Convective Heat Transfer in Gas-Cooled Nuclear Reactors—A Review

Abstract

Gas-cooled reactors are highly sophisticated energy systems in which numerous physical phenomena take place at the same time. Among these, the effective removal of heat from the reactor core is of great importance. In gas-cooled reactors, convective heat transfer and the conditions under which it occurs are critical to both the performance and safety of these reactors. Convective heat transfer in gas-cooled reactors is particularly complex due to the thermo-physical properties of gaseous coolants, high operating temperatures, and diverse flow regimes. It is commonly characterized using empirical and semi-empirical correlations. Each correlation is valid only within specific ranges of operating and geometric conditions, making the appropriate selection of correlations essential for accurate reactor design and reliable safety assessment. The aim of this review is to provide a comprehensive evaluation of the models and correlations applicable to the description and modeling of convective heat transfer in selected types of gas-cooled reactors. For each reactor type, the relevant correlations are categorized and summarized in tables, along with their ranges of applicability and inherent limitations. In total 154 correlations were reviewed. The findings highlight that convective heat transfer in different types of gas-cooled reactors is described differently. This article offer a consolidated reference of correlations useful for engineers and researchers working in the field of heat transfer and nuclear reactor engineering. In addition, remaining challenges are discussed and future research directions are proposed to support improved heat transfer modeling for current and next-generation gas-cooled reactor technologies.

1. Introduction

In an era of rapidly developing civilization and the continuously increasing energy demand of societies, it is exceedingly important to ensure the uninterrupted coverage of the demand for various forms of energy, such as electricity, mechanical work, usable and processed heat, and cooling. Among these, electricity constitutes one of the most essential forms of energy, considering the scale of demand [1]. For this reason, the development of technologies enabling large-scale industrial generation of electricity is one of the most crucial issues in contemporary power engineering.

The traditionally employed technology for large-scale electricity generation is thermal power plant operation on the Clausius–Rankine cycle and the utilization of the energy of fossil fuels, including hard coal, lignite, peat, fuel oil, and natural gas [2]. Although these technologies are capable of generating large quantities of energy, they are characterized by a relatively significant environmental impact due to the necessity of burning fuels and the associated emissions of combustion by-products, including environmentally harmful compounds [3]. Consequently, efforts have been undertaken for many years to develop energy technologies based on renewable energy sources that may serve as an alternative to conventional fossil fuel-based technologies. Such energy sources include solar radiation [4], wind energy [5], geothermal energy [6], and others.

However, some of these energy sources, such as solar and wind energy, exhibit temporal variability, which substantially complicates their use due to the frequent mismatch between demand and supply, the load imposed on the power distribution system, and the still insufficient capacity of energy storage systems [7]. Technologies based on renewable energy sources typically possess limited power output and must be adapted to the variable characteristics of the energy source [8].

Compared with renewable energy-based technologies and traditional fossil fuel power systems, nuclear power plants exhibit many advantageous features. Most importantly, they are characterized by high power output and the absence of emissions of harmful substances into the environment [9]. The main waste product associated with nuclear power plant operation is spent fuel, which is subsequently stored. For these reasons, increased interest in investments in modern nuclear power plants is currently observed [10].

Among the most commonly implemented nuclear reactors, serving as heat sources for nuclear power plants, are light- and heavy-water-cooled reactors (including the widely used PWR, BWR, and CANDU types) [11], as well as gas-cooled reactors (including GCR, AGCR, GFR, PB, PBR, HTGR, HTGR-POLA, HTR-PM, and others) [12,13]. Owing to the power output achieved and the nature of the phenomena associated with heat release from fuel elements into the coolant taking place in the reactor core, it is crucial, regardless of reactor type, to ensure appropriate heat transfer conditions enabling efficient and safe reactor operation [14].

In nuclear reactors, heat is transferred through conduction, convection, and radiation [15]. The contribution of individual mechanisms may vary depending on the reactor type. Convection is one of the most significant mechanisms [16], as its characteristic parameters, such as the value of the convective heat transfer coefficient, substantially influence the efficiency of heat transfer from the surface of fuel elements to the coolant [15]. The value of this coefficient depends on dimensionless numbers characterizing convection, most importantly the Reynolds ($Re=w·d_h/\nu$), Prandtl ($Pr=\nu /a$), Nusselt ($Nu$), and Rayleigh ($Ra=g·\beta ·∆T·d_h^3/\nu ·a$) numbers, as well as on the type and thermophysical properties of the coolant, and the flow configuration of the reactor core system [17]. The equations that relate these parameters are referred to as correlation equations, such as $Nu=C·{Re}^a·{Pr}^b$, and they are obtained via dimensional analysis based on experimental data. An important parameter regarding the criteria numbers is the characteristic dimension (e.g., diameter or length), the definition of which may vary depending on the configuration of the hydraulic system. The characteristic dimension depends on the type of flow geometry. For flow inside a circular pipe, the characteristic dimension is the inner diameter of the pipe, d. In the case of conduits with a non-circular cross-section, the hydraulic diameter is used, defined as $d_h=4·A/P$, where A is the cross-sectional area of the channel and P is the wetted perimeter. For example, for a rectangular duct, the hydraulic diameter is given by $d_h={\displaystyle \frac{2·a·b}{a+b}}$, where a and b denote the side dimensions of the duct.

For external flows around bodies such as spheres or cylinders, the characteristic dimension is taken as the outer diameter D of the object. In the case of flow over a flat plate, the characteristic dimension is the plate length L measured in the direction of the flow.

Their range of applicability is typically limited to the parameter domain in which the given experiment was conducted. For nuclear reactors, different correlation equations are used to describe convective phenomena occurring in different reactor types and with different coolants [15,18].

Convective heat transfer processes in water-cooled and gas-cooled nuclear reactors differ from one another owing to differences in reactor design, operating temperature, and the characteristics of coolant behavior [15,18,19]. The gases most commonly used as coolants in gas-cooled reactors are helium and CO₂. Compared with water, these gases exhibit lower values of density, specific heat, and thermal conductivity. Due to these properties, the resulting heat transfer coefficients may be significantly lower [20] than in water-cooled reactors, which in turn leads to reduced heat transfer intensity. To ensure effective heat removal from the fuel elements in gas-cooled reactors, it is necessary to maintain higher coolant mass flow rates, flow velocities, and elevated pressures relative to those in water-cooled reactors.

Because gas-cooled reactors operate at high temperatures, often reaching several hundred Celsius degrees, the applicability of standard convection correlations may be limited due to their increasing inaccuracy at high temperatures. Furthermore, at such elevated coolant temperatures, thermal radiation plays a significant role in heat transfer. All these characteristics collectively make the description of heat exchange processes in gas-cooled reactors more complex than in water-cooled reactors.

In this article, the authors present a review of the literature concerning correlations that describe convective processes occurring in gas-cooled reactors (i.e., GCR, GFR, PB, and PBRs). For each reactor type, the equations are summarized in tables together with their applicability ranges and limitations.

Section 2 and Section 3 present a review of correlations used in GCR-type reactors and GFR-type reactors. The review is based on publications [21,22,23,24,25,26,27,28,29,30,31,32,33]. In the case of CGRs, specific descriptions concern convective heat transfer in porous media, where specific corrections are used in the correlations to take into account the surface porosity. In the case of GFRs, an important and most frequently studied phenomenon related to the description of heat flow is deorbited heat turbulent transfer (DTHT).

Section 4 presents a review of correlations used in PB-type reactors, based on publications [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58]. In the case of PBs due to the complex design of prismatic blocks, the description of convective heat transfer processes requires a combination of 1D and 3D models due to the homogenization of the core, which is a compromise between the accuracy of calculations and the required computational time and cost.

Section 5 presents a review of correlations used in PBR-type reactors, based on publications [59,60,61,62,63,64,65,66,67,68]. In the case of PBRs, correlations work well for determining convection parameters in macroscopic analysis but fail to predict local thermal stresses.

In

Table 1. A comparison of the most important technical data related to different types of gas-cooled nuclear reactors.

Reactor Type	Coolant	Moderator	Fuel Form	Typical Outlet Temperature (°C)	Pressure (bar)	Core Geometry	Key Design Features	Convective Heat Transfer Characteristics
GCR (Gas-Cooled Reactor)	CO2	Graphite	Natural uranium metal, cladded	ca. 400	ca. 10–20	Channel-type graphite core	Early UK Magnox design; low power density; large core	Forced convection of CO2 through fuel channels; relatively low heat transfer coefficient due to gas coolant and smooth channels
AGR (Advanced Gas-Cooled Reactor)	CO2	Graphite	Enriched UO2 pellets in stainless steel cladding	ca. 640	ca. 40	Channel-type graphite core	Higher temperature and efficiency than GCR; higher enrichment	High-pressure CO2 improves convective heat transfer vs. GCR; turbulent forced convection in narrow fuel channels
GFCR (Gas-Cooled Fast Reactor)	He	None (fast spectrum)	Mixed oxide (MOX) or metal fuel	ca. 850	ca. 70	Compact hexagonal lattice	Fast spectrum; no moderator; high power density	High-pressure helium with high velocity; relies on forced turbulent convection and tight fuel spacing to compensate for helium’s low density
PB (Pebble Bed Reactor)	He	Graphite (fuel pebbles)	TRISO particles in graphite pebbles	ca. 750–950	ca. 70	Randomly packed pebble bed	Online refueling; inherent safety via fuel design	Coolant flows through porous pebble bed; complex mixed forced and local natural convection, enhanced surface area but uneven flow paths
PBR (Prismatic Block Reactor)	He	Graphite blocks	TRISO fuel compacts in prismatic blocks	ca. 750–900	ca. 70	Fixed prismatic graphite blocks	Modular HTGR design; well-defined coolant channels	Helium forced through engineered channels; predictable turbulent convection, easier thermal hydraulic modeling than PB

, the comparison of the most important technical data related to different types of analyzed nuclear reactors is presented.

In addition to compiling existing correlations in tabular form, this article also offers an up-to-date review of ongoing research directions related to heat transfer analysis in gas-cooled nuclear reactors. This article may serve as a compendium of knowledge for engineers and researchers engaged in the design and in experimental and numerical investigations of thermal hydraulic processes occurring in gas-cooled nuclear reactors. This article also summarizes conclusions regarding directions for future research on gas-cooled reactors, such as the development of advanced hybrid methods; the need to develop new, more accurate correlations describing the convective heat transfer processes; the need for comprehensive experimental studies; etc.

2. Gas-Cooled Reactors

In their search for a reactor design that did not require enriched uranium, UK scientists developed the Magnox reactor. This type of reactor uses graphite as a moderator and carbon dioxide as the heat transfer medium. The carbon dioxide removes the heat generated in the moderator and transfers it to a heat exchanger, where it is used to produce steam to drive a turbine. The graphite moderator contains channels lined with magnesium alloy tubes—hence, the name “Magnox”—which hold the uranium fuel.

The United Kingdom built nine Magnox reactors, each with a unique design, and additional units were constructed in Japan and Italy. In the 1960s, the UK developed a second generation of gas-cooled reactors known as Advanced Gas-Cooled Reactors (AGRs), seven of which were built. These reactors retained the graphite moderator and carbon dioxide coolant but used 2% enriched uranium fuel encased in zirconium alloy rods. An AGR’s gas coolant reaches temperatures of around 650 °C before circulating through steam generator tubes located outside the reactor core. The higher core temperatures achieved in AGRs allow for greater power generation efficiency. However, the UK’s gas-cooled reactor fleet has faced operational challenges and has not matched the success of alternative reactor designs. Consequently, no further AGRs are planned, and the most recent reactor to enter service in the United Kingdom was a pressurized water reactor (PWR) [21].

Lim et al. [22] conducted a numerical investigation to analyze steady-state flow and heat transfer characteristics in the multiple-plate porous insulation used in the reactor pressure vessels of Magnox nuclear power plants. A three-dimensional computational model incorporating periodic boundary conditions was developed to simulate the heat transfer behavior within a single flow passage of the insulation pack under conditions of laminar forced convection and constant wall temperatures. The study identified a phenomenon of “mid-dimple peaking” in the Nusselt number distribution, which was attributed to the convective effects caused by distorted velocity profiles around the dimples. These findings improved the understanding of heat transfer mechanisms within the porous insulation structure and provided valuable guidance for evaluating the thermal performance of such systems in nuclear reactor applications.

Trinca, in his PhD thesis [23], focused on improving the accuracy of thermal predictions during refueling operations in Advanced Gas-Cooled Reactors (AGRs), where three-dimensional (3D) flow and heat transfer phenomena become significant but are often oversimplified in traditional one-dimensional analysis codes. The primary goals of his work were to better understand the 3D natural convection flow patterns within the AGR fuel stringer and to develop a computational tool capable of more realistic heat transfer analysis during these scenarios.

Two key contributions of Trinca’s work were highlighted:

• His LES (Large Eddy Simulation) study was, in the author’s opinion, the first simulation of turbulent natural convection within an enclosed concentric pin bundle, providing new Nusselt number correlations of gaps between the fuel pins and subchannels in the stringer (see Equations (4) and (5) in

  Table 2. The convective heat transfer correlations valid for GCR, AGR, and GFCR nuclear reactors.

  Ref	Field of Research and Conditions	Correlation	Remarks/Applicability and Limitations	Equation No.
  [22]	Improved natural convection heat transfer correlations for reactor cavity cooling systems of high-temperature gas-cooled reactors: From computational fluid dynamics to Pronghorn (numerical study)	$Nu=0.2490R{a}^{0.2911}$ $\mathrm{for}x\in \left[0,x_{s,tr}^{*}\right]$	$x_{s,tr}^{*}=e^{-0.4777R{a}^{0.06165}}{x}_{e,tr}^{*}=e^{-0.4349R{a}^{0.05607}}$ $x_{s,tr}^{*},x_{e,tr}^{*}$—nondimensional distances for the start and end of the transition	(1)
  		$Nu=0.1901R{a}^{0.2994}$ $\mathrm{for}x\in \left[x_{s,tr}^{*},x_{e,tr}^{*}\right]$		(2)
  		$Nu=0.08169R{a}^{0.3183}$ $\mathrm{for}x\in \left[x_{e,tr}^{*},1\right]$		(3)
  [23]	Nuclear Fuel Route Thermal Hydraulics Analysis for Advanced Gas-cooled Reactors (AGRs) (experimental study)	${Nu}_L=0.095{·Ra}^{0.322}$	Nusselt number of gaps between the fuel pins	(4)
  		${Nu}_{sc}=0.295·{Ra}_{sc}^{0.315}$	Nusselt number of sub-channels in the stringer	(5)
  		$N{u}_{\mathrm{ax}}=\sqrt[3]{N{u}_{\mathit{ax},\mathit{forced}}^3+N{u}_{\mathit{ax},\mathit{buoy}}^3}$	Nusselt number of forced and buoyant convection along both fuel pins and boundaries	(6)
  		$\mathit{Nu}=\sqrt{{\mathit{Nu}}_{\mathit{ax}}^2{\cos }^2{\varphi }_{\mathrm{z}}+{\left({\mathit{Nu}}_{\mathrm{cf}}{\displaystyle \frac{D_{\mathrm{h}}}{D_{\mathrm{p}}}}\right)}^2{\sin }^2{\varphi }_{\mathrm{z}}}$	Nusselt number along the pins from axial and transversal flow Dh = 4A/P—hydraulic diameter (m) Dp—diameter of the pin (m)	(7)
  		${\mathit{St}}_{\mathit{ax},\mathit{forced}}=H\left({\displaystyle \frac{T_b}{T_w}}\right){\mathit{Re}}^j$	Nuax,forced along rough pins, $St={\displaystyle \frac{Nu}{RePr}}$	(8)
  		${Nu}_{ax,forced}=\psi ·0.023·{Re}^{0.8}·Pr$	For smooth pins	(9)
  		${Nu}_{ax,forced}=K·{Re}^l·{Pr}^{1/3}$	For the boundary surfaces	(10)
  		${Nu}_{ax,buoy}=0.068{·Ra}^{0.37}$	For rough pins and boundary surfaces	(11)
  		${\mathit{log}}_{10}\left({\displaystyle \frac{{\mathit{Nu}}_{\mathit{cf}}}{{\mathit{Pr}}^{0.36}{\left(\frac{\mathit{Pr}}{{\mathit{Pr}}_w}\right)}^{0.25}}}\right)={\displaystyle \sum _{i=0}^4}{M}_i{\left({\mathit{log}}_{10}\mathit{Re}\right)}^i$	Nusselt number due to cross flow	(12)
  [26]	Effect of Cladding Surface Roughness on Thermal-Hydraulic Response of Nuclear Fuel Rod of Advanced Gas-Cooled Reactor (numerical study)	$Nu={\displaystyle \frac{\left(f/8\right)RePr}{1.07+12.7{\left(f/8\right)}^{0.5}\left(P{r}^{2/3}-1\right)}}$	$\mathrm{F}\mathrm{o}\mathrm{r} \mathrm{s}\mathrm{m}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{h} \mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}$ $0.5\le Pr\le 2000$ ${10}^4<Re<5\times {10}^6$ $f={\left(0.790\ln Re-1.64\right)}^{-2}$ $\mathrm{for}3000<Re<5\times {10}^6$	(13)
  		$N{u}_{\alpha }=0.117R{e}^{0.78}P{r}^{0.542}{\left({\displaystyle \frac{k}{d}}\right)}^{0.3}{\left({\displaystyle \frac{p}{d}}\right)}^{-0.22}{\left({\displaystyle \frac{a}{90}}\right)}^{0.215}$	$\mathrm{F}\mathrm{o}\mathrm{r} \mathrm{r}\mathrm{i}\mathrm{b}—\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h} \mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}$ d—pipe diameter (m)	(14)
  [31]	Investigation of Fundamental Thermal Hydraulic Phenomena in Advanced Gas-Cooled Reactors (experimental study)	${Nu}_{MIT1}={\displaystyle \frac{\left(f/8\right)\left(Re-1000\right)Pr}{1+12.7\sqrt{f/8}\left(P{r}^{2/3}-1\right)}}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.45}\left(1+{\left({\displaystyle \frac{x}{D}}\right)}^{-{\displaystyle \frac{2}{3}}}\right)$	If $K_{v,\mathit{inlet}}<2.0\times {10}^{-6}$ ${\mathit{Bo}}_{\mathit{inlet}}^{*}<2.0\times {10}^{-6}$ ${\mathit{Re}}_{\mathit{inlet}}>2300$ D—pipe diameter (m) (Turbulent)	(15)
  		${Nu}_{\mathit{MIT}1-\mathit{temp}}={\displaystyle \frac{\left(f/8\right)\left(Re-{{0.185K}_v}^{-2/3}\right)Pr}{1+12.7\sqrt{f/8}\left(P{r}^{2/3}-1\right)}}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.45}\left(1+{\left({\displaystyle \frac{x}{D}}\right)}^{-{\displaystyle \frac{2}{3}}}\right)$	If $K_{v,\mathit{inlet}}\ge 2.0\times {10}^{-6}$ ${\mathit{Bo}}_{\mathit{inlet}}^{*}<2.0\times {10}^{-6}$ $R{e}_{\mathit{inlet}}>2300$ D—pipe diameter (m) (Kv DTHT) $N{u}_{\mathit{MIT}1}=\mathit{max}\left(N{u}_{\mathit{MIT}1-\mathit{temp}},N{u}_{\mathit{Laminar}}\right)$	(16)
  		${\mathit{Nu}}_{\mathit{MIT}1-\mathit{temp}}={\displaystyle \frac{\left(f/8\right)\left(Re-1.45\times {10}^{-7}{Bo}^{-1.7}\right)}{1+12.7\sqrt{f/8}\left(P{r}^{2/3}-1\right)}}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.45}\left(1+{\left({\displaystyle \frac{x}{D}}\right)}^{-{\displaystyle \frac{2}{3}}}\right)$	If $Bo<2.0\times {10}^3$ $3.5\times {10}^3<{\mathit{Bo}}_{\mathit{inlet}}^{*}<2.0\times {10}^6$ $R{e}_{\mathit{inlet}}>2300$ D—pipe diameter (m) $N{u}_{\mathit{MIT}1}=\mathit{max}\left(N{u}_{\mathit{MIT}1-\mathit{temp}},N{u}_{\mathit{Laminar}}\right)$	(17)
  		${\mathit{Nu}}_{\mathit{MIT}1-\mathit{temp}}={\displaystyle \frac{\left(f/8\right)\left(Re-1.45\times {10}^{-7}{Bo}^{-1.7}\right)}{1+12.7\sqrt{f/8}\left({Pr}^{2/3}-1\right)}}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.45}\left(1+{\left({\displaystyle \frac{x}{D}}\right)}^{-{\displaystyle \frac{2}{3}}}\right)$	If $K_{v,\mathit{inlet}}<2.0\times {10}^{-6}$ $3.5\times {10}^3<{\mathit{Bo}}_{\mathit{inlet}}^{*}<2.0\times {10}^6$ $R{e}_{\mathit{inlet}}>2300$ D—pipe diameter (m) ${Bo}^{*}\ge 6.0\times {10}^{-7}$ $\left(Re—\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{i}\mathrm{n}\mathrm{g}{Bo}^{*}DTHT\right)$ $N{u}_{\mathit{MIT}1}=\mathit{max}\left(N{u}_{\mathit{MIT}1-\mathit{temp}},N{u}_{\mathit{Laminar}}\right)$	(18)
  		${\mathit{Nu}}_{\mathit{MIT}1-\mathit{temp}}={\displaystyle \frac{\left(f/8\right)\left(Re-8.34\times {10}^7{Bo}^{0.69}\right)}{1+12.7\sqrt{f/8}\left({Pr}^{2/3}-1\right)}}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.45}\left(1+{\left({\displaystyle \frac{x}{D}}\right)}^{-{\displaystyle \frac{2}{3}}}\right)$	If $K_{v,\mathit{inlet}}<2.0\times {10}^{-6}$ $3.5\times {10}^{-6}>{\mathit{Bo}}_{\mathit{inlet}}^{*}\ge 2.0\times {10}^{-6}$ $R{e}_{\mathit{inlet}}>2300$ ${Bo}^{*}<6.0\times {10}^{-7}$ D—pipe diameter (m) $\left(Re—\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{i}\mathrm{n}\mathrm{g}{Bo}^{*}DTHT\right)$ $N{u}_{\mathit{MIT}1}=\mathit{max}\left(N{u}_{\mathit{MIT}1-\mathit{temp}},N{u}_{\mathit{Laminar}}\right)$	(19)
  		${\mathit{Nu}}_{\mathit{MIT}1-\mathit{temp}}={\displaystyle \frac{\left(f/8\right)\left(Re-79.4\times {10}^7{Bo}^{-0.28}\right)}{1+12.7\sqrt{f/8}\left({Pr}^{2/3}-1\right)}}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.45}\left(1+{\left({\displaystyle \frac{x}{D}}\right)}^{-{\displaystyle \frac{2}{3}}}\right)$	If $K_{v,\mathit{inlet}}<2.0\times {10}^{-6}$ ${\mathit{Bo}}_{\mathit{inlet}}^{*}>3.5\times {10}^{-6}$ $R{e}_{\mathit{inlet}}>2300$ D—pipe diameter (m) (${Bo}^{*}DTHT$) $N{u}_{\mathit{MIT}1}=\mathit{max}\left(N{u}_{\mathit{MIT}1-\mathit{temp}},N{u}_{\mathit{Laminar}}\right)$	(20)
  		$N{u}_{MIT-Laminar}=\max \left(1,\hspace{0.25em}3.0{\left({\displaystyle \frac{G{r}_q}{R{e}^2}}\right)}^{0.11}\right)N{u}_{Laminar}$	$IfR{e}_{\mathit{inlet}}\le 2300$ (Mixed Convection Laminar and Forced Convection Laminar) $N{u}_{Laminar}={\left[{\displaystyle \frac{1}{N{u}_{\infty }}}-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{m=1}^{10}}\mathit{exp}{\displaystyle \frac{\left(-{\gamma }_m^2{x}^{+}\right)}{A_m{\gamma }_m^4}}\right]}^{-1}$ $f={(1.82{\mathit{log}}_{10}\mathit{Re}-1.64)}^{-2}$ $N{u}_{\infty }=4.364$ $x^{+}={\displaystyle \frac{2x/D}{Re\hspace{0.17em}Pr}};{\gamma }_m=4m+{\displaystyle \frac{4}{3}}$ $A_m=0.4165{\gamma }_m^{-7/3}$	(21)
  		$N{u}_{MTT2}={\displaystyle \frac{\left(\frac{f}{8}\left(Re+500\right)Pr\right)}{1+12.7\sqrt{\frac{f}{8}}\left(P{r}^{2/3}-1\right)}}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.5}\left(1+{{\displaystyle \frac{x}{D}}}^{-2/3}\right)$	If $K_{v,\mathit{inlet}}<2.0\times {10}^{-6}$ ${\mathit{Bo}}_{\mathit{inlet}}^{*}<2.0\times {10}^{-6}$ ${\mathit{Re}}_{\mathit{inlet}}>2300$ (Turbulent) D—pipe diameter (m)	(22)
  		$N{u}_{MTT2-temp}={\displaystyle \frac{\left(\frac{f}{8}\left[Re-3500{\log }_{10}\left(3.8\times 10^5{K}_v\right)\right]Pr\right)}{1+12.7\sqrt{\frac{f}{8}}\left(P{r}^{2/3}-1\right)}}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.5}\left(1+{{\displaystyle \frac{x}{D}}}^{-2/3}\right)$	If $K_{v,\mathit{inlet}}\ge 2.0\times {10}^{-6}$ ${\mathit{Bo}}_{\mathit{inlet}}^{*}<2.0\times {10}^{-6}$ $R{e}_{\mathit{inlet}}>2300$ D—pipe diameter (m) (Kv DTHT) $N{u}_{\mathit{MIT}2}=\mathit{max}\left(N{u}_{\mathit{MIT}2-\mathit{temp}},N{u}_{\mathit{Laminar}}\right)$	(23)
  		$N{u}_{MTT2-temp}={\displaystyle \frac{\left(\frac{f}{8}\left[Re+6500{\log }_{10}\left(4.3\times 10^{5}B{o}^{*}\right)\right]Pr\right)}{1+12.7\sqrt{\frac{f}{8}}\left(P{r}^{2/3}-1\right)}}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.5}\left(1+{{\displaystyle \frac{x}{D}}}^{-2/3}\right)$	If $K_{v,\mathit{inlet}}<2.0\times {10}^{-6}$ $3.5\times {10}^{-6}>{\mathit{Bo}}_{\mathit{inlet}}^{*}\ge 2.0\times {10}^{-6}$ $R{e}_{\mathit{inlet}}>2300$ ${Bo}^{*}\ge 6.0\times {10}^{-7}$ D—pipe diameter (m) $\left(Re—\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{i}\mathrm{n}\mathrm{g}{Bo}^{*}DTHT\right)$ $N{u}_{\mathit{MIT}1}=\mathit{max}\left(N{u}_{\mathit{MIT}2-\mathit{temp}},N{u}_{\mathit{Laminar}}\right)$	(24)
  		$N{u}_{MTT2-temp}={\displaystyle \frac{\left(\frac{f}{8}\left[Re-3900{\log }_{10}\left(2\times 10^{7}B{o}^{*}\right)\right]Pr\right)}{1+12.7\sqrt{\frac{f}{8}}\left(P{r}^{2/3}-1\right)}}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.5}\left(1+{{\displaystyle \frac{x}{D}}}^{-2/3}\right)$	If $K_{v,\mathit{inlet}}<2.0\times {10}^{-6}$ $3.5\times {10}^{-6}>{\mathit{Bo}}_{\mathit{inlet}}^{*}\ge 2.0\times {10}^{-6}$ $R{e}_{\mathit{inlet}}>2300$ ${Bo}^{*}<6.0\times {10}^{-7}$ D—pipe diameter (m) $\left(Re—\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{i}\mathrm{n}\mathrm{g}{Bo}^{*}DTHT\right)$ $N{u}_{\mathit{MIT}2}=\mathit{max}\left(N{u}_{\mathit{MIT}2-\mathit{temp}},N{u}_{\mathit{Laminar}}\right)$	(25)
  		$N{u}_{MTT2-temp}={\displaystyle \frac{\left(\frac{f}{8}\left[Re+2000{\log }_{10}\left(1.6\times 10^{4}B{o}^{*}\right)\right]Pr\right)}{1+12.7\sqrt{\frac{f}{8}}\left(P{r}^{2/3}-1\right)}}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.5}\left(1+{{\displaystyle \frac{x}{D}}}^{-2/3}\right)$	If $K_{v,\mathit{inlet}}<2.0\times {10}^{-6}$ ${\mathit{Bo}}_{\mathit{inlet}}^{*}>3.5\times {10}^{-6}$ $R{e}_{\mathit{inlet}}>2300$ D—pipe diameter (m) (${Bo}^{*}DTHT$) $N{u}_{\mathit{MIT}2}=\mathit{max}\left(N{u}_{\mathit{MIT}2-\mathit{temp}},N{u}_{\mathit{Laminar}}\right)$	(26)
  		$N{u}_{MIT-Laminar}=\max \left(1,\hspace{0.25em}3.0{\left({\displaystyle \frac{G{r}_q}{R{e}^2}}\right)}^{0.11}\right)N{u}_{Laminar}$	$IfR{e}_{\mathit{inlet}}\le 2300$ (Mixed Convection Laminar and Forced Convection Laminar)	(27)
  		$f={\left(1.82{\mathit{log}}_{10}Re-1.64\right)}^{-2}$	$IfRe\ge 10000$	(28)
  		$f=0.314/{Re}^{0.25}$	$If10000>Re\ge 4000$	(29)
  		$f=0.012+6.86{\times 10}^{-6}Re$	$If4000>Re\ge 2300$	(30)
  		$N{u}_{Laminar}={\left[{\displaystyle \frac{1}{N{u}_{\infty }}}-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{m=1}^{10}}\mathit{exp}{\displaystyle \frac{\left(-{\gamma }_m^2{x}^{+}\right)}{A_m{\gamma }_m^4}}\right]}^{-1}$	$If2300>Re$ ${Nu}_{\infty }=4.364,f=64/\mathit{Re}$ $x^{+}={\displaystyle \frac{2x/D}{Re\hspace{0.17em}Pr}};{\gamma }_m=4m+{\displaystyle \frac{4}{3}}$ D—pipe diameter (m) $A_m=0.4165\hspace{0.17em}{\gamma }_m^{-7/3}$	(31)
  		${Nu}_{MIT3}={\displaystyle \frac{\left(f/8\right)\left(Re-1000\right)Pr}{1+12.7\sqrt{f/8}\left(P{r}^{2/3}-1\right)}}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.5}\left(1+{\left({\displaystyle \frac{x}{D}}\right)}^{-{\displaystyle \frac{2}{3}}}\right)$	If $K_{v,\mathit{inlet}}<2.0\times {10}^{-6}$ $B{o}_{\mathit{inlet}}^{*}<2.0\times {10}^{-6}$ ${\mathit{Re}}_{\mathit{inlet}}>2300$ D—pipe diameter (m) (Turbulent)	(32)
  		${Nu}_{MIT3-temp}={\displaystyle \frac{\left(f/8\right)\left(Re-0.011{\left(\frac{q^{+}}{{Re}^{0.44}}\right)}^{-1.16}\right)Pr}{1+12.7\sqrt{f/8}\left(P{r}^{2/3}-1\right)}}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.5}\left(1+{\left({\displaystyle \frac{x}{D}}\right)}^{-{\displaystyle \frac{2}{3}}}\right)$	If $K_{v,\mathit{inlet}}\ge 2.0\times {10}^{-6}orB{o}_{\mathit{inlet}}^{*}<2.0\times {10}^{-6}$ $R{e}_{\mathit{inlet}}>2300$ D—pipe diameter (m) (DTHT) $N{u}_{\mathit{MIT}3}=\mathit{max}\left(N{u}_{\mathit{MIT}3-\mathit{temp}},N{u}_{\mathit{Laminar}}\right)$	(33)
  		$N{u}_{MIT-Laminar}=\max \left(1,\hspace{0.25em}3.0{\left({\displaystyle \frac{G{r}_q}{R{e}^2}}\right)}^{0.11}\right)N{u}_{Laminar}$	$IfR{e}_{\mathit{inlet}}\le 2300$ (Mixed Convection Laminar and Forced Convection Laminar) $N{u}_{Laminar}={\left[{\displaystyle \frac{1}{N{u}_{\infty }}}-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{m=1}^{10}}\mathit{exp}{\displaystyle \frac{\left(-{\gamma }_m^2{x}^{+}\right)}{A_m{\gamma }_m^4}}\right]}^{-1}$ $f={(1.82{\mathit{log}}_{10}\mathit{Re}-1.64)}^{-2}$ $N{u}_{\infty }=4.364$ $x^{+}={\displaystyle \frac{2x/D}{Re\hspace{0.17em}Pr}};{\gamma }_m=4m+{\displaystyle \frac{4}{3}}$ $A_m=0.4165\hspace{0.17em}{\gamma }_m^{-7/3}$	(34)
  [32]	Studies of the deteriorated turbulent heat transfer regime for the gas-cooled fast reactor decay heat removal system (experimental study)	$Nu=CF\times {\displaystyle \frac{\left(f/8\right)\left(Re-1000\right)Pr}{1+12.7{\left(f/8\right)}^{0.5}\left(P{r}^{2/3}-1\right)}}$	$f={\left({\displaystyle \frac{16}{1.5635\mathit{ln}\left(Re/7\right)}}\right)}^2$ $CF={\left({\displaystyle \frac{T_b}{T_{wi}}}\right)}^{0.45}$	(35)

  ).
• His hybrid modeling approach—combining porous media modeling with detailed CFD—laid the groundwork for subsequent research on coarse-grid CFD within the Heat, Flow, and Turbulence Research Group at the University of Sheffield.

In Appendices A and B of [23], the author presents the correlations for Nusselt number used in calculations. As the axial shear stress, the Nusselt number was calculated by blending the values obtained from correlations for forced flow and natural convection, as shown in Equation (6) in Table 2. To derive a correlation for the Nusselt number along the pins, the author blended the contributions from axial and transversal flow components, as shown in Equation (7) in Table 2. The first one, between the forced and buoyant axial contributions, uses a cubic expression, and it is carried out for both the pins and the boundaries. The correlation for the Nusselt number representing the axial forced flow contribution along rough pins was provided by Romero [24] in the form of a Stanton number (see Equation (8) in Table 2). For smooth pins, the correlation used by the author was derived from the Dittus–Boelter correlation [25] (see Equation (9) in Table 2). For the boundary surfaces, the forced convection contribution to the Nusselt number was given by the following correlation (see Equation (10) in Table 2). The contribution from natural convection for both rough pins and boundary surfaces was calculated with the use of Equation (11) in Table 2. The Nusselt number due to cross-flow was determined with the use of Equation (12) in Table 2. Overall, his research significantly advanced the state of heat transfer modeling during AGR refueling, especially in terms of understanding and quantifying 3D convective heat transfer, and produced a practical, validated numerical tool (POSTR) that is both computationally efficient and accurate for engineering use.

Hossain Nishat et al. [26] conducted a thermal hydraulic study of an isolated Advanced Gas-Cooled Reactor (AGR) nuclear fuel rod with smooth and rough cladding surfaces, which was carried out by a computational fluid dynamics simulation and an analytical calculation. Their conclusion was that the parameters of the rough cladding surface show greater values than those for the smooth surface except for surface heat flux.

It was found that only the average surface heat flux increases with an increasing pitch/height ratio. On the other hand, the average values of wall shear stress, Darcy friction factor, skin friction factor, convective heat transfer coefficient, Nusselt number, and thermal hydraulic performance decrease with an increasing pitch-to-height ratio. The simulated results were found to be very close to the values obtained from an analytical calculation. Also, square and circular ribs were compared. The circular ribs showed lower values of convective heat transfer coefficient and wall shear stress but permit high surface heat flux. Hossain Nishat et al. calculated the Nusselt number for the smooth surface from the Petukhov correlation [27] and Gnielinski [28] correlation shown in Equation (13) in Table 2. The temperatures used in the correlation [28] are expressed in Celsius degrees. Ravigururajan and Bergles correlations [29] have been used to calculate the Nusselt number for the rib-rough surface using Equation (14) in Table 2.

3. Gas-Cooled Fast Reactors

The Gas-Cooled Fast Reactor (GFR) system is a high-temperature, helium-cooled, fast-spectrum reactor operating on a closed fuel cycle. Reactors of this type are currently being developed by the European V4G4 Consortium, General Atomics (USA), and Framatome (France). It combines the advantages of fast-spectrum reactors—such as efficient long-term use of uranium resources and reduced radioactive waste through repeated fuel reprocessing and the fission of long-lived actinides—with those of high-temperature systems, which offer high thermal efficiency and the potential for industrial heat applications, including hydrogen production. The GFR shares fuel recycling processes with the Sodium-Cooled Fast Reactor (SFR) and incorporates similar reactor technologies to the Very-High-Temperature Reactor (VHTR). Consequently, its development strategy emphasizes leveraging existing VHTR technologies wherever possible, including materials, structural components, and power conversion systems. However, the GFR also requires additional research and development, particularly in areas such as core design and safety systems.

The reference GFR design features a 2400-MWth reactor core housed within a steel pressure vessel. The core is composed of hexagonal fuel elements containing ceramic-clad, mixed-carbide fuel pins arranged within ceramic hex-tubes. Silicon carbide fiber-reinforced silicon carbide (SiC/SiC) is currently the preferred material for both the fuel cladding and hex-tubes. The reactor core is enclosed within its steel pressure vessel and surrounded by the main heat exchangers and decay heat removal systems. The entire primary circuit is further enclosed by a secondary pressure boundary known as the guard containment. Helium serves as the primary coolant, exiting the core at approximately 850 °C. Heat from the primary helium loop is transferred to a secondary gas circuit containing a helium–nitrogen mixture, which drives a closed-cycle gas turbine. Waste heat from the gas turbine exhaust is then used in a steam generator to produce steam for a secondary steam turbine. This combined-cycle configuration is similar to that used in modern natural gas power plants, with the main distinction being that the GFR employs a fully closed gas turbine system [30].

McEligot et al. [31] conducted a comprehensive experimental study to investigate mixed convection heat transfer in Gas-Cooled Fast Reactor (GFR) core channels, with particular attention to the deteriorated turbulent heat transfer (DTHT) regime that can emerge under post-loss-of-coolant accident (LOCA) conditions. These scenarios are especially relevant for block-type GFR cores, where the flow may deviate from conventional forced convection and enter regimes influenced by buoyancy or flow acceleration, leading to substantial reductions in heat transfer performance.

To address the critical need for accurate Nusselt number correlations in these nonstandard regimes, the authors carried out experiments using nitrogen, helium, and carbon dioxide across a broad range of Reynolds numbers, Jackson’s buoyancy parameter, acceleration parameter, and wall-to-bulk temperature ratios. The resulting dataset captured transitions between forced turbulent convection, mixed convection, and DTHT regimes—including a newly observed phenomenon termed returbulization, where heat transfer recovers from the deteriorated state due to strong temperature-dependent gas property changes.

Upon evaluating numerous existing correlations for forced convection, mixed convection, and DTHT, McEligot et al. [31] concluded that none provided reliable predictions across the full spectrum of observed behavior, particularly for gases under DTHT conditions. In response, the authors developed a new family of Nusselt number correlations built upon the well-established Gnielinski correlation for forced turbulent flow [28]. They introduced empirical modification functions—based on either the acceleration parameter or the buoyancy parameter—to adapt the original formulation for use in acceleration-driven and buoyancy-driven DTHT regimes, respectively. The result was a set of three tailored Nusselt number correlations:

• Type 1: A high-accuracy correlation suited for scientific applications;
• Type 2: A simplified version that removes iteration on wall temperature, balancing accuracy and computational efficiency;
• Type 3: A compact form for industrial applications, sacrificing some accuracy for ease of implementation.

Together, these correlations provided strong agreement with the experimental data, covering the entire heat transfer regime map within approximately ±20%. By explicitly incorporating the physical mechanisms driving DTHT, these formulations offer a much-needed improvement in predictive capability for decay heat removal (DHR) analysis in next-generation GFR systems, where traditional models fall short. The temperatures used in the correlations are expressed in Kelvin.

Lee et al. [32] conducted a comprehensive study addressing the challenges of passive emergency cooling in Gas-Cooled Fast Reactors (GFRs), with a specific focus on the deteriorated turbulent heat transfer (DTHT) regime. As passive decay heat removal (DHR) systems based on natural gas circulation at elevated pressures are a core safety feature of GFR designs, understanding the limits and behavior of heat transfer under low-velocity, high-surface-flux conditions is essential—especially following depressurization accidents. Due to the GFR core’s high power density and low thermal inertia, effective heat removal is critical in such scenarios.

The authors reviewed prior research on these mechanisms and analyzed the DTHT regime by decomposing the governing nondimensional parameters, offering insight into how these factors interact and influence DHR system performance in GFRs. They concluded that the GFR DHR system is likely to operate within the DTHT regime and that this gas-phase regime exhibits distinct characteristics compared to DTHT behavior in liquids or supercritical fluids.

To support their analysis, Lee et al. [32] designed and constructed an experimental facility aimed at investigating the DTHT regime. Initial tests were conducted under forced convection conditions to validate the experimental setup. The results from three test runs at Reynolds numbers of 6700, 8000, and 12,800 showed strong agreement with the Gnielinski correlation (see Equation (35) in Table 2), a widely accepted model for forced convection heat transfer [8]. Despite being in the forced convection regime, the results revealed that variation in fluid properties had a significant impact on heat transfer performance. The authors used a friction factor (valid for 4000 < Re < 107) from Kakaç et al.’s book [33] and a correction factor from Gnielinski’s publication [28].

This work provided both theoretical and experimental foundations for understanding DTHT in gas flows, contributing to the design and validation of passive cooling systems in next-generation GFRs.

The correlations presented in a table provide a comprehensive framework for heat transfer analysis, though their applicability depends strictly on the flow regime and geometry. For low-flow and decay-heat removal scenarios, correlations accounting for mixed convection and buoyancy are essential to maintain accuracy when Re ≤ 2300 (see Equations (21), (27), and (34)). In contrast, fully turbulent conditions are better served by models integrating friction factor dependencies, which offer higher fidelity in high-pressure gas environments (see Equations (13), (15), and (22)). The impact of geometry is particularly evident when comparing smooth cladding (see Equation (13)) to the ribbed surfaces typical of AGRs, where specialized correlations must be used to account for increased turbulence (see Equations (7) and (14)). Furthermore, the transition to deteriorated turbulent heat transfer (DTHT) represents a critical safety limit; the NuMIT series (see Equations (16), (24), and (33)) provides the necessary criteria to predict this phenomenon using acceleration and buoyancy parameters. For high-temperature gradients, the inclusion of property correction factors (see Equation (35)) is vital to avoid overestimating cooling capacity during transient peaks. Ultimately, selecting the appropriate model requires balancing the specific cooling requirements of the reactor core with the prevailing physical phenomena, from buoyancy-driven flow to roughness-enhanced turbulence.

4. Prismatic Block Reactor

Prismatic block (PB) reactors represent an advanced iteration of high-temperature gas-cooled reactors (HTGRs) and are categorized under Generation IV technologies, often specifically classified as Very-High-Temperature Reactors (VHTRs) [34]. These reactors are engineered to reach core outlet temperatures between 750 and 950 °C, thereby maximizing thermal efficiency and facilitating high-grade applications such as hydrogen production, and industrial process heat supply. A defining characteristic of the PB design is its inherent safety, achieved through a low core power density (6.6 W/cm³) and the integration of substantial graphite masses acting as a thermal buffer. This configuration ensures passive heat removal during accidental transients [35].

The PB core features a fixed, geometrically defined structure composed of columns of hexagonal graphite blocks [36]. The graphite serves a dual role as both the moderator (slowing down neutrons) and the reflector [34]. Inside each graphite block, channels are drilled for two primary purposes:

• Fuel channels: These contain stacks of cylindrical fuel compacts, within which TRISO particles are dispersed in a graphite matrix. TRISO particles consist of a fuel kernel surrounded by ceramic silicon carbide coatings, which serve as the primary safety barrier and ensure fuel integrity up to temperatures exceeding 1600 °C [37].
• Cooling channels: These are pathways for coolant circulation [36].

The core operates as a thermal reactor, the PB utilizes graphite to thermalize fast neutrons and sustain the fission chain reaction [34]. The operational principle is based on the circulation of helium gas as the coolant, which is typically maintained at a high pressure of approximately 7 MPa [35]. Helium is chemically inert, which minimizes the creation of radioactive fluids. During the fission process, thermal energy is transferred to the helium as it traverses the core’s cooling channels. The heated helium is then directed to a secondary system (e.g., a steam generator or a turbine in a direct Brayton cycle), where the thermal energy is converted into electricity. Finally, the cooled gas is recirculated back to the core, completing the closed-loop cycle [36].

In a prismatic block reactor, heat transfer is a conjugate process comprising conduction, radiation, and convection. Initially, heat is transferred via conduction within the solid graphite fuel matrix, propagating from the TRISO compacts to the boundaries of the cooling channels. Subsequently, the heat is dissipated into the coolant through convective heat transfer [38].

Normal operation (forced convection): during normal operation, the gaseous coolant (helium) is actively circulated through the core channels. The primary mechanism for thermal energy removal from the channel walls to the helium is forced convection [39]. However, in certain advanced compact designs (e.g., PAHTR), the flow may be in a developing mixed convection regime where buoyant forces are comparable to the forced flow [40].

Accident Mode (Natural Convection and Radiation): In the event of a total loss of active cooling (pressurized conduction cooldown—PCC accident), the reactor relies on passive heat removal. The large graphite mass slows the temperature rise [35]. Decay heat is removed radially via conduction through the graphite reflector to the outer surface of the reactor pressure vessel (RPV). From there, heat is transferred to the reactor cavity cooling system (RCCS) cooling panels, primarily through the coupled effects of thermal radiation and natural convection [41]. Natural convection is the movement of a fluid (in this case, a gas) driven by buoyancy forces resulting from thermal density differences, which transport heat [39]. Due to the high operating temperatures, radiation is a critical and often dominant mechanism in this passive heat removal process [41].

The convective heat transfer correlations valid for prismatic block nuclear reactors are summarized in

Table 3. The convective heat transfer correlations valid for prismatic block nuclear reactors.

Ref.	Field of Research and Conditions	Correlation	Remarks/Applicability and Limitations	Equation No.
[42]	Increasing the helium pressure (from 413.47 to 689.12 kPa) in a dual-channel VHTR enhances heat transfer and reduces wall temperatures, with Ra proportional to p2 (experimental study)	$h_i={\displaystyle \frac{q_i}{{(T}_{s,i}-T_{b,i})}}$	steady-state turbulent helium flow	(36)
		$h_{avg}={\displaystyle \frac{1}{N}}\cdot {\displaystyle \sum _{i=1}^{i=N}}{h}_i$	N = 2000	(37)
		$T_{b,i}={\displaystyle \frac{1}{8}}\cdot {\displaystyle \sum _{j=1}^{j=8}}{T}_{f,i,j}$	Bulk temp. as average of 8 values per segment; uniform mesh	(38)
		$Ra={\displaystyle \frac{g{\rho }^2\beta \Delta T{D}^3{C}_P}{\mu k}}$	413.47 ≤ p ≤ 689.12 kPa	(39)
		$Ra={\displaystyle \frac{g\beta \Delta T{D}^3{C}_P}{k\mu T^2{R}^2}}{p}^2=C$	Ra∝p2; ideal gas behavior confirmed for helium	(40)
		$\lambda ={\displaystyle \frac{K_B{T}_f}{\sqrt{2}\mathrm{\pi }{d}^{2}p}}$	d—helium molecules diameter (9.8 × 10−11 m) $\mathrm{D}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e} \mathrm{g}\mathrm{a}\mathrm{s}:\lambda \ll L,\lambda \approx d$ $\mathrm{D}\mathrm{i}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{g}\mathrm{a}\mathrm{s}:\lambda \ll L,\lambda \gg d$ $\mathrm{R}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{d} \mathrm{g}\mathrm{a}\mathrm{s}:\lambda \approx L$ $\mathrm{K}\mathrm{n}\mathrm{u}\mathrm{d}\mathrm{s}\mathrm{e}\mathrm{n} \mathrm{g}\mathrm{a}\mathrm{s}:\lambda \gg L$	(41)
		$Kn={\displaystyle \frac{\lambda }{L}}$	L—channel length (m) $\mathrm{D}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e} \mathrm{g}\mathrm{a}\mathrm{s}:\lambda \ll L,\lambda \approx d$ $\mathrm{D}\mathrm{i}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{g}\mathrm{a}\mathrm{s}:\lambda \ll L,\lambda \gg d$ $\mathrm{R}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{d} \mathrm{g}\mathrm{a}\mathrm{s}:\lambda \approx L$ $\mathrm{K}\mathrm{n}\mathrm{u}\mathrm{d}\mathrm{s}\mathrm{e}\mathrm{n} \mathrm{g}\mathrm{a}\mathrm{s}:\lambda \gg L$	(42)
		${Nu}_D={\displaystyle \frac{h_{avg}D}{k}}$	D—inside diameter of the flow channel (m) 413.47 ≤ p ≤ 689.12 kPa 0.044 ≤ Z/L ≤ 0.956	(43)
[43]	A loss coefficient correlation for PMR200 cross-gap flow has been developed using a combination of experimental and CFD data. This captures the effect of Reynolds number on the flow and improves the accuracy of bypass flow prediction (experimental and numerical study)	$K_{Gr}={\left({\displaystyle \frac{A_{CG}}{A_{CH}}}\right)}^2\left[3.58{\left({\displaystyle \frac{\delta }{D}}\right)}^{-2.3}\cdot 6.33{\left({\displaystyle \frac{A_{CG}}{\delta \cdot a}}\right)}^{-1.68}\right]$	ACG—inlet cross-sectional area of the crossflow gap ACH—coolant channel area D—channel diameter (m) a—length of one edge of the hexagonal interface at the cross gap For PMR200 geometry; valid for δ/D = 0.01–0.04 Re = 1500–8000; CFD and experimental validation	(44)
		$K={\displaystyle \frac{A_{CG}^2{C}_1}{{\delta }^{3}R{e}_{CG}}}$	Empirical model for gap bypass loss; valid for moderate Re = 0.002–0.004	(45)
		$K={\displaystyle \frac{A_{CG}^2{C}_2}{{\delta }^2}}$	Reynolds-independent loss term; typically used at high Re or simplified cases	(46)
		$K_{ka}={\left({\displaystyle \frac{A_{CG}}{\delta }}\right)}^2\left({\displaystyle \frac{C_1}{\delta R{e}_{CG}}}+C_2\right)$	Unified form; smooth transition between Re-dependent and Re-independent loss; fit for PMR200 blocks	(47)
		$R{e}_{CG}={\displaystyle \frac{4m_{cross}}{\mu p}}$	$P_{wedge}=10·a\hspace{1em}$ $P_{parallel}=12·a$	(48)
		$K_{PMR200}={\displaystyle \frac{C_1}{\delta R{e}_{CG}}}+C_2$	Final loss coefficient correlation for PMR200; recommended for system-scale modeling	(49)
[44]	A validated porous media model for the CFD analysis of prismatic HTGR cores shows good agreement with prototype results and enables efficient, full-core simulations with a greatly reduced mesh size and computation time (numerical study)	${\displaystyle \frac{N_1\mathrm{\pi }{d}_1^2}{\epsilon 4}}={\displaystyle \frac{N_2\mathrm{\pi }{d}_2^2}{4}}$	d—channel diameter (m) Flow area conservation under porosity correction; assumes equivalent thermal cross-section	(50)
		${\displaystyle \frac{\dot{q_2^{″}}}{\dot{q_1^{″}}}}={\displaystyle \frac{h_2}{h_1}}={\displaystyle \frac{N_1{d}_1}{N_2{d}_2}}$	Heat flux density scaling under volume homogenization; applies to channel-based to porous transition d—channel diameter (m)	(51)
		$N{u}_1={\displaystyle \frac{h_1{d}_1}{k_{he}}}={\displaystyle \frac{\left(f/8\right)\left(Re-1000\right)Pr}{1+12.7\sqrt{f/8}\left(P{r}^{2/3}-1\right)}}$	Gnielinski correlation; turbulent flow regime; d—channel diameter (m) Re > 1000; 0.7 ≤ Pr ≤ 500	(52)
		$h_2={\displaystyle \frac{N{u}_2\hspace{0.17em}{k}_{eff}}{d_2}}$	Used for porous domain convection estimation; applicable to homogenized geometry d—channel diameter (m)	(53)
		$N{u}_2=4.364$	Laminar flow	(54)
		$k_3={\displaystyle \frac{k_{eff}-\epsilon k_{he}}{1-\epsilon }}\approx 585 \mathrm{W}/\mathrm{m}\text{·}\mathrm{K}$	Effective solid matrix conductivity; validated for HTGR graphite block with helium porosity ε = 0.39	(55)
		$B{i}_R={\left({\displaystyle \frac{h\delta }{k}}\right)}_{\mathrm{R}}=1$	Biot number at reference point set to unity; ensures thermal resistance parity between models	(56)
		$h_1={\displaystyle \frac{N{u}_1\hspace{0.17em}{k}_{he}}{d_1}}$	Direct application of Nusselt number in channel domain; helium properties	(57)
		$N{u}_1=4.364$	Laminar flow	(58)
		$k_3={\displaystyle \frac{N_1-\epsilon N_2}{N_2\left(1-\epsilon \right)}}\hspace{0.17em}{k}_{he}$	Alternate form of solid conductivity expression with number density adjustment	(59)
[45] We have modified it into superscript format.	A validated porous media model for CFD analysis of a prismatic HTGR core shows good agreement with prototype results and enables efficient, full-core simulations with a greatly reduced mesh size and computation time. Passive flow in a prismatic reactor core was analyzed using a dual-channel facility, which revealed the strong effects of heating and flow reversal. The results provide benchmarks for CFD and thermal hydraulic studies (experimental study)	$G{r}_z={\displaystyle \frac{g\hspace{0.17em}\beta \hspace{0.17em}\Delta T_i\hspace{0.17em}{l}^3}{{\upsilon }^2}}$	Used to characterize buoyancy-driven natural convection; 413.47 ≤ p ≤ 689.12 kPa; Ti = 873–1173 K	(60)
		$\beta ={\displaystyle \frac{1}{{\overline{T}}_{b,i}}}$	Ideal gas approximation; Tb,i in K	(61)
		$\Delta T_i={\overline{T}}_{b,i}-{\overline{T}}_{s,i}$	Wall-to-bulk temperature difference; averaged over channel length	(62)
		$Re={\displaystyle \frac{U_{b,i}{D}_h}{\upsilon }}$	Dh = 0.0125 m, ν = 2.6 × 10−5 m2/s for He	(63)
		$Fr={\displaystyle \frac{U_{b,i}}{\sqrt{g\hspace{0.17em}\frac{T_{b,i}-T_{plenum}}{T_{b,i}}}\hspace{0.17em}l}}$	Froude number	(64)
		${\overline{Nu}}_i={\displaystyle \frac{{\overline{h}}_i\hspace{0.17em}l}{k}}$	l = 1.5 m	(65)
		${\overline{T}}_{b,i}={\displaystyle \frac{1}{9}}{\displaystyle \sum _{j=1}^{j=9}}{\overline{T}}_{f,i,j}$	Mean fluid temp from 9 radial points per cross-section	(66)
		${\overline{U}}_{f,i,j}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{N=1}^{N=6000}}{U}_{f,i,j}$	N = 6000	(67)
		$U_{b,i}={\displaystyle \frac{1}{9}}{\displaystyle \sum _{i=1}^{i=9}}{\overline{U}}_{f,i,j}$	Bulk axial velocity across 9 radial locations	(68)
		${\overline{T}}_{s,i}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{N=1}^{N=6000}}{T}_{s,i}$	N = 6000	(69)
		$h_i={\displaystyle \frac{q_i}{\left({\overline{T}}_{s,i}-T_{b,i}\right)}}$	Local convection coefficient; qi in W/m2	(70)
		${\overline{h}}_i={\displaystyle \frac{1}{N}}{\displaystyle \sum _{N=1}^{N=6000}}{h}_i$	Averaged local heat transfer coefficient	(71)
[46]	The analysis of a prismatic-core VHTR coupled with molten salt TES demonstrates stable neutronic and thermal hydraulic performance, making it ideal for efficient and flexible energy system integration (numerical study)	$L_T=D·\left(a\hspace{0.17em}R{e}_D\hspace{0.17em}Pr\right)$	D—coolant channel diameter (m) LT—length of the developing flow region Entrance length correlation for laminar flow a = 0.1; 0.7 < Pr < 1.0 and a = 0.15; Pr > 1	(72)
		$Nu={\displaystyle \frac{h_{FLiBe}\hspace{0.17em}D}{k}}=4.363$	Fully developed laminar flow	(73)
		$Nu=7.052$	Developing laminar flow	(74)
		$h_{FLiBe}={\left({\displaystyle \frac{Nu·k}{D}}\right)}_{\mathrm{F}\mathrm{L}\mathrm{i}\mathrm{B}\mathrm{e}}=480.1 {\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2\cdot \mathrm{K}}}$	-	(75)
		$h_{He}={\left({\displaystyle \frac{Nu·k}{D}}\right)}_{\mathrm{H}\mathrm{e}}=37.9 {\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2\cdot \mathrm{K}}}$	-	(76)
		$\Delta T_{tot}=\dot{q}\cdot R_{tot}$	-	(77)
		$R_{tot}=R_{con{v}_{in}}+R_{cla{d}_{in}}+R_{LiC{l}_{liquid}}+R_{LiC{l}_{solid}}+R_{cla{d}_{out}}+R_{con{v}_{out}}$	-	(78)
		$R_{con{v}_{in}}={\displaystyle \frac{1}{{\left(hA\right)}_{\mathrm{F}\mathrm{L}\mathrm{i}\mathrm{B}\mathrm{e}}}}$	A—heat transfer surface area	(79)
		$R_{cla{d}_{in}}={\displaystyle \frac{\ln \left(\frac{r_{cla{d}_{in}}}{r_{FLiBe}}\right)}{2\mathrm{\pi }{k}_{Incone{l}_{625}}{L}_{tube}}}$	Ltube—tube length (m)	(80)
		$R_{LiC{l}_{liquid}}={\displaystyle \frac{\ln \left(\frac{r_{LiC{l}_{liquid}}}{r_{cla{d}_{in}}}\right)}{2\mathrm{\pi }{k}_{LiC{l}_{liquid}}{L}_{tube}}}$	-	(81)
		$R_{LiC{l}_{solid}}={\displaystyle \frac{\ln \left(\frac{r_{LiC{l}_{solid}}}{r_{LiC{l}_{liquid}}}\right)}{2\mathrm{\pi }{k}_{LiC{l}_{solid}}{L}_{tube}}}$	-	(82)
		$R_{cla{d}_{out}}={\displaystyle \frac{\ln \left(\frac{r_{cla{d}_{out}}}{r_{LiC{l}_{solid}}}\right)}{2\mathrm{\pi }{k}_{Incone{l}_{625}}{L}_{tube}}}$	-	(83)
		$R_{con{v}_{out}}={\displaystyle \frac{1}{{\left(hA\right)}_{\mathrm{H}\mathrm{e}}}}$	-	(84)
[47]	CFD models (1D, 3D, and coupled 3D/1D) of a prismatic VHTR fuel block show good agreement with detailed simulations, enabling full-core analysis at reduced computational cost (numerical study)	$N{u}_{FD}=0.11{Re}_b^{0.646}P{r}_b^{0.4}$	D—channel diameter (m) z/D ≥ 25	(85)
		$Nu=N{u}_{FD}\cdot \left(1+0.57{\mathrm{e}}^{{\displaystyle \frac{-0.2z}{D}}}\right)$	D—channel diameter (m) z/D < 25	(86)
		$Nu=0.023R{e}_b^{0.8}P{r}_b^{0.4}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.5}$	Dittus–Boelter Nusselt number correlation	(87)
		$Nu=0.021R{e}_b^{0.8}P{r}_b^{0.4}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.5}\left[1+{\left({\displaystyle \frac{z}{D}}\right)}^{-0.7}\right]$	McEligot Nusselt number correlation D—channel diameter (m)	(88)
[48]	The CORONA code, which was developed by KAERI for local thermal analysis in prismatic VHTRs, has been validated against CFD. Despite the fact that the system code grids are coarse, it enables improved prediction of fuel temperature margins (numerical study)	$N{u}_{tur}=0.021\cdot R{e}^{0.8}\cdot P{r}^{0.4}$	Modified Dittus–Boelter Nusselt number correlation	(89)
		$N{u}_{tur}=0.021R{e}^{0.8}P{r}^{0.4}{\left({\displaystyle \frac{T_s}{T_f}}\right)}^{-0.5}{\left(1+{\displaystyle \frac{x}{D}}\right)}^{-0.7}$	D—coolant channel diameter (m) McEligot Nusselt number correlation	(90)
		$N{u}_{tur}={\displaystyle \frac{\left(f/8\right)\cdot \left(Re-1000\right)\cdot Pr}{\left(1+12.7\sqrt{f/8}\cdot \left(P{r}^{2/3}-1\right)\right)}}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.45}{\left(1+{\displaystyle \frac{x}{D}}\right)}^{-2/3}$	Gnielinski Nusselt number correlation	(91)
[49]	The applicability of empirical Nusselt number correlations to prismatic VHTR coolant channels is evaluated using 3D CFD data. The focus is on the influence of different wall heating conditions (numerical study)	$Nu=0.023\cdot R{e}^{0.8}P{r}^{0.4}$	Turbulent pipe flow	(92)
		$Nu=0.021\cdot R{e}^{0.8}P{r}^{0.4}{\left({\displaystyle \frac{T_s}{T}}\right)}^{-0.5}\left[1+{\left({\displaystyle \frac{z}{D}}\right)}^{-0.7}\right]$	D—coolant channel diameter (m) Turbulent pipe flow	(93)
[50]	The FastNet flow network code uses experimental and CFD-based correlations to predict the distribution of flow and temperature in a prismatic VHTR core. It produces results that are in good agreement with those from CORONA and CFD, while significantly reducing computation time (experimental and numerical study)	$h=Nu\cdot {\displaystyle \frac{k_f}{D_h}}$	Dh = 4A/P—hydraulic diameter of the coolant channel (m) Based on Nusselt number correlations implemented in flow network models	(94)
		$Nu=0.021R{e}^{0.8}P{r}^{0.4}$	Assumes turbulent flow in smooth, circular channels with constant wall heat flux and fluid properties	(95)
		$Nu=4.364$	Fully developed laminar flow	(96)
[51]	A 3D computational fluid dynamics (CFD) study of helium flow in a prismatic VHTR channel produced a turbulent heat transfer correlation that captured entrance effects and matched numerical results within ±2% (numerical study)	$h=\left[0.10\left({\displaystyle \frac{k}{D}}\right)R{e}_b^{0.653}P{r}_b^{0.4}\right]\left[1+0.57{\mathrm{e}}^{-0.20\cdot z/D}\right]$	D—coolant channel diameter (m) z/D > 25 ± 2%	(97)
		$N{u}_{FD}=0.10R{e}_b^{0.653}P{r}_b^{0.4}$	z/D ≥ 25 Reb = 2.2 × 104–5.8 × 104	(98)
		$N{u}_{FD}=0.023R{e}_b^{0.8}P{r}_b^{0.4}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.57}$	Taylor correlation for fully developed turbulent convection	(99)
		$Nu=N{u}_{FD}\left[1+0.57{\mathrm{e}}^{{\displaystyle \frac{-0.20z}{D}}}\right]$	Proposed empirical correction to match CFD results in entrance region. D—coolant channel diameter (m) z/D > 25	(100)
		$Nu=N{u}_{FD}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{{\displaystyle \frac{-1.59}{\left(\frac{z}{D}\right)}}}$	Correlation by Taylor for axial heat flux. Underpredicts Nu in entrance region D—coolant channel diameter (m)	(101)
[52]	Using a hybrid CFD system method with 3D conduction and 1D fluid models enables the accurate thermal analysis of prismatic gas-cooled reactors, while reducing computational costs (numerical study)	$Nu=0.021R{e}^{0.8}P{r}^{0.4}$	For turbulent flow	(102)
		$Nu=4.364$	For laminar flow	(103)
		$h=Nu\cdot {\displaystyle \frac{k_f}{D_h}}$	Dh = 4A/P—hydraulic diameter of the coolant channel kf—heat transfer coefficient	(104)
		$Q_{conv,i}=h_j{A}_{i,j}\left(T_{s,i}-T_{f,j}\right)$	-	(105)
		$Q_{conv,j}={\displaystyle \sum _i}{h}_j{A}_{i,j}\left(T_{s,i}-T_{f,j}\right)$	-	(106)
[53]	Correlations for laminar and mixed convection heat transfer in a single vertical coolant channel of a prismatic-core advanced high temperature reactor (PAHTR). Numerical (CFD), steady-state, laminar/mixed convection, FLiBe coolant, high Prandtl Number (numerical study)	$L_T=D·\left(a\cdot R{e}_D\cdot Pr\right)$	LT—length of the developing flow region a—empirical constant dependent on the Prandtl number D—coolant channel diameter (m) $a=0.1;0.7<Pr<1$ $a=0.15;Pr>1$	(107)
		$N{u}_z=\left\{\begin{array}{c}1.302\cdot {z^{*}}^{\left\{-{\displaystyle \frac{1}{3}}\right\}}-1.0,\\ 1.302\cdot {z^{*}}^{\left\{-{\displaystyle \frac{1}{3}}\right\}}-0.5,\\ 4.364+8.68\cdot {\left({10}^3\cdot z^{*}\right)}^{-0.506}\cdot exp\left(-41\cdot z^{*}\right),\end{array}\right.$	$\begin{array}{c}for{z}^{*}\le 0.00005\\ for0.00005<z^{*}\le 0.0015\\ for{z}^{*}>0.0015\end{array}$ where $z^{*}=(Z/D)/({Re}_D\cdot Pr)$	(108)
		${\displaystyle \frac{N{u}_z+1}{5.364·{[1+{\left(\frac{220·z^{*}}{\pi }\right)}^{-\frac{10}{9}}]}^{\frac{3}{10}}}}=\left[1+\left({\displaystyle \frac{\pi /115.2·z^{*}}{{\left.\left(1+{\left(\frac{Pr}{0.0207}\right)}^{\frac{2}{3}}\right.\right)}^{\frac{1}{2}}·{\left(1+{\left(\frac{220·z^{*}}{\pi }\right)}^{-\frac{10}{9}}\right)}^{\frac{3}{5}}}}\right)\right]$	Valid for the higher Prandtl number within the pipe entrance region	(109)
		$N{u}_z=1.24{\left[\left(Re\cdot Pr\cdot D/z\right)+0.025\cdot {\left(Gr\cdot Pr\right)}^{0.75}\right]}^{{\displaystyle \frac{1}{3}}}·{\left({\displaystyle \frac{{\mu }_{bulk}}{{\mu }_{wall}}}\right)}^{0.14}$	$280\le Re\le 3800$ $40\le \mathit{Pr}\le 160$ $1000\le \mathit{Gr}\le 2.8·{10}^4$ $3\le {\displaystyle \frac{z}{D}}\le 192$ D—coolant channel diameter (m) $1.2\le \frac{{\mu }_{bulk}}{{\mu }_{wall}}\le 3.8$ where $Gr=g\cdot \beta \cdot r^2\cdot {\displaystyle \frac{D^3\left(T_{wall}-T_{bulk}\right)}{{\mu }_{bulk}^2}}$	(110)
[54]	The coupling between the three-dimensional solid model and the one-dimensional fluid model utilizes a standard Nusselt number correlation to compute the convective heat transfer between the two domains. For turbulent flow, the correlation of McEligot (numerical study)	$Nu=0.021·{Re}^{0.8}·{Pr}^{0.4}·\left({\displaystyle \frac{T_w}{T_f}}\right)$	For turbulent flow	(111)
		$Nu=4.364$	For laminar flow	(112)
		$q_{s,i}^{conv}={Nu}_j\cdot {\displaystyle \frac{k_{f,j}}{D_{\left\{h,i,j\right\}}}}\cdot A_{s,i,j}·\left.\left(T_{s,i,j}-T_{f,j}\right.\right)$	I represents each surface of the solid contacted with the fluid node j and As,i,j is the heat transfer area between the solid surface i and fluid node j D—coolant channel diameter (m)	(113)
[55,56,57]	Prismatic core, helium coolant; fully developed turbulent flow in fuel channel (3D CFD analysis of a single channel, correlation derived from simulation results) (numerical study)	$N{u}_{FD}=0.11R{e}_b^{0.646}{Pr}_b^{0.4}$	z/D > 25 D—coolant channel diameter (m) ${Re}_b\approx 1.64·{10}^4<Re<6.05·{10}^5$ $0.65<Pr<0.68$ deviation from CFD ≤ +3%	(114)
		$Nu=N{u}_{FD}\left.\left[1+0.57e^{-0.20·z/D}\right.\right]$	Matches full CFD results for uniform and chopped-cosine power profiles within ±2%.	(115)
[58]	General correlation for fully developed turbulent flow of gases and liquids in smooth circular tubes; applied in simplified thermal hydraulic analyses of helium flow in VHTR prismatic block fuel channels, the correlation of Dittus–Boelter	$N{u}_{FD}=0.023R{e}_b^{0.8}{Pr}_b^{0.4}$	$0.7\le {Pr}_b\le 160$ $R{e}_b\ge {10}^4$ ${\displaystyle \frac{z}{D}}\ge 10$	(116)
	Modified correlation for fully developed turbulent convection including viscosity correction. Applied to helium flow in VHTR coolant channels as refinement to Dittus–Boelter	$Nu=0.023R{e}_b^{0.8}{Pr}_b^{{\displaystyle \frac{1}{3}}}·{\left({\displaystyle \frac{{\mu }_b}{{\mu }_w}}\right)}^{0.14}$	${10}^4\le Re\le 5·{10}^5$ $0.67\le Pr\le 100$ $0.0044\le {\displaystyle \frac{{\mu }_b}{{\mu }_w}}\le 9.75$	(117)
	The Taylor correlation for fully developed turbulent convection in uniformly heated circular tubes	${Nu}_{FD}=0.023R{e}_b^{0.8}{Pr}_b^{0.4}·{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.57}$	-	(118)
	Taylor correlation in its entirety, including the entrance mixing effect	$Nu=N{u}_{FD}{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-\left[0.57-{\displaystyle \frac{1.59}{z/D}}\right]}$	D—coolant channel diameter (m) $7.3·{10}^3\le Re\le 1.3·{10}^7$ $0.71\le Pr\le 10$ $1.1\le {\displaystyle \frac{T_w}{T_b}}\le 27.6$	(119)
	The correlation of McEligot for fully developed turbulent convection in uniformly heated circular tubes	$Nu=0.021R{e}_b^{0.8}{Pr}_b^{0.4}·{\left({\displaystyle \frac{T_w}{T_b}}\right)}^{-0.50}$	-	(120)
	The correlation of McEligot in its entirety (numerical study)	$Nu=N{u}_{FD}\left.\left[1+{\left(z/D\right)}^{-0.70}\right.\right]$	$1.64\times {10}^4\le Re\le 6.05\times {10}^5$ $0.65\le Pr\le 0.68$ $1.0\le {\displaystyle \frac{T_w}{T_b}}\le 2.12$ D—coolant channel diameter (m)	(121)

. Said et al. [42] investigated the impact of helium pressure on natural convection heat transfer in a dual-channel loop representing a prismatic Very-High-Temperature Reactor (VHTR). They developed a specialized, scaled-down experimental setup to simulate the riser and downcomer channels under loss-of-flow accident conditions. By employing novel, noninvasive heat flux probes and radial thermocouple adjusters, the researchers obtained local heat transfer coefficients and temperature profiles with high precision. A significant finding was that the Rayleigh number exhibited a quadratic dependence on pressure (Ra∝p²), confirming ideal gas behavior in the helium-filled loop. As the helium pressure increased (from 413.47 to 689.12 kPa), the Nusselt number and the heat transfer coefficient increased by 35% and 30%, respectively, while the wall temperature decreased by up to 18%. Additionally, a reversal of heat and flow was observed at the riser outlet and three characteristic helium temperature profiles (developing, flattened, and reversed) were identified along the upward flow section of the test loop. These measurements provided valuable benchmarking data for the validation of CFD and thermal hydraulic codes.

Lee et al. [43] developed a set of cross-flow loss coefficient correlations for bypass flow between fuel blocks in the Prismatic Modular Reactor (PMR200). The study integrates experimental measurements with CFD simulations to derive formulas tailored to the complex geometry of inter-block gaps that are dependent on Reynolds number. The key contribution is a generalized bypass loss correlation, expressed as a function of gap height δ, flow area A, and Reynolds number Re. For low-Re bypass flows, a nonlinear pressure drop behavior was captured, demonstrating that conventional loss coefficient models underestimate bypass resistance in prismatic HTGR configurations. The proposed correlations, including KHA and KPMR200, significantly improve flow distribution predictions and are crucial for the high-fidelity thermal hydraulic modeling of VHTGR systems.

Wang et al. [44] proposed and validated a porous media model to simulate the thermal hydraulic behavior of the entire core of a prismatic high-temperature gas-cooled reactor (HTGR). Treating the graphite blocks and coolant channels as a homogenized porous domain significantly reduced the computational cost while maintaining excellent agreement with high-fidelity CFD simulations. The effective thermal conductivity and flow resistance were derived based on geometric and material parameters, such as porosity ε, number of coolant holes N_t, and channel dimensions. Under laminar flow assumptions, a constant Nusselt number of 4.364 was used, and for turbulent conditions, a modified Dittus–Boelter correlation was applied. The model was validated using benchmark test cases and enables whole-core CFD predictions to be made using coarse meshes while preserving accuracy in temperature and heat flux distributions. This approach is particularly useful for prismatic block VHTGR design studies.

Zeitoun et al. [45] validated a porous media model and investigated passive natural circulation in a prismatic HTGR using CFD simulations of a dual-channel experimental setup. The study demonstrated how heating intensity and inlet buoyancy effects influence flow reversal and temperature stratification. Key dimensionless groups, such as the Grashof, Reynolds, and Froude numbers, were analyzed to quantify the transition between buoyancy- and inertia-dominated regimes. The computed Nusselt number and heat transfer coefficient profiles revealed that buoyancy effects become significant at low inlet flow rates, causing localized flow stagnation and reverse convection zones. This study provides essential benchmarking data for modeling passive safety systems in VHTGRs with porous domain simplifications.

Alameri et al. [46] evaluated the thermal hydraulic performance of a prismatic-core Very-High-Temperature Reactor (VHTR) integrated with a molten salt thermal energy storage (TES) system. The analysis involved modeling heat transfer across concentric tubes containing molten FLiBe and helium by employing a comprehensive resistance network including convective and conductive terms. Several forms of the Nusselt number were employed, including constant and empirical correlations, to determine the heat transfer coefficients for the salt and helium phases. The total thermal resistance was expressed as the sum of six components, accounting for convection, clad conduction, and salt conduction (in both liquid and solid states). The study confirmed that stable heat removal could be achieved even under transient conditions, ensuring compatibility with flexible energy systems.

GJ Nel [47] performed a detailed comparison of one-dimensional (1D), three-dimensional (3D), and coupled 3D/1D CFD models of a prismatic VHTR fuel block. The aim was to reduce computational cost while maintaining thermal hydraulic accuracy in core-scale simulations. Several heat transfer correlations for the Nusselt number were validated against high-fidelity 3D CFD results. These correlations incorporated the effects of flow development, temperature-dependent properties, and geometric factors such as the channel length-to-diameter ratio. The results demonstrated that the coupled 3D/1D model offered high accuracy with significantly reduced mesh and runtime requirements, providing a promising methodology for full-core simulations.

Lee and Tak [48] validated the CORONA code, which was developed by KAERI for the local thermal analysis of prismatic VHTRs. This was achieved by comparing the results with those of high-fidelity CFD simulations conducted using ANSYS CFX (version 17.2) on one-sixth of a full reactor core, including reflector regions, plenums, and six fuel blocks. Under realistic conditions (a mass flow rate of 26.25 kg/s, an inlet temperature of 259 °C, and a pressure of 7 MPa), the CORONA code achieved a significant reduction in computational time (33 min versus 9 days and 7 h for CFD), while maintaining acceptable accuracy. The maximum temperature deviation in the hottest fuel columns was below 5%. Several Nusselt number correlations, including Dittus–Boelter, McEligot, and Gnielinski-type equations, were applied to characterize convective heat transfer in turbulent flow conditions. This study showed that CORONA could reliably predict local thermal margins even with coarse computational grids, making it suitable for system-scale reactor safety evaluations.

Lee et al. [49] evaluated the applicability of empirical Nusselt number correlations for predicting convective heat transfer within coolant channels of a prismatic VHTR fuel block. Using detailed 3D CFD simulations performed in ANSYS CFX with the standard k–ε model, the authors examined two wall heating conditions—uniform and nonuniform—typical of actual reactor operation. The study compared the classical Dittus–Boelter and McEligot correlations with local Nusselt number distributions obtained from CFD. For uniform heating, both correlations overestimated heat transfer by 17.85% and 5.79%, respectively. For nonuniform heating, the deviations increased significantly to 32% and 19%, respectively, indicating that current empirical models do not adequately reflect thermal behavior under realistic power distribution. The authors concluded that empirical Nusselt correlations require modification or correction factors to account for localized heating effects in prismatic VHTR channels.

Lee et al. [50] employed the FastNet flow network code to predict the thermal hydraulic parameters of a prismatic-core VHTR. FastNet solves the equations of mass, momentum, and energy conservation over a one-dimensional control volume mesh using empirical and CFD-based correlations. The software was benchmarked against CORONA and full-scale CFD results for a typical VHTR fuel column. The code significantly reduced computational time while maintaining high accuracy, with deviations of less than 5% in outlet and peak fuel temperatures. The Dittus–Boelter-based correlation ($Nu=0.021·{Re}^{0.8}{Pr}^{0.4}$) and constant Nusselt number values (e.g., Nu = 4.364 for laminar flow) were used depending on local flow regimes. This model is particularly advantageous for large-scale system simulations where explicit CFD is computationally prohibitive.

Travis et al. [51] conducted a 3D CFD study of helium flow through prismatic VHTR coolant channels in order to derive a turbulent heat transfer correlation that accounts for entrance effects. The study covered a Reynolds number range from 2.2 × 10⁴ to 5.8 × 10⁴ at a Prandtl number of 0.67. The proposed correlation, which was validated against numerical results with a deviation of ±2%, relates the Nusselt number to the Reynolds and Prandtl numbers, as well as the channel length, and the wall/bulk temperature ratio. A generalized form was also presented, incorporating corrections for thermal entrance length and variable wall heating. This work improves the accuracy of predicting convective heat transfer in VHTR core simulations, particularly in the initial channel regions.

In [52], a hybrid CFD system method was developed for thermal analysis of prismatic gas-cooled reactors, combining three-dimensional (3D) conduction with one-dimensional (1D) fluid models. This approach enables accurate, computationally efficient whole-core simulations. Convective heat transfer was characterized using a fixed Nusselt number value (Nu = 4.364) and a simplified correlation for convective power based on local wall temperature differences. This methodology enables convective heat removal to be predicted using only geometric and thermal boundary data, facilitating rapid and reliable reactor thermal assessments. The heat transfer coefficient was calculated using Equation (104), and the total convective power was obtained by summing the heat transfer areas using Equation (105).

Alameri et al. [53] developed correlations for laminar and mixed-convection heat transfer in a single vertical coolant channel of a Prismatic-Core Advanced High-Temperature Reactor (PAHTR), using numerical CFD simulations with FLiBe molten salt. The study looked at steady-state conditions with a high Prandtl number and investigated thermal entrance effects under both forced and mixed convection. By analyzing the evolution of the local Nusselt number along the channel and comparing it with existing models for developing laminar flow, the authors proposed empirical relations for the thermal entry length using Equation (107) and the axial variation of the Nusselt number through Equations (108) and (109). The combined influence of Reynolds, Prandtl, and Grashof numbers is captured by these correlations, which provide accurate predictions for high-Prandtl-number coolants in the entrance region of prismatic reactor channels. Furthermore, these correlations enhance the accuracy of subchannel modeling for PAHTR thermal hydraulic analyses.

Tak et al. [54] developed the CORONA code to perform efficient thermo-fluid analysis of prismatic gas-cooled reactor cores by coupling a three-dimensional (3D) solid heat conduction model with a one-dimensional (1D) fluid flow network. The coupling between the two domains uses a standard Nusselt number correlation to calculate convective heat transfer. This is done by using the McEligot relation for turbulent flow, as defined in Equation (110), and a constant analytical value for laminar conditions, as expressed by Equation (111). These relations were combined within a conjugate heat transfer formulation governed by Equation (112), which links the solid surface heat flux to the local coolant temperature. This implementation in the CORONA code enables the accurate prediction of conjugate heat transfer in prismatic blocks while minimizing computational cost under steady-state reactor conditions.

The correlations presented as Equations (113) and (114) were derived from detailed 3D CFD simulations. These were of helium flow in prismatic fuel channels. The conditions were fully developed turbulent. Travis et al. [55] analyzed the effects of buoyancy and transitional flow regimes in natural-circulation loops, providing validation data for CFD-based modeling. El-Genk and Travis [56] took these studies a step further by expanding them to include forced convection. They developed a helium-specific Nusselt number correlation using Equation (113), which aligns 3D simulation data within a margin of error of ±3%. Koekemoer and du Toit [57] further validated this approach. They used Equation (114) through 1D/3D coupled thermal–fluid simulations. This confirmed its reliability for both uniform and nonuniform wall heating. Collectively, these studies demonstrate that Equations (113) and (114) accurately represent the way heat moves in prismatic VHTR channels.

Travis and El-Genk [58] performed a comprehensive numerical study to establish a reliable turbulent convection correlation for helium flow in prismatic-block VHTR coolant channels. They compared and refined several existing empirical formulations—Dittus–Boelter, Seider–Tate, Taylor, and McEligot—through full 3D CFD simulations of a single heated channel. The Dittus–Boelter relation, as described by Equation (115), provided the baseline for fully developed turbulent convection, while the Seider–Tate modification, as expressed in Equation (116), incorporated viscosity correction for wall-to-bulk temperature differences. Furthermore, Equations (117) and (118) represent Taylor’s formulation, which introduced the influence of the wall-to-fluid temperature ratio and entrance mixing. This meant that its applicability expanded to a wider range of Reynolds numbers. McEligot’s approach, expressed by Equations (119) and (120), further corrected for developing flow and variable thermal properties along the channel length. By employing STAR-CCM+ simulations of a 10 m long helium channel, Travis and El-Genk were able to validate and synthesize these formulations into a unified hybrid correlation that accurately represents both the entrance and fully developed regions. Their results demonstrated that entrance mixing extends up to approximately 25 hydraulic diameters, beyond which the flow becomes fully developed. The resulting hybrid correlation can reproduce 3D CFD data to within ±2%, making it a practical and precise surrogate for detailed CFD in coupled 1D/3D thermal hydraulic analyses of prismatic VHTR cores.

Thermal hydraulic analysis of prismatic block HTGRs and VHTRs relies on a variety of correlations, ranging from simplified analytical values to advanced empirical fits derived from 3D CFD data. A structured quantitative comparison reveals clear differences in accuracy depending on the flow regime and thermal boundary conditions. For laminar flow conditions, the constant Nusselt number (Nu = 4.364) remains the standard benchmark for both channel-based and porous media models (see Equations (54), (58), (73), and (96)), providing a conservative lower-bound estimate of heat transfer in homogenized core analyses. For fully developed turbulent flow, classical correlations such as Dittus–Boelter (Equation (87)) are computationally efficient but deviate significantly from CFD data. Under realistic prismatic VHTR heating conditions, heat transfer is overestimated by between 17.8% and 32%, particularly for nonuniform axial power distributions. To improve predictive capability, modified correlations such as those by McEligot (see Equations (88) and (119)) and Taylor (see Equation (117)) are better suited for system-level codes (e.g., CORONA), as they incorporate variable thermophysical properties and the wall-to-bulk temperature ratio (T_w/T_b). Furthermore, entrance-region effects cannot be neglected: numerical studies show that thermal development extends up to approximately 25 hydraulic diameters. In this regime, the hybrid Travis and El-Genk formulation (see Equations (113) and (114)) provides the highest accuracy, reproducing 3D CFD data with an error margin of ±2% by explicitly accounting for entrance corrections and nonuniform wall heating. For predicting bypass flow and pressure drop, PMR200-specific correlations (see Equations (44) and (49)) represent a significant improvement on conventional loss models, as the latter tend to underestimate resistance in inter-block gaps. Overall, while simplified correlations (e.g., Nu = 0.021Re^0.8 Pr^0.4 remain suitable for preliminary whole-core simulations, high-fidelity safety analysis of peak fuel temperatures requires helium-specific, entrance-corrected, or hybrid correlations to accurately capture localized heating effects.

5. Pebble Bed Reactors (PBRs)

The pebble bed reactor, a variant of the high-temperature gas-cooled reactor (HTGR) concept, is characterized by a core filled with thousands to hundreds of thousands of spherical fuel elements (pebbles), each containing TRISO-coated fuel particles [59]. These pebbles usually consist of a graphite matrix that contains the TRISO particles, and the core is moderated by graphite and cooled by an inert or slightly reactive gas (typically helium) to allow high outlet temperatures [60].

One of the key advantages of the pebble-bed design is its potential for high thermal efficiency due to high coolant outlet temperatures, and its inherent safety features, stemming from the robust fuel design (TRISO) and the passive cooling capabilities. Because the fuel pebbles can be recirculated (or continuously fed) through the core, the design enables online refueling and better utilization of fuel burn-up compared to fixed fuel assemblies.

During operation, the spherical fuel elements traverse the core region while the helium coolant flows past the pebbles, extracting heat. Once the designed burn-up or end-of-life criteria have been achieved, each pebble is removed and fresh pebbles are introduced at the top of the reactor. The reactor structure is usually cylindrical and is surrounded by a reflector and containment vessel. The modular nature of many pebble-bed designs allows for scalability and flexible deployment.

These features make pebble bed reactors widely regarded as promising candidates for the Generation IV class of nuclear reactors, with strong interest worldwide in their development for both electricity production and high-temperature process heat applications.

Chen et al. [61] conducted a numerical investigation of the thermal field and heat transfer characteristics of a hexagonal close-packed (HCP) pebble bed for high-temperature gas-cooled reactor (HTGR) applications. Simulations were performed using a computational fluid dynamics (CFD) methodology with a standard k-ε turbulence model, considering various coolant inlet velocities corresponding to Reynolds numbers in the range of $1.6\times {10}^4$ to $4\times {10}^4$. The thermal and velocity fields were analyzed on designated middle and diagonal planes to identify local heat transfer phenomena. The strongest heat transfer occurred near the right vertices of the top and bottom pebbles, while the weakest heat transfer took place in areas near the inter-pebble contact points. These results were used to propose a new correlation for the overall average heat transfer coefficient as $h_{\mathrm{avg}}=0.1545(k/L)R{e}^{0.8}$. Additionally, it was found that the heat transfer intensity of the HCP structure was weaker than that of a face-centered cubic (FCC) structure, despite both having the same packing density.

Abdulmohsin et al. [62] investigated the local convective heat transfer characteristics in a packed pebble bed reactor, which is a key design feature of Very-High-Temperature Gas-Cooled Reactors (VHTRs). Using a fast-response, non-invasive spherical heat transfer probe, the authors measured local heat transfer coefficients at various radial and axial positions within a cold-flow experimental setup with a diameter of 0.3. The effects of gas velocity (Reynolds number) and bed structure on heat transfer were analyzed. The results showed that the heat transfer coefficients increased from the center of the bed to the wall due to porosity variations and flow channeling. The study highlights the inadequacy of a single overall heat transfer coefficient and emphasizes the need for local correlations. The experimental data were compared with existing empirical correlations, with the Achenbach model [69] showing the best agreement.

Focusing on the specific application of water-cooled pebble bed reactors (WPBR), Liu et al. [63] conducted an experimental investigation into the flow resistance and convective heat transfer within a randomly packed pebble bed featuring internal heat generation. Their study used an experimental facility in which an electrical induction heating system provided a volumetric heat source for carbon steel pebbles, with water acting as the coolant. The effects of key parameters such as input power (12–54 kW), fluid inlet temperature (25–55 °C), porosity (0.3827–0.3946), and fluid velocity were systematically examined. Regarding flow resistance, they found that the Blake-type friction factor could be accurately predicted using the Handley and Heggs correlation with an error margin of 10%. They also proposed a new pressure drop correlation ($f_2=502/R{e}_{dh}+1.11$) based on their data. Regarding heat transfer, their results showed that the Nusselt number increased with the Reynolds number but decreased with a higher inlet fluid temperature and a lower porosity. They found that the Wakao et al. correlation agreed well with the experimental data for Reynolds numbers above 3000, whereas the KTA correlation was more accurate at lower velocities. Ultimately, they proposed a new correlation for the Nusselt number ($N{u}_{dh}={10}^{0.086}{ϵ}^{2.14}R{e}^{0.55}P{r}^{1/3}$) based on the observed similarity in trends between the Colburn j-factor and the friction factor. This correlation predicted their data with a discrepancy of less than 10%.

Dave et al. [64] conducted a numerical assessment of the performance of established packed-bed heat transfer correlations for application in molten fluoride salt-cooled high-temperature reactors (FHRs). Using CFD simulations with both Reynolds-averaged Navier–Stokes (RANS) (k-ε) and Large Eddy Simulation (LES) turbulence models, the authors evaluated Nusselt number predictions across a range of Reynolds numbers (10² to 2.25 × 10³) for body-centered cubic (BCC) and face-centered cubic (FCC) lattice structures. The study reveals that many conventional correlations, developed for coolants with a low Prandtl number such as gases or water, are unable to accurately predict heat transfer for high-Prandtl-number molten salts (Pr ≈ 16). Among the correlations evaluated, only the Meng et al. [70] correlation showed excellent agreement with the numerical results, accurately capturing both the quantitative values and the qualitative trends. The study highlights the inapplicability of many legacy correlations for FHR analysis and identifies the Meng correlation as the most suitable for predicting convective heat transfer in molten salt-cooled pebble-bed cores.

In a numerical study, Zhang et al. [65] used a coupled discrete element method-computational fluid dynamics (DEM-CFD) approach to analyze the heat transfer characteristics of hydrogen flowing through randomly packed beds. The research specifically investigated the impact of the tube-to-particle diameter ratio (D/d), simulating three configurations (D/d = 4, 5, and 6) and accounting for the temperature-dependent thermal properties of hydrogen. The results showed that increasing the D/d ratio significantly speeds up the achievement of thermal equilibrium. The two larger beds were found to reduce the required axial distance by 44% and 58%, respectively, compared to the smallest bed. While this generally enhanced the heat transfer coefficient, the largest bed (D/d = 6) exhibited slightly reduced performance drop at high flow rates due to the formation of large, inefficient fluid channels. A detailed pore-scale analysis further revealed a clear inverse relationship between the local Nusselt number and the porosity distribution along the axial direction. The study also quantified a pronounced “wall effect,” demonstrating that the near-wall region experiences stronger velocity and temperature oscillations and remains consistently cooler than the core region. These insights are crucial for designing and optimizing packed-bed systems utilizing hydrogen, such as those in catalytic reactors and thermal storage units.

Freile et al. [66] conducted a study to improve the modeling of natural convection heat transfer in the reactor cavity cooling system (RCCS) of high-temperature gas-cooled reactors (HTGRs). While existing system-level simulation tools provide accurate models for radiative heat transfer, they often rely on empirical correlations of uncertain accuracy to model natural convection. To address this gap, Freile et al. [66] developed improved correlations for both local and average Nusselt numbers as a function of global and local Rayleigh numbers, as well as the temperature distribution along the hot wall of the RCCS. Due to the lack of experimental data and the high computational cost of direct numerical simulations under realistic conditions, the authors generated their own data using computational fluid dynamics (CFD) simulations based on Reynolds-averaged Navier–Stokes (RANS) models. The results of these simulations were then used to derive new Nusselt number correlations as a function of Rayleigh numbers (see Equations (150)–(152)) in

Table 4. The convective heat transfer correlations valid for pebble bed reactors.

Ref.	Field of Research and Conditions	Correlation	Remarks/Applicability and Limitations	Equation No.
[61]	Thermal Field and Heat Transfer Characteristics of a Hexagonal-Close-Packed Pebble Bed (numerical study)	$Nu=0.177R{e}^{0.8}{Pr}^{0.4}$	$1.6\times {10}^4\le Re\le 4.0\times {10}^4$ $\mathit{Pr}=0.712$	(122)
		$h_{avg}=0.1545\left({\displaystyle \frac{k}{L}}\right){Re}^{0.8}$	L—characteristic length of the pebble in the numerical test section (m)	(123)
[62]	Characteristics of convective heat transport in a packed pebble-bed reactor (experimental study)	$Nu=2+1.1{Pr}^{1∕3}{Re}^{0.1}$	Classic correlation for packed beds [39,41]	(124)
		$Nu=f_{\epsilon }N{u}_{sp}$	$Re/{\epsilon }_b\le 2·{10}^4$	(125)
		$N{u}_{sp}=2+\sqrt{N{u}_{lam}^2+N{u}_{turb}^2}$ $N{u}_{lam}=0.664{\left({\displaystyle \frac{Re}{{\epsilon }_b}}\right)}^{1∕2}{Pr}^{1/3}$ $N{u}_{turb}={\displaystyle \frac{0.037{\left(\mathit{Re}/{\epsilon }_b\right)}^{0.8}Pr}{1+2.4{\left(\mathit{Re}{/\epsilon }_b\right)}^{-0.1}\left({\mathit{Pr}}^{2/3}-1\right)}}$		(126)
		$Nu=1.27\left({\displaystyle \frac{P{r}^{1/3}}{{\epsilon }_b^{1.18}}}\right){\mathit{Re}}^{0.36}+0.033\left({\displaystyle \frac{{Pr}^{1∕2}}{{\epsilon }_b^{1.07}}}\right){Re}^{0.86}$	100$\le Re$ $\le {10}^5$ [39,41]	(127)
		$Nu={\left[{\left(1.18{Re}^{0.58}\right)}^4+{\left(0.23{\left({Re}_h\right)}^{0.75}\right)}^4\right]}^{1∕4}$	${Re}_h={\displaystyle \frac{Re}{1-{\epsilon }_b}}$	(128)
[63]	Experimental analysis of flow and convective heat transfer in the water-cooled packed pebble bed nuclear reactor core (experimental study)	$N{u}_{dh}={100}^{0.086/{ϵ}^{2.14}}{Re}^{0.35}{Pr}^{1/3}$	$350\le {Re}_{dh}\le 9000$ $2.3\le Pr\le 5.9$	(129)
		${\displaystyle \frac{\dot{q_2^{″}}}{\dot{q_1^{″}}}}={\displaystyle \frac{h_2}{h_1}}={\displaystyle \frac{N_1{d}_1}{N_2{d}_2}}$	Heat flux density scaling under volume homogenization; applies to channel-based to porous transition N—pebble number d—pebble diameter	(130)
		$N{u}_1={\displaystyle \frac{h_1{d}_1}{k_{he}}}={\displaystyle \frac{\left(f/8\right)\left(Re-1000\right)Pr}{1+12.7\sqrt{f/8}\left(P{r}^{2/3}-1\right)}}$	Gnielinski correlation; turbulent flow regime; d—pebble diameter Re > 1000; 0.7 ≤ Pr ≤ 500	(131)
		$h_2={\displaystyle \frac{N{u}_2\hspace{0.17em}{k}_{eff}}{d_2}}$	Used for porous domain convection estimation; applicable to homogenized geometry d—pebble diameter	(132)
		$N{u}_2=4.364$	Laminar flow	(133)
		$k_3={\displaystyle \frac{k_{eff}-\epsilon k_{he}}{1-\epsilon }}\approx 585 \mathrm{W}/\mathrm{m}\text{·}\mathrm{K}$	Effective solid matrix conductivity; validated for HTGR graphite block with helium porosity ε = 0.39	(134)
		$B{i}_R={\left({\displaystyle \frac{h\delta }{k}}\right)}_{\mathrm{R}}=1$	Biot number at reference point set to unity; ensures thermal resistance parity between models	(135)
		$h_1={\displaystyle \frac{N{u}_1\hspace{0.17em}{k}_{he}}{d_1}}$	Direct application of Nusselt number in channel domain; helium properties d—pebble diameter	(136)
		$N{u}_1=4.364$	Laminar flow	(137)
[64]	Packed-bed heat transfer correlations for molten salt (numerical study)	$Nu=2+1.8{Re}^{0.5}{Re}^{0.5}$	Evaporation of droplets in air $0\le Re\le 200$	(138)
		$Nu=0.922{Re}^{0.66}{Pr}^{0.33}$	Catalytic packed beds $0\le Re\le 200$	(139)
		$Nu=2+1.1{Re}^{0.6}{Pr}^{0.33}$	Air/water in packed beds $20\le Re$ $\le 3$·${10}^4$	(140)
		$Nu=3.212{Re}^{0.335}{Pr}^{0.034}$	Superheated steam evaporation 1.2·${10}^3\le Re$ $\le 1.3$·${10}^4$	(141)
		$Nu={\epsilon }^{-1}{P}_r^{\mathrm{1,3}}\left(0.0108+{\displaystyle \frac{{Re}^{0.622}}{\mathit{Re}-0.483}}\right)$	Gas/air in packed beds $20\le Re$ $\le 5$·${10}^3$	(142)
		$Nu={\displaystyle \frac{1-\epsilon }{\epsilon }}\left(0.5{\mathit{Re}}^{0.5}P{r}^{0.2}\right)$	Non-spherical particles $20\le Re$ $\le 8$·${10}^4$	(143)
		$Nu={\displaystyle \frac{1-\epsilon }{\epsilon }}\left(0.5{\mathit{Re}}^{0.5}P{r}^{0.2}\right)$	Non-spherical particles $20\le Re$ $\le 8$·${10}^4$	(144)
		$Nu=2+\sqrt{N{u}_{lam}^2+N{u}_{turb}^2}$ ${Nu}_{lam}=0.664{Re}^{0.5}{Pr}^{{\displaystyle \frac{1}{3}}}$ ${Nu}_{turb}={\displaystyle \frac{0.037{Re}^{0.8}Pr}{1+2.443{Re}^{-0.1}({Pr}^{2/3}-1)}}$	Flat-plate analogy $Re>$ ${10}^2$	(145)
		$Nu=1.27{\epsilon }^{-1.18}{Pr}^{0.036}+0.033{\epsilon }^{-1.07}{Re}^{0.86}{Pr}^{0.026}$	German gas-cooled reactors ${10}^2\le Re$ $\le 4$·${10}^5$	(146)
		$N_u=3.212{\left({\displaystyle \frac{Re(1-\epsilon )}{\epsilon }}\right)}^{0.335}{Pr}^{0.034}$	Water in packed beds 1.2·${10}^3\le Re$ $\le 1.3$·${10}^4$	(147)
[65]	Heat Transfer Characteristics for Hydrogen Flow in Randomly Packed Beds (numerical study)	$Nu=\left(7-10\epsilon +{5\epsilon }^2\right)\left(1+0.7R{e}_p^{0.2}{Pr}^{1∕3}\right)+\left(1.33-2.4\epsilon +1.2{\epsilon }^2\right){Re}_p^{0.7}{Pr}^{1/3}$	Porosity (ε) effects on heat transfer in fixed and fluidized beds $0.35\le \epsilon$ $\le 0.75$ ${Re}_p<{10}^5$	(148)
		$Nu=2+0.6{Re}_p^{1/2}{Pr}^{1/3}$	Isolated particles low Rep	(149)
[66]	Improved natural convection heat transfer correlations for reactor cavity cooling systems of high-temperature gas-cooled reactors (numerical study)	$Nu=0.2490R{a}^{0.2911}$ $\mathrm{for}x\in \left[0,x_{s,tr}^{*}\right]$	$x_{s,tr}^{*}=e^{-0.4777R{a}^{0.06165}}$ $x_{e,tr}^{*}=e^{-0.4349R{a}^{0.05607}}$	(150)
		$Nu=0.1901R{a}^{0.2994}$ $\mathrm{for}x\in \left[x_{s,tr}^{*},x_{e,tr}^{*}\right]$		(151)
		$Nu=0.08169R{a}^{0.3183}$ $\mathrm{for}x\in \left[x_{e,tr}^{*},1\right]$		(152)
[67]	Mixed convection in reactor cavity cooling system of HTGR for hydrogen production (experimental study)	$N{u}_{LF}={\left({\displaystyle \frac{{\left(\frac{6.0\times {10}^{-6}}{Bo}\right)}^{2.581}}{1+{\left(\frac{6.0\times {10}^{-6}}{Bo}\right)}^{2.581}}}+{\left(21.81{Bo}^{0.3336}\right)}^{2.581}\right)}^{0.387}$	$Bo={\displaystyle \frac{Gr}{{Re}^3{Pr}^{0.5}}}$	(153)
[68]	Thermal field and heat transfer characteristics of a pebble-bed core in a high-temperature gas-cooled reactor (experimental study)	$Nu=0.194R{e}^{0.8}P{r}^{0.4}$	Pr = 0.712	(154)

) via a sparsity-promoting least squares method. In the final phase of their study, the selected RANS model was used to simulate a PBMR-400 (Pebble-Bed Modular Reactor) cavity undergoing a pressurized loss of forced cooling (PLOFC) transient, using realistic temperature profiles at the RPV wall. These results informed the development of a temperature-dependent correction to the space-varying Nusselt number. Overall, their work provided enhanced heat transfer correlations that enable system-level codes such as Pronghorn to perform more accurate and computationally efficient simulations of RCCS thermal performance in HTGRs.

Shin et al. [67] carried out an experimental study to evaluate the heat transfer behavior of the reactor cavity cooling system (RCCS) in a high-temperature gas-cooled reactor (HTGR). The study focused on the heat transfer performance in the riser ducts of the RCCS.

To investigate this, they built an experimental facility and conducted a series of heat transfer experiments within a riser duct under various heat flux and flow rate conditions. The experimental results revealed that mixed convection occurred under certain conditions, resulting in significant deterioration of heat transfer. Notably, the experimentally evaluated heat transfer coefficients did not align with the predictions of existing mixed convection correlations, which had been derived from test sections with different configurations. To address this discrepancy, Shin et al. proposed a modified heat transfer correlation tailored to the geometry and flow conditions of the RCCS riser duct. This new correlation (see Equation (153) in Table 4) produced an average prediction error of just 6.06% when compared to the experimental data. It is a more accurate tool for verifying the thermal performance of the RCCS, thereby enhancing the safety assessment of HTGRs and increasing confidence in their integration with large-scale hydrogen production systems.

Chen and Lee [68] conducted an experimental study to investigate the local heat transfer characteristics and thermodynamic behavior within a pebble bed reactor (PBR) core. This area has primarily been explored through numerical simulations, with limited experimental validation available in the literature. This study aimed to identify probe locations and analyzes heat transfer variability across the surfaces of pebbles packed in a face-centered cubic (FCC) structure within a controlled test section. The study revealed that heat transfer on the pebble surface depends heavily on location. Experiments were conducted at five different air inlet velocities, and the resulting surface temperature distributions and maximum temperature differences between adjacent pebbles were recorded. Nusselt numbers (see Equation (154) in Table 4) were derived from these measurements and correlated with the Reynolds number. These experimental findings provided valuable insights into the localized thermal behavior within pebble bed reactor cores and offered practical data to support the safe and accurate thermal design of reactors.

A thermal hydraulic analysis of pebble bed reactors (PBRs) requires a careful selection of convective heat transfer correlations because the complex, porous structure of the core results in clear performance differences across various flow regimes and coolants. A structured quantitative comparison reveals that, although macroscopic empirical correlations can accurately predict overall parameters, they frequently fail to account for local thermal stresses and extremes, especially near the reactor walls, where variations in porosity can cause significant velocity and temperature fluctuations. For standard gas-cooled applications, the classical Achenbach (1995) [69] correlation shows the greatest agreement with experimental data in randomly packed beds. However, the packing geometry significantly alters thermal performance. For example, correlations derived for hexagonally close-packed structures $(Nu=0.177{Re}^{0.8}{Pr}^{0.4})$ reveal weaker heat transfer intensity than face-centered cubic arrangements ($Nu=0.194{Re}^{0.8}{Pr}^{0.4})$ despite identical packing densities. When evaluating alternative coolants, conventional correlations formulated for low-Prandtl gases or water generally fail to accurately predict heat transfer. Specifically, for water-cooled pebble beds (WPBRs), the Wakao correlation is effective in turbulent regimes ($Re>3000$), whereas the KTA correlation is more precise at lower velocities. However, a recently proposed geometry-dependent formulation (${Nu}_{dh}={10}^{0.086/{\epsilon }^{2.14}}{Re}^{0.55}{Pr}^{1/3}$) successfully predicts data with an error margin of less than 10%. For fluoride salt-cooled high-temperature reactors (FHRs) utilizing high-Prandtl molten salts ($Pr\approx 16$), the Meng et al. (2012) [70] correlation is uniquely suited, as legacy models exhibit significant quantitative and qualitative predictive inaccuracies. Furthermore, in passive cooling scenarios such as the reactor cavity cooling system (RCCS), standard mixed convection models demonstrate substantial discrepancies, necessitating geometry-specific formulations that reduce average prediction errors to just 6.06%. In conclusion, although global average correlations are computationally efficient, high-fidelity safety analyses that address localized wall effects and thermal equilibrium in PBRs necessitate advanced coupled DEM-CFD approaches or structurally specific empirical fits to guarantee precise thermal predictions.

6. Summary and Conclusions

This article presents the results of an extensive literature review of convective heat transfer in gas-cooled nuclear reactors. These reactors have a different technical structure than water-cooled reactors (for which comprehensive review of correlations describing heat exchange processes was presented in the previous paper of the authors’ research group [15]); thus, the heat transfer processes in such reactors are organized differently. Compared to water-cooled reactors, gaseous coolants do not change phase and have a relatively low specific heat capacity; therefore, the intensity of convective processes occurring in the core has to be increased, for example, by increasing the coolant velocity and flow rate and expanding the heat transfer surface area. In AGRs, CO₂ flows through channels in a graphite moderator, transferring heat to the steam generators, while in GFRs and high-temperature prismatic reactors, high-pressure helium uses forced convection to control high power densities and enable high outlet temperatures. In PBRs, helium flows through a porous bed of spherical TRISO elements, taking advantage of the large contact surface area to maintain uniform fuel temperatures. In all these designs, convective heat transfer ensures safe temperatures for fuel and structural components, and high thermal efficiency of the reactor and power plant and plays a key role in both normal and transient operation, making even coolant flow distribution and reliable circulation systems fundamental to reactor safety and performance.

The following conclusions can be listed basing on this review:

• Correlations used for convection phenomena description in gas-cooled reactors differ from those used for other types of reactors. For example, in gas-cooled reactors, coolant does not undergo phase changes. Therefore correlations describing convection do not include terms and criteria numbers related to boiling and condensation.
• The operating temperature range of gas-cooled reactors is higher than that of water-cooled reactors, which means that the convection phenomenon is largely supplemented by radiation, and the correlations describing convection have their specific limitations and applicability ranges related to the accuracy of prediction in the high temperature range.
• In VHTRs, radiation begins to dominate over convection. A precise description of the interdependence of these heat transfer processes could be the subject of further research, particularly regarding emergency conditions.
• Due to the thermodynamic properties of the applied gaseous coolants (e.g., helium), the correlations require the additional correction factors.
• This review shows that in some cases, due to the limited applicability of correlations, a detailed numerical analysis of the heat transfer process is required.
• In gas-cooled reactors, graphite is applied as the construction material. It has limited durability due to material degradation and operational history, which may have an influence on lowering the maximum fuel temperature. These material constraints may also limit the reactor operating temperature range.
• In the case of GCR reactors, specific descriptions concern convective heat transfer in porous media, where specific corrections are used in the correlations to take into account the surface porosity.
• Due to the complex design of prismatic blocks in PBRs, the description of convective heat transfer processes requires a combination of 1D and 3D models due to the homogenization of the core, which is a compromise between the accuracy of calculations and the required computational time and cost.
• In pebble bed reactors, correlations work well for determining convection parameters in macroscopic analysis but fail to predict local thermal stresses. The most reliable method used for this prediction is coupled DEM-CFD, especially for high-Pr coolants.
• Interblock gap modeling cannot be based on constant loss coefficients but must account for dynamic changes in gap geometry. Basic hydraulic models do not reflect the nonlinear nature of flow in narrow gaps due to the effect of low Re number, which disturbs the coolant balance in the core.
• Empirical Nusselt number correlations, such as Dittus–Boelter or McEligot, significantly lose accuracy under realistic nonuniform heating conditions typical for VHTR operation. In such cases, the deviation in heat transfer coefficient predictions can increase from approximately 18% to over 30%, necessitating the use of specific correction factors for localized power distributions.
• In prismatic block reactors, convective heat transfer is strongly influenced by thermal entrance effects, which can persist for up to 25 hydraulic diameters. Neglecting these effects by assuming fully developed flow throughout the entire channel length leads to significant errors in predicting peak fuel temperatures in the initial core regions.
• In GFRs, an important and most frequently studied phenomenon related to the description of heat flow is deorbited heat turbulent transfer (DTHT).
• In the case of a loss-of-coolant (LOCA) accident, the flow regime often changes to mixed convection, in which buoyancy forces play a significant role, leading to DTHT. Standard correlations have limited applicability in such conditions, so modified correlations that incorporate acceleration and buoyancy parameters (e.g., McEligot) are necessary.
• In pebble beds, stronger temperature oscillations occur near the reactor walls due to changes in surface porosity compared to the central part of the bed. Averaged heat transfer coefficients mask local extremes near the walls, which are important for thermal analysis of the reactor walls.

Future research directions related to convection processes in gas-cooled reactors should focus on:

• The development of advanced hybrid methods.
• Verification of flow correlations in geometries associated with deformed graphite blocks.
• The need to develop new, more accurate correlations describing the convective heat transfer processes.
• Experimental studies on GFR.
• The development of porous media models to increase the accuracy of core modeling calculations.
• The need to verify existing correlations in small-scale experimental setups, which will allow for improved accuracy and the potential introduction of additional correction factors.
• Generating higher-quality experimental results for gases under mixed convection conditions, especially for heat flow analysis during LOCA.