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Quenching and rewetting of rock in liquid nitrogen: Characterizing heat transfer and surface effects
International Journal of Thermal Sciences 148 (2020) 106161
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International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts
Quenching and rewetting of rock in liquid nitrogen: Characterizing heat transfer and surface effects Ran Li, Chengcheng Zhang, Zhongwei Huang * State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum (Beijing), Beijing 102249, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Cryogenic quenching Rock Numerical simulation Surface structuring
Cryogenic quenching of cylindrical rock rodlets in liquid nitrogen pool was experimentally investigated. The sample was 25.4 mm in diameter and 60 mm in length. Quenching was performed under saturated condition at the atmospheric pressure. Thermocouple and high speed camera were employed to attain the transient tem perature history and the two-phase dynamics during quenching. A numerical simulation scheme was developed to model the heat transfer process during quenching and rewetting. The boiling curves of quenching were established for different rock surfaces based on the temperature and visualization data. The rewetting of cy lindrical rock was observed to initiate at the two ends of sample and merged at the upper 2/3 of the sample length. This feature was well captured by our numerical model. It was found that the maximum heat flux during quenching exceeds the stationary boiling limits of liquid nitrogen, the same finding as in the literature. The effect of surface structures on quenching was also explored by experiments and the numerical model demonstrated its applicability for these structured surfaces as well.
1. Introduction
studies concentrated on the change of rock properties before and after liquid nitrogen quenching but no efforts were made to reveal the heat transfer mechanisms. Cai et al. [3] quenched marble and shale rock rodlets in liquid nitrogen pool. Based on scanning electron microscope (SEM) and nuclear magnetic resonance (NMR) techniques, they observed the expansion of micro pores in rock samples after quenching. Wu et al. [5] conducted the same investigations with rock samples at elevated temperatures. They found that the variation of rock properties after liquid nitrogen quenching became more pronounced when the initial temperature of rocks was higher. These authors qualitatively attributed their results to the thermal shock effects of liquid nitrogen, i. e., the thermal stress induced by rapid cooling. However, they never delved into the details of heat transfer and phase change dynamics involved in the process of liquid nitrogen quenching. The rock property change by liquid nitrogen quenching is a thermally-driven phenomenon. It is undeniable that the evolution of temperature fields and thermal characteristics during quenching directly determines the magnitude of the thermal shock effects. In this sense, it is valuable to study the heat transfer during quenching and rewetting of rock surfaces in liquid nitrogen. The thermal-hydraulic features of quenching and rewetting are featured by three separate boiling regimes. At the initial stage of
Quenching is a transient heat and mass transport phenomenon where hot solid surfaces are cooled by a liquid whose boiling point is appre ciably lower than the initial solid temperature. This phenomenon is relevant in many industrial applications such as metallic heat treatment, fuel refilling of space crafts, superconductive magnets and the loss of coolant accidents (LOCA) in nuclear reactors. The performance and ef ficacy of quenching process usually relate to the thermal safety and economic efficiency of energy and power systems. Therefore, a thorough understanding of the physical nature of quenching is desired for optimal design of these engineering applications. In the upstream oil & gas industry, the cryogenic quenching with liquid nitrogen is encountered. Operators have introduced liquid nitro gen into the wellbore and reservoir to increase the oil and gas produc tion rate [1,2]. By quenching reservoir rocks with liquid nitrogen, thermal stresses could be generated and they may increase the pore space in rocks or even induce fractures on rock surfaces. This mechanical change in rocks is beneficial for the oil & gas flow and can contribute to enhanced productions [3–5]. However, a fundamental understanding of the effect of liquid nitrogen quenching on rocks is lacking due to the complex thermal-hydraulic-mechanical coupling in this process. Current
* Corresponding author. E-mail addresses: [email protected] (R. Li), [email protected] (Z. Huang). https://doi.org/10.1016/j.ijthermalsci.2019.106161 Received 18 March 2019; Received in revised form 23 October 2019; Accepted 26 October 2019 Available online 12 November 2019 1290-0729/© 2019 Elsevier Masson SAS. All rights reserved.
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International Journal of Thermal Sciences 148 (2020) 106161
quenching, the solid surface is sufficiently hot to sustain a stable vapor film covering it and the heat transfer rate is relatively low, corre sponding to the film boiling regime. At some point, the vapor film starts to become unstable and eventually collapses which results in liquid-solid contact and the heat removal rate improves significantly. The collapse of vapor film marks the rewetting of solid surface and the start of transition and nucleate boiling. Most research works in the literature focused on the movement of the rewetting front, which is a triple contact line formed by the solid surface, vapor and liquid. This region is transient, oscillated and stochastic in nature, making the fundamental study on it extremely difficult. Yamanouchi [6] presented an analytical solution to the rewetting front velocity. He assigned a constant heat transfer coefficient to the wetted portion of vertical rods and assumed adiabatic conditions in dry areas. This model was referred to as two-region model and subsequent researchers made various refinements by considering precursory cooling [7,8], internal heat generation [9], or by modeling three different re gions [10,11]. An excellent review on these analytical works can be found in Sahu et al. [12]. The analytical models generally assume a steady state where the rewetting front has been well established and its moving velocity does not change with time. As such, the initial forma tion of rewetting front and the effect of geometric boundaries of solids cannot be incorporated. In comparison, the group of Pavlenko et al. [13–15] conducted a series of experiments on the cryogenic quenching of vertical copper plates by liquid nitrogen thin films. The transient surface temperature and rewetting front motion were recorded by thermocouples and high speed camera. Based on these data, the authors presented a numerical simulation of quenching which accounted for the geometric boundary and the initial cooling effects. They successfully reconstructed the boiling curve for liquid nitrogen thin film quenching and compared the results with the quasi-stationary boiling data. It was found that the static boiling data were insufficient in representing the transient quenching heat transfer. Apart from the modeling of quenching front velocity, efforts were also made to improve the quenching performance by modulating the properties of solid surface or quenchants. Superhydrophilic surfaces were found to have the capability of shortening the quenching duration by inducing earlier solid-liquid contact [16–18]. Repeated quenching in water-based nanofluids has analogous effects of quenching enhance ment. The heat transfer rate and rewetting front velocity were both improved in consecutive quenching in alumina nanofluids [19,20]. Takeda et al. [21] and Tsoi et al. [22] found that a layer of low heat conductive coatings on solid surface is capable of improving the Lei denfrost temperature, thus accelerating the overall quenching speed. Hu et al. [23] fabricated alumina nanoporous layers on the surface of an aluminum rodlet and performed quenching experiment in liquid nitro gen pool. The nanoporous layer effectively enhanced all the three boiling regimes with the most remarkable change in transition boiling regime. Hu et al. [24] refined their experiments and analysis on the enhancing effects of the nanoporous alumina layer. In particular, they calculated the active nucleation cavity radius and total nucleation contact line length for different nanoporous surfaces. As a result, the superhydrophilic surface property and excessive nanoscale nucleation sites were identified as two important factors in intensifying the quenching heat transfer rate. Pavlenko et al. [25,26] fabricated a capillary-porous coating on vertical copper plates by means of directed plasma spraying. They reported that the porous coating decreased the total cooling time of copper plate by more than three times comparing to the uncoated surface. He et al. [27] performed systematic cryogenic quenching experiments using aluminum alloy rodlets coated with various surface structures. By selectively opening or closing the nano pores in the anodic aluminum oxide structure, they were able to distinguish between the contributions of low heat conductivity and nanoporous effects to the enhancement of quenching heat transfer. The above mentioned studies provide valuable information for un derstanding the nature of quenching heat transfer and how the surface
topology and properties affect quenching. In particular, the heat con ductivity of solids and micro/nano surface structure seem to have a profound influence on the collapse of vapor film. Comparing to metallic materials, rocks have low values of thermal conductivity coefficient and the rock surface possesses large quantity of cavities and pores to serve as nucleation sites. The present authors [28] demonstrated the ability of rock surfaces in enhancing the Leidenfrost temperature during cryogenic quenching. However, the experimental setup in Ref. [28] does not allow the visualization of the liquid-vapor interface dynamics and the rewet ting front motion. In this work, we modified the experiments in studying the cryogenic quenching heat transfer of rock surfaces in liquid nitrogen. The development of rewetting front on vertical rock surfaces was observed and recorded by high speed camera. Moreover, numerical simulations were conducted to recover the quenching history and to provide the boiling parameters of different surfaces. Different from the horizontal rock surfaces used in Ref. [28], we show that the geometric shape of quenching samples and the vertical orientation are important in determining the rewetting process. 2. Experiments The present experimental setup was designed to study the cryogenic quenching behavior of rock rodlets in liquid nitrogen pool at atmo spheric pressure. Synchronized temperature measurement and high speed video recording were performed during the entire quenching process. Fig. 1a schematically shows the layout of apparatus. The cy lindrical rock samples were fabricated from a sandstone block which consists of fine mineral grains. Fig. 1b shows the detail of the quenching sample assembly. The rock sample was 1 inch (25.4 mm) in diameter and 60 mm in length. A hole with diameter of 2 mm was drilled to the geometric center of the sample to mount the thermocouple wire. For exception of vibration of samples during quenching, a stainless steel pipe with outer diameter of 2 mm was used to reinforce the assembly. The thermocouple wire went through the stainless steel pipe with its tip attached on the bottom of the drilled hole. Silica gel was adopted to bond the thermocouple wire, the stainless steel pipe and the rock surface all together; it also served to seal any gaps at the exit of the hole. The gel was well resistant to the low temperature of liquid nitrogen and it could prevent the leakage of liquid nitrogen into the thermocouple hole. The exposed section of the stainless steel pipe was further fixed onto another bar with a larger diameter to increase the overall stiffness of the quenching sample assembly. Experiments showed that this practice was satisfactory; no appreciable sample vibration and liquid nitrogen leakage were observed. Saturated liquid nitrogen was placed in a double-walled glass dewar with its annular space being evacuated to vacuum. The inner diameter and height of the dewar were 160 mm and 200 mm, respectively. The quenching sample assembly was fixed on a supporting frame which allowed a smooth motion of the sample in vertical direction. During experiments, the rock sample was quenched from room temperature and the transient temperature evolution at the central location was recorded. T-type thermocouple wires (from OMEGA) 1 mm in diameter were used; the precision after calibration was �0.1 K. The signal was collected by a data acquisition board (ZTIC, USB 7410) at 10 Hz sampling rate. For the high speed video visualization, Phantom v310 camera was used and the resolution and frame rate were set to be 800 � 600 (0.17 mm per pixel) and 100 fps, respectively. The frame rate was selected to be relatively low due to a compromise between recording duration and recording speed. We want to record the entire process of quenching from film boiling to single phase convection and this would last for about 100 s. Therefore, the frame rate cannot be too large due to the limited RAM space in the camera. To attain discernible video clips, care must be taken to prevent the formation of frost on the outer surface of glass dewar. We found that the nitrogen vapor coming out of the dewar and sweeping downwards the glass surface was a major cause of frost generation. We thus employed an electrical fan to disperse the 2
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International Journal of Thermal Sciences 148 (2020) 106161
Fig. 1. (a) Schematic of the present experimental setup and (b) Detail of the quenching sample assembly.
undesired cold vapor during quenching. The position of the fan was shown in Fig. 1a; the distance between the fan and the liquid nitrogen dewar was 1.5 m. The wind power of the fan was adjusted so that it could effectively remove the vapor and meanwhile not to affect the stability of quenching sample assembly and glass dewar. Two LED lights were placed at both sides of the glass dewar to pro vide illumination. We found that the light source in the same direction as the camera lens was more useful than that against the camera. With the former light source it is possible to view the rewetting fronts on the rock surface. In contrast, the rock surface would become dark and invisible if only the latter light source was applied. The former light was elevated above the camera and had a downward angle. This is to avoid reflection of light on the glass wall which could create a white spot in the video. To prepare the rock surface in baseline case, we applied 1500 grit sandpaper to polish the rock sample. The rock surface was carefully cleaned with acetone and water and then dried in air before quenching tests. 80–120 grit quartz particles were spread on the rock surface to explore the effect of surface structures on quenching performances. The results are presented in the following sections.
substitute the thermocouple wire, the stainless steel pipe as well as the silica gel. Based on thermal properties of these components and their respective volume in the drilled hole, a volume-weighted average of thermal property was calculated and was assigned to the equivalent mono-substance. Fig. 2 summarized all the results of thermal properties including those of pure rock and of the mono-substance. It is seen that the thermal conductivity of the mono-substance is one order of magni tude larger than that of rock, because of the presence of metallic ma terials in the mono-substance. Low thermal conductivity is one of the features of rock materials compared to common metallic materials. We note that the estimated thermal properties as shown in Fig. 2 are subject to uncertainties that can be large and this is one of the difficulties in thermally related studies on geo-materials. Nevertheless, it will be shown in the next section that our numerical model performed well despite the uncertainty in properties. Fig. 3 compared the computational results for the two cases. The initial condition was a uniform temperature field of 300 K and the boundary condition used is from the next section. As seen from Fig. 3, the two temperature curves agree fairly well, which indicates that the influence of the drilled hole is small. This result should be related to the fact that the volume ratio of the drilled hole to the rock sample is small (0.31%). The maximum discrepancy between the two curves is found to be 4.1 K, less than 2% of the temperature variation range experienced by the rock during quenching.
3. Results 3.1. Validation of experimental data Before further analysis on the heat transfer and rewetting dynamics, it is necessary to evaluate the reliability of the measured temperature data in present experiments. Our first concern is that the drilled hole in the rock sample may influence the heat conduction and temperature measurement. Because the drilled hole was filled with stainless steel pipe, thermocouple wire and the silica gel, the thermal capacity of these substances may make the measured temperature deviate from the in tegrated rock case. Since the rock was presumably considered to be in tegrated in our numerical model, it is necessary to quantify the influence of the introduced substances on the temperature evolution at the rock sample center. In view of this, we calculated the transient temperature curves at the central location of sample in two cases; one is integrated rock and the other is a rock sample with a hole and mounted with thermocouple wire and stainless steel pipe. The calculation has been undertaken using ANSYS Fluent. To determine the various thermal properties that are required in the calculation, we measured the thermal conductivity of the rock sample from room temperature down to 30 � C. The results were then linearly extrapolated to the saturation tempera ture of liquid nitrogen. The specific heat of rock sample was measured at room temperature and the temperature variation trend from the litera ture had been used as a reference, as explained in Ref. [28]. For the region of the drilled hole, we adopted an equivalent mono-substance to
Fig. 2. Thermo-physical properties of the rock sample (solid lines) and the volumetric averaged mono-substance (dash lines) used in present calculations. The densities of them are 2505.8 and 6818.0 kg/m3, respectively. 3
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International Journal of Thermal Sciences 148 (2020) 106161
smooth and relatively calm during this period. Due to the strong reflection of light at the liquid-vapor interface, the rock surface looks bright, as shown in the left-most picture. After approximately 40 s, the rock surface was first wetted by liquid nitrogen at its two ends. The rewetting initiated at the perpendicular corners formed by the lateral face and the two bottom faces of the cyl inder. This is the start of transition boiling and the liquid-vapor interface became violent in this time due to the intense bubble generation. The wetted portion of rock surface became somewhat dim in the video because of the diffuse reflection. This color difference enables us to determine the location of rewetting fronts with naked eyes. In the middle picture of Fig. 4 the two rewetting fronts were marked by red arrows. An interesting finding was that the rewetting fronts consistently initiated at the two ends of the rock sample and merged at the upper half portion. We confirmed this phenomenon by repeated quenching tests and found that the merging point of rewetting fronts was located at ~2/ 3 of the rock sample length. This rewetting phenomenon is unique and is different from Kim et al. [19]’s study where a stainless steel rodlet was quenched in water. The rewetting front only initiated at the rodlet bottom and ascended with time until to the top [19]. In our view, the low thermal conductivity of rocks played an important role in this discrepancy while the property of coolant could also have an effect. More efforts are needed to understand how solids and coolant properties can affect the rewetting front dynamics. Compared with the rewetting of metallic rodlets which typically persists for several seconds, the rewetting of rock surface in our exper iments was slower and lasted for more than 20 s. Again, the low thermal conductivity of rocks was an important reason and this is qualitatively consistent with Yamanouchi’s analysis that the rewetting velocity de creases with the decrease of solid thermal conductivity [6]. In the last stage of quenching the rock surface was completely wetted as shown in the right-most picture in Fig. 4. The original appearance of the rock sample can be recognized in this picture. We examined the rock surface after each quenching test and found no fractures or defects present. The fact that the selected rock sample consisted of fine, dense and uniform mineral grains made it durable in multiple liquid nitrogen quenching tests. In accord with the visualization results, the measured temperature history inside the rock sample exhibited the same heat transfer regimes. The temperature decreases gently just after quenching starts until the occurrence of an inflection point after which the temperature decreases at a faster rate, see the inserted graph of Fig. 3. At the end of quenching the temperature stabilized at the saturation point of liquid nitrogen, which indicates a thorough cooling of the rock sample.
Fig. 3. Influence of the thermocouple hole on temperature measurement and the repeatability of quenching tests (inserted).
Also shown in Fig. 3 (inserted graph) are the measured temperature curves from three consecutive quenching tests. It is seen that the data scattering from repeating experiments is negligible which attests to the robustness of the results. To sum up, the measured temperature data are validated in terms of their repeatability and being insensitive to the drilled hole. 3.2. Numerical simulation of quenching results In this section, we analyze the heat transfer characteristics associated with cryogenic quenching of smooth rock surface in liquid nitrogen. The analysis will be based on the measured temperature history inside the rock sample and the synchronized high speed video results. Through the analysis we expect to obtain a deeper understanding of the quenching and rewetting features of rock surfaces. 3.2.1. Characteristics of quenching results Fig. 4 shows three video snapshots representing typical heat transfer stages during quenching of the rock sample in liquid nitrogen. At the early stage, the rock surface was wrapped by a layer of vapor blanket, which indicates the film boiling regime. The liquid-vapor interface was
Fig. 4. Snapshots showing three typical quenching stages; the figures in brackets signify the time elapsed after the onset of quenching. 4
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International Journal of Thermal Sciences 148 (2020) 106161
3.2.2. Simulation strategies The visualization reveals that the rewetting of rock sample consis tently starts from the perpendicular corners of the cylinder. Since the collapse of vapor film is apt to occur at cold regions, the results indicate that the perpendicular corner was colder than the other areas on rock surface. This is reasonable because the area-to-volume ratio is the largest at the corner region hence temperature in this region drops faster than other regions when subject to the same cooling heat flux. In this way, the geometric shape and boundary exert influences on the rewetting process by controlling the relative temperature distribution over the rock sample surface. The effect of the perpendicular corner on quenching and rewetting can be named as ‘end effects’; it determines the onset location and time of the rewetting process. This kind of boundary effect should definitely not be neglected in the numerical simulation of quenching and rewetting, as similarly stated by Starodubtseva et al. [14]. In this work, a numerical scheme was developed that solves the unsteady heat conduction problem within the cylindrical rock sample along with appropriate initial and boundary conditions. The major assumption was that the condition of rewetting was solely determined by the temperature on the rock surface. When the rock surface tem perature at a location is higher than the rewetting temperature, Tr, the local heat transfer mode is film boiling. For surface region whose tem perature is lower than Tr, the boiling regime turns to transition and nucleate boiling. The purpose of the simulation is to provide quantita tive information regarding the cryogenic quenching heat transfer of the vertical rock surface. Parameters such as rewetting temperature, maximum heat flux and film boiling heat transfer coefficient are of particular interests. These parameters are part of the outputs of the simulation. Fig. 5a illustrates the computational domain of the cylindrical rock. The problem was reduced to two dimensional by virtue of the axissymmetry property. The origin locates at the left-bottom of the domain; the position of a node is described by its radial and axial co ordinates, (r, z). The two dimensional unsteady heat conduction is described by the following governing equation: � � �� ∂T λ ∂2 T 1 ∂ ∂T þ ¼ ⋅ r ; r 2 ½0; R�; z 2 ½0; L�; t > 0 (1) ∂t ρc ∂z2 r ∂r ∂r
room condition, thus the initial condition is given as Tðr; z; 0Þ ¼ T0 ; r 2 ½0; R�; z 2 ½0; L�
(2)
where T0 ¼ 300 K is the room temperature. During quenching, the rock surface was cooled by liquid nitrogen and the cooling heat flux is dependent on the surface temperature head. The functional form between surface temperature and heat flux, q ¼ f (T), is unknown and must be searched according to the measured tem perature history by the thermocouple and the rewetting front dynamics recorded by the camera. Once the boundary condition, q ¼ f(T), is determined, the time-varied temperature field within the rock sample can be calculated. The functional form of f(T) corresponds to the boiling curve of liquid nitrogen on rock surface therefore the general shape of the curve can be known a prior. In film boiling segment, the wall temperature is high and the heat flux is low; the heat flux gently decreases with decreasing temperature. This trend terminates at the point of T ¼ Tr, after which the heat flux rises sharply to the maximum level. With the further decrease of wall temperature, the heat flux monotonically drops until becoming zero. It is evident that the film boiling heat flux, the Tr and the maximum heat flux are three key parameters determining the boiling curve. However, Fig. 4 shows that the two rewetting fronts move at different velocities. The lower front moves faster than the upper front does and they merge at the upper half position rather than at the center. This phenomenon indicates that there is a bias in heat transfer between the upper and lower rewetting fronts. If the two fronts have identical heat transfer patterns, then their merging point cannot deviate from the center. Therefore, it is necessary to devise two different boiling curves for the lower and upper rewetting fronts separately. Nada et al. [29,30] experimentally investigated the effect of orientation on the velocity of rewetting front motion. They found that the propagation of downward moving rewetting front could be adversely affected by the countercur rent vapor flow which moves upward. In present experiments, the generated vapor moves upward due to the buoyancy and this explains the relatively slower rewetting velocity of the upper front. Since the rewetting front is characterized by the direct liquid-solid contact and high local heat flux, we presumed that the countercurrent vapor flow acted to reduce the maximum heat flux associated with the upper front. It is plausible that the countercurrent vapor may exert a retarding force and hamper the liquid-solid contact in vicinity of the upper front.
where R and L are the radius and length of the rock sample, respectively. Before quenching, the temperature in the rock sample was uniform at
Fig. 5. (a) The computational domain of cylindrical rock sample for unsteady heat conduction. (b) Schematic of wall temperature and heat flux distribution along the vertical face of cylinder. 5
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Consequently, the heat transfer will be deteriorated. To this end, two boiling curves with different maximum heat flux were assigned to the two rewetting fronts. The lower front has a larger maximum heat flux than the upper front. In addition, the two boiling curves should overlap in film boiling segment and have the same value of Tr, because these parameters have no connection with the countercurrent vapor flow. The possible distribution of wall temperature and heat flux along the vertical face of rock sample was schematically shown in Fig. 5b where TLN represents the temperature of saturated liquid nitrogen. As shown in Fig. 5b, the temperature is the highest in the middle portion of the face and monotonically decreases towards the two ends. In the simulation, the upper and lower rewetting fronts were identified based on the symbol of local temperature gradient. For node i, if T(zi)>T(ziþ1) then this node is assigned with the boiling curve with smaller maximum heat flux, i.e., the node i is associated with the upper rewetting front. Otherwise, node i is treated as related to the lower rewetting front and the maximum heat flux is larger. Here, zi stands for the axial coordinate of the i-th node on the vertical face. There are two requirements for the determination of boiling curves, q ¼ f(T). First, the computed temperature at the geometric center of rock sample must agree with the measured temperature history with an acceptable accuracy. Second, the computed rewetting front locations should coincide with the experimentally captured rewetting fronts. The onset time of rewetting and the subsequent front locations at various time need to be consistent between simulation and experiments. For film boiling heat flux, the inverse heat transfer technique [31–33] was employed to provide a reference of the order of magnitude. As an example, Fig. 6 presents the inversely calculated heat flux varying with time using the measured temperature data during quenching. The one dimensional radial coordinate system was used in calculation thus the end effects were not considered. As a result, the heat flux data in Fig. 6 are useful only before t ¼ 50s; the maximum heat flux in Fig. 6 is arti ficial and does not correspond to the true condition in quenching. The calculated film boiling heat flux has a magnitude around 2.5e4 W/m2. Based on this result, the film boiling segment in the q ¼ f(T) curve is designed to be close to 2.5e4 W/m2 and having a positive slope. In our practice, a linear shape of the film boiling curve was assumed for simplicity. The heat flux at the ‘middle point’ of the curve was fixed at 2.5e4 W/m2. Then, the curve slope was adjusted until the computed temperature agrees with the measured one in the film boiling phase. The temperature of the ‘middle point’ was roughly 210 K which scales as the average of the initial temperature and the rewetting temperature of our rock surface. Therefore, the procedure of determining film boiling curve was simply fixing the center point of a line and changing its slope. In some cases, slight modifications may be applied on the basis of the
already-determined linear curve to further improve the agreement be tween calculated and measured temperature, although the improvement has been very marginal. With film boiling segment determined, the selection of Tr is straightforward with the help of visualization data. In present experi ments, the thermocouple is distant from the quenching surface (12.7 mm). It is not legitimate to infer the rewetting temperature, Tr, from the measured data because the damping effect of heat conduction creates a large disparity between the temperatures at the center and on the surface. We used the onset time of rewetting as a clue to attain the value of Tr. Assuming that the two ends of the cylindrical rock sample start to be rewetted at time t’ according to the high speed video, then the film boiling segment is held until time t’ in simulation and the computed local temperature at the perpendicular corner at that time will be chosen as Tr. The frame rate of high speed camera (100 fps) is sufficient to discern the onset time of rewetting. With the current location of ther mocouple, the present method to determine Tr is superior to other methods that solely utilize the temperature data, e.g., Refs. [19,34]. When surface temperature drops below Tr, transition boiling occurs and heat flux climbs to the maximum level rapidly. A small temperature range (several degrees) was allocated to the transition boiling. The maximum heat flux for the lower and upper fronts were simultaneously searched to match the experimental rewetting front locations as well as the measured temperature data in transition and nucleate boiling stages. The shape of nucleate boiling segment is also transformed to match the experimental results. Fig. 7 provides a flow-chart to conceptually summarize the numeri cal scheme developed in this work. The numerical simulation was per formed with MATLAB and the governing equation (1) was discretized by fully implicit difference method. 3.2.3. Sensitivity to grid size and time step In present simulations, the time step was 0.05s and the spatial grid size was 0.2 mm (corresponding to 19500 simulation cells). To check the sensitivity of simulation results to the selection of grid size and time step, we calculated the central temperature of the rock domain using different combinations of grid size and time step. The results are given in Fig. 8 where black triangles represent cases whose time step was fixed at 0.05s and grid size was varied. Red squares represent cases whose grid size was fixed at 0.2 mm and time step was varied. The markers that correspond to the case in present simulations were circled out. The ordinate of Fig. 8 is the central temperature sampled at t ¼ 70s. This time was chosen because the temperature drop rate is the highest then, as shown in Fig. 3. It is seen from Fig. 8 that the grid size has no essential influence on the calculated temperature, which attests that 0.2 mm is adequate as the grid size to discretize the governing equation. On the other hand, the calculated temperature slightly decreases with the decrease of time step. Because the heat flux rises rapidly when passing the Tr point, a relatively small time step is necessary to capture this rapid change. Nevertheless, the largest deviation of temperature in Fig. 8 is within 5% of the overall temperature variation during quenching, therefore 0.05s was retained as the time step accounting for the computational costs. 3.2.4. Simulation results The boiling curve was determined for the smooth rock surface using the developed numerical simulation strategy. Fig. 9 presents the two boiling curves for upper and lower rewetting fronts, respectively. The boiling curves had been employed as the boundary condition to numerically check the influence of the drilled hole (as shown in Fig. 3). Also plotted in Fig. 9 are various correlations and experimental data from the literature regarding different boiling parameters. The blue horizontal dash line represents the hydrodynamic theory for critical heat flux (CHF) proposed by Lienhard [35]. This correlation was for static pool boiling on horizontal flat surfaces.
Fig. 6. Measured temperature history and the inversely calculated heat flux for smooth rock surface. 6
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International Journal of Thermal Sciences 148 (2020) 106161
Fig. 7. Conceptual flow chart of the present numerical simulation scheme.
Fig. 9. The boiling curves determined for smooth rock surface using the pro posed numerical simulation scheme.
where h is the boiling heat transfer coefficient. h’fg is the corrected
Fig. 8. The calculated temperature at 70s using different combinations of grid size and time step.
CHF ¼ 0:1492ρg hfg
" gσ ρL
ρg
vaporization heat considering the superheat of vapor film. kg and μg are thermal conductivity and viscosity of vapor, respectively. △Tsat is the wall superheat. The constant coefficient in Bromley’s correlation (Eq. (5)) was given by Hsu and Westwater [38] for the case where vapor and liquid move with the same velocity. The red circular dots in Fig. 9 are the experimental data by Bom bardieri and Manfletti [39] for stationary pool boiling of liquid nitrogen on stainless steel surface. The black rhombus in Fig. 9 shows the quasi-stationary boiling data by Matsekh and Pavlenko [40] for vertical duralumin surface and falling liquid nitrogen film combination. As seen from Fig. 9, the maximum heat flux for the lower rewetting front is significantly larger than that for the upper rewetting front. It seems that the flowing orientation of generated vapor has a profound influence on the heat transfer near the rewetting front. Similar to the findings in Refs. [14,15], the maximum heat flux of the lower front exceeds the experimental CHF limit of the stationary boiling of liquid nitrogen. This is possibly caused by the strong agitation and fluctuation of liquid and vapor near the rewetting front due to the violent bubble
�#0:25
ρ2g
(3)
where ρg ; ρL ; hfg ; σ are vapor density, liquid density, latent heat of vaporization and surface tension force, respectively. The two lines at the right-bottom are correlations by Berenson [36] and Bromley [37] for film boiling heat flux, written as " ’ �#0:25 � � 1=8 hfg gρg k3g ρL ρg σ � Berenson : h ¼ 0:425 (4) μg ΔTsat g ρL ρg " ’ �#0:25 hfg gρg k3g ρL ρg Bromley : h ¼ 0:943 μg LΔTsat
(5)
7
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nucleation. The dynamics of liquid and vapor enhanced the local heat transfer rate near the rewetting front during quenching. On the other hand, the result of the hydrodynamic correlation is far larger than the present numerical result as well as the stationary experimental data. The effect of the heater property was not accounted in the hydrodynamic theory. However, the heat conductive efficiency of boiling surface af fects the heat flux limit appreciably [39]. The rock, stainless steel and duralumin are low heat conductive materials comparing to copper, which could explain the discrepancy between the hydrodynamic cor relation and the other results. For film boiling heat flux, the present numerical result is larger than that of Berenson’s and Bromley’s correlations. This difference should be attributed to the laminar vapor flow assumption adopted in both cor relations. Hsu and Westwater [38] demonstrated that the vapor flow in film boiling on vertical surfaces tends to be turbulent which increased the heat flux considerably. It is noted that the film boiling segment in Fig. 9 has a magnitude close to 2.5e4 W/m2, which was the result from the inverse heat transfer calculation (see Fig. 6). The rewetting temperature for the smooth rock surface was marked in Fig. 9, Tr ¼ 122 K. This value is higher than that in previous experi ments on liquid nitrogen quenching of metal plates. For smooth copper surface, Westwater et al. [41] and Starodubtseva et al. [14] determined Tr to be 97 K and 91 K, respectively. For copper surface coated with capillary-porous structure, Starodubtseva and Pavlenko [15]’s result showed that Tr ¼ 107 K. We have shown in previous work that the rock surface is inherently favorable for enhancing the Leidenfrost tempera ture due to its low thermal conductivity and porous surface structure [28,42]. The present result reinforced our previous conclusion. Fig. 10 shows the comparison between measured temperature his tory at the central position of rock sample and the simulated tempera ture using the determined boiling curves in Fig. 9. The experimental results were presented as a narrow band enclosed by two parallel red dash lines. The width of the band corresponds to the uncertainty of measured temperature data; the uncertainty comes from two sources, including the accuracy of thermocouple as well as the influence of the drilled hole and the heat capacity of the equivalent mono-substance. As seen from Fig. 10, the simulated and measured curves coincide well, which validates the present numerical method and results. Some comments are given regarding the uniqueness of the boiling curves in Fig. 9. Generally, the boiling curves cannot be determined with high authenticity if merely depending on the measured temperature at the center of rock sample, due to the effect of ill-posed problem. How ever, with the assistance of visualization data, the reliability of boiling
curves is greatly improved. For film boiling, our experience was that the heat flux was basically equal to the inversely calculated values. More over, the slope of the film boiling segment cannot be too large, otherwise the measured temperature curve will not be matched well. The rewet ting temperature, Tr, is sought based on film boiling heat flux and the observed onset time of rewetting. This value has a relatively small un certainty because of the clear definition of rewetting onset from the video. Both the maximum heat flux and the nucleate boiling curve shape affect the rewetting front velocity. As shown in Fig. 9, there is a plateau in boiling curve near the maximum heat flux level. It was found that a larger temperature range of this plateau will result in faster rewetting front motion. The combination of this plateau range and the maximum heat flux level was optimized to fit the front locations as well as the temperature data. Therefore, the nucleate boiling segment and the maximum heat flux may be subject to some uncertainties. Nevertheless, this uncertainty should be small compared to the discrepancy between the stationary boiling data [39,40] and present results. Therefore, the basic conclusion and findings of this work were not challenged. In Fig. 11, the snapshots from the high speed video were compared with the temperature field from the numerical simulation. The three dimensional contours shown in Fig. 11 were circumferentially extrap olated from the two dimensional computational domain and are thus axial-symmetric. It is seen that at 22s when no rewetting occurred, the temperature at the two ends of rock sample was lower than in other areas. As a result, the rewetting took place first at the sample ends, as shown by the second group of pictures in Fig. 11. The surface temper ature of wetted parts was dramatically lower than that of non-wetted parts and there was a discernible demarcation between the two parts in the contour. Hence we can directly read the position of rewetting fronts from the temperature contours with no difficulty. The initiated rewetting was circled by a red dash line in Fig. 11 and this is in accord with the temperature contour whose bottom part became deep blue in color, representing an extremely low temperature. At 62s, the two rewetting fronts were about to merge and remarkably, the simulated position of rewetting fronts agreed quite well with that captured by the camera. This result further verifies our present numerical model. During the propagation of rewetting fronts, a large temperature gradient was present on the rock surface. From the mechanical point of view, this would generate considerable thermal stresses on the rock surface and the stress should be at its maxima near the rewetting front. This analysis shed light on how the thermal shock effect was created during cryogenic quenching of rock in liquid nitrogen. Based on the boiling curves determined in Fig. 9, we were able to separate different boiling regimes using a dimensionless parameter: Sp ¼
cpL ⋅ΔTsat hfg
(6)
where Sp and cpL are dimensionless wall superheat and heat capacity of liquid nitrogen, respectively. As a result, the film boiling regime corre sponds to Sp > 0.45 condition; transition boiling corresponds to 0.4
Fig. 10. Comparison of measured temperature data and the simulated results. 8
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Fig. 11. Comparison of calculated temperature contours and high speed video snapshots.
the rock surface before deposition of quartz particles. The manipulating practice was similar to that reported in our previous work [28]. As the glue gets frozen in liquid nitrogen pool, the attached quartz particles stay firmly and no drop-off was observed. Fig. 12 shows the measured temperature histories for three different types of rock surfaces. One is the smooth rock surface covered with the glue layer, which also served as the benchmark here. The other two are surfaces covered with the quartz particles in different number densities. They were labeled as ‘sparser’ and ‘denser’, respectively. As an example, the insert of Fig. 12 shows a photograph of rock surface covered with denser quartz particles. We note that all the three surfaces were made from one individual rock sample so that the surface condition was the sole factor to affect the quenching results. In Fig. 12, the dotted lines represent experimentally measured temperatures. It is evident that the applied quartz particles on rock surface served to enhance the quenching heat transfer. The enhancement became more pronounced with the in crease of the number density of quartz particles. It is seen that the three curves start to deviate from each other at the early stage of quenching. Therefore, the quartz particles acted to improve the film boiling heat flux. Two possible mechanisms can be responsible for this effect. First, the particles enlarged the effective surface area of the rock sample, thus boosting the heat exchange between rock and liquid nitrogen. Second,
the size of quartz particles is large enough (~250 μm) to penetrate the vapor film and to promote local liquid-solid contacts. The intermittent liquid-solid contacts are effective in increasing the heat transfer of quenching, as discussed in Ref. [44]. The proposed numerical simulation scheme was also applied for the structured rock surfaces to derive the boiling curves for these surfaces. Fig. 13 presents the boiling curves of lower rewetting fronts for these three surfaces. The lower rewetting front was more representative of the quenching heat transfer characteristic than the upper front. Fig. 13 in dicates that, comparing to the glue covered surface, the applied quartz particles increased the film boiling heat flux, the rewetting temperature and the maximum heat flux as well. The rewetting temperature for the three surfaces were 110 K, 119.6 K and 125 K, respectively. It is inter esting that the glue covered surface is lower than the clean surface in rewetting temperature (110 vs. 122 K). The glue functioned as an insulating layer to isolate between the liquid nitrogen and the rock surface micro structure. Consequently, the cavities and pores on rock surface cannot contribute to the quenching acceleration by providing nucleation sites. Therefore, the present results corroborated our previ ous analysis on the effect of rock surface properties on quenching. For the three rock surfaces, the simulated temperature history at rock center were plotted in Fig. 12 by black solid lines. The simulated tem perature curves agree well with the experimentally measured ones, demonstrating the good applicability of the proposed numerical model.
Fig. 12. Measured and simulated temperature histories for structured rock surfaces covered with quartz particles. The solid line and dots represent simulated and measured temperature, respectively. The inserted photograph shows the rock surface covered with denser quartz particles.
Fig. 13. Boiling curves of the lower rewetting front determined for structured rock surfaces. 9
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4. Conclusions
[11] S. Sahu, P. Das, S. Bhattacharyya, A three-region conduction-controlled rewetting analysis by the Heat Balance Integral Method, Int. J. Therm. Sci. 48 (2009) 2100–2107. [12] S.K. Sahu, P.K. Das, S. Bhattacharyya, Analytical and semi-analytical models of conduction controlled rewetting: a state of the art review, Therm. Sci. 19 (2015) 1479–1496. [13] I. Starodubtseva, A. Pavlenko, O. Volodin, A. Surtaev, The features of rewetting dynamics of the overheated surface by a falling film of cryogenic liquid, Thermophys. Aeromechanics 19 (2012) 307–316. [14] I. Starodubtseva, A. Pavlenko, A. Surtaev, Heat transfer during quenching of high temperature surface by the falling cryogenic liquid film, Int. J. Therm. Sci. 114 (2017) 196–204. [15] I. Starodubtseva, A. Pavlenko, Quenching by falling cryogenic liquid film of extremely overheated plate with structured capillary-porous coating, J. Eng. Thermophys. 27 (2018) 294–302. [16] L.W. Fan, J.Q. Li, L. Zhang, Z.T. Yu, K.F. Cen, Pool boiling heat transfer on a nanoscale roughness-enhanced superhydrophilic surface for accelerated quenching in water, Appl. Therm. Eng. 109 (2016) 630–639. [17] I.U. Vakarelski, N.A. Patankar, J.O. Marston, D.Y. Chan, S.T. Thoroddsen, Stabilization of Leidenfrost vapour layer by textured superhydrophobic surfaces, Nature 489 (2012) 274–277. [18] J.-Q. Li, L.-W. Mou, Y.-H. Zhang, Z.-S. Yang, M.-H. Hou, L.-W. Fan, Z.-T. Yu, An experimental study of the accelerated quenching rate and enhanced pool boiling heat transfer on rodlets with a superhydrophilic surface in subcooled water, Exp. Therm. Fluid Sci. 92 (2018) 103–112. [19] H. Kim, J. Buongiorno, L.W. Hu, T. Mckrell, Nanoparticle deposition effects on the minimum heat flux point and quench front speed during quenching in water-based alumina nanofluids, Int. J. Heat Mass Transf. 53 (2010) 1542–1553. [20] H. Kim, G. Dewitt, T. Mckrell, J. Buongiorno, L.W. Hu, On the quenching of steel and zircaloy spheres in water-based nanofluids with alumina, silica and diamond nanoparticles, Int. J. Multiph. Flow 35 (2009) 427–438. [21] D. Takeda, K. Fukiba, H. Kobayashi, Improvement in pipe chilldown process using low thermal conductive layer, Int. J. Heat Mass Transf. 111 (2017) 115–122. [22] A.N. Tsoi, A.N. Pavlenko, Enhancement of transient heat transfer at boiling on a plate surface with low thermoconductive coatings, Thermophys. Aeromechþ 22 (2015) 707–712. [23] H. Hu, C. Xu, Y. Zhao, R. Shaeffer, K.J. Ziegler, J.N. Chung, Modification and enhancement of cryogenic quenching heat transfer by a nanoporous surface, Int. J. Heat Mass Transf. 80 (2015) 636–643. [24] H. Hu, C. Xu, Y. Zhao, K.J. Ziegler, J. Chung, Boiling and quenching heat transfer advancement by nanoscale surface modification, Sci. Rep. 7 (2017) 6117. [25] A.N. Pavlenko, D.V. Kuznetsov, Experimental study of the effect of structured capillary-porous coating on rewetting dynamics and heat transfer at film cooling by liquid nitrogen, J. Phys. Conf. Ser. 1105 (2018), 012053. [26] A.N. Pavlenko, A.N. Tsoi, A.S. Surtaev, D.V. Kuznetsov, V.I. Kalita, D.I. Komlev, A. Y. Ivannikov, A.A. Radyak, Experimental study of rewetting of a superheated plate with structured capillary-porous coating by flowing liquid film, J. High Temp. 56 (2018) 404–409. [27] D. He, P. Zhang, F. Lv, S. Wang, D. Shu, Cryogenic quenching enhancement of a nanoporous surface, Int. J. Heat Mass Transf. 134 (2019) 1061–1072. [28] R. Li, Z. Huang, X. Wu, P. Yan, X. Dai, Cryogenic quenching of rock using liquid nitrogen as a coolant: investigation of surface effects, Int. J. Heat Mass Transf. 119 (2018) 446–459. [29] S.A. Nada, Cooling of very hot vertical tubes by falling liquid film in presence of countercurrent flow of rising gases, Int. J. Therm. Sci. 88 (2015) 228–237. [30] S.A. Nada, M. Shoukri, A.F. El-Dib, A.S. Huzayyin, Rewetting of hot vertical tubes by a falling liquid film with different directions of venting the generated steam, Int. J. Therm. Sci. 85 (2014) 62–72. [31] J.V. Beck, B. Blackwell, C.R. St Clair, Inverse Heat Conduction, Ill-Posed Problems, Wiley, New York, 1985. [32] R. Li, Z. Huang, A new CHF model for enhanced pool boiling heat transfer on surfaces with micro-scale roughness, Int. J. Heat Mass Transf. 109 (2017) 1084–1093. [33] R. Li, Z. Huang, Estimating the transient thermal boundary conditions with an improved space marching technique, Int. J. Heat Mass Transf. 127 (2018) 59–67. [34] L.W. Fan, J.Q. Li, D.Y. Li, L. Zhang, Z.T. Yu, Regulated transient pool boiling of water during quenching on nanostructured surfaces with modified wettability from superhydrophilic to superhydrophobic, Int. J. Heat Mass Transf. 76 (2014) 81–89. [35] J.H. Lienhard, A Heat Transfer Textbook, Phlogiston Press, 2003. [36] P.J. Berenson, Film-boiling heat transfer from a horizontal surface, J. Heat Transf. 83 (1961) 351–356. [37] L.A. Bromley, Heat transfer in stable film boiling, Chem. Eng. Prog. 46 (1950) 221–226. [38] Y.-Y. Hsu, J. Westwater, Film boiling from vertical tubes, AlChE J. 4 (1958) 58–62. [39] C. Bombardieri, C. Manfletti, Influence of wall material on nucleate pool boiling of liquid nitrogen, Int. J. Heat Mass Transf. 94 (2016) 1–8. [40] A. Matsekh, A. Pavlenko, Heat transfer and crisis phenomena in the falling films of cryogenic liquid, J. Thermophys. Aeromech. 12 (2005) 99–112. [41] J.W. Westwater, J.J. Hwalek, M.E. Irving, Suggested standard method for obtaining boiling curves by quenching, Ind. Eng. Chem. Fundam. 25 (1986) 685–692.
In this paper, the cryogenic quenching experiments of rock surfaces in liquid nitrogen were reported. The heat transfer characteristics and rewetting dynamics were investigated based on measured temperature data and synchronized high speed video visualization. A numerical scheme was developed to simulate the quenching process by reproduc ing the temperature history and rewetting front positions. Several con clusions were obtained in present study: (1) The rewetting of the cylindrical rock sample starts from its two ends and the merging point of rewetting fronts locates near the upper 2/3 of the sample length. (2) The two rewetting fronts on rock surface bears different heat transfer patterns. The lower front has a larger maximum heat flux and therefore moves faster than the upper front. (3) The maximum heat flux in cryogenic quenching of vertical rock surfaces is higher than the stationary boiling heat flux limits, due to the local fluctuation of liquid and vapor and the intense nucleation and evaporation. (4) The distributed quartz particles on rock surface have the effect of improving film boiling heat flux and the rewetting temperature, as well as the maximum heat flux. (5) The numerical model developed in this paper is robust and widely applicable in simulating the quenching and rewetting of rock surfaces in liquid nitrogen. Declaration of competing interest None Declared. Acknowledgements This work was supported by the National Natural Science Foundation of China (51521063), the National Science Fund for Distinguished Young Scholars (No. 51725404) and Beijing Outstanding Young Scien tist Program (BJJWZYJH01201911414038). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.ijthermalsci.2019.106161. References [1] B.W. McDaniel, S.R. Grundmann, W.D. Kendrick, D.R. Wilson, S.W. Jordan, Field applications of cryogenic nitrogen as a hydraulic fracturing fluid, in: SPE Annual Technical Conference and Exhibition, Society of Petroleum Engineers, San Antonio, Texas, 1997. [2] S. Grundmann, G. Rodvelt, G. Dials, R. Allen, Cryogenic nitrogen as a hydraulic fracturing fluid in the devonian shale, in: SPE Eastern Regional Meeting, Pittsburgh, Pennsylvania, 1998. [3] C. Cai, G. Li, Z. Huang, Z. Shen, S. Tian, J. Wei, Experimental study of the effect of liquid nitrogen cooling on rock pore structure, J. Nat. Gas Sci. Eng. 21 (2014) 507–517. [4] C. Cai, G. Li, Z. Huang, S. Tian, Z. Shen, X. Fu, Experiment of coal damage due to super-cooling with liquid nitrogen, J. Nat. Gas Sci. Eng. 22 (2015) 42–48. [5] X. Wu, Z. Huang, R. Li, S. Zhang, H. Wen, P. Huang, X. Dai, C. Zhang, Investigation on the damage of high-temperature shale subjected to liquid nitrogen cooling, J. Nat. Gas Sci. Eng. 57 (2018) 284–294. [6] A. Yamanouchi, Effect of core spray cooling in transient state after loss of coolant accident, J. Nucl. Sci. Technol. 5 (1968) 547–558. [7] K. Sun, G. Dix, C. Tien, Effect of precursory cooling on falling-film rewetting, J. Heat Transf. 97 (1975) 360–365. [8] S. Dua, C. Tien, Two-dimensional analysis of conduction-controlled rewetting with precursory cooling, J. Heat Transf. 98 (1976) 407–413. [9] S. Sahu, P. Das, S. Bhattacharyya, Rewetting analysis of hot surfaces with internal heat source by the heat balance integral method, Heat Mass Transf. 44 (2008) 1247–1256. [10] M. Sawan, G. Zaki, H. Temraz, A three-regions rewetting model with heat generation and subcooling, Atomkernenerg. Kerntech. 34 (1979) 199–204.
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[42] R. Li, X. Wu, Z. Huang, Jet impingement boiling heat transfer from rock to liquid nitrogen during cryogenic quenching, Exp. Therm. Fluid Sci. 106 (2019) 255–264. [43] T. Palisch, R. Duenckel, B. Wilson, New technology yields ultrahigh-strength proppant, SPE Prod. Oper. 30 (2015) 76–81.
[44] H. Kim, B. Truong, J. Buongiorno, L.W. Hu, On the effect of surface roughness height, wettability, and nanoporosity on Leidenfrost phenomena, Appl. Phys. Lett. 98 (2011) 1–3.
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Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts
Quenching and rewetting of rock in liquid nitrogen: Characterizing heat transfer and surface effects Ran Li, Chengcheng Zhang, Zhongwei Huang * State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum (Beijing), Beijing 102249, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Cryogenic quenching Rock Numerical simulation Surface structuring
Cryogenic quenching of cylindrical rock rodlets in liquid nitrogen pool was experimentally investigated. The sample was 25.4 mm in diameter and 60 mm in length. Quenching was performed under saturated condition at the atmospheric pressure. Thermocouple and high speed camera were employed to attain the transient tem perature history and the two-phase dynamics during quenching. A numerical simulation scheme was developed to model the heat transfer process during quenching and rewetting. The boiling curves of quenching were established for different rock surfaces based on the temperature and visualization data. The rewetting of cy lindrical rock was observed to initiate at the two ends of sample and merged at the upper 2/3 of the sample length. This feature was well captured by our numerical model. It was found that the maximum heat flux during quenching exceeds the stationary boiling limits of liquid nitrogen, the same finding as in the literature. The effect of surface structures on quenching was also explored by experiments and the numerical model demonstrated its applicability for these structured surfaces as well.
1. Introduction
studies concentrated on the change of rock properties before and after liquid nitrogen quenching but no efforts were made to reveal the heat transfer mechanisms. Cai et al. [3] quenched marble and shale rock rodlets in liquid nitrogen pool. Based on scanning electron microscope (SEM) and nuclear magnetic resonance (NMR) techniques, they observed the expansion of micro pores in rock samples after quenching. Wu et al. [5] conducted the same investigations with rock samples at elevated temperatures. They found that the variation of rock properties after liquid nitrogen quenching became more pronounced when the initial temperature of rocks was higher. These authors qualitatively attributed their results to the thermal shock effects of liquid nitrogen, i. e., the thermal stress induced by rapid cooling. However, they never delved into the details of heat transfer and phase change dynamics involved in the process of liquid nitrogen quenching. The rock property change by liquid nitrogen quenching is a thermally-driven phenomenon. It is undeniable that the evolution of temperature fields and thermal characteristics during quenching directly determines the magnitude of the thermal shock effects. In this sense, it is valuable to study the heat transfer during quenching and rewetting of rock surfaces in liquid nitrogen. The thermal-hydraulic features of quenching and rewetting are featured by three separate boiling regimes. At the initial stage of
Quenching is a transient heat and mass transport phenomenon where hot solid surfaces are cooled by a liquid whose boiling point is appre ciably lower than the initial solid temperature. This phenomenon is relevant in many industrial applications such as metallic heat treatment, fuel refilling of space crafts, superconductive magnets and the loss of coolant accidents (LOCA) in nuclear reactors. The performance and ef ficacy of quenching process usually relate to the thermal safety and economic efficiency of energy and power systems. Therefore, a thorough understanding of the physical nature of quenching is desired for optimal design of these engineering applications. In the upstream oil & gas industry, the cryogenic quenching with liquid nitrogen is encountered. Operators have introduced liquid nitro gen into the wellbore and reservoir to increase the oil and gas produc tion rate [1,2]. By quenching reservoir rocks with liquid nitrogen, thermal stresses could be generated and they may increase the pore space in rocks or even induce fractures on rock surfaces. This mechanical change in rocks is beneficial for the oil & gas flow and can contribute to enhanced productions [3–5]. However, a fundamental understanding of the effect of liquid nitrogen quenching on rocks is lacking due to the complex thermal-hydraulic-mechanical coupling in this process. Current
* Corresponding author. E-mail addresses: [email protected] (R. Li), [email protected] (Z. Huang). https://doi.org/10.1016/j.ijthermalsci.2019.106161 Received 18 March 2019; Received in revised form 23 October 2019; Accepted 26 October 2019 Available online 12 November 2019 1290-0729/© 2019 Elsevier Masson SAS. All rights reserved.
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quenching, the solid surface is sufficiently hot to sustain a stable vapor film covering it and the heat transfer rate is relatively low, corre sponding to the film boiling regime. At some point, the vapor film starts to become unstable and eventually collapses which results in liquid-solid contact and the heat removal rate improves significantly. The collapse of vapor film marks the rewetting of solid surface and the start of transition and nucleate boiling. Most research works in the literature focused on the movement of the rewetting front, which is a triple contact line formed by the solid surface, vapor and liquid. This region is transient, oscillated and stochastic in nature, making the fundamental study on it extremely difficult. Yamanouchi [6] presented an analytical solution to the rewetting front velocity. He assigned a constant heat transfer coefficient to the wetted portion of vertical rods and assumed adiabatic conditions in dry areas. This model was referred to as two-region model and subsequent researchers made various refinements by considering precursory cooling [7,8], internal heat generation [9], or by modeling three different re gions [10,11]. An excellent review on these analytical works can be found in Sahu et al. [12]. The analytical models generally assume a steady state where the rewetting front has been well established and its moving velocity does not change with time. As such, the initial forma tion of rewetting front and the effect of geometric boundaries of solids cannot be incorporated. In comparison, the group of Pavlenko et al. [13–15] conducted a series of experiments on the cryogenic quenching of vertical copper plates by liquid nitrogen thin films. The transient surface temperature and rewetting front motion were recorded by thermocouples and high speed camera. Based on these data, the authors presented a numerical simulation of quenching which accounted for the geometric boundary and the initial cooling effects. They successfully reconstructed the boiling curve for liquid nitrogen thin film quenching and compared the results with the quasi-stationary boiling data. It was found that the static boiling data were insufficient in representing the transient quenching heat transfer. Apart from the modeling of quenching front velocity, efforts were also made to improve the quenching performance by modulating the properties of solid surface or quenchants. Superhydrophilic surfaces were found to have the capability of shortening the quenching duration by inducing earlier solid-liquid contact [16–18]. Repeated quenching in water-based nanofluids has analogous effects of quenching enhance ment. The heat transfer rate and rewetting front velocity were both improved in consecutive quenching in alumina nanofluids [19,20]. Takeda et al. [21] and Tsoi et al. [22] found that a layer of low heat conductive coatings on solid surface is capable of improving the Lei denfrost temperature, thus accelerating the overall quenching speed. Hu et al. [23] fabricated alumina nanoporous layers on the surface of an aluminum rodlet and performed quenching experiment in liquid nitro gen pool. The nanoporous layer effectively enhanced all the three boiling regimes with the most remarkable change in transition boiling regime. Hu et al. [24] refined their experiments and analysis on the enhancing effects of the nanoporous alumina layer. In particular, they calculated the active nucleation cavity radius and total nucleation contact line length for different nanoporous surfaces. As a result, the superhydrophilic surface property and excessive nanoscale nucleation sites were identified as two important factors in intensifying the quenching heat transfer rate. Pavlenko et al. [25,26] fabricated a capillary-porous coating on vertical copper plates by means of directed plasma spraying. They reported that the porous coating decreased the total cooling time of copper plate by more than three times comparing to the uncoated surface. He et al. [27] performed systematic cryogenic quenching experiments using aluminum alloy rodlets coated with various surface structures. By selectively opening or closing the nano pores in the anodic aluminum oxide structure, they were able to distinguish between the contributions of low heat conductivity and nanoporous effects to the enhancement of quenching heat transfer. The above mentioned studies provide valuable information for un derstanding the nature of quenching heat transfer and how the surface
topology and properties affect quenching. In particular, the heat con ductivity of solids and micro/nano surface structure seem to have a profound influence on the collapse of vapor film. Comparing to metallic materials, rocks have low values of thermal conductivity coefficient and the rock surface possesses large quantity of cavities and pores to serve as nucleation sites. The present authors [28] demonstrated the ability of rock surfaces in enhancing the Leidenfrost temperature during cryogenic quenching. However, the experimental setup in Ref. [28] does not allow the visualization of the liquid-vapor interface dynamics and the rewet ting front motion. In this work, we modified the experiments in studying the cryogenic quenching heat transfer of rock surfaces in liquid nitrogen. The development of rewetting front on vertical rock surfaces was observed and recorded by high speed camera. Moreover, numerical simulations were conducted to recover the quenching history and to provide the boiling parameters of different surfaces. Different from the horizontal rock surfaces used in Ref. [28], we show that the geometric shape of quenching samples and the vertical orientation are important in determining the rewetting process. 2. Experiments The present experimental setup was designed to study the cryogenic quenching behavior of rock rodlets in liquid nitrogen pool at atmo spheric pressure. Synchronized temperature measurement and high speed video recording were performed during the entire quenching process. Fig. 1a schematically shows the layout of apparatus. The cy lindrical rock samples were fabricated from a sandstone block which consists of fine mineral grains. Fig. 1b shows the detail of the quenching sample assembly. The rock sample was 1 inch (25.4 mm) in diameter and 60 mm in length. A hole with diameter of 2 mm was drilled to the geometric center of the sample to mount the thermocouple wire. For exception of vibration of samples during quenching, a stainless steel pipe with outer diameter of 2 mm was used to reinforce the assembly. The thermocouple wire went through the stainless steel pipe with its tip attached on the bottom of the drilled hole. Silica gel was adopted to bond the thermocouple wire, the stainless steel pipe and the rock surface all together; it also served to seal any gaps at the exit of the hole. The gel was well resistant to the low temperature of liquid nitrogen and it could prevent the leakage of liquid nitrogen into the thermocouple hole. The exposed section of the stainless steel pipe was further fixed onto another bar with a larger diameter to increase the overall stiffness of the quenching sample assembly. Experiments showed that this practice was satisfactory; no appreciable sample vibration and liquid nitrogen leakage were observed. Saturated liquid nitrogen was placed in a double-walled glass dewar with its annular space being evacuated to vacuum. The inner diameter and height of the dewar were 160 mm and 200 mm, respectively. The quenching sample assembly was fixed on a supporting frame which allowed a smooth motion of the sample in vertical direction. During experiments, the rock sample was quenched from room temperature and the transient temperature evolution at the central location was recorded. T-type thermocouple wires (from OMEGA) 1 mm in diameter were used; the precision after calibration was �0.1 K. The signal was collected by a data acquisition board (ZTIC, USB 7410) at 10 Hz sampling rate. For the high speed video visualization, Phantom v310 camera was used and the resolution and frame rate were set to be 800 � 600 (0.17 mm per pixel) and 100 fps, respectively. The frame rate was selected to be relatively low due to a compromise between recording duration and recording speed. We want to record the entire process of quenching from film boiling to single phase convection and this would last for about 100 s. Therefore, the frame rate cannot be too large due to the limited RAM space in the camera. To attain discernible video clips, care must be taken to prevent the formation of frost on the outer surface of glass dewar. We found that the nitrogen vapor coming out of the dewar and sweeping downwards the glass surface was a major cause of frost generation. We thus employed an electrical fan to disperse the 2
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Fig. 1. (a) Schematic of the present experimental setup and (b) Detail of the quenching sample assembly.
undesired cold vapor during quenching. The position of the fan was shown in Fig. 1a; the distance between the fan and the liquid nitrogen dewar was 1.5 m. The wind power of the fan was adjusted so that it could effectively remove the vapor and meanwhile not to affect the stability of quenching sample assembly and glass dewar. Two LED lights were placed at both sides of the glass dewar to pro vide illumination. We found that the light source in the same direction as the camera lens was more useful than that against the camera. With the former light source it is possible to view the rewetting fronts on the rock surface. In contrast, the rock surface would become dark and invisible if only the latter light source was applied. The former light was elevated above the camera and had a downward angle. This is to avoid reflection of light on the glass wall which could create a white spot in the video. To prepare the rock surface in baseline case, we applied 1500 grit sandpaper to polish the rock sample. The rock surface was carefully cleaned with acetone and water and then dried in air before quenching tests. 80–120 grit quartz particles were spread on the rock surface to explore the effect of surface structures on quenching performances. The results are presented in the following sections.
substitute the thermocouple wire, the stainless steel pipe as well as the silica gel. Based on thermal properties of these components and their respective volume in the drilled hole, a volume-weighted average of thermal property was calculated and was assigned to the equivalent mono-substance. Fig. 2 summarized all the results of thermal properties including those of pure rock and of the mono-substance. It is seen that the thermal conductivity of the mono-substance is one order of magni tude larger than that of rock, because of the presence of metallic ma terials in the mono-substance. Low thermal conductivity is one of the features of rock materials compared to common metallic materials. We note that the estimated thermal properties as shown in Fig. 2 are subject to uncertainties that can be large and this is one of the difficulties in thermally related studies on geo-materials. Nevertheless, it will be shown in the next section that our numerical model performed well despite the uncertainty in properties. Fig. 3 compared the computational results for the two cases. The initial condition was a uniform temperature field of 300 K and the boundary condition used is from the next section. As seen from Fig. 3, the two temperature curves agree fairly well, which indicates that the influence of the drilled hole is small. This result should be related to the fact that the volume ratio of the drilled hole to the rock sample is small (0.31%). The maximum discrepancy between the two curves is found to be 4.1 K, less than 2% of the temperature variation range experienced by the rock during quenching.
3. Results 3.1. Validation of experimental data Before further analysis on the heat transfer and rewetting dynamics, it is necessary to evaluate the reliability of the measured temperature data in present experiments. Our first concern is that the drilled hole in the rock sample may influence the heat conduction and temperature measurement. Because the drilled hole was filled with stainless steel pipe, thermocouple wire and the silica gel, the thermal capacity of these substances may make the measured temperature deviate from the in tegrated rock case. Since the rock was presumably considered to be in tegrated in our numerical model, it is necessary to quantify the influence of the introduced substances on the temperature evolution at the rock sample center. In view of this, we calculated the transient temperature curves at the central location of sample in two cases; one is integrated rock and the other is a rock sample with a hole and mounted with thermocouple wire and stainless steel pipe. The calculation has been undertaken using ANSYS Fluent. To determine the various thermal properties that are required in the calculation, we measured the thermal conductivity of the rock sample from room temperature down to 30 � C. The results were then linearly extrapolated to the saturation tempera ture of liquid nitrogen. The specific heat of rock sample was measured at room temperature and the temperature variation trend from the litera ture had been used as a reference, as explained in Ref. [28]. For the region of the drilled hole, we adopted an equivalent mono-substance to
Fig. 2. Thermo-physical properties of the rock sample (solid lines) and the volumetric averaged mono-substance (dash lines) used in present calculations. The densities of them are 2505.8 and 6818.0 kg/m3, respectively. 3
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smooth and relatively calm during this period. Due to the strong reflection of light at the liquid-vapor interface, the rock surface looks bright, as shown in the left-most picture. After approximately 40 s, the rock surface was first wetted by liquid nitrogen at its two ends. The rewetting initiated at the perpendicular corners formed by the lateral face and the two bottom faces of the cyl inder. This is the start of transition boiling and the liquid-vapor interface became violent in this time due to the intense bubble generation. The wetted portion of rock surface became somewhat dim in the video because of the diffuse reflection. This color difference enables us to determine the location of rewetting fronts with naked eyes. In the middle picture of Fig. 4 the two rewetting fronts were marked by red arrows. An interesting finding was that the rewetting fronts consistently initiated at the two ends of the rock sample and merged at the upper half portion. We confirmed this phenomenon by repeated quenching tests and found that the merging point of rewetting fronts was located at ~2/ 3 of the rock sample length. This rewetting phenomenon is unique and is different from Kim et al. [19]’s study where a stainless steel rodlet was quenched in water. The rewetting front only initiated at the rodlet bottom and ascended with time until to the top [19]. In our view, the low thermal conductivity of rocks played an important role in this discrepancy while the property of coolant could also have an effect. More efforts are needed to understand how solids and coolant properties can affect the rewetting front dynamics. Compared with the rewetting of metallic rodlets which typically persists for several seconds, the rewetting of rock surface in our exper iments was slower and lasted for more than 20 s. Again, the low thermal conductivity of rocks was an important reason and this is qualitatively consistent with Yamanouchi’s analysis that the rewetting velocity de creases with the decrease of solid thermal conductivity [6]. In the last stage of quenching the rock surface was completely wetted as shown in the right-most picture in Fig. 4. The original appearance of the rock sample can be recognized in this picture. We examined the rock surface after each quenching test and found no fractures or defects present. The fact that the selected rock sample consisted of fine, dense and uniform mineral grains made it durable in multiple liquid nitrogen quenching tests. In accord with the visualization results, the measured temperature history inside the rock sample exhibited the same heat transfer regimes. The temperature decreases gently just after quenching starts until the occurrence of an inflection point after which the temperature decreases at a faster rate, see the inserted graph of Fig. 3. At the end of quenching the temperature stabilized at the saturation point of liquid nitrogen, which indicates a thorough cooling of the rock sample.
Fig. 3. Influence of the thermocouple hole on temperature measurement and the repeatability of quenching tests (inserted).
Also shown in Fig. 3 (inserted graph) are the measured temperature curves from three consecutive quenching tests. It is seen that the data scattering from repeating experiments is negligible which attests to the robustness of the results. To sum up, the measured temperature data are validated in terms of their repeatability and being insensitive to the drilled hole. 3.2. Numerical simulation of quenching results In this section, we analyze the heat transfer characteristics associated with cryogenic quenching of smooth rock surface in liquid nitrogen. The analysis will be based on the measured temperature history inside the rock sample and the synchronized high speed video results. Through the analysis we expect to obtain a deeper understanding of the quenching and rewetting features of rock surfaces. 3.2.1. Characteristics of quenching results Fig. 4 shows three video snapshots representing typical heat transfer stages during quenching of the rock sample in liquid nitrogen. At the early stage, the rock surface was wrapped by a layer of vapor blanket, which indicates the film boiling regime. The liquid-vapor interface was
Fig. 4. Snapshots showing three typical quenching stages; the figures in brackets signify the time elapsed after the onset of quenching. 4
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3.2.2. Simulation strategies The visualization reveals that the rewetting of rock sample consis tently starts from the perpendicular corners of the cylinder. Since the collapse of vapor film is apt to occur at cold regions, the results indicate that the perpendicular corner was colder than the other areas on rock surface. This is reasonable because the area-to-volume ratio is the largest at the corner region hence temperature in this region drops faster than other regions when subject to the same cooling heat flux. In this way, the geometric shape and boundary exert influences on the rewetting process by controlling the relative temperature distribution over the rock sample surface. The effect of the perpendicular corner on quenching and rewetting can be named as ‘end effects’; it determines the onset location and time of the rewetting process. This kind of boundary effect should definitely not be neglected in the numerical simulation of quenching and rewetting, as similarly stated by Starodubtseva et al. [14]. In this work, a numerical scheme was developed that solves the unsteady heat conduction problem within the cylindrical rock sample along with appropriate initial and boundary conditions. The major assumption was that the condition of rewetting was solely determined by the temperature on the rock surface. When the rock surface tem perature at a location is higher than the rewetting temperature, Tr, the local heat transfer mode is film boiling. For surface region whose tem perature is lower than Tr, the boiling regime turns to transition and nucleate boiling. The purpose of the simulation is to provide quantita tive information regarding the cryogenic quenching heat transfer of the vertical rock surface. Parameters such as rewetting temperature, maximum heat flux and film boiling heat transfer coefficient are of particular interests. These parameters are part of the outputs of the simulation. Fig. 5a illustrates the computational domain of the cylindrical rock. The problem was reduced to two dimensional by virtue of the axissymmetry property. The origin locates at the left-bottom of the domain; the position of a node is described by its radial and axial co ordinates, (r, z). The two dimensional unsteady heat conduction is described by the following governing equation: � � �� ∂T λ ∂2 T 1 ∂ ∂T þ ¼ ⋅ r ; r 2 ½0; R�; z 2 ½0; L�; t > 0 (1) ∂t ρc ∂z2 r ∂r ∂r
room condition, thus the initial condition is given as Tðr; z; 0Þ ¼ T0 ; r 2 ½0; R�; z 2 ½0; L�
(2)
where T0 ¼ 300 K is the room temperature. During quenching, the rock surface was cooled by liquid nitrogen and the cooling heat flux is dependent on the surface temperature head. The functional form between surface temperature and heat flux, q ¼ f (T), is unknown and must be searched according to the measured tem perature history by the thermocouple and the rewetting front dynamics recorded by the camera. Once the boundary condition, q ¼ f(T), is determined, the time-varied temperature field within the rock sample can be calculated. The functional form of f(T) corresponds to the boiling curve of liquid nitrogen on rock surface therefore the general shape of the curve can be known a prior. In film boiling segment, the wall temperature is high and the heat flux is low; the heat flux gently decreases with decreasing temperature. This trend terminates at the point of T ¼ Tr, after which the heat flux rises sharply to the maximum level. With the further decrease of wall temperature, the heat flux monotonically drops until becoming zero. It is evident that the film boiling heat flux, the Tr and the maximum heat flux are three key parameters determining the boiling curve. However, Fig. 4 shows that the two rewetting fronts move at different velocities. The lower front moves faster than the upper front does and they merge at the upper half position rather than at the center. This phenomenon indicates that there is a bias in heat transfer between the upper and lower rewetting fronts. If the two fronts have identical heat transfer patterns, then their merging point cannot deviate from the center. Therefore, it is necessary to devise two different boiling curves for the lower and upper rewetting fronts separately. Nada et al. [29,30] experimentally investigated the effect of orientation on the velocity of rewetting front motion. They found that the propagation of downward moving rewetting front could be adversely affected by the countercur rent vapor flow which moves upward. In present experiments, the generated vapor moves upward due to the buoyancy and this explains the relatively slower rewetting velocity of the upper front. Since the rewetting front is characterized by the direct liquid-solid contact and high local heat flux, we presumed that the countercurrent vapor flow acted to reduce the maximum heat flux associated with the upper front. It is plausible that the countercurrent vapor may exert a retarding force and hamper the liquid-solid contact in vicinity of the upper front.
where R and L are the radius and length of the rock sample, respectively. Before quenching, the temperature in the rock sample was uniform at
Fig. 5. (a) The computational domain of cylindrical rock sample for unsteady heat conduction. (b) Schematic of wall temperature and heat flux distribution along the vertical face of cylinder. 5
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Consequently, the heat transfer will be deteriorated. To this end, two boiling curves with different maximum heat flux were assigned to the two rewetting fronts. The lower front has a larger maximum heat flux than the upper front. In addition, the two boiling curves should overlap in film boiling segment and have the same value of Tr, because these parameters have no connection with the countercurrent vapor flow. The possible distribution of wall temperature and heat flux along the vertical face of rock sample was schematically shown in Fig. 5b where TLN represents the temperature of saturated liquid nitrogen. As shown in Fig. 5b, the temperature is the highest in the middle portion of the face and monotonically decreases towards the two ends. In the simulation, the upper and lower rewetting fronts were identified based on the symbol of local temperature gradient. For node i, if T(zi)>T(ziþ1) then this node is assigned with the boiling curve with smaller maximum heat flux, i.e., the node i is associated with the upper rewetting front. Otherwise, node i is treated as related to the lower rewetting front and the maximum heat flux is larger. Here, zi stands for the axial coordinate of the i-th node on the vertical face. There are two requirements for the determination of boiling curves, q ¼ f(T). First, the computed temperature at the geometric center of rock sample must agree with the measured temperature history with an acceptable accuracy. Second, the computed rewetting front locations should coincide with the experimentally captured rewetting fronts. The onset time of rewetting and the subsequent front locations at various time need to be consistent between simulation and experiments. For film boiling heat flux, the inverse heat transfer technique [31–33] was employed to provide a reference of the order of magnitude. As an example, Fig. 6 presents the inversely calculated heat flux varying with time using the measured temperature data during quenching. The one dimensional radial coordinate system was used in calculation thus the end effects were not considered. As a result, the heat flux data in Fig. 6 are useful only before t ¼ 50s; the maximum heat flux in Fig. 6 is arti ficial and does not correspond to the true condition in quenching. The calculated film boiling heat flux has a magnitude around 2.5e4 W/m2. Based on this result, the film boiling segment in the q ¼ f(T) curve is designed to be close to 2.5e4 W/m2 and having a positive slope. In our practice, a linear shape of the film boiling curve was assumed for simplicity. The heat flux at the ‘middle point’ of the curve was fixed at 2.5e4 W/m2. Then, the curve slope was adjusted until the computed temperature agrees with the measured one in the film boiling phase. The temperature of the ‘middle point’ was roughly 210 K which scales as the average of the initial temperature and the rewetting temperature of our rock surface. Therefore, the procedure of determining film boiling curve was simply fixing the center point of a line and changing its slope. In some cases, slight modifications may be applied on the basis of the
already-determined linear curve to further improve the agreement be tween calculated and measured temperature, although the improvement has been very marginal. With film boiling segment determined, the selection of Tr is straightforward with the help of visualization data. In present experi ments, the thermocouple is distant from the quenching surface (12.7 mm). It is not legitimate to infer the rewetting temperature, Tr, from the measured data because the damping effect of heat conduction creates a large disparity between the temperatures at the center and on the surface. We used the onset time of rewetting as a clue to attain the value of Tr. Assuming that the two ends of the cylindrical rock sample start to be rewetted at time t’ according to the high speed video, then the film boiling segment is held until time t’ in simulation and the computed local temperature at the perpendicular corner at that time will be chosen as Tr. The frame rate of high speed camera (100 fps) is sufficient to discern the onset time of rewetting. With the current location of ther mocouple, the present method to determine Tr is superior to other methods that solely utilize the temperature data, e.g., Refs. [19,34]. When surface temperature drops below Tr, transition boiling occurs and heat flux climbs to the maximum level rapidly. A small temperature range (several degrees) was allocated to the transition boiling. The maximum heat flux for the lower and upper fronts were simultaneously searched to match the experimental rewetting front locations as well as the measured temperature data in transition and nucleate boiling stages. The shape of nucleate boiling segment is also transformed to match the experimental results. Fig. 7 provides a flow-chart to conceptually summarize the numeri cal scheme developed in this work. The numerical simulation was per formed with MATLAB and the governing equation (1) was discretized by fully implicit difference method. 3.2.3. Sensitivity to grid size and time step In present simulations, the time step was 0.05s and the spatial grid size was 0.2 mm (corresponding to 19500 simulation cells). To check the sensitivity of simulation results to the selection of grid size and time step, we calculated the central temperature of the rock domain using different combinations of grid size and time step. The results are given in Fig. 8 where black triangles represent cases whose time step was fixed at 0.05s and grid size was varied. Red squares represent cases whose grid size was fixed at 0.2 mm and time step was varied. The markers that correspond to the case in present simulations were circled out. The ordinate of Fig. 8 is the central temperature sampled at t ¼ 70s. This time was chosen because the temperature drop rate is the highest then, as shown in Fig. 3. It is seen from Fig. 8 that the grid size has no essential influence on the calculated temperature, which attests that 0.2 mm is adequate as the grid size to discretize the governing equation. On the other hand, the calculated temperature slightly decreases with the decrease of time step. Because the heat flux rises rapidly when passing the Tr point, a relatively small time step is necessary to capture this rapid change. Nevertheless, the largest deviation of temperature in Fig. 8 is within 5% of the overall temperature variation during quenching, therefore 0.05s was retained as the time step accounting for the computational costs. 3.2.4. Simulation results The boiling curve was determined for the smooth rock surface using the developed numerical simulation strategy. Fig. 9 presents the two boiling curves for upper and lower rewetting fronts, respectively. The boiling curves had been employed as the boundary condition to numerically check the influence of the drilled hole (as shown in Fig. 3). Also plotted in Fig. 9 are various correlations and experimental data from the literature regarding different boiling parameters. The blue horizontal dash line represents the hydrodynamic theory for critical heat flux (CHF) proposed by Lienhard [35]. This correlation was for static pool boiling on horizontal flat surfaces.
Fig. 6. Measured temperature history and the inversely calculated heat flux for smooth rock surface. 6
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International Journal of Thermal Sciences 148 (2020) 106161
Fig. 7. Conceptual flow chart of the present numerical simulation scheme.
Fig. 9. The boiling curves determined for smooth rock surface using the pro posed numerical simulation scheme.
where h is the boiling heat transfer coefficient. h’fg is the corrected
Fig. 8. The calculated temperature at 70s using different combinations of grid size and time step.
CHF ¼ 0:1492ρg hfg
" gσ ρL
ρg
vaporization heat considering the superheat of vapor film. kg and μg are thermal conductivity and viscosity of vapor, respectively. △Tsat is the wall superheat. The constant coefficient in Bromley’s correlation (Eq. (5)) was given by Hsu and Westwater [38] for the case where vapor and liquid move with the same velocity. The red circular dots in Fig. 9 are the experimental data by Bom bardieri and Manfletti [39] for stationary pool boiling of liquid nitrogen on stainless steel surface. The black rhombus in Fig. 9 shows the quasi-stationary boiling data by Matsekh and Pavlenko [40] for vertical duralumin surface and falling liquid nitrogen film combination. As seen from Fig. 9, the maximum heat flux for the lower rewetting front is significantly larger than that for the upper rewetting front. It seems that the flowing orientation of generated vapor has a profound influence on the heat transfer near the rewetting front. Similar to the findings in Refs. [14,15], the maximum heat flux of the lower front exceeds the experimental CHF limit of the stationary boiling of liquid nitrogen. This is possibly caused by the strong agitation and fluctuation of liquid and vapor near the rewetting front due to the violent bubble
�#0:25
ρ2g
(3)
where ρg ; ρL ; hfg ; σ are vapor density, liquid density, latent heat of vaporization and surface tension force, respectively. The two lines at the right-bottom are correlations by Berenson [36] and Bromley [37] for film boiling heat flux, written as " ’ �#0:25 � � 1=8 hfg gρg k3g ρL ρg σ � Berenson : h ¼ 0:425 (4) μg ΔTsat g ρL ρg " ’ �#0:25 hfg gρg k3g ρL ρg Bromley : h ¼ 0:943 μg LΔTsat
(5)
7
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International Journal of Thermal Sciences 148 (2020) 106161
nucleation. The dynamics of liquid and vapor enhanced the local heat transfer rate near the rewetting front during quenching. On the other hand, the result of the hydrodynamic correlation is far larger than the present numerical result as well as the stationary experimental data. The effect of the heater property was not accounted in the hydrodynamic theory. However, the heat conductive efficiency of boiling surface af fects the heat flux limit appreciably [39]. The rock, stainless steel and duralumin are low heat conductive materials comparing to copper, which could explain the discrepancy between the hydrodynamic cor relation and the other results. For film boiling heat flux, the present numerical result is larger than that of Berenson’s and Bromley’s correlations. This difference should be attributed to the laminar vapor flow assumption adopted in both cor relations. Hsu and Westwater [38] demonstrated that the vapor flow in film boiling on vertical surfaces tends to be turbulent which increased the heat flux considerably. It is noted that the film boiling segment in Fig. 9 has a magnitude close to 2.5e4 W/m2, which was the result from the inverse heat transfer calculation (see Fig. 6). The rewetting temperature for the smooth rock surface was marked in Fig. 9, Tr ¼ 122 K. This value is higher than that in previous experi ments on liquid nitrogen quenching of metal plates. For smooth copper surface, Westwater et al. [41] and Starodubtseva et al. [14] determined Tr to be 97 K and 91 K, respectively. For copper surface coated with capillary-porous structure, Starodubtseva and Pavlenko [15]’s result showed that Tr ¼ 107 K. We have shown in previous work that the rock surface is inherently favorable for enhancing the Leidenfrost tempera ture due to its low thermal conductivity and porous surface structure [28,42]. The present result reinforced our previous conclusion. Fig. 10 shows the comparison between measured temperature his tory at the central position of rock sample and the simulated tempera ture using the determined boiling curves in Fig. 9. The experimental results were presented as a narrow band enclosed by two parallel red dash lines. The width of the band corresponds to the uncertainty of measured temperature data; the uncertainty comes from two sources, including the accuracy of thermocouple as well as the influence of the drilled hole and the heat capacity of the equivalent mono-substance. As seen from Fig. 10, the simulated and measured curves coincide well, which validates the present numerical method and results. Some comments are given regarding the uniqueness of the boiling curves in Fig. 9. Generally, the boiling curves cannot be determined with high authenticity if merely depending on the measured temperature at the center of rock sample, due to the effect of ill-posed problem. How ever, with the assistance of visualization data, the reliability of boiling
curves is greatly improved. For film boiling, our experience was that the heat flux was basically equal to the inversely calculated values. More over, the slope of the film boiling segment cannot be too large, otherwise the measured temperature curve will not be matched well. The rewet ting temperature, Tr, is sought based on film boiling heat flux and the observed onset time of rewetting. This value has a relatively small un certainty because of the clear definition of rewetting onset from the video. Both the maximum heat flux and the nucleate boiling curve shape affect the rewetting front velocity. As shown in Fig. 9, there is a plateau in boiling curve near the maximum heat flux level. It was found that a larger temperature range of this plateau will result in faster rewetting front motion. The combination of this plateau range and the maximum heat flux level was optimized to fit the front locations as well as the temperature data. Therefore, the nucleate boiling segment and the maximum heat flux may be subject to some uncertainties. Nevertheless, this uncertainty should be small compared to the discrepancy between the stationary boiling data [39,40] and present results. Therefore, the basic conclusion and findings of this work were not challenged. In Fig. 11, the snapshots from the high speed video were compared with the temperature field from the numerical simulation. The three dimensional contours shown in Fig. 11 were circumferentially extrap olated from the two dimensional computational domain and are thus axial-symmetric. It is seen that at 22s when no rewetting occurred, the temperature at the two ends of rock sample was lower than in other areas. As a result, the rewetting took place first at the sample ends, as shown by the second group of pictures in Fig. 11. The surface temper ature of wetted parts was dramatically lower than that of non-wetted parts and there was a discernible demarcation between the two parts in the contour. Hence we can directly read the position of rewetting fronts from the temperature contours with no difficulty. The initiated rewetting was circled by a red dash line in Fig. 11 and this is in accord with the temperature contour whose bottom part became deep blue in color, representing an extremely low temperature. At 62s, the two rewetting fronts were about to merge and remarkably, the simulated position of rewetting fronts agreed quite well with that captured by the camera. This result further verifies our present numerical model. During the propagation of rewetting fronts, a large temperature gradient was present on the rock surface. From the mechanical point of view, this would generate considerable thermal stresses on the rock surface and the stress should be at its maxima near the rewetting front. This analysis shed light on how the thermal shock effect was created during cryogenic quenching of rock in liquid nitrogen. Based on the boiling curves determined in Fig. 9, we were able to separate different boiling regimes using a dimensionless parameter: Sp ¼
cpL ⋅ΔTsat hfg
(6)
where Sp and cpL are dimensionless wall superheat and heat capacity of liquid nitrogen, respectively. As a result, the film boiling regime corre sponds to Sp > 0.45 condition; transition boiling corresponds to 0.4
Fig. 10. Comparison of measured temperature data and the simulated results. 8
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International Journal of Thermal Sciences 148 (2020) 106161
Fig. 11. Comparison of calculated temperature contours and high speed video snapshots.
the rock surface before deposition of quartz particles. The manipulating practice was similar to that reported in our previous work [28]. As the glue gets frozen in liquid nitrogen pool, the attached quartz particles stay firmly and no drop-off was observed. Fig. 12 shows the measured temperature histories for three different types of rock surfaces. One is the smooth rock surface covered with the glue layer, which also served as the benchmark here. The other two are surfaces covered with the quartz particles in different number densities. They were labeled as ‘sparser’ and ‘denser’, respectively. As an example, the insert of Fig. 12 shows a photograph of rock surface covered with denser quartz particles. We note that all the three surfaces were made from one individual rock sample so that the surface condition was the sole factor to affect the quenching results. In Fig. 12, the dotted lines represent experimentally measured temperatures. It is evident that the applied quartz particles on rock surface served to enhance the quenching heat transfer. The enhancement became more pronounced with the in crease of the number density of quartz particles. It is seen that the three curves start to deviate from each other at the early stage of quenching. Therefore, the quartz particles acted to improve the film boiling heat flux. Two possible mechanisms can be responsible for this effect. First, the particles enlarged the effective surface area of the rock sample, thus boosting the heat exchange between rock and liquid nitrogen. Second,
the size of quartz particles is large enough (~250 μm) to penetrate the vapor film and to promote local liquid-solid contacts. The intermittent liquid-solid contacts are effective in increasing the heat transfer of quenching, as discussed in Ref. [44]. The proposed numerical simulation scheme was also applied for the structured rock surfaces to derive the boiling curves for these surfaces. Fig. 13 presents the boiling curves of lower rewetting fronts for these three surfaces. The lower rewetting front was more representative of the quenching heat transfer characteristic than the upper front. Fig. 13 in dicates that, comparing to the glue covered surface, the applied quartz particles increased the film boiling heat flux, the rewetting temperature and the maximum heat flux as well. The rewetting temperature for the three surfaces were 110 K, 119.6 K and 125 K, respectively. It is inter esting that the glue covered surface is lower than the clean surface in rewetting temperature (110 vs. 122 K). The glue functioned as an insulating layer to isolate between the liquid nitrogen and the rock surface micro structure. Consequently, the cavities and pores on rock surface cannot contribute to the quenching acceleration by providing nucleation sites. Therefore, the present results corroborated our previ ous analysis on the effect of rock surface properties on quenching. For the three rock surfaces, the simulated temperature history at rock center were plotted in Fig. 12 by black solid lines. The simulated tem perature curves agree well with the experimentally measured ones, demonstrating the good applicability of the proposed numerical model.
Fig. 12. Measured and simulated temperature histories for structured rock surfaces covered with quartz particles. The solid line and dots represent simulated and measured temperature, respectively. The inserted photograph shows the rock surface covered with denser quartz particles.
Fig. 13. Boiling curves of the lower rewetting front determined for structured rock surfaces. 9
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4. Conclusions
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In this paper, the cryogenic quenching experiments of rock surfaces in liquid nitrogen were reported. The heat transfer characteristics and rewetting dynamics were investigated based on measured temperature data and synchronized high speed video visualization. A numerical scheme was developed to simulate the quenching process by reproduc ing the temperature history and rewetting front positions. Several con clusions were obtained in present study: (1) The rewetting of the cylindrical rock sample starts from its two ends and the merging point of rewetting fronts locates near the upper 2/3 of the sample length. (2) The two rewetting fronts on rock surface bears different heat transfer patterns. The lower front has a larger maximum heat flux and therefore moves faster than the upper front. (3) The maximum heat flux in cryogenic quenching of vertical rock surfaces is higher than the stationary boiling heat flux limits, due to the local fluctuation of liquid and vapor and the intense nucleation and evaporation. (4) The distributed quartz particles on rock surface have the effect of improving film boiling heat flux and the rewetting temperature, as well as the maximum heat flux. (5) The numerical model developed in this paper is robust and widely applicable in simulating the quenching and rewetting of rock surfaces in liquid nitrogen. Declaration of competing interest None Declared. Acknowledgements This work was supported by the National Natural Science Foundation of China (51521063), the National Science Fund for Distinguished Young Scholars (No. 51725404) and Beijing Outstanding Young Scien tist Program (BJJWZYJH01201911414038). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.ijthermalsci.2019.106161. References [1] B.W. McDaniel, S.R. Grundmann, W.D. Kendrick, D.R. Wilson, S.W. Jordan, Field applications of cryogenic nitrogen as a hydraulic fracturing fluid, in: SPE Annual Technical Conference and Exhibition, Society of Petroleum Engineers, San Antonio, Texas, 1997. [2] S. Grundmann, G. Rodvelt, G. Dials, R. Allen, Cryogenic nitrogen as a hydraulic fracturing fluid in the devonian shale, in: SPE Eastern Regional Meeting, Pittsburgh, Pennsylvania, 1998. [3] C. Cai, G. Li, Z. Huang, Z. Shen, S. Tian, J. Wei, Experimental study of the effect of liquid nitrogen cooling on rock pore structure, J. Nat. Gas Sci. Eng. 21 (2014) 507–517. [4] C. Cai, G. Li, Z. Huang, S. Tian, Z. Shen, X. Fu, Experiment of coal damage due to super-cooling with liquid nitrogen, J. Nat. Gas Sci. Eng. 22 (2015) 42–48. [5] X. Wu, Z. Huang, R. Li, S. Zhang, H. Wen, P. Huang, X. Dai, C. Zhang, Investigation on the damage of high-temperature shale subjected to liquid nitrogen cooling, J. Nat. Gas Sci. Eng. 57 (2018) 284–294. [6] A. Yamanouchi, Effect of core spray cooling in transient state after loss of coolant accident, J. Nucl. Sci. Technol. 5 (1968) 547–558. [7] K. Sun, G. Dix, C. Tien, Effect of precursory cooling on falling-film rewetting, J. Heat Transf. 97 (1975) 360–365. [8] S. Dua, C. Tien, Two-dimensional analysis of conduction-controlled rewetting with precursory cooling, J. Heat Transf. 98 (1976) 407–413. [9] S. Sahu, P. Das, S. Bhattacharyya, Rewetting analysis of hot surfaces with internal heat source by the heat balance integral method, Heat Mass Transf. 44 (2008) 1247–1256. [10] M. Sawan, G. Zaki, H. Temraz, A three-regions rewetting model with heat generation and subcooling, Atomkernenerg. Kerntech. 34 (1979) 199–204.
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