The Combined Method to Improve Heat Transfer Coefficient on Heat Exchanger

Abstract

The heat transfer process occurs all the time around us, from simple household appliances to equipment used in large industries. Energy efficiency in large-scale use in industry is necessary because it is related to company profits. One way to save energy use in heat exchangers is to change the thermal properties of the cooling fluid. The addition of particles of the nanometer size (nanofluids) in the working fluid will improve the performance of the heat exchanger, and the main goal is the highest efficiency. In addition, there is another method to increase the heat transfer rate, namely, by vibrating the cooling fluid. This chapter will discuss combining nanofluids and ultrasonic vibrations in heat transfer processes in heat exchangers. The application of these two methods simultaneously gives rise to several advantages to the heat transfer system, will promote higher heat transfer, and at the same time function as cleaning of scale/deposits that often appear on the surface of the heat exchanger.

1. Introduction

Energy is an important element in all levels of people’s lives. We live in an interdependent world, and access to easy and reliable energy resources is critical for economic growth to maintain our quality of life. World energy consumption continues to increase, especially in developing countries. Global energy demand has tripled in the last 50 years and may even triple in the next 30 years, while current energy consumption and production levels are unsustainable. The energy sources we used to support the entire range of human activities today come from fossil fuels (coal, oil, and natural gas), supplying about 81%. Fossil-fueled power plants convert heat into mechanical energy, which then operates an electric generator. Changing heat energy, of course, involves heat transfer from one point to another.

The heat transfer process certainly involves an essential tool, including a heat exchanger, which is expected to have high performance with smaller dimensions. For large power, the increase in the performance of the heat exchanger will impact financial reductions in a company. Of course, efforts are needed to increase the heat transfer coefficient of a heat exchanger with various methods.

There are several methods to increase the heat transfer of a heat exchanger, namely, the active method and the passive method or a combination of its (compound). Passive methods increase heat transfer in heat exchangers that do not require an external energy supply. For example, one of the passive method is to change the thermal properties of the cooling fluid by adding solid particles with a much higher thermal conductivity than the thermal conductivity of the liquid. While active methods are ways to increase heat transfer in heat exchangers whose systems require external energy, for example, by vibrating the fluid with ultrasonic frequency in the heat exchanger.

The purpose of writing this article is to combine the two methods mentioned above, namely, the use of nanofluid fluid, at the same time vibrated with ultrasonic frequency to get the maximum increase in heat transfer.

2. Nanofluid

The improvement of the heat transfer coefficient can be achieved by modifying the cooling fluid by adding nanometer-sized particles. Hopefully, the application of nanofluid will save energy and reduce the emissions, global warming potential, and greenhouse-gas effect. It enhances the heat transfer for energy-saving purposes that could contribute to a better quality of human life and fulfill sustainable development. Currently, the production of nano-sized particles is made easy because of the nanotechnology that was predicted 15 years ago. He predicts that energy is one of the “top 10” problems for humankind worldwide in the next 50 years and can only be solved by nanotechnology [1] and it plays a vital role in heat transfer enhancement [2]. Materials commonly used for the manufacture of nanometer-sized solid particles (nanoparticles) from metal materials (Al, Cu, Zn, Ag, and Au), metal oxides, such as SiO₂, TiO₂, Al₂O₃, ZnO, CuO, and organic particles/organic particles, such as carbon nanotubes, graphene oxide, and diamond [3, 4]. Nanoparticles allow the development of cooling fluids called nanofluids. The fluid is a liquid suspension containing nanoparticles smaller than 100 nm and has better thermal conductivity than the base fluid [5]. Adding nanoparticles into the fluid will change the properties of the fluid as a heat transfer fluid (thermophysical properties).

2.1 Thermophysical nanofluid

The properties of nanofluid for density of nanofluid calculated widely used in research of nanofluid for a wide range of volume concentration can be estimated by using the following equations as [6]:

$${\rho }_{\mathit{nf}}={\rho }_b\left(1-\varnothing \right)+{\rho }_{\mathit{np}}\varnothing$$E1

where, ${\rho }_{\mathit{nf}}$, ${\rho }_b,$ and ${\rho }_{\mathit{np}}$ are the nanofluids, base fluid, and particle density, respectively.

The nanofluid volume fraction, $\varnothing ,$can be calculated as follows:

$$\varnothing =\left(\frac{\frac{m_{\mathit{np}}}{{\rho }_{\mathit{np}}}}{\frac{m_b}{{\rho }_b}+\frac{m_{\mathit{np}}}{{\rho }_{\mathit{np}}}}\right)\times 100\%$$E2

where $m_{\mathit{np}}$ and $m_b$ are the mass of nanoparticle and base fluid, respectively.

For hybrid nanofluid, the nanofluid volume fraction, calculated by Eq. (3)

$$\varnothing =\frac{\raisebox{1ex}{$m_{\mathit{np}.1}$}\!\left/ \!\raisebox{-1ex}{${\rho }_{\mathit{np}.1}$}\right.+\raisebox{1ex}{$m_{\mathit{np}.2}$}\!\left/ \!\raisebox{-1ex}{${\rho }_{\mathit{np}.2}$}\right.}{\raisebox{1ex}{$m_{\mathit{np}.1}$}\!\left/ \!\raisebox{-1ex}{${\rho }_{\mathit{np}.1}$}\right.+\raisebox{1ex}{$m_{\mathit{np}.2}$}\!\left/ \!\raisebox{-1ex}{${\rho }_{\mathit{np}.2}$}\right.+V_b}$$E3

where $V_b$ is the volume concentrations of base fluid subscript 1 and 2 indicating the first and the second nanoparticles.

The capacity heat of nanofluid is calculated by Xuan and Roetzel [7] as the following equation is widely used.

$$C_{\mathit{nf}}=\frac{\varnothing {\rho }_{\mathit{nf}}{C}_{\mathit{np}}+\left(1-\varnothing \right){\rho }_b{C}_b}{{\rho }_{\mathit{nf}}}$$E4

where $C_{\mathit{np}}$ and $C_b$ are specific heat of the nanofluid and base fluid, respectively. Nanofluid viscosity was calculated from Brinkman correlations [8], as a shown in Eq. (5)

$${\mu }_{\mathit{nf}}=\frac{{\mu }_b}{{\left(1-\varnothing \right)}^{2,5}}$$E5

where ${\mu }_{\mathit{nf}}$ and ${\mu }_b$ are the viscosity of the nanofluid and base fluid, respectively.

Various models of the thermal conductivity of nanofluid have been published, the earliest of models developed by Maxwell [9] as shown in Eq. (6)

$$k_{\mathit{nf}}=k_b\frac{k_{\mathit{nf}+}2k_b+2\left(k_{\mathit{nf}-k_b}\right)\varnothing }{k_{\mathit{nf}+}2k_b-2\left(k_{\mathit{nf}-k_b}\right)\varnothing }$$E6

where $k_{\mathit{nf}}$ and $k_b$ are thermal conductivity of the nanofluid and base fluid, respectively.

2.2 Preparation of nanofluid

Nanofluid is made by dispersing nanoparticles into a base fluid. Good dispersion is a prerequisite for the use of nanofluids in various fields. Therefore, surfactants are sometimes used to increase the nanofluid’s stability to prevent clogging. In addition, the surface modification of the dispersed particles and the application of strong forces to the clusters of dispersed nanoparticles can improve the stability of nanofluids.

There are two ways to prepare nanofluid: one-step method and two-step method. Each method has advantages and disadvantages. The advantage of the one-step method is that it produces good dispersion, but it is very expensive to manufacture and cannot produce large amounts of nanofluids. While the two-step method produces poor dispersion, the cost of making nanofluids is relatively low and can produce large amounts of nanofluid. Most researchers use this type of method to make nanofluids. Zhu et al. [10] made nanofluids using Al₂O₃ particles with a pure water base fluid using a two-step method. The manufacture of nanofluids using a two-step method is shown in Figure 1. The first step is to prepare the base fluid and nanoparticles by weighing them according to the specified concentration. The next step is to mix the nanoparticles into the base fluid by adding a little surfactant and stirring using magnetic steering. After the nanoparticles and base fluid are well mixed, sonification is carried out, namely, vibrating the mixture with ultrasonic frequency to homogeneous for 1 h.

Figure 1.

Preparation of nanofluids using a two-step method [10].

2.3 Heat transfer characteristics

The heat transfer characteristics of nanofluid depend on various factors, including thermophysical properties of base fluid and nanoparticles, nanoparticle concentration, nanoparticle size, presence of surfactants, temperature, etc. Hence, the functional form of the Nusselt number of nanofluid can be expressed by Xuan and Roetzel [7] in Eq. (7)

$$N_{\mathit{nf}}=f\left(\mathit{Re},\mathit{Pr},k_{\mathit{nf}},{\mathit{\rho C}}_{\mathit{nf},}\varnothing \right)$$E7

The Nusselt number of $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3/\text{water}$ and $\mathrm{Ti}{\mathrm{O}}_3/\text{water}$ nanofluids in turbulent flows proposed by Pak and Cho [6] in Eq. (8)

$$N_{\mathit{nf}}=0.021{\mathit{Re}}^{0.8}{\mathit{Pr}}^{0,5}\text{for}6.54\le Pr\le 12.33\text{and}{10}^4\le Re\le {10}^5$$E8

Xuan and Li [11] studied CuO/water nanofluid in turbulent flow and correlated Eq. (9).

$$N_{\mathit{nf}}=0.0059\left(1+7.628{\varnothing }_p^{0.6886}{\mathit{Pe}}_p^{0.001}\right){\mathit{Re}}^{0.9231}{\mathit{Pr}}^{0,4}$$E9

Several experimental studies have been reported using nanofluid in the heat exchanger. Raei et al. [12] investigated a double-tube heat exchanger with a low volume concentration of particles (0.15%) of Al₂O₃-DI water nanofluids. The nanofluid results in increasing the heat transfer coefficient by 23% without much penalty in pressure drop. Shahrul et al. [13, 14] analyzed nanofluid flow through a shell-and-tube exchanger considering various oxide-based nanofluid at different small volume concentrations (0.01–0.04%) of ZnO, CuO, Fe₃O4, TiO₂, and Al₂O₃ nanoparticles into the water. Heat transfer enhancement was found at about 23–52% compared to the base fluid. In their study, the highest improvement was obtained for ZnO-water nanofluid. Researchers have studied the behavior of nanofluids in automobile radiators due to good heat transfer characteristics. Different nanofluids have been tested to observe their thermal performance and pressure drop to validate their practicality in this application. Heris et al. [15] used base fluid water EG (40/60) to suspend CuO nanoparticles to study the radiator’s performance and the maximum enhancement of heat transfer apprehended about 55% compared to the base fluid. Alosious et al. [16] performed an experimental and numerical analysis of Al₂O₃/Water and CuO/water nanofluid, and they found that the Nusselt number, the total heat transfer coefficient, and convective heat transfer coefficient increased by increasing of Reynolds number. They also demonstrated that the thermal conductivity of alumina oxide is more than copper oxide. However, an increase in pressure drop was not significant.

Subhedar et al. [17] recently researched Al₂O₃/Water-EG in the laminar flow regime and found that for just 0.2% concentration volume of nanoparticles, the Nusselt number increased by 30%. Their research, flow rate, and concentration of volume particles were reported as the key factors influencing heat transfer enhancement. Inlet temperature had a lower effect; an increase of about 26% from 70°C to 85°C was recorded. Heat transfer coefficient enhancement is the process of increasing the effectiveness of the heat exchanger. Adding a small amount of nanoparticles to the base fluid increases the thermal conductivity of the nanofluid with ultrasonic vibrations to increase the performance of a heating system, but the viscosity of the nanofluid is also increased slightly. Working conditions at high temperatures in a heat exchanger is an advantage in itself, which causes the viscosity of the fluid to decrease. The addition of small nanoparticles and high working temperature is an ideal combination for improving efficiency of systems.

The main drawback of nanofluids is their poor stability. Long-term stability of nanofluids is a major concern for engineering applications. Nanoparticles tend naturally to aggregate and sediment in the base fluid, due to the interaction between the particles themselves and between the particles and the surrounding liquid [18]. The main cause is the attraction between the particles called the Van der Waals attraction, which causes the particles to form clusters or agglomerate, then settle to the bottom due to gravity. To obtain stable nanofluids, the repulsion of electrical double layers forces must exceed the Van der Waals attraction.

Overcoming this problem by adding surfactants as a dispersant, is an effective stability enhancement method that prevents the agglomeration of nanoparticles within the nanofluids [19]. It is a simple and economical chemical method, which reduces the surface tension of the base fluid and improves the immersion of nanoparticles. Because surfactants consist of the hydrophobic tail portion (e.g., long-chain hydrocarbons) and the hydrophilic polar head group that tends to increase the hydrophilic behavior between the base fluid and the nanoparticles. The disadvantage of using dispersant as a nanofluid stabilizer is its sensitivity to hot temperature because the rise in temperature causes the bonds between the nanoparticles and the surfactant to be damaged. Some common dispersants are sodium dodecylbenzene sulfonate (SDBS), gum arabic, sodium dodecyl sulfate (SDS), polyvinylpyrrolidone (PVP), dodecyltrimethylammonium bromide (DTAB), oleic acid (OA), etc.

The other method to achieve the long-term stability of nanofluids, without the need for surfactants, is surface modification techniques by modifying the nanoparticles’ surface via functionalization. Functionalization is the process of adding new functions, features, capabilities, or properties to the material by changing the surface chemistry of the material. This is done by introducing functionalized nanoparticles into the base fluid to obtain a self-stabilized nanofluid. Usually, suitable functional organic groups are selected as they tend to attach to the surface of the atom, enabling the nanoparticles to self-organize and avoid agglomeration [20].

3. Ultrasonic vibration

Ultrasonic waves are sound waves whose frequency is above human hearing (>20 kHz), distinguished according to the power and frequency emitted. When viewed from the magnitude of the frequency, it is divided into three categories, namely, low-frequency ultrasonic waves (20–100 kHz) or ultrasonic waves with high power, high-frequency ultrasonic waves (100 kHz–1 MHz), and very high-frequency ultrasonic waves, namely, above 1 MHz. Ultrasonic with a low frequency of 20 kHz–100 kHz can change the medium in which it passes and is widely used in the chemical industry, which aims to change the chemical properties to generate cavitation bubbles and the effect of surface instability.

When sound waves propagate through a liquid, they produce a series of cycles of pressure compression and tensile (rarefaction). In the compression cycle, at the top of the wave position (max), the liquid gets a compressive force, and in the tensile cycle, where the wave position is at the bottom of the valley (min), the liquid experiences a tensile force and so on. Two important phenomena of ultrasonic waves propagating through liquids are acoustic cavitation and acoustic streaming.

3.1 Acoustic cavitation

When a sound wave consists of a maximum cycle (peak) and a minimum cycle (valley), and propagates in a liquid fluid, compression and expansion cycles occur. The compression cycle produces a positive pressure, while the expansion cycle produces a negative pressure causing the liquid molecules to move away [21]. When the magnitude of the tension exceeds the tensile strength between the liquid molecules in the expansion cycle, the liquid will break down, and a cavity will be created to appear as small vapor-filled voids called cavitation bubbles [22], in other words, the phenomenon. The formation, growth, and contraction of bubbles are known as acoustic cavitation.

Acoustic cavitation is divided into stable cavitation and transient cavitation or unstable cavitation/inertia cavitation. Acoustic cavitation consists of three distinct stages: nucleation, rapid bubble growth (expansion) until it reaches a critical size, and bubble collapse (shown in Figure 2.

Figure 2.

The formation, growth, and collapse of bubbles due to ultrasonic vibrations in liquid fluids produce stable and unstable cavitation.

Stable cavitation due to micro-bubbles in a liquid, resulting from changes in liquid pressure (compression and refraction) ultrasonic waves, changing/oscillating in size and shape that causes the surrounding liquid to move is known as non-inertial cavitation. It produces a strong shearing force around the solid surface, and removes particles. But in unstable cavitation, the micro-bubbles oscillate and in the end, the bubbles burst/collapse due to the acoustic wave strength exceeding the cavitation threshold, producing a shock wave.

When the bubble collapses surrounded by liquid, several physical effects will appear, including shock waves, micro-jets, turbulence, and shear forces. This physical effect has been widely applied in emulsification, extraction, and cleaning [23]. In this process, bubble size changes drastically by several hundred times to reach the equilibrium size of the bubble radius before an explosion occurs in just a few microseconds. Figure 3 shows the process of bubble collapse, cavitation releases a large amount of energy and produces local “hotspots” experiencing extreme temperatures and pressures that have a chemical impact because the temperatures and pressures that occur are very high at around 5000°C and 200 atm [24].

Figure 3.

Bubbles size growth due to ultrasonic waves occurs in just microseconds.

3.2 Acoustic streaming

Besides having a chemical impact (sonochemical), ultrasonic waves also impact physical changes (sonophysical). The liquid medium will absorb sound wave energy in the direction of the wave. Physical effects include streaming, micro-jets, and shock waves. Streaming is a term that describes the average mass density at steady conditions or velocity caused by acoustic wave oscillations in a liquid fluid due to a momentum gradient resulting in fluid flow. The sound propagation with large amplitude in the liquid will produce fluid motion. This nonlinear phenomenon is called acoustic streaming. Acoustic streaming caused by ultrasonic vibrations effectively enhances heat transfer and cleans contaminated surfaces.

Unlike bubbles in a liquid, the bursting of bubbles near the solid-liquid interface makes the bubble surface unsymmetrical because the solid surface acts as a flow barrier. The result is a powerful jet coming from the side of the bubble that produces a strong beam toward the surface called micro-jetting, which functions as ultrasonic cleaning or antifouling. This phenomenon is used to clean with the ultrasonic method, while in heat transfer applications, it will interfere with the boundary layer of convection heat transfer, as shown in Figure 4.

Figure 4.

Bubble collapse near the solid surface.

Unlike cavitation bubble collapse in a liquid, cavitation bubble collapse near a solid surface is not symmetrical due to surface barriers impeding fluid flow. The result is a rush of liquid coming from the side of the bubble away from the surface, creating a strong liquid jet directed toward the surface. The collapsing bubble near the solid-liquid interface disrupts the thermal boundary and velocity boundary layers, reducing thermal resistance and creating micro-turbulence due to shock waves. For this reason, ultrasonic vibrations in the liquid enhance convective heat transfer because of the growth and collapse of bubbles near the surface, which disturbs/compresses the thermal boundary layer. The amount of convection heat transfer is inversely proportional to the thickness of the thermal boundary layer. The thinner the thermal boundary layer, the greater the heat transfer rate, as shown in formula (10).

$$q_{\mathit{\text{CONV}}\approx \frac{1}{{\delta }_{\mathit{TBL}}}}$$E10

Another phenomenon due to the bursting of cavitation bubbles near the solid surface is widely applied as a cleaning process due to the presence of liquid jets toward the surface. The study of bubble collapses near the solid wall surface was carried out by Li et al. [25] with numerical and experimental methods on the shape and behavior of dynamic bubbles, which state that there are similarities between experimental results and numerical simulation results. The results show that the velocity of the liquid jet is stronger toward the walls of the spherical bubble compared to the non-spherical shape. Other studies by vibrating the working fluid with ultrasonic frequencies have been carried out by several researchers, including Tajik et al. [26], who showed that the increase in heat transfer is proportional to the ultrasonic power and inversely proportional to the distance between the vibration center and the heat source. The highest increase is due to ultrasonic vibration with a frequency of 18 kHz by 390% with power between 56 and 158 W. Legay et al. [27] investigated convection heat transfer using a double pipe heat exchanger using pure water, and an ultrasonic vibrator was installed in the middle of the heat exchanger with a frequency of 35 kHz, the highest increase in heat transfer was in laminar flow of 150%, whereas in turbulent flow it had no effect. Ultrasonic vibrations also decrease the thermal convection heat transfer resistance. A comparison of the performance of shell-and-tube and double-pipe heat exchangers has also been carried out by Legay et al. [28]. In Gondrexon et al. investigated a shell and tube heat exchanger, the increase in the total heat transfer coefficient with ultrasonic vibrations ranged from 123% to 257% depending on the fluid flow rate [29]. The experiment was carried out by Zhou [30] to test convective heat transfer in a horizontal copper pipe under the influence of an acoustic cavitation field. The fluids used are acetone, ethanol, and pure water at constant heat flux. The results show that the highest heat transfer increase of 395% occurs in acetone fluid. The disturbance of the cavitation bubbles and the collision of the cavitation bubble clusters cause thinning of the thermal boundary layer on the pipe surface, leading to increased convection heat transfer.

These properties in terms of heat exchanger performance are very attractive because two advantages, namely, increasing heat transfer and reducing fouling (dirt on the heat exchanger surface) can be achieved by high-power ultrasonic wave propagation.

4. The combined methods are nanofluid and ultrasonic vibration

As discussed earlier, many studies on the technique of increasing heat transfer by adding nanoparticles to pure water (nanofluid) have been carried out by researchers. There are two main problems in using nanofluid as a cooling fluid: high particle deposition and agglomeration rate of nanoparticles. Practically, this deposition can be avoided by vibrating the fluid with ultrasonic frequency. In addition to antifouling and anti-agglomeration effects, ultrasonic waves can also improve heat transfer. The heat transfer characteristics of nanofluids with vibrations imposed on heat transfer surfaces have not been widely studied until now, which provides an important purpose for writing this book.

Delouei et al. [31] studied nanofluids vibrated by ultrasonic waves to determine the effect of increasing heat transfer and pressure drop on forced convection. It uses nanofluid Al₂O₃ particles passed through a fine pipe with an inner diameter of 17 mm, a pipe thickness of 1 mm, and a length of 1 m, which is vibrated using 28 kHz ultrasonic waves with ultrasonic power variations of 75 W and 100 W. The results show that ultrasonic waves can increase heat transfer by 11.37% and reduce the pressure drop by 15.27%. Furthermore, it shows that ultrasonic vibration can increase heat transfer and reduce pressure drop, and the ultrasonic effect is sensitive at a high percentage of particle volume and a low flow rate. Therefore, ultrasonic waves in a heat exchanger that uses nanofluid fluid are very beneficial, besides being antifouling and anti-agglomeration, the third is increasing heat transfer and reducing pressure drop. The use of ultrasonic waves in nanofluids is the answer to the weakness of nanofluids in practical applications, namely, an increase in pressure drop due to nanoparticles, which cause greater energy consumption. Shen et al. [32] tested the effect of ultrasonic vibrations on natural convection heat transfer by using a platinum rod with a diameter of 2 mm and a length of 8 cm immersed in nanofluid Al₂O₃-water with a volume concentration of 0.01% with variations in the temperature of the fluid (nanofluid) 30°C, 40°C, and 50°C. The results show that ultrasonic vibration can increase the convection heat transfer coefficient by 128% at a temperature of 30°C; at a fluid temperature of 50°C, it increases the convection heat transfer coefficient by 87%, and at a temperature of 50°C, it increases the convection heat transfer coefficient by 25%. In addition, it was found that high heat flux will reduce the heat transfer coefficient and the temperature of the test rod.

Research has been carried out by Sudarmadji et al. [33] using Al₂O₃ particles in radiator coolant (40:60 EG/water) on radiators accompanied by ultrasonic vibrations to increase the rate of heat transfer. The effect of adding nanoparticles to the radiator coolant increased heat transfer by 39.6%, while the ultrasonic effect was only 15.7%. The increase in heat transfer due to both methods is 62.7% when compared to pure radiator coolant without vibration. The drawback of this research is the limitation of ultrasonic power, which is relatively small so that the ultrasonic wave energy absorbed by the cooling fluid is relatively low.

5. Conclusions

One of the important parameters in ultrasonic reactions is the size change of bubbles. Generation, growth, and transient collapse of microbubbles could create intense local energy because these changes produce extremely high temperatures and pressure. The bubble implosion near a solid-liquid interface disrupts thermal and velocity boundary layers, reducing thermal resistance and creating microturbulence. This is one of the reasons why acoustic cavitation is often considered the main reason for heat transfer enhancement by ultrasound. It can also be used as a way to promote or control turbulence, which already suggests some possible use in heat exchange devices. The presence of nanoparticles will strengthen the heat transfer coefficient in the heat transfer systems. The advantage of using this combined method, in addition to increasing heat transfer and reducing pressure drop on the heat transfer system, can also function as a descaling agent on the surface of a heat exchanger (cleaning), which is often called antifouling, and also anti-clogging on nanofluids.

Nomenclature

$\varnothing$

the volume fraction of nanofluid

$C_{\mathit{np}}$

the specific heat of nanofluid

$C_b$

the specific heat of the base fluid

${\rho }_b$

density of the base fluid

${\rho }_{\mathit{nf}}$

density of nanofluids

${\rho }_{\mathit{np}}$

density of nanoparticles

$m_b$

mass of base fluid

$m_{\mathit{np}}$

mass of nanoparticles

${\mu }_b$

viscosity of base fluid

${\mu }_{\mathit{nf}}$

viscosity of the nanofluid

$k_b$

thermal conductivity of the base fluid

$k_{\mathit{nf}}$

thermal conductivity of nanofluid

$N_{\mathit{nf}}$

Nusselt number of nanofluid

$\mathit{Re}$

Reynold number

$\mathit{Pr}$

Prandtl number