Using DIERS Two-phase Equations to Estimate Tube Rupture Flowrates

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August 2001 Full contents

Using DIERS two-phase equations to estimate tube rupture flowrates The method is more accurate than the traditional 'separate' phase approach ASME Section VIII and API 521 discuss the need to design and protect heat exchangers against overpressure due to a tube rupture. API 521 specifically addresses requirements and approaches to solve this problem. The first solution is to raise design pressure of the low-pressure side to a minimum of two-thirds of the design pressure of the high-pressure side. However, this may not be desirable due to economics or other design reasons. The second solution is to add a relieving device, like a relief valve or a rupture disc. To size this relieving device, flow through the tube rupture from the high-pressure side to the low-pressure side of the heat exchanger must be calculated (Fig. 1). Equations developed by the American Institute of Chemical Engineers (AIChE) for this purpose are receiving widespread support.

Fig. 1. Tube rupture diagram. In many instances, high-pressure liquid flashes to a much lower pressure through the tube rupture, thereby requiring a method for calculating two-phase flashing flow. Equations that have been developed in the past two decades for two-phase flashing flow in relief valves and flare header piping can be used for determining flowrate through the tube rupture. Historically, the process industry has sized relief valves whose inlet and/or discharge is two-phase by separating out the phases into pure liquid and pure vapor phases, determining a required area for each phase, and adding the two areas together for a total required relief valve area. While this has no theoretical basis, it was seen as giving a reasonable "ballpark" estimate, without going into complicated mathematical and semiempirical techniques. There is evidence that using this method can lead to incorrectly sizing relief valves.

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Over the past 15 years, The Design Institute for Emergency Relief Systems (DIERS), an arm of AIChE, has been looking into the area of two-phase flow in relief valves and flare headers. It has developed various calculation methods to deal with these problems. These methods are receiving widespread support and use within the industry. Since a tube rupture, as defined in API 521, is just flow through a tube (one side of the break) and flow through an orifice (tubesheet side of the break), the DIERS two-phase mass flux methods can easily be applied to these problems when there are two-phase or flashing fluids. General methodology.DIERS equations for two-phase flow calculate a mass flux based on physical properties of the fluid. A generalized equation for nozzle discharge is used for sizing the relief valves for two-phase flashing flow. Hence, it can be used to approximate flowrate through the tubesheet side of the tube rupture. DIERS mass flux equations have also been developed for pipe flow with choked discharge. These equations can be used for the tube side of the tube rupture. A slightly quicker method would be to simply take twice the flow through the tubesheet, assuming that the flowrates will be reasonably close. This would produce a number that was more conservative (but less accurate) than the other method. It is important to understand that the DIERS equations assume steady-state and equilibrium conditions, and they do not account for transients. An estimation of peak pressures caused by shock waves that occur immediately after the tube rupture would have to be evaluated using a different method. The method for estimating tube rupture flowrate is as follows: ? ? ? ?

Determine mass flux through the tubesheet side of the broken tube. Determine mass flux through the tube side of the broken tube. Calculate tube discharge area. Calculate flowrate through the tube rupture.

To determine inlet composition of the relief valve on the low-pressure side, calculate any additional flashing and mixing that occurs after the rupture. Methodology using DIERS HEM equations.There are various specific DIERS methodologies for calculating two-phase flashing flow. The specific equations presented here were developed by Leung. 1Their strength is ease of use and that they only require inlet conditions. However, they are not necessarily as accurate with fluids close to the critical point. An alternative set of equations was developed by Simpson. 2These equations are stronger when dealing with critical fluids, but require more physical data.* Determine mass flux through the tubesheet side of the broken tube. Step 1: Calculate the compressible flow parameter, ?: The DIERS methods treat the two-phase fluid as a single homogeneous "compressible" fluid. Leung's method does this by calculating a compressible flow parameter, ?. This parameter is a measure of the fluids "compressibility." The larger the value of ?, the more the fluid behaves like a compressible fluid. Values for ? fall into these categories: Flashing flow: ? > 1 Gas/vapor flow: ? < 1 Nonflashing flow: 0 < ? < 1 Liquid flow: ? = 0 The ? is made up of two terms, the first term (x 0?/? v)[1-P 0/(2.7L? v)] describes the compressibility due to presence of vapor in the mixture. The second term {0.18505 C pT oP o? o[(1/? v-1? l)/L] 2} accounts for compressibility due to the phase change upon depressurization. All of the properties are based on the high-pressure side (inlet) conditions.


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where: x 0= mass fraction vapor at the tubesheet ? = overall fluid density, lb/ft 3 ? v= vapor portion density, lb/ft 3 ? l= liquid portion density, lb/ft 3 C p= liquid specific heat, Btu/lb/°F T 0= valve inlet temperature, R P 0= valve inlet pressure, psia L = liquid portion latent heat of vaporization, Btu/lb The first term equals 0 if there is no vapor present at the valve inlet, and all the flashing is in the valve. The second term equals 0 if there is no flashing in the valve. If there is no flashing in the relief valve, the second term should be dropped, and the equation will become:

where: k = the ratio of specific heats, C p/C v This method assumes saturated liquid/vapor on the high-pressure side. For variations of this method dealing with subcooled liquids, consult Leung. 1 Step 2: Calculate pressure ratio, h: The next step is to calculate the ratio between the upstream and downstream pressures. For critical flow, we usually assume that the downstream (critical) pressure is 55% of the upstream pressure, so a quick assumption can be that ? C= 0.55. However, two-phase fluids can choke at a much higher critical pressure, often with ? C= 0.7 to 0.9. A more accurate calculation would be to determine critical pressure drop from the following correlation:

where ? C= the critical pressure ratio Eq. 3 can be approximated by:

at values of ? > 0.6. 3At values < 0.6, Eq. 3 should be solved numerically. Step 3: Calculate the dimensionless mass flux, G*: If the flow is critical:


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If the flow is not critical:

and

where: P b= low-pressure side "relieving pressure", psia P 0= high-pressure side pressure, psia Step 4: Calculate actual mass flux, G(lb/hr/in. 2):

where: P 0= inlet pressure (psia) ? = overall density of the two-phase fluid at inlet, lb/ft 3 Determine the mass flux through the tube side of the broken tube. Flow through the tube side can be approximated by using the DIERS mass flux equations for horizontal pipes (or vertical pipe, if the exchanger is vertical) For consistency, we have continued to use Leung's methods. 4 Use the same ? as was calculated in the previous Step 1. The method consists of three equations, one for tube inlet, one for the flow through the tube and one for tube discharge. If an exchanger doesn't comply with the API 2/3 rule, discharge from the tube will almost certainly be choked, therefore, the discharge nozzle equation can be assumed to be critical flow. Step 1: Inlet "nozzle":

where ? 1= the ratio of the pipe inlet pressure to the "high-pressure side" pressure.

Step 2: Pipe:


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where ? 2= the ratio of the outlet pressure to the "high-pressure side" pressure.

where: f= friction factor L= tube length D= tube diameter Step 3: Outlet nozzle:

These equations are combined and then solved numerically to obtain a G*. Step 4: Calculate actual mass flux, G(lb/hr/in. 2).

where: P 0= inlet pressure, psia ? = overall density of the two-phase fluid, lb/ft 3 Calculate tube discharge area, A t(in. 2)

where: D I= tube inner diameter, in. Or use a cross-sectional area cited in a reference. Calculate flowrate through the tube rupture:

At this point, total flowrate through the rupture has been calculated. The flow then needs to be evaluated at the low-pressure side pressure (presumably relieving pressure) to determine if there is any additional flashing. This will determine inlet conditions of the relief valve being used to relieve the tube rupture flowrate. Sample calculation. As an example, consider a slurry/HP steam generator in an FCC unit. The highpressure (boiler feed water) side has a design pressure of 600 psig. The low-pressure (FCC slurry) has a design pressure of 200 psig (PSV relieving pressure = 220 psig). The tubes are 1-in. 10 BWG tubing (length = 16 ft, inside diameter = 0.732 in., inside cross-section = 0.4208 in. 2, friction factor = 0.00575). Calculate flowrate through a full rupture of a single tube.


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Step 1: Calculate compressible flow parameter, v: (Eq. 1):

Since ? > 1, the flow is flashing (which is as expected). Step 2: Calculate the pressure ratio, ? c(Eq. 4):

Step 3: Calculate dimensionless mass flux through the tubesheet rupture, G 1* (Eq. 5): Since the actual pressure ratio is 234.7/614.7 = 0.382 and the critical pressure ratio is 0.811, the flow is critical. Hence,

Step 4: Calculate actual mass flux through the tubesheet, G(lb/hr/in. 2) (Eq. 8):

So total mass flowrate through the tubesheet would be (Eq. 16):


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Determine mass flux through the tube side of the broken tube. The compressible flow parameter is the same: v = 6.29 Step 1: Inlet "nozzle" (Eq. 9):

Step 2: Pipe (Eq. 10):

Step 3: Outlet Nozzle (Eq. 11):

Substitute Eq. 11 into Eqs. 9 and 11 to eliminate G 2 *, and solve numerically (e.g., on a spreadsheet):

Step 4: Calculate actual mass flux, G(lb/hr/in. 2) (Eq. 14):

So total mass flowrate through the broken tube would be (Eq. 16):

The total mass flowrate is : W = 40,321 + 23,742 = 64,063 lb/hr If flow was estimated by just doubling the tubesheet flow, the total would be 80,642 lb/hr, which is an additional 25%. This would be a conservative answer while still remaining in the ballpark, and most probably not affecting the size of the relief valve that would need to be purchased. To determine the vapor/liquid split for the relief valve inlet, flash the mixture at the relief valve relieving pressure (220 psig). In this example, the mass fraction of flashed steam would be about 13%, and x ofor the relief valve sizing calculation would be 0.13 except that in this example, the hot slurry side will vaporize some of the remaining water, giving a higher x o(perhaps as high as 1.0).


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This will need to be evaluated separately.

Literature cited 1Leung, J. C., "Two-Phase Flow Discharge in Nozzle and Pipes-A Unified Approach," Journal of Loss Prevention Industry, Vol. 3, January 1990, pp. 27-32. 2Simpson, L. L., "Estimate Two-Phase Flow in Safety Devices," Chemical Engineering, August 1991,

pp. 99-102. 3Leung, J. C., "A Generalized Correlation for One-component Homogeneous Equilibrium Flashing

Choked Flow," AIChE Journal, October 1986. 4Leung, J. C., "Easily Size Relief Devices and Piping for Two-Phase Flow," Chemical Engineering

Progress, December 1996, pp. 28-50. 5API Recommended Practice 520, Part I, 6th Edition, 1993. 6API Recommended Practice 521, 3rd Edition, 1990. 7Darby, R., "Perspectives on Safety Relief Valve Sizing for Two-Phase Flow," 2nd Internatioal

Symposium on Runaway Reactions, Pressure Relief Design and Fluid Handling, March 1998. Return to top Copyright © 2007 Hydrocarbon Processing Copyright © 2007 Gulf Publishing Company


Using DIERS Two-phase Equations to Estimate Tube Rupture Flowrates

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