Heat Transfer

Assignment :

Finish Homework 8,
Read the section of the TRACE Theory Manual titled "Wall Heat Transfer Models", (Read pp. 245-270 carefully to understand special considerations necessary to make the correlations useful, scan pp. 271-350 to get a feel for transitions between heat transfer regimes and the scatter in data being correlated)

So far we've talked about two-phase flow, and numerical methods used to model it. How do we get two phase flow?

Dissolved gases in beer or soda
Entrained bubbles in fast moving river
Rain storm
Depressurization of hot liquid
Addition of heat to liquid (as in a coffee pot)

This lecture focuses on the last of these options, and how we model the heat source term to the fluid energy equations. In the context of the differential flow equations, the energy term is written as a source of energy per unit volume. When the source (or sink) of heat is transfer to a structure (pipe wall, core rod, etc.) this term is generated by calculating the metal wall surface area per unit volume and multiplying by the mean surface heat flux. The standard form for obtaining heat flux is to consider it as a product of a heat transfer coefficient and the difference between wall surface temperature and fluid temperature.

heat flux = h (T _wall - T _fluid )

The heat transfer coefficient (h) is generally obtained from a standard set of local assumptions. That is it is a function of the adjacent wall temperature and fluid conditions. This ignores classic entrance effects including the impact of grid spacers in the core. One thing to watch for in reactor safety codes are models requiring input of grid spacer locations. If properly implemented these can improve your predictions of fuel rod temperatures, particularly during reflood. Basic heat transfer correlations also largely ignore the proximity of a boiling liquid surface and the associated field of projected droplets. Review documentation carefully of any safety code that you use to see if such effects are included, and run it against reflood experiments such as FLECHT SEASET and the PSU RBHT experiments.

Pay careful attention to the context within which I use the letter "h" in equations. One major source of confusion to beginning students in thermal-hydraulics in the double use of "h" for both enthalpy and heat transfer coefficients.

The Boiling Curve

The TRACE boiling curve is fairly conventional in form, with specific correlations selected recently (2006) based upon a careful survey of literature on heat transfer experiments.

Convection to subcooled fluid (wall temperature is less saturation temperature) uses the maximum of the Gnielinski correlation (rather than the more standard Dittus-Boelter correlation) for turbulent forced convection, a constant Nusselt number of 4.36 for laminar forced convection and Grashof based Natural Circulation (watch out for the length scale). Results are adjusted for differences between average and near-wall fluid properties. In rod bundles the forced convection correlations are replace be on developed by El-Genk.
Nucleate Boiling occurs when wall temperature is greater than T_ONB (slightly greater than the saturation temperature) but less than the temperature for critical heat flux. TRACE uses the Gorenflo correlation to determine the heat transfer coefficient. The upper bound of the nucleate boiling regime (critical heat flux temperature, T_CHF) is determined from the AECL-IPPE CHF look-up table.
Transition boiling (wall temperature is greater than the temperature for critical heat flux and less than the temperature of minimum stable film boiling ) uses quadratic interpolation between Nucleate and Film boiling
Stable film boiling occurs when the wall temperature is greater than the temperature of minimum stable film boiling (T_min). T_min is obtained from th Groeneveld-Stewart correlation. For film boiling with a liquid core flow a correlation was developed for TRACE by Joe Kelly of the USNRC. For dispersed droplet flow TRACE uses a modified version of the normal convective correlations (Gnielinski etc.), including corrections for turbulence enhancements by droplets and radiative heat transfer to droplets.
Condensation occurs when vapor is present and the wall temperature is less than the saturation temperature. Here TRACE uses an approach based upon calculation of a local film thickness, calculation of the heat transfer coefficient between the liquid film and the wall, and calculation of the interfacial heat transfer coefficients between the liquid film and steam.

Temperatures to Remember

All temperatures referenced in the boiling curve are metal wall temperatures at the metal surface. The temperature of onset of nucleate boiling (T_ONB) must be greater than the saturation temperature. The amount that it is greater is relatively small. See the TRACE theory manual for typical values. The temperature of critical heat flux (T_CHF) is the temperature at which so much vapor is produced at the wall, that the supply of liquid to the hot wall is restricted, reducing the surface heat flux. The temperature of minimum stable film boiling (T_min) is the temperature at which a stable film of vapor blankets the wall. Liquid can no longer boil at the wall, and so the heat flux is at a relatively low value. Boiling takes place at the interface between the vapor film and bulk liquid. Heat flux increases again for higher temperatures simply due to the increase in the difference between wall temperature and the fluid temperature and increased infrared radiation from the wall.

Wall Condensation

Condensation was not a major issue during most of the history of major reactor safety code development, since plant details beyond the inlet to the turbine were not a major consideration in significant accidents. However, the new passive reactor designs place a heavy emphasis on condensation as part of the long term accident control strategy.

The ESBWR (Economic Simplified Boiling Water Reactor) can run steam from the plant primary through the interior of tube banks called isolation condensers (IC's). Steam from the containment can also be condensed in similar tube banks called the Passive Containment Cooling System (PCCS). The heat sink for the exterior of both of these tube banks is a large pool of water (~3650 m3) located near the top of the containment.
The AP1000 (Advanced PWR 1000 Mwatts) can route steam from the primary system through a similar set of condenser tubes called the Passive Residual Heat Removal (PRHR) system. The heat sink for the exterior of these tubes is a 530,000 gallon tank called the In containment Refueling Water Storage Tank (IRWST) located high in the containment. AP600 also has condensation related to the Passive Containment Cooling System (PCCS). This is simply a steel liner to the containment building with external air and water film flow to provide the ultimate heat sink for the system. Water spray for the exterior of the PCCS steel shell is provided by a PCC Water Storage Tank (PCCWST) located above the shell under the roof of the external containment building.
Condensation in both the ESBWR and AP1000 is complicated by the presence of non-condensible gases (generally air). Air is carried with the condensing steam and deposited at the condensing surface. This build-up is balanced by dispersion through natural turbulent diffusion processes. However, the enhanced air content near the condensing surface significantly suppresses the condensation rate. The complexity of this process makes predictions of the condensation rate very difficult.

Reflood Heat transfer

The usual picture is that a water level exists. Somewhere below the water level conditions at the clad change from nucleate boiling below through transition boiling into film boiling over a very short distance. The area of this abrupt change in clad temperature conditions is referred to as the quench front. The distance spanning the quench front is short enough that axial conduction (conduction along the length of the rod's clad) plays a major role in moving heat from the hot clad down to the quenched clad in film boiling. This permits the location of the quench front to move slowly up the rod. Without some continued supply of liquid, this motion will eventually stall out.

Boiling flings drops up from the liquid surface. Larger drops move a short distance and fall back or impact the rod surface near the liquid surface. Some drops are small enough that they are carried with the steam. In either case the drops contribute to the precooling of the clad, and can accelerate the progress of the quench front up through the core. Unfortunately the characteristics of these drops are determined by processes at the liquid surface, and at major obstacles such as grid spacers. The local modeling in TRACE, RELAP, RETRAN, or CATHARE generally determines the properties of the drops (size, drag, etc) based on local conditions at each hydrodynamic mesh cell, and misses key behavior. These codes tend to over predict the arrival time of the quench at high regions of the core. They are conservative, so the USNRC has not been too concerned. TRACE contains an optional reflood heat transfer model that is partially nonlocal. Westinghouse's WCOBRA/TRACE code also include non-local models and improves tracking of drops with a special set of field equations to track mass, energy and momentum of drops in addition to equations that follow any continuous liquid field or falling liquid film. Both of these improved approaches produce more accurate and less conservative results. The result can be that a set of calculations will demonstrate that a plant can be safely operated at a higher power that previously permitted, resulting in significant economic benefits for the plant owners. TRACE should have droplet capabilities in the summer of 2005.

It should be noted that a liquid level need not precede the quench front. If a high enough steam-droplet flow exists, the quench front may move through the upper core while the liquid level remains relatively low in the core.

Key Points

The boiling curve is fundamental to this business. You should know it cold. The concept of reflood heat transfer is also extremely important, and you should have a basic understanding of how it operates. The conduction equation discussed in the next lecture is basic to a wide range of engineering disciplines. You should also know this equation and be familiar with a method to solve it.

The TRACE theory manual is a particularly valuable source of information on two-phase heat transfer. It represents the most recent and most thorough survey of relevant experimental data and available correlations. A review at this level of detail is not likely to occur outside the TRACE project for many years.

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Maintained by John Mahaffy : jhm@psu.edu