Exergy and Entropy Analysis of Heat Exchanger Under Mechanical Vibration and Magnetic Field

Vibrations play a significant role in enhancing heat transfer in various systems [1]. Vibrations make use of oscillating waves and thus improve heat transmission in heat sinks and increase the effectiveness of cooling liquid heat sources like water [2]. Han et al. [3] and Purwono et al. [4] studied the effect of vibrations on heat transfer. As suggested by Purwono et al. [4] ultrasonic vibrations had a greater overall heat transfer coefficient while Han et al. [3] noticed a decline in heat transmission at specific pressures. The impact of vibrations on heat transfer is influenced by factors like frequency, amplitude, and position along the system. Vibrations in a heat exchanger affect tube bundles, resulting in different frequencies and amplitudes [5]. Heat transfer efficiency is affected because of increased vibration amplitudes from higher intake velocities. Vibrations obstruct air movement, alter the distribution of temperature and inhibit the formation of boundary layers, thus increasing the average heat transfer coefficient by up to 66.67% [6]. Heat transfer in pulsating heat pipes is improved by vibrations, particularly during the adiabatic and evaporation portions. This is because vibrations lower thermal resistance and starting-up temperature, which improves heat transfer efficiency [7]. Square waves in particular when compared with sinusoidal waves, cause airflow inter-ruption, which improves heat transmission in heat sinks and increases heat dissipation [8].

By upsetting the thermal boundary layer, vibrations through a pipe enhanced the heat transfer significantly up to 540% more than in a steady-state flow [9]. As per a study by Yuan et al. [10], there exists a positive correlation between convective heat transfer and degree of chaos generated by a vibrating blade. Vibrations affect heat transfer in forced or free convection on vertical plates, changing convection patterns and affecting performance. Vibrations accelerate the natural convection process, thus enhancing overall heat transfer process in a cubic cavity. The Nusselt number increases with increase in vibration frequency leading to quicker steady-states [11]. As presented in a study by Rasangika et al. [12], heat sinks with forced vibrations improves cooling by increasing the Nusselt number. Taller fins decrease heat transfer, but higher frequencies increase it. Heat transmission in concentric and eccentric annular cylinders is impacted by forced vibration [13] whereas Nusselt number fluctuates with eccentricity positions and Rayleigh number. Research conducted by Khan et al. [14] analyses the impact of vibrations on heat transfer from forced convection from tandem cylinders. Heat transfer characteristics are greatly influenced by Nusselt number, which is influenced by flow structures, oscillation amplitude, and diameter ratio. High frequency and low fin height produce the best results. In longitudinal finned tubes, heat transport is improved by vertical vibrations. According to a study, vibrations enhance the value of the average Nusselt number and the greatest value is achieved at an angle of 45 deg [15].

In applications such as nuclear power plants and oil extraction, flow-induced vibration (FIV) modifies the flow structures, shedding frequency, pressure coefficient, and Nusselt number, which in turn impacts heat transfer from tandem cylinders [14]. Fluid flow-induced vibrations in heat exchangers, like vibrations caused by vortexing tube bundles with elastic connections, can increase heat transfer by 11.8% [16]. In devices like refrigerant condensers and radiators, vibrations improve heat transfer [17]. Convection heat transfer coefficient increases with vibration-induced movement, and heat transfer efficiency is enhanced at ideal intensity. As per studies, the overall heat transfer coefficients were improved by up to 44% for radiator-coolant and 31% for Aquades fluids by ultrasonic vibrations at 20 kHz, 30 kHz, and 40 kHz frequencies in a double pipe heat exchanger [4].

In a new type of hollow heat exchanger with helical elastic coiled tubes, vibrations can improve heat transfer by up to 4.37% in terms of overall thermal and hydraulic characteristics [18]. Under fluid-induced vibration, the Nusselt number can increase up to 11.67%, and the impact of this enhancement in heat transfer is connected with tube row spacing [19]. In a double-array helical elastic tube bundle heat exchanger, vibrations boost heat transfer efficiency by increasing turbulent energy and heat absorption. For a shell-side fluid, the effect of vibrations on heat transfer is a heat accumulation process: the greater the flow velocity, the more heat absorbed by the fluid in unit of time, hence better the heat transfer effect [20]. In their study Ji et al. [20], vibrations, an increase of 80.80% was found in the average volume heat transfer coefficient (HTC) when the inlet flow velocity was kept constant, as winding number increases and average volume HTC became 3.6 times when winding number was kept constant, as inlet-flow velocity increases. The vibrations enhance heat transfer in an improved Elastic Bundle Heat Exchanger, with a 41.67% increase in amplitude and 46.47% in heat transfer coefficient as inlet velocity rises [21].

By inhibiting natural convection, lowering the Nusselt number, and increasing the rate of heat transfer, magnetic fields have a substantial impact on heat transfer. This is especially true for greater interspacing distances between embedded cylinders inside a porous enclosure [22]. The magnetic field reduces slippage between fluid layers, easing flow and smoothing isotherm penetration, ultimately enhancing heat transfer in the microchannel with a hydrophobic surface [23]. Up to a 150% increase in average heat transfer has been recorded by in thermomagnetic convection of paramagnetic air due to the strong and precise magnetic field arrangement [24]. By creating swirling flows in ferrofluid zones, applying a magnetic field to a triple tube heat exchanger improves its overall performance, enhances its heat transfer coefficient, and slightly boosts the pumping power [25]. By creating swirling flow, the use of a magnetic field in heat exchangers improves both energetic and exergetic performance [26]. This can result in a 90% improvement in thermal efficiency and a 33% increase in exergy efficiency. A nonuniform magnetic field increases circulation and improves heat transfer efficiency in twin-pipe heat exchangers, in particular triangular tubes exhibit a 15% improvement over smooth tubes [27]. Through the creation of vorticies, the activation of thermal diffusion, and an improvement in convective heat transfer coefficients, external magnetic fields in a fin-tube heat exchanger with ferrofluid coolant improve heat transmission [28].

By rupturing thermal boundary layers, encouraging Brownian and thermophoretic motion, and raising Joule heat under various magnetic field types, the magnetic field improves heat transmission in the Fe₃O₄–water nanofluid [29]. Shafiee et al. [30] proposed that in a metal foam environment, the magnetic field improves heat transfer in nanofluid by improving the heat transfer coefficient with larger Hartmann and Darcy numbers and by enhancing performance assessment criteria. In a ferrofluid inside a helical tube, the magnetic field enhances heat transmission by raising the Nusselt number by about 40% [31]. Heat exchangers in a variety of applications benefit from the enhanced heat transfer that the magnetic field provides in Fe₃O₄/water nanofluids in a heated pipe [32], which can increase Nusselt number by up to 31.3% when compared to water. Lee et al. [33] in their study concluded that with an optimized field strength, the magnetic field increases convective heat transport in Fe₃O₄ and Fe₃O₄-MWCNT nanofluids by up to 3.23% and 5.23%, respectively. Magnetic field presence increases heat transfer significantly in MWCNT-Fe₃O₄/water hybrid nanofluid, with 109.31% higher Nusselt number and 25.02% improved pressure drop [34]. The magnetic field enhances convective heat transfer by up to 35% in a twisted duct with a high twist ratio and NiO/water nanofluid compared to cases without a magnetic field [35].

The present study is a numerical investigation, whereas the study by Ashjaee et al. [36] is experimental. The focus here is on the combined effect of mechanical vibrations and magnetic field on heat transfer in a heat exchanger, analyzed from the perspective of the second law of thermodynamics. In contrast, Ashjaee et al. [36] examined only the effect of the magnetic field on heat transfer due to forced convection, limited to first law of thermodynamics analysis. Both studies use water-based Fe₃O₄ nanofluid as the medium. varied the volume fraction of ferrofluid particles from 0.5% to 3%, while the present study maintains a constant concentration of 2%. Ashjaee [36] considered Reynolds numbers ranging from 200 to 900 in intervals of 100 and magnetic field strengths up to 1200G in intervals of 200G. The present study, however, focuses on Reynolds numbers from 150 to 300 in intervals of 20 and magnetic field strengths from 0 G to 2000 G in intervals of 1000 G.

A two-dimensional heat sink with a rectangular channel of dimensions 40 mm (L) × 4 mm (W) was created in ansys for the present study [36]. The inlet temperature of nanofluid was set at 25 °C, whereas the present study uses an inlet temperature of 20 °C. applied a constant heat flux of 66,000 W/m² from the bottom surface with the top and side surfaces insulated. In the present study, the bottom wall is maintained isothermal at 77 °C, and the top wall is insulated.

The present study explores novel approaches and ideas in the field of heat transfer enhancement using distinctive techniques. With a substantial body of research dedicated to this area, researchers are continually developing innovative methods to maximize the potential uses of magnetic fields and mechanical vibrations. However, the precise effects of vibrations combined with varying magnetic field strengths remain relatively unexplored within the field of transport phenomena. Very limited experimental work has been done in this area so far. After exploring various research databases like Google Scholar, Web of Science and Scopus, we could find only two papers that experimentally study the simultaneous effect of vibrations and magnetic field on heat transfer systems. Both these papers focus on only analysis based on first law of thermodynamics. None of the studies till date cover second law of thermodynamics analysis when mechanical vibrations and magnetic field are simultaneously applied to the system. Shahsavar et al. [37] examined effect of mechanical vibrations on ferrofluid flow inside a tube in presence of rotating magnetic fields. It was found that the cooling efficiency of the ferrofluid increases under the effect of vibrations and magnetic field. An increase of 21.32% in the performance evaluation criteria of the system was noted. Naphon et al. [38] studied the combined effects of pulsating flow, nanofluids, microfind tube and magnetic field on the heat transfer and flow characteristics inside microfins tube. Heat transmission tends to rise as a result of the mixing, accumulation, and sedimentation of nanoparticles in the base fluid, which is significantly influenced by centrifugal force, magnetic force, and microfins tube roughness. Additionally, the Nusselt number is higher for higher magnetic field strengths, nanofluid concentrations, and frequency pulsating the flow than it is for lower ones. At a flow frequency of 20 Hz, the Nusselt number of the nanofluid flowing through the microfins tube with the magnetic field effect is 22.21% greater than that of the working fluid, i.e., water.

Our study aims to conduct a detailed analysis of exergy efficiency and entropy generation to assess their industrial applicability. Through entropy generation analysis, it is possible to identify irreversibility losses, providing insights into minimizing such losses. The findings of this research could significantly impact various industries, including power generation, refrigeration, and chemical processing, where enhancing heat transfer efficiency can lead to substantial energy savings and improved system performance.

In this study, we use a two-dimensional rectangular channel as the geometry in ansysfluent as shown in Fig. 1. The channel measures 4 mm wide and 40 mm long. A magnet with adjustable strength is placed at a distance of 20 mm from the inlet in the direction of flow to act as a virtual baffle. The following boundary conditions were imposed onto the channel

1. Inlet: Nanofluid inflow at 20 °C

2. Top Wall: Heat Flux (q″) = 0 W/m²

3. Bottom Wall: Isothermal at 77 °C

The current investigation utilized ansysfluent version 2022 R1, a commercially available software package, to model the flow dynamics in low Reynolds number laminar conditions. Although the fluid behavior at the inlet of the channel is laminar, the vibration causes vigorous movement of the fluid inside the domain causing lot of deviation from the laminar flow regime hence, the k–ω SST (Shear Stress Transport) turbulence model, which is well- suited for handling laminar, transitional and turbulent flow regimes within ansysfluent was employed to accurately simulate the flow characteristics.

For the coupling of pressure and velocity, the COUPLED technique was utilized. Momentum equations were discretized using a second-order upwind method to enhance accuracy. The energy equation was discretized using the QUICK scheme to improve precision. The convergence criteria were established at 10⁻⁵ for the x velocity, the y velocity, and the momentum, and at 10⁻⁶ for the energy convergence. The simulations were run until the solution converged.

To effectively capture the computational domain's characteristics, meshes comprising 7504, 16441, and 64881 elements were developed. The Nusselt number was calculated for each of these meshes at a Reynolds number of 150, using water as the working fluid without any vibration or magnetic field present.

Among these meshes, the one with 16,441 elements yielded a Nusselt number of 13.79. The results obtained from the other two meshes were within 2% of this value. Consequently, the mesh with 16,441 elements was selected to evaluate all results in this study.

For the remaining simulations, the working fluid within the computational domain consisted of Fe₃O₄ nanoparticles at a concentration of 2% in water. In situations with low Reynolds numbers, it is appropriate to apply no-slip conditions for velocity due to the dominance of viscosity at the walls, resulting in negligible velocity at the walls as particles adhere to them. Figure 2 presents a graph validating the current model in the absence of vibration by comparing it with the results from Ref. [36]. It is evident that the model developed in the present study performs well when compared to the previously established study. Given that vibration is a highly turbulent phenomenon, we have incorporated an additional validation of our model in the presence of vibration. Figure 3 depicts the relationship between the Nusselt number and the frequency of vibration at an amplitude of 1 mm. The findings of this study were compared to those of Ref. [39], and the results exhibit excellent agreement with the previous work.

Here, $vvib$ is the vibration velocity, A is the amplitude of vibration, t is the time, and ω is the frequency of vibration.

The governing equations for the present model have been derived from Ref. [41] as follows:

where L is the length of the channel and $ϕx$ is the local value of $ϕ$ at a distance x from the inlet.

Figure 4 illustrates the increase in the heat transfer coefficient at the heated wall with the enhancement of vibration and magnetic field strength. It is noteworthy that, compared to the base case of water, the mere addition of nanoparticles causes an increase in the heat transfer coefficient due to the higher thermal conductivity of the nanoparticles. Subsequently, with each increment in vibration level, the heat transfer coefficient increases significantly, reaching up to a maximum of 2.8 times at a Reynolds number (Re) of 300, and a vibration level of 5 mm at 25 Hz in the absence of a magnetic field (0 G). The same increment at magnetic field strengths of 1000 G and 2000 G increases up to 3 and 3.1 times, respectively.

Figure 5 depicts the ratio of the Nusselt number of the vibrating channel in the presence of a magnetic field and nanofluid relative to the static channel in the absence of a magnetic field. It can be observed that this ratio consistently reaches a maximum of 3 in the cases of 0 G and 1000 G magnetic field strengths, showing a decreasing trend with increasing Reynolds number. However, in the case of 2000 G, the ratio of the Nusselt number increases with increasing Reynolds number, almost reaching 2.9 at the maximum vibration level at a Reynolds number of 300.

Figure 6 shows the variation of pressure drop across the inlet and outlet of the channel due to frictional losses. It is expected that with increasing vibration levels, the fluid undergoes more collisions with the walls of the channel, thereby losing a significant amount of energy. Consequently, with increasing vibration levels, the pressure drop rises significantly, and a similar trend is observed for all magnetic field intensities.

Figures 7 and 8 depict the velocity vectors and temperature contours, respectively, at 150 Reynolds number for three different cases: (a) in the presence of 5 mm at 25 Hz vibration and no magnetic field, (b) in the presence of 2000 G magnetic field at the middle of the channel and no vibration, and (c) in the presence of both 5 mm at 25 Hz vibration intensity and 2000G magnetic field.

From Fig. 7, it can be inferred that in the presence of only the magnet, the velocity vectors initially follow the flow direction but suddenly exhibit a vertical upward component near the middle of the channel. The magnet acts as a virtual baffle for the fluid, causing it to flow through the constricted space. Subsequently, the fluid collides with the upper wall, then hits the bottom wall with a downward component of velocity, extracting more heat from the bottom wall.

In the presence of only vibration, the fluid velocity takes the shape of a typical sinusoidal curve, alternately colliding with the upper and lower walls, extracting heat from the bottom wall of the channel. In the presence of both magnet and vibration, along with the sinusoidal shape of velocity vectors inside the channel, there is also a bump at the middle of the channel due to the presence of the magnetic field.

The temperature contours in Fig. 8 further support the inferences from the velocity vectors. The magnetic baffle causes the fluid to extract more heat from the middle of the channel when only the magnet is present. In the case where only vibration is present, there are repetitive spikes of temperature rise along the length of the channel, attributed to points where the fluid collides with the bottom wall of the channel in the presence of vibration.

In the case of combined magnetic field and vibration, there is both a large bump of temperature in the middle of the channel and repetitive spikes along the length of the channel. We have also provided a video containing temperature contours and velocity vectors as supplementary material along with this paper, which shows animations of all the different cases, providing further clarity to the above discussion.

There is a possibility that at a specific frequency of vibration, the fluid is supposed to collide with the bottom boundary just at the middle of the channel, but the magnetic field would exert an opposing force to that motion at the middle of the channel, causing destructive interference. However, there is another possibility where the vibration frequency is set such that the fluid collides with the bottom boundary of the channel just before the middle. In this case, both the magnet and the vibration would impart an upward force onto the fluid, leading to constructive interference. A detailed discussion about harmonically synchronizing the Reynolds number, vibration frequency, and the position of the magnet to maximize the combined effect is out of the scope of this paper but can be explored as a future study in this domain.

$η$ represents the second law/exergy efficiency of the system. Exergy efficiency refers to the proportion of useful output exergy to input exergy. Second law efficiency is essential because first law efficiencies don't account for an idealized system comparison. Relying solely on first law efficiencies can be deceptive, potentially leading to an overestimation of a system's efficiency. Thus, second law efficiencies are necessary for a more precise assessment of system performance. The second law dictates that no system can achieve 100 percent efficiency. Figure 9 shows how exergy efficiency varies with Reynolds number (Re) for different intensities of vibration and magnetic fields. It's evident that the efficiency increases as the intensity of vibration rises. For instance, at 150 Re, with 5 mm 25 Hz vibration and a magnetic field of 2000 G, the efficiency reaches 52.81%, compared to the base case without vibration or magnetic field, where it's 31.28%. However, exergy efficiency decreases as Re-increases.

$ϕ$ represents the dimensionless number of irreversibility. In a system where $ϕ$ is less than 1, less energy is dissipated as waste heat due to irreversible processes. This means that more of the energy input into the system is effectively utilized for the intended purpose rather than being lost as waste heat.

Figure 10 shows the effect of various intensities of vibration at different magnetic field intensities as a function of Re. For all the cases of vibration and magnetic field, φ is less than 1. It can be observed that $ϕ$ decreases on increasing the intensity of vibration. That means that entropy generation in the case of vibration is less than that compared to the static case. Moreover, Figs. 10(b) and 10(c) also indicate that the magnetic field causes further reduction in $ϕ$ further reducing the entropy generation. This can be attributed to disturbed thermal layer due to vibration and magnetic field. Both vibration and magnetic field disrupt the boundary layer by making it thin. A thinner boundary layer implies more efficient heat transfer and reduced entropy generation.

The introduction of additional irreversible processes into the system often results in an increase in entropy creation when vibrations and magnetic fields are present. Vibrations lead to frictional losses and fluid turbulence. Vibrations can also alter the internal fluid flow patterns within the heat exchanger. Magnetic fields can induce mechanical stresses or electrical currents (if conductive fluids are involved). Such vibration and magnetic field- induced nonuniformities in the system might enhance entropy generation thus changing values of the heat transfer coefficient. In thermodynamic terms, the increase in entropy generation signifies a decrease in the available work that can be extracted from the system. This reduces overall efficiency of the heat exchanger.

Ultimately, the practical implications of increased entropy generation due to simultaneous application of vibrations and magnetic field are a decrease in the performance and effectiveness of the heat exchanger system. This can manifest as reduced heat transfer rates, higher energy consumption, or the need for more frequent maintenance and operational adjustments to compensate for the effects of vibrations and magnetic fields. Understanding and mitigating these effects are crucial for optimizing the efficiency and longevity of such systems in practical applications, thus serving as an economical advantage for manufacturing companies and their customers.

The entropy generation ratio (S_gen,vib+mag/S_gen,static) is the ratio of entropy generation in the presence of magnetic field and vibration to the case in which both are absent. The ratio of entropy generation increases with increasing intensity of vibration from 0 mm 0 Hz to 5 mm 25 Hz as shown in Fig. 11. Entropy generation indicates the extent of energy dissipation and the degree of disorder within the system which increases on increasing the intensity of vibration. The ratio reaches a maximum of 2.4 at 5 mm 25 Hz vibration and 2000 G magnetic field. It is noteworthy that this ratio lies above 1 for all the cases in the present study showing that the entropy generation increases with increasing vibration and magnetic field intensity in the system.

This observation implies that the system experiences greater energy dissipation and disorder with increasing vibration and magnetic field intensity. The higher ratio suggests that more energy is lost as heat due to increased turbulence and fluid mixing induced by vibration. Additionally, the presence of a magnetic field may further disrupt the flow, contributing to increased entropy generation. Therefore, the results highlight the importance of considering vibration and magnetic field effects on entropy generation for a comprehensive understanding of system behavior.

Table 2 compares the range of Reynolds numbers, concentrations of ferrofluids, frequencies of vibrations, and strengths of magnetic fields applied covered by various authors:

Shahsavar et al. [37] investigated the ferrofluid flow experimentally inside a rifled tube under the influence of vibration and rotating magnetic field (RMF). Variations in Reynolds numbers (Re), nanoparticle concentrations (φ), and rifled tube pitches (P) were taken into account. Highest Nusselt number was recorded when a vibration acceleration of 5 m/s² was given to a 2% concentrated ferrofluid at Reynolds number of 2000—a 35% improvement in Nusselt number as compared to no vibration acceleration case at the same Reynolds number and ferrofluid concentration. Even at lower Reynolds number of 500 for a 2% concentrated ferrofluid, a 24% increment in Nusselt number is found from no vibration acceleration case to the case in which 5 m/s² vibration acceleration is applied to the system. Further comparison, analysis and validation of results of our study with those of Ref. [37] can be seen in Fig. 12.

Naphon et al. [38] performed experiments on nanofluids of different concentrations (0.25%, 0.5% by volume) under varied conditions—Reynolds number in the range of 1000–2400, frequency of the pulsating flow at 10, 15, and 20 Hz, uniform magnetic field of 0.12 μT, inlet nanofluid temperature of 20 C and heat input of 120 W applied in every case. The mobility of suspended nanoparticles in the base fluid is significantly influenced by centrifugal and magnetic forces, leading to increased disturbance in the thermal boundary layer zone. As a result, it was found that nanofluids with magnetic fields have greater Nusselt numbers than those without. Naphon and Wiriyasart [38] also found that in the presence of magnetic fields, Nusselt numbers with higher frequency of the pulsating flow exhibit higher values than those with lower ones . The primary nanofluid flow is more disturbed due to the substantial impact of the frequency pulsing flow on the Brownian motion of suspended nanoparticles in the base fluid. Naphon and Wiriyasart [38] recorded higher Nusselt number of 19.8 in the case when uniform magnetic field of 0.12 μT coupled with a pulsating flow frequency of 20 Hz was applied to the system at a Reynolds number of around 2300. This was a 7% improvement in Nusselt number as compared to the case of continuous flow with no pulsating frequency and magnetic field strength applied at the same Reynolds Number of 2300.

This study investigated the combined effect of vibration and magnetic field on heat transfer, exergy efficiency, irreversibility, and entropy generation in a rectangular channel. Numerical simulations were conducted using ansysfluent, and the following key findings were observed:

• Heat transfer enhancement: The Nusselt number increased with increasing intensity of vibration and magnetic field. At a Reynolds number of 300, the Nusselt number reached a maximum of 55.6 with 5 mm 25 Hz vibration and 2000 G magnetic field, compared to the base case without vibration or magnetic field where it was near 18 showing 225.9% enhancement.

• Exergy efficiency: Exergy efficiency increased with increasing vibration intensity, reaching 52.81% at 5 mm 25 Hz vibration and 2000 G magnetic field. However, it decreased with increasing Reynolds number.

• Irreversibility reduction: The dimensionless number of irreversibility (φ) decreased with increasing vibration and magnetic field intensity.

• Entropy generation: The ratio of entropy generation in the presence of vibration and magnetic field to the static case increased with increasing vibration intensity, reaching a maximum of 2.4 at 5 mm 25 Hz vibration and 2000 G magnetic field. This indicated greater energy dissipation and disorder with higher vibration and magnetic field intensity.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

A  =

amplitude (m)

Dnp =

diameter of nanoparticle (m)

Dh =

hydraulic diameter (m)

$Δp$ =

pressure drop (Pa)

Ec =

Eckert number

Exd =

exergy destruction rate

Exq =

maximum imported exergy rate

Fk =

total force

f =

friction factor

h =

heat transfer coefficient (W/m² K)

k =

thermal conductivity (W/m K)

L =

length (m)

$m˙$ =

mass flow rate (kg/m³)

Ns =

dimensionless number of entropy generation

Nu =

Nusselt number

Qc =

heat flux

Re =

Reynolds number

S_gen,static =

entropy generation for static channel

S_gen,vib+mag =

entropy generation for channel under vibrations and magnet

St =

Stanton number

t =

time (s)

T =

temperature

um =

inlet velocity (m/s)

vm =

mean velocity (m/s)

v_vib =

vibration velocity

Latin Characters

$χm$ =

magnetic susceptibility

$η$ =

second law efficiency

$λ′$ =

ratio of length to hydraulic diameter

$μ$ =

dynamic viscosity (Pa·s)

$ω$ =

angular frequency of vibration (Hz)

$ϕ$ =

volume fraction of nanoparticles

$ρ$ =

density (kg/m³)

Subscripts

b =

bulk

i =

inlet

o =

outlet

static =

in absence of vibration and magnetic field

w =

wall

nf =

nanofluid

vib+mag =

in presence of vibration and magnetic field