Introduction

The microchannel heat exchanger is the ‘heat exchanger of next generation,’ the new technology which can provide the solution for the cooling of high-heat flux problem. Microchannels possess the ability of high heat transfer rate because of their large surface area-to-volume ratio, which makes them the finest choice for the development of the microheat exchangers used for the cooling of electronic and aerospace components. There are many applications of the microchannels in industrial sectors but not limited to electronic, automobile, aerospace and cryosurgery area. Microchannels have attracted the minds of researchers because of its wide applications in various areas like electronic devices, automobile, aerospace, fuel processor and nuclear reactors. Microchannels serve as the sole component in order to build the microheat exchangers which apparently are seen as the heat exchangers of the next generation because of their application advantages as well as their efficiency. To enhance the thermal and hydraulic performance of the microchannels, various studies on both numerical and experimental works have been carried out in the past years for single-phase and multi-phase fluid flows. Basically, the cooling is done in two ways: the air cooling or the water cooling. When the requirement of cooling passes over 100 W cm−2, then the cooling of system is not feasible using air or water for flowing fluids. In several applications, when components of the system require to remove the heat flux of higher order, then it becomes difficult to use a larger heat sink, probably lager than component itself. It has already provided a new platform for researchers to develop a new kind of heat sinks that can be incorporated with the heat source for the removal of the heat flux. These heat sinks are made of silicon material, and a layer of silicon oxide is kept over them for the electrical insulation. Quite narrow microchannels of different shapes such as rectangular, circular or triangular are framed around using fins to increase the rate of heat removal by flowing the cold fluid (coolant) through these microchannels. Tuckerman and Pease [1] explained the idea of using microchannel heat sinks to investigate the single-phase forced convective heat transfer. It was stated that it could potentially remove heat flux up to 1000 W cm−2. The value of the convection heat transfer coefficient (h) was found to be an obstacle to obtain low thermal resistance value. It was seen that for laminar flow, the value of ‘h’ is inversely proportional to the width of the channel. However, the high aspect ratio (w H−1) will, in turn, increase the surface area which reduces the value of thermal resistance to provide a high rate of heat transfer. Philips [2] performed an experiment on microchannel heat sinks that are used for the applications in microelectronics such as laser diodes and for high-energy laser mirrors. To cool the laser diode, heat sinks are made with indium phosphide having a thermal resistance of 0.072 °C W−1 cm−2, which can dissipate the heat flux of 1000 W cm−2. Hahn et al. [3] investigated the packaging of high-powered multi-chip modules, basically focused on the three major areas: (1) fabrication of multi-chip module, (2) development of high-performance microchannel heat sinks and (3) assembling technology having very low thermal performance. A thermostat module having a size of 2 × 2 inches is capable of heat dissipation of hundreds of watts. Thermal resistance value was maintained below 0.6 K cm−2 W−1 at a heat flux of 50 W cm−2. Martin et al. [4] investigated microchannel heat exchangers which were designed for the cooling of heat flux of order 100 W cm−2. Each section had 150 microchannels having dimensions of 100 μm deep, 100 μm wide and space 50–100 μm. It was found that the thermal capacity of the crystal board was 20 W and 15 W at the crystal temperature of 90 °C and 70 °C, respectively. It was observed that at the flow rate of 50 g s−1 of water, the system provides an efficient cooling of 0.6 × 105 W m−2 K−1 [5]. Studies have observed that the heat sink incorporating microchannel has been known for the collective heat flux removal of up to 500 W cm−2 used in the cooling of laser diode array [6,7,8]. The basic governing equations related to fluid flow in conventional channels are given below [9]:

Conservation of mass

$$\nabla \cdot \left( {\rho_{\text{m}} \mathop {V_{\text{m}} }\limits^{ \to } } \right) = 0$$
(1)

Conservation of momentum

$$\nabla \cdot \left( {\rho_{\text{m}} \mathop {V_{\text{m}} }\limits^{ \to } \mathop {V_{\text{m}} }\limits^{ \to } } \right) = - \nabla \cdot P + \nabla \cdot [\tau - \tau_{\text{t}} ] + \nabla \cdot \left[ {\sum\limits_{k - 1}^{n} {\phi_{\text{k}} \rho_{\text{k}} \mathop {V_{{{\text{dr}},{\text{k}}}} }\limits^{ \to } \mathop {V_{{{\text{dr}},{\text{k}}}} }\limits^{ \to } } } \right]$$
(2)

where

$$\tau = \mu_{\text{m}} \nabla \cdot \mathop {V_{\text{m}} }\limits^{ \to } \,{\text{and}}\,\tau_{\rm{t}} = \sum\limits_{\rm{k} - 1}^{n} {\left[ {\phi_{\text{k}} \rho_{\text{k}} \mathop {v_{\text{k}} v_{\text{k}} }\limits^{ \to } } \right]}$$

Conservation of energy

$$\nabla \cdot \sum\limits_{k - 1}^{n} {\left\{ {\phi_{\text{k}} \mathop {V_{\text{k}} }\limits^{ \to } (\rho_{\text{k}} H_{\text{k}} + P)} \right\} = \nabla \cdot \left( {\lambda_{\text{eff}} \nabla T + C_{\text{P}} \rho_{\text{m}} \mathop {vt}\limits^{ \to } } \right)}$$
(3)

The classification criteria of the microchannels were proposed by many researchers over the years according to the various parameters. Serizawa et al. [10] defined the criteria for the microchannel and defined the Laplace constant should be greater than or equal to the microchannel diameter as given below.

$$\lambda \ge D_{\text{h}}$$
(4)

where ‘\(\lambda\)’ is the Laplace constant and the ‘Dh’ is the microchannel hydraulic diameter. Another classification was given by Mehendale et al. [11] and they classified microchannels on the basis of its hydraulic diameter (see Table 1). Kandlikar and Grande [12] also classified the microchannels on the same basis as shown in Table 2.

Table 1 Classification of microchannels [11]
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Table 2 Classification of channels [12]
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Microchannels were defined by Palm [13] as an element or means of heat transfer which does not directly follow the classical theories of heat transfer or fluid flow. Thus, the flow characteristics such as friction factor and the heat transfer characteristics cannot be easily predicted. A study by Stefan [14] used a microscale-sized system which does not show the same typical observations as the macro-sized systems and stated that it is not appropriate to differentiate the microchannels and minichannels on the basis of a specific diameter. The investigation performed by Halelfadl et al. [15] mainly concentrated on the analytical optimization of heat sink incorporated with rectangular microchannels using the water-based solution of nanofluids as a coolant. The results of the study demonstrated that using nanofluids as working fluids has reduced the value of overall thermal resistance and it can increase the thermal performance of the flowing fluid while dealing at large temperature. A study by Warrier et al. [16] investigated a two-phase cooling system incorporating microchannels which also had the side wall for the purpose of reducing the high heat fluxes in semiconductor devices. Yu et al. [17] investigated the thermal characteristics as well as the hydraulic characteristics of the fractal tree-shaped microchannels having various aspect ratios. The range of Reynolds number was set between 150 and 1200. The results of the investigation proposed that microchannels with fractal tree-shaped geometry had much larger heat transfer coefficient as compared to the straight microchannels. The same treelike thermal performance of microchannels was also investigated [18,19,20,21]. Chamkha et al. [22, 23] numerically investigated the fully developed free and mixed conventions in vertically oriented channel. Results of the study show that for a fixed value of R, the material parameter gets reduced to the velocity profile in free convection and velocity and microporation profiles become distorted in developing region of mixed convention. The study also investigated hydromagnetic two-phase flow [24] and reported that the flow rate of fluid and particle phase get reduced due to the presence of the particles in the channel and the volumetric flow rates and the skin friction coefficients decrease as the Hartmann number increases. Chamkha [25,26,27,28] also investigated various areas such as unsteady laminar hydromagnetic flow and heat transfer characteristics in channels, porous channels and vertical channels with wall heating conditions and reviewed the application of nanofluids in microchannels. The magneto hydrodynamic flow in a vertical microchannel using the Al2O3–water-based nanofluid was analyzed by Ibáñez et al. [29]. They concluded that the nanoparticle volume fraction and slip length reach the higher value, when the permeability increases, while the global entropy generation is at its minimum value, and they reach the minimum value when the global entropy generation decreases. Shashi kumar et al. [30] investigated the entropy generation in microchannels using nanofluids with partial slips and convective conditions. It observed that the entropy generation was higher in Ti6Al4VH2O nanofluid as compared to AA7075-H2O. In the study performed by Khodabandeh et al. [31], they showed that microchannel heat sink design with sinusoidal cavities and rectangular ribs can increase the Nusselt number and heat transfer.

The effects of the flow fluctuation during boiling and periodic reversed flow were studied [21, 32,33,34,35,36,37,38] and stated that the thermal performance of the microchannel used as an evaporator in an air-conditioning system may create a problem. Tuo and Hrnjak [21] proposed a bright solution of the problem to decrease these effects to a significant level by using the ventilation system and also by altering the route of backflow vapor collected at the header at the inlet. Moallem et al. [39] performed an experiment on the microchannel heat exchanger incorporated with louvered fins and the effects of frost formation. The results of the experiment concluded that the temperature of the fin surface and the air humidity were the primary parameters which affect mostly the rapid growth rate of frost at a particular dry bulb temperature of the air. The performance of the cyclic frosting and defrosting for two different types of microheat exchangers was also experimented and studied by Xu et al. [40] and Liu et al. [41]. There were several other similar studies that have used microchannels and various conditions to increase the thermal performance of the microchannel heat exchangers [42,43,44,45,46,47,48,49,50,51,52,53]. In another study by Morini et al. [54], they explained a heat sink. Using the numerical investigation, Sarangi et al. [55] developed a numerical model of boiling heat transfer by using microchannels. They focused on the phenomenon of forced convection in two-phase fluid flow in microchannels and used water as working fluid. Flat microtubes made of aluminum materials remained the center of attention for many years. In majority, it was used in designing the air-conditioning systems and in the refrigeration industries where high cooling is required [56,57,58,59,60,61,62,63]. Zhang et al. [56] performed both experimental and numerical works on a bent plane consisting of flat microchannel tubes. The conclusion of the study described that under normal working conditions, the degradation of the microchannels is very small.

In the last past few years, research on the determination of convective heat transfer and pressure drop in mini- and microchannels has been conducted at a rapid rate [64,65,66,67]. Szczukiewicz et al. [64] evaluated the heat transfer coefficient of some refrigerants used in evaporator consist of multiple microchannels. It was concluded that two-phase flow heat sink which has microgap possesses very effective capability to lower the disadvantages associated with the normal heat sink of two-phase flow, mainly of that heat sink which shows some flow instabilities, a reversal inflow and the variation in wall temperatures between the microchannels. Another experiment performed by Alam et al. [65] investigated the characteristics of fluid flow and heat transfer in a microgap heat sink while using deionized water as flowing fluid. The results of this study were compared with the results obtained from the investigation performed over the normal microchannel heat sink. Fani et al. [66] performed an investigation on nanoparticles of spherical shape and studied about the size effect on the thermal performance. The microchannel heat sink consisted of microchannels of trapezoidal shape. This implies that the fluid has more effect on thermal performance as compared to the nanofluids. A detailed review was proposed by Ramezanizadeh [68] on the various approaches for cooling fuel cells. The results of the study show that the use of nanofluids increases the heat transfer and improves the efficiency and also the reduction in size and mass helps in improving the cooling efficiency. Umavathi et al. [69] investigated the mixed convective flow in vertically oriented microchannels which were filled with the electrically conducting viscous fluid. Ramezanizadeh et al. [70] studied the various intelligent methods used for the prediction of thermal conductivity of nanofluids. Abchouyeh [71] studied the heat transfer enhancement using nanofluids between two parallel plates and concluded that with the increase in nanosized particle concentration, the mean Nusselt number increases. The application of nanofluids in thermosyphons was reviewed by Ramezanizadeh et al. [72]. It was explained that the efficiency of thermosyphons increases using nanofluids. Ramezanizadeh et al. [73] investigated nanofluidic thermosyphon heat exchanger using Ni/glycerol–water nanofluid as working fluid in three different concentrations. It was observed that the increase in mass flow rate and inlet temperature of hot stream increases the heat transfer. Chamkha [74] studied the flow of immiscible fluids in porous and non-porous channels in steady laminar regime. It was seen that the increase in the Hartmann number, electrical conductivity ratio and the inverse Darcy number leads to reduction in flow velocities and increases the viscosity ratio. Kumar et al. [75] investigated the fully developed free convective flow in vertical channel using micropolar and viscous fluids. Umavathi et al. [76,77,78,79] investigated unsteady two-fluid flow, Couette flow, mixed convection flow and heat and mass transfer in vertically double-passage channel. In the investigation of unsteady two-fluid flow, it was seen that velocity and temperature decrease when the viscosity ratio increases and increase when the frequency parameter gets increased [76]. It has been seen that the cross-sectional and the structural geometry of microchannel are the most important parameters which affect the flow and heat transfer characteristics. Several studies have been performed by using circular, triangular, trapezoidal, rectangular and square cross sections, and also, researchers have carried out investigations on branched microchannel and fractal-like microchannel heat sink as shown in Fig. 1 [80].

Fig. 1
figure 1

Fractal-like microchannel network [80]

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Single-phase flow

Friction factor and pressure drop

Peiyi and Little [81] executed a number of experiments by using gaseous flow instead of liquid flow in microchannels of trapezoidal cross section made of silicon/glass. The experiments were performed to investigate the characteristics of friction factor and pressure drop inside microchannels. Results of their study showed that the flow transition in microchannel occurs very early as compared to the conventional channels. It was observed that the flow transition in microchannels occurs at Reynolds number range of 400–900. It was concluded that the transition also depends upon the test conditions. They made a suggestion to decrease the transition Reynolds number or the critical Reynolds number for microchannels to advance the heat transfer characteristics. Investigation performed by Pfahler et al. [82] used three microchannels having a rectangular cross section. Area of a cross section of these microchannels was in the range of 80–7200 μm2. In this study, N-propanol was used as a flowing fluid. The results of this study interpreted the characteristics of fluid flow and friction factor. Numerous experiments were performed to investigate friction factor in various conditions by using different methods. Peng et al. [83], Wang et al. [84], Peng and Wang [85] and Peng and Peterson [86] concentrated to investigate the fluid flow and heat transfer characteristics for the various structures of microchannels. Yu et al. [87] used nitrogen gas and water as flowing fluid in microchannels having a hydraulic diameter of 19.52 and 102 μm.

Hydraulic diameter for a different cross section of channels is defined as,

$$D_{\text{h}} = \frac{{A_{\text{C}} }}{{P_{\text{w}} }}$$
(5)

where ‘Ac’ is an area of cross section and ‘Pw’ is wetted perimeter.

For circular channels, the hydraulic diameter is,

$$D_{\text{h}} = D_{\text{i}}$$
(6)

where ‘Di’ is the internal diameter of the channel.

For rectangular channels, the hydraulic diameter is,

$$D_{\text{h}} = \frac{4ab}{{2\left( {a + b} \right)}}$$
(7)

where ‘a’ and ‘b’ are the sides of the channel.

For Isosceles triangular channel, the hydraulic diameter is,

$$D_{\text{h}} = \frac{a}{2\surd 3}$$
(8)

where ‘a’ is the side of the isosceles triangle.

Researchers have used the classical laminar theory as the basis to understand the nature of fluid flow characteristics in microchannels.

$$C^{*} = \frac{{\left( {f*\text{Re} } \right)_{ \exp } }}{{\left( {f*\text{Re} } \right)_{\text{theory}} }}$$
(9)

where ‘f’ is the friction factor and ‘Re’ is Reynolds number, but the multiplication of both quantities gives a non-dimensional number. It is calculated experimentally as well as theoretically and the ratio of it gives the ‘C*’ as the characteristic quantity. Theoretically, the values of friction factor in circular tubes are calculated as:

For laminar flow:

$$f = \frac{64}{\text{Re}}$$
(10)

For turbulent flow:

$$f = 0.3164\text{Re}^{ - 0.25}$$
(11)

A study conducted by Hwang and Kim [88] described the characteristics of pressure drop in the microchannels with internal diameter of 0.244, 0.430 and 0.792 mm, and R134a was used as working fluid to investigate the Reynolds number range of 150–10,000. Yen et al. [89] performed experiment on microchannels having inner diameters of 0.19, 0.30 and 0.51 mm. Flowing fluid used in this study was HCFC123 and FC-72 to observe the variation in heat transfer and fluid flow characteristics. The results of the study showed that when the flow is in the laminar region on the Reynolds number value in the range of 20–265, then the value of friction factor in microchannels was in agreement with theoretical laminar flow. Celata et al. [90] performed an investigation on microchannel using water as flowing fluid. The investigation was performed experimentally as well as analytically in the Reynolds number range of 20–4000, while the hydraulic diameter of the microchannels was considered between 30 and 344 μm (see Table 3).

Table 3 Laminar flow in circular and non-circular channels with diameter (D) and sides a and b [90]
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In laminar flow regime, the Poiseuille number (P0) is a constant and given by the product of friction factor and Reynolds number.

$$P_{0} = f\,\text{Re}$$
(12)

Poiseuille number becomes a very important factor for the rectangular channels as it is a function of aspect ratio. Shah and London [91] proposed a correlation to determine the Poiseuille number. This correlation is as follows,

$$f\,\text{Re} = 24\left( {1 - 1.3553\alpha + 1.9467\alpha^{2} - 1.7012\alpha^{3} + 0.9564\alpha^{4} - 0.2537\alpha^{5} } \right)$$
(13)

The aspect ratio of the channel must be less than 1, and in some cases, it is greater than 1, and the inverse of it will be taken.

Nikuradse [92] also provides a correlation to calculate the value of Poiseuille number in the pure turbulent regime, which is read as:

$$f = \left[ {3.48 - 1.737\ln \left( {{\raise0.7ex\hbox{$e$} \!\mathord{\left/ {\vphantom {e D}}\right.\kern-0pt} \!\lower0.7ex\hbox{$D$}}} \right)} \right]^{ - 2}$$
(14)

where ‘e’ is absolute roughness of the pipe and ‘D’ denotes the internal diameter of the pipe.

The friction factor can be calculated by using a relation given by Colebrook et al. [93] given as follows:

$$\frac{1}{\surd f} = 3.48 - 1.737\ln \left\{ {\left( {\frac{e}{D}} \right) + \frac{9.35}{{\text{Re} \surd f}}} \right\}$$
(15)

The work of Kandlikar and Steinke [94] determined the friction factor by using two components. These two components were, friction factor taken from the classical theory of fully developed flow and the second component was Hagenbach factor.

$$\Delta P = \frac{{2\left( {f\text{Re} } \right)\mu VL}}{{D_{\text{h}}^{2} }} + \frac{{k\left( x \right)\rho \bar{V}^{2} }}{2}$$
(16)

where k(x) is known as the Hagenbach factor and given by,

$$k\left( x \right) = \left( {f_{\text{app}} - f_{\text{FD}} } \right)\frac{4x}{{d_{\text{h}} }}$$
(17)

In this relation, fapp and fFD are known as apparent and fully developed friction factors, respectively. The results of the study showed that this correlation only can predict the friction factor in the laminar region, and for the turbulent region, there is a discrepancy. Since this correlation was developed for the fully developed flow in microchannels and because of temperature variation along with the length of the microchannel, the temperature profile was not completely developed. Thus, this correlation shows deviation from the experimental data. For a rectangular cross section, the Hagenbach factor for fully developed flow is:

$$k\left( \infty \right) = \left( {0.6796 + 1.2197\alpha + 3.3089\alpha^{2} - 9.5921\alpha^{3} + 8.9089\alpha^{4} - 2.9959\alpha^{5} } \right)$$
(18)

Kandlikar and Grande [12] also performed an experiment and presented their findings as correlation to calculate the friction factor in the turbulent regime for fully developed flow.

$$f_{\text{app}} = \left( {0.0929 + \frac{1.01612}{{\frac{L}{{D_{\text{h}} }}}}} \right)\text{Re}^{{*^{{^{{\left( { - 0.268 - \frac{0.3298}{{\frac{L}{{D_{\text{h}} }}}}} \right)}} }} }}$$
(19)

where \(\text{Re}^{*}\) is known as laminar equivalent Reynolds number which was proposed by Jones [95], specifically for rectangular channels only as follows:

$$\text{Re}^{*} = \text{Re} \left\{ {\frac{2}{3} + \frac{11}{24} \propto \left( {2 - \propto } \right)} \right\}$$
(20)

It is very important to acknowledge that this correlation was only proposed for minichannels. Later it was confirmed by Kumar et al. [96] that it also can be used for microchannels successfully.

In the past few years, researchers have done work in single-phase flow in microchannels. They have quite explored areas of both liquid flow and gaseous flow. The work done is shown in Table 4 for liquid flow and in Table 5 for gaseous flow. Morini et al. [54] worked in turbulent flow and reported that the experimental value of friction factor was less than that of calculated by Blasius correlation for flow in smooth tubes, even after taking account of compressibility effects. When there is a gaseous flow, the compressibility effects seem to be more significant as compared to liquid flow. The basic condition for the significance of compressibility effects in gaseous flow occurs when one of these two given conditions as given below are satisfied

$${\text{Ma}} > 0.3$$
(21)
$$\frac{\Delta P}{{P_{\text{in}} }} > 0.05$$
(22)

where ‘Ma’ denotes the Mach number and ‘Pin’ denotes the pressure drop at inlet of channel.

Table 4 Single-phase pressure drop for liquid flow
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Table 5 Single-phase pressure drop for gaseous flow
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In the experiment performed by Vijayalakshmi et al. [206], they concluded that pressure distribution is linear in incompressible flow, when the value of Reynolds number exceeds up to 1600. However, pressure distribution linear in nature when the value of Reynolds number goes beyond 1600. These results were in agreement with the results of Kumar et al. [96]. Later on, another experiment performed by Ding et al. [210] explained that, when the value of pressure drop in the duct is nearly 10 kPa, the compressibility effects in gaseous flow tend to be more significant, while the Mach number is kept less than 0.1. Hrnjak and Tu [143] concentrated in the laminar regime and explained that roughness of the channel is not a very significant factor in laminar flow. However, when it comes to turbulent flow, it changes very chaotically and shows its effect on friction factor. Same observations were performed by Tang et al. [204] and stated that roughness factor of microchannels does not play any role in friction factor ergo friction is solely independent of surface roughness in laminar flow. However, the classical theory of fluid flow does not agree with these results. But, many other researchers [12, 87, 90, 118, 119, 210,211,212] confirmed that the friction factor is closely affected with the change in surface roughness in laminar flow. In some other experiments performed by Morini et al. [54, 205], Kohl et al. [203], Sharp and Adrian [213] and Hrnjak and Tu [143], they observed that there was no transition reported in laminar to the turbulent regime. According to other studies [81, 83, 86, 104, 115, 210, 212, 214], it was seen that there was an early transition as compared to the classical theory of fluid flow.

Heat transfer

The classical theory of fluid flow defines heat transfer on the basis of Nusselt number, a dimensionless quantity which depends upon heat transfer coefficient, hydraulic diameter of the channel and the thermal conductivity of the fluid flowing in the channel. The value of Nusselt number in fully developed laminar flow regime is 4.364, while the constant heat flux boundary condition was used. Experimental investigation performed by Cuta et al. [108] considered the problem of thermal entrance for laminar flow in combination with constant heat flux. The results of this study observed Nusselt number to be a function of axial distance, Reynolds number and Prandtl number.

$${\text{Nu}} = 4.364 + \left\{ {\frac{{0.00668\left( {{\raise0.7ex\hbox{${d_{\text{h}} }$} \!\mathord{\left/ {\vphantom {{d_{\text{h}} } x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)\text{Re} \Pr }}{{1 + 0.04\left[ {\left( {{\raise0.7ex\hbox{${d_{\text{h}} }$} \!\mathord{\left/ {\vphantom {{d_{\text{h}} } x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)\text{Re} \Pr } \right]^{{\frac{2}{3}}} }}} \right\}$$
(23)

An experiment was performed by Schilder et al. [215] on single-phase flow while keeping the diameter of the microchannel as 0.6 mm. In another experiment conducted by Admas et al. [114], they worked on the turbulent flow by considering water as flowing fluid in microchannels of circular cross section having hydraulic diameters of 0.76 and 0.109 mm. They proposed a correlation as given below:

$${\text{Nu}} = {\text{Nu}}_{\text{Gn}} + \left( {1 + F} \right)$$
(24)

where

$${\text{Nu}}_{\text{Gn}} = \frac{{\left( {\frac{f}{8}} \right)\left( {\text{Re} - 1000} \right)\Pr }}{{1 + 12.7\left( {\frac{f}{8}} \right)^{{\frac{1}{2}}} \left( {\Pr^{{\frac{2}{3}}} - 1} \right)}}$$
(25)

where ‘f’ is

$$f = \left\{ {1.82\log \left( {\text{Re} } \right) - 1.64} \right\}^{ - 2}$$
(26)
$$F = 7.6 \times 10^{ - 5} \text{Re} \left\{ {1 - \left( {\frac{{d_{\text{h}} }}{{d_{\text{o}} }}} \right)^{2} } \right\}$$
(27)

NuGn was known as Nusselt number given by Gnielinski [216] in his correlation. Garimella and Singhal [217] proposed both laminar and turbulent conditions and explained that there is a small agreement between experimental results and classical theory. The study concluded that transition from laminar flow to turbulent flow occurs at a very low value of Reynolds number. Mala and Li [115] stated that microchannel flow ceases early transition at low Reynolds number range of 300–900. Peng et al. [83] found the transition to be at 700. Silvério and Moreira [218] contradicted the previous results and stated that the transition does not take place at Reynolds number value less than 1800. These results were in agreement with the results of Sharp and Adrian [213]. Single-phase flow heat transfer is shown in Table 6.

Table 6 Single-phase flow heat transfer
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Che et al. [230] investigated Peclet number effect for liquid–liquid two-phase flow heat transfer for rectangular cross-sectional microchannel. The range of Peclet number was kept from 25 to 800. It was observed that with the increase in Peclet number, Nusselt number increases with respect to time as shown in Fig. 2 and decreases the heat transfer index as shown in Fig. 3.

Fig. 2
figure 2

Nusselt number using varied Peclet numbers with fixed droplet length and the aspect ratio [230]

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Fig. 3
figure 3

Heat transfer index using varied Peclet numbers with fixed droplet length and the aspect ratio [230]

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Fischer et al. [231] carried out investigation on laminar heat transfer with water–5 cS silicone oil or PAO as flowing fluids. The nanoparticles of Al2O3 having the volume fraction of 3% were suspended in water/PAO droplets. In Fig. 4, the average Nusselt number was compared with non-dimensional pressure drop, normalized by pressure drop of single phase of water flow. Figure 5 shows that the 5 cS silicone oil shows good performance in comparison with other fluid. It was also seen that the addition of nanoparticles into PAO oil enhances the heat transfer and reduces the pressure drop slightly. Figure 5 shows that the highly viscous silicone oil had the highest value of Nusselt number, but the pressure increases about 11% for single-phase fluid flow.

Fig. 4
figure 4

Averaged Nusselt number versus pressure drop normalized by the pressure drop for differently shaped droplets, with and without nanoparticles [231]

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Fig. 5
figure 5

Averaged Nusselt number versus pressure drop normalized by the pressure drop for spherical droplets with different viscosity ratios [231]

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Peiyi and Little [81] performed heat transfer characteristics analysis for both laminar flow and turbulent flow and concluded that values of Nusselt number were greater than as compared to the conventional theory. Later Choi et al. [194] confirmed that correlations given by Peiyi and Little were not in agreement with their experimental results. Extensive study of heat transfer characteristics in single-phase flow in microchannels suggested the following important points:

  1. i.

    A study performed by [83, 85, 86, 106, 119, 124, 214, 229] concluded that Nusselt number values obtained by experimental results are much less as compared to the conventional theories.

  2. ii.

    The experimental results of [12, 81, 107, 109, 110, 114, 121, 130, 140, 232,233,234] confirmed that Nusselt number values are greater as compared to the predicted values of conventional theories.

  3. iii.

    The results of a study performed by [16, 111, 127, 135, 235, 236] concluded that conventional theories and experimental results are in very good agreement with each other in correlations given for laminar and turbulent flow regimes.

It is observed that even though researchers have done extensive studies on microchannels, there are still some deviations in results. Finally, it was concluded that these deviations could be because of the following reasons:

  1. i.

    Velocity profile and the entrance region are the most crucial area in microchannel flow. Since velocity profile, temperature and the entrance region are in developing mode, the changes in Nusselt number along with the microchannels take place.

  2. ii.

    The theory of fluid dynamics describes that there are two entrance length: hydrodynamic entrance length and thermal entrance length. When the velocity profile becomes fully developed, then it is necessary to take into account the effects of thermal entrance length.

  3. iii.

    The complications arise when the value of the Prandtl number becomes greater than unity. In that case, Graetz number becomes the criteria to avoid the entrance effects. Morini [237] suggested taking account the effects of entrance length after Graetz number higher than 10, and Rosa et al. [238] reported that entrance effects might be important at high Reynolds number values.

  4. iv.

    The difference between inlet and outlet temperatures could be very high in microchannels. Thus, it is possible that the deviation in Nusselt number takes place because of changes in thermophysical properties.

Multi-phase flow

Multi-phase flow is in which the flow takes place in two or more than forms of state of the matter. When two fluids flow in the channels, it is allowed to change its phase in order to provide maximum heat transfer similar to a case of evaporator and condenser. Most common multi-phase flow exists in the form of two-phase flow. Other types of two-phase flows are as follows: liquid–liquid, liquid–gas and gas–gas flows.

Friction factor and pressure drop

In two-phase flow, the flows are classified into two major categories: (1) homogeneous flow and (2) separated flow. In gas–liquid flow, most of the flows are homogeneous flow and suppose to mix up; thus, the calculation of pressure drop can be done by using standard pressured drop correlations. Since only latent heat is assumed to be exchanged between fluids during phase change, the mean properties are computed as weighted as compared to the vapor and liquid percentage present. A number of correlations for two-phase flow have been suggested. Thoughtful selection of correlation is very important while applying on a model for precise results. Two-phase pressure drop is a function of friction factor, acceleration factor and gravitational factor terms.

$$\frac{{{\text{d}}P}}{{{\text{d}}Z}}_{{{\text{total}}\,{\text{pressure}}\,{\text{drop}}}} = \frac{{{\text{d}}P}}{{{\text{d}}Z}}_{\text{frictional}} + \frac{{{\text{d}}P}}{{{\text{d}}Z}}_{\text{accelaration}} + \frac{{{\text{d}}P}}{{{\text{d}}Z}}_{\text{gravitational}}$$
(28)
$${\text{Horizontal}}\,{\text{ flow}}:\,\frac{{{\text{d}}P}}{{{\text{d}}Z}}_{\text{gravitational}} = 0$$
(29)
$${\text{Adiabatic}}\,{\text{ flow}}:\,\frac{{{\text{d}}P}}{{{\text{d}}Z}}_{\text{accelaration}} = 0$$
(30)

Thus, the total pressure drop,

$$\frac{{{\text{d}}P}}{{{\text{d}}Z}}_{{{\text{total}}\,{\text{pressure}}\,{\text{drop}}}} = \frac{{{\text{d}}P}}{{{\text{d}}Z}}_{\text{frictional}} = \frac{{2f_{\text{TP}} LG^{2} }}{{\rho_{\text{TP}} D_{\text{h}} }}$$
(31)

where ‘fTP’ is denoted as two-phase friction factor and ‘G’ is known as mass velocity and is equivalent to \(G = \rho v\). Also, \(\rho_{\rm{TP}}\) is known as two-phase density.

$$\rho_{\text{TP}} = \left[ {\frac{x}{{\rho_{\text{G}} }} + \frac{1 - x}{{\rho_{\text{L}} }}} \right]^{ - 1}$$
(32)

The two-phase friction factor is described as,

  • For laminar flow,

    $$f_{\text{TP}} = \frac{16}{{\text{Re}_{\text{TP}} }}$$
    (33)

  • For turbulent flow,

    $$f_{\text{TP}} = 0.079\text{Re}_{\text{TP}}^{ - 0.25}$$
    (34)

where two-phase Reynolds number is defined as,

$$\text{Re}_{\text{TP}} = \frac{{{\text{GD}}_{\text{h}} }}{{\mu_{\text{TP}} }}$$
(35)

where \(\mu_{\text{TP}}\) is known as two-phase viscosity.

The study carried out by Cioncolini et al. [211] introduced a new correlation, which was based on the Weber number. This method of observation at macroscale was also meant to sum up the criteria of microscale in both laminar flows. Another correlation given by Costa-Patry et al. [239] performed an experiment on flow boiling in multi-microchannels having a width of 85 μm. It was concluded that correlation given by Cioncolini was in agreement with their results.

$$f_{\text{TP}} = 0.0196\,{\text{We}}_{\text{c}}^{ - 0.0722} \text{Re}_{\text{L}}^{0.318}$$
(36)

Few other viscosity models applicable in two phases are as follows:

  1. i.

    The model is given by McAdams et al. [240]:

    $$\mu_{\text{TP}} = \left[ {\frac{x}{{\mu_{\text{G}} }} + \frac{1 - x}{{\mu_{\text{L}} }}} \right]^{ - 1}$$
    (37)
  2. ii.

    The model is given by Owens [241]:

    $$\mu_{\text{TP}} = \mu_{\text{L}}$$
    (38)
  3. iii.

    The model is given by Cicchitti et al. [242]:

    $$\mu_{\text{TP}} = x\mu_{\text{G}} + \left( {1 - x} \right)\mu_{\text{L}}$$
    (39)
  4. iv.

    The model is given by Dukler et al. [243]:

    $$\mu_{\text{TP}} = \beta \mu_{\text{G}} + \left( {1 - \beta } \right)\mu_{\text{L}}$$
    (40)
  5. v.

    The model is given by Beattie and Whalley [244]:

    $$\mu_{\text{TP}} = \beta \mu_{\text{G}} + \left( {1 - \beta } \right)\left( {1 + 2.5\beta } \right)\mu_{\text{L}}$$
    (41)
  6. vi.

    The model is given by Awad and Muzychka [245]:

    $$\mu_{\text{TP}} = \frac{{2\mu_{\text{G}} + \mu_{\text{L}} - 2\left( {\mu_{\text{G}} - \mu_{\text{L}} } \right)\left( {1 - x} \right)}}{{2\mu_{\text{G}} + \mu_{\text{L}} + \left( {\mu_{\text{G}} - \mu_{\text{L}} } \right)\left( {1 - x} \right)}}$$
    (42)

Venkatesan et al. [246], Cioncolini et al. [218] and Choi and Kim [247] identified the two-phase flow patterns and reported that only in case of bubble flow, HFM is applicable in which flow field is with weak disturbance. Liu et al. [248] reported that bubble flow patterns only occur in high flow velocities of liquids and low velocities of gases. However, Taylor flow is also known as slug flow which also demonstrated the applicability of the HFM assumption. Defined criteria for HFM assumption is to where the tube diameter is less than the diameter of bubble length moving along the capillary. In general, slug flow is not always considered as SFM (separated flow model) because flow patterns do not always come into the picture in SFM. Study on two-phase pressure drop is also represented (see Table 7).

Table 7 Two-phase pressure drop
Full size table

Experiment performed by Choi and Kim [247] described one of the most precise viscosity models, a model of Beattie and Whalley. It is focused on volumetric specification. The results reported that Dukler’s model over-calculates the two-phase pressure drop, but it is under-calculated by rest of the models. Chung et al. [256] reported that when Dukler’s model is used for diameter range of 530–250 μm, it estimates pressure drops. However, when it calculates data in diameter range of 100–50 μm, it shows a good agreement. In another case, it was seen that Beattie and Whalley model was used for the diameter range of 530–250 μm, and it estimates pressure drop very significantly as compared to diameter range of 100–50 μm. Study performed by Choi and Kim at microchannel diameter for Dh = 490 μm reported that Duckler’s model is not very precise, but at diameter Dh = 141 μm it predicts pressure drop more precisely. It was also seen that these results was in agreement with the results of Chung et al. [256], while the model of Beattie and Whalley does not show significant difference for microchannel diameters 490 μm and 141 μm.

Figures 6 and 7 show the pressure drop comparison based on the experimental results of two homogenous flow void fraction models (McAdams and Cicchitti) for flow in fractal-like microchannels. The models are in agreement with the experimental results, and the mean deviations for both models are 12.0% and 12.1% and maximum deviation of 20.3% and 20.8% for McAdams and Cicchitti models, respectively. It was stated that both models can overpredict the measured pressure drop values.

Fig. 6
figure 6

Homogenous flow models given by McAdams [80]

Full size image
Fig. 7
figure 7

Homogenous flow models given by Cicchitti [80]

Full size image

A study conducted by Lee and Mudawar [257] explained the two-phase viscosity model predicted by Cicchitti. They explained that this model was slightly different from the rest of the models and provides a greater estimation of two-phase viscosity at very low quality as compared to rest of the models. They described further that the tendency of overprediction of data at low quality and under-prediction of data at higher quality by Cicchitti method gives reasonable mean deviation. This was also verified by the results of Cioncolini et al. [211], Yue et al. [254] and Kawahara et al. [263]. It is important to understand that the best model calculates the pressure drop in two phases which are different for each and every study. It is also important to notice that the hydraulic diameter of characteristics of the two-phase flowing fluid is the key component for the selection of the best model. It was concluded in various studies that the models of Dukler and Cioncolini are capable of predicting the data precisely for microchannels. On the other hand, the models given by Beattie and Whalley and McAdams are best to predict the pressure drop of two phases in minichannels and macro-channels. Separated flow model (SFM) is a model in which both fluids (liquid and gas) flow separately from each other in a channel. Each phase takes up a certain area of the cross section of the channel. Separated flow model has been studied extensively and experiments have been performed as well considering the combination of air–water as flowing fluid. In most of the cases, triangular geometry of cross section of microchannels has been used. Many researchers have performed experiments by considering various hydraulic diameters. Zhao and Bi [267] conducted a study for hydraulic diameters range of 0.87–2.89 mm. Kawahara et al. [263] studied the SFM for nitrogen and water as flowing fluid in microchannels of circular cross section having a hydraulic diameter of 100 μm. Chung et al. [256] performed experiments on microchannel of square cross section having a hydraulic diameter of 96 μm. A study performed by Lockhart and Martinelli [268] used the concept of the two-phase multiplier for the calculation of pressure drop in liquid flow:

$$\emptyset_{\text{L}}^{2} = 1 + \frac{C}{X} + \frac{1}{{X^{2} }}$$
(43)
$$X = \sqrt {\left[ {\frac{{\left( {\frac{{{\text{d}}P}}{{{\text{d}}z}}} \right)_{\text{L}} }}{{\left( {\frac{{{\text{d}}P}}{{{\text{d}}z}}} \right)_{\text{V}} }}} \right]}$$
(44)

The value of two-phase multiplier, denoted by \(\emptyset_{\text{L}}^{2}\) is calculated with the help of coefficient ‘C’ and a parameter given by Lockhart–Martinelli, X2, which gives the ratio of single-phase liquid to a single-phase gas pressure gradient. Study conducted by Friedel [269] reported a correlation to calculate the pressure gradient multiplier of two-phase flow as follows:

$$\emptyset_{\text{LO}}^{2} = E + \frac{{3.24{\text{FH}}}}{{{\text{Fr}}^{0.045} {\text{We}}^{0.035} }}$$
(45)
$$E = \left( {1 - x^{2} } \right) + x^{2} \frac{{\rho_{\text{L}} f_{\text{vo}} }}{{\rho_{\text{v}} f_{\text{Lo}} }}$$
(46)

where ‘Fr’ and ‘We’ are commonly known as Froude and Weber numbers and also \(f_{\text{vo}}\) and \(f_{\text{Lo}}\) are known as friction factors of gas and liquid at given mass flux ‘G.’

Since many numbers of experimentations were performed below the condition where the value of Reynolds number for gas and liquid (ReL and ReG) was less than 1000, the model of Lockhart and Martinelli estimates the value of ‘C’ to be 5. The findings of other experiments performed by Yue et al. [254, 259], Chung et al. [256] and Fukano and Kariyasaki [270] depicted that results cannot be well estimated by only one value of 5. Thus, they stated further that the value of C tends to get lower as the diameter of the channel is decreased from 530 to 50 μm. This finding was confirmed by the results of Yue et al. [254], who described that the models given by Lockhart and Martinelli cannot predict dependency of C, well the values of mass flux as good as it does for gas flow and liquid flow on superficial velocities. Many researchers have given other correlations to calculate the value of Cin order to further calculate the value of the two-phase multiplier. Experimental results of Cavallini et al. [271] stated that the model is given by Mishima and Hibiki [249] capable of estimating pressure drop in two-phase flow in condensation process of R134a and R-236a in channels having a diameter of 1.4 mm. It was seen that the correlation proposed by Mishima and Hibiki [249] takes account of the dependency of C on the size of the channel, while the channel gap is maintained between 0.4 and 4 mm.

Another equation to estimate the two-phase multiplier was introduced by Chisholm [272] which is as follows:

$$\emptyset_{\text{LO}}^{2} = 1 + \left( {Y^{2} - 1} \right)\left[ {Bx^{{2 - \frac{\rm{n}}{2}}} \left( {1 - x} \right)^{{2 - \frac{\rm{n}}{2}}} + x^{2 - \rm{n}} } \right]$$
(47)

where the value of n is 0.25 and Y is known as Chisholm parameter which is as follows:

$$Y = \frac{{\left( {{\text{d}}P/{\text{d}}Z} \right)_{\text{VO}} }}{{\left( {{\text{d}}P/{\text{d}}Z} \right)_{\text{LO}} }}$$
(48)

If the value of Chisholm parameter lies between 0 and 9.5, then the value of parameter B is as follows:

$$B = {\left\{ {\begin{array}{*{20}l} {4.8 :G \le 500 \,{\text{kg}}\,{\text{m}}^{-2} \,\text{s}^{-1} } \hfill \\ {\frac{2400}{G} :500 < G \le 1900\,{\text{kg}}\,{\text{m}}^{-2}} \hfill \\ {\frac{55}{{G^{0.5} }}:G > 1900 \,{\text{kg}} \, {\text{m}}^{-2} \,{\text{s}^{-1}}} \hfill \\ \end{array} } \right.}$$
(49)

When the values of Y lies between 9.5 and 28,

$$B = \left\{ {\begin{array}{*{20}l} {\frac{520}{{YG^{0.3} }}: G \le 600\,{\text{kg}}\,{\text{m}}^{-2} \,\text{s}^{-1} } \hfill \\ {\frac{21}{Y}: G > 600 \,{\text{kg}}\,{\text{m}}^{-2} \,\text{s}^{-1} } \hfill \\ \end{array} } \right.$$
(50)

When \(Y \ge 28\),

$$B = \frac{15,000}{{Y^{2} G^{0.5} }}$$
(51)

The research was done by Zhang and Webb [250] calculated pressure drop in two-phase flow in adiabatic condition using R404a as flowing fluid in aluminum channel having multi-port extrudes. The hydraulic diameter of the channel was considered as 2.13 mm, and two another tubes made of copper having a hydraulic diameter of 6.25 mm and 3.25 mm were inserted to make it as a concentric channel. Results of this study predicted that correlation given by Friedel cannot estimate two-phase pressure drop precisely. Thus, they proposed a new correlation to calculate the two-phase pressure gradient multiplier as follows:

$$\emptyset_{\text{LO}}^{2} = \left( {1 - x^{2} } \right) + 2.87x^{2} \left( {\frac{P}{{P_{\text{crti}} }}} \right)^{ - 1} + 1.68x^{0.8} \left( {1 - x} \right)^{0.25} \left( {\frac{p}{{P_{\text{crit}} }}} \right)^{ - 1.64}$$
(52)

where critical pressure is denoted by Pcrit and its value remains constant for each fluid.

An experiment performed by Müller-Steinhagen and Heck [273] used the collection of 9300 measurements values of pressure drop for developing a correlation which is as follows:

$$\left( {\frac{{{\text{d}}P}}{{{\text{d}}Z}}} \right)_{f} = F\left( {1 - x} \right)^{{\frac{1}{3}}} + \left( {\frac{{{\text{d}}P}}{{{\text{d}}Z}}} \right)_{\text{VO}} x^{3}$$
(53)

where the value of F is given by,

$$F = \left( {\frac{{{\text{d}}P}}{{{\text{d}}Z}}} \right)_{\text{LO}} + 2\left[ {\left( {\frac{{{\text{d}}P}}{{{\text{d}}Z}}} \right)_{\text{VO}} - \left( {\frac{{{\text{d}}P}}{{{\text{d}}Z}}} \right)_{\text{LO}} } \right]x$$
(54)

In previous years, when there was a focus on analyzing the flow patterns mainly churn flow and annular flow then it was realized that SFM is very realistic. Some studies [274,275,276] proposed that little bubble was present in liquid slug flow or the slug–annular flow which is typically not seen in Taylor flow. Basically, churn flow only comes into the picture where high velocity of gaseous flow takes place. It contains big gas bubbles, but slugs are very small as compared to the bubbles. Annular flow occurs at even higher gas flow velocities but for very low velocities of liquids [277]. A study was conducted by Venkatesan et al. [246] to define the naming structure for the flow patterns based on basic characteristics. Slug flow, annular flow and slug–annular flow come under intermittent flow and bubbly flow, and dispersed bubbly flow comes under dispersed flow regime. Some researchers such as Lee and Mudawar [257], Yue et al. [254, 259] and Chung et al. [256] also worked on flow patterns for the analysis of pressure drop in two-phase flow. However, in another experiment performed by Lee and Mudawar [252] concluded that churn flow cannot be very often found by SFM as stated by Yue et al. [254].

Venkatesan et al. [246] experimented on a tube having a diameter of 3.4 mm by considering a model of Dukler with the bubbly regime and reported that only 10% mean deviation from homogeneous flow model (HFM) takes place bubbly flow. Pressure drop was calculated by the mean deviation of 17%. Prediction of slug flow was made by using the correlation given by Chisholm, and the mean deviation was reported to be 14% as compared to the HFM which had the mean deviation of 43%. It was stated that HFM can very well estimate the bubbly flow regime in a tube having a hydraulic diameter of 1.7 mm with the mean deviation of only 7%. On the other hand, for the same diameter, Taylor flow and slug flow were predicted by the mean deviation of 28% and 22%, respectively. The previous results predicted that SFM is reliable to predict the churning flow and slug low. For the prediction of Taylor flow and bubbly flow, HFM is considered best. A study conducted by Venkatesan et al. [246] on annular flow regime reported that when the gas is flowing at high velocities in a tube having a hydraulic diameter of 0.6 mm, annular flow regime cannot be observed, which was in agreement with the results of Chung and Kawaji. It was concluded that it is possible because of strong surface tension force in narrow channels appearing in microchannels which makes the liquid film to incorporate with the gas core comfortably as compared to the minichannels. This is why the existence of annular flow does not take place. Mean deviation between HFMs and experimental results are shown in Table 8.

Table 8 Mean deviation between HFM and experimental results
Full size table

In two-phase flow in microchannels, viscous forces and inertia forces play a vital role, while the capillary forces are easily negligible [278]. However, it was observed that when the diameter of the tube is reduced then capillary forces comes into the picture and seems to play an important role in the determination of two-phase flow patterns. An experiment performed by Li and Wu [279] described that in theory, there are only four important forces which can be related to the two-phase flow patterns in narrow channels or conventional channels, which are:

  1. (a)

    Surface tension forces

  2. (b)

    Gravitational forces

  3. (c)

    Inertia forces

  4. (d)

    Viscous forces

The bond number is the term which describes the compression of the dimension of channels and the nominal size of the bubble. It is the actual measure of the available body forces to the surface tension forces in a bubble. Reynolds number holds the significance of the ratio of inertia force to the viscous force, for liquid and gases. On the other side, the ratio of inertia to the surface tension is known as Weber number [118]. It is also notable that viscous force-to-surface tension force ratio is known as a capillary number. Many researchers such as Venkatesan et al. [246], Yue et al. [254, 259], Kawahara et al. [252], Lee and Lee [251], Lee and Mudawar [257], Megahed and Hassan [264], Pamitran et al. [258] and Choi et al. [194] worked on the application of inertia forces and surface tension forces for the estimation of pressure drop in two-phase flow.

Heat transfer

Chen [252] experimented and developed a new correlation to be used in saturated boiling. This correlation is old and precise, and it is compatible very well with the water at low pressure. There are some specific conditions to use this correlation which are as follows:

Hydraulic diameter: Dh ≥ 1 mm,

Pressure drop range: P = 0.09–3.45 MPa

Heat flux range: q′′ = 0–2.4 MW m−2

Heat transfer coefficient: h = hNB + hFC

Forced convection is predicted from the following correlation,

$$h_{\text{FC}} = 0.023\text{Re}_{\text{f}}^{0.8} \Pr_{\text{f}}^{0.4} F\left( {K/d_{\text{h}} } \right)$$
(55)

where Re is the Reynolds number and Pr is the Prandtl number:

$${\text{Reynolds}}\,{\text{number}}{:}\,{\text{Re}}_{\text{f}} = \frac{{G\left( {1 - x} \right)d_{\text{h}} }}{{\mu_{\text{f}} }}$$
(56)
$${\text{Prandtl}}\,{\text{number}}{:}\,{ \Pr }_{\text{f}} = \left( {\frac{{\mu C_{\text{p}} }}{K}} \right)_{\text{f}}$$
(57)

‘F’ is known as the enhancement factor and it was given by Chen [252] and its value is given by \(\left( {\text{Re}_{\text{TP}} /\text{Re}_{\text{f}} } \right)^{0.8}\).

The phenomenon of nucleate boiling was studied by Forster and Zuber [280], and they developed a correlation to decrease the average value of superheat incorporated with thermal boundary layer and to enhance bubble nucleation phenomenon in cavities that exist in the wall.

$$h_{\text{NB}} = 0.00122\left[ {\frac{{K_{\text{f}}^{0.79} C_{\text{pf}}^{0.45} \rho_{\text{f}}^{0.49} g_{\text{c}}^{0.43} }}{{\sigma^{0.5} \mu_{\text{f}}^{0.29} h_{\text{fg}}^{0.24} \rho_{\text{g}}^{0.24} }}} \right]\Delta T_{\text{sat}}^{0.24} \Delta P_{\text{sat}}^{0.75} S$$
(58)

where

$$\Delta T_{\text{sat}} = [T_{\text{W}} - T_{\text{sat}} ],{\text{and}}\,\Delta P_{\text{sat}} = \left[ {P_{\text{sat}} (T_{\text{W}} } \right) - P]$$
(59)

In the above equation, ‘S’ is defined as suppression factor and its value is given by \(S = \left[ {\frac{{\Delta T_{\text{eff}} }}{{\Delta T_{\text{sat}} }}} \right]^{0.99} .\)

The study conducted by Kandlikar [281] also proposed a correlation to calculate the heat transfer coefficient, which consists of 1000 different data points and all using water, cryogenic fluids and refrigerants as flowing fluids. This can be used only for hydraulic diameter, Dh ≥ 1 mm.

$$h = \hbox{max} \left( {h_{\text{NBO}} ,h_{\text{CBC}} } \right)$$
(60)

where

$$h_{\text{NBO}} = \left[ {0.6683{\text{Co}}^{ - 0.2} \left( {1 - x} \right)^{0.8} f_{2} \left( {{\text{Fr}}_{\text{fo}} } \right) + 1058BL^{0.7} \left( {1 - x} \right)^{0.8} F_{\text{fL}} } \right]h_{\text{fo}}$$
(61)
$${\text{h}}_{\text{CBC}} = \left[ {1.136{\text{Co}}^{ - 0.9} \left( {1 - {\text{x}}} \right)^{0.8} {\text{f}}_{2} \left( {{\text{Fr}}_{\text{fo}} } \right) + 667.2{\text{BL}}^{0.7} \left( {1 - {\text{x}}} \right)^{0.8} {\text{F}}_{\text{fL}} } \right]{\text{h}}_{\text{fo}}$$
(62)

and

$$h_{\text{fo}} = \frac{{K_{\text{f}} }}{{d_{\text{h}} }}\frac{{\left( {\text{Re}_{\text{f}} - 1000} \right)\left( {\frac{f}{2}} \right)\Pr_{\text{f}} }}{{\left[ {1 + 12.7\left( {\Pr_{\text{f}}^{{\frac{2}{3}}} - 1} \right)\left( {\frac{f}{2}} \right)^{0.5} } \right]}}$$
(63)

In this correlation, ‘Co’ is called convection number, ‘BL’ is known as boiling number and ‘Frfo’ is called the forced number in the saturated condition of all liquids. The equations used for the above given parameters are given as follows:

$${\text{Co}} = (\rho_{\text{g}} /\rho_{\text{f}} )^{0.5} \left[ {\frac{{\left( {1 - x} \right)}}{x}} \right]^{0.8}$$
(64)
$${\text{BL}} = \frac{{q^{''}_{\text{w}} }}{{Gh_{\text{fg}} }}$$
(65)
$${\text{Fr}}_{\text{fo}} = \frac{{G^{2} }}{{\rho_{\text{f}}^{2} gd_{\text{h}} }}$$
(66)

Another study conducted by Gungor and Winterton [282] focused on developing a correlation by using 3700 data points by taking ethylene glycol, water and refrigerants as their flowing fluids. The criteria for using the correlation were dh ≥ 1 mm. Correlation to calculate the heat transfer coefficient was as follows:

$$h = h_{\text{f}} \left\{ {1 + 3000{\text{BL}}^{0.86} + 1.12\left[ {\frac{1 - x}{x}} \right]^{0.75} \left( {\frac{{\rho_{\text{g}} }}{{\rho_{\text{f}} }}} \right)^{0.41} } \right\}E_{2}$$
(67)

where the parameter ‘E2’ is depended upon the following conditions:

$$E_{2} = \left\{ {\begin{array}{*{20}l} {1 :{\text{Fr}}_{\text{fo}} \ge 0.05} \hfill \\ {{\text{Fr}}_{\text{fo}}^{{\left( {1 - 2{\text{Fr}}_{\text{fo}} } \right)}} : {\text{Fr}}_{\text{fo}} < 0.05} \hfill \\ \end{array} } \right.$$
(68)

Few correlations were presented by the study of Shah and London [283]. These correlations were designed to be used for the minichannels and microchannels. They pointed out that both nucleate boiling and convective boiling play a significant role in the study of evaporative heat transfer in two phases. Their correlation was developed to use for both the horizontal and the vertical orientations of tubes. Condition dictates, when N > 1 and BL > 0.0003, in that case, the value of heat transfer coefficient (h) is given by the following correlation:

$$h = 230{\text{BL}}^{0.5} h_{\text{f}}$$
(69)
$$N = \left( {\frac{1 - x}{x}} \right)^{0.8} \left( {\frac{{\rho_{\text{g}} }}{{\rho_{\text{f}} }}} \right)^{0.5}$$
(70)

However, when N > 1 and BL < 0.0003, in this case, the value of heat transfer coefficient (h) is given by,

$$h = \left( {1 + 46{\text{BL}}^{0.5} } \right)h_{\text{f}}$$
(71)

In another condition, when 0.1 < N < 1,

$$h = F_{\text{s}} {\text{BL}}^{0.5} \exp \left( {2.74N - 0.1} \right)h_{\text{f}}$$
(72)

And when the value of N < 0.1, then ‘h’ is given by,

$$h = F_{\text{s}} {\text{BL}}^{0.5} \exp \left( {2.74N - 0.15} \right)h_{\text{f}}$$
(73)

where Fs is known as Shah’s constant and its value is given by,

$$F_{\text{s}} = \left\{ {\begin{array}{*{20}l} {15.43 : {\text{BL}} < 0.0011} \hfill \\ {14.7 : {\text{BL}} > 0.0011} \hfill \\ \end{array} } \right.$$
(74)

A study conducted by Li and Wu [279] used 3744 data points to develop a correlation with the help of parameters such as Reynolds number, boiling number and bond number. This correlation can be used for many different flowing fluids, on various operating conditions and for different geometries, orientations and dimensions of microchannels. They deduced from their results that bond number is a parameter which can be used to define the criteria for estimating the heat transfer coefficient. It was stated that the correlation can be used only for the condition where 0.19 mm ≤ dh ≤ 2.01 mm:

$$h = 334{\text{BL}}^{0.3} \left( {{\text{Bo}}\,\text{Re}_{\text{f}}^{0.36} } \right)^{0.4} \left( {\frac{{K_{\text{f}} }}{{d_{\text{h}} }}} \right)$$
(75)
$${\text{Bo}} = \frac{{g\left( {\rho_{\text{L}} - \rho_{\text{g}} } \right)d_{\text{h}}^{2} }}{\sigma }$$
(76)

A number of experiments have been performed, and the results are published to define the pressure drop and heat transfer characteristics of microchannels. Researchers have worked in both areas—single-phase flow and two-phase flow—and finally, it was concluded that either results were deviating from conventional theory of fluid flow and heat transfer or they simply were not in agreement with others. Two-phase flow heat transfer is shown in Table 9. Heat transfer characteristics of microchannels show much more deviation as compared to the pressure drop. However, in some of the studies, it was seen that results of some researchers were in agreement with the conventional theories, but in few other cases, the results were completely different, while the operating condition, hydraulic diameter and flowing fluid were kept same. Some new correlation was developed in order to predict the friction factor and Nusselt number. Their correlation was developed by using the many numbers of data point collectively, and they were not based on any theoretical assumptions or hypothesis; thus, this correlation cannot be assumed as much reliable and significance.

Table 9 Two-phase flow heat transfer
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Conclusions and recommendations for future studies

The present paper presents an extensive review in the area of fluid flow and heat transfer in microchannels for single-phase as well as multi-phase flows. It was observed that single-phase flow in microchannels having a hydraulic diameter of few hundred micrometers, fluid is considered to be incompressible, while the flow becomes fully developed. However, the flow is taking place in laminar flow regime at constant temperature boundary condition seems to follow the classical theory of fluid flow. Various experiments performed at low value ranges of Reynolds number concluded that the experimental results were in agreement with the theoretical prediction. However, it was seen as the value of Reynolds number becomes higher; then, the experimental results start to deviate from the conventional laminar predictions. It was also noted that experimental results seem to follow the conventional thermal developing flow pattern when the hydraulic diameter of the microchannel is increased. On the other side, the experimental results of pressure drop show a slight agreement for both laminar and turbulent regimes. It is possible that the deviations occurred in pressure drop in laminar and turbulent regimes may be because the correlations were developed by assumed macroscale behavior. Researchers only explored the laminar region for its pressure drop and heat transfer characteristics, while the turbulent flow regime is yet to be explored for pressure drop and heat transfer characteristics. So, it is highly recommended to work in this direction.