**4.1 Local temperature distributions**

Figure 12 shows a velocity field in a microchannel at *Δp*=50 *kPa*. As discussed previously, the thermophysical properties of the water are based upon the estimated liquid bulk temperature. As can be seen by comparison with the results shown in Figure 9, for the same pressure drop along the channel, the thermal properties, velocity profile, mean velocity and Reynolds number are all mainly different.

Fig. 12. X- component velocity field from the numerical calculation, *Δp*=50*kPa*, *Re*=162.68, *Treference*=32°C, *um*=1.44*m/s*,*umax*=2.61*m/s.* 

Fig. 11. Comparison between the numerical and analytical results for temperature

Performance for the Pressure Correction Equations in microchannel heat sinks.

The following four subsections are devoted to the local temperature distributions, the average and bulk characteristics, the local heat flux distributions and the Convergence

Figure 12 shows a velocity field in a microchannel at *Δp*=50 *kPa*. As discussed previously, the thermophysical properties of the water are based upon the estimated liquid bulk temperature. As can be seen by comparison with the results shown in Figure 9, for the same pressure drop along the channel, the thermal properties, velocity profile, mean velocity and

Fig. 12. X- component velocity field from the numerical calculation, *Δp*=50*kPa*, *Re*=162.68,

differences upon the Reynolds number.

**4.1 Local temperature distributions** 

Reynolds number are all mainly different.

*Treference*=32°C, *um*=1.44*m/s*,*umax*=2.61*m/s.* 

**4. Results and discussion** 

As shown, a variation in the reference temperature, *Treference* from 20 to 32°C, changes the mean velocity from 1.1032 to 1.44 *m/s*, and results in a corresponding change in the Reynolds number from 95.38 to 162.68. The numerical results for the temperature distribution in the heat sinks are shown in Figures 13, 14, 15, 16 and 17 for different locations along the channel. Figures 13, 14 and 15 show the local cross-sectional temperature distribution in the *y*–*z* plane at *x=0*, *x=Lx/2* and *x=Lx*, respectively. As shown in Figure 13, the temperature of the liquid at the inlet is initially uniform (at 20°C). The temperature profiles shown in Figures 14 and 15 are identical in shape due to the assumption of hydrodynamic fully developed flow, but the magnitudes of the temperature are different.

Fig. 13. Local temperature distribution in *y–z* plane at *x=0*, (*Δp*=50 *kPa*, *Re*=162.68, *Treference*=32°C, *um*=1.44 *m/s*).

Fig. 14. Local temperature distribution in *y–z* plane at *x=Lx/2*, (*Δp*=50 *kPa*, *Re*=162.68, *Treference*=32°C, *um*=1.44 *m/s*).

Microchannel Simulation 209

Fig. 17. Contour of local temperature inside the channel at the cross-section of the outlet of

The temperature distribution can be showed obviously in Figures 18, 19 and 20, which indicate the local temperature distribution in the *x-y* plane at *z=Lz /2* for the three cases with

Fig. 18. Local temperature distribution in *x-y* plane at *z=Lz/2* (*Δp*=50 *kPa*, *Re*=162.68,

Fig. 19. Local temperature distribution in *x-y* plane at *z=Lz/2* (*Δp*=15 *kPa*, *Re*=85.60,

the channel (*Δp*=50 *kPa*, *Re*=162.68, *Treference*=32*°C*, *um*=1.44 *m/s*).

*qs*=90 *W/cm2*, at *Δp*=50, 15 and 6 *kPa*, respectively.

*Treference*=32°*C*, *um*=1.44 *m/s*).

*Treference*=48*°C*, *um*=0.57 *m/s*).

Fig. 15. Local temperature distribution in *y–z* plane at *x= Lx*, (*Δp*=50 *kPa*, *Re*=162.68, *Treference*=32°C, *um*=1.44 *m/s*).

Figure 16 shows the temperature contours in the heat sink at the outlet of the channel and Figure 17 shows the local temperatures inside the channel. If to be noted in Figure 17 is specified that the temperature is highest at the channel corner. This is due to the low velocity of the flow and the resulting high concentration of heat flux. From these calculations it is apparent that there is a 2–3°C temperature difference between the bottom wall of the substrate and the bottom surface of the channel.

Fig. 16. Contour of temperature in the heat sink at the cross-section of the outlet of the channel (*Δp*=50 *kPa*, *Re*=162.68, *Treference*=32°C, *um*=1.44 *m/s*).

Fig. 15. Local temperature distribution in *y–z* plane at *x= Lx*, (*Δp*=50 *kPa*, *Re*=162.68,

Fig. 16. Contour of temperature in the heat sink at the cross-section of the outlet of the

wall of the substrate and the bottom surface of the channel.

channel (*Δp*=50 *kPa*, *Re*=162.68, *Treference*=32°C, *um*=1.44 *m/s*).

Figure 16 shows the temperature contours in the heat sink at the outlet of the channel and Figure 17 shows the local temperatures inside the channel. If to be noted in Figure 17 is specified that the temperature is highest at the channel corner. This is due to the low velocity of the flow and the resulting high concentration of heat flux. From these calculations it is apparent that there is a 2–3°C temperature difference between the bottom

*Treference*=32°C, *um*=1.44 *m/s*).

Fig. 17. Contour of local temperature inside the channel at the cross-section of the outlet of the channel (*Δp*=50 *kPa*, *Re*=162.68, *Treference*=32*°C*, *um*=1.44 *m/s*).

The temperature distribution can be showed obviously in Figures 18, 19 and 20, which indicate the local temperature distribution in the *x-y* plane at *z=Lz /2* for the three cases with *qs*=90 *W/cm2*, at *Δp*=50, 15 and 6 *kPa*, respectively.

Fig. 18. Local temperature distribution in *x-y* plane at *z=Lz/2* (*Δp*=50 *kPa*, *Re*=162.68, *Treference*=32°*C*, *um*=1.44 *m/s*).

Fig. 19. Local temperature distribution in *x-y* plane at *z=Lz/2* (*Δp*=15 *kPa*, *Re*=85.60, *Treference*=48*°C*, *um*=0.57 *m/s*).

Microchannel Simulation 211

Fig. 22. Numerical predictions of local temperature distribution in the unit cell top wall.

wall.

The temperatures of the side walls vary noticeably in the transverse *y*-direction. As expected, the temperature decreases from the unit cell bottom wall to the unit cell top wall. Figure 23 shows higher temperatures for small *y* values close to the channel bottom

Fig. 23. Numerical predictions of local temperature distribution in the unit cell side walls.

Fig. 20. Local temperature distribution in *x-y* plane at *z=Lz/2* (*Δp*=6 *kPa*, *Re*=47.32, Treference=57*°C*, *um*=0.271 *m/s*).

The water flow is clearly specified in these Figures, which when combined with Figures 13, 14, 15, yield detailed information about the temperature distribution in the micro-heat sink. The temperature at the outlet for the case of *Δp*=6 *kPa* is higher than 100*oC* where boiling may occur. Therefore, the numerical solution for single-phase flow may not be valid there. The temperature increases along the longitudinal *x*-direction over the channel inner walls. There are very slight changes in the temperature gradient in the *x*-direction at the channel bottom wall, Figure 21, and the channel top wall, Figure 22.

Fig. 21. Numerical predictions of local temperature distribution in the unit cell bottom wall.

In fact, a linear temperature rise can be regarded as a good approximation for both planes. The temperature along the transverse *z*-direction is virtually constant for all the *x–z* planes just discussed.

Fig. 20. Local temperature distribution in *x-y* plane at *z=Lz/2* (*Δp*=6 *kPa*, *Re*=47.32,

bottom wall, Figure 21, and the channel top wall, Figure 22.

The water flow is clearly specified in these Figures, which when combined with Figures 13, 14, 15, yield detailed information about the temperature distribution in the micro-heat sink. The temperature at the outlet for the case of *Δp*=6 *kPa* is higher than 100*oC* where boiling may occur. Therefore, the numerical solution for single-phase flow may not be valid there. The temperature increases along the longitudinal *x*-direction over the channel inner walls. There are very slight changes in the temperature gradient in the *x*-direction at the channel

Fig. 21. Numerical predictions of local temperature distribution in the unit cell bottom wall.

In fact, a linear temperature rise can be regarded as a good approximation for both planes. The temperature along the transverse *z*-direction is virtually constant for all the *x–z* planes

Treference=57*°C*, *um*=0.271 *m/s*).

just discussed.

Fig. 22. Numerical predictions of local temperature distribution in the unit cell top wall.

The temperatures of the side walls vary noticeably in the transverse *y*-direction. As expected, the temperature decreases from the unit cell bottom wall to the unit cell top wall. Figure 23 shows higher temperatures for small *y* values close to the channel bottom wall.

Fig. 23. Numerical predictions of local temperature distribution in the unit cell side walls.

Microchannel Simulation 213

Fig. 24. Distribution of heat transfer coefficient along the channel at specified pressure drop.

Fig. 25. Nusselt number variation along the channel at specified pressure drop.

#### **4.2 Average and bulk characteristics**

In order to evaluate the local (averaged on the channel circumference) heat transfer characteristics along the flow direction, the convective heat transfer coefficient and Nusselt number, must be defined. The longitudinal convective heat transfer coefficient is defined as

$$
\overline{h}\_{\chi} = \frac{\overline{q}\_{s,\Gamma}(\chi)}{\Delta \overline{T}(\chi)} \tag{56}
$$

and the averaged longitudinal Nusselt number as

$$\overline{Nu}\_{\mathcal{X}} = \frac{\overline{h}\_{\mathcal{X}} D\_h}{k\_f} \tag{57}$$

The averaged longitudinal local heat flux along the perimeter of the inner wall of the channel in Equation (56) is defined as,

$$\begin{split} \overline{q}\_{s,\Gamma}(\mathbf{x}) &= -k\_{\mathbf{s}} \left( \frac{\partial T\_{\mathbf{s}}(\mathbf{x}, \mathbf{y}, \mathbf{z})}{\partial n} \right) \bigg|\_{\Gamma} \\ &= -k\_{f} \left( \frac{\partial T\_{f}(\mathbf{x}, \mathbf{y}, \mathbf{z})}{\partial n} \right) \bigg|\_{\Gamma} \end{split} \tag{58}$$

and the longitudinal mean temperature difference in Equation (56) is defined as,

$$
\Delta \overline{T}(\mathbf{x}) = \overline{T}\_{\mathbf{s}, \Gamma}(\mathbf{x}) - \overline{T}\_f(\mathbf{x}) \tag{59}
$$

$$\overline{T}\_{s,\Gamma}(x) = \frac{\sum\_{\Gamma} T\_{s,\Gamma}(i\_{\prime}j\_{\prime}k)}{N\_{\Gamma}} \tag{60}$$

$$\overline{T}\_f(\mathbf{x}) = \frac{\left\langle \sum \sum \rho\_f \, \mathbf{u} . c\_p . T . \Delta y \Delta z \right\rangle\_{\mathbf{i} = \text{constant} \mathbf{j}, \mathbf{k}}}{\text{in.c}\_p} \tag{61}$$

where *N*Γ is the total number of nodes along the perimeter of the inner wall (here *N*<sup>Γ</sup> =2×16+2×16). The averaged longitudinal inner wall temperature *T s,Γ* and the averaged local heat flux *q s,Γ* are mathematically averaged along the perimeter of the inner wall, and the longitudinal bulk liquid temperature *T f* is averaged according to energy conservation. With Equations (56)–(59), the longitudinal heat transfer coefficient variation and the longitudinal Nusselt number variation for these three cases can be determined and are shown in Figures 24 and 25, respectively.

From these two Figures it can be concluded that the variations of the heat transfer coefficient and the Nusselt number along the flow direction is quite small for this type of microchannel heat sink after the thermal entrance lengths. Comparisons between the average Nusselt number for the different heat flux and same Reynolds number are shown in Figure 26.

In order to evaluate the local (averaged on the channel circumference) heat transfer characteristics along the flow direction, the convective heat transfer coefficient and Nusselt number, must be defined. The longitudinal convective heat transfer coefficient is defined as

> *<sup>s</sup>*, *<sup>x</sup> q x*

*h*

 

(56)

*<sup>k</sup>* (57)

(58)

*T x*

. *x h <sup>x</sup> f*

*h D Nu*

The averaged longitudinal local heat flux along the perimeter of the inner wall of the

*f*

, ,

 i=costant,j,k .. .. .

*f p*

where *N*Γ is the total number of nodes along the perimeter of the inner wall (here *N*<sup>Γ</sup> =2×16+2×16). The averaged longitudinal inner wall temperature *T s,Γ* and the averaged local heat flux *q s,Γ* are mathematically averaged along the perimeter of the inner wall, and the longitudinal bulk liquid temperature *T f* is averaged according to energy conservation. With Equations (56)–(59), the longitudinal heat transfer coefficient variation and the longitudinal Nusselt number variation for these three cases can be determined and are shown in Figures

From these two Figures it can be concluded that the variations of the heat transfer coefficient and the Nusselt number along the flow direction is quite small for this type of microchannel heat sink after the thermal entrance lengths. Comparisons between the average Nusselt number for the different heat flux and same Reynolds number are shown in Figure 26.

*T xyz <sup>k</sup> n*

*<sup>s</sup> s s*

and the longitudinal mean temperature difference in Equation (56) is defined as,

*f*

*T xyz qx k*

,

*s*

*f*

24 and 25, respectively.

*T x*

*T x*

, , *<sup>s</sup>*

*N* 

*uc T y z*

*m c*

*p*

*T ijk*

, ,

*Tx T x T x s f* , (59)

(61)

(60)

, ,

*n*

**4.2 Average and bulk characteristics** 

channel in Equation (56) is defined as,

and the averaged longitudinal Nusselt number as

Fig. 24. Distribution of heat transfer coefficient along the channel at specified pressure drop.

Fig. 25. Nusselt number variation along the channel at specified pressure drop.

Microchannel Simulation 215

Large temperature gradients near the inlet region are mainly to induce significant thermal stresses and, therefore, must be carefully considered in the practical sink design in order to avoid the mechanical failure. The quasi-linear trend is not validated for the low Reynolds numbers as be shown in Figure 28. In Figure 28 a large portion of heat is conducted into the front part of the heat sink due to the low liquid velocity, and so the temperature gradient

between the top and bottom walls is much small and approaching to the zero value.

Fig. 28. Mean temperature variations along the channel at the top, bottom and side walls

Fig. 29. Average heat transfer coefficient distributions along the channel.

As the flow develops and the boundary layers grow in the longitudinal direction, the average heat transfer coefficients (Figure 29) gradually decrease in magnitude. The heat transfer coefficients are expected and, indeed, are larger at the side walls than at the top and

and in bulk liquid for *Δp* = 6 *kPa.* 

bottom walls.

Fig. 26. Comparisons between the average Nusselt numbers.

These two trends are identical; this is because the Nusselt number for laminar flow is determined solely by the channel geometry and the local flow conditions.

Figure 27 shows the fluid bulk temperature and the average temperatures of the top, bottom and side channel walls, as the functions of the longitudinal distance *x*, for *Δp* = 50 *kPa*. The fluid bulk temperature increases quasi-linearity along the *x*-direction, and it almost reaches the wall temperature at the exit of the microchannel. Overall, the average temperatures of side walls are slightly larger than top wall and smaller than bottom wall, because the convective resistance is much smaller for the close space between the side walls. While, in Figure 28 that is *Δp* = 6 *kPa*, due to low liquid velocity and low convective heat transfer, temperature difference will be increased between the solid and liquid, especially in the inlet region of the channel.

Fig. 27. Mean temperature variations along the channel at the top, bottom and side walls and in bulk liquid for *Δp* = 50 *kPa.* 

These two trends are identical; this is because the Nusselt number for laminar flow is

Figure 27 shows the fluid bulk temperature and the average temperatures of the top, bottom and side channel walls, as the functions of the longitudinal distance *x*, for *Δp* = 50 *kPa*. The fluid bulk temperature increases quasi-linearity along the *x*-direction, and it almost reaches the wall temperature at the exit of the microchannel. Overall, the average temperatures of side walls are slightly larger than top wall and smaller than bottom wall, because the convective resistance is much smaller for the close space between the side walls. While, in Figure 28 that is *Δp* = 6 *kPa*, due to low liquid velocity and low convective heat transfer, temperature difference will be increased between the solid and liquid, especially in the inlet region of the channel.

Fig. 27. Mean temperature variations along the channel at the top, bottom and side walls

and in bulk liquid for *Δp* = 50 *kPa.* 

Fig. 26. Comparisons between the average Nusselt numbers.

determined solely by the channel geometry and the local flow conditions.

Large temperature gradients near the inlet region are mainly to induce significant thermal stresses and, therefore, must be carefully considered in the practical sink design in order to avoid the mechanical failure. The quasi-linear trend is not validated for the low Reynolds numbers as be shown in Figure 28. In Figure 28 a large portion of heat is conducted into the front part of the heat sink due to the low liquid velocity, and so the temperature gradient between the top and bottom walls is much small and approaching to the zero value.

Fig. 28. Mean temperature variations along the channel at the top, bottom and side walls and in bulk liquid for *Δp* = 6 *kPa.* 

As the flow develops and the boundary layers grow in the longitudinal direction, the average heat transfer coefficients (Figure 29) gradually decrease in magnitude. The heat transfer coefficients are expected and, indeed, are larger at the side walls than at the top and bottom walls.

Fig. 29. Average heat transfer coefficient distributions along the channel.

Microchannel Simulation 217

Fig. 32. Numerical predictions of local heat flux distribution for the channel side walls.

For the microchannel heat sink model, illustrated in Figure 1, the total number of control volumes in the heat sink and inside the channel were set to *Nx×Ny×Nz* = 30×82×30 and *Nx×Ny×Nz* = 30×16×16 in the three spatial directions, respectively. The multigrid scheme, as discussed in the previous section, was implemented in the two cross-streamwise directions, which consisted of four grid levels from the finest grid (30×16×16) to the coarsest grid (30×2×2). The multigrid behavior followed the typical V-cycle pattern. The entire residual

**4.4 Convergence performance for the pressure correction equations** 

convergence history with the AC-MG algorithm is depicted in Figure 33.

Fig. 33. Residual convergence history of the pressure Poisson equation.

#### **4.3 Local heat flux distribution**

Figs 30, 31 and 32 illustrate the heat flux distribution along the channel walls for *Δp*=50 *kPa*, *Re*=162.68, *Treference*=32°*C*, *um*=1.44 *m/s*. For all the channel walls, higher heat fluxes are encountered near the channel inlet. This is attributed to the thin thermal boundary layer in the developing region. The heat fluxes vary around the channel periphery, approaching zero in the corners where the flow is weak for a rectangular channel. Figure 32 shows the heat flux along the channel side walls is higher than along the channel top and bottom walls (almost two orders of magnitude larger than those at the top and bottom walls) due to the short distance between the channel side walls and the large velocity gradient present. The local heat fluxes at both the bottom and top walls (Figures 30 and 31, respectively) show significant variation in the transverse *z*-direction, unlike the fluxes at the side walls (Figure 32), which are nearly uniform everywhere but in the inlet and corner regions.

Fig. 30. Numerical predictions of local heat flux distribution for the channel bottom wall.

Fig. 31. Numerical predictions of local heat flux distribution for the channel top wall.

Figs 30, 31 and 32 illustrate the heat flux distribution along the channel walls for *Δp*=50 *kPa*, *Re*=162.68, *Treference*=32°*C*, *um*=1.44 *m/s*. For all the channel walls, higher heat fluxes are encountered near the channel inlet. This is attributed to the thin thermal boundary layer in the developing region. The heat fluxes vary around the channel periphery, approaching zero in the corners where the flow is weak for a rectangular channel. Figure 32 shows the heat flux along the channel side walls is higher than along the channel top and bottom walls (almost two orders of magnitude larger than those at the top and bottom walls) due to the short distance between the channel side walls and the large velocity gradient present. The local heat fluxes at both the bottom and top walls (Figures 30 and 31, respectively) show significant variation in the transverse *z*-direction, unlike the fluxes at the side walls (Figure

32), which are nearly uniform everywhere but in the inlet and corner regions.

Fig. 30. Numerical predictions of local heat flux distribution for the channel bottom wall.

Fig. 31. Numerical predictions of local heat flux distribution for the channel top wall.

**4.3 Local heat flux distribution** 

Fig. 32. Numerical predictions of local heat flux distribution for the channel side walls.

#### **4.4 Convergence performance for the pressure correction equations**

For the microchannel heat sink model, illustrated in Figure 1, the total number of control volumes in the heat sink and inside the channel were set to *Nx×Ny×Nz* = 30×82×30 and *Nx×Ny×Nz* = 30×16×16 in the three spatial directions, respectively. The multigrid scheme, as discussed in the previous section, was implemented in the two cross-streamwise directions, which consisted of four grid levels from the finest grid (30×16×16) to the coarsest grid (30×2×2). The multigrid behavior followed the typical V-cycle pattern. The entire residual convergence history with the AC-MG algorithm is depicted in Figure 33.

Fig. 33. Residual convergence history of the pressure Poisson equation.

Microchannel Simulation 219

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