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Measurement of thermal conductivity of fluid using single and dual wire transient techniques
Measurement 46 (2013) 2746–2752
Contents lists available at SciVerse ScienceDirect
Measurement journal homepage: www.elsevier.com/locate/measurement
Measurement of thermal conductivity of fluid using single and dual wire transient techniques Siddharth Komini Babu 1, K.S. Praveen 1, B. Raja ⇑, P. Damodharan 1 Indian Institute of Information Technology, Design and Manufacturing (IIITD&M) Kancheepuram, Chennai 600 127, India
a r t i c l e
i n f o
Article history: Received 20 August 2012 Received in revised form 7 February 2013 Accepted 13 May 2013 Available online 25 May 2013 Keywords: Transient hot wire Platinum Thermal conductivity Dual wire method
a b s t r a c t A modified measurement device to measure thermal conductivity of fluids using transient hot-wire technique has been designed, developed, tested and presented in this paper. The equipment is designed such that the thermal conductivity could be measured using both single wire sensor of different length and dual wire sensor. The sensor, which is also a heater, is a platinum micro-wire of 50 lm diameter. The influence of wire length on the measurement of thermal conductivity of fluids is tested using two single wires of length 50 mm and 100 mm. The thermal conductivity is also measured using a dual hot wire arrangement; which is achieved by placing the 100 mm and 50 mm wires in a Wheatstone bridge with the 100 mm wire as the sensor and 50 mm wire as a compensation wire. The apparatus requires a 100 ml of test fluid to perform the experiment. The testing temperature of the test fluid during the experimentation can be suitably varied by the choice of heat exchange fluid used in the apparatus. Water is chosen as testing fluids for primary standards. When compared to single wires, the thermal conductivity of the fluids measured is consistent with dual-wire method with an uncertainty of ±0.25%. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The thermal conductivity of fluids is the most essential parameter required for thermal design and its consideration. Laser flash method [1], thermo reflectance techniques [2], 3-omega technique [3–5] etc. techniques are gaining importance since it measures properties based on either steady state or pseudo steady state solutions. Traditionally, transient hot wire techniques; which uses Fourier’s transient heat conduction model, is a well-established method for accurate measurement of thermal conductivity over a wide variety of substances such as solids
⇑ Corresponding author. Tel.: +91 044 27476355; fax: +91 044 27476301. E-mail addresses: [email protected] (S. Komini Babu), [email protected] (K.S. Praveen), [email protected] (B. Raja), [email protected] (P. Damodharan). 1 Tel.: +91 044 27476355; fax: +91 044 27476301. 0263-2241/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.measurement.2013.05.017
[6], polymers [7,8], and molten substances [9]. Among these samples, determining the thermal conductivity of fluid and gas samples is very complicated and delicate. Even though fluids and gases can transfer through conduction, tend to invoke natural convection current which eventually superimposes the desired thermal conductivity value achieved using Fourier heat conduction. A variety of instruments have been modified and improvised in the past for better accuracy of thermal conductivity of fluids using hot wire technique [10–16]. In principle the method can be used to obtain both the thermal conductivity and the thermal diffusivity using the same experiment. It is seen that the wire diameter, length and material, nature of the fluid, material of construction, modes of heat transfer, electrical conductivity of the fluid etc. plays a vital role in measuring the thermal conductivity of the fluid accurately. In this paper a redesign of the hot wire apparatus that could used to measure the thermal conductivity of liquids is presented.
S. Komini Babu et al. / Measurement 46 (2013) 2746–2752
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Nomenclature T r t q k R
final temperature (K) radial distance (m) time (s) heat flux (W m2) thermal conductivity (W m1 K1) resistance (X)
Greek letters a thermal diffusivity (m2 s1) D change c Euler’s constant e emissivity
2. Mathematical model
Stephan Boltzmann constant
Subscript f 0 LONG SHORT g ID PT
fluid initial long wire short wire bridge ideal platinum
DT 2 DT 1 ¼
The platinum wire, which serves as both sensor and heater, is modeled as an infinitely long and thin, ideal continuous line source dissipating heat into an infinite medium, with constant heat generation [17]. Since the wire is very thin, line source is assumed to have an infinite thermal conductivity and zero heat capacity. Thus, the heat conduction process could be represented as the Fourier one dimensional transient heat conduction (Eq. (1))
1 @ @T 1 @T ¼ r r @r @r af @t
ð1Þ
The boundary condition for the above equations is
@T q ¼ t ¼ 0 and r ¼ 0lim r r!0 @r 2pkf
ð2Þ
t P 0 and r ¼ 1limfDTðr; tÞg ¼ 0 r!1
where DT ¼ T T 0 Using the above, the temperature change at a radial distance r, from the heat source is conforms to a Eq. (3) by applying boundary conditions (2). Upon exponential integration the solution is given in Eq. (4).
DTðr; tÞ ¼ Tðr; tÞ T 0 ¼
r
q r2 Ei 4pkf 4af t
ð3Þ
DT ¼ Tðr;tÞ T 0 8 2 2 39 > > r2 r2 < = 4af t 4af t q 6 4af t 7 þ ¼ c þ ln þ :: þ ::::::: 4 5 2 > 1 1! 4pkf > 2 2! r : ; ð4Þ 4af t q DT ¼ Tðr; tÞ T 0 ¼ c þ ln 4pkf r2
ð5Þ
where c ¼ 0:5772 is the Euler constant At any fixed radial distance, in two instances in time the equation, the temperature change can be represented as the following equation:
q t2 ln 4pkf t1
ð6Þ
A plot of temperature against the natural logarithm of time results in a straight line, the slope being propositional to kf
kf ¼
q d lnðtÞ 4p dDT
ð7Þ
Eq. (3) can also be re-written as Eq. (8) such that both thermal conductivity and thermal diffusivity could be achieved from the same equation
4af t q DT ¼ Tðr; tÞ T 0 ¼ c þ ln 4pkf r2 4af t k q ¼ ln e 4pkf r2 4af q q ln 2 ek þ lnðtÞ ¼ 4pkf 4pkf r
ð8Þ
The slope of the equation can give the thermal conductivity and the intercept will give the thermal diffusivity. In this paper thermal conductivity is the matter of interest. 3. Apparatus design and circuitry The schematic layout of the apparatus along with the instrumentation circuitry is shown in Fig. 1a, b and c and the essential apparatus dimensions are shown in Table 1. The apparatus consist of three chambers respectively vacuum chamber (1), coolant chamber (2) and the platinum micro-wire assembly chamber (3). The entire apparatus is constructed using SS 316. The vacuum chamber, which is the exterior compartment, ensures heat infiltration to and from the ambient. It is ensured that the chamber is vacuumed using a vacuum pump (4), prior conducting the experimentation. The next inner concentric chamber is the coolant chamber. Based on the temperature requirement, the coolant can be appropriately chosen to maintain the temperature in the micro-wire chamber which is filled with the test fluid. The coolant is pumped from a temperature controlled bath into the coolant chamber through 5 and leaves the chamber through exit pipe 6. The temperature of the
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Fig. 1. Schematic layout of the apparatus.
Table 1 Essential parameters. S. No.
Description
Dimension
1 2 3 4 5 6
Diameter of the micro wire Length of the long wire Length of the short wire Diameter of the bore that holds micro wire Outer diameter of the coolant chamber Outer diameter of the vacuum chamber
50 lm 100 mm 50 mm 8 mm 100 mm 150 mm
coolant in its chamber is measured using PT100 sensors TS1 and TS2. The inner most chamber is the micro-wire chamber in which the platinum micro-wires are fixed. There are three tubes respectively 7, 8 and 9 for short wire, long wire and common connecting wires. The upper end of three tubes is welded to manifold to which entry (8) and exit (9) valve for test fluid are connected. The PT100 Temperature sensors TS3 and TS4 are used to measure the temperature at the inlet and drain manifolds. The platinum micro-wires are held inside the tube 7 and 9 by using the Teflon holder (10) and soft silicone rubber as shown in Fig. 1b. The platinum wire is exposed to fluid only for 100 mm in the long wire and 50 mm in the short wire. The remaining wire connections are soldered using silver wire, which are Teflon sleeved. The electrical connectivity is carefully brought outside the vessel using a feed through connectors. The most critical operation is the silver soldering of the micro-wires with the connecting wires and
bringing out the apparatus using the feed through connectors. The entire assembly is perfectly insulated with thermo-foam insulation (11). The vacuum system provided is connected to a single stage vacuum pump, which gets its automatic feedback from an absolute pressure gauge. Since the fluid used in the present study is non-volatile, the liquid is manually fed by volume measurement. The data from the circuitry is taken to a computational device using Agilent data acquisition system. In a transient hot wire method, the thermal conductivity of a fluid is measured using the rate of increase of temperature of micro-wire with time for an applied step change of voltage. The overall apparatus is completely isolated from all sources of vibration, such that the test fluid in which the micro-wires are suspended is in absolute static condition. This condition is synonymous to Eq. (1), which mandates that the heat transfer process in the fluid is only through conduction. The supply voltage generates a constant heat flux per unit length throughout the microwire which is transferred to the static fluid medium. The change in resistance of the sensing wire is related to the rate at which the heat is removed from the wire and thus the thermal properties of the fluid. The resistance of the micro-wire during heating could also be measured by placing a resistor in series with a power supply, finding the potential difference across the resistor and dividing by the current through the micro-wire. This would only end in practical difficulties and inaccuracies. For instance, in order to measure resistance, consider that the voltmeter is
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connected across an unknown resistance. A small current is essential to deflect the moving coil in case of an analog voltmeter. This current will then flow through the ammeter in addition to the main current through the resistance. The ammeter reading will be too high and quotient of the two meters readings will be too low. Now consider a technique in which a voltmeter connected across both the resistor and the ammeter. The voltmeter reading will be too large since it measures the potential difference across the ammeter as well, which would result in large resistance. Difficulties in the method attributed to uncertainties to the construction of the meter, which further depend upon such factors as smallest division and the perfectness in the calibration. The uncertainties are compounded when the two meter readings are divided to calculate resistance. For the most accurate measurement of resistance, the Wheatstone bridge circuit is used. This circuit avoids most of the difficulties in the aforementioned methods. This method demands high precision measurement of resistance change making the Wheatstone circuit as the ultimate choice. To construct the Wheatstone bridge, the long wire (L) is placed in one working arm of the bridge and the short wire (S) in the other. An accurate power supply of 5 V DC is applied to the bridge and voltage across the bridge is read as a function of time. Channels are provided in the circuit shown in Fig. 1c is to measure the current and voltages in the circuit. The measurement of thermal conductivity is accomplished in two steps. In the first step the bridge is balanced to null. The switch 1 is turned from the dummy side to the bridge keeping the switch 2 closed. With a very small applied voltage 0.1 V and keeping the cell at constant temperature the leads, hot wires and ballast resistances are measured. The ballast resistance is adjusted until each leg is approximately 100 X. The bridge null is then checked across the channel 6 to ensure the balanced condition. The second step involves the conductivity measurement. Before switching on the power supply, the switch 1 is turned to the dummy side as a precaution to avoid the pre heating of platinum wires keeping the switch 2 closed. The power supply is then set to 5 V and the switch1 is turned to the bridge. Voltages are read on channel 6 using the data logger for 5 s and the values are stored. The power is then switched back to the dummy side to prevent the wires from getting over heated. The circuitry shown in Fig. 1c is capable of accomplishing conductivity measurements using long, short and dual wires arrangement individually. The switches 3 and 4 are provided in the circuit to necessitate the desired measurement by absenting the unwanted wire from the circuit. For instance, if dual wire measurement is to be taken both switches 3 and 4 are closed. For long wire reading an external 100 X is connected to the short wire terminal to switch 4 and similarly for short wire the resistance is externally connected to the long wire terminal and the switch 3. The power supply used provides a constant voltage at the circuit terminals. The instantaneous value of heat flux applied to the circuit is calculated from the Eq. (11). The ideal analysis states that the heat flux q, applied to the wire remains constant during a given run. Under experimental conditions q is assumed to be nearly ideal. However at very
low temperature the q will vary from the beginning to end of a run by values up to several percentages. V (t) is assumed to vary one part in 1000 over a run, and some assume it to vary linearly with ln (t). The voltages measured across the bridge measure the unbalance of the bridge. A typical set of over 1000 readings per sec is used to get the plot. The voltage plot is observed to be logarithmic with time. A certain amount of judgement is required in choosing the required part of the data for calculating the thermal conductivity. The bridge voltage is measured for more than 1 s to enunciate the logarithmic curve but for calculating the thermal conductivity of the fluid, the conduction part is considered. A typical set of reading is used to get the plot between DT and ln (t). The standard for a valid result is that DT values plotted, forms a straight line with ln (t). The range of time over which this linear plot exists is considered for evaluating the thermal conductivity. For most of the experiments the range is between 100 ms and 950 ms. It is also important in identifying the improper results to be scorned. If the value of DT is constant in the run then the fluid sample is interpreted to be in steady state or as the onset of convective currents. A non-uniform temperature field in the wires signals the measurement to be rejected, for this indicates that the segment is subjected to convective cooling. It was observed that measurements on water between 100 and 950 ms were found to be appropriate.
DV g ¼
R1 ðR3 þ DR3 Þ R2 ðR4 þ DR4 Þ V IN ðR1 þ R2 ÞðR3 þ DR3 þ R4 þ DR4 Þ
ð9Þ
R1 ¼ R2 ¼ R3 ¼ R4 ¼ 100 X where R3 = Rbalast1 + Rlong Simplifying
DV g ¼
and
V IN RðtÞ 8 R3
R4 = Rbalast2 + Rshort
ð10Þ
qðtÞ ¼ I2 R ¼ ½VðtÞ=ðR3 ðtÞ þ R4 ðtÞÞ2 ðRLong þ RShort Þ=ðlLong þ lShort Þ ð11Þ
DR3 ¼ DRLONG þ D RLeads DR4 ¼ DRSHORT þ D RLeads Initial Conditions bridgebalanced R1 =R2 ¼ R4 =R3 ¼ 1
DRSHORT ¼ DRLONG =2 Rt ¼ Ro ð1 þ a DTÞ
ð12Þ
Equating (9), (10) yields DR and further substituting in Eq. (12) yields DT seen in Eq. (8). The relation given in the equation implies a straight line for a plot of DT versus ln (t). The slope of the T versus ln (t) relation is obtained over a valid range using the applied power. The numerical slope value is equal to slope part of the Eq. (8), which yields thermal conductivity.
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4. Temperature corrections It is essential to mention about the inherent deviation present in approach. As per the mathematical model, the wire is assumed have a zero radius and infinite length immersed in an infinite medium. But in reality both the diameter and length of the wire have definite values. Also, the sensor, the platinum wire, mathematically has infinite thermal conductivity and zero heat capacity, which is approximate. Due to the heating of the micro-wire, the temperature difference in the test fluid could activate natural convection. Even though Eq. (8) mandates only conduction heat transfer, presence of convection would reduce the temperature and enhance the mathematical value of the thermal conductivity. Still, the presence of such a convective current could be identified by departure from the non-linear trend. The convection current are dominant in gases, and in the presence study the test fluids are liquids and thus effect of convective current is assumed negligible. Most of the fluids are not transparent to infrared and radiation component in those fluids follows the same radiation principle. Considering the radiation between the platinum wire and the cell wall as between diffusive grey surfaces, the radiation is correction is taken as Eq. (13) [13–15].
dT R ¼
8pr ePT rT 3O DT 2ID q
Fig. 2. Transient variation of bridge voltage.
and direct proportionality of resistance with the length, the resistance is less, when compared to other two arrangements. Further, the dual wire arrangement has resulted in less electrical noise when compared to the other two arrangements. Measurements of this type indicate that end correction would be required for single wires whereas in dual wire the compensation is provided by the bridge circuit arrangement. 5.2. Thermal conductivity of water
ð13Þ
5. Results and discussion The test facility to measure thermal conductivity was constructed and the experiments were conducted with water at 1 atm for temperature between 15 and 50 °C. Both the test fluids are infrared absorbing substances and therefore, suitable correction factors are taken into account. Free convection is predominant only in gases and the factors are not considered in the present study. The thermal conductivity for the test condition is measured using the long wire, short wire and the dual wire and compared.
The variation of the temperature rise of the platinum wire with the logarithmic time is studied for all the three configurations are shown in Fig. 3. It is clearly seen from the figure that the long wire holds a plot very close to the dual wire configuration while the short wire being well below the other two configurations. It is observed that the
5.1. Bridge voltage The transient variation of short wire, long wire and dual wire arrangement is shown in Fig. 2. The transient measurements were of close to 5 s and performed for four scans. In all the methods, the logarithmic variation shows two regions of the voltage rises for all the three combination in the Wheat stone bridge, which is demarcated by transient and steady variation of bridge voltage. It is evident that from the figure that as the length of the wire decreases both transient and steady state variation merges, which enhances the error in the value of thermal conductivity. Eq. (8) requires the transient variation with sufficient slope to estimate the thermal conductivity. On the other hand, the long wire yields a slope closer to that of a dual wire. When compared with the dual wire, it could be seen that the steady state region is delayed by 0.1 s in the sample reading shown, which is advantageous to measure the thermal conductivity. Also, due to the short length
Fig. 3. Transient variation of temperature rise.
S. Komini Babu et al. / Measurement 46 (2013) 2746–2752
Fig. 4. Radiation correction on the dual wire.
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observed that both measurement using short and long wire over predicted by +2% and 1.5%. The dual-wire technique very closely matches with an accuracy of ±0.25%. The major practical difficulty in any hot wire apparatus is fixing the platinum micro-wire in the bore and providing a uniform temperature to the test fluids. Both these problems were easily tackled in the present design. Also in the place of electrical heater to rise the temperature of the fluid, as seen in some designs, induction current place a negative role because the entire apparatus is usually preferred to be constructed with stainless steel. The presence of liquid heating and cooling in this present design has completely eradicated the presence of induction. 6. Conclusion
Fig. 5. Comparison of thermal conductivity of water.
A transient hot-wire thermal conductivity instrument is designed and constructed. The equipment is tested with water for the measurement of the isobaric thermal conductivity. The measurement is taken using different length of the wires and their combination in the form of dual wire along with the necessary correction for radiation. It observed that as the length of the wire sensor is decreased, the desired slope of the transient variation of bridge voltage against the time seems to vanish. Thus, the experiment has to be conducted for very small interval of time, which is not really practical in the presence of electrical noise. The dual wire arrangement has delayed the steady state variation when compared to long wire arrangement. The thermal conductivity of the fluids measured is consistent with dual-wire method with an accuracy ±0.25%. Acknowledgements This work is supported by Department of Science and Technology (DST-SERC), India (Grant No. SR/FTP/ETA0017/2010). We thank M/s Delvac Pumps, Chennai, India for extending their facilities to fabricate the dual hotwire device with precise accuracy. References
Fig. 6. Deviation plot.
rise in temperature of the short wire is less than the dual wire configuration by almost 0.25 °C. The slope and the constant of the linear fit, both vary, thereby yielding a small difference in the values of thermal conductivity and diffusivity for different configurations. The influence of correction factors on thermal conductivity is shown in Fig. 4. It is seen that the radiation correction has increases a mean temperature rise of about 0.02 K. However, its importance in the thermal conductivity value is only marginal of around 2%. The variation of measured thermal conductivity for all the three arrangements along with the NIST [18] values for water at different temperature and the deviation plot for the same are shown in Figs. 5 and 6 respectively. It is
[1] F.I. Chu, R.E. Taylor, A.B. Donaldson, Thermal diffusivity measurements at high temperatures by a flash method, J. Appl. Phys. 51 (1) (1980) 336–341. [2] B.M. Zawilski, R.T. Littleton, T.M. Tritt, Description of the parallel thermal conductance technique for the measurement of the thermal conductivity of small diameter samples, Rev. Sci. Instrum. 72 (3) (2001) 1770–1774. [3] D.G. Cahill, R.O. Pohl, Thermal conductivity of amorphous solids above the plateau, Phys. Rev. B 35 (8) (1987) 4067–4073. [4] D.G. Cahill, Thermal conductivity measurement from 30 to 750 K: the 3omega method, Rev. Sci. Instrum. 61 (2) (1990) 802–808. [5] C. Dames, G. Chen, 1 x, 2 x, and 3 x methods for measurements of thermal properties, Rev. Sci. Instrum. 76 (12) (2005) 124902. [6] M.J. Assael, M. Dix, K. Gialou, L. Vozar, W.A. Wakeham, Application of the transient hot-wire technique to the measurement of the thermal conductivity of solids, Int. J. Thermophys. 23 (3) (2002) 615–633. [7] Y. Agari, A. Ueda, S. Nagai, Thermal conductivity of a polyethylene filled with disoriented short-cut carbon fibers, J. Appl. Polymer Sci. 43 (6) (1991) 1117–1124. [8] D.P.H. Hasselman, K.Y. Donaldson, J.R. Thomas Jr., Effective thermal conductivity of uniaxial composite with cylindrically orthotropic carbon fibres and interfacial thermal barrier, J. Compos. Mater. 27 (6) (1993) 637–643.
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[9] S. Nakamura, T. Hibiya, F. Yamamoto, T. Yokata, Measurement of the thermal conductivity of molten InSb under microgravity, Int. J. Thermophys. 12 (5) (1991) 783–789. [10] Y. Nagasaka, A. Nagashima, Absolute measurement of the thermal conductivity electrically conducting liquids by the transient hotwire method, J. Phys. E: Sci. Instrum. 14 (12) (1981) 1435–1440. [11] R.G. Richard, R. Shankland, A transient hot-wire method for measuring the thermal conductivity of gases and liquids, Int. J. Thermophys. 10 (3) (1989) 673–686. [12] J.G. Bleazard, A.S. Teja, Thermal conductivity of electrically conducting liquids by the transient hot-wire method, J. Chem. Eng. Data 40 (4) (1995) 732–737. [13] H.M. Roder, A transient hot wire thermal conductivity apparatus for fluids, J. Res. Natl. Bur. Stand. 86 (5) (1981) 457–480.
[14] R.A. Perkins, H.M. Roders, C.A. Nieto de Castro, A high temperature transient hot-wire thermal conductivity apparatus for fluids, J. Res. Natl. Inst. Stan. 96 (3) (1991) 247–269. [15] H.M. Roder, R.A. Perkins, A. Laesecke, Absolute steady-state thermal conductivity measurements by use of a transient hot-wire system, J. Res. Natl. Inst. Stan. 105 (2) (2000) 221–253. [16] A.I. Johns, A.C. Scott, J.T.R. Watson, D. Ferguson, A.A. Clifford, Measurement of the thermal conductivity of gases by the transient hot-wire method, Philos. Tr. R. Soc. S-A 325 (1585) (1988) 295–356. [17] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, London, 1959 (Chapter 5). [18] REFPROP, NIST Standard Reference Database 23, Version 7.01, 2004.
Contents lists available at SciVerse ScienceDirect
Measurement journal homepage: www.elsevier.com/locate/measurement
Measurement of thermal conductivity of fluid using single and dual wire transient techniques Siddharth Komini Babu 1, K.S. Praveen 1, B. Raja ⇑, P. Damodharan 1 Indian Institute of Information Technology, Design and Manufacturing (IIITD&M) Kancheepuram, Chennai 600 127, India
a r t i c l e
i n f o
Article history: Received 20 August 2012 Received in revised form 7 February 2013 Accepted 13 May 2013 Available online 25 May 2013 Keywords: Transient hot wire Platinum Thermal conductivity Dual wire method
a b s t r a c t A modified measurement device to measure thermal conductivity of fluids using transient hot-wire technique has been designed, developed, tested and presented in this paper. The equipment is designed such that the thermal conductivity could be measured using both single wire sensor of different length and dual wire sensor. The sensor, which is also a heater, is a platinum micro-wire of 50 lm diameter. The influence of wire length on the measurement of thermal conductivity of fluids is tested using two single wires of length 50 mm and 100 mm. The thermal conductivity is also measured using a dual hot wire arrangement; which is achieved by placing the 100 mm and 50 mm wires in a Wheatstone bridge with the 100 mm wire as the sensor and 50 mm wire as a compensation wire. The apparatus requires a 100 ml of test fluid to perform the experiment. The testing temperature of the test fluid during the experimentation can be suitably varied by the choice of heat exchange fluid used in the apparatus. Water is chosen as testing fluids for primary standards. When compared to single wires, the thermal conductivity of the fluids measured is consistent with dual-wire method with an uncertainty of ±0.25%. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The thermal conductivity of fluids is the most essential parameter required for thermal design and its consideration. Laser flash method [1], thermo reflectance techniques [2], 3-omega technique [3–5] etc. techniques are gaining importance since it measures properties based on either steady state or pseudo steady state solutions. Traditionally, transient hot wire techniques; which uses Fourier’s transient heat conduction model, is a well-established method for accurate measurement of thermal conductivity over a wide variety of substances such as solids
⇑ Corresponding author. Tel.: +91 044 27476355; fax: +91 044 27476301. E-mail addresses: [email protected] (S. Komini Babu), [email protected] (K.S. Praveen), [email protected] (B. Raja), [email protected] (P. Damodharan). 1 Tel.: +91 044 27476355; fax: +91 044 27476301. 0263-2241/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.measurement.2013.05.017
[6], polymers [7,8], and molten substances [9]. Among these samples, determining the thermal conductivity of fluid and gas samples is very complicated and delicate. Even though fluids and gases can transfer through conduction, tend to invoke natural convection current which eventually superimposes the desired thermal conductivity value achieved using Fourier heat conduction. A variety of instruments have been modified and improvised in the past for better accuracy of thermal conductivity of fluids using hot wire technique [10–16]. In principle the method can be used to obtain both the thermal conductivity and the thermal diffusivity using the same experiment. It is seen that the wire diameter, length and material, nature of the fluid, material of construction, modes of heat transfer, electrical conductivity of the fluid etc. plays a vital role in measuring the thermal conductivity of the fluid accurately. In this paper a redesign of the hot wire apparatus that could used to measure the thermal conductivity of liquids is presented.
S. Komini Babu et al. / Measurement 46 (2013) 2746–2752
2747
Nomenclature T r t q k R
final temperature (K) radial distance (m) time (s) heat flux (W m2) thermal conductivity (W m1 K1) resistance (X)
Greek letters a thermal diffusivity (m2 s1) D change c Euler’s constant e emissivity
2. Mathematical model
Stephan Boltzmann constant
Subscript f 0 LONG SHORT g ID PT
fluid initial long wire short wire bridge ideal platinum
DT 2 DT 1 ¼
The platinum wire, which serves as both sensor and heater, is modeled as an infinitely long and thin, ideal continuous line source dissipating heat into an infinite medium, with constant heat generation [17]. Since the wire is very thin, line source is assumed to have an infinite thermal conductivity and zero heat capacity. Thus, the heat conduction process could be represented as the Fourier one dimensional transient heat conduction (Eq. (1))
1 @ @T 1 @T ¼ r r @r @r af @t
ð1Þ
The boundary condition for the above equations is
@T q ¼ t ¼ 0 and r ¼ 0lim r r!0 @r 2pkf
ð2Þ
t P 0 and r ¼ 1limfDTðr; tÞg ¼ 0 r!1
where DT ¼ T T 0 Using the above, the temperature change at a radial distance r, from the heat source is conforms to a Eq. (3) by applying boundary conditions (2). Upon exponential integration the solution is given in Eq. (4).
DTðr; tÞ ¼ Tðr; tÞ T 0 ¼
r
q r2 Ei 4pkf 4af t
ð3Þ
DT ¼ Tðr;tÞ T 0 8 2 2 39 > > r2 r2 < = 4af t 4af t q 6 4af t 7 þ ¼ c þ ln þ :: þ ::::::: 4 5 2 > 1 1! 4pkf > 2 2! r : ; ð4Þ 4af t q DT ¼ Tðr; tÞ T 0 ¼ c þ ln 4pkf r2
ð5Þ
where c ¼ 0:5772 is the Euler constant At any fixed radial distance, in two instances in time the equation, the temperature change can be represented as the following equation:
q t2 ln 4pkf t1
ð6Þ
A plot of temperature against the natural logarithm of time results in a straight line, the slope being propositional to kf
kf ¼
q d lnðtÞ 4p dDT
ð7Þ
Eq. (3) can also be re-written as Eq. (8) such that both thermal conductivity and thermal diffusivity could be achieved from the same equation
4af t q DT ¼ Tðr; tÞ T 0 ¼ c þ ln 4pkf r2 4af t k q ¼ ln e 4pkf r2 4af q q ln 2 ek þ lnðtÞ ¼ 4pkf 4pkf r
ð8Þ
The slope of the equation can give the thermal conductivity and the intercept will give the thermal diffusivity. In this paper thermal conductivity is the matter of interest. 3. Apparatus design and circuitry The schematic layout of the apparatus along with the instrumentation circuitry is shown in Fig. 1a, b and c and the essential apparatus dimensions are shown in Table 1. The apparatus consist of three chambers respectively vacuum chamber (1), coolant chamber (2) and the platinum micro-wire assembly chamber (3). The entire apparatus is constructed using SS 316. The vacuum chamber, which is the exterior compartment, ensures heat infiltration to and from the ambient. It is ensured that the chamber is vacuumed using a vacuum pump (4), prior conducting the experimentation. The next inner concentric chamber is the coolant chamber. Based on the temperature requirement, the coolant can be appropriately chosen to maintain the temperature in the micro-wire chamber which is filled with the test fluid. The coolant is pumped from a temperature controlled bath into the coolant chamber through 5 and leaves the chamber through exit pipe 6. The temperature of the
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Fig. 1. Schematic layout of the apparatus.
Table 1 Essential parameters. S. No.
Description
Dimension
1 2 3 4 5 6
Diameter of the micro wire Length of the long wire Length of the short wire Diameter of the bore that holds micro wire Outer diameter of the coolant chamber Outer diameter of the vacuum chamber
50 lm 100 mm 50 mm 8 mm 100 mm 150 mm
coolant in its chamber is measured using PT100 sensors TS1 and TS2. The inner most chamber is the micro-wire chamber in which the platinum micro-wires are fixed. There are three tubes respectively 7, 8 and 9 for short wire, long wire and common connecting wires. The upper end of three tubes is welded to manifold to which entry (8) and exit (9) valve for test fluid are connected. The PT100 Temperature sensors TS3 and TS4 are used to measure the temperature at the inlet and drain manifolds. The platinum micro-wires are held inside the tube 7 and 9 by using the Teflon holder (10) and soft silicone rubber as shown in Fig. 1b. The platinum wire is exposed to fluid only for 100 mm in the long wire and 50 mm in the short wire. The remaining wire connections are soldered using silver wire, which are Teflon sleeved. The electrical connectivity is carefully brought outside the vessel using a feed through connectors. The most critical operation is the silver soldering of the micro-wires with the connecting wires and
bringing out the apparatus using the feed through connectors. The entire assembly is perfectly insulated with thermo-foam insulation (11). The vacuum system provided is connected to a single stage vacuum pump, which gets its automatic feedback from an absolute pressure gauge. Since the fluid used in the present study is non-volatile, the liquid is manually fed by volume measurement. The data from the circuitry is taken to a computational device using Agilent data acquisition system. In a transient hot wire method, the thermal conductivity of a fluid is measured using the rate of increase of temperature of micro-wire with time for an applied step change of voltage. The overall apparatus is completely isolated from all sources of vibration, such that the test fluid in which the micro-wires are suspended is in absolute static condition. This condition is synonymous to Eq. (1), which mandates that the heat transfer process in the fluid is only through conduction. The supply voltage generates a constant heat flux per unit length throughout the microwire which is transferred to the static fluid medium. The change in resistance of the sensing wire is related to the rate at which the heat is removed from the wire and thus the thermal properties of the fluid. The resistance of the micro-wire during heating could also be measured by placing a resistor in series with a power supply, finding the potential difference across the resistor and dividing by the current through the micro-wire. This would only end in practical difficulties and inaccuracies. For instance, in order to measure resistance, consider that the voltmeter is
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connected across an unknown resistance. A small current is essential to deflect the moving coil in case of an analog voltmeter. This current will then flow through the ammeter in addition to the main current through the resistance. The ammeter reading will be too high and quotient of the two meters readings will be too low. Now consider a technique in which a voltmeter connected across both the resistor and the ammeter. The voltmeter reading will be too large since it measures the potential difference across the ammeter as well, which would result in large resistance. Difficulties in the method attributed to uncertainties to the construction of the meter, which further depend upon such factors as smallest division and the perfectness in the calibration. The uncertainties are compounded when the two meter readings are divided to calculate resistance. For the most accurate measurement of resistance, the Wheatstone bridge circuit is used. This circuit avoids most of the difficulties in the aforementioned methods. This method demands high precision measurement of resistance change making the Wheatstone circuit as the ultimate choice. To construct the Wheatstone bridge, the long wire (L) is placed in one working arm of the bridge and the short wire (S) in the other. An accurate power supply of 5 V DC is applied to the bridge and voltage across the bridge is read as a function of time. Channels are provided in the circuit shown in Fig. 1c is to measure the current and voltages in the circuit. The measurement of thermal conductivity is accomplished in two steps. In the first step the bridge is balanced to null. The switch 1 is turned from the dummy side to the bridge keeping the switch 2 closed. With a very small applied voltage 0.1 V and keeping the cell at constant temperature the leads, hot wires and ballast resistances are measured. The ballast resistance is adjusted until each leg is approximately 100 X. The bridge null is then checked across the channel 6 to ensure the balanced condition. The second step involves the conductivity measurement. Before switching on the power supply, the switch 1 is turned to the dummy side as a precaution to avoid the pre heating of platinum wires keeping the switch 2 closed. The power supply is then set to 5 V and the switch1 is turned to the bridge. Voltages are read on channel 6 using the data logger for 5 s and the values are stored. The power is then switched back to the dummy side to prevent the wires from getting over heated. The circuitry shown in Fig. 1c is capable of accomplishing conductivity measurements using long, short and dual wires arrangement individually. The switches 3 and 4 are provided in the circuit to necessitate the desired measurement by absenting the unwanted wire from the circuit. For instance, if dual wire measurement is to be taken both switches 3 and 4 are closed. For long wire reading an external 100 X is connected to the short wire terminal to switch 4 and similarly for short wire the resistance is externally connected to the long wire terminal and the switch 3. The power supply used provides a constant voltage at the circuit terminals. The instantaneous value of heat flux applied to the circuit is calculated from the Eq. (11). The ideal analysis states that the heat flux q, applied to the wire remains constant during a given run. Under experimental conditions q is assumed to be nearly ideal. However at very
low temperature the q will vary from the beginning to end of a run by values up to several percentages. V (t) is assumed to vary one part in 1000 over a run, and some assume it to vary linearly with ln (t). The voltages measured across the bridge measure the unbalance of the bridge. A typical set of over 1000 readings per sec is used to get the plot. The voltage plot is observed to be logarithmic with time. A certain amount of judgement is required in choosing the required part of the data for calculating the thermal conductivity. The bridge voltage is measured for more than 1 s to enunciate the logarithmic curve but for calculating the thermal conductivity of the fluid, the conduction part is considered. A typical set of reading is used to get the plot between DT and ln (t). The standard for a valid result is that DT values plotted, forms a straight line with ln (t). The range of time over which this linear plot exists is considered for evaluating the thermal conductivity. For most of the experiments the range is between 100 ms and 950 ms. It is also important in identifying the improper results to be scorned. If the value of DT is constant in the run then the fluid sample is interpreted to be in steady state or as the onset of convective currents. A non-uniform temperature field in the wires signals the measurement to be rejected, for this indicates that the segment is subjected to convective cooling. It was observed that measurements on water between 100 and 950 ms were found to be appropriate.
DV g ¼
R1 ðR3 þ DR3 Þ R2 ðR4 þ DR4 Þ V IN ðR1 þ R2 ÞðR3 þ DR3 þ R4 þ DR4 Þ
ð9Þ
R1 ¼ R2 ¼ R3 ¼ R4 ¼ 100 X where R3 = Rbalast1 + Rlong Simplifying
DV g ¼
and
V IN RðtÞ 8 R3
R4 = Rbalast2 + Rshort
ð10Þ
qðtÞ ¼ I2 R ¼ ½VðtÞ=ðR3 ðtÞ þ R4 ðtÞÞ2 ðRLong þ RShort Þ=ðlLong þ lShort Þ ð11Þ
DR3 ¼ DRLONG þ D RLeads DR4 ¼ DRSHORT þ D RLeads Initial Conditions bridgebalanced R1 =R2 ¼ R4 =R3 ¼ 1
DRSHORT ¼ DRLONG =2 Rt ¼ Ro ð1 þ a DTÞ
ð12Þ
Equating (9), (10) yields DR and further substituting in Eq. (12) yields DT seen in Eq. (8). The relation given in the equation implies a straight line for a plot of DT versus ln (t). The slope of the T versus ln (t) relation is obtained over a valid range using the applied power. The numerical slope value is equal to slope part of the Eq. (8), which yields thermal conductivity.
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4. Temperature corrections It is essential to mention about the inherent deviation present in approach. As per the mathematical model, the wire is assumed have a zero radius and infinite length immersed in an infinite medium. But in reality both the diameter and length of the wire have definite values. Also, the sensor, the platinum wire, mathematically has infinite thermal conductivity and zero heat capacity, which is approximate. Due to the heating of the micro-wire, the temperature difference in the test fluid could activate natural convection. Even though Eq. (8) mandates only conduction heat transfer, presence of convection would reduce the temperature and enhance the mathematical value of the thermal conductivity. Still, the presence of such a convective current could be identified by departure from the non-linear trend. The convection current are dominant in gases, and in the presence study the test fluids are liquids and thus effect of convective current is assumed negligible. Most of the fluids are not transparent to infrared and radiation component in those fluids follows the same radiation principle. Considering the radiation between the platinum wire and the cell wall as between diffusive grey surfaces, the radiation is correction is taken as Eq. (13) [13–15].
dT R ¼
8pr ePT rT 3O DT 2ID q
Fig. 2. Transient variation of bridge voltage.
and direct proportionality of resistance with the length, the resistance is less, when compared to other two arrangements. Further, the dual wire arrangement has resulted in less electrical noise when compared to the other two arrangements. Measurements of this type indicate that end correction would be required for single wires whereas in dual wire the compensation is provided by the bridge circuit arrangement. 5.2. Thermal conductivity of water
ð13Þ
5. Results and discussion The test facility to measure thermal conductivity was constructed and the experiments were conducted with water at 1 atm for temperature between 15 and 50 °C. Both the test fluids are infrared absorbing substances and therefore, suitable correction factors are taken into account. Free convection is predominant only in gases and the factors are not considered in the present study. The thermal conductivity for the test condition is measured using the long wire, short wire and the dual wire and compared.
The variation of the temperature rise of the platinum wire with the logarithmic time is studied for all the three configurations are shown in Fig. 3. It is clearly seen from the figure that the long wire holds a plot very close to the dual wire configuration while the short wire being well below the other two configurations. It is observed that the
5.1. Bridge voltage The transient variation of short wire, long wire and dual wire arrangement is shown in Fig. 2. The transient measurements were of close to 5 s and performed for four scans. In all the methods, the logarithmic variation shows two regions of the voltage rises for all the three combination in the Wheat stone bridge, which is demarcated by transient and steady variation of bridge voltage. It is evident that from the figure that as the length of the wire decreases both transient and steady state variation merges, which enhances the error in the value of thermal conductivity. Eq. (8) requires the transient variation with sufficient slope to estimate the thermal conductivity. On the other hand, the long wire yields a slope closer to that of a dual wire. When compared with the dual wire, it could be seen that the steady state region is delayed by 0.1 s in the sample reading shown, which is advantageous to measure the thermal conductivity. Also, due to the short length
Fig. 3. Transient variation of temperature rise.
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Fig. 4. Radiation correction on the dual wire.
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observed that both measurement using short and long wire over predicted by +2% and 1.5%. The dual-wire technique very closely matches with an accuracy of ±0.25%. The major practical difficulty in any hot wire apparatus is fixing the platinum micro-wire in the bore and providing a uniform temperature to the test fluids. Both these problems were easily tackled in the present design. Also in the place of electrical heater to rise the temperature of the fluid, as seen in some designs, induction current place a negative role because the entire apparatus is usually preferred to be constructed with stainless steel. The presence of liquid heating and cooling in this present design has completely eradicated the presence of induction. 6. Conclusion
Fig. 5. Comparison of thermal conductivity of water.
A transient hot-wire thermal conductivity instrument is designed and constructed. The equipment is tested with water for the measurement of the isobaric thermal conductivity. The measurement is taken using different length of the wires and their combination in the form of dual wire along with the necessary correction for radiation. It observed that as the length of the wire sensor is decreased, the desired slope of the transient variation of bridge voltage against the time seems to vanish. Thus, the experiment has to be conducted for very small interval of time, which is not really practical in the presence of electrical noise. The dual wire arrangement has delayed the steady state variation when compared to long wire arrangement. The thermal conductivity of the fluids measured is consistent with dual-wire method with an accuracy ±0.25%. Acknowledgements This work is supported by Department of Science and Technology (DST-SERC), India (Grant No. SR/FTP/ETA0017/2010). We thank M/s Delvac Pumps, Chennai, India for extending their facilities to fabricate the dual hotwire device with precise accuracy. References
Fig. 6. Deviation plot.
rise in temperature of the short wire is less than the dual wire configuration by almost 0.25 °C. The slope and the constant of the linear fit, both vary, thereby yielding a small difference in the values of thermal conductivity and diffusivity for different configurations. The influence of correction factors on thermal conductivity is shown in Fig. 4. It is seen that the radiation correction has increases a mean temperature rise of about 0.02 K. However, its importance in the thermal conductivity value is only marginal of around 2%. The variation of measured thermal conductivity for all the three arrangements along with the NIST [18] values for water at different temperature and the deviation plot for the same are shown in Figs. 5 and 6 respectively. It is
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