A Guide to Estimating Tube Rupture Flowrates

Using DIERS Two-Phase Equations

1. Introduction: The Tube Rupture Problem

A tube rupture in a shell-and-tube heat exchanger is a critical overpressure scenario. When the high-pressure (HP) side ruptures, it can rapidly overpressure the low-pressure (LP) side, potentially leading to a catastrophic failure.

Standard guidelines present two main solutions to this problem:

1. Raise the design pressure of the LP side (often uneconomical).
2. Install a properly sized pressure relief device (relief valve or rupture disc).

To size this relief device, we must first calculate the total mass flowrate (W) from the rupture. This is complicated because a high-pressure liquid flashing into a low-pressure environment creates a two-phase (liquid + vapor) flow.

The modern, accepted approach is to use a Homogeneous Equilibrium Model (HEM), such as the one developed by the Design Institute for Emergency Relief Systems (DIERS).

Two Flow Paths

A full tube rupture creates two distinct flow paths from the HP side, both of which must be calculated. The total flow (W) is the sum of the flow from both paths.

• Path 1 (W₁): Orifice Flow
  Flow from the HP side *through the tubesheet break* into the LP side. This behaves like flow through an orifice.
• Path 2 (W₂): Pipe Flow
  Flow from the HP side *into the broken tube* and *through its length* before exiting into the LP side. This behaves like flow through a pipe with friction.

2. The General Calculation Strategy

1. Calculate the Key Parameter (ω): First, determine the fluid's "compressible flow parameter" (omega, ω) using the HP side inlet conditions.
2. Calculate Path 1 (W₁): Determine the mass flux (G₁) for the orifice flow.
3. Calculate Path 2 (W₂): Determine the mass flux (G₂) for the pipe flow.
4. Sum for Total Flow (W): Calculate the total mass flowrate: W_Total = (G₁ × A_tube) + (G₂ × A_tube).
5. Find Relief Valve Conditions: Flash the W_Total at the LP side relieving pressure. The resulting properties (vapor fraction, density, etc.) are the inlet conditions for sizing the relief device itself.

3. Step 1: Find the Compressible Flow Parameter (ω)

The parameter ω (omega) is a measure of the fluid's compressibility and its potential to flash. It is calculated using the HP side inlet conditions.

• ω = Compressible flow parameter (dimensionless)
• x₀ = Mass fraction of vapor at inlet (dimensionless)
• ρ₀ = Overall fluid density at inlet (lb/ft³)
• ρᵥ = Vapor portion density at inlet (lb/ft³)
• ρₗ = Liquid portion density at inlet (lb/ft³)
• P₀ = Inlet pressure (psia)
• T₀ = Inlet temperature (°R)
• L = Latent heat of vaporization (Btu/lb)
• Cₚ = Liquid specific heat (Btu/lb-°F)

4. Step 2: Calculate Path 1 Flow (W₁) - Orifice

A. Find the Critical Pressure Ratio (ηc)

First, determine the critical pressure ratio (ηc), which defines when the flow will "choke." For ω > 0.6, this can be approximated as:

B. Check for Choked (Critical) Flow

Compare the actual pressure ratio (η) to the critical ratio (ηc).

• Plp = Low-pressure side relieving pressure (psia)
• Php = High-pressure side pressure (P₀) (psia)
• Actual Ratio η = Plp / Php

If η < ηc → Flow is Critical (choked).
If η ≥ ηc → Flow is Sub-critical.

C. Calculate Dimensionless Mass Flux (G₁*)

Use the appropriate formula based on the flow condition:

D. Calculate Actual Mass Flux (G₁)

This formula converts the dimensionless flux into real-world units.

5. Step 3: Calculate Path 2 Flow (W₂) - Pipe

Calculating the mass flux (G₂) for the flow *through* the tube is more complex because it must account for frictional losses. This requires numerically solving a set of equations for the tube's inlet, length, and outlet.

6. Step 4: Calculate Total Flow & Final Sizing

1. Find Tube Area (A_t): Calculate the internal cross-sectional area of the tube in square inches.
   A_t (in²) = (π × D_inner²) / 4
2. Calculate Total Flow (W_Total):
   • Rigorous Method: W_Total = (G₁ × A_t) + (G₂ × A_t)
   • Shortcut Method: W_Total = 2 × (G₁ × A_t)
3. Determine Relief Valve Inlet Conditions:

   This is the final step. Take the calculated W_Total and flash this mass flowrate at the LP side relieving pressure. This flash calculation will provide the crucial data (vapor fraction, temperature, density, molecular weight) needed to size the pressure relief device itself.

   Note: You must also evaluate if the hot fluid on the LP side (e.g., hot slurry) will vaporize *more* of the incoming ruptured fluid, further increasing the vapor load on the relief valve.

7. Adapted Example

This example shows how the method is applied.

• Scenario: HP Boiler Feed Water (BFW) ruptures into an LP Slurry.
• HP Side (P₀): 614.7 psia, 948.8 °R (Saturated Liquid, so x₀ = 0)
• LP Side (P_lp): 234.7 psia (220 psig relieving pressure)
• Tube: 0.732 in inner diameter (A_t = 0.4208 in²)
• HP Fluid Properties:
  • ρ₀ = 49.539 lb/ft³
  • ρᵥ = 1.332 lb/ft³
  • ρₗ = 49.539 lb/ft³
  • Cₚ = 1.17 Btu/lb-°F
  • L = 728.3 Btu/lb

Calculation Steps:

1. Calculate ω:
   Since x₀=0, Term 1 is zero.
   Term 2 = 0.18505 × 1.17 × 948.8 × 614.7 × 49.539 × [ (1/1.332 - 1/49.539) / 728.3 ]²
   ω = 6.29
2. Calculate ηc:
   ηc = 0.6055 + 0.1356(ln 6.29) - 0.0131(ln 6.29)²
   ηc = 0.811
3. Check for Choked Flow:
   Actual Ratio η = 234.7 / 614.7 = 0.382
   Since 0.382 < 0.811, the flow is Critical.
4. Calculate G₁* (Orifice Flux):
   G₁* = ηc / (ω)^0.5 = 0.811 / (6.29)^0.5
   G₁* = 0.323
5. Calculate G₁ (Actual Orifice Flux):
   G₁ = 1,700 × 0.323 × (614.7 × 49.539)^0.5
   G₁ = 95,820 lb/hr/in²
6. Calculate Total Flow (W_Total) - Shortcut Method:
   W_Total ≈ 2 × G₁ × A_t
   W_Total ≈ 2 × 95,820 lb/hr/in² × 0.4208 in²
   W_Total ≈ 80,642 lb/hr
7. (Optional) Rigorous Method Result:
   The original document numerically solved for the pipe flow (G₂) and found:
   G₂ = 56,422 lb/hr/in²
   W_Total = (95,820 × 0.4208) + (56,422 × 0.4208) = 40,321 + 23,742
   W_Total (Rigorous) = 64,063 lb/hr