The micro-tube heat transfer and fluid flow of methanol.

Lelea, Dorin ; Cioabla, Adrian Eugen ; Mihon, Liviu 等

1. INTRODUCTION

The recent technological developments and advances in devices that ensure the comfort of everyday life increases the importance of micro-channel heat transfer and fluid flow. As the scale of the systems is decreasing, the effects like thermal properties variation or viscous dissipation influence the thermal and hydrodynamic behavior and could not be neglected.

The first microchannel fluid flow experiment was made by Poiseuille in 1870 on a glass tube with internal diameter ranging from 29 to 140 um with water as the working fluid and non-heating working conditions. Based on these results, the well-known relation for the volume flow rate was established and extended lately to the macrochannels (Lelea, 2009).

Micro Thermal Systems (MTS) (Nishio, 2003), defined as the systems in which the key size has a length scale of a micrometer, could attain the high heat transfer coefficients. For instance, they are used as the cooling devices for LSI chips (Lelea, 2009).

Arici et al. (Arici, 2009) made a numerical study related with thermally developing the laminar forced convection in a pipe including the wall conductance and viscous dissipation. The viscous dissipation is found to affect both the wall and bulk fluid temperature profiles. Significant viscous dissipation effects have been observed for large Br. Its effect becomes more pronounced downstream (Lelea, 2010).

Morini, (Morini, 2005) analyzed theoretically the limit of significance for viscous dissipation effects in microchannel flows. It was found that viscous heating decreases the fluid viscosity, so the friction factor decreases as the Reynolds number increases (Lelea, 2010).

2. EXPERIMENTAL PART AND DISCUSSION

2.1 Establishing the computational relations for Nu and Po criteria

The thermal properties of Methanol are presented below:--Thermal conductivity:

k = 53.694 - 0.65369 x T + 0.0029916 x [T.sup.2] - 6.0761 x [10.sup.-6] x [T.sup.3] + 4.6195 x [10.sup.-9] x [T.sup.4]

--Dynamic viscosity:

[mu] = 0.0008153 - 1.2371 x [10.sup.-5] x T +

9.5 x [10.sup.-8] x [T.sup.2] - 2.9167 x [10.sup.-10] x [T.sup.3]

--Heat capacity:

[c.sub.p] = 42579.0 - 374.19 x T + 1.1487 x [T.sup.2] - 0.0011539 x [T.sup.3]

--Density:

[rho] = 956.01 - 0.21085 x T - 0.0011968 x [T.sup.2]

The following set of partial differential equations is used to describe the studied phenomena, considering the variable thermo-physical properties of the fluid and viscous dissipation:

Continuity equation:

[partial derivative]([rho](T) x u)/[partial derivative]z + 1/r [partial derivative](r x [rho](T) x v)/ [partial derivative]r = 0 (1)

Momentum equation:

[partial derivative]([rho](T)vu)/[partial derivative]r + [partial derivative]([rho](T)uu)/[partial derivative]z = dp/dz + 1/r [partial derivative]/[partial derivative]r ([mu](T)r [partial derivative]u/[partial derivative]r) (2)

Energy equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Where the viscous dissipation term is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Darcy friction factor is defined by the following equation:

f = -(dp/dz) x [D.sub.i]/[rho] x [u.sup.2.sub.m]/2 (5)

and a Re is defined as:

Re = [rho] x [u.sub.m] x [D.sub.i]/[mu] (6)

So, from (5) and (6) the local Po can be obtained in the following form:

Po = f Re = - 2 x (dp/dz) x [D.sup.2.sub.i]/[u.sub.m] x [mu] (7)

While the local Nu number is defined with the following equation:

Nu = h x [D.sub.i]/k([T.sub.m]) (8)

Where: [c.sub.p], J/kg K, specific heat

D, m, tube diameter

f, Darcy friction factor

k, W/mK, thermal conductivity

Nu, Nusselt number

Po, Poiseuille constant

R, m, tube radius

Re, Reynolds number

T, K, temperature

u, v, m/s, velocity components

x, z, spatial coordinates

Greek symbols

[mu], Pa s, viscosity

[rho], kg/[m.sup.3], density

2.2 Nu and Po criteria variation--first case: heating without viscous heating influence

The first case which was analyzed for the Nu and Po criteria variation taking into consideration Methanol as test fluid had as a fundamental basis the absence of viscous heating between the fluid and the walls of the micro-channel.

The graphic representation of the simulations for Nu and Po criteria, considering both the heating and cooling regimes are presented below:

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

2.3 Nu and Po criteria variation--second case: heating with viscous heating influence

The second case which was analyzed for the Nu and Po criteria variation taking into consideration Methanol as test fluid had as a fundamental basis the presence of viscous heating between the fluid and the walls of the micro-channel.

The graphic representation of the simulations for Nu and Po criteria, considering both the heating and cooling regimes are presented below.

4. CONCLUSIONS

The numerical model for heat and fluid flow through the pipes, considering the viscous heating of the fluid and methanol as the working fluid was analyzed. Also the temperature dependent fluid properties are considered.

In the case of the thermal results, the Nu number exhibits the classic boundary layer flow with conventional fully developed value of Nu = 4.36. On the other hand for local Poiseuille number Po=fRe it is not constant with [fRe.sub.fd] = 64 as expected but decreasing in downward the fluid flow.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

5. ACKNOWLEDGMENT

This work was supported by CNCSIS-UEFISCSU, project number 670 PNII--IDEI 938/2008.

6. REFERENCES

Arici, M.E, Aydin, O. (2009). Conjugate heat transfer in thermally developing laminar flow with viscous dissipation effects, Heat and Mass Transfer, Volume 45, pp. 1199-1203, ISSN: 0947-7411

Lelea, D, Cioabla, A.E., Laza, I, Mihon L. (2009). The Nu Number Behavior on Micro-tube Heat Transfer and Fluid Flow of Dielectric Fluid, The Open Thermodynamics Journal, Issue 3, pp. 38-41, ISSN: 1874-396X

Lelea, D, Cioabla, A.E, Laza, I. (2009). The Microtube Heat Transfer and Fluid Flow of Dielectric Fluids. 0965-0967, Annals of DAAAM for 2009 & Proceedings of the 20th International DAAAM Symposium, pp 483, ISBN 978-3-901509-70-4, ISSN 1726-9679, Published by DAAAM International, Vienna, Austria 2009

Morini, G.L. (2005). Viscous heating in liquid flows in micro-channels, International Journal of Heat and Mass Transfer, Volume 48, pp. 3637-3647, ISSN: 0017-9310

Nishio, S. (2003). Single-Phase Laminar Flow Heat Transfer and Two-Phase Osscilating Flow, Proceedings of 1st International Conference on Microchannels and Minichannels, pp. 25-38, 0791836673, ASME, Rochester USA