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On the dynamics of the continuous flow manganese bath system
NUCLEAR
INSTRUMENTS
AND METHODS
76 ([969) 328-332; © N O R T H - H O L L A N D
PUBLISHING
CO.
ON T H E DYNAMICS OF T H E C O N T I N U O U S F L O W MANGANESE BATH SYSTEM T. D. BEYNON and M. C. SCOTT
Department of Physics, University of Birmingham, England Received 13 March 1969 and in revised form 18 August 1969 Exact and approximate solutions of the equation governing the activity growth in continuously cycled assemblies are presented. These solutions are applied to manganese and vanadium systems of practical interest and the implications for neutron yield measurements are discussed.
SIVA((volume=¥]
1. Introduction
Accompanying the growing use of accelerators for neutron production is the need for accurate measurements of the target yields. The anisotropy often associated with neutrons produced in this way, together with high gamma background, seriously restricts the choice of suitable measuring systems. Methods of determining neutron yields which involve measuring the spatial variation of thermal neutrons produced by placing the source at the centre of a large volume of moderator t' 2) suffer from several disadvantages in this respect. Among these are: 1. the system requires calibrating for each neutron source spectrum; 2. the flux has to be monitored at a large number of spatial points; 3. the combination of the effects of the differing neutron and gamma yields and spectra of different targets makes difficult the correction for gamma sensitivity of the detectors. By comparison the manganese bath technique 3-5) suffers from none of these disadvantages: the measurement is an integral one and yields an absolute value for the source strength without reference to external standards. Furthermore, such corrections that are necessary are well documented. Until recently it has only been used occasionally for accelerator sources [compare ref. 6)] because the presence of large source gamma pulses makes in situ determination of 56Mn activity difficult. Other methods of determining the activity, by removing active solution, are slow. The continuous sampling of the solution, proposed in different forms by Adler and Ohnesorge 7) and Axton, Cross and RobertsonS), removes both of these difficulties. The solution is continuously stirred and pumped to a counting chamber which can be as far away from the source as is necessary for shielding purposes. The arrangement is shown schematically in fig. 1. The general equation governing the growth of activity in such a system, which is given in the next section, is similar to that which occurs in many biological and
~.~UI4TING CSOURC(HAHf(lR (vok~ ~
p~s qAc=~=~-_~-~ld C) TiME FORS(X.UTiON TO GO FIIOM A TO B = p Fig. 1. Schematic diagram of the system.
stochastic processes and the approximate and exact solutions for both steady state and time dependent source are therefore of more general interest. 2. Theory 2.1. THE DIFFERENTIAL-DIFFERENCEEQUATION A model s) describing the time dependence of the activity per unit volume, A(t), in the continuously cycled source chamber produces the differential-difference equation,
dA/dt+2]A(t)-yA(t-p)
= SoQ(t).
(1
The following notation has been adopted:
'~1 =
/~-['-OC, ---- 0ee- 2p,
= C/p V (the fraction of the fluid removed per unit time from the source chamber), p = pumping time, the time taken for the active fluid to be pumped to the counting chamber and returned to the source chamber, C = total volume of the counting chamber, pump and associated pipe work, V = volume of source chamber, 2 = decay constant of active atoms in the pumped fluid, Q(t) = source strength per unit volume at time t. So = the fraction of the source neutrons producing activity. Eq. (1) represents the activity balance in the source 328
329
ON THE DYNAMICS OF THE C O N T I N U O U S F L O W MANGANESE BATH SYSTEM
chamber under the assumptions of incompressibility and instantaneous mixing of the fluid. The additional assumption of non-turbulent flow in travelling a pipecontour distance y enables us to write the activity, B(t), at y as
Thus e - sto ,~G(s) and
B(t) = A ( t - yp/Y) e- ayp/r,
AG(t,to) =
where Y is the total distance travelled in the pumping interval p. A necessary initial-value for the solution (1) is
A(t) = O, t<=O (i.e. A ( t - p ) = O, t
where we define
f
= O,
A(t')k(t-t')dt',
(3)
J,
o k(O) = e -a'°,
u(t')K(t-t')dt',
where
K ( t - c) =
c-p),
and H ( t ) i s the unit function. Such equations occur extensively in mathematical physics9), biology~°), branching processes ~~) and chemotherapyl 2). 2.3. ANALYTICAL SOLUTION We may proceed to solve (1) using a Laplace transform. Accordingly, defining
eZJt
SoQ(s)
.
f t e-ZJt°Q(to)dto . o
(6)
oo 1 (21 --22)j= z j(1 + p22e - ~ ' )
--
- 1.
(7)
For SoQ(t) = 1, using (6) and (7), we see that
A(t)
-
1 2t-~'
+
~
e "~'
j=o zj(1-4-p22e -zip)
.
(8)
For values of t greater than about three or four times the reciprocal value of the maximum Re(z j), only the term containing z o = - K o is important, so that (8) becomes
A(t)
1
e - Kot
21-22
Ko(l + p22er°P) '
t,>O.
Similar asymptotic forms can be obtained for other integrable source functions. Of some interest is the source function
SoQ(t) = 1 +asin(wt)
Q(s) = Aa{Q(t)},
such that .4(s) and similar functions converge in a halfplane R e ( s ) > k , we obtain
.4(s) -
(5)
A useful relationship, obtained by integrating (1) with (5) over all time, is
The formulation (3) now describes not the activity balance in the source chamber but the history of activity behaviour for times t' =
for t > to,
= 0, for t
= y e a~p.
A(s) = &a{A(t)},
eZJ(t_to )
"= l+p22e -z~p
Q(t')k(t-t')dt',
t > to,
'
t < t o.
AGO, to ) =
A(t) = Soj~=°
Ao(t ) = S O
u(t) = f ( t ) +
2hi - r-io~ s + z 1-22e --sp
j=o (1 + p),e -zJp)
t--p
o
eS(t-to)ds
1 fr+i~
Hence,
(2)
2.2. INTEGRAL FORMULATION We can reduce (1) to a more familiar form by direct integration. This produces, after a little algebra and the use of (2), the integral equation,
A(t) = Ao(t)+~
s+21-22e -sp '
(4)
s + 2 t -22e -sp The solution is now best obtained in terms of the Green's function At(t, to), for SoO(t) = 6 ( t - to).
This produces
A(t) =
1
+
2t - 7
+i j=o
eZ~t(awzj+ w 2 + z 2) -- a z j r z j s i n ( w t ) +
wcos(wt)]
Z j ( W 2 "~- Z 2 ) (1 + p~ e -z~p)
(9)
330
T. D. B E Y N O N
AND
M. C. S C O T T
A(t) = A.(t), for
t.
Q(t) = Q,(t), 7
This enables us to write (1) as a differential-recurrence r e l a t i o n s h i p for t, < t < t, + 1.
T,
~s
d
d---]A"+AIA"(t)-yA"-l(t-P) = S°Qn(t)"
~4
U s i n g the b o u n d a r y value A.(t.)= grate (11) directly to p r o d u c e
z 2 !
A.(t) = k(t-tn)A._l(tn)+T
0
IO
+So
+ a[K°sin(wt)-wc°s(wt)] (w2+K2)(l+p?e r°z)
A,-?
(10) _~
2.4. N U M E R I C A L SOLUTION A l t h o u g h it is useful to possess the s o l u t i o n to (1) in the a n a l y t i c a l f o r m (6) it is by no m e a n s an easy m a t t e r to o b t a i n a sufficiently large n u m b e r o f the zs's for a g o o d n u m e r i c a l representation. W e n o w p r o c e e d to a n u m e r i c a l s o l u t i o n o f (1) suitable for a digital c o m p u t e r . Firstly, we restrict (1) to a time t such t h a t
t.<=t
t.=np,
n=0,1,2 ....
Q.(t')k(t-t')dt'.
(12)
3
Z.2
A(t) ",~~
f,
tn
2.6-
which, f o r l a r g e values o f t , b e c o m e s
A._l(t.), we inte-
x(t'-p)k(t-t')dt'+
100
Fig. 2. The percentage error in A0*(t), expressed as [Ao*(t)/ /A(t)-l], as a function of tip for a manganese system (T~= 2.573 h) with a pumping timep=20 sec. Curves 1-4 are for a constant source avd curve 5 is for a sinusoidal source.
(11)
1.8 •\
I
-,.0 : \
\ \ \ ~
\
I. C/V=O.05, P=ZOs 2.ClV=0.05, P=200~ 3.C/V=O.Z P=20a 4.C/V=0-2. p=2OOs
\
2
Fig. 4. The percentage error in Ao*(t), expressed as [Ao*(t)/ /A(t)-ll, as a function of tip for a range of C/V values and pumping time, p. The curves are for a vanadium system (T~= 3.72 min) with a constant source.
a n d define I n (12) we recur f r o m the initial c o n d i t i o n (2) such t h a t , as in (11).
04
Ao(t) = So
t CIV=O-I, P=2OJ.
e~i / -
\\
~
~
2~v=o.i, p:12o,
\\
~
4 C/V':O'3, p :1201.
"
'3
I
J
s c/~:o.,, p :2o,.. Q:,..o.s~,,c2.~,o~
1
I
i
l
l
I
10
i
i
I n this f a s h i o n we can easily c o m p u t e A(t) for a n y a r b i t r a r y source function, Q(t). W e note t h a t in obt a i n i n g A.(t) via (12) we are r e q u i r e d to store a c o m p l e t e h i s t o r y o f A , _ 1(0, as we m i g h t guess f r o m the f o r m o f (1).
,
100
tip
Fig. 3. The percentage error in Al*(t), expressed as [Al*(t)/ /A(t)-l], as a function of t/p for a range of C/V values and pumping times, p, for a manganese system (T~ = 2.573 h). Curves 1-4 are for a constant source and curve 5 represents a sinusoidal source.
Q(t') k ( t - t')dt'.
0
sc/v:o.3, p:2o,
~
f,
2 . 5 . A P P R O X I M A T E SOLUTIONS
T h e f o r m o f (1) lends itself to straight f o r w a r d numerical a p p r o x i m a t i o n s . A T a y l o r e x p a n s i o n o f A ( t - p ) a b o u t t reduces (1) to a simple f o r m for the a p p r o x i m a t e d activity, A*(t),
ON T H E D Y N A M I C S OF T H E C O N T I N U O U S
,of o.8[- / /
\
~0.6I /
~
331
t>p
and for
, c/,--0 0,. ,--20, z.c/v:oos, f,:zoo,. 3.c/v:o.z. ~:z0,.
\
F L O W M A N G A N E S E B A T H SYSTEM
A*(t) = Ao(p)g(t-p)+ {So/(l + Tp)}
4.C/V=0.2 . P=2OOs.
f,pg(t-t')Q(t')dt'.
3. Discussion of numerical solution
Fig. 5. The percentage error in Al*(t), expressed as [Al*(t)/ as a function o f tip for a range of C/V values and p u m p i n g time, p. The curves are for a vanadium system ( T ~ = 3 . 7 2 min) with a constant source.
/A(t)-11,
dA*
--+ dt
eA*(t) = SoO(t)/(1 + ~p),
(13)
where = (&-
+
We obtain our solution to (13) by assuming we know the exact value for A(t) for t<=t,. Integrating (13) directly we obtain for our approximate solution
A*(t), (t > t.), A*(t) = A._ ,(t.) g(t- t.) + +{So/(l+~p)}
y, g(t-t')Q(t')dt',
(14)
tn
where g(O) = e-~°.
Two obvious approximations are AN(t ) and since we know A(0) and A(p) exactly. Hence,
f,
A*(t) = { S o / ( l + y p ) } og(t-t')Q(t')dt', 10
A*(t)
(t>0),
The exact solution for the activity can be obtained by direct numerical quadrature in (12) and compared with the approximate solutions A~(t) and A~(t). Such comparisons are made, for a manganese system (T½ = 2.573 h), in figs. 2 and 3 which show, respectively, the percentage error [A*(t)/A(t)-1] and [A*(t)/ /A(t)- 1] as a function of tip for a range of values of C] V of practical interest, assuming a constant source. In fig. 2 the values plotted are for p = 20 sec; the error is, for a given value of C/1/, almost the same for a large, range of values of p when plotted as a function of tip. This is not so for A*(t), however, and fig. 3 shows the errors for two different values of p. In the solution of (12) a finite-difference representation of the integral has been used to produce a numerical accuracy of better than 1 part in 104. It is apparent from fig. 2 that the use of the approximate solution A*(t) will produce large errors when tip is less than 10 and that even for tip = 50 the errors for large values of C/V are still significant. The use of A*(t), which is implicit in the expression given by Axton et al. s) for the predicted asymptotic count rate, gives a substantial improvement as we see from fig. 3 where, for a wide range of (2IV and p values, the error has fallen to below 0.2 percent after about 10 pumping intervals. It has recently been suggested by Volpi and Porges 13) that a suitable vanadium salt be substituted for manganese sulphate because the shorter half-life of S2V
m
0
v
v
10
v
--
20 t/p
v
)
30
Fig. 6. A comparison of manganese activities for sources Q = 1 + 0.1 sin (2nt/5p), A(s), and Q = 1, as a function of time, for p = 20 sec and C~V= 0.5.
40
A(e), expressed as
[A(s)-A(c)]/A(c)
332
T. D. BEYNON AND M. C. SCOTT
(T~ = 3.72 min) would facilitate faster n e u t r o n yield measurements to be made with a shorter delay time between subsequent measurements. With this development in mind the errors in A*(t) and A'~(t) are shown in figs. 4 and 5 with v a n a d i u m as the activated element, again assuming a constant source. In b o t h cases, for the values o f C / V and p chosen, a time o f at least 10 p u m p i n g intervals must elapse before the error falls below 0.1 percent. In particular the A*(t) solution can possess an oscillatory error term for certain combinations o f C / V and p. To simulate the effect o f an accelerator source with a weak time dependence some calculations have been m a d e with a source term Q(t) = 1 + a sin(wt). In figs. 2 and 3 we show some typical errors incurred in the approximations A* and A* for the signal 1 + 0 . 5 sin (27tt/lOp). O u r above remarks are still applicable in that A*(t) m a y well be a sufficiently g o o d approximation in m a n y cases. Finally, as an example o f our analysis, we show the effect of a sinusoidally-varying source on the growth o f manganese activity. In fig. 6 we plot [ A ( s ) - A ( c ) - I / A ( c ) as a function o f time, where A(s) and A(c) denote, respectively, the activities [via (12),l for sources Q ( t ) = 1 +0.1 sin(2nt/5p) and Q ( t ) - - 1 , b o t h with p = 20 sec and C / V = 0.5. We see that a time dependent source o f this kind, n o t by any means untypical, can produce a residual 0.5 percent uncertainty in the measured activity.
The authors wish to acknowledge use o f the D e p a r t ment o f Physics' I B M 360/44 c o m p u t e r and to t h a n k its staff for their assistance. References 1) p. Fieldhouse and E. R. Culliford, Neutron dosimetry 2 (I.A.E.A., Vienna, 1963) p. 565. ~) R. L. Macklin, Nucl. Instr. I (1957) 335. a) F. yon Alder and P. Huber, Helv. Phys. Acta 22 (1949) 368. 4) y. Gurfinkel and S. Amiel, Nucleonics 23 (1965) 76. ~) E. J. Axton, P. Cross and J. C. Robertson, J. Nucl. Energy 19, parts A/B (1965) 409. 6) R. Taschek and A. Hemmendinger, Phys. Rev. 74 (1948) 373. 7) p. yon Adler and A. Ohnesorge, Atomkernenergie 9-13 (1964) 113. 8) A. Capgras, Proc. Colloq. Neutron dosimetry .for radiological protection (I.A.E.A., Vienna, 1967). a) A. D. Miskis, Linear differential equations with a delayed argument (Moscow, 1951). 10) R. Bellman, J. Jacquez and R. Kalaba, Bull. Math. Biophys. 22 (1960) 181. ll) T. E. Harris, The theory of branching processes (SpringerVerlag, Berlin, 1963). 1~) R. Bellman, J. Jacquez and R. Kalaba, Proc. 4 th Berkeley Symp. Mathematical statistics and probability 4 (1961) p. 57. lz) A. De Volpi and. K. G. Porges, Proc. 2 na Conf. Neutron cross sections and technology (Washington, D.C., 1968). 14) M. C. Scott, Proc. 3ra Conf. Accelerator targets for neutron production (Liege, 1967) p. 283 (a more detailed paper is in course of preparation).
INSTRUMENTS
AND METHODS
76 ([969) 328-332; © N O R T H - H O L L A N D
PUBLISHING
CO.
ON T H E DYNAMICS OF T H E C O N T I N U O U S F L O W MANGANESE BATH SYSTEM T. D. BEYNON and M. C. SCOTT
Department of Physics, University of Birmingham, England Received 13 March 1969 and in revised form 18 August 1969 Exact and approximate solutions of the equation governing the activity growth in continuously cycled assemblies are presented. These solutions are applied to manganese and vanadium systems of practical interest and the implications for neutron yield measurements are discussed.
SIVA((volume=¥]
1. Introduction
Accompanying the growing use of accelerators for neutron production is the need for accurate measurements of the target yields. The anisotropy often associated with neutrons produced in this way, together with high gamma background, seriously restricts the choice of suitable measuring systems. Methods of determining neutron yields which involve measuring the spatial variation of thermal neutrons produced by placing the source at the centre of a large volume of moderator t' 2) suffer from several disadvantages in this respect. Among these are: 1. the system requires calibrating for each neutron source spectrum; 2. the flux has to be monitored at a large number of spatial points; 3. the combination of the effects of the differing neutron and gamma yields and spectra of different targets makes difficult the correction for gamma sensitivity of the detectors. By comparison the manganese bath technique 3-5) suffers from none of these disadvantages: the measurement is an integral one and yields an absolute value for the source strength without reference to external standards. Furthermore, such corrections that are necessary are well documented. Until recently it has only been used occasionally for accelerator sources [compare ref. 6)] because the presence of large source gamma pulses makes in situ determination of 56Mn activity difficult. Other methods of determining the activity, by removing active solution, are slow. The continuous sampling of the solution, proposed in different forms by Adler and Ohnesorge 7) and Axton, Cross and RobertsonS), removes both of these difficulties. The solution is continuously stirred and pumped to a counting chamber which can be as far away from the source as is necessary for shielding purposes. The arrangement is shown schematically in fig. 1. The general equation governing the growth of activity in such a system, which is given in the next section, is similar to that which occurs in many biological and
~.~UI4TING CSOURC(HAHf(lR (vok~ ~
p~s qAc=~=~-_~-~ld C) TiME FORS(X.UTiON TO GO FIIOM A TO B = p Fig. 1. Schematic diagram of the system.
stochastic processes and the approximate and exact solutions for both steady state and time dependent source are therefore of more general interest. 2. Theory 2.1. THE DIFFERENTIAL-DIFFERENCEEQUATION A model s) describing the time dependence of the activity per unit volume, A(t), in the continuously cycled source chamber produces the differential-difference equation,
dA/dt+2]A(t)-yA(t-p)
= SoQ(t).
(1
The following notation has been adopted:
'~1 =
/~-['-OC, ---- 0ee- 2p,
= C/p V (the fraction of the fluid removed per unit time from the source chamber), p = pumping time, the time taken for the active fluid to be pumped to the counting chamber and returned to the source chamber, C = total volume of the counting chamber, pump and associated pipe work, V = volume of source chamber, 2 = decay constant of active atoms in the pumped fluid, Q(t) = source strength per unit volume at time t. So = the fraction of the source neutrons producing activity. Eq. (1) represents the activity balance in the source 328
329
ON THE DYNAMICS OF THE C O N T I N U O U S F L O W MANGANESE BATH SYSTEM
chamber under the assumptions of incompressibility and instantaneous mixing of the fluid. The additional assumption of non-turbulent flow in travelling a pipecontour distance y enables us to write the activity, B(t), at y as
Thus e - sto ,~G(s) and
B(t) = A ( t - yp/Y) e- ayp/r,
AG(t,to) =
where Y is the total distance travelled in the pumping interval p. A necessary initial-value for the solution (1) is
A(t) = O, t<=O (i.e. A ( t - p ) = O, t
where we define
f
= O,
A(t')k(t-t')dt',
(3)
J,
o k(O) = e -a'°,
u(t')K(t-t')dt',
where
K ( t - c) =
c-p),
and H ( t ) i s the unit function. Such equations occur extensively in mathematical physics9), biology~°), branching processes ~~) and chemotherapyl 2). 2.3. ANALYTICAL SOLUTION We may proceed to solve (1) using a Laplace transform. Accordingly, defining
eZJt
SoQ(s)
.
f t e-ZJt°Q(to)dto . o
(6)
oo 1 (21 --22)j= z j(1 + p22e - ~ ' )
--
- 1.
(7)
For SoQ(t) = 1, using (6) and (7), we see that
A(t)
-
1 2t-~'
+
~
e "~'
j=o zj(1-4-p22e -zip)
.
(8)
For values of t greater than about three or four times the reciprocal value of the maximum Re(z j), only the term containing z o = - K o is important, so that (8) becomes
A(t)
1
e - Kot
21-22
Ko(l + p22er°P) '
t,>O.
Similar asymptotic forms can be obtained for other integrable source functions. Of some interest is the source function
SoQ(t) = 1 +asin(wt)
Q(s) = Aa{Q(t)},
such that .4(s) and similar functions converge in a halfplane R e ( s ) > k , we obtain
.4(s) -
(5)
A useful relationship, obtained by integrating (1) with (5) over all time, is
The formulation (3) now describes not the activity balance in the source chamber but the history of activity behaviour for times t' =
for t > to,
= 0, for t
= y e a~p.
A(s) = &a{A(t)},
eZJ(t_to )
"= l+p22e -z~p
Q(t')k(t-t')dt',
t > to,
'
t < t o.
AGO, to ) =
A(t) = Soj~=°
Ao(t ) = S O
u(t) = f ( t ) +
2hi - r-io~ s + z 1-22e --sp
j=o (1 + p),e -zJp)
t--p
o
eS(t-to)ds
1 fr+i~
Hence,
(2)
2.2. INTEGRAL FORMULATION We can reduce (1) to a more familiar form by direct integration. This produces, after a little algebra and the use of (2), the integral equation,
A(t) = Ao(t)+~
s+21-22e -sp '
(4)
s + 2 t -22e -sp The solution is now best obtained in terms of the Green's function At(t, to), for SoO(t) = 6 ( t - to).
This produces
A(t) =
1
+
2t - 7
+i j=o
eZ~t(awzj+ w 2 + z 2) -- a z j r z j s i n ( w t ) +
wcos(wt)]
Z j ( W 2 "~- Z 2 ) (1 + p~ e -z~p)
(9)
330
T. D. B E Y N O N
AND
M. C. S C O T T
A(t) = A.(t), for
t.
Q(t) = Q,(t), 7
This enables us to write (1) as a differential-recurrence r e l a t i o n s h i p for t, < t < t, + 1.
T,
~s
d
d---]A"+AIA"(t)-yA"-l(t-P) = S°Qn(t)"
~4
U s i n g the b o u n d a r y value A.(t.)= grate (11) directly to p r o d u c e
z 2 !
A.(t) = k(t-tn)A._l(tn)+T
0
IO
+So
+ a[K°sin(wt)-wc°s(wt)] (w2+K2)(l+p?e r°z)
A,-?
(10) _~
2.4. N U M E R I C A L SOLUTION A l t h o u g h it is useful to possess the s o l u t i o n to (1) in the a n a l y t i c a l f o r m (6) it is by no m e a n s an easy m a t t e r to o b t a i n a sufficiently large n u m b e r o f the zs's for a g o o d n u m e r i c a l representation. W e n o w p r o c e e d to a n u m e r i c a l s o l u t i o n o f (1) suitable for a digital c o m p u t e r . Firstly, we restrict (1) to a time t such t h a t
t.<=t
t.=np,
n=0,1,2 ....
Q.(t')k(t-t')dt'.
(12)
3
Z.2
A(t) ",~~
f,
tn
2.6-
which, f o r l a r g e values o f t , b e c o m e s
A._l(t.), we inte-
x(t'-p)k(t-t')dt'+
100
Fig. 2. The percentage error in A0*(t), expressed as [Ao*(t)/ /A(t)-l], as a function of tip for a manganese system (T~= 2.573 h) with a pumping timep=20 sec. Curves 1-4 are for a constant source avd curve 5 is for a sinusoidal source.
(11)
1.8 •\
I
-,.0 : \
\ \ \ ~
\
I. C/V=O.05, P=ZOs 2.ClV=0.05, P=200~ 3.C/V=O.Z P=20a 4.C/V=0-2. p=2OOs
\
2
Fig. 4. The percentage error in Ao*(t), expressed as [Ao*(t)/ /A(t)-ll, as a function of tip for a range of C/V values and pumping time, p. The curves are for a vanadium system (T~= 3.72 min) with a constant source.
a n d define I n (12) we recur f r o m the initial c o n d i t i o n (2) such t h a t , as in (11).
04
Ao(t) = So
t CIV=O-I, P=2OJ.
e~i / -
\\
~
~
2~v=o.i, p:12o,
\\
~
4 C/V':O'3, p :1201.
"
'3
I
J
s c/~:o.,, p :2o,.. Q:,..o.s~,,c2.~,o~
1
I
i
l
l
I
10
i
i
I n this f a s h i o n we can easily c o m p u t e A(t) for a n y a r b i t r a r y source function, Q(t). W e note t h a t in obt a i n i n g A.(t) via (12) we are r e q u i r e d to store a c o m p l e t e h i s t o r y o f A , _ 1(0, as we m i g h t guess f r o m the f o r m o f (1).
,
100
tip
Fig. 3. The percentage error in Al*(t), expressed as [Al*(t)/ /A(t)-l], as a function of t/p for a range of C/V values and pumping times, p, for a manganese system (T~ = 2.573 h). Curves 1-4 are for a constant source and curve 5 represents a sinusoidal source.
Q(t') k ( t - t')dt'.
0
sc/v:o.3, p:2o,
~
f,
2 . 5 . A P P R O X I M A T E SOLUTIONS
T h e f o r m o f (1) lends itself to straight f o r w a r d numerical a p p r o x i m a t i o n s . A T a y l o r e x p a n s i o n o f A ( t - p ) a b o u t t reduces (1) to a simple f o r m for the a p p r o x i m a t e d activity, A*(t),
ON T H E D Y N A M I C S OF T H E C O N T I N U O U S
,of o.8[- / /
\
~0.6I /
~
331
t>p
and for
, c/,--0 0,. ,--20, z.c/v:oos, f,:zoo,. 3.c/v:o.z. ~:z0,.
\
F L O W M A N G A N E S E B A T H SYSTEM
A*(t) = Ao(p)g(t-p)+ {So/(l + Tp)}
4.C/V=0.2 . P=2OOs.
f,pg(t-t')Q(t')dt'.
3. Discussion of numerical solution
Fig. 5. The percentage error in Al*(t), expressed as [Al*(t)/ as a function o f tip for a range of C/V values and p u m p i n g time, p. The curves are for a vanadium system ( T ~ = 3 . 7 2 min) with a constant source.
/A(t)-11,
dA*
--+ dt
eA*(t) = SoO(t)/(1 + ~p),
(13)
where = (&-
+
We obtain our solution to (13) by assuming we know the exact value for A(t) for t<=t,. Integrating (13) directly we obtain for our approximate solution
A*(t), (t > t.), A*(t) = A._ ,(t.) g(t- t.) + +{So/(l+~p)}
y, g(t-t')Q(t')dt',
(14)
tn
where g(O) = e-~°.
Two obvious approximations are AN(t ) and since we know A(0) and A(p) exactly. Hence,
f,
A*(t) = { S o / ( l + y p ) } og(t-t')Q(t')dt', 10
A*(t)
(t>0),
The exact solution for the activity can be obtained by direct numerical quadrature in (12) and compared with the approximate solutions A~(t) and A~(t). Such comparisons are made, for a manganese system (T½ = 2.573 h), in figs. 2 and 3 which show, respectively, the percentage error [A*(t)/A(t)-1] and [A*(t)/ /A(t)- 1] as a function of tip for a range of values of C] V of practical interest, assuming a constant source. In fig. 2 the values plotted are for p = 20 sec; the error is, for a given value of C/1/, almost the same for a large, range of values of p when plotted as a function of tip. This is not so for A*(t), however, and fig. 3 shows the errors for two different values of p. In the solution of (12) a finite-difference representation of the integral has been used to produce a numerical accuracy of better than 1 part in 104. It is apparent from fig. 2 that the use of the approximate solution A*(t) will produce large errors when tip is less than 10 and that even for tip = 50 the errors for large values of C/V are still significant. The use of A*(t), which is implicit in the expression given by Axton et al. s) for the predicted asymptotic count rate, gives a substantial improvement as we see from fig. 3 where, for a wide range of (2IV and p values, the error has fallen to below 0.2 percent after about 10 pumping intervals. It has recently been suggested by Volpi and Porges 13) that a suitable vanadium salt be substituted for manganese sulphate because the shorter half-life of S2V
m
0
v
v
10
v
--
20 t/p
v
)
30
Fig. 6. A comparison of manganese activities for sources Q = 1 + 0.1 sin (2nt/5p), A(s), and Q = 1, as a function of time, for p = 20 sec and C~V= 0.5.
40
A(e), expressed as
[A(s)-A(c)]/A(c)
332
T. D. BEYNON AND M. C. SCOTT
(T~ = 3.72 min) would facilitate faster n e u t r o n yield measurements to be made with a shorter delay time between subsequent measurements. With this development in mind the errors in A*(t) and A'~(t) are shown in figs. 4 and 5 with v a n a d i u m as the activated element, again assuming a constant source. In b o t h cases, for the values o f C / V and p chosen, a time o f at least 10 p u m p i n g intervals must elapse before the error falls below 0.1 percent. In particular the A*(t) solution can possess an oscillatory error term for certain combinations o f C / V and p. To simulate the effect o f an accelerator source with a weak time dependence some calculations have been m a d e with a source term Q(t) = 1 + a sin(wt). In figs. 2 and 3 we show some typical errors incurred in the approximations A* and A* for the signal 1 + 0 . 5 sin (27tt/lOp). O u r above remarks are still applicable in that A*(t) m a y well be a sufficiently g o o d approximation in m a n y cases. Finally, as an example o f our analysis, we show the effect of a sinusoidally-varying source on the growth o f manganese activity. In fig. 6 we plot [ A ( s ) - A ( c ) - I / A ( c ) as a function o f time, where A(s) and A(c) denote, respectively, the activities [via (12),l for sources Q ( t ) = 1 +0.1 sin(2nt/5p) and Q ( t ) - - 1 , b o t h with p = 20 sec and C / V = 0.5. We see that a time dependent source o f this kind, n o t by any means untypical, can produce a residual 0.5 percent uncertainty in the measured activity.
The authors wish to acknowledge use o f the D e p a r t ment o f Physics' I B M 360/44 c o m p u t e r and to t h a n k its staff for their assistance. References 1) p. Fieldhouse and E. R. Culliford, Neutron dosimetry 2 (I.A.E.A., Vienna, 1963) p. 565. ~) R. L. Macklin, Nucl. Instr. I (1957) 335. a) F. yon Alder and P. Huber, Helv. Phys. Acta 22 (1949) 368. 4) y. Gurfinkel and S. Amiel, Nucleonics 23 (1965) 76. ~) E. J. Axton, P. Cross and J. C. Robertson, J. Nucl. Energy 19, parts A/B (1965) 409. 6) R. Taschek and A. Hemmendinger, Phys. Rev. 74 (1948) 373. 7) p. yon Adler and A. Ohnesorge, Atomkernenergie 9-13 (1964) 113. 8) A. Capgras, Proc. Colloq. Neutron dosimetry .for radiological protection (I.A.E.A., Vienna, 1967). a) A. D. Miskis, Linear differential equations with a delayed argument (Moscow, 1951). 10) R. Bellman, J. Jacquez and R. Kalaba, Bull. Math. Biophys. 22 (1960) 181. ll) T. E. Harris, The theory of branching processes (SpringerVerlag, Berlin, 1963). 1~) R. Bellman, J. Jacquez and R. Kalaba, Proc. 4 th Berkeley Symp. Mathematical statistics and probability 4 (1961) p. 57. lz) A. De Volpi and. K. G. Porges, Proc. 2 na Conf. Neutron cross sections and technology (Washington, D.C., 1968). 14) M. C. Scott, Proc. 3ra Conf. Accelerator targets for neutron production (Liege, 1967) p. 283 (a more detailed paper is in course of preparation).