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["「熱電対による表面温度測定の誤差解析」　解析解シートの説明"]
[]
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["１．概要"]
["・","図１のように，熱電対の測温接点を測定面に接着し，リード線の一部を等温面に這わせた場合の誤差解析を行います．"]
[null,"（解析方法の詳細は，日本機械学会論文集, Vol.84, No.867, DOI: 10.1299/transjsme.18-00216 をご参照ください）"]
[]
["・","簡単のため，測定面は等温（温度 Tw）であり，熱電対の接触により温度が変化しないと仮定しています．"]
[null,"（一般に，測定面の熱容量が大きく熱伝導率が高い場合にはこの近似が成り立ちます）"]
[]
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[null,null,null,null,null,"図１　熱電対による表面温度測定"]
[]
["・","測温接点（junctiuon, 直径 dj）と測定面（measuring surface, 温度 Tw）は，接着剤（adhesive, 熱伝導率 ka）で"]
[null,"接着されています．"]
["・","一対の金属素線（metal wires，直径 dm）が被覆から露出している領域を“素線露出部(exposed region，長さ lm)”，"]
[null,"一対の被覆線（insulated wires，直径 di）がシースから露出している領域を“内側被覆部(inner insulated region, 長さ li)”"]
[null,"と表記しています．また，それ以降のリード線は，2本の被覆線が1本のシース（等価直径 ds）に覆われており，"]
[null,"そのうち加熱面に這わせている領域を“シース加熱部(heated sheathed region, 長さ ls)”，全周が空気流にさらされている"]
[null,"領域を“フィン部(fin region, 長さ ls')”と表記しています．"]
[]
[]
["２．解析で考慮している伝熱経路"]
["・","図１の伝熱経路のうち，図８のように主要な伝熱経路のみを考慮して解析解を導出しています．"]
[null,"（金属素線の熱伝導，および図８に示す熱抵抗 R を介した熱移動のみを考慮しています．ここでは簡単のため，"]
[null,"　図１の内側被覆部とシース加熱部は素線露出部に置き換えています．また，測温接点からの熱伝達（対流+放射）は"]
[null,"　小さいので考慮していません．絶縁テープの熱抵抗は，素線露出部の隙間の熱抵抗 Rgmに加算して考慮可能です）"]
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[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[null,null,null,null,null,"図８　解析解で考慮した伝熱経路"]
[]
["・","各熱抵抗の記号，意味，および論文中の式番号は以下の通りです．"]
[null,"Rj","測温接点と測定面間の熱抵抗",null,null,null,null,"式(10)"]
[null,"Rgm","素線露出部と測定面間の熱抵抗",null,null,null,null,"式(11)"]
[null,"Rhm","素線露出部と空気流間の熱抵抗（対流+放射）",null,null,null,null,"式(7)"]
[null,"Rs'","被覆+シース（フィン部）の厚さ方向の熱抵抗",null,null,null,null,"式(17)'","(式(17)の2lc→ls', ki=ks)"]
[null,"Rhs'","フィン部からの熱伝達による熱抵抗（対流+放射）",null,null,null,null,"式(18)'","(式(18)の2lc→ls')"]
[]
[]
["３．解析解シートの使い方"]
["(1)","「解析解」シートの水色のセルに条件を入力します．"]
[null,"熱電対の種類",null,"T or K",null,"T熱電対の場合は\"T\"を，K熱電対の場合は\"K\"を入力"]
[null,null,null,"素線径 dm",null,"金属素線の直径を入力"]
[null,"測温接点",null,"隙間の熱伝導率 ka",null,"測温接点と測定面間の接着剤の熱伝導率を入力"]
[null,null,null,null,null,"瞬間接着剤は0.1W/(m・K)程度，導電性接着剤は1~4W/(m・K)程度，はんだは50W/(m・K)程度"]
[null,"素線露出部",null,"長さ lm",null,"素線露出部の長さを入力"]
[null,"熱伝達率",null,"周囲風速 u",null,"測定面の周囲風速（時間平均風速）を入力"]
[null,null,null,"測定面辺長 L",null,"測定面の流れ方向辺長（論文の式(32)の Lの値）を入力"]
[null,null,null,"※ 測定面の対流熱伝達率が推定できる場合は，hcv (I24)の値を直接入力（この場合，測定面辺長は入力不要）"]
[null,null,null,"※ フィン表面の対流熱伝達率が推定できる場合は，hcv' (I27)の値を直接入力"]
[null,"測定面温度",null,null,null,"測定面温度 Tw を入力"]
[null,"周囲空気温度",null,null,null,"周囲空気温度 Tf を入力"]
[]
["(1)'","必要に応じて，以下の値を入力します"]
[null,"測温接点",null,"直径 dj",null,"測温接点の直径を入力．デフォルト値として，素線径の2倍の値が入力されています"]
[null,null,null,"測定面との隙間 gj",null,"最小隙間（実効値）を入力"]
[null,null,null,null,null,"デフォルト値として，論文で評価された値(0.01mm)が入力されています"]
[null,"素線露出部",null,"測定面との隙間 gm",null,"最小隙間（実効値）を入力"]
[null,null,null,null,null,"デフォルト値として，論文で評価された値(0.0721mm, 絶縁テープを加味)が入力されています"]
[null,null,null,"隙間の熱伝導率 kg",null,"隙間が接着剤等で充填されている場合は，充填剤の熱伝導率を入力"]
[null,null,null,null,null,"デフォルト値として，空気の熱伝導率が入力されています"]
[null,"フィン部",null,"シース等価直径 ds",null,"シースの長辺と短辺の平均値を入力"]
[null,null,null,null,null,"デフォルト値として，素線径の8倍の値が入力されています"]
[null,null,null,"シースの熱伝導率 ks",null,"シース材の熱伝導率を入力．デフォルト値として，0.2W(m・K)が入力されています"]
[null,null,null,"長さ ls'",null,"フィン部の長さを入力．デフォルト値として，20mmが入力されています"]
[null,null,null,null,null,"(ある程度以上の長さになると誤差率が変化しなくなりますが，グラフの描画範囲が変化します)"]
[null,"熱伝達率",null,"金属素線の放射率",null,"金属素線2本の放射率の平均値を入力．デフォルト値として0.2が入力されています"]
[null,null,null,"シース表面の放射率",null,"シース材表面の放射率を入力．デフォルト値として0.9が入力されています"]
[]
["(2)","「計算実行」のボタンを押す"]
[null,"→　マクロ（SOLVE2）が実行され，解析解を満たす金属素線の温度分布（X21~X121）が計算されると共に，"]
[null,"　　誤差率（X3）が計算されます．"]
[null,"　（マクロでは，ソルバーを用いて熱流量の保存を満たす測温接点温度（Tj, M4）およびフィン部始端温度（T0', M5）を求めています）"]
[null,"　　"]
[null,"誤差率は ηerrは次式で定義されます.",null,null,null,"ηerr = (Tw - Tj)/(Tw - Tf) ×100 (%)"]
[]
[null,"※ 計算実行の際は，あらかじめマクロを利用できる設定にしておく必要があります．"]
[null,"     また，アドインの設定により，ソルバーを利用できる設定にしておく必要があります．"]
[]
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["４．ご利用に際して"]
["・","本解析シートはご自由にお使いください．ご利用しやすいように自由に加工していただいて構いません．"]
[null,"ただし，ご利用に際して，何らかのトラブルや損失・損害等が発生したとしても，製作者は一切責任を負いません．"]
[]
[null,"製作者：　防衛大学校 システム工学群 機械工学科　中村　元"]

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["熱電対による表面温度測定の誤差解析（解析解）"]
[null,"空気の熱伝導率（Tw）",null,null,{"$f":"-0.000000022933*M2^2+0.000072925*M2+0.024175","v":2.9862228799999998E-2},"W/(m・K)",null,null,null,"測定面温度",null,"Tw =",80,"℃","熱流量の保存",null,null,null,null,"熱電対の誤差率"]
[null,"空気の熱伝導率（(Tw+Tf)/2）",null,null,{"$f":"-0.000000022933*((M2+M3)/2)^2+0.000072925*((M2+M3)/2)+0.024175","v":2.7940353418749998E-2},"W/(m・K)",null,null,null,"周囲流体温度",null,"Tf =",25,"℃","x = 0","Qj - Qm(x=0) =",null,{"$f":"$R$8-$R$20","v":7.6395775855235028E-8},null,null,"ηerr = (Tw - Tj)/(Tw - Tf) =",null,null,{"$f":"(M2-M4)/(M2-M3)*100","v":5.3706210533658805},"%"]
[null,"空気の動粘性係数（(Tw+Tf)/2）",null,null,{"$f":"(8.2925*0.00001*((M2+M3)/2)^2+9.4005*0.01*((M2+M3)/2)+13.4885)*0.000001","v":1.8652324531249998E-5},"m2/s",null,null,null,"測温接点温度",null,"Tj =",77.046158420648766,"℃","x = lm","Qm(x=lm) - Qm(x'=0) =",null,{"$f":"$R$21-$R$31","v":1.141696391766811E-8}]
["熱電対の種類",null,null,null,null,null,null,null,null,"フィン部始端温度",null,"T0' =",46.786059505856862,"℃",null,null,null,{"$f":"ABS(R3)+ABS(R4)","v":8.7812739772903138E-8}]
[null,"T or K",null,null,"K",null,null,null,null,null,null,0]
[null,"素線径",null,"dm =",0.2,"mm",null,null,null,"測温接点の熱抵抗"]
["測温接点（形状を球と仮定した場合）",null,null,null,null,null,null,null,null,"式(30)","Qj = (Tw-Tj)/Rj ",null,null,null,null,null,"Qj =",{"$f":"($M$2-$M$4)/$R$9","v":8.1541259211887905E-4},"W"]
[null,"直径",null,"dj =",{"$f":"$E$7*2","v":0.4},"mm","（dm×2）",null,null,"式(10)","Rj = 1/[2πka{(rj+gj)ln((rj+gj)/gj)-rj}]",null,null,null,null,null,"Rj =",{"$f":"1/(2*PI()*$E$11*((($E$9/1000)/2+$E$10/1000)*LN((($E$9/1000)/2+$E$10/1000)/($E$10/1000))-($E$9/1000)/2))","v":3622.5116068855036}]
[null,"測定面との隙間",null,"gj =",0.01,"mm",null,null,null,"素線露出部の温度分布（解析解）"]
[null,"隙間の熱伝導率",null,"ka =",0.1,"W/(m・K)",null,null,null,"式(24)","Tm(x) = B + C1 exp(A x/lm) + C2 exp(-A x/lm)"]
["素線露出部",null,null,null,null,null,null,null,null,"式(24)","A = √{(Rgm+Rhm)Rm/(Rgm Rhm)}",null,null,null,null,null,"A =",{"$f":"SQRT(($R$17+$R$18)*$R$16/($R$17*$R$18))","v":5.3818267665230826}]
[null,"長さ",null,"lm =",17.5,"mm",null,null,null,"式(24)","B = (RgmTf + RhmTw)/(Rgm+Rhm)",null,null,null,null,null,"B =",{"$f":"($R$17*$M$3+$R$18*$M$2)/($R$17+$R$18)","v":75.588916561160019}]
[null,"素線の平均熱伝導率",null,"km =",{"$f":"IF($E$6=\"T\",211,IF($E$6=\"K\",24.5))","v":24.5},"W/(m・K)","（T→211，K→24.5）",null,null,"式(24)","C1 = [Tj exp(-A) - T0' + B{1-exp(-A)}] / {exp(-A) - exp(A)}",null,null,null,null,null,"C1 =",{"$f":"($M$4*EXP(-$R$12)-$M$5+$R$13*(1-EXP(-$R$12)))/(EXP(-$R$12)-EXP($R$12))","v":-0.13250984264980803}]
[null,"測定面との隙間",null,"gm =",7.2099999999999997E-2,"mm","2rm+gm =",{"$f":"$E$7+$E$15","v":0.27210000000000001},null,"式(24)","C2 = [Tj exp(A) - T0' + B{1-exp(A)}] / {exp(A) - exp(-A)}",null,null,null,null,null,"C2 =",{"$f":"($M$4*EXP($R$12)-$M$5+$R$13*(1-EXP($R$12)))/(EXP($R$12)-EXP(-$R$12))","v":1.5897517021385643}]
[null,"隙間の熱伝導率",null,"kg =",{"$f":"$E$2","v":2.9862228799999998E-2},"W/(m・K)","（空気→E2）",null,null,"式(1)","Rm = lm/(km 2πrm2)",null,null,null,null,null,"Rm =",{"$f":"($E$13/1000)/($E$14*2*PI()*(($E$7/1000)/2)^2)","v":11368.210220849669}]
["フィン部（金属素線＋内側被覆＋シース）",null,null,null,null,null,null,null,null,"式(11)","1/Rgm = -πkg(2lm) + 4kg(2lm)(rm+gm)/(2rm+gm)√((2rm+gm)/gm)×atan(√((2rm+gm)/gm))",null,null,null,null,null,"Rgm =",{"$f":"1/(-PI()*$E$16*(2*$E$13/1000)+4*$E$16*(2*$E$13/1000)*($E$7/2+$E$15)/$H$15*SQRT($H$15/$E$15)*ATAN(SQRT($H$15/$E$15)))","v":426.71703272256065}]
[null,"シース直径",null,"ds =",{"$f":"$E$7*8","v":1.6},"mm","（dm×8）",null,null,"式(7)","Rhm = 1/(h πrm 2lm)",null,null,null,null,null,"Rhm =",{"$f":"1/($E$26*PI()*$E$7/1000/2*$E$13/1000*2)","v":4893.843579006144},null,"金属素線の温度分布"]
[null,"シースの熱伝導率",null,"ks =",0.2,"W/(m・K)",null,null,null,"式(25)","dTm(x)/dx = C1(A/lm)exp(A x/lm) - C2(A/lm)exp(-A x/lm)",null,null,null,null,null,null,null,null,"素線露出部",null,"フィン部",null,"素線露出部+フィン部"]
[null,"長さ",null,"ls' =",20,"mm",null,null,null,"式(25)","Qm(x=0) = -(lm/Rm)[dTm(x)/dx](x=0) = -(lm/Rm)[C1(A/lm) - C2(A/lm)]",null,null,null,null,null,"Qm(x=0) =",{"$f":"-($E$13/1000/$R$16)*($R$14*$R$12/($E$13/1000)-$R$15*$R$12/($E$13/1000))","v":8.1533619634302382E-4},"W","x/lm","Tm(x)","x'/ls'","Tm(x')","x (mm)","Tm(x)"]
["熱伝達率（対流＋放射）",null,null,null,null,null,null,null,null,"式(25)","Qm(x=lm) = -(lm/Rm)[dTm(x)/dx](x=lm) = -(lm/Rm)[C1(A/lm)exp(A) - C2(A/lm)exp(-A)]",null,null,null,null,null,"Qm(x=lm) =",{"$f":"-($E$13/1000/$R$16)*($R$14*$R$12/($E$13/1000)*EXP($R$12)-$R$15*$R$12/($E$13/1000)*EXP(-$R$12))","v":1.3642489631126126E-2},"W",0,{"$f":"$R$13+$R$14*EXP($R$12*T21)+$R$15*EXP(-$R$12*T21)","v":77.04615842064878},null,null,{"$f":"T21*$E$13","v":0},{"$f":"U21","v":77.04615842064878}]
[null,"周囲風速",null,"u =",0.7,"m/s",null,null,null,"フィン部の温度分布（解析解）",null,null,null,null,null,null,null,null,null,0.02,76.86887167493424,null,null,0.35000000000000003,{"$f":"U22","v":76.86887167493424}]
[null,"測定面の流れ方向辺長",null,"L =",0.03,"m",null,null,null,"式(28)","Tm(x') = Tf + C3 exp(D x'/ls') + C4 exp(-D x'/ls')",null,null,null,null,"(x' = x + lm)",null,null,null,0.04,76.706428330109702,null,null,0.70000000000000007,76.706428330109702]
[null,null,null,"ReL* =",{"$f":"($E$22/1^2)*$E$23/$E$4","v":1125.8650343991558},"NuL =",{"$f":"0.61*0.71^(1/3)*$E$24^0.5","v":18.259650069104953},"hcv =",{"$f":"$G$24*$E$3/$E$23","v":17.006035874449839},"式(28)","D = √{Rms'/(Rs'+Rhs')}",null,null,null,null,null,"D =",{"$f":"SQRT($R$27/($R$28+$R$29))","v":8.135767927292715},null,0.06,76.556944560982259,null,null,1.05,76.556944560982259]
[null,"金属素線の放射率",null,"ε =",0.2,null,null,"hrd = εσ(Tw4-Tf4)/(Tw-Tf) =",{"$f":"$E$25*0.0000000567*(($M$2+273.15)^4-($M$3+273.15)^4)/(($M$2+273.15)-($M$3+273.15))","v":1.5776561252169901},"式(28)","C3 = [(T0' - Tf)exp(-D)] / {exp(-D) - exp(D)}",null,null,null,null,null,"C3 =",{"$f":"(($M$5-$M$3)*EXP(-$R$24))/(EXP(-$R$24)-EXP($R$24))","v":-1.8687038873579932E-6},null,0.08,76.418686832142882,null,null,1.4000000000000001,76.418686832142882]
[null,"測定面熱伝達率(対流＋放射)",null,"h =",{"$f":"$I$24+$I$25","v":18.583691999666829},"W/(m2・K)","← hcv + hrd",null,null,"式(28)","C4 = [(T0' - Tf)exp(D)] / {exp(D) - exp(-D)}",null,null,null,null,null,"C4 =",{"$f":"(($M$5-$M$3)*EXP($R$24))/(EXP($R$24)-EXP(-$R$24))","v":21.786061374560749},null,0.1,76.290051794479353,null,null,1.75,76.290051794479353]
[null,null,null,"ReD =",{"$f":"$E$22*($E$18/1000)/$E$4","v":60.046135167954979},"NuD =",{"$f":"IF($E$27<4,0.891*$E$27^0.33,IF($E$27<40,0.821*$E$27^0.385,0.615*$E$27^0.466))","v":4.1461895577522396},"hcv' =",{"$f":"G27*$E$3/($E$18/1000)","v":72.403750990455194},"式(16)'","Rms' = ls'/(km 2πrm2) ",null,null,"(式(16)のlc→ls')",null,null,"Rms' =",{"$f":"$E$20/1000/($E$14*2*PI()*(($E$7/1000)/2)^2)","v":12992.240252399621},null,0.12,76.169547691435554,null,null,2.1,76.169547691435554]
[null,"シース表面の放射率",null,"ε' =",0.9,null,null,"hrd' = ε'σ(Ts'4-Tf4)(Ts'-Tf) =",{"$f":"$E$28*0.0000000567*((($M$5+$M$3)/2+273.15)^4-($M$3+273.15)^4)/((($M$5+$M$3)/2+273.15)-($M$3+273.15))","v":5.7136766245642132},"式(17)'","Rs' = ln{rs/((√2)rm)}/(2π ls' ks) ",null,null,"(式(17)の2lc→ls', ki=ks)",null,null,"Rs' =",{"$f":"LN($E$18/(SQRT(2)*$E$7))/(2*PI()*$E$20/1000*$E$19)","v":68.948625047703615},null,0.14000000000000001,76.055777059388632,null,null,2.4500000000000002,76.055777059388632]
[null,"フィン部熱伝達率(対流＋放射)",null,"h' =",{"$f":"$I$27+$I$28","v":78.117427615019409},"W/(m2・K)","← hcv' + hrd'",null,null,"式(18)'","Rhs' = 1/(h' 2πrs ls')",null,null,"(式(18)の2lc→ls')",null,null,"Rhs' =",{"$f":"1/($E$29*2*PI()*(($E$18/1000)/2)*($E$20/1000))","v":127.33629673861589},null,0.16,75.947420521523895,null,null,2.8000000000000003,75.947420521523895]
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[null,null,null,null,null,null,null,null,null,"式(29)","Qm(x'=0) = -(ls'/Rms')[dTm(x')/dx'](x'=0) = -(ls'/Rms')[C3(D/ls') - C4(D/ls')]",null,null,null,null,null,"Qm(x'=0) =",{"$f":"-($E$20/1000/$R$27)*($R$25*$R$24/($E$20/1000)-$R$26*$R$24/($E$20/1000))","v":1.3642478214162208E-2},"W",0.2,75.741971579840992,null,null,3.5,75.741971579840992]
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