Gasdynamic Discontinuities

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NUMBER 3

PHINCETON AERONAUTICAL PAPERBACKS

COLEMAJ\' duP. DONALDSON, GENERAL EDITOR

PRINCETON AERONAUTICAL PAPER.BACKS 1. LIQUID PROPELLANT ROCKETS David Altman, James M. Carter, S. S. Penner, :\>farttn Summerfield. High Temperature Equilibrium, Expansion Processes, Combustion of Liquid Propellants, The Liquid Propellant Rocket Engine. 196 page 2v1) h rmn1mum U - U1 maximum . v maxunum The Ch-J points corresponding to extrema of the slope of the line from the point p 1, v, are lateral extrema (of u1) on a velocity plot, of u vs. u 1. {

II )

D,I · RELATIONS IN A NOR/>1AL l>ISCONTlNUlTY

D · GASDYNAMIC DISCONTINUITIES

This relation between the two plots has a reciprocal property, as extrema of the slope of the line through the origin on the velocity plot, u/ui, are also lateral extrema (of v) on the Hugoniot curve. The point Pi. Vi on the Hugoniot diagram plays an important part in interpreting the properties of the Hugoniot curve, while the point O, 0 does not. The point p., v, is termed the "origin" of the Hugoniot diagram, for ease in reference. The reader may note that if the Hugoniot diagram is considered as a plot of p - pi vs. v - Vi this point is the origin in the conventional sense. Some subsidiary formulas involving derivatives o[ the thermodynamic quantities are of interest, and are expressed in the form

equated in Eq. 1-23 may not vanish as long as there is a change in any one of the variables; this may be considered the differential of a variable 11 defined along the Hugoniot curve, termed the "Hugoniot curve variable," which has no stationary points. If one of the denominators in Eq. 1-23 is zero at a point on the Hugoniot curve, the corresponding variable is stationary there, and conversely. From Eq. 1-14, 1-23a, and 1-23b the following exprelll!ion may be obtained

d'1

~

dv

(l-23a)

1- t(vi - v) ("P)

ile •

Tds = !(p - pi}

+ !(v, -ck

= -

·lr(p

+ p1)

av) ( = i!p ' l =

(

v~ (:):

dp - j(p - p1)

(~~).

T

v,

+ v) + ...'(p

= ( ilv)

CJp • j(v1

-

v) [

(I-23d)

(1-23e)

) (av) i!p •

- Pi

1- (p -

·}d(u 1

-

u)'

Pi)(!~).)

-

j(p -

Pi)(:;),

The derivation of these equations is not presented here, except to indicate the method used. Equations derivable from the Hugoniot relation of the type of Eq. 1-14 involve three first different.in-ls. But between three firat differentials there is a thermodynamic, or state, relation, such as

(ilp)

(~P) as .. ds + av • d»

(1-24)

Elimination of one of the variables gives an equation connecting two first differentials; eliminating dp from Eq. 1-14 and 1-24 gives the relation connecting Eq. 1-23a and 1-23b. Suitable repetition of the method yields the complete set of Eq. 1-23. The differential appearing in various forms (

(~).]

:r

=

(~~). =

(l-26)

2p•a•r

where

J:2 }

+

This dimensionless quantity r is equal to !('Y 1) for a perfect gas with constant specific heats and it is the appropriate replacement for !('Y + l) in the theory of t ransonic similitude in an itnperfect gas. Eq. 1-26 may be derived by differentiating Eq. 1-25 with respect to v and applying the extremal properties of the Ch-J point and those given by Eq. 1-16. In addition to the fundamental thermodynamic inequalities of Eq. and other inequalities are conaidered as conditions which roust be satisfied in order that the Hugoniot curve satisfy certain conditions of proper behavior. These are conditions which cannot be shown to be satisfied by any general thermodynamic argument; they are, on the other hand, conditions which are satisfied by most materials but which may not all be satisfied by certain materials. A symbol a for the coefficient of thermal expansion is introduced

1-11 1-12

(l-23f)

dp =

(1-25)

(1-27)

(2.".) ilh •

dh

i!p , '(

(1-23c)

~).] ( m• + :)

which may be used to relate the geometry of the Hugoniot curve to the question of whether the velocity behind the discontinuity is subsonic or supersonic. Finally, a last property of the Ch-J points is given, with regard to the second derivative of the Hugoniot curve there: [ 1 - !(v, - v)

~·· - -·.

- t(v , ·- v)

ilv)

(l-23b)

m• - p•a• - [ 1 - !(vi - v) (

(1-28) and the following classical thermodynamic relations are

pv"' = P c,.

(iJv) = E (~) iih. ,. T iip ,

=

1

~:a= v(~~). = -;, (~;), = ( 13 }

present~d

+ ~a' (iJp) av , -1 -

~:(~),

(1-29a) (l-29b)

0 · CASDYNAMIC DISCONTINUITIES

D,1 · RELATIONS IN A NORMAL DISCONTINUITY

The conditions which are of interest are: Condition I

(

0

'p)

Cv1

•

= 2p'a'r

>

0

(1-30)

Condition II-weak

~a = l. (~) > c. Top,

Condition II-strong

~a

Condition III -weak

~a=l.(~) -

pv

(1-32)

P T iJp ,

P(0 T) < 1 T iJp,

a•

(1-33) ( 1-34)

It may be noted that condition JI-strong implies condition 11-,veak, and condition III-strong implies condition III-weak. \Vith phase changes the conditions of Eq. 1-31, 1-32, 1-33, and 1-34 are unchanged, except that they may be simplified by replacing (oT / iJp), by dT /dp for a two-phase mixture. For a simple two-phase mixture, condition I is also unchanged, but the quantity 1' or (o2 p/iJv•), is undefined at the point where a substance changes from one-phase to two-phase or vice versa, i.e. under saturation conditions. At such a point the quantity

(iJp)

_ ( 0'1!) = p• = p'a' (1--35) av • op • is discontinuous, and condition I may be replaced by the condition that pa increase with an iscntropic increase in the density. Note that the quantity a• as used here always refers to a thermodynamic derivative which is necessarily identified with the square of the propagation velocity or sound only for sound waves of sulficiently low frequency. Bethe [6] has carried out an investigation of shock waves in an ar bitrary fluid following an approach which has certain features in common with the approach of the first two articles of this section, and specifying certain conditions which are the same as some of those above. Condition I above is Bethe's condition I; condition II-weak is Bethe's condition II; and condition 111-strong is Bethe's condition JU. He investigates in detail t he possibility of violations of his t hree conditions by specific substances; such considerations are not included in t ho present contribution. lt m1>y be said that co1nmon 1naterials satisfy all of Bethe's conditions and also condition II-strong, With respect to condition l Bethe notes t hat at the (thermodynamic state) boundary between a one-phase region and a two-pha~e region the quantity pa is always greater for the single phase than fo~ the mixture of two phases. Thus condition 1 at a phase boundary may be expressed: isentropic expansion must lead to a. phase transition from a one-phase state l o a two-phase slate. Bethe notes that this is

( 14 >

always true if one of the phases is a vapor but that it may be untrue for certain liquid-solid and solid-solid transitions. Of course, the bas.ic theory must be modified in general if any of the material involved is a solid under stress. These various conditions are applied in the next two articles. I n general, it may be seen from Eq. 1-2-3 that in conipression discontinuities condition ill-strong prevents a.n extremum of e and condition III-weak prevents an extremum of p; analogously, in expansion discontinuities condition JI-strong prevents an extl'emum of h a nd condition II-weak preven t.s an extremum of v. D,2. The Normal Shock Wave. A shock wave is a gasdynamic discontinuity in which no chemical reaction is involved, and for which the material on both sides may be considered to obey t he same equation of state (allowing certain discontinuous behavior if a phase change should be involved). In the analysis of this article no phase changes are considered,, although in general the conclusions reached would be valid with such changes present. It is found that conditions I and III-st rong are sufficient to guarantee the good behavior of compre.ssion shocks, by ensuring their existence and uniqueness under appropriate boundary conditions, and by ensuring that the velocity of the fluid is supersonic with respect to the shock in front of it and is subsonic behind it. The point of view which is taken is that the state of the material in front of the shock is known, and that behind the shock is unknown. Weak shocks are considered first. Since the Hugoniot curve passes through the "origin" (1'1, v,) and is presumed to have continuous slope, it satisfies the approximate equation

dp ~ p - p, (2-1) dv = v - v, to first order accuracy. From Eq. 1-14 and 1-26 it may be seen that the origin is not only a stationary point for s, or a Ch-J point, but also a point for which the Hugoniot curve osculates the isentrope. Condition I requires that the curvature of the Hugoniot curve there be positive. This indicates that the entropy change across a weak shock can be no larger than of the third order in the pressure or volume change. I n order to obtain a quantitative expression for the entropy change across a weak shock, Eq. 1-14 is reexpressed and differentiated twice with respect to v T ds dv

=

d., + T d's = dv•

d'T d$ du•

dT dv dv dT d's

db + 2 dV dv' +

d's

~(v, -

v) ~P. dv

+ t(p -

t(v, - v) ~p dv2 d•p

T dv' = f(v, - v) dii'

( 15 }

p,)

(2-2a} (2-2b)

1 d'p

- 2 dv'

(2-2c)

D · GASDYNAMIC DISCONTINUITIES

D,2 · NORM AL SHOCK WA VE

Evaluation of the derivatives of s at the origin gives 7',

(~) du

(d'dv's) 's) Ti (d __. dv'

T, -

=

0

(2-3a)

=

O

(2-3b)

r:s

-

i

1

1

1(d'dv'p) -

2

1(&"p)

= - - ~ 2 &v• •

-

1

p"a 2 r

(2-3c)

where the thermodynamic quantities are evaluated at the origin and r is defined in Eq. 1-27. Using the symbol A to indicate change across the shock, expanding As in a Taylor series gives

Tills = - - I

(&'- p) Av• = -

12 011• •

-.r Au• 6a

(2-4)

with an error of fourth order. Another method involves the use of a variant of Taylor's expansion of a function of two variables, which has been made symmetric with respect to the two argument points. In terms of the function e(v, s) this expansion is expressed Ae(v, s) =

(:~)•• Av + (::).. As

(

D,5 · NAVIER-STOKES S!IOCK STRUCTURE

D · G.dSDYNAMIC DISCONTINUITIES

able. In fact, present studies of the structure of moderately strong shocks using kinetic the-Ory have all used assumptions of varying arbitrarine.~s as to the nature of the solutions to the Boltzmann equation yielding approximations to the shock wave structure. And tbese studies are most.ly lilnited to the case of a monatomic gas with particular postulated laws of interaction. On the other hand, the continuum theory has shown evidence of an en1pirical type indicating a range of validity appreciably wider than that granted to it by most kinetic theorists. Most of the kinetic theory approaches and a sufficiently general continuum theory approach (e.g. Truesdell (1£]) arrive at a set of tensorial relations connecting the stre&i tensor, energy (or best) flow vector, velocity gradient tensor, temperature gradient vector, and the total time and the distance derivatives of these quantities. The Na.vier-Stokes relations (here t.aken to include the Fourier heat conduction relation) appear naturally as the lowest dissipative approximation with either approach. The essential difference between a "classical" kinetic theory approach of this type and a continuu1n theory lies in the determination of the coefficients involved: with a kinetic theory they are to be determined by calculation from a knowledge of the molecular collisions involved, while with a continuum theory they are to be determined by experimental means as arc other material ptopert,ies. Either way of det4J.r1nining these coefficients (besides the shear viscosity and the heat conduction coefficient) is beset with important difficulties. In addition, either of these approaches involves the assignment of a temperature under nonequilibrium conditions for which no temperature may be defined strictly thermodynamically; this difficulty may be obviated by the concept of quasiequilibrium only if the Navier-Stokcs relations are involved. The classical kinetic theory approaches stem primarily from the approximation schemes associatt>d with the names of Hilbert, Chapman, Enskog, and Burnett; the classical exposition of the subject i.s to be found in [13). Analyses of a shock wu.ve by this approach have been made by Wang-Chang (1 4], Zoller (15], Her pin (16], Broer {17), and Travers (18). Grad (1.9) has made a study of shock structure based on a different approximation scheme. ThP.se analyses result in what appear to be reasonable pictures of shock wave structure for moderately weak shocks in a perfect monat-0rnic gas obeying a l\1axwellian or elastic-sphere force law. Mott-Srnith [20] has used a completely different kind of hypothesis, assuming that the velocity distribution function is mu.de up within the shock of a linear combination of that before, and that after, the shock. Tlie state of the gas changes through an assumed law of interaction between the two l\·1ax:wellian distributio.us. l\1ott-Smith's method is open to a number of criticisms and his results are adn1ittedly in error for weak shocks, but his has been the only published attempt on the problem of the structure of a strong shock. The strong shock problem is one for which

neither continuum methods nor the classical kinetic approximatiol\ theory can give even an approximate solution. The approach of l\fott-Smith may well point the way to a satisfactory attack on this problem. The writer is among those of the opinion that for weak and moderately wesk shocks a continuurn approad1 is clearly preferable to a kinetic theory approach. To summarize the reasons for this stand, it may be noted that (cf. Truesdell (21, Sec. 5A)):

The classical Navier-Stokes treatment of the shock wave is due to Becker (26), who obtained the first solutions for nonweak shocks in volving both viscosity and heat conduction. He made the correct but meaningless note that his solution with constant coefficient of viscosity gave shock thickoesses s1ualler than a mean free path in the gas ahead of the shock. Of course, if a correct continuum tlieory gave large changes in physical quantities in a distance small compared with the local mean free path the theory could be dismissed a.~ inapplicable. The poiot of importance here is that the te1nperature variation of the viscosity coefficient and the temperature variation of the mean free path at constant volume are closely connect~d, and the proper comparison to be made is between the shock thickness according to a continuum theory, using the correct temperature variation of viscosity coefficient and an average value of the correct mean free path measured within the shock. Although Becker's

( 36 )

( 37 )

l. The validity of the Boltzmann equation for nooequilibr ium states,

though generally accepted, re1nains somewhat in doubt. This point has been discussed by Grad [22], who points to a "molecular chaos" assumption as the critical one underlying the Boltzmann equation, and also by Jeffreys (23). 2. The validity of the approximation schemes employed has never been established in any case even in an asymptotic sense. On the cont rary, Truesdell [24], in studying his exact solution for shear flow with Maxwellian molecules and his mathematical model for studying the clas, sical approximation scheme by analogy has shown that in certain cases the Na.vier-Stokes solution may be a correct asymptotic solution but that further approximations are worse ones. The disagreement among investigators using various versions of the classical approximation scheme may well be a symptom of this basic difficulty. 3. The kinetic t.heory as presently developed is limited in application to gases which are monatomic and which obey a specified force law. Thus, a polyatomic perfect gas, or even a monatomic gas with a general dependence of viscosity on temperature, cannot be treated. These limitations do not appear in continuum theory . 4. In the only case for which a comparison of experiOlent with theory is available (Sherman (25]) agreement was obtained with the appropriate continuum theory and not with kinetic theory.

D,S · NAVJER-STOKES SHOCK STRUC1'URE

D · GtfSDYNAMIC DISCONTINUITIES

remark may not be taken literally, it is on precisely this point that the Navier-Stokes solutions are most open to criticism for moderately strong shocks, from the continuum point of view as well as from the kinetic theory point of view. ThllS for moderately strong shocks a correction is indicated, involving an altered relaLion between stress and velocity derivatives. It has been claimed by various authors (e.g. Travers f18J) that the correction term proportional to the square of the velocity gradient is the most i1nportant term to include, while terms in high derivativt>.s of t.he velocity and temperature are less important. Gilbarg and Paolucci (27] hi\ve given a proof of the existence of the mathemaHcal solution with a general nonlinear relation between stress and velocity gradient. However, Broer (17) and others have pointed out that in an expansion development in a parameter proportional to shock strength, the linear second derivative terms are of larger order of magnitude than the quadratic first derivative terms. Most classical presentations of Navier-Stokes shock structure have assumed the Stokes relation between shear and second coefficients of viscosity, and have thus assumed zero bulk viscosity. The Stokes rela·tion has not even. been shown to be correct for all monatomic gases and is completely false for most other gases and liquids (see Truesdell [28]). Moreover, for the purpose of the theory of shock structure it is unnecessary and unhelpful. The additional viscous stress in a one-dimensional flow is µ"( ), for which the right side of Eq. 5--0 is zero. These two integral curves on the T, u plot are termed "nonviscous" and "nonconducting," respectively. They play an important part in the geometry of the actual integral curves where neither µ" nor k is zero. Another important special case appears if the enthalpy is a function of the temperature alone, i.e. if pT is a function of pressure alone (this includes a perfect gas), a.nd the longitudinal Prandtl number defined by II

Pr"

= µ

ep

k

(5-7)

is a constant and equal to 1. With the enthalpy a function of temperature alone, Eq. 5-4 may be expressed

~ With Pr"

=

1l' ( l dh

u' m Pr" dx + u du) dz = h + z - ho

(5-8)

1 tins has the integral

h. +

u'

2 =ho

( 39 )

(5-9)

D,5 · NAVIER-ST OKES SHOCK STRUCTURE

D · GASDYNAMIC D ISCONTINUITIES

which, since it satisfies conditions before and after the shock, is the one desired. This special solution was noted by Becker {261, and may be characterized by the pbraae "constant total enthalpy." Since most gases a.re reasonably close to a perfect gas and have a value of P·r " of the order of 1, the constant total enthalpy solution may be considered to be realistic for a gas, while the nonviscous and nonconducting solutions clearly are not. The features of the solution may best be seen from a study of the T, u plot, here illustrated in Fig. D,5a. Integral curves are shown on this figure with an 11.rrow indicating the direction of increasing x . The curYe designated I from point 1 to point 2 is the solution desired. One other

obtained with the simplifying assumption that the gas is a perfect one. Using the subscript , and letting (5-lOa) U = U1 + ou

+ oT

T = T1

(5-lOb)

the equations corresponding to Eq. 5-5 and 5-6 may be expressed, dropping second orde( quantities, µ" u 1 dou

-

m

~-

=

( 55 )

7

6

s

~

•0

I

::i 3 ~

2

0.3 0. 1

0.2

0.3

0.4

0.5

x/ 'A Fig. D,5b.

Decaying saw-tooth wave. After [88, Fig. 19, p. 341.J

0,6 · STRUCTURE OF EXOTflERMIC DISCONTINUITIES

D · GASDYNAMIC DISCONTINUJTIES

onto this space as a "Hugoniot surface," with the understanding, however, that only a single value of m is being considered at one time. The particular line on the Hugo1uot su.rfnce corresponding to the ch0$en value of m is denoted the "equilibrium line." '!'he concept of a space almost identical with the T, v, •one was used by Friedrichs 1441 in a treatment of a special case of the same problem, the case of a perfect gas whose gas constant does not change in the reaction. The concepts of the "unreacted" a.nd "reacted" surfaces, in this case the planes• = 0 and• = 1, are introduced, corresponding to the equations of state for the material before and after the reaction. The structure of the gasdynamic discontinuity is considered to be divided into three zones, the first occurring before the reaction has started and termed the "preheating zone" (on the unreacted surface), the second occurring du.ring the reaction and termed the "reaction zone," and the third occurring after the reaction has finished and termed the "adjustment zone" (on the reacted surface). On a constant • plane of this T, v, • space or on the unreacted or reacted surface, there may be dra.wn a figure similar to fig. D,5a, with the integral curves of Art. 5 in general r epresenting the projections of the actual integral curves. The nonviscous and nonconducting lines become surfaces in the extended space; these are not very significant except in cases of very small or very large Prandtl number. Of particular significa.nce is the reault of Art. 5 that a supersonic point is unapproachable. This result holds under the conditions of this section only for a reaction occurring in a plane• = const, and .so means that there can be no adjustme1tt zone in a weak detonation or a strong defiagr&tion. T here J.llUSt be two degrees of freedom in the parameters which govern the choice of the integral curve in order to en$\lt'e that the curve will go to such a supersonic point. In the development which follows, t he subscripts 1 and 2 are used as before to indicate conditions before and after the reaction. The subscripts u snd b are used to indicate unreacted and fully reacted material, respectively; thus •· = 0 and "' = 1. The equations corresponding to Eq. 5-2, 5-5, and 5-6 1nay be expressed u=mv (6-1) "m dv µ. P dX = (l - •)(pv) .. + .(pv). + m'v' - pot> (6-2)

which must come from chemical kinetic considerations, or in the case of a condensation, from nucleation considerations. The inequality of Eq. 6-4 states that the reaction is irreversible; this condition might need to be altered with an unusual definition of .. The condition

k dT m dx =

(1 - •)e.

+ E€1. -

-km'v'

+ po11 -

h•

(6-3)

with the linear properties assumption. To these equations may be added a fourth, for the time rate of change of •,

d.

mv tk = f(T, v, •) > 0

( 56 >

(6-4)

(6-5) is a result. of this inequality. T he values of the constants 1n, Po, and ho are prc.swned suitably chosen so that the conservation laws permit a solution. The constants are related to conditions in front of the discontinuity by (6-6a) Po= p, + m 1u1 (6-6b) ho = h, + {m'vi as may be noted from Eq. 1-2 and 1-3. Eq. 6-2 and 6-3 may be combined with the variable• eliminated to give

(e, - e.)µ"mv =

!-

(h.

[(pv), - (pv).] ~ ~ = H(T, v)

+ im'v' -

+ im'v' - ho)(pt1), + (m'v• - po11)(h. - h.)

ho)(p11). - (h.

(6-7)

If conditions in front of the discontinuity are fixed the function H may be expressed H(T, v) = F

+

m'-3 are set equal to zero, corresponding to hydrodynamic equilibrium, the result is the equilibrium line mentioned above. This line lies in the invariant cylinder and may be considered as the intersection of the Hugoniot surface and t.he invariant cylinder for the value of 1n of interest. 1'he equilibrium line intersects the unreacted and reacted surfaces in the equilibrium pointa which correspond to conditions in iront of or behind the discontinuity.

( 57 }

D,6 · STRUCTURE OP EXOTHERMIC DISCONTINUITIES

D · GASDYNAMIC DISCONTINUrtLES

If the substances involved in the rea.cLion are perfect gas(.'S with a gas constant that docs not change in the ren.ction, then Eq. 6-2 is independent of< and the nonviscous surface coincides with the invariant cylinder. With an exothermic reaction the unreacted nonviscous curve lies above (i.e. at higher temperatures) or below the invariant cylinder as the gas constant increases or decreases in the reaction.

face. From the same points may be drawn arrows (dotted in Fig. D,6a), indicating the direction of integral curves on the reacted surface. The linear properties assumption requires that the direction of anY integral curve (for which 0 S' • < I) must lie in the angle between the two arrows which is less than 180°. On the it1variant cylinder, a.ud only there, the two arrows have the same angle. Between the point$ .ak detonation or a weak della.gration, apart from the question of stability. A necessa.r y condition for tho existence of a strong dcfiagration ms.y be obtained by demonstrating a con6gure.tion for which this solution is conceivably possible. This configuration is illustrated i.n Fig. D ,6d. I t is necessary I.hat the unreacted integral curve 111 1 p8$$ under the point a. on the T, o plot for the case in question. Since the curve III 1 lies above t he unreacted non viscous curve it is necessary that I.he invariant cylinder also lie above the unreacted nonviscous curve. I t may be seen that the proof of Friedrichs (44) of the nonexistence of strong defiagrations depends upon his assumption of perfect gases with a gas constant that does not change in the reaction, for which the unreacted oonviscous curve lies on the invariant cylinder. With perfect gnsel! it is necessary that the gas

( 62 )

1. 'fhe existence of finite perturbations which can change the strong doBagration to a family of weak or Cb-J deftagrations or to a Ch.J detonation. 2. The impossibility of finding a conceivably possible structure with the Navier-Stokes and linear properties.a.ssumptions, except (or a reaction giving a very particular type of T, v configuration. 3. In case such a !Structure is conceivably possible the limitation on the poesible original states because of the missing degree of freedom in chooeing the integral curve. To the writer's -knowledge there has been no report of a true strong deftagration having been observed in practice. With regard to the weak detonation, which has not been observed (again to the writer's knowledge) in explosion experiments, there is no theoretical evidence for nonexistence. It may only be said that chemical reaction rates are generally much smaller than would be necessary for a weak detonation. As mentioned above, the condensation shock which might be produced in a supersonic wind t11nnel should be of the weak detonation type. D ,7. The Phy5ics of S h ock Wave2. 14. Wang-Chang, C. S. On the theory of the thickness of weak shock waves. Appl. Phy•. Lah., Johna Hopkin.a Univ. Rcpt. CM.-508, 1948. 15. Zoller, K. Z. PhylJik 11;0, l - 38 (1951). 16. ilorpin, A. Mttn. artilltriefran~. $4, 851-897 (1950): 17. BrO