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Numerical study of flow state for a magnetic fluid heat transport device
Journal of Magnetism and Magnetic Materials 201 (1999) 260}263
Numerical study of #ow state for a magnetic #uid heat transport device H. Yamaguchi*, I. Kobori, N. Kobayashi Department of Mechanical Engineering, Doshisha University, Kyoto 630-0321, Japan Received 15 May 1998; received in revised form 31 August 1998
Abstract The #ow state in a magnetic #uid heat transport device is investigated numerically. A simple model geometry of the device is considered when the device is placed vertically. From the results of the calculation a qualitative explanation is made for the #ow state of experimental device when the magnetic "eld is a!ected. 1999 Elsevier Science B.V. All rights reserved. Keywords: Magnetic #uids; Heat transport device; Natural convection; Flow instability
1. Introduction There are some works regarding energy conversion and heat transport devices [1}3] in conjunction with a new engineering application of magnetic #uids. So far, we have designed and developed a heat transport device [3] using a temperature sensitive magnetic #uid. The basic working principle of the device is based on the thermomagnetic cycle [1,2]. However, to date no attempt to examine the #ow state occurring in the device [3] has been made in order to understand #ow phenomena and to estimate the heat transport characteristic.
* Corresponding author. Tel.: #81-774-65-6462; fax: #81774-65-6831. E-mail address: [email protected] (H. Yamaguchi)
In the present investigation, a numerical study is conducted for the #ow state in a simple model geometry of the device in which a con"ned concentric pipe with an annulus passage is considered.
2. Model geometry and numerical procedure Fig. 1 depicts a schematic diagram of the model geometry of the device [3] which is placed vertically. A ring shaped magnet is placed to the outer pipe close to the heating part where the heat is transferred to the #uid at temperature ¹ . At the upper end of the outer pipe the cooling part is situated where the heat is transferred from the #uid to the heat sink at temperature ¹ . The basic dimension of the model geometry used in the present study is that the aspect ratio (ratio of the outer cylinder radius to the length of cylinder) is 5.0 for the outer cylinder and 4.5 for the inner cylinder and thickness of the inner
0304-8853/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 0 2 1 - 9
H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 201 (1999) 260}263
261
where K is the pyromagnetic coe$cient and ¹ is a constant. The magnetic "eld whose components are (H , H ), which were derived from typical data P X measured in an experiment [3], is imposed externally to the magnetic #uid in the annulus passage, H "H erze\CX\X , P H "H ee\CX\X , (5) X where H , e and z are constant to determine the
magnetic "eld distribution. In the present study, the symmetric magnetic "eld given by Eq. (5) is assumed for the center of distribution at z /l"0.4,
where l is the length of the outer cylinder. The nondimensional parameters used in the present study are as follows: o gb (¹ !¹ )l , Ra" ag k H Ml Ra " ,
ag
Fig. 1. Model geometry.
cylinder wall is 0.2 based on the representative length of the outer cylinder radius. In the numerical simulation the system of equations governing the #ow of the magnetic #uid is as follows:
) *"0,
(1)
D* o "! p#g *#k M H!ou, Dt
(2)
D¹ RM o C #k ¹ Dt R¹
DH "j ¹. Dt
(3) 4 & It is noted here that * is the velocity vector, o the density, t the time, p the static pressure, M the magnetization vector, H the magnetic "eld vector, u the gravitational acceleration and ¹ the temperature. Parameters appeared in Eqs. (1)}(3) are commonly used thermophysical properties where g is the viscosity without magnetic "eld, j the thermal conductivity, C the speci"c heat and k the per meability in vacuum. The density of #uid is linearly approximated with the volumetric expansion ratio b . The magnetization is approximated by M"KH(¹ !¹), (4) )
l R¹ Nu" *¹ Rz
, (6) 8 where Ra is the Rayleigh number, Pr the Prandtle number, Ra the magnetic Rayleigh number and
Nu the Nusselt number with *¹"¹ !¹ and a the thermal di!usivity. The system of equations is then solved by HSMAC method [4]. In considering the boundary conditions the #ow is assumed axisymmetric and no-slip condition at all rigid walls is used for the numerical calculation. The lower wall of the outer cylinder is heated while the upper wall is cooled isothermally. The adiabatic condition is used for the inner wall (the inner cylinder).
3. Results and discussion In Fig. 2(A) the steady heat transfer characteristic is depicted as Nu versus Ra, where the representative magnetic "eld strength H is increased from zero (no magnetic "eld as Ra "0) to
H "3.27;10 A/m (Ra "1;10). It is men
tioned that the critical Rayleigh number for Ra "0 is estimated by extrapolating the
262
H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 201 (1999) 260}263
Fig. 2. Results of calculation.
calculated results by the relation Nu& (Ra!Ra ) where Ra is the critical Rayleigh number. As seen in Fig. 2 (A), the data are well "tted by the relation and it is seen that the #ow has the critical value of Ra +450 at which the #ow becomes unstable from the static mode (thermal conduction) to the #ow of the circulation motion. A typical #ow "eld as well as temperature "eld relevant to the post critical #ow mode (Nu'1) for Ra "0 are displayed in Fig. 2(B)-(I) which implies
the state of Ra"1;10 and Nu"0.23;10 in Fig. 2(A). It is noted that in Fig. 2(B), graph (a) shows the #ow "eld and graph (b) shows associated temperature "eld. As seen in Fig. 2(B)-(I), there is a large circulation of #ow from the inner pipe to the annulus passage, transporting from the heat from the heat source (low end) to the heat sink (upper
end). And it is seen from the result of temperature "eld that the heated #uid goes upward #owing in the inner pipe. On the other hand when the magnetic "eld is a!ected as shown in Fig. 2(A) for Ra "5;10 (H "1.64;10 A/m), the data
(solid circles 䢇) cannot be "tted by the critical relation of Nu&(Ra!Ra ) (there exists no critical value) indicating the #ow has no static equilibrium for Ra '0. There is a local circulation
motion (vortex) appeared in the annulus passage under the magnetic "eld distribution as shown in Fig. 2(B)-(II). The circulation direction of the vortex is the clock-wise and by the existance of the vortex the strength (#ow rate) of the large circulation (the #ow passing from the inner pipe to the annulus passage) is enhanced, transporting higher amount of heat in comparison with the case of Ra "0.
H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 201 (1999) 260}263
263
Fig. 3. Heat transfer characteristics.
The vortex acts as if the vortex rotates to pump the #uid to circulate strongly transporting much more heat. Further increase of the magnetic "eld strength to Ra "1;10 as indicated in Fig. 2(A) results in
substantial increase of the Nusselt number particularly at the lower Rayleigh number Ra)1000. In Fig. 3 the heat transport characteristic obtained from the present calculation is qualitatively compared with experimental results [3] for the relation between the strength of magnetic "eld and the heat transported for a given temperature di!erence between the heat source ¹ and heat sink ¹ . Although the model geometry adopted in the present study is di!erent from the actual experimental device [3] (for example the aspect ratio of the outer cylinder for the experimental device is approximately 5 times larger than the model geometry, etc.), the general trend of the results as depicted in Fig. 3a and Fig. 3b shows excellent similarity indicating the increase of the heat transfer rate when the magnetic "eld strength is increased. Thus it is speculated that the #ow phenomena occurring in the device, i.e. the vortex appeared beneath the
magnetic "eld imposition, re#ects the enhancement of the heat transport characteristic, which cannot be thought from merely thermodynamical point of view.
Acknowledgements This work was partly supported by a grant to RCAST at Doshisha University from the Ministry of Education, Japan. The authors are indebted for the grant.
References [1] E.l. Resler Jr., R.E. Rosensweig, J. Eng. Power, Trans. ASME (1967) 399. [2] H. Matsuki, K. Murakami, J. Magn. Magn. Mater. 65 (1987) 363. [3] H. Yamaguchi, I. Kobori, Y. Ishigaki, JSME J. Ser. B 64 (617) (1998) 85. [4] H. Kawamura, K. Hijikata, Numerical Simulation for Flow and Heat, Maruzen, Tokyo, 1995.
Numerical study of #ow state for a magnetic #uid heat transport device H. Yamaguchi*, I. Kobori, N. Kobayashi Department of Mechanical Engineering, Doshisha University, Kyoto 630-0321, Japan Received 15 May 1998; received in revised form 31 August 1998
Abstract The #ow state in a magnetic #uid heat transport device is investigated numerically. A simple model geometry of the device is considered when the device is placed vertically. From the results of the calculation a qualitative explanation is made for the #ow state of experimental device when the magnetic "eld is a!ected. 1999 Elsevier Science B.V. All rights reserved. Keywords: Magnetic #uids; Heat transport device; Natural convection; Flow instability
1. Introduction There are some works regarding energy conversion and heat transport devices [1}3] in conjunction with a new engineering application of magnetic #uids. So far, we have designed and developed a heat transport device [3] using a temperature sensitive magnetic #uid. The basic working principle of the device is based on the thermomagnetic cycle [1,2]. However, to date no attempt to examine the #ow state occurring in the device [3] has been made in order to understand #ow phenomena and to estimate the heat transport characteristic.
* Corresponding author. Tel.: #81-774-65-6462; fax: #81774-65-6831. E-mail address: [email protected] (H. Yamaguchi)
In the present investigation, a numerical study is conducted for the #ow state in a simple model geometry of the device in which a con"ned concentric pipe with an annulus passage is considered.
2. Model geometry and numerical procedure Fig. 1 depicts a schematic diagram of the model geometry of the device [3] which is placed vertically. A ring shaped magnet is placed to the outer pipe close to the heating part where the heat is transferred to the #uid at temperature ¹ . At the upper end of the outer pipe the cooling part is situated where the heat is transferred from the #uid to the heat sink at temperature ¹ . The basic dimension of the model geometry used in the present study is that the aspect ratio (ratio of the outer cylinder radius to the length of cylinder) is 5.0 for the outer cylinder and 4.5 for the inner cylinder and thickness of the inner
0304-8853/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 0 2 1 - 9
H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 201 (1999) 260}263
261
where K is the pyromagnetic coe$cient and ¹ is a constant. The magnetic "eld whose components are (H , H ), which were derived from typical data P X measured in an experiment [3], is imposed externally to the magnetic #uid in the annulus passage, H "H erze\CX\X , P H "H ee\CX\X , (5) X where H , e and z are constant to determine the
magnetic "eld distribution. In the present study, the symmetric magnetic "eld given by Eq. (5) is assumed for the center of distribution at z /l"0.4,
where l is the length of the outer cylinder. The nondimensional parameters used in the present study are as follows: o gb (¹ !¹ )l , Ra" ag k H Ml Ra " ,
ag
Fig. 1. Model geometry.
cylinder wall is 0.2 based on the representative length of the outer cylinder radius. In the numerical simulation the system of equations governing the #ow of the magnetic #uid is as follows:
) *"0,
(1)
D* o "! p#g *#k M H!ou, Dt
(2)
D¹ RM o C #k ¹ Dt R¹
DH "j ¹. Dt
(3) 4 & It is noted here that * is the velocity vector, o the density, t the time, p the static pressure, M the magnetization vector, H the magnetic "eld vector, u the gravitational acceleration and ¹ the temperature. Parameters appeared in Eqs. (1)}(3) are commonly used thermophysical properties where g is the viscosity without magnetic "eld, j the thermal conductivity, C the speci"c heat and k the per meability in vacuum. The density of #uid is linearly approximated with the volumetric expansion ratio b . The magnetization is approximated by M"KH(¹ !¹), (4) )
l R¹ Nu" *¹ Rz
, (6) 8 where Ra is the Rayleigh number, Pr the Prandtle number, Ra the magnetic Rayleigh number and
Nu the Nusselt number with *¹"¹ !¹ and a the thermal di!usivity. The system of equations is then solved by HSMAC method [4]. In considering the boundary conditions the #ow is assumed axisymmetric and no-slip condition at all rigid walls is used for the numerical calculation. The lower wall of the outer cylinder is heated while the upper wall is cooled isothermally. The adiabatic condition is used for the inner wall (the inner cylinder).
3. Results and discussion In Fig. 2(A) the steady heat transfer characteristic is depicted as Nu versus Ra, where the representative magnetic "eld strength H is increased from zero (no magnetic "eld as Ra "0) to
H "3.27;10 A/m (Ra "1;10). It is men
tioned that the critical Rayleigh number for Ra "0 is estimated by extrapolating the
262
H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 201 (1999) 260}263
Fig. 2. Results of calculation.
calculated results by the relation Nu& (Ra!Ra ) where Ra is the critical Rayleigh number. As seen in Fig. 2 (A), the data are well "tted by the relation and it is seen that the #ow has the critical value of Ra +450 at which the #ow becomes unstable from the static mode (thermal conduction) to the #ow of the circulation motion. A typical #ow "eld as well as temperature "eld relevant to the post critical #ow mode (Nu'1) for Ra "0 are displayed in Fig. 2(B)-(I) which implies
the state of Ra"1;10 and Nu"0.23;10 in Fig. 2(A). It is noted that in Fig. 2(B), graph (a) shows the #ow "eld and graph (b) shows associated temperature "eld. As seen in Fig. 2(B)-(I), there is a large circulation of #ow from the inner pipe to the annulus passage, transporting from the heat from the heat source (low end) to the heat sink (upper
end). And it is seen from the result of temperature "eld that the heated #uid goes upward #owing in the inner pipe. On the other hand when the magnetic "eld is a!ected as shown in Fig. 2(A) for Ra "5;10 (H "1.64;10 A/m), the data
(solid circles 䢇) cannot be "tted by the critical relation of Nu&(Ra!Ra ) (there exists no critical value) indicating the #ow has no static equilibrium for Ra '0. There is a local circulation
motion (vortex) appeared in the annulus passage under the magnetic "eld distribution as shown in Fig. 2(B)-(II). The circulation direction of the vortex is the clock-wise and by the existance of the vortex the strength (#ow rate) of the large circulation (the #ow passing from the inner pipe to the annulus passage) is enhanced, transporting higher amount of heat in comparison with the case of Ra "0.
H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 201 (1999) 260}263
263
Fig. 3. Heat transfer characteristics.
The vortex acts as if the vortex rotates to pump the #uid to circulate strongly transporting much more heat. Further increase of the magnetic "eld strength to Ra "1;10 as indicated in Fig. 2(A) results in
substantial increase of the Nusselt number particularly at the lower Rayleigh number Ra)1000. In Fig. 3 the heat transport characteristic obtained from the present calculation is qualitatively compared with experimental results [3] for the relation between the strength of magnetic "eld and the heat transported for a given temperature di!erence between the heat source ¹ and heat sink ¹ . Although the model geometry adopted in the present study is di!erent from the actual experimental device [3] (for example the aspect ratio of the outer cylinder for the experimental device is approximately 5 times larger than the model geometry, etc.), the general trend of the results as depicted in Fig. 3a and Fig. 3b shows excellent similarity indicating the increase of the heat transfer rate when the magnetic "eld strength is increased. Thus it is speculated that the #ow phenomena occurring in the device, i.e. the vortex appeared beneath the
magnetic "eld imposition, re#ects the enhancement of the heat transport characteristic, which cannot be thought from merely thermodynamical point of view.
Acknowledgements This work was partly supported by a grant to RCAST at Doshisha University from the Ministry of Education, Japan. The authors are indebted for the grant.
References [1] E.l. Resler Jr., R.E. Rosensweig, J. Eng. Power, Trans. ASME (1967) 399. [2] H. Matsuki, K. Murakami, J. Magn. Magn. Mater. 65 (1987) 363. [3] H. Yamaguchi, I. Kobori, Y. Ishigaki, JSME J. Ser. B 64 (617) (1998) 85. [4] H. Kawamura, K. Hijikata, Numerical Simulation for Flow and Heat, Maruzen, Tokyo, 1995.