Appendix A: Uncertainty Analysis

Transcription

1 Appendix A: Uncertainty Analysis o compute the uncertainty in the experimental data o this work, error analyses have been conducted according to the principles proposed by aylor [1]. he error analysis procedures are summarized below: Uncertainty in Sums and Dierences Suppose that x,, w are measured with uncertainties δx,, δw and the measured values used to compute = x+¼+ z- ( u+¼+ w) I the uncertainties in x,, w are known to be independent and random, then the uncertainty in is the quadratic sum o the original uncertainties: d dx. dz du d w = ( ) +¼ + ( ) + ( ) +¼+ ( ) In any case, δ is never larger than their ordinary sum: d dx+¼+ dz+ du+¼+ d w Uncertainties in Products and Quotients Suppose that x,, w are measured with uncertainties δx,,, δw and the measured values used to compute = x ¼ z u ¼ w he Author(s) Alam, P.-S. Lee, Flow Boiling in Expanding Microchannels, SpringerBries in Applied Sciences and echnology, DOI /

2 74 Appendix A: Uncertainty Analysis I the uncertainties in x,, w are independent and random, then the ractional uncertainty in is the sum in quadrature o the original ractional uncertainties: d dx dz du d w = æ ö ç x +¼+ æ ö ç è ø è z + æ ö ç ø è u +¼+ æ ö ç ø è w ø In any case, it is never larger than their ordinary sum: d dx dz du d w +¼+ + +¼+ x z u w Uncertainty in Any Function o One Variable I x is measured with uncertainty δx and is used to calculate the unction (x), then the uncertainty δ is d = d dx dx Uncertainty in a Power I x is measured with uncertainty δx and is used to calculate the power = x n (where n is a ixed, known number), then the ractional uncertainty in is n times that in x: d = dx n x Uncertainty in a Function o Several Variables Suppose that x,, z are measured with uncertainties δx,, δz and the measured values used to compute the unction (x,, z). I the uncertainties in x,, z are independent and random, then the uncertainty in is 2 2 æ d d d x x ö = ç z è ø +¼+ æ ö ç è z ø

3 Appendix A: Uncertainty Analysis 75 able A.1 he measurement accuracies and experimental uncertainties associated with sensors and parameters Sensors and parameters Accuracies and uncertainties -type thermocouples ±0.5 C Diode temperature sensors ±0.3 C Flowmeter ±5 ml/min Pressure transducer ±1.8 mbar Dierential pressure ±0.5 - ±1 mbar transducer Voltage measurement ±0.06 V - ±0.62 V Current measurement ±0.035A A Dimension measurement ±10 μm Heat lux 2 15% Heat loss Up to 20% Pressure drop 4 18% Heat transer coeicient 4 20% In any case, it is never larger than their ordinary sum: d dx +¼+ d z x z able A.1 shows the measurement accuracies and experimental uncertainties associated with sensors and parameters.

4 Appendix B: Data Reduction Microgap Data Reduction he eective heat transer rate, q e, to the luid in microgap channel is obtained by q = q- q (1) e where q is input power and q loss is heat loss during low boiling experiment. he eective heat lux q e that the heat sink can dissipate is calculated rom q e loss qe = (2) A where A is the wetted area o silicon heat sink, A = W L. he local heat transer coeicient in microgap is calculated rom h z qe = w- (3) in which is the luid temperature as deined by qewz =, i + ( Single - phase region ) (4) mc p = ( phase region ) (5) sat w is the local wall temperature. his temperature is corrected assuming onedimensional heat conduction through the substrate w qe t = d - K s (6) he Author(s) Alam, P.-S. Lee, Flow Boiling in Expanding Microchannels, SpringerBries in Applied Sciences and echnology, DOI /

5 78 Appendix B: Data Reduction where d is the measured temperature by an integrated diode. t and K s are the substrate thickness and thermal conductivity, respectively. All the heat transer results presented in this work are based on the location last downstream along the center row (as shown in Fig. 2.1(c)) in the microgap as it corresponds to the highest degree o saturated boiling and there is a signiicant progression o heat transer perormance as move downstream [2]. Moreover, the temperature variations o test section in the lateral direction are negligible beore dryout phase [3]. Pressure losses by the sudden contraction and the sudden enlargement are very small compared with the rictional pressure drop in the microgap. hough these values are only 1 6% o total pressure changes, the pressure drop and the pressure recovery at the sudden contraction and the sudden enlargementare considered or calculation o the total pressure drop. hereore, the pressure drops (ΔP gap ) reported below are ( ) DPgap = éë DP- DPc + DPe ù û (7) Further details o data reduction can be ound in Alam et al. [3]. Microchannel Data Reduction For microchannel, the total wetted area o the microchannels is ( ) A = N w + c 2h H L (8) where N is total number o channels; w, H, and L are the width, depth, and length o the channel, respectively; and η is the eiciency o a in with adiabatic tip which is correlated by h = tanh ( mh ) mh (9) and m = 2h Kw s w (10) where K s is the thermal conductivity o the substrate and w w is the width o the channel wall. So, the wall heat lux or microchannel is deined as q w qe = (11) A c

6 Appendix B: Data Reduction 79 he local heat transer coeicient in microchannel is calculated rom w h z qe = A ( ) c w- ( ) qe t- H = tc - K s (12) (13) where t and K s are the substrate thickness and thermal conductivity, respectively. tc is the measured temperature by a thermocouple. Further details o data reduction can be ound in Balasubramanian et al. [4, 5]. Reerences 1. J.R. aylor, An Introduction to Error Analysis, 2nd edn. (University Science Books, New York, 1997) 2.. Alam, P.S. Lee, C.R. Yap, L.W. Jin, A comparative study o low boiling heat transer and pressure drop characteristics in microgap and microchannel heat sink and an evaluation o microgap heat sink or hotspot mitigation. Int. J. Heat Mass rans. 58, (2013) 3.. Alam, P.S. Lee, C.R. Yap, L.W. Jin, Experimental investigation o local low boiling heat transer and pressure drop characteristics in microgap channel. Int. J. Multiphase Flow 42, (2012) 4. K. Balasubramanian, P.S. Lee, L.W. Jin, S.K. Chou, C.J. eo, S. Gao, Experimental investigations o low boiling heat transer and pressure drop in straight and expanding microchannels a comparative study. Int. J. herm. Sci. 50, (2011) 5. K. Balasubramanian, P.S. Lee, C.J. eo, S.K. Chou, Flow boiling heat transer and pressure drop in stepped in microchannels. Int. J. Heat Mass rans. 67, (2013)

7 Index B Bubble dynamics, 71 C Cole Parmer Bench op Analog Drive EW , 63 Critical heat lux (CHF), 4, 53, 71 E Expanding microgap, 7 9, 11 13, 15, 17, 20, 22, 24 F Flow boiling, 1 4, 7, 9, 11, 13, 15, 17, 18, 22, 24 Flow boiling heat transer, 7, 9 11, 13, 17, 18, 20, 22 Flow boiling instabilities in expanding microgap channel heat lux eects, mass lux eects, microgap size eects, Flow boiling microscale heat sinks, 71 Flow visualization, 2, 4, 7, 9, 47, 50, 57 H Heat transer coeicients (HCs), 54 High-amplitude with low-requency (HALF) oscillation, 31, 42 High-speed low visualizations, 7, 9 Hotspot, 1, 3, 20 I IGBs, 61, 67, 68 Inlet pressure oscillation, 27, 28, 30, 32, 33, 35, 37, 39, 40, 42, 45, 46 L Low-amplitude with high-requency (LAHF) oscillation, 35 M McMillan liquid low sensor, 7 Microchannel heat sink, 47, 48, 50, 59 Microchannels, 1 4 Microgap channels, 4 Microgap heat sink, 3, 4 Minichannel heat sink, 61, 62, 64, 68 Mitutoyo 3-axis measuring microscope, 7 O Onset o nucleate boiling (ONB), 12, 13, 15, 55, 65, 66 OriginPro sotware, 20 O-ring, 48, 62 P Photron FASCAM SA5 1000K-M3, 8 Portable Wide Mouth ASME 01 EA Pressure ank 304 SS, 63 he Author(s) Alam, P.-S. Lee, Flow Boiling in Expanding Microchannels, SpringerBries in Applied Sciences and echnology, DOI /

8 82 Index Pressure drop, 1 4, 7, 9 11, 13, 17, 18, 20, 22, 48 50, 56, 57, 59 Printed circuit board (PCB), 7 Pyrex glass, 9 S Silicon test piece, 7 Sloping in microchannels, 47 50, Stepped in microchannels, 61 69, 71 Straight microchannels, 47, 48, 50, 52 57, 59 elon, 61, 64 hermal management, 1 V Very high-amplitude with very low-requency (VHALF), 31 W Wire-cut electro-discharge machining process, 48