(46c) The Fire Case for Pressure Relief – Radiation Exposure Limited By Fuel Supply

In design of pressure relief systems, fire is one of the most common relief cases.  But in many situations, it is one of the most complicated cases when determining the relief rate.  

 Models for the Fire Case

 Both simple and robust models are used.  One classification groups the models into three categories.

• Rigorous fire models come in many variations ranging from models used in computational fluid dynamics to simpler engineering models.  Most are unsteady state.  They utilize the Stefan-Boltzmann equation to relate temperature of the fire to the heat flux absorbed by the equipment.  The correlations are often included with three-dimensional computational fluid dynamics (CFD) models which solve unsteady-state flow, heat and mass transfer.  Although powerful tools, their detail is a disadvantage in process engineering relief system design. 

• Heat Flux is limited by Equipment Exposed Area.  This model is the workhorse of the hydrocarbon process industries and is detailed in API Standard 521.  Implicitly developed for hydrocarbon pool fires on liquid filled vessels, this is a steady-state model that assumes ample fuel supply.  Based on correlations from actual pool fire data, the fire heat input (energy/time) used for relief rate generation is calculated from the wetted area of the liquid filled vessels times a pseudo heat flux (energy/time-area).   
• Heat Flux is limited by Fuel Supply.  This assumption is familiar to those using the traditional pipe flare radiation calculations, such as those published in API Standard 521.  Radiation (energy/time) is determined from the pipe flare mass flow (mass/time) at the lower heating value (energy/mass) as adjusted by a factor representing the amount of energy radiated.  Additionally, the model has been adapted for hydrocarbon pool fires on liquid filled vessels and is published in handbooks from the SFPE (Society of Fire Protection Engineers) and NFPA (National Fire Protection Association).  This steady-state model assumes that the fuel supply limits the heat flux to the equipment.  This paper describes application of this model for pool fires.  

 The Sizing of the Relief Valve for Fire

 The scenario is a liquid filled vessel exposed to a pool fire.  The radiant heat (heat/time) from the fire is absorbed by the equipment, and boils the liquid, generating vapor which is relieved through the pressure relief valve.  For this example, the vapor generation rate is assumed a simple function of heat input.

       Relief Rate Vapor Generation (mass/time) = heat input (heat energy/time)  / latent heat of the liquid (heat/mass)

 Pool Fire Model Description

 The assumption for the pool fire is that the surface area of the equipment available for heat absorption is large and the heat absorbed is limited by the amount of fuel in the pool.  The pool of flammables is formed from a spill at grade.  Some of the spilled material is drained away from the pool and some is burned. 

 At steady-state, the material balance:

                   Spill Rate [volume/time ] = Burn Rate [volume/time] + Drain Rate [volume/time]

 Since the amount of liquid fuel burning and the amount of draining are a function of the pool size, the pool will increase in size until the "equilibrium" pool size is reached which satisfies the material balance.

 The presentation will discuss the parameters used to calculate the spill rate, burn rate and drain rate.  Since both burn rate and drain rate are a function of pool size, determining the pool is a primary focus of this presentation.

•  The spill rate used in the balance is the liquid to ground.  There are many commercial programs that calculate spill rates. There are several considerations that are to be included in the model basis.  To provide a spill rate, one has to postulate a leak scenario.  Typically postulated is a hole or flange leak, which may be from adjacent equipment or the equipment exposed to the fire.  If the spill originates at the equipment exposed to the fire, and depending on the release rate and duration, there may be different cases.

---- A large leak will empty the equipment in a short period of time in an "instantaneous spill".   With no internal fuel to vaporize and a large open pipe, there will be no relief.

---- With a smaller leak but continuous flow of liquid fuel to the ground, the inventory will be depleted, but not until the equipment pressures and relieves.  A smaller leak will produce a small relief load but for a longer period of time. 

 The second issue is the type of flow through the rupture and how that affects the amount of liquid to grade.  The flow through the hole is calculated and the liquid-to-grade is the flow through the hole minus fuel evaporated or entrained as mist.  In reality, the flow through the hole could be in any direction, and consist of fine spray or a full stream.  The commercial program used in this example assumes that the rupture is an effective round hole with the specified area.   

• The total amount of liquid drained is correlated to the pool size.  The correlation was developed in-house but is a variation based on the traditional empirical Manning equation for channel flow. The correlation uses the pool area and the rate of change in liquid level, also called the "drain-down". The drain-down is a function of pool diameter.   The general description is:

                Volume liquid lost by draining [volume/ time] = rate of change in liquid level [length /time] X pool area

 The correlation assumes that the liquid has been pooled with the pool effective diameter as the “width of flow” and “depth of flow” deep.  The flammable liquids will drain down a channel with constant width and depth and a specified slope.   For the calculation, the drainage stays at this depth and width for the entire length of the channel apron, with no spreading.  In reality it is expected that the liquid would spread and that as it spreads, the depth will change.

• The total amount of liquid burned is correlated to the pool size.  The correlations are presented in the handbooks from the SFPE and NFPA and use the pool area and the rate of change in liquid level, also called the "burn-down" or vertical liquid consumption velocity.  The general description is:

               Volume liquid lost by fire [volume/ time] = rate of change in liquid level [length /time] X pool area

 As an assumption, only the pool is considered to burn and provide heat to the equipment. The liquid that is draining is assumed not to ignite, or if ignited, the burn is located just far from the equipment that heat input is not significant.  The burn-down rate is different depending on whether the fluid boiling point is above or below ambient temperature.  If below ambient temperature, additional heat is provided from the ground to the pool which enhances the burn-down rate. 

 Heat Radiated and Absorbed by Equipment

 The amount of heat radiated from the fire (energy/time) is based on the amount of fuel burned

             Heat radiated (energy/time)  = Mass Burned [mass/time]  X LHV [energy/mass ] X F [% radiated]

 The F factor is the fraction that of heat that is actually radiated from the pool fire. Descriptions and values are

provided in SFPE and NFPA handbooks. 

 For this presentation, the amount of fire heat absorbed is equal to the heat radiated. This equation may be adjusted on several bases, including view angle or wetted surface. 

 Conclusion

 An example is presented for a steady-state pool fire model which assumes that the fuel supply limits the heat flux to the equipment.  The size of the pool is determined by the spill rate to grade, the amount of flammable liquids drained away from the pool and the amount burned.  Conversely the amount burned and drained is a function of the pool size.  Solving for the amount burned yields the heat radiated.   The relief rate for the fire case is thus determined from the heat absorbed by the equipment from the heat radiated from the pool fire.