Heat transfer of temperature-sensitive magnetic fluids around single heating pipe

1. Introduction

Magnetic fluids are a particle suspension in which nm-sized ferromagnetic particles are dispersed in a solvent such as water or hydrocarbon. Because the Brownian motion of the particles is dominant, the magnetic particles are stably dispersed in the solvent [1]. The magnetic fluid can be regarded macroscopically as a fluid that responses to a magnetic field. The magnetic fluid dealt with in this study is a temperature-sensitive magnetic fluid (TSMF). TSMF is composed of magnetic particles with low Curie temperature, such as Mn-Zn ferrite, and has strong temperature-dependent magnetization. When the fluid temperature rises even at room temperature, the magnetization of TSMF decays dramatically. This temperature-sensitive magnetization enables energy conversion from thermal energy to fluid kinetic energy [2,3]. When a non-uniform magnetic field is applied to TSMF, magnetic body force, which depends on magnetization and magnetic field gradient, acts on the fluid. By partially heating TSMF, TSMF self-drives because the balance of the magnetic body force is broken due to the temperature-dependent magnetization. Consequently, TSMF is self-driven by the magnetic body force, and simultaneously absorbs and transports heat only by applying a magnetic field and heat. The advantage of this heat transport system is that no mechanical pump is required to drive the refrigerant [4]. Since the structure is simple, it is easy to miniaturize the system [5]. The most interesting feature is a long-distance heat transport [6]. Taking these advantages, the heat transport system is expected to be applied to automotive thermal management systems, solar cells and satellite cooling systems.

The authors have conducted researches related to magnetically-driven heat transport devices using TSMF [4–9]. For long-distance heat transport, the long-distance heat transport of 6,000 mm was successfully achieved with a channel diameter of 1.54 mm. However, the heat input to this system is limited up to 5.0 kW/m² [6]. In order to enhance the heat transport capability, it is effective to drastically expand the heat exchange surface area using a porous material or a heat sink. However, the temperature field becomes complicated on such a complicated heat exchange surface. Especially in TSMF, the thermal flow inside of such heat exchangers becomes more complicated due to the implication between the complicated temperature field and temperature-sensitive magnetization. Fumoto et al., visualized the thermal flow of TSMF between heating parallel plates. When a magnetic field is applied to TSMF, the magnetic body force acts on the fluid. Because the magnitude of body force depends on the temperature of the fluid, the flow velocity increases near the heating plate. This thermal flow behavior is clearly different from that of a Newtonian fluid [10,11]. Although many studies on the thermal flow of TSMF in single heating surface such as a heating circular tube [4–10,12,14,15] or a rectangular tube [11, 13], there is no research on the thermal flow inside heat exchangers with complex geometry. Therefore, in the present study, the heat transfer of TSMF around a single heating pipe under a magnetic field was investigated as a basic study to obtain the basic knowledge of the thermal flow of TSMF on a complex-shaped heating surface.

2. Experiments

The fluid to be tested in the present study was a temperature-sensitive magnetic fluid (TS-50K, ICHINEN CHEMICALS Co., LTD), which was composed of the low-Curie temperature magnetic nanoparticles of Mn-Zn ferrite dispersed in carrier liquid of Kerosene with the weight fraction of 50 wt.%. The magnetic and thermal properties of the test fluid under 298 K and 1 atm are introduced in reference [4].

Figure 1 is a schematic diagram of the test section (heating part). The test section was composed of a heater, a heated body, a flow path and a magnet. A stainless steel cylindrical heated body with a diameter of 50 mm was placed inside the flow path. The flow path was designed to be a Hele–Shaw cell in which the fluid flow could be assumed as two-dimensional flow. This enables an experimental visualization of temperature-distribution in two-dimension, which will be compared with the result of numerical simulation (details will be described in Section 3) to investigate the effect of the magnetic field on the thermo-magnetic flow. The Hele-Shaw cell was with a height of 1 mm, a width of 100 mm, and a length of 300 mm. A cartridge heater (HI-SD ROD Cartridge heater, Sakaguchi E.H Voc Corp.) with a diameter of 5 mm was vertically immersed into the center of the heated body with a diameter of 50 mm. Thermocouples were installed at position 1∼ 8 (see Fig 2). The magnet consisted of two Neodymium magnets of 10 × 10 × 50 mm and a magnetic yoke (SS400). Figure 3 is the magnetic field distribution at z = 0 mm. The distance between the magnet surface and the center of the Hele-Shaw cell was set to be 3.5 mm. The magnet position c, which is the distance between centers of the magnet and the heater, changed from −5 to 5 mm (see Fig. 1). Although the original concept of this work is the magnetically-drive of the TSMF, a mechanical pump (NDR, IWAKI CO., LTD.) was connected to the test section to operate the test fluid because enough driving pressure could not be obtained by a single heating pipe. The temperature at the inlet of the test section was adjusted by a pre-heater. These components of the test section, mechanical pump and pre-heater were connected with Teflon tubes with an inner diameter of 1.54 mm. During experiments, the heater temperature was controlled to be constant of 80, 100, and 120°C. The test fluid was operated by a mechanical pump, and the temperature at the inlet of the Hele-Shaw cell was adjusted to be 25^°C. The experiments were performed with varying the temperature of the heater, the magnet position c, and Reynolds number Re = DU∕𝜐, where D is the diameter of heated body, U is the reference velocity, and 𝜐 is the kinetic viscosity. The heat transfer was evaluated by the Nusselt number, Nu = hD∕𝜆, where, h is the heat transfer coefficient, D is the characteristic length, and 𝜆 is the thermal conductivity. The heat transfer coefficient is given by h = q∕(T _W − T _IN ), where q is the heat flux, T _W is the temperature at heating wall, and T _IN is the temperature at the inlet of the test section. The heat flux q and the temperature at heating wall T _W were estimated by Fourier’s law.

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Fig. 1.

Schematic diagram of the test section composed of a heater, a heated body, a magnet, and a flow path. The flow path is a Hele-Shaw cell to assume two-dimensional flow.

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Fig. 2.

Schematic diagram of a heating pipe.

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Fig. 3.

Magnetic field distribution of the magnet.

Because magnetic fluids are opaque liquid, it is difficult to visualize the velocity fields experimentally. In the present study, to understand the variation of the heat transfer with and without the magnetic field, the temperature and flow fields of the TSMF was simulated by the hybrid lattice Boltzmann method [5]. The magnetic hydrodynamics can be described by the following governing equations: (1) Momentum equation considering magnetic body force: (2)(3) In the above equations, 𝜌 is the density, u is the velocity, p is the pressure, Pr = 𝜂c _p 𝜆⁻¹ is the Prandtl number, G r m = ρ D 2 μ 0 χ 0 H max 2 η − 2 is defined as the magnetic Grashof number. Here, in the above dimensionless parameters, 𝜂 is the viscosity, c _p is the specific heat, μ₀ is the magnetic permeability in vacuum, 𝜒₀ is the magnetic susceptibility.

Equations ((1)) and ((2)) were solved by the lattice Boltzmann method and Eq. ((3)) was solved by the finite volume method. The calculation domain and boundary conditions are depicted in Fig. 4. The interaction between the fluid and the solid body of the heating pipe was complemented by the immersed boundary method [16]. The bounce back boundary conditions were employed when the velocity on the walls. The inlet and outlet boundary conditions were a Poiseuille flow and a developed flow, respectively. When the temperature on the wall was calculated, the constant temperature on the heating wall was calculated. The simulation was made on a uniform grid of 100 × 500. In the calculation, the Reynolds number Re, the Prandtl number Pr, the magnetic Grashof number Grm were set to be 1.0, 166.4, 1.6 × 10³, respectively. The magnetic field distribution was used the same one in experiment, and the non-dimensional magnet position was set to be 0.1.

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Fig. 4.

Calculation domain and conditions.

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Fig. 5.

Average Nusselt number with varying Reynolds number in the absence of the magnetic field at the heater temperature of 100°C and magnet position c = 0 mm. The solid line is the theoretical value of the Zukauskas equation.

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Fig. 6.

Influence of the magnetic field on average Nusselt number.

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Fig. 7.

Local Nusselt number at (a) north Nu₁₋₂, (b) south Nu7 −8, (c) west Nu₅₋₆, and (d) east Nu₃₋₄ (See the position 1 8 in Fig. 2) at the Reynolds number Re = 1.0 and the heater temperature of 100°C.

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Fig. 8.

Temperature distributions visualized by infrared thermography.

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Fig. 9.

Numerical simulation results by hybrid lattice Boltzmann method.

Figure 5 shows the average Nusselt number in the absent magnetic field. The Reynolds number changes from 0.5 to 1.5. The solid line in Fig. 5 is the correlation for heat transfer over a heating pipe proposed by Zukauskas [17] as the following equation: (4) where the values of constant C₁ and exponent n changes depending on the Reynolds number Re. In the present study, the constant C₁ and exponent n was set to be 0.75 and 0.4. With taking the liquid-single phase flow, Pr∕Pr _w should be 1. Although there is a small quantitative difference between the experimental and theoretical values due to the heat loss to the atmosphere, the average Nusselt number gradually increases with the Reynolds number. This result is qualitatively consistent with the Zukauskas equation, indicating that the experiments performed in this work can describe the heat transfer of TSMF around the single heating pipe.

Of particular interesting is the influence of the magnetic field on the heat transfer. Figure 6 shows the influence of the magnetic field on average Nusselt number (a) when the Reynolds number changes, and (b) when the heater temperature changes. When the magnetic field is applied to the system, the average Nusselt number is dramatically enhanced. From Fig. 6(b), when the heater temperature is higher, which means that the fluid is well heated, the enhancement of the average Nusselt number by applying the magnetic field is found to be more significant. Figure 7 shows the local Nusselt number at (a) north Nu_1→2, (b) south Nu_7→8, (c) west Nu_5→6 and (d) east point Nu_3→4 (see Fig. 2). When the magnetic field is applied, all local Nusselt numbers are enhanced. Comparing the local Nusselt numbers at the north and south points, Nu_1→2 and Nu_7→8, the enhancement of these Nusselt numbers by applying the magnetic field is almost symmetric. On the other hand, comparing the local Nusselt number at west and east point, Nu_5→6 and Nu_3→4, the local Nusselt number at the west point, Nu_5→6, is more enhanced by applying the magnetic field. In order to investigate this heat transfer variation by applying the magnetic field, the temperature distribution around the single heating pipe was visualized by infrared thermography in the absence and presence of the magnetic field (Fig. 8). In the absence of the magnetic field (Fig. 8(a)), the temperature boundary develops toward the downstream side. This is a typical phenomenon in heat transfer. Of particular interesting is the temperature distribution in the presence of the magnetic field (Fig. 8(b)). When the magnetic field is applied, the thermal boundary layer at north, south, west and east points are dramatically developed. This phenomenon seems like flares of the sun. This result implies that heat transfer is dramatically enhanced by applying the magnetic field.

In order to realize why the thermal boundary layers are dramatically developed by applying the magnetic field, the numerical simulation was performed to investigate the temperature and the flow fields. Because the magnetic fluid is opaque, it is too difficult to visualize the flow field experimentally. Figure 9 shows the numerical simulation results relating to (a) temperature distribution and (b) velocity distribution. The results of temperature distribution are qualitatively consistent with the experimental visualization result (Fig. 8(b)). From the results of velocity distribution (Fig. 9(a)), it is found that the thermal boundary is developed by the large vorticities occurring nearby the heating pipe. The vorticities occur due to the temperature-dependent magnetization of TSMF. Because the temperature close to the heating body is higher, the distribution of magnetization becomes non-equilibrium, and then the magnetic body force induces the large vorticities. The present study revealed that when the TSMF is utilized as the refrigerant in the heat transport device, the heat transfer has been dramatically enhanced by applying the magnetic field.

Temperature-dependent magnetization induces thermomagnetic convection if the temperature gradient in the TSMF. In the present study, the influence of thermomagnetic convection on heat transfer around a single cylindrical heating pipe was investigated experimentally. The obtained results show that the averaged Nusselt number increases significantly by applying a magnetic field. The heat transfer is further enhanced with the temperature gradient larger. This is because of the several large vorticities occurs around the heating pipe due to the effect of the temperature-dependent magnetization of the TSMF. TSMF is expected as a refrigerant in a magnetically-driven heat transport device that can transfer the heat with a significantly long distance. The results obtained in this work imply that, for further development of the heat transport device, enlarging the heat exchange surface area has great potential.