Apparent Slip Enhanced Magnetohydrodynamic Pump

Ryan Enright1,†, Aritra Ghosh1,2, Anne Gallagher1,3, Shenghui Lei1 and Tim Persoons3

1Efficient Energy Transfer (ηET) Department, Bell Labs Ireland, Nokia, Dublin 15, Ireland 2Mechanical Engineering Department, University of Illinois, Chicago, USA

3School of Engineering, Trinity College Dublin, Dublin 2, Ireland

†Corresponding author: ryan.enright@alcatel-lucent.com

Abstract Reliable thermal management is a key aspect in the operation of telecommunications equipment, both photonic and electronic. In liquid-cooled thermal management, conventional pump designs incorporate moving parts, introducing reliability concerns. The magnetohydrodynamic (MHD) phenomenon can be used to fabricate highly reliable, compact, solid-state pumps that avoid moving parts and make it suitable for rack level integration in telecommunications equipment. However, pumping efficiency and performance is limited by pressure losses in the pump. Here we explore the performance of an MHD pump demonstrating apparent hydrodynamic and thermal slip on its major walls (perpendicular to the magnetic field) that can be achieved using micro-nano structured surfaces to reduce pressure losses due to friction. Our initial analysis suggests that, while the pump section performance can be significantly enhanced by apparent slip, the overall pump performance is strongly dictated by the nature of pressure losses associated with the fringing magnetic field at the pump inlet and outlet such that enhancement due to apparent slip is only manifested for relatively long pump sections.

Introduction Device and component integration is critical to enable next generations of efficient and scalable electronic and telecommunications products. The level of integration required has severe implications for hardware design in general but even more considerable challenges from a thermal perspective. The thermal challenge grows with ever-increasing levels of integration, as the designer struggles to build more functionality into shrinking package space. Packing more functionality (e.g., devices and components) into smaller package footprints leads to increased thermal densities, which in turn, drives the development of new thermal solutions [1]. In particular, Galinstan, an electrically conductive liquid metal (melting point ~ -19 °C, thermal conductivity ~ 16× of water, viscosity ~ 3× of water), has recently received interest for use as a high performance coolant for thermal management applications [2]–[6]. While demonstrating useful thermophysical properties, it also has the advantage that it can be pumped reliably using the magnetohydrodynamic (MHD) effect. Similar to other liquid metals, Galinstan also has a large surface tension (~ 7× of water) and does not readily wet the electrically insulating wall material of a typical MHD pump, displaying advancing contact angles of ~150° in the smooth surface limit [3]. This leads us to consider the potential for introducing apparent hydrodynamic slip into the

design of an MHD pump using surfaces structured at capillary length scales. Indeed, surface structuring and the manipulation of surface forces in the context of heat transfer has recently become a widely investigated area for applications including microchannel heat transfer [3], [7], boiling [8] and condensation [9], [10]. Here, the efficacy of creating apparent slip at the flow boundaries is analyzed for a simple Hartmann-type MHD pump design [11]. Of particular interest is the fact that the modified velocity profile associated with Hartmann flow allows relatively small slip lengths (with respect to the channel dimension) to significantly decrease the fluid resistance [12]. Specifically, we consider a nominally parallel MHD flow between two plates (Hartmann flow) separated by 2h (see Figure 1) and characterized by a Hartmann number of Ha ≤ 100. For pump walls demonstrating moderate scaled apparent slip length of l/h ≤ 0.2, we calculate the resulting enhanced maximum flow rate (based on modified Hartmann flow theory with slip boundary conditions) of > 2×. However, assessment of the overall pump performance when considering pressure losses due to fringing magnetic fields shows that the apparent slip effect only becomes significant when the relative length of the pump section becomes large.

KEY WORDS: MHD pump, apparent slip

NOMENCLATURE b induced magnetic field strength (T) B0 imposed uniform magnetic field strength (T) B magnetic field (T) E electric field (V) h channel half-height (m) L channel length (m) l slip length (m) p pressure (Pa) Δp pressure difference (Pa) Q flow rate (m3/s) t thickness (m) v stream-wise fluid velocity (m/s) w channel half-width (m) x stream-wise coordinate y span-wise coordinate in direction of magnetic field z span-wise coordinate in direction of electric field Greek symbols ϵ channel span wise aspect ratio (= h/w) ϵ’ channel stream wise aspect ratio (= L/w) μ fluid dynamic viscosity (Pa.s) ρ fluid density (kg/m3) σ fluid electrical conductivity (S/m)

978-1-4673-8121-5/$31.00 ©2016 IEEE 1030 15th IEEE ITHERM Conference

Subscripts c channel f fringing field m mean ns no slip s slip Superscripts * dimensionless

Figure 1. Schematic of a MHD Hartmann pump section where flow is generated by the interaction of a fixed magnetic flux density (Bo) transverse to a current density (j) imposed by a electric field (E). The walls of the pump parallel to the current flow direction demonstrate a slip velocity (vw) characterized by a slip length (l). Adapted from ref. [13].

Apparent Slip MHD Pump Theory When an electrically conductive fluid in a rectangular

channel is exposed to a magnetic (B) and electrical (E) field perpendicular to each other, the fluid experiences a force: Lorentz force, mutually perpendicular to both the E and B, and flows in the direction of induced force. A pump operating with the above principle is known as a magnetohydrodynamic (MHD) pump [13], [11]. Here, we consider the case of fully-developed, steady, incompressible, laminar MHD flow in a parallel plate geometry (ϵ = h/w → 0). We assume that the flow can be described using the inductionless formulation such that the magnetic Reynolds number, Rem ( )

1, where the fluid magnetic permeability is approximately that of free-space μ0 (= 4π ×10-7 H.m-1) and σ is the fluid electrical conductivity. The competition between electromagnetic and viscous forces in an MHD flow are described by the Hartmann number

Eq. 1 where the geometric scale, h, is the half-channel width, Bo is the magnitude of the imposed magnetic flux density, σ is the fluid electrical conductivity and μ is the fluid dynamic viscosity [13]. We choose as the velocity scale, where dp/dx is the pressure gradient in the stream-wise direction, and consider the major walls of the pump to be electrically insulating such that the wall conductance ratio c = 0.

Thus, the dimensionless governing equations for flow momentum and induced magnetic field over the domain

are, respectively,

Eq. 2

and

Eq. 3

where v* is the dimensionless stream-wise velocity and b* is the dimensionless induced magnetic field strength. The boundary conditions applied at are

Eq. 4

where l* is the dimensionless slip length and I* is the dimensionless electrical current ( ) imposed by an external power supply. Solution of the above system of equations gives the following expressions for the velocity and induced magnetic field profiles

Eq. 5

and

Eq. 6

We then integrate Eq. 5 over the channel cross-section to obtain the volumetric flow rate, which corresponds to twice the dimensionless mean velocity and will be negative when the fluid is being pumped against an adverse pressure gradient,

Eq. 7

The portion of the flow rate that is independent of current is called the Hartmann flow rate , which can be used to define the ratio [13]

Eq. 8

Thus, in order to generate flow against an adverse pressure gradient (pump operation) requires that . Note that this ratio holds regardless of slip length so long as the hydrodynamic boundary conditions are the same for both. We are further interested in comparing the MHD pumping flow rate between the slip and no-slip case

Eq. 9

which we find is independent of the injected current. Thus, the enhancement due to slip in MHD pumping flow rate is identical to that of a simple pressure driven MHD flow [12]. Next, we consider the efficiency of the pump defined generally as

Eq. 10

where and are the pressure and volumetric flow rate developed by the pump, respectively. The Joule component of the electrical power dissipated by the pump is where and are the injected electrical current and pump section electrical

Bo

v

y

x z

E

j vw = l(dv/dy)w

2h

resistance, respectively. For convenience, we return Eq. 7 to dimensional quantities and then re-scale using the pressure gradient scale and the velocity scale to arrive at

Eq. 11

where is the ratio of the back pressure to the pressure developed by the pump (stall pressure) in the absence of end losses. Note, this form of the dimensionless volumetric flow rate scales out the injected current density. We can then define the pump efficiency in terms of dimensionless quantities as

Eq. 12

Figure 2 plots the dimensionless pump curve and efficiency of the pump section (no end losses) for several values of dimensionless slip length and Ha. For , the enhancement in pumping section flow rate is rather modest (< 2×) reflecting the fact that the drag reduction is defined by a shear layer thickness set by the channel dimension. However, at larger Ha the flow rate enhancement is more pronounced (>4×) since the shearing layer thickness scales as Ha-1 [12]. Conversely, the pump section efficiency sees a broader and larger peak enhancement at smaller Ha coinciding with the fact that the viscous and Lorentz forces are closer in magnitude.

Figure 2. MHD pump section curves and efficiencies for (a, b) Ha = 1, (c, d) Ha = 10 and (e, f) Ha = 100. The solid, dashed and dot-dashed curves are for l* = 0, 0.02 and 0.2, respectively.

Clearly, when the predominant loss in the MHD pump is frictional apparent slip can play a significant role in improving pump performance. However, in the presence of a spatially varying (i.e., fringing) magnetic field strength, an MHD flow undergoes additional three-dimensional forces that introduce a fundamental pressure loss that has a disadvantageous scaling with pump size, which we discuss next.

As the fluid enters and leaves the pump section, it is subject to a large change in magnetic field strength. The voltage induced upstream between the walls is higher than the downstream induced voltage, resulting in a current flowing in stream-wise and counter stream-wise direction near the walls [13]. To close the electrical path, current also flows in different cross-stream directions before and after the transition region. The resulting Lorentz forces act to repel the fluid from the core, first accelerating and then decelerating the near wall flow, and vice versa for the core flow.

Moreau et al. (2010) proposes a new simplified model, which includes inertia terms and the effect of current exchange between the core of the flow and the Hartmann layers [14]. The governing 3D equations are reduced to a quasi 2D form for vorticity, stream function and electric potential. A scaling parameter is introduced to characterize the effect of inertia

, where , the duct aspect ratio is and w is the pump section half width. They give the

following expression for pressure drop in the presence of an exponentially decaying fringing magnetic field strictly valid for , , (inertialess limit)

Eq. 13

where is the channel aspect ratio. Note, for the remainder of the analysis, we fix the channel aspect ratio at a maximum of ϵ = 0.1 to be reasonably consistent with the 1D behavior of the hydrodynamics modeled. We also apply Eq. 13 for cases where Ha < 100. Considering the additional end loss, we can show that the dimensionless back-pressure gradient is now

Eq. 14

where is the ratio of the pump section’s length to half-width. The first term on the right hand side of Eq. 14 is the pressure gradient generated by the Lorentz force, the second term is the parasitic pressure gradient due to frictional pressure losses through the pump section and the third term is the inlet and outlet fringing field pressure loss given by twice Eq. 13 divided by the pump section length. Evident from the fringing field term is the fact that, for fixed channel aspect ratio, its relative contribution can be minimized by making the pump section longer [15]. The maximum flow rate is then found by setting the left hand side of Eq. 14 equal to zero and re-arranging to give

Eq. 15

Figures 3 - 5 plot the pump curves and efficiencies including end losses due to the fringing magnetic field with the channel aspect ratio fixed at ϵ = 0.1. For ϵ’ = 1000 (Figure 3) and Ha =

0 0.5 1 1.5

p* p

0

0.2

0.4

0.6

0.8

1

0 0.5 10

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6

p* p

0

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

Q0 0.2 0.4

p* p

0

0.2

0.4

0.6

0.8

1

Q/Qmax

0 0.5 10

0.2

0.4

0.6

0.8

1

a b

c d

e f

ΔP

ΔP

ΔP

Q∗ Q∗/Q∗max

1, there is a modest reduction in the flow rate and efficiency compared to the case without end losses (Figure 2) and apparent slip still results in a significant improvement in performance. However, as Ha increases there is a marked reduction in the efficacy of apparent slip and, generally, the pump flow rate as a result of the Ha3/2 dependency in the end loss term.

Figure 3. MHD pump section curves and efficiencies for ϵ = 0.1 and ϵ’ = 1000 with (a, b) Ha = 1, (c, d) Ha = 10 and (e, f) Ha = 100. The solid, dashed and dot-dashed curves are for l* = 0, 0.02 and 0.2, respectively. Further decrease in the stream-wise aspect ratio to ϵ’ = 100 (Figure 4) results in almost no effect due to apparent slip for Ha = 100, while some flow rate enhancement is still possible for lower values of Ha. This is contrast to the case with no end losses where the largest gains in pumping flow rate due to apparent slip are observed at large Ha. Also observable is a change in the characteristic shape of the pump efficiency curve from linear to increasingly parabolic at larger Ha. Thus, the shift towards a more parabolic efficiency curve at relatively large Ha (≥ 10) is indicative of the importance of end losses on the overall pump performance.

Figure 4. MHD pump section curves and efficiencies for ϵ = 0.1 and ϵ’ = 100 with (a, b) Ha = 1, (c, d) Ha = 10 and (e, f) Ha = 100. The solid, dashed and dot-dashed curves are for l* = 0, 0.02 and 0.2, respectively. As the relative channel length is shortened further to ϵ’ = 10 (Figure 5), the performance curves for a pump section demonstrating apparent slip become almost indistinguishable from the no-slip case and, at the highest Ha, the flow rate is now three orders smaller than in the case with no end losses. Interestingly, in the range of Ha = 10 - 100, the pump peak efficiency is between 10% and 20% and the shape is parabolic, which compares well with the ~10% pump efficiency found at Q/Qmax ≈ 0.5 by Ghoshal et al. for Ha ≈ 40 [5]. Thus, despite the simplistic nature of our analysis, we replicate the salient features of MHD pump performance in a parameter range consistent with compact devices suitable for electronic thermal management applications. This point also serves to highlight the unfortunate scaling of MHD pump performance for compact, low aspect ratio form factors most suited for electronic thermal management applications.

Summary & Conclusions The effect of hydrodynamic apparent slip on the

performance of a MHD Hartmann pump (quasi 1D flow) has been analyzed. The results demonstrate that, while significant enhancements in the MHD pump section performance can be achieved for modest apparent slip lengths, pressure losses due to fringing magnetic field effects significantly reduce the impact of apparent slip on overall device performance. Future work will explore the impact of apparent slip on a wider range

0 0.5 1

p* p

0

0.2

0.4

0.6

0.8

1

0 0.5 10

0.05

0.1

0.15

0.2

0 0.1 0.2 0.3

p* p

0

0.2

0.4

0.6

0.8

1

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

Q0 0.01 0.02

p* p

0

0.2

0.4

0.6

0.8

1

Q/Qmax

0 0.5 10

0.2

0.4

0.6

0.8

a b

c d

e f

ΔP

ΔP

ΔP

Q∗ Q∗/Q∗max

0 0.2 0.4 0.6

p* p

0

0.2

0.4

0.6

0.8

1

0 0.5 10

0.05

0.1

0.15

0 0.02 0.04

p* p

0

0.2

0.4

0.6

0.8

1

0 0.5 10

0.1

0.2

0.3

0.4

Q 10-30 1 2

p* p

0

0.2

0.4

0.6

0.8

1

Q/Qmax

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

a b

c d

e f

ΔP

ΔP

ΔP

Q∗ Q∗/Q∗max

of pump geometries by implementing the above analysis in two dimensions to asses under what conditions apparent slip may improve the performance of compact MHD pumps for thermal management applications. Furthermore, detailed studies of MHD flows at apparent slip boundaries are required. Finally, there is the need to develop structured surface designs that are compatible with liquid metal working fluids and robust, in terms of wetting state stability, to ensure reliable MHD pump operation. More generally, our analysis serves to restate the issue of end losses due to fringing field effects that lead to a significant performance trade-off for compact devices of low aspect ratio. In the context of thermal management applications, this would suggest that there is a need to further explore MHD pump designs that maintain reliability, but with an improved performance/form factor trade-off.

Figure 5. MHD pump section curves and efficiencies for ϵ = 0.1 and ϵ’ = 10 with (a, b) Ha = 1, (c, d) Ha = 10 and (e, f) Ha = 100. The solid, dashed and dot-dashed curves are for l* = 0, 0.02 and 0.2, respectively.

Acknowledgments Bell Labs Ireland acknowledges the financial support of the Industrial Development Agency (IDA) Ireland. References [1] R. Enright, S. Lei, K. Nolan, I. Mathews, A. Shen, G.

Levaufre, R. Frizzell, G.-H. Duan, and D. Hernon, “A Vision for Thermally Integrated Photonics Systems,” Bell Labs Tech J, vol. 19, pp. 31–45, 2014.

[2] M. Hodes, Rui Zhang, L. S. Lam, R. Wilcoxon, and N. Lower, “On the Potential of Galinstan-Based Minichannel and Minigap Cooling,” Compon. Packag. Manuf. Technol. IEEE Trans. On, vol. 4, no. 1, pp. 46–56, Jan. 2014.

[3] L. S. Lam, M. Hodes, and R. Enright, “Analysis of Galinstan-Based Microgap Cooling Enhancement Using Structured Surfaces,” J. Heat Transf., vol. 137, no. 9, pp. 091003–091003, Sep. 2015.

[4] A. Miner and U. Ghoshal, “Cooling of high-power-density microdevices using liquid metal coolants,” Appl. Phys. Lett., vol. 85, no. 3, pp. 506–508, 2004.

[5] U. Ghoshal, D. Grimm, S. Ibrani, C. Johnston, and A. Miner, “High-performance liquid metal cooling loops,” in Semiconductor Thermal Measurement and Management Symposium, 2005 IEEE Twenty First Annual IEEE, 2005, pp. 16–19.

[6] Rui Zhang, M. Hodes, N. Lower, and R. Wilcoxon, “Water-Based Microchannel and Galinstan-Based Minichannel Cooling Beyond 1 kW/cm Heat Flux,” Compon. Packag. Manuf. Technol. IEEE Trans. On, vol. 5, no. 6, pp. 762–770, Jun. 2015.

[7] R. Enright, M. Hodes, T. Salamon, and Y. Muzychka, “Isoflux Nusselt number and slip length formulae for superhydrophobic microchannels,” J. Heat Transf., vol. 136, no. 1, p. 012402, 2014.

[8] K.-H. Chu, R. Enright, and E. N. Wang, “Structured surfaces for enhanced pool boiling heat transfer,” Appl. Phys. Lett., vol. 100, no. 24, p. 241603, 2012.

[9] N. Miljkovic, R. Enright, Y. Nam, K. Lopez, N. Dou, J. Sack, and E. N. Wang, “Jumping-Droplet-Enhanced Condensation on Scalable Superhydrophobic Nanostructured Surfaces,” Nano Lett, vol. 13, no. 1, pp. 179–187, 2012.

[10] R. Enright, N. Miljkovic, A. Al-Obeidi, C. V. Thompson, and E. N. Wang, “Condensation on superhydrophobic surfaces: The role of local energy barriers and structure length scale,” Langmuir, vol. 28, no. 40, pp. 14424–14432, 2012.

[11] J. Hartmann, “Hg-Dynamics I - Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field,” K Dan Vidensk Selsk Mat Fys Medd, no. XV, pp. 1–28, 1937.

[12] S. Smolentsev, “MHD duct flows under hydrodynamic ‘slip’ condition,” Theor. Comput. Fluid Dyn., vol. 23, no. 6, pp. 557–570, Nov. 2009.

[13] U. Müller and L. Bühler, Magnetofluiddynamics in Channels and Containers. Springer Berlin Heidelberg, 2013.

[14] R. Moreau, S. Smolentsev, and S. Cuevas, “MHD flow in an insulating rectangular duct under a non-uniform magnetic field,” PMC Phys. B, vol. 3, no. 1, pp. 1–43, Oct. 2010.

[15] G. W. Sutton, H. Hurwitz, and H. Poritsky, “Electrical and pressure losses in a magnetohydrodynamic channel due to end current loops,” Am. Inst. Electr. Eng. Part Commun. Electron. Trans. Of, vol. 80, no. 6, pp. 687–695, Jan. 1962.

0 0.05 0.1 0.15

p* p

0

0.2

0.4

0.6

0.8

1

0 0.5 10

0.01

0.02

0.03

10-30 2 4

p* p

0

0.2

0.4

0.6

0.8

1

0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

Q 10-40 1 2

p* p

0

0.2

0.4

0.6

0.8

1

Q/Qmax

0 0.5 10

0.05

0.1

0.15

0.2

0.25

a b

c d

e f

ΔP

ΔP

ΔP

Q∗ Q∗/Q∗max