting
disparities in energy-level populations at
a given temperature imply differences in
the thermodynamic and transport properties
of the two types of hydrogen. Thus, for
example, the enthalpies and thermal con-
ductivities of PH2-oH 2 mixtures differ ap-
preciably from _hose of pure parahydrogen
in the 36°R-500 R range, not only in abso-
lute magnitude but also in temperature
dependence. Aside from thermodynamic and
transport properties, para- and orthohydro-
gen may differ on a purely hydrodynamical
basis. The latter possibility, which has
been given little attention, derives from
the fact that the differences in accessible
rotational energies for the two types of
molecule are associated with differences in
angular momenta.
The liquid hydrogen that is stored for
use in nuclear rocket propulsion is usually
over 99% pH 2. However, the high-tempera-
ture hydrogen gas that emerges from the noz-
zle of the nuclear rocket is invariably a
mixture consisting of 75% oH 2 and 25% pH 2.
This change in proportions results from the
facts that (I) certain chemical or magnetic
476
https://ntrs.nasa.gov/search.jsp?R=19720010012 2020-03-17T03:45:47+00:00Z
reactions can convert parahydrogen to
orthohydrogen when sufficient energy is
available, and (2) in the presence of a
suitable conversion catalyst, hydrogen tends
to a PH2/oH 2 ratio that is determined by
the temperature. Catalysts which convert
parahydrogen to its equilibrium proportions
are available in the reactor core (in the
form of H-atoms from thermal dissociation),
if not elsewhere. When catalysts are pre-
sent in large enough quantities that a
shift in temperature is accompanied, prac-
tically Instantaneously, by a shift in PH2/
oH 2 ratio to a value corresponding to the
new temperature, the para-orthohydrogen
mixture is called "reacting" or "equilibrat-
ing." When no such catalysts are available,
the pH2/oH2 ratio is not affected by a
temperature change, and the mixture is
called "frozen." In the temperature range
from 60°R to 300°R the specific heats of
a given nominal PH2/oH 2 mixture differ
radically according to whether the hydro-
gen is frozen or reacting.
In nuclear rocket design, a question a-
rises as to what type of hydrogen should
be assumed to exist in the relatively cold
regions between the propellant tank and
the core inlet. If the hydrogen is chemi-
cally frozen throughout this region, then
it obviously remains in almost pure pH_
form., and thermodynamic and heat transZer
calculations can be based on parahydrogen
properties. If, on the other hand, the
flowing hydrogen i_ catalysed by radiation
or by paramagnetic surfaces, then the hy-
drogen may be chemically reacting over at
least part of the region below the reactor
core, and pure parahydrogen properties may
not apply. From the standpoint of an over-
all design study, the effect of catalytic
conversion is not limited to the values of
thermodynamic "and transport properties of
hydrogen at a prescribed temperature and
pressure. The sudden equilibration of
hydrogen due to catalysis at a certain
point in the flow-path can reduce the tem-
perature by as much as 40°R, through a
thermochemical process that is analogous
to the cooling of paramagnetic salts in a
decreasing magnetic field. Such "conver-
sion-cooling" can reduce the available
energy for a turbine operation or alter the
dimensions of various orifices for an op-
timum design.
Whether hydrogen conversion in the sub-
core region is of design significance de-
pends on three qu_stlons. These are: (i)
does an effective mechanism for conversion
actually exist in relatlvely cold hydrogen
subject to irradiation and possible en-
counters with paramagnetic sites, and how
much conversion Can it produce? (2) What
are the quantitative thermodynamic and
transport propertlesof frozen and reacting
PH2/oH 2 mixtures? (3) What is the overall
effect of conversion on a specific design?
In regard to the first question, it will
only be noted that data obtained during
practical tests indicate that relatively
told parahydrogen can be converted by
radiation, and that reasonable chemical
models have been advanced to explain this
effect. The third question is in the pro-
vince of rocket designers. The results
given here pertain only to the second ques-
tion, viz., information on the basic pro-
perties of pH2-oH2 mixtures.
Equilibrium Orthohydrogen
Fraction
The equilibrium orthohydrogen fraction
at a giventemperature is independent of
pressure, at least for pressures in the
range of nuclear rocket application and
lower. For temperatures below 500°K (900°R)
the equilibrium orthohydrogen fraction, We,
is given to an accuracy of 0.01% by
7 (odd) IT/_ -Oj IT
w e = _ gj e "0_ i -- gj e (i)
j-I / j-0
where j is the number of a rotational level
Even and odd values of j correspond to pH 2
and oH 2 rotational levels, respectively.
The parameter @i represents the molecular
rotational energy of a particular level
divided by Boltzmann's constant. Using the
spectroscopic term values cited in the
definitive work of Woolley, et al. at NBS
(Ref. 1), it is found that a suitable ap-
proximation is
9j(°K) = 85.3757 j(J+l) - 0.065769 j2(j+l)2
+ 0.0000712 j3(j+l)3 (2)
for values of J less than or equal to 8.
This prescription ignores the role of vi-
brational excitation, which is negligible
below 500°K. The parameter gj in Equation
(I) is a statistical weight glven by
gj - 3(2J+I), j odd
- (2J+l), j even (3)
At temperatures above T - 500°K, the equi-
llbrlumorthohydrogen fraction is m e -
0.750.
477
Evaluation of Equation (1) yie lds  t h e  de- 
pendent e o f equi li b r  ium o r  t hoh ydrogen f rac - 
t i o n  on temperature shown i n  Figure 1. 
is  seen t h a t  t h e  temperature range over 
which oe varies s i g n i f i c a n t l y  is 3OoK t o  
22OoK, o r  54OR t o  396'R. 
It 
Calculation of Enthalpy and Entropy 
The calculat ion of enthalpies  and entro-  
pies  i s  obviated by expressing t h e  r e s u l t s  
as ideal gas values,  augmented by "real- 
gas" terms t h a t  a r i s e  from intermolecular 
reac t ions .  The la t ter  are obtained from 
an equation of state,  which i s  usua l ly  
w r i t t e n  i n  t h e  form of a temperature and 
dens i ty  dependent f o r  t h e  compressibi l i ty  
f a c t o r  2 = pv/RT. I n  obtainirg t h e  pre- 
s e n t  r e s u l t s  i t  has been assumed t h a t  the  
equation of s t a t e  
2 = Z(T,P) (4 1 
is  the  same for e i t h e r  para- o r  ortho- 
hydrogen. The s t a t e  equation used is  t h a t  
given by Wooley et a l .  f o r  normal hydrogen 
(75% ortho) .  
was  based on ra ther  o l d  normal hydrogen 
data, has been compared with d a t a  reported 
more recent ly  by Michels e t  a l .  (Ref. 2). 
Typical comparisons are shown i n  Figure 2. 
An a n a l y s i s  of t h e  d i f fe rence  between t h e  
o l d  and new data, which is taken as a 
measure of  the present uncer ta in ty  i n  t h e  
state equation f o r  -normal hydrogen, shows 
t h a t  t h e  corresponding u n c e r t a i n t i e s  i n  
enthalpy and energy of normal hydrogen are 
respec t ive ly  only about 1 Btu/lb and 0.002 
Btu/lb OR. 
Hermans e t  al .  (Ref. 3), on t h e  d i f fe rence  
between the  second v i r i a l  c o e f f i c i e n t s  of 
para- and normal hydrogen, ind ica te  t h a t  
t h e  use of a normal hydrogen state equa- 
t i o n  on pH2-0H2 mixtures w i l l  not  lead t o  
an e r r o r  grea te r  than t h a t  corresponding 
t o  t h e  present  uncertainty i n  t h e  normal- 
hydrogen state data.  
The real-gas enthalpies  and entropies  
of a number of mixtures have been calcu- 
l a t e d  using tabular  da ta  given by Woolley 
e t  a l .  t o  obtain t h e  i n t e g r a l s  involving 2. 
Ideal-gas values of t h e  enthalpy and en- 
tropy f o r  both frozen and reac t ing  pH2-0H2 
m i x t u r s h a v e  been obtained from t h e  8 and 
This state equation, yhich 
Furthermore, recent  da ta  by 
g j  defined i n  Equations (2) and (3). j 
The en tha lp ies  of frozen pH2, frozen oH2 
and e q u i l i b r a t i n g  hydrogen a t  0 and 800 
psi are shown i n  Ftgure 3. 
ing s p e c i f i c  hea ts ,  obtained by d i f fe ren-  
t i a t i o n  of H with respec t  t o  temperature, 
are shown i n  Figure 4. 
EH2 behaves as a mixture with an interme- 
d i a t e  s p e c i f i c  heat  only above 100%; a t  
lower temperatures t h e  s p e c i f i c  heat  of 
reac t ing  hydrogen is appreciably higher 
than t h a t  of  e i t h e r  frozen component. By 
c o n t r a s t ,  t h e  s p e c i f i c  heat  of a frozen 
mixture is  obtained by simply c a l c u l a t i n g  
t h e  average of t h e  frozen pH2 and 0H2 val- 
ues ,  weighted by t h e i r  respec t ive  concen- 
t r a t i o n s  i n  the  mixture. 
The correspond- 
It i s  seen t h a t  
478 
Transport Propert ies  
The v i s c o s i t y  of  a pH2-0H2 mixture i s  
determined by t r anspor t  of momentum alone, 
oH2 composition in so fa r  as pH2 and 0H2 con- 
form t o  t h e  same equation of state. 
the  o the r  hand, t h e  thermal conduct ivi ty  of 
an ideal-gas mixture of para- and ortho- 
hydrogen i s  r e l a t e d  t o  t h e  v i s c o s i t y ,  v ,  
by Eucken ' s r e l a t i o n  
l and i s  the re fo re  independent of t h e  pH2- 
~ 
On 
KO = (9Y-5 )Ct  77/(4M) ( 5 )  
where & is t h e  s p e c i f i c  heat  o f  t h e  i d e a l  
gas ,  Y is t h e  r a t i o  of s p e c i f i c  hea t s ,  
and M i s  t h e  molecular weight. 
gas thermal conduct ivi ty  K i s  r e l a t e d  t o  
KO by t h e  following Enskog r e l a t i o n :  
K/Ko = 1 + 0.5753 + 0.5017E2 - 0.204E3 
The real- 
(6) 
where E i s  given by 
E = Z - 1 + T (E) (7 )  6'T p 
Figure 5 shows t h e  thermal conduct ivi ty  
f o r  pH , oH2 and a r e a c t i n g  mixture a t  0 
p s i  an% 800 p s i .  As i n  t h e  case of SpeCi- 
f i c  h e a t ,  t h e  thermal conduct ivi ty  depends 
s t rong ly  on whether a given mixture is 
f rozen o r  react ing.  
Cooling Due t o  Conversion 
I f  a chemically frozen parahydrogen gas 
i s  suddenly exposed t o  a conversion-catalyst 
such as r a d i a t i o n  o r  a paramagnetic su r -  
f ace ,  t h e  r e s u l t i n g  e q u i l i b r a t i o n  t o  a 
pH2-oH2 mixture i s  an i r r e v e r s i b l e  process.  
Hence, t h e  r eac t ion  cannot be i s e n t r o p i c  
even i f  it happens t o  be ad iaba t i c .  
general ,  t h e  enthalpy change due t o  a d i f -  
f e r e n t i a l  conversion process is 
I n  
dH = TdS + Vdp + Pldxl  + p2dx2 (8) 
where x1 and x2 are t h e  pH2 and oH2 mole 
f r a c t i o n s ,  r e spec t ive ly ,  and where pl and 
p2 are t h e  chemical p o t e n t i a l s .  I n  t h e  
case where no heat  e n t e r s  o r  leaves t h e  
system and t h e  pressure is  kep t  a t  a con- 
s t a n t  value,  Equation (8) s i m p l i f i e s  t o  
TdS E ( P ~ - P Z ) ~ X ~  ( 9  1 
479 
This increase i n  entropy is  accompanied by 
a decrease i n  temperature, which can, f o r  
example, subsequently reduce t h e  amount of 
ava i lab le  energy from a turb ine  operating 
between two prescribed pressures .  The 
poss ib le  e f fec t  of such cooling is seen 
from t h e  approximate H-S c h a r t  shown i n  
Figure 6,  which has been constructed from 
Figure 3 and ca lcu la ted  entropy data .  
When parahydrogen a t  297’R i s  converted t o  
60% 0% a t  constant enthalpy and constant 
pressure,  the temperature decreases from 
297% t o  269%. 
occurred, the energy a v a i l a b l e  t o  a t u r -  
bine operating between 975 p s i  and 675 p s i  
would be 111 Btu/lb. However, it is  seen 
t h a t  609. conversion would reduce t h e  a v a i l -  
a b l e  energy t o  98 Btu/lb. The magnitudes 
of such losses  depend upon t h e  i n l e t  en- 
thalpy and, of course,  t h e  amount of con- 
vers ion t h a t  i s  assumed. Figure 7 shows 
t h e  percent loss i n  i d e a l  energy f o r  an 
assumed conversion of 60% as  a function of 
i n l e t  enthalpy . 
I f  conversion had not  
Possible  Hydrodynamic Differences 
2 Between pH2 and OH 
The difference between t h e  thermodynamic 
and t ranspor t  propert ies  of parahydrogen 
and those of orthohydrogen a r e  completely 
s p e c i f i e d  by t h e  r o t a t i o n a l  energy l e v e l s  
access ib le  t o  each, a t  least insofar  as 
t h e  two modifications conform t o  t h e  same 
equation of s ta te .  However, from t h e  hy- 
drodynamical standpoint,  another type of 
d i f fe rence  might be worth exploring. 
Parahydrogen molecules may take i n t r i n -  
s i c  angular momenta of 0,  2fi, 4fi, etc. ; 
whereas orthohydrogen molecules may take 
i n t r i n s i c  angular momenta of .K, 3ti, 5K, 
etc. A t  temperatures below 180°R, over 
97% of a l l  parahydrogen molecules are i n  
a s ta te  of zero angular momentum. 
p r a c t i c a l  purposes one may say t h a t ,  a t  
these  temperatures, parahydrogen is a 
f l u i d  composed of molecules with no i n t r i n -  
s ic  angular momentum, whereas orthohydro- 
gen is  a f l u i d  whose molecules have an 
i n t r i n s i c  angular momentum of 5 .  
For 
Hence, 
of such phenomena a6 t h e  t r a n s i t i o n  from 
laminar t o  turbulent  flow might d i f f e r  i n  
t h e  two types of f l u i d s ,  independently of 
any e f f e c t s  associated with thermodynamic 
and t ranspor t  p roper t ies .  
References 
1. Woolley, H. W.,  Sco t t ,  R. B., and 
Brickwedde, F. G. ,  NBS Research Paper 
RP1932, Volume 41, November 1948, 
pp. 379-475. 
On a rigorous hydrodynamical basis (Ref 4 ,  9 2. Michels, A * ,  et Physics 25, 1959, 
the  equations f o r  energy and momentum con- 
servat ion,  which a re  s u f f i c i e n t  t o  descr ibe 
be augmented by an equation of angular 
momentum i n  t h e  case of orthohydrogen. 
the  language of transport  theory, c o l l i -  
s ions between orthohydrogen molecules in-  
volve an e x t r a  surrnnational invarient .  It  
pp. 25-42. 
t h e  behavior of a parahydrogen f l u i d ,  must 3.  Hemans, j. F., et al., physics 31, 
1965, pp. 1567-1574. In  
4 .  Chapman, S . ,  and Cowling, T. G. ,  The 
Mathematical Theory of Non-Unifom- 
Gases (Cambridge), 1953, p. 202. 
is  therefore  conceivable t h a t  t h e  d e t a i l s  -
480 


Thermodynamic and transport properties of frozen and reacting pH2-oH2 mixtures

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