USING NUMERICAL
METHODS TO ANALYZE
THE SIZING OF
PRESSURE RELIEF
DEVICES
CHEN 320
{Numerical
Analysis}
Group 8:
Abbey Reisz
Jacob Darst
Gracie Rogers
Daniel Ir vin
Alan Schultz
Figure 1: A pressure relief system in action.
1
Image Credit:
http://www.equityeng.com/sites/default/files/pressure-relief.jpg
OUTLINE






Figure 2: Alka-Seltzer boy.




Figure 3: An inside look at a pressure relief valve.
2
Impor tance
Objective
Fundamental Principles
Real World Examples
Numerical Methods Methodology
Numerical Methods: Numerical
Integration to Find Flow
Numerical Methods: Sizing
Analysis Decision Tree with
Iteration Graphs
Validating Results and How Did
We Improve the Paper?
Conclusions
Future Research
Recommendations
Image Credit:
http://www.debiderryberry.com/wp-content/uploads/2012/01/speedy.jpg
http://www.ctgclean.com/tech-blog/wp-content/uploads/Backpressure-Limiting-Valve.jpg
IMPORTANCE
 Though pressure relief devices
may never be activated, they
must be designed and sized to
function correctly no matter what
the operational situation in order
t o s a v e c o m p a ny a s s e t s , e n s u r e
operational
excellence,
and
maintain
facility
and
worker
safety.
 Equipment must be protected
against being subjected to an
internal vacuum that is lower
than
the
equipment
can
withstand. This protects the
system
from
low
pressure
suction forces.
 Can be used as a secondar y
relief source called a bypass
valve that returns all or part of
the fluid back to a storage
reser voir or the inlet of a pump
or gas compressor. This protects
the equipment from excessive
pressure.
3
Figures 4/5/6: Tanks that have
partially collapsed due to a failed
pressure relief valve. The negative
pressure led to a vacuum that
sucked the sides of the tank
inward.
Info and Image Credit:
Sizing Pressure-Relief Devices (original article provided)
http://blog.iqsdirectory.com/wp-content/uploads/files/pressure%20vessels%206.jpg
http://www.forensic.cc/images/04728m.jpg
http://www.pveng.com/ASME/ASMEComment/ExternalPressure/image002.png
IMPORTANCE
Figures 7/8:
Typical Piping and Instrumentation Diagrams with pressure relief valves (labeled
PSV for pressure safety valves). These act as barriers until there is a deviation in
pressure, then they will open to be released into the flare for gases and drain for
liquids. Important for operational integrity and safety.
4
Image Credit:
http://www.enggcyclopedia.com/wp-content/uploads/2011/04/PID-typical-arrangement-for-pressure-safety-valves.jpg
http://www.enggcyclopedia.com/wp-content/uploads/2011/04/PID-typical-arrangement-for-3-phase-separator-vessels.jpg
OBJECTIVE
 T h e p urp o s e o f r el i ef s i z i ng i s to
d eter m i n e t h e p ro p er d i s c h ar g e
a r e a o f t h e r e l i ef dev i c e an d
d i am eter o f t h e a s s o c i a ted i nl et
a n d o u t l et p i p i n g .
 Relief
devices
cannot
be
undersized because
 high pressure and equipment
failure may occur.
 Relief
devices
cannot
be
oversized because
it may become unstable during
operation and will fail. This cost for
an oversized relief valve is also
more than for the appropriate size
of relief valve.
Figure 9: Like in the
Goldilocks story,
pressure, relief valves
need to be the right
size.
Figure 10: Different
sizes of apparatus in
pressure relief
devices..
 S i z i ng b a s ed o n o v erpr e ssur e s c e n ar i o s , s p ec i al c o n s i d er at i o n s ( suc h a s pl ug g i ng ) ,
v o l umet r ic a n d m a s s f l o w r a te , m a te r i a l s , a n d t y p e o f p h a s e .
 T hro ug h t h e u s e o f num er i c al m et ho d s , w e c an a n al y z e t h e p ro c ed ur e a nd d ete r m i n e
h o w to g a t h e r d a t a to s i z e a r e l i e f s y s tem f o r a l i q ui d o r a g a s .
5
Info and Image Credit:
http://www.generant.com/images/products/High-Pressure-Relief-Valves.jpg
http://www.enggcyclopedia.com/wp-content/uploads/2011/04/PID-typical-arrangement-for-pressure-safety-valves.jpg
http://judithcurry.com/2012/12/22/the-goldilocks-principle/
FUNDAMENTAL PRINCIPLES
 Pressure
relief
systems
(usually involves a valve) i s
used to control the limit or
pressure in a system or
vessel whi ch can build up by
a process upset, instrument,
equipment failure, or fire.
 Spring-l oaded pressure relief
valve will be analyzed in this
case.
 Pressure
is
relieved
by
allowing
the
pressuri zed
fluid (may be liquid or gas)
to fl ow from an auxiliar y
passage out of the system.
 Designed to open at a
predetermined set pressure
to protect equipment from
being subjected to pressures
that exceed their design
limit.
6
Figures 11/12: Schematic diagrams of a conventional
spring-loaded pressure relief valve.
Info and Image Credit:
Sizing Pressure-Relief Devices (original article provided
http://en.wikipedia.org/wiki/Relief_valve
http://www.askmehelpdesk.com/attachments/plumbing/18576d1239819237-cold-water-pressure-relief-valve-leaking-pressure-relief-valve-2.jpg
FUNDAMENTAL PRINCIPLES
 Maximum allowable working
pressure is a primar y parameter
when sizing a pressure vessel.
Typically, relief device’s at set to
open at the MAWP.
 Maximum allowable pressure at
the top of the vessel at a
designated temperature.
 MAWT and MAWP related by
thermodynamics; strength of
metal is reduced. These
parameters are in place to
ensure the most severe case
possible has be considered.
7
Figure 13: ASME Boiler and Pressure Vessel Code Section
VIII sets out requirements for standard pressure vessels
(left) and the relief valves protecting them (right) as a
percentage of the maximum allowable working pressure.
Info and Image Credit:
Sizing Pressure-Relief Devices (original article provided
http://www.enggcyclopedia.com/wp-content/uploads/2011/04/PID-typicalarrangement-for-pressure-safety-valves.jpg
FUNDAMENTAL PRINCIPLES
Figure 14: The relief device sizing procedure
involves these steps.
8
Figure 15: The relief device sizing procedure
involves these guidelines.
Info and Image Credit:
Sizing Pressure-Relief Devices (original article provided
http://www.enggcyclopedia.com/wp-content/uploads/2011/04/PID-typical-arrangement-for-pressure-safety-valves.jpg
REAL WORLD EXAMPLES
Pop Corn
 As the kernel heats up,
water begins to expand
 At 212 °F the water
turns
into
steam
building up pressure
inside the pericarp
 The kernel continues to
heat to about 347 °F.
 The pericarp is much
stronger than that of
all other corn kernels
and is able to retain
this pressurized steam
up to 135 psi, bursting
the hull open.
 As it explodes, steam
inside the kernel is
released
9
Figure 16: Pericarp is the tough outer shell surrounding a popcorn
kernel; Endosperm contains the trapped water. (Own creation).
“Popcorn in Slow Motion” Video.
http://www.popcorn.org/AboutUs/Media/PopVideos/tabid/114/Default.aspx
Info and Image Credit:
http://www.carolina.com/teacher-resources/Interactive/the-science-of-popcorn/tr23952.tr
REAL WORLD EXAMPLES
Pressure Cookers
 Pressure cooker s were extremely popular af ter WWII .
 Unfor tunately the bulk of the manufacturer s were shady.
 safety features
 used thin metals
 were cheaply made
Figure 17: Piece of a pressure cooker
after it has exploded.
10
Figure 18: Photo of a pressure cooker disaster.
Info and Image Credit:
http://gdb.voanews.com/578C9967-6B9A-4249-BE63-827AB899FC46_mw1024_n_s.jpg
http://theladyisachef.com/2013/01/07/food-science-pressure-cooking/
REAL WORLD EXAMPLES
Pressure Cookers
 C u r r e n t p r e s s ur e c o o ke r s h av e a t l e a s t a
t r i p le s a f et y f e a t u r e s y s tem s et u p
 The first line of defense is the
i n te r l o c ki n g l i d t h a t m a ke s i t i m po s si bl e
to o p e n t h e l i d w h i l e t h e p r e s s ur e c o o ke r
s t i l l h a s p r e s s ur e .
 Po p u p P r e s s ur e I n d i c a to r i s a d ev i c e o n
m o d e r n p r e s s ur e c o o ke r s t h a t s h o w
ex a c t l y w h e n t h e s e l e c te d p r e s s ur e
s et t i ng h a s b e e n r e a c h e d.
Figure 19: Schematic diagram of pressure cooker.
 I n t h e w o r s e c a s e s c e n a r i o - - s u c h a s ex t r e m e
o v e r h e a t i n g o r o v e r p r e s s u r in g - - t h e g a s ket
will be pushed out from an open slot in the
r i m o f t h e l i d a l l ow i n g b u i l t - up s te a m to
e s c a p e s a f e l y.
11
Figure 20: Pressure being
released once gasket is
open. 
Info and Image Credit
http://theladyisachef.com/2013/01/07/food-science-pressure-cooking/
http://missvickie.com/images/cutout.jpg
REAL WORLD EXAMPLE
Water Heater
 Most residential tanks hold 40 to 60
gallons
 Steel tanks are tested to handle 300
psi
 Other water heater parts include:
 A dip tube to let cold water into the
tank
 A pipe to let hot water out of the tank
 A thermostat to control the
temperature of the water inside the
tank
 Heating elements similar to those
inside an electric oven
 A drain valve that allows you to drain
the tank to replace the elements, or to
move the tank
 A temperature or pressure relief valve
that keeps the tank from exploding
 A sacrificial anode rod to help keep
the steel tank from corroding
Figure 21/22: The Temperature /Pressure relief valves used on residential
water heaters are designed relieve on pressure at 150 psig and/or
temperature at 210 °F. The causes of discharge can be thermal expansion,
excess system pressure, low temperature relief, too high a setting on the
water heater, or something in the water heater causing excess
temperatures in the heater.
WARNING: Temperature and Pressure Relief Valves should be inspected AT LEAST ONCE EVERY THREE YEARS, to ensure that the product has not been
affected by corrosive water conditions. Certain naturally occurring conditions may corrode the valve or its components over time, rendering the valve
inoperative. FAILURE TO REINSPECT THIS VALVE AS DIRECTED COULD RESULT IN UNSAFE TEMPERATURE OR PRESSURE BUILD-UP WHICH CAN RESULT IN
SERIOUS INJURY OR DEATH AND/OR SEVERE PROPERTY DAMAGE.
Info and Image Credit:
12
Http://www.merchantcircle.com/blogs/JC.Huggins.Home.Inspections.Tucson.AZ.520-777-9558/2008/9/Check-Your-Water-Heater-Pressure-Relief-ValveAnnually/114726
http://www.watts.com/pages/support/tp.asp?catId=64
REAL WORLD EXAMPLES
Fukushima
 11th M arc h 2011: 2:46 The
Ear thquake struck . Diesel
generator s turned on and star ted
circulating water to keep reactor
cores cool.
 11th M arc h 2011: 3:41 The
tsunami arrives. The plant is
disconnected from mains
electricity, and the diesel
generator s are destroyed. The
batter y powered cooling system
turns on.
 Reactor#1 4:36: The batteries
failed. The remaining cooling
method was to discharge steam
into the ‘wet well’. This provides
cooling, but lower s the level of
water in the reactor vessel,
eventually exposing the core
material.
13
Figure 23: Inside look at Fukushima nuclear reactor.
Info and Image Credit:
http://protonsforbreakfast.wordpress.com/2011/04/19/fukushima-what-happened/
REAL WORLD EXAMPLES
Fukushima
 The liquid water in the core
becomes a boiling mass, the foam
provides some cooling. So even at
50% exposure the core is safe.
Fur ther loss of coolant is critical:

At 33% exposure, the temperature of the
central part of the core exceeds 900 °C

At 25% exposure, the temperature of the
central part of the core exceeds 1200 °C

The core was exposed for 27 hours and
the temperature rose to 2700 °C
Figure 24:
Schematic of
nuclear reactor
Figure 25: Water level decreasing, core becoming exposed
14
Info and Image Credit:
http://protonsforbreakfast.wordpress.com/2011/04/19/fukushima-what-happened/
REAL WORLD EXAMPLES
Fukushima
 The pressure is over 8 bar in a
c o n t ai n e r d e s i g n e d f o r 4 b a r. T h e
o p e r a to r s d e c i d e t o r e l e a s e t h e g a s a n d
s o l o w e r t h e p r e s s ur e.
 This will: release short-lived isotopes
into the atmosphere (b)
 Result in an explosion as the hydrogen
mixes with air.
 The core will remain contained with
little release of the long-lived
radioactive elements in the core.
 The pressure was release at 4:00 on
1 2 t h a n d t h e hy d r o g e n ex p l o s i o n
f o l l owe d s h o r t l y a f te r. T h e
s u p e r s t r uct ur e o f t h e r e a c to r b u i l d in g
w a s b l ow n a p a r t . T h e r e w a s n o d a m a g e
t o t h e c r it i c a l c o n t a in m e n t s y s te m s .
E v e n t u al l y t h e e n t i r e s y s te m w a s
c o o l e d b y f l o o d i n g w i t h s e aw a te r.
Figure 26: Pressure relief system.
15
Info and Image Credit:
http://protonsforbreakfast.wordpress.com/2011/04/19/fukushima-what-happened/
NUMERICAL METHODS
METHODOLOGY
 Numerical Integration to find
flow
 The mass flux will be
calculated with numerical
integration (Simpson’s
Rule) and related to mass
flow rate.
 Cross-sectional area of pipe
is not necessarily crosssectional area of the
pressure relief system.
 Sizing Analysis Decision Tree
 Correction factors will be
found using iterative graph.
 Mass flux from numerical
integration will aid in
finding sizing area for
pressure relief device.
16
Figure 27/28/29: Typical safety
valves used to relieve pressure.
Image Credit:
http://upload.wikimedia.org/wikipedia/en/b/ba/Valvole_di_sicurezza_di_grandi_dimensioni.jpg
http://www.netherlocks.com/wp-content/uploads/2012/07/applications-psv.jpg
http://upload.wikimedia.org/wikipedia/commons/9/93/Relief_valve01.jpg
NUMERICAL METHODS WITH ANALYSIS:
NUMERICAL INTEGRATION FOR NON-IDEAL FLOW
𝑷𝒊+𝒉
𝑷𝒊
𝒗𝒅𝑷~
𝑮𝟐 = 𝟐
𝒉
𝟔
𝒗| 𝑷𝒊 + 𝟒𝒗| 𝑷𝒊+ 𝒉 + 𝒗| 𝑷𝒊+𝒉
Simpsons Rule
𝟐
𝑷𝒊+𝒉
𝒗𝒅𝑷
𝑷𝒊
𝒗𝟐
P e r r y ’s 7 t h E d . ( G i s m a s s f l ux )
 F i r s t S te p : E s t i ma te i n te g r a l a n d c a l c ul a te m a s s f l u x .
 S e c o n d S te p : E s t i m a te i n te g r a l , a d d to p r ev i o us s te p i n te g r a l , t h e n c a l c ula te m a s s
f l ux .
 Re p e a t u n t i l n ex t s te p r e s u l t s i n l o w e r m a s s f l ux
 C o nv e r t G ( m a s s f l ux ) to W ( m a s s f l o w r a te )
Figure 30: a) Graphical depiction of Simpson’s 1/3 Rule: It
consists of taking the area under a parabola connecting
three points. b) Graphical depiction of Simpson’s 3/8 rule:
It consists of taking the area under a cubic equation
connection four points.
17
Info Credit:
http://www.aiche.org/resources/chemeondemand/webinars/pressure-relief-valve-sizing-equations-basis
Applied Numerical Methods for Engineers with MatLab, 3rd Edition; Steven Chapra
NUMERICAL METHODS WITH ANALYSIS:
NUMERICAL INTEGRATION STEP ONE
•
𝑃𝑖+ℎ
𝑣𝑑𝑃
𝑃𝑖
~
ℎ
6
𝑣|𝑃𝑖 + 4𝑣|𝑃𝑖+ℎ + 𝑣|𝑃𝑖+ℎ ~
2
60
6
.009949|5176 + 4 ∗
• EXAMPLE DATA: Area of pipe is .003278 m2
Pressure in kPa Specific Volume in m^3/kg Integral m^2/s^2 Mass Flux kg/s.m^2
5500
0.009477
5338
0.009706
5176
0.009949
3145.54
7972.34
Figure 31 (Table 1): Calculation table for the first step of integration (own creation).
18
Info Credit:
http://www.aiche.org/resources/chemeondemand/webinars/pressure-relief-valve-sizing-equations-basis
Applied Numerical Methods for Engineers with MatLab, 3rd Edition; Steven Chapra
NUMERICAL METHODS WITH ANALYSIS:
NUMERICAL INTEGRATION STEP TWO
•
𝑃𝑖+ℎ
𝑣𝑑𝑃
𝑃𝑖
60
6
~
ℎ
6
𝑣|𝑃𝑖 + 4𝑣|𝑃𝑖+ℎ + 𝑣|𝑃𝑖+ℎ ~
2
.010984|4852 + 4 ∗.010984|4852+60 +.010984|4852+60 ~ 𝟑𝟑𝟑𝟓. 𝟕
2
𝑃𝑖+ℎ
𝑣𝑑𝑃
𝑃𝑖
𝑣2
=10364.8
•
𝐺2
•
Integral from the second step (3335.17) must be added to the integral from the
first step to yield the total integral. (3335.17+3145.54)=6480.71
=2
= 2(6480.71/.010984^2)
Pressure in kPa Specific Volume in m^3/kg Integral m^2/s^2 Mass Flux kg/s.m^2
5500
0.009477
5338
0.009706
5176
0.009949
3145.54
7972.34
5014
0.010207
4852
0.010984
6480.71
10364.8
Figure 32 (Table 2): Calculation table for the second step of integration (own creation).
19
Info Credit:
http://www.aiche.org/resources/chemeondemand/webinars/pressure-relief-valve-sizing-equations-basis
Applied Numerical Methods for Engineers with MatLab, 3rd Edition; Steven Chapra
NUMERICAL METHODS WITH ANALYSIS:
NUMERICAL INTEGRATION ALL STEPS
Pressure in kPa Specific Volume in m^3/kg Integral m^2/s^2 Mass Flux kg/s.m^2
5176
0.009949
3145.54
7972.34
4852
0.010984
6480.71
10364.8
4528
0.011091
10000.6
12750.8
4204
0.011791
13704.9
14040.6
3880
0.012604
17653.6
14908.1
3556
0.013558
21887.5
15432
3232
0.014703
26459.1
15645.8
2908
0.016527
31496.2
15186.2
Figure 33 (Table 3): Calculation table for the all steps of integration (own creation).
𝑊 = 𝐺 ∗ 𝐴𝑝𝑖𝑝𝑒 ∗ 𝐾
= 15645.8 * .003278 * .975
W ~ 50.0048 kg/s
20
Info Credit:
http://www.aiche.org/resources/chemeondemand/webinars/pressure-relief-valve-sizing-equations-basis
Applied Numerical Methods for Engineers with MatLab, 3rd Edition; Steven Chapra
NUMERICAL METHODS WITH ANALYSIS:
SIZING ANALYSIS DECISION TREE
•
Pmax Equations:
•
Pi is absolute maximum pressure
Abs. max pressure=MARP+ Patm
•
Backpressure found by:
•
Constants are defined as…
Figure 34: Part 1 of Matlab Program (own creation)
21
Info and Image Credit:
Sizing Pressure-Relief Devices (original article provided)
NUMERICAL METHODS WITH ANALYSIS:
SIZING ANALYSIS DECISION TREE
• Kb varies with different values of gamma.
22
Figure 35/36: Use the plot above to determine the
backpressure correction factor, Kb, for conventional
spring-operated relief devices in a vapor service. It is
drawn using the equation and constants in Figure 35
(left).
Info and Image Credit:
Sizing Pressure-Relief Devices (original article provided)
NUMERICAL METHODS WITH ANALYSIS:
SIZING ANALYSIS DECISION TREE
• The following equations give the steps in calculating
the pressure relief size diameter by hand. They are
the basis for MatLab calculations.
Figure 37: Part 2 of Matlab Program (own creation)
23
Info and Image Credit:
Sizing Pressure-Relief Devices (original article provided)
VALIDATING RESULTS
HOW DID WE IMPROVE THE PAPER?
• Answer is in meters (.14m=14cm)
• Matches results in article
• Error could be due to rounding in
MatLab
• This numerical integration in
MatLab program improves on the
information in the article selected
because it relates mass flux to the
equations in the paper. It also takes
all the formulas in the paper and
gives a quick and easy program to
size pressure relief valves in an
efficient and optimized way.
Figure 38: Command window
input. (own creation)
24
Info Credit:
Sizing Pressure-Relief Devices (original article provided)
CONCLUSIONS
 Through the use of numerical methods, we can determine the mass flux of flow through
Simpson’s Rule. We can then relate this value to the mass flow rate with the cross-sectional
area of the pipe and correction factors.
 This mass flow rate was plugged into a decision tree implemented in MatLab (which includes
iteration from graphs and tables) to generate an initial approximation of the minimum area for
a spring-operated relief valve in single phase gas flow.
 N o w t h e p r e s s u r e r e l i e f d e v i c e c a n b e u s e d i n a m o r e c o m p l e x s y s t e m ( s h o w n b e l o w ) i n o r d e r to
ensure safety and integrity of processes!
Info and Image Credit:
Sizing Pressure-Relief Devices (original article provided)
http://www.plastomatic.com/relief-diagram.gif
http://www.extension.org/sites/default/files/Bottom%20heat%20system%20piping%20schematic.gif
Figure 39/40: Pressure relief systems in action.
25
FUTURE RESEARCH RECOMMENDATIONS
 T h e s a m e a n a l y s i s c o ul d b e u s e d to s i z e a p r e s s u r e r e l i e f d ev i c e f o r l i q ui d s ( s a m e
o r i g in a l a r t i c l e c o u l d b e u s e d ) .
 A n u m e r i c a l a n a l y s i s f o r d i f fe r e n t k i n d o f f l u i d s ( i e N ew to n i a n a n d N o n - N ew to ni a n
fluids)
 A n u m e r i c a l a n a l y s i s f o r n o n - i d e al f l u i d s
 Re s e a r c h o n t h e d y n a m i c b e h av i o r o f a p r e s s u r e r e l i ef v a l v e
 The article below could be used with the Runge-Kutta method to start with this
research; figuring out how the relief device behaves could optimize the type and size
of relief device chosen.
 http://www.simdut.com.br/Trabalhos/ENCIT-2008.pdf
Figure 41/42/43: A spring-loaded pressure relief system; starting point for finding the
dynamic behavior of a pressure relief valve.
26
Image Credit:
http://www.simdut.com.br/Trabalhos/ENCIT-2008.pdf