Two-phase flow

The BWRX-300 is the reason Rohan's group needs to be fluent in two-phase flow. In a boiling water reactor the coolant boils in the core by design, and in a CANDU some boiling occurs at the outlet of the hottest channels at full power, so the outlet feeders carry a steam-water mixture. "In single-phase flow, you have one fluid and one velocity," Rohan says. "Once it boils you have two fluids, two velocities, a shape to the flow, and a density that can change by a factor of ten along one channel. Most of the difficult questions in reactor thermal hydraulics live here."

Describing a two-phase mixture

Several quantities are needed to describe a flowing mixture of steam and water, and confusing them is the commonest mistake in two-phase calculations.

Quality is a mass ratio and void fraction is an area or volume ratio. Because steam is much less dense than water, a small quality gives a large void fraction. At 7 MPa, near BWR conditions, saturated water has a density of about 740 kg/m³ and saturated steam about 36.5 kg/m³, a ratio of about 20. A flow quality of only 5 percent corresponds to a void fraction of roughly 45 to 50 percent. At 15.5 MPa, PWR pressure, the ratio is only about 6, so the same quality gives much less void.

The equilibrium quality \(x_e\) comes straight from the energy balance of Lesson 3: once the coolant enthalpy is known, \(x_e = (h - h_f)/h_{fg}\). It is negative for subcooled liquid. The actual flow quality \(x\) can differ from \(x_e\), because near the onset of boiling, steam bubbles form on a hot wall while the bulk liquid is still subcooled. Lesson 6 explains how that subcooled boiling region is handled.

The relation between quality and void

If the two phases moved at the same velocity, the void fraction would follow from quality and densities alone. This is the homogeneous model:

In reality the vapour usually moves faster than the liquid, because it is lighter and is pushed by buoyancy and by the pressure gradient. With a slip ratio \(S\),

so slip greater than 1 lowers the void fraction for a given quality. The homogeneous model therefore overestimates void and underestimates mixture density, which matters for pressure drop, for natural circulation driving head and, in a boiling water reactor, for neutron moderation and the void reactivity feedback of NUC-300.

The drift-flux model

The drift-flux model of Zuber and Findlay is the workhorse for void fraction in vertical channels. It writes the average vapour velocity as

where \(j = j_g + j_f\) is the total volumetric flux. The distribution parameter \(C_0\), typically about 1.1 to 1.2, accounts for the tendency of bubbles to gather where the velocity is highest, near the channel centre. The drift velocity \(V_{gj}\) accounts for the local rise of vapour relative to the liquid because of buoyancy. For churn-turbulent bubbly flow,

which is about 0.15 to 0.2 m/s for water at BWR pressures. Rearranging gives the void fraction directly from the superficial velocities, \(\alpha = j_g / (C_0 j + V_{gj})\). Boiling water reactor design codes use refined drift-flux correlations fitted to bundle data across the full range of pressure, flow and void.

# Quality, void fraction and two-phase density: homogeneous model vs drift flux (Zuber-Findlay).
import math
rho_f, rho_g, sigma, g = 740.0, 36.5, 0.0174, 9.81    # water/steam near 7 MPa (rounded)
G = 1400.0                                             # kg/m2-s, illustrative BWR-type channel
C0 = 1.13
Vgj = 1.41 * (sigma * g * (rho_f - rho_g) / rho_f**2) ** 0.25   # churn-turbulent drift velocity
print(f"drift velocity Vgj = {Vgj:.3f} m/s, C0 = {C0}")
def alpha_hom(x): return 1 / (1 + (1 - x) / x * rho_g / rho_f)
def alpha_df(x):
    jg = G * x / rho_g; jf = G * (1 - x) / rho_f
    return jg / (C0 * (jg + jf) + Vgj)
for x in (0.01, 0.05, 0.10, 0.15, 0.20):
    ah, ad = alpha_hom(x), alpha_df(x)
    S = (x / (1 - x)) * (rho_f / rho_g) * ((1 - ad) / ad)
    rho_m = ad * rho_g + (1 - ad) * rho_f
    print(f"x = {x:4.2f}: void hom {ah:4.2f}, drift flux {ad:4.2f}, slip {S:4.2f}, mixture density {rho_m:5.0f} kg/m3")
# Homogeneous two-phase friction multiplier (same friction factor)
for x in (0.05, 0.10, 0.20):
    phi2 = 1 + x * (rho_f / rho_g - 1)
    print(f"x = {x:4.2f}: homogeneous multiplier phi_lo^2 = {phi2:4.1f}")

drift velocity Vgj = 0.172 m/s, C0 = 1.13
x = 0.01: void hom 0.17, drift flux 0.14, slip 1.25, mixture density   641 kg/m3
x = 0.05: void hom 0.52, drift flux 0.44, slip 1.36, mixture density   431 kg/m3
x = 0.10: void hom 0.69, drift flux 0.60, slip 1.52, mixture density   320 kg/m3
x = 0.15: void hom 0.78, drift flux 0.68, slip 1.70, mixture density   263 kg/m3
x = 0.20: void hom 0.84, drift flux 0.73, slip 1.90, mixture density   228 kg/m3
x = 0.05: homogeneous multiplier phi_lo^2 =  2.0
x = 0.10: homogeneous multiplier phi_lo^2 =  2.9
x = 0.20: homogeneous multiplier phi_lo^2 =  4.9

Flow regimes

Steam and water arrange themselves in recognizable patterns, or flow regimes, that depend on the flow rates, pressure, channel size and orientation. In a vertical heated channel, the sequence is usually as follows.

In horizontal channels such as CANDU fuel channels and feeders, gravity separates the phases, and stratified and wavy regimes appear at low flows, with vapour along the top. At the high flows of normal CANDU operation the mixture stays well mixed, but during some accidents and in low-flow conditions, stratification can leave the upper elements of a bundle exposed to vapour, which is a specific concern of CANDU safety analysis.

Flow regime maps, plotted in terms of superficial velocities or of mass flux and quality, summarise where each regime occurs. System codes use them to choose the right correlations for interfacial friction, heat transfer and wall drag in each region of the plant.

Two-phase pressure drop

Friction in two-phase flow is usually larger than in liquid flow at the same mass flux, because the mixture has a higher average velocity and the interfaces add turbulence. The most common approach multiplies the friction pressure drop that the whole flow would have as liquid by a two-phase multiplier \(\phi_{lo}^{2}\):

For homogeneous flow with the same friction factor, \(\phi_{lo}^{2} = 1 + x(\rho_f/\rho_g - 1)\), which at 7 MPa and 10 percent quality is about 2.9. The Martinelli-Nelson correlation, based on separated-flow ideas and adjusted for pressure, and later correlations such as Friedel's and Chisholm's, are used in design. Acceleration pressure drop, small in liquid flow, becomes significant as the density falls along the channel. Two-phase form losses at grids and orifices are also larger than single-phase values and use their own multipliers.

Instabilities

Two-phase flow in heated channels can oscillate. Two kinds matter in reactors. A static instability, the Ledinegg excursion, occurs when the channel's pressure drop curve has a region where pressure drop falls as flow rises; the flow can then jump suddenly to a much lower value. Dynamic density-wave oscillations occur because changes in inlet flow take time to travel up the channel as changes in void, so the pressure drop responds with a delay; with the wrong combination of delay and gain, the flow oscillates with a period of one to a few seconds. Inlet orifices, which add single-phase resistance, are the main design cure. In boiling water reactors, density-wave oscillations couple with void reactivity feedback and can produce power oscillations, so BWR operating maps exclude regions of low flow and high power, and stability detection and suppression systems are installed.

Rohan notes that the BWRX-300's natural-circulation design, with a tall chimney above the core, has been analysed carefully for stability. Natural-circulation boiling water reactors are susceptible to density-wave oscillations at startup, when the chimney fills with void for the first time, and their startup procedures are designed to pass through that region safely.

Critical flow

When a pipe carrying high-pressure water breaks, the discharge flow rises as the downstream pressure falls, until it reaches a maximum and stops responding to further drops in back pressure. This choked or critical flow limits how fast coolant can leave a broken reactor system. For saturated or subcooled water, flashing to steam in the break limits the flow, and models such as the homogeneous equilibrium model, the Moody model and the Henry-Fauske model estimate it. Critical flow sets the blowdown rate in a loss-of-coolant accident, the subject of Lesson 11.

Measuring void fraction

Void fraction is hard to measure in a hot, high-pressure channel, and every two-phase correlation is only as good as the measurements behind it. Gamma densitometers send a narrow beam of gamma rays through the channel and infer the mixture density from the attenuation, using the same exponential law as in shielding. Impedance probes and wire-mesh sensors measure the electrical conductivity of the mixture across a grid of points, giving cross-sectional pictures of the flow. Quick-closing valves trap a section of flow so that the liquid inside can be weighed. Neutron and X-ray radiography give pictures of rod bundle flows in test sections. For BWR bundles, large tests with electrically heated full-scale bundles at reactor pressure have produced void data that vendors use to tune their drift-flux correlations.

In operating reactors, void cannot be measured directly. In a BWR it is inferred from the agreement between measured and calculated power distributions, since the in-core detectors respond to the local neutron flux, which depends on void. In a CANDU, channel outlet temperatures and pressures, along with the heat balance, show which channels are boiling at the outlet.

Subcooled void and the bulk energy balance

A careful designer distinguishes three quantities near the start of boiling: the equilibrium quality \(x_e\) from the energy balance, which is still negative; the true flow quality \(x\), which is slightly positive because steam is being generated at the wall; and the void fraction, which is larger than either would suggest because the bubbles hug the wall where the liquid is slow. Profile-fit models, such as Levy's or the one derived from Saha-Zuber, give the true quality as a smooth function of \(x_e\) that starts at zero at the onset of significant void and approaches \(x_e\) further downstream. Codes use this relation to compute void in the subcooled boiling region of PWR hot channels, BWR bundle entrances and CANDU channel outlets.