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

Abstract

The heat and mass transfer to solid particles in one-dimensional oscillating flows are investigated in this work. A meta-correlation for the calculation of the Nusselt number (Sherwood number) is derived by comparing 33 correlations and data point sets from experiments and simulations. These models are all unified by their dependencies on the amplitude parameter ${10}^{-3}\le ϵ\le {10}^3$ and the Reynolds number ${10}^{-1}\le Re\le {10}^6$, while the $ϵ$-$Re$ plane is applied as a framework in order to graphically display the various models. This is the first study to consider this problem in the entire $ϵ$-$Re$ plane quantitatively while taking preexisting asymptotic models for various areas of the $ϵ$-$Re$ plane into account.

1. Introduction

The heat and mass transfer (HMT) to (or from) a rigid, spherical particle suspended in a harmonically oscillating one-dimensional flow is considered. This configuration can be found, for example, in ultrasonic levitators [1,2] and pulsation reactors [3,4,5]. In case the particle is fixed in position (or the particle executes harmonic oscillations in a fluid at rest), the interaction between the fluid and the particle as well as the resulting flow state is defined by two dimensionless numbers: the particle Reynolds number $Re$ and the amplitude parameter $ϵ$. The particle Reynolds number $Re=U{\rho }_{f}d/\eta$ expresses the ratio between inertial and viscous forces with the slip velocity amplitude U, the density of the fluid ${\rho }_f$, the particle diameter d, and the dynamic viscosity of the fluid $\eta$. The amplitude parameter $ϵ=A/d$ sets the particle displacement A in relation to the particle diameter, while the particle displacement $A=U/\omega$ itself is the ratio of slip velocity amplitude to the angular frequency $\omega$ of the oscillating fluid. An overview of these relations is provided in Appendix B. The Reynolds number and amplitude parameter span a plane in which all flow states can be pinpointed, as shown in Figure 1. The bisectors of this plane are the Womersley number $W{o}^2=Re/ϵ$[6], which is important for relaxation considerations, and the streaming Reynolds number $R{e}_S=Reϵ$[7], which is central for the modeling of steady streaming. Note that the origin of Figure 1 has the coordinates $ϵ=1,Re=1$. The $ϵ$-$Re$ plane was introduced by Wang [8] and then adopted by Chong et al. [9]. A comprehensive discussion of particle motion, particle relaxation and slip velocity in this configuration with the $ϵ$-$Re$ plane as a basis was then conducted by Heidinger et al. [10]. Going forward, the $ϵ$-$Re$ plane will simply be referred to as “plane” in this context. Not only is the particle motion and flow state of the described configuration defined by the two central dimensionless numbers $ϵ$ and $Re$, but also the resulting heat and mass transfer. This means that pinpointing the location on the plane also delivers the intensity of the HMT from the particle. The intensity of the HMT is expressed via the Nusselt number $Nu$ or the Sherwood number $Sh$, respectively. Many experimental and theoretical studies have already been conducted, but as they each cover only a small part of the plane, a comprehensive analysis of the HMT over the entire plane is still lacking. In the following, 33 data sets and correlations, each of which covers only a small part of the plane, are utilized in order to derive a meta-correlation, which covers the entire plane. In Section 2, an overview of the utilized works is presented, along with the general taxonomy and the approach adopted in this work. Then, in Section 3.1, the models for the quasi-steady sub-area of the plane are discussed and a steady meta-correlation is derived, which is later utilized for the general meta-correlation. The method for evaluating model deviations is introduced and the fit of the steady meta-correlation is evaluated. After that, Section 3.2 deals with data sets from oscillating flows, from which two general meta-correlations, one for fluid and one for gaseous environments, are derived. This is followed by a discussion of the design of the two meta-correlations in Section 4.1, and a discussion of the deviations between the meta-correlations and the utilized data sets in Section 4.2. Section 4.3 closes the discussion by evaluating the limits of the often applied quasi-steady assumption under the derived meta-correlations. The work is concluded with Section 5, in which the main features of the meta-correlations are highlighted.

2. Method

A Nusselt (Sherwood) number correlation for the entire plane is derived. For this purpose, a structured literature review is conducted in order to cover large parts of the plane with experimental, numerical, and analytical data for the occurring HMT. A list of the considered works can be found in Table 1 and a graphical overview in Figure 2. The Reynolds analogy (Prandtl analogy) is considered applicable in oscillating flows [9,11,12], leading to the interchangeability of correlations for Nusselt number and Sherwood number by exchanging the Schmidt number for the Prandtl number and vise versa. Therefore, the correlations for HMT to particles suggested by many authors can be expressed generally as

$$Nu\left(orSh\right)=A+\left(BR{e}^i+CR{e}^j\right)P{r}^k\left(orS{c}^k\right){ϵ}^l$$

(1)

while the parameters ($A,B,C,i,j,k,l$) differ for individual correlations. For steady HMT, this approach has been used by Yavuzkurt et al. [12] and it is here extended with the term ‘${ϵ}^l$’ in order to incorporate correlations for oscillating flows. All correlations in Table 1 fit this general ansatz except one.

Several authors concluded that, for a large enough amplitude parameter $ϵ$, the HMT in oscillating flow can be described by correlations for steady flow with sufficient accuracy. In these cases, the quasi-steady assumptions hold true.

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 × 104	7.0 (21.9)	5.2 (46.1)	2.000	0.400	0.060	0.500	0.667	0.400	0.000	[16]
Gnielinski	1–104	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	102–5 × 104	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 × 104–1.5 × 105	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 × 105	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 × 102–8.4 × 103	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 × 103–2.0 × 105	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 × 102–1.4 × 104	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 × 102	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 × 102–2.9 × 103	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 × 104–6.1 × 104	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. $††$	35.4–1.4 × 106	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 × 102–5 × 103	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 × 102–4 × 103	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)	10–106	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)	10–106	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 × 103	1.48 × 10−2	55.9	data points— no given correlation							[37]
Gopinath and Mills	2.87 × 102	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 × 103	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)	10−1–106	10 × 10−3–103	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{ϵ}^{-1/2}+1}+\frac{1}{2.50exp{(log\left(ϵ\right))}^2-1.25}\right]$							-
Meta correlation (liquid)	10−1–106	10 × 10-3–103	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{ϵ}^{-1/2}+1}+\frac{1}{2.50exp{(log\left(ϵ\right))}^2-1.85}\right]$							-

†, ‡, †† The conducted data preparation for some marked works can be found in Appendix A.

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 × 104	7.0 (21.9)	5.2 (46.1)	2.000	0.400	0.060	0.500	0.667	0.400	0.000	[16]
Gnielinski	1–104	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	102–5 × 104	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 × 104–1.5 × 105	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 × 105	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 × 102–8.4 × 103	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 × 103–2.0 × 105	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 × 102–1.4 × 104	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 × 102	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 × 102–2.9 × 103	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 × 104–6.1 × 104	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. $††$	35.4–1.4 × 106	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 × 102–5 × 103	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 × 102–4 × 103	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)	10–106	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)	10–106	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 × 103	1.48 × 10−2	55.9	data points— no given correlation							[37]
Gopinath and Mills	2.87 × 102	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 × 103	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)	10−1–106	10 × 10−3–103	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{ϵ}^{-1/2}+1}+\frac{1}{2.50exp{(log\left(ϵ\right))}^2-1.25}\right]$							-
Meta correlation (liquid)	10−1–106	10 × 10-3–103	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{ϵ}^{-1/2}+1}+\frac{1}{2.50exp{(log\left(ϵ\right))}^2-1.85}\right]$							-

†, ‡, †† The conducted data preparation for some marked works can be found in Appendix A.

This threshold was suggested as $ϵ>0.75$ by Gibert and Angelino [35] and supported by Al Taweel and Landau [29], while Drummond and Lyman [39] derived it to be $ϵ>0.25$. The general quasi-steady assumption for large $ϵ$ is also supported by the theoretical work of Ha and Yavuzkurt [36] and the work of Subramaniyam et al. [29]. Additionally, the experimental work of Xu et al. [41] fit the quasi-steady assumption quite well, for $ϵ>1$ but also $ϵ<1$. The simulation results of Blackburn [25] and Alassar et al. [40] match each other well and also match the steady-state assumption for $ϵ\gg 1$. In light of the review of the bulk of recent data, the approach of modeling the HMT for large $ϵ$ validly with the quasi-steady assumption is substantiated. Nevertheless, the exact quasi-steady limit remains unclear, but it is approximately $ϵ\approx 1$. For the sake of a comprehensive approach and in order to avoid model discontinuities (which other authors accepted), $ϵ\ge 3$ is chosen as the quasi-steady limit in this work. Figure 2 shows all the available data, while colored patches indicate a correlation given by the respective authors and dots indicate single data points without a correlation provided. Additionally, a black rectangle marks the quasi-steady area, where a multitude of steady data is available. This quasi-steady area is handled first in the following Section 3.1.

3. Results

3.1. Steady HMT Models

In accordance with the explanations in the previous section, the HMT for $ϵ\ge 3$ is modeled under the quasi-steady assumption. Therefore, a structured review of steady models for the HMT to particles was conducted. While meta-studies on this topic by Whitaker [16] and Gnielinski [17] provide good insights and also correlate a large number of works, they arrive at somewhat different correlations. So these two studies, along with central preceding studies, as well as newer experimental and numerical works are plotted in Figure 3. Solid lines indicate correlations given by the respective authors in the stated ranges of validity, or implicitly by the investigated ranges, while dots represent data points without a correlation provided by the authors. The dependencies of the investigated correlations on the Prandtl number (Schmidt number) are not considered in this work in order to achieve a better comparison of the Reynolds number dependencies. A meta-correlation of the steady Nusselt number (Sherwood number) averaged over the surface of the particle

$$\overline{Nu}\left(\overline{Sh}\right)=2+0.5R{e}^{1/2}P{r}^{1/3}\left(S{c}^{1/3}\right)$$

(2)

is suggested for its simplicity, while fitting the data well. The normalized root-mean-square deviation ($NRMSD$) of each correlation or data set from the steady meta-correlation is provided in Table 1. The $NRMSD$ is calculated via

$$NRMSD=\sqrt{\frac{{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{n{\overline{y}}^2}}$$

(3)

with the value of the respective correlation or single data point $y_i$, the value of the meta-correlation ${\widehat{y}}_i$, the value range of the respective correlation $\overline{y}$, and the number of sample points n. Note that the R² value (coefficient of determination) is not a good measure for the correlation of nonlinear regressions [42] and is therefore not used here.

The meta-correlation fits most of the correlations very well with each $NRMSD$$\le 10\%$. The only outlier is Witte (26.2%), which represents a special case with the investigation of the HMT from a tantalum sphere in liquid sodium. Based on the deviation from the other experimental setups and investigated systems, a somewhat different correlation is expected. Since the steady meta-correlation of this work over-predicts the lowest values of the meta-study by Whitaker (7.0%) to a similar degree as it under-predicts the highest values of the meta-study by Gnielinski (9.8%), it is deemed acceptable at this point based on the lack of any further data for spheres in high Reynolds number flows. Meta-Correlation (2) is now used in order to model the HMT to particles in oscillating flows for $ϵ\ge 3$, together with the data from oscillating flows.

3.2. HMT Models for Oscillating Flows

Several works exist in the literature which have investigated the HMT to spheres in an oscillating flow with an $ϵ\ge 3$. Their data fit the quasi-steady assumption often well or they suggested a steady correlation themselves, as discussed in Section 3.1. This is not the case for $ϵ\ll 1$, where the amplitude parameter has to be taken into account, especially if, in addition to a small $ϵ$, also large Womersley numbers $W{o}^2\gg 1$ occur. In these cases, the streaming Reynolds number $R{e}_S=Reϵ$ is significant, leading to an increased influence of steady streaming in the HMT process [7]. Most investigated authors account for the influence of steady streaming by correcting the steady HMT correlations with a term ‘${ϵ}^l$’, or sometimes directly working with the streaming Reynolds number. Taxonomy (1) was updated to incorporate these models accordingly. The meta-study by Al Taweel and Landau [29], confined to the mass transfer due to steady streaming, laid the groundwork for this updated meta-study. On the one hand, unfortunately, some of their referenced papers are no longer available, leaving only the data as stated by Al Taweel and Landau. On the other hand, even more data could be retrieved from some of the works investigated by Al Taweel and Landau with the help of the relations presented in Figure A1. A detailed explanation of the conducted data preparation can be found in Appendix A. However, much more data from various recent works are available now, especially for a gaseous environment: Gopinath and Mills [38] based their work on the flow formulations by Riley [43] for the region of impactful steady streaming with $ϵ\ll 1$ and $W{o}^2\gg 1$. They derived the governing equation of energy for this case, which is numerically solved along with the equation of motion in order to provide several data points for a regression. A simple experiment was conducted for the validation of the resulting correlation. Kawahara et al. [37] investigated the mass transfer from a camphor-covered sphere placed in a USL. They found good agreement with the experiments of Gopinath and Mills and Burdukov and Nakoryakov. Drummond and Lyman [39] applied a pseudospectral method in order to solve the NSEs and mass transport equations. They found a decreasing HMT for an increasing amplitude parameter, which is a unique result within the literature. Additionally, they suggested a quasi-steady limit of $ϵ\ge 0.25$. Alassar et al. [40] solved the NSEs and energy equations for a Boussinesq fluid. The Prandtl number was assumed constant, $Pr=0.71$, as was done in this work. While the authors investigated the forced and mixed convection regimes, only data for forced convection were utilized in this work for better comparability. Ha and Yavukurt [36] solved the two-dimensional, unsteady, laminar conservation equations for mass, momentum, and energy transport numerically in order to investigate the heat transfer to a particle. They found that for $ϵ\gg 1$ it can be approximated well with the steady HMT approach. Xu et al. [41] investigated the heat transfer from a coal particle in a power plant boiler in the presence of an acoustic field. The mathematical framework of Ha and Yavukurt was utilized, while the particle size was kept constant at 100 $\mathsf{\mu }$$\mathrm{m}$ and the flue gas properties were kept constant at a temperature of 1200 ${}^{\circ }\mathrm{C}$. The oscillation frequency and oscillation amplitude were varied and they found a decrease in heat transfer intensity at $ϵ\ll 1$, an increase at $ϵ\approx 1$, and a decreasing dependency on the amplitude parameter for $ϵ\gg 1$. The numerical calculations were validated by experiments with copper spheres in an acoustic field. Blackburn [25] investigated the heat and mass transfer to a particle in an oscillating flow numerically. The author also utilized the two defining dimensionless numbers of amplitude parameter $0.05\le ϵ\le 5$ and oscillating Reynolds number $1\le Re\le 100$. Additionally, the steady case was calculated for comparison. Blackburn found a slightly decreased HMT intensity for the oscillating case compared with the steady case. Even though the author’s values for the steady case are slightly higher than in many other works, this is a unique result.

The HMT in gases and liquids are quite similar to each other since it is subject to the same physical phenomena and therefore often modeled similarly. Although the Prandtl numbers (Schmidt numbers) are commonly several magnitudes greater in gases than in liquids, the basic models and correlations are the same. In fact, the statements in this work so far can be applied to gases and liquids alike. This is not the case for steady streaming, where different asymptotic behavior could be observed for gases and liquids, necessitating a distinction of cases [29]. Therefore, similar to the approach for the steady models in the previous section, two meta-correlations were found covering the entire plane, each of which being applicable to either a gaseous or liquid environment, respectively. The Nusselt number (Sherwood number) is averaged over the particle surface and averaged over one oscillation cycle in this case:

$$\begin{array}{c}\mathrm{Meta}\text{-}\mathrm{correlation}\mathrm{for}\mathbf{gaseous}\mathrm{environments}:\\ \hfill \overline{\overline{Nu}}\left(\overline{\overline{Sh}}\right)=2+& 0.5R{e}^{1/2}P{r}^{1/2}\left(S{c}^{1/3}\right)\left[\frac{1}{0.45{ϵ}^{-1/2}+1}+\frac{1}{2.50exp{(log\left(ϵ\right))}^2-1.25}\right]\hfill \end{array}$$

(4)

$$\begin{array}{c}\mathrm{Meta}\text{-}\mathrm{correlation}\mathrm{for}\mathbf{liquid}\mathrm{environments}:\\ \hfill \overline{\overline{Nu}}\left(\overline{\overline{Sh}}\right)=2+& 0.5R{e}^{1/2}P{r}^{1/3}\left(S{c}^{1/3}\right)\left[\frac{1}{0.78{ϵ}^{-1/6}+1}+\frac{1}{2.50exp{(log\left(ϵ\right))}^2-1.85}\right]\hfill \\ \hfill \underset{A}{\underbrace{}}& \underset{B}{\underbrace{}}\underset{C}{\underbrace{}}\underset{D}{\underbrace{}}\hfill \end{array}$$

(5)

Figure 4 displays all the investigated models in the plane together with Meta-Correlation (4) for gaseous environments. The Prandtl number is set to $Pr=0.71$ as can be found with air at STP.

4. Discussion

4.1. Meta-Correlation Design

The plane captures the extremes of very small and very large amplitude parameters and Reynolds numbers. It was ensured that Meta-Correlations (4) and (5) follow all the various asymptotic behaviors that the experimental and theoretical works in the literature derived. For small Reynolds numbers, the conductive (diffusive) limit of $Nu\left(Sh\right)=2$ for a sphere is ensured with term ‘A’. The quasi-steady behavior of $ϵ\gg 1$ is modeled with term ‘B’, which reflects the Steady Meta-Correlation (2). While term ‘C’ tends towards unity for large $ϵ$, term ’D’ tends towards zero. The steady Term ‘B’ is then corrected for $ϵ\ll 1$ with Term ‘C’. This term brings the dependency on $ϵ$ and corrects the factor accordingly. Here, a distinction needs to be drawn between the asymptotic behavior in gases and liquids, hence the two correlations differ here. The enhancement of the HMT at $ϵ\approx 1$, which is suggested by several models, is described by term ‘D’. By form, the standard normal distribution of a probability density function was utilized here. It describes the enhancement of the HMT at $ϵ\approx 1$ well, especially for $ϵ<1$. The expectation and the standard deviation were fitted well to values of zero and unity, respectively. This term also differs for gases and liquids, mostly in order to offset the difference in Term ‘C’ and to ensure model consistency. The split of correlations in the steady streaming area for gases and liquids in Figure 4 is noticeable, especially with their different asymptotic behavior for $limϵ\to 0$, while the asymptotic behavior for gases is captured well with Meta-Correlation (4).

4.2. Deviations

Figure 5 shows the deviations of Meta-Correlations (4) and (5) from the respective individual models.

The colored bars give the $NRMSD$, calculated with Equation (3), while the white bars give the maximum normalized deviation. Meta-Correlations (4) and (5) deviate insignificantly from Steady Meta-Correlation (2) and from the investigated steady models in general, as pointed out in Section 3.1. While the correlations for steady flow behave very similarly, this is not so clear for oscillating flows. Besides the mentioned necessity for a distinction between gaseous and liquid flows, due to their different asymptotic behavior, the overall measured and simulated data diverge substantially. Drummond and Lyman are the obvious outlier here since they are the only authors who predicted an increase of HMT intensity with a decreasing amplitude parameter $ϵ$, which is also reflected in a large deviation of their data. Blackburn [25] commented on this study and suggested that the mesh resolution near the surface of the sphere might be too low. This would result in the inability to capture the flow behavior for small oscillation amplitudes properly. The other models for oscillating flow, which often cover the same region of the plane, deliver scattered data. This implies an insufficient experimental or simulation design by many authors, or additional dependencies beyond solely the oscillation Reynolds number and the amplitude parameter, contrary to the consensus in the literature. Nevertheless, based on the currently available data, Meta-Correlations (4) and (5) are viable, especially with regard to the large parameter ranges covering several orders. This is also reflected in their $NRMSD$ from all combined data of 0.8% for gases and 3.7% for liquids. Additionally, Meta-Correlations (4) and (5) only slightly deviate from the previous meta-analyses in their respective part of the plane (1.2% for gases and 8.3% for liquids).

4.3. Quasi-Steady Assumption

Often in the literature, the quasi-steady assumption is applied for the HMT to particles in oscillating flows. Figure 6 shows a comparison of Meta-Correlation (4) with the derived Steady Correlation (2). It shows that the quasi-steady approach is only valid for $ϵ\gg 1$ or $Re<1$. For $ϵ\approx 1$, it under-predicts the intensity of the HMT, while it over-predicts the intensity of the HMT for $ϵ\ll 1$ and $Re\gg 1$.

5. Conclusions

Overall, the derived Meta-Correlations (4) and (5) can be considered viable under consideration of:

• the high number of 33 correlated data sets;
• the large size of the covered parameter space of amplitude parameter $ϵ$ 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 $ϵ\approx 1$, and the quasi-steady HMT for $ϵ\gg 1$.

The presented meta-correlations enable a direct prediction of the heat (and mass) transfer to a single, solid particle in an oscillating gas or liquid flow, with a known oscillating Reynolds number $Re$, amplitude parameter $ϵ$, and Prandtl number $Pr$ (or Schmidt number $Sc$).