Heat transfer in particle-laden turbulence
Preferential concentration of inertial particles by turbulence is a well recognized phenomenon. This study investigates how this phenomenon impacts the mean heat transfer between the fluid phase and the particle phase. Using direct numerical simulation for turbulent flows and Lagrangian point particle tracking, we explore this phenomena over wide range of input parameters. Among the nine independent dimensionless numbers defining this problem, we show that particle Stokes number and a new identified number called heat mixing parameter have the most significant effect on particle to gas heat transfer, while variation in other non-dimensional numbers can be ignored. Using our numerical results we propose an algebraic reduced order model for heat transfer in particle-laden turbulence.
A broad range of natural and industrial processes involve interaction of particles and background turbulent flows —formation of clouds (Shaw, 2003; Grabowski and Wang, 2013), dispersion of pollutant in urban areas (Britter and Hanna, 2003), planetary accretion (Cuzzi et al., 2001), spray combustion (Sahu et al., 2014), and particle-based solar receivers (Pouransari and Mani, 2017) to name a few.
Particle-turbulence interaction results in a range of well-studied phenomena. A particle immersed in turbulent flow experiences a centrifugal force from high vorticity regions toward high strain regions. This results in inhomogeneous distribution of particles, known as preferential concentration (Lazaro and Lasheras, 1989; Squires and Eaton, 1991). Where gravity is present particles exhibit preferential sweeping (Wang and Maxey, 1993). In wall-bounded particle-laden flows turbophoresis, which refers to the tendency of particles to concentrate close to the wall, is expected (Reeks, 1983).
In many of the particle-laden flow scenarios, a primary interest is in understanding of thermal exchanges between the two phases. For example, in particle-based solar receivers, particles are the primary absorbers of external radiation, which then conductively transfer their absorbed heat to the carrier fluid. The heated particles absorb fraction of the received flux and transfer the rest to the surrounding fluid. In the case of heated particle-laden flows, additional phenomena are observed. Pouransari et al. (2017) showed that under sufficiently large thermal flux, hot particles can modify turbulence spectra through pressure-dilatation. When gravity is present, heated particles give rise to non-uniform buoyant forcing of the flow, resulting in a sustained turbulence (Zamansky et al., 2014, 2016). Frankel et al. (2016) showed that when particles are heated the preferential sweeping can be supressed or even reversed. In a previous study (Pouransari and Mani, 2017) we investigated a specific regime of particle-laden flows and showed that preferential concentration of particles by turbulence can adversely impact the heat transfer efficiency. To obtain a fundamental understanding of impact of particle clustering on heat transfer, in the present study we consider a canonical setting involving heat transfer from inertial particles to a turbulent flow. By considering a combination of DNS data and a simple phenomenological model, we develop and verify an algebraic model for heat transfer in particle-laden turbulent flows.
Ii Model problem
We consider direct numerical simulations of homogenous isotropic turbulence (HIT) laden with heated point particles in a triply periodic box with length . The simulation code (Pouransari et al., 2015) is fourth order in time and second order in space using uniform staggered grid. A linear forcing scheme Rosales and Meneveau (2005); Lundgren (2003) is used to maintain a statistically stationary turbulence with zero-mean velocity.
The cold mixture (with temperature ) is simulated for sufficiently large time with no external heating to obtain a fully developed particle-laden turbulence. By monitoring the fluid kinetic energy and particle segregation (see for example Vié et al., 2016) versus time, we verify statistically stationarity of fluid and particle phases. This is achieved after 50 large-eddy turnover times defined below. Particle heating is then activated with constant heat flux, and the heated mixture is allowed to be developed. The statistically stationary heated state is verified by monitoring the mean particle to fluid heat flux versus time. All statistics are collected after these two transition stages over a period of order 100 large-eddy turnover times. Note that when statistically stationary state is achieved, the ensemble-averaged particle and fluid temperatures grow linearly with time, while the ensemble-averaged temperature difference and particle-to-fluid heat transfer are constant.
The fluid phase is assumed to be variable density governed by the ideal gas equation of state , subject to a low-Mach flow. The dynamic viscosity , constant-volume and constant-pressure specific heat coefficients , , and heat conductivity coefficient of the gas are assumed to be constant and independent of the temperature. Conservation of mass implies constant averaged gas density .
Mono-dispersed spherical particles with density and a constant diameter much smaller than the Kolmogorov micro scale are suspended in the fluid. Particle specific heat coefficient is constant and independent of the temperature. We assume particle temperature to be a lumped quantity (constant along one particle) justified by large particle Biot number. The slip velocity between particle and the surrounding gas is assumed to be a finite small value. Thus, we assume that the particle momentum and heat exchange with the fluid can be expressed, respectively, in terms of drag and heat exchange laws derived in the low Reynolds and Péclet limits. We ignore the convective effects at the scale of particles justified by low thermal Péclet number based on slip velocity, and particle diameter.
The particle-fluid mixture is assumed to be very dilute (volume fraction ). We use a simplified version of the Maxey and Riley (1983) equations describing the dynamics of an immersed particle, and model particle motion through the Lagrangian point particle framework. In the regimes considered in our study, Stokes drag is the only significant force experienced by particles. We ignore momentum two-way coupling (i.e., particles do not modify the fluid through momentum equation).
ii.2 Non-dimensional equations
Based on the aforementioned assumptions, we introduce a set of dimensionless equations describing heated particle-laden flows. We use flow integral length scale as the reference length scale, where is the root mean square of the single-component velocity fluctuations, and is the averaged dissipation rate. Large-eddy turnover time is used as the reference timescale. , , and are used to non-dimensionalize velocities, gas density, and temperatures, respectively.
In the above equations , , , and denote density, velocity, hydrodynamic pressure, and temperature of the gas phase, respectively. The last term in eq. 2 is the non-dimensionalized linear forcing term (Rosales and Meneveau, 2005) to maintain turbulence (with dimensionless dissipation rate 1). The last term in eq. 3 is the heat transfer from particles to gas. and , respectively, represent temperature and position of a particle with index , where is the total number of Lagrangian particles. is dimensionless three-dimensional Dirac delta function.
The non-dimensionalized equation of state for ideal gas is . The thermodynamic pressure is non-dimnesionalized with , where is the gas constant. Note that given the low Mach number assumption, the thermodynamic pressure is assumed to be constant in space.
In the above equations denotes velocity of a particle with index . The first and second terms in the right hand side of eq. 5 are the constant heat flux absorbed by a particle and the heat transfer from particle to gas, respectively.
Next, we introduce the dimensionless factors denoted in eqs. 5, 4, 3, 2 and 1. is the Reynolds number. is the ratio of gas heat capacities. is the Prandtl number. is the particle Stokes, which is the ratio of particle momentum relaxation time to the gas large-eddy turnover time.
is the heat mixing parameter defined as the ratio of gas thermal relaxation time to the large-eddy turnover time, where is the mean particle concentration. Nu is the Nusselt number for particle to gas heat transfer. Note that we selected the fluid thermal relaxation time, as opposed to the commonly used particle thermal relaxation time to form the dimensionless heat mixing parameter. For reasons that are discussed in section III.5 we shall see the former choice results in minimal number of significant dimensionless parameters, while the latter choice would not.
is the ratio of dispersed phase total heat capacity to the gas phase total heat capacity. is the non-dimensional heat source, where is the external heat flux received by each particle. can be interpreted as the ratio of large-eddy turnover time to the gas warmup time .
Iii Heat-transfer model
iii.1 Reduced order equations
In this section, we develop a reduced order model to describe the evolution of the averaged particle and gas temperature. We start with definition of the averaging operator.
For given scalar fields and in domain , the weighted average is defined as
If we use gas density as weight, is the Favre-average (Wilcox et al., 1998). We define dimensionless particle local concentration as . Therefore, . In case of particle temperature .
iii.2 Correction factor
Equation 7 is not closed due to appearance of on the right hand sides. Similar to Pouransari and Mani (2017), we define a correction factor for the heat transfer term as follows to close the equations.
Therefore, eq. 7 transforms to the following equations.
is the average gas temperature seen by particles, similar to the concept of average gas velocity seen by particles when the drift velocity is concerned (Jung et al., 2008). Since particles are directly heated, it is expected that the average temperature of the gas at the location of particles to be greater than the volume-averaged temperature of the gas, . Hence, we expect . The closure question is then to determine in terms of known input parameters. This is similar to the work by Sumbekova et al. (2017), in which they investigate the parameter space of unheated particle-laden turbulence experimentally to explore the effect of each parameter on preferential clustering.
iii.3 Parameter study
We investigate the dependence of the parameter in eq. 9 on the dimensionless numbers governing the problem as introduced in section II.2. Considering common gas-solid mixtures, we assume , , and . We sweep the parameter space by changing the remaining dimensionless number(s) of interest at a moment, while all other dimensionless numbers are kept constant. In table 1 we list the nominal value and sweeping range of each dimensionless number. For each dimensionless number, we run a simulation of the full 3D equations for sufficiently long time, and average the value of when the heated HIT is developed in time by post-processing the data to compute terms on the right-hand-side of eq. 8.
The heat transfer equations are linearly dependent on , and thus one expects that the correction factor, , to be independent of this parameter. This expectation is justifiable as long as thermal flux is not too strong to modify the turbulence itself (Pouransari et al., 2017). Figure 1 depicts as a function of , while other dimensionless numbers are kept at their nominal values, verifying is independent of . In addition, in the limit of sufficiently large and negligible particle-particle collisions it is expected that to be independent of . Figure 1 shows versus confirming its independence on . Increasing and/or gives rise to higher total heat transfer from particles to gas, yet the correction factor is constant.
In fig. 1 we plot as a function of particle to gas total heat capacity . This figure suggests weak dependence of on particularly in the limit of large or small . In section III.5 we provide a phenomenological model justifying this observation. Therefore, only three remaining dimensionless numbers , and may significantly affect the correction coefficient .
Figure 1 illustrates variation of as a function of Reynolds number. The bottom x-axis shows , and the top x-axis shows , the Reynolds number based on the Taylor’s micro-scale. For HIT we have . The value of other dimensionless numbers are the same as table 1, except the particle Stokes number which is kept at . Our results suggest that when Stokes number and heat mixing parameter are defined based on the large eddy turnover time, is a weak function of the Reynolds number. We further discuss implications of alternative choices of reference time scale in the definition of Stokes number and heat mixing parameter in section III.4.
Note that we found the correction factor to be independent of , , and assuming no turbulence modification by the particles. In general, particles can modify the background turbulence either through the momentum exchange or through local expansion resulting from heat transfer. The former is significant when particle mass loading ratio is high (Squires and Eaton, 1990; Elghobashi and Truesdell, 1993; Sundaram and Collins, 1999), and the latter is significant in case of high heating that results in high dilatational modes quantified by (Pouransari et al., 2017). This is the case in most turbulent combustion applications, for example. Here we assume low particle mass loading ratio and low total heating resulting in negligible turbulence modification.
In fig. 2 the heat transfer correction factor is plotted as a function of particle Stokes number. The bottom x-axis is Stokes number based on large-eddy turnover time , and the top x-axis shows the Stokes number based on the Kolmogorov time scale , where is the Kolmogorov timescale. For HIT we have .
For very small and large Stokes numbers particle distribution is close to homogeneous. Therefore, the weights in eq. 6 are almost uniform, and is close to one. For moderate values of Stokes number the highest level of preferential concentration is expected. In this case particle to gas heat transfer occurs at the location of particle clusters that inevitably introduce spatial heterogeneity. Hence, when preferential concentration is high, the effective volume of cold gas seen by particles is reduced resulting in less heat transfer from particles to gas (i.e., ).
Figure 2 shows as a function of heat mixing parameter . is a strictly increasing function of , such that as , , and as , . The value of quantifies the rate of heat mixing by turbulence in terms of gas thermal relaxation time scale. Small values of means heat mixing due to turbulence is weak, and large values of represents strong heat mixing due to turbulence. We study the effects of simultaneous variations of and in section III.5.
iii.4 The choice of reference timescale
In this study we use large-eddy turnover time as the flow reference timescale. Alternatively, we could use the Kolmogorov time scale as the reference flow timescale. Using the Kolmogorov time scale, in particular, is appealing in order to define the Stokes number as done in numerous previous studies characterizing preferential concentration.
In fig. 1 we showed that the correction factor has small dependence on Reynolds number, when all other non-dimensional numbers are kept constant. Note that as the Reynolds number increases the ratio between large-eddy turnover time and the Kolmogorov time also increases as well. Therefore, the choice of reference time scale for Stokes number and the heat mixing parameter , that are kept constant while Reynolds is varied, is important.
In fig. 3 variation of with Reynolds number is shown when different timescales are used for Stokes number and/or heat mixing parameter . Subscript denote normalizing with the large-eddy turnover time, and subscript denote normalizing with the Kolmogorov time. Note that is the same as defined in section II.2. Figure 3 demonstrates that our choice of large-eddy turnover time for both Stokes number and heat mixing parameter results in the least dependency of on Reynolds. Further studies are required to investigate variations of for larger values of Reynolds number.
iii.5 Closure model
We concluded that Stokes number and heat mixing parameter, defined based on , are the most relevant dimensionless numbers determining . Therefore, we sweep the parameter space in two dimensions ( and ) to discover the full dependency. In fig. 3 variation of as a function of Stokes number is plotted for different values of . Other dimensionless numbers are kept at their nominal values as reported in table 1.
The non-monotonic dependence of as a function of can be observed for all values of . However, as dependence of on particle Stokes number vanishes. For large values of the heat mixing due to turbulence is strong, therefore, even for high level of preferential concentration the heat transferred from particles to gas is quickly mixed uniformly. This fast mixing makes the gas temperature uniform despite heterogeneity of the source, thus brings closer to leading to . This effect is visually evident in fig. 4 where particle distribution and normalized fluid temperature deviations, , is illustrated for various combinations of St and . Note that particles have high level of preferential concentration when irrespective of value of (the middle column in fig. 4). However, only for small values of the particle preferential concentration affect particle to gas heat transfer.
Here, we introduce a simple phenomenological heat-transfer model consistent with the observed dependency of on the dimensionless parameters. We use the result of this approach to provide a closed algebraic form for the correction coefficient .
Consider the cloud of gas in vicinity of particles as shown in fig. 5. Assume this cloud occupies volume fraction of the total gas. The coefficient is only a function of particle spatial distribution. In this approach we consider one averaged temperature for cloud of gas near particles, and one for the rest of the gas, denoted by and , respectively. We assume particles receive energy from an external heat source, and transfer energy conductively to the surrounding cloud of gas. This cloud then transfers heat convectively to the rest of the gas. In the former, heat-transfer is dominated by gas thermal relaxation time augmented by factor to account for smaller mass fraction. In the latter, large-eddy turnover time is the dominant time scale in heat transfer as the mixing by large eddies is the main heat-transfer mechanism (see Prandtl, 1925; Dimotakis, 2005; Vassilicos, 2015). Using the same non-dimensionalization as in section II.2 the governing equations are as follows.
Neglecting spatial variation of gas density we can write:
Combining eqs. 11 and 10, and noting that under the fully developed conditions all temperatures increase linearly with time with the same slope (, to satisfy conservation of energy for the full system) we can derive the following equation:
Comparing against equation eq. 5 suggests the correction factor introduced in eq. 8 is . This analysis also provides insights on the observed weak dependence of on , as the derived expression does not involve parameter in this simplified limit.
Next, assuming being only function of St, we consider a generalized form for . Based on our phenomenological model we expect a value of close to 1. Reverse engineering of this expression indicates that versus St must result in plots independent of . As shown in fig. 5, results in collapse of curves corresponding to different values of confirming the usefulness of our phenomenological model.
We consider as a generalization to the expression suggested by Esmaily-Moghadam and Mani (2016) to quantify the level of particle preferential concentration. We found the optimal values of , , , and by fitting to our data. Finally, we propose the following algebraic form for particle-to-gas heat transfer correction coefficient:
In this paper, we developed a non-dimensional set of equations describing heated point particles suspended in a variable-density turbulent flow. We studied the averaged particle to fluid heat transfer using direct numerical simulations for fluid and Lagrangian point particle tracking for the dispersed phase. In the presented formulation, nine non-dimensional numbers appear. Considering wide range of applications involving small solid particles in gas, we investigated the regime in which , , and , while we systematically varied all other non-dimensional numbers.
We showed that in the limits studied in this paper the number of particles, dimensionless heating rate, ratio of particle to fluid heat capacities, and Reynolds number have minor effect on mean heat transfer coefficient. However, our results indicate that particle Stokes number () and the newly introduced non-dimensional number, the heat mixing parameter (), have significant effect on particle to fluid heat transfer. The former parameter governs the topology of particle distribution, which controls the spatial distribution of heat sources to the fluid. The latter parameter determines how fast the background fluid can mix the heat received by particles. Therefore, even for high level of preferential concentration we can expect almost uniform fluid temperature distribution due to turbulent mixing when is high.
Using our parameter study, in section III.5 we introduced an algebraic closure formula to model macroscopic particle to gas heat transfer for general conditions. The inputs of our model are and that found to be the most relevant non-dimensional numbers for particle to fluid heat transfer. Furthermore, we explored different timescales to define the Stokes number and heat mixing parameter, namely large-eddy turnover time and Kolmogorov time. Our results indicate that macroscopic-averaged heat transfer correction coefficient, , is best described by these parameters when they are nondimensionalized based on large eddy time.
A direct application of this study is to provide closure to heat transfer terms in macroscopic models that do not directly capture particle clustering. For example, Reynolds-averaged Nervier Stokes (RANS) models only represent the ensemble-averaged velocity fields. Therefore, even the most accurate particle solver can provide mean particle number density, while missing the clustering effect. This is a suitable situation for application of the proposed model, by which the heat transfer terms can be closed using the available mean particle number density and mean turbulence dissipation rate (provided by the turbulence model). Future researches can investigate wether the presented approach can be adopted in the context of large-eddy simulations (LES), where the input parameters are defined based on sub-grid turbulence dissipation rate.
This work can be extended by relaxing some of our assumptions such as considering temperature dependent thermodynamical properties, inhomogeneous flows, and compressibility effects in case of extreme heating. Furthermore, dependence of our model parameter on the Reynolds number can be further studied by considering flows with larger values of Reynolds number.
We would like to acknowledge the support by the US Department of Energy under the Predictive Science Academic Alliance Program 2 (PSAAP2) at Stanford University.
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