A Comprehensive Review of Mixed Convective Heat Transfer in Tubes and Ducts: Effects of Prandtl Number, Geometry, and Orientation

Abstract

This paper presents a comprehensive review of mixed convective heat transfer phenomena involving fluids with varying Prandtl numbers, specifically focusing on their behavior in different geometries and orientations. This study systematically explores heat transfer characteristics for fluids with low, medium, and high Prandtl numbers across a range of tube geometries, including circular, rectangular, triangular, and elliptical cross-sections, and examines their effects in both horizontal and vertical tube orientations. By consolidating existing research findings and analyzing various experimental and numerical studies, this review elucidates the complex interactions between fluid properties, tube geometry, and flow orientation that influence mixed convection heat transfer. Key insights are provided into the mechanisms driving heat transfer enhancements or degradations in different scenarios. In view of the findings from this paper, more than 84% of studies were conducted in a horizontal orientation and circular cross-section with a tendency to use medium-to-high Prandtl numbers as the working fluid for the past 10 years. This paper also identifies critical gaps in current knowledge and suggests future research directions to advance the understanding and application of mixed convective heat transfer in diverse engineering systems. Furthermore, apart from having different geometries applied in industrial applications, there is still room for improvement through the addition of passive methods to the heat transfer system, including helical coils, corrugations, swirl generators, and ribs. Overall, from the literature review, it is found that there are few relevant numerical simulations and experimental studies concentrating on middle Prandtl number fluids. Hence, it is recommended to perform more research on medium Prandtl number fluids that can be used as energy storage systems (ESS) in concentrating solar power plants, nuclear reactors, and geothermal systems.

1. Introduction

In heat transfer engineering, mixed convection is a fluid thermodynamics phenomenon and it occurs when natural or free convection and forced convection mechanisms act together to transfer heat. The dynamic transition between both convections is correlated and can be plotted to visualize flow regime maps. The maps will determine which convections dominate the flow: either free or forced convection or both (a mixture of both convections) [1]. This thermodynamics phenomenon also exhibits its fluid behavior associated with the fluid velocity and thermal boundary layer thickness. Understanding its effect is crucial for optimizing thermal systems and engineering processes. Fundamentally, there are three primary modes of thermodynamics phenomenon: conduction, convection, and radiation [2]. Conduction occurs when there is an interaction (physical touching) between two objects; however, it is different with radiation as the heat transfer occurs without direct physical contact with each other, and convection occurs when a gas or liquid goes around an object, as shown in Figure 1. Figure 2 shows the different types of heat transfer mechanisms.

In order for the heat to transfer efficiently, apart from the influence of the convection flow form (natural, forced, or mixed) through mapping visualization and hydrodynamic behavior as the fluid rubbing with the inner wall surface, the geometry configurations of the enclosure for the heat transfer process to be taken place are also vital as both cooling and heating processes very much depend upon the fluid flow scattered area of interest [4]. This heat exchange is either by bringing the two fluids condition in direct or indirect contact.

According to ‘Newton’s law of cooling,’ the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings, as shown in Equation (1):

$$Q=hA\mathsf{\Delta }T.$$

(1)

where h = convection heat transfer coefficient (W/m²·K)

A = surface area through which convection heat transfer takes place (m²)

ΔT = difference between surface temperature and fluid temperature sufficiently far from the surface (K).

It is deduced that heat transfer performance can be further reinforced through three aspects: first, by growing the temperature difference; second, by enlarging the heat transfer area A; third, by improving the heat transfer coefficient. Liu et al. [5] investigated the heat transfer properties of a triple-tube latent heat storage system when oriented both horizontally and vertically. They discovered that the system’s heat storage performance markedly declines as the tube length increases and that the release performance of the vertically positioned triple-tube phase change heat storage system is notably reduced. Additionally, the vertical orientation facilitates more effective solidification and melting of lauric acid, as well as improved axial heat transfer, compared with the horizontal orientation.

The reinforcement of heat transfer performance is vital in the design and application of heat exchangers. The heat transfer coefficient can be further improved by optimizing the turbulence flow in the system. There is an empirical correlation by Dittus–Boelter in the analysis of turbulent tube flow [6].

Nu = 0.023.Re^0.8. Prⁿ

(2)

0.6 ≤ Pr ≤ 160,  Re ≥ 10⁴,  L/d_w ≥ 60

where Nu = hd_w/k, Re = u_md_w/v, Pr = C_pμ/k, d_{w = 2rw}—inner tube diameter, L—tube length, u_m—fluid mean velocity, h—heat transfer coefficient, C_p—specific heat at constant pressure, ρ—density, k—thermal conductivity, v = μ/ρ—kinematic viscosity, μ—dynamic viscosity.

From the above Dittus–Boelter formula, D. Taler and J. Taler [7] carried out an analysis and came out with an improvised power-type formula for turbulent flow for three different Prandtl numbers groups: high, medium, and low.

Nu = 0.02155.Re^0.8018. Pr^0.7095

(3)

for low Pr 0.1 ≤ Pr ≤ 1 3.10³ ≤ Re ≤ 10⁶

Nu = 0.01253.Re^0.8413. Pr^0.6179

(4)

for medium Pr 1 ≤ Pr ≤ 3 3.10³ ≤ Re ≤ 10⁶

Nu = 0.02155.Re^0.8018. Pr^0.7095

(5)

for high Pr 3 ≤ Pr ≤ 1000 3.10³ ≤ Re ≤ 10⁶

As initiated by the Dittus–Boelter equation, the further outcome shown by the improvised three Equations (3)–(5) above demonstrates the significance impact of Prandtl number selection in heat exchanger design. If there is a low Prandtl number, power exponents with Prandtl and Reynolds numbers are almost identical. If it moves towards a high Prandtl number, the power with the Prandtl number decreases; therefore, continuous experimental research is carried out to find straightforward and accurate heat transfer correlations in the design and performance calculations of heat exchangers, HVAC, and thermal management systems.

The Prandtl number is a dimensionless number, a ratio of the fluid momentum diffusivity to the fluid thermal diffusivity [8].

P_r = μ_f. C_p
k

(6)

where μ_f = fluid dynamic viscosity (kg/m·s)

C_p = specific heat capacity (J/kg·K)

k = thermal conductivity (W/m·K)

Momentum diffusivity (kinematic viscosity) represents the material’s resistance to shear flows in relation to density [8]. The occurrence of shear flows is due to dissimilar layers within the flow that travel with different velocities of adjacent walls. The geometry of the walls does not influence the Prandtl number as the number is dependent solely on the fluid and the fluid state [8]; therefore, the dependency between both the Prandtl number and the fluid state helps to tell how the heat and momentum interact within a fluid, making it a critical parameter in heat transfer analysis [9].

A wide range of investigations were conducted by researchers to study the consequences of different Prandtl numbers towards the development of a mixed convective heat transfer mechanism. Different working fluids react contrarily to heat flow [10].

Table 1. Typical Prandtl number values for certain working mediums.

		Working Medium	Pr	Applications
Prandtl Number classification	Low	Liquid Metals Mercury	0.0015	▪ Nuclear reactor cooling system. ▪ Cooled fast breeder reactors.
		Binary Helium Mix [11] He + CH4 He + N2 He + O2 He + CO2 He + SF6 He + CF4 He + Xe	0.60 0.48 0.48 0.43 0.28 0.20 0.12	▪ Cryogenic systems. ▪ Superconducting devices. ▪ Industrial gas applications. ▪ Gas dynamics and mixing. ▪ Energy applications. ▪ Environmental monitoring.
		Air	0.699	▪ Aerospace engineering. ▪ HVAC systems. ▪ Combustion processes.
		Oxygen (O2)	0.63	▪ Cryogenics ▪ Environmental engineering.
		Hydrogen (H)	0.684	▪ Hydrogen production processes. ▪ Environmental and Safety considerations.
		Carbon dioxide (CO2)	0.76	▪ Carbon capture and Storage (CCS) ▪ Environmental impact studies. ▪ Food and beverage industry. ▪ Climate control and HVAC systems
	Medium -to- High	Molten salts [10,11] ▪ LiF-BeF2-ThF2-UF4 ▪ FLiBe ▪ FLiNaK ▪ KF-ZrF4 ▪ NaNO3-KNO3 ▪ MgCl2-KCl	6–28 7–31 3–21 5–27 4–15 3–5	▪ Solar salt in solar power plant ▪ Thermal energy storage ▪ Nuclear reactors ▪ Thermal management in electronics.
		Ammonia	1.38	▪ Refrigerant system, heat exchangers, combustion processes.
		Water (H2O)	5–10	▪ Natural convection in heat exchangers.
		Glycerol	1000	▪ Heat exchangers in engines and industrial processes.
		Glycerin	2450	▪ Heat transfer analysis, food and pharmaceutical industries, cosmetics, and personal care products.
		Polymer melts	10,000	▪ Heat transfer analysis, thermal conductivity measurement, polymer processing.

shows typical Prandtl number values for certain working mediums.

Based on the information provided, glycerol, along with its mixtures with water and ethylene glycol, is considered a fluid with a high Prandtl number. In contrast, air and nitrogen are classified as fluids with a low Prandtl number, whereas water and ethyl alcohol have moderate Prandtl numbers [8]. Based on the information provided, glycerol, along with its mixtures with water and ethylene glycol, is considered a fluid with a high Prandtl number. In contrast, air and nitrogen are classified as fluids with a low Prandtl number, whereas water and ethyl alcohol have moderate Prandtl numbers [8]. All the above-mentioned fluids, at different Prandtl numbers, deliver different impacts to the heat transfer process, such as heat transfer characteristics, flow regimes, natural convection, heat exchangers application, and thermal stability. Understanding its impact is essential as it allows engineers to predict fluid flow behaviors, design optimization, and improve energy efficiency in systems ranging from cooling systems to chemical reactors [8].

In mixed convective heat transfer analysis, most of the previous studies had correlated with another parameter, the Nusselt number. Both Nusselt and Prandtl numbers are dimensionless numbers used in fluid mechanics and heat transfer and play crucial roles in characterizing convective heat transfer. Nusselt numbers (Nu) is defined as the ratio of convective to conductive heat transfer across a boundary and is given by the formula:

$$Nu={\displaystyle \frac{hL}{k}}$$

(7)

Nu depends on both Re and Pr, as expressed by:

$$Nu=f(Re,Pr)$$

(8)

where Re is the Reynolds number, which characterizes the flow regime (laminar or turbulent).

f is a function that represents a relationship that depends on the Reynolds number (Re) and the Prandtl number (Pr).

Both numbers are interconnected through empirical correlations that relate convective heat transfer performance to fluid properties and flow characteristics. In general, a higher Prandtl number indicates that the fluid has a greater ability to retain thermal energy compared with momentum. This often leads to higher Nusselt numbers, signifying enhanced convective heat transfer. Understanding this relationship is crucial for designing and analyzing systems involving heat transfer, such as heat exchangers, cooling systems, and various engineering applications.

The main focus of this review paper is to attain as much information as possible from studies conducted by researchers on the effectiveness of mixed convection heat transfer through various Prandtl number fluids, geometry setup, and orientation applied, as shown in Figure 3. Studies with different cross-sections (circular, rectangular, triangle, and elliptical) will be the main focus of this paper, as shown in Figure 4. The Prandtl number (Pr) is one of the indicators as it embodies one of the thermodynamic properties (the mass heat capacity at constant pressure) and two transport properties (viscosity and thermal conductivity). The ratio of the fluid momentum diffusivity to the thermal diffusivity influences the convection heat transfer process. Several studies have shown that changes in the aspect ratio of the geometry significantly affect the flow distribution pattern [12].

This paper has systematically explored previous studies on mixed convective heat transfer related to the fluid property of Prandtl number, tube geometry, and fluid flow orientation. These three elements represent three different objectives. In a nutshell, the objectives are set based on the target of having a better insight into the selection of Prandtl number fluid, tube geometry, and flow orientation in the tube used previously by the researchers as an indicator for future research.

2. Low Prandtl Number

In heat transfer problems, the Prandtl number is essential for determining the relative thickness of the fluid’s momentum and thermal boundary layers, primarily in the thermal entrance (developing) region. For low Prandtl number fluid, both momentum and thermal element behave inversely as the smaller value of Pr (Pr < 1) indicates that the heat diffuses quickly compared with momentum [8]. Hence, the thermal boundary layer (δ_T) will be much thicker (dominates more) compared with the velocity boundary layer (δ), as shown in Figure 5.

2.1. Low Prandtl Number—Vertical Circular Tube

Mohammed and Salman [13] investigated the interplay between natural and forced convection heat transfer in a vertical circular cylinder, utilizing laminar air flows (Pr = 0.7) under steady wall heat flux conditions. Their experimental setup, illustrated in Figure 6, featured a cylindrical test section integrated into an open-air circuit and mounted on a wooden board. This setup could be tilted around a horizontal spindle, allowing for adjustments in the cylinder’s angle of inclination.

During the experiment, the incoming air passed through the calming section and the test section. At the entrance of the section, the flow is hydrodynamically developed, thermally developing, and thermally fully developed, using both long and short aluminum pipes. Thermocouples are positioned along the test section at four different distances: 20 mm, 25 mm, 40 mm, and 50 mm, as depicted in Figure 7.

The effect of varying the cylinder inclination angle on the mixed convective heat transfer process was also investigated as summarized in

Table 2. Summary findings on the effect of varying cylinder inclination angle on the mixed convective heat transfer process.

Inclination angle change	90° → 0°		0° → 90°
Position	Vertical → Horizontal		Horizontal → Vertical
Heat flux	Constant	Constant	Constant	Constant
Convection type	Free convection dominant	Forced convection dominant	Free convection dominant	Forced convection dominant
Surface temp.	↓	↑	-	-
Nusselt number	↑	-	-	↑
Reynolds number	↓	-	-	↑

. The results indicate that as the cylinder angle changes from Ѳ = 90° to Ѳ = 0°, the surface temperature decreases when free convection dominates. Conversely, when forced convection dominates, the surface temperature increases as the inclination angle shifts from a vertical to a horizontal position. Additionally, for the same heat flux, the surface temperature is higher at lower Reynolds numbers compared with higher Reynolds numbers due to the predominance of free convection. This study also demonstrated that as the heat flux rises and the cylinder’s inclination angle decreases from Ѳ = 90° to Ѳ = 0°, the Nusselt number increases.

2.2. Low Prandtl Number—Horizontal Circular Tube

Maughan and Incropera [14] conducted calculations for low Prandtl number air (Pr = 0.7) to analyze fully developed laminar and mixed convection flow in a horizontal parallel plate channel, with uniform heating applied to both the top and bottom plates. Their objective was to investigate mixed convection heat transfer in the thermal entry region. As depicted in the schematic in Figure 8, the experimental setup was designed to achieve boundary layer growth of less than 10% of the channel height for Reynolds numbers greater than 1500. This was facilitated by a nozzle that effectively prevented flow separation; however, studying the simultaneous development of thermal and hydrodynamic boundary layers at low Reynolds numbers (i.e., Re < 500) was not feasible.

Spanwise conduction within the plates was taken into account, and calculations were conducted for Rayleigh numbers ranging from 0 to 2.5 × 10⁴ and non-dimensional conductance ratios between 10⁻⁵ and 10³. The results demonstrated that the enhancement of mixed convective heat transfer is limited to the lower surface. Additionally, the secondary flow generated causes significant spanwise variations in surface temperature, with the maximum temperature on the lower surface exceeding that of the upper surface.

Additional research involving air (Pr = 0.7) as the working fluid is covered by Hishida et al. [15] and Chou and Hwang [16]. They investigated mixed convection flow in the thermal entrance region of an isothermally heated horizontal pipe. Prior to their work, most studies in this area focused on high Prandtl number fluids and fell into two categories: those examining mixed convection in the thermal entrance region and those exploring hydrodynamically and thermally developing flow regions. Their study involved numerical solutions of mixed convection in the entrance region of an isothermally heated horizontal pipe, resulting in steady-state solutions. Their findings highlighted how buoyancy forces (Grashof number) affect the developing velocity and temperature fields, as well as the related heat transfer characteristics in the entrance region. Specifically, an increase in the Grashof number reduces the entrance length before noticeable free convection effects begin and enhances the local maximum Nusselt number. Additionally, Eiamsa-ard et al. [17] investigated the thermal performance characteristics of cold air using a combination of twisted tape and wire coil inserts, focusing on how these non-uniform inserts affect thermal performance.

Another study on convective heat transfer in both horizontal and vertical circular pipes was conducted by Wang et al. [18]. They used liquid mercury (Pr << 0.1) and liquid sodium (Pr << 0.1) as working fluids to systematically examine the effects of low Prandtl numbers on mixed convection in these liquid metals within horizontal and vertical circular tubes. This study covered parameters including Peclet numbers (Pe) ranging from 0.001 to 700 and Rayleigh numbers (Ra) from 0.05 to 35,000. The local Nusselt number was plotted against the Graetz number for various Rayleigh numbers. With the target of studying the axial conduction effect, the flow field in both horizontal and vertical pipes is partitioned into two regions, as shown in Figure 9a,b. Insulation was added to the upstream region while keeping the temperature in the downstream region. After all, the main objective is to look for the effects of both transverse conduction and buoyancy force towards thermal entrance regions on both horizontal and vertical pipes numerically for mixed convection at low Pr. The results are summarized in

Table 3. Effects of both transverse conduction and buoyancy force towards thermal entrance regions on both horizontal and vertical pipes numerically for mixed convection at low Pr.

	Parameter	Horizontal Pipe	Vertical Pipe
1	Distortion of transverse temperature and velocity profiles.	Begin further upstream. Axial conduction and free convection intensify as Pr decreases. Effects become more pronounced with increasing Rayleigh number (Ra).	Due to the buoyancy effect, it becomes larger with │Ge/Re│ increase. Weakens at very low Pe flows.
2	Secondary flow	Development occurs in the upstream region at low Prandtl numbers.	-
3	Reversal flow	Negative velocity is observed near the top pipe wall. The flow region shifts upstream as Pr decreases.	Produced at pipe center (heating case) Produced near the wall (cooling case) at high │Ge/Re│ Regions move upstream due to the axial conduction effect.
4	Wall shear stress	The minimum velocity occurs at the top pipe wall during reverse flow. The maximum velocity is found at the bottom pipe wall during reverse flow. The positions of maximum and minimum velocities shift further upstream as the Peclet number (Pe) decreases.	Larger in the heating case. Lower in the cooling case at increasing │Ge/Re│.
5	Local Nusselt number	There is a significant decrease in velocity with the onset of reverse flow.	Increases with increasing │Ge/Re│ (heating case) Local maximum at high Pe. Decreases with increasing │Ge/Re│ (cooling case). Lowest value at high Ra.

.

2.3. Low Prandtl Number—Horizontal Rectangular Duct

Another geometry investigation was performed on a square duct by Smith et al.. [19]. The square duct is fitted with a tandem wire coil element to create a turbulent regime in the forced convection heat transfer at a Pr of 0.7, and Re range from 4000 to 25,000. The wired coil length was varied to look for a pressure drop effect whereby it caused a higher pressure drop; however, as the Re increased, Nu decreased. Figure 10 shows the square test duct fitted with a wired coil. For twisted tapes, Zhang et al. [20] utilized them in numerical studies of heat transfer and flow characteristics for laminar flow in square ducts.

2.4. Low Prandtl Number—Horizontal Triangular Duct

Saleh et al. [21] experimentally compared three different ducts: circular, triangular, and rectangular cross-sections to look for the local temperature distribution and local Nusselt number. An electric current outside the ducts was used to provide constant heat flux, as shown in Figure 11 and Figure 12. Porous media (glass sphere of 12 mm diameter) was filled up in the test section. The heater properties are listed in

Table 4. Properties of the heater [21].

Cross Section	Material	Number of Heaters	Resistance (Ohm)	Diameter of Coil (mm)	Number of Coils
Circular	Stainless steel	4	27.027	8	23
Rectangular	Nickel chrome	2	20.8	0.25	52
Triangular	Nickel chrome	2	18.2	0.25	54

. The results showed that both were impacted by Reynolds number and cross-section. Finally, the triangle duct showed a better heat transfer, followed by the rectangular duct and circular tube. Similarly, Rasheed et al. [22] carried out experiments on those three geometries (triangular, rectangular, and circular tubes) but in porous media forced heat transfer convection, and the results are similar to the results obtained by Saleh et al. [21].

2.5. Low Prandtl Number—Horizontal Elliptical Tube

Ibrahim et al. [23] found out that the best thermal performance of the elliptic tube could be achieved with lower Re, axis ratio, and angle of attachment. The investigation was conducted using a thermofluid in an elliptical tube bundle in a crossflow; however, the maximum deviation of 28.4% was concluded by Yang et al. [24]. Based on their numerical simulation results, they managed to acquire an empirical correlation of the turbulent convective heat transfer in an elliptical tube.

In a nutshell, for low Prandtl numbers, most of the research is conducted on a horizontal axis in uniformly heated conditions using diverse cross-sections, including circular, horizontal, triangular, and elliptical shapes, as shown in

Table 5. (a): Findings of mixed convective heat transfer for low Prandtl number in the vertical axis. (b): Findings of mixed convective heat transfer for low Prandtl number in the horizontal axis.

(a): Vertical Axis
Authors	Working Fluids	Tube Type	Methodology	Heat Transfers Correlation	Findings
Mohammed; Salman (2007) [13]	Air (Pr:0.7)	Vertical Circular Cylinder	▪ Inclined [ɵ = 90° to 0°]; ▪ Uniformly heated short tube [L/Di:30]	$\overline{Nu}$= 3.7151 ($\overline{Ra}$/$\overline{Re}$)0.11868 ▪ Kumar, Baig, Asrar [25] $Nu = 1.86 {\mathrm{Gz}}^{1/3}({\displaystyle \frac{\mu f}{\mu w}}$)0.14 ▪ Brown, Gauvin [26] ▪ Sieder, Tate [27]	When cylinder inclination θ moves: ▪ Vertical to Horizontal ○ Surface Temp. drop when free conv. dominants. ○ Surface Temp. up when forced conv dominants.
(b): Horizontal Axis
Authors	Working Fluids	Tube Type	Method	Heat Transfer Correlation	Findings
Maughan and Incropera (1990) [14]	Air (Pr:0.7)	Horizontal, Parallel Plate Channel	Uniformly heated (top and bottom plates)	For smaller Ra: Nu = Convective Force Limit For larger Ra: Nu = 0.20Ra1/3	▪ The onset of instability moves upstream for Gr increase and delayed for Re increase. ▪ Inclining the channel for buoyancy-assisted flow result: ○ Heat transfers up prior to the onset of instability. ○ Delay in the formation of secondary flow.
Chou and Hwang (1988) [16]	Air (Pr:0.7)	Horizontal Tube	Uniformly heated	Re.Ra$\tau$ = (${\displaystyle \frac{2}{\pi }})$ Gr+ [25] Ra*= ${\displaystyle \frac{16Ra}{Nu}}{\displaystyle \frac{4PrReRa\tau }{Nu}}$ [26]	▪ At given Ra ○ Pr decrease will increase the distortion of axial velocity and temperature profile. ▪ Axial distance shortened as: ▪ Increase Ra when Pr is fixed. ▪ Increase Pr at given Ra.
Wang et al. (1994) [18]	Hg (Pr < < 0.1) Na (Pr < < 0.1)	Horizontal and Vertical Circular Tubes	Uniformly heated	Nu vs. Abscissa Gz for Ra	▪ Ra increase = Nu increase (Pr > 0.1) ▪ Ra increase =Nu decrease (Pr = 0.003–0.025) ▪ Nu decreases as X0 increase (Pe ≤ 2.5) [28]

a,b.

3. High Prandtl Number

Once the outcome of the ratio between momentum diffusivity and thermal diffusivity is more than the equilibrium state (Pr = 1), this tendency shows that momentum diffuses quickly compared with thermal diffusivity. The larger Pr values mean that momentum diffusivity dominates and the heat transfer process is more influenced by fluid motion (convection) [8]. Hence, the velocity boundary layer (δ) will be much thicker compared with the thermal boundary layer (δ_T), as shown in Figure 13.

3.1. High Prandtl Number—Vertical Circular Tube

Extended studies were carried out by Thomas and Sobhan [29] to examine and compare the measurements of convective heat transfer and phase change in nanofluids from the perspective of thermal conductivity and thermal performance of nanofluids system. By using transient and steady-state methods, of which the transient hot wire method produced better accuracy, the outcome of thermal conductivity measurement has shown that nanofluid performance in forced convection is better than that of the base fluid. Hence, nanofluids are expected to provide boosted convective heat transfer coefficients. As most of the investigation had proven capacity of convective heat transfer in affecting thermophysical properties of the nanofluid, including both viscosity and thermal capacity, there is still controversy on its effects by natural convection. Even though some experimentation had shown an interaction between the Rayleigh number and free convection in heat transfer growth, there is also uncertainty at certain Rayleigh numbers, Ra = 104, as it was not sensitive to nanoparticle concentration.

3.2. High Prandtl Number—Horizontal Circular Tube

Apart from water, nanofluids are also being used by researchers for experimenting with the high Pr fluid in convective heat transfer. The term “nano-fluids” was likely coined by Choi [30], who demonstrated the feasibility of this concept. The successful estimation of thermal conductivity for various nanofluids has led to significant energy and cost savings and has enabled the design of more compact and lightweight heat exchanger systems. Nanofluids are created by mixing ultrafine particles with conventional base fluids, which greatly enhances the heat transfer characteristics of the original fluid. These fluids typically contain small volumetric concentrations (around 0.0001–10%) of nanosized solid particles (100 nm or smaller). The effectiveness of nanofluids is achieved when these nanoparticles are uniformly dispersed throughout the fluid, resulting in a higher thermal conductivity due to the enhanced thermal conduction coefficient provided by the nano additives. Initial research into the use of nanometer-sized particles began in the early 1990s by scientists at Argonne National Laboratory (ANL).

Most of the researchers who studied the impact of nanofluids towards heat transfer performance either through forced or natural convection had used Al₂O₃ to make suspensions, and water acts as the base liquid since Al₂O₃ is proven and capable of providing good thermal conductivity, viscosity, and the flow properties of it is well established. Their research findings demonstrated the importance of Al₂O₃ in the cooling process [31,32,33,34,35,36,37]. Moghaieb et al. [37] specifically focused on improving engine cooling performance with Al₂O₃/water nanofluid. The vehicle engine cylinder head is used to act out the rectangular duct. The outcome showed promising results as 78.67% was recorded as the highest heat transfer coefficient throughout the experimentation, which was achieved at 1% nanoparticle volume concentration. Hence, Al₂O₃ is a good choice of cooling mixture for the cast iron components of the engine. Rashidi et al. [38] highlighted the significance and impact of an accurate Prandtl number model by experimentally investigating the laminar flow of incompressible Al₂O₃–H₂O and Al₂O₃–C₂H₆O₂ nanofluids over a vertical stretching sheet. Their findings revealed that increasing the volume fraction of Al₂O₃ nanoparticles led to a decrease in the temperature of the nanofluids when an effective Prandtl number was considered, whereas, in the absence of an effective Prandtl number, the temperature of the nanofluids increased. Vishnu Ganesh [39] also carried out a study on similar solutions with different base fluids over a stretching sheet. With an effective Prandtl number, Ganesh had carried out further investigation on the non–linear thermal radiation effects on the Marangoni boundary layer of Al₂O₃ nanofluids. Marangoni convection is defined as an interfacial liquid flow phenomenon driven by the surface tension gradient, which can accelerate the renewal of the liquid surface and enhance the transfer coefficient of the heat and mass transfer process. The effects of the magnetic field on the Marangoni boundary layer of Al₂O₃ nanofluids were analyzed in the presence of an effective Prandtl number.

The main objective of implying this suspension is to ensure proper dispersion of nanoparticles in the liquid. Ultrasonic vibration is used to achieve this proper dispersion. Nanoparticles were produced by the physical vapor synthesis method by Putra et al. [40]. The synthesis process allowed these particles to form loose agglomerates under atmospheric conditions. A transmission electron microscope (TEM) is used to observe the breaking of the agglomerates, as shown in Figure 14.

However, there are other suspensions used by other researchers that are qualitatively different from the usual ones. Okada and Suzuki [41], Kang et al. [42], and Okada et al. [43], in their studies on suspensions of these micrometer size particles, concluded that nanofluids are different from common suspensions with regards to natural convection because of their nature of interface formation due to particles sedimentation.

Colla et al. [44] chose to study nanofluid capability to increase heat transfer by using TiO₂, a water-based nanofluid (Pr = 6.7–6.9), as the working fluid. The evaluation was on convective heat transfer under laminar forced and mixed flow conditions in a horizontal circular uniformly heated pipe. The general form of the Nusselt number is estimated over the function of Prandtl number, Reynolds numbers, heat capacity, Brownian diffusion, thermophoresis and Dufour-effect.

Apart from improving both thermal conductivity and performance, nanofluids are also beneficial in stretching both thermal and momentum layer surfaces as both layers are features of Prandtl number significantly. Mahmood et al. [45] studied the matter by using a nanoliquid, alumina Al₂O_3, as an additive to pure water (H₂O) and ethylene glycol (C₂H₆O₂) (see

Table 6. Properties value of water, ethylene glycol, and alumina [45].

	ρ (kg/m3)	Cp (J/kgK)	K (W/mK)	Pr
Pure water (H2O)	998.3	4182	0.60	6.96
ethylene glycol (C2H6O2).	1116.6	2382	0.249	204
Al2O3	3970	765	40	-

).

Maiga et al.’s [46] numerical investigation of mixing nanofluids of Al₂O₃ with ethylene glycol produced a remarkable increase in heat transfer performance compared with a mixture of water (H₂O) and alumina oxide (Al₂O₃) in forced convection flows even though it gave adverse effect on the wall shear stress. Each opposing outcome is to be anticipated as the addition of nanoparticles, apart from affecting the fluid viscosity (increase), also increases friction/pressure losses. The correlations have been derived to calculate the average Nusselt number of nanofluids as a function of Reynolds and Prandtl numbers.

Higher Prandtl numbers have also been used to study mixed convection heat transfer. Coutier and Greif [47] examined the laminar mixed convection of a propylene–glycol solution with water (Pr = 28.1 to 46.0) in a horizontal circular tube under isothermal conditions. They explored the correlation between the Nusselt number and the Graetz number. Their findings showed that nanoparticles could either reduce the effectiveness of natural convection in the overall heat transfer for a given heat flux or enhance it under specific Grashof numbers.

Li and Feng [48,49] conducted an investigation to correct the fluid dynamic viscosity and the Grashof number. This study was divided into two parts. In Part I, they analyzed the laminar mixed convection of a SiO₂-ethylene glycol–water nanofluid (Pr = 20 to 300) flowing through a horizontal tube at low Reynolds numbers (9 < Re < 450). The investigation utilized two base fluids: ethylene glycol (EG) and a mixture of 50 vol% de-ionized water (EG/H₂O), with nanoparticles of two sizes, 15 nm and 50 nm. This study focused on four types of nanofluids—EG-15 nm, EG-50 nm, EG/H₂O-15 nm, and EG/H₂O-50 nm—to examine how nanoparticle size and concentration affect convective heat transfer in a laminar mixed flow regime. The experimental setup is illustrated in Figure 15.

During the experiment, the fluid is pushed out from the reservoir and passes through a test section at constant heat flux boundary conditions. All the desired parameter readings are collected, including inlet and outlet temperature, wall temperatures, flowmeter reading, voltage, and current.

The summary of the procedures is shown in

Table 7. Process flow of laminar mixed convection of a SiO₂-ethylene glycol–water nanofluid.

	Apparatus	Description	Process Flow
I	Reservoir tank	Storage of fluid (Water)	The compressed Nitrogen gas pushed the fluid from the reservoir tank. The fluid flows through a flowmeter, a needle valve, and a constant temperature bath before reaching the test section, and is then collected in a container. To achieve a constant-heat-flux boundary condition, the heat transfer test section was electrically heated using a constant DC power supply, adjustable within the ranges of 0–10 A and 0–250 V. Inlet and outlet fluid temperatures were measured with two 4-wire RTDs, while wall temperatures were determined using five T-type thermocouples, which were axially and uniformly soldered to the outer wall. Flowmeter readings, as well as the current and voltage from the DC power supply, were recorded, and instantaneous temperature measurements were collected by a data acquisition system.
2	High pressure Nitrogen tank	Contain compressed nitrogen gas to push the fluid from the reservoir tank.
3	Flowmeter	To measure fluid volume/mass for flow rate determination.
4	Needle valve	To regulate the fluid flowing through the circuit effectively and safely.
5	Constant temperature valve	To maintain a set-point in the primary circuit of a hot water-based heating system.
6	Test Section	A straight copper tube (L = 0.9 m, Di = 0.01 m, thickness of 0.002 m). Hydraulic calming section with L = 0.4 m. Entire test section is heavily insulated with foam and glass fiber insulation.
7	Container	The container was placed on a balance, and the average mas flow rate was determined by calculating the mass increment per unit time.
8	Pressure indicator	To measure the state of the fluid in the system (indicate pressure)
9	Pressure regulator	Control the pressure of a fluid by reducing a high input pressure to a lower output pressure.
10	Safety valve	To protect other equipment/parameters by opening automatically at a certain pressure and prevent damage due to excessive pressure in the process and storage system.
11	Data acquisition system	To collect and convert analog signals from sensors or instruments into digital values for processing.
12	DC power supply	To supply a one-directional flow of electric charge that flows through a conductor (i.e., wire).

.

It was observed that, for a given Grashof number, nanoparticles could either reduce or enhance the contribution of natural convection to the overall heat transfer. In Part II, the results from Part I were compared with existing correlations for mixed convection involving high-Prandtl number nanofluids. The comparison led to the conclusion that nanofluids can be considered homogeneous mixtures with effective thermophysical properties. The outcome obtained relates the connection between the flow inside the duct towards the convection heat transfer. H.-K. Park, B.-J. Chung [50] employed fins in the cavity to study its effect on heat transfer of the cooling system. Their plan was to optimize the fins plate spacing to generate optimal tip clearance in order to maximize the overall heat transfer, as shown in Figure 16 and Figure 17.

The investigation lies between laminar forced convection heat transfer, the Prandtl number (Pr: 0.7–2014), and the Reynolds number (Re: 500–1000) by using a cupric acid-copper sulfate (H₂SO₄-CuS0₄) electroplating system.

The effects of varying the gap between the fins (tip clearance), Prandtl number, and Reynolds number on the rate of heat transfer were investigated. It was found that the optimal tip clearance decreased with increasing Reynolds and Prandtl numbers. This is because the Reynolds number influences the bypass flow, while the Prandtl number affects the boundary layer thickness (see Figure 18).

For the circular geometry, A. Barletta and E. Magyari [51] investigated forced convection heat transfer in a circular duct with a constant heat flux applied to the wall surface while maintaining laminar flow inside the duct (see Figure 19). Their findings revealed that the temperature distribution at the duct entrance was non-uniform. Consequently, this study was divided into two cases: one with constant heat flux along the wall and another with varying heat flux along the wall’s axis.

A comprehensive comparison of heat transfer between circular and rectangular tube heat exchangers was numerically studied by Bisht et al. [52]. The CFD results between both circular and rectangular tubes under constant wall temperature conditions of the same length are compared and validated by the numerical correlations used by different researchers. The geometry settings for both circular and rectangular models are the same; both have a 3 m tube length and a 0.015 m diameter with water as a working fluid as per

Table 8. Geometry setting (Circular and Rectangular model) and properties of water.

Geometry Setting	Circular Model	Rectangular Model
Tube length (m)	3.0	3.0
Diameter (m)	0.015	0.015
Working fluid	Water	Water
Inlet water temperature (K)	323	323
Outlet water temperature (K)	343	343
Constant wall temperature (K)	373	373
Properties of Water	Circular Model	Rectangular Model
Density (kg/m3)	990	990
Specific heat Cp (J/kg-k)	4184	4184
Thermal conductivity (W/m-K)	0.65	0.65
Kinematic viscosity (m2/s)	0.516 × 10−6	0.516 × 10−6
Prandtl number (Pr)	3.15	3.15

.

The results specified circular tube is better compared with the rectangular tube by way of increment in heat transfer performance (2.5%) and increment in Nusselt number (8.5%), as shown in Figure 20 and Figure 21. The pressure drop in a circular tube is also higher compared with a rectangular tube. A mass flow rate is directly proportional to fluid velocity, density, and cross-sectional area, as shown in Equation (3). A bigger cross-sectional area in a circular tube compared with a rectangular tube provides better mass flow rates [8].

ṁ = ρ × A × V

(9)

Hence, as a result of a better mass flow rate, the circular tube produced a higher Nusselt number and heat transfer coefficient compared with a rectangular tube.

In terms of pressure drop, the circular tube induced more compared with the rectangular tube because of the emergence of secondary flow, which in turn heightens the flow resistance; however, for a circular tube, increasing the mass flow rate will exponentially increase the pressure drop compared with the rectangular tube which varies linearly, shown in Figure 22.

The phenomena of convective heat transfer in a circular tube are further studied by Dang and Wang [53], this time with the insertion of the twined coil inside as mechanism enhancement, shown in Figure 23.

As a result, the tube with the twined coil insert produced better heat transfer performance compared with the smooth tube. The outcome differences between both setups are due to:

(a)

the longitudinal vortices produced (from rotating fluid after guiding of spiral coin)

(b)

the secondary flow intensity increases greatly.

(c)

mixing of fluid with different temperatures.

(d)

enhancement of heat transfer performance in the tube.

The results shown in Figure 24 reflect the findings after validating these numerical data with experimental measurements, including the fluid flow and temperature distributions, Nusselt number (Nu), friction characteristics (f), and thermal performance factor (JF) in the tube with the twined coil insert.

There are many more studies conducted by other researchers with almost similar approaches to tube insertion to enhance heat transfer performance in a circular tube. Apart from the twined coil insert style, Lin et al. [54] used a twisted tape inside the tube in their numerical study of the laminar flow. Saysroy and Eiamsa-ard [55,56] also used the same material with a square-cut twisted tape in their investigation to enhance heat transfer in a convective flow of laminar and turbulent flow regions and a periodically fully-developed heat and fluid flow behavior in a turbulent flow. The same twisted tapes also were applied by Agarwal and Rao [57] and Li et al. [58]. All inquiries were using circular or round tubes. Another parameter apart from twined coil insertion and twisted tape is a wire-coil insertion [59,60,61,62,63,64,65].

Ultimately, a larger Prandtl number fluid of sulfuric acid—copper sulfate (H₂SO₄-CuSO₄) has been used by Chae and Chung [66] in their experiment on laminar mixed convection in a horizontal tube. The fixed Prandtl number of 2094 has been correlated with Grashof numbers of 1.4 × 10⁶ and 2.6 × 10⁶ and Reynolds numbers of 58 to 1270. Another parameter includes the length-to-diameter ratio L/D ranging from 0.9 to 19.2.

3.3. High Prandtl Number—Horizontal (Circular/Square/Triangular Ducts)

Sutthivirode and Suparos [67] used air to investigate the forced convection heat transfer in three different geometries (circular, square, and triangle) but with identical cross-sections for all ducts (0.002 m²). The detailed parameters used are shown in

Table 9. Experimental parameters.

Duct Types	Circular, Square, and Triangle
Heat rate (W)	920
Heat flux (W/m2)	16 to 250
Air velocity (m/s)	8 to 24
Correlations	Reynolds (Re), Nusselt (Nu), and Stanton (St) numbers

and Figure 25 below:

A series of test experiments were conducted, and the heat transfer performance outcomes from the three different geometries were summarised in

Table 10. Series of experiments on different test conditions.

Test Condition	Parameters	Findings
(i) Same air speed (Q = 10–14 W/s2)	Low air speed (8 m/s)	When Re increased, Nu increased. ErrorAvg < 10%. (The least is circular duct).
	High air speed (24 m/s)	From low to high air speed (8 m/s to 24 m/s): Nu increased both theoretically and practically. ErrorAvg < 10% (The least is Circular tube)
(ii) Constant Heat Flux	Q = 16 W/m2 Air speed 8–24 m/s	When airflow increased, Re correlated with Nu. Re increased from 22,300 to 72,000.

.

Based on the findings in Table 10, among the three ducts, a circular pipe is the best shape compared with square and triangle ducts. With the smallest average error at both low speed and high air speed, the shape exhibited the highest average Nusselt and larger heat transfer performance difference.

In a nutshell, both Suttivirode and Suparos [67] had concluded that:

i.

Higher transfer of heat by convection at the pipe entry rather than the exit.

ii.

Correlation between Re, Nu, and St regards air flow speed inside the pipe.

iii.

Heat flux is directly proportional to Re and Nu (Q increased, both Re and Nu increased).

iv.

At high air speed, Nu is higher, but St is lower.

v.

At low air speed, St is higher, but Nu is lower.

4. Medium Prandtl Number

Since a low Prandtl number signifies a value lower than 1 (ratio of momentum diffusivity to thermal diffusivity in a fluid) whereas a high Prandtl number signifies a value higher than 1, the case of medium Prandtl number epitomizes the equilibrium of both diffusivities (momentum and thermal) [8].

4.1. Medium Prandtl Number—Vertical Circular Tube

Bernier and Baliga [68] used the following parameters Pr = 5, Re = 1, Gr = 5000, K = 50, and ∆ = 0.05 to prove the significance of axial wall conduction on mix convection flows in vertical pipes with upward flow and uniform wall heat flux. The heat flux applied on the outside surface of the heated section could be redistributed by this axial conduction within the heated, upstream, and downstream sections of the pipe. Meanwhile, Khan and Bera [69] investigated the effect of the Prandtl number and gap between cylinders on the linear instability of concentric annular flow. The schematic diagram of the physical problem is illustrated in Figure 26. In this cylinder, the gradient of external pressure and the gradient of temperature between the inner and adiabatic nature of the outer had caused mixed convective flow vertically.

The value of Pr strongly affects two types of oscillatory and stationary bifurcation modes. At moderate Pr, the bifurcation is oscillatory when the aspect ratio is small, while in the case of the aspect ratio, the stationary bifurcation mode arises [70,71]; therefore, the basic flow becomes unstable because of hydrodynamic inertial instability [72]. In summary, different Pr had impacted the flow instability into a few categories, shown in

Table 11. Pr impacts towards flow instability.

Range	0 < Pr < 0.02	0.02 < Pr < 0.5	0.5 < Pr
Flow Instability	Oscillatory flow	Stationary bifurcation	Hopf bifurcation [73]

.

Table 12. Findings of mixed convective heat transfer for medium Prandtl number in the vertical axis.

Authors	Working Fluids	Tube Type	Method	Findings
Bernier, Baliga (1992) [68]	Water	Vertical circular tube	Uniform wall heat flux	▪ Axial wall conduction can greatly affect mixed convection flows in vertical pipes with upward flow and uniform wall heat flux. ▪ As the solid-to-fluid thermal conductivity ratio (K) or the pipe thickness-to-diameter ratio (A) increases, the intensity and range of both upstream and downstream heating increase steadily.
Khan, Bera (2020) [69]	Water	Vertical circular tube	Upward flow	▪ When C is large, a non-axisymmetric disturbance is most unstable. ▪ When Pr is large, the axisymmetric disturbance is most unstable. Increasing the value of C increases the critical value of Ra.

summarizes some of the studies for medium Prandtl number fluids in the vertical orientation.

4.2. Medium Prandtl Number—Horizontal Circular Tube

Meyer and Everts [74] investigated the impact of free convection on local heat transfer characteristics in smooth horizontal circular tubes heated with constant heat flux, using water as a medium-Prandtl-number fluid (Pr: 3–7) at Reynolds numbers ranging from 500 to 10,000. They conducted 1046 mass flow rate measurements, 89,459 temperature readings, and 2906 pressure drop measurements across two smooth circular test sections. In the laminar flow regime, they identified three regions: forced convection developing (FCD), mixed convection developing (MCD), and fully developed (FD) flow. In the transitional flow regime, they found three other regions: transition, quasi-turbulent, and turbulent flow. Meyer and Everts [75] also named and studied these flow regimes, creating correlations to estimate thermal entrance lengths for the transitional flow regime, as well as local and average Nusselt numbers for both developing and fully developed flows under mixed convection. They observed that the effects of free convection became more pronounced as the tube diameter and heat flux increased, leading to higher Nusselt numbers and shorter thermal entrance lengths, allowing the flow to reach a fully developed state more quickly. The experimental setup is shown in Figure 27.

With a similar set-up and water as a test fluid (Pr = 3–7), another work was carried out by Meyer and Everts [76] to investigate this transitional flow regime behavior further. This time, the focus was on the relationship between heat transfer and pressure drop, which is beneficial in obtaining the best heat transfer coefficients. The said relationship, as well as the average Nusselt number, were determined via the development of a correlation. As a result, the different among these four flow regimes (transitional, laminar, quasi-turbulent, and turbulent) can be differentiated through this relationship between heat transfer and pressure drop. In a nutshell, the boundaries of the different flow regimes were the same for pressure drop (friction factors) and heat transfer.

Sheng Zhang [77] used molten salts for liquid fuel Molten Salt Reactors (MSRs) in the range of 2–32 at their respective potential working temperatures, as shown in Figure 28.

In order to develop the axial profile of the local Nusselt number, it is needed to locate a few different axial locations, and the author pointed out 42 various locations throughout the 5 m length tube (refer to Figure 29). The tube is set at five different diameters of 5, 10, 15, 20, and 25 mm for this study.

The outcome proves that the Prandtl number has a positive impact on the separation location and thermal entrance length as the mixed convection curve deviates from the force convection curve earlier for conditions of smaller Prandtl numbers, as shown in Figure 30.

Meanwhile, for wire-coil inserts, Feng et al. [78] carried out a numerical investigation on laminar flow and heat transfer in rectangular microchannel heat. S. K. Saha, with transverse ribs and wire coil inserts, focused on exploring the thermal and friction characteristics of laminar [79] and turbulent [80] flow through rectangular and square ducts. Findings of mixed convective heat transfer for medium Prandtl number in the horizontal axis are shown in

Table 13. Findings of mixed convective heat transfer for medium Prandtl number in the horizontal axis.

Authors	Working Fluids	Tube Type	Method	Findings
Meyer and Everts (2018) [75]	Water	Horizontal circular tubes	Constant heat flux	-The flow transitioned faster with increasing free convection effects and Reynolds number.
Taher et al. (2021) [81]	Water	Horizontal Rect. channel	Uniformly heated	-Numerical simulations of heat transfer characteristics for Rayleigh number and Reynolds number in the range of 104 ≤ Ra ≤ 106 and 25 ≤ Re ≤ 100, respectively, have shown that the mixed convection flow offers higher heat transfer enhancement compared with pure forced convection flow.
Colla et al. (2015) [44]	TiO2 water nanofluid	Horizontal circular tubes	Uniformly heated	-Nanofluid thermophysical properties closely equivalent to base fluid (i.e., thermal conductivity, dynamic viscosity, heat capacity)
Wang et al. (1994) [18]	Super critical CO2	Horizontal circular tubes	Uniformly heated.	-Avg Nu decreased with increasing Ra at low Pr (occurrence of reverse flow)

.

4.3. Medium Prandtl Number—Horizontal Elliptical Tube

There are also other non-circular ducts, such as elliptical pipes exist in many heat transfer applications. Hermany et al. [82] investigated convective heat transfer in non-Newtonian fluids flowing over elliptical tubes. They used a structured design and an exhaustive search approach to determine the optimal ellipse aspect ratios for maximizing the Nusselt number and minimizing the dimensionless pressure.

5. Findings

In this paper, the objectives of exploring the research trend of mixed convective heat transfer in tubes and ducts from the selection perspective of Prandtl number, geometry, and fluid flow orientation have been made. The challenges in mixed convective heat transfer is summarized as per

Table 14. Challenges in mixed convective heat transfer.

Modes	No.	Challenges	Explanation
Experimental	1.	Material limitations	- Limited thermal conductivity in the material used affects heat transfer efficiency. - Materials may corrode and lead to altering properties and flow characteristics over time. Discrepancies between numerical predictions and experimental data. - Results sensitivity to input parameters.
	2.	Measurement accuracy	- Traditional temperature sensors. - Existing sensors have slow response time (impact heat flux measurement). - Calibration issue.
	3.	Flow instabilities	- Exertion during the transition from laminar to turbulent flow. - Existence of flow separation. - Existence of buoyancy effect.
	4.	Scaling effects	- Differences in geometry when scaling up lab experiments. - Understanding non-dimensional parameters (Grashof number, Reynolds number). - Technique effectiveness at small scales (nanofluids) might be different to larger scale.
Numerical	1.	Computational complexity	- Selection of the right resolutions for mesh requirements. - Longer computational times lead to longer solution times.
	2.	Turbulence modeling	- Choosing the right turbulence models, which have varying strengths and limitations based on the flow regime. - Issue on closure approximations leads to inaccurate flow characteristics representation.
	3.	Convergence Issues	- Non-linear settings of buoyancy and inertia lead to convergence in simulations. - Divergence due to unstable numerical methods under specific conditions (high gradient, steep temperature profiles).
	4.	Validation against experimental data	- Discrepancies between numerical predictions and experimental data. - Results sensitivity to input parameters.

and the distribution of papers is as per

Table 15. Distribution of papers according to three reviewed parameters: Prandtl number, geometry, and orientation.

Prandtl No		Low		Medium		High		Σ
Orientation		Ver	Hor	Ver	Hor	Ver	Hor
Geometry	Circular	Ref.13,2007 Σ = 1	Ref.14,1990 Ref.15,1982 Ref.16,1988 Ref.17,2010 Ref.18,1994 Σ = 5	Ref.69,1992 Ref.70,2020 Ref.71,2018 Ref.72,2019 Ref.73,2018 Ref.74,2019 Σ = 6	Ref.75,2018 Ref.76,2018 Ref.77,2018 Ref.78,2022 Ref.79,2017 Ref.80,2010 Ref.81,2010 Σ = 7	Ref.29,2011 Σ = 1	Ref.30,1995 Ref 31,2005 Ref.32,2004 Ref.33,2007 Ref.34,2010 Ref.35,2012 Ref.36,2013 Ref.37,2014 Ref.38,2016 Ref.39,2016 Ref.40,1997 Ref.41,1997 Ref.42,2001 Ref.43,1998 Ref.44,2015 Ref.45,2023 Ref.46,2019 Ref.47,2010 Ref.48,1985 Ref.49,2013 Ref.50,2013 Ref.51,2015 Ref.52,2007 Ref.53,2015 Ref.54,2021 Ref.55,2017 Ref.56,2017 Ref.57,2017 Ref.58,1996 Ref.59,2015 Ref.60,2005 Ref.61,2015 Ref.62,2018 Ref.63,2019 Ref.64,2018 Ref.65,2008 Ref.66,2018 Ref.67,2014 Ref.68,2018 [Σ = 39]	59
	Rectangular	-	Ref.19,2012 Ref.20,2012 [Σ = 2]	-	Ref.68,2018 [Σ = 1]	-	Ref.68,2018 [Σ = 1]	4
	Triangular	-	Ref.21,2018 Ref.22,2017 [Σ = 2]	-	Ref.68,2018 [Σ = 1]	-	Ref.68,2018 [Σ = 1]	4
	Elliptical	-	Ref.23,2009 Ref.24,2018 [Σ = 2]	-	Ref.82,2018 [Σ = 1]	-	-	3
	Σ	1	11	6	10	1	41
Remark (Σ)		Low Pr = 12		Med Pr = 16		Hi Pr = 42
		Vertical = 8;			Σ Horizontal = 62

.

Data in Table 15 summarize the frequencies of papers published by the researcher into three categories: Prandtl number, geometries, and orientations. In terms of Prandtl number, with 42 out of 70 (60%), high Prandtl number fluids dominate the studies on mixed convective heat transfer; however, the research trend started to shift towards the medium Prandtl number of fluids for the past 10 years. For the geometries, the circular channel was the most preferred, with 84%. Meanwhile, for the orientation, the horizontal axis fully dominates the selection with 89%; therefore, it can be deduced that mixed convective heat transfer it performed optimally using a horizontal circular tube with a medium Prandtl number as the working fluid.

6. Limitations

There present several challenges across various domains throughout the course of reviewing and understanding all the previous studies on mixed convective heat transfer analysis pertaining to the selection of Prandtl number fluids, geometry setup and flow orientation. The detailed overview of these challenges could be grouped into two modes: experimentally and numerically.

7. Future Recommendations

According to the literature review, the cavities have numerous shapes and geometries for heat transfer analysis. The flow orientation could also be affected by this tube geometry apart from the selection of working fluid properties. The following are suggestions for future research:

(a)

According to the reviewed literature, limited research and the potential of having a medium Prandtl number as a working fluid parameter in convective heat transfer provide optimism for future development.

(b)

From the observation of previous studies, most of the researchers had chosen circular geometry as a working tube over rectangular, triangular and elliptical. Most of the findings also showed better results in terms of Nusselt number, heat transfer coefficient, and pressure drops.

(c)

In terms of flow orientation, horizontal ducting had become the most preferred choice in carrying out mixed convective heat transfer analysis over the vertical due to lower gravitational effects, good flow uniformity, lower pressure drops, less thermal stratification, and better accessibility for maintenance.

(d)

As an agreement to statements (2) and (3) above, most of the industrial players also carried out mixed convective heat transfer applications using circular ducts with horizontal orientation; however, to some extent, few also applied extra measures by adding passive methods to these circular horizontal ducting settings, such as helical coils, corrugations, swirl generators, and ribs; therefore, it is recommended to aid this passive method in a combination to the favored circular horizontal ducting in the future research.

(e)

In addition, apart from the geometry concern, it is also recommended to give more attention to the effectiveness of secondary flow creation and its impact on increasing the rate of heat transfer. Certain fine tunes related to the design of geometry as an aid to the occurrence of this secondary flow should be taken into consideration. Passive method design, including the introduction of baffles, dimples, and grooves, might assist in creating more turbulence in the flow and, ultimately, the secondary flow.

8. Conclusions

The effects of Prandtl number, geometry, and orientation on the mixed convective heat transfer performance in tubes and ducts were reviewed comprehensively. The findings show that diverse combinations between these three (3) parameters produced different heat transfer performances. The key conclusions from this review are as follows:

(a)

Most researchers conducted experiments in circular tubes with a horizontal axis for mixed heat convective heat transfer.

(b)

Lower Prandtl numbers brought different dynamics to the heat transfer process as thermal diffusivity dominates, leading to a thinner thermal boundary layer. This scenario resulted in higher heat transfer coefficients.

(c)

Higher Prandtl numbers influenced the mixing (convection) in turbulent flow and improved the heat transfer as the momentum diffusivity dominates.

(d)

Research on mixed convective heat transfer in horizontal circular tubes typically focuses on fluids with either very low or very high Prandtl numbers, such as air or oil; however, there is limited research on fluids with medium Prandtl numbers, such as molten salts used in liquid-fuel and solid-fuel Molten Salt Reactors (MSRs).

(e)

Nanofluids have been shown to enhance thermal performance across all types of geometries studied. They offer significant potential for improving cooling, thermal storage, solar energy applications, heat exchangers, and related cooling technologies.

(f)

Experimental investigations into the convective heat transfer characteristics of tubes with four different cross-sectional shapes under constant heat flux reveal that variations in tube geometry influence heat transfer efficiency.

(g)

In convective heat transfer applications, the shape and design of the cavity play a crucial role in achieving optimal results. The effectiveness of the cavities depends on their intended use, making it essential to select the appropriate cavity design for thermal systems.

(h)

Various geometries, including circular, square, triangular, and elliptical, are used in heat transfer studies. The literature suggests that circular cavities generally offer the best performance due to their high heat transfer rate, low pumping power requirements, and efficiency at low Reynolds numbers.