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A note on the stress distribution at great depth in a silo
Shorter Communications
[l] [2] [3] [4] [5] [6]
REFERENCES SZEKELY J. and MENDRYKOWSKI J., Chem. Engng Sci. 1972 27 959. SHERWOOD T. K., SHIPLEY G. H. and HOLLOWAY F. H. L., Ind. Engng Chem. 1938 30 765. STANDISH N. and DRINKWATERJ. B.,J. Mefals 1972 2443. KUKARIN A. S. and KITAEV B. l., lzo. V.U.Z. Chern. Met. 1962 12 20. STANDISH N., Chem. Engng Sci. 1968 23 5 1. HOWKINS J. E. and DAVIDSON J. F.,A./.Ch.EJI 1958 4 324.
Chemical
Engineering
Science,
1973, Vol. 28, pp. 1907-1908
Pergamon
Press.
Printed in Great Britain
A note on the stress distribution at great depth in a silo (Received
9 November
IN HIS extension of the classical Janssen analysis of the stress distribution in a silo, Walker [ 1] introduces the factor D, the ratio of the axial stress (u& at the wall to the mean value across the bunker. This can be calculated at great depths on the assumption that the radial stress, ur. is constant. Walters [2] has presented extensive calculations on the value of D for a wide range of variables, again on the assumption that or is constant. The present communication questions this assumption and presents a more rigorous analysis for D. The resulting numerical values seem however not to differ significantly from those predicted by the simple theory except in extreme cases. In a system with axial symmetry the shear stresses T,~and r,,, are zero and uB is therefore a principal stress. At great depth in a vertically sided silo all derivatives with respect to z become zero and the stress equations therefore reduce to: ;$
or,*) = pg
do, x+-----or-ok_
r
o.
(1)
(2)
and E is the angle between the r axis and the direction of the major principal stress. For axisymmetric failure, two of the three Mohr’s circles for any point must touch the Coulomb yield locus and hence ue must be equal to one of the other principal stresses. At the centre line rr2 = 0 and hence U, and ur are principal stresses and furthermore duddr is finite. Hence from Eq. (2) uB = ur at the centre line. Two cases, an Active and Passive type of solution, will be considered, the Mohr’s circles for which are given in Fig. 1. From the geometry of the circles u,=p(l+sin6cos2e) u,=p(l-sin6cos2e) u,=p(l-Ksin6)
3 dr (
(3)
(4)
where K is + 1 in the active case and - 1 in the passive case. It should be noted at this stage that at r = 0, E = n/2 in the active case and 0 in the passive case. Substituting for u, and uB in Eq. (2) and eliminating p by means of Eq. (3) gives.
Since 7,.*= 0 at r = 0, Eq. (1) integrates to, 7rz=y=psinSsin2e
1972)
pgr(l+sin6cos2e) 2 sin 6 sin 2e
which can be rearranged
where p is the arithmetic mean of the major and minor principal stresses, 6 is the effective angle of internal friction
+(K+cos2~)pg=~
)
2 sin 2~
to give
dr (sin 6+ cos Ze)d(cos 2~) ~=(l+Ksin6+2sin6cos2~)(cos~2~-1)’
Fig. 1. (a) Active case; (b) Passive case. EYL = Effective Yield Locus. Angle @varies between zero at the centre and I#Iat the wall.
1907
Shorter Communications This can be separated into partial fractions and integrated subject to the boundary condition that at the wall, r = rm and z = l,, where E, is given by
and 4 is the angle of wall friction, as shown in Fig. 1. The integral (after some manipulation) is, ;=
(E)(I’.I:z;J x
l+Ksin6+2sin6cos2c ( l+Ksin6+2sin8cos2elc
-(*+I) >
(5)
bunker in the active case and a 14.7 per cent variation in the passive case. The values of D were 0.970 (active) and 1.520 (passive) which may be compared with the values of 09798 (active) and 1.539 (passive) given by Walters. The error in this case is seen to be small. However the error is likely to be larger when 4 is close to 6 as this results in a greater variation in E. The case 6 = do= 20” was also considered, the resulting values of D being 068 (active) and 1.37 (passive) compared with Walters’ values of 0.710 (active) and 1.690 (passive). The error in the active case is again small but a considerable difference is found in the passive case. Thus it is seen that Walker’s assumption that (T,is constant, though wrong in principle, gives negligible errors in D for most situations. The errors can however be significant in extreme cases.
where A=
-sin 6 3 sin6+K’
Certain special cases occur in this integration, the only one of any physical significance being K = - 1, sin 6 = 4for which Eq. (5) takes an exponential form. Substitution of Eq. (5) into Eqs. (3) and (4) gives p and hence (T,, and u, as functions of l . a=
l+sin6cos2r ( l+sin6cos2c, x
x
* >
(6)
1-KCOS~E A > I I-Kcos2e,
l+Ksin6+2sin6cos2r l+KsinS+2sinScos2eU
+I+‘)
>
R. M. NEDDERMAN Department of Chemical University of Cambridge, England
Engineering Cambridge
NOTATION =-sin6/(3 sin6+K) Walker’s correction factor, (uz),/Fz g acceleration due to gravity K = + 1 in active case, = - 1 in passive case p mean of the major and minor principal stresses r radial co-ordinate z axial co-ordinate
.
(7)
Greek symbols
effective angle of interval friction angle between r axis and the direction principal stress density normal stress mean normal stress shear stress angle of wall friction
D can then be found from the relationships,
pr,% 2 = hrw 2rrru, dr,
No analytical Numerical 6 = 50”. r$ = paper. These
J. K. WALTERS
A D
1-t K sin 6 + 2 sin 6 cos 2~ --(“+I) l+Ksin6+2sin6cos2c, 1-sin6cos2e 1-singcos2s,
&=
I-KCOS~E I( I-KCOS~E,
Department of Chemical Engineering University of Nottingham, Nottingham England
solution of this integral has been found. calculations have been made for the case 25” as this is the case considered hr Walters’ predict a 1.9 per cent variation of u, across the
of the major
Subscript w value at the wall
REFERENCES
[ 1J WALKER D. M., Chem. Engng Sci. 1966 21975. [21 WALTERS J. K., Chem. Engng Sci. 1973 28 13. ChemicalEngine~ringScience.1973,Vol.Z8,pp.
1908-1910 PergamonPress. PrintedinGreatBtitain
A note on second and third order reactions isothermally axial dispersion model THE AXIAL dispersion model is of great importance for ‘the calculation of the conversion of chemical reactions performed in non-ideal flow reactors. In a recent paper[ll Deckwer, Kolbel and Langemann derive an equation which
performed according to the
is, in principle, applicable to a chemical reaction of any order and stoichiometry and at different initial concentrations of two reacting species. In the paper referred to [I] Deckwer et al. present a somewhat lengthy method for calculating the
1908
[l] [2] [3] [4] [5] [6]
REFERENCES SZEKELY J. and MENDRYKOWSKI J., Chem. Engng Sci. 1972 27 959. SHERWOOD T. K., SHIPLEY G. H. and HOLLOWAY F. H. L., Ind. Engng Chem. 1938 30 765. STANDISH N. and DRINKWATERJ. B.,J. Mefals 1972 2443. KUKARIN A. S. and KITAEV B. l., lzo. V.U.Z. Chern. Met. 1962 12 20. STANDISH N., Chem. Engng Sci. 1968 23 5 1. HOWKINS J. E. and DAVIDSON J. F.,A./.Ch.EJI 1958 4 324.
Chemical
Engineering
Science,
1973, Vol. 28, pp. 1907-1908
Pergamon
Press.
Printed in Great Britain
A note on the stress distribution at great depth in a silo (Received
9 November
IN HIS extension of the classical Janssen analysis of the stress distribution in a silo, Walker [ 1] introduces the factor D, the ratio of the axial stress (u& at the wall to the mean value across the bunker. This can be calculated at great depths on the assumption that the radial stress, ur. is constant. Walters [2] has presented extensive calculations on the value of D for a wide range of variables, again on the assumption that or is constant. The present communication questions this assumption and presents a more rigorous analysis for D. The resulting numerical values seem however not to differ significantly from those predicted by the simple theory except in extreme cases. In a system with axial symmetry the shear stresses T,~and r,,, are zero and uB is therefore a principal stress. At great depth in a vertically sided silo all derivatives with respect to z become zero and the stress equations therefore reduce to: ;$
or,*) = pg
do, x+-----or-ok_
r
o.
(1)
(2)
and E is the angle between the r axis and the direction of the major principal stress. For axisymmetric failure, two of the three Mohr’s circles for any point must touch the Coulomb yield locus and hence ue must be equal to one of the other principal stresses. At the centre line rr2 = 0 and hence U, and ur are principal stresses and furthermore duddr is finite. Hence from Eq. (2) uB = ur at the centre line. Two cases, an Active and Passive type of solution, will be considered, the Mohr’s circles for which are given in Fig. 1. From the geometry of the circles u,=p(l+sin6cos2e) u,=p(l-sin6cos2e) u,=p(l-Ksin6)
3 dr (
(3)
(4)
where K is + 1 in the active case and - 1 in the passive case. It should be noted at this stage that at r = 0, E = n/2 in the active case and 0 in the passive case. Substituting for u, and uB in Eq. (2) and eliminating p by means of Eq. (3) gives.
Since 7,.*= 0 at r = 0, Eq. (1) integrates to, 7rz=y=psinSsin2e
1972)
pgr(l+sin6cos2e) 2 sin 6 sin 2e
which can be rearranged
where p is the arithmetic mean of the major and minor principal stresses, 6 is the effective angle of internal friction
+(K+cos2~)pg=~
)
2 sin 2~
to give
dr (sin 6+ cos Ze)d(cos 2~) ~=(l+Ksin6+2sin6cos2~)(cos~2~-1)’
Fig. 1. (a) Active case; (b) Passive case. EYL = Effective Yield Locus. Angle @varies between zero at the centre and I#Iat the wall.
1907
Shorter Communications This can be separated into partial fractions and integrated subject to the boundary condition that at the wall, r = rm and z = l,, where E, is given by
and 4 is the angle of wall friction, as shown in Fig. 1. The integral (after some manipulation) is, ;=
(E)(I’.I:z;J x
l+Ksin6+2sin6cos2c ( l+Ksin6+2sin8cos2elc
-(*+I) >
(5)
bunker in the active case and a 14.7 per cent variation in the passive case. The values of D were 0.970 (active) and 1.520 (passive) which may be compared with the values of 09798 (active) and 1.539 (passive) given by Walters. The error in this case is seen to be small. However the error is likely to be larger when 4 is close to 6 as this results in a greater variation in E. The case 6 = do= 20” was also considered, the resulting values of D being 068 (active) and 1.37 (passive) compared with Walters’ values of 0.710 (active) and 1.690 (passive). The error in the active case is again small but a considerable difference is found in the passive case. Thus it is seen that Walker’s assumption that (T,is constant, though wrong in principle, gives negligible errors in D for most situations. The errors can however be significant in extreme cases.
where A=
-sin 6 3 sin6+K’
Certain special cases occur in this integration, the only one of any physical significance being K = - 1, sin 6 = 4for which Eq. (5) takes an exponential form. Substitution of Eq. (5) into Eqs. (3) and (4) gives p and hence (T,, and u, as functions of l . a=
l+sin6cos2r ( l+sin6cos2c, x
x
* >
(6)
1-KCOS~E A > I I-Kcos2e,
l+Ksin6+2sin6cos2r l+KsinS+2sinScos2eU
+I+‘)
>
R. M. NEDDERMAN Department of Chemical University of Cambridge, England
Engineering Cambridge
NOTATION =-sin6/(3 sin6+K) Walker’s correction factor, (uz),/Fz g acceleration due to gravity K = + 1 in active case, = - 1 in passive case p mean of the major and minor principal stresses r radial co-ordinate z axial co-ordinate
.
(7)
Greek symbols
effective angle of interval friction angle between r axis and the direction principal stress density normal stress mean normal stress shear stress angle of wall friction
D can then be found from the relationships,
pr,% 2 = hrw 2rrru, dr,
No analytical Numerical 6 = 50”. r$ = paper. These
J. K. WALTERS
A D
1-t K sin 6 + 2 sin 6 cos 2~ --(“+I) l+Ksin6+2sin6cos2c, 1-sin6cos2e 1-singcos2s,
&=
I-KCOS~E I( I-KCOS~E,
Department of Chemical Engineering University of Nottingham, Nottingham England
solution of this integral has been found. calculations have been made for the case 25” as this is the case considered hr Walters’ predict a 1.9 per cent variation of u, across the
of the major
Subscript w value at the wall
REFERENCES
[ 1J WALKER D. M., Chem. Engng Sci. 1966 21975. [21 WALTERS J. K., Chem. Engng Sci. 1973 28 13. ChemicalEngine~ringScience.1973,Vol.Z8,pp.
1908-1910 PergamonPress. PrintedinGreatBtitain
A note on second and third order reactions isothermally axial dispersion model THE AXIAL dispersion model is of great importance for ‘the calculation of the conversion of chemical reactions performed in non-ideal flow reactors. In a recent paper[ll Deckwer, Kolbel and Langemann derive an equation which
performed according to the
is, in principle, applicable to a chemical reaction of any order and stoichiometry and at different initial concentrations of two reacting species. In the paper referred to [I] Deckwer et al. present a somewhat lengthy method for calculating the
1908