# Heat and Mass Transfer to Particles in One-Dimensional Oscillating Flows

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

**Table 1.**A list of investigated works in the literature dealing with the HMT to spherical particles in steady (upper part) and oscillating flows (lower part).

Authors | $\mathit{Re}$ ( - ) | Steady $\mathit{NRMSD}$ (%) (Max) | Meta $\mathit{NRMSD}$ (%) (Max) | A | B | C | i | j | k | l | Source |
---|---|---|---|---|---|---|---|---|---|---|---|

Mori et al. | 4–24 | 8.4 (12.3) | 8.7 (15.1) | 2.000 | 0.550 | 0.000 | 0.500 | 0.000 | 0.333 | 0.000 | [13] |

Ranz andMarshall | 0.1–2 × 10^{−2} | 3.8 (10.3) | 3.8 (24.3) | 2.000 | 0.600 | 0.000 | 0.500 | 0.000 | 0.333 | 0.000 | [14] |

Hsu et al. | 60–320 | 6.4 (9.0) | 8.4 (30.2) | 2.000 | 0.544 | 0.000 | 0.500 | 0.000 | 0.333 | 0.000 | [15] |

Whitaker | 3.5–7.6 × 10^{4} | 7.0 (21.9) | 5.2 (46.1) | 2.000 | 0.400 | 0.060 | 0.500 | 0.667 | 0.400 | 0.000 | [16] |

Gnielinski | 1–10^{4} | 9.8 (32.9) | 7.2 (28.1) | 2.000 | 0.664 | 0.000 | 0.500 | 0.000 | 0.333 | 0.000 | [17] |

Ke et al. | 10–200 | 9.7 (19.2) | 9.0 (21.7) | 1.910 | 0.545 | 0.019 | 0.500 | 0.667 | 0.333 | 0.000 | [18] |

Richter and Nikrityuk | 10–250 | 6.9 (15.5) | 6.8 (24.6) | 1.760 | 0.550 | 0.014 | 0.500 | 0.667 | 0.333 | 0.000 | [19] |

Sayegh and Gauvin | 0.2–100 | 3.4 (5.1) | 3.8 (28.3) | 2.000 | 0.473 | 0.000 | 0.552 | 0.000 | 0.780 | 0.000 | [20] |

Melissari and Argyropoulos | 10^{2}–5 × 10^{4} | 2.9 (7.2) | 3.5 (36.7) | 2.000 | 0.470 | 0.000 | 0.500 | 0.000 | 0.360 | 0.000 | [21] |

Witte | 3.5 × 10^{4}–1.5 × 10^{5} | 26.2 (35.8) | 20.9 (72.2) | 2.000 | 0.386 | 0.000 | 0.500 | 0.000 | 0.500 | 0.000 | [22] |

Chuchottaworn et al. | 1–200 | 7.8 (26.5) | 6.3 (23.4) | 2.000 | 0.370 | 0.000 | 0.610 | 0.000 | 0.510 | 0.000 | [23] |

Bagchi et al. | 50–500 | 10.2 (15.1) | 11.7 (23.1) | data points—no given correlation | [24] | ||||||

Blackburn | 1–100 | 6.0 (11.2) | 6.0 (13.8) | data points—no given correlation | [25] | ||||||

Acrivos and Taylor | 0–1 | 6.4 (7.4) | 6.2 (14.0) | $\overline{\overline{Nu}}=2+\frac{1}{2}RePr+\frac{1}{4}{\left(RePr\right)}^{2}log\left(RePr\right)+\frac{1}{16}{\left(RePr\right)}^{3}log\left(RePr\right)$ | [26] | ||||||

Steady meta-correlation |
10^{−1}–1.5 × 10^{5} | 4.3 (27.1) | 1.7 (24.0) | 2.000 | 0.500 | 0.000 | 0.500 | 0.000 | 0.333 | 0.000 | - |

Authors | $\mathit{Re}$ (-) | ϵ (-) | Meta$\mathit{NRMSD}$(%) (Max) | A | B | C | i | j | k | l | Source |

Fiklistov and Aksel’rud | 10.5–93.5 | 0.24–0.7 | 21.1 (38.4) | 0.000 | 0.490 | 0.000 | 0.700 | 0.000 | 0.333 | 0.130 | [27] |

Burdukov and Nakoryakov † | 5.5 × 10^{2}–8.4 × 10^{3} | 2 × 10^{−3}–4.5 × 10^{−2} | 8.2 (20.8) | 0.000 | 1.300 | 0.000 | 0.500 | 0.000 | 0.500 | 0.500 | [28] |

Subramaniyam et al. | 4.5 × 10^{3}–2.0 × 10^{5} | 1–2.5 | 7.4 (33.5) | 0.000 | 0.259 | 0.000 | 0.620 | 0.000 | 0.333 | 0.000 | [29,30] |

Burdukov and Nakoryakov ‡ | 2 × 10^{2}–1.4 × 10^{4} | 3.2 × 10^{−2}–0.18 | 23.1 (62.2) | 0.000 | 0.640 | 0.000 | 0.500 | 0.000 | 0.333 | 0.167 | [31] |

Noordzij andRotte | 16–2.6 × 10^{2} | 3 × 10^{−2}–6 × 10^{−2} | 29.7 (61.8) | 0.000 | 0.096 | 0.000 | 0.500 | 0.000 | 0.500 | 0.000 | [29,32] |

Padamanabha andRamachandran | 4 × 10^{2}–2.9 × 10^{3} | 0.2–0.87 | 27.5 (107.4) | 0.000 | 0.505 | 0.000 | 0.640 | 0.000 | 0.000 | 0.630 | [33] |

Hara et al. | 5.5 × 10^{4}–6.1 × 10^{4} | 4.4 × 10^{−3}–0.11 | 26.8 (50.1) | 0.000 | 7.500 | 0.000 | 0.500 | 0.000 | 0.333 | 0.167 | [29] |

Boldarev et al. $\u2020\u2020$ | 35.4–1.4 × 10^{6} | 3.1 × 10^{−4}–0.25 | 15.4 (52.5) | 0.000 | 0.640 | 0.000 | 0.500 | 0.000 | 0.333 | 0.167 | [34] |

Gibert and Angelino | 2 × 10^{2}–5 × 10^{3} | 0.2–0.75 | 10.3 (29.0) | 0.000 | 0.592 | 0.000 | 0.538 | 0.000 | 0.333 | 0.269 | [35] |

Gibert andAngelino | 3 × 10^{2}–4 × 10^{3} | 0.75–2 | 23.9 (40.0) | 0.000 | 0.558 | 0.000 | 0.538 | 0.000 | 0.333 | 0.000 | [35] |

Ha and Yavuzkurt | 16–94 | 12.5–500 | 7.9 (16.0) | 2.000 | 0.420 | 0.000 | 0.500 | 0.000 | 0.333 | 0.000 | [36] |

Al Taweel and Landau (gas) | 10–10^{6} | 10^{−4}–1 | 1.2 (10.5) | 0.000 | 1.100 | 0 | 0.500 | 0 | 0.500 | 0.500 | [29] |

Al Taweel and Landau (liquid) | 10–10^{6} | 10^{−4}–1 | 8.3 (51.6) | 0.000 | 0.640 | 0 | 0.500 | 0 | 0.500 | 0.500 | [29] |

Kawahara et al. | 1.9 × 10^{3} | 1.48 × 10^{−2} | 55.9 | data points— no given correlation | [37] | ||||||

Gopinath and Mills | 2.87 × 10^{2} | 0.54 | 15.8 | data points— no given correlation | [38] | ||||||

Drummond and Lyman | 1–150 | 10^{−4}–1 | 64.0 (190.7) | data points— no given correlation | [39] | ||||||

Alassar et al. | 10–200 | 0.16–5 | 39.3 (76.7) | data points— no given correlation | [40] | ||||||

Xu et al. | 1.25–18 | 0.22–2.7 × 10^{3} | 11.5 (39.5) | data points— no given correlation | [41] | ||||||

Blackburn | 1–100 | 5 × 10^{−2}–5 | 30.6 (73.1) | data points— no given correlation | [25] | ||||||

Meta correlation (gas) | 10^{−1}–10^{6} | 10 × 10^{−3}–10^{3} | 0.8 (10.4) | $\overline{\overline{Nu}}\left(\overline{\overline{Sh}}\right)=2+$$0.5$$R{e}^{1/2}P{r}^{1/3}\left(S{c}^{1/3}\right)\left[\frac{1}{0.45{\u03f5}^{-1/2}+1}+\frac{1}{2.50exp{(log\left(\u03f5\right))}^{2}-1.25}\right]$ | - | ||||||

Meta correlation (liquid) | 10^{−1}–10^{6} | 10 × 10^{-3}–10^{3} | 3.7 (51.5) | $\overline{\overline{Nu}}\left(\overline{\overline{Sh}}\right)=2+$$0.5$$R{e}^{1/2}P{r}^{1/3}\left(S{c}^{1/3}\right)\left[\frac{1}{0.78{\u03f5}^{-1/2}+1}+\frac{1}{2.50exp{(log\left(\u03f5\right))}^{2}-1.85}\right]$ | - |

## 3. Results

#### 3.1. Steady HMT Models

^{2}value (coefficient of determination) is not a good measure for the correlation of nonlinear regressions [42] and is therefore not used here.

#### 3.2. HMT Models for Oscillating Flows

## 4. Discussion

#### 4.1. Meta-Correlation Design

#### 4.2. Deviations

#### 4.3. Quasi-Steady Assumption

## 5. Conclusions

- the high number of 33 correlated data sets;
- the large size of the covered parameter space of amplitude parameter $\u03f5$ and Reynolds number $Re$ and the comprehensive nature of the correlations;
- the carefully modeled asymptotic behavior for extreme values as the relevant literature suggests by theoretical considerations and a multitude of experiments;
- them highlighting the substantiated characteristics of a decreased HMT in the steady streaming region, an increased HMT for $\u03f5\approx 1$, and the quasi-steady HMT for $\u03f5\gg 1$.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | displacement amplitude |

m | mass |

d | particle diameter |

n | number of sample point |

y | sample point |

$Re$ | oscillation Reynolds number |

$Stk$ | oscillation Stokes number |

$Nu$ | Nusselt number |

$Sh$ | Sherwood number |

$R{e}_{S}$ | Streaming Reynolds number |

U | slip velocity amplitude |

$W{o}^{2}$ | Womersley number |

$Pr$ | Prandtl number |

$Sc$ | Schmidt number |

$\gamma $ | density ratio |

$\u03f5$ | amplitude parameter |

$\delta $ | boundary layer thickness |

$\Delta $ | difference |

$\eta $ | dynamic viscosity |

$\rho $ | density |

$\omega $ | angular frequency |

## Abbreviations

$HMT$ | heat and mass transfer |

$NRMSD$ | normalized root mean square deviation |

$STP$ | standard temperature and pressure |

## Indices

p | particle |

f | fluid |

## Appendix A. Conducted Data Preparation

- †
- Al Taweel and Landau did not provide a value range for the displacement amplitude A and stated only that the amplitude parameter is much smaller than unity for the work of Burdukov and Nakoryakov [28]. While this statement agrees with the respective paper, also the applied decibel range of the utilized levitator is stated in that paper: 150 dB to 163 dB. With the standard reference sound pressure level of ${p}_{0}=20\mu \mathrm{Pa}$[44] and the linear relation between pressure amplitude P and velocity amplitude ${U}_{f}$, $P={\rho}_{f}c{U}_{f}$, with the speed of sound c in the fluid, the fluid velocity amplitude was calculated. Since no significant particle relaxation occurs [10], the slip velocity amplitude was set equal to the fluid velocity amplitude as given in Table A1. Subsequently, the displacement amplitude and the amplitude parameter were calculated according to the relations in Figure A1.
- ‡
- Al Taweel and Landau did not provide a value for the spheres’ diameter, but in the original paper of Burdukov and Nakoryakov [31] is stated that the glass spheres with mount weighed about ${m}_{sphere}=1.3\text{}\mathrm{g}$. Additionally, the thickness of benzoic acid coating was about ${d}_{acid}-{d}_{sphere}=\Delta d=$ 0.6 $\mathrm{m}$$\mathrm{m}$ to 1 $\mathrm{m}$$\mathrm{m}$, with a weight of about ${m}_{acid}=150\text{}\mathrm{m}\mathrm{g}$. This information opens up two ways of estimating the diameter of the glass sphere:
- Neglecting the weight of the mount and assuming the utilization of borosilicate glass, which is often used in a scientific environment, with a density of about ${\rho}_{sphere}=2235\text{}\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$ [45], the diameter can be calculated to about$$\begin{array}{cc}\hfill {V}_{sphere}& ={\rho}_{sphere}{m}_{sphere}=\frac{\pi}{6}{d}_{sphere}^{3}\hfill \end{array}$$$$\begin{array}{cccc}\hfill {d}_{sphere}& ={\left(\frac{6{m}_{sphere}}{\pi {\rho}_{sphere}}\right)}^{1/3}\hfill & \hfill \phantom{\rule{1.em}{0ex}}& \approx 1\mathrm{c}\mathrm{m}\hfill \end{array}$$In case ordinary glass (soda–lime) was used with a density of ${\rho}_{sphere}=2520\text{}\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$, the spheres would be insignificantly smaller.
- Calculating the dimensions of the coating by assuming a benzoic acid density of ${\rho}_{acid}=1.260\text{}\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$ [46], the diameter can be calculated$$\begin{array}{cc}\hfill {V}_{acid}& ={\rho}_{acid}{m}_{acid}=\frac{\pi}{6}\left({d}_{acid}^{3}-{d}_{sphere}^{3}\right)\hfill \end{array}$$$$\begin{array}{cccc}\hfill {d}_{sphere}& =-\frac{\Delta d}{2}+\sqrt{{\left(\frac{\Delta d}{2}\right)}^{2}+\frac{{\Delta d}^{2}}{3}+\frac{2{m}_{acid}}{\pi {\rho}_{acid}\Delta d}}\hfill & \hfill \phantom{\rule{1.em}{0ex}}& \approx 1\mathrm{c}\mathrm{m}\hfill \end{array}$$Even though Equation (A4) has two theoretical solutions due to its quadratic nature, only the physically plausible solution with a positive diameter was chosen.In approaches 1 and 2, the diameter of the spheres can be estimated to be $d=1\text{}\mathrm{c}\mathrm{m}$. Therefore, this value is used in this work. Still, this problem is undetermined and the determination of another parameter is necessary in order to calculate all the parameters listed in Table A1. In the original paper is stated that the frequency varied from 10 $\mathrm{Hz}$ to 125 $\mathrm{Hz}$ translating to angular frequencies of 63 s
^{−1}to 785 s^{−1}. All other input properties listed in the first row of Figure A1 are kept constant except the velocity amplitude U, which is dependent on the frequency. The parameter $K={\left[\left({U}^{2}d\right)/\left(2\sqrt{\omega \nu}D\right)\right]}^{1/3}$ in the original paper was varied between 100 and 1200. The Schmidt number $Sc=\nu /D$ was estimated by Al Taweel and Landau to be approximately 1000 in this setup. Adopting this value and linking low oscillation frequencies to low-velocity amplitudes and high oscillation frequencies to high-velocity amplitudes delivers an investigated velocity amplitude window of approximately $U=$ 0.02 m s^{−1}to 1.43 m s^{−1}. With these values, all the parameters listed in Table A1 can be determined via the relations in Figure A1.

- $\u2020\u2020$
- One parameter is missing in the original paper [34] in order to calculate all the values listed in Table A1. The velocity amplitude is calculated via the same approach as in the previous paragraph. The parameter $b=\left({U}^{2/3}{(d/2)}^{1/3}\right)/\left({\left(\omega \nu \right)}^{1/6}{D}^{1/3}\right)$ is varied between 25 and 100 in the original paper while keeping all parameters except the oscillation frequency constant. This translates with $Sc=\nu /D\approx 2200$ to a velocity amplitude window of about $U=$ 0.24 m s
^{−1}to 239 m s^{−1}.

**Table A1.**A list of investigated works by Al Taweel and Landau [29]. Some of their stated data have been marked ($\u2020,\u2021,\u2020\u2020$) and expanded with information from the original papers.

Authors | $\mathbf{\omega}$ [s${}^{-\mathbf{1}}$] | $\mathit{A}$ (m) | $\mathit{d}$ (m) | $\mathit{U}$ (m s${}^{-\mathbf{1}}$) | $\mathit{\nu}$ (m${}^{\mathbf{2}}$ s${}^{-\mathbf{1}}$) | $\mathit{\u03f5}$ ( - ) | $\mathit{Re}$ ( - ) | Source |

Fiklistov and Aksel’rud | 3.8 33.3 | 1.2 × 10^{−3}3.5 × 10 ^{−3} | 5× 10^{−3} | 2.1× 10^{−3} 1.9× 10 ^{−2} | 10^{−6} | 0.24 0.7 | 10.5 93.5 | [27] |

Burdukov and Nakoryakov † | 7.2 × 10^{4}1.1 × 10 ^{5} | 2.0 × 10^{−5}1.4 × 10 ^{−4} | 3.5 × 10^{−3}10 ^{−2} | 2.2 9.8 | 1.2 × 10^{−5} | 2 × 10^{−3} 4.5 × 10 ^{−2} | 5.5 × 10^{2}8.4 × 10 ^{3} | [28] |

Subramaniyam et al. | 25 126 | 2.7 × 10^{−2}3.7 × 10 ^{−2} | 1.3 × 10^{−2}2.5 × 10 ^{−2} | 0.3 8.0 | 10^{−6} | 1 2.5 | 4.5 × 10^{3}2.0 × 10 ^{5} | [29] [30] |

Burdukov and Nakoryakov ‡ | 63 785 | 3.2 × 10^{−4} 1.8 × 10 ^{−3} | 1 × 10^{−2} | 2 × 10^{−2} 1.4 | 10^{−6} | 3.2 × 10^{−2} 0.18 | 2 × 10^{2}1.4 × 10 ^{4} | [31] |

Noordzij and Rotte | 0 220 | 7.8 × 10^{−}4 1.6 × 10 ^{−3} | 2.5 × 10^{−2} | 0.17 0.33 | 10^{−6} | 3 × 10^{−2} 6 × 10 ^{−2} | 16 2.6 × 10 ^{2} | [29] [32] |

Padamanabha and Ramachandran | 19 63 | 1 × 10^{−2} 2.2 × 10 ^{−2} | 2.5 × 10^{−2} 5 × 10 ^{−2} | 0.19 1.38 | 1.2 × 10^{−5} | 0.2 0.87 | 4 × 10^{2} 2.9 × 10 ^{3} | [33] |

Hara et al. | 1.2 × 10^{4} 1.2 × 10 ^{5} | 4.4 × 10^{−5} 7.2 × 10 ^{−4} | 6.8 × 10^{−3} 10 ^{−2} | 5.5 9.0 | 10^{−6} | 4.4 × 10^{−3} 0.11 | 5.5 × 10^{4} 6.1 × 10 ^{4} | [29] |

Boldarev et al. $\u2020\u2020$ | 1.3 × 10^{5} 6.3 × 10 ^{6} | 1.9 × 10^{−6} 3.8 × 10 ^{−5} | 1.5 × 10^{−4} 6 × 10 ^{−3} | 0.24 239 | 10^{−6} | 3.1 × 10^{−4} 0.25 | 35.4 1.4 × 10 ^{6} | [34] |

Gibert and Angelino | 5 25 | 1.6 × 10^{−3} 2.3 × 10 ^{−2} | 8 × 10^{−3} 3 × 10 ^{−2} | 6.6 × 10^{−3} 6.3 × 10 ^{−1} | 10^{−6} | 0.2 0.75 | 2 × 10^{2} 5 × 10 ^{3} | [35] |

Gibert and Angelino | 5 25 | 6 × 10^{−3} 6.1 × 10 ^{−2} | 8 × 10^{−3} 3 × 10 ^{−2} | 9.8 × 10^{−3} 0.5 | 10^{−6} | 0.75 2 | 3 × 10^{2} 4 × 10 ^{3} | [35] |

## Appendix B. Central Dimensionless Numbers in Oscillating Flows

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**Figure 1.**$\u03f5$-$Re$ plane by Heidinger et al. [10] in which all flow states of a single particle in an oscillating flow can be defined. Note that the origin has the coordinates $\u03f5=1,Re=1$.

**Figure 2.**Overview of considered experimental and numerical data and correlations for the HMT to particles in

**oscillating**flows. Colored patches indicate a provided correlation by the authors, while single points indicate individual measurement or simulation points without a correlation provided by the respective authors. All the sources of the data are listed in Table 1.

**Figure 3.**Overview of considered experimental and numerical HMT data for single spherical particles in

**steady**flow. Solid lines indicate correlations provided by the respective authors, while single points indicate individual measurements or simulation points without correlations provided by the authors. Additionally,

**Steady**Meta-Correlation (2) is plotted, which is fitted to the listed data. The Prandtl number is set to $Pr=0.71$ as it can be found with air at STP. All the sources of the data are listed in Table 1.

**Figure 4.**Nusselt number averaged over the particle surface and averaged over one oscillation cycle predicted by various investigated models plotted in the $\u03f5$-$Re$ plane. Additionally, Meta-Correlation (4) for gaseous environments is plotted, while the Prandtl number is set to $Pr=0.71$ as can be found with air at STP.

**Figure 5.**Deviations of Meta-Correlation (4) and (5) to the respective individual models investigated and listed in Table 1. The colored bars give the $NRMSD$, calculated with Equation (3), while the white bars give the maximum normalized deviation. The deviation presented for Meta-Correlations (4) and (5) is for all investigated models and data combined.

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**MDPI and ACS Style**

Heidinger, S.; Unz, S.; Beckmann, M. Heat and Mass Transfer to Particles in One-Dimensional Oscillating Flows. *Processes* **2023**, *11*, 173.
https://doi.org/10.3390/pr11010173

**AMA Style**

Heidinger S, Unz S, Beckmann M. Heat and Mass Transfer to Particles in One-Dimensional Oscillating Flows. *Processes*. 2023; 11(1):173.
https://doi.org/10.3390/pr11010173

**Chicago/Turabian Style**

Heidinger, Stefan, Simon Unz, and Michael Beckmann. 2023. "Heat and Mass Transfer to Particles in One-Dimensional Oscillating Flows" *Processes* 11, no. 1: 173.
https://doi.org/10.3390/pr11010173