Long-distance heat transport based on temperature-dependent magnetization of magnetic fluids

Abstract

Mn-Zn ferrite-based magnetic fluid is temperature-sensitive. Because of the low Curie temperature of Mn-Zn ferrite particles, magnetization is dramatically attenuated as the temperature rises. This temperature-dependent magnetization can be used to transport heat by magnetically driving a fluid via the application of a non-uniform magnetic field and heat. A potential application of this scheme is long-distance heat transport without the use of mechanical pumps. In the present work, a miniaturized magnetically-driven heat transport device with an inner diameter of 𝜙1.54 mm and a flow path length of 1,500 to 6,000 mm was fabricated. The influence of flow path length on the flow rate induced by the magnetic body force was investigated experimentally and theoretically. Although the flow rate decreased with increasing flow path length due to friction loss, the driving pressure increased with rising heater temperature because of the temperature-dependent magnetization. The influence of flow path length on the driving pressure is relatively insignificant. An increase in the flow path length leads to effects that both attenuate and enhance the driving pressure. Substantial long-distance (6,000 mm) heat transport was realized by applying a non-uniform magnetic field and heat.

Keywords

Magnetic nanofluid heat transfer magnetic fluid temperature-dependent magnetization energy conversion

1. Introduction

Nanofluids have received widespread interest because their exceptional properties can enhance heat transfer [1–3]. Magnetic fluids are nanofluids in which magnetic nanoparticles are stably dispersed in a carrier liquid such as water or a hydrocarbon liquid. The size of the magnetic particles is generally 3 to 12 nm. Such particles have a single magnetic domain, resulting in superparamagnetism. To mitigate the aggregation of particles due to magnetic dipole interactions, surfactants are adsorbed on the particle surfaces. A magnetic fluid responds strongly to magnetic fields. Rheological properties such as viscoelasticity only slightly change under a magnetic field. A magnetic fluid thus retains its liquid-like behavior in the presence of a magnetic field. Heat transfer can be regulated and enhanced by applying magnetic fields [4–6].

For the application of a magnetic fluid for heat transfer, energy conversion using a temperature-sensitive magnetic fluid (TSMF) has received interest. Magnetization drastically attenuates as the temperature rises because the Curie temperature of magnetic particles such as Mn-Zn ferrite particles, dispersed in a TSMF is low. This temperature-dependent magnetization can be applied to energy conversion from thermal energy to fluid kinetic energy. When a non-uniform magnetic field and heat are applied to a TSMF, the magnetic force acts on the fluid, and the fluid transports the heat (the details of the operation principle are described in Section 2).

This heat transport concept was originally proposed by Resler and Rosensweig in 1964 for a new thermal engine (power cycle) [7]. In this cycle, the refrigerant TSMF is magnetically driven and transports heat from a heat source to a radiator. Resler and Rosensweig also proposed an energy recovery system using magnetohydrodynamic power generation. Adding electrical conductivity to a TSMF enables energy conversion from thermal energy to fluid kinetic energy, which is then converted to electrical energy by the magnetohydrodynamic power generator [8]. After the original report on energy conversion, several experimental and theoretical studies have been conducted [9–26]. To enhance the magnetic driving force and heat transfer, the use of boiling gas-liquid two-phase flow has been proposed. Kamiyama et al. [9,10] heated a TSMF using a laser and investigated the effect of boiling gas-liquid two-phase flow on the driving force. They visualized the distribution of boiling bubbles in the presence of a non-uniform magnetic field and found that the bubbles efficiently enhance the driving force. The authors proposed a binary TSMF in which a low-boiling-point organic solution was mixed into a TSMF and realized effective energy conversion using boiling gas-bubble two-phase flow [12–17]. The boiling point can be lowered by adding a low-boiling-point organic solution such as n-hexane to the TSMF. We previously investigated the effect of boiling gas-liquid two-phase flow on heat transfer [11–13,16] and driving force [14–16]. Miniaturized magnetically driven heat transport devices based on TSMFs have been reported [15,16,18,19] for cooling small electrical devices with significantly high heat flux, such as large-scale integrated circuits and central processing units. Fumoto et al. [18,19] designed such a device and confirmed its stable operation and heat transport using TSMFs. They also visualized the flow behavior of the TSMFs in the presence of a non-uniform magnetic field and found that the flow is accelerated near the heating wall due to temperature-dependent magnetization. The authors [15,16] also fabricated a miniaturized magnetically driven heat transport device that uses binary TSMFs and tested its operation under boiling gas-bubble two-phase flow. They found that optimization of the magnetic field distribution is necessary for achieving stable fluid mobility.

Although many reports related to energy conversion using a TSMF have been published, the maximum heat transport distance is unknown. In the present study, we designed a miniaturized magnetically driven heat transport device with a flow tube diameter of 𝜙1.54 mm. The length of the tube was varied from 1,500 to 6,000 mm. The effect of flow path length on the flow rate induced by the magnetic body force was investigated experimentally and theoretically.

2. Magnetic driving force based on temperature-dependent magnetization of magnetic fluid

When a TSMF is subjected to a non-uniform magnetic field, the magnetic body force F _m, given by Eq. ((1)), acts on the fluid per unit volume. $\begin{eqnarray}\displaystyle \boldsymbol{F}_{m}={\mu}_{0}\boldsymbol{M}\cdot {\nabla}\boldsymbol{H} & & \displaystyle\end{eqnarray}$ (1) where μ₀ is the magnetic permeability, M is the magnetization, and H is the magnetic field. Because the magnetic fluid exhibits superparamagnetism, magnetization M can be considered to be parallel to applied magnetic field H , and expressed as $\begin{eqnarray} \displaystyle \boldsymbol{M}={\chi}_{T}K_{H}\boldsymbol{H}, & & \displaystyle \end{eqnarray}$ (2) where 𝜒_T is the magnetic susceptibility and K _H is the thermomagnetic constant. To describe the operation principle, Fig. 1 shows a driving part (heating part) to which a solenoidal magnetic field is applied. Although the magnetic field distribution of permanent magnets is more complicated according to Maxwell’s law 𝛻 × B = 0, the solenoidal field is convenient for explaining the operation principle. As shown in Fig. 1, a straight flow tube is filled with the TSMF. The downstream side is heated with respect to the center of the magnet (x > 0). The TSMF absorbs the heat and its temperature rises, resulting in an attenuation of its magnetization. This decrease in magnetization induces a difference in the magnetic body force strengths in the driving part; a positive magnetic body force acts on the TSMF at x < 0 and a negative magnetic body force acts at x > 0. The intensity of the negative magnetic body force is smaller than that of the positive one due to the temperature-dependent magnetization. This magnetic body force difference is the driving force that acts on the working fluid.

Fig. 1.

Diagram of operation principle for magnetically driven heat transport device based on temperature-sensitive magnetic fluid.

3. Experiments

3.1. Test fluid

The TSMF employed in the present study was an Mn-Zn ferrite-based magnetic fluid (TS-50K, Ichinen Chemicals Co., Ltd.). Magnetic particles with an average diameter of 9.7 × 10⁻⁹ m were dispersed in the carrier liquid (kerosene) at a weight fraction of 49.86 wt.%. The thermophysical and magnetic properties of the TSMF at 25 °C and 1 atm are listed in Table 1.

3.2. Experimental apparatus

The effect of flow path length on flow rate was investigated using our self-designed magnetically driven heat transport device. The experimental setup is shown in Fig. 2. The apparatus mainly consisted of a driving part (heating part), a flow rate measurement part, and a cooling unit. The parts were connected with Teflon tubes with an inner diameter of 𝜙1.54 mm to construct a closed loop. Thermal insulators covered the tubes to mitigate heat loss to the atmosphere. The total length of the flow path was changed by varying the length of the Teflon tube between A and A’. The whole system was horizontally placed and tested.

Table 1
Thermophysical and magnetic properties of temperature-sensitive magnetic fluid (TS-50K, Ichinen Chemicals Co., Ltd.) used in this study at 25 °C and 1 atm (data from manufacturer)

Density [kg/m³] 1.37 × 10³

Viscosity [Pa ⋅ s] 1.23 × 10⁻²

Specific heat [J/(kg ⋅ K)] 1387

Thermal conductivity [W/(m ⋅ K)] 0.175

Thermal expansion [1/K] 6.90 × 10⁻⁴

Curie temperature [K] 528

Saturated temperature [K] 377--523

Saturated magnetization [mT] 41.1

Mass fraction of magnetic particle [wt.%] 49.86

Density of magnetic particle [kg/m³] 5.00 × 10³

Averaged diameter of magnetic particle [m] 9.7 × 10⁻⁹

Number of magnetic particle [1/m³] 5.45 × 10²³

Fig. 2.

Schematic diagram of experimental setup.

Figure 3 shows the details of the driving part (heating part), where the TSMF was heated to obtain the magnetic body force. The driving part was mainly composed of permanent magnets, a heated pipe and a heater. These components were placed inside a vacuum chamber to mitigate heat loss to the atmosphere. The heated pipe was made of alumina with an inner diameter of 𝜙1.6 mm and a length of 40 mm. The heater was a polyimide-heater (Showa Manufacturing Co., Ltd.), in which a thin stainless steel film with a thickness of 30 μm was sandwiched between polyimide sheets. The heater was flexible and installed so as to cover the heated pipe. DC current was directly supplied to the heater by a DC power supply. The permanent magnets were placed so as to sandwich the heated pipe to apply a non-uniform magnetic field. Figure 4 shows the magnetic field distribution. The magnetic field strength was varied by adjusting the distance between the two permanent magnets. The magnet position c (relative position between the centers of the magnets and heated pipe) was changed from 0 to 30 mm to investigate the influence of magnetic field distribution on the flow rate. Multiple measurements were performed under similar conditions using different fresh samples to take the measurement error into account. The experimental uncertainty is indicated in the experimental results.

Fig. 3.

Schematic diagram of driving part (heating part).

Fig. 4.

Distribution of magnetic field. The magnetic field strength was regulated by changing the distance between two permanent magnets.

4. One dimensional analysis of magnetic body force (pressure) considering temperature-dependent magnetization

To analytically investigate the influence of magnetic field strength, heat flux and flow path length on the driving force (flow rate), a model of magnetic driving pressure that considers temperature-dependent magnetization was derived. When the TSMF is exposed to a non-uniform magnetic field, the magnetic body force F _m acts on the fluid, as given by Eq. (1). The magnetic moments of the magnetic nanoparticles immediately orient in the magnetic field direction. This rotational relaxation time is on the order of ∼10⁻⁷ s. Because of this extremely short relaxation time, the magnetization of the magnetic fluid shows superparamagnetism, and assuming M ∥ H , it can be expressed using the Langevin function [4]. $\begin{eqnarray}\displaystyle M=M_{S}\left(\cosh \frac{mH}{k_{B}T}-\frac{mH}{k_{B}T}\right) & & \displaystyle\end{eqnarray}$ (3) where M _S is the saturated magnetization, m is the magnetic moment, and k _B is the Boltzmann constant. If the magnetization M linearly depends on the temperature T, the thermomagnetic constant K _H can be given as $\begin{eqnarray}\displaystyle K_{H}=1-\left(\frac{T-T_{ref}}{T_{C}-T_{ref}}\right) & & \displaystyle\end{eqnarray}$ (4) where T _C and T _ref are the Curie and reference temperatures, respectively. To simplify the analysis, the one-dimensional magnetic driving pressure Δp _m in the flow direction is given by integrating the magnetic body force F _{m, x} within the range of the magnetic field, as shown below. $\begin{eqnarray}\displaystyle {\rm\Delta}p_{m} & = & \displaystyle \int F_{m,x}dx\nonumber\\ \displaystyle & = & \displaystyle {\mu}_{0}\int M_{x}\frac{\partial H_{x}}{\partial x}dx.\end{eqnarray}$ (5) To analyze the flow rate Q at various flow path lengths, the energy equation in the experimental system is expressed as $\begin{eqnarray}\displaystyle \frac{{\rm\Delta}p_{m}}{{\rho}g}=h_{f} & & \displaystyle\end{eqnarray}$ (6) where 𝜌 is the density, g is the gravitational acceleration, and h _f is the friction loss. Taking the circular pipe and laminar flow into account, the loss can be expressed using the Darcy-Weisbach equation. $\begin{eqnarray}\displaystyle h_{f}=\frac{32{\mu}LV}{D^{2}{\rho}g} & & \displaystyle\end{eqnarray}$ (7) where μ is the viscosity, L is the flow path length, V is the velocity, and D is the tube diameter. Substituting Eq. (7) into Eq. (6), yields the flow rate Q as $\begin{eqnarray}\displaystyle Q=\frac{AD^{2}}{32{\mu}L}{\rm\Delta}p_{m} & & \displaystyle\end{eqnarray}$ (8) where A is the flow path cross-sectional area.

In the present analysis, the thermophysical and magnetic parameters listed in Table 1 were used. The temperature upstream of the heating pipe was set to 295.4 K. After heating, the fluid temperature was assumed to be the temperature of the heater measured directly by a thermocouple.

5. Results and discussion

5.1. Flow rate variation with time and effect of magnet position

When a non-uniform magnetic field is applied to the TSMF, the magnetic body force distribution should be symmetric across the center of the magnet to follow Maxwell’s law 𝛻 × B = 0. When the TSMF is heated, its temperature gradually rises and the symmetry of the magnetic body force distribution is broken, causing the TSMF to transport heat. Figure 5 shows the typical flow rate variation with time for heat flux q = 5.0 kW∕m², magnetic field strength H _{x, max} = 136 mT, magnet position c = 20 mm, and flow path length L = 1,500 mm. The TSMF was heated from t = 0 s. The TSMF began to move in response to heating immediately. The flow rate rapidly increased with time, reaching a steady state after 200 s. A steady state is reached because the magnetization of the TSMF decayed due to the increase in temperature, resulting in induced magnetic pressure (Eq. (5)) being the driving pressure for the TSMF. This result means that the TSMF can transport heat without any mechanical pumps; that is, placing permanent magnets and supplying heat can drive the TSMF and transport heat.

Fig. 5.

Variation of measured flow rate with time for heat flux q = 5.0 kW∕m², magnetic field strength H _{x, max} = 136 mT, magnet position c = 20 mm, and flow pass length L = 1,500 mm.

Because the magnetic driving pressure depends on a local attenuation of magnetization due to an increase in temperature, the magnet position with respect to the heating area (heated pipe) is a crucial parameter for its optimization. Figure 6 shows the flow rate at various magnet positions with respect to the heated pipe c (see Fig. 3). For magnet position c = 0 mm, the centers of the magnets and the heated pipe are coincident. For magnet position c = 20 mm, the TSMF is heated from the center of the magnets. As shown in Fig. 6, the flow rate strongly depends on the magnet position; the flow rate was zero for c = 0 mm and maximum for c = 20 mm. A magnetic driving pressure cannot be obtained for c = 0 mm because the magnetic body force distribution should be symmetric across the center of the magnet. Even under heating, the symmetry does not collapse because the centers of the magnets and the heated pipe are coincident.

Fig. 6.

Measured flow rate at various heat fluxes and magnet positions. The magnetic field strength is 136 mT and the flow path length is 1,500 mm.

The maximum flow rate was obtained for c = 20 mm because at this position the symmetry collapses the most due to the temperature-dependent magnetization.

In the following sections, the effects of flow path length, heat flux, and magnetic field strength on the magnetic driving pressure (flow rate) are discussed in detail for magnet position c = 20 mm.

5.2. Effect of flow path length

For the present heat transport device, the maximum heat transport distance is of interest. To determine this, the flow rate was investigated by varying the flow path length from 1,500 to 6,000 mm. Figure 7 shows the flow rate for various flow path lengths and heat flux q = 5.0 kW∕m², magnetic field strength H _{x, max} = 136 mT, and magnet position c = 20 mm. The dotted line is the result of the one-dimensional analysis described in Section 4. The flow rate decreased with flow path length due to friction loss. As the flow rate decreased, the temperature of the heater rose because the heat transfer was attenuated. This temperature rise increased the driving pressure due to the temperature-dependent magnetization. Therefore, an increase in flow path length led to effect that both attenuated and increased the driving pressure.

Fig. 7.

Flow rates obtained from experiment and one-dimensional analysis and temperatures at heater for various flow path lengths. The magnetic field strength is 136 mT, the heat flux is 5.0 kW∕m², and the magnet position is 20 mm.

Fig. 8.

Magnetic body force distribution for various flow path length. The magnetic field intensity is 136 mT, the heat flux is 5.0 kW∕m², and the magnet position is 20 mm.

Fig. 9.

Magnetic driving pressure distribution for various flow path lengths. The magnetic field intensity is 136 mT, the heat flux is 5.0 kW∕m², and the magnet position is 20 mm.

To clearly explain the driving pressure generation, the one-dimensional analysis described in Section 4 was carried out. As shown in Fig. 7, the analytical results qualitatively agree with the experimental results. The quantitative difference may have resulted from the one-dimensional assumption. To accurately analyze real thermal flow, a three-dimensional simulation is required. Figure 8 shows the distribution of the magnetic body force F _{m, x} acting on the TSMF. The magnetic body force is the inner product of the magnetization and the gradient of the magnetic field (Eq. (1)). In the magnetic field distribution (Fig. 4), the magnetic poles are inverted, and one peak and two valleys appear. This complex magnetic field distribution induces a complex magnetic body force distribution, with three peaks and three valleys, as shown in Fig. 8. Therefore, along the x-axis, the magnetic body force is reversed several times. Because the fluid is heated from the center of the magnet (x = 0 mm), the magnetic body force strength at x > 0 decreases due to the temperature-dependent magnetization. Because the temperature rises with flow path length, the magnetic body force at x > 0 further decreases with the flow path length. The magnetic driving pressure Δp _m can be expressed as the integral of the magnetic body force (Eq. (5)). Figure 9 shows the magnetic pressure distribution, given by $\begin{eqnarray}\displaystyle {\rm\Delta}p_{m}=\int _{x=-30∼\text{mm}}^{x}F_{m,x}dx. & & \displaystyle\end{eqnarray}$ (9) In the above equation, the net magnetic driving pressure Δp _m is obtained for x = 30 mm. Because the intensity of the magnetic body force at x > 0 decreases, the magnetic driving pressure increases with flow tube length, even though a longer flow path leads to larger friction loss. Consequently, in the present work, exceptional long-distance (6,000 mm) heat transfer without any mechanical pumps was achieved with magnetic field strength H _x = 136 mT and heat flux q = 5.0 kW∕m² was achieved.

Fig. 10.

Magnetic body force distribution for various heat fluxes. The magnetic field intensity is 136 mT, the magnet position is 20 mm and the flow path length is 1,500 mm.

Fig. 11.

Magnetic driving pressure distribution for various heat fluxes. The magnetic field intensity is 136 mT, the magnet position is 20 mm and the flow path length is 1,500 mm.

5.3. Effect of heat flux

As shown in Fig. 6, the flow rate increased with heat input. The magnetic body force and the magnetic pressure (Eq. (9)) for heat flux q = 0.0, 3.0, and 5.0 kW∕m² are shown in Fig. 10 and 11, respectively. Without heating (q = 0.0 kW∕m²), the distribution of the magnetic body force is symmetric across the center of the magnet (x = 0 mm) (see Fig. 10), resulting in no magnetic driving pressure (Δp _m = 0 Pa) at x = 30 mm (see Fig. 11). With increasing heat flux, the intensity of the magnetic body force at x > 0 decreases due to the temperature-dependent magnetization (Fig. 10), resulting in increased magnetic driving pressure.

5.4. Effect of magnetic field strength

The TSMF is driven by the magnetic body force. It is thus essential to investigate the effect of the magnetic field strength on the TSMF’s mobility. Figure 12 shows the flow rate and the heater temperature for magnetic field strength H _{x, max} = 96, 108 and 136 mT and heat flux q = 5.0 kW∕m², magnet position c = 20 mm, and flow tube length L = 1,500 mm. The dotted line is the results of the one-dimensional analysis at the same conditions. With increasing magnetic field strength, the flow rate increased and the temperature dropped. This means that applying a magnetic field effectively drives the TSMF, which acts as a refrigerant. The analytical results qualitatively agree with the experimental results. Figures 13 and 14 show the analytical results of the distribution of the magnetic body force and the magnetic pressure, respectively. In Fig. 13, the magnetic body force at x > 0 decreases because of the temperature-dependent magnetization. The intensity of the magnetic body force increases with increasing magnetic field strength. As shown in Fig. 14, the net magnetic driving pressure Δp _m at x = 30 mm dramatically increases.

Fig. 12.

Flow rates obtained from experiment and one-dimensional analysis and temperatures at heater for various magnetic field strength. The magnet position is 20 mm, the heat flux is 5.0 kW∕m², and the flow path length is 1,500 mm.

Fig. 13.

Magnetic body force distribution for various magnetic field strength. The magnet position is 20 mm, the heat flux is 5.0 kW∕m², and the flow path length is 1,500 mm.

Fig. 14.

Magnetic driving pressure distribution for various magnetic field strength. The magnet position is 20 mm, the heat flux is 5.0 kW∕m², and the flow path length is 1,500 mm.

A comparison of the effects of flow path length, heat flux, and magnetic field strength on the magnetic driving pressure (see Figs 9, 11 and 14) indicates that the heat flux and the magnetic field strength significantly affect the magnetic driving pressure, but the flow path length does not. In the present study, long-distance (6,000 mm) heat transport based on the temperature-dependent magnetization of a TSMF was realized.

6. Conclusions

The magnetization of a TSMF strongly depends on the magnetic field and the fluid temperature. By utilizing a TSMF as a refrigerant, heat can be magnetically transported without any mechanical pumps. In the present study, a miniaturized magnetically driven heat transport device that uses a TSMF was fabricated and the influence of flow path length on the fluid flow rate was experimentally and theoretically investigated. Long-distance (6,000 mm) heat transport was achieved with an inner tube diameter of 𝜙1.54 mm, a magnetic field strength of 136 mT, and a heat flux of 5.0 kW∕m². Moreover, the effects of flow path length, heat flux, and magnetic field strength on the magnetic driving pressure were investigated using a one-dimensional analysis that considered the magnetic body force. The results showed that heat flux and the magnetic field strength significantly influence the magnetic driving pressure because the temperature-dependent magnetization effectively breaks the symmetry of the magnetic body force distribution. Although the flow rate decreased with increasing flow path length due to friction loss, the driving pressure increased with rising heater temperature because of the temperature-dependent magnetization. The influence of flow path length on the driving pressure is relatively insignificant. An increase in the flow path length leads to effects that both attenuate and enhance the driving pressure.

Footnotes

Acknowledgements

The work was supported by a Grant-in-Aid for Scientific Research (C) from the Ministry of Education, Culture, Sports, Science and Technology (grant no. 26420126), Japan.

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Density [kg/m³]	1.37 × 10³
Viscosity [Pa ⋅ s]	1.23 × 10⁻²
Specific heat [J/(kg ⋅ K)]	1387
Thermal conductivity [W/(m ⋅ K)]	0.175
Thermal expansion [1/K]	6.90 × 10⁻⁴
Curie temperature [K]	528
Saturated temperature [K]	377--523
Saturated magnetization [mT]	41.1
Mass fraction of magnetic particle [wt.%]	49.86
Density of magnetic particle [kg/m³]	5.00 × 10³
Averaged diameter of magnetic particle [m]	9.7 × 10⁻⁹
Number of magnetic particle [1/m³]	5.45 × 10²³