Pool Boiling Heat Transfer of Ethanol on Surfaces with Minichannels

Abstract

In this paper, the pool boiling of ethanol was analyzed. The experiments were carried out at atmospheric pressure. Heat transfer surfaces in the form of deep minichannels were made of copper. The channels with a depth of 0.2 to 0.5 mm were milled in parallel. The width of the minichannels was 0.6–1.2 mm, and the depth was 5.5, 6, and 10 mm. The highest heat transfer coefficient, 52 kW/m²K, was achieved for the minichannels with a depth of 6 mm and a width of 0.8 mm. The maximum heat flux of 953 kW/m² was produced using minichannels 5.5 mm deep and 0.5 mm wide. An over threefold increase in the heat transfer coefficient and over a twofold increase in the maximum heat flux in relation to the plain surface were obtained. In the heat flux range 21.2–1035 kW/m², the influence of channel width and depth on the heat exchange process was determined. The diameters of the detaching vapor bubbles were determined on the experimental setup using a high-speed camera. An analytical model was developed to determine the diameter of the departing bubble for the analyzed enhanced surfaces. The model correctly represented the changes in bubble diameter with increasing heat flux.

1. Introduction

Nucleate pool boiling is a complex physical phenomenon. Bubble dynamics plays a key role in the development of analytical or semi-analytical models that allow for the determination of heat flux. Bubble dynamics includes the following phenomena:

• Bubble growth
• Bubble departure
• Bubble departure frequency

There are two basic periods of bubble growth [1,2]:

• Isothermal period (duration on the order of milliseconds), in which the growth rate is controlled by the momentum equation. The most important are the effects of buoyancy, surface tension, inertia, and drag, as well as the pressure difference between the vapor and the liquid.
• The asymptotic period (tens to hundreds of milliseconds) occurs after the radius has increased by about twice its initial value. The surface tension and inertia effects can be negligible, and bubble growth is a result of heat transfer between the bubble, the surrounding liquid, and the heating wall.

Mahmoud and Karayiannis [3] drew attention to two categories of bubble growth: (1) symmetric growth (homogeneous boiling) referred to as uniform superheating liquid—the vapor bubble is in contact with the superheated liquid—and (2) asymmetric growth (heterogeneous boiling) referred to as nonuniformly superheated liquid. The authors described three stages of bubble growth:

• Stage 1—the growth of the surface tension is driven by the pressure difference p_v − p_l = 2 σ/r, which is called the “surface tension dominated stage”;
• Stage 2—the isothermal growth to a diameter slightly larger than the initial diameter in stage 1—“inertia-controlled growth”, which is dominated by liquid inertia, the pressure difference can be determined according to the equation ρ_l[r(d²r/dt²) + (3/2)(dr/dt)²] = p_v − p_l∞;
• Stage 3—evaporation occurs on the surface of the bubble and the vapor pressure p_v becomes equal to the liquid pressure p_l∞ (system pressure); a thin thermal boundary layer is formed around the bubble; additionally, the “heat diffusion mechanism” which was commonly called “heat transfer-controlled growth” or “asymptotic growth” begins to work. This mechanism is driven by the temperature difference T_l∞ − T_v and ρ_l[r(d²r/dt²) + (3/2)(dr/dt)²] = 0 because p_v − p_l∞ = 0.

Rayleigh [4] simplified the bubble growth analysis neglecting the surface tension stage, assuming that the bubble grows without heat transfer, i.e., p_v −p_l∞ remains constant with time. This led to the Rayleigh solution for “inertia-controlled growth”, which shows that the bubble grows at a constant rate, i.e., the radius increases linearly with time:

r = [(2/3)(p_v − p_l∞)/ρ_l]^1/2t

Fritz and Ende [5] proposed a model for bubble growth with temperature decrease across a boundary layer around the bubble. Their asymptotic growth model used a transient heat conduction equation in a semi-infinite plate, and finally they obtained the equation that showed that the radius r increased with time as t^1/2. A similar bubble growth equation for heat diffusion-controlled growth was created by Plesset and Zwick [6].

The justification for the choice of ethanol as a working fluid was presented, among others, by Chen et al. [7] and Kalani and Kandlikar [8]. Ethanol is characterized by dielectric properties and an appropriate saturation temperature, 77–78 °C, which can be reduced using decreased pressure. It has a higher heat of vaporization (963 kJ/kg) at a low density (717 kg/m³) compared to classical refrigerants, which allows for obtaining higher heat transfer coefficients. In recent years, the boiling of ethanol on structured surfaces has been analyzed, among others, in publications by Deng et al. [9], Shen et al. [10], and Hożejowska et al. [11].

The diameter of the departing bubble is used to analyze boiling on both plain and extended surfaces. Determining this diameter is related to the use of the balance of positive forces that tend to separate the bubble and negative forces that keep the bubble on the tested surface.

Beer [12] analyzed the balance of forces acting on the bubble during boiling on a smooth surface. He assumed the following positive forces: differential pressure force, buoyancy force, and negative forces connected with surface tension, viscous drag, and liquid inertia. Wang et al. [13] proposed a similar system of forces in the analysis of boiling FC-72, where negative forces were supplemented with the Marangoni force. Kumar et al. [14] assumed a negative force connected with surface tension and identical positive forces. Zeng et al. [15] conducted studies for water and methanol at reduced gravity. They considered two forces holding the bubble (surface tension force and unsteady growth force) and three forces that tend to detach the bubble (contact pressure force, buoyancy force, and lift force). In turn, Bucci et al. [16] and Iyer et al. [17] took into account two forces that tend to detach the bubble, i.e., contact pressure force and buoyancy force, while the forces that maintain the bubble were surface tension and growth forces. Furthermore, Iyer et al. [17] took into account the viscous drag force, which allowed the determination of the diameter of the bubble with an error below 9% for boiling water and methanol.

For the enhanced surface with microchannels between micropin fins, the authors analyzed different arrangements of forces acting on the bubble: 1 or 2 positive forces tending to detach the bubble and 1 to 4 negative forces keeping the bubble on the surface. Zhou et al. [18] and Kong et al. [19] considered microchannels of 0.03–0.05 mm width and 0.06–0.12 mm depth [18] and 0.3–0.5 mm width and 0.06 mm depth in the form of a bistructured surface [19] when analyzing boiling on surfaces with microchannels created between micro-pin-fins and strip microfins. The authors assumed buoyancy force and differential pressure force as positive forces, while the forces keeping the bubble were the liquid inertia force, surface tension force, and viscous drag force. Additionally, Zhou et al. supplemented the force balance with Marangoni force, and Kong et al. with additional virtual force-channel pressure. Zhao et al. [20] adopted a balance of three forces acting on a bubble that departs from a surface with microchannels of width and height in the range of 0.2 to 0.8 mm, formed between micropin fins; the capillary pressure force and the buoyancy force balanced the surface tension force. Zhou et al. [21] presented the pool boiling experiments on vertical surfaces with micropin fins that form microchannels of width 0.03 to 0.05 mm and depth 0.03 to 0.12 mm. The forces acting on the bubble in the case of bubbles departing from the bottom of the microchannel and from the tops of the microfins were determined. It was found that for the bubbles nucleated at the base of the micropin fins, the initial growth rate was significantly higher than that at the top. For a bubble nucleated at the top of the microfins, the forces acting on the bubble were the same as those on the plain surface (drag force, inertial force, differential pressure force, and surface tension). For a bubble nucleated at the bottom of the microchannel, a reaction force was additionally taken into account.

Hu et al. [22] analyzed bubble growth on straight and expanding microchanneled surfaces and took into account buoyancy force, surface tension force, and net evaporation momentum force. The authors presented the dependence of the bubble diameter on the saturation pressure. Dong et al. [23] analyzed shallow liquid boiling in rectangular open microchannels with a channel width of 0.4 mm and a height of 0.5 mm. The authors discussed the balance of six forces that act on the bubble (related to buoyancy, pressure difference, surface tension, bubble inertia, liquid inertia, and drag), consistent with the balance of forces presented by Pastuszko et al. [24]. Similarly to Pastuszko et al. [24] and Pastuszko [25], they used the modified Chien and Webb relationship [26] to calculate the mean bubble growth rate. Long et al. [27] studied pool boiling and bubble dynamics in open V-shaped microchannels with channel widths of 0.1 and 0.05 mm and depth-to-width ratios in the range of 0.5 to 2. They analyzed the balance of five forces acting on a bubble: two negative forces (surface tension and drag force) and three positive forces (buoyancy, internal pressure, and inertial force). The authors did not provide data on the diameters of departing bubbles, but focused on the analysis of changes, among others contact angles and wickability. Berght et al. [28] studied the single bubble growth for refrigerants R32 and R1234yf on a structured surface in the form of triangular microchannels with a height of 0.15 mm. Using an OpenFOAM multiphase library, they concluded that the growth rate for the surface with microchannels was slower than that for the smooth surface. They assumed that the attaching forces (surface tension) were in the negative direction, whereas the detaching forces (buoyancy and pressure) were positive. Walunj and Sathyabhama [29] for variable-width microchannels (top width 0.25–0.5 mm and base width 0.5–0.8 mm) and depth 0.5 mm applied the balance of two positive forces (buoyancy and lift forces) and three negative forces (surface tension, unsteady growth, and bubble inertia), obtaining a prediction error of the bubble diameter determination of 5.6% for boiling water.

Analyzing boiling on surfaces with microchannels of 0.2–0.4 mm width and 0.2–0.5 mm depth, Kaniowski and Pastuszko [30] applied a simplified balance of two forces (surface tension and buoyancy forces) for ethanol and FC-72, obtaining an error of approximately 25% in determining bubble departure diameter. In the case of boiling FC-72 and Novec-649 in deep minichannels, assuming the balance of differential pressure and buoyancy forces versus drag, surface tension, and inertia forces, Pastuszko et al. [24] obtained greater errors in computing the bubble departure diameter, that is, 18–32%. The proposed models for calculating the diameter of the departing bubble for boiling ethanol on surfaces with microchannels [31] and water on surfaces with deep minichannels [25] allowed the representation of changes in bubble diameters with increasing heat flux.

The purpose of the study was to determine the surface with deep minichannels for the highest critical heat flux (CHF) and heat transfer coefficient (HTC). Additionally, a computational model for the bubble departure diameter for ethanol nucleate pool boiling on a deep minichannel system was proposed.

2. Materials and Methods

Ethanol is a relatively inexpensive and widely available liquid. It is often used as a working agent for nucleate pool boiling. Thermodynamic properties are shown in

Table 1. Thermophysical properties of ethanol.

Parameters at 1013.25 hPa	Ethanol
Tsat, °C	78.3
ρl, kg/m3	717
ρv, kg/m3	1.43
λl, W/(mK)	0.17
ilv, kJ/kg	963
σl, N/m	0.0177
μl, Pas	0.00044
cpl, J/(kgK)	723

.

Figure 1 shows the experimental setup, which allowed the temperatures necessary to determine the quantities characterizing boiling heat transfer, i.e., heat fluxes, surface superheating, heat transfer coefficient, and determining the diameter of departing vapor bubbles depending on the heat flux. The main module of the setup is shown in Figure 2. A water cooler operating in a thermosiphon system allowed condensation of ethanol vapor. A glass vessel is closed at the bottom with a Teflon substitute flange and the test sample is connected to a copper heating cylinder. To avoid optical distortions, a rectangular vessel with flat glass walls was used. The distance between the walls was 50 mm. The copper bar had a diameter of 45 mm. The distance between the upper side of the heater and the boiling site at the bottom of the minichannels was 3.5 mm (δ_bs + 0.25 mm, Figure 3).

A 1500 W cartridge heater with a diameter of 16 mm and a length of 100 mm was placed in the heating cylinder. The distance between the edge of the sample and the lens of the camera was approximately 30–40 mm. The camera axis is parallel to the side surfaces of the minichannels, which allows images of growing and detaching bubbles. Figure 2 shows the camera directed perpendicularly to the walls of the minichannels, which is a drawing simplification—in reality the sample is rotated by 90 degrees.

The camera axis was tilted at an angle of 20–30° to the horizontal, so the diameters of the bubble were determined in a plane perpendicular to the camera axis. The diameter of the departure bubble was determined as the arithmetic mean of the maximum and minimum bubble diameter measurements. Taking into account the constant minichannel pitch (2 ± 0.1 mm) and relating it to the measurements of the pitch and diameter in the images, the actual average diameter of the departure bubble was determined. Assuming that the bubble diameter was approximately twice the minichannel pitch, the uncertainty in the maximum and minimum diameters was assumed to be 0.2 mm. The combined uncertainty in the diameter of the departure bubble was 0.14 mm.

The autotransformer was used to regulate the heat flux supplied to the copper bar. The time required for the temperature to stabilize the heating cylinder and the sample was approximately 15 min with increasing heat flux and 20 min with decreasing heat flux. The FLUKE Hydra Series II measurement data acquisition system was used to read and record the temperature of eight thermocouples. This time is required to reach steady state after a step change in power supplied to the copper bar.

To measure the temperature in the heating cylinder and the boiling liquid, K-type thermocouples (NiCr-NiAl) with a diameter of 0.5 mm were used. The arrangement of the thermocouples is shown in Figure 3a. The sample with minichannels was connected to the copper bar with a layer of tin. Before actual measurements, the thermocouples were calibrated using the Altek 422 calibrator. The calibration procedure was carried out by the calibrator with all thermocouples used in the range of 70–170 °C. For subsequent analysis, the calibrated thermocouples were connected to a data logger. The temperature differences observed did not exceed 0.1 K. Given the sensitivity of the logger (0.1 K) and the accuracy of the thermocouples for direct measurement of the temperature difference (assumed to be ±0.2 K), the uncertainty of the heat flux and the heat transfer coefficient were calculated taking this value as the uncertainty of the independent variable.

The measurements were repeated if the temperature difference under the sample (reading from thermocouples T3 and T4, Figure 3) was greater than 0.2 K. It was assumed that the difference in the heat flux and HTC values obtained during the increase and decrease in the heat flux should not exceed the calculated combined uncertainties in the measurement of the heat flux and heat transfer coefficients.

An example graph of temperature changes read for thermocouples placed in the cylinder axis is shown in Figure 3b. The linear distribution with a coefficient of determination R² close to 1 confirms one-dimensional heat flow.

The thermal conductivity of the copper samples was 380 W/(mK). Dimensions and photos of the samples are shown in Figure 4. The minichannels were made with a disc milling cutter with a width of 0.5, 0.6, 0.8, 1.0, and 1.2 mm and a constant pitch of 2 mm. The geometric parameters of the minichannels and the values characterizing the extended surface, i.e., the aspect ratio of the fin (w/h) and the surface extension coefficient φ are given in

Table 2. Surface codes and specifications.

Sample code	W, mm	h, mm	p, mm	φ	w/h	Bo	αmax, kW/m2K
DMC-0.5-5.5	0.5	5.5	2.0	6.5	0.09	0.10	45.7
DMC-0.8-6.0	0.8	6.0	2.0	7	0.13	0.27	52
DMC-0.6-10	0.6	10	2.0	11	0.06	0.15	42.4
DMC-1.0-10	1.0	10	2.0	11	0.10	0.42	47.8
DMC-1.2-10	1.2	10	2.0	11	0.12	0.60	42.4

. The active surface of the sample was 27 × 27 mm².

Creating channels in copper using milling is associated with the possibility of inaccuracies, especially in the case of deep minichannels. After the surface was fabricated, the channel depths were checked with a caliper and their widths were checked with a feeler gauge. A tolerance of 0.03 to 0.05 mm was assumed.

Roughness measurements were performed using a Leica (Wetzlar, Germany) DCM8 confocal microscope. The arithmetic mean deviation of the roughness profile, Ra, was 0.46–0.47 μm for the minifin tops, while the values for the minichannel bottoms ranged from 0.50 to 0.69 μm. Increasing roughness increases the density of nucleation centers, which contributes to an increase in HTC at lower heat fluxes and an increase in CHF. These changes are significant at roughnesses greater than 2–3 μm. For the analyzed DMC surfaces, it can be assumed that the mechanism of bubble nucleation and initial bubble growth in the corner at the bottom of the minichannel dominates, resulting in the formation of bubbles with diameters of 2–6 mm. The few small bubbles with diameters of 0.2–0.3 mm visible in the visualization photos are formed as a result of nucleation in microcavities on the surface of the bottom of the minichannels, without contact with the minifin walls.

To develop the pool boiling mechanism for DMC surfaces, it is important to visualize the dynamics of bubble growth. The images were recorded using a digital monochrome camera PHOT MV-D1024-160-CL (Photonfocus, Lachen, Switzerland), which recorded images at a speed of 428 frames per second at a resolution of 500 × 250 pixels. Visualization was also performed using the RX-10 camera (Sony Corp., Tokyo, Japan).

The copper bar was surrounded by a thick insulating layer, which ensured one-dimensional heat conduction. Relationship (1) takes into account the increase in heat flux caused by the reduction in the sample base cross-section compared to the copper bar cross-section:

$$q={\lambda }_{Cu}{\displaystyle \frac{T_{\mathrm{T}8}-T_{\mathrm{T}5}}{{\delta }_{\mathrm{T}8-\mathrm{T}5}}}\cdot {\displaystyle \frac{\mathsf{\pi }{d}_{cyl}^2}{4{w_s}^2}}$$

(1)

The wall superheat was referred to the surface of the bottoms of the minichannels; therefore, it was necessary to extrapolate the temperature read from the thermocouples T3 and T4 to the distance δ_bs (Figure 3).

$$∆T={\displaystyle \frac{T_{\mathrm{T}3}+T_{\mathrm{T}4}}{2}}-q{\displaystyle \frac{{\delta }_{bs}}{{\lambda }_{Cu}}}-{\displaystyle \frac{T_{\mathrm{T}1}+T_{\mathrm{T}2}}{2}}$$

(2)

The heat transfer coefficient was defined according to Newton’s law:

$$\alpha ={\displaystyle \frac{q}{∆T}}$$

(3)

where

• T_T1 to T_T4—thermocouple temperatures,
• δ_bs—the thickness between the base of the sample and the bottom of the minichannel,
• λ_Cu—a copper heat conductivity coefficient,
• δ_T8–T5—the distance between thermocouples T8 and T5,
• d_cyl—the external diameter of the copper cylinder to which the sample is soldered.

The diameters of the bubbles d_b were measured in two directions x and y. They were determined for at least 30 consecutive departing bubbles with an average value of 8–10 nucleation sites. Diameters were determined with constant heat flux q. The departing bubble was treated as two rotating semi-ellipsoids with a vertical and a horizontal rotation axis. The bubble diameter can be determined according to the relationship [32]:

$$d_b={\left(x^{2}y\right)}^{0.33}$$

(4)

The dimensionless number relating to the flow resistance in the minichannel space is the Bond number, which can be defined as

$$\mathrm{Bo}={\left({\displaystyle \frac{w}{L}}\right)}^2$$

(5)

where L is the capillary length:

$$L=\sqrt{{\displaystyle \frac{\sigma }{g\left({\rho }_l-{\rho }_v\right)}}}$$

(6)

The increase in the heat transfer surface is one of the factors that increases the heat flux. The surface enhancement factor (or extension coefficient) can be defined as the ratio of the total extended surface area to the base (smooth) surface area [33]:

$$\phi ={\displaystyle \frac{A_{ext}}{A_{bs}}}={\displaystyle \frac{2h}{p}}+1$$

(7)

Experimental studies were performed in the range from ONB (onset of nucleate boiling) to CHF (critical heat flux). The tested samples contained surfaces with deep minichannels with dimensions presented in Table 1. The boiling curves were determined for increasing and decreasing heat flux. Before starting the measurement series, a power of approximately 300 W was supplied to the heater for 20–30 min, which gave a heat flux of approximately 400 kW/m² in order to degas the tested surfaces and the vessel.

The analysis of measurement uncertainty was performed in a way similar to the work of Pastuszko [25] using the total differential error.

The combined standard uncertainties in the measurement of the heat flux and heat transfer coefficients were defined as the total differential error dependent on the following:

• The uncertainty in determining the thermal conductivity coefficient u(λ) = 1 W/mK,
• The uncertainty in determining the temperature difference on the heating cylinder u(∆T_T5–T8) = 0.2 K;
• the uncertainty in determining the temperature difference u(∆T_sat) = 0.2 K, where ∆T_sat = (T_T3 + T_T4)/2 − (T_T1 + T_T2)/2;
• The uncertainty in determining the distance between the temperature sensors u(δ_T5–T8) = 0.25 mm;
• The uncertainty in determining the diameter of the heating cylinder u(d_cyl) = 0.25 mm;
• The uncertainty in determining the side of the sample u(a) = 0.25 mm.

The following equation shows the relationship:

$$u(q)=\sqrt{{\left({\displaystyle \frac{\partial q}{\partial \lambda }}u(\lambda )\right)}^2+{\left({\displaystyle \frac{\partial q}{\partial ∆T_{\mathrm{T}5-\mathrm{T}8}}}u(∆T_{\mathrm{T}5-\mathrm{T}8})\right)}^2+{\left({\displaystyle \frac{\partial q}{\partial {\delta }_{\mathrm{T}5-\mathrm{T}8}}}u({\delta }_{\mathrm{T}5-\mathrm{T}8})\right)}^2+{\left({\displaystyle \frac{\partial q}{\partial d_{cyl}}}u(d_{cyl})\right)}^2+{\left({\displaystyle \frac{\partial q}{\partial a}}u(a)\right)}^2}$$

(8)

The uncertainty of the heat transfer coefficient was determined as follows:

$$u(\alpha )=\sqrt{{\left({\displaystyle \frac{\partial \alpha }{\partial q}}u(q)\right)}^2+{\left({\displaystyle \frac{\partial \alpha }{\partial ∆T}}u(∆T)\right)}^2}$$

(9)

where the uncertainty in the superheat measurement was calculated according to the equation:

$$u(∆T)=\sqrt{{\left({\displaystyle \frac{\partial ∆(∆T)}{\partial \lambda }}u(\lambda )\right)}^2+{\left({\displaystyle \frac{\partial ∆(∆T)}{\partial ∆T_{sat}}}u\left(∆T_{sat}\right)\right)}^2+{\left({\displaystyle \frac{\partial ∆(∆T)}{\partial {\delta }_{bs}}}u({\delta }_{bs})\right)}^2+{\left({\displaystyle \frac{\partial ∆(∆T)}{\partial q}}u(q)\right)}^2}$$

(10)

Changes in measurement uncertainties with increasing heat flux and HTC are shown in Figure 5 and Figure 6, respectively. The relative combined standard uncertainties in the measurement of the heat flux and heat transfer coefficients are inversely proportional to the heat flux and HTC. For increasing heat flux in the range of 17–950 kW/m², the relative uncertainty of the determination of the heat flux varied between 17% and 2% with increasing heat flux (Figure 5), while the relative uncertainty of the heat transfer coefficient changed from 35 to 5% with increasing heat transfer coefficient (Figure 6). The uncertainties of HTC determination are high at the lowest heat flux and heat transfer coefficients because of the small superheats at these values.

3. Results—Boiling Curves

The onset of nucleate boiling occurs at the moment of reaching the saturation temperature corresponding to the atmospheric pressure and observing the separation of steam bubbles. The high-speed camera was used as an indicator of ONB. The surface of the DMC wall superheated at ONB was 2.8–3.6 K. The heat flux of 20.3 kW/m² for the surface of DMC-0.5-5.5 corresponds to a superheat of 2.8 K. These are significantly lower values than those obtained during ethanol boiling, e.g., on the perforated micromesh surface (superheat of 10–12 K at ONB [7]).

Figure 7 and Figure 8 show the influence of minichannel geometry on the heat transfer process. For surfaces with minichannels of depths 5.5 to 6 mm and 10 mm, an approximately twofold increase in the critical heat flux was observed. Changes in the depth and width of the minichannels have a small effect on the values of the obtained CHF. The increase in heat transfer intensity on extended surfaces occurs with an increase in the heat exchange surface, as well as with an increase in the frequency of bubble detachment and the density of nucleation centers. The maximum heat transfer coefficients obtained are in the range of 42 to 52 kW/m²K, with higher values (46 to 52 kW/m²K) obtained for surfaces with shallower minichannels (5.5 to 6 mm). This can be explained by a smaller temperature drop in the case of lower fins, which results in obtaining greater superheat at the fin tips, i.e., at the minichannel outlets.

The small space between the minifins that form the walls of the minichannel causes an increase in capillary pressure, allowing the liquid to be sucked into the lower part of the channel, which helps to replenish the liquid and prevents the minichannel spaces from drying out. For heat fluxes below 500 kW/m², both in the case of minichannels with a depth of 5.5–6 mm and 10 mm, it is more beneficial to use the smallest width of minichannels, i.e., 0.6 mm. The high capillary pressures occurring at this small width will contribute to the intensive inflow of liquid into the minichannel space, allowing for the obtained heat transfer coefficients of about 40 kW/m²K at a heat flux of 300 kW/m². For surfaces with minichannels of this width, after exceeding the heat flux of 500 kW/m², a gradual decrease in HTC of 20–25% can be observed. A more advantageous solution, guaranteeing an increase in the heat transfer coefficient at higher heat fluxes, is the use of minichannels with a width increased to 0.8–1 mm. In the case of DMC with a depth of 6 mm, an HTC of up to 52 kW/m²K can be obtained at a heat flux of about 700 kW/m². Slightly lower HTC values (approximately 47 kW/m²K) at the same heat flux were obtained for minichannels with a depth of 10 mm. Using a channel with a width of 0.6 mm at a depth of 10 mm results in a significant decrease in the heat transfer coefficient after exceeding 500 kW/m². Despite the large superheat of the fins forming the channel walls, the significant friction flow resistance that is formed blocks the release of vapor [34], which leads to drying out of the minichannel space. A significant increase in superheat, especially at the bottom of the minichannel, i.e., at the base of the minifins, with a small increase in the heat flux will cause a gradual decrease in the heat transfer coefficient.

According to Jaikumar and Kandlikar [35], increasing the channel width results in better liquid flow to the channel, which limits drying out; however, excessive increases in width, despite the improvement in the intensity of the vapor flow from the bottom of the minichannel, also result in a decrease in the thickness of the minifins. The reduced thickness of the fins that limit the minichannel causes an intensive temperature drop at their height, because the superheat at the channel outlet may not be sufficient to maintain nucleate boiling. According to Winter and Weibel [36], with the decrease in the width of the interfin space, i.e., at the same time as the width of the minichannel, a stable vapor layer will not be formed on the surface of the minifin base until larger heat fluxes appear. In turn, Huang et al. [37] analyzed the fin array with fins of constant thickness of 1 mm predicted critical heat flux for varying heights (3–10 mm) and spacings (0.3–2 mm). Maximum CHF values were obtained for fins of about 4 mm height with the smallest interfin space widths (0.3–0.6 mm). However, it should be noted that, with a constant fin thickness, reducing the spacing between the fins or increasing the fin height increases the total available heat transfer surface. In the case of the analyzed DMC surfaces with a constant minichannel pitch, i.e., simultaneous fins, reducing the minichannel width results in an increase in the aspect ratio of the fin, (p-w)/h, while the surface extension coefficient φ remains constant. With a similar minichannel width and increasing depth, the liquid flow resistance in the minichannel spaces increases, which can be observed as a slight decrease in CHF.

When studying heat transfer during boiling on extended surfaces, fins are used to form minichannels with widths ranging from 0.5 to 8.5 mm and depths between 0.5 and 15 mm. According to Winter and Wiebel [36], at channel widths smaller than the capillary length, there is confinement of vapor between the fins. The capillary length for ethanol is 1.55 mm; therefore, the author assumed w < L. Additionally, a smaller width increases capillary pressure and the intensity of liquid inflow into the minichannel. According to Chan et al. [34], greater channel depths limit the coalescence of large bubbles, which prevents the entire developed surface from being covered by a “vapor blanket”. The author of this paper wanted to determine whether, in the case of 10 mm deep minichannels, this leads to increased HTC and CHF.

The analysis of surfaces with minichannels with depths of 5–10 mm is related to the reference of the channel dimensions to the capillary length L, which for ethanol has a value of approximately 1.55 mm. If both dimensions related to the space between the fins, that is, the height of the fin (minichannel depth) and the width of the interfin space (channel width) exceed the dimension L, then the fins forming a given minichannel system can be treated as isolated [36]. For DMC surfaces h > L and w < L, the height of the boiling regimes along the minifin and superheat of the channel bottom are strongly affected by the confinement of the vapor. The difference in the obtained HTC values with the increase and decrease in the heat flux is within the range of the calculated measurement uncertainties. The exception is the DMC-0.8-6.0 surface, for which after reaching maximum heat flux and decreasing q from 940 to 690 kW/m², hysteresis related to the increase in HTC by approximately 8 kW/m²K was obtained. Hysteresis disappeared with a further decrease in the heat flux below 690 kW/m². Hysteresis is generally considered an unfavorable phenomenon, which prevents thermal stabilization of systems that emit high heat fluxes. The effect of hysteresis is significant in the case of boiling on surfaces with a porous structure. In the case of surfaces with deep minichannels, residual hysteresis related to the increase in HTC was observed only for the DMC-0.8-6 sample in the first stage of heat flux reduction after reaching CHF. Because this phenomenon did not occur for the other tested surfaces and the differences in HTC during heat flux increases and decreases were within the measurement uncertainty range, it can be assumed that in the case of boiling on surfaces with deep minichannels, it is of negligible significance.

The line of the critical heat flux shown in Figure 7 and Figure 8 is calculated according to the Zuber formula [38]:

$$q_{\mathrm{CHF}}=0.131{{\rho }_v}^{0.5}{i}_{lv}{\left[\sigma \mathrm{g}({\rho }_l-{\rho }_v)\right]}^{0.25}$$

(11)

A distinctive highest value of CHF was obtained with DMC surfaces—approximately a doubling of CHF was achieved. A large channel flow cross-section contributes to easy flooding with liquid and vapor removed even at high superheats and heat fluxes.

The differences in the maximum heat flux obtained for individual surfaces are small, with a slight advantage of minichannels with a depth of 5.5–6 mm. A smaller channel width (0.5 mm) gives a greater fin thickness at a constant pitch, which in combination with a smaller height increases the fin efficiency. Additionally, the higher capillary pressure in the narrower channel ensures an intensive inflow of liquid into the minichannel space. As a result, a surface with such a combination of depth and width has the ability to remove the largest heat fluxes.

The author compared the performance of DMP surfaces and other enhancement structures for ethanol boiling using data with the maximum HTC and CHF presented by Chen et al. [7]. The comparison presents data for 14 micro/nanostructured surfaces. Three of the surfaces presented allow for obtaining higher HTCs (perforated micromesh surface, microchannel, and nanofiber-covered surface), one has a similar maximum HTC value, while ten have lower values. With respect to CHF, DMC surfaces provide values approximately 10–15% lower than the three best and at least 45% higher than the remaining ten.

Different trends are observed for the DMC surfaces tested during boiling of working fluids with low saturation temperatures, i.e., FC-72 and Novec-649. The best performance is observed using deeper minichannels (10 mm) with thicknesses of 0.6 and 1.0 mm [24]. Reducing the depth to 5.5–6 mm is unfavorable for these fluids. It can be assumed that similar phenomena will occur in the case of ethanol boiling at reduced pressure, but this requires a series of measurements on a modified experimental setup.

4. Results—Bubble Departure Diameter

4.1. Bubble Growth Rate

Figure 9 shows boiling images for four heat fluxes for surface DMC-0.5-5.5. Figure 10, Figure 11 and Figure 12 show bubble growth for three samples with minichannel depths of 10 mm for selected heat fluxes.

Determining the diameter of a detaching bubble based on the balance of forces acting on it requires knowledge of the bubble growth rate dr/dt and the acceleration d²r/dt². On plain surfaces, the asymptotic bubble growth rate is proportional to the square root of the growth time t^1/2. This relationship was presented by Plesset and Zwick [6] and Han and Griffith [2]. Ghazvini et al. [39] concluded that traditional growth models were established based on a plain surface, while pool boiling on the fins was not considered. For the surface with microchannels of width 0.25–0.8 mm and depth 0.5 mm, Walunj and Sathyabhama [29] assumed a linear dependence of the bubble growth rate on time for the inertia-driven mechanism. They assumed a simple relationship r(t) = at, where a was a function of the fluid property and the wall temperature. In the case of flat surfaces, the intensity of bubble growth decreases towards the end of the growth period. For surfaces with minichannels, a constant linear increase in the radius may be caused by the intensive inflow of vapor from the channel space between the fins to the growing bubble. This article analyzes changes in the radius of the growing bubble for the surfaces listed in Table 2, in the superheat range of up to 3.9–5.6 K. The limitation of the maximum superheat results from the difficulty of observing individual bubbles at heat flux exceeding 100–200 kW/m². Example frames of the bubble growth recording for minichannels with a depth of 10 mm are shown in Figure 10, Figure 11 and Figure 12. Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17 show the experimentally obtained changes in the size of the radius of the growing bubble for four superheats. For the data presented, linear interpolation was used: the obtained coefficient of determination obtained for linear regression ranged from approximately 0.85 to 1. The graphs show the values of the coefficients of determination and the equation of the trend line. The slope of the line (gradient) is also the average value of the growth rate, which was used in the calculation model to determine the drag force and the pressure difference force. The growth rate took values from about 0.02–0.03 m/s at the smallest superheats (1.4–3.1 K) to about 0.12 m/s at the largest analyzed superheats (3.9–5.6 K). The best fit (coefficients of determination) was obtained for larger superheats.

The three phases of bubble growth described in the introduction refer to boiling on a plain surface. In the case of pool boiling in vertical minichannels, analyzed in this article, the measurements showed bubble growth linearly with time, i.e., the dominance of stage 2 (“inertia controlled growth”). For boiling on a flat surface, this is the initial stage of bubble growth. This growth can be described by the Reyleigh equation [3,4] in the following form:

$$r={\displaystyle \frac{2}{3}}{\left({\displaystyle \frac{{\rho }_v{h}_{fg}∆T}{{\rho }_l{T}_{sat}}}\right)}^{1/2}t$$

(12)

For example, the growth rate calculated from the above equation for superheating of 4.2 K and 5.1 K is 0.12 m/s and 0.133 m/s. Using measurement interpolation, 0.119 m/s and 0.122 m/s were obtained, respectively (DMC-0.5-5.5, Figure 13). Greater discrepancies appear for smaller superheats. At higher superheats, due to the merging of bubbles (coalescence) and their irregular shapes, it is impossible to determine the changes in diameter in relation to a single, isolated bubble.

4.2. Forces Balance

In this paper, the balance of four forces was adopted to compute the diameter of the departing bubble on the surface with minichannels, two of which tend to detach (the buoyancy force and pressure difference force), and two forces hold the bubble (drag force and surface tension force), Figure 18.

$$F_{bu}+F_p=F_{st}+F_d$$

(13)

4.2.1. Buoyancy Force

The departing bubble has the shape of an elongated vertically truncated spheroid, but, assuming a spherical shape for simplicity, the buoyancy force is determined according to the following formula:

$$F_{b_u}={\displaystyle \frac{\mathsf{\pi }{d}_b^3}{6}}\mathrm{g}\left({\rho }_l-{\rho }_v\right)$$

(14)

4.2.2. Pressure Difference Force

The pressure difference force [12,13] depends on the capillary pressure and the vapor pressure:

$$F_p=\left({\displaystyle \frac{C_d{\rho }_l}{8}}{\left({\displaystyle \frac{dr}{dt}}\right)}^2+{\displaystyle \frac{4\sigma }{d_b}}\right)A_{\mathrm{c}}$$

(15)

The bubble growth rate dr/dt was assumed based on experimental data (Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17). The drag coefficient, C_d, was calculated according to Beer [12]:

$${\mathit{C}}_{\mathit{d}}={\displaystyle \frac{5360}{{\mathrm{Re}}^{0.79}}}$$

(16)

This relationship was also used by Wang et al. [13] and Wu et al. [40]. The Reynolds number was determined using the following formula:

$$\mathrm{Re}={\displaystyle \frac{\frac{dr}{dt}{d}_b}{\nu }}$$

(17)

The contact surface area between the minifin’s top and the bubble is the area of two segments of a circle with a radius corresponding to half the diameter of the bubble base d_bs. Due to the small width of the interfin space (channel) compared to the width of the fin, it was assumed that the length of the segment chord corresponds to its diameter. Finally, the contact surface area is the surface of a circle with diameter d_bs, reduced by the area of a rectangle with sides corresponding to the diameter d_bs and the width of the minichannel w:

$$A_c={\displaystyle \frac{\mathsf{\pi }{d}_{bs}^2}{4}}-2d_{bs}w$$

(18)

The diameter of the bubble base corresponds to the sum of the microchannel width and the thickness of two microfins, giving d_bs = 2p-w (Figure 18).

4.2.3. Surface Tension Force

The surface tension force can be calculated according to the following formula:

$$F_{st}=\sigma L_c\sin \Theta$$

(19)

The contact line L_c consists of two horizontal arcs with radii corresponding to half the diameter of the bubble base (d_bs), placed at the tops of adjacent microfins, closed by two chords with a length approximately corresponding to the diameter of d_bs (Figure 18):

$$L_c=\mathsf{\pi }{d}_{bs}-2w+2d_{bs}$$

(20)

The diameter of the bubble base d_bs was determined according to the explanation in Formula (12). Furthermore, it was assumed that the contact angle Θ = 13°. A static contact angle value of 13° was entered based on measurements using the Biolin Scientific’s Attension Theta Tensiometer. Since the surface tension force depends on the value of the contact angle, the bubble departure diameter calculation results are sensitive to changes in θ. For DMC-0.5-5.5 surfaces in the superheat range of 3.5 to 5.1 K, which corresponds to a heat flux range of 74 to 170 kW/m², reducing the contact angle value to 8° results in a reduction of the determined d_b value from 2.4 to 1.1%. Increasing the contact angle value to 18° results in an increase in the calculated d_b value of 2.2 to 1%.

4.2.4. Drag Force

The drag force is related to the resistance of the liquid to the expansion of the vapor bubble. The drag coefficient C_d and the bubble growth rate, dr/dt, were calculated similarly to the pressure difference force:

$$F_d={\displaystyle \frac{C_d{\rho }_l}{2}}{\left({\displaystyle \frac{\mathrm{d}r}{\mathrm{d}t}}\right)}^2{\displaystyle \frac{\mathsf{\pi }{d}_b^2}{4}}$$

(21)

4.3. Bubble Departure Diameters

Figure 19 shows a comparison of the bubble diameters measured experimentally and computed based on the theoretical model presented, for individual surfaces. The graphs also show models developed previously by the author and coworkers [24] and a simple computational model by Cooke and Kandlikar [41]. For the balance of dynamic forces, the authors [24] assumed that the calculation of the diameter of the bubble is based on a balance of six forces (buoyancy, surface tension, pressure difference, drag, liquid inertia, and bubble inertia). In the case of the static force balance analysis, only the surface tension force and the buoyancy force were taken into account. The original relationship given by Cooke and Kandlikar is the following:

$$\begin{array}{l}{d}_b={\displaystyle \frac{w}{\cos \frac{{180}^{\circ }+{\Theta }_v}{2}}}\end{array}$$

(22)

which gives d_b values less than zero. It has been corrected in the following form:

$$\begin{array}{l}{d}_b={\displaystyle \frac{w}{\cos \frac{{90}^{\circ }+{\Theta }_v}{2}}}\end{array}$$

(23)

The corrected relationship presented was used to calculate the bubble departure diameter for the contact angle Θ_v referring to the vertical sides of the minichannel. Calculations were performed for the contact angle values of Θ_v = 90° − Θ.

A small bubble diameter can be observed, 2–3.5 mm at the lowest heat flux (22–30 kW/m²). An increase in the heat flux to approximately 70–190 kW/m² results in stabilization of the bubble diameter in the range of 4.2–5.6 mm. The proposed calculation model correctly reflects the trend related to the increase in the bubble departure diameter for the heat flux range considered. The best agreement of the calculations with the experimental data was obtained for a greater depth of the minichannels, i.e., 10 mm. The relative error in determining the diameter does not exceed 17% for q > 40 kW/m². The model of Pastuszko et al. [24] is consistent with the experimental data in the range of the highest heat flux and for minichannels with widths of 0.5–1 mm, it is close to the current model. The Cooke and Kandlikar model [41] presents the experimental data well and can be used for fast calculations only for the smallest minichannel widths, that is, 0.6 mm for heat fluxes greater than 70–80 kW/m². For larger minichannel widths, a significant (3–4 times) overprediction is obtained. However, the model of Pastuszko et al. related to dynamic forces gives good accuracy in the range of the lowest heat flux (20–30 kW/m²). For the DMC-1.0-10 surface, almost perfect agreement of the new model with the dynamic forces model was obtained in the range of low and medium heat flux, 20 to 80 kW/m². A similar trend occurs for the largest minichannel width (sample DMC-1.2-10), but in the range of medium and largest heat fluxes, i.e., from about 70 to 120 kW/m².

For similar DMC surfaces spreading heat during water boiling, the proposed model using the balance of five forces [25] enabled the determination of the diameter of departing bubbles with errors of up to 25%. Also, the models developed by the author and coworkers for these structures [24] gave bubble departure diameter prediction errors of up to 32% (boiling FC-72) and 24% (boiling Novec-649).

HTC does not appear in Formulas 13-21 used to determine the diameter of the departure bubble. When comparing the calculated and experimental values, which are related to the heat flux, the uncertainty of the heat flux determination is important. For the analyzed range q = 20–180 kW/m² (Figure 19), ∆q = 5.6–7 kW/m².

The possible sources of discrepancy between the predicted bubble diameters and the experimental data are as follows:

• Changes in the bubble departure diameters at different locations (outlets) of the minichannel
• Non-spherical shape of bubbles
• Failure to meet the assumption of a linear dependence of the radius of the growing bubble on time.

5. Conclusions

This paper presents the results of experimental and theoretical studies of ethanol nucleate pool boiling on surfaces with deep minichannels (DMC). Experiments were performed from the onset of boiling (ONB) to the boiling crisis (CHF) at atmospheric pressure. For the tested surfaces, an increase in heat transfer was observed in comparison to that for a plain smooth surface, with the heat transfer coefficient (HTC) and the critical heat flux (CHF) depending on the depth and width of the minichannel. The maximum HTCs obtained were more than three times higher than for the plain surface for the Chen et al. data [7], reaching a value of 52 kW/m²K for the DMC-0.8-6.0 sample. The highest HTC was obtained for the sample with a low surface extension φ = 7, that is, for minichannels with a width of 0.8 mm and a depth of 6 mm. The critical (maximum) heat fluxes for all tested surfaces were similar and about twice as high as those for the smooth plain surface.

On the basis of the measurements carried out, the following general recommendations can be given to obtain the highest heat transfer coefficients during ethanol pool boiling of ethanol:

• At high heat fluxes (above 500 kW/m²K), the analyzed minichannels with a depth of 5.5 to 10 mm should be 0.8 to 1 mm wide, that is, approximately 1/2 to 2/3 of the capillary length for ethanol;
• At lower heat fluxes (below 500 kW/m²K), it is more beneficial to use narrower minichannels (0.5–0.6 mm), that is, about a third of the capillary length;
• Due to similar values of the heat transfer coefficients in the case of lower and higher HTC values, it is more optimal to use minichannels with a depth of 5.5–6 mm in terms of lower material consumption;
• There are two main possible applications for the analyzed DMC surfaces in terms of heat removal from electronic components or electronic subassemblies:
• Direct cooling, also known as two-phase immersion cooling—cooling of electronic systems connected to the DMC surface, immersed in a dielectric fluid;
• Indirect cooling—the use of a heat pipe or thermosyphon with a DMC surface as an evaporator to remove large heat fluxes from electronic systems.

For the analyzed DMC surfaces, one can assume inertia-controlled growth of the bubble: the author presented a method of bubble departure calculation assuming a constant bubble growth rate. The developed model allowed determining the diameter of the departing bubble on surfaces with deep minichannels of width 0.6–1.2 mm with an average relative error of 23.5% in the heat flux range of 22–190 kW/m² and below 17% in the range of 40–190 kW/m². The model showed satisfactory agreement with the experimental data and a correct trend of increasing bubble diameter with increasing heat flux. The increase in the growth rate influenced the increase in the diameter of the departing bubble.