It is frequently necessary to displace the contents of a pipeline or umbilical tube (fluid B) with another fluid (fluid A). If we don’t use a pig to separate the liquids, there will be mixing at the interface (axial mixing). The mixing zone requires us to overflush the line to effectively remove fluid B from the line.

Below, we address a method of calculating the length of the mixing zone in order to determine the effective overflush requirement for a given pipeline or umbilical tube.

**The Taylor Equation—Axial Dispersion in Turbulent Flow**

Taylor1 developed Equation 1 to describe axial dispersion in turbulent flow:

The error function (erf) is defined by Equation 3, and is illustrated in Figure 1. The curve illustrates the concentration profile of fluid B across the mixing zone. Inspecting the Taylor equation, we see that the erf is zero when x—vt is zero. This is the time (t) at which the center of the mixing zone arrives at the end of the line. Where the erf equals zero, the Taylor equation gives the mixture concentration of 50% of fluid B.

**Characteristic Length, Mixing Zone Length**

We can now define the characteristic length as the denominator of the erf, as illustrated in Equation 2.

Per the erf, we see that, with a distance of 1 characteristic length from the center of the mixing zone, the concentration of fluid B is 84.5% at the leading edge or 15.5% at the trailing edge. At a characteristic length of 2, the concentration of fluid B is 99.5% leading edge or 0.5% at the trailing edge. If this is adequate displacement, then we can define the mixing zone length as 4 * characteristic length which is equal to 8 * sqrt (Dl t).

**Axial Dispersion Coefficient, Dl**

To calculate characteristic length, we need to determine the axial dispersion coefficient, Dl. Richardson and Coulson2 provide a plot (Figure 2) relating Reynolds Number (Re), Schmidt Number (Sc), and the dimensionless form of dispersion coefficient, Dl/vd. There are several things worth noting on Figure 2.

- In turbulent flow, Dl is only a function of Re. At Re > 100,000 Dl/vd is fixed.
- In laminar flow, the Schmidt number dominates.
- The Dl for gases is relatively low at all values of Re.
- The highest Dl (worst displacement) will occur at the transition from laminar to turbulent flow.

**Schmidt Number Estimation**

In laminar flow, the Sc dominates. Sc is described via Equation 7. Sc can be calculated via Equation 7, if the diffusivity is known. Diffusivity can be estimated via Equation 5. A simpler approach, which has proven effective, is to estimate Sc based on viscosity alone. Note that Sc is a function of viscosity squared because viscosity is in both the Sc equation and the diffusivity equation.

The following values of Sc should be used for the fluids described below:

- For low viscosity chemicals, such as methanol, use Sc = 250.
- For water, use Sc = 1,000.
- For higher viscosity chemicals, such as MEG, use Sc = 2,500.

**Sample Calculation 1**

A 10 inch ID (0.254 m) 20 km gas export line is filled with nitrogen and is to be displaced with natural gas. Displacement velocity is 8 m/sec, the Re number is greater than 1,000,000. What is the length of the mixing zone? From Figure 2, Dl/vd = 0.2, hence:

This agrees with field experience that gas line displacements have small mixing zones.

**Sample Calculation 2**

A 4 mile (6439 m) 1” ID (0.025 m) umbilical tube containing storage fluid of EG/Water mixture was displaced with diesel with a viscosity of 4 cP. The maximum displacement rate was 2 gpm. This yields Re = 1,270, Velocity = 0.82 ft/sec = 0.25 m/sec, Displacement time = 25,800 sec.

Assuming a Schmidt Number of 2,000 (for pure MEG we would have used 2,500). From Figure 2, Dl/vd is approximately = 7,000, hence:

In this case, it took between 1.5 and 2 times the volume of the umbilical to effectively displace the storage fluid.

**Conclusions and Cautions**

The following conclusions and cautions can be applied to the method described above:

- The method yields rough estimates only. Opportunities to collect field data to validate the model are rare (If you have data we would like to see it.)
- The Taylor equation was developed for miscible fluids in turbulent flow. The method proposed here extends it to laminar flow. Sample Calculation 2, shown above, confirms successful application to immiscible fluids.
- Two fluids provide different Reynolds numbers. Use the one giving the highest value of Dl.
- These equations do not take elevation effects into account. For fluids with different densities, elevation effects may be important.
- If the displaced fluid adheres to the pipe wall, then the tail may be extended.

**References**

- Taylor, “The Dispersion of Matter in Turbulent Flow through a Pipe”, Proceedings of the Royal Society of London, Vol 223, May 20, 1954
- Coulson, Richardson, Chemical Engineering, Elsevier, 1991