Abstract

Particle to fluid heat transfer in supercritical carbon dioxide (sCO2) is encountered in energy technologies and in materials synthesis. Near the critical point, the extreme pressure and temperature sensitivity of sCO2’s thermal conductivity will change the expected heat transfer in these systems. The current work combines the Kirchoff transformation for thermal conductivity with the conduction shape factor for a sphere, allowing prediction of heat transfer in these systems and quantification of the impact of these property changes. Results show that the heat transfer is non-linear for supercritical heat transfer, with the non-linearity particularly significant near the critical point. The results also show that approaches such as an average thermal conductivity based on film temperature are unlikely to accurately predict heat transfer in this region. The methods described in this paper can be applied to fluid–particle heat transfer at low Reynolds number in other fluids with large variations in thermal conductivity.

Introduction

Power systems using supercritical carbon dioxide (sCO2) are of growing engineering interest due to their increased thermodynamic efficiency and relative compactness [1]. Concentrated solar power (CSP) [2,3] and combustion systems operating in sCO2 [4] have emerged as particular applications of interest. Studies of individual power system components, including compressors [5] and heat exchangers [6,7], show that the extreme variability in fluid properties of sCO2 must be accounted for in both simulation and analysis of experimental data. Similar challenges may be encountered in other engineering systems using sCO2, including thermal protection systems for planetary exploration [8], electronics cooling [9], and material synthesis [1012].

Solid particle-to-fluid heat transfer is relevant in many of these systems, particularly CSP heat exchanger design, combustion systems, and material synthesis. In these systems, the heat transfer from solid particles to sCO2 fluid will be a major factor in performance. Because small particles in a flow often have Reynolds numbers well below one, particle-to-fluid heat transfer can be modeled based on conduction [13] and will be sensitive to changes in the thermal conductivity of the fluid. Just above the critical temperature Tc of 30.98 °C (304.13 K) and the critical pressure Pc of 7.377 MPa [14], sCO2 has large variations in thermal conductivity k, as shown in Fig. 1 [14,15]. The thermal conductivity has a large peak above the critical temperature. At a pressure of 7.5 MPa, for instance, the thermal conductivity doubles and then returns to its original value within 3 °C. Similar peaks appear, although smaller in magnitude and at higher temperatures, at higher pressures. The current work incorporates these highly variable transport properties into predictions of particle heat transfer.

Problem Formulation and Solution Method

At the limit of a Reynolds number well below one, particle-to-fluid heat transfer can generally be found using solutions based on conduction for steady-state conduction from the geometry in a semi-infinite medium. For a sphere with a diameter D and a constant surface temperature Ts in a medium with a constant thermal conductivity k at a temperature T, allows the analytic shape factor to be used to compute the heat lost from the sphere q
q=kS(TsT)=k(2πD)(TsT)=k(2πD)ΔT
(1)
where S is a conduction shape factor and ΔT is equal to Ts minus T [13]. If the heat transfer at the surface is modeled using a convective heat transfer coefficient h for the sphere surface area AS, then Eq. (1) can be re-written as
q=hAs(TsT)=h(πD2)(TsT)=(2kD)(πD2)(TsT)
(2)
giving a value for h of 2k/D. This result corresponds to a Nusselt number NuD of 2.0, which in the limit of Reynolds number approaching zero, leads to the most commonly used correlation for heat transfer from a sphere [16].

However, if the thermal conductivity is not constant, Eqs. (1) and (2) will not be accurate predictors of the heat lost from the sphere. A correction for variable thermal conductivity can be obtained using the Kirchhoff transformation [17,18], which has been successfully used to model one-dimensional conduction heat transfer in engineering applications with large changes in thermal conductivity, including electronics heat transfer [19], LED-devices [20], nuclear fuels [21], and automobile braking systems [22].

The Kirchhoff transformation variable θ is defined as
θ(T)=1krTrTk(T)dT
(3)
where Tr is a reference temperature and kr is the thermal conductivity at the reference temperature. When the Kirchoff transformation is used, Eq. (1) can be re-derived [23] as
q=kr(2πD)(θsθ)
(4)
The convection coefficient h will be given by
h=(2krD)(θsθTsT)
(5)
The variable θ is found by numerically integrating the available thermal conductivity data, using a reference temperature Tr equal to the critical temperature of 30.98 °C and setting kr to the thermal conductivity at Tr. The step size for the numerical integration using the trapezoidal rule is 0.01 °C. The results for pressures of 7.4, 7.5, 8.0, 8.5, 9.0, and 10.0 MPa are shown in Fig. 2, with the reference thermal conductivities shown in Table 1. A constant slope indicates a constant value of k(T), while a slope of 1 indicates that k(T) is equal to kr. As would be expected from the thermal conductivity data in Fig. 1, the non-linearity in θ is largest at pressures near the critical pressure, and at temperatures near the critical point, where the spikes in thermal conductivity are the largest.

Results for Heat Transfer of a 50 µm Particle With a Constant Surface Temperature

Equations (3)(5), and the Kirchoff transformation data shown in Fig. 2 and Table 1 were used to characterize how the extreme temperature and pressure dependence impact heat transfer from a solid spherical particle with a diameter of 50 μm. While Eqs. (4) and (5) show that both q and h vary with diameter, these results can be directly scaled to varying particle diameters.

Figures 3 and 4 show the heat flux q and the effective heat transfer coefficient h for a particle with a surface temperature Ts of 31.0 °C for pressures ranging from 7.4 MPa to 10.0 MPa, for values of ΔT of 0.5, 1.0, 2.0, 4.0, 8.0, and 16.0 °C. These values correspond to fluid temperatures of 31.5, 32.0, 35.0, 39.0, and 47.0 °C. The results show that the large variations in thermal conductivity cause three significant effects for particle-to-fluid heat transfer in sCO2. The first is that the heat flux and effective heat transfer coefficient are both extremely pressure-sensitive, which will not be encountered in most other engineering power systems. The second is that the heat transfer coefficients will also vary by large amounts based on the value of ΔT. The final effect is that both of these changes are extremely non-linear. Instead of a relatively constant value, h can vary by more than 50% for pressures near the critical pressure, and by 20–30% at higher pressures. The peak heat transfer rate for any particular pressure depends on ΔT. At a pressure of 7.5 MPa, the highest heat transfer coefficient will be at a value of ΔT of 1 °C. However, at 8.0 MPa, the highest heat transfer coefficient will be at a value of ΔT of 4 °C. The result is an extremely non-linear response that has the ability to frustrate the design of thermal systems, especially the ones designed to operate over a range of pressures and temperatures near the critical point.

Figures 5 and 6 explore this phenomenon more systematically, showing the heat flux q and the effective heat transfer coefficient h for a particle with a surface temperature Ts of 31.0 °C as the fluid temperature T increases from 31.0 °C to 60.0 °C for pressures ranging from 7.4 MPa to 10.0 MPa. The results for heat flux show non-linear behavior at temperatures near the critical point but appear to approach a linear slope as the temperature increases.

The results for effective heat transfer coefficient, however, show the non-linear behavior due to changing thermal conductivities much more dramatically. For pressures of 8.0 MPa and below, there are peaks in the heat transfer coefficients. These occur, as expected, in the regions where the sCO2 has a peak in the thermal conductivity. However, the results do not exactly correspond to the peaks in fluid thermal conductivity.

Scaling Results

Solid to fluid heat transfer is generally scaled using the Nusselt number NuD based on diameter
NuD=hD/k
(6)
However, there are multiple values of the thermal conductivity that can be, and are, used in scaling the Nusselt number. Depending on the correlation, previous researchers have scaled the heat transfer coefficient using the thermal conductivity at the surface ks evaluated at Ts, the thermal conductivity of the free-stream k evaluated at T, or the thermal conductivity kf evaluated at a film temperature Tf. The film temperature Tf is generally found by averaging T and Ts [13]. A fourth approach is to average the thermal conductivity over the entire range from Ts to T to find k(T)¯. Figure 7 shows the Nusselt number obtained for each of these approaches for a sphere at a pressure of 7.5 MPa.

As noted earlier, for a constant thermal conductivity, the Nusselt number of a sphere at low Reynolds numbers is 2.0. Using ks, k, or kf to scale these values leads to an extremely unpredictable scaling, with the result as much as 50% higher or lower than the theoretical value. Therefore, engineering predictions made using these scalings may result in extremely inaccurate predictions of particle–fluid heat transfer. The only way to predictably recover a Nusselt number of 2.0 is to use k(T)¯. Because this integration is encapsulated in the Kirchoff integral, the results will be consistent with this scaling.

Constant Heat Flux Results

If the particle is undergoing a chemical reaction, such as combustion, or is receiving energy through radiative transfer and then transferring it to the surrounding fluid at Ts through conduction, using a constant heat flux condition is more realistic. Equation (4) can be re-written to find the value of θ at the surface
θs=θ+(q2πkrD)
(7)
The surface temperature can then be found by a reverse table-lookup to find the temperature Ts based on θs. Figures 8(a) and 8(b) show the surface temperature as a function of heat flux for particles in fluids with a pressure of 7.5 and 8.5 MPa, and fluid temperatures of 31, 35, and 40 °C. These results show that the system remains non-linear for these cases. The non-linearity is strongest when the fluid temperature is just above the critical point. The non-linear behavior is strongest when the heat flux results in surface temperatures where the peaks in the thermal conductivity are between Ts and Tf. However, the non-linearity remains a feature of the system even at higher heat fluxes.

Conclusions

These results show that for sCO2 even the simplest thermal transport case imaginable, one-dimensional conduction, yields non-linear results that cannot be accurately accounted for without incorporating integral averages of the thermal conductivity. The large variations in thermal conductivity near the critical temperature and pressure, in particular, make using a film temperature approach inaccurate. In all circumstances, the linear behavior expected for heat transfer over small perturbations in temperature is not seen for supercritical carbon dioxide.

The results are also relevant to convective heat transfer in sCO2 at higher Reynolds numbers. In boundary layers near the critical point, the property changes are likely to cause the same non-linear behavior as seen in a one-dimensional conduction. These effects may even be magnified, since the density, specific heat, and dynamic viscosity all have their own non-linearities in this regime. For instance, the specific heat of sCO2 has a peak similar to, but at different temperatures and magnitudes, compared with the peaks seen in thermal conductivity. The combination of non-linearity and difficulty in determining film averages means that convective heat transfer coefficients for supercritical fluids are extremely unlikely to fall within the values predicted by empirical correlations developed from experiments and simulations performed under ideal gas conditions.

The same characteristics that lead to the extreme non-linearity seen in supercritical carbon dioxide: a large change in thermal conductivity of more than a factor of two within a few degrees, and a local maximum above the critical temperature, may appear in other supercritical fluids. The approach outlined for particle-to fluid heat transfer in sCO2 can be applied to other supercritical fluids as needed. The only requirement is the numeric integration of NIST or other thermal conductivity data for these fluids to generate accurate predictions of heat transfer.

Acknowledgment

This work was authored by the National Renewable Energy Laboratory (Funder ID: 10.13039/100006233), operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308; Funder ID: 10.13039/100000015. Funding provided by the Exascale Computing Project (17-SC-20-SC), a collaborative effort of two U.S. Department of Energy (DOE) organizations, the Office of Science and the National Nuclear Security Administration. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes.

Nomenclature

     
  • D =

    diameter

  •  
  • H =

    convection heat transfer coefficient

  •  
  • K =

    thermal conductivity

  •  
  • P =

    pressure

  •  
  • Q =

    heat transfer rate

  •  
  • S =

    shape factor

  •  
  • T =

    temperature

  •  
  • As =

    surface area

  •  
  • NuD =

    Nusselt number based on diameter

Greek Symbols

     
  • ΔT =

    temperature change

  •  
  • θ =

    Kirchoff transformation variable

Subscripts

     
  • C =

    critical point

  •  
  • F =

    film property

  •  
  • R =

    reference property

  •  
  • S =

    surface property

  •  
  • =

    free-stream property

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