- Email: [email protected]
Liquid and gas discharge rates through holes in process vessels
Liquid and gas discharge rates through holes in process vessels John L. Woodward and Krishna S. Mudan Technica Inc., 355 East Campus View Boulevard, USA
Suite I70, Columbus,
Ohio, 43235,
Risk analysis often addresses the problem of calculating the time-varying discharge rate from an accidental puncture of a storage tank or process vessel. For selected cases, reasonably simple analytical solutions are presented for discharge rates. In addition to helping to validate more complex numerical solutions, these analytical solutions help in estimating a single representative discharge rate to drain completely a partially-filled vessel. To select a representative discharge rate, it is useful to first find the average discharge rate. Then, a useful rate for risk analysis can be chosen, typically around midway between the average and the initial discharge rate. Formulas are given for average rates as a function of the initial liquid level and vessel head pressure for spheres, horizontal cylinders and right cylinders. For gas and vapour discharge, the average discharge rate is given in terms of the initial pressure. (Keywords:
discfarge
rates; gas; liquid)
For risk analysis, a scenario which is frequently postulated is an accident which would drain the entire contents of a partially liquid filled or gas filled storage tank or process vessel. This event is inherently time varying since the discharge rate decays as the vessel pressure and/or liquid head above the puncture decreases. For screening level studies, one representative discharge rate is desired. This representative rate should be chosen to make an idealized steady state analysis roughly match the actual transient event A complex set of events must be consequences. modelled, including possible aerosol formation and rainout, pool spread and vaporization, vapour dispersion and toxic dosage or fire radiation dosage. After performing such transient analyses for a limited number of materials, our experience has been that a reasonable steady state discharge rate to use is roughly midway between the initial and the average discharge rates. Thus, it is useful to have simple analytic formulas for the average discharge rate, then apply the above rule of thumb to select a representative discharge rate.
Discharge rates for liquids
rir PL
Integrating
AP,
-ADCD 2gh(t) + 2 -
differential
095@4230/91/030161-06 @)I991 Butteworth-Heinemann
Ltd
VL = aR+h
(2)
dVL = nR:dh
(3)
Spheres VL = _$rh*(3R, - h)
(4)
dVL = nh(DT - h) dh
(5)
Horizontal
cylinders (h - RT)(DTh - h*)@ + Rising’
x(y)+;R;] dVL = 2L(DTh - h*)“* dh Equate Equation give:
J
(1) with Equations
(3), (5) a
i)to
PL
-rR+dh
dt =
1 (1)
AD&, 2gh +2-
‘@
volumes to obtain a relationship
Received4 September 1990
Right cylinders
Right cylinders
For liquids, the time-varying volumetric discharge rate, neglecting entrance effects? is given by the Bernoulli formula: dVL ~=-_=
between liquid volume and liquid head and then differentiating the resulting equations gives a geometric relationship for dVL/dh and h as follows:
APr PL
Spheres dt =
-ah(
1
(8)
1
(9)
ln
DT - h) dh
A P, A,CD 2gh + 2 PL
U2
J. Loss Prev. Process Ind., 799 1, Vol4, April
161
Liquid For t <
and gas
and K. S. Mudan
IF/C
l/2
ADCD
fi
liquids is the drain are availE, which rate or:
dVL _ z(h)=?
dpv
Pv
P,PvY
(28)
1
where:
(z-$yl _ (I-L)‘k-l)‘k]
y =
J. L. Woodward
The average volumetric discharge rate for simply Vu/t,, the initial volume divided by time. More usefully, since analytic solutions able for tr, we can find the liquid level, produces a discharge rate equal to the average
t > tc, subsonic flow
t-tt,=-VT
rates:
Average discharge rate for liquids
tcr sonic flow
(27) For
discharge
(39)
Substituting into Equation (39), Equation (1) evaluated at the average liquid head, E, the volume Equations (2), (4) and (6) evaluated at ho. and the results for tf, Equations (14), (15) and (16) with H = 0, gives the desired expressions for the average head as follows: Right cylinders
Equation (27) is simplified by transforming PT. and py to tank temperature TT using Equation (20) giving: For r <
t=
t, YM, ?fz,(k
I
1)
-
-312 du
U
TT __ = [F(t)]2 TTO
where Denom 2(Q
-
I)(I
(40)
= + R)@
E _=_
ho
t)]W--L)
(32)
2(2 X
= fi,[F(r)](k+‘)l(k-‘)
(33)
F(r)
= [l + At]-’
(34)
A =
%(k
- 4,QU + W3/2
+ ;QR~~
+ ;;R”’ Horizontal
PVO h(r)
-R
(41)
(30)
PT _ [F(t)]ww” PTU [F(
ho
(29)
where ti,, is the initial discharge rate given by Equation (23) at PLO, pvIo. Equation (29) is integrated to give:
!? =
[Cl + R) 1;_ R1/2]2
Spheres
rr/ro
1
1
E -=
cylinders
- ;;(R - ;)(l
+ R)3’2
(R = 0)
3 16 Q)(Q
-
l)‘/’
+ Q2
QW _
(Q
_
1)3i2
where:
(42)
-
1)
2M, The total amount
discharged
is:
M(t) = j-i c%(A)djl
(35)
M(t) = Mo[l - F(t)2”k-“]
(36)
or
The mass remaining
MT(t)
in the tank is:
= MO - M(t)
(37)
The time at which sonic flow begins, tc. is found by equating PT in Equation (31) with P, in Equation (22) and substituting F(t) from Equation (34) to give:
t, =
a[ (k+l)‘:2(&)““-;k - ‘1
(38)
Equation (40) is plotted in Figure I. This shows that for right cylinders, the average liquid head ranges between 1 and i of the initial height as the ratio R ranges from 0 b 0~. That is, for atmospheric pressure storage where R = 0, use i; = h,/4. For pressurized storage where the top pressure is large relative to the pressure contributed by the liquid head, the discharge rate is essentially constant as the liquid head varies and 6 = h,&?. For spheres, Equation (41) is plotted in Figure 2. In this case, the average liquid head is near $ over the entire range of R and Q values. For low initial levels, say Q -’ = 0.25, the average head ratio is always above 1 ranging from 0.54 to 0.675. For high initial levels, say $1 = 0.90, t he average head ratio ranges between 0.43 and 0.54. For horizontal cylinders at low top pressure, Equation (42) is plotted in Figure 3. This shows that the average head ratio in this case ranges between 0.35 and 0.45. Although it is not possible to obtain an analytic solution for the horizontal right cylinder with R # 0, it is possible to make use of a computer model to calculate the drainage time for this case. Figure 4 shows the results of such modelling, using the LEAKER
J. Loss Prev. Process
Ind.,
1991,
Vol4,
April
163
Liquid
and gas discharge
rates:
J. L. Woodward
and K. S. Mudan Avetage/lnitM
Liquid Ht
0.5
/ 0.4
t--_---
/
1
0.3
0.2 1
..
.. .- ..’
...
‘. .. ...
..
I
I --.....‘...-..
0.1
““‘.
0
0.1
0.2
0.3
0.4
...,....
0.5
0.6
0.7
0.8
0.9
1
HWTankDlpnwtsr Figure1 Liquid cylinders
0.7
head
for
average
discharge
rates
for
right Figure3 Liquid head for average discharge cylinders
rates for horizontal
Average/lnitiai Liquid Ht
,
o.6 /vera9eMtiai
Liquid Ht
I, -i 0.4 -
0.3 -..
,,,,,,_,,.,.,
0.3 t
0.2 _.......
0.2
t
0.1 -
0 Figure2 Liquid head for average discharge H,/&: 1.0.90; +, 0.75; *,0.50; 0.0.25
rates for spheres.
Average discharge rate for gases The average mass discharge rate for gases over the period of sonic flow is the total mass discharged up to time tC, divided by t,, or
numerator
of
Equation
164
J. Loss Prev. Process
(43)
Ind.,
is Equation
1991, Vol4,
1
1.5
2 RatfoR
2.5
3
3.5
4
Figure4 Liquid head for average discharge rates for horizontal cylinders determined by LEAKER model. H,/&: n , 0.75; +, 0.50; *, 0.25
model*. Figure 4 illustrates that the use of 0.5 as the ratio of average to initial liquid height is well justified for draining horizontal cylinders subject to at least a small amount of head pressure.
The
0.5
April
(36)
(38). evaluated at rer and t, is given by Equation Substituting these equations into Equation (43) gives the average rate as a ratio of the initial discharge rate as:
ii -zz
Liquid where
PR
is the
ratio
of ambient
to initial
and gas discharge
pressure,
pa/p,. Equation (44) is plotted in Figure 5, which shows that the average discharge rate varies between approximately $ of the initial discharge rate at high initial pressures to identically equal to the initial discharge rate if the initial discharge rate is just becoming subsonic (at the critical pressure ratio). For initial pressure ratios near the critical pressure ratio, the approximation that the average rate over the sonic flow period represents the average over the entire period becomes poor, but for P&P, > 10 the approximation is reasonable. For a specific case, M(r,) should be evaluated to verify the validity of the approximation of treating only the sonic flow regime.
Recommendations The analytic ations (14),
solutions for mass discharge rate, Equ(lS), (16) and (33), provide a simple
Avoragm/lnitlal
Discharge
1 ‘Handbook of Fluids in Motion’, (Eds N. P. Cheremisinoff Gupta) Ann Arbor Science, 1983, pp. 211-234 2 Woodward, J. L., 1. Loss Prevention 1990.3,33
AD B CD CPR DT F g h
I 1 l/PI3
10 = Pjo/Pamb
100
Figure5 Effect of initial pressure ratio on average/initial discharge rate (average over sonic flow period). k = : n , 1.2; +. 1.4; +, 1.6
and R
Nomenclature
g RT t tf 7-T. ” VL VT Z Y
Y
and K. S. Mudan
References
z MT. P, PT. APT PR
0.2
J. L. Woodward
method of calculating time-dependent discharge rates for several important special cases. Use Equations (40), (41), (42) and (44) or Figures 1-5 to find the average conditions for complete discharge of vessel contents. For screening level studies, find a representative discharge rate midway between the average and the initial rate.
h H k L
Rate
rates:
m. Pv
Area of orifice at puncture (I$) Defined by Equations (15). (16), (17) Discharge coefficient (dimensionless) Critical pressure ratio defined by Equation Tank diameter (m) Defined by Equation (34) Acceleration due to gravity (m s-*) Height of liquid (m)
(22)
Average height of liquid(m) Defined by Equation (13) Coefficient of non-ideal adiabatic expansion Length of horizontal cylinder(m) Discharge rate (kg s-l) Integral mass discharged (kg) Mass remaining in vessel (kg) Ambient pressure (Pa) Tank pressure above liquid (Pa) PT - P, Pressure ratio defined for Equation (44) Reciprocal initial level DT/ho Defined by Equation (11) or ideal gas constant Tank radius (m) Time(s) Time to drain vessel (s) Temperature of gas or liquid in vessel (K) Specific volume of vapour (m3 kg-‘) Volume of liquid in vessel (m3) Vessel volume (m3) Compressibility factor (dimensionless) Ratio of heat capacities at constant pressure and wnstant volume, C,/C, , for gas or vapour Liquid density (kg m-3) Vapour or gas density (kg mm3)
Subscripts 0 9
Initial value Critical point at which discharge becomes Vena contracta conditmns
J. Loss Prev.
Process
Ind.,
1991,
Vol4,
subsonic
April
165
Suite I70, Columbus,
Ohio, 43235,
Risk analysis often addresses the problem of calculating the time-varying discharge rate from an accidental puncture of a storage tank or process vessel. For selected cases, reasonably simple analytical solutions are presented for discharge rates. In addition to helping to validate more complex numerical solutions, these analytical solutions help in estimating a single representative discharge rate to drain completely a partially-filled vessel. To select a representative discharge rate, it is useful to first find the average discharge rate. Then, a useful rate for risk analysis can be chosen, typically around midway between the average and the initial discharge rate. Formulas are given for average rates as a function of the initial liquid level and vessel head pressure for spheres, horizontal cylinders and right cylinders. For gas and vapour discharge, the average discharge rate is given in terms of the initial pressure. (Keywords:
discfarge
rates; gas; liquid)
For risk analysis, a scenario which is frequently postulated is an accident which would drain the entire contents of a partially liquid filled or gas filled storage tank or process vessel. This event is inherently time varying since the discharge rate decays as the vessel pressure and/or liquid head above the puncture decreases. For screening level studies, one representative discharge rate is desired. This representative rate should be chosen to make an idealized steady state analysis roughly match the actual transient event A complex set of events must be consequences. modelled, including possible aerosol formation and rainout, pool spread and vaporization, vapour dispersion and toxic dosage or fire radiation dosage. After performing such transient analyses for a limited number of materials, our experience has been that a reasonable steady state discharge rate to use is roughly midway between the initial and the average discharge rates. Thus, it is useful to have simple analytic formulas for the average discharge rate, then apply the above rule of thumb to select a representative discharge rate.
Discharge rates for liquids
rir PL
Integrating
AP,
-ADCD 2gh(t) + 2 -
differential
095@4230/91/030161-06 @)I991 Butteworth-Heinemann
Ltd
VL = aR+h
(2)
dVL = nR:dh
(3)
Spheres VL = _$rh*(3R, - h)
(4)
dVL = nh(DT - h) dh
(5)
Horizontal
cylinders (h - RT)(DTh - h*)@ + Rising’
x(y)+;R;] dVL = 2L(DTh - h*)“* dh Equate Equation give:
J
(1) with Equations
(3), (5) a
i)to
PL
-rR+dh
dt =
1 (1)
AD&, 2gh +2-
‘@
volumes to obtain a relationship
Received4 September 1990
Right cylinders
Right cylinders
For liquids, the time-varying volumetric discharge rate, neglecting entrance effects? is given by the Bernoulli formula: dVL ~=-_=
between liquid volume and liquid head and then differentiating the resulting equations gives a geometric relationship for dVL/dh and h as follows:
APr PL
Spheres dt =
-ah(
1
(8)
1
(9)
ln
DT - h) dh
A P, A,CD 2gh + 2 PL
U2
J. Loss Prev. Process Ind., 799 1, Vol4, April
161
Liquid For t <
and gas
and K. S. Mudan
IF/C
l/2
ADCD
fi
liquids is the drain are availE, which rate or:
dVL _ z(h)=?
dpv
Pv
P,PvY
(28)
1
where:
(z-$yl _ (I-L)‘k-l)‘k]
y =
J. L. Woodward
The average volumetric discharge rate for simply Vu/t,, the initial volume divided by time. More usefully, since analytic solutions able for tr, we can find the liquid level, produces a discharge rate equal to the average
t > tc, subsonic flow
t-tt,=-VT
rates:
Average discharge rate for liquids
tcr sonic flow
(27) For
discharge
(39)
Substituting into Equation (39), Equation (1) evaluated at the average liquid head, E, the volume Equations (2), (4) and (6) evaluated at ho. and the results for tf, Equations (14), (15) and (16) with H = 0, gives the desired expressions for the average head as follows: Right cylinders
Equation (27) is simplified by transforming PT. and py to tank temperature TT using Equation (20) giving: For r <
t=
t, YM, ?fz,(k
I
1)
-
-312 du
U
TT __ = [F(t)]2 TTO
where Denom 2(Q
-
I)(I
(40)
= + R)@
E _=_
ho
t)]W--L)
(32)
2(2 X
= fi,[F(r)](k+‘)l(k-‘)
(33)
F(r)
= [l + At]-’
(34)
A =
%(k
- 4,QU + W3/2
+ ;QR~~
+ ;;R”’ Horizontal
PVO h(r)
-R
(41)
(30)
PT _ [F(t)]ww” PTU [F(
ho
(29)
where ti,, is the initial discharge rate given by Equation (23) at PLO, pvIo. Equation (29) is integrated to give:
!? =
[Cl + R) 1;_ R1/2]2
Spheres
rr/ro
1
1
E -=
cylinders
- ;;(R - ;)(l
+ R)3’2
(R = 0)
3 16 Q)(Q
-
l)‘/’
+ Q2
QW _
(Q
_
1)3i2
where:
(42)
-
1)
2M, The total amount
discharged
is:
M(t) = j-i c%(A)djl
(35)
M(t) = Mo[l - F(t)2”k-“]
(36)
or
The mass remaining
MT(t)
in the tank is:
= MO - M(t)
(37)
The time at which sonic flow begins, tc. is found by equating PT in Equation (31) with P, in Equation (22) and substituting F(t) from Equation (34) to give:
t, =
a[ (k+l)‘:2(&)““-;k - ‘1
(38)
Equation (40) is plotted in Figure I. This shows that for right cylinders, the average liquid head ranges between 1 and i of the initial height as the ratio R ranges from 0 b 0~. That is, for atmospheric pressure storage where R = 0, use i; = h,/4. For pressurized storage where the top pressure is large relative to the pressure contributed by the liquid head, the discharge rate is essentially constant as the liquid head varies and 6 = h,&?. For spheres, Equation (41) is plotted in Figure 2. In this case, the average liquid head is near $ over the entire range of R and Q values. For low initial levels, say Q -’ = 0.25, the average head ratio is always above 1 ranging from 0.54 to 0.675. For high initial levels, say $1 = 0.90, t he average head ratio ranges between 0.43 and 0.54. For horizontal cylinders at low top pressure, Equation (42) is plotted in Figure 3. This shows that the average head ratio in this case ranges between 0.35 and 0.45. Although it is not possible to obtain an analytic solution for the horizontal right cylinder with R # 0, it is possible to make use of a computer model to calculate the drainage time for this case. Figure 4 shows the results of such modelling, using the LEAKER
J. Loss Prev. Process
Ind.,
1991,
Vol4,
April
163
Liquid
and gas discharge
rates:
J. L. Woodward
and K. S. Mudan Avetage/lnitM
Liquid Ht
0.5
/ 0.4
t--_---
/
1
0.3
0.2 1
..
.. .- ..’
...
‘. .. ...
..
I
I --.....‘...-..
0.1
““‘.
0
0.1
0.2
0.3
0.4
...,....
0.5
0.6
0.7
0.8
0.9
1
HWTankDlpnwtsr Figure1 Liquid cylinders
0.7
head
for
average
discharge
rates
for
right Figure3 Liquid head for average discharge cylinders
rates for horizontal
Average/lnitiai Liquid Ht
,
o.6 /vera9eMtiai
Liquid Ht
I, -i 0.4 -
0.3 -..
,,,,,,_,,.,.,
0.3 t
0.2 _.......
0.2
t
0.1 -
0 Figure2 Liquid head for average discharge H,/&: 1.0.90; +, 0.75; *,0.50; 0.0.25
rates for spheres.
Average discharge rate for gases The average mass discharge rate for gases over the period of sonic flow is the total mass discharged up to time tC, divided by t,, or
numerator
of
Equation
164
J. Loss Prev. Process
(43)
Ind.,
is Equation
1991, Vol4,
1
1.5
2 RatfoR
2.5
3
3.5
4
Figure4 Liquid head for average discharge rates for horizontal cylinders determined by LEAKER model. H,/&: n , 0.75; +, 0.50; *, 0.25
model*. Figure 4 illustrates that the use of 0.5 as the ratio of average to initial liquid height is well justified for draining horizontal cylinders subject to at least a small amount of head pressure.
The
0.5
April
(36)
(38). evaluated at rer and t, is given by Equation Substituting these equations into Equation (43) gives the average rate as a ratio of the initial discharge rate as:
ii -zz
Liquid where
PR
is the
ratio
of ambient
to initial
and gas discharge
pressure,
pa/p,. Equation (44) is plotted in Figure 5, which shows that the average discharge rate varies between approximately $ of the initial discharge rate at high initial pressures to identically equal to the initial discharge rate if the initial discharge rate is just becoming subsonic (at the critical pressure ratio). For initial pressure ratios near the critical pressure ratio, the approximation that the average rate over the sonic flow period represents the average over the entire period becomes poor, but for P&P, > 10 the approximation is reasonable. For a specific case, M(r,) should be evaluated to verify the validity of the approximation of treating only the sonic flow regime.
Recommendations The analytic ations (14),
solutions for mass discharge rate, Equ(lS), (16) and (33), provide a simple
Avoragm/lnitlal
Discharge
1 ‘Handbook of Fluids in Motion’, (Eds N. P. Cheremisinoff Gupta) Ann Arbor Science, 1983, pp. 211-234 2 Woodward, J. L., 1. Loss Prevention 1990.3,33
AD B CD CPR DT F g h
I 1 l/PI3
10 = Pjo/Pamb
100
Figure5 Effect of initial pressure ratio on average/initial discharge rate (average over sonic flow period). k = : n , 1.2; +. 1.4; +, 1.6
and R
Nomenclature
g RT t tf 7-T. ” VL VT Z Y
Y
and K. S. Mudan
References
z MT. P, PT. APT PR
0.2
J. L. Woodward
method of calculating time-dependent discharge rates for several important special cases. Use Equations (40), (41), (42) and (44) or Figures 1-5 to find the average conditions for complete discharge of vessel contents. For screening level studies, find a representative discharge rate midway between the average and the initial rate.
h H k L
Rate
rates:
m. Pv
Area of orifice at puncture (I$) Defined by Equations (15). (16), (17) Discharge coefficient (dimensionless) Critical pressure ratio defined by Equation Tank diameter (m) Defined by Equation (34) Acceleration due to gravity (m s-*) Height of liquid (m)
(22)
Average height of liquid(m) Defined by Equation (13) Coefficient of non-ideal adiabatic expansion Length of horizontal cylinder(m) Discharge rate (kg s-l) Integral mass discharged (kg) Mass remaining in vessel (kg) Ambient pressure (Pa) Tank pressure above liquid (Pa) PT - P, Pressure ratio defined for Equation (44) Reciprocal initial level DT/ho Defined by Equation (11) or ideal gas constant Tank radius (m) Time(s) Time to drain vessel (s) Temperature of gas or liquid in vessel (K) Specific volume of vapour (m3 kg-‘) Volume of liquid in vessel (m3) Vessel volume (m3) Compressibility factor (dimensionless) Ratio of heat capacities at constant pressure and wnstant volume, C,/C, , for gas or vapour Liquid density (kg m-3) Vapour or gas density (kg mm3)
Subscripts 0 9
Initial value Critical point at which discharge becomes Vena contracta conditmns
J. Loss Prev.
Process
Ind.,
1991,
Vol4,
subsonic
April
165