Proceedings of the 22nd National and 11th International

ISHMT-ASME Heat and Mass Transfer Conference December 28-31, 2013, IIT Kharagpur, India

HMTC1300201

AXIAL WALL CONDUCTION IN PARTIALLY HEATED MICROTUBES

Motish Kumar Department of Mechanical Engineering

National Institute of Technology Rourkela Rourkela, Odisha, 769008

India motish86@gmail.com

Manoj Kumar Moharana Department of Mechanical Engineering

National Institute of Technology Rourkela Rourkela, Odisha, 769008

India mkmoharana@gmail.com

ABSTRACT A two dimensional numerical study is carried out to study

the effect of axial wall conduction in a partially heated microtube in conjugate heat transfer situations. The flow of fluid through the microtube (inner radius 0.2 mm, total length 60 mm) is laminar, and simultaneously developing in nature. 6 mm each at the inlet and the outlet end of the microtube is insulated and the remaining 48 mm is subjected to constant wall temperature boundary condition over its outer surface, and the cross-sectional solid faces are considered adiabatic. The tube wall thickness, material, and liquid flow rate is varied and simulations have been performed for a wide range of tube wall to convective fluid conductivity ratio (ksf ≈ 2.26 - 646), tube thickness to inner radius ratio (δsf ≈ 1, 10), and flow Reynolds number (Re ≈ 100, 500). The case of fully heated microtube is also considered and a comparison is presented between partially heated and fully heated microtube. The results show that wall conductivity (ksf) and wall thickness (δsf) plays a dominant role in the conjugate heat transfer process. In the fully heated microtube the average Nusselt number (Nuavg) increases with decreasing wall conductivity. Secondly, thicker walls provide higher Nuavg. Due to higher flow development length, higher flow Re increases magnitude of average Nusselt number (Nuavg) for any value of ksf and δsf. In partially heated microtube, the average Nusselt number (Nuavg) for thicker wall microtube is found to be less compared to thinner wall microtube except at very low wall conductivity (ksf) at which it is higher than thinner wall microtube. Thus, the curves for the thin and the thick wall microtube intersect each other at lower ksf. KEYWORDS

Microtube, Axial conduction, Conjugate heat transfer, Partially heated. NOMENCLATURE Cp Specific heat at constant pressure, J/Kg-K D Diameter (inner) of the tube, m hz Local heat transfer coefficient, W/m2-K kf Thermal conductivity of working fluid, W/m-K ks Thermal conductivity of tube wall, W/m-K

ksf Ratio of ks and kf, (-) L Length of the tube, m Nuz Local Nusselt number (hzD/kf), (-) Nuavg Average Nusselt number over the channel length, (-) Pr Prandtl number (Cpμ/kf), (-) r Radius, m Re Reynolds number (ρuD/μ), (-) T Temperature, K u Average fluid velocity at the microtube inlet, m/s z Axial coordinate

z* Dimensionless axial distance along the microtube (z/Re·Pr·D), (-)

Greek symbols δf Inner radius of the tube, m δs Thickness of the tube wall, m δsf Ratio of δs and δf, (-) μ Dynamic viscosity, Pa-s ρ Density, kg/m3 Θ Non-dimensional temperature, (-) Subscripts f Fluid fi Fluid inlet fo Fluid outlet i Inlet, inner o Outlet, outer q Constant heat flux boundary condition s Solid T Constant wall temperature boundary condition w Inner wall of microtube z Axial length along the channel INTRODUCTION

With developments in micro manufacturing, there has been an increasing trend towards miniaturization of appliances. Microchannels including circular microtubes are frequently used in many engineering applications. In a conventional circular tube, the ratio of the thickness of the tube wall to the inner radius is very small but in microtubes this ratio is very high. This is because the diameter of a microtube is of the order of few microns and the physical need of certain

minimum thickness of the tube wall. From practical standpoint, the thickness of the microtube cannot be less than the micron size inner radius. This makes a microtube wall relatively thick compared to its inner radius. This causes multi-dimensional conjugate heat transfer situation where factors such as flow rate of the working fluid and the thermo-physical properties of the working fluid, and microtube wall material influence the heat transfer process.

Because of relatively thin wall and higher flow rate, axial wall conduction has negligible effect on heat transfer in a conventional tube. But negligence of axial heat conduction along the solid walls of microtubes will lead to erroneous conclusions and inconsistencies in the interpretation of heat transfer results. In reality, the distribution of heat flux and temperature at the conjugate wall of the microtube depends on (i) dimensions of both the solid and the fluid domain, (ii) thermal properties of solid wall and working fluid used, and (iii) flow characteristics of working fluid. Practically, boundary conditions are applied on the outer surface whereas the boundary condition experienced at the solid-fluid interface control the heat transfer process. So, the objective should be to regulate the boundary condition at the solid-fluid interface, rather than on the outer surface. LITERATURE REVIEW

Axial back conduction is not a new phenomenon; it was explored as early as in sixties when Bahnke and Howard [1] studied performance of periodic-flow heat exchangers subjected to longitudinal heat conduction. Afterward many studies [2-4] had been carried out to understand the phenomena of axial wall conduction. Due to its negligible role in conventional channels, the focus on exploring this concept of axial wall conduction diminished with time. Again with developments in micro manufacturing and its widespread use in heat transfer devices, the study on axial conduction has again gained momentum, considering its relative importance in small geometries.

Peterson [5] numerically studied effect of axial wall conduction on thermal performance of microscale counter flow heat exchangers. Maranzana et al. [6] introduced axial conduction number, M (defined as the ratio of the conductive heat transfer to the convective heat transfer) and stated that axial conduction in the solid substrate can be neglected if M < 0.01. Li et al. [7] and Zhang et al. [8] studied conjugate heat transfer in thick circular tubes and found that the criteria proposed by Maranzana et al. [6] for neglecting axial conduction is not true for every case. Depending on the geometrical parameters and boundary conditions, the criteria for judging the effect of axial wall conduction may vary depending on the situation.

By considering wide parametric variations, Moharana et al. [9] numerically studied axial wall conduction in square shaped microchannel carved on a solid substrate and the bottom of the substrate is subjected to constant heat flux. Moharana et al. [9] found that there exists an optimum value of conductivity ratio (conductivity of the solid substrate material to conductivity of working fluid) which maximizes the average Nusselt number over the channel length. This observation was based on an extensive parametric variation of conductivity ratio, flow Re and substrate thickness, while the heating perimeter was kept constant. This can be explained as follows: Higher conductivity ratio leads to severe axial back conduction, thus decreases average Nu. Very low value of conductivity ratio leads to a situation which is qualitatively similar to the case of zero thickness substrate with constant heat flux applied on one

wall only (the other three sides being adiabatic). This again lowers the average value of Nusselt number. Similar phenomena were also observed in a microtube subjected to constant wall heat flux on its outer surface. Moharana and Khandekar [10] numerically studied axial wall conduction in a microtube subjected to constant wall temperature boundary condition on its outer surface. Though the study by Moharana and Khandekar [10] is similar to the one by Moharana et al. [9], no optimum conductivity ratio was observed, which could maximize the average Nusselt number. It was found that the average Nusselt number is continuously increasing with decreasing value of conductivity ratio and the slope is becoming steep. This is because a constant temperature boundary condition applied on the outer surface of the microtube can manifest itself as a constant heat flux boundary condition on the actual solid-fluid interface, depending on the controlling parameters, i.e. ksf, δsf, and Re.

Review of literature indicate that a number of theoretical [11], experimental [7, 11-15], and numerical [7, 15-17] studies have been performed to explore the discerning parameters for explicitly isolating the effect of axial conduction on transport coefficient. A more detailed review in this direction can be found in Moharana et al. [9]. To a certain extent, now the concept of axial wall conduction in microchannels is explored. But most of the studies reported in the literature had considered heating over the full length of the microchannels. However, in industrial applications, the heating length is not always equal to the full length of the channels used. Lelea [18] numerically studied conjugate heat transfer in partially heated microtube where half of the tube length was heated (either upstream or downstream) and the remaining half was insulated. The heated portion was subjected to constant heat flux on the outer surface. Lelea [18] found that axial wall conduction has negligible influence on thermal characteristics for low conductive material (steel) compared to higher conductive materials (silicon and copper). Lelea [19] numerically studied conjugate heat transfer in partially heated rectangular microchannel similar to their earlier study [18], and found that upstream heating has a lower thermal resistance compared to central or downstream heating.

Review of literature reveals that only a very limited number of studies on thermal performance of partially heated microchannels are available. In this background, a two-dimensional numerical investigation is carried out using commercial Ansys-Fluent® platform. The objective of this study is to understand and highlight the effect of tube thickness to inner radius ratio (δsf), solid wall to working fluid conductivity ratio (ksf) and flow Re on the axial wall conduction in partially heated microtubes subjected to constant wall temperature boundary condition in the heated length. The detailed numerical procedure is outlined in next section. NUMERICAL ANALYSIS

This numerical study has been carried out with the following prior assumptions:

a. Flow is single phase, laminar and incompressible. b. Heat transfer takes place at steady state. c. Thermo-physical properties are constant. d. Heat loss by natural convection and radiation is

negligible. Considering angular symmetry, two-dimensional Cartesian

coordinate system (here named as r-z) is used to solve the computational domain. A microtube and its details are shown schematically in Fig. 1. The dimensions of the microtube are

inner radius ri = δf, tube thickness ro – ri = δs, and total length L. The total length (L) of the microtube is divided in to three parts, L1 = 6 mm, L2 = 48 mm, and L3 = 6 mm as shown in Figure 1(c). The relative thickness of the microtube wall is represented by δsf, which is equal to ratio of δs and δf. The thickness of the microtube is varied (δsf = 1, 10) in the computational model while the other two dimensions (δf and L) are kept constant at 0.2 mm, and 60 mm respectively.

Two different thermal boundary conditions are considered in this analysis; (i) fully heated and, (ii) partially heated. For fully heated microtube, its outer surface is maintained at constant temperature over its total length L, as shown in Fig. 1(b). For partially heated microtube, its outer surface is maintained at constant temperature over the length L2, and the remaining portion i.e. length L1 and L3 near the inlet and the outlet respectively are insulated on its outer surface, as shown in Fig. 1(c). The value of L1 and L3 are taken to be 10 % each of the total length of the microtube. Secondly, the cross-sectional solid faces of the microtube are insulated, as shown in Fig. 1(b,c). Water is considered as the working fluid, and it enters the microtube at 300K with a slug velocity profile. Thus, the flow is hydrodynamically as well as thermally developing in nature at the microtube inlet.

The continuity, momentum, and energy equations are solved in commercial platform Ansys-Fluent®. The ‘standard’ scheme was used for pressure discretization, and the SIMPLE algorithm was used for velocity-pressure coupling in the multi-grid solution procedure. ‘second-order upwind’ was used for solving the momentum and energy equations. An absolute convergence criterion for continuity and momentum equations is taken as 10-6 and for energy equation it is 10-9.

Figure 1. (a) MICROTUBE, AND COMPUTATIONAL DOMAIN

OF (b) FULLY HEATED MICROTUBE (c) PARTIALLY HEATED MICROTUBE.

Figure 2. LOCAL NUSSELT NUMBER CALCULATED ALONG THE STREAMWISE DIRECTION OF A MICROTUBE (δsf ≈ 0)

FOR THREE DIFFERENT MESH SIZES.

The computational domain was meshed using rectangular elements. The grid size for all geometry considered in the analysis was decided based on individual grid independence test. For an example, local Nusselt number, calculated for a microtube with negligible wall thickness (δsf ≈ 0), for three mesh sizes of 30×3600, 35×4200 and 40×4800 (for the computational domain as shown in Fig. 1(b.)), at Re = 100, is shown in Fig. 2. The local Nusselt number at the fully developed flow regime (near the tube outlet) changed by 0.7% from the mesh size of 30×3600to 35×4200, and changed by less than 0.5% on further refinement to mesh size of 40×4800. Moving from first to the third mesh, no appreciable change is observed. So, the intermediate grid (35×4200) was selected. It can also be observed in Fig. 2. that the local Nusselt number values in the fully developed region coincides with NuT = 3.66 where NuT is the Nusselt number for fully developed flow in a tube subjected to constant wall temperature. Finer meshing was used at the tube entrance and at the boundary layer. DATA REDUCTION

The parameters of interest are (a) local wall temperature (b) local bulk fluid temperature, and (a) local heat flux. These parameters are essential to find the local Nusselt number, from which it can be inferred the effect of axial conduction on heat transfer. The conductivity ratio (ksf) is defined as the ratio of thermal conductivity of the microtube wall (ks) to that of the working fluid (kf). The axial coordinate, z, in dimensionless form is given by:

*

Re Pr

zz

D (1)

The microtube inner wall and bulk fluid temperature in dimensionless form are given by:

( )

( )w fi

wfo fi

T T

T T

(2)

( )

( )f fi

ffo fi

T T

T T

(3)

where, Tfi and Tfo are the average bulk fluid temperature at the microtube inlet and outlet respectively; Tf is the average bulk

fluid temperature at any location, and Tw is the wall temperature at the same location. The local Nusselt number is given by:

zz

f

h DNu

k

(4)

where the local heat transfer coefficient is:

( )z

zw f

qh

T T

(5)

The average Nusselt number over the total channel length

is given by:

0

L

avg zNu Nu dz (6)

RESULTS AND DISCUSSION

The microtube length (L) and inner radius (δf) are kept constant while the outer radius (δs + δf) is varied to change the wall thickness (δs). Secondly, different materials for the microtube are considered to vary the conductivity ratio (ksf). Finally, the fluid velocity is also varied to change the flow Re. Thus, the parametric variations used in the study are δsf = 1 and 10, ksf = 2.26 – 646, Re = 100 and 500. To study the effect of heating length, two cases are considered; Case-I: heating over the full length of the tube (see Fig. 1(b)), Case-II: insulation of 10% of total length each from the inlet and the outlet are insulated while the remaining central portion is heated (see Fig. 1(c)). The heating is such that constant wall temperature is imposed on the outer surface in the heated portion. The parameters of interest are (a) local wall temperature at the inner surface (b) local bulk fluid temperature, and (a) local wall heat flux at the inner surface.

The axial variation of dimensionless inner wall temperature and average bulk fluid temperature are shown in Fig. 3, which corresponds to a microtube subjected to constant wall temperature over its full length at its outer surface. As per the definition as in Eq. 3 and 4, the fluid temperature will be 0 and 1 at inlet and outlet respectively, and its axial variation should be parabolic while the wall temperature is constant at a value higher than 1. Similarly, if a constant wall heat flux is applied, the fluid temperature should linearly vary between 0 at inlet and 1 at outlet (shown by a dotted line), and the wall temperature will be higher than the fluid temperature and parallel to it in the fully developed region. In Fig. 3(a), the axial variation of bulk fluid temperature is parabolic and the wall temperature is almost constant throughout its length except near the inlet. The effect of conductivity ratio (ksf) is also minimal as the curves for different ksf are overlapping with each other. This is in line with conventional theory. As the wall thickness is increased to δsf = 10 (see Fig. 3(b)), the conductivity ratio started playing its role. For the fluid temperature, the lower the value of ksf, the more it is close to the dotted line i.e. linear variation. Secondly, for the wall temperature, the lower the value of ksf, the more it is away from uniform temperature i.e. horizontal line and vice versa. This indicates that for higher wall thickness, constant wall temperature is experienced at the solid-fluid interface at higher ksf and it drifts away towards constant wall heat flux at lower ksf.

As the flow is increased to Re = 500 (see Fig. 3(c-d)), it can be seen that the bulk fluid temperature is close to linearly varying between 0 and 1. Secondly, the lower the ksf, the more it is close to the dotted line. At higher ksf, the wall temperature is almost constant throughout the length of the tube except near the inlet. At lower ksf, the axial variation of both bulk fluid and wall temperature resemble to that of constant heat flux boundary condition at the solid-fluid interface.

The axial variation of dimensionless inner wall temperature and average bulk fluid temperature are shown in Fig. 4 for a microtube which is partially heated and partially insulated near the inlet and the outlet as was shown in Fig. 1(c). Ideally, the wall temperature at the solid-fluid interface should be equal to the temperature of local bulk fluid temperature along the length of insulation (L1 and L3 as shown in Fig. 1(c)). Accordingly, the wall temperature along the length L1 ideally should be equal to inlet fluid temperature (here equal to 0) and along the length L2 equal to the applied constant temperature (here some constant value higher than 1). In Fig. 4(a), it can be seen that for lower ksf the value of wall temperature almost follows this trend along the length L1 but it deviates drastically along the length L2. For higher ksf, the wall temperature approximately constant along length L2 while the temperature is increasing from some lower value to applied wall temperature along the length of the flow. This is due to some axial back conduction at higher ksf as thermal resistance is low. Along the insulated length L3 near the outlet, the bulk fluid temperature remains almost constant along the direction of flow while there is small gradual drop in wall temperature such that at the outlet the bulk fluid and the wall temperature are almost equal. Secondly, the bulk fluid temperature along the heated length L2 is parabolic and it is comparatively more towards linear variation at lower ksf.

Figure 4(b) shows the corresponding plot of Fig. 4(a) for higher wall thickness of δsf = 10, where the trend is similar to Fig. 4(a). In the insulated region L1, both the fluid and the wall temperature are higher compared to its counterpart in Fig. 4(a). This indicates more flow of heat (compared to Fig. 4(b)) in the backward direction from the region of L2 to the region of L1 by conduction. Secondly, the bulk fluid temperature in the heated region L2 is more scattered and comparatively closer to linear variation for lower ksf. Finally, the decrease in wall temperature along the direction of fluid flow in the insulated region L3 is comparatively less. This is due to conductive heat flow from the heated region L2 to the insulated region L3.

When the flow is increased to Re = 500 (see Fig. 4(c-d)), the fluid carried more heat with it from the interface wall, thus comparatively reducing axial back conduction in the wall along the insulated length L1. The bulk fluid temperature variation becomes almost linearly varying in the heated length L2. The wall temperature in the insulated region L3 starts decreasing in a parabolic manner as more heat is carried by the fluid due to higher flow Re. At higher wall thickness (δsf = 10, in Fig. 4(d)), conductivity starts playing its role and the wall temperature deviates much from the constant value as the ksf decreases.

Figure 5 shows the axial variation of local Nusselt number Nuz in the fully heated microtube which corresponds to Fig. 3. The limiting values Nuq = 4.34, and NuT = 3.66 (represented by horizontal lines) are fully developed Nu values for a tube subjected to constant heat flux and constant wall temperature respectively. For low flow Re and smaller wall thickness (see Fig. 5(a)), Nuz values in the fully developed region approach the theoretical value of NuT = 3.66. This is in line with the

Figure 3. DIMENSIONLESS INNER WALL AND BULK FLUID TEMPERATURE VARYING AXIALLY FOR FULLY HEATED

MICROTUBE.

Figure 4. DIMENSIONLESS INNER WALL AND BULK FLUID TEMPERATURE VARYING AXIALLY FOR PARTIALLY

HEATED MICROTUBE.

Figure 5. LOCAL NUSSELT NUMBER VARYING AXIALLY FOR FULLY HEATED MICROTUBE.

Figure 6. LOCAL NUSSELT NUMBER VARYING AXIALLY FOR PARTIALLY HEATED MICROTUBE.

conventional theory, which indicates negligible axial wall conduction due to lower ksf and δsf. As the wall thickness is increases (δsf = 10, see Fig. 4(b)) the corresponding fully developed Nu values increased slightly, indicating axial wall conduction causing shifting of the boundary condition experienced at the solid-fluid interface towards a constant wall heat flux. This conclusion becomes stronger by revisiting Fig. 4(b) where the axial variation of bulk fluid temperature comparatively becomes closer to linear variation along the length L2. Increasing flow Re from 100 to 500, increases the flow development length by five times, because of which the Nu values in Fig. 5(c-d) are still developing near the outlet.

The ensuing axial variation of local Nu is presented in Fig. 6 for partially heated microtube, which corresponds to Fig. 4. Back conduction in the insulated region L1 causes the local Nu to be higher than its counterpart in Fig. 5. Due to insulation near the outlet end along L3, the Nusselt number starts decreasing in this region. For low conductive material the decrease in local Nu is high and vice versa (see Fig. 6(a)). For conductive wall material, there will be axial wall conduction in the forward direction from region L2 to the region L3. Thus, the Nusselt number again starts increasing in the direction of flow. Even for thicker wall (δsf = 10) and higher conductivity (ksf), the wall temperature do not get ample opportunity to decrease any more in the insulated region L3 due to energy balance by forward axial conduction and convection from the wall from. Thus, the wall temperature remains almost flat even in the region L3, as can be seen in Fig. 6(b, d).

Figure 7. AVERAGE NUSSELT NUMBER VARYING WITH CONDUCTIVITY RATIO IN (a) FULLY HEATED MICROTUBE

(b) PARTIALLY HEATED MICROTUBE.

Figure 7 presents the variation of average Nusselt number (Nuavg) as a function of conductivity ratio ksf, while varying wall thickness δsf and flow Re. For the fully heated microtube (see Fig. 7(a)), the average Nusselt number (Nuavg) is found to be increasing with decreasing value of ksf and the slope increases at very low value of ksf. This indicate that lower wall conductive material is better for higher thermal performance when the tube is subjected to constant wall temperature on its outer surface over its full length. Secondly, thicker wall leads to comparatively higher thermal performance especially at lower ksf. Increasing flow Re, increases flow development length, and thus Nuavg. Finally, wall thickness has more effect at higher flow Re than lower flow Re.

For partially heated microtube (see Fig. 7(b)) with thick wall (δsf = 10), the average Nusselt number (Nuavg) is found to be increasing with decreasing value of ksf and the slope increases at very low value of ksf. This trend is exactly same as in Fig. 7(a). For thin wall (δsf = 1), the Nuavg increases with decreasing ksf but beyond a certain value of ksf, it again starts to decrease suddenly. Thus, for thinner wall microtube, there exists an optimum ksf at which Nuavg is maximum. The Nuavg values which are to the right of the peak Nuavg value are higher than its counterpart values of the thick wall microtube. Thus, the curves for δsf = 1 and δsf = 10 for each flow Re intersect at very low value of ksf.

CONCLUSION

A numerical study has been carried out to understand and highlight the effect of axial wall conduction in a partially heated microtube in conjugate heat transfer situations. Water flows through a microtube of 60 mm in length and 0.2 mm inner diameter and the flow is laminar, and simultaneously developing in nature. 6 mm each at the inlet and the outlet end of the microtube is insulated and the remaining 48 mm is subjected to constant wall temperature boundary condition over its outer surface. The cross-sectional solid faces are considered adiabatic. Simulations have been carried out for a wide range of parameters: (i) tube wall to convective fluid conductivity ratio (ksf ≈ 2.26 - 646), (ii) tube thickness to inner radius ratio (δsf ≈ 1, 10), and (iii) flow Reynolds number (Re ≈ 100, 500). Secondly, fully heated microtube (subjected to constant wall temperature) is also considered and a comparison is presented between partially heated and fully heated microtube. The results show that wall conductivity (ksf) and wall thickness (δsf) of the microtube plays a dominant role in the conjugate heat transfer process. In the fully heated microtube the average Nusselt number (Nuavg) increases with decreasing wall conductivity. Secondly, thicker walls provide higher Nuavg. Due to higher flow development length, higher flow Re increases magnitude of average Nusselt number (Nuavg) for any value of ksf and δsf. In partially heated microtube, the average Nusselt number (Nuavg) for thicker wall microtube is found to be less compared to thinner wall microtube except at very low wall conductivity (ksf) at which it is higher than thinner wall microtube. Thus, the Nuavg curves for the thin and the thick wall microtube (δsf = 1, and 10) intersect each other at lower value of ksf. REFERENCES

[1] Bahnke, G.D., and Howard, C.P., 1964. “The Effect of Longitudinal Heat Conduction on Periodic-Flow Heat Exchanger Performance”. Journal of Engineering Power, 86(2), April, pp. 105–117.

[2] Chiou, J.P., 1980. “The Advancement of Compact Heat Exchanger Theory Considering the Effects of Longitudinal Heat Conduction and Flow Nonuniformity”. Symposium on Compact Heat Exchangers-History, Technological Advancement and Mechanical Design Problems. Book no. G00183, HTD Vol. 10, ASME, New York.

[3] Faghri, M., and Sparrow, E.M., 1980. “Simultaneous Wall and Fluid Axial Conduction in Laminar Pipe-Flow Heat Transfer”. Journal of Heat Transfer, 102(1), February, pp. 58–63.

[4] Cotton, M.A., and Jackson, J.D., 1985. “The Effect of Heat Conduction in a Tube Wall upon Forced Convection Heat Transfer in the Thermal Entry Region”. In: Numerical Methods in Thermal Problems, vol. (iv), Pineridge Press, Swansea, pp. 504–515.

[5] Peterson, R.B., 1999. “Numerical Modeling of Conduction Effects in Microscale Counter Flow Heat Exchangers”. Microscale Thermophysical Engineering, 3(1), pp. 17–30.

[6] Maranzana, G., Perry, I., and Maillet, D., 2004. “Mini-and Micro-channels: Influence of Axial Conduction in the Walls”. International Journal of Heat and Mass Transfer, 47(17-18), August, pp. 3993–4004.

[7] Li, Z., He, Y.L., Tang, G.H., and Tao, W.Q., 2007. “Experimental and Numerical Studies of Liquid Flow and Heat Transfer in Microtubes”. International Journal of Heat and Mass Transfer, 50(17–18), August, pp. 3447–3460.

[8] Zhang, S.X., He, Y.L., Lauriat, G., and Tao, W.Q., 2010. “Numerical Studies of Simultaneously Developing Laminar Flow and Heat Transfer in Microtubes with Thick Wall and Constant outside Wall Temperature”. International Journal of Heat and Mass Transfer, 53(19–20), September, pp. 3977–3989.

[9] Moharana, M.K., Singh, P.K., and Khandekar, S., 2012. “Optimum Nusselt Number for Simultaneously Developing Internal Flow under Conjugate Conditions in a Square Microchannel”. Journal of Heat Transfer, 134(7), May, pp. 071703 (1-10).

[10] Moharana, M.K., and Khandekar, S., 2012. “Numerical Study of Axial Back Conduction in Microtubes”. In Proceedings of the Thirty Ninth National Conference on Fluid Mechanics and Fluid Power, December 13-15, SVNIT Surat, Gujarat, India, Paper number FMFP2012 - 135.

[11] Hetsroni, G., Mosyak, A., Pogrebnyak, E., and Yarin, L.P., 2005. “Heat Transfer in Micro-channels: Comparison of Experiments with Theory and Numerical Results”. International Journal of Heat and Mass Transfer, 48(25-26), December, pp. 5580–5601.

[12] Celata, G.P., Cumo, M., Marconi, V., McPhail, S.J., and Zummo, G., 2006. “Microtube Liquid Single-Phase Heat Transfer in Laminar Flow”. International Journal of Heat and Mass Transfer, 49(19-20), September, pp. 3538–3546.

[13] Liu, Z., Zhao, Y., and Takei, M., 2007. “Experimental Study on Axial Wall Heat Conduction for Convective Heat Transfer in Stainless Steel Microtube”. Heat and Mass Transfer, 43(6), April, pp. 587–594.

[14] Liu, Z., Liang, S., Zhang, C., and Guan, N., 2011. “Viscous Heating for Laminar Liquid Flow in Microtubes”. Journal of Thermal Science, 20(3), September, pp. 268–275.

[15] Moharana, M.K., Agarwal, G., and Khandekar, S., 2011. “Axial Conduction in Single-phase Simultaneously Developing Flow in a Rectangular Mini-channel Array”. International Journal of Thermal Sciences, 50(6), June, pp. 1001–1012.

[16] Avci, M., Aydin, O., and Arici, M.E., 2012. “Conjugate Heat Transfer with Viscous Dissipation in a Microtube”. International Journal of Heat and Mass Transfer, 55(19-20), September, pp. 5302–5308.

[17] Moharana, M.K., and Khandekar, S., 2013. “Effect of Aspect Ratio of Rectangular Microchannels on the Axial Back Conduction in its Solid Substrate”. International Journal of Microscale and Nanoscale Thermal and Fluid Transport Phenomena (in press).

[18] Lelea, D., 2007. “The Conjugate Heat Transfer of the Partially Heated Microchannels”. Heat and Mass Transfer, 44(1), November, pp. 33–41.

[19] Lelea, D., 2009. “The Heat Transfer and Fluid Flow of a Partially Heated Microchannel Heat Sink”. International Communications in Heat and Mass Transfer, 36(8), October, pp. 794–798.