Keywords

56.1 Introduction

A multi-parametric analysis involves the collective study of various investigations on drift flux models as applied to both vertical and horizontal pipelines. Drift flux models were first developed by Zuber in 1965. The Drift flux models fall into the category of Computational Fluid Dynamics CFD used for particle transport prediction equations. A drift flux model is employed to represent slip between fluid phases [4]. Brethour and Hirt [7] were of the opinion that the concept behind the Drift Flux models is that the relative motion between these components can be described as a continuum rather than by discrete elements. A continuum considers the entire process as a whole with no distinct exclusive or conclusive attention given to the elements or parts separately. It is uniquely used for the study of sedimentation, fluidised beds and other flow processes that involve relative motion (interaction) between phases that are controlled by buoyancy and fluid drag forces. The relative flow moves in a slip pattern propagated by Kinematic Shocks or Expansion Waves mostly caused by turbulent fluid motion aided by external and internal forces such as the inward radial pressure generated by interfacial surface tension of a fluid in a stable high thermal environment as in the case of a viscoplastic slurry or paste transport situation.

Applications range from scenarios and processes that see the occurrence of elements as simple as bubbles and slug particles, to more complex and subtly devastating affects such as pipeline erosion and the attendant wear rate. Drift Flux models consider the different densities and sizes of the volume fraction of particles assumed to be continuously slipping; in other words it considers the relative motion between and within the fluids or fluid/particle or fluid/particle/pipe wall at constant velocity due to gravitational and/or centrifugal forces [9].

The aforementioned Zuber considered a one dimensional flow of a mixture of two components, A and B. The volumetric fluxes of the two components, jA and jB, were related to the total volumetric flux, j, the drift flux, jAB and the volume fraction, α A + α B = 1. To determine the relative motion (Drift Flux) by applying the theory of dynamics to the forces on the individual phases, the momentum and energy equations would have to be understood although not exclusively or conclusively.

Drift flux models are not without limitations in that some multiphase flows can not be approximated especially when the relative motion is intimately connected with the pressure and velocity gradients in the two or more phases. However, since Zuber, many researchers have applied the model to a two phase flow with success [2, 3, 8].

A fluid carrying pipeline is rife with opportunities for applying drift flux models. In this research work, the application of the various drift flux models to pipeline engineering is examined.

56.2 Application to a Vertical Pipe Considering the Buoyancy Effect

Since their introduction in the 1960s the Drift Flux models have proven very adaptable to various engineering challenges encountered. This versatility is of immense value in the prediction of expected and anticipated engineering failures, it is also important for examining post failure root cause analysis. The need for a full fledged parametric analysis of various drift flux models cannot be overemphasised.

The governing equations describing one dimensional two phase drift flux transport equations in vertical pipes are considered by stating the mass conservation, momentum and internal energy conservation equations; this is a basic start:

For Mass Conservation

$\frac{\partial {p}_{j}} {\partial t} + \frac{\partial } {\partial z}\left ({P}_{j}{U}_{j}\right ) = 0$
(56.1)

For Drift Flux Momentum Conservation

${\rho }_{j}\frac{\partial {v}_{j}} {\partial t} + {\rho }_{j}{V }_{j}\frac{\partial {V }_{j}} {\partial z} + \frac{\partial } {\partial z}\left (\frac{{\rho }_{v}{\rho }_{1}\alpha {V }_{\mathit{vj}}^{2}} {{\rho }_{j}\left (1 - \alpha \right )} \right ) = -{\sum }_{i=1}^{N}{F}_{\mathit{ gl}} -\frac{\partial {P}_{j}} {\partial z} - {\rho }_{j}g\cos \theta $
(56.2)

where θ is the angle of contact between the surface of the liquid and the surface of the pipe which is assumed to be 0∘.

Drift flux internal Energy conservation

$\begin{array}{rcl} & & \frac{\partial } {\partial t}\left ({\rho }_{j}{u}_{j}\right ) + \frac{\partial } {\partial z}\left ({\rho }_{j}{u}_{j}{v}_{j}\right ) + \frac{\partial } {\partial z}\left [\frac{\alpha {\rho }_{l}{\rho }_{v}\left ({u}_{v} - {u}_{l}\right ){V }_{vj}} {{\rho }_{j}} \right ] \\ & & \qquad + P \frac{\partial {V }_{j}} {\partial z} + {P}_{m} \frac{\partial } {\partial z}\left [\frac{\alpha \left ({\rho }_{l} - {\rho }_{v}\right ){V }_{vj}} {{\rho }_{j}} \right ] \\ & & \quad ={ \sum }_{i=1}^{N}{q}_{ l}^{n}\frac{{P}_{1}} {{A}_{l}} + {V }_{j}\left ({\sum }_{i=1}^{N}{F}_{ wi}\right ) \end{array}$
(56.3)

where A is the surface area of the pipe; uv and ul are the vapor and liquid velocities respectively, V is the volumetric flow rate, Fg is the gravitational force, P is the operating pipe pressure, Z is the height of the pipe, ρl and ρv are the liquid and vapor density respectively, q is the heat flux whereas, Fwi is the wall shear force.

This study was done by examining the works of Holman [13] and DF Models [9]. In nucleate boiling, Holman [13] observed that bubbles are created by the expansion of entrapped gas or vapor at small cavities in the surface. The bubbles increase in size depending on the surface tension at the liquid vapor interface and the temperature and pressure.

In this scenario, a superheated liquid at its boiling point would have bubbles of vapor form on the heating element surface. These bubbles collapse as the heat increases, and the entrapped gases escape through the liquid to the surface of the vertical pipe being investigated. The volumetric drift flux of bubbles as they move through the liquid is represented by Eq. (56.4)

${J}_{\mathit{VL}} = \alpha \left (1 - \alpha \right ){U}_{\mathit{VL}}$
(56.4)

where JVL is the drift flux, UVL is the relative velocity and α is the volume fraction.

The relative velocity can also be represented in Eq. (56.5)

${U}_{\mathit{VL}} = {U}_{\mathit{VLO}}\left (1 - \alpha \right )$
(56.5)

In terms of the terminal velocity of single bubble in the dispersed vapor phase, UVLO, as represented in Eq. (56.6) and the corresponding drift flux written in Eq. (56.7)

$\begin{array}{rcl} & & {U}_{\mathit{VL}} = {U}_{\mathit{VLO}}{\left (1 - \alpha \right )}^{b-1}\end{array}$
(56.6)
$\begin{array}{rcl} & & {J}_{\mathit{VL}} = {U}_{\mathit{VLO}}{\left (1 - \alpha \right )}^{b}\end{array}$
(56.7)

The term b is some constant of the order of 2 or 3. b takes on values from 3 for very minute bubbles to 2 for somewhat larger bubbles.

To determine the terminal velocity of individual bubbles rising, UVLO, the first step here is to determine the radius of the bubble, R. The buoyancy force, Fb which propels the gas through the liquid is considered and expressed in Eq. (56.8)

${F}_{b} = \frac{4} {3}\pi {R}^{3}g\left ({\rho }_{ L} - {\rho }_{V }\right )$
(56.8)

where R is the radius of the bubble, g is the acceleration due to gravity and ρ L , ρ V is the density of the liquid and vapor, respectively. The surface tension force F σ, is also considered

${F}_{\sigma } = 2\pi R\sigma $
(56.9)

where σ is the surface tension of the liquid and vapor interface

$\sigma = \frac{1} {2}\left [\left ({\rho }_{L} - {\rho }_{V }\right )\mathit{gR}\left (H + \frac{R} {3} \right )\right ]$

where R is the radius of the bubble and H is the rise of the bubble [11].

Equating the two forces of Eqs. (56.8) and (56.9) gives a formula for R, to be

$R ={ \left [ \frac{3\sigma } {2g\left ({\rho }_{L} - {\rho }_{V }\right )}\right ]}^{\frac{1} {2} }$
(56.10)

The second step here is to determine UVLO. This is achieved by equating the drag force. FD to the buoyancy force, Fb in Eq. (56.8)

${F}_{D} = \frac{{C}_{D}\pi {R}^{2}{\rho }_{L}{U}_{\mathit{VLO}}^{2}} {2}$
(56.11)

where CD is the drag coefficient.

Equating Eqs. (56.8) to (56.11) generated Eq. (56.12)

${U}_{\mathit{VLO}} ={ \left [\frac{8\mathit{Rg}\left ({\rho }_{L} - {\rho }_{v}\right )} {3{\rho }_{L}{C}_{D}} \right ]}^{\frac{1} {2} }$
(56.12)

When Eq. (56.12) is substituted into Eq. (56.6) and Eq. (56.7), the values of the volumetric drift flux and the relative velocity of the bubble and liquid interface would be obtained.

However, Holman [13] was of the opinion that when a liquid is heated above the saturation temperature, boiling occurs and the heat flux will depend on the difference in temperature between the surface and the saturation temperature. Zuber and Findlay [21] proposed an equation to determine the peak heat flux in nucleate boiling as expressed in Eq. (56.13)

${ \left ( \frac{q} {A}\right )}_{\max } = \frac{\pi } {24}{h}_{\mathit{fg}}{\rho }_{v}{\left [\frac{\sigma g\left ({\rho }_{L} - {\rho }_{V }\right )} {{\rho }_{v}^{2}} \right ]}^{\frac{1} {4} }{\left (1 + \frac{{\rho }_{V }} {{\rho }_{L}} \right )}^{\frac{1} {2} }$
(56.13)

where q is the heat flux, A is the surface area of the pipe, represented in Eq. (56.14)

$A = \pi \mathit{dL}$
(56.14)

d and L is the diameter and length of the pipe respectively. The heat transfer coefficient, hfg is expressed in Eq. (56.15) as

${h}_{\mathit{fg}} = 2.54{\left ({T}_{V } - {T}_{l}\right )}^{3}{e}^{ \frac{p} {1.551} }(W\left /\right.{m{}^{2}}^{\circ }C)$
(56.15)

(5 < P < 170 atm)

where P is the pressure in meganewtons per square meter. Tv, TL are the vapor and liquid temperature, respectively.

Holman [13] stated that in saturated boiling, when the bubbles break away from the surface because of the buoyancy action, the bubbles move back into the body of the liquid. This results when the temperature of the surrounding liquid is lower than the saturated temperature in the bubble. This can be explained by deriving an expression for the pressure gradient that exists between the interface of the vapor and liquid phase.

The pressure force Fp and the surface tension force, F σ are considered at equilibrium

${F}_{\rho } = \pi {R}^{2}\left ({P}_{ V } - {P}_{l}\right )$
(56.16)

where, Pv is the vapor pressure inside the bubble and PL is the liquid pressure.

Equating Eqs. (56.9) to (56.16) generated Eq. (56.17)

${P}_{v} - {P}_{L} = \frac{2\sigma } {R}$
(56.17)

Holman [13] was of the opinion that Eq. (56.17) indicates that when the pressure inside the bubble is reduced, the corresponding vapor temperature will also reduce. This implies that the bubble will rise and move further away from the heat source to where the liquid temperature is lower. This means that heat is conducted out of the bubble and the vapor inside the bubble condenses and collapses back to the liquid especially in a forced convective condition.

In this condensed state, to determine the drift flux and relative velocity of the vapor–liquid interface. The terminal velocity, UVLO should be obtained.

Here, the net gravitational force, Fg is equated to the drag force, FD

${F}_{g} = \frac{4} {3}\pi {R}^{3}g\left ({\rho }_{ L} - {\rho }_{V }\right )$
(56.18)

Eq. (56.18) is same as Eq. (56.8)

${F}_{D} = \frac{{C}_{D}\pi {R}^{2}{\rho }_{v}{U}_{\mathit{VLO}}^{2}} {2}$
(56.19)

Equating Eqs. (56.18) to (56.19) generated Eq. (56.20)

${U}_{\mathit{VLO}} ={ \left [\frac{8\mathit{Rg}\left ({\rho }_{L} - {\rho }_{v}\right )} {3{\rho }_{v}{C}_{D}} \right ]}^{\frac{1} {2} }$
(56.20)

The value for R, as determined by Zuber et al. [20] is expressed in Eq. (56.21)

$R \approx \lambda \alpha {\left [ \frac{\sigma } {g\left ({\rho }_{L} - {\rho }_{V }\right )}\right ]}^{\frac{1} {2} }$
(56.21)

where λ is the wavelength in its unstable state related to Rayleigh–Taylor unstable surface and it is assumed to be equal to the size of water droplets at the vapor/liquid interface [9].

Sun and Lienhard [16] proposed an equation for determining qL, to be

${q}_{L} = \frac{0.061} {K}$
(56.22)

where

$K = \frac{d} {{\left [ \dfrac{\sigma } {g\left ({\rho }_{l} - {\rho }_{v}\right )}\right ]}^{\frac{1} {2} }}$
(56.23)

where d is the diameter of the tube. Equation (56.23) should be used when K < 2. 3, however where K < 0. 24, there is no nucleate boiling.

56.3 Drift Flux Models as Applied to Wear Rate in Horizontal Pipelines

In this case, the wear rate effect on the interface between the volume fraction of particles immersed in a transport fluid and the internal walls of a pipeline have been studied [1]. However, the Eulerian continuum flow model, the particle equation of motion and the erosion prediction equation are explained here in detail.

56.3.1 The Continuous Model

This model describes the behaviour of fluid flow patterns in a continuous phase. In this phase the conservation equations for mass and momentum in combination with transport equations for a turbulence model are applied. Tian [17] was of the opinion that in CFD model equations, governing equations are fundamentally based on fluid dynamics, which represents the mathematical statements of the conservation law of physics. These laws have been derived from the fact that certain measures must be conserved in a particular volume, known as a control volume. The governing equations for axisymmetric turbulent flow were expressed as follows [4, 18].

$\frac{\partial } {\partial {x}_{j}}\left (\rho {u}_{j}\right ) = 0$
(56.24)

where Uj is the average or mean velocity component and ρ is the fluid density.

Equation (56.24) is expanded as expressed in Eq. (56.25)

$\frac{\partial } {\partial {x}_{j}}\left ({\rho }_{f}{u}_{\mathit{if }}{u}_{\mathit{jf }}\right ) = \frac{-\partial {P}_{t}} {\partial {x}_{i}} + \frac{\partial } {\partial {x}_{j}}\left (\mu \frac{\partial {u}_{i}} {\partial {x}_{j}}\right ) - \frac{\partial } {\partial {x}_{j}}\left (\rho {u}_{i}{u}_{j}\right )$
(56.25)

where P is the static pressure and the stress tensor was further expanded as written in Eq. 56.26 as proposed by Hinze [12]

$-\rho {u}_{i}{u}_{j} = \left [{\mu }_{f,t}\left (\frac{\partial {u}_{i}} {\partial {x}_{j}} + \frac{\partial {u}_{j}} {\partial {x}_{i}} \right )\right ] -\frac{2} {3}\rho k{\delta }_{\mathit{ij}}$
(56.26)

where δ ij is the Kronecker delta and μ gt is the eddy viscosity or turbulent viscosity. The turbulent quantity, K which is the Kinetic energy of turbulence expressed in Cartesian tensor notation as

$K = \frac{1} {2}\overline{{u}_{i}{u}_{i}}$
(56.27)

Can be simply expressed as

$K = 0.01{u}_{f}^{2}$
(56.28)

The rate of production of turbulent kinetic energy, Pk is given by

${P}_{k} = -{\rho }_{f}\overline{{u}_{i}^{1}{u}_{j}^{1}}\frac{\partial {u}_{\mathit{if }}} {\partial {x}_{j}} \ \mathrm{and}$

the rate of dissipation of turbulent energy, ε is expressed as

$\epsilon = \frac{{\mu }_{ft}} {{\rho }_{t}} \overline{\left (\frac{\partial {u}_{i}^{1}} {\partial {x}_{j}} \right )\left (\frac{\partial {u}_{i}^{1}} {\partial {x}_{j}} \right )}$
(56.29)

ε can simply be calculated from Eq. (56.30)

${\mu }_{ft} = {\rho }_{f}{C}_{\rho }\frac{{K}^{2}} {\epsilon }$
(56.30)

where C ρ is the specific heat capacity of the fluid and was given by Reynolds [14] as 0.0845.

The turbulent viscosity, if not given can be determined from Eq. (56.31)

${\mu }_{f,t} = \upsilon {\rho }_{t}$
(56.31)

where υ is the Kinematic viscosity of the fluid. Considering Eqs. (56.28)–(56.31), the term K can be obtained and substituting the value of K into Eq. (56.28), Um would be obtained.

The mass rate of flow, \dot{m} is calculated from Eq. (56.32)

$\dot{m} = {\rho }_{t}{u}_{f}A$
(56.32)

where A is the cross sectional area of the pipe given as πdl, where d and l have been previously defined.

The mass flow velocity, G is given in Eq. (56.33) as

$G = \frac{\dot{m}} {A} = {\rho }_{f}{u}_{m}$
(56.33)

The term Uf is the fluid terminal velocity.

56.3.2 Particle Equation of Motion

In deriving this equation, two assumptions were employed

  1. 1.

    The solid particles do not interact with each other.

  2. 2.

    The influence of particle motion on the fluid flow field is very small and could be neglected.

These assumptions were also adopted by Edwards et al. [10] and Wallace et al. [19] in their various research works.

The governing particle equation of motion is given as

$\frac{d{u}_{p}} {\partial t} = {F}_{D}({u}_{f} - {u}_{p}) + \frac{g\ ({\rho }_{\rho } - {\rho }_{f})} {{\rho }_{\rho }} + \sum \nolimits {F}_{x} + \Delta P + {F}_{d}$
(56.34)

where F D (u f − u p ) is the drag force per unit particle mass and F D is given by

${F}_{D} = \frac{3{C}_{D}\mu {\mathrm{Re}}_{p}} {4{\rho }_{p}{d}_{p}^{2}}$
(56.35)

where ρ p is the density of particle material, dp is the particle diameter, up is the particle velocity and Rep is the relative Reynolds number written as expressed in Eq. (56.36)

${ \mathrm{Re}}_{p} = \frac{{\alpha }_{f}{\rho }_{f}{d}_{p}({u}_{p} - {u}_{f})} {{\mu }_{f,t}}$
(56.36)

where α f is the volume fraction of the fluid, Up is the particle velocity and Cd is the drag coefficient, this is defined in Eq. (56.37)

${ C}_{D} = \left \{\begin{array}{@{}l@{\quad }l@{}} 0.44 \quad &R{e}_{p} > 1000 \\ \dfrac{24} {R{e}_{p}}\left (1.0 + \dfrac{1} {6}{\mathrm{Re}}_{p}^{0.66}\right )\quad &{\mathrm{Re}}_{p} \leq 1000 \\ \quad \end{array} \right.$
(56.37)

\(\frac{g({\rho }_{p}-{\rho }_{f})} {{\rho }_{p}}\) represents the particle buoyancy force that keeps the particles in continuous flow suspension when it is at equilibrium with the pressure force Fd is the Saffman lift force proposed by Saffman [15].

∑F x is the increase in momentum flux in the fluid around the particles. This could be represented as

$\sum \nolimits {F}_{x} = \frac{\partial (m{u}_{T})x} {\partial \tau }$
(56.38)

where UT is the friction velocity and τ is the shear force due to flow.

Most of the energy loss takes place during the algebraic particle-wall collision at the interface. This causes the disintegration of particles and pipewall deformation. However, large energy loss due to molecular level forces, such as adhesion is not reversible and occurs primarily during rebound [6, 17].

The friction velocity or relative velocity at the interface can be calculated from Eq. (56.39)

${U}_{T} = \sqrt{\frac{{\tau }_{w } } {{\rho }_{f}}}$
(56.39)

where τ w is the wall shear stress

$\mbox{ The pressure gradient,}\Delta \!\,P = f \frac{L} {d} {\rho }_{f}\frac{{U}_{T}^{2}} {2g}$
(56.40)

where f is the frictional force, L is the length of the pipe and d is the diameter of the pipe.

Here Eqs. (56.36)–(56.41) is used to determine the velocity of the particle, Up.

The value of Up is used to compute the Erosion wear rates of the pipeline depending on the angle of contact between the particle transported by the fluid and the internal pipewall, α.

56.3.3 The Erosion Prediction Equation

The erosion prediction equations suggested by Wallace et al. [19] were used because of their simplicity and high level of accuracy for the prediction of erosion rates. These equations are given in Eqs. (56.41) and (56.42).

$E = \left \{\frac{\frac{1} {2}{U}_{p}^{{2}\cos }{}^{2}\alpha \sin 2\alpha } {\Upsilon } + \frac{\frac{1} {2}{U}_{p}^{{2}\sin}{}^{2}\alpha } {\sigma } \right \}$
(56.41)

For α ≤ 45∘

And

$E = \left \{\frac{\frac{1} {2}{U}_{p}^{{2}\cos }{}^{2}\alpha }{\Upsilon } + \frac{\frac{1} {2}{U}_{p}^{{2}\sin }{}^{2}\alpha }{\sigma } \right \}$
(56.42)

For α > 45∘.

Where Y and σ are the cutting wear and deformation wear coefficients having the values 33,316.9 and 77,419.7, respectively.

From the study made by Bitter [5] peak erosion rates have been measured to occur at impact angles of 25–30∘, indicating that cutting wear dominates. The difference between Up and Uf gives the drift flux velocity. The term of U f − U p can be used to replace the term U p in Eqs. (56.41) and (56.42) to obtain erosion rates based on the relative motion of fluid/particle interface.

56.4 Conclusion

The parametric analysis of these select drift flux models has introduced a wide range of applications. The application of drift flux models of fluid/fluid flow in a stagnant position and fluid/particle flow in a continuum phase to pipeline have been clearly expressed. The effect of interfacial motion between the fluid-particle and pipe wall as related to the relative motion of the fluid velocity and the particle velocity was applied to the erosion wear equation to determine the resistance of the pipe wall material to wear. These models therefore act like templates for solving pipeline engineering problems whenever they arise or when anticipated.