SCMHTCClosureBase

Overview

This is the base class from which all the convective heat transfer coefficient closure models inherit. It provides the toolset for the calculation of the Nusselt number and finally the heat transfer coefficient.

Channel-to-Pin and Channel-to-Duct Heat Transfer Modeling

The pin/duct surface temperature are computed via the convective heat transfer coefficient as follows:

where:

  • is the surface temperature for the pin at a height

  • is the number of subchannels neighboring the pin

  • is the bulk temperature for a subchannel neighboring the pin at a height

  • is the linear heat generation rate for the pin at a height

  • is the pin diameter at a height

  • is the convective heat transfer coefficient for a subchannel neighboring the pin at a height .

For the duct, the duct surface temperature is defined as follows:

where:

  • is the duct surface temperature at a height

  • is the bulk temperature of the subchannel next to the duct node

  • is the heat flux at the duct at a height

  • is the convective heat transfer coefficient for the subchannel next to the duct node at a height

In both cases, the heat exchange coefficients are computed using the Nusselt number (Nu) as follows:

where:

  • is the thermal conductivity of the subchannel neighboring the structure

  • is the hydraulic diameter of the subchannel neighboring the structure

commentnote

Currently there is no subchannel-to-duct heat transfer model implemented for square assemblies (square ducts). It only exists for hexagonal assemblies (hexagonal ducts). The subchannel-to-pin model is available for both hexagonal and square assemblies.

General Expression for the Nusselt number

The laminar, turbulent and transition regimes use different coefficients for the Nusselt number.

The bounding laminar and turbulent Reynolds numbers for the turbulent transition are defined as follows Chen et al. (2018):

where:

  • P is the pitch

  • D is the pin diameter

The flow is laminar if , turbulent if , and in the transition regime if it lies in between. The modeling of each regime is explained below.

Laminar Nusselt Number

The following relation is used depending on the subchannel:

The values defined here are chosen based on engineering judgement and the values for laminar flow in a circular/square tube with constant surface temperature (3.66/3.091) and laminar flow in a circular/square tube subjected to constant surface heat flux (4.36/3.54). The heat transfer boundary condition for the interface area between subchannels is considered to be that of constant temperature. The heat transfer boundary condition for the fuel-pin surface and duct surface is considered to be that of constant heat-flux.

For a center subchannel that has approximately half contact with circular fuel pins and half with flat interface, the laminar Nusselt number is chosen to be the average of the two: . Similar logic has been applied for the edge and corner subchannels. For the edge: . For the corner:

Correlation for Turbulent Nusselt Number

The modeling of the Turbulent Nusselt number and consequently of the convective heat transfer coefficient h is defined by the user. The closure models available to the user that are implemented in SCM are the following:

According to Todreas and Kazimi (2021), the correlations of Borishanskii and Schad-modified yield the best agreement over the entire range of P/D values. The Graber and Rieger correlation appears to significantly overpredict the heat transfer coefficient if extended beyond the published range of applicability P/D ≤ 1.15. The Kazimi and Carelli correlation underestimates Nu at high values of P/D. A summary of the closure models available in SCM with the range of validity, is presented in Table Table 1 below:

Table 1: Convective heat transfer coefficient, closure models availabel in SCM.

NameGeometryFluid TypeRange of ValidityRod SpacingEntrance EffectsReferences
Dittus-BoelterCircular tubesOrdinary non-metallic fluidsBare (no wire grid or spacers)NoneDittus (1930), McAdams and McAdams (1954)
Dittus-Boelter/PresserRod bundles triangular and squareOrdinary non-metallic fluidstriangular: , square: Bare (no wire grid or spacers)NonePresser (1967)
Dittus-Boelter/WeismanRod bundles triangular and squareWatertriangular: , square: Bare (no wire grid or spacers)NoneWeisman (1959)
Modified GnielinskiConcentric annular ducts / Circular tubesExtended to most fluidsExtended from original to: Bare (no wire grid or spacers)NoneGnielinski (1975)
Kazimi-CarelliTriangular Rod BundleMetallic Fluids, Wire-wrapped (no grid or spacers)NoneKazimi and Carelli (1976)
Schad-ModifiedTriangular Rod BundleMetallic Fluids and ,Bare (no grid or spacers)NoneKazimi and Carelli (1976)
Graber-RiegerTriangular Rod BundleMetallic Fluids and ,Bare (no grid or spacers)NoneGraber et al. (1972)
BorishanskiiTriangular Rod BundleMetallic Fluids and ,Bare (no grid or spacers)NoneBorishanskii et al. (1969)
commentnote

The following features are not currently implemented: spacer grids in quad lattices, entrance region effects. Though it is important to point out that, the value of the turbulent Nu is insensitive to the boundary conditions for Pr > 0.7 Todreas and Kazimi (2021).

Transition Regime

In the transition region we use a linear interpolation. The linear interpolation weight is defined as follows:

Then, the Nusselt number in the transition regime is defined by linearly interpolating the laminar Nusselt number and the turbulent one, which is defined with the chosen correlation for the pin or duct, as follows:

References

  1. VM Borishanskii, MA Gotovskii, and EV Firsova. Heat transfer to liquid metals in longitudinally wetted bundles of rods. Soviet Atomic Energy, 27(6):1347–1350, 1969.[BibTeX]
  2. SK Chen, YM Chen, and NE Todreas. The upgraded cheng and todreas correlation for pressure drop in hexagonal wire-wrapped rod bundles. Nuclear Engineering and Design, 335:356–373, 2018.[BibTeX]
  3. Frederick William Dittus. Heat transfer in automobile radiators of the tubular type. Univ. of California Pub., Eng., 2(13):443–461, 1930.[BibTeX]
  4. Volker Gnielinski. Neue gleichungen f"ur den w"arme-und den stoff"ubergang in turbulent durchstr"omten rohren und kan"alen. Forschung im Ingenieurwesen A, 41(1):8–16, 1975.[BibTeX]
  5. H. Graber, M. Rieger, and O. E. Dwyer. Experimental study of heat transfer to liquid metals flowing in-line through tube bundles. In Fourth International Seminar on Heat and Mass Transfer in Liquid Metals. Trogir, Yugoslavia, 1972. EURATOM, Ispra, Italy, Pergamon Press. Held 6 September 1971. URL: https://www.osti.gov/biblio/4398107.[BibTeX]
  6. MS Kazimi and MD Carelli. Clinch river breeder reactor plant. heat transfer correlation for analysis of crbrp assemblies. Technical Report, Westinghouse Electric Corp., Madison, PA (United States). Advanced Reactors Div., 1976.[BibTeX]
  7. William Henry McAdams and William H McAdams. Heat transmission. Volume 3. McGraw-hill New York, 1954.[BibTeX]
  8. K Presser. Waermeuebergang und druckverlust an reaktorbrennelementen in form laengsdurchstroemter rundstabbuendel.(heat transfer and pressure loss of reactor fuel elements in the form of longitudinal flow through round rod bundles). Technical Report, Kernforschungsanlage, Juelich (West Germany). Institut fuer Reaktorbauelemente, 1967.[BibTeX]
  9. Neil E Todreas and Mujid S Kazimi. Nuclear systems volume I: Thermal hydraulic fundamentals. CRC press, 2021.[BibTeX]
  10. Joel Weisman. Heat transfer to water flowing parallel to tube bundles. Nuclear Science and Engineering, 6(1):78–79, 1959.[BibTeX]