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Investigation of heat transfer around microwire in air environment using 3ω method
International Journal of Thermal Sciences 64 (2013) 145e151
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Investigation of heat transfer around microwire in air environment using 3u method Z.L. Wang a, *, D.W. Tang b a b
Thermal Engineering and Power Department, China University of Petroleum, Tsingtao 266555, China Institute of Engineering Thermophysics, Chinese Academy of Science, Beijing 100080, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 April 2012 Received in revised form 29 July 2012 Accepted 3 August 2012 Available online 21 September 2012
The 3u principle is presented for the measurement of the heat transfer coefficient (h) based on natural convection model and heat conduction model. The 3u technique is used at room temperature to measure h over the surfaces of microwires of the diameters 10e100 mm at horizontal and vertical orientations. The fitted results show that the heat loss from the microscale platinum wire to the air is dominated by heat conduction and the natural convection contribution is negligible. The comparison of the measured third harmonics for horizontal and vertical wires justifies that the orientation effect is negligible at microscales. The measured value of h is nearly two orders larger than that at macro scale and of the similar order to those from other literatures. Based on the 3u principle, an explicit expression with a heat conduction shape factor is introduced and can predict the heat transfer coefficient reasonably in the validated range of frequency. Both the experimental results and the theoretical analysis conclude that the scale effect of heat transfer may be contributed to two factors: the effect of buoyancy, the driving force at microscales may be negligible; the heat loss is enhanced mainly by high ratio of surface to volume at microscales. It also shows the validation of the 3u principle for thin wire if the heat loss to surrounding gas is dominated by heat conduction at microscale. Ó 2012 Elsevier Masson SAS. All rights reserved.
Keywords: 3u method Natural convection Heat conduction Driving force Microwire
1. Introduction With the fast development of microstructures and integrated circuits, heat loss to air through natural convection brings an important effect on microstructures. The effect of scaling below 1 mm on the natural convective coefficient is still a topic of debate. The complexity lies in the interplay of the very thin boundary layers and the relative change of importance of driving forces at microscale. Several promising theoretical studies argue that natural convection may be less important than thermal conduction for microscale devices [1e4]. Peirs et al. [1] have proposed a scaling law for natural convective coefficient and suggested h w 100 W m2 K1 for air when the scale is less than 100 mm, which is 5e10 times larger than that at macro scales, and compared shape memory actuators with electrostatic, magnetic and piezoelectric actuators based on thermal aspects and scaling effects on both convective and conductive heat transfer, which is a quite novel
* Corresponding author. E-mail address: [email protected] (Z.L. Wang). 1290-0729/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ijthermalsci.2012.08.002
approach. Jafarpur and Yovanovich [2] presented a simple but accurate approximate analytical method based on a linearization of the energy equation for the area mean Nusselt number (Nu) for free convection heat transfer from isothermal spheres for the range of Rayleigh number (Ra) 0 < Ra < l08 and all Prandtl numbers. Guo and Li [3] argued that natural convection should be less significant at the microscale because buoyancy, the driving force for natural convection becomes very small. Kim and King [4] investigate transient heat conduction between a heated microcantilever and its air environment with continuum finite element simulations, and the simulated effective heat transfer coefficients around the heater and around the leg are considerably large and on the order of 1 kW m2 K1. Up to now, only a little published experimental work has investigated heat flow from the microscale heater to its nearby air environment. Hu et al. [5] fabricated a microheater on a silicon nitride membrane and characterized the natural convection on the microheater precisely using the 3u method, with the heater oriented at different angles to the gravitational field and h reaches 30.4 W m2 K1; the temperature oscillation of the thin film at different angle has little difference; the heat loss to the surrounding air is dominated by heat conduction and the natural convection
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Nomenclature A cp d h I k Kn l P qV r r0 R R0 Rth S t
heat transfer area around the microwire, m2 specific heat capacity, J/K wire diameter, m heat transfer coefficient, W/m2 K amplitude of oscillating current, A thermal conductivity, W/m K Knudsen number, L=d length of wire, m heating power, W/m power intensity, W/m3 radial direction, m radius of wire, m electric resistance at temperature T, U electric resistance at initial temperature, U thermal resistance of the platinum wire, l/kS, K/W cross section area, m2 time, s
contribution can be negligible. Kim et al. [6] showed that the orientation effect is negligible at microscales by investigating experimentally the natural convection around microfin arrays on vertical and horizontal surfaces. Hou et al. [7] investigated the natural convection between air and microwires ranging from 39.9 mm to 350.1 mm placed horizontally and vertically, the experimental results show that the deviation between the experimental Nusselt number and the values of the classical correlations increases with decrease of the wire diameter. Lee et al. [8] performed measurement and modeling of heat flow between the cantilever and a partial vacuum environment, showing remarkably high thermal conductance between the cantilever and its partial vacuum environment, with an effective heat transfer coefficient near 2000 W m2 K1. Frequency-dependent thermal responses of the microcantilevers were studied and the reported effective heat transfer coefficients around the microcantilever were in the range 1000e3000 W m2 K1 [9]. From the limited measurements, it is not clear that these effective convection coefficients are the best possible estimates or what the underlying physical mechanism is that results in their values. These experimentally determined values for heat transfer coefficient are well in excess of what would be expected for natural convection, and it is likely that natural convection is not the dominant heat transfer mode. Therefore, it is desirable that more work be needed to understand natural convection at small scales, microscale thermal conduction between a solid and a quiescent fluid, and the relative magnitudes of these heat transfer mechanisms under various conditions. Besides the wide application in thermal characterization of thin films and thin wires [10e12], the 3u method based on Fourier law has been applied to determination of the thermal conductivity or heat capacity of liquids [13,14] and gases [15e17] with an ultra thin wire both as a heater and sensor based on different thermal conduction models. Through the use of a high vacuum and a different theoretical formulation, the 3u technique is also available to measure the properties of the heating element itself; in order to ensure a stable third harmonic signal and to eliminate the heat loss due to radiation and convection, the measured samples are usually placed in a vacuum chamber [12]. For an individual single-wall carbon nanotube on a substrate in vacuum, neglecting convection and radiation losses, introducing a thermal contact resistance term into the one-dimensional heat conduction equation, a linear relationship between the third harmonics and current still validates [18]. Wang et al. [13] and Chen et al. [14] applied the
T V3u x
temperature, K voltage component at 3u, V axial direction, m
Greek letters f phase angle, a thermal diffusivity, m2/s b coefficient of volume expansion, K1 q temperature difference, K L mean free path, m n kinematics viscosity, m2/s r density, kg/m3 s thermal diffusion characterization time, s ¼ l2/a, s u angular frequency, s1 Subscript 1 properties for platinum wire 2 properties for air a air
3u method to measure thermal conductivities of liquids using a several micrometer platinum wire. Recently, Yusibani et al. [15e17] derived a two-dimensional analytical solution for the 3u method for measurement of thermal conductivity of materials with a fine wire; the analytical solution includes the wire heat capacity and the effect of heat losses from the ends of the wire. Although many studies have been done on the characterization of the thermal conductivity and diffusivity of the wire specimen itself under different thermal boundary conditions, application of the 3u method to the measurement of heat transfer coefficient is still difficult especially when the wire specimen is immersed in gas medium. One important reason for this is that we do not obtain a straight line when we plot the in-phase 3u voltage component against the logarithm of the frequency; the other is that we do not obtain an analytical solution including the natural convection. Successful characterization of natural convection heat transfer on the microheater in air [5] and thermal conductivities of gases [15e17] and liquid [13,14] in frequency-domain makes it promising to precisely measure the heat transfer coefficient around microwire in air using the 3u method if the heat transfer process is dominated by thermal conduction. The major hypothesis of the 3u method is the description of the heat transport using the Fourier equation to obtain the value of the thermal conductance. At small dimensions non Fourier effects may come into play. Heron et al. [19] demonstrated that the 3u cannot work anymore when the mean free path is increasing up to the size of the nanosystems; great care has to be taken when using the 3u method at very low temperature; the better situation is when the sample to be measured is also the thermometer. Shenoy et al. [20] presented the 3-omega method for thermal conductivity measurement using the hyperbolic heat conduction equation (HHCE) and mathematical expressions representing the conditions when non-Fourier effects cannot be neglected are formulated; non-Fourier effects need to be taken into consideration when measuring low temperature thermal conductivity for sub micron thick dielectric films with high thermal diffusivity at short timescales or at higher frequency than wMHz. Fortunately, even if the non-Fourier effects need to be taken into consideration, the 3u method can still characterize the thermal properties by introducing a correction term [19] or using the HHCE [20]. But for most cases especially at high temperature and at relatively low frequency, the 3u method neglecting non-Fourier effects holds well [10e18,20].
Z.L. Wang, D.W. Tang / International Journal of Thermal Sciences 64 (2013) 145e151
In this paper, firstly by introducing the natural convection into the heat conduction equation, a one-dimensional analytical solution is derived for the 3u voltage components including heat capacity, thermal resistance of the wire and air natural convection effects, and the measurement principle is given; secondly, a 3u experimental system is used to measure the heat transfer coefficients for platinum wires of the diameter, 10e100 mm; finally, an analytical model is developed for the heat transfer coefficients for a microwire based on 3u principle and the scale effect is investigated.
qð0; tÞ ¼ qðl; tÞ ¼ qðx; NÞ ¼ 0
The measurement structure for heat transfer coefficient between air and microwire using 3u method is shown in Fig. 1(a). The platinum holders with diameter of 1.5 mm serve as heat sink. The whole structure is placed into a vacuum chamber. An alternating current (ac) at frequency u is passed through the wire generating heat flux oscillation at 2u frequency due to Joule heating. Because of the thermoresistance effect, the electrical resistance of the wire oscillates with the surface temperature of the sample at the same 2u frequency. The resulting voltage drop across the heating wire at 3u frequency is detected by a two-terminal measurement using a lock-in amplifier. The natural convection model and heat conduction model neglecting non-Fourier effects are separately presented. The corresponding coordinate system is shown in Fig. 1(b). 2.1. Natural convection model Assuming that the temperature in the radial direction is uniform and the heat loss to air is contributed to natural convection, the one-dimensional (1D) heat conduction equation in the longitudinal direction for a microwire is
C
vT v2 T I 2 sin2 ðutÞ wðx; tÞ ¼ k 2þ R0 ½1 þ aCR ðT T0 Þ vt lS S vx
(1)
Neglecting heat loss due to radiation, the heat loss w(x,t) to air is
wðx; tÞ ¼ hpdðT T0 Þ
C
pdh I2 R0 aCR sin2 ðutÞ vq v2 q I 2 R0 sin2 ðutÞ q ¼ k 2þ vt S lS lS vx
(3)
Assuming that the heat capacity of the holders at two ends is large enough and the boundary condition can be treated as isothermal condition, the initial and boundary conditions are
Ceramic plate Platinum holder
0
# (5)
where mn ¼ ½1 ð1Þn 2 =n2 p2 , x ¼ phd=CS. According to the relationship between the third harmonic response V3u and temperature change fluctuating at 2u
q ¼
2 V3u
(6)
aCR V1u
Then the relationship between V3u and I can be described as
V3u ðtÞ ¼ I 3 R0 RaCR Rth
N X
1 cosð3ut þ fn Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n2 p2 þ xs 2m n n¼1 1 þ cot2 f
(7)
n
Seen from Eq. (7), the third harmonics is relative to thermal conductivity, heat capacity of the platinum wire and h, and the linear relationship between V3u and I3 validates at low frequency. Therefore, the measured third harmonics in the air can be used to determine h after the thermal properties of the platinum wire are experimentally determined through the third harmonics in the vacuum. 2.2. Heat conduction model For the experimental structure used in Fig. 1, and for the case of l [ d, the temperature oscillation in the axial direction is uniform and the heat transfer by natural convection of air is negligible. The temperature oscillations in the frequency domain in the platinum wire and air are expressed as follows: In the platinum wire,
r
Platinum wire
x
wire Platinum holder
b Coordinate system
Fig. 1. Measurement cell and coordinate system.
(8)
In the air,
d2 q2 1 dq2 i2uq2 þ ¼ 0 a2 r dr dr 2
(9)
Neglecting the contact resistance between air and the platinum wire, the temperature continuity and heat flux continuity at the surface are assumed to hold, the corresponding boundary conditions at r ¼ r1
q1 jr¼r1 ¼ q2 jr¼r1 ;
holder
a Measurement cell
"
sinð2ut þ fn Þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ I R0 Rth m2 n2 p2 þ xs n¼1 n 1 þ cot2 fn 1
d2 q1 1 dq1 i2uq1 q þ ¼ V a1 r dr k1 dr 2
wire
N X
(2)
Let q(x,t) ¼ T T0
(4)
Eq. (3) can be solved by Fourier transformation. Generally, the frequency is larger than 0.1 Hz and the length is larger than 10 mm, then us > 0.5 for the platinum wire. Therefore, retaining only the first term of a series expansion will result in large deviation, the mean temperature change q responding to heating is found to be from Eq. (3) as 2
2. Experimental principles
147
k1
dq1 dq2 ¼ k a dr r¼r1 dr r¼r1
(10)
The temperature oscillation amplitude changes could be very small in the radial direction in the platinum wire due to the very low heating power by the ac signal and the large thermal conductivity of the platinum wire. Therefore, it is reasonable to substitute the temperature oscillation amplitude, q1(u,r1), on the surface of the platinum wire for the average value of the temperature oscillation amplitude across the platinum wire in the radial direction. Considering the natural boundary conditions, at low frequency, the thermal penetration length in the platinum wire is
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Z.L. Wang, D.W. Tang / International Journal of Thermal Sciences 64 (2013) 145e151
much larger than d, the solution of the temperature oscillation of the air in the frequency domain is
q2 ðu; rÞ ¼
4I 2 Rr2 =Sl ka qa dK1 ðqa rÞ þ k1 ðq1 rÞ2 K0 ðqa rÞ
(11)
I 3 R2 aCR # " ka qa r1 K1 ðqa r1 Þ k1 ðq1 r1 Þ2 4pl þ 4 K0 ðqa r1 Þ
(12)
Therefore, the heat conduction model, Eq. (12) shows the validation of the linear relationship between V3u and I3 at low frequency. 2.3. Thermal penetration depth The thermal penetration depth is important for the validation of thermal models of Eq. (12) and the selection of experimental conditions. The thermal penetration depth is usually defined as a multiple of the square root of the thermal diffusivity divided by the square root of the frequency. Here we take the penetration depth to be given by Eq. (13). This definition is similar to that used by Cahill [21] which is derived from a one-dimension model. In the one-dimensional analysis, the penetration depth must be sufficiently larger than the radius of the wire for validity of the truncated series approximations to the Bessel functions K0 and K1 especially when the analytical model is developed for the heat transfer coefficients for a microwire based on 3u principle in Section 4.3. In addition, for Eq. (9) to be valid, the penetration depth must be smaller than the radius of the measurement cell shown in Fig. 1(a).
qffiffiffiffiffiffiffiffiffiffiffiffiffi lTPD ¼ j1=qj ¼ a=i2u
(13)
To neglect end effects, an even tighter restriction Eq. (14) is suggested as a guide for the minimum length and the allowable penetration depth for neglecting end effects [17].
l=lTPD >30
4. Results and analysis 4.1. Mechanism of heat transfer around platinum microwire
The relationship between V3u and I can be described as
V3u ¼
1 Hz, the relationship between V3u and I3 is fitted by Eq. (7) and letting x ¼ 0 in vacuum, and the thermal conductivity of the platinum wires is 66.5W m1 K1.
(14)
In our experiment at room temperature, the thermal penetration depth lTPD in air ranges from 1.3 mm to 4.13 mm at frequency from 0.1 Hz to 1.0 Hz. To satisfy the restriction on the thermal penetration depth, the minimum length of the platinum wire is chosen as 35 mm. The thermal penetration depth at 1.0 Hz is much larger than the maximum diameter of the platinum wires and is much smaller than the effective diameter of the measurement cell.
The measured amplitude of the third harmonics in the 10.6 mm platinum wire at different air pressure is shown in Fig. 2 as a function of frequency. From top to bottom, the data represent air pressure ranging from vacuum, 58 kPa to one atmosphere (1 atm). After the structure is exposed to atmosphere, the heat loss through the surrounding air reduces the amplitude of the third harmonics, as shown in Fig. 2. Therefore, the sensitivity of heat loss to the third harmonics is large. At a low frequency smaller than about 10 Hz, the measured values of the third harmonics have a relatively larger difference at different air pressure. Fig. 3 shows the relationship between V3u and I3 for the 10.6 mm platinum wire. Firstly, heat loss through the surrounding air is modeled using Eq. (7), the natural convection model, as expected from heat transfer at large scales. The third harmonics generated by the combination of the platinum wire and the air. The value of h that best fits the experimental data at 1 Hz is 960.5 W m2 K1. The fitted third harmonics are given by the solid curve in Fig. 3. The value of h is substantially bigger by two orders than 5e 10 W m2 K1 that are generally encountered for natural convection in air at large scales. The fitted heat transfer coefficient seems to indicate that natural convection is enhanced at small scales. Secondly, heat loss through the surrounding air is modeled using Eq. (12), the heat conduction model. This model yields the dashed curve in Fig. 3 with ka ¼ 0.0261 W m1 K1. The good agreement between the model and the experimental data suggests that the heat loss to air from the platinum wire is actually dominated by heat conduction and that natural convection is negligible. The experimental amplitudes of 3u voltage components in atmosphere are shown as Fig. 4 for the 10.6 mm and 32.6 mm platinum wires placed horizontally and vertically, respectively. It can be seen that the measured third harmonics are almost independent of the platinum wire orientation to the gravitational field. This cannot be explained if the natural convection is the dominant heat transfer mode to air, and then another possible heat transfer mode is heat conduction. Furthermore, with the horizontal orientation, the Ra number is less than 105 when the diameter of the
8 Vacuum 58 kPa 1 atm
7 6
3. Experimental conditions
V3 ( µV)
The main components of apparatus in the 3u system include a lock-in amplifier and a well controlled constant amplitude current source based on an amplifier with minimum overshoot current. The ac voltage is supplied by the oscillator in the Lock-in amplifier. After suppressing the first harmonic voltage component using a Wheatstone bridge, V3u is determined by measuring the nonequilibrium voltage. The platinum wire is enclosed in a stainless steel cylindrical vessel with an inside diameter of 160 mm and height of 270 mm. The wire is spot-welded onto 1.5 mm diameter platinum lead terminals to reduce the thermal contact resistance at the ends. At room temperature, the resistance-temperature coefficient aCR of the platinum wire is measured to be 0.00375 K1. At frequency of
5 4 3 2 1 0
1
10
f ( Hz) Fig. 2. Experimental amplitudes of 3u voltage components of the microwire as a function of frequency.
Z.L. Wang, D.W. Tang / International Journal of Thermal Sciences 64 (2013) 145e151
149
500
18
Meas. Natural convection model Heat conduction model
15
Vertical Horizontal
450
h ( Wm-2K-1)
V3 ( µV)
12 9 6
400
350
3
d=32.6 µm
0
5
10
15
20
25
300
30
500
1000
I 3( mA3)
1500
2000
l/d
Fig. 3. The current dependence of the amplitudes of 3u voltage in atmosphere at 1 Hz compared with analytical models.
wire becomes smaller than about 100 mm and the thermal parameters of air at room temperature are involved; however, with the vertical orientation, the Ra number is about 4100 when the length of the wire is taken to be 35 mm. Therefore, the Ra number seems to lose its important role at microscale, and the classical NusselteRayleigh correlation at large scale for natural convection cannot well explain the abnormal phenomenon of no significant difference with variable orientations in Fig. 4 at microscale. The comparison of the measured third harmonics for horizontal and vertical wires shows that the orientation effect caused by the buoyancy, the driving force at microscales may be negligible at microscales. To further demonstrate the effect of the length on heat transfer characteristics around the microwires with variable orientations, the measurements of heat transfer coefficients are made on the 32.6 mm platinum wires with lengths from 16.3 mm to 65.2 mm. Fig. 5 shows the dependence of heat transfer coefficients at different length-to-diameter ratio for the 32.6 mm platinum wire placed horizontally and vertically, respectively. It is physically interesting that the measurements in Fig. 5 do not exhibit any clear dependence on the length-to-diameter ratio, so it is the same with the length. Besides, there exists no clear dependence of the heat transfer coefficients on orientation for the 32.6 mm platinum wire with different length. It is concluded that the length or the length-
Fig. 5. Experimental heat transfer coefficients as a function of lengthediameter ratio with variable orientations.
to-diameter ratio may have no direct effect on the heat transfer process and the buoyancy-induced natural convection may be neglected at microscales. In order to investigate the scale effect of heat transfer around the platinum wire, seven samples of different diameters, 10.6 mm, 20.4 mm, 32.6 mm, 40.2 mm, 57.0 mm, 76.8 mm and 95.6 mm, are measured for horizontal case. The measured heat transfer coefficients at room temperature fitted by heat conduction model are shown in Fig. 6 for different diameter platinum wires. The measured heat transfer coefficients seem to be inverse proportional to size for small dimensions (<100 mm) and constant for larger sizes. The measured results by Hou et al. [7] are also shown in Fig. 6. Although the measurement technique is quite different, it can be seen that the results from Hou et al. with DC electric current heating agree well with the values in this work. The compared result also shows the validation of the 3u method at rather low frequency for the characterization of heat transfer process around an electrical microwire immersed in air. Overall, the measured effective heat transfer coefficient in the present study is in similar order of 1 kW m2 K1 to the measurement [8,9] and estimation using the continuum finite element simulations [4]. More importantly, it shows that the 3u principle for thin wires [12] still validates if the microwire is immersed in gas environment and the heat loss to the surrounding gas is dominated by heat conduction. Microscale heat flow at the boundary between a solid and a quiescent fluid is in general not well understood. The Rayleigh
15 d=32.6 µm horizontal d=32.6 µm vertical d=10.6 µm horizontal d=10.6 µm vertical
1200
Hou[7] this work 900
9 h (Wm-2K-1)
V3 (µV)
12
6 3
600
300
0 1
10 f (Hz)
100
Fig. 4. Experimental amplitudes of 3u voltage components in atmosphere with variable orientations as a function of frequency.
0
20
40
60
80
100
d (µm) Fig. 6. Experimental heat transfer coefficients as a function of diameter.
Z.L. Wang, D.W. Tang / International Journal of Thermal Sciences 64 (2013) 145e151
number, Ra, for the microwires is in the range 106 to 105. The correspondingly low value of Gr suggests that the buoyancy force is much smaller than viscous reaction forces in the air. Thus conduction may dominate over convection for heat flow from the wire to the surrounding air. Due to the small value of Ra, we are skeptical about buoyancy-driven convection around the ultra thin wires. Generally, the time constant of the lock-in amplifier is set to be about 20 ms for a stable measurement, and the effective radius of the heated region is less than 1 mm. Even for sufficiently long time, the temperature oscillation in heated volume at a given frequency still remains unchanged such that there would be no buoyancydriven flow. It is entirely possible that the convection correlations for ultra thin wires are also dominated by thermal conduction and are not affected by air convection.
The total uncertainty in the heat transfer coefficient measurement is obtained by calculating the root mean square of the uncertainties of all the contributing measured quantities including diameter d and length l, electric current I and electric resistance R and temperature coefficient of electric resistance, aCR, which can be expressed as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 ½dhd þ ½dhl þ ½dhI þ ½dhR þ ½dhaCR
qa ¼
(15)
The uncertainties in the dimensions are estimated by comparing results from independent measurements. In the present analysis, the maximum deviations from the average values are taken as uncertainties. The diameter of the platinum wire is determined by scanning the electron micrograph (SEM). SEM measurements of the diameter of a sample of the wire suggested a departure of less than 0.5 mm from the nominal value. It is difficult to determine the length with a resolution more accurately than 0.5 mm due to uncertainty in the locations of the weld positions. The uncertainties of the heat transfer coefficient caused by the uncertainties in diameter and length are estimated to be 5.0% and 1.4%, respectively. The uncertainty in aCR is estimated by considering uncertainties in the measured electrical resistance and in the temperature reading from a thermocouple attached to the platinum holder. The relative uncertainties in the electrical resistance and the ac current are estimated to be 0.06% and 0.18% by taking into account both statistical fluctuations and instrument resolution. The uncertainty in the temperature reading is near 0.2 K. The uncertainty in aCR is estimated by perturbing the electrical resistance and the temperature values with their respective uncertainties and comparing the linear regression results of the resulting ReT curves. The total uncertainty in aCR, taken to be one-half the difference between the maximum and the minimum values obtained in this fashion, is about 3% of the average value of aCR. Accordingly, the relative uncertainty in aCR, which is relevant for the interconnect thermometry, is estimated to be 3%. The uncertainty of the measured heat transfer coefficient is estimated to be 2.5%. The total relative uncertainty of the heat transfer coefficient is obtained to be 6.0%. Therefore, the main contributors to the experimental uncertainty in the present measurements are the uncertainty in the sample dimensions and the uncertainty in the calibration coefficient of the platinum wire. The estimated uncertainties do not include the approximations made to the analytical model.
aa P 1 1 ip ln 2 ln 2u g pka l 2 d 2 2
aa 1 1 1 ip ln 2 ln 2u g pka l 2 d 2 2
Even though h can still be phenomenally used for microwires by natural convection model, Eqs. (1)e(7), the value of h can be
(17)
The effective heat transfer coefficient h is defined as
h ¼
1 AjZa j
(18)
The air thermal conductivity ka may decrease due to the Knudsen effect when the microwire characteristic size (d) close to the air molecule mean free path (L), which is about tens of nanometers at room temperature and one atmosphere. Letting h takes the following form,
h ¼
ka C =A 1 þ Kn S
(19)
where CS ¼ 1/(kaZa) is the heat conduction shape factor. For a microwire in the air, CS is given by
1 aa 1 ip 1 CS ¼ pl ln 2 ln 2u g 2 2 2 d
(20)
Finally, the explicit form of the end result of the present analysis based on 3u principle is given by
h ¼
aa g ip 1 ka 1 1 1 u ln ln 2 1 þ Kn d 4 2 4 4 d2
(21)
Fig. 7 shows the dependence of the heat transfer coefficients on the frequency calculated from Eq. (21) for the platinum wires
1200 10.6 µm 1000
800 20.4 µm
600
32.6 µm 40.2 µm
400 57 µm 76.8 µm
200 0.0
4.3. Prediction of h based on heat conduction model
(16)
Then the thermal impedance of the air around the microwire is given by
Za ¼
4.2. Uncertainty analysis
dh ¼
estimated just based on the analysis of heat conduction to surrounding air in the similar way as Hu et al. [5] has done for a 3u heater immersed in a thin film. For a microwire immersed in an infinite fluid (air), the temperature oscillation in the air in the 3u measurement has been given by Eq. (11). To simplicity, by retaining only the first term of a series expansion, the mean temperature change responding to heating is found to be from Eq. (11) as
h ( Wm-2K-1)
150
0.2
0.4
0.6
0.8
1.0
1.2
f (Hz) Fig. 7. Predicted heat transfer coefficients as a function of frequency with variable diameters.
Z.L. Wang, D.W. Tang / International Journal of Thermal Sciences 64 (2013) 145e151
of different diameters. In Fig. 7, the fitted values of the heat transfer coefficient (h) are 960.5, 558, 410, 342, 244, 215 and 156 W m2 K1 for the samples 10.6 mm, 20.4 mm, 32.6 mm, 40.2 mm, 57 mm, 76.8 mm and 95.6 mm respectively. As the explicit form of the heat transfer coefficient based on 3u principle is a simplified form from the heat conduction model given by Eq. (12), the validated range of the explicit form is between 0.1 and 0.3 Hz based on the measured results if the relative error is limited to 10%, then the expression of the heat transfer coefficient based on 3u principle takes the following form at the optimized frequency:
h ¼
1=2 aa ka 1 1 2 aa 0:292ln þ 0:958 ln 1 þ Kn d 16 d2 d2
(22)
Eq. (22) shows that h is considered to be inversely proportional to size d at microscale while at macro scale, h is considered constant and independent of the scaling effects. It is no surprise that h is larger than that for macro scales. This is because heat conduction is relatively more significant for microscale wires, owing to the increased ratio of surface to volume or the increased curvature, indicating less significance of the volume force, buoyancy at microscale. From the perspective of geometry of the thin wire, both the measurements shown in Figs. 5 and 6 and theoretical analysis by Eq. (22) show that the heat transfer around microscale wire seems to have clear dependence on the ratio of surface to volume or curvature. 5. Conclusions The natural convection model and heat conduction model are presented for the heat transfer coefficient (h) measurement based on 3u method. The measured results of the microscale platinum wires with the diameters from 10.6 mm to 95.6 mm show that the heat loss to air from the platinum wire is actually dominated by heat conduction and that natural convection is negligible. The measured effective heat transfer coefficients show that the 3u principle for thin wires still validates if the microwire is immersed in gas environment and the heat loss to the surrounding gas is dominated by heat conduction. In addition, an explicit expression for h is derived through analysis based on the 3u principle. The fact that h is larger than that for macro scales is because heat conduction is relatively more significant for microscale wires, owing to the increased ratio of surface to volume. The heat transfer process around the microwires in the air does appear to show the microscale effect of heat conduction. The 3u measurements in this study are performed at room temperature, and the heat transfer characteristics in the high- and low-temperature regimes warrants investigation. In such temperature regions, thermal radiation may play an important role. Further, the end effects of the microwires on the thermal boundary condition and the heating frequency also remain an area for further study. It is suggested that microscale natural convection is an area including measurements, simulations, and optical diagnostics of microscale air convection.
151
Acknowledgments We acknowledge funding supports from National Natural Science Foundation of China (No. 51176205), Promotive Research Fund for Excellent Young and Middle-aged Scientists (No. BS2009CL017) of Shandong Province. References [1] J. Peirs, D. Reynaerts, H.V. Brussel, Scale effects and thermal considerations for micro-actuators, in: IEEE International Conference on Robotics and Automation, Leuven, Belgium, 1998. [2] K. Jafarpur, M.M. Yovanovich, Laminar free convective heat transfer from isothermal spheres: a new analytical method, International Journal of Heat and Mass Transfer 35 (1992) 2195e2201. [3] Z.Y. Guo, Z.X. Li, Size effect on microscale single-phase flow and heat transfer, International Journal of Heat and Mass Transfer 46 (2003) 149e159. [4] K.J. Kim, W.P. King, Thermal conduction between a heated microcantilever and a surrounding air environment, Applied Thermal Engineering 29 (2009) 1631e1641. [5] X.J. Hu, A. Jain, K.E. Goodson, Investigation of the natural convection boundary condition in microfabricated structures, International Journal of Thermal Sciences 47 (2008) 820e824. [6] J.S. Kim, B.K. Park, J.S. Lee, Natural convection heat transfer around micro fin arrays, Experimental Heat Transfer 21 (2008) 55e72. [7] Y.L. Hou, X.C. Wang, C.W. Zhang, Z.G. Liu, Experimental study on heat transfer of natural convection between microwire and air, Journal of Engineering Thermophysics 28 (2007) 460e462. [8] J. Lee, T.L. Wright, M.R. Abel, E.O. Sunden, A. Marchenkov, S. Graham, W.P. King, Thermal conduction from microcantilever heaters in partial vacuum, Journal of Applied Physics 101 (2007) 014906. [9] K. Park, J. Lee, Z.M. Zhang, W.P. King, Frequency-dependant electrical and thermal response of heated atomic force microscope cantilevers, Journal of Microelectromechanical Systems 16 (2007) 213e222. [10] D.G. Cahill, W.K. Ford, K.E. Goodson, G.D. Mahan, A. Majumdar, H.J. Maris, R. Merlin, S.R. Phillpot, Nanoscale thermal transport, Journal of Applied Physics 93 (2003) 793e818. [11] D.W. Tang, Z.L. Wang, Characterization of Thermophysical Characteristics of Micro and Nano Materials, Science Press, Beijing, 2010. [12] L. Lu, W. Yi, D.L. Zhang, 3u method for specific heat and thermal conductivity measurements, Review of Scientific Instruments 72 (2001) 2996e3003. [13] Z.L. Wang, D.W. Tang, N. Araki, Thermal properties measurements of nanofluids by 3u method and prediction analysis, International Journal of Thermophysics 28 (2007) 1255e1268. [14] F. Chen, J. Shulman, C.W. Chu, Thermal conductivity measurement under hydro-static pressure using the 3u method, Review of Scientific Instruments 75 (2004) 4578e4584. [15] E. Yusibani, P.L. Woodfield, M. Fujii, K. Shinzato, X. Zhang, Y. Takata, Application of the three-omega method to measurement of thermal conductivity and thermal diffusivity of hydrogen gas, International Journal of Thermophysics 30 (2009) 397e415. [16] E. Yusibani, P.L. Woodfield, S. Moroe, K. Shinzato, M. Kohno, Y. Takata, M. Fujii, A procedure for application of the three-omega method to measurement of gas thermal conductivity, Journal of Thermal Science and Technology 4 (2009) 146e158. [17] E. Yusibani, P.L. Woodfield, M. Kohno, K. Shinzato, Y. Takata, M. Fujii, End effects in the three-omega method to measure gas thermal conductivity, International Journal of Thermophysics 30 (2009) 833e850. [18] Z.L. Wang, D.W. Tang, W.G. Zhang, Length-dependent thermal conductivity of an individual single-wall carbon nanotube, Applied Physics Letters 91 (2007) 119e123. [19] J.S. Heron, G.M. Souche, F.R. Ong, P. Gandit, T. Fournier, O. Bourgeois, Temperature modulation measurements of the thermal properties of nanosystems at low temperatures, Journal of Low Temperature Physics 154 (2009) 150e160. [20] S. Shenoy, E.V. Barrera, Y. Bayazitoglu, Non-Fourier 3-omega method for thermal conductivity measurement, Nanoscale and Microscale Thermophysical Engineering 14 (2010) 123e136. [21] D.G. Cahill, Thermal conductivity measurement from 30 to 750 K: the 3u method, Review of Scientific Instruments 61 (1990) 802e808.
Contents lists available at SciVerse ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Investigation of heat transfer around microwire in air environment using 3u method Z.L. Wang a, *, D.W. Tang b a b
Thermal Engineering and Power Department, China University of Petroleum, Tsingtao 266555, China Institute of Engineering Thermophysics, Chinese Academy of Science, Beijing 100080, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 April 2012 Received in revised form 29 July 2012 Accepted 3 August 2012 Available online 21 September 2012
The 3u principle is presented for the measurement of the heat transfer coefficient (h) based on natural convection model and heat conduction model. The 3u technique is used at room temperature to measure h over the surfaces of microwires of the diameters 10e100 mm at horizontal and vertical orientations. The fitted results show that the heat loss from the microscale platinum wire to the air is dominated by heat conduction and the natural convection contribution is negligible. The comparison of the measured third harmonics for horizontal and vertical wires justifies that the orientation effect is negligible at microscales. The measured value of h is nearly two orders larger than that at macro scale and of the similar order to those from other literatures. Based on the 3u principle, an explicit expression with a heat conduction shape factor is introduced and can predict the heat transfer coefficient reasonably in the validated range of frequency. Both the experimental results and the theoretical analysis conclude that the scale effect of heat transfer may be contributed to two factors: the effect of buoyancy, the driving force at microscales may be negligible; the heat loss is enhanced mainly by high ratio of surface to volume at microscales. It also shows the validation of the 3u principle for thin wire if the heat loss to surrounding gas is dominated by heat conduction at microscale. Ó 2012 Elsevier Masson SAS. All rights reserved.
Keywords: 3u method Natural convection Heat conduction Driving force Microwire
1. Introduction With the fast development of microstructures and integrated circuits, heat loss to air through natural convection brings an important effect on microstructures. The effect of scaling below 1 mm on the natural convective coefficient is still a topic of debate. The complexity lies in the interplay of the very thin boundary layers and the relative change of importance of driving forces at microscale. Several promising theoretical studies argue that natural convection may be less important than thermal conduction for microscale devices [1e4]. Peirs et al. [1] have proposed a scaling law for natural convective coefficient and suggested h w 100 W m2 K1 for air when the scale is less than 100 mm, which is 5e10 times larger than that at macro scales, and compared shape memory actuators with electrostatic, magnetic and piezoelectric actuators based on thermal aspects and scaling effects on both convective and conductive heat transfer, which is a quite novel
* Corresponding author. E-mail address: [email protected] (Z.L. Wang). 1290-0729/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ijthermalsci.2012.08.002
approach. Jafarpur and Yovanovich [2] presented a simple but accurate approximate analytical method based on a linearization of the energy equation for the area mean Nusselt number (Nu) for free convection heat transfer from isothermal spheres for the range of Rayleigh number (Ra) 0 < Ra < l08 and all Prandtl numbers. Guo and Li [3] argued that natural convection should be less significant at the microscale because buoyancy, the driving force for natural convection becomes very small. Kim and King [4] investigate transient heat conduction between a heated microcantilever and its air environment with continuum finite element simulations, and the simulated effective heat transfer coefficients around the heater and around the leg are considerably large and on the order of 1 kW m2 K1. Up to now, only a little published experimental work has investigated heat flow from the microscale heater to its nearby air environment. Hu et al. [5] fabricated a microheater on a silicon nitride membrane and characterized the natural convection on the microheater precisely using the 3u method, with the heater oriented at different angles to the gravitational field and h reaches 30.4 W m2 K1; the temperature oscillation of the thin film at different angle has little difference; the heat loss to the surrounding air is dominated by heat conduction and the natural convection
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Nomenclature A cp d h I k Kn l P qV r r0 R R0 Rth S t
heat transfer area around the microwire, m2 specific heat capacity, J/K wire diameter, m heat transfer coefficient, W/m2 K amplitude of oscillating current, A thermal conductivity, W/m K Knudsen number, L=d length of wire, m heating power, W/m power intensity, W/m3 radial direction, m radius of wire, m electric resistance at temperature T, U electric resistance at initial temperature, U thermal resistance of the platinum wire, l/kS, K/W cross section area, m2 time, s
contribution can be negligible. Kim et al. [6] showed that the orientation effect is negligible at microscales by investigating experimentally the natural convection around microfin arrays on vertical and horizontal surfaces. Hou et al. [7] investigated the natural convection between air and microwires ranging from 39.9 mm to 350.1 mm placed horizontally and vertically, the experimental results show that the deviation between the experimental Nusselt number and the values of the classical correlations increases with decrease of the wire diameter. Lee et al. [8] performed measurement and modeling of heat flow between the cantilever and a partial vacuum environment, showing remarkably high thermal conductance between the cantilever and its partial vacuum environment, with an effective heat transfer coefficient near 2000 W m2 K1. Frequency-dependent thermal responses of the microcantilevers were studied and the reported effective heat transfer coefficients around the microcantilever were in the range 1000e3000 W m2 K1 [9]. From the limited measurements, it is not clear that these effective convection coefficients are the best possible estimates or what the underlying physical mechanism is that results in their values. These experimentally determined values for heat transfer coefficient are well in excess of what would be expected for natural convection, and it is likely that natural convection is not the dominant heat transfer mode. Therefore, it is desirable that more work be needed to understand natural convection at small scales, microscale thermal conduction between a solid and a quiescent fluid, and the relative magnitudes of these heat transfer mechanisms under various conditions. Besides the wide application in thermal characterization of thin films and thin wires [10e12], the 3u method based on Fourier law has been applied to determination of the thermal conductivity or heat capacity of liquids [13,14] and gases [15e17] with an ultra thin wire both as a heater and sensor based on different thermal conduction models. Through the use of a high vacuum and a different theoretical formulation, the 3u technique is also available to measure the properties of the heating element itself; in order to ensure a stable third harmonic signal and to eliminate the heat loss due to radiation and convection, the measured samples are usually placed in a vacuum chamber [12]. For an individual single-wall carbon nanotube on a substrate in vacuum, neglecting convection and radiation losses, introducing a thermal contact resistance term into the one-dimensional heat conduction equation, a linear relationship between the third harmonics and current still validates [18]. Wang et al. [13] and Chen et al. [14] applied the
T V3u x
temperature, K voltage component at 3u, V axial direction, m
Greek letters f phase angle, a thermal diffusivity, m2/s b coefficient of volume expansion, K1 q temperature difference, K L mean free path, m n kinematics viscosity, m2/s r density, kg/m3 s thermal diffusion characterization time, s ¼ l2/a, s u angular frequency, s1 Subscript 1 properties for platinum wire 2 properties for air a air
3u method to measure thermal conductivities of liquids using a several micrometer platinum wire. Recently, Yusibani et al. [15e17] derived a two-dimensional analytical solution for the 3u method for measurement of thermal conductivity of materials with a fine wire; the analytical solution includes the wire heat capacity and the effect of heat losses from the ends of the wire. Although many studies have been done on the characterization of the thermal conductivity and diffusivity of the wire specimen itself under different thermal boundary conditions, application of the 3u method to the measurement of heat transfer coefficient is still difficult especially when the wire specimen is immersed in gas medium. One important reason for this is that we do not obtain a straight line when we plot the in-phase 3u voltage component against the logarithm of the frequency; the other is that we do not obtain an analytical solution including the natural convection. Successful characterization of natural convection heat transfer on the microheater in air [5] and thermal conductivities of gases [15e17] and liquid [13,14] in frequency-domain makes it promising to precisely measure the heat transfer coefficient around microwire in air using the 3u method if the heat transfer process is dominated by thermal conduction. The major hypothesis of the 3u method is the description of the heat transport using the Fourier equation to obtain the value of the thermal conductance. At small dimensions non Fourier effects may come into play. Heron et al. [19] demonstrated that the 3u cannot work anymore when the mean free path is increasing up to the size of the nanosystems; great care has to be taken when using the 3u method at very low temperature; the better situation is when the sample to be measured is also the thermometer. Shenoy et al. [20] presented the 3-omega method for thermal conductivity measurement using the hyperbolic heat conduction equation (HHCE) and mathematical expressions representing the conditions when non-Fourier effects cannot be neglected are formulated; non-Fourier effects need to be taken into consideration when measuring low temperature thermal conductivity for sub micron thick dielectric films with high thermal diffusivity at short timescales or at higher frequency than wMHz. Fortunately, even if the non-Fourier effects need to be taken into consideration, the 3u method can still characterize the thermal properties by introducing a correction term [19] or using the HHCE [20]. But for most cases especially at high temperature and at relatively low frequency, the 3u method neglecting non-Fourier effects holds well [10e18,20].
Z.L. Wang, D.W. Tang / International Journal of Thermal Sciences 64 (2013) 145e151
In this paper, firstly by introducing the natural convection into the heat conduction equation, a one-dimensional analytical solution is derived for the 3u voltage components including heat capacity, thermal resistance of the wire and air natural convection effects, and the measurement principle is given; secondly, a 3u experimental system is used to measure the heat transfer coefficients for platinum wires of the diameter, 10e100 mm; finally, an analytical model is developed for the heat transfer coefficients for a microwire based on 3u principle and the scale effect is investigated.
qð0; tÞ ¼ qðl; tÞ ¼ qðx; NÞ ¼ 0
The measurement structure for heat transfer coefficient between air and microwire using 3u method is shown in Fig. 1(a). The platinum holders with diameter of 1.5 mm serve as heat sink. The whole structure is placed into a vacuum chamber. An alternating current (ac) at frequency u is passed through the wire generating heat flux oscillation at 2u frequency due to Joule heating. Because of the thermoresistance effect, the electrical resistance of the wire oscillates with the surface temperature of the sample at the same 2u frequency. The resulting voltage drop across the heating wire at 3u frequency is detected by a two-terminal measurement using a lock-in amplifier. The natural convection model and heat conduction model neglecting non-Fourier effects are separately presented. The corresponding coordinate system is shown in Fig. 1(b). 2.1. Natural convection model Assuming that the temperature in the radial direction is uniform and the heat loss to air is contributed to natural convection, the one-dimensional (1D) heat conduction equation in the longitudinal direction for a microwire is
C
vT v2 T I 2 sin2 ðutÞ wðx; tÞ ¼ k 2þ R0 ½1 þ aCR ðT T0 Þ vt lS S vx
(1)
Neglecting heat loss due to radiation, the heat loss w(x,t) to air is
wðx; tÞ ¼ hpdðT T0 Þ
C
pdh I2 R0 aCR sin2 ðutÞ vq v2 q I 2 R0 sin2 ðutÞ q ¼ k 2þ vt S lS lS vx
(3)
Assuming that the heat capacity of the holders at two ends is large enough and the boundary condition can be treated as isothermal condition, the initial and boundary conditions are
Ceramic plate Platinum holder
0
# (5)
where mn ¼ ½1 ð1Þn 2 =n2 p2 , x ¼ phd=CS. According to the relationship between the third harmonic response V3u and temperature change fluctuating at 2u
q ¼
2 V3u
(6)
aCR V1u
Then the relationship between V3u and I can be described as
V3u ðtÞ ¼ I 3 R0 RaCR Rth
N X
1 cosð3ut þ fn Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n2 p2 þ xs 2m n n¼1 1 þ cot2 f
(7)
n
Seen from Eq. (7), the third harmonics is relative to thermal conductivity, heat capacity of the platinum wire and h, and the linear relationship between V3u and I3 validates at low frequency. Therefore, the measured third harmonics in the air can be used to determine h after the thermal properties of the platinum wire are experimentally determined through the third harmonics in the vacuum. 2.2. Heat conduction model For the experimental structure used in Fig. 1, and for the case of l [ d, the temperature oscillation in the axial direction is uniform and the heat transfer by natural convection of air is negligible. The temperature oscillations in the frequency domain in the platinum wire and air are expressed as follows: In the platinum wire,
r
Platinum wire
x
wire Platinum holder
b Coordinate system
Fig. 1. Measurement cell and coordinate system.
(8)
In the air,
d2 q2 1 dq2 i2uq2 þ ¼ 0 a2 r dr dr 2
(9)
Neglecting the contact resistance between air and the platinum wire, the temperature continuity and heat flux continuity at the surface are assumed to hold, the corresponding boundary conditions at r ¼ r1
q1 jr¼r1 ¼ q2 jr¼r1 ;
holder
a Measurement cell
"
sinð2ut þ fn Þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ I R0 Rth m2 n2 p2 þ xs n¼1 n 1 þ cot2 fn 1
d2 q1 1 dq1 i2uq1 q þ ¼ V a1 r dr k1 dr 2
wire
N X
(2)
Let q(x,t) ¼ T T0
(4)
Eq. (3) can be solved by Fourier transformation. Generally, the frequency is larger than 0.1 Hz and the length is larger than 10 mm, then us > 0.5 for the platinum wire. Therefore, retaining only the first term of a series expansion will result in large deviation, the mean temperature change q responding to heating is found to be from Eq. (3) as 2
2. Experimental principles
147
k1
dq1 dq2 ¼ k a dr r¼r1 dr r¼r1
(10)
The temperature oscillation amplitude changes could be very small in the radial direction in the platinum wire due to the very low heating power by the ac signal and the large thermal conductivity of the platinum wire. Therefore, it is reasonable to substitute the temperature oscillation amplitude, q1(u,r1), on the surface of the platinum wire for the average value of the temperature oscillation amplitude across the platinum wire in the radial direction. Considering the natural boundary conditions, at low frequency, the thermal penetration length in the platinum wire is
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Z.L. Wang, D.W. Tang / International Journal of Thermal Sciences 64 (2013) 145e151
much larger than d, the solution of the temperature oscillation of the air in the frequency domain is
q2 ðu; rÞ ¼
4I 2 Rr2 =Sl ka qa dK1 ðqa rÞ þ k1 ðq1 rÞ2 K0 ðqa rÞ
(11)
I 3 R2 aCR # " ka qa r1 K1 ðqa r1 Þ k1 ðq1 r1 Þ2 4pl þ 4 K0 ðqa r1 Þ
(12)
Therefore, the heat conduction model, Eq. (12) shows the validation of the linear relationship between V3u and I3 at low frequency. 2.3. Thermal penetration depth The thermal penetration depth is important for the validation of thermal models of Eq. (12) and the selection of experimental conditions. The thermal penetration depth is usually defined as a multiple of the square root of the thermal diffusivity divided by the square root of the frequency. Here we take the penetration depth to be given by Eq. (13). This definition is similar to that used by Cahill [21] which is derived from a one-dimension model. In the one-dimensional analysis, the penetration depth must be sufficiently larger than the radius of the wire for validity of the truncated series approximations to the Bessel functions K0 and K1 especially when the analytical model is developed for the heat transfer coefficients for a microwire based on 3u principle in Section 4.3. In addition, for Eq. (9) to be valid, the penetration depth must be smaller than the radius of the measurement cell shown in Fig. 1(a).
qffiffiffiffiffiffiffiffiffiffiffiffiffi lTPD ¼ j1=qj ¼ a=i2u
(13)
To neglect end effects, an even tighter restriction Eq. (14) is suggested as a guide for the minimum length and the allowable penetration depth for neglecting end effects [17].
l=lTPD >30
4. Results and analysis 4.1. Mechanism of heat transfer around platinum microwire
The relationship between V3u and I can be described as
V3u ¼
1 Hz, the relationship between V3u and I3 is fitted by Eq. (7) and letting x ¼ 0 in vacuum, and the thermal conductivity of the platinum wires is 66.5W m1 K1.
(14)
In our experiment at room temperature, the thermal penetration depth lTPD in air ranges from 1.3 mm to 4.13 mm at frequency from 0.1 Hz to 1.0 Hz. To satisfy the restriction on the thermal penetration depth, the minimum length of the platinum wire is chosen as 35 mm. The thermal penetration depth at 1.0 Hz is much larger than the maximum diameter of the platinum wires and is much smaller than the effective diameter of the measurement cell.
The measured amplitude of the third harmonics in the 10.6 mm platinum wire at different air pressure is shown in Fig. 2 as a function of frequency. From top to bottom, the data represent air pressure ranging from vacuum, 58 kPa to one atmosphere (1 atm). After the structure is exposed to atmosphere, the heat loss through the surrounding air reduces the amplitude of the third harmonics, as shown in Fig. 2. Therefore, the sensitivity of heat loss to the third harmonics is large. At a low frequency smaller than about 10 Hz, the measured values of the third harmonics have a relatively larger difference at different air pressure. Fig. 3 shows the relationship between V3u and I3 for the 10.6 mm platinum wire. Firstly, heat loss through the surrounding air is modeled using Eq. (7), the natural convection model, as expected from heat transfer at large scales. The third harmonics generated by the combination of the platinum wire and the air. The value of h that best fits the experimental data at 1 Hz is 960.5 W m2 K1. The fitted third harmonics are given by the solid curve in Fig. 3. The value of h is substantially bigger by two orders than 5e 10 W m2 K1 that are generally encountered for natural convection in air at large scales. The fitted heat transfer coefficient seems to indicate that natural convection is enhanced at small scales. Secondly, heat loss through the surrounding air is modeled using Eq. (12), the heat conduction model. This model yields the dashed curve in Fig. 3 with ka ¼ 0.0261 W m1 K1. The good agreement between the model and the experimental data suggests that the heat loss to air from the platinum wire is actually dominated by heat conduction and that natural convection is negligible. The experimental amplitudes of 3u voltage components in atmosphere are shown as Fig. 4 for the 10.6 mm and 32.6 mm platinum wires placed horizontally and vertically, respectively. It can be seen that the measured third harmonics are almost independent of the platinum wire orientation to the gravitational field. This cannot be explained if the natural convection is the dominant heat transfer mode to air, and then another possible heat transfer mode is heat conduction. Furthermore, with the horizontal orientation, the Ra number is less than 105 when the diameter of the
8 Vacuum 58 kPa 1 atm
7 6
3. Experimental conditions
V3 ( µV)
The main components of apparatus in the 3u system include a lock-in amplifier and a well controlled constant amplitude current source based on an amplifier with minimum overshoot current. The ac voltage is supplied by the oscillator in the Lock-in amplifier. After suppressing the first harmonic voltage component using a Wheatstone bridge, V3u is determined by measuring the nonequilibrium voltage. The platinum wire is enclosed in a stainless steel cylindrical vessel with an inside diameter of 160 mm and height of 270 mm. The wire is spot-welded onto 1.5 mm diameter platinum lead terminals to reduce the thermal contact resistance at the ends. At room temperature, the resistance-temperature coefficient aCR of the platinum wire is measured to be 0.00375 K1. At frequency of
5 4 3 2 1 0
1
10
f ( Hz) Fig. 2. Experimental amplitudes of 3u voltage components of the microwire as a function of frequency.
Z.L. Wang, D.W. Tang / International Journal of Thermal Sciences 64 (2013) 145e151
149
500
18
Meas. Natural convection model Heat conduction model
15
Vertical Horizontal
450
h ( Wm-2K-1)
V3 ( µV)
12 9 6
400
350
3
d=32.6 µm
0
5
10
15
20
25
300
30
500
1000
I 3( mA3)
1500
2000
l/d
Fig. 3. The current dependence of the amplitudes of 3u voltage in atmosphere at 1 Hz compared with analytical models.
wire becomes smaller than about 100 mm and the thermal parameters of air at room temperature are involved; however, with the vertical orientation, the Ra number is about 4100 when the length of the wire is taken to be 35 mm. Therefore, the Ra number seems to lose its important role at microscale, and the classical NusselteRayleigh correlation at large scale for natural convection cannot well explain the abnormal phenomenon of no significant difference with variable orientations in Fig. 4 at microscale. The comparison of the measured third harmonics for horizontal and vertical wires shows that the orientation effect caused by the buoyancy, the driving force at microscales may be negligible at microscales. To further demonstrate the effect of the length on heat transfer characteristics around the microwires with variable orientations, the measurements of heat transfer coefficients are made on the 32.6 mm platinum wires with lengths from 16.3 mm to 65.2 mm. Fig. 5 shows the dependence of heat transfer coefficients at different length-to-diameter ratio for the 32.6 mm platinum wire placed horizontally and vertically, respectively. It is physically interesting that the measurements in Fig. 5 do not exhibit any clear dependence on the length-to-diameter ratio, so it is the same with the length. Besides, there exists no clear dependence of the heat transfer coefficients on orientation for the 32.6 mm platinum wire with different length. It is concluded that the length or the length-
Fig. 5. Experimental heat transfer coefficients as a function of lengthediameter ratio with variable orientations.
to-diameter ratio may have no direct effect on the heat transfer process and the buoyancy-induced natural convection may be neglected at microscales. In order to investigate the scale effect of heat transfer around the platinum wire, seven samples of different diameters, 10.6 mm, 20.4 mm, 32.6 mm, 40.2 mm, 57.0 mm, 76.8 mm and 95.6 mm, are measured for horizontal case. The measured heat transfer coefficients at room temperature fitted by heat conduction model are shown in Fig. 6 for different diameter platinum wires. The measured heat transfer coefficients seem to be inverse proportional to size for small dimensions (<100 mm) and constant for larger sizes. The measured results by Hou et al. [7] are also shown in Fig. 6. Although the measurement technique is quite different, it can be seen that the results from Hou et al. with DC electric current heating agree well with the values in this work. The compared result also shows the validation of the 3u method at rather low frequency for the characterization of heat transfer process around an electrical microwire immersed in air. Overall, the measured effective heat transfer coefficient in the present study is in similar order of 1 kW m2 K1 to the measurement [8,9] and estimation using the continuum finite element simulations [4]. More importantly, it shows that the 3u principle for thin wires [12] still validates if the microwire is immersed in gas environment and the heat loss to the surrounding gas is dominated by heat conduction. Microscale heat flow at the boundary between a solid and a quiescent fluid is in general not well understood. The Rayleigh
15 d=32.6 µm horizontal d=32.6 µm vertical d=10.6 µm horizontal d=10.6 µm vertical
1200
Hou[7] this work 900
9 h (Wm-2K-1)
V3 (µV)
12
6 3
600
300
0 1
10 f (Hz)
100
Fig. 4. Experimental amplitudes of 3u voltage components in atmosphere with variable orientations as a function of frequency.
0
20
40
60
80
100
d (µm) Fig. 6. Experimental heat transfer coefficients as a function of diameter.
Z.L. Wang, D.W. Tang / International Journal of Thermal Sciences 64 (2013) 145e151
number, Ra, for the microwires is in the range 106 to 105. The correspondingly low value of Gr suggests that the buoyancy force is much smaller than viscous reaction forces in the air. Thus conduction may dominate over convection for heat flow from the wire to the surrounding air. Due to the small value of Ra, we are skeptical about buoyancy-driven convection around the ultra thin wires. Generally, the time constant of the lock-in amplifier is set to be about 20 ms for a stable measurement, and the effective radius of the heated region is less than 1 mm. Even for sufficiently long time, the temperature oscillation in heated volume at a given frequency still remains unchanged such that there would be no buoyancydriven flow. It is entirely possible that the convection correlations for ultra thin wires are also dominated by thermal conduction and are not affected by air convection.
The total uncertainty in the heat transfer coefficient measurement is obtained by calculating the root mean square of the uncertainties of all the contributing measured quantities including diameter d and length l, electric current I and electric resistance R and temperature coefficient of electric resistance, aCR, which can be expressed as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 ½dhd þ ½dhl þ ½dhI þ ½dhR þ ½dhaCR
qa ¼
(15)
The uncertainties in the dimensions are estimated by comparing results from independent measurements. In the present analysis, the maximum deviations from the average values are taken as uncertainties. The diameter of the platinum wire is determined by scanning the electron micrograph (SEM). SEM measurements of the diameter of a sample of the wire suggested a departure of less than 0.5 mm from the nominal value. It is difficult to determine the length with a resolution more accurately than 0.5 mm due to uncertainty in the locations of the weld positions. The uncertainties of the heat transfer coefficient caused by the uncertainties in diameter and length are estimated to be 5.0% and 1.4%, respectively. The uncertainty in aCR is estimated by considering uncertainties in the measured electrical resistance and in the temperature reading from a thermocouple attached to the platinum holder. The relative uncertainties in the electrical resistance and the ac current are estimated to be 0.06% and 0.18% by taking into account both statistical fluctuations and instrument resolution. The uncertainty in the temperature reading is near 0.2 K. The uncertainty in aCR is estimated by perturbing the electrical resistance and the temperature values with their respective uncertainties and comparing the linear regression results of the resulting ReT curves. The total uncertainty in aCR, taken to be one-half the difference between the maximum and the minimum values obtained in this fashion, is about 3% of the average value of aCR. Accordingly, the relative uncertainty in aCR, which is relevant for the interconnect thermometry, is estimated to be 3%. The uncertainty of the measured heat transfer coefficient is estimated to be 2.5%. The total relative uncertainty of the heat transfer coefficient is obtained to be 6.0%. Therefore, the main contributors to the experimental uncertainty in the present measurements are the uncertainty in the sample dimensions and the uncertainty in the calibration coefficient of the platinum wire. The estimated uncertainties do not include the approximations made to the analytical model.
aa P 1 1 ip ln 2 ln 2u g pka l 2 d 2 2
aa 1 1 1 ip ln 2 ln 2u g pka l 2 d 2 2
Even though h can still be phenomenally used for microwires by natural convection model, Eqs. (1)e(7), the value of h can be
(17)
The effective heat transfer coefficient h is defined as
h ¼
1 AjZa j
(18)
The air thermal conductivity ka may decrease due to the Knudsen effect when the microwire characteristic size (d) close to the air molecule mean free path (L), which is about tens of nanometers at room temperature and one atmosphere. Letting h takes the following form,
h ¼
ka C =A 1 þ Kn S
(19)
where CS ¼ 1/(kaZa) is the heat conduction shape factor. For a microwire in the air, CS is given by
1 aa 1 ip 1 CS ¼ pl ln 2 ln 2u g 2 2 2 d
(20)
Finally, the explicit form of the end result of the present analysis based on 3u principle is given by
h ¼
aa g ip 1 ka 1 1 1 u ln ln 2 1 þ Kn d 4 2 4 4 d2
(21)
Fig. 7 shows the dependence of the heat transfer coefficients on the frequency calculated from Eq. (21) for the platinum wires
1200 10.6 µm 1000
800 20.4 µm
600
32.6 µm 40.2 µm
400 57 µm 76.8 µm
200 0.0
4.3. Prediction of h based on heat conduction model
(16)
Then the thermal impedance of the air around the microwire is given by
Za ¼
4.2. Uncertainty analysis
dh ¼
estimated just based on the analysis of heat conduction to surrounding air in the similar way as Hu et al. [5] has done for a 3u heater immersed in a thin film. For a microwire immersed in an infinite fluid (air), the temperature oscillation in the air in the 3u measurement has been given by Eq. (11). To simplicity, by retaining only the first term of a series expansion, the mean temperature change responding to heating is found to be from Eq. (11) as
h ( Wm-2K-1)
150
0.2
0.4
0.6
0.8
1.0
1.2
f (Hz) Fig. 7. Predicted heat transfer coefficients as a function of frequency with variable diameters.
Z.L. Wang, D.W. Tang / International Journal of Thermal Sciences 64 (2013) 145e151
of different diameters. In Fig. 7, the fitted values of the heat transfer coefficient (h) are 960.5, 558, 410, 342, 244, 215 and 156 W m2 K1 for the samples 10.6 mm, 20.4 mm, 32.6 mm, 40.2 mm, 57 mm, 76.8 mm and 95.6 mm respectively. As the explicit form of the heat transfer coefficient based on 3u principle is a simplified form from the heat conduction model given by Eq. (12), the validated range of the explicit form is between 0.1 and 0.3 Hz based on the measured results if the relative error is limited to 10%, then the expression of the heat transfer coefficient based on 3u principle takes the following form at the optimized frequency:
h ¼
1=2 aa ka 1 1 2 aa 0:292ln þ 0:958 ln 1 þ Kn d 16 d2 d2
(22)
Eq. (22) shows that h is considered to be inversely proportional to size d at microscale while at macro scale, h is considered constant and independent of the scaling effects. It is no surprise that h is larger than that for macro scales. This is because heat conduction is relatively more significant for microscale wires, owing to the increased ratio of surface to volume or the increased curvature, indicating less significance of the volume force, buoyancy at microscale. From the perspective of geometry of the thin wire, both the measurements shown in Figs. 5 and 6 and theoretical analysis by Eq. (22) show that the heat transfer around microscale wire seems to have clear dependence on the ratio of surface to volume or curvature. 5. Conclusions The natural convection model and heat conduction model are presented for the heat transfer coefficient (h) measurement based on 3u method. The measured results of the microscale platinum wires with the diameters from 10.6 mm to 95.6 mm show that the heat loss to air from the platinum wire is actually dominated by heat conduction and that natural convection is negligible. The measured effective heat transfer coefficients show that the 3u principle for thin wires still validates if the microwire is immersed in gas environment and the heat loss to the surrounding gas is dominated by heat conduction. In addition, an explicit expression for h is derived through analysis based on the 3u principle. The fact that h is larger than that for macro scales is because heat conduction is relatively more significant for microscale wires, owing to the increased ratio of surface to volume. The heat transfer process around the microwires in the air does appear to show the microscale effect of heat conduction. The 3u measurements in this study are performed at room temperature, and the heat transfer characteristics in the high- and low-temperature regimes warrants investigation. In such temperature regions, thermal radiation may play an important role. Further, the end effects of the microwires on the thermal boundary condition and the heating frequency also remain an area for further study. It is suggested that microscale natural convection is an area including measurements, simulations, and optical diagnostics of microscale air convection.
151
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