Introduction

The traditional heat exchange equipment can no longer meet the current heat exchange intensity, so it is urgent to find new methods to overcome this problem; at present, it is mainly to change the heat transfer equipment structure or seek heat transfer working medium with higher thermal conductivity; for example, water is a traditional heat transfer fluid, which is often used in solar evaporators [1, 2]; however, the available water resources are limited, and it is urgent to find a more perfect heat transfer fluid, so the concept of nanofluids is introduced; the thermal conductivity of nanofluids is relatively high, and its application field is very wide; for example, in the inserted band heat exchanger [3,4,5], fin radiator [6,7,8], solar collector [9,10,11] and microchannel radiator [12,13,14], the addition of nanofluids enhances the efficiency of the heat exchanger, and also improves the service life of the equipment; nanofluids are also used in heat pipes [15], battery thermal management [16], energy storage [17], bionic structures [18,19,20], etc. Not only that some researchers combined nanoparticles with phase change materials [21, 22], which improved the recycling rate of phase change materials and saved costs, but also the appearance of nanofluids greatly increases the service life of heat exchanger, and also provides a certain reference value for the effective utilization of coal resources.

Many scholars have made many contributions to the research field of nanofluids; for example, Fan et al. [23, 24] studied the thermal hydraulic performance of the working medium through experiments, which confirmed that the staggered arrangement of magnetic field makes the heat exchange effect the best. Talebizadehsardari et al. [25] studied the rheological behavior of nanofluids under the interference of magnetic field through experiments. It was found that the addition of magnetic field increases the viscosity of the fluid, and the higher the magnetic field force, the greater the viscosity. Through experiments, it was found that the existence of magnetic field has a huge impact on the behavior or mechanism of nanofluids.

Many scholars have also made outstanding achievements in numerical simulation; for example, Taghipour et al. [26, 27] used dissipative particle dynamics to analyze the behavior of ions in microchannels under magnetic field and obtained the maximum particle density of Ar/O2 as 0.042/0.043, and the particle temperatures of Ar and O2 can reach 390 ℃ and 489 ℃, respectively. Sajjadi et al. [28,29,30] simulated the heat exchange of the working medium in the porous cavity, results presented that the influence of Ra is the greatest, and simulation results confirmed that temperature is positively correlated with HTR. Farzinpour et al. [31] concluded that the magnetic field shortens the aggregation time of nanoparticles through the flow of nanofluids by molecular dynamics, and the mass of nanoparticles, constant magnetic field and time-varying magnetic field are proportional to the temperature of nanofluids. Reddy et al. [32,33,34] identified that when the m% of single-walled carbon nanotubes is 0.05 and the m% of silver nanoparticles is 0.05, the HTR increases from 6.2 to 15.6% and 10.4%, respectively; compared with cobalt oxide, magnetite, manganese zinc ferrite and cobalt ferrite, the HTR is increased by 30%, 18%, 15% and 12%, respectively. Al-Kouz et al. [35] discussed the three-dimensional free convection and TEG of conductive water–Fe3O4/CNT mixed nanofluids under magnetic field force, and the simulation result showed that the Ha and Ra are inversely proportional to the average Be. It was found that both the influence of nanofluids themselves and external factors such as magnetic fields will have different conclusions on the simulation results. Li et al. [36,37,38] simulated the heat transfer behavior of Al2O3/water nanofluids in a model; it was known that the Be is inversely proportional to the Ra, but is proportional to the Ha; it was also noted that with the increase of the Ra, the HTR and TEG are increased by 39% and 90%, respectively. These scholars draw simulation conclusions by changing the internal structure of the model under the action of magnetic force, which enriches the scope of research on nanofluids in the fields of phase change materials or optimizing the structure of heat exchangers.

Alsarraf et al. [39, 40] researched the influence of thermostatic tubes on the working medium of heat exchange under uniform magnetic field and learned that the greater the spacing between thermostatic tubes, the greater the influence of B on the velocity domain compared with the influence of m% of nanoparticles. Li et al. [41,42,43] conducted a numerical study on the radiation of nanofluids and learned that the entropy generation has a great relationship with the aspect ratio of the pipe; when there is no radiation, as Ha increases gradually, the HTR and TEG are reduced by 30% and 25%, respectively; when there is radiation, the corresponding reduction is 29%. Rosca et al. [44, 45] studied the HTR and TEG of mixed nanoheat transfer fluids in a model under magnetic field excitation, and found that when the magnetic field force is small, the average Nu is positively correlated with the m% of nanoparticles, and the Ha is inversely proportional to the variables such as fluid motion intensity, average Nu and average kinetic energy. Mousavi et al. [46,47,48] studied the hydrodynamic behavior of ferromagnetic fluid flow in a wavy channel under non-uniform magnetic field excitation, it can be concluded that compared with the flat wall, the wavy wall structure can enhance the heat transfer rate at the bottom of the wavy channel, and it was also known that the magnetic field enhances the flow and heat transfer of the fluid in the corrugated wall channel compared to the flat wall channel.

Innovation

In the publicly published literature, it has been found that scholars have conducted extensive research on changing the structure of heat exchangers and external environmental factors. Most scholars still start with numerical simulations, and although the conclusions obtained from numerical simulations are objective, they are only approximate solutions. Therefore, they still need to be verified by a large amount of experimental data, and there are relatively few experiments to explore the free thermal convection of nanometer fluid in a sinusoidal curved cavity under magnetic field excitation. To our knowledge, there have been no reports in the existing literature on the heat transfer effects of nanofluids in sinusoidal cavities under horizontal magnetic fields and different vertical magnetic field strengths and directions, and the uniqueness of this paper is mainly to study the free convection heat transfer characteristics of nanofluids in a sinusoidal curved cavity under a magnetic field excitation. There are two innovations: Firstly, a square cavity structure with sinusoidal surface was developed, and secondly, the thermodynamic behavior of heat transfer medium in a new square cavity under horizontal, unilateral vertical, bilateral corresponding vertical and bilateral staggered vertical magnetic fields was studied. Finally, the impact law of entropy generation was summarized.

Novelty and objective

One of the highlights of this study is the use of Fe3O4 nanofluids instead of traditional water with low thermal conductivity as a heat transfer working medium, the choice of this working medium is novel, and because of its stability and ferromagnetism, it is very suitable for application in production sites. Another highlight is the use of a new type of heat transfer cavity and magnetic field arrangement coupling to explore a new set of high-heat-transfer systems, and these two highlights distinguish this study from the previous research results and make it reflect the potential application ability and novelty. The importance of this research work is self-evident; first of all, it can provide some theoretical guidance for the selection of heat exchangers under magnetic field, again, it provides some guidance for design of cavity and the application of magnetic nanofluids in the field of thermal management of electronic components, and finally, the results of this study provide the possibility for developing more efficient thermal management systems in the future. Figure 1 shows the organizational structure of the paper.

Fig. 1
figure 1

Organization of the paper

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Methodology

Experimental materials

The main material selected is Fe3O4 nanoparticle, which is produced and prepared by Shanghai Naiou Nanoparticle Co., Ltd, the particle diameter is about 20 nm, and Fe3O4 nanoparticles have a small particle size, and possess characteristics such as iron magnetism; this is the reason why it is chosen as the working medium; in addition, the dispersion of gum arabic can be uniformly mixed in deionized water and other base fluids; however, if the dispersion dose of gum arabic is added too much, the viscosity of nanofluids will increase to a certain extent. The process of preparing nanofluids includes two different steps; the nanofluids prepared by one-step method is very stable; however, the amount of preparation is relatively small, and the two-step method is simple and the amount of preparation is large, which can meet the needs of industrial production: First, take a certain amount of deionized water as the base solution, and then, pour the gum arabic with m% = 0.8 into deionized water for mixing; finally, Fe3O4 nanoparticles with different m% are added into the mixed solution to prepare Fe3O4 nanofluids with three different m%, and the m% is 0.1, 0.3 and 0.5, respectively; beyond a certain range of m%, the viscosity of the nanofluids increases; research has also found that when the m% of nanofluids exceeds 0.5, the increase in heat transfer enhancement is not significant, making research meaningless. NaOH solution is used to adjust the pH value of the mixed solution, the main effect is that the same charge on the surface of the nanoparticles repels each other and prevents the nanoparticles from collecting, deionized water is used as control group, the preparation process of the nanofluid is shown in Fig. 2, and the instruments required for the experiment are shown in Table 1.

Fig. 2
figure 2

Preparation process

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Table 1 Equipment used in the experiment
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Introduction to system

The core part of this system is the heat exchange chamber, which is a square cavity with a size of 80 mm × 80 mm × 80 mm, the inside surface of the model is a symmetrical sine curved surface, the oscillation amplitude is about 3 mm, the wave number spectrum is 2, the left end of the heat exchange chamber is a heating system, and a silicon heater and a DC power supply are attached outside the copper plate to provide the heat required for the experiment; at the right end is a small cavity which is managed by a low-temperature thermostatic bath to maintain a constant low temperature, five thermocouples are attached to the surface of the hot and cold copper sheet for temperature measurement, the other end of the thermocouple is linked to the data acquisition instrument, and the measured temperature is displayed on the computer screen; to reduce heat loss, the entire chamber is wrapped with insulating cotton to make it in an adiabatic state. Because this research studies the natural heat exchanging transfer effect of nanoscale fluid under magnetic field, electromagnets are arranged around the insulation cotton to provide different B and direction, the horizontal magnetic field is that the electromagnet is arranged on the outside of the hot end of the heat transfer model, and the vertical magnetic field is arranged on the outside of the upper and lower ends of the heat transfer model. Gauss meter (CH-15, error: ± 1G) is applied to survey the B. Figure 3 shows the system diagram.

Fig. 3
figure 3

System diagram. a Experimental device diagram, b Magnetic field layout

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The formula used to process the data

The Q is expressed by Eq. (1):

$$ Q = U \cdot I $$
(1)

where U is the voltage value and I is the current value.

The Qnet is shown by Eq. (2):

$$ Q_{{\text{net}}} = Q_{\text{z}} - Q_{{\text{loss}}} $$
(2)

where Qz is the total heat absorbed and Qloss is the heat lost.

The TH’ is expressed by Eq. (3):

$$ T_{\text{H}}^{\prime} = \frac{(T_1 + T_2 + T_3 + T_4 + T_5 )}{5} $$
(3)

where T1, T2,…, T5 refer to the hot end temperature measured by the thermocouples.

On the basis of the heat conductance theorem, the TH is expressed by Eq. (4):

$$ T_{\text{H}} = T_{\text{H}}^{\prime} - \frac{{Q_{{\text{net}}} \delta }}{{A_{\text{m}} \lambda_{\text{w}} }} $$
(4)

where δ is the cavity thickness determination, Am is cavity cross-sectional area and λw denotes the temperature conductivity of the cavity.

The TC’ is expressed by Eq. (5):

$$ T_{\text{C}}^{\prime} = \frac{{(T_6 + T_7 + T_8 + T_9 + T_{10} )}}{5} $$
(5)

where T6, T7,…, T10 indicate the cold end temperature measured by the thermocouples.

The TC is expressed by Eq. (6):

$$ T_{\text{C}} = T_{\text{C}}^{\prime} - \frac{{2Q_{{\text{net}}} \delta }}{{A_{\text{m}} \lambda_{\text{w}} }} $$
(6)

Td is displayed by Eq. (7):

$$ T_{\text{d}} = \frac{{T_{\text{H}} + T_{\text{C}} }}{2} $$
(7)

The h is used with Eq. (8):

$$ h = \frac{{Q_{{\text{net}}} }}{{A_{\text{m}} (T_{\text{H}} - T_{\text{C}} )}} $$
(8)

Nu is used by Eq. (9):

$$ {\text{Nu}} = \frac{h \cdot W}{{\lambda_{\text{f}} }} $$
(9)

where λf represents the thermal coefficient of the heat exchange medium.

To make the experimental more convincing, ε is introduced, and the corresponding equation is shown in Eq. (10):

$$ \varepsilon = \frac{{{\text{Nu}} - {\text{Nu}}_{({\text{mass\%}} = 0,{\rm{B}} = 0)} }}{{{\text{Nu}}_{({\text{mass\%}} = 0,{\rm{B}} = 0)} }} $$
(10)

The local entropy production is expressed by Eq. (11):

$$ S_{\text{l}} = \frac{{\lambda_{\text{f}} }}{{T_{\text{d}}^2 }}\left[ {{\left( {\frac{\partial T}{{\partial x}}} \right)}^2 + {\left( {\frac{\partial T}{{\partial y}}} \right)}^2 + {\left( {\frac{\partial T}{{\partial z}}} \right)}^2 } \right] + \frac{\mu }{{T_{\text{d}} }}\left[ {2{\left( {\frac{\partial u}{{\partial x}}} \right)}^2 + 2{\left( {\frac{\partial v}{{\partial y}}} \right)}^2 + {\left( {\frac{\partial u}{{\partial y}} + \frac{\partial v}{{\partial x}}} \right)}^2 } \right] $$
(11)

The first term is due to the entropy generation caused by thermal convection, and the second term is due to the entropy generation caused by fluid flow. Because in this experiment, the entropy generation caused by fluid flow relative to heat transfer can be neglected, Eq. (11) can be simplified to:

$$ S_{\text{l}} = \frac{{\lambda_{\text{f}} }}{{T_{\text{d}}^2 }}\left[ {{\left( {\frac{\partial T}{{\partial x}}} \right)}^2 + {\left( {\frac{\partial T}{{\partial y}}} \right)}^2 + {\left( {\frac{\partial T}{{\partial z}}} \right)}^2 } \right] $$
(12)

The expression of total entropy generation is shown in Eq. (13):

$$ S_{\text{T}} = \int_v {S_{\text{l}} } {\text{d}}v $$
(13)

Error calculation

The error calculation of h is expressed by Eq. (14) [49]:

$$ \begin{aligned} \frac{\Delta h}{h} & = \left| {\frac{\partial \ln h}{{\partial Q_{{\text{net}}} }}} \right|\Delta Q_{{\text{net}}} + \left| {\frac{\partial \ln h}{{\partial A}}} \right|\Delta A_{\text{m}} + \left| {\frac{\partial \ln h}{{\partial (T_{\text{H}} - T_{\text{C}} )}}} \right|\Delta (T_{\text{H}} - T_{\text{C}} ) \\ & = \frac{{\Delta Q_{{\text{net}}} }}{{Q_{{\text{net}}} }} + \frac{{\Delta A_{\text{m}} }}{{A_{\text{m}} }} + \frac{{\Delta (T_{\text{H}} - T_{\text{C}} )}}{{(T_{\text{H}} - T_{\text{C}} )}} \\ \end{aligned} $$
(14)

Error of Nu is expressed by Eq. (15) [49]:

$$ \begin{aligned} \frac{{\Delta {\text{Nu}}}}{{{\text{Nu}}}} & = \left| {\frac{{\partial \ln {\text{Nu}}}}{\partial h}} \right|\Delta h + \left| {\frac{{\partial \ln {\text{Nu}}}}{\partial W}} \right|\Delta W + \left| {\frac{{\partial \ln {\text{Nu}}}}{{\partial \lambda_{\text{f}} }}} \right|\Delta \lambda_{\text{f}} \\ & = \frac{\Delta h}{h} + \frac{\Delta W}{W} + \frac{{\Delta \lambda_{\text{f}} }}{{\lambda_{\text{f}} }} \\ \end{aligned} $$
(15)

The error of ST is Eq. (16):

$$ \frac{\Delta S_T }{{S_T }} = \left| {\frac{{\partial \ln S_{\text{T}} }}{{\partial \lambda_{\text{f}} }}} \right|\Delta \lambda_{\text{f}} + \left| {\frac{{\partial \ln S_{\text{T}} }}{{\partial T_{\text{d}} }}} \right|\Delta T_{\text{d}} + \left| {\frac{{\partial \ln S_{\text{T}} }}{\partial u}} \right|\Delta u = \frac{{\Delta \lambda_{\text{f}} }}{{\lambda_{\text{f}} }} + \frac{{\Delta T_{\text{d}} }}{{T_{\text{d}} }} + \frac{\Delta u}{u} $$
(16)

According to Eqs. (14), (15) and (16), the error of h is about 5.65%, the error of Nu is about 6.34%, and the error of ST is about 3.45%; it demonstrates that the system is authentic.

It can be seen from the above sixteen formulas that in order to obtain the corresponding Nu under each working condition, the average temperature of the cold and hot ends is firstly understood, and the temperature difference is further obtained; then, according to the law of heat conduction, the formula of h and the Nu, the Nu under each working condition can be calculated. The new contribution of these programs is to introduce a new variable ε to make the experimental data diagram more clearly present, and the last two formulas are used to verify the error of the experimental results.

System-level verification

To guarantee the reliability of the experimental platform, it is necessary to analyze and calculate the experimental system, compare the data of deionized water measured in this paper with the data in [50] and [51], and find that the maximum error is within 10%, which demonstrates the feasibility of the experimental platform. Figure 4 shows the system verification diagram.

Fig. 4
figure 4

System verification

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Results and discussion

Horizontal magnetic field

The effect of m%

This section studies the dynamics of Fe3O4–H2O nanofluids in a square model under the excitation of a parallel magnetic field. In Fig. 5, the Nu changes of heat exchange working fluids under different Q are studied. With the increase of Q, the Nu also increases; the cold side of the heat exchange chamber maintains a constant low-temperature environment; as the power of the hot end increases, the hot side temperature also increases; the temperature difference at both ends increases, and the driving force becomes stronger; the Nu will raise accordingly. Compared with water, when B from 0.0 to 0.04 T, the Nu is enhanced by 4.24%, 3.42%, 2.83%, 2.16% and 1.57% at most; as the m% continues to increase, the corresponding viscosity will also increase. In this study, the m% is 0.0, 0.1, 0.3 and 0.5, respectively; through the previous research results, it is found that when the m% exceeds 0.5, the increase of Nu is very small; because the m% always increases, the viscosity becomes larger, which will lead to the increase in the flow resistance and a negative effect on the heat transfer enhancement.

Fig. 5
figure 5

Effect of Q on Nu

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In Fig. 6, the B and Nu show a negative effect; the reason is that the nanofluids are subject to the horizontal right magnetic field force in addition to its own gravity, which shows that the resultant force direction of nanofluids is downward to the right, which is not conducive to thermal convection; therefore, with the increase of B, the Nu has a downward trend. Under the action of horizontal magnetic field, when m% is 0.1, 0.3, 0.5, Nu is decreased by 2.1%, 2.18%, 2.57% at most. Since deionized water does not contain ferromagnetic particles, it is also seen from the figure that no matter how the B changes, the Nu of the working medium with m% = 0.0 is constant at the same Q, this is a group of control groups, although the horizontal magnetic field has a weakening effect on the results of the experimental model, the obtained Nu is larger than that of the control group, the reason is that although the magnetic field weakens the heat transfer, as long as there is a magnetic field, it will more or less interfere with the flow trend of the working medium in the model, and this is why the Nu of the control group is relatively low.

Fig. 6
figure 6

Effect of B on Nu

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To explore the specific effect of nanofluids in the cavity, the ε is introduced to represent the results of the experimental comparison, and ε represents the ratio of the Nu obtained under various working conditions to the Nu of water, which is shown in Fig. 7. According to the previous analysis, the horizontal magnetic field has a weakening effect; as the B increases, the value of heat transfer enhancement decreases, indicating that the greater the horizontal magnetic field strength, the smaller the ε, the worse the heat transfer effect; the figure also shows that the m% = 0.5 is larger than that of m% = 0.1 at the same Q, the reason may be that the m% of the working medium is large within the reasonable research range, and the corresponding magnetic field also has a large interference on it, so the corresponding heat transfer is slightly larger.

Fig. 7
figure 7

Effect of B on ε

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The impact of B

This section mainly describes the corresponding relationship between B and Nu. Figure 8 indicates that the horizontal magnetic field has a certain weakening effect; compared with the reported literature [52], it is found that the horizontal magnetic field is more likely to cause the agglomeration of nanoparticles, which will increase the thermal resistance of heat transfer and weaken the heat transfer. Compared to deionized water, when m% is 0.1, the maximum Nu can be increased by 2.68%, 1.99%, 1.56%, 1.07% and 0.52% when the B from 0.0 to 0.04 T, respectively, when m% is 0.3, the maximum Nu can be increased by 3.21%, 2.74%, 2.16%, 1.57% and 0.98%, and when m% is 0.5, the maximum Nu can be increased by 4.24%, 3.42%, 2.83%, 2.16% and 1.57%. The figure shows that at the same m%, by increasing the B, it is found that the increase of the Nu is decreasing, which also confirms the weakening effect of the horizontal magnetic field.

Fig. 8
figure 8

Effect of Q on Nu

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Influence of vertical magnetic field

Influence of m%

Nanofluids under the excitation of horizontal magnetic field have a certain weakening effect, and B or magnetic field layout type in different directions will affect the heat transfer effect [53]. In order to verify the authenticity of the experimental results, considering the limitation of journal length, the results under the bilateral staggered magnetic field and m% = 0.5% are compared with the experimental data in the published literature [53], as shown in Table 2; the maximum error is 4.64%, and this also confirms the rationality of the experimental data. Therefore, this section mainly studies the heat transfer effect of nanofluids under the unilateral perpendicular magnetic field, the double-sided corresponding vertical magnetic field and the double lateral interlace vertical magnetic field. Figure 9 shows the heat transfer effect of nanofluids under a certain B. Regardless of the type of magnetic field layout, under a certain B, when Q raises, the Nu will increase. Compared with deionized water, under unilateral vertical magnetic field, when B is from 0.0 to 0.04 T, Nu is increased by 4.24%, 4.47%, 4.65%, 4.93% and 5.12% at most, the Nu under the bilateral vertical corresponding magnetic field is increased by 4.24%, 4.59%, 4.82%, 5.03% and 5.26% at most, and the Nu under the bilateral vertical staggered magnetic field is increased by 4.24%, 4.65%, 4.92%, 5.1% and 5.37% at most. Under the same Q, m% = 0.5 shows better heat transfer effect; as the m% increases, the corresponding viscosity will also increase, so the m% cannot be increased indefinitely.

Table 2 Comparative analysis of Nu under bilateral staggered magnetic field (m% = 0.5)
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Fig. 9
figure 9

Relationship between Q and Nu. a Unilateral, b bilateral correspondence, c bilateral staggered

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Figure 10 shows that the Nu of different heat transfer fluids changes with the increase of B at a certain Q, and it is found that the Nu increases with the increase of B at any Q. The possible reasons are as follows: In a unilateral vertical magnetic field, the nanoparticles migrate to the lower end of the square cavity due to the action of the magnetic field force, resulting in a relatively large m% at the inferior of the square cavity, and the local thermal coefficient of the corresponding nanofluids will also increase, so the overall heat transfer effect of the entire square cavity has been improved. Under the corresponding magnetic field strength on both sides, the nanoparticles are affected by the upper and lower magnetic forces at the same time, and it is shown that the chain structure formed by more nanoparticles on the upper–lower inner surfaces of the square model increases the local convection heat transfer coefficient of the upper and lower end faces, so the overall heat transfer effect of the heat exchange cavity is better. In the double-sided staggered magnetic field, compared with the corresponding magnetic field on both sides, it is equivalent to extending the chain structure; at the same time, the nanoparticles are subject to the magnetic force in the upper and lower directions, which makes the nanofluids mix more evenly in the square cavity, and the effect of heat transfer enhancement is more prominent.

Fig. 10
figure 10

Relationship between B and Nu. a Unilateral, b bilateral correspondence, c bilateral staggered

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In Fig. 11, at a certain Q, with the increase of B, no matter what the m% is, ε increases, which also proves that the vertical magnetic field has a promoting effect; for example, when Q is 10 W and m% is 0.3, the ε values corresponding to B in the range of 0.01 to 0.04 T are 0.02817, 0.03037, 0.03286 and 0.03551, respectively; it is clarified that with the raise of vertical B, ε becomes larger and larger, and the corresponding heat transfer effect becomes better and better.

Fig. 11
figure 11

Relationship between B and ε. a Unilateral, b bilateral correspondence, c bilateral staggered

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The influence of B

In Fig. 12, the influence of B on Nu at a specific m% of nanofluids is shown; from the perspective of three different magnetic field arrangements, no matter which arrangement, Nu will be reinforced with the improve of B, the enhancement of B is equivalent to exerting greater magnetic force on the nanoparticles, and the resultant force of the nanofluids will be greater, so the disturbance in the cavity will be enhanced, and the heat transfer effect will be better. Compared with water, when the m% is from 0.1 to 0.5, the maximum Nu is increased by 3.63%, 4.07% and 5.12%, respectively, in a unilateral vertical magnetic field, and increased by 3.8%, 4.23% and 5.26%, respectively, in a bilateral vertical corresponding magnetic field, and increased by 3.88%, 4.3% and 5.37%, respectively, in a bilateral vertical staggered magnetic field.

Fig. 12
figure 12

Relationship between Q and Nu. a Unilateral, b bilateral correspondence, c bilateral staggered

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Total entropy production

This section mainly studies the influence of magnetic field direction and strength on entropy generation at Q = 25 W. As shown in Fig. 13, at a specific concentration of nanofluid, when the magnetic field direction is constant, entropy generation increases with the increase in the magnetic field strength. The possible reasons are as follows: In the previous section, we have concluded that a vertical magnetic field can enhance heat transfer; with the increase of B, the force acting on nanoparticles increases, and the motion trajectory of nanoparticles becomes from ordered to disordered, resulting in increased heat transfer and entropy generation. When at a specific B, the staggered magnetic field exhibits higher entropy generation compared to the other two magnetic fields; the reason for this is that the staggered magnetic field, on the one hand, makes the molecules of nanoparticles more disordered to each other, the chainlike structure becomes longer and the turbulence intensity is higher. On the other hand, the staggered magnetic field leads to a greater increase in temperature and a more pronounced heat transfer effect, resulting in higher entropy generation.

Fig. 13
figure 13

Influence of magnetic field direction and strength on entropy generation. a m% = 0.1%, b m% = 0.3%, c m% = 0.5%

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Conclusions

This article innovatively proposes a sinusoidal cavity heat transfer model with an amplitude of 3 mm and two wave numbers, and studies the free convection heat transfer characteristics of nanofluids in the heat transfer model under magnetic field excitation, and the conclusions are as follows:

  1. (1)

    Horizontal magnetic force weakens the convective heat transfer; when m% is 0.1, 0.3, 0.5, the Nu is decreased by 2.1%, 2.18% and 2.57% at most.

  2. (2)

    When the B is from 0.0 to 0.04 T, compared with water, when m% is 0.1, the Nu can be increased by 2.68%, when m% is 0.3, the Nu can be increased by 3.21%, and when m% is 0.5, the Nu can be increased by 4.24%.

  3. (3)

    Compared with water, in the unilateral vertical magnetic field, when B is from 0.0 to 0.04 T, the Nu is increased by 5.12% at most, the Nu in a bilateral vertical corresponding magnetic field is increased by up to 5.26%, and in a bilateral vertical staggered magnetic field, the Nu is increased by up to 5.37%.

  4. (4)

    For the four magnetic field arrangements studied, the double staggered magnetic field has better heat transfer effect; within the scope of this study, the greater the B, the higher the Nu.

  5. (5)

    Under a vertical magnetic field, a double-sided alternating magnetic field has greater entropy generation. As the B increases, entropy generation increases, and as the m% of nanofluids increases, entropy generation also increases.

  6. (6)

    Limitation: The model in this paper is a symmetrical sine curved surface, the oscillation amplitude is about 3 mm, and the wave number spectrum is 2. However, due to the limitation of the length of this paper, surfaces with more structural parameters have not been studied. These contents will be studied in the following work.