System description

An experimental setup based on a coaxial-jet model was designed and built to study the temperature fluctuations. The setup included hot fluid, cold fluid, mixing, and measurement systems, as shown in Fig. 1(a). Both hot- and cold-fluid systems consist of a thermostatic water tank, booster pump, electromagnetic flowmeter, storage tank, valves, and pipes, which can provide steady water flow at specified temperatures and flow rates. The mixing system included coaxial-jet nozzles, a mixing zone, valves, and pipes. The coaxial-jet nozzle was designed using two independent passages. Cold fluid flows through the central passage, whereas hot fluid flows through the outer passage. The two fluids are mixed coaxially in the mixing zone. The measurement system is mainly used for the temperature measurement of the fluid and plate surface in the mixing zone and comprises thermocouples, brackets, a data acquisition card, and a computer.

Fig. 1Full-screen View

(Color online) Experimental setup and measurement points. (a) Schematic of the experimental setup (b) Photograph of the experimental setup (c) Test zone (d) Coaxial-jet nozzle (e) Schematic of measurement points

During the experiment, hot and cold water were pumped into the mixing zone from the thermostatic water tank via the valves. The mixed water flows into the storage tank and then recycled into the thermostatic water tank. Figure 1(b) shows a photograph of the experimental setup.

Test zone and inlet temperature test point

The test zone was a transparent acrylic box with dimensions 250 mm × 250 mm × 100 mm, as shown in Fig. 1 (c). The coaxial jet inlet is located at the bottom of the test zone. The nozzle was placed 10 mm above the bottom to reduce the impact of the bottom plate on the inlet fluid. The outlet was located on the two sides above the box, away from the test zone, to reduce its impact on the flow field. The brackets were movable and can be placed at certain heights within the box to enable temperature measurements at different heights.

The cold-fluid inlet was circular with a diameter of 10 mm. The hot-fluid inlets were annular with 24.5 mm and 32 mm. Three measurement points were evenly arranged on the circumferences of the hot and cold-fluid nozzles to measure the inlet temperatures of the fluids, as shown in Fig. 1(d).

Uncertainty analysis

The temperature was measured using a T-type thermocouple with a diameter of 0.08 mm. The time interval for temperature sampling was 0.01 s. The velocity was calculated using the hot and cold flow rates and the cross-sectional area of the coaxial-jet nozzle. The hot and cold flow rates were measured using electromagnetic flow meters. The setup temperature measurement range was 20-90 ℃, and the flow rate was 0.141-2.802 m³/h. The temperature measurement uncertainty was determined using a thermocouple and a data acquisition card. An electromagnetic flow meter was used to directly determine the uncertainty in the flow measurement.

The measurement range and accuracy of the equipment were as follows. The measurement range of the T-type thermocouple was 0-150 ℃, and the expanded uncertainty was U(T_thermocouple) = 0.2 K with an inclusion factor k = 2. The standard uncertainty of the thermocouple u(T_thermocouple) = 0.1 K. The interval half-width of the data acquisition card (model, Hioki) a(T_Hioki) = 0.5 K for the T-type thermocouple within 0-100 ℃. The electromagnetic flowmeter measurement range was 0-3 m³/h with a half-width a(V_flowmeter) = 0.0102 m³/h. Considering uniform distribution, the inclusion factor was taken as $k=\sqrt{3}$, and the standard uncertainty of the data acquisition card u(T_Hioki) = 0.28 K. The standard uncertainty of flowmeter u(V_flowmeter) = 0.00589 m³/h. The combined standard uncertainty of the temperature measurement u_c(T) can be obtained using Eq. (1) ^[35]. $u_{\text{c}}\left(T\right)\text{ =}\sqrt{u{\left(T_{\text{thermocouple}}\right)}^{\text{2}}\text{+}u{\left(T_{\text{Hioki}}\right)}^{\text{2}}}$ (1)

Thus, the combined standard uncertainty of the temperature u_c(T) = 0.311 K. Similarly, the standard uncertainty u(V) and relative expanded uncertainty range u_crels(V) of the flow measurement can be obtained using Eqs. (2) and (3), respectively: $u\left(V\right)=u\left(V_{\text{flowmeter}}\right),$ (2) $u_{\text{crels}}\left(V\right)\text{ = }k\left(\frac{u_{\text{s}}\left(V\right)}{V_{\text{max}}}\sim \frac{u_{\text{s}}\left(V\right)}{V_{\text{min}}}\right).$ (3)

The standard uncertainty u(V) = 0.00589 m³/h, and the expanded uncertainties of the flow measurement u_crels(V)= 0.42%-8.4%.

The inlet temperature T_in is the average of the values at the three measurement points.

The combined standard uncertainty of the inlet temperature measurement u_c(T_in) can be calculated using Eq. (4). $u_{\text{c}}\left(T_{\text{in}}\right)\text{=}\sqrt{\frac{{\left[u\left(T_{\text{thermocouple}}\right)\right]}^{\text{2}}\text{+}{\left[u\left(T_{{}_{\text{Hioki}}}\right)\right]}^{\text{2}}}{\text{3}}}$ (4)

The combined uncertainty of inlet temperature measurement $u_{\text{c}}\left(T_{\text{in}}\right)$ was 0.180 K. This shows that the setup has high measurement stability and can be used for experimental measurements.

Data processing method

For the convenience of discussion, the temperature and radius values were nondimensionalized. The dimensionless temperature is expressed as $T^{\text{*}}\text{=}\frac{T\text{-}{T}_{\text{Cold}}}{T_{\text{Hot}}\text{-}{T}_{\text{Cold}}},$ (5) where T_Cold and T_Hot denote the inlet temperatures of the cold and hot fluids, respectively. The average values measured at the test points at the inlet were used. The dimensionless average temperature was calculated as $T_{\text{Avg}}^{\text{*}}\text{=}\frac{\text{1}}{n}{\displaystyle \sum _{i=1}^n{T}^{*}}.$ (6)

The dimensionless radius is the ratio of the diameter to the cold-fluid inlet diameter. $r^{\text{*}}\text{=}\frac{r}{r_{\text{Cold}}}$ (7) where r_Cold is the radius of the cold-fluid inlet (5 mm). The root-mean-square (RMS) temperature can be used to express the intensity of temperature fluctuations. $T_{\text{RMS}}^{\text{*}}\text{=}\sqrt{\frac{\text{1}}{n}{{\displaystyle \sum _{i=1}^n\left(T_{\text{i}}^{\text{*}}\text{-}\overline{T^{\text{*}}}\right)}}^{\text{2}}}$ (8)

Transient temperature analysis

Subsequent average temperature analysis revealed that the dimensionless average temperature of the fluid was symmetric. Therefore, an analysis of the transient temperature was performed along the radius (P15-P18 in Fig. 1(e)). The temperature fluctuations were entirely random, as shown in Fig. 2(a). When the plate height z_p = 50 mm, the temperature at each radial position significantly fluctuated; r^* = 0 showed the most pronounced fluctuations. In addition, there were small fluctuations overall temperature increase and decrease. For example, at r^* = 0, the temperature shows a decreasing trend between 0 and 0.2 s; however, this process is accompanied by two minor temperature increases.

Fig. 2Full-screen View

(Color online) Dimensionless transient temperature, average temperature, and intensity of temperature fluctuations of fluid on the plate surface (a) Transient temperature on plate surface with plate height z_p = 50 mm (b) Transient temperature of fluid with plate height z_p = 50 mm (c) Average temperature in the radial direction (d) Average temperature in the axial direction (e) RMS temperature in the radial direction (f) RMS temperature in the axial direction

The cause of the temperature fluctuation can be analyzed by comparing the temperature fluctuation in Fig. 2(a) with the transient temperature distribution on the plate surface at different times shown in Fig. 3(a). These temperature fluctuations can mainly be attributed to the fluid instability. As indicated by the transient temperature distribution on the plate structure, this instability is mainly reflected in the following three aspects. First, the cold fluid would expand and contract as time progresses. For example, the cold fluid expands during 0-0.2 s and contracts during 0.2-0.4 s. Correspondingly, the center (r^* = 0) temperature decreases in 0-0.2 s, and the temperature increases in 0.2-0.4 s (Fig. 2(a)). Second, the jet center would swing over time. At 0.2 s, the center of the cold fluid was at r^* = 0, as shown in Fig. 3(a). At 0.5 s, the center moved downward. At 0.6-0.7 s, the cold-fluid center appeared on the left. The transient temperatures are presented in Fig. 2(a). The temperature at r^* = 2 increases when the cold-fluid center swings to the left at 0.4 s. Finally, the shape of the jet also changes significantly. Occasionally, the temperature distribution on the plate surface presents a relatively regular circle, for instance, that at 1.1 s. It may also extend in different directions, for example, the observations corresponding to 0.1 s, 0.2 s, and 0.9 s. Mostly, the shape of the jet is relatively irregular. Under the combined influence of these three factors, the temperature would fluctuate at each point on the plate surface.

Fig. 3Full-screen View

(Color online) Dimensionless transient temperature distribution on the plate surface (a) z_p = 50 mm (b) z_p = 30 mm (c) z_p = 40 mm (d) z_p = 60 mm

In the flow field between the jet nozzle and the plate, the transient temperatures at z_f^* = 0.2 and z_f^* = 0.6 were as shown in Fig. 2(b). The position at which the temperature fluctuation is relatively prominent changes along the axial direction. The position with notably evident temperature fluctuation is at r^* = 1 when z_f^* = 0.2 and at r^* = 0 when z_f^* = 0.6. Compared to plate surface, temperature changes were more frequent in the flow field z_f^* = 0.2 and z_f^* = 0.6.

Average temperature distribution

Over a long period, the temperatures at various radial positions fluctuated around a fixed value. This is reflected in the dimensionless average temperature values. The dimensionless average temperature of the plate surface at z_p = 50 mm is shown in Fig. 4(a). The average temperature was the lowest at the center and continued to increase along the radial direction. The temperature distribution was almost circular. Therefore, the temperature along the radial direction was selected for further analysis. The dimensionless average temperature of the fluid in the radial direction was symmetrical, as shown in Fig. 2(c). The blue and red blocks on the horizontal axis represent the positions of the cold- and hot-fluid nozzles, respectively. The average temperature was the lowest at the center. In the positive radial direction, it increased rapidly and reached the maximum value above the hot-fluid nozzle. Subsequently, the average temperature decreased slowly and finally reached a constant value of approximately 0.836. Moreover, the lower the height, the smaller would be the minimum value and the larger would be the maximum value. The average dimensionless temperature on the plate surface showed a considerably different behavior. It increased continuously in the positive radial direction. There was no longer a prominent high-temperature peak above the hot-fluid nozzle. This phenomenon may be attributed to the cold fluid in the center flowing in all directions and rapidly mixing with the hot fluid when the jet reaches the plate surface.

Fig. 4Full-screen View

(Color online) Dimensionless average temperature and RMS dimensionless temperature on the plate surface (a) Average temperature (b) RMS temperature

To examine the temperature variation in the axial direction, the center position (P15 in Fig. 1(e), r^* = 0), mixing zone between the hot and cold fluids (P16 in Fig. 1(e), r^* = 2), and outer position (P18 in Fig. 1(e), r^* = 6) were considered for the analysis. Fig. 2(d) shows the average dimensionless temperature of the fluid in the axial direction. The average temperature increased at the center (r^* = 0) in the axial direction. It decreases at r^* = 2 and remains constant at r^* = 6. This variation is linear between z_f^* = 0.2-0.8.

Intensity of temperature fluctuation

Intensity of the temperature fluctuations can better characterize their severity. This can be expressed as the RMS temperature. For z_p = 50 mm, the RMS dimensionless temperature on the plate surface was as shown in Fig. 4(b). The temperature evidently fluctuated above the jet nozzle. The RMS temperature along the radial direction is shown in Fig. 2(e). At z_f^* = 0.2, 0.4, and 0.8, the RMS temperature initially increased and then decreased, with the maximum value observed at r^* = 1. At z_f^* = 0.6, the RMS temperature decreased continuously from r^* = 0. At the plate surface, the RMS temperature was relatively higher in the range of r^* < 4.

Figure 2(f) shows the variation of the intensity of the temperature fluctuations in the axial direction. At r^* = 0, the RMS temperature increased when z_f^* < 0.6 and gradually decreases when z_f^* ≥ 0.6. It shows an increasing trend at r^* = 2 and remains constant at r^* = 6. Severe temperature fluctuations indicated that the hot and cold fluids constantly alternated at this position. This point was the junction of the hot and cold fluids. Therefore, the intensity of the temperature fluctuation at r^* = 0 reflected the amplitude of the jet swing. When z_f^* < 0.6, the jet swing amplitude increased in the axial direction. The junction between the hot and cold fluids can swing toward the center. Therefore, the RMS temperature at r^* = 0 continued to increase. When z_f^* ≥ 0.6, the plate makes the jet more stable, and the swing amplitude becomes small. The RMS temperature decreases at r^* = 0. This phenomenon was also reflected in the maximum RMS temperature, as shown in Fig. 2(f). When z_f^* < 0.6, the maximum RMS temperature was observed at r^* = 1. With increasing height, it moved to the center (r^* = 0). Evidently, the RMS temperature is the highest at r^* = 0 when z_f^* = 0.6 in Fig. 2(f). When the position approaches the plate surface, the maximum RMS temperature was restored to r^* = 1 at z_f^* = 0.8. In the flow field, hot and cold fluid flowed along the positive axial direction. When the fluid reached the plate surface, the direction of fluid flow changed. The fluid flowed along the radial directions. Therefore, the junction of the cold and hot fluids moves outward. The maximum RMS temperature was observed at r^* = 2 on the plate surface. Furthermore, the maximum RMS temperature at the position 0.2 < z_f^* < 0.8 is much larger than that on the plate surface; this implies that the temperature-fluctuation attenuation is rapid near the plate surface.

Transient temperature changes with height

Compared with z_p = 50 mm, the cold fluid expanded and contracted with time at different plate heights. However, changes in the swing and shape of the jet were insignificant when the plate height z_p = 30 mm. As the height increases, the jet swing and temperature distribution changes become increasingly pronounced, as shown in Fig. 3.

The transient temperature at z_p = 30 mm and its frequency domain representation in the radial direction are shown in Fig. 5. The temperature fluctuation was the most severe at r^* = 2 between the hot and cold fluids at this height. The fluctuation range was approximately 0.2-0.9. The frequency domain (Fig. 5(a)) representation shows that the temperature fluctuation has no prominent peak frequency. However, the temperature fluctuations were more severe at lower frequencies. At r^* = 2, the frequency of the temperature fluctuation was mainly distributed within 10 Hz. At r^* = 0, the amplitude above 5 Hz is almost zero.

Fig. 5Full-screen View

(Color online) Transient dimensionless temperature in time and frequency domains (a) z_p = 30 mm (b) z_p = 40 mm (c) z_p = 50 mm (d) z_p = 70 mm (e) z_p = 90 mm

The transient temperature variations at plate heights z_p = 40, 50, 70, and 90 mm are shown in Fig. 5. The ranges of temperature fluctuations at different positions exhibited different changes in the axial direction. At the center (r^* = 0), the temperature-fluctuation range changed from 0-0.1 at z_p = 30 mm to 0.5-0.7 at z_p = 50 mm and 0.7-0.8 at z_p = 90 mm. The temperature-fluctuation range at r^* = 2 decreased from 0.2-0.9 at z_p = 30 mm to 0.7-0.8 at z_p = 90 mm. The temperature fluctuation at r^* = 5 was from 0.9-1 at z_p = 30 mm to 0.75-0.85 at z_p = 50 mm and finally reached 0.7-0.8 at z_p = 90 mm. During the entire process, the fluctuation range remained almost unchanged. Only the average temperature decreased slightly. The variation in the temperature-fluctuation range was the result of cold and hot fluids mixing and jet instability. The variation in temperature fluctuation with height was analyzed using the average temperature and temperature fluctuation intensity.

Average temperature changes with height

The average temperature of the center gradually increased in the positive axial direction. The outside temperature gradually decreased, as shown in Fig. 4(a). The temperature distribution was almost circular. Therefore, the temperature in the radial direction was used for further analysis.

The dimensionless average temperature of the plate surface in the radial direction is shown in Fig. 6(a). The temperature in the radial direction was symmetrical. However, because of the non-coaxial nature of the testing bracket, its symmetry region was between r^* = 0 and r^* = -1. The average temperature at the center was low and increased gradually with the radius. It reached a constant value at r^* = 5. As the height increased, the temperature at the center increased. The temperature at each radial point was identical. This indicated that the mixture became uniform.

Fig. 6Full-screen View

(Color online) Dimensionless average temperature and intensity of temperature fluctuations on the plate surface (a) Average temperature in the radial direction (b) Average temperature in the axial direction (c) RMS temperature in the radial direction (d) RMS temperature in the axial direction

The dimensionless average temperature variation along the axial direction was as shown in Fig. 6(b). The temperature at r^* = 0 increased continuously with increasing height. The position at r^* = 2 showed a relatively slow increase. However, the temperature at r^* = 6 decreases in the range of 30 mm < z_p ≤ 70 mm. When z_p > 70, it remained almost constant. This demonstrates the effect of height on fluid mixing. At z_p = 30 mm, the average temperature at r^* = 0 is approximately 0. This implies that the cold fluid at the center did not start to mix. The temperature at each point changed rapidly within the range 30 mm < z_p ≤ 70 mm. This implies that the fluid mixes rapidly at this height. When z_p > 70 mm, the temperature gradually varied. The temperatures at each point tended to be the same, indicating that the mixing process was complete.

Temperature-fluctuation intensity changes with height

The position of the maximum temperature-fluctuation intensity on the plate surface evidently changed with height. When z_p = 30 mm, an annular shape was observed with a strong temperature fluctuation between the hot and cold fluids. The RMS temperature at the center was relatively low, as shown in Fig. 4(b). However, when z_p = 40 mm, the RMS temperature above the nozzle showed little difference and only decreased significantly when r^* > 2. As the height increased, the RMS temperature gradually decreased, and the temperature-fluctuation intensity at each point gradually became uniform.

The RMS temperature variation in the radial direction are shown in Fig. 6(c). The points with higher temperature-fluctuation intensity at different heights are mainly distributed in the circular area above the nozzle in the range r^* ≤ 4. The temperature-fluctuation intensity is determined by the jet instability and hot and cold fluid mixing with height. As the height increases, the amplitude of the jet sway increases. For low values of height, the point of the most severe temperature fluctuation is between the hot and cold fluids (r^* = 2). However, when the height is higher, the intensity of temperature fluctuation in the range of r^* ≤ 4 tends to be consistent. However, with an increase in the axial direction, the jet mixing becomes more uniform, and the amplitude of the temperature fluctuation decreases gradually. Under the combined action of these two factors, the temperature-fluctuation intensity first increases and then decreases at r^* = 0, and it decreases continuously at r^* = 2. It remains almost unchanged when r^* = 6, as shown in Fig. 6(d). This implies that when the plate is located at different heights above the jet, the points affected by the temperature fluctuations that need to be analyzed are entirely different. When the height is low, the analysis must focus on the locations between the hot and cold-fluid nozzles. When the height is high, a series of positions from the center of the cold-fluid nozzle to the position of the hot fluid nozzle must be comprehensively considered. The analysis of the temperature-fluctuation range in Subsection 5.1 also highlights this phenomenon.