Heat Transfer Coefficient (HTC)

Heat Transfer Coefficient (HTC) is a measure of heat transfer across a solid-fluid interface. HTC was conceived as a model reduction approach to calculate heat flux or temperature on solid surfaces for predefined flow geometries via empirical formulas based on dimensional and dimensionless parameters of the case without solving for the fluid flow. Currently, HTC is predominantly employed as a means of offline coupling between a fluid and solid solver. Most empirical HTC formulas assume constant temperature or constant heat flux and feature several dimensional and dimensionless variables that allow for active modification of the HTC values by the solid solver during calculations. In contrast, most HTC coefficients extracted from CFD simulations are constant and only apply to the specific scenario solved to obtain them. In this respect, HTC coefficients extracted by CFD solvers are more akin to plain heat flux rather than the form represented by empirical HTC formulas.

To obtain the most accurate form of HTC representing heat flux for a simple heat transfer case through numerical simulation, you must solve the coupled solid and fluid problems by conducting a full scale simulation, considering temperature dependent fluid and solid properties.

All boundary conditions and source terms must be fully defined and the simulation must reach a steady state for both thermal and flow fields. The heat fluxes and temperature values obtained from this simulation are then used to calculate HTC values on the solid surfaces. Naturally there are drawbacks to this approach. Not all problems reach a steady state and the ones with a transient nature should either rely on a snapshot of HTC per output or use a time averaging method which may reduce the accuracy of the obtained HTC. Furthermore, even for inherently steady cases, the difference in thermal and flow timescales bind the simulation time of the coupled flow and thermal problem to the longer of the two timescales. Satisfying the spatial and temporal resolution requirements of the coupled problem might put further strain on the computational resources. In some cases, the thermal domain may extend further beyond the flow domain, putting additional strain on the coupled solver. In short, solving the full problem is the most accurate method for obtaining HTC representing heat flux for steady state flows. However, it may not be feasible for most practical cases.

While nanoFluidX does not offer the aforementioned option of extracting HTC representing heat flux in its most accurate form, as of 2022, it offers two other forms of HTC by introducing simplifying assumptions. This means the obtained HTC values are accurate, however, only within the constraints of the assumptions made and when particle distribution is satisfactory. To give a brief overview, nanoFluidX offers HTC in the following forms:

• Non-reset partial energy equation (nrpee): The solids are kept isothermal while temperature varies in fluids by solving the (partial) energy equation. The obtained HTC based on temperature gradient at the fluid-solid interface is then averaged over a number of snapshots. The isothermal solid assumption brings this form closer to one of the basic definitions of HTC.
• Flat plate based empirical (emp): The empirical formula for an isothermal flat plate heating a fluid is used to calculate an HTC value based on a locally interpolated velocity. The obtained HTC is then averaged over a number of snapshots.

Calculating HTC is a part of the extractor feature. Extractors offer two forms of output for HTC values. The first output option is the regular VTK polydata format shared with other extractor outputs. To facilitate using the HTC values in other solvers, extractors offer a space delimited text file output option separately.

The text files are placed in extractor_name_txt under EXTRACT directory with a file generated for each output time named surfext_timestepnumber.txt, similar to vtp outptut format. Each text output contains coordinates of the extractor surface centers as well as instantaneous (i) and periodically time averaged (pta) output. A separate text file containing header information is also generated to help in identifying column content. The header file is not regenerated when a recon run is carried out in the same directory as the clean run, that is the header file will not be updated even if the number of columns in the text outputs change after the recon run.

Note that even if all vtp outputs are turned off, vtp, vtm (in case of sync output), and pvd files will still be generated. The vtp files in this case will only contain coordinates, but no other data. pvd files can be used to associate file names with physical simulation times for both vtp and txt files. An easy way to see these values is to open the pvd file in a text editor or in ParaView.

Non-reset Partial Energy Equation (NRPEE)

HTC based on solving non-reset partial energy equation, non-reset partial energy equation HTC (NRPEE HTC), uses the temperature field obtained from the simulation to calculate the HTC based on the following formulas:

$\mathrm{q"}=\mathrm{-\kappa }\left(\frac{\partial T}{\partial n}\right)$

Where $h$ is HTC, $\mathrm{q"}$ is heat flux, $\kappa$ is thermal conductivity, $n$ is wall normal direction and $T$ is temperature while subscripts $w$ and $\inf$ denote wall and bulk, respectively. In essence, this form of HTC represents a heat flux divided by a set of temperature difference.

To calculate NRPEE HTC, the simulation setup must satisfy the following:

• Energy equation must be solved to produce a temperature field in fluid phases. This means simulationParameters.energy_transport must be set to true. Naturally this introduces some overhead in the simulation compared to the isothermal fluid simulations.
• Each solid may have a different temperature value. However, it is recommended for all solids to be made isothermal by setting phases.phase.evolve_temp to false. While being isothermal is not required for NRPEE HTC evaluation procedure to function, it is recommended as the output text files do not contain temperature values. Additionally, non-isothermal solids deactivate subphasing and require complete discretization of the solids (beyond three layers), thus imposing substantial performance penalties.
• Bulk temperature must be defined for each extractor via extractors.extractor. extractor_htc_nrpee_bulktemp keyword. This value should be sufficiently different from the temperature of the solid phase the extractor is attached to.

Additional notes:

• When setting solids to isothermal, thermal conductivity and specific heat capacity of the solids no longer affect the simulation. As mentioned before, this is reflected in the log file by displaying a dash in PHASE PROPERTIES under relevant columns.
• As NRPEE HTC relies on gradient of a variable, it is much more sensitive to irregularities in particle distribution. While this applies to other postprocessed values depending on derivatives of primitive variables such as force and torque as well, NRPEE HTC is even more sensitive as it is not summed over entire phases or extractor surfaces and is a local value of each extractor surface.

Flat Plate Based Empirical Approach (EMP)

Nusselt number ( $\mathrm{Nu}$ ) of an isothermal flat plate subject to uniform parallel flow where the fluid is heated as it passes over the solid has a well known empirical relation in literature of the form.

$\mathrm{Nu}=\mathrm{0.644}{\mathrm{Re}}^{\mathrm{0.5}}{\mathrm{Pr}}^{\mathrm{0.33}}$

Where $\mathrm{Nu}$ , $\mathrm{Re}$ , and $\mathrm{Pr}$ are the well known Nusselt, Reynolds, and Prandtl non-dimensional numbers, respectively. HTC is then calculated from the Nusselt number. When calculating flat plate based empirical HTC (EMP HTC), nanoFluidX calculates these non-dimensional numbers with the following approximations:

• When calculating velocities:
  • Tangential velocity calculated from fluid and solid particles on the surface of the extractor will be assumed as $u_{\inf }$ for moving extractors.
  • Tangential velocity calculated from fluid particles on the surface of the extractor will be assumed as $u_{\inf }$ for stationary extractors.
• $L$ is assumed to be equal to dx. Note that this is separate from extractor facet sizes.
• Fluid properties are assumed to be constant per fluid and only depend on volume fraction calculated from the resolved fluid phases.

There is no way to know where the actual modeled plate starts and ends which translates into an issue of specifying the exact value of $L$ . We therefore relegate to assuming the extractor surface is made of a collection of free floating dx long platelets, for the lack of a better term. One of the consequences of this approximation is that the expected fall in heat transfer coefficient for longer plates may not be captured. Another side effect of this approximation is that changing resolution for the same problem affects the calculated EMP HTC directly.