JP3568271B2 - Method and apparatus for measuring thermal constant using laser flash method - Google Patents

Description

【００３８】
以下、試料裏面における理論温度応答を与える基本式である前記の数式４を用いて、温度応答との２乗偏差Ｒについての熱拡散率α、比熱ｃあるいはビオー数ｈに関する偏微分をゼロとおくことにより得られる方程式を満足する熱拡散率α、比熱ｃあるいはビオー数ｈをニュートン法により求める手順について示す。
熱特性不明層の熱拡散率α、比熱ｃあるいはビオー数ｈを変数ｘと表し、Δｘ＝−（δＲ／δｘ）／（δ² Ｒ／δｘ² ）を求め、ｘ_new ＝ｘ＋Δｘにより、ｘを更新する。この手順を繰り返すことにより、δＲ／δｘ＝０を満たすｘ、即ち熱拡散率α、比熱ｃあるいはビオー数ｈを求めることができる。
【００３９】
上述したように各層の理論温度応答には熱拡散率α及び単位体積当たりの熱容量（ρ・Ｃ_p ）が現れる。従って直接的に求められるものは、この２つの熱特性値（α、ｃ）とその積である熱伝導率であり、密度が既知であれば定圧比熱も求められることになる。但し、解析上は熱特性不明層の密度を定数、定圧比熱を変数として取り扱い、計算終了後この定圧比熱に密度を乗じて単位体積当たりの熱容量を求めることも可能である。本数値解析法においては、（１）１次元の熱伝導方程式が成立し、（２）試料の表裏面におけるビオー数はそれぞれ等しい、（３）第ｉ層以外の全ての熱特性値及び全ての層厚みは既知であるとする前提条件の下で計算を行うものとする。さらに以下に示す時定数法を適用する場合には、（４）多層材料の定常状態における各層の熱伝導率λ_i と全体の熱伝導率λとの関係式が近似的に成立するものとして取り扱う。
【００４０】
図３に示すような多層材料１０の場合には未知変数として熱特性不明層の（１）熱拡散率α、（２）比熱ｃ、（３）ビオー数ｈの３変数が未知数であり、ｈ＝ｇ（α、ｃ）の関係式により変数を一つ減らして、かつ理論温度応答と温度応答との偏差を最小にする条件によりこの変数（α、ｃ、ｈ）を定めることができる。
【００４１】
（時定数τの計算）
厚みがＬである単層材料の場合、特性時間ｔ₀ をｔ₀ ＝Ｌ² ／（π² ・α₀ ）として定義すると、ｍを定数として、ｔ＞＞ｍ・ｔ₀ を満足する時間領域ｔにおいては、数式５の第２項以降（ｎ≧１）は初項に比較して無視できるほど小さくなるために、Ｔ（Ｌ、ｔ）／Ｔ_m ＝Ａ₀ ｅｘｐ（−ａ₀ ・ｔ）∫ｅｘｐ（ａ₀ ・ｔ′）ｆ（ｔ′）ｄｔ′式により裏面温度の理論解を近似することができる。ここでＡ₀ ＝２β₀ ²／（β₀ ²＋２ｈ＋ｈ² ）、ａ₀ ＝β₀ ²・α／Ｌ² である。
【００４２】
【数２】

【００４３】
従って、温度応答データの対数を時間に対してプロットしその傾きから時定数τを求めると、時定数τはａ₀ の逆数に等しいとみなすことができる。即ちτ＝１／ａ₀ ＝Ｌ² ／（β₀ ²・α）であり、この関係が多層材料１０の平均熱拡散率α_avに対しても成立するものとして、α_av＝αとしてτ＝１／ａ₀ ＝Ｌ² ／（β₀ ²・α_av）が得られる。
【００４４】
単層材料の場合、固有値β_n についてのｔａｎ（β_n ）＝２ｈ・β_n ／（β_n ² −ｈ² ）式よりビオー数ｈをβ₀ の関数として求め、ｈの値として正のものを選ぶことによりｈ₀ ＝β₀ ・（１／ｓｉｎβ₀ −１／ｔａｎβ₀ ）を得ることができる。そして、この関係が同じく多層材料１０についても成立することから、β₀ は多層材料１０の平均熱拡散率α_avと時定数τを用いて、β₀ ＝Ｌ／ｓｑｒｔ（α_av・τ）で与えられる。即ち、ビオー数ｈは平均熱拡散率α_avと時定数τの関数として上記のように表記することができ、多層材料１０の平均熱拡散率α_avは多層材料１０中の未知の熱拡散率αと比熱ｃとの関数であるから、全体としてはｈ＝ｇ（α、ｃ）としてビオー数ｈを設定することが可能となる。
【００４５】
多層材料１０の中の任意の１層に含まれる未知の熱定数を、以下に示す直接法と時定数法によって測定することができる。
直接法とは、求める３つの変数（熱拡散率α、比熱ｃ、ビオー数ｈ）を直接的に設定して理論温度応答と温度応答との２乗偏差Ｒを最小とするような（α、ｃ、ｈ）の最適解に到達する方法であり、時定数法とは、レーザ照射後試料の特性時間に対して充分時間が経過した後における時定数τを求め、その時定数τを介して熱拡散率αとビオー数ｈが特定の関係にあることを利用して変数を減らす方法である。即ち、α、ｃ及びｈ間に成立する関係式により１変数を減じて、最小の２乗偏差Ｒに至る方法である。
【００４６】
ここで、偏差としては、下式に定義されるような（イ）順２乗偏差Ｉ、（ロ）順２乗偏差II、（ハ）逆２乗偏差Ｉ、（ニ）逆２乗偏差II、（ホ）対数２乗偏差等の各偏差Ｒを適宜採用することができる。
（イ） Ｒ＝Σ（Ｑ・Ｔ_j −Ｅ_j ）²
（ロ） Ｒ＝（ΣＥ_j ² ）・（ΣＴ_j ² ）−（ΣＥ_j ・Ｔ_j ）²
（ハ） Ｒ＝Σ｛（１／（Ｑ・Ｔ_j ））−（１／Ｅ_j ）｝²
（ニ） Ｒ＝（Σ１／Ｅ_j ² ）・（Σ１／Ｔ_j ² ）−｛Σ１／（Ｅ_j ・Ｔ_j ）｝²
（ホ） Ｒ＝Σ｛ｌｎ（Ｑ・Ｔ_j ）−ｌｎ（Ｅ_j ）｝²
【００４７】
ここでＱは多層材料１０に照射するレーザパルスの試料に吸収された全エネルギーである。そして、上記の各偏差について計算を行う場合、以下のようなそれぞれの特徴を有している。
（イ）一般的な定義である。δＲ／δＱ＝０により求まるＱを（イ）式に代入してＱを消去したものを偏差として使用する。
（ロ）一般的な偏差である（イ）式を用いてδＲ／δＱ＝０により求まるＱを（イ）式に代入し、その分子部分を零とおくことにより得られる偏差であり、（イ）に比較して計算式が簡単になる利点がある。
（ハ）理論温度の分母に変数が集中しているため、逆数にすることにより計算式が簡単となる。δＲ／δＱ＝０により求まるＱを（ハ）式に代入してＱを消去したものを偏差として使用する。
（ニ）（ハ）式を用いてδＲ／δＱ＝０より求まるＱを（ハ）式に代入し、その分子部分を零とおくことにより得られる偏差であり、（ハ）に比較して計算式が簡単となる利点がある。
（ホ）対数形式の２乗偏差とすることにより極小値が容易に判定できる。δＲ／δＱ＝０により求まるＱを（ホ）式に代入してＱを消去したものを偏差として使用する。
なお、（イ）〜（ホ）式においてラプラス変換形式を採用する場合には、ラプラス変数ｐを複数個設定し、その複数個設定したｐに対して温度応答データをラプラス変換することにより、対応するＥ_j を求め、これらを（イ）〜（ホ）式に対して適用するものとする。
【００４８】
（時定数法による熱定数の測定方法）
以下に時定数法における測定方法の手順を示す。
（１）平均熱拡散率α_avの初期値計算
通常の半価値到達時間ｔ_1/2 法に従ってレーザパルス入熱曲線の重心位置から最高温度上昇値Ｔ_m の半分の温度まで上昇する時間を求め、この値より平均熱拡散率の初期値α_avを計算する。あるいは、熱定数不明層の比熱、熱拡散率を設定し、それに基づいて多層材料の平均熱拡散率α_avの初期値を計算してもよい。
ここで、多層材料における平均比熱ｃ_av、及び平均熱拡散率α_avは以下の式で与えられる。
ｃ_av＝（Σρ_i Ｃ_i Δｌ_i ）／ｌ_n
α_av＝ｌ_n ／（（ΣΔｌ_i ／（ρ_i Ｃ_i α_i ））・ｃ_av）
但し、Δｌ_i はｎ層からなる多層材料における第ｉ層の厚さ（＝ｌ_i −ｌ_i-1 であり、多層材料の全体厚みｌ_n はｌ_n ＝ΣΔｌ_i である。
【００４９】
（２）時定数τの計算
レーザパルス入熱後、充分時間が経過して単層材料と仮定した場合の試料裏面温度の時間空間理論解の初項のみによる近似式が成立する時間領域において、測定試料温度の減衰から時定数τを求める。
（３）ビオー数初期値ｈ₀ の計算
平均熱拡散率α_av、時定数τ及び固有値式からビオー数の初期値ｈ₀ を求める。
（４）比熱初期値設定
熱定数不明層に対して複数の比熱を設定する。
（５）ニュートン法による熱定数不明層の熱拡散率計算
複数の比熱に対して、レーザパルスを用いて求めた多層材料１０の裏面の理論温度応答と測定温度の２乗偏差Ｒを用いてニュートン法により熱定数不明層の熱拡散率を更新する。更新した熱拡散率を用いて多層材料１０の平均熱拡散率、及びビオー数を更新する。そして、これらの値を用いて繰り返し計算し、Δα／αが充分小さくなるまで繰り返して計算を行う。
【００５０】
（６）２乗偏差Ｒの最小値の判定と複数比熱の再設定
前記（５）にて計算した比熱ｃ、熱拡散率α、及びビオー数ｈにより２乗偏差を計算し、最小値を与える比熱を求める。複数設定した比熱の間隔Δｃとｃとの比率Δｃ／ｃが充分小さくなるまで比熱をその最小値を与える比熱の近傍に再設定し、これを繰り返して実行する。
上記手順では比熱ｃを複数設定し、それに対応する熱拡散率αをニュートン法で求めたが、熱拡散率αを比熱ｃで交換してもよい。
即ち、熱拡散率αを複数設定し、それに対応する比熱ｃをニュートン法で求める方法としてもよい。この場合、比熱ｃに関しては代数方程式となるのでその根を解析的に求める方法としてもよい。
【００５１】
図６は上記時定数法により求めた３層からなる多層材料１０中の未知層の厚みを変化させた場合における計算結果である。（ａ）は最下層の第３層が未知の熱定数を含む場合を、（ｂ）は中間層である第２層に未知の熱定数を含む場合の例を示している。同図から明らかなように、（ａ）、（ｂ）いずれの場合においても計算誤差即ち測定誤差を概ね±１％以下の範囲に納めることができることが分かる。
【００５２】
（直接法による熱定数の測定方法）
直接解析法とは、熱拡散率α、比熱ｃ、及びビオー数ｈの３個の独立変数で表記される偏差Ｒについて、前述した最小２乗法及びニュートン法を適用して、偏差Ｒの最小値を満たす熱拡散率α、比熱ｃ、及びビオー数ｈを求めるものである。この手順の２例を以下に示す。
【００５３】
直接法１
図７に従い複数のビオー数を同時に設定して所望の熱定数を得る手順を以下に説明する。
（１）比熱の初期値ｃ₀ を設定する（Ｓ−１）。
（１）ビオー数を複数設定する（Ｓ−２）。
（２）各ビオー数に対して、２乗偏差Ｒの最小値Ｒ_j を与える熱拡散率α及び比熱ｃを次の手順で求める。
（２）−１複数設定された比熱のそれぞれに対して、最小２乗法及びニュートン法を適用して偏差Ｒの最小値Ｒ_j を与える熱拡散率を求める（Ｓ−３）、（Ｓ−４）。あるいはＳ−３において熱拡散率を複数設定して、偏差Ｒを極小とする比熱を求める方法も可能である。
（２）−２比熱と熱拡散率の各組合わせに対して求めた偏差Ｒを比較し、最小値Ｒ_j を与える比熱を選択する（Ｓ−５）、（Ｓ−６）。
（２）−３最小値を与えた比熱の近傍に偏差Ｒをより小さくする方向に新たな複数の比熱を設定し（Ｓ−３）、（２）−１より繰り返す。Δｃ／ｃが充分小さくなった時のα及びｃが求めるものである。
【００５４】
（３）偏差Ｒの極小値Ｒ_j を与えるビオー数を次の手順で求める。
（３）−１各ビオー数及び工程（２）により求めた熱拡散率及び比熱の組合わせに対し、偏差Ｒを求め偏差Ｒの最小値Ｒ_i を与えるビオー数を選択する（Ｓ−７）。
（３）−２最小値Ｒ_i を与えたビオー数の近傍に偏差Ｒをより小さくする方向に新たなビオー数を複数設定し（Ｓ−２）、工程（２）を実施する。Δｈ／ｈが充分小さくなった時のｈ、α、ｃが求めるものである（Ｓ−８）。
（４）前記所望の値ｈ_m 、α_m 、ｃ_m を出力する。（Ｓ−９）
以上の偏差Ｒの極小値を求める手順においては、ビオー数、比熱及び熱拡散率は同等の独立変数であるので変数を交換しても同様に成立する。例えば、複数の比熱を設定した中に複数ビオー数を設定し、極小値を与える熱拡散率を求める方法としてもよい。また、複数変数に対するニュートン法によっても偏差Ｒの極小値を与える熱拡散率、比熱及びビオー数を求めることが可能である。
【００５５】
直接法２
（１）最初に、図３に示すような多層材料１０中の一層における未知数である比熱ｃ、及びビオー数ｈを適当に定めて、それぞれの初期値ｃ₀ 、ｈ₀ をｈ−ｃ平面上の（ｃ₀ 、ｈ₀ ）点に設定する。
（２）前記で設定した各初期値（ｃ₀ 、ｈ₀ ）を理論温度応答Ｆ（α、ｃ、ｈ、ｔ）、Ｆ′（α、ｃ、ｈ、ｐ）に代入して、未知数αを含む理論温度応答Ｆ（α、ｃ₀ 、ｈ₀ 、ｔ）、Ｆ′（α、ｃ₀ 、ｈ₀ 、ｐ）と実際の温度応答Ｅ（ｔ）、Ｅ′（ｐ）との偏差Ｒを最小とするαの値をδＲ／δα＝０とすることにより求める。
【００５６】
（３）δＲ／δα＝０式を満たすαの値を前述した最小２乗法、及びニュートン法等の数値解析法により求める。
このとき得られる熱拡散率αの初期値をα₀ 、偏差Ｒの初期値をＲ₀ として、これら初期値の組（Ｒ₀ 、α₀ 、ｃ₀ 、ｈ₀ ）を仮の最小値の組（Ｒ_m 、α_m 、ｃ_m 、ｈ_m ）として設定する。
（４）最小値の組（ｃ_m 、ｈ_m ）＝（ｃ₀ 、ｈ₀ ）で表示される二次元平面上の点の近傍に新たな値の組（ｃ₁ 、ｈ₁ ）を設定する。この新たな点の設定には前記二次元平面上の点を中心点として、予め設定した基準半径内の円周内を乱数法によって指定する等の操作をコンピュータ上で演算して処理することもできる。またこの基準半径を漸次縮小していくことにより精度を高められる。更に、複数の既に設定された熱定数の組を用いて、偏差あるいは更新される熱定数の差が小さくなる方向に、新たな複数の更新値を設定することも可能である。
【００５７】
（５）前記ｃ₁ 、ｈ₁ の値を理論温度応答Ｆ（α、ｃ、ｈ、ｔ）、Ｆ′（α、ｃ、ｈ、ｐ）に代入して、未知数αを含む理論温度応答Ｆ（α、ｃ₁ 、ｈ₁ 、ｔ）、Ｆ′（α、ｃ₁ 、ｈ₁ 、ｐ）と実際の温度応答Ｅ（ｔ）、Ｅ′（ｐ）との偏差Ｒが最小となるようなαの値をδＲ／δα＝０とすることにより求める。このときの得られる熱拡散率αの値をα₁ 、偏差Ｒの値をＲ₁ として、これらの組（Ｒ₁ 、α₁ 、ｈ₁ 、ｃ₁ ）を定める。
（６）次に、最小値として既に設定されているＲ_m （＝Ｒ₀ ）と、Ｒ₁ とを比較してＲ₁ がＲ_m より大きい場合には、前記した最小値の組（ｃ_m 、ｈ_m ）＝（ｃ₀ 、ｈ₀ ）の近傍に新たな（ｃ₂ 、ｈ₂ ）の組を再設定する。Ｒ₁ がＲ_m より小さい場合には、Ｒ_m ＝Ｒ₁ とすることにより最小値Ｒ_m を更新する。
（７）そして、予め設定してある許容基準値εよりも最小値Ｒ_m が小さくなるまで以上の操作を繰り返すことにより、最小値Ｒ_m を次第に減少させて、最小値Ｒ_m が許容基準値εより小さくなった時点で最小値の組（Ｒ_m 、α_m 、ｃ_m 、ｈ_m ）を所望のα、ｃ、ｈとして出力する。
なお、以上の操作は上記の操作手順のプログラムを入力したコンピュータにより、レーザフラッシュ法を用いて得られる温度応答のデータを演算処理することで実行される。
【００５８】
図８は上記直接法２により求めた３層からなる多層材料１０の未知層の厚みを変化させた場合における計算結果である。（ａ）は中間層である第２層に未知の熱定数を含む場合の例を、（ｂ）は最下層の第３層が未知の熱定数を含む場合を示している。同図から明らかなように、（ａ）、（ｂ）いずれの場合においても計算誤差即ち測定誤差は一般的に必要とされる測定精度の範囲を満たしていることがわかる。
【００５９】
以上、本発明の実施例を説明したが、本発明はこれらの実施例に限定されるものではなく、要旨を逸脱しない条件の変更等は全て本発明の適用範囲である。
例えば、本実施例においては、理論温度応答、及び温度応答のデータをラプラス変換して処理する場合について説明したが、通常の時間空間において処理してもよい。また、本発明に適用する理論温度応答と温度応答との偏差には、必要に応じて特徴の異なる順２乗偏差、逆２乗偏差、及び対数２乗偏差等を採用することができる。
更に、理論温度応答と温度応答との偏差の式中に含まれる未知数である熱拡散率、比熱、及びビオー数の３つの変数は数学的にはいずれも同等の変数であるため、上記実施例で説明したような関係式中で各変数を任意に交換した関係式を使用する場合も同様に本発明の権利範囲内であるとみなす。
【００６０】
【発明の効果】
請求項１〜６記載のレーザフラッシュ法におけ熱定数の測定方法及びその装置においては、熱拡散率α、比熱ｃ、ビオー数ｈを設定して得られる理論温度応答と現実の温度応答との偏差を比較して、未知数の値を逐次効率的に更新し、漸近的に実際の温度応答の測定データを満たす未知数の値を高精度で決定することができると共に、従来のように温度応答の絶対値を必要とせず、相対的な温度応答の測定データに基づいて多層材料中の未知の熱定数を求めることができる。
さらに、理論温度応答と実際に得られる温度応答との偏差を指標として、例えばビオー数ｈ及び比熱ｃを最小偏差Ｒ_m を与える値の近傍に逐次設定して、最小偏差の更新を行うため、未知のビオー数、比熱及び熱拡散率の真値を単純な繰り返し計算手段の構成により求めることができ、コンピュータによる演算処理が高速で行える。
【００６１】
特に、請求項２記載のレーザフラッシュ法を用いた熱定数の測定方法においては、熱拡散率、比熱及びビオー数を変数として偏差Ｒを求め、該偏差の極小値を与えるものとしてα、ｃ、ｈの値を正確かつ迅速に求めることができる。
さらに、請求項３記載のレーザフラッシュ法を用いた熱定数の測定方法においては、ビオー数の初期値が温度応答における時定数、及び半価値到達時間ｔ_1/2 の値との関係等に基づいて設定されるので、真値の近傍に予め設定でき、計算回数を大幅に削減することができると共に、求める熱定数が発散してしまうことがない。
【００６２】
また、請求項５記載のレーザフラッシュ法を用いた熱定数の測定装置においては、理論温度応答と実際に得られる温度応答との偏差Ｒを指標として、例えばビオー数ｈ及び比熱ｃを最小偏差Ｒ_m を与える値の近傍に逐次設定して、最小偏差Ｒ_m の更新を行って、未知のビオー数、比熱及び熱拡散率の真値を単純な計算手段の構成により求めることができるので、絶対温度を測定する必要がなく、試料に接触することなく試料温度を測定できる放射温度計等を使用して高温の温度域まで測定が可能であり、測定の環境条件に影響されることが少ない。
さらに、請求項６記載のレーザフラッシュ法を用いた熱定数の測定装置においては、熱拡散率及び比熱を基にしてビオー数が更新されるので、理論温度応答と実際に得られる温度応答との偏差Ｒを指標として、熱拡散率αあるいは比熱ｃを最小偏差Ｒ_m を与える値の近傍に逐次設定して、最小偏差Ｒ_m の更新を行って、未知のビオー数、比熱及び熱拡散率の真値を少ない演算計算手段の繰り返しにより求めることができるため、計算誤差を減少させることができる。
【図面の簡単な説明】
【図１】本発明の一実施例に係るレーザフラッシュ法を用いた熱定数の測定方法を適用したレーザフラッシュ装置の構成図である。
【図２】レーザフラッシュ法を用いた熱定数の測定方法の説明図である。
【図３】多層材料を用いるレーザフラッシュ法の説明図である。
【図４】レーザフラッシュ法を用いた熱定数の測定方法における直接法のフロー図である。
【図５】レーザフラッシュ法を用いた熱定数の測定方法における時定数法のフロー図である。
【図６】３層からなる多層材料中の未知層の厚みを変化させた場合における時定数法の計算誤差を示す図である。
【図７】レーザフラッシュ法を用いた熱定数の測定法における直接法のフロー図である。
【図８】３層からなる多層材料中の未知層の厚みを変化させた場合における直接法の計算誤差を示す図である。
【符号の説明】
１０ 多層材料
１１ ハーフミラー[0001]
[Industrial applications]
The present invention relates to a method and apparatus for measuring a thermal constant using a laser flash method, which is capable of determining a thermal diffusivity, a specific heat and a Biot number of a sample with high accuracy.
[0002]
[Prior art]
The laser flash method is a method that has rapidly spread in recent years as a method for measuring the thermal diffusivity and specific heat of a homogeneous material.Compared to other methods, a small number of measurement samples are required, and highly accurate measured values can be obtained over a wide temperature range. It has features where it can be obtained. This method irradiates the sample surface with a pulse generated by a laser, measures the back surface temperature of the sample using a radiation thermometer, etc., and obtains the thermal diffusivity and specific heat of the sample from the data of the temperature response characteristics. Things.
[0003]
When there is an unsteady heat flow inside the one-dimensional material, δT / δt = α · (δ^Two T / δx^Two ) Expression. T is a temperature, α is a thermal diffusivity, and x is a one-dimensional coordinate variable, and a solution of T can be obtained by setting initial conditions and boundary conditions. The thermal diffusivity α in this equation is α = κ / (C_p Ρ) = κ / c, C_p , Ρ, and c are constant pressure specific heat, density, and heat capacity per unit volume (hereinafter referred to as specific heat), respectively. Therefore, if the specific heat at constant pressure and the density of the measurement sample are known, the thermal conductivity κ of the sample can be obtained using the thermal diffusivity α.
In recent years, an attempt has been made to apply such a laser flash method to a multilayer material to obtain an unknown thermal diffusivity in the multilayer material. However, since the calculation formula becomes complicated, many calculations are repeated. Therefore, satisfactory results have not been obtained due to problems such as accumulation of calculation errors. Here, a method of analyzing the thermal diffusivity of a single-layer material will be described.
[0004]
A general method for measuring the thermal diffusivity in the laser flash method is obtained by making the theoretical solution coincide with the measured value data of the measured sample back surface temperature.
That is, the theoretical temperature response T of the back surface temperature of the sample according to the one-dimensional heat conduction equation assumes the thermal diffusivity α, the specific heat c, which are unknown variables in one layer in the multilayer material, and the biot number h, which is a parameter related to heat loss, respectively. Assuming that the laser pulse is Q (t), it can be strictly expressed by the relational expression of T = F (α, c, h, t) using time t. F (α, h, t). However, under the condition that the theoretical temperature response F (α, c, h, t) matches the pattern based on the actual back surface temperature response E (t), the above-mentioned unknowns α, h are calculated by the trigonometric function , And implicit functions including exponential functions are included, and actual numerical calculations are extremely difficult. Therefore, in general, approximate values are used for specific variables, and data used for calculations is limited to data within a specific area. For example, a method for simplifying the calculation is adopted. At present, as a method of analyzing the thermal diffusivity by such a laser flash method, the following (1) half value arrival time t_1/2 As methods for analyzing the specific heat, (2) the area method, and (3) the specific heat analysis method, there is known a method of measuring the temperature rise of a measurement sample to obtain the specific heat.
[0005]
(1) Half value arrival time method
Half value arrival time t_1/2 Is the maximum rise temperature T from the initial temperature value on the temperature response curve on the back surface of the material._m Is the time required for the temperature to rise to half the value of Then, on the temperature response curve, the maximum rise temperature T_m Time t until the temperature rises to half the value of t_1/2 Which is established when there is no heat loss using₀ = 1.36975L^Two / (Π^Two ・ T_1/2 ), The thermal diffusivity α by the half-value time-of-flight method₀ Is calculated. Here, L is the thickness of the measurement sample. The maximum rise temperature T_m Is obtained by extrapolating the decay curve after the temperature rise to the pulse irradiation time. Next, the relaxation coefficient k of the decay curve (the reciprocal of the time constant τ of the decay curve) and the half-value arrival time t_1/2 The heat loss correction coefficient K is obtained by multiplying the heat loss correction coefficient K by the product of₀ To determine the corrected thermal diffusivity α.
[0006]
(2) Area method
This is a method of calculating a deviation R between an average temperature in a specific section region set as a temperature response and an average temperature obtained from a theoretical formula, thereby obtaining a thermal diffusivity α. Extrapolating the decay curve to the laser pulse irradiation time, the maximum rise temperature T_m The time constant τ of the decay curve is used to express the Biot number h as a function of the thermal diffusivity α, and the deviation R is described. Then, the thermal diffusivity α is expressed by the conditional expression R = 0 for the deviation R. Is a method of calculating
[0007]
(3) A method of calculating the specific heat of the measurement sample by irradiating the measurement sample with a certain amount of laser flash and measuring the temperature rise value ΔT of the measurement sample is performed according to the following procedure.
(A) Measurement of absorbed energy Q
At room temperature, a glassy carbon plate is adhered to a plate having a known specific heat (usually a sapphire plate or the like) with silicon grease, and a laser pulse is irradiated to measure the absorption energy Q. Assuming that the respective weights and specific heats are known, the absorbed energy Q is expressed by the following equation from the measured value of the temperature rise value ΔT ′.
Q = (M₁ C₁ + M_Two C_Two + M_Three C_Three ) · ΔT '
Where M₁ , M_Two , M_Three Are the weights of glassy carbon, silicon grease, and sapphire, respectively,₁ , C_Two , C_Three Is the specific heat of glassy carbon, silicon grease, and sapphire, respectively.
(B) Measurement of sample specific heat at room temperature
Next, at the room temperature, a laser pulse is irradiated in the same manner as in (a) of attaching a glassy carbon plate to the sample with silicon grease, and the temperature rise value ΔT is measured.
Assuming that the emissivity of glassy carbon is equal, the absorbed energy is equal and the following equation is established.
Q = (M₁ C₁ + M_Two C_Two + M_s C_s0) · ΔT
Where M_s Is the weight of the sample, C_s0Is the specific heat of the sample at room temperature. Therefore, the specific heat C of the sample at room temperature_s0Is given by the following equation using Q obtained in (a).
C_s0= (Q · ΔT^-1-M₁ C₁ -M_Two C_Two ) / M_s
(C) Specific heat measurement of sample at high temperature
A laser pulse is irradiated to the sample alone, and the temperature rise value ΔT at room temperature and high temperature₀ , ΔT₁ Are respectively measured.
At this time, Q '= C_s0M_s ΔT₀ , Q ″ = C_s1M_s ΔT₁ If the absorbed energy of the sample does not change with temperature, Q ′ = Q ″, and the specific heat of the sample is given by the following equation.
C_s1= C_s0ΔT₀ / ΔT₁
When the absorption rate of the laser pulse is small, a material having a high absorption rate (such as carbon spray) is applied to the sample and the measurement is performed.
[0008]
[Problems to be solved by the invention]
However, since the method for analyzing the thermal diffusivity and specific heat in the laser flash method described above targets a single-layer material, the above-described method is applied when an unknown thermal diffusivity or specific heat in a multilayer material is determined. In such a case, the calculation becomes extremely complicated, so that calculation errors accumulate and the factors involved in the calculation errors increase, and it becomes extremely difficult to obtain a highly accurate value.
As shown in FIG. 2, the intensity distribution of the laser pulse is represented as Q (t) as a function of the time t, and the laser pulse is irradiated from the front side of the multilayer material, so that the temperature response E on the back side is obtained. (T) is obtained.
FIG. 3 is a cross-sectional view of an n-layered multi-layer material. The thermal diffusivity α in the i-th layer, the specific heat c, and the biot number h on the front and back surfaces of the multi-layer material are unknown. It is assumed that all thermal constants and thicknesses for are known.
[0009]
The theoretical temperature response T on the back side is strictly expressed as T = F (α, c, h, t), and specific values of the thermal characteristic values α, c, h in the multilayer material are given. By giving a numerical value, a deviation between the theoretical temperature response T and the temperature response E can be calculated. The smaller the deviation, the higher the degree of conformity between the theoretical temperature response T and the temperature response E, and the higher the accuracy of the values of α, c, and h set at this time. That is, F (α ′, c ′, h ′, t) has a larger deviation from the temperature response E (t) than F (α, c, h, t) shown in FIG. (c, h) is a numerical value with higher accuracy than (α ′, h ′, c ′). However, since the equation of the theoretical temperature response T = F (α, c, h, t) in a multilayer material becomes complicated, it is extremely difficult to strictly determine such a minimum value of the deviation by mathematical analysis means. is there.
[0010]
The method (3) of irradiating the measurement sample with a certain amount of laser flash and measuring the temperature rise ΔT of the measurement sample to calculate the specific heat of the measurement sample has the following problems. is there.
(A) The measurement condition is that the pulse energy is not changed by repeated irradiation, but this is not guaranteed.
(B) The absorption rate of the sample needs to be constant with respect to temperature, but there is no guarantee. In addition, it is necessary that the coating material for increasing the absorption energy does not deteriorate to a high temperature and the absorption energy needs to be constant, but this is not guaranteed.
(C) Since the absolute value of the temperature rise value is required, the temperature is measured with a thermocouple. When the sample is a ceramic or the like, the sample is attached to the sample with an adhesive, but the temperature at which the adhesive functions is usually up to about 1000K, and the specific heat cannot be measured at a temperature higher than 1000K.
(D) Installation of the thermocouple is complicated and requires skill.
Normally, the measurement of thermal diffusivity requires a fast response and the absolute value of temperature is not required, so it is measured with a radiation thermometer. It is necessary to measure in pairs. Therefore, when the thermal conductivity is to be obtained, the measurement must be performed twice.
[0011]
The present invention has been made in view of such circumstances, and can determine a heat constant such as an unknown thermal diffusivity or a specific heat in one layer of a multilayer material with high accuracy and efficiency, and furthermore, is capable of measuring the measurement environment. It is an object of the present invention to provide a method and an apparatus for measuring a thermal constant using a laser flash method which are less affected by conditions.
[0012]
[Means for Solving the Problems]
A method for measuring a thermal constant using a laser flash method according to claim 1 in accordance with the above object is characterized in that a thermal diffusivity and a specific heat of a back surface side obtained by irradiating a laser flash from a front surface side of a multilayer material including one layer whose unknown specific heat is applied.Laplace converted temperatureIncludes temperature response, thermal diffusivity, specific heat, and biot number as variables.Given by the solution of the Laplace transformed heat conduction equationThe thermal diffusivity, specific heat, and Biot number are determined by comparing the thermal diffusivity with the theoretical temperature response.
[0013]
The method for measuring a thermal constant using the laser flash method according to claim 2 is a method for measuring a thermal constant using the laser flash method according to claim 1, wherein the thermal diffusivity,SaidSpecific heat, andSaidAssuming that one of the biot numbers is A, the other is B, and the other is C,SaidA first step of setting an initial value of B, obtaining C that minimizes a deviation between the back surface temperature response and the theoretical temperature response of the multilayer material, and setting the deviation as an initial value of a minimum deviation; New A and B of the material give the minimum deviationSaidA andSaidA second step of setting the vicinity of B, a third step of obtaining C and a deviation from A and B of the multilayer material that minimize the deviation between the theoretical temperature response and the back surface temperature response of the multilayer material, and the deviation WhenSaidCompared with the minimum deviation, the deviationSaidIf the difference is larger than the minimum deviation, the second step to the third step are repeated.SaidBy newly setting the minimum deviation and repeating the second to third steps,SaidAnd a fourth step of outputting the thermal diffusivity, specific heat, and biot number when the minimum deviation reaches the vicinity of the minimum value.
[0014]
The method for measuring a thermal constant using the laser flash method according to claim 3 is the method for measuring a thermal constant using the laser flash method according to claim 1,SaidSpecific heat, andSaidSet each initial value of the biot number, determine the deviation between the theoretical temperature response including the respective initial value and the back surface temperature response, the thermal diffusivity,Thespecific heat,TheBiot number, andTheA first step of setting the deviation to a set thermal diffusivity, a set specific heat, a set biot number, and a set deviation, and setting any one of the set heat diffusivity and the specific heat to A and setting the other to B, in the vicinity of A And a theoretical temperature response including the set A and the set biot number.SaidB is obtained by a calculation step of obtaining B that minimizes the deviation from the temperature response, and A and B are obtained.TheA third step of updating the biot number by B to determine a new set biot number, and repeating the calculation steps until reaching the vicinity of the deviation minimum value in a state where A is fixed to determine the thermal diffusivity, specific heat, and biot number And the difference between the theoretical temperature response including the thermal diffusivity, the specific heat, and the Biot number, and the temperature response, determine whether the difference has reached near the minimum value, and if not, the second From the processSaidRepeat up to the third step,TheWhen the deviation reaches near the minimum value, a fourth step of outputting the thermal diffusivity, specific heat, and biot number at that time is provided.
A method for measuring a thermal constant using a laser flash method according to claim 4 is a method for measuring a thermal constant using a laser flash method according to claim 3, wherein the updating of the biot number is performed by means of the average thermal diffusion of the multilayer material. Rate α _av The total thickness of the multilayer material l _n , And eigenvalue β ₀ The time constant l of the multilayer material expressed by _n ^Two / (Β ₀ ^Two ・ Α _av ) Plots the logarithm of the temperature response data against time and determines the eigenvalue β equal to the time constant determined from the slope. ₀ And β ₀ ・ (1 / sinβ ₀ −1 / tanβ ₀ ) Is set.
[0015]
Claim5The thermal constant measuring device using the laser flash method described is a thermal diffusivity and specific heat of the back side obtained by irradiating a laser flash from the front side of a multilayer material including a single layer with unknown specific heat.Laplace converted temperatureTemperature response,TheThermal diffusivity,TheSpecific heat, andTheIncluding Biot number as a variableGiven by the general solution of the Laplace transformed heat conduction equationCompare with the theoretical temperature response,TheThermal diffusivity,TheSpecific heat, andTheAn apparatus for measuring the thermal constant of the multilayer material using a laser flash method for obtaining a Biot number,
The thermal diffusivity,SaidSpecific heat, andSaidAssuming that one of the biot numbers is A, the other is B, and the other is C,SaidFirst calculating means for setting an initial value of B, obtaining C that minimizes the deviation between the temperature response on the back surface of the multilayer material and the theoretical temperature response, and setting the deviation as an initial value of the minimum deviation; ,
New A and B of the multilayer material,SaidFrom the first calculation set near A and B giving the minimum deviation and A and B of the multilayer material, C and the deviation that minimize the deviation between the theoretical temperature response and the back surface temperature response of the multilayer material are obtained. Do a second calculation,TheDeviation andSaidCompare with the minimum deviation,TheDeviationSaidIf the difference is larger than the minimum deviation, the first and second calculations are repeated.SaidSecond calculating means for newly setting as the minimum deviation;
SaidIt is determined whether the minimum deviation has reached the vicinity of the minimum value, and if not, the first and second calculations are repeated.SaidWhen the minimum deviation value reaches the vicinity of the minimum value, there is provided a third calculating means for outputting the thermal diffusivity, specific heat, and biot number.
[0016]
Claim6The thermal constant measuring device using the laser flash method described is a thermal diffusivity and specific heat of the back side obtained by irradiating a laser flash from the front side of a multilayer material including a single layer with unknown specific heat.Laplace converted temperatureTemperature response,TheThermal diffusivity,TheSpecific heat, andTheIncluding Biot number as a variableGiven by the solution of the Laplace transformed heat conduction equationCompare with the theoretical temperature response,TheThermal diffusivity,TheSpecific heat, andTheAn apparatus for measuring the thermal constant of the multilayer material using a laser flash method for obtaining a Biot number,
SaidThermal diffusivity,SaidSpecific heat, andSaidSet each initial value of the biot number, determine the deviation between the theoretical temperature response including the respective initial value and the back surface temperature response, the thermal diffusivity,Thespecific heat,TheBiot number, andTheFirst calculating means for setting the deviation as a set thermal diffusivity, a set specific heat, a set biot number, and a set deviation, respectively;
A second computing means for setting any one of the set thermal diffusivity and specific heat as A and the other one as B, and setting a new A near the A,
A set by the second calculating means, and a theoretical temperature response including a set biot number;SaidCalculate B that minimizes the deviation from the back surface temperature response, update the biot number with the thermal diffusivity and specific heat obtained from the calculation result, determine a new set biot number, and determine the deviation in the state where A is fixed. A third calculating means for determining the thermal diffusivity, specific heat, and biot number by repeating the above calculation until reaching the local minimum value;
Theoretical temperature response including the thermal diffusivity, specific heat, and Biot number obtained by the third arithmetic means;SaidA fourth calculating means for determining a deviation from the back surface temperature response and determining whether the deviation has reached near the minimum value;
When the deviation obtained by the fourth arithmetic means does not reach the vicinity of the minimum value, a new A set in a direction for giving a smaller minimum value is input to the second arithmetic means, and the minimum value is input to the second arithmetic means. And a fifth calculating means for outputting the thermal diffusivity, specific heat, and biot number at that time when the value reaches the vicinity of the value.
[0017]
Here, as the deviation between the theoretical temperature response and the back surface temperature response, a forward squared deviation which is a sum of squares of the difference between them, and an inverse squared deviation which is a sum of squares of the difference of each reciprocal, respectively Logarithmic squared deviation obtained by summing the squares of the logarithmic difference of the above, and a deviation consisting of a combination thereof. Further, the arithmetic processing can be performed by performing Laplace transform on each of the measured data and the theoretical temperature response data, or the arithmetic processing can be performed with the actual data as it is.
The difference and the difference refer to a so-called absolute value of the difference. Further, the specific allowable value, the specific reference value, and the allowable reference value are determination values for updating a deviation or the like, and are values that are appropriately set according to an allowable limit of a required calculation error. .
The time when the deviation reaches the vicinity of the minimum value refers to a state in which the difference before and after the update remains within a predetermined allowable reference value even if the update calculation of the deviation is repeated.
[0018]
[Action]
The theoretical temperature response T of the back surface of the sample according to the one-dimensional heat conduction equation is obtained by setting the thermal diffusivity α, the specific heat c in one layer in the multilayer material, and the biot number h, which is a parameter relating to the heat loss in the front and back surfaces of the multilayer material, respectively. , T = F (α, c, h, t) or T = F ′ (α, c, h, p) as a function of time t or Laplace variable p.
Claim 1~ 4In the method of measuring the thermal constant using the laser flash method described, the theoretical temperature response F (α, c, h, t) or F ′ (α, c, h, p) and the time of the actually obtained temperature response The thermal diffusivity α, the Biot number h, and the specific heat c are defined as the minimum deviation R using the deviation R from the function E (t) using t as a variable or the function E ′ (p) using the Laplace variable p as a parameter._m Is set in the vicinity of the value giving the minimum deviation R_m Can be obtained to obtain the true values of the required Biot number h, the specific heat c, and the thermal diffusivity α.
[0019]
In the method of measuring a thermal constant using a laser flash method according to claim 2, any one of the thermal diffusivity, specific heat, and biot number is A, the other is B, and the other is C, and A that gives a minimum deviation is used. , And one or more sets of values of A and B are set in the vicinity of B, C and the deviation that minimize the deviation for the combination of A and B are obtained, and the minimum deviations are compared with each other, and the smaller minimum is sequentially determined. A and B are updated in the direction in which the deviation is given, and when the minimum deviation reaches the vicinity of the minimum value, the thermal diffusivity, the specific heat, and the Biot number are output. By appropriately selecting two unknowns, a desired value can be obtained accurately and quickly. In the following description, the case where the specific heat is assigned as A, the biot number and the thermal diffusivity as B and C, respectively, is shown. Hereinafter, a procedure for obtaining the thermal diffusivity α and the specific heat c will be described in detail with reference to a flowchart shown in FIG.
[0020]
First, the specific heat c and biot number h, which are unknowns in one layer in the multilayer material as shown in FIG. 3, are appropriately determined from literature values and the like, and the respective initial values c₀ , H₀ Is set (S-1).
Then, each initial value (c₀ , H₀ ) Is substituted for the theoretical temperature response F (α, c, h, t) or F ′ (α, c, h, p) to obtain the actual temperature response E (t) or E ′ (p). The value of α that minimizes the square deviation R is obtained by setting δR / δα = 0.
Here, the expression δR / δα = 0 is a one-dimensional equation of the form g (α) = 0 including only α as a variable, and the form in which the root α can be accurately obtained by a numerical analysis method such as Newton's method. belongs to.
The initial value of the thermal diffusivity α thus obtained is α₀ , The initial value of the deviation R is R₀ As a set of these initial values (R₀ , Α₀ , C₀ , H₀ ) Is replaced with a tentative minimum value set (R_m , Α_m , C_m , H_m ) (S-2).
[0021]
Subsequently, a set of minimum values (R_m , Α_m , C_m , H_m ), (C_m , H_m ), A new set of values (c_i , H_i ) Is set (S-3).
And c_i , H_i Is substituted into the theoretical temperature response F (α, c, h, t) or F ′ (α, c, h, p) to obtain the actual temperature response E (t) or E ′ (p). The value of α that minimizes the square deviation R is determined by setting δR / δα = 0. The value of the obtained thermal diffusivity α is α_i , The value of the deviation R_i As these sets (R_i , Α_i , H_i , C_i Is set (S-4).
In S-5, R which has already been set as the minimum value_m And R obtained in the above S-4_i And R_i Is R_m If it is larger, the process returns to S-3, and R_i Is R_m If less than R_m = R_i And the minimum value R_m To R at that time_i (S-6).
Subsequent to S-6, a preset allowable reference value ε and a minimum value R_m (S-7), the minimum value R_m Is smaller than the allowable reference value ε, the process returns to S-3. When the value is smaller than the allowable reference value ε, the set of minimum values (R_m , Α_m , C_m , H_m ) Is output as desired α, c, h (S-8), and the entire procedure for obtaining the thermal diffusivity α and specific heat c in the multilayer material is completed.
[0022]
Hereinafter, (1) to (3) show h in S-3._i , C_i An example of the setting method will be described.
(1) The biot number is fixed.
(2) The thermal diffusivity and the deviation are obtained by sequentially changing the specific heat, and the specific heat is moved in a direction that gives a smaller deviation by comparing the magnitude of the deviation. The transfer amount of the specific heat is reduced as it approaches the minimum deviation value in a state where the Biot number is fixed. By this procedure, the specific heat, the thermal diffusivity, and the deviation for one biot number are obtained.
(3) Change the set value of the biot number and execute the procedures (1) and (2). By comparing the magnitude of the deviation determined for each biot number according to (2), the biot number is moved in a direction that gives a smaller deviation. The moving amount of the biot number is reduced as the deviation approaches the minimum value. By this procedure, the biot number, specific heat, and thermal diffusivity that give the minimum deviation value are obtained.
In the above procedure, it is also possible to set a plurality of biot numbers and specific heats respectively. In this case, the set interval ΔX between the biot number and the specific heat is set to be smaller as the deviation approaches the minimum value.
[0023]
In addition, as the determination of S-7, one of the thermal diffusivity, specific heat, biot number or deviation is determined as X_i And the value at that time is X_m As | (X_i -X_m ) / X_m | <Ε or | X_i -X_m The determination may be made under a condition satisfying | <ε.
[0024]
In the method for measuring the thermal constant using the laser flash method according to the third aspect, either the thermal diffusivity or the specific heat can be set first. Hereinafter, the case where the specific heat is set first and the thermal constant is determined will be described.
After setting a new specific heat in the vicinity of the specific heat that gives the minimum deviation, the Bioe number is updated based on a relational expression determined by the specific heat and the value of the thermal diffusivity that gives the minimum deviation, so only the specific heat c is used as a variable. Desired values of α, c, and h can be obtained accurately and quickly.
This measurement procedure will be described in detail below with reference to the flowchart shown in FIG. First, as the initial values of the specific heat c and the thermal diffusivity α of the unknown layer in the multilayer material, c₀ , Α₀ Is appropriately determined with reference to literature values and the like (S-1).
Regarding the thermal diffusivity, t when there is no radiation loss_1/2 And the average thermal diffusivity.
Next, the initial value h of the biot number h₀ Is a time constant τ obtained from a temperature response curve, and a half-value arrival time t_1/2 Equation h using the value of₀ = F (t_1/2 , Τ) (S-2), and the initial values are set as a set specific heat, a set thermal diffusivity, and a set biot number, respectively. The first step is completed by S-1 and S-2. This initial value setting method is not limited only to the measuring method described in claim 3.
Subsequently, a new specific heat c is set near the set specific heat._i Is set (S-3), and the second step is terminated.
Then, the values (c) set in S-2 and S-3 are set._i , H₀ ) Is substituted into the theoretical temperature response F (α, c, h, t) or F ′ (α, c, h, p) obtained by Laplace transform to obtain the actual temperature response E (t), Alternatively, the value of α that minimizes the deviation R from E ′ (p) is determined by setting δR / δα = 0, and this is determined by α_i (S-4).
[0025]
In S-5, the set c_i , And α_i H using the value of_i = G (α_i , C_i Biot number h updated by equation_i Is obtained (S-5).
Where h_i Can be calculated based on the following equation, for example.
c_av= (Σ (ρ_j ・ C_pj.DELTA.l_j )) / L_n
α_av= L_n / ((Σ (Δl_j / (Ρ_j ・ C_pj・ Α_j ))) ・ C_av)
β_0i= L_n / (Sqrt (α_av・ Τ)
h_i = Β_0i・ ((Sin β_0i)^-1− (Tanβ_0i)^-1)
Where ρ_j ・ C_pj.DELTA.l_j , L_n Are the density in the j-th layer, the specific heat at constant pressure, the thickness of the j-th layer, and the thickness of all the layers, respectively. C_avIs the average value of the heat capacity per unit volume of the multilayer material, α_avRepresents the average thermal diffusivity of the multilayer material, and Σ represents the sum of the terms from the first layer to the n-th layer which is the final layer.
Next, the difference between any one of the values (thermal diffusivity, biot number, deviation) obtained in S-4 and S-5 and each set value already set or the ratio of each value of the difference is calculated. The determination is made by comparing based on the allowable reference value (S-6), and the above S-4 and S-5 are repeated until the determination conditions are satisfied, and finally the desired heat constant (thermal diffusivity, bio (Number, specific heat), and the third step is completed.
The thermal diffusivity used in S-6 is the thermal diffusivity of the unknown layer or the average thermal diffusivity of all or some layers including the unknown layer.
[0026]
In S-7, the thermal diffusivity, the specific heat, and the deviation between the theoretical temperature response and the temperature response including the Biot number are determined, and the difference between the specific heat or the deviation and the set specific heat and the set deviation, the set specific heat and A ratio or a difference from the set deviation is calculated (S-8). If the ratio or the difference is larger than the specified allowable value, the process returns to S-3 and repeats from S-3 to S-7. Alternatively, when the difference becomes equal to or less than the specific allowable value, the thermal diffusivity, specific heat, and biot number at that time are output (S-9), and the fourth step is completed and the entire procedure is completed.
[0027]
An example of the method of setting the specific heat in S-3 is shown below.
First, when the specific heat set value is changed, the biot number, thermal diffusivity, and deviation are determined for each specific heat. The magnitude of this deviation is compared, and the next specific heat is set in a direction that gives a small deviation. The amount of change in the specific heat is reduced as it approaches the minimum deviation value. In the above procedure, a plurality of specific heats can be set. In this case, the set interval Δc of the specific heat is reduced as approaching the deviation minimum value.
[0028]
In addition, in the determination of S-8, | (X) where X is one of the thermal diffusivity and biot number._i -X_m ) / X_m | <Ε or | X_i -X_m The determination may be made under a condition satisfying | <ε. When a plurality of specific heats are set in S-3, the thermal diffusivity, biot number, and deviation are calculated for each specific heat using S-4, S-5, and S-6. The specific heat that gives the minimum deviation from is selected, and the same determination is made in S-8. If the condition is not satisfied, a plurality of new specific heats are set in S-3 in a direction that gives a smaller deviation.
[0029]
In the thermal constant measuring apparatus using the laser flash method according to claim 5, one of the thermal diffusivity, specific heat, and biot number is A, the other is B, and the other is C, and A and B First calculating means for setting an initial value, obtaining C that minimizes the deviation between the temperature response and the theoretical temperature response on the back surface of the multilayer material, and setting the deviation as an initial value of the minimum deviation; A first calculation that sets new A and B of the material near A and B that gives the minimum deviation, and deviations of A and B of the multilayer material from the theoretical temperature response and the backside temperature response of the multilayer material Performs a second calculation to find the C and the deviation that minimizes, compares the deviation with the minimum deviation, and if the deviation is greater than the minimum deviation, repeats the first and second calculations, When the deviation is less than the minimum deviation, A second calculating means for newly setting the deviation as a minimum deviation; determining whether the minimum deviation has reached near a local minimum value; if not, repeating the first and second calculations to minimize the deviation; When the value reaches the vicinity of the minimum value, there is provided a third calculating means for outputting the thermal diffusivity, the specific heat, and the biot number, so that the time function F (α, c, h, t) of the theoretical temperature response is provided. A and B are defined by using a deviation R between F ′ (α, c, h, p), which is the Laplace transform form, and an actually obtained temperature response E (t) or E ′ (p) as an index. Minimum deviation R_m Is set in the vicinity of the value giving the minimum deviation R_m Is updated, and the true values of the unknown Biot number, specific heat and thermal diffusivity can be obtained by a simple calculation means. Each of the first to third calculation means includes a computer configured to perform an actual calculation according to a preset program.
[0030]
In the thermal constant measuring apparatus using the laser flash method according to claim 6, the thermal diffusivity, the specific heat, and the initial value of the biot number are set, and the theoretical temperature response and the back surface temperature response including each initial value. A first calculating means for determining the thermal diffusivity, specific heat, biot number, and the deviation as a set thermal diffusivity, a set specific heat, a set biot number, and a set deviation, respectively, and the set thermal diffusivity, Including one of the specific heats as A and the other one as B, the second arithmetic means for setting a new A in the vicinity of the A, the A set by the second arithmetic means, and the set biot number B that minimizes the deviation between the theoretical temperature response and the temperature response was calculated, the biot number was updated with the thermal diffusivity and specific heat obtained from the calculation result, a new set biot number was determined, and A was fixed. Near the deviation minimum in the state A third calculating means for determining the thermal diffusivity, specific heat and Biot number by repeating the above calculation, and a theoretical temperature response and a back surface temperature response including the thermal diffusivity, specific heat and Biot number obtained by the third calculating means And a fourth calculating means for determining whether the deviation has reached the vicinity of the minimum value. When the deviation obtained by the fourth calculating means has not reached the vicinity of the minimum value, A new A set in a direction to give a small minimum value is input to the second calculating means, and when the value reaches near the minimum value, a heat diffusivity, specific heat, and biot number at that time are output. 5, the Biot number is updated based on the thermal diffusivity and the specific heat, and the time function F (α, c, h, t) of the theoretical temperature response, or the Laplace transform type F '(Α, c, h, p) and the temperature actually obtained Answer E (t), or E 'the deviation R of the (p) as an index, the minimum specific heat c deviation R_m Is set in the vicinity of the value giving the minimum deviation R_m Is updated, and the true values of the unknown Biot number, specific heat and thermal diffusivity can be obtained by repetition of a simple arithmetic calculation means.
[0031]
【Example】
Next, embodiments of the present invention will be described with reference to the accompanying drawings to provide an understanding of the present invention.
Here, FIG. 1 is a configuration diagram of a laser flash device to which a method of measuring a thermal constant using a laser flash method according to an embodiment of the present invention is applied, and FIG. 2 is a diagram of a method of measuring a thermal constant using a laser flash method. FIG. 3 is an explanatory diagram of a laser flash method using a multilayer material, FIG. 4 is a flowchart of a direct method in a method of measuring a thermal constant using a laser flash method, and FIG. 5 is a diagram of a thermal constant using a laser flash method. FIG. 6 is a flow chart of the time constant method of the measurement method, FIG. 6 is a diagram showing a calculation error of the time constant method when the thickness of the unknown layer in the multilayer material having three layers is changed, and FIG. FIG. 8 is a flow chart of the direct method in the method of measuring the thermal constant, and FIG. 8 is a diagram showing a calculation error of the direct method when the thickness of the unknown layer in the multilayer material having three layers is changed.
[0032]
First, an apparatus for measuring thermal diffusivity, specific heat, and thermal constant such as Biot number using a laser flash method according to an embodiment of the present invention shown in FIG. 1 will be described. This is an apparatus for measuring an unknown thermal constant in one layer included in the multilayer material 10 using a laser pulse generator as a measurement sample as the multilayer material 10. The laser pulse generator can generate a pulse having an arbitrary waveform as required. The calculation can be simplified or specified by selecting a special waveform according to the range of the physical property values of the multilayer material 10 and the method of analyzing the thermal constant. It is also possible to improve the measurement accuracy in the area. The temperature response of the back surface of the sample is measured by a radiation thermometer or a thermometer such as a thermocouple.
[0033]
The laser pulse waveform signal applied to the multilayer material 10 is captured by a laser pulse detector via a half mirror 11 provided between the multilayer material 10 and the laser pulse generator. Further, it is also possible to measure the pulse waveform by measuring the leakage light in the middle of the optical path of the laser light.
Then, the signal data from the temperature measuring device and the laser pulse detecting device are taken into a computer, and an arithmetic process for determining thermal constants such as a thermal diffusivity, a specific heat, and a Biot number is performed based on the signal data. The whole is configured so that the results can be displayed on an output device connected to a computer.
[0034]
In order to analyze thermal characteristics such as thermal diffusivity or specific heat of the multilayer material 10 using the laser flash method, first, a theoretical temperature response when the multilayer material 10 is irradiated with the laser pulse Qf (t) is obtained. Here, f (t) uses a function normalized with the integral value in the entire time domain set to 1. This analysis is performed on the Laplace space obtained by the Laplace transform, and the theoretical solution of the temperature response can be obtained as follows.
[0035]
First, assuming that the multilayer material 10 is composed of n layers, the thermal diffusivity α in the i-th layer_i , Constant pressure specific heat C_pi, Density ρ_i , And thickness ΔL_i Is determined. Therefore, the thermal conductivity k_i Is k_i = Α_i ・ C_pi・ Ρ_i Is represented by The thickness L from the surface to the boundary between the i-th layer and the (i + 1) -th layer_i Is L_i = ΔL₁ + ΔL_Two + ... + ΔL_i It becomes.
The heat conduction equation for the i-th layer in the laminated composite material is δT_i (X, t) / δt = α_i ・ Δ^Two T_i (X, t) / δx^Two And the initial condition is T_i (X, t) = 0 and the boundary condition is T_i (L_i , T) = T_{i + 1} (L_i , T) and k_i ・ ΔT_i (L_i , T) / δx = k_{i + 1} ・ ΔT_{i + 1} (L_i , T) / δx, respectively.
Therefore, by applying Laplace transform to these, the Laplace variable is defined as p, and the above heat conduction equation becomes δ^Two T_i (X, p) / δx^Two = (P / α_i ) ・ T_i (X, p) and the boundary condition is T_i (L_i , P) = T_{i + 1} (L_i , P) and k_i ・ ΔT_i (L_i , P) / δx = k_{i + 1} ・ ΔT_{i + 1} (L_i , P) / δx. Then, by applying the same operation sequentially from the first layer to the final n-th layer, a set of Laplace-transformed heat conduction equations for all the multilayer materials 10 is obtained.
[0036]
As described above, the general solution of the heat conduction equation in the i-th layer is given by T_i (X, p) = A_i exp (r_i ・ X) + B_i exp (-r_i X). Where r_i Is r_i = Sqrt (p / α_i ) And A_i , B_i Is an integration constant.
Then, based on the boundary condition between the i-th layer and the (i + 1) -th layer, the integral constant A of the temperature equation of the i-th layer_i , B_i And the integral constant A of the temperature equation of the (i + 1) th layer_{i + 1} , B_{i + 1} Equation 1 can be obtained by finding the relationship Where G_{i, i + 1} Is a matrix represented by Expression 2. Therefore, the coefficient of the first layer temperature and the coefficient of the n-th layer temperature have a relationship represented by Expression 3.
A similar analysis is performed using the conditions of heat input and heat radiation on the front and back surfaces, and finally the temperature response in the n-th layer, that is, the sample back surface temperature Ts, is given by Equation 4, and three variables (thermal diffusivity) α_i , Specific heat c_i , Biot number h) is represented by a relational expression T = F ′ (α, c, h, p). (However, the value h on the surface side of the biot number)₀ And the value h on the back side₁ Are assumed to be equal. h = h₀ = H₁ )
[0037]
(Equation 1)

[0038]
In the following, the partial differential of the thermal diffusivity α, the specific heat c or the Biot number h with respect to the square deviation R with respect to the temperature response is set to zero using the above-mentioned formula 4 which is a basic formula for giving the theoretical temperature response on the back surface of the sample. The procedure for obtaining the thermal diffusivity α, specific heat c or Biot number h that satisfies the equation obtained by the Newton method will be described.
The thermal diffusivity α, specific heat c or Biot number h of the layer with unknown thermal characteristics is represented by a variable x, and Δx = − (δR / δx) / (δ^Two R / δx^Two ) And x_new = X + Δx, x is updated. By repeating this procedure, x that satisfies δR / δx = 0, that is, thermal diffusivity α, specific heat c, or biot number h can be obtained.
[0039]
As described above, the theoretical temperature response of each layer includes the thermal diffusivity α and the heat capacity per unit volume (ρ · C_p ) Appears. Therefore, what is directly obtained is the thermal conductivity which is the product of these two thermal characteristic values (α, c). If the density is known, the specific heat at constant pressure is also obtained. However, it is also possible to treat the density of the thermal characteristics unknown layer as a constant and the constant pressure specific heat as variables in the analysis, and to calculate the heat capacity per unit volume by multiplying the constant pressure specific heat by the density after the calculation. In the present numerical analysis method, (1) a one-dimensional heat conduction equation is established, (2) the biot numbers on the front and back surfaces of the sample are respectively equal, and (3) all the thermal characteristic values except for the i-th layer and all the The calculation is performed under the precondition that the layer thickness is known. Further, when the following time constant method is applied, (4) the thermal conductivity λ of each layer in the steady state of the multilayer material_i And the relational expression between the heat conductivity and the overall thermal conductivity λ is approximately satisfied.
[0040]
In the case of the multilayer material 10 as shown in FIG. 3, three variables of unknown layer of thermal characteristics, (1) thermal diffusivity α, (2) specific heat c, and (3) Biot number h, are unknown variables. = G (α, c), this variable (α, c, h) can be determined by the condition of reducing the variable by one and minimizing the deviation between the theoretical temperature response and the temperature response.
[0041]
(Calculation of time constant τ)
In the case of a single-layer material having a thickness of L, the characteristic time t₀ To t₀ = L^Two / (Π^Two ・ Α₀ ), M is a constant and t >> mt₀ In the time domain t that satisfies the following condition, since the second and subsequent terms (n ≧ 1) of Equation 5 become negligibly smaller than the first term, T (L, t) / T_m = A₀ exp (-a₀ ・ T) @exp (a₀ The theoretical solution of the back surface temperature can be approximated by the equation (t ′) f (t ′) dt ′. Where A₀ = 2β₀ ^Two/ (Β₀ ^Two+ 2h + h^Two ), A₀ = Β₀ ^Two・ Α / L^Two It is.
[0042]
(Equation 2)

[0043]
Therefore, when the logarithm of the temperature response data is plotted against time and the time constant τ is obtained from the slope, the time constant τ is a₀ Can be considered equal to the reciprocal of That is, τ = 1 / a₀ = L^Two / (Β₀ ^TwoΑ), and this relationship is the average thermal diffusivity α of the multilayer material 10._avHolds for α_av= Α and τ = 1 / a₀ = L^Two / (Β₀ ^Two・ Α_av) Is obtained.
[0044]
For single-layer materials, the eigenvalue β_n Tan (β_n ) = 2h · β_n / (Β_n ^Two -H^Two From the equation, the biot number h is β₀ , And by choosing a positive value for h, h₀ = Β₀ ・ (1 / sinβ₀ −1 / tanβ₀ ) Can be obtained. Then, since this relationship also holds for the multilayer material 10, β₀ Is the average thermal diffusivity α of the multilayer material 10_avAnd time constant τ, β₀ = L / sqrt (α_av・ Τ). That is, the biot number h is the average thermal diffusivity α_avAnd the time constant τ as a function of the average thermal diffusivity α of the multilayer material 10._avIs a function of the unknown thermal diffusivity α in the multilayer material 10 and the specific heat c, so that the biot number h can be set as h = g (α, c) as a whole.
[0045]
The unknown thermal constant contained in any one layer in the multilayer material 10 can be measured by the following direct method and time constant method.
The direct method is such that the three variables (thermal diffusivity α, specific heat c, biot number h) are set directly to minimize the square deviation R between the theoretical temperature response and the temperature response (α, c, h) is a method to reach the optimum solution. The time constant method is to find a time constant τ after a sufficient time has elapsed with respect to the characteristic time of the sample after laser irradiation, and to obtain a thermal constant through the time constant τ. This is a method of reducing variables using the fact that the diffusion rate α and the biot number h have a specific relationship. That is, this is a method in which one variable is reduced by a relational expression established between α, c and h to reach the minimum square deviation R.
[0046]
Here, as the deviation, (a) forward square deviation I, (b) forward square deviation II, (c) inverse square deviation I, and (d) inverse square deviation II as defined by the following equations , (E) Each deviation R such as a logarithmic square deviation can be appropriately adopted.
(B) R = Σ (Q · T_j -E_j )^Two
(B) R = (ΣE_j ^Two ) ・ (ΣT_j ^Two )-(ΣE_j ・ T_j )^Two
(C) R = Σ ｛(1 / (Q · T)_j ))-(1 / E_j )｝^Two
(D) R = (Σ1 / E)_j ^Two ) ・ (Σ1 / T_j ^Two )-｛Σ1 / (E_j ・ T_j )｝^Two
(E) R = Σ ｛ln (Q · T_j ) -Ln (E_j )｝^Two
[0047]
Here, Q is the total energy absorbed by the sample of the laser pulse applied to the multilayer material 10. When the calculation is performed for each of the above-described deviations, the following characteristics are provided.
(A) This is a general definition. The Q obtained by δR / δQ = 0 is substituted into the equation (A), and the value obtained by eliminating Q is used as the deviation.
(B) The deviation obtained by substituting Q obtained by δR / δQ = 0 into the equation (A) using the general equation (A) and setting the numerator to zero, There is an advantage that the calculation formula is simpler than that of ()).
(C) Since the variables are concentrated in the denominator of the theoretical temperature, the calculation formula is simplified by making the reciprocal. The Q obtained by δR / δQ = 0 is substituted into equation (c), and the value obtained by eliminating Q is used as the deviation.
(D) A deviation obtained by substituting Q obtained from δR / δQ = 0 using the equation (c) into the equation (c) and setting the numerator to zero, which is calculated in comparison with the equation (c) There is an advantage that the formula is simplified.
(E) The minimum value can be easily determined by setting the logarithmic square deviation. The value obtained by substituting Q obtained by δR / δQ = 0 into equation (e) and eliminating Q is used as the deviation.
In the case where the Laplace transform format is adopted in the equations (a) to (e), a plurality of Laplace variables p are set, and the temperature response data is subjected to the Laplace transform for the plurality of set p. E_j And these are applied to the equations (a) to (e).
[0048]
(Method of measuring thermal constant by time constant method)
The procedure of the measurement method in the time constant method is shown below.
(1) Average thermal diffusivity α_avInitial value calculation
Normal half-value arrival time t_1/2 The maximum temperature rise value T from the center of gravity of the laser pulse heat input curve according to the method_m The time required for the temperature to rise to half the temperature is calculated, and the initial value of the average thermal diffusivity α_avIs calculated. Alternatively, the specific heat and thermal diffusivity of the layer whose thermal constant is unknown are set, and the average thermal diffusivity α of the multilayer material is set based on the specific heat and thermal diffusivity._avMay be calculated.
Here, the average specific heat c in the multilayer material_av, And average thermal diffusivity α_avIs given by the following equation.
c_av= (Σρ_i C_i Δl_i ) / L_n
α_av= L_n / ((ΣΔl_i / (Ρ_i C_i α_i )) ・ C_av)
Where Δl_i Is the thickness of the i-th layer (= l_i −l_i-1 And the total thickness l of the multilayer material_n Is l_n = ΣΔl_i It is.
[0049]
(2) Calculation of time constant τ
The time constant from the decay of the measured sample temperature in the time domain in which the approximate expression based on only the first term of the time-space theoretical solution of the sample back surface temperature is satisfied when sufficient time has passed after the laser pulse heat input and the single-layer material is assumed. Find τ.
(3) Biot number initial value h₀ Calculation
Average thermal diffusivity α_avFrom the time constant τ and the eigenvalue equation₀ Ask for.
(4) Initial value of specific heat
Set multiple specific heats for the layer with unknown thermal constant.
(5) Calculation of thermal diffusivity of layer with unknown thermal constant by Newton method
For a plurality of specific heats, the thermal diffusivity of the layer with unknown thermal constant is updated by Newton's method using the theoretical temperature response of the back surface of the multilayer material 10 obtained by using a laser pulse and the square deviation R of the measured temperature. The average thermal diffusivity and the biot number of the multilayer material 10 are updated using the updated thermal diffusivity. Then, the calculation is repeatedly performed using these values, and the calculation is repeatedly performed until Δα / α becomes sufficiently small.
[0050]
(6) Determination of minimum value of squared deviation R and resetting of multiple specific heats
A square deviation is calculated from the specific heat c, the thermal diffusivity α, and the Biot number h calculated in the above (5), and the specific heat that gives the minimum value is obtained. Until the ratio Δc / c of the plurality of set specific heat intervals Δc and c becomes sufficiently small, the specific heat is reset to the vicinity of the specific heat giving the minimum value, and this process is repeated.
In the above procedure, a plurality of specific heats c are set, and the corresponding thermal diffusivity α is obtained by the Newton method. However, the thermal diffusivity α may be replaced with the specific heat c.
That is, a method may be used in which a plurality of thermal diffusivities α are set and the specific heat c corresponding thereto is obtained by the Newton method. In this case, since the specific heat c is represented by an algebraic equation, the root thereof may be analytically obtained.
[0051]
FIG. 6 is a calculation result in the case where the thickness of the unknown layer in the multilayer material 10 composed of three layers obtained by the time constant method is changed. (A) shows an example in which the lowermost third layer contains an unknown thermal constant, and (b) shows an example in which the intermediate layer, the second layer, contains an unknown thermal constant. As is clear from FIG. 7, it can be seen that in both cases (a) and (b), the calculation error, that is, the measurement error can be kept within a range of about ± 1% or less.
[0052]
(Method of measuring thermal constant by direct method)
The direct analysis method is a method of applying the least squares method and the Newton method described above to a deviation R expressed by three independent variables of a thermal diffusivity α, a specific heat c, and a Biot number h to obtain a minimum value of the deviation R. To determine the thermal diffusivity α, specific heat c, and biot number h that satisfy the following conditions. Two examples of this procedure are shown below.
[0053]
Direct method 1
A procedure for simultaneously setting a plurality of biot numbers to obtain a desired thermal constant according to FIG. 7 will be described below.
(1) Initial value of specific heat c₀ Is set (S-1).
(1) Set a plurality of biot numbers (S-2).
(2) For each biot number, the minimum value R of the squared deviation R_j Is determined by the following procedure.
(2) -1 For each of a plurality of set specific heats, the minimum value R of the deviation R by applying the least squares method and the Newton method._j (S-3), (S-4). Alternatively, a method is also possible in which a plurality of thermal diffusivities are set in S-3 and a specific heat that minimizes the deviation R is obtained.
(2) -2 The deviation R obtained for each combination of specific heat and thermal diffusivity is compared, and the minimum value R_j (S-5) and (S-6).
(2) -3 A plurality of new specific heats are set in the direction of decreasing the deviation R near the specific heat giving the minimum value (S-3), and the process is repeated from (2) -1. Α and c when Δc / c is sufficiently small are obtained.
[0054]
(3) Minimum value R of deviation R_j The number of Biot that gives is calculated by the following procedure.
(3) -1 For each biot number and the combination of the thermal diffusivity and the specific heat determined in the step (2), the deviation R is determined and the minimum value R of the deviation R is determined._i Is selected (S-7).
(3) -2 Minimum value R_i A plurality of new biot numbers are set in the direction of decreasing the deviation R near the biot number given (S-2), and the step (2) is performed. H, α, and c when Δh / h is sufficiently small are obtained (S-8).
(4) The desired value h_m , Α_m , C_m Is output. (S-9)
In the above-described procedure for obtaining the minimum value of the deviation R, the biot number, the specific heat, and the thermal diffusivity are the same independent variables, so the same holds even if the variables are exchanged. For example, a method may be used in which a plurality of biot numbers are set while a plurality of specific heats are set, and a thermal diffusivity that gives a minimum value is obtained. The thermal diffusivity, specific heat and Biot number that give the minimum value of the deviation R can also be obtained by the Newton method for a plurality of variables.
[0055]
Direct method 2
(1) First, the specific heat c, which is an unknown in one layer of the multilayer material 10 as shown in FIG. 3, and the biot number h are appropriately determined, and the respective initial values c₀ , H₀ On the hc plane (c₀ , H₀ ) Set point.
(2) Each initial value (c₀ , H₀ ) Is substituted for the theoretical temperature response F (α, c, h, t) and F ′ (α, c, h, p), and the theoretical temperature response F (α, c₀ , H₀ , T), F ′ (α, c₀ , H₀ , P) and the actual temperature responses E (t), E ′ (p) are determined by setting the value of α that minimizes the deviation R to δR / δα = 0.
[0056]
(3) The value of α that satisfies the equation δR / δα = 0 is obtained by a numerical analysis method such as the least square method and Newton method described above.
The initial value of the thermal diffusivity α obtained at this time is α₀ , The initial value of the deviation R is R₀ As a set of these initial values (R₀ , Α₀ , C₀ , H₀ ) Is replaced with a tentative minimum value set (R_m , Α_m , C_m , H_m ).
(4) Minimum value set (c_m , H_m ) = (C₀ , H₀ ), A new set of values (c₁ , H₁ ) Is set. The setting of this new point may be performed by performing an operation on a computer, such as designating a circle within a preset reference radius by a random number method with the point on the two-dimensional plane as a center point. it can. The accuracy can be improved by gradually reducing the reference radius. Furthermore, it is also possible to set a plurality of new update values in a direction in which the deviation or the difference between the updated heat constants becomes smaller using a plurality of already set heat constant sets.
[0057]
(5) The above c₁ , H₁ Is substituted for the theoretical temperature response F (α, c, h, t) and F ′ (α, c, h, p), and the theoretical temperature response F (α, c₁ , H₁ , T), F ′ (α, c₁ , H₁ , P) and the actual temperature responses E (t), E ′ (p) are determined by setting δR / δα = 0 such that α is minimized. The value of the obtained thermal diffusivity α is α₁ , The value of the deviation R₁ As these sets (R₁ , Α₁ , H₁ , C₁ ).
(6) Next, R already set as the minimum value_m (= R₀ ) And R₁ And R₁ Is R_m If it is larger, the minimum value set (c_m , H_m ) = (C₀ , H₀ ), A new (c)_Two , H_Two ) Is reset. R₁ Is R_m If less than R_m = R₁ And the minimum value R_m To update.
(7) Then, the minimum value R is smaller than the preset allowable reference value ε._m By repeating the above operation until the value becomes smaller, the minimum value R_m Is gradually reduced to a minimum value R_m Is smaller than the allowable reference value ε, the set of minimum values (R_m , Α_m , C_m , H_m ) Is output as desired α, c, h.
Note that the above operation is performed by a computer that has input the program of the above-described operation procedure, by performing arithmetic processing on temperature response data obtained using the laser flash method.
[0058]
FIG. 8 is a calculation result in the case where the thickness of the unknown layer of the multilayer material 10 composed of three layers obtained by the direct method 2 is changed. (A) shows an example in which the second layer, which is the intermediate layer, contains an unknown thermal constant, and (b) shows a case in which the lowermost third layer contains an unknown thermal constant. As is clear from FIG. 7, in both cases (a) and (b), the calculation error, that is, the measurement error satisfies the generally required range of measurement accuracy.
[0059]
As described above, the embodiments of the present invention have been described. However, the present invention is not limited to these embodiments, and all changes in conditions without departing from the gist are within the scope of the present invention.
For example, in the present embodiment, a case has been described in which the theoretical temperature response and the data of the temperature response are subjected to the Laplace transform for processing, but the processing may be performed in a normal time space. Further, as the deviation between the theoretical temperature response and the temperature response applied to the present invention, a forward squared deviation, an inverse squared deviation, a logarithmic squared deviation, and the like having different characteristics can be adopted as needed.
Further, since the three variables of thermal diffusivity, specific heat, and Biot number, which are unknowns included in the equation of the deviation between the theoretical temperature response and the temperature response, are all mathematically equivalent variables, The use of a relational expression in which each variable is arbitrarily exchanged in the relational expression described above is also considered to be within the scope of the present invention.
[0060]
【The invention's effect】
Claim 16In the method and apparatus for measuring the thermal constant in the laser flash method described, the deviation between the theoretical temperature response obtained by setting the thermal diffusivity α, the specific heat c, and the biot number h is compared with the actual temperature response. It is possible to update the value of the unknown value sequentially and efficiently, determine the value of the unknown value that asymptotically satisfies the measured data of the actual temperature response with high accuracy, and also requires the absolute value of the temperature response as in the past. Instead, the unknown thermal constant in the multilayer material can be determined based on the measured data of the relative temperature response.
Further, for example, using the deviation between the theoretical temperature response and the actually obtained temperature response as an index, for example, the biot number h and the specific heat c are reduced to the minimum deviation R_m , The minimum value is updated and the true value of the unknown biot number, specific heat, and thermal diffusivity can be obtained by a simple repetitive calculation means. Can be performed at high speed.
[0061]
In particular, in the method for measuring the thermal constant using the laser flash method according to claim 2, the deviation R is obtained by using the thermal diffusivity, specific heat and Biot number as variables, and α, c, The value of h can be determined accurately and quickly.
Further, in the method for measuring a thermal constant using the laser flash method according to the third aspect, the initial value of the biot number is a time constant in the temperature response, and a half-value arrival time t._1/2 Is set on the basis of the relationship with the value, and so on, so that it can be set in advance in the vicinity of the true value, the number of calculations can be greatly reduced, and the required thermal constant does not diverge.
[0062]
Claims5In the apparatus for measuring a thermal constant using the laser flash method described above, for example, using the deviation R between the theoretical temperature response and the actually obtained temperature response as an index, for example, the minimum deviation R_m Is set in the vicinity of the value giving the minimum deviation R_m And the true values of the unknown biot number, specific heat and thermal diffusivity can be obtained by a simple calculation means.Therefore, there is no need to measure the absolute temperature, and the sample temperature can be measured without contacting the sample. It is possible to measure up to a high temperature range using a radiation thermometer or the like that can measure the temperature and is less affected by the environmental conditions of the measurement.
Claims6In the thermal constant measuring apparatus using the laser flash method described, the biot number is updated based on the thermal diffusivity and the specific heat, so the deviation R between the theoretical temperature response and the actually obtained temperature response is used as an index. , Thermal diffusivity α or specific heat c to minimum deviation R_m Is set in the vicinity of the value giving the minimum deviation R_m Is updated, and the true values of the unknown Biot number, the specific heat and the thermal diffusivity can be obtained by repetition of a small number of calculation calculation means, so that the calculation error can be reduced.
[Brief description of the drawings]
FIG. 1 is a configuration diagram of a laser flash apparatus to which a method for measuring a thermal constant using a laser flash method according to an embodiment of the present invention is applied.
FIG. 2 is an explanatory diagram of a method for measuring a thermal constant using a laser flash method.
FIG. 3 is an explanatory diagram of a laser flash method using a multilayer material.
FIG. 4 is a flowchart of a direct method in a method of measuring a thermal constant using a laser flash method.
FIG. 5 is a flowchart of a time constant method in a method of measuring a thermal constant using a laser flash method.
FIG. 6 is a diagram illustrating a calculation error of a time constant method when the thickness of an unknown layer in a multilayer material including three layers is changed.
FIG. 7 is a flowchart of a direct method in a method for measuring a thermal constant using a laser flash method.
FIG. 8 is a diagram illustrating a calculation error of the direct method when the thickness of an unknown layer in a multilayer material including three layers is changed.
[Explanation of symbols]
10 Multilayer materials
11 Half mirror

Claims (6)

The temperature response of the back side obtained by irradiating the laser flash from the front side of the multilayer material including the one layer whose thermal diffusivity and specific heat are unknown is Laplace-transformed, and the thermal diffusivity, specific heat, and biot number are included as variables. A method for measuring a thermal constant using a laser flash method, comprising comparing the thermal diffusivity, specific heat, and biot number with a theoretical temperature response given by a solution of a heat conduction equation obtained by Laplace transform . The thermal diffusivity, the specific heat, and one of the Biot number A, B and the remaining one, the other as C, and the initial values of the A and the B, a backside temperature response of the multi-layer material A first step of obtaining C that minimizes a deviation from the theoretical temperature response and setting the deviation as an initial value of the minimum deviation;
The new A and B of the multilayer material, a second step of setting in the vicinity of the A and the B gives the minimum deviation,
A third step of determining C and a deviation from A and B of the multilayer material that minimize the deviation between the theoretical temperature response and the back surface temperature response of the multilayer material;
By comparing the minimum deviation and the deviation, when the deviation is larger than the minimum deviation from the second step by repeating the third step, when the deviation is below the minimum deviation, the deviation by repeating the newly set to the second step from the third step as the minimum deviation, at the time when the minimum deviation reaches its minimum value near the thermal diffusivity, specific heat, and a fourth step of outputting the Biot's number 2. A method for measuring a thermal constant using a laser flash method according to claim 1, comprising: The thermal diffusivity, the specific heat, and sets the initial values of the Biot number, a deviation between the theoretical temperature response and backside temperature response including respective initial value, the heat diffusivity, the specific heat, the Biot number , and each set thermal diffusivity of the deviation, a first step to set specific heat, set Biot number and configuration deviation,
A second step of setting any one of the set thermal diffusivity and specific heat as A and the other as B, and setting a new A near the A,
Seeking B by calculation step of calculating a B to minimize the deviation between said temperature responsive to the theoretical temperature response including the set A, and set Biot number, new updates the Biot's number by the A and the B A third step of determining the thermal conductivity, specific heat, and the number of the biot by repeating the calculation process until reaching the vicinity of the deviation minimum value in a state where A is fixed.
The thermal diffusivity, specific heat, and the deviation between the theoretical temperature response and the temperature response, including the Biot number, to determine whether the deviation has reached near the local minimum, if not, from the second step the third repeated until step, when the deviation reaches its minimum value near the thermal diffusivity at that point, specific heat, and laser flash method according to claim 1, further comprising a fourth step of outputting the Biot's number Measurement method of thermal constant using. The update of the biot number is based on the average thermal diffusivity α of the multilayer material. _av The total thickness of the multilayer material l _n , And eigenvalue β ₀ The time constant l of the multilayer material expressed by _n ^Two / (Β ₀ ^Two ・ Α _av ) Plots the logarithm of the temperature response data against time and determines the eigenvalue β equal to the time constant determined from the slope. ₀ And β ₀ ・ (1 / sinβ ₀ −1 / tanβ ₀ 4. The method for measuring a thermal constant using a laser flash method according to claim 3, wherein the value is set to a value obtained from (1). And temperature response of the temperature from the surface side of the back surface side obtained by irradiating a laser flash multilayer material Laplace transform thermal diffusivity and specific heat containing unknown one layer, the thermal diffusivity, the specific heat, and the Biot's number comparing the theoretical temperature response given by general solution of unrealized Laplace transform thermal conduction equation as a variable, the thermal diffusivity, the specific heat, and the heat of the multilayer material using a laser flash method for obtaining the Biot's number A constant measuring device,
The thermal diffusivity, the specific heat, and one of the Biot number A, B and the remaining one, the other as C, and the initial values of the A and the B, temperature response in the rear surface of the multilayer material First calculating means for determining C that minimizes the deviation between the deviation and the theoretical temperature response, and setting the deviation as an initial value of the minimum deviation;
The new A and B of the multilayer material, a first and a calculation to be set in the vicinity of A and B gives the minimum deviation, from A and B of the multilayer material, the theoretical temperature response and backside temperature response of the multi-layer material performs deviation second calculation for obtaining the C and deviation which minimizes the, by comparing the with the deviation minimum deviation, the deviation is when the larger minimum deviation, the first and second Repeat calculations, when the difference is below the minimum deviation, second calculating means for newly setting the deviation as the minimum deviation,
It said minimum deviation is determined has been reached in the vicinity of the minimum value, at the time when the minimum deviation value by repeating said first and second calculation when it has not reached reaches its minimum value near the thermal diffusivity And a third calculating means for outputting a specific heat and a Biot number. A measuring apparatus for a thermal constant using a laser flash method. And temperature response of the temperature from the surface side of the back surface side obtained by irradiating a laser flash multilayer material Laplace transform thermal diffusivity and specific heat containing unknown one layer, the thermal diffusivity, the specific heat, and the Biot's number comparing the theoretical temperature response given by the solution of unrealized Laplace transform thermal conduction equation as a variable, the thermal diffusivity, the specific heat, and thermal constant of the multilayer material using a laser flash method for obtaining the Biot's number Measuring device,
The thermal diffusivity, the specific heat, and sets the initial values of the Biot number, a deviation between the theoretical temperature response and backside temperature response including respective initial value, the heat diffusivity, the specific heat, the Biot number , and each set thermal diffusivity of the deviation, the first calculation means to set the specific heat, set Biot number and configuration deviation,
A second computing means for setting any one of the set thermal diffusivity and specific heat as A and the other one as B, and setting a new A near the A,
The second A is set by the operation means, and said deviation of the back surface temperature response to calculate the B to minimize the theoretical temperature responses including setting Biot number, thermal diffusivity and specific heat obtained by the calculation result The third calculation means for determining the thermal diffusivity, specific heat and biot number by repeating the above calculation until the biot number is updated to determine a new set biot number and reaching the vicinity of the deviation minimum value in the state where A is fixed. When,
The third thermal diffusivity was obtained by the calculation means, specific heat, and the theoretical temperature response including the Biot's number and a deviation between the backside temperature response, the deviation is the fourth determining whether reaches the vicinity of the minimum value Arithmetic means;
When the deviation obtained by the fourth arithmetic means does not reach the vicinity of the minimum value, a new A set in a direction for giving a smaller minimum value is input to the second arithmetic means, and the minimum value is input to the second arithmetic means. And a fifth calculating means for outputting the thermal diffusivity, specific heat, and Biot number at the time when the value reaches the vicinity of the value, and a thermal constant measuring apparatus using a laser flash method.

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