Review of the Fluid Dynamics and Heat Transport Phenomena in Packed Pebble Bed Nuclear Reactors

Abstract

Knowledge and proper safety analyses of the gas coolant and heat transport mechanism in the dynamic core of packed pebble bed nuclear reactors pose challenges to the reliable design and efficient operation of these reactors. Therefore, this paper carefully reviews most of the gas coolant mixing and heat transport studies performed for the fluid flow and heat transfer processes in packed pebble bed reactors (PBRs). It begins with a brief introduction and description of nuclear PBRs. The second section summarizes the physical characteristics of packed bed reactors in terms of the bed structure (porosity) and its radial and axial distributions. The next section examines in detail the characteristics of fluid flow in terms of flow regime identification and pressure drop measurements and correlations. The fourth section considers the investigations and quantifications of the gas dispersion and mixing phenomena of packed bed reactors. The next section deals with the current state of the heat transfer characteristics, measurements, and predictions including both empirical correlations and semiempirical model-based studies. Tables summarize the reported experimental studies along with their operating condition ranges. Comprehensive comparisons with the empirical correlations and available models are presented with significant findings. The content and findings of the present work could provide a thorough understanding and useful information and advance knowledge of the pressure drop, gas coolant mixing, and convective heat transport phenomena in packed pebble bed nuclear reactors.

Keywords:

• Packed pebble bed
• nuclear reactor
• gas coolant
• pressure drop
• heat transfer

I. INTRODUCTION

To achieve sustainability, high economics and efficiency, enhanced safety, reliability, waste minimization, and proliferation resistance and still be environmentally friendly, the Generation IV International Forum (GIF) was initiated in 2000 for the development of fourth generation [Generation IV (Gen IV)] nuclear power plants.¹ The 13 current members of the GIF guide the collaborative efforts of the world’s leading nuclear technology nations to develop these nuclear energy systems. The technology roadmap produced by the GIF (Ref. 2) for long-term research projects resulted in proposals for six nuclear reactor technologies called Gen IV nuclear reactors, as listed in Table I. The six most promising reactor concepts were selected based on their ability to provide a reliable and safe energy system together with reduced nuclear waste production and increased economic competitiveness.³

TABLE I Gen IV Nuclear Reactors Selected by GIF*

Gen IV System	Acronym	Neutron Spectrum	Coolant	Temperature (^oC)	Fuel Cycle	Size [MW(electric)]
Gas-cooled fast reactor	GFR	Fast	Helium	850	Closed	1200
Lead-cooled fast reactor	LFR	Fast	Lead	480 to 800	Closed	300 to 1200 600 to 1000
Sodium-cooled fast reactor	SFR	Fast	Sodium	550	Closed	30 to 150 300 to 1500 1000 to 2000
Molten salt reactor	MSR	Thermal/fast	Fluoride salts	700 to 800	Closed	1000
Supercritical-water-cooled reactor	SCWR	Thermal/fast	Water	510 to 625	Open/ Closed	300 to 700 1000 to 1500
Very high-temperature reactor	VHTR	Thermal	Helium	900 to 1000	Open	250 to 300

*Abdulmohsin.¹

The next-generation nuclear plants (NGNP), or the Gen IV nuclear reactors, are supposed to fulfill future energy demand and environmental needs. In addition, they can be used to produce hydrogen and process heat for industrial needs. The very high-temperature reactor (VHTR) is one of these six advanced concepts for Gen IV nuclear reactors that are being considered for electric power, process heat, and hydrogen production. The VHTR is a continuation and optimization of the present high-temperature gas-cooled reactor (HTGR) designs, with the aim of reaching a coolant outlet temperature of around 1000°C or above, which would increase reactor performance. The core configuration of the VHTR can be a pebble bed type or a prismatic block type, according to the fuel element assembly. An annulus filled with mobile fuel spheres is used in the core of the pebble bed reactor (PBR) while a hexagonal prismatic fuel block core configuration is used for the prismatic block reactor. Both the pebble fuel type and prismatic block type are still considered for the NGNP design with a once-through low-enriched uranium fuel cycle at a high burnup value. Recently, various studies have been conducted on these reactors. Thus, in this work, we focus on reviewing the recent advances that have been made in fluid dynamics and heat transport phenomena related to only packed pebble bed nuclear reactors.

I.A. Description of the Packed Pebble Bed Nuclear Reactor

The pebble bed nuclear reactor gets its name from the type of nuclear fuel it consumes, and it offers many advantages over conventional reactors. A pebble bed type of very high-temperature gas-cooled reactor (VHTR) is one of the most probable solutions⁴ and promising concepts⁵ of the six classes of Gen IV advanced nuclear technologies. The PBR concept has been adopted by many test and demonstration reactors, including the modular pebble bed reactor (MPBR) in the United States⁶ and the prototype reactor of the pebble bed modular reactor (PBMR) in South Africa^5,7; the 10-MW(thermal) high-temperature reactor (HTR-10) in China^8,9; and the prototype PBR at Jülich research center in Germany that is known as Arbeitsgemeinschaft Versuchsreaktor (AVR), which translates to experimental reactor consortium, from the early 1960s (Refs. 10, 11, and 12). Recently, Xe‐100, a HTGR is being developed by X-Energy under the U.S. Department of Energy’s Advanced Reactor Demonstration Program. It is designed to provide a secure future for the global energy and process heat markets.

In general, the PBR is a pyrolytic graphite-moderated and helium gas-cooled nuclear reactor that achieves a requisite high outlet temperature while retaining the passive safety and proliferation resistance requirements of Gen IV designs.¹³ A schematic of a PBR is shown in Fig. 1 from Rycroft.¹⁴ In these reactors, graphite blocks are stacked to serve as neutron reflectors and to form a cylindrical reactor core, The core has a “double-zone” configuration; i.e., there are two cores, an inner blind core of graphite spheres (without uranium fuel) at the center and an outer annular active core with fuel spheres. The inner graphite reflectors serve to flatten the neutron flux across the annular fuel region without placing any fixed structures inside the core vessel. The fuel and reflector spheres, called pebbles, are approximately the size of a tennis ball (usually about 6 cm in diameter). Both the fuel and graphite reflector pebbles are almost the same in terms of shape and average density, except that the fuel pebbles in a graphite matrix contain a large amount of uranium particles (about 11 000 particles).¹⁵

Fig. 1. A schematic diagram of the pebble bed nuclear reactor (Rycroft).¹⁴

In the core of the nuclear PBR, hundreds of thousands of microspheres of coated particles (about 900 to 950 μm in diameter) known as tri-structural isotropic (TRISO) fuel particles are embedded within a graphite matrix to form a final fuel pebble and provide two barriers for the fission products. The TRISO fuel particles cause fission in a graphite pebble,¹⁶ and because of their high surface-to-volume ratio, TRISO fuel particles easily transfer heat from fuel to matrix graphite. A schematic sketch of a typical microstructure of the TRISO fuel particle is shown in Fig. 2 from Ortensi and Ougouag.¹⁷

Fig. 2. A schematic sketch of the typical microstructure for the TRISO fuel particles in PBR (Ortensi and Ougouag).¹⁷

Each TRISO fuel particle consists of a spherical fuel kernel (~0.5 mm) composed of high-assay low-enriched uranium, which is uranium dioxide, and sometimes uranium oxycarbide (UCO) or a mixture of uranium and platinum oxides in the center, coated with four concentric layers of three isotropic materials. The four layers are (1) a porous buffer layer made of carbon of low density that serves to capture any fission product particles emitted from the fuel kernel, (2) a dense inner layer of pyrolytic carbon (PyC) of high density, (3) a ceramic layer of polycrystalline silicon carbide to retain fission products at elevated temperatures and to give the TRISO particles more structural integrity, and (4) another dense outer layer of PyC. Microspheres of TRISO fuel particles are designed not to crack because of stress from processes (such as differential thermal expansion or fission gas pressure), even at temperatures beyond 1600°C.

The fuel and graphite pebbles move downward by gravitational force through the reactor core in the form of a very slowly moving pebble bed. The pebbles stack randomly inside the reactor, where the older ones are removed from the bottom, inspected for burnup and mechanical integrity, and recirculated into the top of the reactor core until it achieves the specified high discharge burnup. Because of this unique online fueling feature of the moving pebbles and dynamic core, the transport phenomena and physical processes involved are extremely complex mechanisms in this type of reactor.^1,18

In the annular active core, heat generated from the nuclear fission reaction and decay heat from fission products inside the fuel spheres are removed by the forced circulation of the pressurized (typically up to 8.5 MPa) coolant helium gas (~500°C inlet core temperature), which is pumped into the reactor mainly through the inlet riser channels shaped in the side graphite block reflectors to an upper plenum (cold plenum), where it flows down the reactor core to bring out the fission heat and exits the reactor through a lower plenum (hot plenum) at a very high temperature (900°C to 1000°C). The elevated static pressure and the large pebble diameter cause high values of the Reynolds number (up to about 4.5 × 10⁴), under normal operating conditions, which exceed those usually occurring in the conventional packed bed technology by one order of magnitude.

Helium gas is chosen as a coolant in VHTRs because of its excellent heat exchange properties, and because it is both chemically and radiologically inert, it is used as supercritical gas and does not condensate in the considered range of operation. In addition, it is naturally available in sufficient quantities.¹⁹ It is worth mentioning here that an axial core downflow of the coolant removes the problem of bed levitation that would limit the power density of the reactor.²⁰

In a PBR core, the coolant flow structure and hence the heat removed appear to be strongly dependent on the distribution of the moving fuel pebbles. As the helium gas flows downward under high flow conditions (relatively high Reynolds numbers of about 50 000) through the reactor core and over these heated, randomly, and closely distributed pebbles, combined with the high-temperature integrity of the fuel and structural graphite, the coolant gas attains a very high temperature at the core outlet (~900°C). This is one of the attractive features because the high operating temperature allows a higher thermal efficiency to be yielded (it is possible to extract up to about 50% or a little more) than what would be rendered by conventional nuclear plants (typically between 35% to 40%), and it can be used in various industrial processes. For example, high-temperature helium gas can be used to produce industrial process heat and hydrogen providing the needed energy by splitting water, which in turn can be used for treating metals, processing food, and creating an alternative fuel source in the form of hydrogen fuel cells.

One of the most essential characteristics of pebble bed nuclear reactors is that the reactor core is designed such that a maximum fuel element temperature of 1600°C is not surpassed during any accident. Active core cooling is not required for decay heat removal during accidents. It is sufficient to discharge the decay heat to the cavity coolers located outside the reactor pressure vessel using passive heat transport processes such as heat conduction, radiation, and natural convection.²¹ The bed structure, coolant flow dynamics, pressure drop, and heat transport, which determine the thermal-hydraulic characteristics of the PBR, are among the essential phenomena that need to be well understood for the proper design and safe performance of these reactors. The advantages and disadvantages of nuclear PBRs are summarized in Table II.

TABLE II Advantages and Disadvantages of PBRs*

Pros	Cons
The moving fuel pebbles provide variations in packing, physics, and heat removal.	Moving bed of complex-flow structure and path.
Inherent safety due to fuel type and gas coolant; hence, a negative temperature coefficient is achieved, which means that if the temperature rises, the nuclear reaction is slowed, and the power is reduced.	Because of the system’s complexity, extremely complex transport and processes are involved.
High outlet gas temperature yields higher thermal efficiency.	Accurate analyses of the flow field and heat transport in the dynamic core pose an extreme challenge to efficient design and safe operation.
High heat capacity and low power density
Unlike conventional nuclear reactors, PBRs do not need to be shut down in order to check on the integrity and consumption of uranium and to be refueled; this is due to online refueling.
Promises to generate less nuclear waste
The design produces a small reactor that can be built cheaply with a short construction time and operated safely.
The pebbles are supposed to survive temperatures of 1600°C, far hotter than the worst foreseeable accident.

*Abdulmohsin.¹

I.B. Motivation for the Present Study

To reliably simulate the thermal-hydraulic phenomena and hence the performance in the core of nuclear packed PBRs, the coolant gas dynamics and heat transport processes must be characterized.^1,18,22,23 In addition, the experimental investigation of the thermal-hydraulic characteristics of pebble beds is an issue of high importance when selecting the core geometry and evaluating the performance and safety of such kind of reactors.²⁴ The efficiency of the PBR is strongly dependent upon how the coolant removes the generated heat from the dynamic core of this reactor. Furthermore, the knowledge of dispersion and mixing in the longitudinal direction is most important when temperatures are rapidly changing with respect to time or axial coordinates due to nuclear reaction and interphase heat transport. Recently, some computational fluid dynamics (CFD) studies²⁵ were reported in the literature about the knowledge and quantification of the complex coolant gas flow structure in the pebble bed nuclear reactor. As CFD is expensive, there is still a need for the correlations to perform transient analysis in the PBRs. On the other hand, the local fuel temperatures depend not only on the local power generation but also on the point heat removal rate. In other words, besides reactivity control and radioactive retention, heat removal has been considered one of those three fundamental safety functions in HTGRs. Hence, detailed information and proper understanding of the transport of heat generated during nuclear fission from slowly moving hot fuel pebbles to the flowing coolant gas are crucial for the safe design and efficient operation of packed pebble bed nuclear reactors. All three modes of heat transport (i.e., conduction, convection, and radiation) are important for modeling and predicting the pebble bed core temperature distribution. During the nominal operation of the reactor at relatively high Reynolds numbers, the heat transfer mechanism is governed by the forced convection mode. At low Reynolds numbers (in the case of an accident), the effects of free convection, thermal radiation, heat conduction, and heat dispersion come into the same order of magnitude as the contribution of forced convection.²⁶ However, limited information related to the pebble-to-coolant gas heat transfer is available in the literature.^{1,18,22,23,27}

II. THEORETICAL BACKGROUND AND LITERATURE REVIEW

Engineers and scientists have been studying packed beds of small particles since before the turn of the 20th century, and extensive literature exists regarding the flow of gases, the transfer of heat and mass, and the pressure drop in fluids flowing through packed beds. However, for beds with large particles like those encountered in PBRs, still few studies have been conducted. The key phenomena of interest for the randomly packed PBRs involve the variability in the packing structure throughout the bed, pressure drop across the bed, dispersion and mixing, and heat transport processes. Therefore, this work discusses and analyzes the background, literature, and our recent advances related to the existing knowledge of the bed structure, fluid flow, pressure drop, coolant gas dispersion and mixing phenomena, and heat transfer characteristics of packed PBRs.

II.A. Physical Characteristics of Packed Pebble Bed Nuclear Reactors

It is well known that the statistical parameter of porous media is the porosity or void fraction (voidage). Therefore, the principal physical quantities of a randomly packed PBR must combine this statistical structural parameter (porosity) with the characteristics of particle size and mean interstitial velocity.

The thermal design of a packed PBR is based upon the mechanisms of heat transport and the flow and pressure drop of the coolant throughout the pebble bed.²⁸ The mechanisms in turn are all sensitive to the porous structure or porosity variations of the packed bed.²⁹ Therefore, before any rigorous analysis of the fluid flow and heat transfer is attempted, it is important to have a thorough understanding of the structural arrangement of the packed bed under consideration.

The bed voidage could be broadly categorized by two terms, that is, the average (mean) porosity of the bed ${\epsilon }_b$ and the local voidage [radially $\epsilon (r)$ or axially $\epsilon \left(x\right)$]. Traditionally, investigators have defined the local porosity or void fraction as the ratio of the void volume to the volume of the bed packing structure at a localized position within the packed bed,³⁰ and it has a numerical value between 0 (no voidage) and 1 (no bed). For randomly packed pebble beds, the void fraction can be expressed as

(1a) $\epsilon (r)=\frac{Localvolumeofvoidsinpacking}{Localbulkvolumeofpacking}=\frac{V_T-V_S}{V_T}=1-\frac{V_S}{V_T},$(1a)

where $V_S$ is the volume of the solid particles (pebbles) and $V_T$ is the total volume of the bed.

The average cross-sectional porosity of the bed ${\epsilon }_b$ can be azimuthally averaged based on the cross-sectional area defined as

(1b) ${\epsilon }_b=\frac{2}{{\mathrm{R}}^2}\underset{0}{\overset{R}{\int }}\epsilon (\mathrm{r})\mathrm{r}\mathrm{d}\mathrm{r}$(1b)

while the axially averaged porosity along the height can be defined as

(1c) ${\epsilon }_b=\frac{2}{L}\underset{0}{\overset{L}{\int }}\epsilon \left(x\right)\mathrm{d}\mathrm{x},$(1c)

where R is the radius of the packed pebble bed and L is the height of the bed.

In a packed bed, the porosity varies sharply near the wall since at that location, the geometry of the packing is interrupted.²⁹ As a result, the velocity profile inside a packed bed can be severely distorted near the wall, reaching a maximum in the near-wall region. This phenomenon is known as flow or wall channeling. Wall channeling can have a significant impact on heat and mass transfer in packed beds.²⁹ However, in the case of the pebble bed nuclear reactor, this might lead to a reduction in wall temperature and also lead to a nonuniform temperature distribution at the outlet of the bed.³¹ Knowledge of the porosity distribution within a packed bed is thus important to any proper analysis of the transport phenomena in the bed,³² and this analysis must be made before any design changes can be recommended, for example, to improve the temperature distribution at the outlet of the reactor. Experimental investigation and characterizing the bed structure of PBRs have been studied and quantified for the first time in terms of solids and voids in our laboratory using gamma-ray computed tomography as a part of another study.³³

II.A.1. Mean Bed Porosity

As mentioned before, the total average (mean) porosity is a useful structural parameter in the design and guide to characterizing of packing fixed-packed systems. In the gas-cooled PBR, the cylindrical core consists of randomly packed same-size spherical pebbles with a homogeneous porosity except at the wall region. Near the wall, the porosity is higher due to the presence of the wall, and the porosity fluctuates toward the core region of the bed, where it becomes uniform if the bed is large enough due to the wall effect. The following formula was recommended by Fenech²⁶ and Achenbach^34,35 to estimate the mean bed porosity ε_b:

(2) ${\epsilon }_b=\frac{0.78}{{\left(\mathrm{D}/{\mathrm{d}}_p\right)}^2}+0.375\mathrm{f}\mathrm{o}\mathrm{r}\left(\mathrm{D}/{\mathrm{d}}_p\right)>2,$(2)

where D is the diameter of the bed and d_p is the pebble diameter. The above formula represents the experimental results of Carman³⁶ and Barthels^37,38 as quoted by Achenbach.³⁵ It is worth mentioning that the mean porosity is independent of the pebble diameter itself but depends on the aspect ratio or the tube–to–pebble diameter ratio $D/d_p$. It decreases as the aspect ratio increases, and it levels out to an average value of about 0.375 for a very high value of the aspect ratio, $D/d_p\to \mathrm{\infty }$. The voidage varies radially through the bed toward the core region due to the wall effect, and the extent of this variation depends on the aspect ratio.

The distribution of the spherical pebbles in a packed PBR is no longer random near the wall because of the orientation forced by the presence of the wall. The high values of voidage near the wall, of course, cause a nonuniform velocity distribution across the core of the pebble bed. In the center of the bed, the gas velocity is lower than the mean velocity calculated from the overall mass flow, while close to the wall, the velocity is higher than the mean velocity. To estimate the magnitude of the wall effect, it is assumed that the core of the packed pebble bed consists of two parts of different void fractions.²⁶ The near-wall region and the central region of the corresponding porosities can be expressed, respectively, as follows:

Near-wall region:

(3a) ${\epsilon }_w=\frac{63.6}{{\left[\left(\mathrm{D}/{\mathrm{d}}_p\right)+15\right]}^2}+0.43\mathrm{f}\mathrm{o}\mathrm{r}\left(\mathrm{D}/{\mathrm{d}}_p\right)>2$(3a)

and

Central region:

(3b) ${\epsilon }_c={\epsilon }_w-\frac{{\epsilon }_w-{\epsilon }_b}{{\left[1-\left({\mathrm{d}}_p/\mathrm{D}\right)\right]}^2}\mathrm{f}\mathrm{o}\mathrm{r}\left(\mathrm{D}/{\mathrm{d}}_p\right)>2.$(3b)

The near wall-region voidage ε_w correlation was developed based on approximating the experimental results of Benenati and Brosilow³⁹ while the central voidage ε_c correlation was developed based on the calculated values by means of the equation of conservation of mass.²⁶

II.A.2. Radial Distribution of Bed Porosity

Several empirical correlations and mathematical models to describe the radial variation in the porosity of packed beds of small particles have been proposed by various researchers. Du Toit³¹ stated that the correlations to predict the variation in the porosity of packed beds can be classified into two categories, i.e., those that attempt to describe the oscillatory behavior of the variation in the porosity and those that attempt to describe the variation in the average porosity using an exponential expression. It should be noted that the porosity is uniform in the tangential direction, i.e., an axially symmetric approach. The correlations of the approaches are presented in the next sections:

II.A.3. Oscillatory Porosity Correlations

Various attempts at modeling the voidage variations are presented in the literature. Most of the more recent models describe both the oscillatory nature and damping of the voidage variations. Using the experimental data of Benenati and Brosilow,³⁹ Martin⁴⁰ proposed the following correlation:

(4a) $\epsilon \left(x\right)=\left\{\begin{array}{c}{\epsilon }_{min}+\left(1+{\epsilon }_{min}\right)x^2\mathrm{f}\mathrm{o}\mathrm{r}-1\le x<0\\ {\epsilon }_b+\left({\epsilon }_{min}-{\epsilon }_b\right)\\ \mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{x}{4}\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\pi }{\mathrm{C}}x\right)\mathrm{f}\mathrm{o}\mathrm{r}x\ge 0\end{array}\right.$(4a)

with

(4b) $\begin{array}{rl}& x=2\frac{\mathrm{R}-\mathrm{r}}{{\mathrm{d}}_p}-1\\ & \mathrm{C}=\left\{\begin{array}{c}0.816\mathrm{D}/{\mathrm{d}}_p\to \mathrm{\infty }\\ 0.876\mathrm{D}/{\mathrm{d}}_p=20.3\end{array}\right.,\end{array}$(4b)

where ε_min is the minimum porosity within the range from 0.20 to 0.26, ε_b is the bulk porosity of the packed bed undisturbed by wall effects, and C is a constant.

Based on the findings of Roblee et al.⁴¹ and other investigators, Cohen and Metzner⁴² fitted the following set of correlations to represent the oscillatory variation of the porosity in the radial direction away from the wall of a cylindrical packed bed:

(5a) $\begin{array}{rl}& \frac{1-\epsilon \left(x\right)}{1-{\epsilon }_b}=4.5\left[x-\frac{7}{9}{x}^2\right]\mathrm{f}\mathrm{o}\mathrm{r}x\le 0.25\\ & \frac{\epsilon \left(x\right)-1}{1-{\epsilon }_b}={\mathrm{a}}_1\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\mathrm{a}}_{2}x\right)\\ & \mathrm{c}\mathrm{o}\mathrm{s}\left({\mathrm{a}}_{3}x-{\mathrm{a}}_4\right)\pi \mathrm{f}\mathrm{o}\mathrm{r}0.25<x<8\\ & \epsilon \left(x\right)={\epsilon }_b\mathrm{f}\mathrm{o}\mathrm{r}8<x<\mathrm{\infty }\end{array}$(5a)

with

(5b) $x=\frac{\mathrm{R}-\mathrm{r}}{{\mathrm{d}}_p},$(5b)

where ${\epsilon }_b$ is the average porosity of the bed. The authors determined the constants a₁ through a₄ to be a₁ = 0.3463, a₂ = 0.4273, a₃ = 2.4509, and a₄ = 2.2011 while R refers to the outer radius of a cylindrical bed.

It is worth mentioning here that the models suggested by Martin⁴⁰ and Cohen and Metzner⁴² are similar in the sense that they both contain a cosine term to describe the oscillations and an exponential term to describe the dampening. In addition, the influence of the column–to–particle diameter ratio on the period of oscillation was recognized and included in their models.

Mueller^43,44 modeled the oscillations of the voidage with a zero-order Bessel function of the first kind and described the dampening with an exponential term. Using his results and other existing data, Mueller⁴⁴ derived an empirical correlation that can be used to predict the variation in the porosity in the radial direction for fixed-packed beds of uniformly sized spheres in cylindrical containers. The effect of the column–to–particle diameter ratio on the period of the oscillations was considered as the following⁴⁴:

(6a) $\begin{array}{rl}& \epsilon (\mathrm{r})={\epsilon }_b+\left(1-{\epsilon }_b\right){\mathrm{J}}_0\left(\mathrm{a}\frac{\mathrm{r}}{{\mathrm{d}}_p}\right)\\ & \mathrm{e}\mathrm{x}\mathrm{p}\left(-b\frac{\mathrm{r}}{{\mathrm{d}}_p}\right)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{D}/{\mathrm{d}}_p\ge 2.02\end{array}$(6a)

with

(6b) $\begin{array}{rl}& \mathrm{a}=\left\{\begin{array}{c}7.45-\frac{3.15}{\mathrm{D}/{\mathrm{d}}_p}\mathrm{f}\mathrm{o}\mathrm{r}2.02\le \mathrm{D}/{\mathrm{d}}_p\le 13\\ 7.45-\frac{11.25}{\mathrm{D}/{\mathrm{d}}_p}\mathrm{f}\mathrm{o}\mathrm{r}13<\mathrm{D}/{\mathrm{d}}_p\end{array}\right.\\ & b=0.315-\frac{0.725}{\mathrm{D}/{\mathrm{d}}_p}\\ & {\epsilon }_b=0.365+\frac{0.22}{\mathrm{D}/{\mathrm{d}}_p},\end{array}$(6b)

where ${\epsilon }_b$ is the average porosity of the bed.

Various versions of the correlation proposed by Mueller⁴⁴ exist.^45–47 Mueller⁴⁸ also formulated the local radial porosity area based on analytical equation formulas for a cylindrical system with mono-sized spherical particles. Lately, Mueller developed a new and simple method for calculating the radial porosity profile for mono-sized spheres in cylindrical containers. The new method was derived from geometrical and analytical analyses and uses arc lengths to calculate the radial porosity profile.

The same exponentially damped sinusoidal form that Martin⁴⁰ posited was used by De Klerk⁴⁹ in the development of his model. De Klerk⁴⁹ determined the constants of the model by fitting the form of the correlation to the porosity data found in the literature. The constants were then adjusted so that the correlation yielded sensible average bed porosities. The radial variation of porosity through a cylindrical packed bed of spherical particles can be written as follows:

(7a) $\epsilon \left(x\right)=\left\{\begin{array}{c}2.14x^2-2.53x+1\mathrm{f}\mathrm{o}\mathrm{r}x\le 0.637\\ {\epsilon }_b+0.29\mathrm{e}\mathrm{x}\mathrm{p}\left(-0.6x\right)\\ \mathrm{c}\mathrm{o}\mathrm{s}\left[2.3\pi \left(x-0.16\right)\right]\\ +0.15\mathrm{e}\mathrm{x}\mathrm{p}\left(-0.9x\right)\mathrm{f}\mathrm{o}\mathrm{r}x>0.637\end{array}\right.$(7a)

with

$\begin{array}{rl}& x=\frac{\mathrm{r}-{\mathrm{R}}_i}{{\mathrm{d}}_p}{\mathrm{R}}_{\mathrm{i}}\le \mathrm{r}\le \frac{{\mathrm{R}}_o+{\mathrm{R}}_i}{2}\\ & x=\frac{{\mathrm{R}}_o-\mathrm{r}}{{\mathrm{d}}_p}\frac{{\mathrm{R}}_o+{\mathrm{R}}_i}{2}\le \mathrm{r}\le {\mathrm{R}}_o.(7\mathrm{b})\end{array}$

It is important to note that in Eq. (7b$\begin{array}{rl}& x=\frac{\mathrm{r}-{\mathrm{R}}_i}{{\mathrm{d}}_p}{\mathrm{R}}_{\mathrm{i}}\le \mathrm{r}\le \frac{{\mathrm{R}}_o+{\mathrm{R}}_i}{2}\\ & x=\frac{{\mathrm{R}}_o-\mathrm{r}}{{\mathrm{d}}_p}\frac{{\mathrm{R}}_o+{\mathrm{R}}_i}{2}\le \mathrm{r}\le {\mathrm{R}}_o.(7\mathrm{b})\end{array}$), R_i refers to the inner radius of the annulus and R_o refers to the outer radius of an annular packed bed.

It is obvious that various authors have performed experiments to obtain different porosity correlations for the variation in the voidage of packed beds in the bulk and near-wall regions. Although many different experimental techniques have been used, the results in general agree. A good overview of the experimental methods used by the various authors is given by De Klerk.⁴⁹ However, porosity results obtained from the analysis of numerically generated annular packed beds and physical experimental data obtained by Du Toit³¹ were used to evaluate the different porosity correlations.

Van Antwerpen et al.⁵⁰ made an evaluation based on the comparison between the relevant correlations with the numerical results of Du Toit³¹ for the heat transfer test facility (HTTF), as shown in Fig. 3. Du Toit emphasized that in the case of Cohen and Metzner,⁴² the correlation between the dimensionless distance $x$ from both walls in the middle of the annulus is less than eight, and the correlation therefore never achieves the bulk value for the porosity.

Fig. 3. Comparison between radial oscillatory porosity correlations.⁵⁰

Theuerkauf et al.⁵¹ stated that because of the nature of the Bessel function employed by Mueller,⁴⁴ the predicted variation in the porosity next to the wall was not correct, which led to a significant overprediction of the porosity in the near-wall region. Thus, the correlation by Mueller⁴⁴ was not included in the comparison by Du Toit³¹ and was also not considered in the evaluation by Van Antwerpen et al.⁵⁰ Du Toit³¹ stated that the correlation proposed by Martin⁴⁰ was the most representative of Du Toit’s numerical results. However, it was reported by Van Antwerpen et al.⁵⁰ that the correlation proposed by De Klerk⁴⁹ gave an even better prediction of the variation in the radial porosity than that of Martin.⁴⁰

II.A.4. Exponential Porosity Correlations

In some simplified models, such as the model of Vortmeyer and Schuster,⁵² it is assumed that the “average” porosity decays exponentially from unity at the wall to the bulk value farther away from the wall. Following Cheng and Hsu,⁵³ Hunt and Tien,⁵⁴ and Sodre and Parise,⁵⁵ the radial porosity distribution for an annular packed bed can be written as follows:

(8a) $\epsilon \left(r\right)=\left\{\begin{array}{c}{\epsilon }_o\left[1+C\mathrm{e}\mathrm{x}\mathrm{p}\left(-N\frac{\mathrm{r}-{\mathrm{R}}_{\mathrm{i}}}{{\mathrm{d}}_p}\right)\right]\mathrm{f}\mathrm{o}\mathrm{r}{\mathrm{R}}_{\mathrm{i}}\le \mathrm{r}\le \frac{{\mathrm{R}}_o+{\mathrm{R}}_i}{2}\\ {\epsilon }_o\left[1+C\mathrm{e}\mathrm{x}\mathrm{p}\left(-N\frac{{\mathrm{R}}_o-\mathrm{r}}{{\mathrm{d}}_p}\right)\right]\mathrm{f}\mathrm{o}\mathrm{r}\frac{{\mathrm{R}}_o+{\mathrm{R}}_i}{2}\le \mathrm{r}\le {\mathrm{R}}_o\end{array}\right.,$(8a)

where R_i is the inner radius of the annulus and R_o is the outer radius of the annulus. Vortmeyer and Schuster,⁵² Cheng and Hsu,⁵³ and Hunt and Tien⁵⁴ used the expression ε_o = ε_b to represent the bulk porosity of the bed while Sodre and Parise⁵⁵ used ε_o = ε_∞ to represent the porosity of an infinite bed. Most researchers use a value of C that gives a porosity of 1 at the wall, but Cheng and Hsu⁵³ used C = 1. For spherical particles, Vortmeyer and Schuster⁵² and Cheng and Hsu⁵³ used 2 as the value of N, but Hunt and Tien⁵⁴ used N = 6. Sodre and Parise⁵⁵ proposed that the value of N be obtained from the following:

(8b) $N=\frac{2C{\epsilon }_{\mathrm{\infty }}{\mathrm{d}}_p\left[1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-N\left({\mathrm{R}}_o-{\mathrm{R}}_i\right)/2{\mathrm{d}}_p\right)\right]}{\left(\bar{\epsilon }-{\epsilon }_{\mathrm{\infty }}\right)\left({\mathrm{R}}_o-{\mathrm{R}}_i\right)},$(8b)

where $\bar{\epsilon }$ is the average bed porosity for the annulus given by the following:

(8c) $\bar{\epsilon }=0.3517+0.387\frac{{\mathrm{d}}_p}{2\left({\mathrm{R}}_o-{\mathrm{R}}_i\right)}.$(8c)

Du Toit³¹ noted that the correlation derived by Sodre and Parise⁵⁵ failed to fit with the results obtained by the other correlations and proposed that ${\epsilon }_{\mathrm{\infty }}$ be substituted by ${\epsilon }_b$ in the bulk region of the annulus and substituted $\bar{\epsilon }$ with the average porosity for the annulus obtained from the numerical results.

Equation (8a)(8a) $\epsilon \left(r\right)=\left\{\begin{array}{c}{\epsilon }_o\left[1+C\mathrm{e}\mathrm{x}\mathrm{p}\left(-N\frac{\mathrm{r}-{\mathrm{R}}_{\mathrm{i}}}{{\mathrm{d}}_p}\right)\right]\mathrm{f}\mathrm{o}\mathrm{r}{\mathrm{R}}_{\mathrm{i}}\le \mathrm{r}\le \frac{{\mathrm{R}}_o+{\mathrm{R}}_i}{2}\\ {\epsilon }_o\left[1+C\mathrm{e}\mathrm{x}\mathrm{p}\left(-N\frac{{\mathrm{R}}_o-\mathrm{r}}{{\mathrm{d}}_p}\right)\right]\mathrm{f}\mathrm{o}\mathrm{r}\frac{{\mathrm{R}}_o+{\mathrm{R}}_i}{2}\le \mathrm{r}\le {\mathrm{R}}_o\end{array}\right.,$(8a) must be solved using an iterative procedure. In contrast, White and Tien²⁹ proposed a radial porosity distribution of this form:

(9) $\epsilon \left(r\right)=\left\{\begin{array}{c}{\left[\begin{array}{c}1+\left(\frac{1-{\epsilon }_b}{{\epsilon }_b}\right)\\ \sqrt{1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-2\frac{\mathrm{r}-{\mathrm{R}}_i}{{\mathrm{d}}_p}\right)}\end{array}\right]}^{-1}\mathrm{f}\mathrm{o}\mathrm{r}{\mathrm{R}}_i\le \mathrm{r}\le \frac{{\mathrm{R}}_o+{\mathrm{R}}_i}{2}\\ {\left[\begin{array}{c}1+\left(\frac{1-{\epsilon }_b}{{\epsilon }_b}\right)\\ \sqrt{1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-2\frac{{\mathrm{R}}_o-\mathrm{r}}{{\mathrm{d}}_p}\right)}\end{array}\right]}^{-1}\mathrm{f}\mathrm{o}\mathrm{r}\frac{{\mathrm{R}}_o+{\mathrm{R}}_i}{2}\le \mathrm{r}\le {\mathrm{R}}_o\end{array}\right.,$(9)

Van Antwerpen et al.⁵⁰ have evaluated and made a comparison between the exponential porosity correlations, Eqs. (8)(8a) $\epsilon \left(r\right)=\left\{\begin{array}{c}{\epsilon }_o\left[1+C\mathrm{e}\mathrm{x}\mathrm{p}\left(-N\frac{\mathrm{r}-{\mathrm{R}}_{\mathrm{i}}}{{\mathrm{d}}_p}\right)\right]\mathrm{f}\mathrm{o}\mathrm{r}{\mathrm{R}}_{\mathrm{i}}\le \mathrm{r}\le \frac{{\mathrm{R}}_o+{\mathrm{R}}_i}{2}\\ {\epsilon }_o\left[1+C\mathrm{e}\mathrm{x}\mathrm{p}\left(-N\frac{{\mathrm{R}}_o-\mathrm{r}}{{\mathrm{d}}_p}\right)\right]\mathrm{f}\mathrm{o}\mathrm{r}\frac{{\mathrm{R}}_o+{\mathrm{R}}_i}{2}\le \mathrm{r}\le {\mathrm{R}}_o\end{array}\right.,$(8a) and (9(9) $\epsilon \left(r\right)=\left\{\begin{array}{c}{\left[\begin{array}{c}1+\left(\frac{1-{\epsilon }_b}{{\epsilon }_b}\right)\\ \sqrt{1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-2\frac{\mathrm{r}-{\mathrm{R}}_i}{{\mathrm{d}}_p}\right)}\end{array}\right]}^{-1}\mathrm{f}\mathrm{o}\mathrm{r}{\mathrm{R}}_i\le \mathrm{r}\le \frac{{\mathrm{R}}_o+{\mathrm{R}}_i}{2}\\ {\left[\begin{array}{c}1+\left(\frac{1-{\epsilon }_b}{{\epsilon }_b}\right)\\ \sqrt{1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-2\frac{{\mathrm{R}}_o-\mathrm{r}}{{\mathrm{d}}_p}\right)}\end{array}\right]}^{-1}\mathrm{f}\mathrm{o}\mathrm{r}\frac{{\mathrm{R}}_o+{\mathrm{R}}_i}{2}\le \mathrm{r}\le {\mathrm{R}}_o\end{array}\right.,$(9) ), and the numerical results of Du Toit³¹ for the HTTF, as shown in Fig. 4. After a careful examination by Du Toit,³¹ it was found that the correlation proposed by Hunt and Tien⁵⁴ gave the best representation of the “average” variation of porosity in the radial direction.

Fig. 4. Comparison between radial exponential porosity correlations.⁵⁰

II.B. Identification of Fluid Flow Regimes in Packed Bed Reactors

Resistance to fluid flow is usually obtained from pressure drop measurements in randomly packed beds. It is possible to distinguish four different flow regimes in packed PBRs, based on the effective Reynolds number, which is defined as

(10a) ${Re}_h=\frac{\rho \mathrm{V}{\mathrm{d}}_h}{\mu }=\frac{1}{\left(1-{\epsilon }_b\right)}Re,$(10a)

where $d_h$ is the equivalent hydraulic (effective) diameter, which is the characteristic length of the packed pebble bed and defined as follows:

(10b) ${\mathrm{d}}_h={\mathrm{d}}_p\frac{{\epsilon }_b}{\left(1-{\epsilon }_b\right)},$(10b)

while $V$ is the interstitial velocity, which is the characteristic or the mean velocity in the gaps between the pebbles and defined as follows:

(10c) $\mathrm{V}=\frac{{\mathrm{V}}_g}{{\epsilon }_b},$(10c)

where $V_g$ is the superficial gas velocity based on the cross section of the empty column. Re is the Reynolds number and is defined on the basis of the total mass flow rate through the total cross-sectional area of the packing and on the diameter of the pebbles as follows:

(10d) $Re=\frac{\rho {\mathrm{V}}_g{\mathrm{d}}_p}{\mu }.$(10d)

The physical significance of these four different flow regimes is as follows:

1. For ${Re}_h<1$, a creeping-flow regime that is purely viscous. It follows Darcy’s law; therefore, it is called Darcian flow. In this regime, the viscous forces dominate over the inertia forces, and only the local (pore-level) geometry influences the flow.⁵⁶ This regime is also characterized by a linear relationship between pressure drop and mass flow.³⁵ Therefore, it is sometimes referred to as the linear-laminar-flow regime.⁵⁷

2. For $1\mathrm{t}\mathrm{o}10<{Re}_h<150$, a steady laminar-flow regime in which the inertia effects begin to play an important role in the flow condition; therefore, it is called the inertial-flow regime.

3. For $150<{Re}_h<300$, an unsteady laminar-flow regime in which both viscous and inertia forces are important. In this regime wake instability might be responsible for the transition from the laminar steady flow to unsteady flow. In this regime, the deviation from Darcy’s law begins; hence, this is sometimes called the nonlinear-laminar-flow regime.⁵⁷

4. For ${Re}_h>300$, a turbulent-flow regime in which viscous effects are negligible. It is a highly unsteady chaotic flow; therefore, it is called an unsteady- and chaotic-flow regime. There is a failure of Darcy’s law to describe the flow through fixed beds in this regime.

III. CHARACTERISTICS OF FLUID FLOW IN PACKED BED REACTORS

It is well known that the fluid flow problem in porous media is caused by transition between flow in channels and flow around submerged objects. According to the discontinuity of this system, an exact representation of the fluid flow distribution in porous media is impossible.⁵⁸ For flow through packed bed reactors, it is desirable to be able to predict the flow rate obtainable for a given energy input (usually measured as pressure drop) or to be able to predict the pressure drop necessary to achieve a specific flow rate. Practically, the complexity of the flow pattern rules out a rigorous analytic solution to the problem; hence, an empirical or semiempirical correlation has been suggested. Generally, in packed PBRs, the resistances of flow are usually described in terms of total pressure drop $\mathrm{\Delta }P$ or the pressure drop coefficient, which is defined as dimensionless pressure drop:

(11) $\psi =\frac{\mathrm{\Delta }\mathrm{P}}{\left(\rho /2\right){\mathrm{V}}^2}\frac{{\mathrm{d}}_h}{\mathrm{L}}.$(11)

The pressure loss due to friction between solid (pebbles) and gas phases in the core of the pebble bed can be expressed as the following²⁶:

(12) $\mathrm{\Delta }\mathrm{P}=\psi \frac{\mathrm{L}}{{\mathrm{d}}_h}\frac{\rho }{2}{\mathrm{V}}^2=\psi \frac{\mathrm{L}}{{\mathrm{d}}_p}\frac{\rho }{2}{\mathrm{V}}_g^2\left(\frac{1-{\epsilon }_b}{{\epsilon }_b^3}\right).$(12)

On the one hand, there are two main approaches for developing friction factor expressions for packed beds.^59,60 In the first approach, the packed bed is visualized as a bundle of tubes. In the second approach, the packed bed is regarded as a collection of submerged objectives. Based on these two approaches, the pressure drop in fixed-packed beds has been described by two different models.⁶¹ The first one is the model of the hydrodynamic diameter, and the second is the model of the flow around a single particle. The first model is older and leads to the relatively easy pressure drop equations, such as the classical Ergun-type equation.⁶² It is more useful to mention here that this model assumes the packing is statistically uniform, so there are no channeling or bypassing effects (although in the actual situation of a PBR, channeling, bypassing, etc., would occur). Thus, the development given here does not apply to the randomly packed PBRs. The second model is newer,⁶³ and it overcomes the assumption of statistical uniformity; therefore, it is more appropriate for randomly packed PBRs.

On the other hand, the dimensionless pressure drop or the pressure drop coefficient $\psi$ is a function of the effective Reynolds number ${Re}_h$; therefore, several correlations were developed and verified using experimental data.⁶⁴ The well-known Ergun equation expresses the friction factor in a packed bed as follows⁶²:

(13) $\psi =\frac{150}{{Re}_h}+1.75\mathrm{f}\mathrm{o}\mathrm{r}{Re}_h\le 5\times {10}^4,$(13)

where ${Re}_h$ is a modified or effective Reynolds number that is based on the average interstitial velocity $V$ and on the characteristic length scale of the pores (an equivalent hydraulic diameter$d_h$) follows by recalling Eq. (10a)(10a) ${Re}_h=\frac{\rho \mathrm{V}{\mathrm{d}}_h}{\mu }=\frac{1}{\left(1-{\epsilon }_b\right)}Re,$(10a) :

(13a) ${Re}_h=\frac{\rho \mathrm{V}{\mathrm{d}}_h}{\mu }=\frac{1}{\left(1-{\epsilon }_b\right)}Re.$(13a)

Equation (10a)(10a) ${Re}_h=\frac{\rho \mathrm{V}{\mathrm{d}}_h}{\mu }=\frac{1}{\left(1-{\epsilon }_b\right)}Re,$(10a) is formed by adding the Carmen-Kozeny^36,65 equation for purely laminar flow (viscous effect, ${Re}_h<1$) through a porous medium modeled as an assembly of capillaries, to the Burke-Plummer⁶⁶ equation derived for the fully turbulent (inertia effect, ${Re}_h\ge 300$) limit in a capillaric medium.⁵⁹ The first term in the expression, Eq. (13(13) $\psi =\frac{150}{{Re}_h}+1.75\mathrm{f}\mathrm{o}\mathrm{r}{Re}_h\le 5\times {10}^4,$(13) ), refers to viscous energy losses, of importance at low flow rates (i.e., streamline flow), and the second term refers to kinetic energy losses, of importance at high flow rates (i.e., turbulent flow).

Eisfeld and Schnitzlein⁶⁷ compared their measurements with predictions of 24 different pressure drop correlations from the literature, and they pointed out that Reichelt’s approach⁶⁸ of correcting the Ergun equation for the wall is the most promising one. Eisfeld and Schnitzlein⁶⁷ developed an improved correlation that accounted for the effect of the wall as follows:

(14a) $\psi =\frac{308{\mathrm{A}}_w^2}{{Re}_h}+\frac{2{\mathrm{A}}_w}{{\mathrm{B}}_w}\mathrm{f}\mathrm{o}\mathrm{r}{Re}_h\le 2\times {10}^4$(14a)

with the wall correction terms

(14b) ${\mathrm{A}}_w=\left[1+\frac{2}{3}\frac{\left({\mathrm{d}}_p/\mathrm{D}\right)}{(1-{\epsilon }_b)}\right]$(14b)

and

(14c) ${\mathrm{B}}_w={\left[1.15{\left(\frac{{\mathrm{d}}_p}{\mathrm{D}}\right)}^2+0.87\right]}^2.$(14c)

In fact, this is an Ergun-type equation where the contribution of confining walls to the hydraulic radius was accounted for analytically by the coefficient A_w. Additionally, the coefficient $B_w$ is introduced, describing empirically the porosity effect of the walls at the high Reynolds number.

The German Nuclear Safety Standard Commission [Kerntechnischer Ausschuss (KTA)] has considered and analyzed about 30 papers relevant to the results of the randomly packed bed with spherical particles.²⁶ The KTA adopted the following empirical correlation for the applications of the high-temperature packed pebble bed nuclear reactors⁶⁹:

(15) $\psi =\frac{320}{{Re}_h}+\frac{6}{{Re}_h^{0.1}}\mathrm{f}\mathrm{o}\mathrm{r}{Re}_h\le 5\times {10}^4.$(15)

The first term of Eq. (15)(15) $\psi =\frac{320}{{Re}_h}+\frac{6}{{Re}_h^{0.1}}\mathrm{f}\mathrm{o}\mathrm{r}{Re}_h\le 5\times {10}^4.$(15) represents the asymptotic solution for laminar flow while the second term represents the same for turbulent flow.

The Association of German Engineers [Verein Deutscher Ingenieure (VDI)] Heat Atlas provides the following correlation for the coefficient of loss of pressure through friction in fixed beds⁶¹:

(16) $\psi ={\left(\frac{0.4}{{\epsilon }_b}\right)}^{0.78}\frac{317}{{Re}_h}+\frac{6.17}{{Re}_h^{0.1}}.$(16)

Table III summarizes the most important correlations in packed bed reactors.

TABLE III Summary of Selected Correlations for Pressure Drop in Packed Bed Reactors*

Author	Correlation	Range
Ergun^Citation62	$\psi =\frac{150}{{Re}_h}+1.75$${Re}_h=\frac{\rho \mathrm{V}{\mathrm{d}}_h}{\mu }=\frac{1}{\left(1-{\epsilon }_b\right)}Re$	${Re}_h\le 5\times {10}^4$
Eisfeld and Schnitzlein^Citation67	$\psi =\frac{308{\mathrm{A}}_w^2}{{Re}_h}+\frac{2{\mathrm{A}}_w}{{\mathrm{B}}_w}$${\mathrm{A}}_w=\left[1+\frac{2}{3}\frac{\left({\mathrm{d}}_p/\mathrm{D}\right)}{(1-{\epsilon }_b)}\right]$${\mathrm{B}}_w={\left[1.15{\left(\frac{{\mathrm{d}}_p}{\mathrm{D}}\right)}^2+0.87\right]}^2$	${Re}_h\le 2\times {10}^4$
KTA Standards^Citation69	$\psi =\frac{320}{{Re}_h}+\frac{6}{{Re}_h^{0.1}}$	${Re}_h\le 5\times {10}^4$
Wirth^Citation61	$\psi ={\left(\frac{0.4}{{\epsilon }_b}\right)}^{0.78}\frac{317}{{Re}_h}+\frac{6.17}{{Re}_h^{0.1}}$	${Re}_h\le 5\times {10}^4$

*Abdulmohsin¹ and Abdulmohsin and Al-Dahhan.⁸⁰

Finally, it is very useful for modeling purposes to address here that the total pressure drop phenomenon within the flow due to the presence of the pebble bed can also be characterized by the dimensionless Euler number⁷⁰ as follows:

(17) $\mathrm{E}\mathrm{u}=\frac{\mathrm{\Delta }\mathrm{P}}{\rho {\mathrm{V}}^2}=\frac{1}{2}\psi \frac{\mathrm{L}}{{\mathrm{d}}_h}=\frac{1}{2}\psi \frac{\mathrm{L}}{{\mathrm{d}}_p}\left(\frac{1-\epsilon }{\epsilon }\right).$(17)

It can be interpreted as a measure of the ratio of pressure to inertial forces; a perfect frictionless flow corresponds to the Euler number of unity. Rousseau and Van Staden⁷⁰ also illustrated the relation between the Euler number and momentum transport via the momentum conservation equations for the axial and radial gas flow paths within the packed pebble bed nuclear reactor.

III.A. Comparison with the Predictions of the Empirical Correlations

In the work of Abdulmohsin¹ and Abdulmohsin and Al-Dahhan,^18,71 the predictions of the Ergun,⁶² KTA Standards,⁶⁹ Eisfeld and Schnitzlein,⁶⁷ and Wirth⁶¹ correlations were evaluated against the experimental results. As discussed in Sec. III, these corrections are the most promising and recommended ones for the prediction of the high-temperature packed pebble bed nuclear reactors.

The measured pressure drop along the packed pebble¹ and hence pressure drop coefficients $\psi$ are plotted against the effective Reynolds number Re_h, as shown in Fig. 5 (Ref. 1). In general, for Re_h > 10³, the friction factor decreases slightly with the Reynolds number of coolant gas, and its values range between 3 and 2. This is because with the increase in the coolant flow rate, the laminar sublayer size becomes so small that it is comparable with the surface roughness and the pressure losses are mainly produced by the protruding roughness. In other words, at the laminar-flow regime, the friction is highly affected while at high Reynolds number (turbulent-flow regime), the friction effect is less dominating.⁶⁷

Fig. 5. Coefficient of loss of pressure through friction ψ as a function of the effective Reynolds number (Re/(1 − ε) (Refs. 1 and 71).

Figure 5 also illustrates the predictions obtained by the aforementioned four different correlations and their comparisons with the obtained experimental results for the case of a uniform size spherical packed pebble bed of D/d_p = 6 and void fraction of about 0.397 (Abdulmohsin).¹ From Fig. 5, it has been observed that the measured pressure drop values agreed with the KTA and VDI correlations of average errors of about 1.79% and 2.81%, respectively. Hassan and Kang⁷² verified that the KTA correlation could be used for a gas-cooled PBR. The comparison between their experiment results and the KTA correlations showed that the pressure drop of large bed–to–particle diameter ratios, D/dp = 19, 9.5, and 6.33, matched very well with the original KTA correlation. However, the authors claimed that the published KTA correlations cannot be expected to predict accurate pressure drop for certain conditions, especially for pebble beds of very low aspect ratio, D/dp < 5.

From Fig. 5, it has been observed that the dimensionless pressure drop $\psi$ is proportional to the reciprocal of effective Reynolds number Re_h in the laminar-flow regime and becomes independent of Re_h at higher values for both Ergun-type equations, i.e., Refs. 62 and 67. Contrary to the prediction of other correlations, i.e., KTA (Ref. 69) and VDI (Ref. 61), the dependence of the pressure drop coefficient on the Reynolds number changes gradually with the increasing Reynolds number, indicating a smooth transition from laminar- to turbulent-flow regimes.

The empirical correlation of Eisfeld and Schnitzlein⁶⁷ overpredicts the pressure drops within the range of 300 > Re_h > 1500. The deviation from the measured pressure drops varies dramatically from acceptable average error (9.4%) for low effective Reynolds number (Re_h < 300) to well prediction average error (3.3%) for intermediate Reynolds number (300 < Re_h < 1500) to considerable average error (19%) for high Reynolds number (Re_h > 1500). Although Eisfeld and Schnitzlein⁶⁷ made an improved correlation that accounts for the wall effects where they manipulated the coefficients of the wall correction factor for the inertial pressure loss term, their correlation cannot properly predict the pressure drop coefficients and the trend for certain conditions. This is because the Eisfeld and Schnitzlein⁶⁷ wall correction factor for the inertial pressure loss term does not come from physical reasoning, and it is based on the curve-fitting model.⁷² Although Ergun’s correlation was proven to be valid for most of the gas-solid applications in the chemical industry, such as chemical/catalytic packed bed reactors, the pressure drop across the core of PBR is overpredicted (underpredicted in terms of dimensionless pressure drop coefficient $\psi$) by this correlation, as shown in Fig. 5. The deviations from the measured pressure drops vary considerably from an average error of about 48.51% for low effective Reynolds number (Re_h < 1000) to an average error of about 35.69% for high Reynolds number (Re_h > 1000).

However, early pressure drop studies through PBRs (Refs. 72 through 75) have reported that the Ergun equation considerably overpredicts the pressure drop in the high Reynolds number range of practical interest. This is because of the mass flow rates, static pressure, and particle diameter, and hence, the Reynolds numbers in chemical industrial applications are relatively small compared to those used in packed pebble bed nuclear reactors. In addition to that, Ergun’s correlation was based on the model assuming the packing is statistically uniform, so that there are no channeling or bypassing effects (in an actual situation, channeling would occur). Therefore, Ergun’s correlation does not predict very well for randomly packed pebble bed nuclear reactors. Hence, the obtained experimental results demonstrate the applicability of the VDI and KTA correlations for randomly packed PBRs.

n a recent study, Reger et al.⁷⁶ investigated the accuracy of the KTA correlation compared to the high-fidelity large eddy simulation (LES) in a bed packed with 1568 pebbles and an aspect ratio D_bed/D_p of 13 operating at Reynolds numbers between 625 and 10 000. The bed was packed using the Discrete Element Method (DEM), and it was split into five concentric subdomains to compare the average velocity, friction losses, and form losses between the KTA correlation and the LES code. The NekRS spectral element CFD code was utilized to create high-fidelity simulation results while the Pronghorn porous media code was used to obtain comparative intermediate-fidelity results utilizing the KTA correlation.

The study revealed that the KTA correlation significantly underpredicts the form losses in the near-wall region and overpredicts the form losses in the bulk of the bed, which results in an overprediction of the velocity near the wall by about 30%. The authors reported that an accurate description of porosity in a porous media model does not result in an accurate prediction of velocity near the wall. To produce accurate velocity profiles, the pressure drop correlation near the wall must be modified to account for the variation in flow patterns. Based on a detailed analysis of the results from this study, the authors suggested modifying the inertial form loss term in the KTA correlation from $\frac{6}{{\mathrm{R}\mathrm{e}}_m^{0.1}}$ to $\frac{8.9}{{\mathrm{R}\mathrm{e}}_m^{0.1}}$ for the near-wall region and from $\frac{6}{{\mathrm{R}\mathrm{e}}_m^{0.1}}$ to $\frac{5.1}{{\mathrm{R}\mathrm{e}}_m^{0.1}}$ for the bulk region. The results also showed that the KTA correlation tends to overpredict the friction factor compared to NekRS. However, the pressure drop from friction is very small compared to the form losses since these studies were performed at high Reynolds number. Thus, the friction factor was not modified for this study.

The study shows that use of these suggested modifications for the near-wall and bulk regions in the KTA correlation displayed significantly improved agreement with the NekRS simulation results. In order to provide a second case for comparison, a second bed of 1700 pebbles was also simulated in NekRS and Pronghorn. The suggested modifications in the KTA correlation resulted in an improvement over the original KTA correlation; however, the agreement was not as good as what was observed in the 1568-pebble scenario. Therefore, this study should be expanded to include different sizes of packed beds and low Reynolds number flows to develop a more generalized correlation for pressure drop. A thorough description of the governing equations; the relations used for material, heat transfer, and fluid flow parameters; and the numerical method utilized in Pronghorn can be found in the “Pronghorn Theory Manual.”⁷⁷ A validation study of the Pronghorn porous media model with pressure drop measurements has been done by Lee et al.⁷⁸

IV. GAS DISPERSION AND MIXING PHENOMENA OF PACKED BED REACTORS

Dispersion is a well-known phenomenon in porous media primarily for heat and mass transfer processes. The dispersion coefficient is a property valid only under continuum assumptions. This is similar to viscosity in momentum transfer, heat conductivity in heat transfer, and the diffusion coefficient in mass transfer. The axial dispersion phenomenon in a pebble bed is a consequence of the combined contributions of both the molecular diffusion and the hydrodynamic mixing (convection) mechanisms in the spaces between the pebbles along the length of the pebble bed. At the macroscopic level, the individual contribution of each mechanism to the overall dispersion phenomenon depends mainly on the gas flow conditions and bed structure. Typically, the axial dispersion and degree of mixing in the packed bed are characterized and quantified in terms of axial dispersion coefficients and dispersive Peclet numbers, respectively.

It is well known that the phenomenon of axial dispersion is indicated by the spread of residence times of the individual elements of a fluid stream passing through a packed bed. Even if it is possible theoretically in unpacked tubular reactors to quantify deviations from an ideal plug flow model by measuring fluid velocities to obtain a complete velocity distribution profile, this approach is never used in packed PBRs because it is physically impossible to realize it in practice. Therefore, simple knowledge of the residence time distribution (RTD) is necessary. The RTD can be obtained by studying the response of the system to a tracer impulse. Different approaches are available in the literature to obtain the parameters from the RTD (Ref. 79).

The main problem with the RTD method comes from possible interactions between process dynamical behavior and the dynamics of the sensor. As a result, the obtained measurements are the time convolution of the desired phenomenon and of an unexpected one. From a mathematical point of view, the time response of the sensors cannot be subtracted from the RTD since these are two dynamical systems in series. Therefore, convolution and deconvolution integral methods are used to analyze the RTD.

No detailed experimental measurements, knowledge, and quantification of coolant gas dispersion and its extent of mixing for pebble bed nuclear reactors were available prior to the work of Abdulmohsin¹ and Abdulmohsin and Al-Dahhan.^22,23,80 However, there were studies reported in the literature related to the dispersion of the gas and liquid phases and their mixing in the chemical/catalytical packed bed reactor of smaller particles (1 to 3 mm in diameter) (Refs. 81 through 90). Delgado⁹¹ summarized and reviewed the literature on the phenomenon of dispersion (longitudinal and transverse) in packed beds. The author stated that there are several variables that need to be considered in the analysis of the dispersion in packed beds, such as the length of the packed bed, viscosity and density of the fluid, ratio of the column diameter to the particle diameter (aspect ratio), ratio of the column length to the particle diameter, particle size distribution, particle shape, velocity of the fluids, and operating temperature.

IV.A. Available Empirical Correlations for Axial Gas Dispersion Coefficients

Despite this large number of studies, the correlations reported in the literature for predicting the axial gas dispersion coefficient in packed beds of large particles are still not reliable. There are some correlations that predict the axial gas dispersion coefficient of chemical/catalytic packed bed systems of small particles in terms of dispersive Peclet numbers, as summarized in Table IV.

TABLE IV Summary of Selected Correlations for Axial Gas Dispersion in Chemical Packed Bed Reactors*

Author	Correlation	Range
Gunn^Citation87,Citation92,Citation94	$\begin{array}{rl}& \frac{1}{\mathrm{P}{\mathrm{e}}_D}=\left[\frac{\mathrm{P}{\mathrm{e}}_M}{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}\right]{\left(1-\mathrm{p}\right)}^2+{\left[\frac{\mathrm{P}{\mathrm{e}}_M}{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}\right]}^2\\ & p{\left(1-p\right)}^3\left\{\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}{p\left(1-p\right)\mathrm{P}{\mathrm{e}}_M}\right]-1\right\}+\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}\end{array}$$p=0.17+0.33\times \mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{24}{\mathrm{R}\mathrm{e}}\right)$	$\begin{array}{rl}& 0.4\le Re\le 420\\ & \mathrm{S}\mathrm{c}=0.88\\ & 0.368\le \epsilon \le 0.37\end{array}$
Bischoff and Levenspiel^Citation105	$\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{{\epsilon }_b/\tau }{\mathrm{P}{\mathrm{e}}_M}+\frac{0.45}{\left[1+7.3{\left(\mathrm{P}{\mathrm{e}}_M\right)}^{-1}\right]}$	$\begin{array}{rl}& 0.05\le Re\le 44000\\ & \epsilon =0.40\end{array}$
Edwards and Richardson^Citation86	$\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{0.73}{\mathrm{P}{\mathrm{e}}_M}+\frac{0.5}{\left[1+9.7{\left(\mathrm{P}{\mathrm{e}}_M\right)}^{-1}\right]}$	$\begin{array}{rl}& 0.008\le Re\le 50\\ & \mathrm{S}\mathrm{c}=0.72\\ & 0.368\le \epsilon \le 0.41\end{array}$
Wen and Fan^Citation106	$\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{0.3}{\mathrm{P}{\mathrm{e}}_M}+\frac{0.5}{\left[1+0.38{\left(\mathrm{P}{\mathrm{e}}_M\right)}^{-1}\right]}$	0.008 < Re < 400 0.28 < Sc < 2.2
Wakao and Kaguei^Citation108	$\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{0.7}{\mathrm{P}{\mathrm{e}}_M}+\frac{1}{2}$	$Re\ge 5$
Gunn^Citation99	$\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}+\frac{1}{2}$	$\begin{array}{rl}& 0.4\le Re\le 420\\ & \mathrm{S}\mathrm{c}=0.88\\ & 0.368\le \epsilon \le 0.37\end{array}$
Tsotsas and Schlünder^Citation90	$\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{0.3}{\mathrm{P}{\mathrm{e}}_M}+\frac{1}{1.14\left[1+10{\left(\mathrm{P}{\mathrm{e}}_M\right)}^{-1}\right]}$	$\begin{array}{rl}& 0.05\le Re\le 44000\\ & \epsilon =0.40\end{array}$
Guedes de Carvalho and Delgado^Citation85	$\begin{array}{c}\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{\mathrm{P}{\mathrm{e}}_M}{5}{\left(1-p\right)}^2+\frac{{\mathrm{P}\mathrm{e}}_M^2}{25}p{\left(1-p\right)}^3\\ \left\{\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{5}{p\left(1-p\right)\mathrm{P}{\mathrm{e}}_M}\right]-1\right\}+\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}\end{array}$$p=\frac{0.48}{\mathrm{S}{c}^{0.15}}+\left(\frac{1}{2}-\frac{0.48}{\mathrm{S}{c}^{0.15}}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{75\mathrm{S}\mathrm{c}}{\mathrm{P}{\mathrm{e}}_M}\right)$	$\begin{array}{rl}& 0.02\le Re\le 89.1\\ & 57\le \mathrm{S}\mathrm{c}\le 1938\\ & 0.37\le \epsilon \le 0.385\end{array}$

*Abdulmohsin¹ and Abdulmohsin and Al-Dahhan.⁸⁰

Early attempts to correlate and predict the dispersion coefficients in a packed bed of smaller particles were performed by Gunn and Pryce⁹² and Gunn⁸⁷ using different approaches.⁹³ Gunn described dispersion in a randomly packed bed as a stochastic process, and the author also used the probability theory to incorporate both diffusion and mixing effects. The early analysis of Gunn⁸⁷ of the tracer motion led to the following expression for the dispersive Peclet number:

(18) $\begin{array}{c}\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\left[\frac{\mathrm{P}{\mathrm{e}}_M}{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}\right]{\left(1-\mathrm{p}\right)}^2+{\left[\frac{\mathrm{P}{\mathrm{e}}_M}{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}\right]}^2\mathrm{p}\left(1-\mathrm{p}\right){}^3\\ \left\{\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}{\mathrm{p}\left(1-\mathrm{p}\right)\mathrm{P}{\mathrm{e}}_M}\right]-1\right\}+\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}.(18)\end{array}$(18)

The dimensionless groups are given by the following:

(18a) $\left.\begin{array}{c}\mathrm{P}{\mathrm{e}}_D=\frac{{\mathrm{V}}_g{\mathrm{d}}_p}{{\epsilon }_b{\mathrm{D}}_{ax}}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{P}\mathrm{e}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\\ \mathrm{P}{\mathrm{e}}_M=\frac{{\mathrm{V}}_g{\mathrm{d}}_p}{{\epsilon }_b{\mathrm{D}}_{\mathrm{A}\mathrm{B}}}=\mathrm{R}{\mathrm{e}}_P\mathrm{S}\mathrm{c}\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}(\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s})\mathrm{P}\mathrm{e}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\\ \mathrm{R}{\mathrm{e}}_P=\frac{\rho {\mathrm{V}}_g{\mathrm{d}}_p}{{\epsilon }_b\mu }=\frac{\mathrm{R}\mathrm{e}}{{\epsilon }_b}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{R}\mathrm{e}\mathrm{y}\mathrm{n}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\\ \mathrm{S}\mathrm{c}=\frac{\mu }{\rho {\mathrm{D}}_{AB}}\mathrm{S}\mathrm{c}\mathrm{h}\mathrm{m}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\end{array}\right\}.$(18a)

In Eqs. (18)(18) $\begin{array}{c}\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\left[\frac{\mathrm{P}{\mathrm{e}}_M}{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}\right]{\left(1-\mathrm{p}\right)}^2+{\left[\frac{\mathrm{P}{\mathrm{e}}_M}{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}\right]}^2\mathrm{p}\left(1-\mathrm{p}\right){}^3\\ \left\{\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}{\mathrm{p}\left(1-\mathrm{p}\right)\mathrm{P}{\mathrm{e}}_M}\right]-1\right\}+\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}.(18)\end{array}$(18) and (18a(18a) $\left.\begin{array}{c}\mathrm{P}{\mathrm{e}}_D=\frac{{\mathrm{V}}_g{\mathrm{d}}_p}{{\epsilon }_b{\mathrm{D}}_{ax}}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{P}\mathrm{e}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\\ \mathrm{P}{\mathrm{e}}_M=\frac{{\mathrm{V}}_g{\mathrm{d}}_p}{{\epsilon }_b{\mathrm{D}}_{\mathrm{A}\mathrm{B}}}=\mathrm{R}{\mathrm{e}}_P\mathrm{S}\mathrm{c}\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}(\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s})\mathrm{P}\mathrm{e}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\\ \mathrm{R}{\mathrm{e}}_P=\frac{\rho {\mathrm{V}}_g{\mathrm{d}}_p}{{\epsilon }_b\mu }=\frac{\mathrm{R}\mathrm{e}}{{\epsilon }_b}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{R}\mathrm{e}\mathrm{y}\mathrm{n}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\\ \mathrm{S}\mathrm{c}=\frac{\mu }{\rho {\mathrm{D}}_{AB}}\mathrm{S}\mathrm{c}\mathrm{h}\mathrm{m}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\end{array}\right\}.$(18a) ), D_ax is the axial gas dispersion coefficient, D_AB is the molecular diffusion coefficient, ${\alpha }_1$ is the first root of the first-order Bessel function, and p is the fluid mechanical probability. According to the hypothesis of Gunn,⁸⁷ p is a function of only the Reynolds number (Re = ρV_gd_p/µ). Hence, later, Gunn⁸⁷ proposed a correlation for p as outlined in the following discussion.

Gunn⁹⁴ proposed two limits for the dispersive Peclet number Pe_D by expanding the exponential term in Eq. (18)(18) $\begin{array}{c}\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\left[\frac{\mathrm{P}{\mathrm{e}}_M}{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}\right]{\left(1-\mathrm{p}\right)}^2+{\left[\frac{\mathrm{P}{\mathrm{e}}_M}{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}\right]}^2\mathrm{p}\left(1-\mathrm{p}\right){}^3\\ \left\{\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}{\mathrm{p}\left(1-\mathrm{p}\right)\mathrm{P}{\mathrm{e}}_M}\right]-1\right\}+\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}.(18)\end{array}$(18) based on the value of the product of the particle Reynolds number; the Schmidt number, which is called the molecular (mass); and the Peclet number Pe_M. These two limits are the following.

For small values of Pe_M,

(19) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M};$(19)

i.e., the dispersive Peclet number is due to molecular diffusion alone.

For large values of Pe_M,

(20) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{1-p}{2p};$(20)

i.e., the dispersive Peclet number is due to convection alone.

In Eq. (18(18) $\begin{array}{c}\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\left[\frac{\mathrm{P}{\mathrm{e}}_M}{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}\right]{\left(1-\mathrm{p}\right)}^2+{\left[\frac{\mathrm{P}{\mathrm{e}}_M}{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}\right]}^2\mathrm{p}\left(1-\mathrm{p}\right){}^3\\ \left\{\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}{\mathrm{p}\left(1-\mathrm{p}\right)\mathrm{P}{\mathrm{e}}_M}\right]-1\right\}+\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}.(18)\end{array}$(18) ), $\tau$ is the tortuosity factor, which is defined as the ratio that compares the average length of the actual fluid flow paths through the packing to the packing heights.⁹⁵ This parameter was originally introduced to account for the sinuosity of the flow paths in the permeability model.⁹⁶ The tortuosity factor is also lumped with both tortuous zigzag flow paths and constricted points and can be approximated as $\tau \approx \sqrt{2}$ for a packed bed of spherical particles^96–98 correlating the tortuosity factor to readily measure porosity through this simple relation:

(21a) $\tau =\sqrt{1-ln{\left({\epsilon }_b\right)}^2}.$(21a)

Lanfrey et al.⁹⁵ developed a theoretical model for the tortuosity of a fixed bed randomly packed with identical spherical particles. They found that the tortuosity was proportional to a packing structure factor, which could well capture the balancing effect between porosity and particle sphericity, as follows:

(21b) $\tau =\frac{{\epsilon }_b}{\left[{\left(1-{\epsilon }_b\right)}^{4/3}\right]}.$(21b)

As porosity decreased, the tortuosity increased, and it did not depend on the particle size.

Gunn⁹⁹ proposed a correlation for the fluid mechanical probability p needed for Eq. (18)(18) $\begin{array}{c}\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\left[\frac{\mathrm{P}{\mathrm{e}}_M}{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}\right]{\left(1-\mathrm{p}\right)}^2+{\left[\frac{\mathrm{P}{\mathrm{e}}_M}{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}\right]}^2\mathrm{p}\left(1-\mathrm{p}\right){}^3\\ \left\{\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}{\mathrm{p}\left(1-\mathrm{p}\right)\mathrm{P}{\mathrm{e}}_M}\right]-1\right\}+\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}.(18)\end{array}$(18) as a function of Re for packing of spherical particles as follows:

(22) $p=0.17+0.33\times \mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{24}{\mathrm{R}\mathrm{e}}\right),\mathrm{R}\mathrm{e}=\frac{\rho {\mathrm{V}}_g{\mathrm{d}}_p}{\mu }.$(22)

Equation (22)(22) $p=0.17+0.33\times \mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{24}{\mathrm{R}\mathrm{e}}\right),\mathrm{R}\mathrm{e}=\frac{\rho {\mathrm{V}}_g{\mathrm{d}}_p}{\mu }.$(22) suggests that p should have the value of 0.5 for $\mathrm{R}\mathrm{e}\to \mathrm{\infty }$.

Gunn⁹⁹ also proposed another simplified correlation for Pe_D by assuming that diffusive and mixing components of dispersion are additive and rewrote Eq. (18)(18) $\begin{array}{c}\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\left[\frac{\mathrm{P}{\mathrm{e}}_M}{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}\right]{\left(1-\mathrm{p}\right)}^2+{\left[\frac{\mathrm{P}{\mathrm{e}}_M}{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}\right]}^2\mathrm{p}\left(1-\mathrm{p}\right){}^3\\ \left\{\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{4{\alpha }_1^2\left(1-{\epsilon }_b\right)}{\mathrm{p}\left(1-\mathrm{p}\right)\mathrm{P}{\mathrm{e}}_M}\right]-1\right\}+\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}.(18)\end{array}$(18) in the following form:

(23) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}+\frac{1}{2}.$(23)

Delgado⁹¹ evaluated Gunn’s correlation, Eq. (23(23) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}+\frac{1}{2}.$(23) ), with available experimental data, as shown in Fig. 6, and he pointed out that the experimental values of the dispersive Peclet number are generally higher than predicted by Eq. (23)(23) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}+\frac{1}{2}.$(23) . Delgado⁹¹ also pointed out that Eq. (23)(23) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}+\frac{1}{2}.$(23) is inaccurate over part of the intermediate range of Pe_M and that there are significant deviations observed only in the range from 0.6 < Pe_M < 60. It is important to state here that Fig. 6 shows that for low values of Pe_M (creeping flow regime), there seems to be a tendency for Pe_D to become independent of Sc. It has been reported that several correlations^86,100–104 have been proposed to represent the data reasonably in this intermediate range (see Fig. 6).

Fig. 6. Some experimental data points for axial dispersion in gaseous systems (Delgado⁹¹), where Pe_m = Pe_M.

Bischoff and Levenspiel¹⁰⁵ developed this semiempirical correlation for dispersion in a packed bed as

(24) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{{\epsilon }_b/\tau }{\mathrm{P}{\mathrm{e}}_M}+\frac{0.45}{\left[1+7.3{\left(\mathrm{P}{\mathrm{e}}_M\right)}^{-1}\right]}.$(24)

Edwards and Richardson⁸⁶ proposed an empirical correlation for axial dispersion of gases flowing through a fixed bed of small particles expressed as the following:

(25) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}+\frac{0.5}{\left[1+\left(\beta /\mathrm{P}{\mathrm{e}}_M\right)\right]}.$(25)

The term $[1+(\beta /\mathrm{P}{\mathrm{e}}_M)]$ on the right-hand side of Eq. (25)(25) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}+\frac{0.5}{\left[1+\left(\beta /\mathrm{P}{\mathrm{e}}_M\right)\right]}.$(25) is an empirical correction factor that takes into account that the radial (transverse) dispersion might take place at a low Reynolds number that reduces the axial (longitudinal) dispersion as introduced by the authors. $\beta$ is a constant, and it increases as the diffusivity of gas D_AB increases.

The best fit of their experimental results was obtained with a value of 9.7 for $\beta$ and using the value of approximately $\sqrt{1.87}$ for $\tau$. Equation (25)(25) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}+\frac{0.5}{\left[1+\left(\beta /\mathrm{P}{\mathrm{e}}_M\right)\right]}.$(25) then becomes the following:

(26) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{0.73}{\mathrm{P}{\mathrm{e}}_M}+\frac{0.5}{\left[1+9.7{\left(\mathrm{P}{\mathrm{e}}_M\right)}^{-1}\right]}.$(26)

Wen and Fan¹⁰⁶ and Tsotsas and Schlünder⁹⁰ deduced alternative correlations for the prediction of the dispersive Peclet number Pe_D of gas flowing in packed beds of spherical particles as follows.

The correlation by Wen and Fan¹⁰⁶ is expressed as follows:

(27) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{0.3}{\mathrm{P}{\mathrm{e}}_M}+\frac{0.5}{\left[1+0.38{\left(\mathrm{P}{\mathrm{e}}_M\right)}^{-1}\right]}.$(27)

The correlation by Tsotsas and Schlünder⁹⁰ is expressed as

(28a) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{\gamma }{\mathrm{P}{\mathrm{e}}_M}+\frac{1}{1.14\left[1+10{\left(\mathrm{P}{\mathrm{e}}_M\right)}^{-1}\right]}.$(28a)

The quantity $\gamma$ is a function of bed porosity and can be approximated empirically by the following¹⁰⁷:

(28b) $\gamma =\frac{{\epsilon }_b}{\tau }=1-{\left(1-{\epsilon }_b\right)}^{1/2}.$(28b)

Tsotsas and Martin used $\gamma \approx 0.3$ for ${\epsilon }_b=0.4$.

Wakao and Kaguei¹⁰⁸ gave an overview of the different experimental data and proposed the following correlation for axial dispersion in the packed bed of spherical particles:

(29) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{0.7}{\mathrm{P}{\mathrm{e}}_M}+\frac{1}{2}.$(29)

Guedes de Carvalho and Delgado⁸⁵ and Delgado⁹¹ developed a mathematical expression that would represent their experimental data with good accuracy for the longitudinal dispersion in a chemical packed bed as the following:

(30a) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{\mathrm{P}{\mathrm{e}}_M}{5}{\left(1-\mathrm{p}\right)}^2+\frac{{\mathrm{P}\mathrm{e}}_M^2}{25}\mathrm{p}{\left(1-\mathrm{p}\right)}^3\left\{\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{5}{\mathrm{p}\left(1-\mathrm{p}\right)\mathrm{P}{\mathrm{e}}_M}\right]-1\right\}+\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}$(30a)

with

(30b) $p=\frac{0.48}{\mathrm{S}\mathrm{c}{\hspace{0.17em}}^{0.15}}+\left(\frac{1}{2}-\frac{0.48}{\mathrm{S}\mathrm{c}{\hspace{0.17em}}^{0.15}}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{75\mathrm{S}\mathrm{c}}{\mathrm{P}{\mathrm{e}}_M}\right).$(30b)

It is important to bear in mind that Eq. (30a)(30a) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\frac{\mathrm{P}{\mathrm{e}}_M}{5}{\left(1-\mathrm{p}\right)}^2+\frac{{\mathrm{P}\mathrm{e}}_M^2}{25}\mathrm{p}{\left(1-\mathrm{p}\right)}^3\left\{\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{5}{\mathrm{p}\left(1-\mathrm{p}\right)\mathrm{P}{\mathrm{e}}_M}\right]-1\right\}+\frac{1}{\tau }\frac{1}{\mathrm{P}{\mathrm{e}}_M}$(30a) is recommended only for random packings of spherical particles that are well packed⁹¹ and that it covers a wide range of values of Pe_M and Sc.

It is clear from the above correlations that dispersive Peclet numbers Pe_D for gases flowing through packed beds depend on the variations in the molecular Peclet numbers Pe_M and, hence, on the Schmidt number. Under extremely low flow rate conditions (creeping flow regimes) of coolant gas, there are no reliable measurements because of experimental difficulties, and the dispersion phenomenon is related to the pure molecular diffusion mechanism. In other words, at the limit $\mathrm{P}{\mathrm{e}}_M\to 0$, axial dispersion takes place by molecular diffusion alone. At high flow rate conditions (turbulent-flow regimes), dispersion occurs purely by turbulent mixing, and it is obvious that upon increasing the velocity of the gas, the dispersive Peclet number tends to reach the limiting value of about 2. This value can be estimated theoretically using the equivalence of a packed bed (at $\mathrm{P}{\mathrm{e}}_D\to \mathrm{\infty }$) with a series of perfect mixers.⁹⁰

IV.B. Comparison with Empirical Correlations of Axial Gas Dispersion Coefficients

As mentioned earlier, before the work of Abdulmohsin¹ and Abdulmohsin and Al-Dahhan,⁸⁰ there were no detailed experimental measurements, knowledge and quantification of the gas phase dynamics, and its extent of dispersion and mixing for PBRs. However, there are a large number of studies reported in the literature related to the dispersion of the gas phase in the chemical packed bed reactor of smaller particles (1- to 3-mm diameter). Despite this large number of studies, the correlations reported in the literature for predictions of the axial gas dispersion in gas-solid packed beds are very few, as they were reported in Sec. IV.A, in Table IV. Three different types of correlations developed by Edwards and Richardson,⁸⁶ Gunn,⁹⁴ and Guedes de Carvalho and Delgado⁸⁵ were selected and used for the comparison in this work.

Although these correlations were based on experimental data for upward flow of gas and have been developed for small particles used as catalysts in chemical packed bed reactors, they are evaluated in this work for packed pebble beds of pebbles with diameters of 1.25, 2.5, and 5 cm as a first attempt.

In the work of Abdulmohsin¹ and Abdulmohsin and Al-Dahhan,⁸⁰ the theoretical model developed by Lanfrey et al. has been used, recalling Eq. (21b(21b) $\tau =\frac{{\epsilon }_b}{\left[{\left(1-{\epsilon }_b\right)}^{4/3}\right]}.$(21b) ), to calculate the tortuosity of a fixed bed randomly packed with identical spherical particles as

(31a) $\tau =\frac{{\epsilon }_b}{\left[{\left(1-{\epsilon }_b\right)}^{4/3}\right]}.$(31a)

Based on the average absolute relative error (AARE), the predictions of the correlations were assessed against the experimental data. AARE between the measured and the predicated Peclet numbers is expressed as

(31b) $\mathrm{A}\mathrm{A}\mathrm{R}\mathrm{E}=\frac{1}{\mathrm{N}}\sum _1^{\mathrm{N}}\left|\frac{\mathrm{P}{\mathrm{e}}_{pred\left(i\right)}-\mathrm{P}{\mathrm{e}}_{exptl\left(i\right)}}{\mathrm{P}{\mathrm{e}}_{exptl\left(i\right)}}\right|,$(31b)

where N is the number of the data points.

Figures 7 and 8 show values of gas phase dispersion in terms of dispersive Peclet numbers Pe_D and dispersion numbers (reciprocal of Peclet numbers) with respect to molecular Peclet numbers Pe_M and particle Reynolds number Re_P, respectively. Abdulmohsin¹ and Abdulmohsin and Al-Dahhan⁸⁰ have compared their obtained experimental data with those predicted by the selected correlations of Edwards and Richardson,⁸⁶ Gunn,⁹⁹ and Guedes de Carvalho and Delgado.⁸⁵ The correlation developed by Gunn⁹⁹ seems to provide a good prediction at both low and high superficial gas velocities where the value of AARE is about 2.2%. The prediction based on the Edwards and Richardson⁸⁶correlation is shown in Figs. 7 and 8. At low superficial gas velocities, the trends and the values do not match well, while the prediction of the correlation is better at high superficial gas velocities with AARE of about 1.1%. However, at low superficial gas velocities, there is a relatively larger deviation in the prediction, but it is still acceptable (AARE is about 16.7%). In the predictions of the Guedes de Carvalho and Delgado⁸⁵ correlation, the trends and the values do not match well for both low and high gas flow conditions. This can be attributed to uncertainties in different measurement techniques used and different operating and design conditions used in the development of correlations,⁸⁵ such as particle size, tracer type of experiment design, etc. In the work of Abdulmohsin¹ and Abdulmohsin and Al-Dahhan,⁸⁰ pebble diameters (1.25, 2.5, and 5 cm) have been used that yield higher values of the average bed porosity besides the high molecular diffusivity of helium gas in air of about 0.65 cm²/s, which leads to a low value of the Schmidt number (Sc ~ 0.24). However, the trend of the measured axial dispersion number is still qualitatively similar to the experimental findings of the dispersion of the gas phase flowing in fixed beds.^105,109 The results indicate that more investigation of mechanisms that govern the axial dispersion coefficient and the wide range of data at various relevant conditions in the pebble bed would be needed to further improve the predictions of the axial dispersion coefficients.

Fig. 7. Comparison of the measured dispersive Peclet number Pe_D with those estimated by empirical correlations.^1,80

Fig. 8. Comparison of the measured dispersion number 1/Pe_D with those estimated by empirical correlations.^1,80

V. HEAT TRANSFER CHARACTERISTICS OF PACKED PBR

Heat transport in packed pebble beds is an extremely complex phenomenon where the contributions of the three modes of conduction, convection, and thermal radiation need to be accounted for. Moreover, the heat transfer modes might interact with each another. Therefore, the phrase “packed pebble bed heat transfer” is used to describe a variety of mechanisms where the following might occur¹:

1. Heat conduction through the solid pebble itself from one side of the pebble through to the other side.

2. Forced convection heat transfer due to the bulk flow and turbulent mixing of the coolant gas.

3. Conduction heat transfer through the point of physical contact between the individual pebbles in the bed. This mode can be further subdivided into the axial and radial directions that refer to the radial pebble-to-pebble conduction and axial pebble-to-pebble conduction, respectively.

4. Heat transfer by conduction across the stagnant gas surrounding the point of contact between pebbles.

5. Thermal radiation heat transfer between the surfaces of adjacent pebbles within the pebble bed.

6. Forced convection heat transfer from the hot pebbles to the coolant gas flowing through the bed, sometimes is referred to as the pebble-coolant heat transfer mode. In packed PBRs, at normal operating conditions of elevated temperatures, this mode will be an important process.

7. Radiation absorption by the coolant gas.

8. Heat transfer by natural convection in the coolant gas, this mode will be dominant at extremely low flow rates, which are the case when an accident occurs within the reactor.

All these modes of heat transport phenomena are illustrated schematically in Fig. 9. In the normal operation of the nuclear PBR, two or more of the modes listed above might take place simultaneously in parallel or in series. It is obvious that the heat transport mechanism in packed pebble beds is extremely dependent on the fluid flow distribution in the core. As a result, the foregoing list involves the following, broadly¹¹⁰: first, mechanisms that are independent of fluid flow¹¹¹ (numbers 1, 3, and 4 above) and second, mechanisms that depend on fluid flow (numbers 2, 5, 6, 7, and 8 above).

Fig. 9. Schematic sketch of the heat transfer modes in the core of packed PBRs (Ref. 111).

V.A. Conductive and Radiative Heat Transfer in PBR

Thermal radiation is an important part of heat transfer in pebble bed HTGR nuclear reactors. Particle radiation is a rather complicated issue that should be treated carefully. Compared to conduction and convection, the radiation of particles increases significantly with temperature, and it has long-range impacts that are highly reliant on the regional structure and material characteristics of the bed and packed particles. Moreover, researchers have discovered that when a particle’s solid conductivity is significantly higher than its effective thermal conductivity (ETC), the particle’s surface temperature distribution will be uniform, and ETC is directly proportional to the third power of the average temperature in kelvins (Refs. 112, 113, and 114).

Numerous numerical models have been developed to simulate particle radiation in large-scale granular systems, including the subcell radiation model,¹¹⁵ boundary element method,¹¹⁶ Monte Carlo method,^117,118 and empirical correlations based on experimental data.^119–121

The short-range radiation model¹²² (SRM) and the local average model,¹²³ which are used in CFD-DEM simulations, are effective methods for estimating particle radiation flux. However, in these models, only the radiation between particles at partial integral scales is considered. This means that in these numerical radiation models, only the nearby particles at a distance of one particle diameter^124,125 or only a single neighboring Voronoi cell are taken into account. In contrast, the long-range model takes into account all possible radiation between surrounding spheres, even those that are not in direct contact. Reference 126 performed a three-dimensional (3D) simulation study to investigate thermal radiation heat exchange in a 3D packed pebble bed. In this study, the results of the long-range and the short-range models were analyzed and compared with reference to the existing correlations. The study concluded that the SRM is not suited for estimating high-temperature particle radiation because it overestimates the ETC, radiation exchange factor, and surface emissivity at high temperatures. On the other hand, it was shown that the long-range radiation model was a reliable method for predicting radiative heat exchange when compared to the existing correlations. Wu et al.¹²² developed a comprehensive CFD-DEM simulation that considers particle radiation, convection and conduction heat, fluid flow, and pebble-fluid interactions under several situations with various initial conditions or physical properties and was carried out in order to investigate how particle thermal radiation affects the flow and heat transport processes in packed pebble beds. The SRM was used in the particle radiation model. Additionally, HTR-10’s performance at steady-state full power and under decay heat removal was examined.

The findings demonstrated that particle thermal radiation dramatically rises at high temperatures. The particle temperature field in a packed pebble bed tends to be considerably more uniform, and radiation significantly improves heat transfer.

Moreover, the particle radiation factor was found to decrease gradually as the fluid’s thermal conductivity or heat storage capacity increased. Furthermore, even while other fluid characteristics that depend on different temperatures may also affect particle radiation in HTGRs via the Reynolds number or Prandtl number, they are not as significant as the fluid density and thermal conductivity. The ETC of the radiation results for the HTR-10 benchmark study obtained with a SRM and the existing empirical and semiempirical correlations were in good agreement as shown in Fig. 10. Additionally, the results indicate the power needed by the fans to drive the convective flow will grow dramatically when the reactor is operated above full power, and it may be difficult to assure the nuclear safety of the control systems. Particle radiation is crucial in removing the decay heat from HTR-10 and keeping the temperature below the permitted upper limit.

Fig. 10. The ETC for radiation of the equivalent HTR-10 core as a function of temperature.¹²²

The models of multiple-body radiation were examined and compared on full integral scales in an analysis and evaluation study within a pebble bed using CFD-DEM simulation backed by experimental data.¹²⁷ The study showed that for particle radiation with black surfaces, the black radiation model was shown to be strictly valid and to coincide with existing correlations and models. Moreover, the particle scale radiation model was found to be efficient in predicting local radiative heat transfer in pebble beds and therefore is feasible to be coupled with descript element method simulations.

A new analytical expression of ETC is derived based on the continuum model to solve conduction and radiation heat transfer between pebbles in pebble beds filled with mono-sized spheres.¹²⁸ Based on the concept of continuum, this model predicts the conduction and thermal radiation in a nuclear pebble bed through a uniform framework.

In this model, the conductive and the radiative heat transfer in a packed bed are determined by the number density, heat transfer coefficient, and radial distribution function as follows:

(32a) $k_{eff}=\frac{1}{6}n{T}^{n-1}{\rho }_0^2{\displaystyle \int \text{}}\text{}\text{}{\displaystyle \int \text{}}\text{}\text{}\int R^{3}H\left(r_{ij}\right)g\left(r_{ij}\right)r_{ij}^{2}d{x}_{j}d{y}_j{d}_j,$(32a)

where

$k_{eff}$	=	= ETC
${\rho }_0$	=	= number density in the packed bed
$R$	=	= thermal contact resistance and is a function of contact force, surface roughness, and material properties
$X\left(r_{ij}\right)$	=	= obstructed view factor and is assumed to be a function of the distance.
${\rho }_0$	=	= number density in the packed bed
$r_{ij}$	=	= distance between the two particle centers
$H\left(r_{ij}\right)$	=	= heat transfer coefficient and is determined as

(32b) $H\left(r_{ij}\right)={\epsilon }_r{A}_i\sigma X\left(r_{ij}\right),$(32b)

where $\sigma$, ${\epsilon }_r$, and $A_i$ are the Stefan-Boltzmann constant, emissivity, and particle surface area, respectively, and n is 1 for particle-particle conduction at contact and 4 for the radiative heat transfer. The radial distribution functions are given by

(32c) $g\left(r_{ij}\right)=\mathrm{\Delta }n\left(r_{ij}\right)/\left({\rho }_0\mathrm{\Delta }{V}_j\right).$(32c)

Results from the proposed continuum model and those from the discrete particle simulation were in good agreement.

V.B. Convective Heat Transport of Packed Pebble Bed Reactor

As mentioned earlier, for the proper modeling and predicting of the pebble bed core temperature distribution, all three modes of heat transport (i.e., conduction, convection, and radiation) are important. However, during nominal operation of the reactor (relatively high Reynolds numbers), the heat transfer mechanism is governed by forced convection between the hot pebbles to the coolant gas flowing through the bed. This heat convection can be quantified and characterized in terms of the pebble coolant convective heat transfer coefficient or nondimensional Nusselt number. At low Reynolds numbers (the case of an accident), the effects of free convection, thermal radiation, heat conduction, and heat dispersion come into the same order of magnitude as the contribution of the forced convection.²⁶ Thermal radiation heat transfer inside the core is a complex mechanism and very difficult to characterize. The ETC is a lumped parameter that characterizes the conduction and radiation heat transfer mechanisms in a packed bed.

In the open literature, heat transfer data have been obtained by direct measurements (in which the component particles are separately heated) and indirect means (by involving transient heating of fluid or mass transfer experiments). On the other hand, the measurement techniques applied for packed pebble bed heat transfer are the electrically heated single sphere buried in the unheated packing,^{1,24,35,70,129–132} analogy and simultaneous heat and mass transfer,^35,130 and a regenerative heating technique that is based on the concept of unsteady heat transfer of a heated sphere in a packed pebble bed through which a cooling fluid flows.¹³³

Semiempirical methods^134,135 and recently computational and theoretical models^27,136–141 have been used to predict the heat transfer rate and coefficients in PBRs. Based on the predetermined criteria or model, it is worthwhile to mention that these experimental/computational determinations of heat transfer coefficients have been made under steady-state and/or transient conditions.

Unfortunately, in these previous studies, it was found that the experimental results are quite different and show considerable departures from one another, particularly at low Reynolds number. Reference 35 claimed that the reported results cannot be generalized to represent the convective heat transfer in a randomly packed bed. Schröder et al.¹³² pointed out that inhomogeneous interstitial flow velocities are responsible for the scattering of the heat transfer experimental data of other investigators. In fact, this is due to convective heat transfer influenced by many parameters such as local flow condition, bed characteristics, etc. In addition to that, there are inaccuracies in the heat flux and temperature-measuring techniques. For instance, the method of a single heated sphere requires that the local heat flux and sphere surface temperature be measured accurately beside the local gas flow temperature in the gap between the pebbles. In all previous studies, except the work of Abdulmohsin¹ and Abdulmohsin and Al-Dahhan,¹²⁹ the heat flux is based on the directed energy input method, and the boundary condition of the constant surface temperature was assumed. This assumption is unreliable for the boundary condition. Kaviany⁵⁶ pointed out that the thermal conductivity of the solid is not large enough to lead to an isothermal surface temperature; the thermal conductivity of the solids also influences the temperature field around it. Another important issue is that the surface temperature is approximately obtained, and this is due to the uncontrolled heat losses via the points of contact with unheated neighboring spheres and the influence of heat transfer by the radiation. The surface temperature was taken to be the arithmetic average of the readings of three or four thermocouples, where their tips were flushed with the sphere (Hoogenboezem TA, 2006). In addition to that, the mass transfer analogy experiments are difficult and not as accurate as direct heat transfer measurement. Also, the ideal plug flow model was generally assumed in the computational and theoretical approaches, although gas dispersion occurs even at high gas velocities and the actual velocity profile is nonuniform with a pronounced slip at the wall. All these crucial limitations in previous studies inevitably reduce the accuracy of the experimental results. Thus, the selected measurement technique has an important influence on the generated heat transfer data.¹

It is obvious that extensive investigations are required to further advance the knowledge of heat transport occurring in pebble beds, which will provide information for safe and efficient design and operation of packed PBRs.

V.C. Available Empirical Correlations for Convective Heat Transfer Predictions

Generally, in packed beds, the convective heat transfer is from the particles to the fluid flowing through the bed; sometimes, it is referred to as the fluid-to-particle mode. The basic idea for the treatment of particle-to-fluid heat transfer is to consider the situation of the individual particle. In the literature, considerable efforts have been made to evaluate the heat transfer coefficient in chemical/catalytic packed bed reactors because of the importance of this parameter. An extensive review of experimental/theoretical works on particle-to-fluid heat transfer in the packed beds can be found in Abdulmohsin,¹ Abdulmohsin and Al-Dahhan,¹²⁹ Rimkevicius et al,¹³¹ Tsotsas,¹⁴² and Wakao and Kaguei.¹⁰⁸ In fact, the heat transfer in packed beds is an extremely complex process, and there is, of course, no exact theory that satisfactorily describes this phenomenon.

However, there are some correlations reported in the literature related to the convection heat transfer coefficient in gas-solid packed bed systems in terms of Nusselt numbers, as summarized in Table V.

TABLE V Summary of Selected Correlations for the Heat Transfer Coefficient in Packed PBRs*

Author	Correlation	Range
Ranz^Citation143	$\mathrm{N}\mathrm{u}=2+0.6\stackrel{1/3}{Pr}{Re}^{0.5}$	$\begin{array}{rl}& Re\ge 100\\ & 0.6\le Pr\le 400\end{array}$
Rowe and Claxton^Citation159	$\begin{array}{rl}& \mathrm{N}\mathrm{u}=\mathrm{A}+\mathrm{B}{Pr}^{1/3}{Re}^{\mathrm{n}}\\ & \mathrm{A}=\frac{2}{1-{(1-{\epsilon }_b)}^{1/3}};\mathrm{B}=\frac{2}{3{\epsilon }_b}\\ & \frac{2-3\mathrm{n}}{3\mathrm{n}-1}=4.65{Re}^{-0.28}\end{array}$	$\begin{array}{rl}& 1\le Re\le 1\times {10}^5\\ & 0.71\le Pr\le 7.18\\ & 0.26\le {\epsilon }_b\le 0.632\end{array}$
Gupta et al.^Citation147	$\mathrm{N}\mathrm{u}=2.876\left(\frac{{Pr}^{1/3}}{{\epsilon }_b}\right)+0.3023\left(\frac{{Pr}^{1/3}}{{\epsilon }_b}\right){Re}^{0.65}$	$\begin{array}{rl}& 10\le Re\le 1\times {10}^5\\ & 0.71\le Pr\le 7.18\\ & 0.26\le {\epsilon }_b\le 0.935\end{array}$
Gnielinski^Citation134,Citation135	$\mathrm{N}\mathrm{u}=f_{\epsilon }\mathrm{N}{\mathrm{u}}_{sp}$$f_{\epsilon }=1+1.5(1-{\epsilon }_b)$$\mathrm{N}{\mathrm{u}}_{sp}=2+\sqrt{{\mathrm{N}\mathrm{u}}_{lam}^2+{\mathrm{N}\mathrm{u}}_{turb}^2}$$\mathrm{N}{\mathrm{u}}_{lam}=0.664{\left({\epsilon }_b\right)}^{\frac{1}{2}}\stackrel{\frac{1}{3}}{Pr}$$\mathrm{N}{\mathrm{u}}_{turb}=\frac{0.037{\left(\mathrm{R}\mathrm{e}/{\epsilon }_b\right)}^{0.8}Pr}{1+2.443{\left(\mathrm{R}\mathrm{e}/{\epsilon }_b\right)}^{-0.1}\left({Pr}^{\frac{2}{3}}-1\right)}$	$\begin{array}{rl}& Re/{\epsilon }_b\le 2\times {10}^4\\ & 0.71\le Pr\le {10}^4\\ & {\epsilon }_b=0.387\end{array}$
Wakao and Kaguei^Citation108	$\mathrm{N}\mathrm{u}=2+1.1\stackrel{\frac{1}{3}}{Pr}\mathrm{R}{\mathrm{e}}^{0.6}$	$\begin{array}{rl}& 15\le Re\le 8500\\ & {\epsilon }_b=0.4\end{array}$
KTA Standards^Citation146	$\mathrm{N}\mathrm{u}=1.27\left(\frac{{Pr}^{\frac{1}{3}}}{{{\epsilon }_b}^{1.18}}\right)\mathrm{R}{\mathrm{e}}^{0.36}+0.033\left(\frac{{Pr}^{\frac{1}{2}}}{{{\epsilon }_b}^{1.07}}\right)\mathrm{R}{\mathrm{e}}^{0.86}$	$\begin{array}{rl}& 100\le Re\le {10}^5\\ & Pr=0.70\\ & 0.36\le {\epsilon }_b\le 0.42\end{array}$
Hahn and Achenbach^Citation34	$\mathrm{N}{\mathrm{u}}_W=\left(1-\frac{1}{\mathrm{D}/{\mathrm{d}}_p}\right)\mathrm{R}{\mathrm{e}}^{0.61}\stackrel{\frac{1}{3}}{Pr}$	$100\le Re\le 2\times {10}^4$
Achenbach^Citation35	$\mathrm{N}\mathrm{u}={\left[{\left(1.18\mathrm{R}{\mathrm{e}}^{0.58}\right)}^4+{\left(0.23{\left(\mathrm{R}{\mathrm{e}}_h\right)}^{0.75}\right)}^4\right]}^{\frac{1}{4}}$	$\begin{array}{rl}& Re/{\epsilon }_b\le 7.7\times {10}^5\\ & Pr=0.71\\ & {\epsilon }_b=0.387\end{array}$
Bird^Citation60	$\mathrm{N}\mathrm{u}=2.19\stackrel{\frac{1}{3}}{Pr}\mathrm{R}{\mathrm{e}}^{\frac{1}{3}}+0.78\stackrel{\frac{1}{3}}{Pr}\mathrm{R}{\mathrm{e}}^{0.62}$	$\begin{array}{rl}& 1\le Re\le 1\times {10}^5\\ & Pr>0.70\end{array}$

*References 1 and 129.

Wakao and Kaguei¹⁰⁸ gave an overview of the different experimental data existing at that time and proposed the following semiempirical correlation for heat transfer in a packed bed:

(33) $\mathrm{N}\mathrm{u}=2+1.1Pr{ }^{\frac{1}{3}}\mathrm{R}{\mathrm{e}}^{0.6},$(33)

where the nondimensional Prandtl number Pr is defined as follows:

(33a) $Pr=\frac{\mu {\mathrm{C}}_p}{\mathrm{k}},$(33a)

where $\mathrm{N}{\mathrm{u}}_h$ is an effective Nusselt number that is defined based on the average interstitial velocity and on the characteristic length scale for the pores (an equivalent hydraulic diameter $d_h$) as follows:

(33b) $\mathrm{N}{\mathrm{u}}_h=\frac{h{\mathrm{d}}_h}{\mathrm{k}}=\frac{{\epsilon }_b}{\left(1-{\epsilon }_b\right)}\mathrm{N}\mathrm{u},$(33b)

where the Nusselt number is defined based on pebble diameter d_p and is given by

(33c) $\mathrm{N}\mathrm{u}=\frac{h{\mathrm{d}}_p}{\mathrm{k}},$(33c)

where h is the average convective solid-gas heat transfer coefficient in the pebble bed and k is the thermal conductivity of flowing coolant gas.

Ranz¹⁴³ and Claxton²⁰ earlier suggested alternative correlations for the prediction of the Nusselt number in packed beds, listed in Table V. Kaviany⁵⁶ stated that the above correlation, Eq. (33(33) $\mathrm{N}\mathrm{u}=2+1.1Pr{ }^{\frac{1}{3}}\mathrm{R}{\mathrm{e}}^{0.6},$(33) ), is a reliable one because it is based on a rigorous selection and adaptation of relevant experimental data. It is worthwhile to mention that the minimum Nusselt number, Nu = 2, of the single sphere as the Reynolds number goes to zero, $Re\to 0$, represents the heat transfer by conduction only. This asymptotic value results from the solution of the unsteady-state heat conduction equation for chemical packed bed reactors, and it is subject to discussion in nuclear PBRs. Nelson and Galloway¹⁴⁴ argued that for $Re\to 0,$ the heat transfer from spheres in the pebble bed cannot be related to that of a single sphere in an infinite surrounding since the boundary conditions are different. They showed that for dense packed systems, which is the case of pebble bed nuclear reactors, the Nusselt number Nu grows linearly with Re and declines to zero as Re approaches zero. Nelson and Galloway suggested the following correlation in densely packed beds:

(34) $\mathrm{N}\mathrm{u}=\frac{0.18}{{\left(1-{\epsilon }_b\right)}^{\frac{1}{3}}}\left[\frac{1}{{\left(1-{\epsilon }_b\right)}^{\frac{1}{3}}}-1\right]\mathrm{R}\mathrm{e}Pr{ }^{\frac{2}{3}}.$(34)

The average void fraction of the bed ${\epsilon }_b$ occurs as a parameter in Eq. (1b)(1b) ${\epsilon }_b=\frac{2}{{\mathrm{R}}^2}\underset{0}{\overset{R}{\int }}\epsilon (\mathrm{r})\mathrm{r}\mathrm{d}\mathrm{r}$(1b) .

For high-temperature packed pebble bed nuclear reactors, the theory explaining the convective solid-gas heat transfer coefficient assumes that the heat transfer of heated pebbles can be related to the heat transfer from a single sphere (pebble) by introducing an arrangement or form factor $f_{\epsilon }$, which depends on the void fraction.¹⁴⁵ Hence, Gnielinski,^134,135 evaluated the experimental results of about 20 authors and established a relationship among the Nusselt number, Reynolds number, Prandtl number, and porosity of the packed pebble bed, in the following form:

(35) $\mathrm{N}\mathrm{u}=f_{\epsilon }\mathrm{N}{\mathrm{u}}_{sp}\mathrm{f}\mathrm{o}\mathrm{r}Re/{\epsilon }_b\le 2\times {10}^4,$(35)

where

(35a) $f_{\epsilon }=1+1.5(1-\epsilon )$(35a)

and $\mathrm{N}{\mathrm{u}}_{sp}$ is the Nusselt number of a single sphere (pebble), which can be calculated according to Eq. (35b)(35b) $\mathrm{N}{\mathrm{u}}_{sp}=2+\sqrt{{\mathrm{N}\mathrm{u}}_{lam}^2+{\mathrm{N}\mathrm{u}}_{turb}^2},$(35b) :

(35b) $\mathrm{N}{\mathrm{u}}_{sp}=2+\sqrt{{\mathrm{N}\mathrm{u}}_{lam}^2+{\mathrm{N}\mathrm{u}}_{turb}^2},$(35b)

where $\mathrm{N}{\mathrm{u}}_{lam}$ and $\mathrm{N}{\mathrm{u}}_{turb}$ are the Nusselt numbers of the single sphere for laminar flow and turbulent flow, respectively. They can be obtained from the equations valid for the flat plate by introducing a length scale as a characteristic streaming length that is equal to the sphere diameter in the case of spherical pebbles; thus,

(35c) $\mathrm{N}{\mathrm{u}}_{lam}=0.664{\left(Re/{\epsilon }_b\right)}^{1/2}\stackrel{1/3}{Pr}$(35c)

and

(35d) $\mathrm{N}{\mathrm{u}}_{turb}=\frac{0.037{\left(\mathrm{R}\mathrm{e}/{\epsilon }_b\right)}^{0.8}Pr}{1+2.443{\left(\mathrm{R}\mathrm{e}/{\epsilon }_b\right)}^{-0.1}\left({Pr}^{\frac{2}{3}}-1\right)}.$(35d)

In their modular PBR project,⁷ the Idaho National Engineering and Environmental Laboratory, the Massachusetts Institute of Technology, and the Association of German Engineers (VDI) Heat Atlas,¹⁴⁵ provided Eqs. (35a)(35a) $f_{\epsilon }=1+1.5(1-\epsilon )$(35a) through (35d(35d) $\mathrm{N}{\mathrm{u}}_{turb}=\frac{0.037{\left(\mathrm{R}\mathrm{e}/{\epsilon }_b\right)}^{0.8}Pr}{1+2.443{\left(\mathrm{R}\mathrm{e}/{\epsilon }_b\right)}^{-0.1}\left({Pr}^{\frac{2}{3}}-1\right)}.$(35d) ) as recommended correlations for the predication of pebble-to-gas heat transfer in the core of high-temperature packed pebble bed nuclear reactors.

Based on experimental data from several independent studies of heat convection in randomly packed pebble beds, the German Nuclear Safety Standard Commission (KTA) proposed a correlation to determine the heat transfer coefficient of solid to flowing gas for a German high-temperature reactor (HTR), as follows¹⁴⁶:

(36) $\begin{array}{rl}& \mathrm{N}\mathrm{u}=1.27\left(\frac{{Pr}^{\frac{1}{3}}}{{{\epsilon }_b}^{1.18}}\right)\mathrm{R}{\mathrm{e}}^{0.36}+0.033\left(\frac{{Pr}^{\frac{1}{3}}}{{{\epsilon }_b}^{1.07}}\right)\mathrm{R}{\mathrm{e}}^{0.86}\\ & \mathrm{f}\mathrm{o}\mathrm{r}100\le \mathrm{R}\mathrm{e}\le {10}^5.\end{array}$(36)

The above correlation, Eq. 34(34) $\mathrm{N}\mathrm{u}=\frac{0.18}{{\left(1-{\epsilon }_b\right)}^{\frac{1}{3}}}\left[\frac{1}{{\left(1-{\epsilon }_b\right)}^{\frac{1}{3}}}-1\right]\mathrm{R}\mathrm{e}Pr{ }^{\frac{2}{3}}.$(34) , is very similar to the one developed by Gupta et al.¹⁴⁷ and to that correlation recommended for the flow of gases through packed beds by Bird,⁶⁰ which are listed in Table IV. According to Gougar,¹⁴⁸ the Idaho National Laboratory has adopted the KTA’s correlation in its multiscale and multidimensional simulation and optimization code for the design and analysis of pebble bed HTRs, which is called the PEBBED code. A similar empirical heat transfer correlation was developed by Achenbach³⁵ for a pebble bed heat transfer coefficient in which the Reynolds number range exceeds ranges used by other researchers by one order of magnitude, as follows:

(37) $\begin{array}{rl}& \mathrm{N}\mathrm{u}=\left[{\left(1.18\mathrm{R}{\mathrm{e}}^{0.58}\right)}^4+{\left(0.23{\left(\mathrm{R}{\mathrm{e}}_h\right)}^{0.75}\right)}^4\right]{ }^{\frac{1}{4}}\\ & \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{R}\mathrm{e}/{\epsilon }_b\le 7.7\times {10}^5.\end{array}$(37)

Finally, the convection heat transfer at the wall, in terms of the wall Nusselt number Nu_w, for fluid flow in a packed pebble bed can be expressed as follows³⁴:

(38) $\begin{array}{rl}& \mathrm{N}{\mathrm{u}}_{\mathrm{W}}=\left(1-\frac{1}{\mathrm{D}/{\mathrm{d}}_p}\right)\mathrm{R}{\mathrm{e}}^{0.61}Pr{ }^{\frac{1}{3}}\\ & \mathrm{f}\mathrm{o}\mathrm{r}100\le \mathrm{R}\mathrm{e}\le 2\times {10}^4.\end{array}$(38)

V.D. Comparison of the Available Empirical Correlations for Convective Heat Transport

As mentioned earlier, numerous studies have been conducted around heat transfer in packed beds of spheres of small particles (catalyst). However, for the convective heat transfer coefficients, a number of correlations have been reported in the literature for packed pebble beds. The literature shows a great scattering in the heat transfer coefficient predictions of the reported correlations, especially when it comes to the fluid of high Prandtl and extremely low flow conditions. This is because the experiments have been mainly conducted with air and the results are mapped to high Prandtl fluids and to extremely low flow conditions that have been done through analogy with mass transfer experiments.

In the work of Abdulmohsin¹ and Abdulmohsin and Al-Dahhan,¹²⁹ the local heat transfer coefficients have been measured experimentally. Four correlations were used to predict the overall average convection heat transfer coefficient in packed pebble bed nuclear reactors. The correlations have been selected because they were developed based on a large experimental database, as discussed in Sec V.C.

Based on AARE, a statistical test was performed to check the fitting of prediction. AARE between the measured and the predicated Nusselt numbers is expressed as

(39) $\mathrm{A}\mathrm{A}\mathrm{R}\mathrm{E}=\frac{1}{\mathrm{N}}\sum _1^{\mathrm{N}}\left|\frac{\mathrm{N}{\mathrm{u}}_{Pred(i)}-\mathrm{N}{\mathrm{u}}_{Exptl(i)}}{\mathrm{N}{\mathrm{u}}_{Exptl(i)}}\right|,$(39)

where N is the data point number.

Figure 11 shows the values of pebble-to-gas heat transfer coefficients in terms of Nusselt numbers $\left(\mathrm{N}\mathrm{u}=h{d}_p/k\right)$ at different effective Reynolds numbers. In Fig. 11, the experimental values of the averaged local heat transfer coefficients obtained by Abdulmohsin¹ and Abdulmohsin and Al-Dahhan¹²⁹ are compared with those predicted based on the above-selected correlations. The prediction of the Achenbach³⁵ correlation is relatively better for all flow conditions, and AARE with the experimental data of this work is about 4.4%. The correlations developed by Gnielinski¹³⁵ and KTA Standards¹⁴⁹ seem to provide a reasonable prediction for turbulent-flow regimes of high flow conditions (Re_h > 300), where the values of AAREs are about 3.37% and 2.23%, respectively. However, both correlations overpredict laminar-flow conditions, i.e., Re_h < 200. There is a relatively larger deviation in the prediction based on the correlation of Wakao and Kaguei¹⁰⁸ (AARE is about 13%) for low flow conditions, i.e., Re_h < 200. The correlation also gives AARE of about 9% for high flow conditions of the turbulent-flow regime.

Fig. 11. Comparison of the measured average heat transfer coefficient with the empirical correlations.^1,129

At low superficial gas velocities, the trends and the values do not match well for all correlations. This can be attributed to uncertainties in different measurement techniques used to validate these correlations and different operating and design conditions of the experimental data used to assess them. In addition to that, the forced convective heat transfer coefficient is influenced by several parameters, for instance, Reynolds number, Prandtl number, local porosity, aspect ratio, local flow conditions, etc. However, the variation of the local porosity and hence local flow conditions remains an important issue for the local heat transfer coefficient. In the work of Abdulmohsin¹ and Abdulmohsin and Al-Dahhan,¹²⁹ a large pebble diameter, 5 cm, is been used that yields a higher value of the average bed porosity.

Variations in local values of the heat transfer coefficient indicate that more investigations on the mechanisms that govern heat transfer using a wide range of relevant conditions in the pebble bed are needed to develop correlations capable of properly predicting the local heat transfer coefficients and to further improve such predictions of the local convective heat transfer coefficients in these reactors.

It is obvious that to obtain more accurate results and a proper understanding of the local heat transfer coefficients and related mechanisms, detailed qualitative and quantitative information of local gas velocity fields and local porosity is needed. Therefore, special investigations of local gas velocity fields and local porosity are necessary in packed PBRs. Since such investigations are not an easy task, the CFD are important for predictions of the local flow field for measured or computed local porosity to estimate the local heat transfer coefficient using one of the aforementioned correlations. Hence, developing a correlation that is capable of predicting the local heat transfer coefficient will facilitate using proper integration of hydrodynamics (CFD) and heat transfer computation.

V.E. Effect of Pebble Surface Temperature on Heat Transfer

For PBRs to be designed and operated safely, it is essential to locate hot areas in the reactor core and examine their thermodynamic properties. High pebble surface temperatures could compromise their structural integrity and let fuel element fission products escape into the reactor system or environment. Therefore, pebble surface temperatures and heat transfer characteristics of regular pebble structures, e.g., simple cubic, body-centered cubic, and face-centered-cubic (FCC) structures as well as their combinations, have been investigated using CFD (Refs. 139 and 150 through 155). Chen and Lee¹⁵⁶ developed a test section, shown in Fig. 12a, to experimentally investigate the local and average heat transfer characteristics in a test section including two full spheres, four hemispheres, and eight quarter spheres, each with a diameter of 12 cm, packed in an FCC structure. A 3D view of the pebble structure and a top view of the test section are shown in Fig. 2. It was found that the strongest heat transfer occurs in the regions of ϕ = 36 deg and ϕ = 117 deg, whereas the weakest heat transfer occurs in the regions where ϕ = 0.90 deg and ϕ = 180 deg, where ϕ is the angle from the z-axis to the hole. The results also showed that the maximum temperature difference between the pebbles decreases as the coolant mass flow rate increases, which implies that the heat transfer becomes stronger as the Reynolds number increases.

Fig. 12. (a) Schematic of the FCC structure and temperature measurement locations and (b) comparison of Nusselt numbers (Chen and Lee¹⁵⁷).

Chen and Lee¹⁵⁷ simulated a single FCC pebble bed using hybrid Reynolds-Averaged Navier-Stokes, the standard k-ε turbulence model, and a thermal energy model in CFX to investigate the impact of pebble diameter on the heat transfer characteristics and flow field of a pebble bed. Pebble diameters of 10, 12, and 14 cm were studied. The pebbles were two times scaled up so that the fluid velocity could be reduced to half (to keep the Reynolds number the same) to achieve the observation of the flow regime formed in the void using particle image velocimetry; therefore, the pebble diameters described in this study, 10, 12, and 14 cm, represent the fuel elements with diameters of 5, 6, and 7 cm in a reactor core, respectively. A schematic of the temperature measurement locations is presented in Fig. 12b.

The results showed that decreasing the pebble size improves the overall heat transfer and decreases the pebble surface temperature. An enhancement of 10.4% was observed in the heat transfer coefficient when the 10-cm pebbles were used compared to 12-cm pebbles while the 14 cm-diameter pebble bed weakens the heat transfer by 18.6%. The bed with 10-cm pebbles was found to have the lowest surface temperature compared to the 12-cm pebbles: a maximum 5 K and an average 3 K lower than the 12-cm pebble bed. This means that using smaller pebble sizes reduces the possibility of local hot spots and hence safer reactor operation.

The heat transfer coefficient was investigated at several heat inlet gas velocities of 2.3, 2.52, 3.36, 4.2, and 5.16 m/s, corresponding to Reynolds numbers of 1.46 × 10⁴, 1.61 × 10⁴, 2.15 × 10⁴, 2.69 × 10⁴, and 3.3 × 10⁴, respectively. It was found that the average heat transfer coefficient increases with the gas velocity. It was found that the fluid velocity gradually increased and the solid surface temperature decreased as the pebble size reduced.

Chen and Lee¹⁵⁷ proposed a correlation between h_av and the Re number for the 10-cm pebbles as follows:

(40) $h_{avg}=0.1588\left(k/L\right)\mathrm{R}{\mathrm{e}}^{0.8}.$(40)

Considering the effect of pebble diameter on the Nusselt number for all three pebble sizes, a correlation containing the pebble diameter and suitable Re range was obtained as follows:

(40a) $\begin{array}{rl}& \mathrm{N}\mathrm{u}=0.194\mathrm{R}{\mathrm{e}}^{0.8}\mathrm{P}{\mathrm{r}}^{0.4}-0.3226{\left(\frac{L}{D}-1.027\right)}^2\mathrm{R}{\mathrm{e}}^{0.8}\mathrm{P}{\mathrm{r}}^{0.4}\\ & \mathrm{f}\mathrm{o}\mathrm{r}1.46\times 104\le \mathrm{R}\mathrm{e}\le 3.3\times 104,\end{array}$(40a)

where D equals 12 cm and L is the pebble diameter. Based on the similarity principle, the above correlation can be applied to a reactor core when D is taken as the diameter of the fuel elements. Comparison of the developed Nusselt number model with other models is presented in Fig. 12b.

In the same study, the effect of adding small spheres to the bed was also investigated. Spheres with diameters of 2, 3, and 4.14 cm were added, and the velocity field, pebble surface temperature, and average heat transfer coefficient were analyzed. Compared to no spheres, the heat transfer coefficient was enhanced by 27% when the sphere with a diameter of 4.14 was added to the bed loaded with 10-cm pebbles.

VI. EFFECT OF POROSITY ON PRESSURE DROP AND FORCED CONVECTIVE HEAT TRANSFER

In randomly packed PBRs, the value of porosity appreciably influences the absolute magnitude of the pressure drop across the bed, the axial dispersion process, and the convective heat transfer coefficient between the solid and flowing coolant gas. In order to explain analytically the effect of voidage on pressure drop for a randomly packed pebble bed of spherical particles (pebbles), the KTA’s empirical correlation, Eq. (15(15) $\psi =\frac{320}{{Re}_h}+\frac{6}{{Re}_h^{0.1}}\mathrm{f}\mathrm{o}\mathrm{r}{Re}_h\le 5\times {10}^4.$(15) ), is rewritten for the dimensionless pressure drop form (or it is called friction force coefficient ψ) in terms of the Reynolds number as follows⁶⁹:

(41) $\psi =\frac{320}{\left[Re/\left(1-{\epsilon }_b\right)\right]}+\frac{6}{{\left[Re/\left(1-{\epsilon }_b\right)\right]}^{0.1}}\mathrm{f}\mathrm{o}\mathrm{r}\left[Re/\left(1-{\epsilon }_b\right)\right]\le 5\times {10}^4.$(41)

As mentioned earlier, the first term of Eq. (41)(41) $\psi =\frac{320}{\left[Re/\left(1-{\epsilon }_b\right)\right]}+\frac{6}{{\left[Re/\left(1-{\epsilon }_b\right)\right]}^{0.1}}\mathrm{f}\mathrm{o}\mathrm{r}\left[Re/\left(1-{\epsilon }_b\right)\right]\le 5\times {10}^4.$(41) represents the asymptotic solution for laminar flow while the second term represents the solution for turbulent flow. Each of the terms can be written as follow^26,35:

(42) $\psi =\mathrm{A}{\left(\frac{Re}{1-{\epsilon }_b}\right)}^{-\mathrm{n}}=\mathrm{A}{\left(1-{\epsilon }_b\right)}^{\mathrm{n}}{Re}^{-\mathrm{n}},$(42)

where n = 1 represents the low Reynolds number range and n = 0 represents the high one. The variation of pressure drop with porosity has been expressed by Fenech as per the following:

(43) $\frac{\mathrm{d}\left(\mathrm{\Delta }\mathrm{P}\right)}{\mathrm{\Delta }\mathrm{P}}=\frac{\mathrm{\partial }\left(\mathrm{\Delta }\mathrm{P}\right)}{\mathrm{\partial }{\epsilon }_b}\frac{\mathrm{d}{\epsilon }_b}{\mathrm{\Delta }\mathrm{P}}.$(43)

Combining Eqs. (12)(12) $\mathrm{\Delta }\mathrm{P}=\psi \frac{\mathrm{L}}{{\mathrm{d}}_h}\frac{\rho }{2}{\mathrm{V}}^2=\psi \frac{\mathrm{L}}{{\mathrm{d}}_p}\frac{\rho }{2}{\mathrm{V}}_g^2\left(\frac{1-{\epsilon }_b}{{\epsilon }_b^3}\right).$(12) and (42(42) $\psi =\mathrm{A}{\left(\frac{Re}{1-{\epsilon }_b}\right)}^{-\mathrm{n}}=\mathrm{A}{\left(1-{\epsilon }_b\right)}^{\mathrm{n}}{Re}^{-\mathrm{n}},$(42) ) with Eq. (43)(43) $\frac{\mathrm{d}\left(\mathrm{\Delta }\mathrm{P}\right)}{\mathrm{\Delta }\mathrm{P}}=\frac{\mathrm{\partial }\left(\mathrm{\Delta }\mathrm{P}\right)}{\mathrm{\partial }{\epsilon }_b}\frac{\mathrm{d}{\epsilon }_b}{\mathrm{\Delta }\mathrm{P}}.$(43) yields

(44) $\frac{\mathrm{d}\left(\mathrm{\Delta }\mathrm{P}\right)}{\mathrm{\Delta }\mathrm{P}}=-\frac{\left[3-{\epsilon }_b(2-\mathrm{n})\right]}{\left(1-{\epsilon }_b\right)}\frac{\mathrm{d}{\epsilon }_b}{{\epsilon }_b},$(44)

where n = 1 for laminar-flow conditions and n = 0 for turbulent-flow conditions.¹⁴⁹ These values of the exponent n come from the KTA correlation that is used for determining the friction force coefficient or the dimensionless pressure drop form, Eq. (15)(15) $\psi =\frac{320}{{Re}_h}+\frac{6}{{Re}_h^{0.1}}\mathrm{f}\mathrm{o}\mathrm{r}{Re}_h\le 5\times {10}^4.$(15) .

It can be shown from Eq. (44)(44) $\frac{\mathrm{d}\left(\mathrm{\Delta }\mathrm{P}\right)}{\mathrm{\Delta }\mathrm{P}}=-\frac{\left[3-{\epsilon }_b(2-\mathrm{n})\right]}{\left(1-{\epsilon }_b\right)}\frac{\mathrm{d}{\epsilon }_b}{{\epsilon }_b},$(44) that a positive relative variation of the void fraction dε_b/ε_b causes a negative relative variation of the pressure drop, d(ΔP)/ΔP, multiplied by a factor that is dependent on the porosity ε_b and on the slope n of the Reynolds number. In other words, it is greater by a factor of 3 − ε_b(2 − n)/(1 − ε_b).

For a randomly packed bed of spherical particles, the values for real packings typically fall into the range ε_b = 0.36 to 0.42 (Ref. 158). Therefore, the normal packing of typical voidage ε_b of around 0.4 represents a separate line between loose packing, ε_b > 0.4, and dense packing, ε_b < 0.4.

Using Eq. (44(44) $\frac{\mathrm{d}\left(\mathrm{\Delta }\mathrm{P}\right)}{\mathrm{\Delta }\mathrm{P}}=-\frac{\left[3-{\epsilon }_b(2-\mathrm{n})\right]}{\left(1-{\epsilon }_b\right)}\frac{\mathrm{d}{\epsilon }_b}{{\epsilon }_b},$(44) ), Fig. 13 has been plotted to show the effect of void fraction on pressure drop. For example, at ε_b = 0.4, the percentage of error with respect to pressure drop is approximately four times. The error is defined as undergone for the determination of porosity. In other words, an error of 1% in ε_b causes errors of ~4% in ΔP as per Eq. 44(44) $\frac{\mathrm{d}\left(\mathrm{\Delta }\mathrm{P}\right)}{\mathrm{\Delta }\mathrm{P}}=-\frac{\left[3-{\epsilon }_b(2-\mathrm{n})\right]}{\left(1-{\epsilon }_b\right)}\frac{\mathrm{d}{\epsilon }_b}{{\epsilon }_b},$(44) and Fig. (11).

Fig. 13. Effect of void fraction (porosity) on the pressure drop in the laminar and turbulent-flow regimes (Abdulmohsin¹).

The literature of the axial dispersion phenomenon, as discussed earlier, shows that the dispersive Peclet number Pe_D is a function of the Reynolds number Re, the Schmidt number Sc, and the porosity ε_b created by the packing. Therefore, the functional dependence of these groups can be expressed by the following:

(45) $\mathrm{P}{\mathrm{e}}_D=f\left({\epsilon }_b;\mathrm{R}\mathrm{e};\mathrm{S}\mathrm{c}\right).$(45)

To explain the trend of the influence of porosity on the axial dispersion process, the early correlations of axial dispersion in packed beds, as shown in Table IV, have been analyzed, and accordingly, Abdulmohsin¹ suggested the following correlation that describes the relationship of Pe_D with respect to Re, Sc, and ε_b:

(46) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\mathrm{A}{\left(\frac{1-{\epsilon }_b}{{\epsilon }_b}\right)}^{-\mathrm{n}}{\left(\frac{\mathrm{R}\mathrm{e}}{1-{\epsilon }_b}\right)}^{-\mathrm{n}}\mathrm{S}{c}^{-\mathrm{n}}.$(46)

This form represents a sum of the contribution of the diffusion and convection terms.

As mentioned earlier, at low flow rates, axial dispersion is a function of the diffusion coefficient modified by a factor that accounts for the tortuosity and porosity created by the packing. As the flow velocity increases, dispersion becomes a function of the hydrodynamics using the same packing. Therefore, the exponent can be considered n = 1 for low flow rate and n = 0 for high flow rate.

By following the same approach of variation of pressure drop with porosity, the variation of axial dispersion with porosity can be presented in this work as follows:

(47) $\frac{\mathrm{d}\left(1/\mathrm{P}{\mathrm{e}}_D\right)}{\left(1/\mathrm{P}{\mathrm{e}}_D\right)}=\frac{\mathrm{\partial }\left(1/\mathrm{P}{\mathrm{e}}_D\right)}{\mathrm{\partial }{\epsilon }_b}\frac{\mathrm{d}{\epsilon }_b}{\left(1/\mathrm{P}{\mathrm{e}}_D\right)}.$(47)

Combining Eq. (47)(47) $\frac{\mathrm{d}\left(1/\mathrm{P}{\mathrm{e}}_D\right)}{\left(1/\mathrm{P}{\mathrm{e}}_D\right)}=\frac{\mathrm{\partial }\left(1/\mathrm{P}{\mathrm{e}}_D\right)}{\mathrm{\partial }{\epsilon }_b}\frac{\mathrm{d}{\epsilon }_b}{\left(1/\mathrm{P}{\mathrm{e}}_D\right)}.$(47) together with Eq. (46)(46) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\mathrm{A}{\left(\frac{1-{\epsilon }_b}{{\epsilon }_b}\right)}^{-\mathrm{n}}{\left(\frac{\mathrm{R}\mathrm{e}}{1-{\epsilon }_b}\right)}^{-\mathrm{n}}\mathrm{S}{c}^{-\mathrm{n}}.$(46) yields

(48) $\frac{\mathrm{d}\left(1/\mathrm{P}{\mathrm{e}}_D\right)}{\left(1/\mathrm{P}{\mathrm{e}}_D\right)}=-\frac{\left(3-2\mathrm{n}{\epsilon }_b\right)}{\left(1-{\epsilon }_b\right)}\frac{\mathrm{d}{\epsilon }_b}{{\epsilon }_b},$(48)

where n = 1 for laminar-flow conditions and n = 0 for turbulent-flow conditions.

Regarding the effect of the voidage on the forced convective heat transfer, similar to those effects on the pressure drop, Eq. (43(43) $\frac{\mathrm{d}\left(\mathrm{\Delta }\mathrm{P}\right)}{\mathrm{\Delta }\mathrm{P}}=\frac{\mathrm{\partial }\left(\mathrm{\Delta }\mathrm{P}\right)}{\mathrm{\partial }{\epsilon }_b}\frac{\mathrm{d}{\epsilon }_b}{\mathrm{\Delta }\mathrm{P}}.$(43) ), and the axial dispersion and mixing in terms of Pe_D, Eq. (45(45) $\mathrm{P}{\mathrm{e}}_D=f\left({\epsilon }_b;\mathrm{R}\mathrm{e};\mathrm{S}\mathrm{c}\right).$(45) ), Fenech²⁶ reported the following expression:

(49) $\frac{\mathrm{d}\left(\mathrm{N}\mathrm{u}\right)}{\mathrm{N}\mathrm{u}}=-\frac{\left(1-\mathrm{n}{\epsilon }_b\right)}{\left(1-{\epsilon }_b\right)}\frac{\mathrm{d}{\epsilon }_b}{{\epsilon }_b},$(49)

where n = 0 for laminar-flow conditions and n = 0.6 for turbulent-flow conditions.

It can be shown from Eqs. (47)(47) $\frac{\mathrm{d}\left(1/\mathrm{P}{\mathrm{e}}_D\right)}{\left(1/\mathrm{P}{\mathrm{e}}_D\right)}=\frac{\mathrm{\partial }\left(1/\mathrm{P}{\mathrm{e}}_D\right)}{\mathrm{\partial }{\epsilon }_b}\frac{\mathrm{d}{\epsilon }_b}{\left(1/\mathrm{P}{\mathrm{e}}_D\right)}.$(47) and (48(48) $\frac{\mathrm{d}\left(1/\mathrm{P}{\mathrm{e}}_D\right)}{\left(1/\mathrm{P}{\mathrm{e}}_D\right)}=-\frac{\left(3-2\mathrm{n}{\epsilon }_b\right)}{\left(1-{\epsilon }_b\right)}\frac{\mathrm{d}{\epsilon }_b}{{\epsilon }_b},$(48) ) that a positive relative variation of the void fraction dε_b/ε_b causes a negative relative variation of both the reciprocal Peclet number, d(1/Pe_D)/(1/Pe_D), and the Nusselt number, d(Nu)/Nu, multiplied by the factors of (3 – 2nε_b)/(1 – ε_b) and (1 – nε_b)/(1 – ε_b), respectively. Using Eqs. (45)(45) $\mathrm{P}{\mathrm{e}}_D=f\left({\epsilon }_b;\mathrm{R}\mathrm{e};\mathrm{S}\mathrm{c}\right).$(45) and (46(46) $\frac{1}{\mathrm{P}{\mathrm{e}}_D}=\mathrm{A}{\left(\frac{1-{\epsilon }_b}{{\epsilon }_b}\right)}^{-\mathrm{n}}{\left(\frac{\mathrm{R}\mathrm{e}}{1-{\epsilon }_b}\right)}^{-\mathrm{n}}\mathrm{S}{c}^{-\mathrm{n}}.$(46) ), Figs. 12a and 12b show the effect of void fraction on axial dispersion and convective heat transfer, respectively. For example, at ε_b = 0.4, the percentage of error with respect to the reciprocal Peclet number and the Nusselt number are ~4.3 times and ~1.5 times, respectively, the error undergone for the determination of porosity. In other words, an error of 1% in ε_b causes errors of ~4.3% and ~1.5% for 1/Pe_D and Nu, respectively.

Based on Figs. 14a and 14b, the percentage of error for all relative variations rises with increasing porosity ε_b and decreases as the exponent n increases. Hence, the strong dependence of the pressure drop, axial dispersion and mixing process, and heat transfer on the void fraction underline the importance of packing and refueling pebble beds carefully to avoid bypass and channeling coolant flow due to local variations in the packing density.

Fig. 14. Effect of void fraction (porosity) in the laminar- and turbulent-flow regimes on the following: (a) convective heat transfer and (b) axial dispersion and mixing process (Abdulmohsin¹).

It is obvious that the fluid flow, pressure drop, axial dispersion and mixing, and heat transport mechanisms are all sensitive and influenced by the porous structure of the packed PBR. Therefore, a proper understanding and characterization of the porous structure of the bed is of great importance for safe design and efficient operation of packed PBRs.

VII. CONCLUDING REMARKS

The following findings and concluding remarks may be drawn from the reported work of pressure drop, gas dispersion and mixing, and convective heat transport phenomena in packed pebble bed nuclear reactors:

1. To test the accuracy of the predictions of pressure drop by the empirical correlations in the literature, a comparison was made with the experimental data of Abdulmohsin¹ and Abdulmohsin and Al-Dahhan.⁷¹ The obtained experimental results of pressure drop confirm that the classical Ergun-type equations, commonly used to estimate pressure drop through chemical packed beds, considerably overpredict the pressured drop of the pebble beds at high Reynolds number.

2. The obtained experimental results of pressure drop by Abdulmohsin¹ and Abdulmohsin and Al-Dahhan⁷¹ demonstrate the applicability of the VDI and KTA correlations for prediction of pressure drop in randomly packed pebble bed nuclear reactors.

3. A comparison was made by Abdulmohsin¹ and Abdulmohsin and Al-Dahhan⁸⁰ between the measured axial gas dispersion coefficients in terms of Peclet numbers and dispersion numbers (reciprocal of Peclet numbers) at different gas velocities with those predicted by selected correlations. The correlation of Gunn⁹⁹ predicts well the obtained experimental data. However, additional investigations and more data are needed to reach a sound conclusion and to possibly develop a new correlation for packed pebble bed nuclear reactors.

4. The convective pebble-gas heat transfer coefficients in terms of Nusselt numbers have been compared by Abdulmohsin¹ with those predicted based on published correlations. The results showed that the classical Wakao equation of chemical packed bed reactors cannot predict accurate convective heat transfer coefficients for certain conditions, especially for packed pebble bed nuclear reactors of the turbulent-flow regime.

5. The obtained experimental results of heat transfer by Abdulmohsin¹ and Abdulmohsin and Al-Dahhan¹²⁹ demonstrate the applicability of the Ref. 35 correlation for randomly packed pebble bed nuclear reactors.

6. The variations in the local values of the heat transfer coefficient by Abdulmohsin¹ and Abdulmohsin and Al-Dahhan¹²⁹ indicate that more investigations on the mechanisms that govern heat transfer using a wide range of relevant conditions in the pebble bed are needed to develop correlations capable of properly predicting the local heat transfer coefficients and to further improve such predictions of the local convective heat transfer coefficients in these reactors.

7. Accordingly, measuring the variation of the local bed structure and the local gas velocity along with the heat transfer coefficient was studied by Al Falahi and Al-Dahhan.³³

Nomenclature

DAB	=	= molecular diffusion coefficient (m2/s molecular)
Dax	=	= effective axial dispersion coefficient of the gas phase (m2/s)
Dc	=	= column diameter (m)
dh	=	= effective (hydraulic) pebble diameter (m)
dp	=	= pebble diameter (m)
L	=	= length of the bed (not the column) (m)
N	=	= data point number, Eq. (39(39) $\mathrm{A}\mathrm{A}\mathrm{R}\mathrm{E}=\frac{1}{\mathrm{N}}\sum _1^{\mathrm{N}}\left|\frac{\mathrm{N}{\mathrm{u}}_{Pred(i)}-\mathrm{N}{\mathrm{u}}_{Exptl(i)}}{\mathrm{N}{\mathrm{u}}_{Exptl(i)}}\right|,$(39) )
n	=	= total number of experimental data points
t	=	= time (s)
tm	=	= mean residence time of the bed (s)
V	=	= interstitial gas velocity (= Vg/ε) (m/s)
Vg	=	= superficial gas velocity based on empty column (m/s)
VT	=	= total bed volume (cm3)
Z	=	= axial distance along the bed (m)
z	=	= axial distance along the bed (m) Greek
εb	=	= average (mean) voidage of bed, dimensionless
µ	=	= dynamic viscosity of the fluid, Kg.m/s
ρ	=	= density of fluid, kg/m3 Dimensionless Groups
Nu	=	= Nusselt number (= hdp/k), dimensionless
Nuh	=	= effective Nusselt number [= hdh/k= εNu/(1 − ε)], dimensionless
PeD	=	= dispersive Peclet number (= Vgdp/εDax), dimensionless
PeM	=	= molecular Peclet number (= RePSc = Vgdp/εDAB), dimensionless
Pr	=	= Prandtl number (= µCp/k), dimensionless
Re	=	= Reynolds number (= ρVgdp/µ), dimensionless
Reh	=	= effective Reynolds number [= ρVdh/µ = Re/(1 − ε)], dimensionless
ReP	=	= particle Reynolds number (= ρVgdp/εµ = Re/ε), dimensionless
Sc	=	= Schmidt number (= µ/ρD), dimensionless

Acronyms

AARE:	=	average absolute relative error
CFD:	=	computational fluid dynamics
DEM:	=	Discrete Element Method
ETC:	=	effective thermal conductivity
FCC:	=	face-centered cubic
Gen IV:	=	fourth generation of nuclear reactors
GFR:	=	gas-cooled fast reactor
GIF:	=	Generation IV International Forum
HTGR:	=	high-temperature gas-cooled reactor
HTR:	=	high-temperature reactor
HTTF:	=	heat transfer test facility
KTA:	=	German Nuclear Safety Standard Commission (Kerntechnischer Ausschuss)
LES:	=	large eddy simulation
LFR:	=	lead-cooled fast reactor
MSR:	=	molten salt reactor
NGNP:	=	next-generation nuclear plant
PBR:	=	pebble bed reactor
PEBBED:	=	Multi-Scale Simulation and Optimization of BPRs code
PyC:	=	pyrolytic carbon
RTD:	=	residence time distribution
SCWR:	=	supercritical-water-cooled reactor
SFR:	=	sodium-cooled fast reactor
SRM:	=	short-range radiation model
VDI:	=	Association of German Engineers (Verein Deutscher Ingenieur)
VHTR:	=	very high-temperature reactor
3D:	=	three-dimensional

Acknowledgments

This work was supported by the U.S. Department of Energy Nuclear Energy Research Initiative under grant NERI-08-043.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Office of Nuclear Energy [NERI-08-043].