## Abstract

Experiments were conducted on a cold flat aluminum plate to characterize the variation of frost roughness over both time and location on the surfaces. The testing conditions included air temperatures from 8 to 16 °C, wall temperatures from −20 to −10 °C, relative humidities from 60 to 80%, and air velocities from 0.5 to 2.5 m/s. Each test lasted 2 h. A 3D photogrammetric method was employed to measure the variation in frost root-mean-square height and skewness by location and time. These data were used to develop the equivalent sand-grain roughness for the frost at different locations and time. The experimental results showed that frost roughness varied by location and changed with time. For the environmental conditions in this study, relative humidity and air temperature were the most important factors determining changes in the peak frost roughness. For example, at an air temperature of 12 °C and a surface temperature of −15 °C, the frost roughness peaked at about 40 min for a relative humidity of 80% and 90 min for a relative humidity of 60%. Empirical correlations were provided to describe the relationships between the environmental conditions and the appearance of the peak frost roughness.

## Introduction

Frost is a common phenomenon in nature as well as in engineering applications. When a cold surface with a temperature below 0 °C is surrounded with moist air, frost may grow on the cold surface. In the past 70 years, researchers [112] have extensively investigated the dependence of frost growth under different atmospheric conditions. They indicated that air humidity, air velocity, wall temperature, and air temperature had primary impacts on the properties of frost.

Frost formation on a cold surface usually starts with the condensation of water droplets and the subsequent freezing of these super-cooled droplets [13]. Initial frost crystals then emerge from the frozen condensate droplets. These crystals vary in shapes that correspond to the ambient cold wall temperature and air humidity [1315]. As the frost crystals continue to expand and interact with each other, a porous frost layer with a rough surface eventually forms. During the process of frost growth, moist air either deposits on the frost surface to increase the frost thickness or diffuses into the frost layer to increase the frost density [1621].

Frost crystal structure can play an import role on frost formation. The initial porosity of the frost layer directly influences the frost layer growth characteristics [22]. The variation of frost crystal growth direction contributes to the changes in frost growth rate [23]. Moreover, frost crystal shape was considered as a key factor affecting frost thermal conductivities [2]. Specifically, column shape crystals had higher heat conduction and led to a higher frost thermal conductivity compared with plate-type crystals. In addition, the frost morphology changes were also associated with the variation of frost crystal structure. As time increased, frost crystals on the top of the frost layer might change from feather-shaped to needle-shaped, then flake-shaped or even irregular-shaped as the frost grows. The different shapes of frost crystals propagating from the frost surface can produce a range in roughness [24]. Generally, frost crystals growing normal to the frost surface tended to form a rough layer; while plate-type frost crystals were likely to create a smoother frost layer [25].

The roughness of the frost surface can cause many undesired impacts. It could increase turbulence levels of the air on the surface to generate an additional frictional force restricting air flow across the surface [26]. If frost is formed on a wing surface of aircraft, the additional roughness of frost could cause a reduction in the lift and a rise in drag [27]. For heat pump applications, frost with different frost morphologies has been shown to cause different air pressure drops even though the amounts of frost were equal [28].

Frost properties can present an uneven distribution along a plate when frost formed on a cold flat plate under forced convection [6,12,2932]. Cheng and Shiu [12] studied the spatial variation of the frost thickness on a cold plate. They found that at the leading edge of the cold plate, frost crystals grew in the radial direction to form a “round head” of the frost layer which had a smoother profile. In the downstream region, the frost crystals grew at approximately the same rate to form a homogeneous frost layer of uniform thickness. Some modeling studies [2932] have also shown that frost density was non-uniform along the cold surface. In the frontal region, the water vapor mass fraction was higher, and the cold surface cooled the humid air to create a faster phase change mass transfer, which led to denser frost. In the rear region of the frost layer, the gradient of the air humidity in the frost surface was smaller than the values in the frontal region, and the frost density was smaller. Frost thermal conductivity can also be uneven along the cold surface due to the spatial variation of frost density [32].

Although the variation of frost properties, such as thickness, density, and thermal conductivity, have been investigated extensively, none of them have been related to frost surface roughness. The variation of frost surface roughness along the length of the cold plate has been not characterized. Frost surface roughness, as characterized by equivalent sand-grain roughness of frost, is another variable that is important in understanding the potential impact of frost on an engineered system such as a heat pump or refrigeration evaporator or an aviation airfoil. Therefore, new data on frost roughness can potentially help engineers in designing these systems.

Conventional methods of frost roughness measurement [25,33] have used lateral side two-dimensional (2D) frost surface profiles. Frosting usually presents nonhomogeneous nucleation along the edges of test surfaces. Moreover, the lateral side profiles include an overlapped view of frost formed at different distances from the camera. These profiles might not represent the actual distribution of frost crystals over the whole frost surface. Another approach of frost roughness measurement is to apply 3D photogrammetry. In 1985, Mikkelsen et al. [34] used two cameras to capture peaks and valleys of ice formed on the wing of an aircraft. Image data were further processed by using an analytical plotter to derive the spatial and geometrical information of the ice surface. A similar concept of 3D close-range photogrammetry was later applied by Collier et al. [35] to build a 3D model of accreted ice. Recently, Miyauchi et al. [36] developed a 3D photogrammetric method for nonintrusive and in situ measuring of the frost roughness in a closed-loop psychrometric wind tunnel. Because the process was nonintrusive, it did not interfere with the growth of the frost on the cold surface.

In this study, the 3D photogrammetry method was applied for the measurement of frost roughness. The overall objective of this study was to characterize the surface roughness of frost formed on a flat plate under forced convection as a function of location and time. Several environmental conditions were varied, including air temperature, air humidity, air velocity, and wall temperature.

## Methodology

### Closed-Loop Psychrometric Wind Tunnel.

A closed-loop psychometric wind tunnel system was used to provide conditioned airflow over a cold test surface. Figure 1 shows the schematic drawing of the wind tunnel system. An ultrasonic humidifier and a dry gas (pure nitrogen) injection system were applied to precisely control the air humidity level in the wind tunnel. An adjustable electronically commutated motor blower was employed to control the air velocity in the closed loop. Two cooling coils and a duct heater were used to control the air temperature. In addition, a thermal stage mounted in the test section was utilized to precisely control wall temperature as needed.

Fig. 1
Fig. 1
Close modal

A detailed schematic of the test section is presented in Fig. 2. The test section was made of an optical transparent acrylic glass with 0.2 W/(m K) thermal conductivity. This panel allowed for in situ, vertical and horizontal observations of the frost growth process without any intrusions into the frost layer. Conditioned air entering the test section was split into two flow paths by a rectangular annulus (15 mm by 60 mm) and a bypass duct: the upper portion flowed into the acrylic panel and crossed over a chilled polished aluminum test surface (3 cm by 3 cm with an average roughness arithmetic mean of 0.0042 µm), while the rest of the air crossed through the lower duct and by-passed the test section. The air temperature was measured by a type-T thermocouple located upstream from the test surface with a measurement uncertainty of ±0.5 °C. A Vaisala HMT-333 series relative humidity sensor was used for the measurement of relative humidity (±2%). A TSI 1750 constant temperature hot-wire anemometer (model 1201 sensor probe) was applied to measure the air velocity (±0.05 m/s).

Fig. 2
Fig. 2
Close modal

At the center of the test section, a thermal stage was bonded with the aluminum test surface (0.672 mm thick) through a thermally conductive material (with 0.254 mm thickness and 3.0 W/(m K) thermal conductivity). The thermal stage consisted of a type-T thermocouple mounted to the underside of the test surface, a thermoelectric cooler (Peltier module), and a copper heat sink. Figure 3 provides an exploded view of the thermal stage. During testing, the wall surface temperature was precisely adjusted as needed by the thermal stage using a NI labview program.

Fig. 3
Fig. 3
Close modal

### Frost Equivalent Sand-Grain Roughness Measurement.

Equivalent sand-grain roughness was the parameter used to represent the surface morphology of frost formed on the tested plate. Equivalent sand-grain roughness has been widely employed in studies of the effects of roughness on surface aerodynamic performance [37]. To measure frost surface roughness, a 3D photogrammetric method developed by Miyauchi et al. [36] was applied. This approach was nonintrusive and did not impose effects on the frost surface, structure, or temperature when frost was grown over a wide range of environmental conditions.

Generally, the 3D photogrammetric method was used to generate a 3D digital model of the frost surface based on multiple images of the frost surface. An example of an image layout is given in Fig. 4. To capture these images of frost morphology, a Nikon D3400 DSLR camera with an AF Micro Nikkor 60 mm f/2.8D lens was applied in this study. The whole camera system, supported by a two degrees-of freedom telescoping boom stand, was fixed above the top of the frost surface to observe the changes in the characteristics of the frost surface. The camera was set at 1.0x reproduction ratio, ISO 200, and an aperture of f/22. At the beginning of data acquisition, the camera was manually traversed approximately 5 mm along the positive x-direction for acquisition of seven images, and then crossed approximately 2 mm along the positive y-direction for one repetition. Next, the image capturing cycle was repeated in the negative x-direction. At the end of the image data acquisition, a total of five rows of seven images were captured. The approximate traversing distances, 5 mm and 2 mm, were chosen based on the distances between adjacent autofocus dots of the camera viewfinder in x-direction and y-direction, respectively. These autofocus dots provided a reference to follow during capturing images. The whole image capturing procedure took about 45 s. During the procedure, an illuminator with two fiber optic guides was employed to provide the lighting source from the lateral side of the test surface.

Fig. 4
Fig. 4
Close modal

The collected images of the frost surface needed to be processed to generate a digital model of the frost surface by applying a commercial photogrammetric software, agisoft photoscan pro [38]. Figure 5 shows an example of a frost surface model produced by using the image data of frost morphology. A 60-deg sloped scale bar with specks of black spray paint was placed near the x-direction edge. The specks of black spray paint on the scale bar were used to create a reference for the horizontal xy plane of the digital frost surface model in the software of agisoft photoscan pro. On the other hand, the 60-deg slope of the scale bar was applied to correct units of the digital model in the vertical z-direction. The scaling process of the z-direction is described later.

Fig. 5
Fig. 5
Close modal
A text file with a point cloud of the frost surface was exported from the frost surface model generated by agisoft photoscan pro. The point cloud was a set of data points of the external frost surface in space. This file was then processed by a matlab program to scale the model units in the z-direction through the reference information of the 60-deg slope of the scale bar. The scaling magnitude was determined by comparing the physical slope of the scale bar to the computed slope of the scale bar in the digital model. Thus, a detailed height distribution of the frost surface was achieved from the correctly scaled txt data file, which would be used to calculate the root-mean-square (RMS) height and skewness of the frost surface. The RMS height (Rq), as described in Eq. (1), represented the standard deviation of height from the mean surface value
$Rq=[1NP∑i=1NP(Zi−Z¯)2]1/2$
(1)
The skewness (Skw), as described in Eq. (1), represented the symmetric distribution of the surface where a positive Skw indicated a frost surface had more frost peaks than valleys
$Skw=1Rq3[1NP∑i=1NP(Zi−Z¯)3]$
(2)
In the last part, the calculation of equivalent sand-grain roughness height (ks) was done by using the empirical formula (Eq. (3)) proposed by Flack and Schultz [37]:
$ks=4.43Rq(1+Skw)1.37$
(3)

A detailed description of this 3D photogrammetric method can be found in Miyauchi et al. [36].

The 3D photogrammetry can overcome the difficulties generated by conventional 2D roughness measurements [25,33] using a surface profile from the lateral edge of the flat plate. Figure 6 gives an example conducted by using our wind tunnel system to show the difference of frost roughness observed from a lateral side 2D frost surface profile (Fig. 6(a)) and a 3D frost surface elevation map generated by the photogrammetric method (Fig. 6(b)). The lateral side profiles were an overlapped view of frost formed in different distances, which made the frost surface looked smoother. The 3D frost surface elevation map could rebuild the spatial morphology of frost to present these actual frost peaks and valleys. Hence, the data obtained from 3D photogrammetry should be closer to the true roughness of the frost surface. The photogrammetry validation was done by Miyauchi et al. [36] through measuring and comparing artificial roughness of a set of artificial plastic matte-white surfaces by a Keyence VR 3000 structured-light profilometer and the 3D photogrammetry. The difference in the two measurement methods was less than 7%.

Fig. 6
Fig. 6
Close modal

### Testing Procedures.

The experiments were carried out over the following ranges: surface temperatures (Ts) from −20 to −10 °C, freestream air temperatures (Tair) from 8 to 16 °C, relative humidities (RH) from 60 to 80%, and air velocities (Vair) from 0.5 to 2.5 m/s. Table 1 gives the testing conditions as a function of surface temperature used in this study.

Table 1

List of experimental test conditions

Case numberTs (°C)Tair (°C)RH (%)Vair (m/s)Absolute humidity (kg/kga)
1−20
2−1512701.00.006086
3−10
4160.007913
5−158701.00.004644
6600.005209
7−1512801.00.006965
80.5
91512701.50.006086
102.5
Case numberTs (°C)Tair (°C)RH (%)Vair (m/s)Absolute humidity (kg/kga)
1−20
2−1512701.00.006086
3−10
4160.007913
5−158701.00.004644
6600.005209
7−1512801.00.006965
80.5
91512701.50.006086
102.5

The polished aluminum test surface was washed with acetone and de-ionized water subsequently through an ultrasonic bath before first using. Between frosting tests, the aluminum substrate was cleaned with acetone to remove any residual contamination and then gently dried with a laboratory wipe. To avoid any condensation before frosting tests, the thermoelectric cooler in the thermal stage was maintained in the heating mode before environmental conditions reached the target setting. Once the steady-state was achieved in the air temperature, humidity, and velocity, the thermoelectric cooler was then switched to the cooling mode to quickly decrease the wall surface temperature down to the demanded value within 10 s. A NI labview program was used to monitor and control all the environmental parameters. Each frosting test lasted for 2 h. Frost surface images were captured every 15 min and then used for the calculation of frost equivalent sand-grain roughness height.

## Results and Discussion

### Frost Roughness Uniformity.

For all frosting tests, the dew point temperatures of air were above the freezing point. Hence, frost formation initially started with water droplet condensation, then freezing of super-cooled water droplets, and eventually frost crystals grew from the frozen water droplets. Frost crystals of different shapes grew on the top of a frost layer, and thus, produced a range in frost roughness.

Similar to the non-uniformed distribution of other frost properties found by previous investigations [6,12,2932], the uneven types of frost crystals along the cold surface were observed in this study. Figure 7 gives an example of different frost crystal structure (or morphology) obtained at 45 min at an air temperature of 16 °C, a surface temperature of −15 °C, a relative humidity of 70%, and an air velocity of 1.0 m/s. The conditioned air was blown from left to right. The entire frosted surface was divided into three parts: front region, center region, and rear region. The frost morphology shown in the frontal region was quite different compared with that on the rest of the frost surface. Frost crystals near the leading edge had a cluster shape. These crystals were shorter and smoother compared with these needle-shaped crystals that appeared in the center and rear regions. This phenomenon was potentially caused by the variation of the air temperature and humidity gradients close to the frost surface. The gradient of air temperature and humidity in the frontal region should be greater than those in the middle and back parts. The high gradient either melted the peaks of previously formed frost crystals or forced frost crystals to form a more clustered structure. As a result, a smoother frost layer was shown in the front region of Fig. 7. Moreover, the type of frost crystal in the center region in Fig. 7, as well as the amount of frost crystals, were similar to these in the rear region.

Fig. 7
Fig. 7
Close modal

The smoother frost surface nearer the front was observed in all the frosting tests. To quantitatively represent frost surface roughness, frost equivalent sand-grain roughness height (ks) was used. Table 2 lists all data of frost ks heights with time under different testing conditions and has the same case number with Table 1.

Table 2

Frost ks heights of front (F), center (C), and rear(R) regions with time for all cases listed in Table 1

Case numberLocationFrost ks heights (mm) at
15 min30 min45 min60 min75 min90 min120 min
1F1.81872.32032.17611.61401.21601.62541.2271
C2.90762.87692.99022.96732.74203.08001.4920
R3.25033.08803.28133.25532.36692.88062.0444
2F1.89282.67251.76961.57511.22451.53021.3952
C1.34823.21083.10643.54742.77862.25181.5332
R1.33552.74723.36123.78903.45492.76591.9442
3F1.52752.05762.16172.57072.25691.62291.6674
C0.23471.65983.52514.04084.62622.86541.7904
R0.19590.88423.69454.41084.41103.37862.3380
4F2.33241.78531.12661.62681.34880.73020.7575
C2.59613.35103.31902.37851.61561.66242.0421
R1.75793.85603.55592.21761.73171.54052.2912
5F1.12931.49541.66232.35661.84441.88431.7546
C1.78711.64631.48012.14802.14322.64562.5437
R1.70951.81871.46802.37571.79542.17462.8245
6F1.14051.89602.29921.80211.80591.48731.5501
C1.12541.93012.83632.76053.05073.14432.5685
R1.27331.93722.56673.09913.73773.54743.1634
7F2.39222.62362.25412.25491.37851.39581.5878
C1.64943.14404.70214.46462.95782.51192.0958
R1.56952.78755.31014.82053.49502.73792.8402
8F2.45652.45953.09342.41692.45022.67252.2409
C2.70962.13952.24103.19914.34843.19102.9222
R2.46811.91382.40722.47893.68273.27373.4522
9F1.77362.28011.53921.78321.21071.17761.3146
C1.76123.87773.05662.14331.40611.19761.7143
R1.53793.62962.89392.16611.94171.96891.7201
10F1.42831.21870.94720.96071.08931.06510.7842
C2.43282.13261.67731.66501.62341.72501.7590
R2.68812.54372.07831.27431.54861.74121.8748
Case numberLocationFrost ks heights (mm) at
15 min30 min45 min60 min75 min90 min120 min
1F1.81872.32032.17611.61401.21601.62541.2271
C2.90762.87692.99022.96732.74203.08001.4920
R3.25033.08803.28133.25532.36692.88062.0444
2F1.89282.67251.76961.57511.22451.53021.3952
C1.34823.21083.10643.54742.77862.25181.5332
R1.33552.74723.36123.78903.45492.76591.9442
3F1.52752.05762.16172.57072.25691.62291.6674
C0.23471.65983.52514.04084.62622.86541.7904
R0.19590.88423.69454.41084.41103.37862.3380
4F2.33241.78531.12661.62681.34880.73020.7575
C2.59613.35103.31902.37851.61561.66242.0421
R1.75793.85603.55592.21761.73171.54052.2912
5F1.12931.49541.66232.35661.84441.88431.7546
C1.78711.64631.48012.14802.14322.64562.5437
R1.70951.81871.46802.37571.79542.17462.8245
6F1.14051.89602.29921.80211.80591.48731.5501
C1.12541.93012.83632.76053.05073.14432.5685
R1.27331.93722.56673.09913.73773.54743.1634
7F2.39222.62362.25412.25491.37851.39581.5878
C1.64943.14404.70214.46462.95782.51192.0958
R1.56952.78755.31014.82053.49502.73792.8402
8F2.45652.45953.09342.41692.45022.67252.2409
C2.70962.13952.24103.19914.34843.19102.9222
R2.46811.91382.40722.47893.68273.27373.4522
9F1.77362.28011.53921.78321.21071.17761.3146
C1.76123.87773.05662.14331.40611.19761.7143
R1.53793.62962.89392.16611.94171.96891.7201
10F1.42831.21870.94720.96071.08931.06510.7842
C2.43282.13261.67731.66501.62341.72501.7590
R2.68812.54372.07831.27431.54861.74121.8748

To better show the variation of frost roughness over locations on the frost surface, Fig. 8 provides an example of frost ks variations with time under different air temperatures. The corresponding data for this example were listed in Table 2 (see case 5, 2, and 4). At the front, center, and rear regions of the frost surface, frost ks values might present similar growth trends. For example, in the air temperature case of 8 °C, the frost roughness increased over time for the front, center, and rear regions with the frost roughness being largest at the end of 120 min for the center and read regions. In contrast at air temperatures of 12 °C and 16 °C (the bottom two plots in Fig. 8), the frost in the front section was significantly smoother than either the center or rear and, with the exception at the beginning of the 12 °C test, tended to decrease over time during the most of the tests. For both the 12 °C and 16 °C tests, both the center and rear regions increased in roughness as measured in the ks values reaching peaks of over 3 mm at 12 °C and nearly 4 mm at 16 °C. Although frost roughness varied across the test surface, the frost ks values in the center and rear regions tended to follow similar trends. This observed trend was consistent for all cases listed in Table 2, which meant that the frost in the downstream region was nearly homogenous and formed a nearly uniform roughness. However, near the leading edge, the frost tended to be much smooth than in the middle or rear sections. The smoother frost layer produced a smaller ks value. We hypothesize that the frost was smoother in the leading edge because of two factors. First, at the leading edge, the temperature and moisture gradients between the air and frost surface were larger. Second, both the heat and mass transfer coefficients should be larger at the leading edge. Both factors would drive the frost surface faster to 0 °C where the frost layer would experiences melting before the layers in the middle or rear sections. A third reason for the differences could be because the frost crystals near the front tended to be of the cluster shape which are smoother than the frost crystals at the other locations. The ks value differences among these various regions reflect the uneven distribution of roughness along the length of the frost surface.

Fig. 8
Fig. 8
Close modal

The unevenness of the frost surface was also affected by variations in environmental conditions, such as air temperature, air velocity, air humidity, and wall temperature. To estimate the non-uniformity in the frost roughness, the standard deviation of the frost equivalent sand-grain roughness heights at the front, center, and rear regions, Std_ks, was calculated. The Std_ks value represented the amount of variation of frost ks at three different regions. A high standard deviation indicated that the frost roughness heights were spread out over a wider range, while a small value of standard deviation corresponded to a more even or smoother frost surface. The results are presented in the Growth Behavior of Environmental Parameters on Frost Roughness Unevenness section.

### Growth Behavior of Environmental Parameters on Frost Roughness Unevenness.

The standard deviation of the frost equivalent sand-grain roughness (Std_ks) was computed in this section to evaluate the unevenness in the frost roughness. Figure 9 shows the results of the Std_ks values with time under different environmental conditions. As time increased, the Std_ks increased until reaching its maximum value, and then started decreasing. The timing for the appearance of the maximum frost Std_ks showed a strong relationship with environmental conditions. Within our testing ranges, the frost Std_ks reached its maximum value sooner when air temperature, relative humidity, or air velocity were increased, or wall temperature was decreased. To predict when the maximum frost Std_ks appeared, an empirical correlation with dimensionless parameters was generated, as shown in Eq. (4):
$Fostd=αtD2=0.019(wa)−2.926(ReD)−0.607(Tair−ToTo)1.127(To−TsTo)−0.705$
(4)
where Fostd is the Fourier number to represent the dimensionless time when the frost Std_ks reaches its maximum height, α is the thermal diffusivity of air, D is the hydraulic diameter, t is the real time when the frost Std_ks reaches its maximum height, wa is the absolute air humidity, ReD is the Reynolds number calculated by the hydraulic diameter of the test duct, and To is the triple point temperature. The R2 of this empirical correlation was 0.732. Figure 10 shows the comparison of the measured and correlated data of the dimensionless appearance time. The potential reasons limiting the accuracy of the correlation may be from two aspects: one is the small amount of data points (ten frosting cases), and the other is the large time-step (15 min) for data collections. Although this correlation has some room for improvement, it is a good tool to describe the growth behavior of the frost Std_ks. This empirical correlation was developed under the following assumptions: (1) the frost formation begins after the process of condensed water droplet nucleation and (2) the water vapor near the cold wall surface was saturated. These assumptions were also applied for other empirical correlations presented in this study.
Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
Close modal

From Fig. 9, we can also qualitatively identify the importance of each environmental parameter on the growth of Std_ks heights. Changing air temperature or relative humidity led to considerable variations on the Std_ks, while the effects of adjusting wall temperature or air velocity were limited. To quantitatively evaluate the importance of each parameter, an analysis of variance (ANOVA) test was performed on the resulting Std_ks values.

By varying the environmental conditions, ten cases were used to perform a one-way ANOVA test to identify the most important parameters determining the unevenness of the frost roughness. To remove the time parameter from the ANOVA analysis, the growth rates of Std_ks were analyzed for two time spans: (1) over the 0−45 min test span and (2) over the 45–90 min test span. The slopes of regression lines through the resulting points were evaluated as shown in Eq. (5), where the bar on the top of the $Std_ks¯$ term represented a time derivative of the Std_ks
$Std_ks¯=dStd_ksdt=N∑tjStd_ks,j−∑tj∑Std_ks,jN∑tj2−(∑tj)2$
(5)

The statistical variance of all the growth rates, for the Std_ks, was compared with the statistical variance of the growth rates from each subgroup with each of the four environmental parameters. Based on the comparison of the statistical variance of the group growth rates to the statistical variance of the entire set of growth rates, a probability, or P-value, can be evaluated to assess the importance of each parameter on the resulting Std_ks growth rates. This part of work was done by applying jmp software. The lower the P-value, the more important the variable was in determining the growth rate of roughness unevenness.

Table 3 lists the results of ANOVA analysis for the Std_ks growth rates. The P-value for relative humidity and air temperature at both test spans was smaller than 0.1 or even 0.01, which indicated that relative humidity and air temperature were the most important factors governing the formation of frost roughness unevenness. Increasing air humidity or air temperature provided more water vapor for the growth of frost in the center and rear regions. As a result, frost with the rougher surface were formed downstream the cold surface. In the meanwhile, a higher gradient of air temperature and air humidity in the front region either melted the peaks of previously formed frost crystals or forced frost crystals forming a clustered structure, which created a smoother frost layer in the front. All these impacts led to a larger Std_ks value over the frost surface.

Table 3

P-value of the Std_ks growth rates

ParameterP-value (0–45 min)P-value (45–90 min)
Vair0.377600.79541
Ts0.979910.44209
RH0.012330.00398
Tair0.006830.02584
ParameterP-value (0–45 min)P-value (45–90 min)
Vair0.377600.79541
Ts0.979910.44209
RH0.012330.00398
Tair0.006830.02584

To further investigate the importance of each environmental parameter on the formation of frost ks height at different locations (the front, center, and rear regions), ANOVA tests, similar with the one for Std_ks growth rates, were conducted. Table 4 lists all the P-values of frost ks growth rates at different locations under two time spans. For the formation of frost ks in the front region, the P-value of air velocity had the smallest value for both time spans, which indicated that air velocity had the most significant impact on frost ks growth near the leading edge. The Reynolds number in this study was calculated by the hydraulic diameter of the cross section of the test channel and ranged from 710 to 3550 corresponding to the air velocity varied from 0.5 to 2.5 m/s. The airflow was changed from laminar to turbulent as the air velocity increased, and then, more eddies were generated to mix the air flow. Therefore, increasing air velocity produced higher heat and mass transfer in the front region which can help melt the sharp peaks of frost crystals to form a smoother frost layer.

Table 4

P-value of frost ks growth rates at different locations

P-value (0–45 min) FrontP-value (45–90 min) FrontP-value (0–45 min) CenterP-value (45–90 min) CenterP-value (0–45 min) RearP-value (45–90 min) Rear
Ts0.995540.894100.557380.761950.638320.91507
Vair0.051920.221380.451370.667180.466360.59195
RH0.802460.572910.253250.130350.089620.01993
Tair0.359590.381860.245140.132560.105390.08385
P-value (0–45 min) FrontP-value (45–90 min) FrontP-value (0–45 min) CenterP-value (45–90 min) CenterP-value (0–45 min) RearP-value (45–90 min) Rear
Ts0.995540.894100.557380.761950.638320.91507
Vair0.051920.221380.451370.667180.466360.59195
RH0.802460.572910.253250.130350.089620.01993
Tair0.359590.381860.245140.132560.105390.08385

For the formation of frost ks in the center and rear regions, air velocity was no longer the most critical factor. The heat and mass transfer produced by increasing air velocity became limited in the center and rear regions due to the thicker and thicker velocity boundary layer formed in that area. Instead, relative humidity and air temperature became dominated. In the region far away from the leading edge, the growth of frost ks was more dependent on the amount of water vapor as measured by the relative humidity. Higher relative humidity created frost crystals with larger sizes, which led to a rougher frost surface.

### Average Frost ks Height.

Since the frost far away from the leading edge grew in a way to form a nearly uniform roughness, an average value of the frost ks heights in the center and rear regions was used to represent the general ks value for the frost surface. Figure 11 lists the results of average frost ks (kave) values with time under various environmental conditions. The frost kave increased with time. Once it reached its maximum height, the frost kave value then started decreasing. It can be noticed that the maximum value of frost kave (kmax), as well as the timing it appeared, showed a strong dependence on environmental conditions. Within our testing ranges, the frost kave had a larger maximum value when air temperature, relative humidity, or wall temperature were higher, or air velocity was lower. On the other hand, the maximum height appeared earlier when air temperature, relative humidity, or air velocity were higher, or wall temperature was lower.

Fig. 11
Fig. 11
Close modal
Equations (6) and (7) give the empirical correlations of the dimensionless appearance time of frost kmax and the value of the frost kmax, respectively. These two empirical correlations should be useful for describing the growth behavior of frost equivalent sand-grain roughness and further investigating the effects of frost roughness on the aerodynamic performance of the frosted surface. The correlation of Eq. (6) had a 0.911 R2 while that for Eq. (7) was 0.634. The measured and correlated data of the dimensionless appearance time of frost kmax and the value of the frost kmax were compared and shown in Figs. 12 and 13, respectively. The outlier in Fig. 12 was the only test with an air velocity of 0.5 m/s. The predictive accuracy of Eq. (6) was reduced in this case due to the lack of data obtained at the minimum tested air velocity (0.5 m/s) when generating the empirical correlation
$Foave=1.003(wa)−1.084(ReD)−1.616(Tair−ToTo)−0.731(To−TsTo)−0.774$
(6)
$kmax=208.9(wa)−0.253(ReD)−0.880(Tair−ToTo)−0.268(To−TsTo)−0.436$
(7)
Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal

## Conclusion

The variation of frost roughness across a flat plate was investigated by conducting frosting experiments under a range of environmental conditions covering air temperatures from 8 to 16 °C, wall temperatures from −20 to −10 °C, relative humidities from 60 to 80%, and air velocities from 0.5 to 2.5 m/s. The frost equivalent sand-grain roughness was measured by applying a 3D photogrammetric method. The experimental results revealed that frost roughness was uneven from the front to the rear of the frost surface. The frost layer in the front region was smoother and had a smaller equivalent sand-grain roughness than the frost formed in the center and rear regions. The frost roughness in the center and rear regions of the cold plate behaved similarly.

The standard deviation of the frost equivalent sand-grain roughness was used to represent the unevenness of frost roughness over the frost surface. The timing for the appearance of the largest frost roughness unevenness appeared sooner when the air temperature, relative humidity, or air velocity were higher, or the wall temperature was lower. An empirical correlation was developed to estimate the timing of the maximum frost roughness unevenness.

An ANOVA test was conducted on the data of the standard deviation of the frost equivalent sand-grain roughness over the frost surface. Results indicated that within our ranges of environmental conditions, relative humidity and air temperature were the most important factors determining the formation of frost roughness unevenness. Similar ANOVA tests were also conducted on the data of frost equivalent sand-grain roughness at different locations of the frost surface. The results showed that within our ranges of environmental conditions, air velocity was the dominating factor for the growth of frost equivalent sand-grain roughness in the front region of the cold plate. However, relative humidity and air temperature took over the domination when frost formed in the center and rear regions.

During the 2-h duration of the experiments, the average frost equivalent sand-grain roughness over the frost surface generally increased with time until it reached its maximum height, and then started decreasing. The timing for the appearance of the largest average frost equivalent sand-grain roughness was shorter when air temperature, relative humidity, or air velocity were higher, or wall temperature was lower. The maximum frost equivalent sand-grain roughness became greater when air temperature, relative humidity, or wall temperature were higher, or air velocity was lower. Two empirical correlations were provided for the prediction of the maximum frost equivalent sand-grain roughness and its appearance time.

## Acknowledgment

The work in this paper was supported as part of FAA Grant 17-G-011. Any opinions presented in this paper are those of the authors and do not reflect the any official position of the U.S. Federal Aviation Administration or the United States government.

## Nomenclature

• t=

time

•
• D=

hydraulic diameter

•
• $Z¯$=

the mean surface elevation, $=1/NP∑i=1NPZi$

•
• kave=

average of frost equivalent sand-grain roughness

•
• kmax=

maximum value of equivalent sand-grain roughness

•
• ks=

equivalent sand-grain roughness

•
• wa=

absolute air humidity

•
• Np=

the number of points in a surface point cloud

•
• Rq=

root-mean-square roughness height

•
• Tair=

air temperature

•
• Ts=

wall surface temperature

•
• To=

triple point temperature

•
• Vair=

air velocity

•
• Fo=

Fourier number

•
• ReD=

Reynolds number calculated by hydraulic diameter

•
• Skw=

skewness

•
• Std_ks=

standard deviation of frost equivalent sand-grain roughness

•
• Z(x,y)=

surface elevation at a point

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper. No data, models, or code were generated or used for this paper.

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