Analytic Modeling of Heat Transfer to Vertical Dense Granular Flows

Watkins, Megan F.; Chilamkurti, Yesaswi N.; Gould, Richard D.

doi:10.1115/1.4045311

Abstract

The high packing fractions of dense granular flows make them an attractive option as a heat transfer fluid or thermal energy storage medium for high temperature applications. Previous works studying the heat transfer to dense flows have identified an increased thermal resistance adjacent to the heated surface as a limiting factor in the heat transfer to a discrete particle flow. While models exist to estimate the heat transfer to dense flows, no physics-based model describing the heat transfer in the near-wall layer is found; this is the focus of the present study. Discrete element method (DEM) simulations were used to examine the near-wall flow characteristics, identifying how parameters such as the near-wall packing fraction and number of particle-wall contacts may affect the heat transfer from the wall. A correlation to describe the effective thermal conductivity (ETC) of the wall-adjacent layer (with thickness of a particle radius) was derived based on parallel thermal resistances representing the heat transfer to particles in contact with the wall, particles not in contact with the wall, and void spaces. Empirical correlations based on DEM results were developed to estimate the near-wall packing fraction and number of particle-wall contacts. The contribution from radiation was also incorporated using a simple enclosure analysis. The ETC correlation was validated by incorporating it into dense flow models for chute flows and cylindrical flows and comparing with the experimental data for each.

Introduction

The heat transfer to particulate flows has been studied for decades due to its various industrial applications, including drying and heating of particulates such as food grains and coal. More recently, the heat transfer characteristics of particle-based heat transfer fluids are of interest for concentrated solar power tower applications, as particles offer many intriguing characteristics over the current fluids. The present work studies the heat transfer behavior of internal gravity-driven dense granular flows.

Internal granular flows are characterized by shearing at the wall boundaries and are driven by particle collisions. Dense flows demonstrate high packing fractions of approximately 60%. Studies of the heat transfer in a static particle bed indicate that the packing fraction plays a primary role in the heat transfer capabilities of the bed, as the primary pathway for heat transfer is through the interstitial gas [1]. Since the interstitial gas typically exhibits low thermal conductivity compared with the particle material, the effective thermal conductivity of a bed decreases with decreasing packing fraction. Therefore, the high packing fraction of dense flows makes them a viable option as a heat transfer fluid.

While the heat transfer characteristics of static particle beds have been studied quite extensively both experimentally (for example, Refs. [2–4]) and theoretically (as summarized in Ref. [1]), the heat transfer to dense flows has been less widely studied. Sullivan and Sabersky [5] provided one of the pioneering works in the field with their experimental and theoretical analysis of the heat transfer to a vertical chute flow. Using a small rectangular cross section with a single heated wall, they examined the variation in the heat transfer measured for different particles types, particles sizes, and flowrates. In an attempt to understand the significance of the discrete nature of the flow, they compared their experimental results with an analytic solution for a continuum plug flow in their setup. Their experimental results did not agree with the results predicted by the continuum solution; instead, the granular flows demonstrated inferior heat transfer performance, which was attributed to the discrete nature of the flow and the discrete interaction of the particles with the heated wall. The analytic solution was modified to incorporate an increased thermal contact resistance at the heated wall, which accounted for reduced heat transfer due to a more structured arrangement of particles adjacent to a surface. The contact resistance was represented as a thin air gap a fraction of a particle diameter thick (0.085 in their case) and was assumed constant for all test cases. The resulting semi-empirical correlation relating the Nusselt number with the Péclet number showed better agreement with their experimental results. Natarajan and Hunt [6] expanded upon the work of Sullivan and Sabersky, studying longer heated lengths and higher flowrates. Their experimental results demonstrated similar trends to those observed by Sullivan and Sabersky for low velocity, high density flows; namely, an increasing Nusselt number with increasing Péclet number. At higher flowrates, however, Natarajan and Hunt observed a critical point after which the Nusselt number decreased and plateaued. The semi-empirical model developed by Sullivan and Sabersky captured the results observed for low velocity flows but was unable to capture the decrease observed at higher flowrates. The deviation from the model at higher flowrates was explained by recognizing that the density of the flow adjacent to the wall varies with flowrate, rendering the constant contact resistance assumption of Sullivan and Sabersky invalid when considering a wide range of flowrates.

Spelt et al. [7], Patton et al. [8], and Ahn [9] studied the heat transfer to inclined chute flows over a wide range of flowrates. Similar to Natarajan and Hunt, their results displayed a critical point after which the Nusselt number began to decrease with increasing Péclet number. Once again, the model developed by Sullivan and Sabersky captured the trends observed at lower flowrates but was unable to capture the decrease observed at higher flowrates. It is important to note that the best agreement with the model was found when using an air gap thickness of approximately 1/40th of a particle diameter. The thinner air gap was attributed to the difference in system configuration; gravity pushes particles toward the heated wall in an inclined chute, whereas gravity has no effect on the compactness of particles at the wall in a vertical chute. Ahn attempted to modify the model developed by Sullivan and Sabersky to incorporate the variation in the contact resistance with packing fraction; their model, however, was successful only for certain particle materials.

Denloye and Botterill [10] also studied the heat transfer to vertical chute flows, comparing their experimental results with the “packet renewal” model originally developed by Mickley and Fairbanks [11] for fluidized beds. The modified version of the model developed by Baskakov [12], which incorporates a contact resistance at the wall, was also compared with the experimental data. The thermal contact resistance was approximated using the near-wall thermal conductivity model developed by Yagi and Kunii [13] for static beds. The modified model showed better agreement with the experimental data, once again indicating the importance of the wall-adjacent layer on the overall heat transfer to the flow.

More recently, Morris et al. [14] sought to characterize the thermal contact resistance for inclined chute flows using the same physics applied in discrete element method (DEM) simulations. Examination of the heat transfer from a heated wall to a single particle indicated that the heat transfer decreases quickly when the particle loses contact with the wall, as the small layer of air between the wall and the particle acts as an insulation layer. This phenomenon was incorporated into their model to estimate the contact resistance using a particle-wall distribution function. Using DEM simulations to examine the particle-wall distribution function for different flow parameters and angles of incline, an empirical correlation to describe the Nusselt number in the wall-adjacent layer was developed in the form of a seventh-order polynomial. Incorporation of this model into a two-phase continuum model as a boundary condition showed reasonable agreement with results from DEM flow simulations [15]. The model has not yet been compared with the experimental results.

Our previous work [16] experimentally investigated the heat transfer to gravity-driven dense granular flows of ceramic particles in cylindrical tubes. The effect of flowrate and other flow parameters on the heat transfer to the flow at low operating temperatures was studied, along with the effect of temperature on the heat transfer for a single flowrate. Comparison of the experimental results with a continuum solution for the system studied emphasized the importance of the discrete nature of the flow, as the granular flows demonstrated inferior heat transfer performance. This work seeks to develop a model to predict the heat transfer characteristics observed experimentally and to gain a better understanding of the near-wall heat transfer behavior, examining how the contact resistance varies with different flow properties. DEM simulations were used to examine different flow characteristics in the wall-adjacent layer, assessing how near-wall particle structure may influence the heat transfer behavior. These observations were used to develop a physics-based model to predict the heat transfer through the wall-adjacent layer in the form of an effective thermal conductivity (ETC). The ETC correlation developed was incorporated into a generalized heat transfer model derived for the cylindrical flow system studied experimentally and compared with the experimental results. The ETC correlation was also incorporated into the model developed by Sullivan and Sabersky [5] and compared with the experimental results of both Sullivan and Sabersky and Natarajan and Hunt [6] to emphasize the applicability and versatility of the model developed.

Modeling Approach

A common approach used to model the heat transfer in static particle beds entails dividing the cross section into two different regions: a bulk region that encompasses the majority of the bed and a thin layer adjacent to the wall characterized by an increased thermal resistance, which captures the reduced heat transfer resulting from the discrete arrangement of particles adjacent to a surface. A similar method was employed in the present work to model the cylindrical flow system studied experimentally [16]. Figure 1 depicts the two layers employed and the corresponding dimensions. An analytic solution was derived for a thermally developed flow using the two-dimensional energy equation in cylindrical-axial coordinates

U \frac{\partial T_{j} (r, z)}{\partial z} = \frac{α_{j}}{r} \frac{\partial}{\partial r} (r \frac{\partial T_{j} (r, z)}{\partial r})

(1)

Fig. 1

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Schematic of layers used to model dense flow in a cylindrical tube

Here,

T

represents the temperature,

U

represents the axial flow velocity,

α

represents the thermal diffusivity,

r

represents the radial position,

z

represents the axial position, and subscript

j

denotes the jth layer of the composite. Note that Eq. models the flow as a plug flow; this is a common assumption employed for internal granular flows, as such flows display an essentially uniform velocity profile with a thin shear layer [17,18]. Assuming constant thermal properties in each layer, Eq. was solved subjected to the following boundary conditions:

T_{b} (r = 0, z) is finite

(2)

k_{wa} {\frac{\partial T_{wa}}{\partial r} |}_{r = R_{i}, z} = q_{wall}^{''}

(3)

T_{b} (r = δ, z) = T_{wa} (r = δ, z)

(4)

k_{b} {\frac{\partial T_{b}}{\partial r} |}_{r = δ, z} = k_{wa} {\frac{\partial T_{wa}}{\partial r} |}_{r = δ, z}

(5)

T_{b} (r, z = 0) = T_{wa} (r, z = 0) = T_{in}

(6)

where

k

represents the effective thermal conductivity,

δ

represents the radius of the bulk layer,

R_{i}

represents the inner tube radius,

q_{wall}^{″}

represents the constant wall heat flux applied at the outer boundary, and subscripts

b

and

wa

denote the bulk and wall-adjacent layers, respectively. Note that a constant wall heat flux boundary condition was employed at the tube wall to align with the conditions tested experimentally [16]. The radius of the bulk layer, as defined in Fig. 1, is equivalent to

δ = R_{i} - 0.5 d_{p}

⁠, where

d_{p}

represents the particle diameter. Solving Eqs. –, assuming that the density and the specific heat of the bulk and wall-adjacent layers are the same, yields the following Nusselt number correlation:

N u_{k_{b}} = \frac{h D_{i}}{k_{b}} = \frac{8}{W^{4} + \frac{k_{b}}{k_{w a}} (1 - W^{4})}

(7)

W = (\frac{δ}{R_{i}}) = (1 - \frac{1}{(D_{i} / d_{p})})

(8)

Here, $h$ represents the convective heat transfer coefficient of the flow, $D_{i}$ represents the inner tube diameter, and $W$ is a new parameter defined to represent the ratio of the outer radii of the two layers. Note that $W$ depends solely on the system configuration; therefore, according to Eq. (7), the Nusselt number depends solely on the ratio of the bulk-to-wall-adjacent layer thermal conductivities for a given system configuration.

Examination of Eq. (7) for a single layer (i.e., when $k_{b} = k_{wa}$ ⁠) indicates that the Nusselt number converges to a constant value of 8, as expected for a thermally developed plug flow subjected to a constant wall heat flux [19]. It is interesting to note that Eq. (7) suggests that the Nusselt number varies with the system configuration (i.e., $D_{i} / d_{p}$ ⁠), increasing with an increase in the tube-to-particle diameter ratio. As $D_{i} / d_{p} \to \infty (i . e ., W \to 1)$ ⁠, the Nusselt number again approaches the continuum solution of 8. This is reasonable, as the relative size of the bulk layer (⁠ $W$ ⁠) approaches a value of 1. This observation suggests that the relative size of the bulk layer should be considered when designing dense flow systems.

Effective Thermal Conductivity of Wall-Adjacent Layer.

In order to approximate the Nusselt number using Eq. (7), an understanding of the effective thermal conductivities of the bulk and wall-adjacent layers is required. As summarized by Van Antwerpen et al. [1], numerous correlations exist in the literature to estimate the bulk effective thermal conductivity of a static particle bed, while fewer exist for estimating the effective thermal conductivity of the wall-adjacent layer. While the existing correlations vary in complexity, all correlations depend on three primary parameters: the packing fraction of the bed, the interstitial gas thermal conductivity, and the solid material thermal conductivity. As noted earlier, previous works modeling the heat transfer to dense granular flows relied on empirical constants and static bed correlations to approximate the increased thermal resistance at the wall. Morris et al. [14] noted, however, that static bed correlations, which are derived for densely packed beds, do not capture the intermittent nature of particle contacts in a flow. Our work, therefore, sought to characterize the effective thermal conductivities of the bulk and wall-adjacent layers for a flowing case.

Discrete element method simulations were used to examine the particle flow mechanics and their variation with flowrate, inferring how the observed characteristics may affect the heat transfer to the flow. A description of the DEM simulation methodology is beyond the scope of this work; however, a summary of the DEM simulation setup and validation studies performed can be found in Refs. [20–22]. The DEM simulations were used to calculate the packing fraction and instantaneous number of particle-wall contacts for various flowrates (a description of calculation procedure is found in Ref. [23]).

An example of the variation in the packing fraction observed in both the bulk and wall-adjacent layers is displayed in Fig. 2 for 300 μm spherical particles in a 7.75 mm ID tube. The packing fraction of the bulk layer varies minimally from the static bed value of 62%, decreasing by less than 4% for the range of flowrates tested. Static bed values for the bulk layer effective thermal conductivity were therefore deemed a reasonable approximation for the flowing case. The packing fraction of the wall-adjacent layer displays a slightly greater variation with flowrate, decreasing by up to 14%. A more interesting observation comes from examining the variation in the number of particle-wall contacts. On average, 10–60% of the particles in the wall-adjacent layer are not in contact with the wall, depending on the flowrate, particle diameter, and tube diameter. This variation is depicted pictorially in Fig. 3 using DEM images of the particle arrangements in the wall-adjacent layer for different flowrates. The red particles in Fig. 3 represent particles in contact with the wall, while the blue particles represent particles not in contact with the wall, and the yellow represents void spaces where no particle is present. These images indicate that the number of particle-wall contacts decrease with increasing flowrate. The variation in packing fraction with flowrate is also highlighted by the increased amount of yellow space at faster flowrates (Fig. 3(b)).

Fig. 2

$Variation of packing fraction with flowrate in both the bulk and wall-adjacent layers of dense flow$

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Variation of packing fraction with flowrate in both the bulk and wall-adjacent layers of dense flow

Fig. 3

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Particle arrangements in the wall-adjacent layer (as viewed from outside of the tube, looking through the wall) for mean flow velocities of (a) 0.023 m/s and (b) 0.25 m/s (using 7.75 mm ID tube and 300 μm particles). Red (or gray for print version) particles denote particles in contact with the wall, blue (or black for print version) particles denote particles not in contact with the wall, and yellow (light background for print version) denotes void spaces. Particle flow is in the +z-direction.

The significance of this decrease in particle-wall contacts on the heat transfer through the wall-adjacent layer is demonstrated by Morris et al. [14]. Looking at a single particle, they noted that the heat transfer from a heated wall to the particle decreases significantly as soon as the particle loses contact with the wall, as the thin gas layer that forms between the wall and the particle acts as an insulation layer. Consequently, when estimating the heat transfer through the wall-adjacent layer, it is important to consider the heat transfer to both particles in contact with the wall and those not in contact with the wall. This work sought to incorporate these observations into a simple correlation to estimate the effective thermal conductivity of the wall-adjacent layer for a dense particle flow.

Conduction in the Wall-Adjacent Layer.

The work of Legawiec and Ziolkowski [24] was used as a guide in the derivation of an effective thermal conductivity for the wall-adjacent layer. They first characterized the heat transfer to a single particle, considering the heat transfer through both the interstitial gas and the particle, then calculated the total ETC of the wall-adjacent layer based on the number of particles present in the wall-adjacent layer. Since they were considering a static particle bed, they assumed that all particles were in contact with the wall and that the amount of void spaces where no particle was present was small enough that the heat transfer through these spaces was negligible.

Our model assumes that the heat transfer through the wall-adjacent layer is composed of three thermal resistances: a resistance due to particles in contact with the wall

(R_{C})

⁠, a resistance due to particles not in contact with the wall

(R_{NC})

⁠, and a resistance due to void spaces where no particle is present

(R_{void})

⁠. The three resistances act in parallel; therefore, the total resistance to heat transfer through the wall-adjacent layer can be described by

\frac{1}{R_{wa}} = \frac{1}{R_{C}} + \frac{1}{R_{NC}} + \frac{1}{R_{void}}

(9)

R_{wa} = \frac{\ln (R_{i} / δ)}{k_{wa}} \frac{R_{i}}{A_{wa}}

(10)

R_{C} = \frac{\ln (R_{i} / δ)}{k_{C}} \frac{R_{i}}{A_{C} N_{C}}

(11)

R_{NC} = \frac{\ln (R_{i} / δ)}{k_{NC}} \frac{R_{i}}{A_{NC} N_{NC}}

(12)

R_{void} = \frac{\ln (R_{i} / δ)}{k_{void}} \frac{R_{i}}{A_{wa} - N_{C} A_{C} - N_{NC} A_{NC}}

(13)

Here,

R_{n}

represents the thermal resistance of the nth heat transfer pathway,

k_{n}

represents the thermal conductivity of the nth pathway,

A_{n}

represents the area of the tube wall associated with the nth pathway, the subscripts

wa

⁠,

C

⁠,

NC

⁠, and

void

denote the heat transfer pathway associated with the wall-adjacent layer, contact particles, no-contact particles, and void spaces, respectively, and

N_{C}

and

N_{NC}

represent the number of particles in contact with the wall and not in contact with the wall, respectively. The thermal conductivity of the void spaces is equivalent to the thermal conductivity of the interstitial gas

(k_{g})

⁠. The “contact” and “no-contact” thermal conductivities are determined for a single particle. Similarly, the contact and no-contact areas represent the tube wall area considered in the calculation of the heat transfer to a single particle of each type; they are calculated based on the projected area of a single particle on the wall. The area associated with the wall-adjacent layer (⁠

A_{wa}

⁠) represents the total tube wall area of the tube length under consideration

(A_{wa} = A_{void} + N_{c} A_{C} + N_{NC} A_{NC})

⁠. Note that the model was derived for axial flow through a cylindrical tube to mimic our experimental studies. Equations – were rearranged to give the ETC of the wall-adjacent layer, yielding

k_{wa} = k_{void} (1 - n_{C} A_{C} - n_{NC} A_{NC}) + k_{C} n_{C} A_{C} + k_{NC} n_{NC} A_{NC}

(14)

Here,

n_{C}

and

n_{NC}

represent the number of particles with and without contact per unit wall area, respectively. All particles not in contact with the wall were assumed to be at the same distance from the wall, with the distance represented as a fraction of a particle radius

(a r_{p})

⁠. The number of contact particles (⁠

n_{C}

⁠) is estimated from the DEM simulations, while the number of particles not in contact with the wall is estimated from the packing fraction of the wall-adjacent layer (⁠

ϕ_{wa}

⁠) and the number of contact particles using the expression for the packing fraction

ϕ_{wa} = \frac{2 A (\forall_{C} n_{C} + \forall_{NC} n_{NC})}{r_{p} (2 A - 1)}

(15)

Here, $\forall_{C}$ and $\forall_{NC}$ represent the volume of a single contact and no-contact particle intersected by the wall-adjacent layer (i.e., partial volume of a sphere), respectively, and $A$ represents the ratio of the tube-to-particle diameters $(A = D_{i} / d_{p})$ ⁠. Note that contact particles are assumed to have a point contact and the packing fraction is estimated from the DEM simulations. Since the correlation was derived to describe experiments using ceramic particles, heat transfer through the small contact areas is negligible. This assumption becomes invalid for high particle-to-gas thermal conductivity ratios (>1000) [1]. The volume and the wall projected area for a single particle of both groups (i.e., contact and no-contact) were determined based on geometry.

The thermal conductivity of a single particle was derived using an approach similar to that of Legawiec and Ziolkowski [24]. The heat transfer from the wall to the edge of the wall-adjacent layer was represented by two thermal resistances connected in series, capturing the heat transfer through the interstitial gas and the particle. Initially, the thermal conductivities were derived for a cylindrical tube wall; the resulting solutions, however, could not be reduced to a simple algebraic expression, which was desired for the present modeling effort. A simplified solution was therefore obtained by deriving the thermal conductivity for a flat plate and incorporating a correction factor to account for the curved surface of a cylindrical wall. Using Fig. 4 as a guide, the infinitesimal thermal resistance for the gas

(d R_{g})

and particle

(d R_{s})

can be represented by

d R_{g} = \frac{D_{g} (r)}{k_{g} 2 π rdr} = \frac{(1 + a) r_{p} - \sqrt{r_{p}^{2} - r^{2}}}{k_{g} 2 π rdr}

(16)

d R_{s} = \frac{D_{s} (r)}{k_{s} 2 π rdr} = \frac{\sqrt{r_{p}^{2} - r^{2}} - a r_{p}}{k_{s} 2 π rdr}

(17)

where

k_{s}

represents the thermal conductivity of the solid particle material and

r

represents the radial position as defined in Fig. 4. The total thermal conductivity for a single particle can then be calculated by integrating the sum of the thermal resistances for

0 \leq r \leq r_{p} \sqrt{1 - a^{2}}

yielding

k_{particle} = \frac{2 k_{g}}{(1 - a^{2})} {(\frac{K}{K - 1})}^{2} [\frac{a (K - 1) + K}{K} \ln (\frac{K}{a K + 1 - a}) - \frac{(1 - a) (K - 1)}{K}]

(18)

Fig. 4

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Schematic of a single particle in the wall-adjacent layer with a flat plate

Here, $K$ represents the ratio of the solid-to-gas thermal conductivities $(K = k_{s} / k_{g})$ ⁠. The thermal conductivities calculated using Eq. (18) show minimal variation from the curved wall solution (<1.5%). The largest variation is observed for small tube-to-particle radii ratios, where the effect of curvature is greatest.

According to our DEM simulations, the average distance of the no-contact particles from the wall is greater than 0.2

r_{p}

for all flowrates and system configurations tested (see Fig. 3.8 of Ref. [25]). The thermal conductivity shows minimal variation for distances greater than

a = 0.2

⁠; therefore, the average value of a from all system configurations (⁠

a = 0.22)

was implemented in our correlations. Substituting Eqs. and into Eq. and letting

a = 0

for

k_{C}

and

a = 0.22

for

k_{NC}

⁠, the ETC of the wall-adjacent layer can be written in its simplified form

k_{wa} = k_{g} {1 + π r_{p}^{2} n_{C} (1 + \frac{1}{A}) [2 Φ_{C} - 1] + (0.74 + \frac{1}{A}) [3 (1 - \frac{0.5}{A}) ϕ_{wa} - 2 π r_{p}^{2} n_{C}] (2 Φ_{NC} - 0.95)}

(19)

Φ_{C} = {(\frac{K}{K - 1})}^{2} [\ln K - \frac{(K - 1)}{K}]

(20)

Φ_{N C} = {(\frac{K}{K - 1})}^{2} [(1.22 - \frac{0.22}{K}) \ln (\frac{K}{0.22 K + 0.78}) - \frac{0.78 (K - 1)}{K}]

(21)

Radiation in the Wall-Adjacent Layer.

Experimental and theoretical studies of the heat transfer through static particle beds indicate that the heat transfer increases with temperature due to the increased radiative heat transfer contribution [1,2,26]. Similar observations are expected for dense granular flows at high operating temperatures. While numerous works have sought to model the radiation within a static particle bed, a clear modeling approach has not been determined due to the complexity of radiation heat transfer and the random packing arrangement of particles in a bed. Furthermore, the radiation heat transfer in the wall-adjacent layer has received minimal attention. The present work therefore sought to develop a simple model for the radiation contribution in the wall-adjacent layer to provide a first-order approximation for the heat transfer to dense flows at high operating temperatures.

An enclosure analysis was used as the basis for the model development. The particle flow near the wall was classified into different particle layers, as defined in Fig. 5. The first layer (i.e., the wall-adjacent layer) has a thickness of a particle radius; the remaining two layers are each one particle diameter thick. Each layer of particles was considered a surface of the enclosure, with the surface areas of all particles in the layer summed to find the total surface area (as indicated by the different colors in Fig. 5). According to Pitso [27], the radiation from a single sphere in a randomly packed bed penetrates no more than 2–3 particle diameters. A similar assumption was employed in the present model; the enclosure was therefore assumed to consist of four layers: the wall and three particle layers.

Fig. 5

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Schematic of particle layers used to define radiation from the wall

For simplicity, the wall and particles were taken as blackbodies. The heat flux leaving the wall in the negative radial direction can thus be written as

q_{wall, rad}^{''} = F_{w - 1} σ (T_{w}^{4} - T_{1}^{4}) + F_{w - 2} σ (T_{w}^{4} - T_{2}^{4}) + F_{w - 3} σ (T_{w}^{4} - T_{3}^{4})

(22)

where

q_{wall, rad}^{″}

represents the radiation heat flux leaving the wall,

F_{w - y}

represents the view factor from the wall to the yth particle layer,

T_{y}

represents the average temperature of the yth layer, and

σ

represents the Stefan–Boltzmann constant. If

(T_{w} - T_{y}) ≪ 4 {\bar{T_{w y}}}^{2}

(where

\bar{T_{w y}}

represents the average temperature of

T_{w}

and

T_{y}

⁠), then

(T_{w}^{4} - T_{y}^{4}) \approx 4 {\bar{T_{w y}}}^{3} (T_{w} - T_{y})

⁠. This simplification seems appropriate for the flows studied since the temperature drop across a few particle diameters is significantly less than the wall temperature. A linear temperature distribution was assumed across layers 2 and 3 (i.e.,

T = Δ T, (T_{2} - T_{3}) = Δ T),

while the temperature gradient across layer 1 (i.e., the wall-adjacent layer) was assumed twice that of the other layers due to the increased thermal resistance adjacent to the wall (i.e.,

T = Δ T) .

These temperature gradient assumptions were guided by our experimental results, which showed a 2× to 3× temperature gradient near the wall [16]. Finally, assuming

Δ T ≪ T_{w}, \bar{T_{w y}} \approx T_{w}

⁠. Incorporating the above assumptions, Eq. can be rewritten as

q_{wall, rad}^{''} = 4 σ T_{w}^{3} Δ T [F_{w - 1} + 2 F_{w - 2} + 3 F_{w - 3}]

(23)

To model the total heat transfer through the wall-adjacent layer, the radiation and conduction contributions must be combined, necessitating an expression for the effective thermal conductivity of the wall-adjacent layer due to radiation. By comparing Eq. with Fourier's law, the thermal conductivity due to radiation can be defined as

k_{wa, rad} = 4 σ r_{p} T_{w}^{3} [F_{w - 1} + 2 F_{w - 2} + 3 F_{w - 3}]

(24)

In order to calculate the radiation ETC, the view factors from the wall to each layer must be approximated. The ray-tracing algorithm of starccm+ was employed to examine the radiation heat transfer between the surfaces and extract the view factors to each surface. The centroid data from the DEM flows discussed in the Effective Thermal Conductivity of Wall-Adjacent Layer section were used to create a solid model of the particles in the near-wall region of the cylinder using solidworks. The surface of each particle was split according to the cylindrical surface “cutting” the particle at the boundary of each layer; the total surface area of each layer was then assigned to the sum of the individual surfaces in the associated layer. The process of splitting particle surfaces was a very time intensive and difficult process, as each particle had to be manually selected; consequently, ensuring that all particles were selected was a challenge due to the random packing. A small section of the near-wall layers was therefore used in the modeling process. As a result, there was a small amount of radiation “leakage” to the top, bottom, and side surfaces. Table 1 summarizes the system configuration modeled and the resulting view factor calculations. A slow flowrate was modeled to match the flowrates tested experimentally at high operating temperatures in our previous work [16]. The view factors may change with increasing flowrate due to the variation in the wall-adjacent layer packing fraction; however, due to the time-intensive nature of view factor calculation, further examination into the effect of flowrate on view factors was beyond the scope of the present study. Note that the leakage was incorporated into the primary view factors (proportionately to each of the layers based on the view factors) such that the view factors of the enclosure summed to one.

Table 1

Properties of view factor simulation

Property	Value
Tube diameter (mm)	7.75
Particle diameter (mm)	0.60
Mean flow velocity (m/s)	0.016
$F_{w - 1}$	0.80
$F_{w - 2}$	0.18
$F_{w - 3}$	0.02

Effective Thermal Conductivity Results

Flow Parameters.

The effective thermal conductivity of the wall-adjacent layer due to conduction can be determined using Eqs. (19)–(21); however, an understanding of the wall-adjacent layer packing fraction and the number of particle-wall contacts is required. A series of DEM simulations were performed using starccm+ to study these parameters and how they varied with tube diameter, particle diameter, and flowrate. Six different configurations, using different combinations of tube and particle diameter, were tested at 15 different flowrates, ranging from creeping flow (U = 0.002 m/s) to the upper limit of choked flow (which ensures a dense flow and varies for each system configuration). Two different tube diameters were employed (7.75 and 10.92 mm ID), along with three different particle diameters (300, 450, and 600 μm). These values were selected as they reflect the systems tested experimentally in our previous work. Future work may examine a broader range of parameters to verify the trends observed here. The centroids of all particles were tracked at each time-step, for a total of 1000 time steps, and used to calculate packing fraction and number of particle-wall contacts at a given instant; the parameter values reported are time-averaged values.

The variation in the wall-adjacent layer packing fraction with flowrate as a function of particle diameter is displayed in Fig. 6(a). In general, the packing fraction decreases with increasing flowrate, which agrees with the observation made in Fig. 3. The slight increase observed with increasing flowrate at slow flowrates is curious; however, since our work focused primarily on the dense flow regime, further investigations into the slow flowrate behavior were not pursued. The variation in pressure with flowrate may be one possible explanation for the trends observed. The pressure inside the tube may manifest itself by pushing particles against the wall, forcing particles to fill in available void spaces near the wall, thereby increasing the packing fraction. The pressure inside the tube was examined from the DEM simulations and is plotted in Fig. 6(b). The pressure decreases with increasing flowrate, similar to the trends observed in the wall-adjacent layer packing fraction. The pressure, however, shows an opposite trend with particle diameter compared to the packing fraction. Since the DEM simulations have been validated in our previous work [20–22], these trends are believed to be correct. An empirical correlation was developed to capture the trends observed, with the goal of predicting the packing fraction for different system configurations and flow velocities. The correlation that best captures the trends observed takes the form of

ϕ_{wa} = 0.43 - \frac{2}{3 A^{2}} - (0.003 + \frac{1}{4 A}) \frac{U}{\sqrt{g d_{p}}}

(25)

where $U$ represents the average flow velocity and $g$ represents the gravitational acceleration. The correlation agrees with the simulation results for all test cases to within 5%.

Fig. 6

$Variation in the (a) wall-adjacent layer packing fraction and (b) pressure with mean flow velocity as a function of particle diameter (using 7.75 mm ID tube). Note that y-axes do not start at zero.$

View large Download slide

Variation in the (a) wall-adjacent layer packing fraction and (b) pressure with mean flow velocity as a function of particle diameter (using 7.75 mm ID tube). Note that y-axes do not start at zero.

The variation in the number of particle-wall contacts with flowrate as a function of particle diameter is displayed in Fig. 7. The number of contacts was normalized using the maximum number of possible contacts, which was defined assuming a hexagonal packing of particles along the wall

(n_{C, \max} = (1 - (1 / A)) / 2 \sqrt{3} r_{p}^{2})

⁠. For all test cases, the number of particle-wall contacts decreases with increasing flowrate. If this behavior resulted solely from the decrease in packing fraction, a linear correlation between the two would exist; however, no such relationship was observed. A secondary phenomenon, such as increased velocity fluctuations in the wall-adjacent layer, is therefore likely at play. In our previous studies [22], the velocity fluctuations (quantified using the granular temperature) were observed to increase with increasing flow velocity. Increased fluctuations likely make particle-wall contact events more intermittent, yielding fewer contact points at a given moment. An empirical correlation was developed to capture the trends observed in the DEM data

\frac{n_{C}}{n_{C, \max}} = (\frac{1.6}{A^{0.09}} - 0.6) \exp (- (0.07 + \frac{175}{A^{2.55}}) \frac{U}{\sqrt{g d_{p}}})

(26)

Fig. 7

View large Download slide

Variation in the normalized number of particle-wall contacts per unit wall area with flowrate and particle diameter (using 7.75 mm ID tube)

The correlation agrees with the simulation results to within 16%. Although this error may seem large, the error observed in $k_{wa}$ as a result of error in both the packing fraction and the number of particle-wall contacts is less than 7%.

It is important to note that the simulations were performed using particle material properties representative of the ceramic zirconia–silica particles used in our experiments. We recognize that the behavior of granular flows depends on many structural and contact properties. We suspect that the general trends discussed above are true regardless of particle material; however, we are uncertain whether the relative magnitudes will change. Since no universal rheological model for granular flows exists, it is difficult to find an all-encompassing model. Future work may seek to test a wider range of system parameters.

Effect of Flowrate and Temperature.

Incorporating the empirical correlations discussed in the Flow Parameters section into the effective thermal conductivity correlation (Eqs. (19)–(21)), the variation in the heat transfer at the wall with flowrate at low operating temperatures was examined (Fig. 8). As expected, the ETC of the wall-adjacent layer decreases with increasing flowrate due to the decreasing packing fraction and number of particle-wall contacts. The relative contribution of the packing fraction and the number of contacts to the decrease observed was examined by varying one parameter while keeping the other constant. Each case is plotted in Fig. 8, using the slow flowrate value for the parameter held constant. It is clear that the decrease in the number of particle-wall contacts is the driving force behind the decrease in the wall-adjacent layer ETC. This makes sense, as the number of particle-wall contacts decreases more significantly with flowrate than the packing fraction (see Figs. 6(a) and 7). Finally, the significance of including the heat transfer to both particles with and without contact is also examined in Fig. 8 by studying the ETC assuming all particles contact the wall (i.e., when $a = 0$ ⁠). The thermal conductivity is greater than that predicted with NC particles included due to the superior heat transfer exhibited by contact particles.

Fig. 8

View large Download slide

Effective thermal conductivity of the wall-adjacent layer as a function of mean flow velocity for A = 17.2, calculated using (1) variable $n_{C}$ and $ϕ_{wa}$ ⁠, (2) constant $n_{C}$ and variable $ϕ_{wa}$ ⁠, (3) variable $n_{C}$ and constant $ϕ_{wa}$ ⁠, and (4) $a = 0$ and variable $ϕ_{wa}$

The effect of temperature on the heat transfer through the wall-adjacent layer is displayed in Fig. 9 for various particle diameters (assuming $D_{i} / d_{p}$ is constant). As expected, the heat transfer in the wall-adjacent layer increases with increasing temperature due to enhanced thermal properties and an increased radiation contribution at higher temperatures. Larger particle diameters exhibit superior heat transfer at elevated temperatures due to increased radiation contributions resulting from larger void spaces and therefore penetration distances. This is an encouraging result for the validity of the ETC correlation developed, as the same phenomenon was observed in studies of the ETC in the bulk of a static bed. A similar result is observed when considering the heat transfer to a dense flow, as seen on the right-hand axes (i.e., dashed lines) of Fig. 9. Here, the two-layer model (Eq. (7)) and the wall-adjacent layer ETC correlation (Eqs. (19) and (24)) were used to predict the heat transfer coefficients for a dense flow through a cylindrical tube for the same particle diameters. Note that the y-axis plots $h D_{i}$ ⁠, such that the variation due to tube diameter is excluded. This result suggests that larger particle diameters may be more beneficial for high temperature applications.

Fig. 9

View large Download slide

Effective thermal conductivity of wall-adjacent layer and heat transfer to dense flow in a cylindrical tube as a function of temperature and particle diameter

Model Validation

The effective thermal conductivity correlation developed for the wall-adjacent layer was validated by implementing it into dense flow models and comparing with the experimental data. In particular, the experimental data from our low and high temperature studies in cylindrical tubes and data published in the literature for vertical chute flows were considered. The models were found to have good agreement with the experimental results, as discussed in the Experimental Results and Published Data sections.

Experimental Results.

The heat transfer to the dense flows through cylindrical tubes studied in our previous work [16] was modeled using the two-layer Nusselt number correlation presented in Eq. (7). Our experimental studies were subdivided into two parts: investigation into the effect of flowrate and other system parameters, such as tube and particle diameter, on the heat transfer to the flow at low operating temperatures, and investigation into the effect of temperature on the heat transfer to the flow. Results from the low temperature studies using the 7.6 mm and 10.8 mm ID tubes with 270 μm zirconia–silica particles (as presented in Fig. 8 of Ref. [16]) are compared with the two-layer model in Fig. 10. The two-layer model results were calculated using thermal properties at the mean flow temperature of the tube (T = 105 °C). The error bars and dashed lines denote the 95% confidence interval for the experimental data and two-layer model, respectively. Note that the radiation term in the wall-adjacent layer ETC was neglected for low-temperature results presented in Fig. 10. A similar comparison for the remaining three data sets displayed in Fig. 8 of Ref. [16] can be found in Ref. [25].

Fig. 10

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Comparison of experimental data [16] and two-layer model for the 270 μm zirconia-silica particles in the (a) 7.6 mm ID and (b) 10.8 mm ID tubes. Error bars and dashed lines denote 95% confidence intervals. Black symbols denote values measured at heated length of 0.64 m and gray symbols denote values measured at heated length of 1.26 m.

The experimental results, apart from a couple stray values, show good agreement with the two-layer model within the experimental uncertainty. The worst agreement is observed at the two extremes of the flowrates tested. The slow flowrates (large inverse Graetz numbers) demonstrate the largest experimental uncertainty due to small temperature differences across the flow; the fast flowrates at shorter heated lengths (smallest inverse Graetz numbers) approach the thermal entrance region, where the thermally developed solution plotted is no longer appropriate. Close examination of Fig. 10 indicates that the 10.8 mm ID tube yields slightly larger Nusselt numbers than the 7.6 mm ID tube (direct comparison can be seen in Fig. 7 of Ref. [16]). This variation may potentially be explained by the two-layer model, which suggests that the heat transfer decreases slightly with a decreasing tube-to-particle diameter ratio (decreasing $W$ ⁠).

Results from the high temperature studies, again using the 7.6 mm and 10.8 mm ID tubes with 270 μm zirconia–silica particles, are presented in Fig. 11, along with the corresponding values predicted by the two-layer model. Note that the results are plotted as a function of the wall temperature here. The effective thermal conductivity of both the bulk and wall-adjacent layers is required to estimate the heat transfer using the two-layer model. Ideally, the bulk layer ETC would be calculated using the mean temperature and the wall-adjacent layer ETC would be calculated using the wall temperature. The model, however, was developed to be used as a predictive tool; as such, knowledge of both the mean and wall temperatures is unlikely. A relationship between the two is not straight forward; therefore, a single temperature to approximate the ETC of both layers is required. Using a single temperature will lead to minor inaccuracies in the value calculated; implementation of the wall temperature resulted in smaller errors and was therefore selected as the reference temperature. From a design standpoint, the expected heat transfer to a flow at a desired wall operating temperature may be more valuable; for example, if designing based on the maximum operating temperature of a tube. Due to the smaller temperature differences present in the low temperature studies and absence of radiation, accurate selection of a reference temperature was less of a concern.

Fig. 11

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Heat transfer coefficients calculated for the 270 μm zirconia–silica particles in the 7.6 and 10.8 mm ID tubes as a function of the wall temperature. Black symbols denote values measured at heated length of 0.64 m and gray symbols denote values measured at heated length of 1.26 m. Solid lines denote two-layer calculation with radiation included; dashed lines denote two-layer calculation without radiation included.

The two-layer model shows good agreement with the experimental data within experimental uncertainty, with the largest variation observed at the highest temperatures. The relative contribution from radiation is also displayed in Fig. 11 by plotting the model prediction without radiation included. As expected, the contribution from radiation is negligible at low temperatures and begins to grow at higher temperatures. For the systems tested, the inclusion of radiation increases the heat transfer coefficient by up to 29%.

Published Data.

The ETC correlation developed for the wall-adjacent layer was further validated using experimental results published by Sullivan and Sabersky [5] and Natarajan and Hunt [6] for vertical chute flows. Sullivan and Sabersky and Natarajan and Hunt studied the heat transfer to chute flows (rectangular cross section) with a single heated wall at a constant temperature, each using different heated lengths. Both found good agreement between their experimental results and the single-resistance model developed by Sullivan and Sabersky for low flowrates.

Sullivan and Sabersky's model assumed that the flow consisted of a thin air gap adjacent to the heated wall; the rest of the flow was assumed to maintain bulk effective properties. Unlike the two-layer model developed in the present study, their solution employed a temperature jump boundary condition in the energy equation solution, using a conductance term to represent the increased thermal resistance at the wall. The model developed by Sullivan and Sabersky is reprinted below for convenience:

\bar{N u_{L}} = \frac{1}{\frac{χ d_{p} k_{b}}{k_{g} L} + \frac{\sqrt{π}}{2} \sqrt{\frac{1}{P e_{L}}}}

(27)

Here,

χ

represents the relative thickness of the air gap adjacent to the heated wall and

P e_{L}

represents the Péclet number based on the length of the heated section,

L

⁠. To calculate

χ

using the ETC correlation developed in this study, our wall-adjacent layer modeling approach was compared with that of Sullivan and Sabersky. The wall-adjacent layer utilized in this study has a thickness of one particle radius, with a single ETC associated with the whole domain. This domain was divided into two regions to account for the modeling approach used by Sullivan and Sabersky: a small air layer with thickness

χ d_{p}

and a bulk layer in the remaining thickness. Using thermal resistances, the thickness of the air layer used by Sullivan and Sabersky can be approximated by

χ = \frac{0.5}{\frac{k_{b}}{k_{g}} - 1} (\frac{k_{b}}{k_{wa}} - 1)

(28)

To approximate

k_{wa}

for the chute geometry, the wall-adjacent layer ETC correlation was rewritten in a more general form

k_{wa} = k_{g} {1 + π r_{p}^{2} n_{C} B_{C} (2 Φ_{C} - 1) + 0.74 B_{NC} [3 (1 - \frac{0.5}{A_{m}}) ϕ_{wa} - 2 π r_{p}^{2} n_{C}] (2 Φ_{NC} - 0.95)}

(29)

A_{m} = \frac{P}{β d_{p}} with β = \{\begin{matrix} π; cylinder \\ 4; chute \end{matrix}

(30)

Here, $B_{C}$ and $B_{NC}$ represent the correction terms for the contact and no-contact particles for a cylindrical system, respectively $(B_{C} = 1 + (1 / A); B_{NC} = 1 + (1.35 / A))$ ⁠, $A_{m}$ represents a modified tube-to-particle diameter ratio, $P$ represents the perimeter of the cross section, and $β$ represents a shape-dependent correction factor. Note that the flat plate solution is applicable for chute flows $(D_{i} / d_{p} \to \infty)$ ⁠, which is captured by the correction terms evaluating to 1.

While the correlations developed in the present study for the packing fraction and number of particle-wall contacts are based on data obtained for flows through cylindrical tubes, they were used to estimate values for the chute flows under consideration by utilizing the modified tube-to-particle diameter ratio. This may not be the best parameter to compare the flow physics in a cylinder and chute; however, without any chute data, the modified tube-to-particle diameter ratio was used in an attempt to stay consistent. The hydraulic diameter was also discussed as a potential controlling parameter; however, incorporation of the hydraulic diameter yielded negative values for the number of NC particles in a chute. A wider range of DEM data for different cross-sectional shapes would help clarify the uncertainties present in extrapolating the empirical correlations to different system configurations. An understanding of the physics controlling the packing fraction and number of particle-wall contacts in the wall-adjacent layer would also help explain the trends observed.

The experimental results for the glass particles tested by Sullivan and Sabersky are plotted in Fig. 12, along with the Nusselt numbers predicted by their single-resistance model, using their value of $χ = 0.085$ and the value calculated using $k_{wa}$ ⁠. Note that the bulk ETC was calculated using the correlation developed by Zehner and Schlünder [28], which has been shown to provide reasonably accurate predictions over a wide range of solid–gas thermal conductivity ratios [1]. The Nusselt numbers calculated using $k_{wa}$ show good agreement with the experimental data, and therefore, the empirical value of $χ$ implemented by Sullivan and Sabersky. This suggests that for the small range of slow flowrates tested, it is reasonable to assume that the thermal contact resistance remains essentially constant.

Fig. 12

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Variation of the Nusselt number with Péclet number and length of heated section. Data points represent the experimental data from Sullivan and Sabersky [5] for (a) 0.33 mm glass traffic beads and (b) 1.35 mm glass impact beads. Solid lines denote single resistance model prediction using $k_{wa}$ for $χ$ ⁠. Dashed lines denote single resistance model prediction with $χ = 0.085$ ⁠.

Natarajan and Hunt expanded upon the work of Sullivan and Sabersky by testing a wider range of flowrates and a longer heated length. Their experimental results for the glass particles tested are displayed in Fig. 13. Once again, the Nusselt numbers predicted using a constant $χ = 0.085$ and a variable $χ$ calculated using $k_{wa}$ are also plotted. Both model predictions show good agreement with the experimental data for slow flowrates (U < 9.0 cm/s). At faster flowrates, however, a constant resistance does not capture the decrease in Nusselt number observed. Incorporating the wall-adjacent layer ETC correlation, which shows a decrease in ETC with increasing flowrate (corresponding to an increase in $χ$ ⁠), captures the decrease in Nusselt number observed experimentally quite well.

Fig. 13

View large Download slide

Variation of the Nusselt number with Péclet number. Data points represent the experimental data from Natarajan and Hunt [6] for 3.0 mm glass particles in a vertical chute with L = 1.0 m. Solid lines denote single resistance model prediction using $k_{wa}$ for $χ$ ⁠. Dashed lines denote single resistance model prediction with $χ = 0.085$ ⁠.

Conclusion

This work sought to gain a better understanding of the near-wall heat transfer characteristics of internal dense granular flows and develop a physics-based model to predict the heat transfer to such a flow. Experimental results from our companion study [16] corroborated the observation of numerous studies in the literature that dense granular flows demonstrate inferior heat transfer compared with a continuum flow. It is well-accepted that the decrease in heat transfer observed in discrete flows results from an increased thermal resistance adjacent to the heated surface due to a more structured arrangement of particles. While previous works have modeled the reduced heat transfer to dense flows, they rely on empirical parameters or correlations derived for static beds to estimate the near-wall contact resistance. The present work sought to overcome this deficiency by deriving an ETC correlation for the wall-adjacent layer based on the fundamental physics, considering potential variation for a flowing bed.

Discrete element method simulations were employed to examine the near-wall flow physics and understand how parameters such as the near-wall packing fraction and number of particle-wall contacts may affect the heat transfer from the wall; both parameters were shown to decrease with increasing flowrate. Since the heat transfer to a single particle decreases drastically upon losing contact with the wall, it is important to consider heat transfer to both particles in contact with the wall and particles that are not. Using a parallel circuit of thermal resistances, an ETC correlation for the wall-adjacent layer was derived. The resulting correlation depends on the tube diameter, particle diameter, thermal conductivities of the solid and gas materials, the packing fraction of the wall-adjacent layer, the number of particles in contact with the wall, and the wall temperature. Empirical correlations were developed based on DEM simulation results to estimate the near-wall packing fraction and number of particle-wall contacts. The wall temperature allows the radiation contribution to be incorporated.

The ETC correlation was validated by incorporating it into dense flow models for different system configurations. A two-layer Nusselt number correlation was developed in this study to estimate the heat transfer to dense flows in vertical cylindrical tubes; the resulting correlation was compared with the experimental results from our companion study. The two-layer model showed good agreement with the experimental results for different tube diameters, particle diameters, flowrates, and temperatures. The wall-adjacent layer ETC correlation was also used to estimate the near-wall contact resistance described in the single-resistance model developed by Sullivan and Sabersky [5] for vertical chute flows. The modified model showed good agreement with the experimental results of both Sullivan and Sabersky and Natarajan and Hunt [6]. Unlike the original model, which assumed that the contact resistance did not vary with flowrate, the modified version was able to capture the decrease in heat transfer observed at faster flowrates.

The agreement between the model developed and the experimental results shows promise for the ETC correlation developed. In its current form, the ETC correlation is limited by an understanding of the variation in the near-wall packing fraction and the number of particle-wall contacts with system parameters; extrapolation of the empirical correlations outside of the regime of parameters for which they were developed presents an uncertainty currently and must be considered with care. An understanding of the mechanics that control the behavior of each parameter as well as a study into the variation across a wider range of system parameters would greatly expand the applicability of the ETC correlation derived. Ultimately, we anticipate our model to serve as a gateway to analyzing the overall heat transfer behavior of dense granular flows and to serve as a means to identify other parameters that can be modified to improve the overall performance of dense flow systems.

Funding Data

Advanced Research Projects Agency—Energy (ARPA-E), U.S. Department of Energy (Grant No. DE-AR0000414; Funder ID: 10.13039/100000015).

Nomenclature

a =: fraction of a particle radius that “no-contact” particles are from wall
A =: tube-to-particle diameter ratio
A_n =: tube wall area of the nth heat transfer pathway, m²
B_C =: correction term for contact particles in cylindrical system
B_NC =: correction term for “no-contact” particles in cylindrical system
d_p =: particle diameter, mm
D_i =: inner tube diameter, m
dR =: infinitesimal thermal resistance, K/W
F_w−y =: view factor from wall to yth layer
g =: gravitational acceleration, m/s²
h =: heat transfer coefficient, W/m²-K
k =: thermal conductivity, W/m-K
K =: ratio of solid-to-gas thermal conductivities
k_particle =: thermal conductivity of single particle in wall-adjacent layer, W/m-K
k_wa,rad =: thermal conductivity of wall-adjacent layer due to radiation, W/m-K
L =: length of heated section, m
n_C =: number of contact particles per unit wall area, m^–2
n_C,_max =: Max. number of contact particles per unit wall area, m^–2
n_NC =: number of “no-contact” particles per unit wall area, m⁻²
N_C =: number of particles in contact with the wall
N_NC =: number of particles not in contact with the wall
$N u_{k_{b}} =$ =: Nusselt number using bulk thermal conductivity
$\bar{N u_{L}} =$ =: length-averaged Nusselt number
P =: perimeter of flow cross section, m
Pe_L =: Péclet number based on length
$q_{wall}^{″}$ =: wall heat flux, W/m²
$q_{wall, rad}^{″}$ =: radiation heat flux leaving the wall, W/m²
R_i =: inner tube radius, m
R_n =: thermal resistance of nth heat transfer pathway, K/W
r =: radial position, m
r_p =: particle radius, mm
T =: temperature, °C or K
$\bar{T_{w y}}$ =: average temperature of wall and yth layer, K
U =: mean axial flow velocity, m/s
W =: ratio of bulk-to-wall-adjacent layer radii
z =: axial position, m

$α$ =: thermal diffusivity, m²/s
$β$ =: shape-dependent correction factor
$δ$ =: radius of bulk layer, m
$Δ T$ =: temperature difference, K
$σ$ =: Stefan–Boltzmann constant, W/m²-K⁴
$ϕ_{wa}$ =: packing fraction of wall-adjacent layer
$Φ_{C}$ =: intermediate parameter for contact particles
$Φ_{NC}$ =: intermediate parameter for “no-contact” particles
$χ$ =: relative thickness of air gap adjacent to wall in Sullivan and Sabersky model
$\forall$ =: volume of single particle in wall-adjacent layer, m³

b =: bulk layer
C =: contact particles
g =: gas
j =: layer of composite for two-layer model
NC =: No-contact particles
s =: solid material
void =: void spaces
w =: wall
wa =: wall-adjacent layer
y =: layer of near-wall for radiation enclosure analysis

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Analytic Modeling of Heat Transfer to Vertical Dense Granular Flows

Abstract

Introduction

Modeling Approach

Effective Thermal Conductivity of Wall-Adjacent Layer.

Conduction in the Wall-Adjacent Layer.

Radiation in the Wall-Adjacent Layer.

Effective Thermal Conductivity Results

Flow Parameters.

Effect of Flowrate and Temperature.

Model Validation

Experimental Results.

Published Data.

Conclusion

Funding Data

Nomenclature

References

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Analytic Modeling of Heat Transfer to Vertical Dense Granular Flows

Abstract

Introduction

Modeling Approach

Effective Thermal Conductivity of Wall-Adjacent Layer.

Conduction in the Wall-Adjacent Layer.

Radiation in the Wall-Adjacent Layer.

Effective Thermal Conductivity Results

Flow Parameters.

Effect of Flowrate and Temperature.

Model Validation

Experimental Results.

Published Data.

Conclusion

Funding Data

Nomenclature

References

Get Email Alerts

Cited By

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