Flow of SWCNT and MWCNT based hybrid nanofluids in a semi-circular enclosure with corrugated wall

Abstract

Heat transfer mechanisms participate to fulfill the necessities of modern technologies. The phenomenon of heat transport has implementations in multiple fields including metal working, heating systems, thermal management in spacecraft, solar energy, and automobile engines. In the current novel work, heat transfer mechanism in a semi-circular enclosure with corrugated circular wall is studied numerically. A hybrid nanofluid consisting of water as the base fluid and solid particles of single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) is considered. Prevailing mathematical equations are explained by finite element method with the help of COMSOL Multiphysics. The study examines various volume fractions of solid particles over a broad range. The SWCNT volume fraction is adjusted from 0.02% to 0.1%, while the MWCNT volume fraction ranged from 0.01% to 0.04%. Velocity, temperature, and pressure contours are imagined. Local and average Nusselt numbers are studied for each of the cases. The results indicate that heat transfer in the semi-circular enclosure improves with higher volume fractions of solid particles. As the volume fraction of solid particles increases, the average Nusselt number over the heated surface decreases.

Introduction

The study of semi-circular lid-driven cavities has strengthened significantly over the years. The usage variety of lid-driven cavities comprises insulation materials, solar ponds, crystal growth, cooling of electronic devices, etc. Thermal engineers are particularly concerned about the transfer of heat in several geometries. Heat transfer should be as economical as feasible, according to engineers. For this purpose, various mathematical models were established. Supporting numerical problems by analyzing heat transfer is common and results are generated from software using a commercially accessible tool such as COMSOL Multiphysics. ¹

In recent years, there has been a lot of interest in researching ways for improving heat transmission in heat exchangers to meet the growing need for improved efficiency in the devices. One effective method for enhancing heat transfer is the use of corrugated surface geometry. As fluid flows through a corrugated channel, the expanding recirculation zones near the corrugated walls disrupt the flow, enhancing fluid mixing and heat transfer. ² Pandey and Nema ³ evaluated the effectiveness of an alumina-water nanofluid in an offset flow corrugated plate heat exchanger. There are numerous strategies for enhancing heat transfer technologies. A common approach to reducing thermal resistance involves increasing the surface area of the heat exchanger or minimizing the thickness of the thermal boundary layer on its surface. However, these methods can lead to higher material costs and an increase in the heat exchanger’s weight. The technique of designing vortices was established to shrink the thickness of the boundary layer. Fluid extracts (nanofluids) and vortex flow devices (corrugated surface) were used in the current study to boost heat transfer performance. ⁴

Significant research has focused on mixed convection in semi-circular enclosures. By using constant heat flux values, Blasiak and Kolasinski ⁵ assumed a lid-driven cavity to investigate the mixed convective phenomenon. They performed analysis for a large collection of $\mathrm{Pr}$ and $\mathrm{Ri}$ numbers. According to their outcomes, $\mathrm{Pr}$ changes the temperature and heat transfer but has no impact on flow behavior. For fixed $\mathrm{Gr},\mathrm{Ri}$ number decreases, and $\mathrm{Pr}$ number rises by enhancing the rate of heat transfer. The viscosity of the temperature boundary layer worsens by the increase of $\mathrm{Pr}$ . The inclusion of nanoparticles in fluids can develop convectional heat transfer. The concept of nanofluids was introduced by Choi in 1995. ⁶ Under different boundary conditions, Nanofluids have been applied in various flow regimes defining their capability to improve heat conduction. ⁷ Nanofluids demonstrate a hopeful perspective as heat transfer fluids because of their improved stability and extremely high heat conduction. The higher the thermal conductivities of Nanofluid, the smaller the particle size. Nanofluids are colloids made up of nanoparticles and base fluids. Under constant wall temperature, Akhavan-Behabadi ⁸ examined heat exchange of nanofluid flow within vertical helically coiled pipes. It was scrutinized that using helical coil tubes and nanofluid flow can result in a 3–10-fold perfection in heat conductivity in contrast to using a straight duct with a base fluid flow. The initiation of nanoparticles significantly builds up the heat conduction behavior of base fluids. The nanoparticle materials contain Carbon in various forms (graphite, CNT, diamond). Karami ⁹ presented CNT nanofluids as a fantastic working fluid for direct absorbing of solar collectors because of dispersion stability. It can be observed that both temperature and volume fraction increase as thermal conductivity rises. CNT has many special characteristics such as chemical stability, physical strength, mechanical resistivity, and very high specific conduction. Within microchannels having different shaped ribs, Behnampour et al. ¹⁰ disclosed the dynamics of nanofluid. An experimental study within a rectangular medium was conducted by et al. Alawi ¹¹ to explore the flow characteristics of fluid containing nanoparticles. On numerous occasions, the lid-driven enclosure movement has been utilized as a normal test case for evaluating numerically outcomes for the Navier-Stokes equations. Experimental outcomes for combined convection in deep lid-driven cavities heated upon bottom were reported by Prasad and Koseff. ¹² They found that the Richardson number had a minimal impact on heat transfer. Imura et al. ¹³ measured the heat transfer on a horizontal heated plate to investigate the conversion from laminated to unstable flows. The study of material properties at the nanoscale has gotten a lot of attention because nanoparticles behave differently than macroscale particles. Nanoparticle’s dispersal into standard fluids such as ethylene glycol, oil and water may alter the features of colloids and enhance exchange of the heat. Nanofluids have superior thermal characteristics to the base fluids used in traditional particle–fluid suspensions. Fang et al. ¹⁴ reported the most significant increase in diffusibility. According to researchers, Rhodamine B flow rate is 26 times higher in Cu-water nanofluid with 0.5% Cu particle concentration than that in base fluid at 25°C. They explored the shape, the goods of size, the volume fraction, and the nanoparticles thermophysical characteristics. More investigations concerning to nanofluid are disclosed in Refs. [15–19].

Mansour et al. ²⁰ researched combined convection in a lid-driven chamber with different nanofluids cooling it. The impact of the inclusion of solid particles was investigated. In accordance with their outcomes, the flowing liquid decreased, by maximizing the nanoparticle volume fraction. The specific material as well stated to play a major role, as $A{l}_2{O}_3$ nanoparticles generated a higher increase in Nusselt number than $\mathrm{Ti}{O}_2$ nanoparticles. The addition of metal nanoparticles (e.g. aluminum, silver, silicon, and copper) improves the heat conduction of such compounds. Nanofluids have charmed interest as a modern trend of thermal systems in plants, heat exchangers, and automobiles cooling systems owing to their unique thermal efficiency. Enhanced heat transmission, reduced heat transfer system size, low clogging, and microchannel cooling are all advantages of using nanofluids. Oztop and Abu-Nada ²¹ used several types of nanoparticles to examine thermal performance and fluid movement employing buoyancy forces in a partially heated cavity. They claimed that at low aspect ratios, heat transmission was improved more than at large aspect ratios. Numerous studies on convective heat transfer using nanofluids have been published in recent years which is highly encouraging by Amani et al. ²² Selimefendigil ²³ assumed a trapezoidal-shaped cavity for nanofluids to analyze the heat transfer characteristics in a lid-driven, non-square enclosure. Radhakrishnan ²⁴ conducted combined natural and forced convection in an aired cavity with a heat source through both numerically and experimentally methods. They determined the relationship between the average Nusselt number and the maximum non-dimensional temperature inside a heat generator. By varying the position of the supply of heat, the thermal transfer position was enhanced. They were in good approval by contrasting their numerical results with experimental tests. By applying finite element technique, Basak ²⁵ examined effect of regular and irregular lower wall heating on forced and natural convection lid-driven movements in a square hollow. It was concluded that the thermal conductivity for homogeneous heating was consistently greater than that for non-homogeneous heating.

From the above-mentioned review, no information has been reported for mixed convection in a semi-circular enclosure with a heated corrugated wall. Therefore, the present work is considered in a lid-driven enclosure along a top movable wall that slides in a flat direction uniformly. The mixed convection has been induced and corrugated wall is subjected to a steady heat exchange. Recirculation flow patterns were described using velocity measurements, and flow visualization. A parametric study is conducted to examine the effects of relevant variables, such as nanoparticle volume fraction, on fluid flow and heat transfer within the enclosure. Through thermal physical characteristics of multi-walled carbon nanotubes of Hybrid nanofluid calculations are obtained.

Mathematical development of the problem

A two-dimensional, steady-state, incompressible, and laminar flow of hybrid nanofluid in a semi-circular enclosure is considered. Geometry of the problem is portrayed in Figure 1. Circular wall of the enclosure is corrugated and is described by the equation (1)

$$\mathrm{\Xi }\left(\theta \right)=-r-\mathrm{hsin}\left(\mathrm{\Lambda }\theta \right),\theta \in [0,\pi ]$$

(1)

Where $r,\theta$ are the polar coordinates and describes radial and angular displacements respectively. “h” is the small displacement describing the variations in radius and is responsible for the amplitude of the corrugations in circular wall. “ $\mathrm{\Lambda }$ ” is the frequency of the corrugations.

Figure 1.

Geometry of the problem.

Solid particles of SWCNT and MWCNT are mixed into liquid water ( $H_{2}O$ ) to form a hybrid nanofluid. Fluid motion is governed by moving top wall at constant velocity $U_t$ from left to right in its own reference plane. The enclosure’s top wall is exposed to a steady temperature $T_c$ . Constant heat flux $q$ is applied inward at corrugated wall. No slip of velocity is assumed at the top wall.

Governing incompressible continuity and momentum equations ²⁶ are,

$$\frac{1}{r}(u_{\theta }{)}_{\theta }=-\frac{1}{r}(r{u}_r{)}_r,$$

(2)

$$\begin{array}{c}{\rho }_{\mathrm{hnf}}({(u_r)}_r{u}_r-\left(\frac{1}{r}\right)u_{\theta }^2+u_{\theta }{(u_r)}_{\theta }\left(\frac{1}{r}\right))\hfill \\ =-(p{)}_r+{\mu }_{\mathrm{hnf}}({(u_r)}_r\left(\frac{1}{r}\right)+\left(\frac{1}{r^2}\right){(u_r)}_{\theta \theta }\hfill \\ -2\left(\frac{1}{r^2}\right){(u_{\theta })}_{\theta }+{(u_r)}_{\mathrm{rr}}-u_r\left(\frac{1}{r^2}\right)),\hfill \end{array}$$

(3)

$$\begin{array}{c}{\rho }_{\mathrm{hnf}}(u_{\theta }\left(\frac{1}{r}\right){(u_{\theta })}_{\theta }+u_r{(u_{\theta })}_r+u_r\left(\frac{1}{r}\right)u_{\theta })\hfill \\ =-\left(\frac{1}{r}\right)(p{)}_{\theta }+{\mu }_{\mathrm{hnf}}(\left(\frac{1}{r}\right){(u_{\theta })}_r+\left(\frac{1}{r^2}\right){(u_{\theta })}_{\theta \theta }\hfill \\ -\left(\frac{1}{r^2}\right)u_{\theta }+{(u_{\theta })}_{\mathrm{rr}}+2\left(\frac{1}{r^2}\right){(u_r)}_{\theta }),\hfill \end{array}$$

(4)

where ${\rho }_{\mathrm{hnf}}$ and ${\mu }_{\mathrm{hnf}}$ are the density and viscosity of the hybrid nano-fluid.

Now energy equation is given by,

$${\left(\rho C_p\right)}_{\mathrm{hnf}}(u_r{(T)}_r+\frac{u_{\theta }}{r}{(T)}_{\theta })=k_{\mathrm{hnf}}(\frac{1}{r}{(T)}_r+{(T)}_{\mathrm{rr}}+{(T)}_{\theta \theta }).$$

(5)

Boundary conditions:

Top wall:

$$U=1,T=T_c,V=0,\mathrm{at}\lambda \in [-r,r]\mathrm{and}\theta =0$$

$\lambda$ is a real parameter.

Corrugated wall:

$$U,V=0,q=-k_{\mathrm{hnf}}\frac{\partial T}{\partial n},\mathrm{at}\mathrm{\Xi }\left(\theta \right),\mathrm{for}\theta \in [0,\pi ]$$

In (2–5) equations ²⁷

$$u_{\mathrm{hnf}}=u_f\left\{{(1-{\varphi }_1)}^{-2.5}{(1-{\varphi }_2)}^{-2.5}\right\}$$

$${\rho }_{\mathrm{hnf}}={\varphi }_2{\rho }_{s1}+\{{\rho }_f(1-{\varphi }_1)+{\varphi }_1{\rho }_{s2}\}(1-{\varphi }_2)$$

$$\begin{array}{c}\multicolumn{1}{c}{(\rho C_p{)}_{\mathrm{hnf}}={\varphi }_2{\left(\rho C_p\right)}_{s1}+\{(\rho C_p{)}_f(1-{\varphi }_1)}\\ +{\varphi }_1(\rho C_p{)}_{s2}\}(1-{\varphi }_2)\hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{k_{\mathrm{hnf}}=k_{\mathrm{nf}}\left\{\frac{1}{k_{s2}+2k_{\mathrm{nf}}+2{\varphi }_2(k_{\mathrm{nf}}-k_{s2})}\right\}}\\ \{k_{s2}+2k_{\mathrm{nf}}-2{\varphi }_2(k_{\mathrm{nf}}-k_{s2})\}\hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{k_{\mathrm{nf}}=\left\{\frac{1}{k_{s1}+2k_f+2{\varphi }_1(k_f-k_{s1})}\right\}{k}_f}\\ \{k_{s1}+2k_f-2{\varphi }_1(k_f-k_{s1})\}\hfill \end{array}$$

Solution procedure

Mathematical equations including boundary conditions are computed by Galerkin finite element technique in COMSOL. Mesh independent study is conducted for different number of elements and average temperature on a vertical line inside the geometry is calculated. Mesh with 291,804 elements is chosen for simulations (Table 1).

Table 1.

Data for mesh independent test.

Number of elements	Average temperature gradient
85911	1.07751278916654
112,809	1.07497354404332
155,905	1.06902943564787
187,105	1.06585066656138
231,462	1.06469330951814
291,804	1.06357068638511

Results and discussion

Numerical simulations are performed for a wide range of values for ${\varphi }_1$ and ${\varphi }_2$ . Thermophysical characteristics of the base fluid and solid particles are provided in Table 2.

Table 2.

Thermophysical properties of the base fluid and solid particles.

Thermal properties	Liquid water	SWCNT	MWCNT
$\rho \left(\frac{\mathrm{kg}}{m^3}\right)$	997.1	2600	1600
$C_p\left(\frac{J}{\mathrm{kgK}}\right)$	4179	425	796
$k\left(\frac{W}{\mathrm{mK}}\right)$	0.613	6600	3000

The present simulation analyzed the transfer of heat in semi-circular enclosure with the thermophysical parameters of the base fluid and solid nanoparticles. To physically visualize the results, values of different variables are taken as $1\le p\le 33$ , $0.02\%\le {\varphi }_1\le 0.1\%$ , $0.05\le T\le 0.95$ , and $0.01\%\le {\varphi }_2\le 0.04\%$ . Figure 2 shows streamlines for ${\varphi }_2=0.01\%$ . As the fluid is moving in right direction and because of no slip condition it create two recirculating vortexes. One is formed at left corner and other is slightly far away from center. By increasing the volume fraction of all varied values of SWCNT, we conclude that all streamlines become dense, so circulation of the vortexes expands. Streamlines for ${\varphi }_2=0.02\%$ are shown in Figure 3. By considering these figures, the streamlines appears as two vortexes for all varied values of ${\varphi }_1$ . Semi-circular enclosure influences the initiatory vortex. This initially vortex has sufficient velocity to flow fluid in the cavity’s left hand corner. Due to the effect of this velocity, secondary vortex is created. As the streamlines become more dense, the vortexes expands. Figure 4 shows streamlines for ${\varphi }_2=0.03\%$ . With the increase of all varied values of ${\varphi }_1$ we conclude that streamlines become thinner from the center of the both vortexes and with the increment of the layers of streamlines a new vortex is generated at ${\varphi }_1=0.1\%$ .

Figure 2.

Streamlines for ${\varphi }_2=0.01\%$ .

Figure 3.

Streamlines for ${\varphi }_2=0.02\%$ .

Figure 4.

Streamlines for ${\varphi }_2=0.03\%.$

The streamlines for ${\varphi }_2=0.04\%$ are represented in Figure 5. By the increment of volume fraction of MWCNT, near the heated enclosure the streamlines become more dominant and away from heated enclosure the layers become squeezed.

Figure 5.

Streamlines for ${\varphi }_2=0.04\%$ .

Pressure contours for the volume fractions of MWCNT are given in the Figures 6 to 9. It can be observed from pressure contours that by raising the volume fraction of all varied values of ${\varphi }_1$ pressure point increases, and new vortexes are generated in the enclosure. It can be seen in all figures that pressure point starts from negative value and ends at positive value. It can also be noticed that the contour of pressure point remains same for first varied values of ${\varphi }_1$ . For second varied values of ${\varphi }_1$ , the contour of pressure decreases by increasing volume fraction. For third varied value of ${\varphi }_1$ ,the pressure contour remains same only for $0.01\%$ volume fraction of MWCNT but when volume fraction of MWCNT increases the pressure contour increases for remaining values so, with the increase value of volume fractions the vortex size increases, and it makes new zones inside the enclosure. We can conclude that by increasing the volume fractions of MWCNT, the pressure contour increases as well as remains same for all varied values of ${\varphi }_1=0.1\%$ .

Figure 6.

Pressure contours for ${\varphi }_2=0.01\%$ and varied values of ${\varphi }_1$ .

Figure 7.

Pressure contours for ${\varphi }_2=0.02\%$ and varied values of ${\varphi }_1$ .

Figure 8.

Pressure contours for ${\varphi }_2=0.03\%$ and varied values of ${\varphi }_1$ .

Figure 9.

Pressure contours for ${\varphi }_2=0.04\%$ and varied values of ${\varphi }_1$ .

Figure 10 shows the contours of temperature changes in semi-circular enclosure with the volume fraction of SWCNT ranging from $0.02$ % to $0.1\%$ . In temperature contours the thermal conductivity rises in addition with solid nanoparticles to base fluids. The layers away from heated surface have smaller temperature gradients. Similarly, Figure 11 shows the temperature contours for ${\varphi }_2=0.02\%$ . The temperature of the fluid layer rises as the heat exchange surface and fluid velocity increase. Figure 12 shows the isotherms of the flow domain where the heated source surface is located at semi-circular enclosure. As heat is released to fluid and the heat flux to fluid is dependent on fluid parameters, five SWCNT cases were chosen for analysis. These contours show that the fluid temperature is higher at the heated wall in all cases. When the volume fraction increases, the fluid near the hot zone turns more heated. Thus, it can be included that volume fraction tend to enhance the heat transfer. ²⁸ Figure 13 represent the contours for higher volume fraction of MWCNT. Temperature distribution in fluid layers is pretentious by heated area. The temperature dispersal can be seen in a uniformly manner with improved heat absorption.

Figure 10.

Temperature contours for ${\varphi }_2=0.01\%$ .

Figure 11.

Temperature contours for ${\varphi }_2=0.02\%$ .

Figure 12.

Temperature contours for ${\varphi }_2=0.03\%$ .

Figure 13.

Temperature contours for ${\varphi }_2=0.04\%$ .

It is notable from Figures 14 to 17 that when the volume fraction is maximum, the velocity field is found higher so these contours denote the increase in velocity magnitude. By increasing fluid velocity, the temperature contours level can be decreased. By the increment of the velocity of fluid at top wall, flow diverges from its direction and circulates throughout the enclosure and by combining with cold fluid there exists cavity. This cavity shifts from left to right side and the fluid become motionless at corrugated enclosure and a part of heated fluid exists at the top of the lid. Increased fluid velocity reduces the impact of backflows while boosting heat transfer.

Figure 14.

Velocity magnitude contours for ${\varphi }_2=0.01\%$ .

Figure 15.

Velocity magnitude contours for ${\varphi }_2=0.02\%$ .

Figure 16.

Velocity magnitude contours for ${\varphi }_2=0.03\%$ .

Figure 17.

Velocity magnitude contours for ${\varphi }_2=0.04\%$ .

Table 3 shows tabulated values for local Nusselt number for every volume fraction of MWCNT varying from $0.01$ % to $0.04\%$ . The conclusions show that Nusselt number reduces because of increased volume fraction of SWCNT. Heat transfer between fluid and surface happens due to temperature difference of surfaces. When heat transfer in hot areas is more consistent, Nusselt decreases. The results show that Nusselt number reduces by maximizing volume fraction of SWCNT. In all cases by increasing all the varied values of ϕ₂, Nusselt number decreases.

Table 3.

Average Nusselt number for top wall for different values of volume fractions of SWCNT and MWCNT.

${\varphi }_1$	${\varphi }_2$
	0.01%	0.02%	0.03%	0.04%
0.02%	13.0984	12.9357	12.776	12.6029
0.04%	12.68	12.5103	12.3376	12.1623
0.06%	12.2315	12.0572	11.8819	11.7075
0.08%	11.7663	11.5944	11.4249	11.2561

For different values of volume fraction of SWCNT, the average Nusselt number drops by rising the volume percentage of MWCNT.

In Table 4, a comparison with study ²⁹ is added. As the Reynolds number increased, the average Nusselt number consistently rose with a decrease in aspect ratio. The data can also be verified from the current studies.^30,31 It is also observed that as the aspect ratio increases, the average Nusselt number (represented by the vertical lines in the table) decreases for even small values of Reynolds number. Physically, with the increment of the aspect ratio of Reynolds number, the turbulent flow behavior is observed where the efficiency of heat transport becomes lower. As a result, a reduction in the Nusselt number is developed. In the absence of Reynolds number, creeping viscous flow is acquired. The solution of creeping flow can be analytically computed as given in Turkyilmazoglu and Alotaibi. ³²

Table 4.

Average Nusselt number for different values of Reynolds number.^29–31

Cases	$\mathrm{Re}=100$	$\mathrm{Re}=500$	$\mathrm{Re}=1000$	$\mathrm{Re}=1500$	$\mathrm{Re}=1800$
0.133C	36.073	37.617	40.493	43.129	44.497
0.288C	16.204	20.881	24.638	27.442	28.941
0.5C	11.553	17.760	22.655	26.516	28.565

Figure 18 shows that by increased volume fraction of MWCNT varied with $0.01$ %– $0.04\%$ , the Area weighted average values of temperature on corrugated wall decreases. Here we see that for a small volume fraction average temperature on the corrugated wall gives a small decrease.

Figure 18.

Area weighted average values of temperature for $0.01\%\le {\varphi }_2\le 0.04\%$ .

Conclusion

Heat transfer in semi-circular enclosure is studied numerically with the heated corrugated wall. Hybrid nano-fluid with base fluid and solid particles are examined. The volume fraction of SWCNT is varied for volume fraction of MWCNT. The contours of pressure, temperature and velocity are visualized.

• Increasing the volume percentage of nanoparticles enhanced heat transfer.

• As the volume fraction of nanoparticles increases, pressure contours rise and the temperature gradient decreases.

• As the volume fraction of the fluid increases, the fluid velocity also increases. This leads to a reduction in the effects of backflows, resulting in a higher rate of heat transfer.

• As the volume fraction of SWCNT increases, the local Nusselt number decreases.

• Elevating the volume percentage of MWCNT lowers the temperature on the corrugated surface.

• The average Nusselt number decreases with the increment of the volume fraction of MWCNT.