Abstract

An enabling advantage of carbon-based conductors is their low density and high thermal conductivity. To put this in the perspective of applications, current rating of carbon-based and copper nanocomposite conductors of different lengths are modeled. For comparison, the current and current density required to raise the maximum temperature of studied conductors to 150°C are calculated with a joule heating model. The model is validated with an experimental setup equipped with a thermal camera. It is shown that while doped carbon nanotube (CNT) conductors may potentially result in improved performance compared with copper on a weight basis, ultra-conductive copper (UCC) can outperform copper on both volume and weight bases. Additionally, a hypothetical copper-matrix composite conductor with different volume fractions of high thermal conductivity and lightweight graphene fibers (Cu–C composite) is included in the analysis. The properties of the Cu–C composite are evaluated based on the Lewis–Nielson and rule of mixture models, as inputs for the joule heating model. The results show that while the improved thermal conductivity of the composite is beneficial for improving the current rating in micro-electronics applications, the tradeoff for the decreased electrical conductivity results in lower current carrying capacity in applications that use longer conductors.

Introduction

The increasing demand for improved electrical, mechanical, thermal, and anticorrosion properties of conductors has driven the development of advanced electrical conductors, including metal matrix nanocomposites and lightweight carbon-based conductors [1,2]. Applications for such conductors span from small-scale conductors for micro-electronics and interconnects to busbars, electric motors, and high voltage cables [3,4]. In certain applications, such as electrical and thermal management systems in aviation, the main goal is to reduce the overall weight of the components without sacrificing other useful properties [5]. Many studies have, therefore, focused on using lightweight conductors such as graphene and carbon nanotubes (CNTs). Others have used these nanocarbons as additives to enhance metallic conductors (metal-matrix nanocarbon composites) [1]. Recently, ultra-conductive copper, containing minimal graphene loadings, has emerged [6].

Depending on their specific application, conductors require a high electrical conductivity (s) in conjunction with other attributes such as high thermal conductivity (k), mechanical properties, density, and cost. The previously developed models for maximum current rating of a conductor can be generalized into two categories. The first category deals with the maximum current that can be applied to a conductor before it fails/breaks at a given duration. Fusing current is the common term used for this phenomenon and the models attempting to predict the fusing current date back to 1883, an empirical model developed by Preece. Later models for calculating fusing current such as Schwartz-James [7] and Onderdonk [8] tried to establish more accurate estimations in general or for shorter fusing times. These formulae correlate the diameter of the wires to the fusing current for 10 s of applied current and follow the form of i=adb, where i and d are the fusing current and the diameter of the conductor, respectively, and a and b are constants. The shortcoming of these models lies within their empirical derivation, relying on constants a and b to be found for every known material, used for electrical conduction, by experimental measurements. More recent models such as the one presented by Mertol [9] try to resolve this issue by presenting a theoretical approach for calculating fusing times and currents. Others have also introduced more realistic models for calculating fusing thresholds in special cases [10]. The second category for a conductor's current rating deals with the ampacity rating. For most applications, conductors require to carry higher current densities without overheating above a threshold temperature. Ampacity is an established metric for comparing similar-sized conductors based on this criterion. The most commonly used model for calculating the ampacity of a conductor is the Neher-McGrath method [11]. The Neher-McGrath model considers steady-state equations that were originally developed for underground power cables. Later attempts expanded this model to cables installed in free air by implicitly accounting for natural convection and radiation [12]. While the Neher-McGrath and its derived models consider AC and DC resistances of the conductor, the disadvantages of these models are twofold: (i) They assume an infinitely long cable with uniform dissipation of heat along the cable's length; hence not considering the dissipated heat through the termination points that has a substantial effect in shorter wires and cables [13], and (ii) they do not explicitly consider the temperature dependency of the physical properties of the conductor material.

To overcome the deficiencies of the current models, a heat transfer model considering Joule heating, effects of termination points, and temperature-dependent physical properties of the conductor, and dissipation through natural convection and radiation is developed in this study and numerically solved. Step-by-step development and validation of the ampacity prediction model are discussed, and results from several case studies are provided. A thorough discussion of the input model parameters is also provided. Input parameters to the model include both reported experimental values as well as theoretically calculated ones. The model is validated with an experimental setup equipped with a thermal imaging system. The model is then used to provide a direct comparison between various advanced electrical conductors and copper.

An enabling advantage of carbon-based conductors is their low density and sometimes higher thermal conductivity. To put this in the perspective of applications, conductors of different length scales are modeled based on equivalent volume and equivalent weight. Finally, a hypothetical composite conductor made from copper (high s and relatively high k) and graphene fibers (relatively high s and high k) is optimized for different applications.

Methodology: Model Development and Finite Difference Analysis

The model consists of a Joule heating term and three terms for heat dissipation including conduction, convection, and radiation. Existing models and experimental values for each input parameter are investigated in this study, and the significance of each term over a range of conductor dimensions is explored. In the second part of this section, the experimental setup for model validation is described.

Heat Generation and Dissipation Terms.

From conservation of energy, heat generation = heat storage + heat dissipation. Heat generation in a conductor depends on the electrical resistivity and the applied current in steady-state condition. For a copper conductor, dimensional variation may impose other limitations for developing a general model. As such, the average grain size of wires of less than 100μm in diameter and the ratio of oxidized layer to pristine copper for smaller wire diameters can affect the conductor performance. Hence, discussion here is limited to wires that are larger than 100μm in diameter for all conductors considered. The International Annealed Copper Standard (IACS) metric corresponding to an electrical resistivity of (1.72×108Ωm=100%IACS)is used. The temperature coefficient of resistance of copper as a reference for comparisons is
ρCu(T)=ρ0(1+α(TT0))Ωm
(1)

where ρ0 is the electrical resistivity of copper at room temperature, α is the copper's temperature coefficient of resistance, and T0=300K is considered [14].

Heat dissipation through end contacts is governed by the thermal conductivity of the conductor material and the interface contact resistance. The interface contact resistance is a function of conductor material, pressure, roughness, temperature, and contact type (welded contacts, pressed, bolted, etc.) and should be treated on a case-by-case basis. Thermal conductivity on the other hand is a temperature-dependent material property and for copper is provided by Abu-Eishah [15]
k=aTbecTedT
(2)
a=82.56648,b=0.262301,c=0.00041,d=59.72934
Above the room temperature, 300 K, this expression can be simplified to the following, with a goodness of fit of 99.16% as shown in Fig. 1 
κCu(T)=k0(1+β(TT0))Wm1K1
(3)

where the model constants are provided in Fig. 1.

There are various models for calculating the effective thermal conductivity of particle/fiber filled composite materials. The early models from Maxwell and Rayleigh assumed no thermal resistance and flow through boundaries between the phases. Later, models with better predictions, simplicity, and accounting for various particle shapes in the empirical model, such as the Lewis–Nielsen, were developed [16]. It is noteworthy that for transient conditions, a dynamic model is needed while the aforementioned models are derived for steady-state conditions.

Thermal conductivity of metal-matrix nanocomposites is discussed in detail by Batish et al. [3]. On the other hand, the thermal conductivity of carbon-matrix conductors, which are usually porous, requires a different treatment since the heat transfer through solid–gas boundary is significantly different from solid-solid boundaries and involves porous media approaches. As a result, the thermal conductivity of carbon-matrix composites was extracted form experimentally reported values, while the Lewis–Nielsen model is utilized for estimating the thermal conductivity of a hypothetical copper-matrix composite with carbon inclusions (Cu–C composite). The hypothetical composite in this scenario consists of 2–20 vol. % (ϕ=0.02,0.05,0.1,0.2) of carbon fillers in a copper matrix. The filler properties are based on doped graphene fibers with relatively high electrical and thermal conductivities as follows [17,18]. The carbon inclusions have the following properties:
κC(T)=1575(10.00279(T300))Wm1K1
(4)
ρC(T)=96(10.001(T300))μΩcm
(5)
For this hypothetical composite, electrical and thermal conductivities were estimated using the Lewis–Nielsen model, as previously mentioned. The empirical approach of Lewis–Nielsen is chosen due to the following reasons: (a) it provides an explicit relation to calculate conductivities of a composite material, (b) multiple reports have confirmed the accuracy of this model [19,20], and (c) to avoid using an unknown Lorentz relationship for thermal to electrical conductivity conversion; a unified Lorentz number to correlate electrical and thermal conductivity cannot be used due to the phonons-driven thermal properties of carbon materials. According to different studies, the Lewis–Nielsen model is a useful tool for calculating both electrical and thermal conductivities [3,16,19]. The Lewis–Nielsen equation is given as
κκC=1+ABϕ1BψϕB=(λ1λ+A)λ=κdκCψ=1+(1ϕϕ2)ϕ
(6)

where ϕ is the filler volume fraction and ϕis the maximum packing volume fraction of the fillers. Constants A and ϕ depend on packing type and aspect ratio (length/diameter) of the particles and their values are tabulated in multiple Refs. [16 and 21].

Figure 2 demonstrates the effect of particle aspect ratio and ϕ on the relative thermal conductivity of the matrix material. The chosen value of ϕ=0.82 corresponds to uni-directional fiber type fillers in the matrix. Using the known thermal conductivities of the copper matrix and the carbon fillers in the range of 300425K, the following relationship for thermal conductivity of a given composite withϕ=0.05 is found by solving the Lewis–Nielsen equation:
κCuC=447×(10.000471×(T300))Wm1K1
(7)

Similarly, the effective thermal conductivity of the hypothetical Cu–C composites withϕ=0.02,0.1,0.2, and ϕ=0.82 is calculated and plotted in Fig. 2. The composite with 20 vol. % of carbon fillers achieves a distinguishably higher conductivity for uni-directional embeded fibers.

The corresponding Lewis–Nielsen model for the relative electrical conductivity of a given composite is
σσC=1+ABϕ1Bψϕ
(8)
Using the same approach, which was used for calculating the effective thermal conductivity of a Cu–C composite, the electrical conductivity of the 2,5,10,and20vol.% Cu–C composites across the 300–425 K range is calculated and plotted in Fig. 3. As an example, the following relationship for electrical conductivity of a given composite withϕ=0.05 is found
ρCuC=1.82×(1+0.00392(T300))μΩ·cm
(9)

Figure 3 shows the effect of filler loading and aspect ratio on the relative electrical conductivity of the composite.

Finally, the rule of mixture can be used for calculating the effective density of this theoretical composite, based on the densities of the constituents (PC=2260kg/m3 and PCu=8960kg/m3) and volume fraction of the fillers (ϕ=0.02,0.05,0.1,0.2).

Heat dissipation through natural convection for a wire includes orientation, geometry, buoyancy, kinematic viscosity, conduction, and the surrounding medium. The expression introduced by Churchill and Chu has been widely accepted for calculating the effective convection coefficient from surfaces. Recently, it has been reported that with the decrease of the wire diameter below 120μm, the heat transfer coefficient for natural convection deviates from predictions by Churchill and Chu [22]. However, the ratio of thickness of the boundary layer (film) to the diameter does not change for natural convection in air [22]. Here, wires with diameters >100μm, as mentioned previously, were considered. A simple model that agrees with Churchill and Chu [22] and is more suitable for the numerical implementation was used. This model, developed by Kakac et al. [23], considers natural convection from a horizontal cylinder (wire) as
Nu=0.436Pr(PrGr)14=hdkair
(10a)
Gr=g(TT0)d3Tfν2
(10b)
This expression can be further simplified using appropriate properties of air, g=9.81m2s1,ν=1.66×105m2s1,kair=0.027Wm1K1,Pr=0.72, and assuming that TfT0 to achieve a simple expression that correlates the convection coefficient of the conductor to the geometrical factor, d, and the temperature of the conductor
h(d,T)=0.806(TT0)14d14Wm2K1
(11)
Whether a radial asymmetry approach (two-dimensional formulation) or a one-dimensional (1D) approach is appropriate, depends on the geometry of the system under study. Biot number (Bi) can be used to determine whether a 1D model is sufficiently accurate. For a wire conductor with homogeneous properties (kr=kCu(T))
Bi=hd2kr=0.806d34(TT0)142×400.93(10.00016(TT0))
(12)

From this expression, a Biof 6.9×104 is evaluated for a conductor with a diameter of 0.1m at a final temperature of 500K (which can be considered as the upper-bound temperature for which most conductors are used at). For an inhomogeneous conductor where the radial thermal conductivity is even 100-fold lower than a copper conductor, the 1D assumption remains valid (Bi<0.1). Hence, for all conductors considered in this study, a one-dimensional model with 104<d<102m is employed.

The final mechanism for heat dissipation is radiation from the surface of the conductor to the surrounding medium
Qrad=ϵσP(T4T04)
(13)
To evaluate the significance of this term over the dimensions (104<d<102m) and temperatures of interest (300500K) in this study, the ratio of radiation versus convection is calculated. The expression is simplified to only incorporate the geometrical factor d and the final temperature of the section
Ratio(ϵ,d,T)=QradQconv=ϵσP(T4T04)hP(TT0)=ϵσd0.25(T4T04)0.806(TT0)1.25
(14)

Figure 4 is the plot of Eq. (14) for emissivity values of ϵ=0.5and 0.8. These results show that radiation heat transfer is significant for the temperature range and wire diameters considered and, therefore, cannot be neglected.

The energy balance of an infinitesimal element, with a length of dx, of a given conductor at steady-state condition can be written as
kx(T(x))AdT(x)dxxkx(T(x))AdT(x)dxx+dx+h(T)Pdx(T(x)T0)+QraddxQgen.Adx=0
(15)
Dividing this equation by dxand substituting Qrad from Eq. (14) and the heat generation term by Qgen=i2R(T(x))Adx=i2ρ(T(x))AdxdxA=i2ρ(T(x))A2, the final partial lumped expression for the steady-state condition is derived as
ddx(κx(T(x))AdT(x)dx)h(T)P(T(x)T0)ϵσP(T(x)4T04)+i2ρ(T(x))A=0
(16)
andκ(T(x))=k0(1+β(T(x)T0))Wm1K1h(T)=0.806(TT0)14d14Wm2K1ρ(T(x))=ρ0(1+α(T(x)T0))Ωm

kx(T(x)) and ρ(T(x)) terms are either extracted from experimental reports or evaluated by the Lewis–Nielsen model for the hypothetical Cu–C composite. Equation (16) is used here to evaluate and compare the electrical current required for different conductors to reach a Tmax=150°C. A numerical approach with constant temperature boundary conditions on contacts was used to solve the variable coefficient second-order differential Eq. (16).

Verification and Contact Calibration.

To verify the results of the numerical approach, the analytical solution to the following equation is used
kx,T0AT2x2hP(TT0)+i2ρ0(1+α(TT0))A=0h=0.806(TestT0)14d14
(17)
where Testis the estimated average temperature of the wire. The solution to Eq. (17) considering the end contacts remaining at T(l2)=T(l2)=T0 is as follows
T(x)=T0ghαg(cosh(mx)cosh(ml2)1)m=hαgkx,T0A,g=i2ρ0A
(18)

For the experimental calibration, the emissivity of a CNT wire sample was estimated with the setup shown in Fig. 5. Samples were mounted on a hot plate using a carbon tape (with known emissivity) and heated to 100°C. The temperatures were confirmed using an additional k-type thermocouple attached on the surface of the hot plate in the proximity of the samples. After achieving thermal equilibrium, the radiance incident of each pixel collected with a FLIR Systems (Wilsonville, OR) A655sc thermal camera (7.514μm spectral range) equipped with a microscope lens is measured. An emissivity value is given to the sample based on the readings from the k-type thermocouple and the temperature profile of the carbon tape. The samples are then cooled down and the accuracy of the emissivity value is confirmed at 70°C, 40°C, and room temperature consecutively upon cooling down.

To estimate the end contact interface thermal conductance, the temperature profile of the conductor was acquired. A sample was mounted between copper blocks with carbon tapes as the end contact (Fig. 5). The suspended length of the sample is 14.2mm in the presented setup. A known current is applied to the copper blocks to induce a temperature change in the suspended wire and the temperature variation is monitored with respect to time. The known properties of the material (density, dimensions, electrical conductivity, temperature coefficient of resistance, and thermal conductivity) are then used to compare the model with the experimental temperature distribution along the wire. For this calibration step, the following boundary condition was applied:
kwiredTdxx=±l2=hint(TT0)
(19)

where hint is the interface thermal conductance between the wire and pads, which depends on material, roughness, and temperature. The hint was then estimated from the measured temperature profile of the wire.

Results and Discussion

Analytical solutions for a copper conductor with a diameter of 200μm based on Eqs. (16) and (17) were considered for the verification of the numerical model. An applied current of 0.55A was chosen to eliminate the end contact effects; reach a constant temperature far from the contacts. Figure 6 shows the numerical solution to Eq. (17) compared with the analytical solution. Figure 6 also includes the numerical solution to Eq. (16) where temperature dependency of convection coefficient, temperature-dependent thermal conductivity, and radiation dissipation are considered. The analytical and numerical solutions of Eq. (17) agree well and, therefore, the model is verified. Numerical solutions of Eqs. (16) and (17) confirm that the temperature-dependency of properties plays a key role in the performance of the wire.

To calibrate the experimental measurements obtained from the model, from Eq. (16), a configuration with a suspended CNT fiber between two copper blocks (as heat sinks and electrical contacts) is used. An emissivity of 0.82±0.03 is estimated with a measurement described in the methodology section. The measured values for density, electrical resistivity, temperature coefficient of resistance, and thermal conductivity are provided in Table 1. The average diameter of the CNT conductor is 180μm. Figure 7 shows the thermal image of the CNT fiber under an applied current of 0.1A after one second. The known parameters of the CNT fiber are used to fit the coefficient of thermal conductance through contacts (hint) in Eq. (19), and the results of this analysis are shown in Fig. 8. Figure 8 also includes the temporal evolution of temperatures measured at the center and the end contacts (TmaxandTmin,respectively) to show the convergence of the temperature profile to steady-state conditions after 0.8 s.

Input properties for the selected materials in this study are shown in Table 1. Properties of the reference copper were discussed in the methodology section. The emissivity values suggested for copper in the literature vary between 0.3 and 0.8. Emissivity was estimated to be 0.52±0.02, using the method described in the verification section. Here, the emissivity value of 0.52 is used for the analysis. Properties of wet-spun CNT yarns, ultraconductive copper (UCC), intercalated carbon fibers (CF), and graphene fibers are extracted from Refs. [6] and [2431]. The chosen materials for this section have shown great potential for electronics and electrical applications with high electrical and thermal conductivities. Lastly, the results of our hypothetical Cu–C conductor, described in the methodology section, are summarized in the last four rows of Table 1. For Cu–C conductor, the density was calculated based on the rule of mixture; electrical resistivity, thermal conductivity, α, and β coefficients were determined based on the Lewis–Nielsen model; and ϵ was assumed to be the same as copper (matrix material). Equation (16) is solved with the values from Table 1 as inputs. The analyses are based on equivalent volume and equivalent weight. For both analyses, a diameter of 200μm is chosen for the reference copper conductor. For the analysis based on equivalent weights, an equivalent diameter for other conductors is calculated based on the density values provided in Table 1 
d=dCuPCuP
(20)

where PCu is the density of copper and P is the density of the conductor.

To compare the different conductors, Eq. (16) is solved for an ampacity rating of 150°C; current required for a wire conductor of certain diameter to reach 150°C. The choice of 150°C is a reasonable upper limit of service temperature of a given wire conductor in emerging applications. It should be noted that this temperature depends on the conditions in which the wire is being used or more generally, it is governed by the temperature rating of its insulating polymer coating. For the analysis based on equivalent volume, a diameter of 200μm and three different lengths of l=5mm,l=5cm,and l=50cmare chosen for all the conductors listed. The chosen lengths are representative of different scales at which the conductors could be used in applications from micro-electronics to small electrical motors. However, it is paramount that for every application, a specific study based on the geometry of the wiring be performed. Here, the goal is to gain an understanding of ampacity trends across different length scales with respect to material properties without any applications in mind. Depending on the application, the ampacity model should be adjusted to incorporate proper boundary conditions. It is noteworthy that conductor lengths longer than ∼0.5 m resulted in negligible differences in the calculated current densities. This is due to the minimization of thermal conductivity effect for long wires. Figure 9 depicts the differences in temperature gradients of all conductors across the three length scales when Joule heated to the upper limit of Tmax=150°C. This gradient difference is attributed to the difference in the thermal conductivity of the materials and is more pronounced at higher lengths.

To better understand the effect of carbon filler volume fraction on the ampacity, current was calculated for the theoretical Cu–C conductors with a diameter of 200μm, ϕ=0.02,0.05,0.1,0.2, and l=0.5,5,50cm. The results are shown in Fig. 10. While the increase in volume fraction of carbon fillers has a positive influence on ampacity of a short (5mm) composite wire, the positive effect is dampened at higher length scales. This increase in ampacity at shorter lengths can be attributed to the significance of heat dissipation term as a result of the high thermal conductivity of the carbon fillers. On the other hand, the lower electrical conductivity of the carbon fillers plays a significant role at higher length scales, where the effect of heat dissipation through thermal conductance is reduced, and Joule heating dominates the response. This figure shows that the conductor's length, electrical conductivity, and thermal conductivity control the ampacity and can be optimized for specific applications. As such, higher thermal conductivities are critical for smaller aspect ratio conductors while lower electrical resistivities govern the ampacity in longer wires.

The current and current density values required for samples from Table 1 to reach 150°C are shown in Figs. 11 and 12. Analysis at each length is separated into two columns. The column on the left at each length scale shows the results for conductors that have the same weight (thus different diameters) as the copper conductor. The column on the right is dedicated to the equivalent volume (same diameter but different weight) analyses. Results for the copper conductor with a diameter of 200μm are placed at the center as the reference.

It can be observed that for CNT fibers, UCC, and Intercalated CFs, the current (I) required to increase the temperature of the conductor is higher compared to the reference copper conductor. This is true across all length scales, and only when an equivalent weight of the conductor is considered. However, when a conductor with the same diameter as the reference copper conductor is used, only UCC and intercalated CFs prove to be competitive with copper. Moreover, UCC and intercalated CFs showed higher current carrying capacities (Fig. 12) than copper at lower length scales. The higher current density can be attributed to their higher thermal conductivities. A more thermally conductive material is more effective in dissipating the generated heat to the surrounding and end contacts. It is understandable that this desirable phenomenon is more prominent at shorter length wires. It is noteworthy that at l=0.5m, copper is only outperformed by UCC (for current density) due to both the higher electrical and thermal conductivities of the UCC. Finally, the hypothetical Cu–C composite conductor discussed in the methodology section shows comparable results with a copper conductor in all cases. This is due to the competing balance between thermal conductivity gains and the electrical conductivity losses resulting from the Lewis–Nielsen model.

Conclusions

Analytical treatment of any given conductor requires simultaneous consideration of its electrical and thermal properties. Here, a simple numerical model based on joule heating is used to compare the performance (ampacity rating at 150°C) of several advanced conductors with copper. Conductors chosen for this investigation include carbon-based and copper–matrix composites based on the interest devoted to them in the past decade. Based on the case studies, the emergence of ultraconductive copper materials may result in improved current carrying performance or equivalently a reduction in the weight of electrical components. Similarly, intercalated carbon-based conductors may result in ampacity rating improvements compared with copper. However, their competitiveness is limited to micro-electronic applications, where shorter conductors are more prevalent. Additionally, theoretical analysis based on Lewis–Nielsen model shows that the thermal conductivity improvement due to carbon filler addition in a copper matrix is beneficial for conductors at lower length scales.

Acknowledgment

The authors would like to thank Professor Cullinan and his group for their help in thermal imaging.

Funding Data

  • Office of Naval Research (Grant No. N00014-18-1-2441; Funder ID: 10.13039/100000006).

Nomenclature

A =

cross-sectional area of the conductor

Bi =

Biot number

d =

diameter of the conductor

f =

bulk density

g =

acceleration of gravity

Gr =

Grashof number

h =

convection coefficient

i =

electrical current

kair=

thermal conductivity of air

kr =

radial thermal conductivity of the conductor

kx =

longitudinal thermal conductivity of the conductor

l =

length of the conductor

Nu =

Nusselt number

P =

density of the conductor

P =

perimeter of the conductor

Pr=

Prandtl number

R =

resistance of the conductor

T =

temperature

Tf =

film temperature

T0 =

temperature of surrounding / initial temperature of the conductor

x =

length variable

X =

dimensionless length variable

α =

temperature coefficient of resistivity

β =

temperature coefficient for thermal conductivity

ϵ =

emissivity

θ =

dimensionless temperature

λ =

thermal or electrical conductivity ratio

ν =

kinematic viscosity

ρ =

resistivity of the conductor

σ =

Boltzmann constant

ϕ =

volume fraction of the filler in the composite

ϕ =

maximum packing volume fraction of the fillers in the composite

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