1 Introduction

With the advancements in aerospace technology, micro-electromechanical systems, hybrid data centres and microfluidics, the miniature size electronic chips in such applications are the need of the century. According to the Moore’s law, the increase in operating temperature of the electronic boards results in detoriation of component’s durability. Also, International technology Roadmap for semiconductors (ITRS, 2009) alarmed that the thermal management in high performace chip packages will be crucial for intermittent and efficient operation of the parent device. Innovative as well as effective thermal solution is demanded by the growing technical advancements. Though the worldwide research has been conducted on different thermal heat dissipation, microchannel heat sinks are very promising in terms of high heat flux dissipation capacity and compatibility with the electronics system. However, the performance of microchannel heat sinks is quite unsatisfactory and lot of research have been done in improving the microchannel system.

The research on Microchannels were began by Tuckerman et al. [1]. Successive researches have been done to increase the heat transfer coefficients using single phase flow in microchannel with different aspect ratio. Singh et al. [2] demonstrated the successful application of microchannel cooling for moderation of high heat flux transients such as p-n diodes, Metal-oxide semiconductor Field-Effect transistor (MOSFET) and resitance termperature detectors (RTDs) [3]. Flow boiling in microchannels displays varying heat distribution in channel walls for constant heat flux boundary at the outer wall of microchannel. The problem of flow boiling, flow reversal, reduced critical heat flux, thermal stresses, pressure and temperature fluctuation in linear microchannels urged for different topology [4–7]. Many researches were done on microchannels with micro pin fins, surface roughness and tortuous channels [4, 8–12]. Nucleate boiling in the above mentioned modifications limits its applications. In general, only sinusoidal wall shape were considered in the design of microchannels [12]. Lee and Pan [6] used diverging microchannels to mitigate the effects of flow instablilites that causes nucleate boiling. The presence of critical angle which can reduce the pressure drop in single phase microchannel were reported by [7].

Relatively few studies on the diverging/converging microchannels were done. Liang Gong et al. used microchannel with wavy walls to improve the heat transfer performance in low Reynolds number laminar fluid flow for high heat flux [13]. They reported 55% increase in overall performance in comparison to straight microchannels. Particle motions in converging–diverging microchannels were experimentally investigated by Xiangchun et al. [14]. They studied the motion of polystyrene particles in converging–diverging section of microchannel and partially successful in concluding the dependability of velocity ratio and particle size. Louisos and Hitt [14] and Duryodhan et al. [16] studied the convective heat transfer in diverging and converging microchannels. The dominance of conjugate effect in heat transfer were also reported. Duryodhan et al. [17] proposed a methodology for calculating the characteristic length for varying cross section in microchannels. The selection of hydraulic diameter at the characteristic dimension for the converging/diverging microchannels is important to analyze the flow conditions. In their further studies [18], experimental demonstration of the isothermal wall condition for a constant heat flux condition were developed. Later, two cross sections, converging and diverging as separate microchannels were considered. The pressure drop and Nusselt number were compared in both the models. Yong et al. simulated the heat transfer performance of Microchannels with converging–diverging passages [19]. Their study was validated by calculating the Enhancement factors (En) and performance factor (PF). They reported an overwhelming improvement of thermal–hydraulic performance to 60% in comparison to straight microchannels. In summary, the microchannels with varying cross section increases the heat transfer enhancement in electronic devices with high heat flux applications. The flow instability in the microchannels were reduced thereby reducing the friction factor and increasing the heat transfer coefficient. To the best of author’s knowledge, the research involving the converging–diverging microchannels is very minimal. Thorough investigation in converging–diverging microchannels is necessary for its substantial applications.

In this study, a combination of converging and diverging microchannel heat sink were considered. Converging diverging (CD) microchannels were designed by FloEFD 15 and a three-dimensional numerical study was conducted for optimizing the performance of CD microchannel. Experiments were conducted to understand the heat transfer phenomenon in CD microchannel. De-ionized water was used as the working fluid and the measurements were made for heat input and mass flow rate range of 10–50 W/cm2 and 0.001232–0.01848 kg/s respectively. The pressure drop and heat transfer coefficient were measured and plotted for different correlations. To validate the experimental measurements, CFD simulations were performed. The simulation results were found to be in good agreement with the experimental results.

2 Experimental details

2.1 Experimental setup and procedure

The schematic layout of the experimental setup employed in this study is shown in the Fig. 1 which is similar to that used earlier [18, 27]. The experimental setup consists of fluid tank, pump, flow meter, differential pressure gauge, test section (Microchannel under test and header), heater, digital thermocouples and DC power supply. De-ionized water is used as the working fluid. The pump used is QB-60 whose maximum power is 0.37 kW. A pre-calibrated flow meter is used to control the mass flow rate of the liquid in the setup. The pre-calibrated pressure gauge is used to measure the pressure across the microchannel. Two digital thermocouples are used to measure the inlet and outlet temperatures on the surface of the microchannel. A custom-made header is designed and manufactured to study the performance of the converging–diverging microchannel. The header have inlet and outlet ports for the fluid to flow in and flow out. The header have 3 mm holes for the thermocouple beads to be slotted for measuring temperature. A 5 W heater is placed on top of the microchannel which acts as the heat source supplying constant heat flux. Insulation is provided between the header and microchannel surface to prevent any sort of heat transfer and unnecessary heat losses. Figure 2 shows the exploded assembly view of the header, CD microchannel heat sink and heater. A DC power supply is connected to the heater for constant power supply. All the devices are connected using Teflon tubing which is capable of withstanding high temperature.

Fig. 1
figure 1

Schematic layout of experimental setup used in this study

Full size image
Fig. 2
figure 2

Exploded assembly view of header, CD microchannel and heater

Full size image

The experimental procedure adopted is similar to that employed by Duryodhan et al. [16]. The de-ionized water was boiled for 30 min to evaporate the dissolved air (if any) and then cooled to room temperature. The fluid tank was filled with the boiled de-ionized water. The pump flow was started and the flow meter has been adjusted to provide a steady flow in the experimental circuit. The input and output temperature of the fluid and the pressure drop was continuously monitored until the flow in the circuit is steady. This means that, the steady state flow was assumed to be achieved when the fluctuation in temperature and pressure drop measurement is almost zero. Once the steady state condtion was achieved, all the required parameters such as flow rate, pressure drop, inlet and outlet temperature, heat flux were measured. Aforementioned procedure was repeated for different measurement conditions.

2.2 Converging–diverging microchannels

The converging–diverging microchannels and straight microchannels employed in this study are shown in the Fig. 3a. The microchannels are designed using Solidworks® software and fabricated by high precision micro-milling machine. Aluminium 5051 material was chosen for this study. The surface roughness of each microchannel was measured using Axio CSM700 Controller Zeiss microscope which posses 10× magnification factor. The average surface roughness of the microchannel was measured to be 0.853 µm. The detailed geometrical specification of the microchannels employed in this study are given in Table 1.

Fig. 3
figure 3

Microchannel heat sink considered in this study a 3D CAD model of CD Microchannel heat sink, b 2D Top view of CD Microchannel heat sink with detail A showing Sects. 1, 2, 3, 4 and 5

Full size image
Table 1 Geometrical parameters
Full size table

The length (L), width (B) and thickness (t) of the CD microchannel heat sinks are 120, 40 and 1.2 mm respectively. The dimensions of the CD microchannel was optimized for higher heat dissipation. The bigger width and smaller width of CD microchannel dimensions used in this study are 3000 µm (Maximum) and 500 µm (Minimum) with 51.34° converge–diverge angular constraint. For better understanding and easy calculation of aspect ratio and hydraulic diameter, the single unit of CD microchannel was divided into five sections as shown in Fig. 3b. Sections 1 and 5 comprises minimum aspect ratio while the Sect. 3 has maximum aspect ratio. To be noted, Sects. 2 and 3 are the converging and diverging sections respectively and hence possess varying aspect ratio.

2.3 Data reduction

The heat transfer coefficient, Reynolds number and Nusselt number calculation procedures are explained in this section. The heat supplied to the surface of the microchannel is not completely carried away by the flowing water. At the instance of power supply to the heater, the water at the inlet reservoir utilizes a small fraction of heat. Another fraction of the heat supplied gets dissipated to the ambient air in the surrounding through convection and radiation. Duryodhan et al. conducted a heat loss analysis and reported a maximum of 12% loss to the surrounding in the form of convection and radiation [16]. However, the inclusion of header in this study requires a different heat loss analysis.

The heat loss curve determines the amount of heat lost (Q loss ) to the atmosphere. To generate the heat loss curve, the following procedure was adopted. Firstly, power was supplied to the microchannel with no fluid-flow in it and the corresponding surface temperature of the microchannels was measured. Secondly, fluid was allowed to flow at constant flow rate and input power to the heater was adjusted to obtain the same surface temperature as that of no fluid flow. The difference in power obtained against the surface temperature was plotted as heat loss curve as shown in Fig. 4. In this case, the maximum heat loss was found to be 16% for the mass flow rate of 0.01355 kg/s. The increase in mass flow rate reduces the heat loss through convection and radiation.

Fig. 4
figure 4

Heat loss curve for CD microchannel

Full size image

The following are the equations to determine the heat carried away by the fluid in the microchannels: Total heat in the tank is given by:

$${\text{Q}}_{\text{tank}} = {\text{ Q}}_{\text{in}} {-} \, \left( {{\text{Q}}_{\text{channel}} + {\text{ Q}}_{\text{loss}} } \right)$$
(1)

Heat in the channel is calcualted by:

$${\text{Q}}_{\text{channel}} = \dot{m}{\text{c}}_{\text{p}} \left( {{\text{T}}_{\text{out}} - {\text{ T}}_{\text{in}} } \right)$$
(2)

The heat carried away by the channel is given by:

$${\text{h}}_{\text{out}} \,\, = \,\,\frac{{Q_{heat} }}{{\left( {T_{f} {-} T_{s} } \right)}}$$
(3)

where Tf and Ts is the average fluid temperature and Surface temperature respectively. Tf and Ts are given by:

$${\text{T}}_{\text{f}} = \frac{{T_{fi} + T_{fo} }}{2}$$
$${\text{T}}_{\text{s}} = \frac{{T_{si} + T_{so} }}{2}$$

where Tfi, Tfo, Tsi and Tso are the inlet fluid temperature, outlet fluid temperature, inlet surface temperature and outlet surface temperature respectively.

The heat flux in the channel is given by:

$$Q_{heat} \, = \,\frac{{Q_{channel} }}{{A_{s} }}$$
(4)

where As is the surface area of the microchannel.

For ease of calculation and comparison of previous studies, the mass flow rate and pressure drop are converted to their respective non-dimensional forms as Reynolds number (Re) and friction factor (f) given by:

$$Re\,\, = \,\,\frac{{\rho U_{avg} D_{h} }}{\upmu}$$
(5)
$$f\, = \,\frac{2 \Delta P}{{\rho U_{avg}^{2} \frac{L}{{D_{h} }}}}$$
(6)

where ρ and µ are the density and viscosity of fluid repectively. U avg is the streamline velocity of fluid in flow direction and L is the length of the microchannel under test. D h is the hydraulic diameter calculated at the charecteristic location. The viscosity of water is a function of temperature and calculated at average fluid temperature as given by Eq. 7. The average fluid temperature measured at reference temperature of 25 °C and viscosity as 8.55 × 10−4 kg/m.s.

$$\upmu\left( {\text{T}} \right) \, = \,\upmu\left( {{\text{T}}_{\text{ref}} } \right) \cdot \left( {{\text{T}}/{\text{T}}_{\text{ref}} } \right)^{8.9} \exp \left[ {4700\left( {1/{\text{T }}{-} \, 1/{\text{T}}_{\text{ref}} } \right)} \right]$$
(7)

From previous study of Duryodhan et al. [17], the location of characteristic dimension for the diverging microchannel is at L/3 distance from the narrow end. Similarly, the characterisic dimension for the converging microchannel is located at L/3.6 distance from the narrow end. These characteristic dimensions are independent of Re, divergence or convergence angle, D h and L. To the best of author’s knowledge, not much research were carried out on location of hydraulic diameter in converging–diverging microchannels. Hence, to minimize the complications in understanding, the above characteristic location were chosen to calculate the average geometrical and flow parameters such as: hydraulic diameter (D h ), aspect ratio (γ), velocity (U avg ) and Reynolds number (Re).

To validate our findings, the experimental f.Re value is compared with the theoritical value calculated by Morini et al. [19]. The theoritical correlation is a function of aspect ratio (γ) given by:

$$f.\text{Re} = \, 24 \, {-} \, 42.267 \, \gamma^{2} + \, 64.272 \, \gamma^{3} {-} \, 118.42 \, \gamma^{4} + \, 242.12 \, \gamma^{5} {-} \, 178.79 \, \gamma^{6}$$
(8)

Further, Nusselt number is calculated to show the heat transfer performance of the microchannels. The Nusselt number (Nu) is given by:

$$Nu\, = \,\frac{{h_{avg} D_{h} }}{{K_{f} \left( T \right)}}$$
(9)

where Kf(T) is the fluid thermal conductivity calculated at average fluid temperature Tf. The thermal conductivity of fluid at a constant temperature is given by:

$${\text{K}}_{\text{f}} \left( {\text{T}} \right) \, = \, \left( {3.3 \, + \, 1.9{\text{T}}} \right) \, \times \, 10^{ - 3}$$
(10)

To validate the experimental Nusselt number, the correlation from different researchers were used. Shah and London [21] used rectangular microchannels and calculated Nusselt number (as a function of aspect ratio) for a fully developed laminar flow with constant heat flux boundary condition and constant surface temperature condition as shown in Eqs. 11 and 12 respectively.

$$Nu_{{{\text{Shah}},{\text{Q}}}} = \, 8.235 \, \left( {1 \, {-} \, 2.0421\gamma \, + \, 3.0853\gamma^{2} {-} \, 2.4765\gamma^{3} + \, 1.0578\gamma^{4} {-} \, 0.1861\gamma^{5} } \right)$$
(11)
$$Nu_{{{\text{Shah}},{\text{ T}}}} = \, 7.541 \, \left( {1 \, {-} \, 2.610\gamma + \, 4.970\gamma^{2} {-} \, 5.119\gamma^{3} + \, 2.702\gamma^{4} {-} \, 0.548\gamma^{5} } \right)$$
(12)

Furthermore, Lee and Garimella [22] calculated Nu for a thermally developing flow in rectangular microchannels with constant heat flux boundary condition as shown in Eq. 13. The flow condition used was laminar.

$$Nu = \, \left[1.766 \, \left(\text{Re} \, \Pr \frac{{D_{h} }}{L}\right)^{0.378} \left(\frac{H}{W}\right)^{0.1224} \right]$$
(13)

where Pr is the Prandtl number, H and W are the height and width of the microchannel respectively. Pr is given by:

$$Pr\, = \,\frac{\nu }{\alpha }$$
(14)

where ν and α are the kinematic viscosity and thermal diffusivity of the working fluid.

The uncertainity analysis in measurement data were conducted based on the procedures explained by Figliola and Beasley [23]. The uncertainity in various parameters are shown in Table 2.

Table 2 Uncertainity in experimental measurements
Full size table

3 Numerical method

3.1 Numerical model and governing equations

In a microchannel system, it is important to understand the phenomenon of conjugate heat transfer as the conduction and convection takes place simulataneously. According to the literature, if the solid wall to fluid thickness goes beyond 2–3, the conjugate effect is predominant [24]. Numerical simulations were performed on the converging–diverging microchannels to estimate its increase in heat transfer coefficient. The simulations results were validated with the experimental results to analyze the predictability of the numerical methods.

A three-dimensional (3D) steady state simulation were conducted using FloEFD 15. The computational domain and boundary conditions for numerical study is shown in Fig. 5. The computational domain was set as 0.35 m × 0.35 m × 0.35 m as shown in Fig. 5a. The boundary conditions applied to the model is shown in Fig. 5b. The inset of Fig. 5b shows the distinction between the solid domain and fluid domain. Prior to assumptions on the nature of heat transfer in the computational domain, either natural convection or forced convection, evaluation of Rayleigh (Ra) number is mandatory [25]. The Rayleigh number is given by:

$$Ra = \frac{{g\beta\Delta TL_{r}^{3} }}{\nu \alpha }$$
(15)

where, \(\Delta T\) is the temperature difference of the package and ambient. The \(\Delta T\), under maximum operating conditions, was assumed to be 100 °C, L r is the characteristic length of the microchannel, which is 0.12 m. The other parameters were the standard thermo physical parameters of air. Solving Eq. 15 results in Ra ≈ 1.7 × 107, which was just below the typical transition value 108 ~ 109 for the turbulent regime. Hence, natural convection under laminar flow was assumed for present numerical analysis.

Fig. 5
figure 5

a Computational domain and b boundary conditions in FloEFD 15

Full size image

The Navier–Stokes equations for the steady state conditions are: mass conservation equation, the momentum equation, and the thermal energy conservation equation as follows:

Continuity equation:

$$\frac{\partial }{{\partial {\text{x}}}}\left( {\uprho{\text{u}}} \right) + \frac{\partial }{{\partial {\text{y}}}}\left( {\uprho\vartheta } \right) + \frac{\partial }{{\partial {\text{z}}}}\left( {\uprho{\text{w}}} \right) = 0 .$$
(16)

Momentum equations:

$$\frac{\partial }{{\partial {\text{x}}}}\left( {\uprho{\text{uu}}} \right) + \frac{\partial }{{\partial {\text{y}}}}\left( {\uprho{\text{vu}}} \right) + \frac{\partial }{{\partial {\text{z}}}}\left( {\uprho{\text{wu}}} \right) = - \frac{{\partial {\text{P}}}}{{\partial {\text{x}}}} +\upmu \left( {\frac{{\partial^{2} {\text{u}}}}{{\partial {\text{x}}^{2} }} + \frac{{\partial^{2} {\text{u}}}}{{\partial {\text{y}}^{2} }} + \frac{{\partial^{2} {\text{u}}}}{{\partial {\text{z}}^{2} }}} \right)$$
(17a)
$$\frac{\partial }{{\partial {\text{x}}}}\left( {\uprho{\text{uv}}} \right) + \frac{\partial }{{\partial {\text{y}}}}\left( {\uprho{\text{vv}}} \right) + \frac{\partial }{{\partial {\text{z}}}}\left( {\uprho{\text{wv}}} \right) = - \frac{{\partial {\text{P}}}}{{\partial {\text{y}}}} +\upmu \left( {\frac{{\partial^{2} {\text{v}}}}{{\partial {\text{x}}^{2} }} + \frac{{\partial^{2} v}}{{\partial {\text{y}}^{2} }} + \frac{{\partial^{2} {\text{v}}}}{{\partial {\text{z}}^{2} }}} \right)$$
(17b)
$$\frac{\partial }{{\partial {\text{x}}}}\left( {\uprho{\text{uw}}} \right) + \frac{\partial }{{\partial {\text{y}}}}\left( {\uprho{\text{vw}}} \right) + \frac{\partial }{{\partial {\text{z}}}}\left( {\uprho{\text{ww}}} \right) = - \frac{{\partial {\text{P}}}}{{\partial {\text{z}}}} +\upmu \left( {\frac{{\partial^{2} {\text{w}}}}{{\partial {\text{x}}^{2} }} + \frac{{\partial^{2} {\text{w}}}}{{\partial {\text{y}}^{2} }} + \frac{{\partial^{2} {\text{w}}}}{{\partial {\text{z}}^{2} }}} \right) -\uprho{\text{g }} .$$
(17c)

Thermal energy conservation equation:For fluid:

$$\frac{\partial }{{\partial {\text{x}}}}\left( {\uprho{\text{uT}}} \right) + \frac{\partial }{{\partial {\text{y}}}}\left( {\uprho{\text{vT}}} \right) + \frac{\partial }{{\partial {\text{z}}}}\left( {\uprho{\text{wT}}} \right) = \frac{\text{k}}{{{\text{C}}_{\text{p}} }} \left( {\frac{{\partial^{2} {\text{T}}}}{{\partial {\text{x}}^{2} }} + \frac{{\partial^{2} {\text{T}}}}{{\partial {\text{y}}^{2} }} + \frac{{\partial^{2} {\text{T}}}}{{\partial {\text{z}}^{2} }}} \right) + {\text{S}}_{\text{e}}$$
(18)

where \({\text{S}}_{\text{e }}\) indicates the heat source term for the radiative heat transfer which is not considered in this study.

For solid:

$$\left( {\frac{{\partial^{2} {\text{T}}}}{{\partial {\text{x}}^{2} }} + \frac{{\partial^{2} {\text{T}}}}{{\partial {\text{y}}^{2} }} + \frac{{\partial^{2} {\text{T}}}}{{\partial {\text{z}}^{2} }}} \right) = 0 .$$
(19)

The following assumptions and boundary conditions (as shown in Fig. 5b) were made for the simulation:

  1. 1.

    Fluid flow will be laminar in the computational domain.

  2. 2.

    All outer surfaces were adiabatic and non-radiant.

  3. 3.

    The process of heat transfer is under steady state condition.

  4. 4.

    Mass flow rate was specified as the Inlet wall boundary.

  5. 5.

    Outlet wall was prescribed with environmental pressure condition.

  6. 6.

    Constant heat flux was applied on the top surface of the microchannel.

  7. 7.

    Numerical simulation excludes the header. Bottom wall of microchannel was specified with adiabatic wall condition to replicate the insulated header in the experimental setup.

  8. 8.

    Materials assumed were homogeneous and isotropic.

  9. 9.

    No-slip conditions were assumed at the solid–liquid bondary interface.

A Semi Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm was used to couple the pressure and velocity. A second order upwind scheme was used for the discretization of the convection terms in the governing equations. In general, a relative error for the convergence criterion was considered to be less than 10−4. FloEFD 15 possesses the ability for different meshing functions. In this simulation, every component in the assembly were given different meshing grids depending on the thickness of the individual layers. Table 3 shows simulation results with different number of solid cells, fluid cells and partial cells for grid independence study. From the inference, approximately 2.1 million grids were found to have good estimation [16, 26].

Table 3 Grid independence
Full size table

4 Results and discussion

4.1 Experimental results

This section focuses on the study of fluid flow and heat transfer characteristics in CD microchannel. Experimental pressure drop, temperature difference and heat transfer coefficient were calcualted in their respective non-dimensional forms. The mass flow rate and heat flux used ranges from 1.2 × 10−2 to 1.848 × 10−2 kg/s and 10–50 W/cm2 respectively.

4.1.1 Pressure drop

Figure 6 shows the variation of pressure drop for different mass flow rate at different heat inputs. The pressure drop is measured for microchannel in non-heating (0 W) condition in order to analyze its performance at higher heat inputs. For all heat inputs, the pressure drop varies non-linearly with the increase in mass flow rate unlike in straight microchannels where pressure drop varies linearly [27]. The non-linearity in pressure drop suggests the influence of viscous forces over the pressure and inertial forces [17]. It also shows that the pressure drop decreases with increase in heat input. As the heat input increases, the viscosity of water decreases owing to decreased viscous forces. The reduction in viscous force reduces pressure drop adversely.

Fig. 6
figure 6

Variation of pressure drop with the mass flow rate for different heat input

Full size image

Figure 7 reveals similar findings in correlation to viscosity of water. The average fluid temperature measured at the outlet of the header were 28.3, 47.7, 61.3 °C for corresponding heat input of 10, 20 and 30 W/cm2 respectively. The percentage reduction in viscosity can be calculated by Eq. 7 with reference to viscosity at room temperature as already explained in Sect. 2.3. The reduction in viscosity were reported to be decreasing for increase in heat input [15]. Hence, the reduction in friction factor was almost consistent with reduction in viscosity. To validate, friction factor (f) was plotted against Reynolds number (Re) as shown in Fig. 7. Exponential decrease in curve shows the reduction in viscous force within the CD microchannel. This suggests that further investigation regarding the location of charateristic dimension is needed to normalize the friction factor.

Fig. 7
figure 7

Variation of a friction factor with Reynolds number b Poiseulle number with Reynolds number for different heat input

Full size image

Furthermore, Fig. 7b asserts the variation of poiseulle number (f.Re) with respect to theoretical correlation. The theoretical f.Re calcuated from Eq. 8 is 293.65. Experimentally calculated f.Re shows wide deviation to theoretical value. This is due to the fact that the theoretical value was calculated based on the aspect ratio of the microchannels measured at the characteristic dimension (L 2 /3.6 and L 4 /3). Whereas, the experimental f.Re were calculated by the viscosity as a function of temperature. It is interesting to note that, for all the cases of heating condition as well as non-heating condition, f.Re narrowed down to theoretical value for high Re. This means that, the viscous force and entrance effect of CD microchannel need to be further investigated [28]. To be noted, the error percentage on comparing the theoretical f.Re to experimental f.Re are 25% for non heating condition. On heating condition, the error rises to 27.7 and 34.8% for 10 and 20 W heat input respectively. It is important to note that these errors are calculated at Re = 300. The uncertainity in calculation of viscosity of water contributes to this error [16].

4.1.2 Heat transfer coefficient

Figure 8a shows the variation of local heat transfer coefficient (h x ) along the axial distance of CD microchannel. The measurements were made for three different mass flow rate: 0.001232, 0.01355 and 0.01602 kg/s. As already explained in Sect. 2, single unit of CD microchannel comprises of three different cross sections which is validated by heat transfer coefficient curve. Figure 8a shows that h x varies non-linearly along the direction of flow in microchannel. It is interesting to note that along the straight to converging section (L 1 to L 2 ), h x decreases due to decrease in Reynolds number due to change in cross-section. This happens till [L 1  + (L 2 − L 2 /3.6)] length of single CD microchannel unit since the flow is always in developing region for converging cross-section [17]. For L 3 in CD microchannel, h x remains almost linear due to uniform Re. Subsequently, when there is a change in cross-section from straight-divergent in (L 3 to L 4 ) section, the Reynolds number increases which results in increase in h x . It should be noted that the maximum h x in L 5 section is lesser than in L 1 section. Over the entire length, the heat transfer coefficient reduces in comparison to the previous CD unit as acknowledged in Fig. 8a. As expected, heat transfer ability of the working fluid reduces along the length of the microchannel. In other words, h x decreases as the fluid temperature increases. Since water is low-viscous fluid, the reduction in h x is obvious. Additionally, when the mass flow rate were increased, the heat transfer coefficient increases along the length of the microchannel. Although, similar trend of non-linear variation in h x were observed when mass flow rate were increased. Equivalent results were obtained for the corresponding local Nusselt number plot as shown in Fig. 8b.

Fig. 8
figure 8

Variation of a local heat transfer coefficient with the axial distance b local Nusselt number with axial distance for different mass flow rate

Full size image

Figure 9 shows the variation of average heat transfer coefficient (h avg ) against mass flux flow rate (ṁ). The figure shows non-linear increase in h avg for the mass flux range of 0.001232–0.01848 kg/s. The heat transfer coefficient for straight microchannels increased linearly as witnessed by Kumaraguruparan et al. [27]. Aforementioned comparison supports the heat transfer trend in the CD microchannel and conversely, it is inappropriate to compare two experimental data with dissimilar input conditions. However, it is important to understand that the straight microchannel with uniform cross-section have relatively constant flow-regime and hot solid–fluid boundary remain intact once the flow is fully developed. In case of CD microchannel, as already mentioned, the flow-regime is still in developing region. Hence, the solid–fluid boundary is constantly perturbed which results in significant thinning of hot fluid boundary adjacent to microchannel wall [13]. Therefore, CD microchannel is reported to maintain non-linear h avg variation. This can be observed by Nusselt number variation as explained in next section.

Fig. 9
figure 9

Variation of average heat transfer coefficient with mass flow rate

Full size image

In order to validate experimental data in CD microchannel, the experiments were repeated for different values of heat flux at constant mass flow rate as suggested by Duryodhan et al. [16]. Figure 10 shows the measurements taken for heat flux range of 10–50 W/cm2 for different mass flow rates: 0.001232, 0.01355 and 0.01848 kg/s. Figure 10 also shows that the h avg is independent of heat flux which were reported for conventional microchannels with uniform cross-sections. As expected in forced convection flow regime, the independence of heat transfer coefficient on heat flux was observed for CD microchannels.

Fig. 10
figure 10

Variation of average heat transfer coefficient with Input heat flux for different mass flow rate

Full size image

4.1.3 Correlations

Figure 11 shows the variation of experimental Nu with Re. The figure shows that there is repititive linearity rather than consistent linearity in Nu plot unlike in straight microchannels [27]. This affirms that the converging–diverging section in CD microchannel contributes for increased Nu value. The concept of solid–fluid boundary perturbation can be correlated for affirming increased Nu value as explained earlier [13]. To validate the above-mentioned, correlations were made in order to compare the standard results of other researchers [16].

Fig. 11
figure 11

Nu Vs Re plot for CD microchannel

Full size image

Figure 12 shows the comparison of experimental Nu in this study with results from various other researchers as given in Eqs. (10–13). It should be noted that the trend is similar in all cases but the linearity differs. The two horizontal plots indicates the Nu calculated based on Eq. 11. The upper horizontal plot denotes the Nu calculated for aspect ratio at converging section (L 2 ) while the lower horizontal plot denotes Nu at diverging section (L 4 ) Results of Lee and Garimella was found to deviate much to CD microchannel results because it does not account for the conjage heat transfer effect [22]. The results of sole converging microchannel and diverging microchannel does not concur exactly to CD microchannel results since the cross-section was either expanding or narrowing [16]. Also the geometry of the test sections considered were different. However, the results of CD microchannel was found to establish connection with converging microchannel in early slope of Nu curve. This upholds the consideration of characteristic dimesion at L/3.6 and L/3 (from the narrow end) for the converging and diverging section respectively. Straight microchannel reported to have Nu value in order for 0.1–0.35 for mass flow rate ranging from 0.002 to 0.003 kg/s [27]. Experimental Nu value of CD microchannel shows significant increase than the straight microchannel. For correlation with equation of Shah and London [20], the experimental Nu value is 0.9% lower than the theoretical value calculated for aspect ratio at converging section and 3.7% higher than Nu calculated for aspect ratio at diverging section. For correlation with Lee and Garimella [21], experimental Nu shows 15.8% higher. All these results indicates, the evaluation of heat transfer coefficients at the characteristic dimension is good for calculating the thermo-hydraulic performance of CD microchannel. Additionally it was noted, that the percentage increase in experimental Nu value is due to the consideration of conjugate effect in heat transfer in this study. Hence, the deviation in experimental value should not be misunderstood as error.

Fig. 12
figure 12

Nu Vs Re correlations for CD microchannel

Full size image

4.2 Numerical results

CFD simulation was conducted by specifying the boudary conditions as explained in Sect. 3.1. Initially, the simulations were commenced for the standard geometries and unifrom cross-section microchannels for which the results were known. This was done to validate the numerical results. Further, the numerical simulations were done for CD microchannels that replicates the experimental conditions. Figure 13 shows the temperature distribution in CD microchannel. As it is colossal to present data for all the cases studied, results of microchannel subjected to uniform heat flux of 20 W/cm2 and three mass flow rates: 0.01232 kg/s, 0.01355 kg/s and 0.01848 kg/s were shown in Fig. 13(a–c) respectively. Apparently, the surface temperature of the heat sink reduces due to superior thermal performance in CD section. At low mass flow rate, the surface temperature was high at the outlet end of the microchannel as shown in Fig. 13a. As the mass flow rate increases, the kinetic energy of the fluid increases resulting in enhanced heat transfer in microchannel. As an effect, the surface temperature of the CD microchannel heat sink becomes almost uniform barring the surface near outlet as shown in Fig. 13c [18]. The Nusselt number can be calculated from the numerical data using Eq. 9. The average Nusselt number computed was plotted against the Reynolds numbers for experimental results and numerical results as shown in Fig. 14. From the figure, it was observed that the experimental results and simulation predictions were in good agreement. The maximum error was observed to be 7.8% which was due to experimental uncertainities. Henceforth, the simulation data appears to be a valuable resource in predicting the experimental results [24].

Fig. 13
figure 13

Temperature distribution in CD microchannel for inlet mass flow rate of a 0.001232 kg/s b 0.01355 kg/s c 0.01848 kg/s

Full size image
Fig. 14
figure 14

Comparison of Nu Vs Re obtained from experiment and simulation

Full size image

5 Conclusions

Microchannels with varying cross-section are being extensively used for cooling in this study. Converging–diverging microchannels were fabricated on Aluminium heat sink using micro milling technique. Experimental and three dimensional CFD investigations were carried out for CD microchannels. Experimental setup was created and measurements were conducted for mass flow rate ranging from 0.001232 to 0.01848 kg/s and heat flux ranges of 10–50 W/cm2. Pressure drop and heat transfer coefficients were measured for the subjected experiments. On experiments, it was found that the pressure drop in CD microchannels vary linearly while the friction factor observed to decrease due to the influence of viscous forces. The heat transfer coefficient was detected to be higher than the straight microchannel. Re-circulation zones in CD microchannel enhances the perturbation of solid–fluid boundary in the channel wall to enhance the heat transfer coefficient. The predicted Nusselt number affirms the same. On further correlations, theoretical Nu decreased 0.9% and increased 3.7% when the aspect ratio was measured at converging section and diverging section respectively. Also, the experimental Nu was 15.8% higher than parallel microchannels. Numerical simulations were performed for similar experimental conditions in order to validate the measurement data. Numerical results adheres to experimental measurements and found to have maximum error of 8%. Therefore, CD microchannels exhibits better thermal–hydraulic performace than the straight microchannels. This affirms the application of CD microchannels in thermal management of microdevices.