Adaptable thermal conductivity characterization of microporous membranes based on freestanding sensor-based 3ω technique

International Journal of Thermal Sciences 89 (2015) 185e192

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Adaptable thermal conductivity characterization of microporous membranes based on freestanding sensor-based 3u technique L. Qiu a, X.H. Zheng a, *, P. Yue a, J. Zhu a, D.W. Tang a, *, Y.J. Dong b, Y.L. Peng b a b

Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China Beijing University of Technology, Beijing 100124, People's Republic of China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 16 March 2014 Received in revised form 5 November 2014 Accepted 6 November 2014 Available online

Adaptable thermal conductivity measurements of microporous membranes based on freestanding sensorbased 3u technique is proposed. The adaptability and flexibility of the freestanding sensor enable microporous membranes with various mechanical and thermal properties to be easily sandwiched between the freestanding sensor and a semi-infinite thermally conductive substrate, assembling into a five-layer substrate-membrane-sensor-membrane-substrate configuration. A theoretical model for the calculation of membrane's thermal conductivity is provided by comparing the temperature difference between the fivelayer configuration and a three-layer substrate-sensor-substrate configuration. The well agreed experimental results with the theoretically calculated values indicate that the present strategy can be widely applied to the thermal properties characterization of microporous membranes in membrane distillation. © 2014 Elsevier Masson SAS. All rights reserved.

Keywords: Adaptable thermal conductivity characterization Freestanding sensor-based 3u technique Microporous membrane Membrane distillation

1. Introduction In membrane distillation (MD), the accurate thermal conductivity measurements for the microporous membranes have attracted great interest as it determines the thermal efficiency h of the MD process [1e3]. In order to obtain the effective thermal conductivity of microporous membranes, several models have been exploited, such as parallel thermal resistance model, cascade thermal resistance model, and combination of both of them with a distribution factor, which takes into account the structural effects of the material [1,3,4]. Although the thermal resistance technique based on an improved Lees' disc apparatus for thermal conductivity characterization of several commercial membranes has been reported [4], the experimental techniques for measuring the thermal conductivity of microporous membranes has rarely been reported as the thickness of the microporous membranes is only about 100 mm, which cannot fulfill the experimental condition using conventional techniques. Therefore, exploring practical techniques for the accurate characterization of the thermal conductivity of microporous membranes with tens of micrometers thickness is urgently expected and needed.

* Corresponding authors. E-mail addresses: [email protected] (X.H. Zheng), [email protected]. cn (D.W. Tang). http://dx.doi.org/10.1016/j.ijthermalsci.2014.11.005 1290-0729/© 2014 Elsevier Masson SAS. All rights reserved.

A promising measurement technique for thermal conductivity named freestanding sensor-based 3u technique has been pioneered in our previous work, and it has been demonstrated as a useful method for the bulk and wafer samples, which are applicable to the semi-infinite media assumption [5,6]. For the microporous membrane typically with low thermal conductivity, the thermal resistance introduced by the membrane when subject to harmonic thermal waves can be accurately indentified in a certain range of frequencies, providing the basis for the feasibility of this technique in membrane measurement. Moreover, different from the traditional 3u technique [7,8], this modified technique exhibits higher accuracy and unique adaptability. On one hand, it is not necessary to deposit the electrodes on the samples for the measurement, thus effectively avoiding destruction to the microporous membranes. On the other hand, the flexible sensor enables excellent contact with the surface of microporous membranes with different morphology and roughness, thus effectively promoting the accuracy of the measurement. Therefore, it is rational to believe that this adaptable freestanding sensor-based 3u technique can become a competent candidate for the thermal conductivity characterization of thermal insulating films, such as microporous membranes in MD applications. In this work, based on this adaptable technique stemming from the differential 3u method [8e11], the thermal conductivity measurements of three kinds of microporous membranes are reported. Significantly, we deduced the theoretical model in the

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L. Qiu et al. / International Journal of Thermal Sciences 89 (2015) 185e192

symmetrical five-layer system can be described as a reduced form of the universal two-dimensional multilayer heat conduction formulation derived by Borca-Tasciuc et al. [11]

p DT ¼  2plky;1

Z∞ 0

1 sin2 ðblÞ dl; A1 B1 ðblÞ2

(1)

where

Ai1 ¼

symmetrical five-layer substrate-membrane-sensor-membranesubstrate system and investigated its applicability for thermal conductivity characterization of thermal insulation microporous membranes with tens of microns thickness. Furthermore, three factors in this measurement, the loading pressure, the thickness of membrane and the thermal contact resistance at adjacent interfaces, have been also investigated in detail. 2. Theory It should be indicated that for the freestanding sensor-based 3u technique the test structure can be regarded as a symmetrical threelayer configuration, sample-sensor-sample, when the sample (bulk and wafer) thickness is far larger than the thermal penetration depth [5,6]. However, for membrane specimens, which are typically tens of micrometers thick, the theoretical model is different from that for the bulk and wafer samples. In order to identify the thermal conductivity of the membrane, a symmetrical five-layer substratemembrane-sensor-membrane-substrate system was built up, in which the membranes were symmetrically sandwiched between the freestanding sensor and substrates. Since the substrate is used for providing an isothermal boundary condition as well as flattening the membrane, we use the high thermally conductive materials, such as the stainless steel 304. The temperature responses in the symmetrical five-layer system and the symmetrical three-layer substrate-sensor-substrate system can be obtained, respectively (Fig.1). According to the differential 3u method, the difference in the temperature response between the two configurations should be attributed to the presence of the membranes. Therefore, the thermal conductivity of the membrane can be calculated by comparing the temperature difference between the two systems. Based on our previous research [5,6], the solution of the complex temperature rise of the freestanding sensor for the

1 1  ¼" A1 A*1 k

y;3 B3

ky;1 B1

  tanhð42 Þ 1  tanh2 ð41 Þ k

B

k

B

1

y;2

2

1

 tanhð4i1 Þ

k B Ai k y;i Bi y;i1 i1

; i ¼ 2; 3 tanhð4i1 Þ

!1=2 i2u ; i ¼ 1; 2; 3 ay;i

(2)

(3)

4i ¼ Bi di ;

(4)

 kxy;i ¼ kx;i ky;i :

(5)

In the above expressions, subscript x, y corresponds to the in-plane and cross-plane directions, respectively. Subscript 1, 2, 3 corresponds to the Kapton film (the flexible insulating layer of the freestanding sensor), the membrane specimen and the substrate, respectively. p/l is the electrical power per unit length, k is the thermal conductivity, kxy is the ratio of the in-plane to cross-plane thermal conductivity, b is the strip half width, l is the integrating factor, u is the angular frequency of the alternating current, a is the thermal diffusivity, and d is the thickness. As the substrate is semi-infinite, A3 ¼ 1 [11]. Similarly, the temperature rise of the freestanding sensor for the symmetrical three-layer system (marked by superscript *) is described as

p DT * ¼  2plky;1 k

A*1

¼

Z∞ 0

1 sin2 ðblÞ dl; A*1 B1 ðblÞ2

(6)

B

A*3 ky;3 B3  tanhð41 Þ y;1

1

k

B

1  A*3 ky;3 B3 tanhð41 Þ y;1

:

(7)

1

Here the semi-infinite substrate assumption is still valid, A*3 ¼ 1. Thus, the temperature difference between two systems is obtained as,

p DT  DT * ¼  2plky;1

Z∞ 0

1 B1

1 1  A1 A*1

!

sin2 ðblÞ dl; ðblÞ2

(8)

A detailed derivation of ðð1=A1 Þ  ð1=A*1 ÞÞ was listed in the Supplementary materials (See supplementary material for detailed derivation of temperature difference between two test systems). And the final expression is,

! k2y;3 B23 y;2 B2 ky;1 B1

k

k

B

þ ky;2 B2 y;1

#

1

þ ky;2 B2 tanhð42 Þ þ tanhð41 Þ þ ky;3 B3 tanhð41 Þtanhð42 Þ  y;1

ky;i Bi

y;i1 Bi1

kxy;i l2 þ

Bi ¼ Fig. 1. Schematic diagram for (a) five-layer substrate-membrane-sensor-membranesubstrate characterization configuration, (b) three-layer substrate-sensor-substrate configuration as a control, and the temperature responses of the freestanding sensor at a wide range of frequencies.

Ai k

"

#; ky;3 B3 ky;1 B1

þ tanhð41 Þ

(9)

L. Qiu et al. / International Journal of Thermal Sciences 89 (2015) 185e192

If the frequency of the alternating current is very low, the thermal penetration depth is relatively large compared to the total thickness of membrane and Kapton layer. In this case, a group of approximate relations are valid, which yields,

tanhð41 Þ < < 1;

(10a)

tanhð41 Þz41 ;

(10b)

tanhð42 Þ < < 1;

(10c)

tanhð42 Þz42 :

(10d)

Combining with the thermally conductive substrate assumption as well as the thermal insulating membrane assumption, which means ky,3B3 >> ky,1B1 and ky,2B2 << ky,1B1, Eq. (9) can be approximated as k2y;3 B23 d2

ky;1 B1 d2 1 1 k k B  * z y;2 y;1!12 ¼  ; A1 A1 ky;2

Z∞ 0

sin2 ðblÞ ðblÞ2

dl;

(12)

As the integral in Eq. (12) equals to p/2b, this yields,

DT  DT * ¼

pd2 : 4blky;2

DT ¼

2U3u;rms : aCR U1u;rms

(14)

where aCR is the temperature coefficient of resistance (TCR) of the nickel strip in the freestanding sensor, U1u and U3u are the measured 1u and 3u voltage harmonics. All voltages are rms. The temperature rise of the symmetrical three-layer structure is calculated by using Eq. (6). Finally, the thermal conductivity of the microporous membrane can be extracted by utilizing the average values of DT  DT* at the corresponding low frequencies (typically below 0.06 Hz) via the Eq. (13).

3. Experimental 3.1. Membranes

Thus, Eq. (8) is reduced to

pd2 2plky;2

The temperature rise in the symmetrical five-layer system can be obtained by directly measuring the 1u and 3u voltage harmonics using the following relationship [7,12,13],

(11)

ky;3 B3 ky;1 B1

DT  DT * ¼

187

(13)

This equation indicates that the presence of microporous membranes simply introduces a frequency independent temperature rise to the thermal response of the thermally conductive substrate DT* when the frequency is very low.

There commercial types of flat sheet microporous membrane: one sort, of polytetrafluoroethylene (PTFE), with 45 mm thickness (PTFE45) and the second one, of polyvinylidenedifluoride (PVDF), with 177 mm thickness (PVDF177), and the third one, of polypropylene (PP), with 44 thickness (PP44) were used. Membranes were tested as received without any pretreatment. As shown in Fig. 2, the PTFE45 membrane shows a fibrous structure with large surface area; the PP44 membrane shows a separated lamellae structure with a lot of voids and crystalline connecting bridges; while the PVDF177 membrane shows a spongy structure with holetype pores. Their experimentally determined thickness d and void volume fraction ε together with matrix thermal conductivity kp provided by the manufacturers are tabulated in Table 1. It is to be noted that the matrix thermal conductivity here is the thermal conductivity for the fully-dense polymer not for the porous membranes. The experimentally obtained thermal conductivity keff for porous membranes will be given later in the Results and discussion

Fig. 2. SEM images of (a) PTFE45 membrane surface, (b) PP44 membrane surface, (c) PVDF177 membrane surface and (d) PVDF177 membrane cross-section.

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L. Qiu et al. / International Journal of Thermal Sciences 89 (2015) 185e192

Table 1 Membrane characteristics with their standard errors. Membrane Thickness, Void volume fraction, ε d (mm) micrometer PTFE45 PVDF177 PP44 a

45 (±3) 177 (±8) 44 (±3)

Matrix thermal Manufacturer conductivity, kp (W/m K)

0.828 (±0.006) 0.27a 0.730 (±0.004) 0.19a 0.581(±0.004) 0.11a

HHET HHET Yi Teng

Manufacturers' data.

section. The membrane thickness is determined using a micrometer. The void volume fraction is determined by a comparative weighting method using isopropyl alcohol (IPA) as the wetting liquid. The specific operation steps are as follows: First, a piece of dry membrane with suitable sizes was soaked in IPA for more than 10 h; After that the IPA residues on the surface of the membrane was quickly wiped away, and the remained membrane was put into a enclosed weighing bottle for weighting to obtain the weight of the wetted membrane (W1); Then, the wetted membrane was placed into a vacuum drying chamber for drying, when a constant weight was observed, the dry membrane was quickly taken out and weighted for the weight of the dry membrane (W2). The void volume fraction was calculated according to the following formula:

ε¼

ðW1  W2 Þ=rIPA : ðW1  W2 Þ=rIPA þ W2 =rM

(15)

where ε is the void volume fraction; W1, W2 are the weight of wetted membrane and dry membrane, respectively; rIPA, rM represent the density of IPA and membrane, respectively. To obtain a reliable and accurate result, 10 samples were used for each membrane. 3.2. Adaptable freestanding sensor-based 3u technique for thermal conductivity measurement The traditional freestanding sensor-based 3u method is only applicable for the semi-infinite specimens by performing a multi-

Fig. 3. Schematic representation of the freestanding sensor-based 3u technique for measurement of microporous membranes. The Signal Generator is an Agilent model 33220A, it also output a reference signal for the commercial lock-in amplifier, Ameteck Signal Recovery 7265. Power Amplifier (EG&G Model 5113), Operational Amplifier (AMP03) and low temperature drift resistors R1~R8 (5 PPM/ C) are mounted in a Model SYN-35-D15A power supply. The electrical resistances for the resistors are: R1 ¼ R2 ¼ R5 ¼ R6 ¼ 1 kU, R3 ¼ R7 ¼ 100 U, R4 ¼ R8 ¼ 10 kU. The Signal Generator and Lock-in amplifier are controlled by a computer.

parameter least-square fitting algorithm to the experimental data based on Eq. (1) [5,6]. Different from the traditional 3u method, there is no need to perform the complicated fitting algorithm process in this adaptable freestanding sensor-based 3u technique. Fig. 3 shows the instrumentation. The membranes were cut into rectangle shape with sizes larger than those of the freestanding sensor. No further coating was implemented to improve the thermal contact, as coating process would vary the thermal properties of the membranes when the amount of air entrapped into the membrane surface pores is changed [4]. The symmetrical five-layer structure was pressed tight via a loading screw mechanism to minimize contact resistances. The whole test structure together with the sample holder was mounted in a chamber at room temperature and atmosphere pressure. Radiation loss was minimized by shielding, and conduction loss through the four pads adjoining to the strip heater by using 30 mm diameter SieAl alloy wires for connection leads. For the electrical circuit part, an Operational Amplifier Circuit is added to convert the voltage output of the Signal Generator into the current [12]. A sinusoidal current in the form of I0sin(ut) drives through the 150 nm thick, 10 mm long and 300 mm wide nickel strip encapsulated in the freestanding sensor from the outward two bare pads. The thermal wave at 2u frequency symmetrically penetrates the Kapton layer of the freestanding sensor, then the membrane, and finally goes into the semi-infinite substrate at very low frequencies. The voltage response extracted from the two inner bare pads connecting the nickel strip (occurs at the first and third harmonics: U1u and U3u, respectively) can provide rich information about thermophysical properties of the specimen. Since U1u is typically 300e1000 times as large as U3u, allowing for higher signal-to-noise ratio and more precise measurements, U1u is subtracted before U3u signal is measured by a commercial Lock-in Amplifier with high dynamic reserve in our experiments. To accomplish the subtraction, a Resistance box with adjustable electrical resistance is placed in series with the sensor resistance. By adjusting its resistance, the u voltage from the series Resistance box can be made equal to the U1u from the sensor. The differential input of the commercial Lock-in Amplifier (via A-B mode) can then greatly reduce the u component and thus realize reliable 3u component detection. For PTFE45 with relatively small thickness, measurement frequencies below 0.3 Hz (corresponding to a thermal penetration depth larger than 126 mm) are used to extract the membrane's thermal conductivity based on Eq. (13). For PVDF177 with a larger thickness and PP44 with small thickness but lower thermal conductivity, frequencies below 0.05 Hz (corresponding to a thermal penetration depth larger than 252 mm for PVDF177 measurement and 193 mm for PP44 measurement) are preferred to realize a reliable thermal conductivity measurement. To ensure a reliable measurement results, three samples for each kind of membrane were used and five trials were performed for each sample. In the calibration procedures of this sensor, a copper-constantan thermocouple and a Pt thermometer were used to monitor the temperature. The TCR values of the nickel strip are obtained by evaluating the derivative of the resistivity equation as a function of temperature at the specific temperature of interest. And we have obtained a nearly temperature-independent TCR (0.0064  C1) for the temperature range in which we are interested (50 to 120  C). In addition, considering the temperature tolerance of Kapton film, the temperature upper limit for the present method is near 270  C. For the thermal conductivity measurement of tens of micrometers thickness membranes, the uncertainty is mainly related to the uncertainties in the first harmonic voltage (0.16%), thickness of the membrane (no more than 6.7% for PTFE45, only 4.5% for PVDF177), width (2.6%) and length (2.3%) of nickel strip, electric resistance (0.07%), temperature coefficient of resistance (3.2%,

L. Qiu et al. / International Journal of Thermal Sciences 89 (2015) 185e192

which determines the uncertainty in experimental temperature rise according to Eq. (14) as the uncertainties in U3u and U1u are much smaller). After calculating the root mean square of all the related uncertainties mentioned above via Eq. (16), the total uncertainty in thermal conductivity measurement is estimated to be within 8.2%.

189

running parallel to the direction of heat conduction (Fig. 5(a)). Then the material can be viewed as a parallel combination of conductances (Fig. 5(b)). If we construct a unit-cube (i.e. a 1  1  1 cube) with multi-cross-sections of which the air area is ε and polymer area is (1  ε), then the equivalent parallel conductivity keff can be computed as above.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u      2    2    2   2   2   2 dky;2 u vln ky;2 vln ky;2 vln ky;2 vln ky;2 vln ky;2 t vln ky;2 *Þ : db þ dl þ dR þ dðDT  DT ¼ dU1;u þ dd2 þ ky;2 vU1;u vd2 vb vl vR vðDT  DT * Þ (16)

4. Results and discussion 4.1. Experimental results Fig. 4 shows the temperature rise of the freestanding sensor as a function of the frequency for the cases of PTFE45 and PVDF177. It can be clearly seen that there exist three slopes for the temperature rise curve. The measured thermal conductivity values for PTFE45 and PVDF177 are (0.071 ± 0.006) W/m K and (0.077 ± 0.006) W/ m K, respectively. These values agree well with the calculated effective thermal conductivity by the parallel thermal resistance model (Eq. (17)) [14,15]: 0.068 W/m K for PTFE45 and 0.070 W/m K for PVDF177.

keff ¼ ð1  εÞkp þ εka :

(17)

where kp and ka correspond to the thermal conductivities of polymer and air, and ε is the porosity. The values of kp and ε are summarized in Table 1. The value of ka is reported as 0.026 W/m K at 297 K [16,17]. This model assumes the microporous membrane consists of a matrix of membrane material with cross-sections in air

Fig. 4. Temperature amplitudes of the freestanding sensor measuring PTFE45 and PVDF177 membranes. The solid dots and open circles are representing the experimental data. The curves in black type represent calculation regarding stainless steel 304-membrane-sensor -membrane-stainless steel 304 configuration while the curves in red type is calculation regarding stainless steel 304-sensor-stainless steel 304 configuration under the same heating power condition. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

To validate the repeatability of the proposed technique, experiments to measure the membrane thermal conductivity were performed ten times for PTFE45 and PVDF177. The results are summarized in Fig. 6, exhibiting the deviations from the mean value (the effective thermal conductivity). The maximum and the standard deviations of the measured data are 6.8% and 4.8%, respectively, which demonstrates that this method is capable of measuring the thermal conductivity of typical microporous membranes with acceptable repeatability.

4.2. Effect of loading pressure The loading screw mechanism for fixing the symmetrical fivelayer structure inevitably induces a pressure loading membrane samples. It is intuitive that this pressure will reduce the thickness, the porosity and air entrapped into the membrane pore and consequently, the thermal conductivity will vary. To study the influence of the pressure on thermal conductivity measurements, a Preset Mechanical Torque Wrench (Model NB-5: Range 1e5 N m) was used to quantitatively adjust the pressure loading on the sample [18]. For variable loading pressures, the membrane thickness and thermal conductivity varies with pressure for both PVDF177 and PTFE45, but the changing amplitudes are different. For PVDF177 with large thickness (~177 mm), an increasing loading pressure slightly decreases the membrane thickness and consequently the thermal conductivity is overestimated due to reduced thermal resistance difference from observation (Fig. 7(a)). If the loading pressure is relatively small, such as less than 1.1 MPa, the variation is within 5.0% which could be neglected. When the loading pressure reaches 3.3 MPa, the variation is as high as 30.3% and thus this effect must be taken into account. For PTFE45 with small thickness (~45 mm) and low compression strength, the effect of loading pressure is remarkable. When three pieces of PTFE45 membranes were placed on both sides of the freestanding sensor for the test, even a moderate pressure (such as the pressure of 0.6 MPa used herein) would make the thickness of PTFE45 membranes greatly decrease and thus the deduced thermal conductivity value presents a larger error (Fig. 7(b)). With the increasing loading pressure, the thermal resistance difference further decreases and results in increased error in thermal conductivity estimation. Given the above analysis, the measurement should be performed with the extra loading pressure from the screw mechanism set to zero, that is the membrane-sensor-membrane is pressed only be the weight of the top stainless steel 304 bulk and the tip screw just touch the top surface of stainless steel 304 bulk and no extra pressure is loaded.

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Fig. 5. (a) Schematic for parallel model: heat conduction is assumed in the x direction and the matrix material is illustrated as transparent, (b) thermal conductance diagram.

!

4.3. Thickness limit of membrane samples The measured result for PVDF177 has a relatively larger deviation. This might be attributed to the large thickness of the membrane which does not satisfy the assumption that the presence of the membrane just introduces a frequency independent temperature rise [19]. Thus, it is rational to believe that there is a thickness upper limit for the thermal conductivity measurement of membranes using the new adaptable technique. The purpose of this section is to discuss the thickness limit of membrane samples that is applicable to thermal conductivity measurement based on the theory described by Eq. (13). The starting point is Eq. (9) which has not been simplified using any assumptions. If the thermal penetration depth is not large enough compared to the thickness of membrane, then the assumption that the membrane equals to a frequency independent thermal resistance is invalid. Thus, only Eq. (10a) and Eq. (10b) are still valid. As ky,3B3 >> ky,1B1, ky,2B2 << ky,1B1 and 4i is much smaller than ky,3B3/ky,1B1, the second and third terms in the first square bracket of the denominator is neglected. Likewise, the second term in the second square bracket of the denominator and the second term in the square bracket of the numerator are also neglected. And thus Eq. (9) reduces to

tanhð42 Þ

1 1  z" A1 A*1 k



k2y;3 B23 ky;2 B2 ky;1 B1

#

y;3 B3

ky;1 B1

þ

ky;3 B3 ky;2 B2

tanhð41 Þtanhð42 Þ 

"

#;

(18)

ky;3 B3 ky;1 B1

Through rearrangement, the above expression can be written as,

tanhð42 Þky;1 B1 1 1 ;  z A1 A*1 ky;2 B2 þ ky;1 B1 tanhð41 Þtanhð42 Þ

(19)

The thermal resistance (DZ ¼ DT/p) difference is therefore expressed as

DZ DZ * ¼

1 2pl

Z∞ 0

tanhð42 Þ sin2 ðblÞ dl; ky;2 B2 þky;1 B1 tanhð41 Þtanhð42 Þ ðblÞ2 (20)

As the heating power is not rigidly the same for each test, here we use the thermal resistance difference instead of the temperature difference for discussion. The presence of air between the consecutive interfaces in the test structure, such as membraneemembrane interface, membrane-stainless interface and membrane-sensor interface, can be regarded as an additional thermal resistanceda/4blka, where da and ka are the thickness and the thermal conductivity of the air layers, respectively. For the low frequency region (<0.1 Hz), a new equation can be obtained as

1 DZ  DZ ¼ 2pl *

Z∞ 0

tanhð42 Þ sin2 ðblÞ dl ky;2 B2 þ ky;1 B1 tanhð41 Þtanhð42 Þ ðblÞ2

da : þ 4blka (21)

Fig. 6. Deviation of the experimental data for PTFE45 and PVDF177 membranes.

where d2 ¼ Ndm, dm is the thickness of one piece of the membrane to be tested; da ¼ (N  1)dmem, dmem is the air thickness between two adjacent membranes. In order to further confirm the applicability of the Eq. (21), membranes with small thickness are used to perform a control experiment. Due to fact that the thickness of PTFE45 will easily decrease when subject to a loading pressure and the change in thickness is irreversible (verified in Fig. 7(b)), PP44 membranes

L. Qiu et al. / International Journal of Thermal Sciences 89 (2015) 185e192

191

Fig. 7. Observed thermal resistance difference versus different loading pressures for (a) one-layer PVDF177 and (b) three-layer PTFE45 as the sample.

were used to examine the thickness limit. The thermal resistance difference values as a function of the number of PP44 membranes are exploited, as shown in Fig. 8. The well agreed experimental data with the model prediction indicates the high reliability and accuracy of Eq. (21) for describing the thickness dependent thermal resistance difference for membrane measurement. The thermal resistance brought by the air layer between two adjacent membrane interfaces is fitted to be 15.8 K/W. As to four pieces of PP44 membrane piled up for measurement, the contribution of the existence of air layers to the total thermal resistance accounts for 8% when the signals at 0.01 Hz are used. The extracted thermal conductivity for PP44 membrane is 0.063 W/m K with the effect of air layers taken into account. Furthermore, as can be seen from Fig. 8, the value of DZ  DZ* increases with increasing number of membranes (or thickness) while it deviates from the linear trend implied in Eq. (13). Three different frequencies of 0.1 Hz, 0.06 Hz, and 0.01 Hz were chosen to investigate the trend of discrepancy. As shown in Fig. 8, the discrepancy between the true DZ  DZ* curve and the ideal linear trend is even larger for frequencies above 0.1 Hz, while the discrepancy becomes smaller with the decreasing frequencies, such as 0.06 Hz. For further decreased frequencies, such as 0.01 Hz, the discrepancy further becomes smaller at the cost of time-consuming test as the scanning time constant needs to be large enough. Therefore, taking PP44 membrane as an example, based on Eq. (13) in order to ensure a within 10% measurement error of the temperature difference at a fixed frequency of 0.06 Hz

(this frequency just needs a much smaller scanning time constant compared to 0.01 Hz, and thus is more time-saving and convenient in actual operation), the thickness should be no larger than about 64 mm as deduced from Fig. 8. Similarly, by comparing the accurate form for the total thermal resistance (Eq. (21)) with the simplified form (Eq. (13)) and simultaneously ensuring a no more than 10% error when the measurement frequency is fixed at 0.06 Hz, the minimum thickness for PTFE and PVDF membranes are deduced to be 73 mm and 78 mm, that is about 14.1% and 21.8% higher than PP44, respectively. It is to be noted that the higher thickness limits for PTFE and PVDF membranes investigated in this work are attributed to their higher thermal conductivity and thermal diffusivity.

Fig. 8. Thermal resistance difference as a function of the number of PP44 membranes at 0.01 Hz, 0.06 Hz and 0.1 Hz. Open pentagon are the experimental data at 0.01 Hz, solid squares at 0.06 Hz and open squares at 0.1 Hz; the short dashed line is the prediction based on Eq. (13); while the three long dashed lines represent the model prediction based on Eq. (21) at 0.01 Hz, 0.06 Hz and 0.1 Hz. Each datum indicates an average value from five measurements, and each error bar indicates a standard deviation.

b  DT b* ¼ DT

4.4. Thermal contact resistance of sensor-membrane and membrane-substrate interfaces Besides the additional thermal resistance brought by entrapped air layers, softness of membranes brings a contact problem at sensor-membrane and membrane-substrate interfaces, which is a serious issue in the case of multi-layer membranes measurement. Although the entrapped air on those interfaces is extremely small when the test structure is firmly held, thermal contact resistance will unavoidably affect the temperature response, resulting in affecting the measured value of thermal conductivity. It is intuitively believed that the contact condition would be greatly improved by applying a higher pressure between the substrates. However, this might result in an inaccurate thermal conductivity value due to the seriously attenuated membrane thickness during the measurement, which has been pointed by García-Payo et al. [4] and verified in the Section 4.2. Therefore, in order to obtain the true value of the thermal conductivity, it is also necessary to discuss the effect of the thermal contact resistance on the obtained temperature response. Considering the interlayer thermal contact resistances at the sensor-membrane interface (Rc2) and the membrane-substrate interface (Rc3), a detailed derivation of the temperature rise of the freestanding sensor for the two systems was also listed in the Supplementary materials (See supplementary material for detailed derivation of temperature amplitude considering the thermal contact resistances at interfaces). It is indicated that the thermal contact resistance can be separated and the temperature change is expressed as

pd2 p Rcn þ 4blky;2 4bl

ðn ¼ 2; 3Þ:

(22)

The over-caret denotes quantities considering the thermal contact resistance. R*c2 is the thermal contact resistance at sensorsubstrate interface in the control test. For (i) R*c2 z Rc2, which

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Table 2 Surface roughness and hardness of the related materials at room temperature. Material

Vicker's hardness (MPa)

Surface roughness (mm)

Kapton MT sheet Stainless steel 304 PTFE45 PVDF177 PP44

22.7 [22] 129 [23] 14.0a 16.2a 12.7a

0.390 [21] 0.631a 0.170a 0.162a 0.155a

a

Manufacturers' data.

assumes the thermal contact resistance for sensor-substrate interface is close to that for sensor-membrane interface, n ¼ 3 from the derivation (see Supplementary material, content 2); Similarly, for (ii) R*c2 z Rc3, which assumes the thermal contact resistance for sensor-substrate interface is more close to that for membranesubstrate interface, n ¼ 2. It is to be noted that Eq. (22) is exact in the limit of low membrane thermal conductivity (to satisfyky,2B2 << ky,1B1), low thermal contact resistance, wide heater lines compared with the film thickness (the thickness limit), and very low frequency (to satisfy Eqs. (10a)e(10d)). It has been proved that the softness of Kapton film (the insulating layer for the freestanding sensor) can increase the effective contact area and distinctly reduce the Rc when squeezed into the voids between the actual contact spots at the Kapton film-metal interface [20]. The resulted thermal contact resistance is on the order of 1  104 K m2/W [20]. In addition, the thermal contact resistance at the interface between two adjacent dry Kapton films is measured to be 4.2  107 K m2/W (0.14 K/W for the interface between two adjacent dry Kapton films [21]) for the heating area of 300 mm (wide)  10 mm (length) of the freestanding sensor. As the hardness and roughness of the microporous membrane is more close to that of the Kapton film instead of that of stainless steel 304 (shown in Table 2), it is more likely to be R*c2 z Rc3, which yield n ¼ 2 from this above mentioned analysis and means the thermal contact resistances at the sensor-membrane interface Rc2 should be considered. The change of the temperature response for considering the thermal contact resistance in the five-layer configuration is pRc2/4bl , which is obviously smaller than the temperature response neglecting the thermal contact resistance pd2/4blky,2. The deviation is estimated to be no more than 0.065% (Rc2 ¼ 4.2  107 K m2/W, here we use the thermal contact resistance at the interface between two adjacent dry Kapton films for estimating Rc2, d2 ¼ 45 mm, ky,2 ¼ 0.07 W/m K). Therefore, the effect of the thermal contact resistance of the Kapton-membrane interface can be neglected for the present technique. 5. Conclusion In this study, we present an adaptable freestanding sensorbased 3u technique for the membrane measurement. Based on the mathematical derivation, the effective thermal conductivity of tens of micrometers thick microporous membranes can be accurately calculated by comparing the experimental temperature response of the symmetrical five-layer configuration with the temperature response of a symmetrical three-layer configuration. The measured thermal conductivities of PTFE45 and PVDF177 are consistent with their reference values from the parallel model. Furthermore, the effect of loading pressure and thickness limit of membrane samples for this technique are discussed. The thermal contact resistances at the sensor-membrane interface and membrane-substrate interface are also addressed. Most importantly, the high adaptability allows the present technique as a

promising candidate for characterization of thermophysical properties of various microporous membranes in membrane distillation filed. Acknowledgments The authors acknowledge financial support from Projects 51306183 and 51336009 supported by National Natural Science Foundation of China. Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.ijthermalsci.2014.11.005. References [1] M. Khayet, Membranes and theoretical modeling of membrane distillation: a review, Adv. Colloid Interface Sci. 164 (2011) 56e88. [2] M. Qtaishat, T. Matsuura, B. Kruczek, M. Khayet, Heat and mass transfer analysis in direct contact membrane distillation, Desalination 219 (2008) 272e292. [3] J. Phattaranawik, R. Jiraratananon, A.G. Fane, Heat transport and membrane distillation coefficients in direct contact membrane distillation, J. Membr. Sci. 212 (2003) 177e193. [4] M.C. García-Payo, M.A. Izquierdo-Gil, Thermal resistance technique for measuring the thermal conductivity of thin microporous membranes, J. Phys. D Appl. Phys. 37 (2004) 3008e3016. [5] L. Qiu, D.W. Tang, X.H. Zheng, G.P. Su, The freestanding sensor-based 3u technique for measuring thermal conductivity of solids: principle and examination, Rev. Sci. Instrum. 82 (2011) 045106. [6] L. Qiu, X.H. Zheng, G.P. Su, D.W. Tang, Design and application of a freestanding sensor based on 3u technique for thermal-conductivity measurement of solids, liquids, and nanopowders, Int. J. Thermophys. 34 (2013) 2261e2275. [7] D.G. Cahill, Thermal conductivity measurement from 30 to 750 K: the 3u method, Rev. Sci. Instrum. 61 (1990) 802e808. [8] D.G. Cahill, M. Katiyar, J.R. Abelson, Thermal conductivity of a-Si: H thin films, Phys. Rev. B 50 (1994) 6077e6081. [9] T. Borca-Tasciuc, D. Song, J.L. Liu, G. Chen, K.L. Wang, X. Sun, M.S. Dresselhaus, T. Radetic, R. Gronsky, Anisotropic thermal conductivity of a Si/Ge superlattice, Mater. Res. Soc. Symp. Proc. 545 (1998) 473. [10] R. Venkatasubramanian, Lattice thermal conductivity reduction and phonon localization like behavior in superlattice structures, Phys. Rev. B 61 (2000) 3091e3097. [11] T. Borca-Tasciuc, A.R. Kumar, G. Chen, Data reduction in 3u method for thinfilm thermal conductivity determination, Rev. Sci. Instrum. 72 (2001) 2139e2147. [12] C. Dames, G. Chen, 1u, 2u, and 3u methods for measurements of thermal properties, Rev. Sci. Instrum. 76 (2005) 124902. [13] B.W. Olson, S. Graham, K. Chen, A practical extension of the 3u method to multilayer structures, Rev. Sci. Instrum. 76 (2005) 053901. [14] M. Kaviany, Conduction heat transfer, in: Frederick L. Ling (Ed.), Principles of Heat Transfer in Porous Media, second ed., Springer, New York, 1995, p. 130. [15] J.H. Tavman, Effective thermal conductivity of isotropic polymer composites, Int. Commun. Heat. Mass Transfer 25 (1998) 723e732. [16] K. Kadoya, N. Matsunaga, A. Nagashima, Viscosity and thermal conductivity of dry air in the gaseous phase, J. Phys. Chem. Ref. Data 14 (1985) 947e970. [17] D.R. Lide, CRC Handbook of Chemistry and Physics, 81st ed., CRC Press, New York, 2000, pp. 6e175. [18] L. Qiu, Y.M. Li, X.H. Zheng, J. Zhu, D.W. Tang, J.Q. Wu, C.H. Xu, Thermal-conductivity studies of macro-porous polymer-derived SiOC ceramics, Int. J. Thermophys. 35 (2014) 76e89. [19] S. Shin, H.N. Cho, B.S. Kim, H.H. Cho, Influence of upper layer on measuring thermal conductivity of multilayer thin films using differential 3-u method, Thin Solid Films 517 (2008) 933e936. [20] L. Zhao, P.E. Phelan, Thermal contact conductance across filled polyimide films at cryogenic temperatures, Cryogenics 39 (1999) 803e809. [21] D.L. Rule, D.R. Smith, L.L. Sparks, Effect of Thermal Contact Resistance. in: Thermal Conductivity of a Polymide Film between 4.2 and 300 K, with and without Alumina Particles as Filler, U.S. Department of Commerce, National Institute of Standards and Technology, Boulder, 1990, pp. 53e54. [22] J. Megusar, Low temperature fast-neutron and gamma irradiation of Kapton polyimide films, J. Nucl. Mater. 245 (1997) 185e190. [23] M. Milada, N. Zreibaa, F. Elhalouanib, C. Baradaib, The effect of cold work on structure and properties of AISI 304 stainless steel, J. Mater. Process. Technol. 203 (2008) 80e85.