Tuesday, June 11, 2019

For overpressure scenarios involving heat input to a multi-component fluid, with subsequent vaporization of that fluid and relief of the vapors generated, a question arises: what is the effective heat of vaporization that should be used?

The most precise, yet most time consuming, method is to perform successive dynamic vaporization and sizing steps using detailed thermophysical properties, potentially decoupling time from the analysis.^1,2,3 Note we are hesitant to label this technique the “most accurate” as is sometimes done, since there are many low resolution variables in the analysis (heat flux, as the prime example) that affect overall accuracy. For most cases, the use of detailed dynamic analysis is not commensurate with the precision of the other input in the analysis. Reactive systems would be the one exception in which a detailed dynamic analysis is generally needed.

One alternative in the CCPS Guidelines³ advocates taking each individual component in the mixture, determining its latent heat of vaporization at the relief conditions, and using the most stringent relief requirement. This approach sounds nice in theory, but is difficult in practice for most multicomponent mixtures, especially those with dissolved gases, minor quantities of light components (particularly when these components are not even liquid at the relief conditions), or significant amounts of hypothetical pseudo-components.

Recognizing that continued exposure to elevated fire temperatures will weaken vessel walls, and a pressure relief valve may not provide protection over extended periods of time, some will focus on the very initial conditions. This approach uses the initial latent heat at the bubblepoint at relief pressure, perhaps first removing light components.

The most common, and practical, method is a compromise of sorts – select a percentage of the fluid vaporized, determine the enthalpy change due to that vaporization, and use that single effective heat of vaporization to size a relief device. Most will select a percentage vaporized in the initial 25%. Recognizing that the initial heat input will vaporize fluids that contribute to pressuring up the system, and not to the overpressure itself, some will select an initial vaporization point that is non-zero; most commonly, this is 5 or 10%. As a result, many companies will use something like 10-25% vaporization. Of course, we could debate the basis of the percentages (mass, mole, volume, even temperature) – mass percent is most common, and given that we are not looking for high precision values, is appropriate.

One may recognize that this approach may result in some heat input attributed to sensible heat changes in the vapor and liquid. At the very least, the sensible heat changes associated with the vapor should be removed, since the vapor is leaving the system through the relief device. For a slightly more conservative basis, the liquid sensible heat can also be removed. A detailed step wise approach to this calculation is provided below.

To illustrate the effects of the various methods and parameters, we provide two test cases.

Example case – fuel gas knockout drum. A vertical fuel gas knockout drum (4’ dia × 10’ len, MAWP = 200 psig), containing methane, ethane, propane, and n-butane, is exposed to an external fire. The knocked out liquids (2’) are assumed to be in equilibrium with the gas. Based on the initial wetted surface area, the heat input from fire is 460,000 BTU/hr determined based on API Standard 521 for pressure vessels.¹ The ideal required area is calculated using different techniques (dynamic analysis, individual components, initial bubblepoint, or effective heat of vaporization over different ranges) for comparison purposes.

Note the primary reason for lower relief areas for the lighter components appears to be the significantly lower equilibrium temperatures (resulting in a more dense relief gas).

Focusing on just the results of the dynamic analysis and effective heat of vaporization at 0 – 25%:

In this example, the range of required areas in the dynamic analysis is very small: 0.177 to 0.182 in², and the required areas using the effective heat of vaporization (regardless of the minima / maxima) are all within this range.

For this example, the starting relief temperature is 150°F, while the ending relief temperature is 200°F, so the total boiling point range is 50°F. Future work to evaluate comparisons at wider temperature ranges would be interesting to see.

Example case – CCPS Guidelines 2nd Edition §D.1.1. An example problem provided in the CCPS Guidelines 2nd Edition³ provides the analysis for the second case.

A nominal 3000 gallon vertical tank containing a mixture of 50% acetone, 30% ethanol, and 20% water (by weight) is exposed to an external fire. The tank is a vertical, cylindrical vessel with a 7-foot inside diameter, 10-foot straight side height and 2:1 elliptical heads. The vessel has approved drainage and contains a water-miscible liquid mixture having an average heat of combustion not exceeding that of ethanol and which is the only potential fuel for a fire. Using the NFPA 30 Code the fire protection environmental factor is 0.25 and the resulting heat input rate is 1,250,000 BTU/hr.

Not stated in the description is that the vessel has an MAWP of 50 psig, and 21% allowable accumulation is used.

Various parameters for the individual component method in the guidelines are given; interestingly enough, the actual results are missing. Using the equations in the guidelines with the volumetric correction factor, the following results are calculated:

	Water	Ethanol	Acetone
Required Rate [lb/hr]	1,377	3,997	6,513
Required Area [in²]	0.418	0.784	1.117

The example problem in the guidelines then provides eight pages of text on how to input the data into SuperChems™ for DIERS, and finally reports “The simulation with a 1½H3 relief valve (0.785 in² orifice) gives a peak venting pressure of 55.4 psig”. The actual ‘dynamic’ result (that is, the relief requirement as a function of time or % vaporized) is not provided in the CCPS Guidelines example. To provide some differentiation with the ethanol-only result, this result of 0.785 in² is plotted at an arbitrary mass percentage in the graph below. One will note that a slightly lower required relief area could be computed to yield the allowable accumulation pressure of 60.5 psig; however, this is not reported and is not expected to be significantly different given that the example also indicates “A second simulation done exactly the same except using a valve size of 1½G3 (0.503 in² orifice), gave a peak venting pressure of 85.4 psig…”

The effective heat of vaporization for the mixture, using the various ranges as before, and subsequent required relief areas were calculated.

In this example, there is not much difference in results between the various minima/maxima for the effective heat of vaporization. Interestingly, there is also not much difference in the results for the effective heat of vaporization versus the dynamic analysis. The results for the individual components do vary significantly, which is directly attributable to the properties of those components – the high latent heat of water compared to the other components being an obvious example of this.

Calculation assumptions and settings. Fluid properties are obtained from the NIST REFPROP program⁴. Sizing calculations were performed based on API Standard 520 Part I⁵ Annex B.3.1 vapor sizing for real gases, assuming an ideal nozzle (K_d=1) and choked flow (backpressure = 0 psig).

For the dynamic analysis, the start of relieving conditions is based on an isochordic flash from the initial overall bulk density to the relief pressure. This essentially accounts for the pressuring up of the vessel from operating conditions to relief conditions. Steps of 1% quality were taken, and time was decoupled from the analysis. The theoretical mass flux was based on the vapor properties at the end point of the step.

The latent heat of each component is taken at the relief pressure, regardless of the saturation temperature required. This is clearly a drawback of the individual component method when minor light (or even heavy for that matter) components are present.

For the initial bubblepoint, the normally residing liquid was flashed at bubblepoint at relief pressure. The vapor that was calculated to be in equilibrium with the liquid at the bubblepoint was used to calculate the theoretical mass flux.

For the effective heat of vaporization, the normally residing liquid was flashed at the relief pressure, and the calculations performed as described below.

Effective heat of vaporization calculation. We will use a mass basis, and an arbitrary total amount of mass of 1 lb, allowing us to specify the initial and final flashes based on the mass quality of the stream. Take the residing liquid and perform a pressure-quality flash at the relief pressure and initial quality (x_min); and then at the relief pressure and final quality (x_max). At each flash, the following properties are needed: Bulk enthalpy (H), temperature (T), and heat capacity at constant pressure (C_p). If the enthalpies of the liquid and vapor are more readily available at the flashed state, then the bulk enthalpy can be calculated:

For a phase change, the total change of enthalpy is a function of sensible heats and the effective heat of vaporization (ΔH.eff):

If we assume an average heat capacity is acceptable, the integration can be simplified:

Solving for the effective heat of vaporization:

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[1] American Petroleum Institute. “API Standard 521-Pressure-relieving and Depressuring Systems”. 6th Edition, April 2014.
[2] Ouderkirk R. “Rigorously Size Relief Valves for Supercritical Fluids”. Chemical Engineering Progress. 2002 August; 98(8): 34-43.
[3] AIChE Center for Chemical Process Safety. “CCPS Guidelines for Pressure Relief and Effluent Handling Systems”. 2nd Edition, 2017; New Jersey: John Wiley & Sons, Inc.
[4] Lemmon, E.W., Huber, M.L., McLinden, M.O. NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 9.1, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, 2013.
[5] American Petroleum Institute. “API Standard 520: Sizing, Selection, and Installation of Pressure-relieving Devices in Refineries; Part I—Sizing and Selection”. 9th Edition, 2014.