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Density correction for a boiling liquid hydrogen target
Nuclear Instruments and Methods 204 (1983) 295-298 North-Holland Publishing Company
DENSITY
CORRECTION
FOR A BOILING
295
LIQUID
HYDROGEN
TARGET
*
M a r k W . T A T E a n d M i c h a e l E. S A D L E R
Department of Physics, Abilene Christian University, Abilene, Texas 79699, U,S.A. Received 22 March 1982
A method to determine the density correction of a boiling liquid target is presented. The correction is determined from the time required to empty the target and from the bubble velocity. The case of a cylindrical liquid hydrogen target is evaluated.
1. Introduction
2. Method of calculation
M e a s u r e m e n t of absolute differential cross sections to a n accuracy of a few percent requires a knowledge of the target density to the same or b e t t e r accuracy. The boiling of a cooled liquid target, such as liquid hydrogen (LH2), will decrease the target density. This density correction can be kept below 1% by adequate insulation a n d has typically b e e n ignored or estimated by previous experimenters. A m e t h o d to evaluate this correction for more precise m e a s u r e m e n t s is presented. Cylindrical targets with diameters of 10.16 a n d 12.7 cm were used in a recent m e a s u r e m e n t [1] of differential cross sections for ~r + p elastic scattering. The measurem e n t s were completed at the Clinton P. A n d e r s o n Meson Physics Facility of the Los Alamos N a t i o n a l L a b o r a t o r y ( L A M P F experiment 363). These targets were c o n t a i n e d within a v a c u u m and were insulated with aluminized mylar to minimize heat flow into the target. A closed L H 2 system was utilized; the gas was taken from the top of the target to a refrigeration system, liquified, a n d returned to the b o t t o m of the target. Resistors were placed near the top a n d b o t t o m of the target, which showed different resistances when b a t h e d in L H 2 t h a n when surrounded by the warmer gas. The emptying of the target was accomplished by closing the gas outlet. As the gas formed the liquid was displaced into a reservoir outside the target. The rate of boiling a n d the density correction are determined from the time observed to empty the target.
The rate at which the gas is formed is assumed to be p r o p o r t i o n a l to the surface area of the liquid hydrogen in contact with the cylinder.
* Supported in part by the United States Department of Energy and the Research Council of Abilene Christian University. Work performed in part at the Clinton P. Anderson Meson Physics Facility (LAMPF) of the Los Alamos National Laboratory, Los Alamos, New Mexico, 87545. 0 1 6 7 - 5 0 8 7 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 N o r t h - H o l l a n d
d V / d t = aqrr2 + fl2~rry. T h e proportionality constants a a n d /3 d e p e n d on the rate of heat flow through unit areas on the b o t t o m a n d side of the target, respectively. Each has units of velocity. The p a r a m e t e r r is the radius of the target, a n d y is the height of the liquid in the emptying target, as shown in fig. 1. The volume of gas in the emptying target is
V = ( h - - y ) , r r r 2, where h is the height of the target. The initial volume of the gas in the target is neglected. Differentiating this equation,
d V / d t = -~rr2d y / d t Thus,
-~rrZd y / d t = a~rr 2 +/32~rry,
OUTLET
Fig. 1. The emptying cylindrical liquid hydrogen target. In emptying, the gas outlet is closed, so as gas is formed the liquid hydrogen is displaced into a reservoir. The height of the liquid hydrogen in the target is y, while a and b are the heights of the two level-sensing resistors.
M. I/E Tate, M.E. Sadler / Density correction
296
surface area r d O d y is B r d O ( y - - y ) d y / V b , where y is the height of the differential area. Integrating from 0 to 3' gives the volume of gas below 3' in a window of width rd 0.
or
dy
dt
(ar+2fly)
r
Integrating b o t h sides, dV2 _-- flrfo r3' t~b -y
ln( ar + 2fly ) = - - -2fir +C, r
The volume of gas between two points h e + % and h c -- o v is
or
ar+ 2fly=ke
2~,/,,
dV2=flrdO[(hc+ov)2-(hc
where C and k are integration constants. At t = 0, the target is full ( y = h). Therefore,
k = ar + 2flh. A t t = t, (the time for the LH 2 to be low enough to change the first resistance), y = a
ar+ 2fla=(ar+
dO d y = ½fir d03'2//)b .
2 f l h ) e 2flt~/r
%)2]/Vb
de[4
--
To find the total volume encountered by the beam, d V 2 is integrated from 0 = - s i n 1 % / r to + s i n l o~/r which yields V2
z
4flrhc°v" s i .n _ U b
1 03, , r
Likewise, at t = t b, y = b
ar + 2fib = ( ar + 2flh ) e - 2Bq'/r. Eliminating a from these equations and simplifying gives
(h-b)e
2~t"/r-(h-a)e
2Bt~'/r+b-a=O.
The roots may be obtained numerically using a standard N e w t o n - R a p h s o n iteration. After finding fl, a may be found from OL
2fl h e -2Btb/r -- b r 1 -- e 2titbit
~ f + o~ , f~+ ~ f<+o, V1 = l.)b ~ g "]h o d x 12 b
"
2.0.5
)
V2 ~ 4flOxoyhc/v b .
The total volume of gas seen by the b e a m for this limiting case is simply
ar + flh ~ Vga~= I71 + 2V2 = 8 % % - 19 b
T h e ratio, Vg~s/Vtot~1 where VtotaI ~ 8%Oyr, is,
The item of interest is the volume of gas intersected by a beam in a full cylindrical target. First consider the c o n t r i b u t i o n to this volume from the gas rising from the b o t t o m surface. This volume is assumed to be c o n s t a n t throughout the target. The volume of gas in a differential volume d r is a d a d y / v b, where d a is the projected area on the b o t t o m of the target, d y is the incremental length a b u b b l e travels in d r , and v b is the velocity of the bubble, d y / v b is therefore the time interval that the b u b b l e remains in d r . Integrating over the volume encountered by the beam,
__ 4 a [ % ( r 2 _ %
which should be multiplied by 2 due to the b e a m b o t h entering a n d leaving the target. Again for r >> o~, this expression simplifies to,
+r2sin
dz d y
~o~ ] Oy, r
Vgas 1 Vtota,- v-b ( a + f l h c / r ) . T h e density correction, C O R R , is this ratio multiplied by a density factor
1 (a+flhJr)(1-pg/p,), C O R R = VTb where pg and Pl are the densities of gaseous a n d liquid hydrogen, respectively.
Table 1 Evaluation of a, fl, and CORR obtained from measuring a and b to the bottom, middle and top of the resistors. Other parameters are h=19.1 cm, r=5.08 cm, ta=18S, tb=410 S and v t --7.7 cm/s. Note the unphysical negative value for a in the last column.
where % and oy are the b e a m half-width a n d half-height, respectively. For the case where r > > % , sin - l O x / r ~ o u r , and the integral reduces to, V 1 ~- 8OtOxOyr/V b.
The contribution from the side is assumed to remain close to the side as it bubbles up. The concentration of gas will increase as the height up the target increases. T h e volume of gas below a point 3' produced by a
a b a fl CORR
Bottom of resistors
Middle of resistors
Top of resistors
17.8 cm 0.8 cm 0.0069 cm/s 0.022 c m / s 0.50%
18.1 cm 1.1 cm 0.0032 cm/s 0.032 cm/s 0.46%
18.4 cm 1.4 cm --0.0013 cm/s 0.049 c m / s 0.60%
297
M. 144 Tate, M.E. Sadler / Density correction The inverse d e p e n d e n c e on r is deceiving as 1 / r occurs with fl in the equation from which fl was determined. This ratio of f l / r is c o n s t a n t for a given set of e m p t y i n g conditions, a n d is also used in finding a, making the correction i n d e p e n d e n t of radius. F o r the L H 2 targets used in the experiment the resistances were placed vertically, making measurements for a a n d b more uncertain t h a n if they were horizontal. Calculations for the correction for a a n d b at the top, middle, a n d b o t t o m of the resistors showed large changes for a a n d fl, but an increase in one p a r a m e t e r caused a c o r r e s p o n d i n g decrease in the other, with little change in the total correction (see table 1).
7O ~s0 © o
RIGID S P H E R E ~ J
450
IO o'3
3. Determination of bubble velocity A major u n c e r t a i n t y arises in the d e t e r m i n a t i o n of the velocity of the b u b b l e s rising in L H 2. The drag force o n a particle (a b u b b l e in this case) r i s i n g i n a liquid is given by [2] Fd - ½ C d A p V 2 , where Ca is the drag coefficient, A is the cross sectional area, and v is the velocity of the particle. W h e n this force equals the b u o y a n t force, a terminal velocity is reached. This velocity is given by [2] o,=[4gO(ol -
o g / 3 o , c a ], 0.5 ,
where g is the acceleration due to gravity a n d D is the d i a m e t e r of the particle. The Reynold's n u m b e r is useful in describing the flow of fluids. This dimensionless n u m b e r is given by Re = Do~vt/l~, where v t is the velocity of the particle a n d ~ is the viscosity of the liquid. For a very small Reynold's n u m b e r , less than 0.3, the rise of bubbles in fluid is in the l a m i n a r flow region (drag force proportional to the velocity). Cd in this region is given by [2]
ds & BUBBLE DIAMETER(cm)
+-
Fig. 2. Terminal velocity versus bubble diameter for hydrogen bubbles rising in liquid hydrogen. The rigid sphere approximation should be valid for small bubble diameters and the drag coefficient, Ca, should approach 2.6 for large diameters. The dashed curve represents a transition between these extremes assuming that Ca and Re are releated in the same way as for air bubbles rising in distilled water as shown in the next figure. The dotted curve uses the relationship between Ca and Re for air bubbles in tap water [2].
Cd for 100 < R e < 2 0 0 0 must be experimentally det e r m i n e d for each liquid a n d corresponding particles. Fig. 2 shows the velocity d e p e n d e n c e on b u b b l e diameter for L H 2. The dotted curve represents a transition between the rigid-sphere a p p r o x i m a t i o n a n d the constant Cd of 2.6, and would be accurate if the relationship between Ca a n d Re for L H 2 were close to that for tap
Ca = 2 4 / R e . This region applies to very small bubbles, < 0.05 m m in the case of liquid hydrogen. For Re greater than 2000 ( D > 3 m m for l . H a ) , the flow becomes turbulent (drag force p r o p o r t i o n a l to the square of the velocity). In this region, the drag coefficient is very nearly constant; 0.44 for rigid spheres a n d 2.6 for fluid b u b b l e s [3]. However, m o s t of the b u b b l e s e n c o u n t e r e d in the LH 2 target were in the range 0.3 < Re < 2000. This region encompasses the transition of velocity d e p e n d e n c e on the force. The drag coefficient in this region is highly d e p e n d e n t u p o n the physical properties of the liquid. For Re less t h a n 100, Ca is described very well by a rigid-sphere model. In this transition region, Ca for a rigid sphere is given approximately by [2] Cd = 1 8 . 5 / R e °.6.
"', /
2 I
,o
02
R,eO
E
,04
_
f~
Fig. 3. Drag coefficient versus Reynold's number in the transition region of force dependence on velocity [2]. The solid curve is for a rigid sphere, while the dashed and dotted curves are for bubbles rising in distilled and tap water, respectively, as in the previous figure.
298
M.W. Tate, M.E. Sadler / Density correction
water [2]. The dashed curve, shown for comparison, is found from the dependence of Cd vs. Re for distilled water [2].
4. Bubbling corrections for LAMPF experiment 363 F r o m a picture of a boiling LH 2 target, though not the target for experiment 363, a mean b u b b l e diameter of 0.4 m m was found. This target was not as well insulated as the experiment 363 targets, which resulted in an increased boiling rate. Bubble diameter is probably related to the boiling rate, but an idea of b u b b l e diameter was obtained. The picture showed that most of the bubbles were in the transition region of velocity dependence. If the curve for tap water is used for the transition, a correction of 0.5% + 0.2% is found. This correction region encompasses diameters from 0.3 to 3 mm. If, however, the curve for distilled water is used, the correction is a b o u t 0.30%-+0.15%. Table l gives
values for a, fl, a n d the percent correction for given emptying conditions. We wish to t h a n k J. N o v a k and group MP-7 at L A M P F who designed and built the L H 2 targets. We also t h a n k W.J. Briscoe, D.H. Fitzgerald, B.M.K. Nefkens and other members of the U C L A Particle Physics G r o u p for their helpful discussions and critique.
References [1] M.E. Sadler, F.O. Borcherding, W.J. Briscoe, D.H. Fitzgerald, P.F. Glodis, R.P. Haddock, N. Matz, A. Mokhtari, B.M.K. Nefkens and C. Seftor, Proc. 9th Int. Conf. on High energy physics and nuclear spectroscopy, Versailles (1981) and to be published. [2] R.H. Perry and C.H. Chilton, Chemical Engineer's Handbook, 5th ed. (McGraw-Hill, New York, 1973) pp. 5-61. [3] N.M. Aybers, Two-phase flows and heat transfer, eds., S. Kakac and F. Mayinger (Hemisphere, Washington, 1977).
DENSITY
CORRECTION
FOR A BOILING
295
LIQUID
HYDROGEN
TARGET
*
M a r k W . T A T E a n d M i c h a e l E. S A D L E R
Department of Physics, Abilene Christian University, Abilene, Texas 79699, U,S.A. Received 22 March 1982
A method to determine the density correction of a boiling liquid target is presented. The correction is determined from the time required to empty the target and from the bubble velocity. The case of a cylindrical liquid hydrogen target is evaluated.
1. Introduction
2. Method of calculation
M e a s u r e m e n t of absolute differential cross sections to a n accuracy of a few percent requires a knowledge of the target density to the same or b e t t e r accuracy. The boiling of a cooled liquid target, such as liquid hydrogen (LH2), will decrease the target density. This density correction can be kept below 1% by adequate insulation a n d has typically b e e n ignored or estimated by previous experimenters. A m e t h o d to evaluate this correction for more precise m e a s u r e m e n t s is presented. Cylindrical targets with diameters of 10.16 a n d 12.7 cm were used in a recent m e a s u r e m e n t [1] of differential cross sections for ~r + p elastic scattering. The measurem e n t s were completed at the Clinton P. A n d e r s o n Meson Physics Facility of the Los Alamos N a t i o n a l L a b o r a t o r y ( L A M P F experiment 363). These targets were c o n t a i n e d within a v a c u u m and were insulated with aluminized mylar to minimize heat flow into the target. A closed L H 2 system was utilized; the gas was taken from the top of the target to a refrigeration system, liquified, a n d returned to the b o t t o m of the target. Resistors were placed near the top a n d b o t t o m of the target, which showed different resistances when b a t h e d in L H 2 t h a n when surrounded by the warmer gas. The emptying of the target was accomplished by closing the gas outlet. As the gas formed the liquid was displaced into a reservoir outside the target. The rate of boiling a n d the density correction are determined from the time observed to empty the target.
The rate at which the gas is formed is assumed to be p r o p o r t i o n a l to the surface area of the liquid hydrogen in contact with the cylinder.
* Supported in part by the United States Department of Energy and the Research Council of Abilene Christian University. Work performed in part at the Clinton P. Anderson Meson Physics Facility (LAMPF) of the Los Alamos National Laboratory, Los Alamos, New Mexico, 87545. 0 1 6 7 - 5 0 8 7 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 N o r t h - H o l l a n d
d V / d t = aqrr2 + fl2~rry. T h e proportionality constants a a n d /3 d e p e n d on the rate of heat flow through unit areas on the b o t t o m a n d side of the target, respectively. Each has units of velocity. The p a r a m e t e r r is the radius of the target, a n d y is the height of the liquid in the emptying target, as shown in fig. 1. The volume of gas in the emptying target is
V = ( h - - y ) , r r r 2, where h is the height of the target. The initial volume of the gas in the target is neglected. Differentiating this equation,
d V / d t = -~rr2d y / d t Thus,
-~rrZd y / d t = a~rr 2 +/32~rry,
OUTLET
Fig. 1. The emptying cylindrical liquid hydrogen target. In emptying, the gas outlet is closed, so as gas is formed the liquid hydrogen is displaced into a reservoir. The height of the liquid hydrogen in the target is y, while a and b are the heights of the two level-sensing resistors.
M. I/E Tate, M.E. Sadler / Density correction
296
surface area r d O d y is B r d O ( y - - y ) d y / V b , where y is the height of the differential area. Integrating from 0 to 3' gives the volume of gas below 3' in a window of width rd 0.
or
dy
dt
(ar+2fly)
r
Integrating b o t h sides, dV2 _-- flrfo r3' t~b -y
ln( ar + 2fly ) = - - -2fir +C, r
The volume of gas between two points h e + % and h c -- o v is
or
ar+ 2fly=ke
2~,/,,
dV2=flrdO[(hc+ov)2-(hc
where C and k are integration constants. At t = 0, the target is full ( y = h). Therefore,
k = ar + 2flh. A t t = t, (the time for the LH 2 to be low enough to change the first resistance), y = a
ar+ 2fla=(ar+
dO d y = ½fir d03'2//)b .
2 f l h ) e 2flt~/r
%)2]/Vb
de[4
--
To find the total volume encountered by the beam, d V 2 is integrated from 0 = - s i n 1 % / r to + s i n l o~/r which yields V2
z
4flrhc°v" s i .n _ U b
1 03, , r
Likewise, at t = t b, y = b
ar + 2fib = ( ar + 2flh ) e - 2Bq'/r. Eliminating a from these equations and simplifying gives
(h-b)e
2~t"/r-(h-a)e
2Bt~'/r+b-a=O.
The roots may be obtained numerically using a standard N e w t o n - R a p h s o n iteration. After finding fl, a may be found from OL
2fl h e -2Btb/r -- b r 1 -- e 2titbit
~ f + o~ , f~+ ~ f<+o, V1 = l.)b ~ g "]h o d x 12 b
"
2.0.5
)
V2 ~ 4flOxoyhc/v b .
The total volume of gas seen by the b e a m for this limiting case is simply
ar + flh ~ Vga~= I71 + 2V2 = 8 % % - 19 b
T h e ratio, Vg~s/Vtot~1 where VtotaI ~ 8%Oyr, is,
The item of interest is the volume of gas intersected by a beam in a full cylindrical target. First consider the c o n t r i b u t i o n to this volume from the gas rising from the b o t t o m surface. This volume is assumed to be c o n s t a n t throughout the target. The volume of gas in a differential volume d r is a d a d y / v b, where d a is the projected area on the b o t t o m of the target, d y is the incremental length a b u b b l e travels in d r , and v b is the velocity of the bubble, d y / v b is therefore the time interval that the b u b b l e remains in d r . Integrating over the volume encountered by the beam,
__ 4 a [ % ( r 2 _ %
which should be multiplied by 2 due to the b e a m b o t h entering a n d leaving the target. Again for r >> o~, this expression simplifies to,
+r2sin
dz d y
~o~ ] Oy, r
Vgas 1 Vtota,- v-b ( a + f l h c / r ) . T h e density correction, C O R R , is this ratio multiplied by a density factor
1 (a+flhJr)(1-pg/p,), C O R R = VTb where pg and Pl are the densities of gaseous a n d liquid hydrogen, respectively.
Table 1 Evaluation of a, fl, and CORR obtained from measuring a and b to the bottom, middle and top of the resistors. Other parameters are h=19.1 cm, r=5.08 cm, ta=18S, tb=410 S and v t --7.7 cm/s. Note the unphysical negative value for a in the last column.
where % and oy are the b e a m half-width a n d half-height, respectively. For the case where r > > % , sin - l O x / r ~ o u r , and the integral reduces to, V 1 ~- 8OtOxOyr/V b.
The contribution from the side is assumed to remain close to the side as it bubbles up. The concentration of gas will increase as the height up the target increases. T h e volume of gas below a point 3' produced by a
a b a fl CORR
Bottom of resistors
Middle of resistors
Top of resistors
17.8 cm 0.8 cm 0.0069 cm/s 0.022 c m / s 0.50%
18.1 cm 1.1 cm 0.0032 cm/s 0.032 cm/s 0.46%
18.4 cm 1.4 cm --0.0013 cm/s 0.049 c m / s 0.60%
297
M. 144 Tate, M.E. Sadler / Density correction The inverse d e p e n d e n c e on r is deceiving as 1 / r occurs with fl in the equation from which fl was determined. This ratio of f l / r is c o n s t a n t for a given set of e m p t y i n g conditions, a n d is also used in finding a, making the correction i n d e p e n d e n t of radius. F o r the L H 2 targets used in the experiment the resistances were placed vertically, making measurements for a a n d b more uncertain t h a n if they were horizontal. Calculations for the correction for a a n d b at the top, middle, a n d b o t t o m of the resistors showed large changes for a a n d fl, but an increase in one p a r a m e t e r caused a c o r r e s p o n d i n g decrease in the other, with little change in the total correction (see table 1).
7O ~s0 © o
RIGID S P H E R E ~ J
450
IO o'3
3. Determination of bubble velocity A major u n c e r t a i n t y arises in the d e t e r m i n a t i o n of the velocity of the b u b b l e s rising in L H 2. The drag force o n a particle (a b u b b l e in this case) r i s i n g i n a liquid is given by [2] Fd - ½ C d A p V 2 , where Ca is the drag coefficient, A is the cross sectional area, and v is the velocity of the particle. W h e n this force equals the b u o y a n t force, a terminal velocity is reached. This velocity is given by [2] o,=[4gO(ol -
o g / 3 o , c a ], 0.5 ,
where g is the acceleration due to gravity a n d D is the d i a m e t e r of the particle. The Reynold's n u m b e r is useful in describing the flow of fluids. This dimensionless n u m b e r is given by Re = Do~vt/l~, where v t is the velocity of the particle a n d ~ is the viscosity of the liquid. For a very small Reynold's n u m b e r , less than 0.3, the rise of bubbles in fluid is in the l a m i n a r flow region (drag force proportional to the velocity). Cd in this region is given by [2]
ds & BUBBLE DIAMETER(cm)
+-
Fig. 2. Terminal velocity versus bubble diameter for hydrogen bubbles rising in liquid hydrogen. The rigid sphere approximation should be valid for small bubble diameters and the drag coefficient, Ca, should approach 2.6 for large diameters. The dashed curve represents a transition between these extremes assuming that Ca and Re are releated in the same way as for air bubbles rising in distilled water as shown in the next figure. The dotted curve uses the relationship between Ca and Re for air bubbles in tap water [2].
Cd for 100 < R e < 2 0 0 0 must be experimentally det e r m i n e d for each liquid a n d corresponding particles. Fig. 2 shows the velocity d e p e n d e n c e on b u b b l e diameter for L H 2. The dotted curve represents a transition between the rigid-sphere a p p r o x i m a t i o n a n d the constant Cd of 2.6, and would be accurate if the relationship between Ca a n d Re for L H 2 were close to that for tap
Ca = 2 4 / R e . This region applies to very small bubbles, < 0.05 m m in the case of liquid hydrogen. For Re greater than 2000 ( D > 3 m m for l . H a ) , the flow becomes turbulent (drag force p r o p o r t i o n a l to the square of the velocity). In this region, the drag coefficient is very nearly constant; 0.44 for rigid spheres a n d 2.6 for fluid b u b b l e s [3]. However, m o s t of the b u b b l e s e n c o u n t e r e d in the LH 2 target were in the range 0.3 < Re < 2000. This region encompasses the transition of velocity d e p e n d e n c e on the force. The drag coefficient in this region is highly d e p e n d e n t u p o n the physical properties of the liquid. For Re less t h a n 100, Ca is described very well by a rigid-sphere model. In this transition region, Ca for a rigid sphere is given approximately by [2] Cd = 1 8 . 5 / R e °.6.
"', /
2 I
,o
02
R,eO
E
,04
_
f~
Fig. 3. Drag coefficient versus Reynold's number in the transition region of force dependence on velocity [2]. The solid curve is for a rigid sphere, while the dashed and dotted curves are for bubbles rising in distilled and tap water, respectively, as in the previous figure.
298
M.W. Tate, M.E. Sadler / Density correction
water [2]. The dashed curve, shown for comparison, is found from the dependence of Cd vs. Re for distilled water [2].
4. Bubbling corrections for LAMPF experiment 363 F r o m a picture of a boiling LH 2 target, though not the target for experiment 363, a mean b u b b l e diameter of 0.4 m m was found. This target was not as well insulated as the experiment 363 targets, which resulted in an increased boiling rate. Bubble diameter is probably related to the boiling rate, but an idea of b u b b l e diameter was obtained. The picture showed that most of the bubbles were in the transition region of velocity dependence. If the curve for tap water is used for the transition, a correction of 0.5% + 0.2% is found. This correction region encompasses diameters from 0.3 to 3 mm. If, however, the curve for distilled water is used, the correction is a b o u t 0.30%-+0.15%. Table l gives
values for a, fl, a n d the percent correction for given emptying conditions. We wish to t h a n k J. N o v a k and group MP-7 at L A M P F who designed and built the L H 2 targets. We also t h a n k W.J. Briscoe, D.H. Fitzgerald, B.M.K. Nefkens and other members of the U C L A Particle Physics G r o u p for their helpful discussions and critique.
References [1] M.E. Sadler, F.O. Borcherding, W.J. Briscoe, D.H. Fitzgerald, P.F. Glodis, R.P. Haddock, N. Matz, A. Mokhtari, B.M.K. Nefkens and C. Seftor, Proc. 9th Int. Conf. on High energy physics and nuclear spectroscopy, Versailles (1981) and to be published. [2] R.H. Perry and C.H. Chilton, Chemical Engineer's Handbook, 5th ed. (McGraw-Hill, New York, 1973) pp. 5-61. [3] N.M. Aybers, Two-phase flows and heat transfer, eds., S. Kakac and F. Mayinger (Hemisphere, Washington, 1977).