The Forced Convection Analysis of Water Alumina Nanofluid Flow through a 3D Annulus with Rotating Cylinders via $\kappa -\epsilon$ Turbulence Model

Abstract

We investigated the dynamics of nanofluid and heat transfer in a three-dimensional circular annular using the $\kappa -\epsilon$ turbulence model and energy equations. The pipe contained two concentric and rotating cylinders with a constant speed in the tangential direction. A heat flux boundary condition was executed at the inner cylinder of the annular. The pipe was settled vertically, and water alumina nanofluid was allowed to enter, with the initial velocity depending on the Reynolds number, ranging from 30,000 to 60,000. The volume fraction of the solid particles was tested from 0.001 to 0.1. The speed of the rotation of the cylinders was tested in the range from 0.5 to 3.5. The simulations were developed using COMSOL Multiphysics 5.6, adopting the finite element procedure for governing equations. The results were validated using the mesh independent study and the average Nusselt number correlations. We found that the average Nusselt number in the middle of the channel decreases linearly with the increase in the volume fraction of the water alumina nanofluid. The novelty of the present work is that various correlations between the average Nusselt number and volume fraction were determined by fixing the Reynolds number and the rotation of the inner cylinder. We also found that fixing the Reynolds number and the volume fraction improves the average Nusselt number at the outlet linearly. In addition, it was stated that the increase in the total mass of the nanofluid would decrease the average temperature at the outer cylinder of the annular. Moreover, the maximum average improvement percentage in the average Nusselt number, which is about 21%, was observed when the inner cylinder rotation was changed from 1.5 to 2.5 m/s.

1. Introduction

It has been more than 100 years since the field of fluid mechanics expanded the tactics to broaden heat transfer in industrial applications. Therefore, when employing fluids with lower thermal conductivity, such as water, glycols, and oil, it is difficult to overcome the requirements on the industrial side. Researchers, through various experiments, affirm that this difficulty can be rectified by using a fluid with an outstanding level of thermal conductivity. The experimental study [1] indicates that utilizing solid particles in the base fluid with some volume fraction will improve the thermal conductivity of the subsequent combination. These particles can even be microscopic and sized in millimeters or nanometers. However, they can be influenced by dilemmas such as pressure expense, clogging, and resistance.

Many decades have passed to promote the technology of rendering such components using a mixture of solid and fluid [2]. These nano-sized particles are adjourned in equilibrium in base fluids, which can transmit them to expand in the domain. Mixing these particles in the base fluid, called conventional fluid, will make the thermal conductivity of the resulting mixture superb. Various fluid dynamics experts have put their effort into checking or measuring the thermal conductivity of the nanofluid by changing the volume fraction of the nanoparticles in the base fluid. They found that the exhibition of such a mixture would heighten the thermal conductivity tremendously. In [3], it was demonstrated that when an oxide ceramic is mixed in either water or glycol, it will increase the thermal conductivity of the resulting nanofluid. It was explained in [4] that when Al₂O₃ particles of 13 nm in diameter are mixed in a base fluid such as water in a 0.043 volume fraction, this will stimulate the thermal conductivity by about 30% under normal conditions. In [5,6], it was illustrated that a justification for increased nanofluid thermal conductivity might be that mixing nanoparticles will heighten the thermal extent instigated by the wild motion of the nanoparticles and raise the energy exchange in the fluid. Such a procedure will improve the thermal conductivity of nanofluid. Four methods were explained in [7] that boost heat transfer when nanoparticles are added to the base fluid. The researchers explained that improved heat transfer could be caused by Brownian motion, multi-level layering, and ballistic heat transfer of solid particles and clusters. Furthermore, in [8], it was explained that adding a liquid molecule interface to a nanoparticle cluster would improve heat transfer rapidly.

Numerous hypothetical and experimental investigations have been expanded to compute the thermal conductivity of nanofluids. The thermal conductivity observed in experiments was found to be more significant than the hypothetical explorations [9,10]. Various studies are based on the theoretical investigation of the nanofluid’s thermal conductivity to enhance the channel’s heat transfer rate [11,12,13,14,15]. Moreover, many researchers explained the relationship between temperature, the size of nanoparticles, and volume fraction with the thermal conductivity of the nanofluids [16,17]. Most researchers have worked with laminar as well as turbulent fluid, and heat transfer to develop critical correlations for future predictions. Among them, a well-known study was carried out by [18,19]. Their work established the first correlation of the average Nusselt number with the Reynolds number and the Prandtl number while examining the different types of nanofluids. A laminar nanofluid flow was investigated to analyze the convection and the pressure loss due to viscous influences in a vertical test tube [20]. Later, the trapezoidal channel was used to evaluate the thermal execution [21]. It was determined that an enhancement in the thermal performance could be achieved by increasing the nanofluid’s pumping power from the channel’s entrance. Moreover, the volume fraction has a favorable role in heightening the thermal performance, but some extra benefits were lost due to the decline in pressure loss. A square cavity was investigated [22] with a cooled water copper nanofluid with laminar and natural convection conditions. It was determined that the square cavity’s heat distribution decreases with the volume fraction increase when the Rayleigh number is fixed. In addition, the heat transfer is positively influenced by increasing the Rayleigh number for a fixed volume fraction. The mean diameter of the solid particles plays a vital role in heat transmission. It was revealed by [23] that the heat transfer coefficient is improved when the mean diameter of the nanomaterials is increased in the mixed convection case. Applications of nanofluids are also presented in the field of thermoelectric sciences. A thermoelectric module was investigated using a nanofluid-based heat exchanger [24]. It was found that there was a time break for the thermal connection between the two components. There are two ways to observe the heat transfer with the nanofluid dynamics; the first is the single-phase flow, and the second is the two-phase flow. Because the single-phase flow is easy and takes less computation time to give outstanding precision, various theoretical studies [25,26,27,28,29] have been conducted with a single-phase approach. However, as indicated in [30], two-phase fluid can be assumed when the Peclet number is greater than 10, and the mixing can be assumed as non-uniform or two-phase. With a numerical scheme of the finite element method, the boundary layer flow was investigated for mass and heat transfer in the presence of a chemical reaction and unsteady magnetic effect [31]. The results illustrated that the Nusselt number increases the Brownian motion and the thermophoresis parameter. With the implication of Cattaneo–Christov characteristics, the observation of temperature fluctuation in the domain was studied with microbes engrossed in the water-based nanofluid [32]. Mainly, it was elaborated that increasing the melting factor significantly enhances the density of the resultant fluid. The consequence of the magnetic dipole movement for the heat transfer was investigated when two nanoparticles, Fe and Fe₃O₄, were mixed in a water–ethylene glycol solution [33]. It was found that the thermal conductivity of Fe was more improved than that of Fe₃O₄. A comparative study to enhance the heat transfer rate with rotational flow under the influence of the magnetic effect was conducted to transport two separate hybrid nanofluids of Al₂O₃-Cu and Al₂O₃-TiO over a plane sheet [34]. It was indicated that the temperature and concentration of nanoparticles are higher in the case of Al₂O₃-Cu than Al₂O₃-TiO.

Conserving large applications in the heat exchanger, annular geometry is widely explored for the heat transfer phenomenon and is often called Isothermal Furnace Liner (IFL). An experiment [35] was performed to observe the thermodynamics between the unheated length of the annular and the annular ratio by changing the heat transfer coefficient from the inlet of the pipe. An annular was also investigated using an implicit finite difference scheme [36]. The research revealed substantial changes in the heat distribution with a relatively small annular ratio. In [37], laminar forced convection was examined at the annular entrance. With distinct initial thermal conditions, it was depicted that exploring the heat transfer in a pipe-like annular is more significant than circular geometry. A three-dimensional cavity was also investigated to observe the convection effect via two rotating cylinders embedded in the cavity [38]. Three different types of nanofluids were observed to obtain the correlation for the average Nusselt number relating to the Rayleigh number and the angular rotation of the cylinder.

2. Problem Formulation, Boundary Condition, and Thermo-Physical Properties

For many decades, the heat transfer process with nanofluid has been observed for various two-dimensional channels. In this research article, we observe a three-dimensional annular constructed by fitting two cylinders. One of the cylinders is an inner cylinder, and the other is an outer cylinder. To make the observed geometry interesting, we produced both cylinders rotating in a tangential direction/along the y-axis with the same velocity $\omega$ (see Figure 1). These two cylinders are fixed so that their centerlines coincide with each other. We are interested in settling these cylinders so that the aspect ratio from the radius of the inner cylinder to the outer cylinder will be 0.5. Let R₁ be the radius of the inner cylinder and R₂ be the radius of the outer cylinder, then R₁/R₂ = 0.5.

The length of these two cylinders is the same as tradition. This annular pipe is settled vertically, and the top of this channel is recognized as the inlet channel, whereas the opening from the bottom is selected as an outlet. The inner cylinder is heated with a constant heat flux magnitude of 100 [W/m²]. The initial or reference temperature in this tube is 293.15 K.

Let U_in be the average inlet velocity entered from the top of the pipe; then, it must be the function of the Reynolds number. In the current problem, the Reynolds number ranged from 30,000 to 60,000. The nanofluid volume fraction is chosen in the range of 0.001–0.1. We develop the simulations for the current problem with only $\varphi$ = 0.001, 0.009, 0.01, 0.09, and 0.1. Moreover, the nanofluid flow inside the tube will be dispersed further when the inner and outer cylinders rotate in the same tangential direction of the channel at the same speed. The current simulations are developed using the speeds of these cylinders in the range of 0.5 to 3.5 m/s. The fluid entering the tube is assumed to be laminar and turbulent flow with $\kappa -\epsilon$, and the turbulent dissipation rate is kept from 0.23403 to 9.2401 m²/s³. The Prandtl number is used between 0.85 and 0.97. Taking T_ref = 293.15 as the cool temperature and the updated temperature T as the hot temperature, the Grashof number in the current problem varies from 40.368 to 1391.6.

With a wide range of applications on the industrial side to enhance the heat transfer rate in the domain of interest, nanofluid is traditional. It will follow the same steps in almost all research articles. When a mixture of the base fluid and the nanomaterial occurs in the channel, the empirical formulas to determine the density ${\rho }_{nf}$, thermal expansion ${\beta }_{nf}$, and heat capacitance ${(\rho c_p)}_{nf}$ are uniquely used and are given in

Table 1. Parameters and thermophysical properties [40].

Description	Property Name	Value/Range
Density of water	${\rho }_f$	997.1 [kg/m3]
Density of aluminum oxide	${\rho }_s$	3970 [kg/m3]
Heat capacity of water at constant pressure	${(c_p)}_f$	4179 [J/(mol K)]
Heat capacity of aluminum oxide	${(c_p)}_s$	765 [J/(mol K)]
Thermal conductivity of water	${\kappa }_f$	0.613 [W/(m K)]
Thermal conductivity of alumina	${\kappa }_s$	40 [W/(m K)]
Thermal expansion of water	${\beta }_f$	21 × 10−5 [1/K]
Thermal expansion of water	${\beta }_s$	0.85 × 10−5 [1/K]
Volume fraction	$\varphi$	0.001, 0.01, 0.1, 0.2
Density of nanofluid	${\rho }_{nf}$	$(1-\varphi ){\rho }_f+\varphi {\rho }_s$
Viscosity of water	${\mu }_f$	0.00081 [Pa s]
Dynamic viscosity of nanofluid [8]	${\mu }_{nf}$	${\mu }_f(1+7.3\varphi +123{\varphi }^2)$
Acceleration due gravity	g	9.8 m/s2
Thermal expansion of the nanofluids	${\beta }_{nf}$	$(1-\varphi ){\beta }_f+\varphi {\beta }_s$
Thermal conductivity of nanofluid Maxwell Garnet model [39]	${\kappa }_{nf}$	${\kappa }_f\frac{{\kappa }_s+2{\kappa }_f-2\varphi ({\kappa }_f-{\kappa }_s)}{{\kappa }_s+2{\kappa }_f+\varphi ({\kappa }_f-{\kappa }_s)}$
Heat capacitance of fluid	${(\rho c_p)}_f$	${\rho }_f{(c_p)}_f$
Heat capacitance of solid	${(\rho c_p)}_s$	${\rho }_s{(c_p)}_s$
Heat capacitance of nanofluid	${(\rho c_p)}_{nf}$	$(1-\varphi ){(\rho c_p)}_f+\varphi {(\rho c_p)}_s$
Thermal diffusivity of nanofluid	${\alpha }_{nf}$	$\frac{{\kappa }_{nf}}{{(\rho c_p)}_{nf}}$
Reference temperature of the square cavity	$T_{ref}$	293.15 [K]
Hydraulic diameter	$D_h$	4 Area of the channel/Perimeter
Inlet velocity	$u_{in}$	$\frac{\mathrm{Re}{\mu }_{nf}}{D_h{\rho }_{nf}}$
Reynolds number	Re	35,000
Radius of cylinder	Ar	0.4, 0.5, 0.6

. Moreover, to determine the effective viscosity and thermal conductivity, many models are available. We used the models described in [8,39]. Further details of the selected parameters and the thermophysical properties of the nanofluid are given in Table 1.

Let $u=<u_r,u_{\theta },u_z>$ be the velocity field; then, governing Navier–Stokes equations and the $\kappa -\epsilon$ RANS model of turbulence in the vector form are displayed as (1)–(5) and the boundary conditions as (6)–(9).

• Continuity equation:

  $${\rho }_{nf}(\nabla ·u)=0$$

  (1)

• Momentum equation:

  $${\rho }_{nf}(u·\nabla )u=\nabla ·[-pI+(\mu +{\mu }_T)(\nabla u+{(\nabla u)}^T]$$

  (2)

• $\mathbf{\kappa }-\mathbf{\epsilon }$ turbulence model:

  $${\rho }_{nf}(u·\nabla )\kappa =\nabla ·\left[(\mu +\frac{{\mu }_T}{{\sigma }_{\kappa }})\nabla \kappa \right]+p_{\kappa }-{\rho }_{nf}\epsilon$$

  (3)

  $${\rho }_{nf}(u·\nabla )\epsilon =\nabla ·\left[(\mu +\frac{{\mu }_T}{{\sigma }_{\kappa }})\nabla \epsilon \right]+c_{\epsilon 1}\frac{\epsilon }{\kappa }-c_{\epsilon 2}{\rho }_{nf}\frac{{\epsilon }^2}{\kappa }$$

  (4)

  where

  $${\mu }_T={\rho }_{nf}{c}_{\mu }\frac{{\kappa }^2}{\epsilon }\mathrm{and}{p}_{\kappa }={\mu }_T\left[\nabla u:(\nabla u+{(\nabla u)}^T\right]$$

  (5)

• Boundary conditions:

  $$\mathrm{Inlet}:\mathrm{z}=\mathrm{L},0\le \mathrm{r}\le {2\mathrm{R}}_1,0\le \theta \le 2\pi :u=-U_{in}n,\kappa =\frac{3}{2}(U_{in}{I}_T),\epsilon {=\mathrm{c}}_{\mu }^{3/2}\frac{\kappa }{L_T},\frac{\partial \mathrm{T}}{\partial n}=0$$

  (6)

  $$\mathrm{Outlet}:\mathrm{z}=0,0\le \mathrm{r}\le {2\mathrm{R}}_1,0\le \theta \le 2\pi :u·n=0,\mathrm{p}=0,\nabla \mathrm{k}·n=0\mathrm{and}\nabla \epsilon ·n=0$$

  (7)

  $${\mathrm{At}\mathrm{outer}\mathrm{cylinder}:\mathrm{r}=\mathrm{R}}_2,0\le \mathrm{z}\le \mathrm{L},0\le \theta \le 2\pi :u·n=\omega ·n,\nabla \mathrm{k}·n=0,\nabla \epsilon ·n=0,\frac{\partial \mathrm{T}}{\partial n}=0$$

  (8)

At the inner cylinder:

$${\mathrm{r}=\mathrm{R}}_1,0\le \mathrm{z}\le \mathrm{L},0\le \theta \le 2\pi :u·n=\omega ·n,\nabla \mathrm{k}·n=0,\nabla \epsilon ·n=0,{\mathrm{q}}_0{=\mathrm{k}}_{nf}\nabla T=100$$

(9)

where n is the normal vector to the selected boundary, I_T = 0.05 cm, and L_T = L represents the turbulence intensity and length of the pipe, respectively.

• Turbulence variable:

The turbulence variables are given in

Table 2. Turbulence variables.

${\mathit{c}}_{\mathit{\epsilon }1}$	${\mathit{c}}_{\mathit{\epsilon }2}$	${\mathit{c}}_{\mathit{\mu }}$	${\mathit{\sigma }}_{\mathit{\kappa }}$	${\mathit{\sigma }}_{\mathit{\epsilon }}$
1.44	1.92	0.09	1	1.3

.

• Computational Parameters:

Finally, we enlist the computational parameters (10)–(16):

⮚

Grashof number:

$$Gr=\frac{g{\beta }_f(T_h-T_c)D_h^3}{{\mu }^2}$$

(10)

⮚

Prandtl number:

$$\mathrm{Pr}=\frac{{\mu }_{nf}{(c_p)}_{nf}}{{\kappa }_{nf}}$$

(11)

⮚

Heat flux:

$$Q=\frac{{\kappa }_{nf}}{{\kappa }_f}\nabla T$$

(12)

⮚

Heat transfer coefficient:

$$h=\frac{Q}{A(T-T_b)}$$

(13)

⮚

Bulk temperature:

$$T_b=\frac{{\displaystyle \underset{\Omega }{\int }uTd\Omega }}{{\displaystyle \underset{\Omega }{\int }ud\Omega }}$$

(14)

⮚

Local Nusselt number along the ith direction:

$$Nu=\frac{h{\mathrm{D}}_h}{k_{nf}}$$

(15)

⮚

Average Nusselt number:

$$N{u}_{avg}=\frac{1}{A}{\displaystyle {\int }_{\Omega }^{}NudA}$$

(16)

⮚

Rayleigh number:

$$Ra=Gr\mathrm{Pr}$$

(17)

• COMSOL Workflow:

The current physical problem based on the governing equations subject to boundary conditions is solved using the software COMSOL Multiphysics 5.6, which applies the methodology of the finite element method to obtain the simulation. In the current investigation, we developed about 80 different simulations using parameters such as Reynolds number, volume fraction, and the rotational speed of the cylinders. The COMSOL Wagon Wheel contained the following steps:

 Step 1:

To make the observed channel and material used in the channel, we will first select the parameters;

 Step 2:

Develop geometry using the tools of COMSOL Multiphysics 5.6;

 Step 3:

Apply the boundary conditions by selecting the premises of the selected channel;

 Step 4:

Perform the grid or mesh independency test to achieve high accuracy;

 Step 5:

Validate or compare the results with the available results or experimental correlations;

 Step 6:

Compute the numerical results by post-processing.

3. Mesh Independent Study and Validation

To scrutinize the computational results, we execute the independent mesh study in this section. We also attempt to validate the results with the correlation for the average Nusselt number at the exit of this geometry. Since a finite element method is functional to solve the current problem with COMSOL Multiphysics 5.6, we used tetrahedral mesh elements for the current channel (see Figure 2). The present channel is meshed by using various elements from 10,000 to 70,000. The objective of the independent mesh study is to fix the required number of elements to achieve high accuracy for the designated derived variable(s). It is acknowledged that for the numerical methods, good precision for one element will give good precision for the next element. Therefore, using a large and sufficient number of elements is necessary to achieve suitable simulation accuracy. We performed the independent mesh study for the current problem for the average Nusselt number in the middle of the channel z = L/2. We plotted the graph in Figure 3a for the average Nusselt number and the increasing number of elements. It is obvious from the graph that increasing the number of elements will improve the accuracy of the computational results. An independent mesh study is achieved when the number of elements exceeds 30,000 for the computational result, correct to two decimal places.

For the current simulation, the computational results for the average Nusselt number against the increasing Reynolds number at the exit of the channel were validated by various correlations [8,38,39,40]. The results are displayed in Figure 3b and

Table 3. Comparison table for the average Nusselt number in inner cylinder with $\omega$ = 0.5 and $\varphi$ = 0.001.

Re	Present Work	Dittus and Boelter [41] $0.023{\mathrm{Re}}^{0.8}{\mathrm{Pr}}^{1/3}$	Sieder and Tate [42] $0.027{\mathrm{Re}}^{0.8}{\mathrm{Pr}}^{1/3}$	Li and Xuan [43] $0.0059{\mathrm{Re}}^{0.9238}{\mathrm{Pr}}^{0.4}$	Dawid and Jan [44] $0.02155{\mathrm{Re}}^{0.8018}{\mathrm{Pr}}^{0.7095}$
3 × 104	81.606	83.272	97.754	75.736	74.886
4 × 104	102.674	104.770	122.990	98.734	94.216
5 × 104	122.712	125.217	146.994	121.303	112.620
6 × 104	141.964	144.861	170.055	143.533	130.312

. It can be seen that the numerical results for the average Nusselt number at the outlet of the channel are comparable to the results obtained by the Dittus–Boelter equation [41].

4. Results Discussion

In this section, the numerical results are displayed using the graphs and tables for the average Nusselt number at the exit of the channel by fixing the Reynolds number and the inner cylinder rotation, mass flow rate, and average temperature. In this study, we sought to establish the correlation between the average Nusselt number and nanofluid volume fraction by fixing the inner cylinder’s rotation. We successfully demonstrated the correlation between the total mass flow rate against the volume fraction of the nanofluids.

4.1. Average Nusselt Number at the Exit of the Channel

Figure 4a–e demonstrates the variation in the average Nusselt number against the increasing Reynolds number at the channel outlet by fixing the volume fraction. The present channel is a three-dimensional annular constructed by two concentric cylinders. The inner and upper cylinders are rotated along the tangential direction with the same speed $\omega$. A heat flux condition is applied to the inner cylinder. We developed the graphs in Figure 4a–e for the average Nusselt number with the fixed volume fraction of the nanomaterial, and the pattern was checked by varying the rotational speed of the inner cylinder. These graphs reveal that the average Nusselt number at the outlet of the channel increases linearly with the increasing Reynolds number for a fixed rotation and volume fraction. The average Nusselt number significantly improved by increasing the rotation of both cylinders. By definition, the Nusselt number is the ratio from the convection to conduction process. By increasing the Nusselt number, it can be determined that the convection process is conquered over the conduction process and vice versa. The increment in the average Nusselt number at the outlet of the channel against the Reynolds number was found in a linear pattern. Moreover, with an increase, the volume fraction rate of the increment is slightly affected. This may be the reason why the mass or density of the nanofluid increased due to an increment in the volume fraction. Therefore, the fluid cannot be dispersed within the same rotation value. It can be noted that when the volume fraction has increased, the difference in the average Nusselt number is reduced by changing the rotation from 0.5 to 1.5 m/s. This behavior can be seen in Figure 4c–e.

Table 4. Average Nusselt number at the outlet of the channel and with percentage increments when the rotation of the cylinders changes by one unit.

Re	$\mathit{\varphi }$	$\mathit{\omega }=0.5$	$\mathit{\omega }=1.5$	${\%}_1$	$\mathit{\omega }=2.5$	${\%}_2$	$\mathit{\omega }=3.5$	${\%}_3$
30,000	0.001	62.17	91.16	46.63	120.93	32.66	151.06	24.92
30,000	0.009	61.90	89.36	44.37	117.83	31.86	146.66	24.47
30,000	0.01	61.86	89.11	44.04	117.39	31.74	146.04	24.41
30,000	0.09	59.72	66.60	11.52	82.33	23.61	96.20	16.85
30,000	0.1	59.60	65.05	9.13	79.32	21.95	92.14	16.15
40,000	0.001	76.47	100.84	31.87	128.44	27.37	156.46	21.82
40,000	0.009	76.30	99.02	29.78	125.56	26.80	152.39	21.37
40,000	0.01	76.28	98.76	29.48	125.15	26.72	151.82	21.31
40,000	0.09	74.75	78.93	5.59	91.35	15.74	105.76	15.77
40,000	0.1	74.66	78.12	4.63	88.06	12.73	101.91	15.72
50,000	0.001	90.48	109.23	20.72	136.86	25.30	163.36	19.36
50,000	0.009	90.34	107.26	18.72	134.18	25.10	159.50	18.87
50,000	0.01	90.33	106.99	18.45	133.81	25.07	158.96	18.80
50,000	0.09	89.11	92.23	3.49	100.16	8.60	114.79	14.61
50,000	0.1	89.05	91.68	2.96	97.83	6.71	110.48	12.93
60,000	0.001	104.08	117.46	12.86	145.80	24.13	170.92	17.23
60,000	0.009	103.97	115.86	11.44	143.18	23.58	167.24	16.80
60,000	0.01	103.95	115.65	11.26	142.81	23.48	166.72	16.74
60,000	0.09	102.95	105.49	2.47	110.94	5.17	122.63	10.54
60,000	0.1	102.90	105.06	2.10	109.42	4.15	118.55	8.34

%₁ = When rotation changes from $\omega =0.5\mathrm{to}\omega =1.5$. %₂ = When rotation changes from $\omega =1.5\mathrm{to}\omega =2.5$. %₃ = When rotation changes from $\omega =2.5\mathrm{to}\omega =3.5$.

details the precise impacts of altering the Reynolds number, cylinder rotation, and nanofluid volume fraction.

Table 4 shows that for a fixed rotation speed and Reynolds number, the average Nusselt number at the outlet of the channel decreases with the increase in volume fraction. Moreover, it can be concluded that the average Nusselt number increases with the increase in Reynolds number for fixed rotation and volume fraction of the nanofluids. Moreover, with the fixed Reynolds number and volume fraction, the average Nusselt number is improved by increasing the rotation of the cylinders. Furthermore, in Table 4, the percentage change is calculated when the simulation is obtained by altering the speed of the cylinders by one unit. These percentages are visualized in the fifth, seventh, and ninth columns. By fixing any other parameters, the percentage change in the average Nusselt number decreases with the increase in volume fraction, decreases with the increase in rotation of cylinders, and decreases with the increase in Reynolds number. The maximum percentage change in the Nusselt number can be seen when Re = 30,000, $\varphi$ = 0.001, and the cylinders’ rotation is from 0.5 to 1.5. Moreover, the minimum percentage can be seen when Re = 60,000 and $\varphi$ = 0.001 when the rotation turns from 0.5 m/s to 1.5 m/s.

4.2. Average Nusselt Number, Volume Fraction, and the Correlations

We examined the fluid flow and heat transfer in a three-dimensional annular with water alumina nanoparticles, where the solid particles are present in the base fluid in a ratio from 0.001 to 0.1. The inner and outer cylinders are rotated tangibly at the same speed. We tested the impact of volume fraction on the average Nusselt number by fixing the rotation speed of the cylinders. In addition, we endeavored to clarify the correlations between them.

In Figure 5a–d, we show the numerical results for the average Nusselt number at the end of the annular against the increasing volume fraction by fixing the rotation of the cylinder at $\omega$ = 0.5 and the Reynolds number, Re. A clear message is that the average Nusselt number decreases with the increasing volume fraction of the nanoparticles in the base fluid for all the cases. An increment in the average Nusselt number demonstrates the domination of the conduction process over the conduction. Moreover, at the volume fraction’s zero value, we found that the maximum Nusselt number increases with an increase in the Reynolds number. In Figure 6a–d, the same pattern of a decreasing average Nusselt number with an increasing volume fraction was found by fixing the cylinders’ rotational speed at $\omega$ = 3.5 and the Reynolds number. In both graphs, a clear linear pattern appears. Therefore, to predict the behavior of the average Nusselt number against the volume fraction, we developed several correlations by using the present conditions of the problem.

Table 5. Correlations for average Nusselt number against the increasing volume fraction of water alumina nanofluid.

Correlations	Conditions
$N{u}_{avg}=62.14770875-26.1240635\varphi$	$\omega$ = 0.5, Re = 30,000
$N{u}_{avg}=76.47099849-18.56012862\varphi$	$\omega$ = 0.5, Re = 40,000
$N{u}_{avg}=90.47993473-14.728676\varphi$	$\omega$ = 0.5, Re = 50,000
$N{u}_{avg}=104.0795444-12.1610582\varphi$	$\omega$ = 0.5, Re = 60,000
$N{u}_{avg}=91.6519204-271.375263\varphi$	$\omega$ = 1.5, Re = 30,000
$N{u}_{avg}=101.0916353-237.0862955\varphi$	$\omega$ = 1.5, Re = 40,000
$N{u}_{avg}=109.0113258-179.3960469\varphi$	$\omega$ = 1.5, Re = 50,000
$N{u}_{avg}=117.1541-125.0397317\varphi$	$\omega$ = 1.5, Re = 60,000
$N{u}_{avg}=121.5389985-428.083536\varphi$	$\omega$ = 2.5, Re = 30,000
$N{u}_{avg}=129.1209901-414.4750866\varphi$	$\omega$ = 2.5, Re = 40,000
$N{u}_{avg}=137.6074661-405.7243489\varphi$	$\omega$ = 2.5, Re = 50,000
$N{u}_{avg}=146.4162574-380.5897152\varphi$	$\omega$ = 2.5, Re = 60,000
$N{u}_{avg}=151.9178705-607.143654\varphi$	$\omega$ = 3.5, Re = 30,000
$N{u}_{avg}=157.2588622-561.667193\varphi$	$\omega$ = 3.5, Re = 40,000
$N{u}_{avg}=164.1914757-542.241229\varphi$	$\omega$ = 3.5, Re = 50,000
$N{u}_{avg}=171.8464556-538.982427\varphi$	$\omega$ = 3.5, Re = 60,000

details several correlations made with the present data.

In these correlations, the multiple constants of the volume fraction $\varphi$ are slopes. It can be interpreted that for a fixed rotation of the inner cylinder, these slopes decrease with the increase in Reynolds number. Moreover, for a fixed rotation and Reynolds number, the rate of these slopes increases by the rotational speed of the cylinders.

4.3. Mass Flow Rate and the Average Temperature on the Outer Cylinder

In the present work, we investigated water alumina nanofluid and heat transfer through an annular with two concentric cylinders. In this three-dimensional annular, the inner and outer cylinders rotate in the tangential direction. A heat flux condition was applied to the inner cylinder. We measured the average temperature at the outer cylinder related to mass entering the channel’s inlet. The reference temperature is T_ref = 273.15 K. We can obtain the mass entering the annular with the volume integration of the fluid density.

Mass and volume fraction correlation:

$$M_{nf}=0.004698632063+0.01400918897\varphi$$

(18)

In Figure 7, the graph is plotted with an increasing volume fraction and mass of the nanofluid. It can be seen that there is a direct relationship between the mass and the volume fraction. Therefore, applying a linear regression process [45], we can drive a linear regression equation that relates the average mass and the volume fraction of the nanofluid, given as (17).

Mass is the physical quantity that depends on the volume fraction and the density of the nanofluid. It cannot be altered with the other parameters used in the problem, such as the Reynolds number and the inner and outer cylinder rotation. We produced a graph which presents the average temperature against the mass of the nanofluids. From Figure 8a,b, it can be understood that the average temperature decreases with an increase in the mass of the nanofluid up to a certain limit, and then increases. The mass flow rates where an alteration in the average temperature can be determined are about 0.006 and 0.0025 kg/s. Moreover, the alteration can be controlled by increasing or decreasing the Reynolds number, which suggests that increasing the mass flow rate will not always support the average temperature in the domain. It may depend on other factors such as the Reynolds number or the rotation of the inner cylinder. However, it can be understood that keeping a high mass flow rate compared to the low mass flow rate always favored decreasing the average temperature at the outer surface in our study. Additionally, the average temperature can be further improved against the mass flow rate when the rotation of the inner cylinder is increased (see Figure 8a,b). Observing these values for the same Reynolds number can be checked for the average temperature against the mass flow rate.

5. Conclusions

We investigated the heat transfer through water alumina oxide nanofluid using the $\kappa -\epsilon$ turbulence model. A three-dimensional annular consisting of two concentric inner cylinders was kept under observation. A heat flux condition of 100 W/m² was applied to the inner cylinder of the annular, and both cylinders were movable at the same speed $\omega$. The three-dimensional annular was settled vertically, and the water alumina nanofluid was allowed to enter with inlet U_in, the Reynolds number function. The nanoparticle fraction to the base fluid was taken at 0.001, 0.009, 0.01, 0.09, and 0.1. The Reynolds number was tested in the range of 30,000 to 60,000. The cylinders were moved with the same speed from 0.5 to 3.5 m/s in the tangential direction. The Prandtl number, turbulent dissipation rate, and Grashof number were 0.85–0.97, 0.23403–9.2401 m²/s³, and 40.368–1391.6, respectively. All the simulations were performed using COMSOL Multiphysics 5.6. The outcomes were validated using the mesh independent study and by comparing the average Nusselt number at the outlet of the channel with the Dittus–Boelter equation. The results are displayed in graphs and tables. We conclude with the following points.

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The average Nusselt number at the exit of the channel is increased in a linear pattern against the increasing Reynolds number when the rotation of the inner and outer cylinders is fixed. Moreover, by increasing the rotation of the cylinders, the same behavior can be seen between the average Nusselt number and the Reynolds number, but the increment rate is gradually increased.

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From the tables, it is apparent that by fixing the Reynolds number and volume fraction, the average Nusselt number at the exit of the channel is increased by increasing the rotation of the circular cylinders. By fixing both the Reynolds number and the rotation of the circular cylinder, the average Nusselt number is reduced by increasing the volume fraction.

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The average Nusselt number in the middle of the channel decreases linearly with the increased volume fraction for the fixed rotation rate and the Reynolds number. Therefore, we determined several correlations by applying a linear regression process. These linear regression equations can be used for future predictions based on the conditions.

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The mass of the nanofluid entering the region was determined based on the computational results. The mass increased linearly with the increase in volume fraction. Moreover, a linear regression was given between the mass and volume fraction of the nanofluid.

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The average temperature against the mass flow rate decreased under certain limitations. Two critical values were found for the mass flow rate where the average temperature increases or decreases. Therefore, we suggest that this increment or decrement could be controlled by using the Reynolds number or the rotation of the inner cylinder, or both.