Source: http://www.google.com/patents/US7629558?dq=6,928,433
Timestamp: 2017-07-21 04:07:07
Document Index: 606766721

Matched Legal Cases: ['Application No. 2', 'Application No. 03808185', 'Application No. 200680003031', 'Application No. 2003', 'Application No. 10', 'Application No. 10', 'Application No. 10', 'Application No. 10', 'Application No. 2004127250', 'Application No. 2004127250', 'Application No. 20040907418']

Patent US7629558 - Systems and methods for modifying an ice-to-object interface - Google PatentsSearch Images Maps Play YouTube News Gmail Drive More »Sign inPatentsA method for controlling a coefficient of friction between an object and ice includes steps of (1) pulsing power to an interface between the object and the ice to melt an interfacial layer of ice at the interface and decrease the coefficient of friction, (2) facilitating refreezing of the interfacial...http://www.google.com/patents/US7629558?utm_source=gb-gplus-sharePatent US7629558 - Systems and methods for modifying an ice-to-object interfaceAdvanced Patent SearchTry the new Google Patents, with machine-classified Google Scholar results, and Japanese and South Korean patents.Publication numberUS7629558 B2Publication typeGrantApplication numberUS 11/409,914Publication dateDec 8, 2009Filing dateApr 24, 2006Priority dateFeb 11, 2002Fee statusPaidAlso published asCA2476202A1, CA2476202C, CA2667789A1, CA2667789C, CN1647584A, DE60322846D1, EP1483939A1, EP1483939B1, US6870139, US7034257, US20030155467, US20050035110, US20070045282, US20100084389, WO2003069955A1Publication number11409914, 409914, US 7629558 B2, US 7629558B2, US-B2-7629558, US7629558 B2, US7629558B2InventorsVictor PetrenkoOriginal AssigneeThe Trustees Of Dartmouth CollegeExport CitationBiBTeX, EndNote, RefManPatent Citations (108), Non-Patent Citations (67), Referenced by (14), Classifications (35), Legal Events (3) External Links: USPTO, USPTO Assignment, EspacenetSystems and methods for modifying an ice-to-object interface
Q = Q i + Q S ≈ W · t = ( T m - T ) 2 W [ ρ i c i λ i + ρ s c s λ s ] ( Eq . 0 - 5 ) where W is density of heating power on the interface.
Q min =l i ·q i·ρi d heater C heaterρheater(T m −T), where (Eq. 1-3)
An example of the operative characteristics of system 70 is now described. Consider a de-icer environment in which the ambient temperature T is about −10° C., air speed is about 320 km/hour, and thickness of aircraft wing 80 is about 10 cm, with a convective heat exchange coefficient h, of about 1200 watt/K·m2 (based on experimental data).
Modern chemical batteries are known for high density of stored electric energy (e.g., about 60 kJ/kg for a lead battery). However, chemical batteries have a relatively low power density. For example, a car battery can deliver up to about 1000 A at twelve volts for about ten seconds, corresponding to a power of about 12 kW. A typical car battery has a large capacity of about Q≈12V×100 A×3600 sec=4.32·106 J. Therefore, for use in pulse de-icer systems and methods, the car battery may effectively de-ice areas up to about 1.5 m2, which is ideal for automobile windshields.
C i ≅ 1.2 × 10 - 11 D ( m ) F m 2 ( Eq . 12 - 1 ) and a HF-conductance per square meter of:
G i = 0.53 · 10 - 4 D ( m ) · ⅇ 6670 ( 1 273 - 1 T ( k ) ) ( 1 ohm · m 2 ) , ( Eq . 12 - 2 ) where
V B≈1.7×106 D(m). (Eq. 12-4)
εo is free space permittivity (e.g., εo=8.85·10−12 F/m), ε is a relative permittivity of ice, and σ is a conductivity of ice. Assuming a=b, the following can be concluded:
C ∝ G ∝ 1 ℓ · b a · ɛɛ o ∝ 1 ℓ ∝ σ ℓ , ( Eq . 13 - 2 ) where l is equal to a plus b, also known as the structure period. The mean electric field E is:
G ∝ 1 ℓ · G ′ , ( Eq . 14 - 1 ) where
W ∝ GV 2 ∝ G ′ ℓ · V 2 ∝ V 2 ℓ · ℓ n ( a + ℓ 2 a ) ( Eq . 14 - 3 ) W ∝ V 2 ℓ ℓ n ( a + ℓ 2 a ) = E 2 ℓ max · a 2 · ℓ n ( a + ℓ 2 a ) , ( Eq . 14 - 4 ) where
In one embodiment, interdigitated circuit 180 modifies a coefficient of friction of an object's surface-to-ice interface in cooperation with natural friction change between an object and ice or snow over temperature. For example, a steel object “slider” slides on ice when at a velocity of 3.14 m/s, the friction coefficient of the slider on the ice drops from 0.025 at −15° C. to 0.01 at −1° C. To increase the temperature of ice that is in direct contact with the slider, interdigitated circuit 180 can either heat the ice directly using HF electric fields or heat a surface of the slider.
W h ≈ σ i · V 2 d 2 , ( Eq . 16 - 2 ) V is the rms AC voltage. While the power Wh of Eq. 16-2 relates to electric power per unit volume, power per square meter Ws of an ice/slider interface is of greater concern. To estimate the power per square meter Ws, the power Wh is multiplied by the thickness of the heated layer, approximately d, as previously indicated. Therefore, the power per square meter Ws follows the equation:
l D=√{square root over (D·t)}, where (Eq. 16-8)
D = λ C · ρ , ( Eq . 16 - 10 ) where
ρ = 3 · 10 2 kg m 3 , ( Eq . 16 - 12 ) the change in temperature of the interfacial layer of snow ΔT is
W speed - fraction = W H · a · L , ( Eq . 16 - 16 ) where
W speed=134W×0.066≈9W. (Eq. 16-18)
In graph 190, at −10.1° C., ice has an electrical conductivity of about 0.1 μS/m at approximately-10 kHz. Ice conductivity decays exponentially when temperature decreases. Accordingly, the conductivity of ice at −30° C. would be about one order of magnitude less than conductivity of ice at −10° C.
TABLE 19-1 ε0 := 8.85 · 10−12 f := 10, 100 . . . 1 · 105 ω(f) := 2 · π · f T := 243, 244 . . . 273 τ D ( T ) := 1.5 · 10 - 4 · exp [ 6670 ( 1 T - 1 253 ) ] ɛ s ( T ) := 25047 T εinf := 3.2 σ inf ( T ) := 1.8 · 10 - 5 · exp [ 6670 ( 1 253 - 1 T ) ] σ0 := 10−8 ɛ ( f , T ) := ɛ inf + ( ɛ s ( T ) - ɛ inf ) 1 + ( τ D ( T ) · ω ( f ) ) 2 σ ( f , T ) := [ [ ( σ inf ( T ) - σ 0 ) · ( τ D ( T ) · ω ( f ) ) 2 ] 1 + ( τ D ( T ) · ω ( f ) ) 2 ] + σ 0 d := 10−7, 2 · 10−7 . . . 3 · 10−5 εd := 9.9 C d ( d ) := ɛ 0 · ɛ d 8 d l := 2.5 · 10−4 V = 500 RS := 0 R i ( f , T , d ) := 4 ( 3 2 1 - 2 · d ) σ ( f , T ) C i ( f , T , d ) := ɛ 0 · ɛ ( f , T ) 4 ( 3 2 1 - 2 · d ) Z i ( f , T , d ) := R i ( f , T , d ) 2 π · f · i · C i ( f , T , d ) · ( R i ( f , T , d ) + 1 2 · π · f · i · C i ( f , T , d ) ) Z ( f , T , d ) := Z i ( f , T , d ) + 1 2 · π · f · i · C d ( d ) I ( f , T , d ) := V R S + Z ( f , T , d ) π Pi(f, T, d) := V · Re(I(f, T, d)) εw := 80 σw = 5 · 10−4 R w ( d ) := 4 ( 3 2 1 - 2 · d ) σ w C w ( d ) := ɛ 0 · ɛ w 4 ( 3 2 1 - 2 · d ) Z ( f , d ) := R w ( d ) 2 π · f · i · C w ( d ) · ( R w ( d ) + 1 2 · π · f · i · C w ( d ) ) Z w ( f , d ) := Z ( f , d ) + 1 2 · π · f · i · C d ( d ) I w ( f , d ) := V R S + Z w ( f , d ) Pw(f, d) := V · Re(Iw(f, d)), where ε0 is free space permittivity, f is incremental frequency, ω is radial frequency as a function of f, T is incremental ambient temperature in K, τD is an ice dielectric relaxation time, εs is a static dielectric permittivity of ice, εinf is a high-frequency permittivity of ice, σinf is a high-frequency conductivity of ice, GO is a static conductivity of ice, σinf is an ice permittivity (e.g., as a function of frequency f and temperature T), σ is an ice conductivity (e.g., as a function of frequency f and temperature T), d is a thickness of the protective dielectric layer, εd is a permittivity of the protective dielectric layer l, V is voltage, Zi is impedance of ice (e.g., as a function of frequency f, temperature T, and distance d), Z(f,T,d) is a total circuit impedance with ice covering the electrodes (e.g., as a function of frequency f, temperature T, and distance d), I is applied current (e.g., as a function of frequency f, temperature T, and distance d), Pi is power delivered to heat the ice (e.g., as a function of frequency f, temperature T, and distance d), εw is a permittivity for water, σw is a conductivity for water, Rw is a water resistance, Cw is a water capacitance, Z(T,d) is a total circuit impedance with water covering the electrodes (e.g., as a function of frequency f and distance d), Zw is impedance for water (e.g., as a function of frequency f and distance d), Iw is applied current (e.g., as a function of frequency f and distance d), and Pw is power delivered to the water (e.g., as a function of frequency f and distance d). Electric power was calculated for both of the following cases: when ice covers the electrodes, and when ice was melted and water is in contact with the electrodes.
TABLE 19-2 v := 1, 1.1 . . . 30 L := 0.1, 0.2 . . . 1 Retr := 105 v := 1.42 · 10−5 Re L ( v , L ) := v · L v k := 0.0235 Pr := 0.69 ReL(20, 0.5) = 7.042 × 105 h ( v , L ) := k L · [ 0.664 Re tr 0.5 · Pr 1 3 + 0.036 Re L ( v , L ) 0.8 · Pr 0.43 · [ 1 - ( Re tr Re L ( v , L ) ) 0.8 ] ] , where v is air velocity, L is a length of the windshield surface, Re is a range of Reynolds number from 105 to 107, h(v, L) is a heat transfer coefficient (e.g., as a function of voltage and L), k is on air thermal conductivity, and Pr is air Prandtl number, and ν is the air kinematic viscosity coefficient. In this embodiment, the heat transfer coefficient h(v, L) at about 30 m/s and a length of about 0.5 meters was 89.389 w/m2K. Accordingly, FIG. 22 graphically shows (in plot 240) the relationship of the heat transfer coefficient h(v, L) to air velocity.
FIG. 23 illustrates one dependence of minimum HF power Wmin of circuit 200 on outside temperature T (in °) for vehicle velocities of 10 m/s (plot 252), 20 m/s (plot 251), and 30 m/s (plot 250). In FIG. 23, Y-axis 253 represents minimum HF power Wmin (watt/m2) and X-axis 254 represents temperature T. The minimum heating power Wmin to maintain the outer surface of the windshield at about 1° C. is shown in the following Table 19-3 (MathCad file):
TABLE 19-3 S := 0, 0.1 . . . 2 T := 0, −1 . . . −30 Wmin(v, L, T, S) := h(v, L) · S · (1 − T), where
TABLE 24-1 ε0 := 8.85 · 10−12 f := 10, 100 . . . 1 · 105 ω(f) := 2 · π · f T := 243, 244 . . . 273 τ D ( T ) := 1.5 · 10 - 4 · exp [ 6670 ( 1 T - 1 253 ) ] ɛ s ( T ) := 25047 T εinf := 3.2 σ inf ( T ) := 1.8 · 10 - 5 · exp [ 6670 ( 1 253 - 1 T ) ] σ0 := 10−8 ɛ ( f , T ) := ɛ inf + ( ɛ s ( T ) - ɛ inf ) 1 + ( τ D ( T ) · ω ( f ) ) 2 σ ( f , T ) := [ [ ( σ inf ( T ) - σ 0 ) · ( τ D ( T ) · ω ( f ) ) 2 ] 1 + ( τ D ( T ) · ω ( f ) ) 2 ] + σ 0 d := 10−7, 2 · 10−7 . . . 3 · 10−5 εd := 9.9 C d ( d ) := ɛ 0 · ɛ d 8 d l := 2.5 · 10−4 V = 500 RS := 0 R i ( f , T , d ) := 4 ( 3 2 1 - 2 · d ) σ ( f , T ) C i ( f , T , d ) := ɛ 0 · ɛ ( f , T ) 4 ( 3 2 1 - 2 · d ) Z i ( f , T , d ) := R i ( f , T , d ) 2 π · f · i · C i ( f , T , d ) · ( R i ( f , T , d ) + 1 2 · π · f · i · C i ( f , T , d ) ) Z ( f , T , d ) := Z i ( f , T , d ) + 1 2 · π · f · i · C d ( d ) I ( f , T , d ) := V R S + Z ( f , T , d ) π Pi(f, T, d) := V · Re(I(f, T, d)) εw := 80 σw = 5 · 10−4 R w ( d ) := 4 ( 3 2 1 - 2 · d ) σ w C w ( d ) := ɛ 0 · ɛ w 4 ( 3 2 1 - 2 · d ) Z ( f , d ) := R w ( d ) 2 π · f · i · C w ( d ) · ( R w ( d ) + 1 2 · π · f · i · C w ( d ) ) Z w ( f , d ) := Z ( f , d ) + 1 2 · π · f · i · C d ( d ) I w ( f , d ) := V R S + Z w ( f , d ) Pw(f, d) := V · Re(Iw(f, d)), where the variables are the same as those found in Table 19-1, but with different values. For example, σw is the conductivity for water with the same value of 5×10−4 S/m
FIGS. 24-26 graphically illustrate a dependence of heating power generated in distilled water (plots 261, 270, 281 of respective FIGS. 24, 25 and 26) at 20° C. and in ice (plots 260, 271, 280 of respective FIGS. 24, 25 and 26) at −10° C., which differ in the thickness of the dielectric layer: 10−5 m (FIG. 24), 10−6 m (FIG. 25), 2·10−5 m (FIG. 26). The heating power as shown in FIGS. 24, 25 and 26 depends on a frequency of the AC power. As frequency increases, the amount of applied power used to melt an interfacial layer of ice levels off. The AC voltage was about 500 v. At a coating thickness of about 10 μm (10−5 m), the respective heating power for water and ice are substantially equal, as is shown from FIG. 24.
TABLE 27-1 ρ := 300 kg m 3 x := 0, 0.0001 . . . 0.1 m C := 2 · 103 J/kg K λ := 0.2 2 m . K W := 1 · 10 3 w m 2 D := λ C · ρ D = 3.333 × 10−7 y ( x , t ) := x 4 · D · t Δ ( x , t ) := W λ · 4 · D · t · ∫ y ( x , t ) ∞ ( 1 - erf ( z ) ) d z t := 0, 0.001 . . . 1 s a := 0.1 m L := 1.5 m v := 1, 2 . . . 30 W speed ( Δ , v ) := a · Δ · v · λ · C · L · ρ Wspeed(1, 10) = 134.164 watt, where
FIG. 27 illustrates dependence of snow overheating temperature Δ (e.g., degrees C.) with respect to the distance from a slider. In FIG. 27, Y-axis 295 represents overheating temperature Δ (° C.) and X-axis 294 represents distance from the slider (in meters). With a heating power W of about 1 kwatt/m2, plots 290, 291, 292, and 293 illustrate temperature dependences for heating pulses having approximate durations of t=0.1 s, 0.2 s, 0.5 s, and 1 s, respectively. FIG. 28 illustrates the snow-slider interface-temperature dependency with respect to time (plot 300) when HF-power of density 1000 watt/m2 was applied. In FIG. 28, Y-axis 301 represents overheating temperature Δ (° C.) and X-axis 302 represents time (in seconds).
FIG. 29 illustrates the heating power required to increase the interface temperature by 1° C. when the slider is traveling at velocity v of about 30 m/s. In FIG. 29, Y-axis 311 represents heating power Wspeed and X-axis 312 represents velocity ν. In the example, as the slider travels at about 5 m/s, the heating power is about 100 watts. The heating power Wspeed is plotted with respect to velocity ν (plot 310).
TABLE 30-1 (MathCad file) v := 89 D := 0.03 v := 10, 11 . . . 100 h c ( v , D ) := v 0.63 D 0.37 · 0.19 · 0.024 · 0.69 0.36 ( 1.2 · 10 - 5 ) 0.63 watt / m 2 K hc(89, 0.254) = 141.057 watt/m2 K hc(89, 0.0254) = 330.669 watt/m2 K, where
FIG. 31 shows a dependence of the steady-state (stationary solution) overheating Δ in ° C. on ice thickness in meters. In FIG. 31, Y-axis 335 represents overheating Δ and X-axis 336 represents thickness L. Plot 330 shows a dependence of steady-state overheating in ° C. on ice thickness in meters assuming a theoretically perfect insulating layer between the de-icer and the aerofoil, while plot 331 shows the dependence for a 2 mm thick Teflon film between the de-icer and the aerofoil. De-icing performance is maximized when ice thickness exceeds approximately 1 mm (point 333 for the theoretically perfect insulating layer, and point 334 for the 2 mm thick Teflon film).
FIG. 32 shows a dependence of the steady-state overheating Δ in ° C. on electrode size in meters (plot 340), assuming a perfect insulating layer and a 1 cm thickness of ice. In FIG. 32, Y-axis 341 represents overheating Δ and X-axis 342 represents electrode size l. In the example, bubbling on the interfacial layer of ice may be seen. Bubbling is the result of ice evaporation (e.g., steam) and is evidence of overheating by more than 110° C.
FIG. 33 shows a dependence of the steady-state (stationary solution) overheating Δ in ° C. on ice thickness in meters. In FIG. 33, Y-axis 355 represents overheating Δ and X-axis 356 represents thickness L. Plot 350 shows a dependence of steady-state overheating in ° C. on ice thickness in meters assuming a theoretically perfect insulating layer between the de-icer and the aerofoil, while plot 351 shows the dependence for a 2 mm thick Teflon film between the de-icer and the aerofoil. De-icing performance is maximized when ice thickness exceeds approximately 1 mm (point 352 for the theoretically perfect insulating layer and point 353 for the 2 mm thick Teflon film).
TABLE 30-4 (MathCad file) ρ := 920 kg m 3 C := 2 · 10 3 J kg · K x := 0, 0.0001 . . . 0.1 m λ := 1 w mK W := 4.5 · 10 3 w m 2 D := λ ρ · C y ( x , t ) := x 4 · D · t Δ ( x , t ) := W λ · 4 · D · t · ∫ y ( x , t ) ∞ ( 1 - erf ( z ) ) ⅆ z t := 0, 0.1 . . . 1000 s D = 5.435 × 10−7, where
ρ is ice density, C is ice heat capacity of the ice, λ is a thermal conductivity coefficient of the ice, x is a distance from the heater, W is an applied power per square meter, D is a coefficient of heat diffusivity, and t is the duration in which power is applied (e.g., as a heat pulse). FIG. 34 illustrates plots 360, 361, 362 and 363 for respective time values of 200 s, 100 s, 25 s and 5 s as the power W of about 4.5 kwatt/m2 is applied to an atmospheric ice mixture of solid ice, unfrozen water and gas bubbles with a thermal conductivity coefficient λ of 1 W/m·K. In FIG. 34, Y-axis 365 represents overheating Δ and X-axis 366 represents distance from the heater x.
FIG. 35 illustrates how the interface temperature depends on time by showing a dependence of interfacial overheating temperature Δ in ° C. on time. In FIG. 35, Y-axis 371 represents overheating Δ and X-axis 372 represents time. When a short pulse of heating is applied, thermal energy can be minimized and still melt the interfacial layer of ice. For example, thermal energy may be calculated according to the following Table 30-6:
TABLE 30-6 (MathCad file) Δ ( t ) := W λ · 4 · D · t · ∫ 0 ∞ ( 1 - erf ( z ) ) d z Δ ( t ) := 2 · W λ · D · t π t ( Δ ) := ( Δ · λ 2 · W ) 2 · π D Q ( W ) := ( Δ · λ 2 ) 2 · π D · W , where t is the time it takes to reach a desired overheating temperature Δ of the interfacial layer of ice, and Q is the total thermal energy needed to reach that temperature. As in FIG. 1, total thermal energy Q may be substantially inversely proportional to applied power W, to employ a de-icer with a higher power output that conserves overall electric power.
t cool ≈ [ Q S · 1 ( T m - T ) ( λ snow · ρ snow · c snow + λ ski · ρ ski · c ski ) ] 2 , ( Eq . 40 - 1 ) where
At about −10° C., and with ice grown on disc 562 and thermal transfer system 560 in a vertical position, a power of approximately 10-25 watts heats disc 563 to about 20° C. when applied to heating element 565. When the vacuum pump withdraws air from chamber 573, such that discs 562 and 563 contact one another, ice 577 is removed from disc 562, e.g., by gravity. While air is typically used in chamber 573, other thermally insulating gases may alternatively be used in chamber 573.
TABLE 42-1 (MathCad file) v := 1.57 · 10−5 L := 0.0125 g := 9.8 β := 1 273 Pr := 0.69 Tm := 273 Ts := 243, 244 . . . 273 Th(Ts) := 2 · Tm − Ts + 5 Δ(Ts) := Th(Ts) − Ts (i.e., the temperature difference between the heater and the environment) Ra L ( T s ) := g · β · L 3 · Pr · ( Δ ( T s ) ) v 2 RaL(243) = 1.276 × 104 Nu1(Ts) := 0.0605 RaL(Ts)1/3 Nu 2 ( T s ) := [ 1 + [ 0.104 Ra L ( T s ) 0.293 1 + ( 6310 Ra L ( T s ) ) 1.36 ] 3 ] 1 3 where Ts is the temperature of the substrate material (disc 562), Th is the temperature of the heating element (disc 563), ν is air kinematic viscosity, L is a distance between discs 562 and 563, g is gravity acceleration, β is air thermal expansion coefficient, Pr is air Prandtl number, Tm is ice melting temperature, Ts is an incremental temperature of disc 562, Δ is temperature difference, Ra is air Rayleigh number, Nu1 and Nu2 are Nusselt number.
TABLE 42-2 (MathCad file) λa := 0.025 W c ( T s ) := λ a · Nu ( T s ) · Δ ( T s ) L W c ( 243 ) 2 = 91.887 watt m 2 where λa is a thermal conductivity coefficient of the air, and Wc/2 is a mean heat transfer rate when the heater heats disc 563 from Ts to Th. In FIG. 42, Y-axis 581 represents convection Nu and X-axis 582 represents temperature Ts of the substrate material. A mean loss of heat Wc through the air gap is shown in FIG. 43 (plot 590). In FIG. 43, Y-axis 591 represents convection heat transfer Wc/2 and X-axis 592 represents temperature Ts of the substrate material.
TABLE 42-4 (MathCad file) ε := 0.04 σ := 5.67 · 10−8 WR(Ts) := ε · σ · (Th(Ts)4 − Ts 4) W R ( 243 ) = 12.502 watt m 2 where ε is the emittance of discs 562 and 563 emittance, and σ is the Stefan-Boltzmann constant. From Table 42-4, the radiative heat transfer WR can be plotted (plot 600) as a function of temperature Ts in FIG. 44 (Ts and Tm being defined above). In FIG. 44, Y-axis 601 represents radiative heat transfer WR and X-axis 602 represents temperature Ts of the substrate material.
W(Ts) := ¾ · ((Wc(Ts) + Win(Ts) + WR(Ts)))
TABLE 42-6 (MathCad file) d := 0.001 t := 1, 2 . . . 300 Cs:= 900 λs := 170 ρs := 2700 Ci := 2000 ρi := 920 λi := 2 Q1(Ts) := d · Cs · ρs · (Th(Ts) − Tm) Q 1 ( 243 ) = 8.505 × 10 4 Joul m 2 Q2(Ts, t) := W(Ts) · t, where
W := 104, 2 · 104 . . . 106 λski := 0.2
Cski = 1.54 × 103 ρsnow := 300
Csnow := 2.2 · 103 λsnow := 0.2
C := 10−4, 2 · 10−4 . . . 2 · 10−2 F
L D ( t ) := D snow · t LD(10−2) = 5.505 × 10−5 LD(1) = 5.505 × 10−4 LD(0.1) = 1.741 × 10−4 LD(0.01) = 5.505 × 10−5 V := 100 S := 0.0025 W ( R ) := V 2 2 · R · S dheater := 1.25 · 10−5 Cheater := 523 ρheater := 4.5 · 103 lmelt := 1 × 10−5 qlatent := 3.33 · 105 Q = πΔ 2 S 4 W ( R ) [ ρ snow c snow λ snow + ρ ski c ski λ ski ] 2 + d i · q i · ρ i + d heater C heater ρ hea C ( Δ , R ) := 2 · Q ( Δ , R ) V 2 C(20, 2.5) = 8.464 × 10−4 Δ := 20 dheater · S · ρheater · Δ · Cheater = 1.471 lmelt · ρsnow · S · qlatent = 2.498 where S is heater area, Tm is melting temperature, T is ambient temperature, λ is a thermal conductivity coefficient, ρ is the material density, and C is the material heat capacity (subscript “ice” denotes ice and/or snow, subscript “ski” denotes substrate material, such as a ski or a snowboard, subscript “heater” denotes a heating element), Q is thermal energy, D is a heat diffusivity coefficient, Δ denotes temperature change, t is time, V is voltage, d is thickness, R is resistance, W is a power per square meter, lmelt is thickness of melted layer, and q is latent heat of melting. Accordingly, for very short pulses, nearly all thermal energy Q is used to melt a thin layer of snow (plot 890, FIG. 66); snow and ski heat capacitance contributes little to Q. A calculation of refreezing time for the melted layer is shown by the following Table 65-2:
watt · hour
t ≈ 0.1 m · s 3 · 10 1 m ≈ 3.3 · 10 - 3 s , and the heated zone φo acquires an energy density of:
d · q · ρ i = w ′ A · υ 0 , ( Eq . 67 - 5 ) and, therefore,
w′=A·υ 0 ·d·q·ρ i. (Eq. 67-6)
t = 2 · 10 - 1 m 6 m / s = 30 m sec . This time is available for melting and refreezing actions, and is long enough to accomplish such actions.
FIGS. 68 and 69 illustrate experimental results in which ice friction was reduced by either application of HF-power, as in FIG. 68, or by application of low-energy heating pulses, as in FIG. 69. In FIG. 68, Y-axis 915 represents frictional force and X-axis 914 represents time in seconds. For example, FIG. 68 shows a frictional force N versus time for the slider in motion on ice with an ambient temperature T of about −5° C., a normal pressure P of about 42 kPa, and a sliding velocity ν of about 1 cm/s. In this embodiment, the system modifying the friction includes an interdigitated circuit attached to a base of the slider that interfaces with ice. The interdigitated circuit includes a copper clad Kapton polyimid film. The interdigitated circuit also includes copper electrodes having an inter electrode spacing of about 75 μm. A power supply provided HF AC voltage of about 30V rms at about 20 kHz to the electrodes. The electrodes generated heat in ice of about 100 watts/m2 density. When the slider moves at a velocity of about
FIG. 69 shows a frictional force N versus time for the slider in motion on snow with an ambient temperature T of about −10° C., a normal pressure P of about 215 kPa, and a sliding velocity ν of about 3 mm/s. In FIG. 69, Y-axis 925 represents frictional force and X-axis 926 represents time in seconds. In this embodiment, the system modifying the friction includes a thin titanium-foil heater. Short heating pulses of DC power are applied to the heater at time moments 922 and 923 causing decrease in snow friction, as opposed to the braking effect by the same system described earlier. The main difference of this experiment is the pulse braking; as shown in FIG. 69, the magnitude of heating energy is not sufficient to melt snow. Without a melted layer, refreezing does not occur and there is no braking action. Nevertheless, since the heater warms snow, the friction decreases. In the experiment of FIG. 69, the snow surface is heated by the pulses from −10° C. to about −10° C. The slider experiences a rapid increase in static friction between the ice and the slider at time point 921 (e.g., about time t equal to 31 s). The power supply provides pulse power at time points 922 and 923 (time t equal to 38 s and 42 s, respectively) to the electrodes. In this embodiment, the slider stops at time point 924, when time t equals 50 s.
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