Appendix A Uncertainty Analysis for Experimental Data

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1 Appendix A Uncertainty Analysis for Experimental Data To compute the uncertainty in the experimental data of this work, error analyses have been conducted according to the principles proposed by Taylor [1]. The error analysis procedures are summarized below: Uncertainty in Sums and Differences Suppose that x,, w are measured with uncertainties dx,, dw, and the measured values used to compute f ¼ x þþz ðu þþwþ If the uncertainties in x,, w are known to be independent and random, then the uncertainty in f is the quadratic sum of the original uncertainties. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi df ¼ ðdxþ 2 þþðdzþ 2 þðduþ 2 þþðdwþ 2 In any case, df is never larger than their ordinary sum, df dx þþdz þ du þþdw Uncertainties in Products and Quotients Suppose that x,, w are measured with uncertainties dx,, dw, and the measured values used to compute f ¼ x z u w If the uncertainties in x,..., w are independent and random, then the fractional uncertainty in f is the sum in quadrature of the original fractional uncertainties, T. Alam et al., Flow Boiling in Microgap Channels, SpringerBriefs in Thermal Engineering and Applied Science, DOI: / , Ó The Author(s)

2 76 Appendix A: Uncertainty Analysis for Experimental Data s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi df jf j ¼ dx 2 þþ dz 2 þ du 2 þþ dw 2 jj x jj z juj jwj In any case, it is never larger than their ordinary sum, df jf j dx jj x þþdz jj z þ du jj u þþdw jwj Uncertainty in Any Function of One Variable If x is measured with uncertainty dx and is used to calculate the function f(x), then the uncertainty df is df ¼ df dx dx Uncertainty in a Power If x is measured with uncertainty dx and is used to calculate the power f = x n (where n is a fixed, known number), then the fractional uncertainty in f is jnj times that in x, df jf j ¼ jnjdx jj x Uncertainty in a Function of Several Variables Suppose that x,, z are measured with uncertainties dx,, dz, and the measured values used to compute the function f(x,, z). If the uncertainties in x,, zare independent and random, then the uncertainty in f is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of 2 df ¼ ox dx þþ of 2 oz dz In any case, it is never larger than their ordinary sum,

3 Appendix A: Uncertainty Analysis for Experimental Data 77 Table A.1 The measurement accuracies and experimental uncertainties associated with sensors and parameters Sensors and parameters Accuracies and uncertainties T-type thermocouples ±0.5 C Diode temperature sensors ±0.3 C Flow meter ±5 ml/min Pressure transducer ±1.8 mbar Differential pressure transducer ±1 mbar Voltage measurement ±0.06 V Current measurement ±0.15 A Dimension measurement ±10 lm Heat flux 2 8 % Pressure drop 4 18 % Heat transfer coicient 4 10 % df of of oxdx þþ oz dz Table A.1 shows the measurement accuracies and experimental uncertainties associated with sensors and parameters. Table A.2 shows a set of uncertainty values in different parameters calculated based on the above equations for 300 lm depth microgap at mass flux, G = 390 kg/m 2 s and heat flux, q 00 ¼ 52:5 W=cm2 :

4 78 Appendix A: Uncertainty Analysis for Experimental Data Table A.2 Sample uncertainty calculation for 300 lm depth microgap at mass flux, G = 390 kg/m 2 s and heat flux, q 0 ¼ 52:5W=cm2 Width (W) Length (L) Thickness (t) Voltage (V) Current (I) T f T d K s 1.27 ± cm 1.27 ± cm ± cm 10.7 ± 0.06 V 8.5 ± 0.15 A ± 0.5 C C 1.21 W/cm C ±0.3 C Calculation dq 00 ¼ q 00 s q 00 ¼ q A ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ di 2 þ dw 2 þ dl 2 dv jv j Tf ¼ Tsat dtf ¼ dtsat ¼0:5 C jj I jw j jl j ¼ fðv; I; W; L Þ ¼ 52:5 W/cm2 ¼ 0:01855 ¼ 1:855 % 2 % Tw ¼ q00 t ¼ 116:77 C Td s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ks otw dtw ¼ :dtd þ ot w otd oq 00 :dq 00 þ ot w :dt þ ot w :dks ot oks s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ¼ ð1:dtd Þ 2 þ t :dq 00 þ q00 :dt þ0 ¼ 0:3 C Ks Ks Tw Tf DT ¼ ¼ 15:005 C q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ðtw Tf Þ ¼ ddt ¼ ðdtw Þ 2 þ ðdtf Þ 2 ¼0:6 C dhz j j ¼ hz v uut hz ¼ q00 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DT! 2 dq 00 þ ddt 2 ¼ 0:038 ¼ 3:8 % 4 % q 00 jdt j

5 Appendix B Nomenclature A Footprint area (cm 2 ) A c Wetted area of microchannel (cm 2 ) A gap Microgap cross-sectional area (cm 2 ) A man Manifold cross-sectional area (cm 2 ) Bl Boiling number Bo Bond number Co Confinement number c p Specific heat, (J/kg C) d Depth of microchannel (lm) D Microgap depth (lm) g Gravitational acceleration G Mass flux (kg/m 2 s) h Heat transfer coicient (W/m 2 K) h fg Heat of vaporization (J/kg) k s Thermal conductivity, W/cm C K c Loss coicient L Length of the substrate (cm) _m Mass flow rate (kg/s) N Number of microchannels P Pressure (bar) DP Pressure drop (bar) q Total heat dissipation (W) q Effective heat dissipation (W) q 00 Effective heat flux (W/cm 2 ) q loss Heat loss (W) R a Roughness parameter (arithmetic mean value) Re Reynolds number R t Roughness parameter (maximum peak to valley height) t Substrate thickness (cm) T Temperature ( C) T. Alam et al., Flow Boiling in Microgap Channels, SpringerBriefs in Thermal Engineering and Applied Science, DOI: / , Ó The Author(s)

6 80 Appendix B: Nomenclature V d w W x z Voltage drop across diode (V) Channel width (lm) Width of the substrate (cm) Vapor quality z-coordinate (axial distance) (cm) Greek Symbols q Density (kg/m 3 ) l Dynamic viscosity (Ns/m 2 ) r Surface tension (N/m) g Fin iciency Subscripts c d e f g gap i o man s sat sp w z Contraction Diode Expansion Liquid Vapor Microgap Manifold inlet Manifold outlet Manifold Substrate Saturated Single-phase Wall Local

7 Appendix C Data Reduction Heat Transfer Data Reduction The ective heat supplied, q to the fluid in each test piece and the ective heat flux q 00 is calculated as given. The ective heat transfer rate, q to the fluid in microgap channel is obtained by: q ¼ q q loss ðc:1þ Where q is input power and q loss is heat loss. The ective heat flux q 00 that the heat sink can dissipate is calculated from: q 00 ¼ q ðc:2þ A where A is the base area of heat sink, A = W 9 L. For microchannel, the total wetted area of the microchannels is: A c ¼ Nðw þ 2gdÞL ðc:3þ where N is total number of channels; w, d and L are the width, depth, and length of the channel respectively and g is the iciency of a fin with adiabatic tip which is correlated by: g ¼ tanhðmdþ ðc:4þ md and rffiffiffiffiffiffiffiffiffiffiffi 2h m ¼ ðc:5þ K s w w where K s is the thermal conductivity of the substrate and w w is the width of the channel wall. So, the wall heat flux for microchannel is defined as q 00 w ¼ q A c ðc:6þ T. Alam et al., Flow Boiling in Microgap Channels, SpringerBriefs in Thermal Engineering and Applied Science, DOI: / , Ó The Author(s)

8 82 Appendix C: Data Reduction The test section is divided into two regions: an upstream subcooled inlet region and a downstream saturated region as subcooled (T f,i \ T sat ) water is supplied into the heat sink for all test conditions; the location of zero thermodynamic equilibrium quality (x = 0) serves as a dividing point between the two regions. The local heat transfer coicient in microgap is calculated from, h z ¼ q00 T w T f The local heat transfer coicient in microchannel is calculated from, q h z ¼ A c ðt w T f Þ ðc:7þ ðc:8þ in which T f is the fluid temperature as defined by T f ¼ T f;i þ q00 Wz _mc p ðsingle phase regionþ ðc:9þ where z, _m and c p are the axial distance, mass flow rate, and specific heat respectively. T f ¼ T sat ðsaturated regionþ ðc:10þ T w, is the local wall temperature. This temperature is corrected assuming one dimensional heat conduction through the substrate T w ¼ T d q00 t K s ðfor microgapþ ðc:11þ T w ¼ T d q00 ðt dþ ðfor microchannelþ ðc:12þ K s where t and K s are the substrate thickness and thermal conductivity respectively. T d is the measured temperature by an integrated diode. Bond number is defined as the ratio of buoyancy force to surface tension force. Bo ¼ gðq f q g Þ D 2 ðc:13þ r where r is the surface tension, g is the gravitational acceleration, q f and q g are liquid and vapor densities of fluid respectively. D is the gap depth. Some other non-dimensional parameter like Boiling number, Bl which is non-dimensional heat flux and Reynolds number, Re are defined as follows: Bl ¼ q00 Gh fg Re ¼ GD l f ðc:14þ ðc:15þ

9 Appendix C: Data Reduction 83 where h fg and l f are the heat of vaporization and dynamic viscosity of fluid respectively. Pressure Drop Data Reduction Pressure taps are located across the microgap and microchannel inlet and outlet plenum. These taps are positioned as close as possible to the test die. Pressure losses by the sudden contraction (DP c ) and the sudden enlargement (DP e ) were very small compared with the frictional pressure drop. Though these values are very small of total pressure changes, the pressure drop and the pressure recovery at the sudden contraction and the sudden enlargement were considered for calculation of the total pressure drop. Pressure losses are calculated based on the methods described in Blevins [2], Chislom and Sutherland [3] and Collier and Thome [4]. As mentioned earlier, subcooled water (T f, i \ T sat ) is supplied into the heat sink for all test conditions. The pressure drop associated with the liquid flow at the sudden contraction in microgap channel is calculated as " DP c ¼ G2 1 A # 2 gap þk c ðc:16þ 2q f A man where G is mass flux in the microgap, q f is liquid density and K c is the nonrecoverable loss coicient for laminar flow given by K c ¼ 19 l f þ 0:64 ðc:17þ GD The pressure recovery at the sudden enlargement at the exit is calculated as DP e ¼ G2 A gap 1 A "! # gap 1 þ q f 1 x ðc:18þ q f A man A man q g The microchannel pressure drops (DP) are calculated as follows. The pressure drop associated with the liquid flow at the sudden contraction is calculated as " DP c ¼ G2 1 NA # 2 ch þk c ðc:19þ 2q f A man where G is mass flux in the microgap, q f is liquid density and K c is the nonrecoverable loss coicient for laminar flow given by K c ¼ 0:0088 d 2 0:1785 d þ 1:6027 ðc:20þ w w

10 84 Appendix C: Data Reduction The pressure recovery at the sudden enlargement at the exit is calculated as DP e ¼ G2 NA ch 1 NA "! # ch 1 þ q f 1 x ðc:21þ q f A man A man q g Therefore, the pressure drops (DP) reported below are DP ¼ ½ðP i DP c Þ ðp o þ DP e ÞŠ ðc:22þ References 1. Taylor JR (1997) An introduction to error analysis: The study of uncertainties in physical measurements, University Science Books, 2nd edn. US 2. Blevins RD (1991) Applied fluid dynamics handbook. Krieger Publishing Co., Berlin, pp Chislom D, Sutherland LA (1969) Prediction of pressure gradients in pipeline systems during two-phase flow. Symposium in two-phase flow systems. University of Leeds 4. Collier JG, Thome JR (1994) Convective boiling and condensation. Clarendon Press, Oxford

Appendix A: Uncertainty Analysis

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