# Figure 1: A pressure relief system in action.

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Figure 1: A pressure relief system in action.
Using Numerical Methods to Analyze the Sizing of Pressure Relief Devices CHEN 320 {Numerical Analysis} Group 8: Abbey Reisz Jacob Darst Gracie Rogers Daniel Irvin Alan Schultz Figure 1: A pressure relief system in action. Image Credit:

Outline “Oh, what a relief it is!” 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 Figure 2: Alka-Seltzer boy. Figure 3: An inside look at a pressure relief valve. Image Credit:

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 to save company assets, ensure 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 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. 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)

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:

Objective The purpose of relief sizing is to determine the proper discharge area of the relief device and diameter of the associated inlet and outlet 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. 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.. 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 relief system for a liquid or a gas. Info and Image Credit:

Fundamental Principles
Pressure relief systems (usually involves a valve) is used to control the limit or pressure in a system or vessel which can build up by a process upset, instrument, equipment failure, or fire. Spring-loaded pressure relief valve will be analyzed in this case. Pressure is relieved by allowing the pressurized fluid (may be liquid or gas) to flow from an auxiliary 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. Figures 11/12: Schematic diagrams of a conventional spring-loaded pressure relief valve. Info and Image Credit: Sizing Pressure-Relief Devices (original article provided

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

Fundamental Principles
Figure 14: The relief device sizing procedure involves these steps. Figure 15: The relief device sizing procedure involves these guidelines. Info and Image Credit: Sizing Pressure-Relief Devices (original article provided

“Popcorn in Slow Motion” Video.
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 Figure 16: Pericarp is the tough outer shell surrounding a popcorn kernel; Endosperm contains the trapped water. (Own creation). “Popcorn in Slow Motion” Video. Info and Image Credit:

Real WORLD Examples Pressure Cookers
Pressure cookers were extremely popular after WWII. Unfortunately the bulk of the manufacturers were shady. safety features used thin metals were cheaply made Figure 17: Piece of a pressure cooker after it has exploded. Figure 18: Photo of a pressure cooker disaster. Info and Image Credit:

Real WORLD Examples Pressure Cookers
Current pressure cookers have at least a triple safety feature system set up The first line of defense is the interlocking lid that makes it impossible to open the lid while the pressure cooker still has pressure. Pop up Pressure Indicator is a device on modern pressure cookers that show exactly when the selected pressure setting has been reached. Figure 19: Schematic diagram of pressure cooker. Figure 20: Pressure being released once gasket is open.  In the worse case scenario -- such as extreme over heating or over pressuring -- the gasket will be pushed out from an open slot in the rim of the lid allowing built-up steam to escape safely. Info and Image Credit

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:

Real WORLD Examples Fukushima
11th 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:

Figure 25: Water level decreasing, core becoming exposed
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. 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:

Real WORLD Examples Fukushima
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 followed shortly after. The superstructure of the reactor building was blown apart. There was no damage to the critical containment systems. Eventually the entire system was cooled by flooding with seawater. Figure 26: Pressure relief system. Info and Image Credit:

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 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. Figure 27/28/29: Typical safety valves used to relieve pressure. Figure 21 Image Credit:

𝑮 𝟐 =𝟐 𝑷𝒊 𝑷𝒊+𝒉 𝒗𝒅𝑷 𝒗 𝟐 Perry’s 7th Ed. (G is mass flux)
Numerical Methods with Analysis: Numerical Integration for Non-Ideal Flow 𝑷𝒊 𝑷𝒊+𝒉 𝒗𝒅𝑷~ 𝒉 𝟔 𝒗 | 𝑷𝒊 +𝟒 𝒗| 𝑷𝒊+ 𝒉 𝟐 +𝒗 | 𝑷𝒊+𝒉 Simpsons Rule 𝑮 𝟐 =𝟐 𝑷𝒊 𝑷𝒊+𝒉 𝒗𝒅𝑷 𝒗 𝟐 Perry’s 7th 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 until next step results in lower mass flux Convert G (mass flux) to W (mass flow rate) 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. Info Credit: Applied Numerical Methods for Engineers with MatLab, 3rd Edition; Steven Chapra

Numerical Methods with Analysis: Numerical Integration Step One
𝑃𝑖 𝑃𝑖+ℎ 𝑣𝑑𝑃 ~ ℎ 6 𝑣 | 𝑃𝑖 +4 𝑣| 𝑃𝑖+ ℎ 2 +𝑣 | 𝑃𝑖+ℎ ~ | ∗ | | | ∗ | | ~ 3145 𝐺 2 =2 𝑃𝑖 𝑃𝑖+ℎ 𝑣𝑑𝑃 𝑣 2 = 2(3145/.00949^2) = EXAMPLE DATA: Area of pipe is m2 Figure 31 (Table 1): Calculation table for the first step of integration (own creation). Info Credit: Applied Numerical Methods for Engineers with MatLab, 3rd Edition; Steven Chapra

Numerical Methods with Analysis: Numerical Integration Step two
𝑃𝑖 𝑃𝑖+ℎ 𝑣𝑑𝑃 ~ ℎ 6 𝑣 | 𝑃𝑖 +4 𝑣| 𝑃𝑖+ ℎ 2 +𝑣 | 𝑃𝑖+ℎ ~ | ∗ | | ~ 𝟑𝟑𝟑𝟓.𝟕 𝐺 2 =2 𝑃𝑖 𝑃𝑖+ℎ 𝑣𝑑𝑃 𝑣 2 = 2( / ^2) = Integral from the second step ( ) must be added to the integral from the first step to yield the total integral. ( )= Figure 32 (Table 2): Calculation table for the second step of integration (own creation). Info Credit: Applied Numerical Methods for Engineers with MatLab, 3rd Edition; Steven Chapra

Numerical Methods with Analysis: Numerical Integration All Steps
Figure 33 (Table 3): Calculation table for the all steps of integration (own creation). 𝑊=𝐺∗ 𝐴 𝑝𝑖𝑝𝑒 ∗𝐾 = * * .975 W ~ kg/s Info Credit: 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… function [ output ] = sizer(w,pbub,pmawp,tproc,ps,mw,gamma,ispring,isfired) %w = mass flow rate (kg/s) %pbub = built up backpressure (%) %pmawp = maximum allowable working pressure (barg) %tproc = process temp (K) %ps = set pressure (barg) %mw = molecular weight (g/mole) %gamma = heat capacity ratio %z= real gas compressibility factor %isspring(value is 1 if spring operated, value 0 if balanced-bellows) %isfired (value is 0 if piping, 1 if unfired, 2 if fire rated) % patm = 1.013; gc=1; R=8314; Kd=.975;%assumed z=1;%assumed if isfired == 0 pmax = 1.33*pmawp; elseif isfired == 1 pmax = 1.1*pmawp; else pmax = 1.21*pmawp; end pi = pmax+patm; backpress = (pbub/100)*ps pbubabsperc= (backpress+patm)/(pi) pbubabsperc = pbubabsperc * 100 if gamma ==1.1 if pbubabsperc>0 && pbubabsperc<60 kb=1; kb= E-6*pbubabsperc^3 elseif gamma==1.3 kb= E-6*pbubabsperc^3; elseif gamma==1.5 kb= E-6*pbubabsperc^3; kb= E-6*pbubabsperc^3; c1=((2/(gamma+1))^((gamma+1)/(gamma-1))); c2=c1*(gamma*gc)/R c=sqrt(c2); area = (w/((c*Kd*pi*kb))*sqrt((tproc*z)/mw)); area=area/100000; d=sqrt(4*area/ ); output=d; Figure 34: Part 1 of Matlab Program (own creation) 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. 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) 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. sizer(50,.7,8,473,7,100,1.3,1,1) Figure 38: Command window input. (own creation) Info Credit: Sizing Pressure-Relief Devices (original article provided)

Conclusions Figure 39/40: Pressure relief systems in action.
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. Now the pressure relief device can be used in a more complex system (shown below) in order to ensure safety and integrity of processes! Info and Image Credit: Sizing Pressure-Relief Devices (original article provided) Figure 39/40: Pressure relief systems in action.

Future Research Recommendations
The same analysis could be used to size a pressure relief device for liquids (same original article 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 relief 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. Figure 41/42/43: A spring-loaded pressure relief system; starting point for finding the dynamic behavior of a pressure relief valve. Image Credit: