Using Numerical Methods to Analyze the Sizing of Pressure Relief Devices

CHEN 320 {Numerical Analysis}

Group 8: Abbey Reisz Jacob Darst Gracie Rogers Daniel IrvinAlan Schultz

USING NUMERICAL METHODS TO ANALYZE THE

SIZING OF PRESSURE RELIEF

DEVICES

Figure 1: A pressure relief system in action.Image Credit:

http://www.equityeng.com/sites/default/files/pressure-relief.jpg1

Importance 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

OUTLINE

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

Figure 2: Alka-Seltzer boy.

Figure 3: An inside look at a pressure relief valve.2

Though pressure re l ie f devices may never be act ivated, they must be designed and sized to function correctly no matter what the operat ional s i tuat ion in order to save company assets, ensure operational excellence, and maintain faci l ity 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 secondary relief source called a bypass valve that returns all or part of the fluid back to a storage reservoir or the inlet of a pump or gas compressor. This protects the equipment from excessive pressure.

IMPORTANCE

Info and Image Credit:Sizing Pressure-Relief Devices (original article provided)

http://blog.iqsdirectory.com/wp-content/uploads/files/pressure%20vessels%206.jpghttp://www.forensic.cc/images/04728m.jpg

http://www.pveng.com/ASME/ASMEComment/ExternalPressure/image002.png

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.

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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. 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

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The purpose of re l ief s iz ing is to determine the proper discharge area of the rel ief device and diameter of the associated inlet and out let piping. 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.

OBJECTIVE

Info and Image Credit:http://www.generant.com/images/products/High-Pressure-Relief-Valves.jpg


http://judithcurry.com/2012/12/22/the-goldilocks-principle/

Figure 9: Like in the Goldilocks

story, pressure, relief valves need

to be the right size.

Sizing based on overpressure scenarios, special considerations (such as plugging), volumetric and mass flow rate, materials, and type of phase.

Through the use of numerical methods, we can analyze the procedure and determine how to gather data to size a rel ief system for a l iquid or a gas.

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Figure 10: Different sizes of

apparatus in pressure relief

devices..

Pressure rel ief systems (usually involves a valve) is used to control the l imit or pressure in a system or vessel which can build up by a process upset, instrument, equipment fai lure, or fire.

Spring-loaded pressure rel ief valve wil l be analyzed in this case.

Pressure is rel ieved by allowing the pressurized fluid (may be l iquid or gas) to flow from an auxil iary 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 l imit.

FUNDAMENTAL PRINCIPLES

Info and Image Credit:Sizing Pressure-Relief Devices (original article provided

http://en.wikipedia.org/wiki/Relief_valvehttp://www.askmehelpdesk.com/attachments/plumbing/18576d1239819237-cold-water-pressure-relief-valve-leaking-pressure-

relief-valve-2.jpg

Figures 11/12: Schematic diagrams of a conventional spring-loaded pressure relief valve.

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Maximum allowable working pressure is a primary 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.


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.


http://www.enggcyclopedia.com/wp-content/uploads/2011/04/PID-typical-arrangement-for-pressure-safety-valves.jpg7




Figure 14: The relief device sizing procedure involves these steps.

Figure 15: The relief device sizing procedure involves these guidelines.

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REAL WORLD EXAMPLES

Figure 16: Pericarp is the tough outer shell surrounding a popcorn kernel; Endosperm contains the trapped water. (Own creation).

“Popcorn in Slow Motion” Video.

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

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

Pop Corn

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REAL WORLD EXAMPLES

Pressure cookers were extremely popular after WWII. Unfortunately the bulk of the manufacturers were shady.

safety features used thin metals were cheaply made

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/

Figure 17: Piece of a pressure cooker after it has exploded.

Figure 18: Photo of a pressure cooker disaster.

Pressure Cookers

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Current pressure cookers have at least a triple safety feature system set up

The first l ine of defense is the interlocking l id that makes i t impossible to open the l id whi le the pressure cooker st i l l has pressure.

Pop up Pressure Indicator is a device on modern pressure cookers that show exactly when the selected pressure sett ing has been reached.

Info and Image Credithttp://theladyisachef.com/2013/01/07/food-science-pressure-cooking/

http://missvickie.com/images/cutout.jpg

REAL WORLD EXAMPLES

In the worse case scenario -- such as extreme over heating or over pressuring -- the gasket wi l l be pushed out from an open slot in the r im of the l id al lowing bui l t-up steam to escape safely.

Figure 19: Schematic diagram of pressure cooker.

Pressure Cookers

Figure 20: Pressure being released once

gasket is open.

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Most res ident ia l tanks ho ld 40 to 60 ga l lons

Stee l tanks are tested to hand le 300 ps i

Other water heater par ts inc lude: 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 re l ie f va lve that keeps the tank f rom exp lod ing A sacrificial anode rod to help keep

the steel tank from corroding

REAL WORLD EXAMPLE

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:

Http://www.merchantcircle.com/blogs/JC.Huggins.Home.Inspections.Tucson.AZ.520-777-9558/2008/9/Check-Your-Water-Heater-Pressure-Relief-Valve-Annually/114726

http://www.watts.com/pages/support/tp.asp?catId=64

Water Heater

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REAL WORLD EXAMPLES11th March 2011: 2:46 The Earthquake struck. Diesel generators turned on and started circulating water to keep reactor cores cool. 11th March 2011: 3:41 The tsunami arrives. The plant is disconnected from mains electricity, and the diesel generators are destroyed. The battery 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 lowers the level of water in the reactor vessel, eventually exposing the core material.

Figure 23: Inside look at Fukushima nuclear reactor. Info and Image Credit:

http://protonsforbreakfast.wordpress.com/2011/04/19/fukushima-what-happened/

Fukushima

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The liquid water in the core becomes a boiling mass, the foam provides some cooling. So even at 50% exposure the core is safe. Further 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 Info and Image Credit:http://protonsforbreakfast.wordpress.com/2011/04/19/fukushima-what-happened/

FukushimaREAL WORLD EXAMPLES

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The pressure is over 8 bar in a container designed for 4 bar. The operators decide to release the gas and so lower the pressure. 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 12th and the hydrogen explosion fol lowed shortly after. The superstructure of the reactor bui lding was blown apart. There was no damage to the critical containment systems. Eventual ly the entire system was cooled by flooding with seawater.

Info and Image Credit:http://protonsforbreakfast.wordpress.com/2011/04/19/fukushima-what-happened/

Figure 26: Pressure relief system.

FukushimaREAL WORLD EXAMPLES

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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 cross-sectional 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.

NUMERICAL METHODS METHODOLOGY

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.jpghttp://upload.wikimedia.org/wikipedia/commons/9/93/Relief_valve01.jpg

Figure 27/28/29: Typical safety valves used to relieve pressure.

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Simpsons Rule

Perry’s 7 t h Ed. (G is mass flux)

First Step: Estimate integral and calculate mass flux. Second Step: Estimate integral , add to previous step integral , then

calculate mass flux. Repeat unti l next step results in lower mass flux Convert G (mass flux) to W (mass flow rate)

NUMERICAL METHODS WITH ANALYSIS:NUMERICAL INTEGRATION FOR NON-IDEAL FLOW

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

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.

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NUMERICAL METHODS WITH ANALYSIS:NUMERICAL INTEGRATION STEP ONE

Pressure in kPa Specific Volume in m^3/kg Integral m^2/s^2 Mass Flux kg/s.m^25500 0.0094775338 0.0097065176 0.009949 3145.54 7972.34

• ~ 3145

• ==7972.24



• EXAMPLE DATA: Area of pipe is .003278 m2

Figure 31 (Table 1): Calculation table for the first step of integration (own creation).

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NUMERICAL METHODS WITH ANALYSIS:NUMERICAL INTEGRATION STEP TWO• ~ • = =10364.8

• 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

Pressure in kPa Specific Volume in m^3/kg Integral m^2/s^2 Mass Flux kg/s.m^25500 0.0094775338 0.0097065176 0.009949 3145.54 7972.345014 0.0102074852 0.010984 6480.71 10364.8



Figure 32 (Table 2): Calculation table for the second step of integration (own creation).

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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^25176 0.009949 3145.54 7972.344852 0.010984 6480.71 10364.84528 0.011091 10000.6 12750.84204 0.011791 13704.9 14040.63880 0.012604 17653.6 14908.13556 0.013558 21887.5 154323232 0.014703 26459.1 15645.82908 0.016527 31496.2 15186.2

Info Credit:http://www.aiche.org/resources/chemeondemand/webinars/pressure-relief-valve-sizing-equations-

basisApplied Numerical Methods for Engineers with MatLab, 3rd Edition; Steven Chapra

= 15645.8 * .003278 * .975

W ~ 50.0048 kg/s

Figure 33 (Table 3): Calculation table for the all steps of integration (own creation).

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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)

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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).

• Kb varies with different values of gamma.

Info and Image Credit:Sizing Pressure-Relief Devices (original article provided)22



Figure 37: Part 2 of Matlab Program (own creation)23

• The following equations give the steps in calculating the pressure relief size diameter by hand. They are the basis for MatLab calculations.

VALIDATING RESULTSHOW 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.

Info Credit:Sizing Pressure-Relief Devices (original article provided)

Figure 38: Command window input. (own creation)24

Through the use of numerical methods, we can determine the mass flux of flow through Simpson’s Rule. We can then re late this value to the mass flow rate with the cross-sect ional area of the pipe and correct ion factors.

This mass flow rate was p lugged into a decis ion tree implemented in MatLab (which inc ludes i terat ion f rom graphs and tables) to generate an in i t ia l approximat ion of the minimum area for a spr ing-operated re l ie f valve in s ingle phase gas flow.

Now the pressure re l ie f device can be used in a more complex system (shown below) in order to ensure safety and integr i ty of processes!

CONCLUSIONS


http://www.plastomatic.com/relief-diagram.gifhttp://www.extension.org/sites/default/files/Bottom%20heat%20system%20piping%20schematic.gif

Figure 39/40: Pressure relief systems in action.25

The same analysis could be used to size a pressure rel ief device for l iquids (same original art icle could be used).

A numerical analysis for different kind of fluids ( ie Newtonian and Non-Newtonian fluids)

A numerical analysis for non-ideal fluids Research on the dynamic behavior of a pressure rel ief valve

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

FUTURE RESEARCH RECOMMENDATIONS

Image Credit: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.

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Using Numerical Methods to Analyze the Sizing of Pressure Relief Devices

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