JP2004251843A - Method, apparatus, and computer program for analyzing inverse problem, and computer-readable recording medium - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To stabilize inverse problem analysis that uses a non-steady state heat conduction equation. <P>SOLUTION: The temperature or thermal flux on the surface of a reactive vessel is acquired by inverse problem analysis, using non-steady heat conduction equation, based on the temperature data measured with a thermocouple embedded in the wall of the reactive vessel that accompanies temperature change reaction. In that case, the temperature data are collected at a sampling time which is shorter than, for example, a calculation time step at inverse problem analysis. The data are averaged over the time, to increase the number of decimals of the temperature data measured with the thermocouple, and the temperature data in which the number of decimals is increased are used for inverse problem analysis. <P>COPYRIGHT: (C)2004,JPO&NCIPI

Description

【００２２】
数１において、ρは反応容器の材料の密度、Ｃ_ｐは反応容器の材料の比熱、Ｔは反応容器内部の温度の計算値、ｔは時間、ｋは反応容器の材料の熱伝導度を表す。
【００２３】
熱伝導逆問題解析というのは、計算領域を支配する非定常熱伝導方程式を基にして、領域内部の温度を既知として、領域境界での温度や熱流束等の境界条件を推定することをいう。これに対して、熱伝導順問題解析というのは、既知である領域境界での温度や熱流束等の境界条件から領域内部の温度を推定することをいう。
【００２４】
２次元逆問題解析の手法の例としては、例えば、本願出願人が特開２００２−２０６９５８号公報に開示したものがあり、この手法はそのまま１次元逆問題解析へも適用できる。また、１次元逆問題解析の例として、Ｂｅｃｋらにより提案された解析手法が知られている（Ｂｅｃｋその他、Ｉｎｖｅｒｓｅ Ｈｅａｔ Ｃｏｎｄｕｃｔｉｏｎ，１９８５，Ｗｉｌｅｙ，Ｎｅｗ Ｙｏｒｋ）。
【００２５】
また、逆問題解析の最近の手法として、カルマンフィルター理論や、射影フィルタ理論等の確率的推定法を適用することも考えられる。この手法は、現状では、上記数１の左辺をゼロと置いた、定常熱伝導方程式（観測方程式）への適用が検討されているが、非定常項を含めて適切に観測行列を構成できれば、同様の逆問題解析ができる可能性がある。この定常微分方程式への、確率推定法の適用例としては、登坂その他、「逆問題の数理と解法・偏微分方程式の逆解析」（東京大学出版会（１９９９））に詳しい。
【００２６】
本実施の形態では、逆問題解析の手法として上記特開２００２−２０６９５８号公報に開示した考え方を用いる。すなわち、下記の数２に示すように、ある１次元方向（図２に示す１ａ→１ｂ→１ｃや１ｄ→１ｅ等）に配置された各熱電対で測定された温度Ｙと、反応容器の内表面及び外表面における熱流束の仮定値から非定常熱伝導方程式により算出された各熱電対位置での温度Ｔとの差の二乗の和が最小となる仮定値を反応容器の内表面及び外表面における熱流束として求める。なお、Ｊは熱電対の数を表す。
【００２７】
【数２】

【００２８】
このように複数の熱電対位置での温度Ｔ、Ｙを完全に一致させるような解（反応容器の内表面及び外表面における熱流束）を求めるのではなく、最小二乗的に満たすような解を求めることにより、現実的な熱流束変化の推定が可能となる。その理由は、測定温度には様々な測定誤差要因が含まれるため、完全に一致させることは実用的に意味がないといえるからである。
【００２９】
なお、計算を安定化させるために、正則化項を付加するようにしてもよい。下記の数３には、０次の正則化項の例を示す。ｐは推定熱流束の分割数の数であり、α_０は経験値から得られる正則化パラメータである。
【００３０】
【数３】

【００３１】
以下に、より具体的に、複数の熱電対位置での温度Ｙを既知として、反応容器の内表面及び外表面における熱流束を推定する定式化と、計算手続きの一例を示す。
【００３２】
下記の数４のＳ_ｍは全体の目的関数を表し、下記の数５は、実測温度Ｙと計算温度Ｔの偏差を表す目的関数を示す。下記の数６は、計算を安定化するために付加した目的関数であり、空間分割方向の値の急激な変化を抑える働きがある。数６中のα_０やα_１は、一定の経験値から得られる正則化パラメータである。
【００３３】
【数４】

【００３４】
【数５】

【００３５】
【数６】

【００３６】
上記数５では、ある熱電対で測定された温度Ｙと、熱流束の仮定値から熱伝導方程式モデルにより算出された温度Ｔの差の二乗が最小となるように目的関数を設定している。また、上記数６では、温度測定誤差があっても解が安定するように空間方向の正則化を施す目的関数を設定している。そして、数４を全体の目的関数として、下記の数７に示すように、未知である熱流束分割領域に対して極小点を探す。
【００３７】
【数７】

【００３８】
ここで、数８に示すように、解を安定させる目的で、各時間ステップの熱流束値が、一定の未来時間まで不変であると仮定する。時間ステップは、対象とする材料の熱物性・形状等によって変わる。数８のｑは熱流束を示し、ｍ時間ステップにおける熱流束ｑ_ｍから、将来時間ｍ＋ｒ−１時間ステップにおける熱流束ｑ_{ｍ＋ｒ−１}が一定であると仮定している。
【００３９】
【数８】

【００４０】
そして、数７の極小化を、数８の仮定を用いて展開すると、数９に示すように、マトリクス形に展開することができる。
【００４１】
【数９】

【００４２】
数９のＸ^ＴＸは数４の右辺第１項から導かれ、Ｘ^ＴＸに続く２項（α_０Ｈ_０ ^ＴＨ_０＋α_１Ｈ_１ ^ＴＨ_１）は、数４の右辺第２項から導かれる（上付のＴは、転置行列を表す）。Ｘの構成は、補足式数１０として下部に、Ｘ_{ｊ，ｉ，ｋ}として示している。ここで、時間方向の分割数を示すｉは、最大Ｍ時間ステップまで変化し、熱電対の数を示すｊは、最大Ｊ個まで変化して、熱流束分布の分割数を示すｋは、最大ｐまで変化する。なお、数９の上付の＊は、繰り返し収束計算での参照値であることを示しており、Ｔ^＊は温度参照値、ｑ^＊は熱流束参照値である。１次元の場合は、両端の境界条件を推定するので、熱流束分布の分割数ｋは、最大ｐ＝２である。
【００４３】
【数１０】

【００４４】
数９は、温度変化が起きた場合の熱流束の変化を推定する連立方程式であり、各時間ステップにおいて、この数９を用いて両端の熱流束ｑを求める。まずは、前時間ステップでの熱電対位置での計算温度を初期Ｔ^＊とし、数９によりｑを求める。このｑを、並行して計算している順問題熱伝導方程式モデルの境界条件として与え、温度分布を計算する。ここで求めた温度計算値を、次の温度参照値Ｔ^＊として、ｑを再修正する（数９に代入してｑを再び求める）。この操作を、数５が一定残差以下になる（収束）まで、ｑとＴ^＊の修正を繰り返し、各時間ステップにおける両端の熱流束（最終的なｑ）を求めていく。この計算手続きを繰り返すことにより、両端の熱流束ｑの変化を、２つ同時に推定することが可能となる。
【００４５】
数１０は、一種の感度行列を表しており、端的に言うと、境界端点での熱流束ｑの単位変化に対する熱電対位置での計算温度Ｔの変化の大きさの比率を示している。数１０は、逆解析と同時に計算している順問題計算によって、各時間ステップにおいて、単位時間ステップあたりの値の計算が可能である。
【００４６】
以下、１次元の逆問題解析を例にして、より望ましい解法について説明する。上述のように、２つの端面（反応容器の内表面及び外表面）の熱流束を未知の境界条件とした１次元逆問題を構成（定式化）しても、原理上は解を求めることができる。
【００４７】
ただし、熱電対の数や材料の熱物性条件等によって多解となる場合があり、計算が不安定となる可能性がある。その理由の一つは、「未知両端面の熱流束差」の組み合わせを適当に選ぶことができれば、離散的な温度測定点の温度変化を表現する熱流束の組み合わせは無数に存在する可能性があるためである。特に、熱伝導度の低い物質の場合、表面温度が極端に大きくなったり、小さくなったりしてしまうような境界条件を推定してしまう場合でも、離散測定点の温度の変化だけを再現すれば、一つの解として認識してしまうことも起こり得る。これは、現実の現象としてはあり得ないことであるばかりでなく、逆問題計算を非常に不安定なものとする。
【００４８】
また、実際の問題として、逆問題解析を開始する時の熱電対の温度（離散測定点の温度）は既知として与えられるが、その他の解析領域での温度分布の初期条件は不明であることが一般的である。このため、任意に与えた仮初期温度分布から計算を始め、計算ステップを進める中で、実際の温度分布を探索・推定し、妥当な温度分布へと徐々に修正しながら、安定的に計算を進めていけるような計算ロジックにすることが求められる（ここで言う温度分布とは、例えば、逆問題解析の計算手続きの中で、上記数９の解を修正するために並行して計算している順問題熱伝導方程式モデルの計算値である）。このように、初期温度分布が不確定であることも、逆問題計算を不安定なものとする大きな要因の一つとなる。
【００４９】
以上のことは、逆問題を安定化するためには、逆問題解析の過程で、ある程度の表面温度の目安（拘束条件）を与える必要性があることを示しているといえる。この考え方に基づき、拘束条件を適当に与える手法を、図３のフローチャートを参照して説明する。
【００５０】
まず、反応容器の内表面及び外表面のいずれか片側、ここでは外表面における熱流束として仮の熱流束ｑを与える。この仮の熱流束ｑの与え方として、熱伝達率ｈと参照温度Ｔ_ｂとを用いて、
ｑ＝ｈ（Ｔ_ｓｕｒｆ−Ｔ_ｂ）
として与える（ステップＳ２０１）。
【００５１】
Ｔ_ｓｕｒｆは未知境界、ここでは反応容器の外表面における温度を示している。この表面温度Ｔ_ｓｕｒｆは、逆問題解析の過程で熱流束の値を修正するために、通常は順問題解析も同時に行うが、この順問題解析で求めた表面温度に相当する。
【００５２】
また、参照温度Ｔ_ｂは反応容器の内部及び内外表面以外での温度である。本実施の形態では、反応容器の冷却条件、例えば、水冷ならば水温等に基づいて定めるようにしている。
【００５３】
結果として、上式の左辺である熱流束ｑをあたかも既知の熱流束情報として与えることができる。このように仮の熱流束情報を与えることで、熱伝達率ｈと参照温度Ｔ_ｂという２つの拘束条件を与えることとなり、任意の熱流束を与えるのに比べて物理的な妥当性を確保して、極端な温度分布が生じることを防ぐことが可能となる。
【００５４】
次に、反応容器の外表面における仮の熱流束ｑ（＝ｈ（Ｔ_ｓｕｒｆ−Ｔ_ｂ））を与えて、上記数２、又は、数５に示した温度Ｔ、Ｙの差の二乗の和が最小となる反応容器の内表面における熱流束を、反応容器の内表面における仮の熱流束として算出する（ステップＳ２０２）。このステップは、逆問題解析のメインの計算手続きであり、具体的な解法の一つとして、数４から数９に示した定式化と計算手続きが、そのまま適用できる。この場合では、数９を解く際に、反応容器の外表面における仮の熱流束ｑ（＝ｈ（Ｔ_ｓｕｒｆ−Ｔ_ｂ））は既知として与え、反応容器の内表面における仮の熱流束を未知として解くことを意味する。
【００５５】
ここで、上記のように片側（反応容器の外表面）の熱流束情報を与えて、逆問題解析により求めた反対側（反応容器の内表面）の熱流束は、一つの解の可能性を示しているに過ぎない。また、既知と仮定した熱伝達率ｈや参照温度Ｔ_ｂも概算値であり、本来ならば未知の値である。
【００５６】
そこで、熱伝達率ｈ及び外部参照温度Ｔ_ｂの両方或いはいずれかを数点変化させて、すなわち、反応容器の外表面における仮の熱流束ｑの値を数点（Ｋ点）振って、反応容器の外表面における仮の熱流束ｑと、各仮の熱流束情報ｑを与えたとき温度Ｔ、Ｙの差の二乗の和が最小となる反応容器の内表面における熱流束との組み合わせをＫ個得る（ステップＳ２０３）。
【００５７】
そして、下記の数１１に示すように、反応容器の外表面における仮の熱流束ｑと、各仮の熱流束情報ｑに対応して得られた反応容器の内表面における熱流束とのＫ個の組み合わせのうち、温度Ｔ、Ｙの差の二乗の値が最も小さくなる組み合わせを選び出し、その組み合わせを反応容器の内表面及び外表面における熱流束とする（ステップＳ２０４）。
【００５８】
【数１１】

【００５９】
上式の大括弧の中は、片側の熱流束を既知として逆問題解析した１ケースの計算結果を示し、その計算をＫケース計算した中から更に最小二乗差の最も小さな結果を選び出すことを意味する。
【００６０】
この手続を、各時間ステップにおいて繰り返し行うことにより、反応容器の内表面及び外表面における熱流束経時変化を逐次同時計算していくことができる。
【００６１】
以上述べたように、反応容器の内表面及び外表面における熱流束変化を同時に求めるような１次元逆問題解析を安定して実行することができる。そして、反応容器の内表面及び外表面における温度変化や熱流束変化を同時推定できれば、例えば、ある温度測定点における温度変動が、反応容器の内表面における熱流束変化によるものなのか、反応容器外に設置された冷却装置の接触不良等によって引き起こされるような反応容器の外表面における熱流束変化によるものかを区別するようなことが可能となる。
【００６２】
上記手法は１次元逆問題解析に適用すると簡便であり、実際問題として有効である場合が多い。その理由は、一般的には、反応容器の上端と下端とは断熱条件（対称）とする場合が多く、実用的にも問題ないからである。
【００６３】
したがって、図２の破線で区切られた範囲での厚み方向１次元を仮定して逆問題解析し、その結果を上下方向に組み合わせることで、２次元化することも可能である。
【００６４】
より厳密に図２の上下方向の熱流れも考慮したい場合には、２次元逆問題解析が必要である。このような２次元解析は、図２の左右両端部の熱流束分割を上方向に細かくして、これらの熱電対位置での温度を最小二乗的に最小な熱流束分布を求めることと等価であり、上述した特開２００２−２０６９５８号公報に開示した逆問題定式化と同様の手法に従って本発明を適用すればよいこととなる。
【００６５】
この場合に、図２の上端下端の熱流束に関しては、未知としても、既知としても構わないが、計算の安定性を考慮すると、物理的な考察から適当な熱流束（例えば、断熱等）を与えて既知とした方が望ましい。
【００６６】
同様の考えに基づいて、３次元解析への拡張も容易に行うことができる。
【００６７】
ここで、例えば上述した逆問題解析により反応容器の表面における熱流束変化の「現在」の変化を推定するためには、できるだけ近い過去の反応容器内の状況を推定する必要がある。
【００６８】
そのためには、逆問題計算の単位時間ステップを短くすることが考えられる。逆問題計算の単位時間ステップが長いと、少なくともその時間ステップ分、大きく過去に遡った変化を推定してしまうことになる。また、逆問題計算は、計算時間ステップで時間平均した計算結果であるので、復元した熱流束変化も、鈍った変化を捉えることになり、時間ステップ以下の急激な変化は捉えられない。
【００６９】
ところが、逆問題計算の単位時間ステップを短くするということは、その短い時間の間に、熱流束の変化位置（例えば、溶銑と接触する高炉の内表面）から熱電対位置まで伝わってくる小さな温度変化を捉えなければならないことを意味する。特に、熱電対位置と熱流束の変化位置とが離れていて、その間の物質の熱伝導率が小さい場合、短時間での温度の動きは非常に小さなものとなる。
【００７０】
このような小さな温度変化から、熱流束変化による温度変化を理論的に抽出して、その熱流束変化を復元する（短時間の逆問題解析）のためには、温度測定方法や温度データ処理方法に工夫が必要である。
【００７１】
そこで、本願発明者らは、熱電対で測定される温度データを活用しつつ、短い時間ステップで、できるだけ近い過去の熱流束分布を推定できるようにすべく、鋭意検討を重ねた。
【００７２】
高炉等の操業の場合、温度の変化だけを見ていることもあって、大局的な温度の流れを見るに際しては、小数点以下の温度データを切り捨てた値で評価することが一般的である。
【００７３】
また、現在の温度の時間変化を細かく見る場合でも、熱電対の接点電圧変化から温度に変換する温度測定装置のメーカ保証範囲がほとんどの場合小数点１桁までであることもあり、小数点１桁までの温度データを使用することが多い。すなわち、採取して、記録している温度データは、小数点０桁か１桁といった粗いものである。
【００７４】
ところが、非常に微妙な温度変化を引き起こす原因となる、遠く離れた位置での境界条件変化（熱流束変化）を推定する（逆問題解析）ためには、温度変化の小数点以下の挙動を推定することが非常に重要となる。
【００７５】
すなわち、逆問題解析においては、温度そのものの精度よりは、温度の時間推移・温度変化の流れが重要である。温度そのものの精度だけを追求して、階段状に温度が変化してしまうと、逆問題解析を不安定なものとしてしまう。特に、短い時間ステップでの逆問題解析を指向すると、その時間ステップでは考えられないような階段状の温度変化が生じてしまうこともある。そのため、この階段状の温度変化で逆問題解析して熱流束解が得られた場合でも、物理的にありえない解に至る可能性を秘めている。実際、そのほとんどの場合、解が発散してしまい、逆問題解析の続行が不可能となる。
【００７６】
合理的な方法で、小数点以下の桁数の少ない温度データから、人工的に小数点以下の桁数を増やした温度データを生成することは、階段状の時間変化を滑らかにする上で有効な手法である。
【００７７】
その一つの手法として、演算部１０２において、逆問題解析での時間ステップより短いサンプリング時間で小数点以下０桁や１桁の温度データを採取し、それらを時間平均して、逆問題解析での計算時間ステップで用いる温度データの代表値とする。例えば、逆問題解析での時間ステップ（１時間）より短い５分間隔でサンプリングした小桁数の温度を、１時間範囲で単純平均して、１時間ステップの温度データの代表値とする。これにより、見かけ上、１時間温度データの小数点数を増やすことが可能となる。この手法を用いると、熱電対で測定される精度が保証されている小数点以下０桁、小数点以下１桁の温度データ（熱電対で測定されたもの）に基づいて、逆問題計算に用いる温度データの小数点以下の桁数を増やすことが可能である。
【００７８】
また、別の手法により、逆問題計算に用いる温度データの小数点以下の桁数を増やすようにしてもよい。例えば、熱電対において、温度データそのものの保証範囲は小数点１桁である場合でも、実際には熱電対接点での電圧から温度へと換算する変換式に基づいて計算するか、キャリブレーションを取った電圧−温度換算表から温度に変換するかの方法が一般的である。
【００７９】
前者の方法では、多数桁の温度計算値を四捨五入等して小数点以下１桁にしているわけであり、実際には計算機の能力に応じた小数点桁数まで求めることが可能である。例えば、Ｋタイプの熱電対での電圧から温度に変換する変換式として、下式に示すようなものがある。
Ｔ＝ａ＋ｂ・Ｘ＋ｃ・Ｘ^２＋ｄ・Ｘ^３＋ｅ・Ｘ^４＋ｆ・Ｘ^５＋ｇ・Ｘ^６＋ｈ・Ｘ^７＋ｉ・Ｘ^８
Ｔ：温度 ［℃］
Ｘ：接点電圧 ［Ｖ］
ａ＝０．２２６５８４６０２
ｂ＝２４１５２．１０９００
ｃ＝６７２３３．４２４８
ｄ＝２２１０３４０．６８２
ｅ＝−８６０９６３９．９
ｆ＝４．８３５０６×１０^１０
ｇ＝−１．１８４５２×１０^１２
ｈ＝１．３８６９０×１０^１３
ｉ＝−６．３３７０８×１０^１３
上記のような変換式を用いて、接点電圧から温度に直接換算するようにすれば、簡単に温度データの小数点以下の数値を求めることが可能である。なお、ここでいう接点電圧とは、冷接点温度補償を経た真の接点電圧であることはいうまでもない。
【００８０】
（第２の実施の形態）
上記第１の実施の形態で説明したように小数点以下の桁数を増やす以外にも、図４に示すように、入力部１０１にフィルタ１０１ａを設けておき、温度データ前処理にフィルタ処理を施すことが挙げられる。すなわち、フィルタ１０１ａを用いて、熱電対で測定された温度データを修正し、その修正後の温度データを逆問題解析に用いる。フィルタを用いると、元の温度データが小数点以下の桁数の少ないものであっても、見かけ上、温度データに小数点以下の桁を生成することができるので、温度の変化が滑らかになって逆問題解析の計算が非常に安定する。
【００８１】
理論的に最も有効と思われるのは、ローパスフィルタを用いることである。ローパスフィルタは、等時間間隔でサンプリングした実測データに混入している高周波ノイズによる変動の影響を除去するために、実測データを補正するものである。
【００８２】
逆問題解析においてローパスフィルタを用いて温度データを修正することが有効な理由は、熱電対から遠く離れた位置での熱流束の変化は、耐火物内を熱拡散現象によって伝わってくる間に鈍化して、熱電対位置まで伝わった時には、温度変化が非常になまっている（低周波の変化が伝達される）と考えられるためである。
【００８３】
これは、遠く離れた炉内内部の変化によって引き起こされた熱電対位置での温度変化は、時間的に低周波な変化であることを意味する。ところが、実際の熱電対の接点電圧変化（温度変換前の元信号）には、他の原因による高周波のノイズが重畳されている。そこで、実際の測定温度変化から高周波ノイズを取り除く処理を施した温度データを逆問題解析で用いることは、物理的に非常に意味のある手法となる。したがって、この手法により、逆問題解析の計算安定性を向上させることができるのみならず、熱流束の推定精度を向上させることもできる。
【００８４】
ローパスフィルタ処理は、先に述べた時間平均をとったり、変換式を使ったりして桁数を増やした温度データは勿論のこと、その他なんらかの手法で小数点以下の桁数を増やした温度データや、元の小数点以下の桁数の少ない測定温度データに対して付与しても非常に有効である。すなわち、第１の実施の形態で説明した小数点以下の桁数を増やすことと、ローパスフィルタ処理を施すことを併用してもよく、それにより相乗的に温度変化の平滑化を図ることができる。
【００８５】
フィルタについては多種多様な種類が考えられるが、例えば、Ｒ．Ｗ．ＨａｍｍｉｎｇのＤｉｇｉｔａｌ ｆｉｌｔｅｒｓ，Ｄｏｖｅｒ ｐｕｂｌｉｃａｔｉｏｎｓ Ｉｎｃ．１９９８等を参照するとよい。その他、参考資料は多数存在する。
【００８６】
ローパスフィルタの古典的な手法としては、等時間間隔でサンプリングした測定温度データＹを用いて中間点の温度を修正するものが知られている。その修正のための補正式として、下記の数１２に示す式（１）〜（３）の式が導出される。下付き文字のｉは時間のサンプリング回数を示しており、現在時刻ｉに対して、前後の時間の測定温度により修正を掛けることを意味する。
【００８７】
【数１２】

【００８８】
式（１）〜（３）はそれぞれ３項法、５項法、７項法の式を示しており、修正に用いる測定温度データ数が増えるほど、ローパスフィルタとしての性能が向上する。すなわち、より低周波の信号のみを抽出することができる。基本的には、各項の係数で表現できるので、ウインドウズ表示で表現すると、それぞれ式（４）〜（６）のようになる。
【００８９】
一般的に、項数が増えれば増えるほど、ローパスフィルタとしての性能は向上し、フィルタ設計の自由度が増える。つまり、多項フィルタにより、高周波の外乱が広範囲に除去することができるので、逆問題解析の計算は極めて安定する。その反面、項数が増えると、ある時刻点の修正温度値を得るために、その前後の複数時間点の測定温度を使用しなければならず、これは実質的に過去に遡っていることになるため、「逆問題解析で現在の熱流束を推定する」という場合の「現在」そのものが、ローパスフィルタによって引き戻されることを意味する。したがって、ローパスフィルタによる逆問題解析の安定性を確保することと、「現在」の推定時間をできるだけ今に近づけることの関係はトレードオフなので、両者の関係を鑑みながら、適当な組み合わせを選ぶ必要がある。
【００９０】
こういう観点からいえば、上記第１の実施の形態で説明した手法により、できるだけ小数点以下の桁数を増やした温度データを用い、階段状の温度変化を抑えた上で、項数の少ないローパスフィルタを用いることが、「現在」の熱流束を推定する上で有効となる。
【００９１】
（実施例）
図５には、高炉炉底の底盤に設置されているカーボン煉瓦内に埋め込まれた熱電対を用いて、１次元の非定常熱伝導モデルを仮定して、逆問題解析を試みた例を模式的に示す。熱電対は、冷却面側に偏って２本埋め込まれており、ＴＣ１は高温側熱電対、ＴＣ２は低温側熱電対を示す。これら熱電対ＴＣ１、ＴＣ２で測定される温度変化から、上述した逆問題解析により、高温熱流束面での非定常熱流束ｑ１と、冷却面での非定常熱流束ｑ２とを同時に推定する。高温熱流束面位置は冷却面から４．０ｍ奥の定点である。仮定したカーボン煉瓦の熱物性値は、比熱Ｃｐ＝７１２Ｊ／（ｋｇ・Ｋ）、密度ρ＝２３００ｋｇ／ｍ^３、熱伝導度ｋ＝２１．２Ｗ／（ｍ・Ｋ）である。
【００９２】
図６（ａ）に、本実施例で得られた高温側熱流束ｑ１の推定結果を示す。図６の横軸は日付を表し、９月１目から１０月１目までの結果を示している。実際には、９月１目以前より計算を始めて、計算終了は１０月１日である。なお、逆問題解析の時間ステップは、８時間ステップである。
【００９３】
ケース１は、熱電対により測定される温度データの小数点１桁の数値を四捨五入して、小数点以下０桁としたデータに対して、フィルタ処理を実行した場合の計算結果である。ケース２は、温度データの小数点２桁の数値を四捨五入して、小数点以下１桁としたデータに対して、フィルタ処理を実行した場合の計算結果である。
【００９４】
図６（ｂ）、（ｃ）には、高温側熱電対ＴＣ１及び低温側熱電対ＴＣ２のフィルタ処理後の温度データ（逆問題解析に用いる温度データ）をプロットしたグラフを示す。また、図６（ｄ）には、熱電対ＴＣ１、ＴＣ２それぞれの位置でのケース１とケース２の温度差（ケース１−ケース２）を示す。
【００９５】
フィルタについては、過去の温度データには極力多くの項数のローパスフィルタをかけて計算を安定化すべく、最大２１項のスペンサーの式（式（１０））を用い、現在に近づくにつれて徐々に項数を減らしていく手法を用いている。具体的には、ウインドウズ形式の表記で、式（４）〜（１０）を用いている。即ち、１０月１日に近づくにつれて項数が減少し、最終的には式（４）を用いており、最後の温度データにはフィルタ処理を行っていない。
【００９６】
ケース１、ケース２のいずれの場合でも、過去の解析に関してはフィルタ効果で非常に安定化しているが、現在に近づくにつれて、徐々に不安定化して、最終的な推定熱流束は、現実的には考えられないほどの大きな振動を起こしている。これは、ローパスフィルタの効果が徐々に低下して、計算が不安定になっていることを示すと考えられる。
【００９７】
ケース１とケース２を比較すると、ケース２の方が最終的な振動の大きさを小さく抑えることができていることが分かる。これは、小数点以下の桁数が多い温度データの方が、項数の少ないローパスフィルタでも、比較的安定した計算が可能であることを示している。
【００９８】
更に、いずれの場合も、最後の推定結果で振動を起こしていることから、同じ桁数の温度データで比較した場合、ローパスフィルタの項数を増やして、温度データを極力平滑化することが、安定した解を得る上で有効な手段であることも示している。特に、式（４）〜（１０）の形式のローパスフィルタを用いた場合、式（４）を用いても、最後のデータにはフィルタがかからないので、元の温度データの小数点以下の値は多い方が、見かけ上平滑化されやすいので、逆問題解析解が安定する。
【００９９】
ケース１及びケース２において、温度データにフィルタを全くかけないで逆問題解析を試みたが、解が発散してしまい、計算を進めることができなかった。
【０１００】
図７には、熱電対の設置深さの異なるケースについて、図５と同様の模式図を示す。図８（ａ）に、逆問題解析の計算時間ステップを８時間とした場合と、６時間とした場合とについて、高温側熱流束ｑ１の推定結果を示す。これらの計算結果ついては、もっと長い期間の計算結果の中での９月１目から１０月１日までを抽出して示す。
【０１０１】
計算に用いた温度データは小数点以下１桁のものを用い、更にフィルタとして式（１０）を用いて平滑化を行っている。図８（ｂ）、（ｃ）には、その平滑化後の高温側熱電対ＴＣ１及び低温側熱電対ＴＣ２の温度データをプロットしたグラフを示す。また、図８（ｄ）は、実質的には、定常の熱流束に対応する、高温側熱電対温度と低温側熱電対温度との差（ＴＣ１−ＴＣ２）を示す。
【０１０２】
図８（ｄ）の疑似定常熱流束の推移と、図８（ａ）の逆問題解析により推定した非定常熱流束のピーク同志の対応関係を比較して見ると、直感的ながら、逆問題解析が遅れ時間を補正して、非定常熱流束の方が早めにピークが表れていることが分かる（図８（ａ）中の破線で示す）。
【０１０３】
図８（ｄ）のグラフの疑似定常熱流束は、熱電対位置周辺を通過している平均的な熱流束と解釈でき、４ｍ奥の高温面での値を推定した非定常熱流束の値と比較して必ずしもピーク同志が対応しているとは限らないが、この場合は、概略２〜３日のピーク遅れが観察される。換言すると、逆問題解析による非定常熱流束は、従来の定常法よりも、概略２〜３日早めに炉内の動きを検知していることになる。
【０１０４】
また、図８（ａ）のグラフでの８時間ステップ計算と６時間ステップ計算を比較すると、６時間ステップの計算の方がより輪郭がはっきりした動きを示している。時間ステップを長くすると、その時間内の平均化した動きしか捉えることができないので、極力時間ステップを短くとった方が、実際の内部の動きを表現できる可能性が高い。
【０１０５】
（他の実施の形態）
以上説明した逆問題解析装置は、コンピュータのＣＰＵ或いはＭＰＵ、ＲＡＭ、ＲＯＭ、ＲＡＭ等で構成されるものであり、上述のようにＲＡＭやＲＯＭ等に記憶されたプログラムが動作することによって実現される。
【０１０６】
したがって、プログラム自体が上述した実施の形態の機能を実現することになり、本発明を構成する。プログラムの伝送媒体としては、プログラム情報を搬送波として伝搬させて供給するためのコンピュータネットワーク（ＬＡＮ、インターネット等のＷＡＮ、無線通信ネットワーク等）システムにおける通信媒体（光ファイバ等の有線回線や無線回線等）を用いることができる。
【０１０７】
さらに、上記プログラムをコンピュータに供給するための手段、例えばかかるプログラムを格納した記憶媒体は本発明を構成する。かかる記憶媒体としては、例えばフレキシブルディスク、ハードディスク、光ディスク、光磁気ディスク、ＣＤ−ＲＯＭ、磁気テープ、不揮発性のメモリカード、ＲＯＭ等を用いることができる。
【０１０８】
なお、上記実施の形態において示した各部の形状及び構造は、何れも本発明を実施するにあたっての具体化のほんの一例を示したものに過ぎず、これらによって本発明の技術的範囲が限定的に解釈されてはならないものである。すなわち、本発明はその精神、又はその主要な特徴から逸脱することなく、様々な形で実施することができる。例えば、本発明をネットワーク環境で利用すべく、一部のプログラムが他のコンピュータで実行されるようになっていてもかまわない。
【０１０９】
【発明の効果】
以上述べたように本発明によれば、反応容器の壁内部に配置された温度測定点で測定される温度データに基づいて、非定常熱伝導方程式を用いた逆問題解析を行うことにより、上記反応容器の表面における温度或いは熱流束を求めるに際して、例えば、できるだけ近い過去の温度或いは熱流束の変化を推定するために逆問題解析での時間ステップを短くするような場合でも、温度データの小数点以下の桁数を増やしたり、フィルタ処理を施したりすることにより、小さな温度変化を捉えることが可能となり、逆問題解析を安定化させることができる。
【図面の簡単な説明】
【図１】第１の実施での形態の逆問題解析装置の構成を示す図である。
【図２】複数の熱電対が埋め込まれた反応容器（高炉）の炉壁近くの２次元断面を示す図である。
【図３】逆問題解析における処理を説明するためのフローチャートである。
【図４】第２の実施での形態の逆問題解析装置の構成を示す図である。
【図５】実施例における熱電対ＴＣ１、ＴＣ２の配置関係を示す図である。
【図６】実施例における解析結果を説明するための図である。
【図７】他の実施例における熱電対ＴＣ１、ＴＣ２の配置関係を示す図である。
【図８】他の実施例における解析結果を説明するための図である。
【符号の説明】
１０１ 入力部
１０１ａ フィルタ
１０２ 演算部
１０３ 逆問題解析部
１０４ 出力部[0001]
TECHNICAL FIELD OF THE INVENTION
INDUSTRIAL APPLICABILITY The present invention provides a method, an apparatus, and a computer, which are suitable for estimating a temperature or a heat flux at a surface of a reaction vessel accompanied by a temperature change reaction such as a blast furnace, a steel heating furnace by combustion, a coal gasification reactor, and the like. The present invention relates to a program and a computer-readable storage medium.
[0002]
[Prior art]
For example, as a technique for monitoring the flow of molten iron at the bottom of a blast furnace, a thermocouple is embedded in a brick at the bottom of the furnace, and judgment is made based on a temperature measured by the thermocouple.
[0003]
However, since this temperature change has passed a delay time after the change of the furnace flow, it does not reflect the current flow. This is because the temperature abnormality in the blast furnace is transmitted to the surface of the blast furnace as a heat flux change, and then heat is accumulated inside the bricks of the blast furnace, and then gradually transmitted by the heat conduction effect, and finally the temperature changes to the thermocouple. In principle, the heat conduction phenomenon in a solid having a heat capacity has a time delay (unsteadiness).
[0004]
On the other hand, for example, as disclosed in Patent Document 1, the heat conduction phenomenon inside the wall of a reaction vessel such as a blast furnace is considered to be an unsteady heat conduction inverse problem, and the reaction is performed based on the temperature change in the thermocouple. A method for estimating a heat flux change on the inner surface of a container has been proposed.
[0005]
[Patent Document 1]
JP 2001-234217 A
[0006]
[Problems to be solved by the invention]
By the way, when estimating the heat flux change on the surface of a reaction vessel such as a blast furnace by inverse problem analysis, it is necessary to stabilize the inverse problem analysis.
[0007]
The present invention has been made in view of the above points, and performs an inverse problem analysis using an unsteady heat conduction equation based on temperature data measured by a thermocouple or the like buried inside a wall of a reaction vessel. Accordingly, the object of the present invention is to stabilize the inverse problem analysis when obtaining the temperature or heat flux at the surface of the reaction vessel.
[0008]
[Means for Solving the Problems]
In the present invention, based on the temperature data measured at the temperature measurement points arranged inside the wall of the reaction vessel accompanied by a temperature change reaction, by performing an inverse problem analysis using an unsteady heat conduction equation, the reaction vessel When calculating the temperature or heat flux at the surface of the surface, increase the number of decimal places of the temperature data measured at the temperature measurement point, and use the temperature data with the increased number of decimal places for the inverse problem analysis. Has features. In this case, in order to increase the number of digits after the decimal point of the temperature data, for example, temperature data is collected with a sampling time shorter than the calculation time step in the inverse problem analysis, and the time data are averaged to obtain the inverse problem. A representative value of the temperature data used in the calculation time step in the analysis.
[0009]
In the present invention, the inverse problem analysis using an unsteady heat conduction equation is performed based on temperature data measured at a temperature measurement point arranged inside the wall of a reaction vessel accompanied by a temperature change reaction, whereby When the temperature or heat flux at the surface of the reaction vessel is determined, the temperature data measured at the temperature measurement point is filtered, and the temperature data after the filtering is used for the inverse problem analysis. In this case, for example, low-pass filtering is used as the filtering.
[0010]
In the present invention, the inverse problem analysis using an unsteady heat conduction equation is performed based on temperature data measured at a temperature measurement point arranged inside the wall of a reaction vessel accompanied by a temperature change reaction, whereby When obtaining the temperature or heat flux at the surface of the reaction vessel, the number of digits after the decimal point of the temperature data measured at the temperature measurement point is increased, and the temperature data with the increased number of digits after the decimal point is subjected to a filtering process. It is characterized in that the temperature data after the filter processing is used for the inverse problem analysis. In this case, for example, low-pass filtering is used as the filtering.
[0011]
BEST MODE FOR CARRYING OUT THE INVENTION
Hereinafter, preferred embodiments of an inverse problem analysis method, an apparatus, a computer program, and a computer-readable storage medium according to the present invention will be described with reference to the drawings.
[0012]
(First Embodiment)
FIG. 1 shows a configuration of an inverse problem analysis apparatus according to the present embodiment. In the figure, reference numeral 101 denotes an input unit to which temperature data measured by a thermocouple (see FIG. 2) embedded in a wall of the reaction vessel is input.
[0013]
Reference numeral 102 denotes an arithmetic unit, which performs an operation of increasing the number of digits after the decimal point using the temperature data input to the input unit 101, as described later.
[0014]
Reference numeral 103 denotes an inverse problem analysis unit that performs an inverse problem analysis using an unsteady heat conduction equation based on the temperature data obtained by increasing the number of digits after the decimal point by the calculation unit 102, thereby obtaining the temperature or the temperature at the surface of the reaction vessel. Find the heat flux.
[0015]
Reference numeral 104 denotes an output unit which displays a change in temperature or heat flux on the surface of the reaction vessel calculated by the inverse problem analysis unit 103 on, for example, a display (not shown).
[0016]
First, the inverse problem analysis used in the present embodiment will be described. As described in Patent Document 1 described above, considering the heat conduction phenomenon inside the wall of the reaction vessel as an unsteady one-dimensional inverse heat conduction problem, one thermocouple temperature change or a plurality of thermocouples arranged in one-dimensional direction are considered. A method for estimating a heat flux change on the inner surface of a reaction vessel from a thermocouple temperature change has been proposed.
[0017]
FIG. 2 shows a two-dimensional cross section near a furnace wall of a reaction vessel (blast furnace) in which a plurality of thermocouples “×” are embedded. Although the boundary is indicated by a broken line in the furnace wall, one-dimensional means that only the heat flow in the direction along the broken line is considered. That is, for example, assuming heat conduction in the directions of 1a → 1b → 1c and 1d → 1e, the heat flux on the furnace inner surface is estimated. At this time, assuming that the cooling condition of the outer furnace surface is known, it is general to obtain the heat flux on the unknown inner furnace surface. Of course, it is also possible to reverse the known and unknown boundary conditions.
[0018]
However, the original unsteady one-dimensional heat conduction inverse problem is to simultaneously estimate the boundary conditions on the inner and outer surfaces of the furnace, and the inverse problem solving method on the assumption that the boundary conditions on one side are known is unknown. We can only get an approximate answer to the given boundary condition. For example, whether the temperature fluctuation measured by a certain thermocouple is caused by a change in the heat flux on the inner surface of the reaction vessel as described above, or a reaction caused by a poor contact of a cooling device installed outside the reaction vessel or the like. It will not be possible to tell if it is due to a change in the heat flux on the outer surface of the container.
[0019]
Further, in order to evaluate more strictly, the heat conduction phenomenon should occur in the upward direction across the broken line shown in FIG. 2, and it is necessary to solve the inverse heat conduction problem in two dimensions. In this case, even when the upper and lower boundaries in FIG. 2 are assumed to be adiabatic, it is necessary to construct a two-dimensional inverse problem for estimating a fine heat flux distribution on the left and right boundaries.
[0020]
Therefore, the present applicant has proposed an inverse problem analysis for simultaneously estimating a heat flux change and a temperature change on the inner surface and the outer surface of the reaction vessel. Hereinafter, the inverse problem analysis proposed by the applicant of the present application will be described in detail. An unsteady heat conduction equation used for the inverse problem analysis is expressed as shown in Equation 1 below.
[0021]
(Equation 1)

[0022]
In Equation 1, ρ is the density of the material of the reaction vessel, C_pRepresents the specific heat of the material of the reaction vessel, T represents the calculated value of the temperature inside the reaction vessel, t represents time, and k represents the thermal conductivity of the material of the reaction vessel.
[0023]
Heat conduction inverse problem analysis refers to estimating boundary conditions such as temperature and heat flux at the region boundary, based on the unsteady heat conduction equation governing the calculation region, with the temperature inside the region known. . On the other hand, the heat conduction order problem analysis refers to estimating the temperature inside the region from known boundary conditions such as the temperature at the region boundary and the heat flux.
[0024]
As an example of the two-dimensional inverse problem analysis method, for example, there is one disclosed by the present applicant in Japanese Patent Application Laid-Open No. 2002-206958, and this method can be directly applied to one-dimensional inverse problem analysis. Further, as an example of the one-dimensional inverse problem analysis, an analysis method proposed by Beck et al. Is known (Beck et al., Inverse Heat Construction, 1985, Wiley, New York).
[0025]
In addition, as a recent method of inverse problem analysis, it is conceivable to apply a probabilistic estimation method such as Kalman filter theory or projection filter theory. At present, application of this method to the steady-state heat conduction equation (observation equation) with the left-hand side of Equation 1 set to zero has been studied. However, if the observation matrix including the non-stationary term can be appropriately constructed, A similar inverse problem analysis may be possible. Examples of the application of the probability estimation method to this steady-state differential equation are detailed in Nobosaka and others, "Mathematics and Solution of Inverse Problems / Inverse Analysis of Partial Differential Equations" (The University of Tokyo Press (1999)).
[0026]
In the present embodiment, the concept disclosed in Japanese Patent Application Laid-Open No. 2002-206958 is used as a method of analyzing the inverse problem. That is, as shown in the following Expression 2, the temperature Y measured by each thermocouple arranged in a certain one-dimensional direction (1a → 1b → 1c or 1d → 1e shown in FIG. 2) and the temperature Y in the reaction vessel The assumed value that minimizes the sum of the squares of the difference from the temperature T at each thermocouple position calculated from the assumed value of the heat flux on the surface and the outer surface by the unsteady heat conduction equation is the inner surface and outer surface of the reaction vessel. As the heat flux at J represents the number of thermocouples.
[0027]
(Equation 2)

[0028]
As described above, a solution that satisfies the least squares is not obtained as a solution (heat flux on the inner surface and the outer surface of the reaction vessel) that completely matches the temperatures T and Y at a plurality of thermocouple positions. By obtaining, it is possible to estimate a realistic heat flux change. The reason is that the measurement temperature includes various measurement error factors, so that it can be said that it is practically meaningless to completely match them.
[0029]
Note that a regularization term may be added to stabilize the calculation. Equation 3 below shows an example of a zero-order regularization term. p is the number of divisions of the estimated heat flux, α₀Is a regularization parameter obtained from empirical values.
[0030]
(Equation 3)

[0031]
Hereinafter, more specifically, an example of a formulation for estimating the heat flux on the inner surface and the outer surface of the reaction vessel assuming the temperatures Y at a plurality of thermocouple positions as known and an example of a calculation procedure will be described.
[0032]
S in Equation 4 below_mRepresents an overall objective function, and the following equation 5 represents an objective function representing a deviation between the measured temperature Y and the calculated temperature T. Equation 6 below is an objective function added for stabilizing the calculation, and has a function of suppressing a rapid change in the value in the space division direction. Α in Equation 6₀And α₁Is a regularization parameter obtained from a certain empirical value.
[0033]
(Equation 4)

[0034]
(Equation 5)

[0035]
(Equation 6)

[0036]
In Equation 5, the objective function is set so that the square of the difference between the temperature Y measured by a certain thermocouple and the temperature T calculated from the assumed value of the heat flux by the heat conduction equation model is minimized. In the above equation (6), an objective function for performing regularization in the spatial direction is set so that the solution is stable even if there is a temperature measurement error. Then, using Equation 4 as the overall objective function, a minimum point is searched for the unknown heat flux division area as shown in Equation 7 below.
[0037]
(Equation 7)

[0038]
Here, as shown in Equation 8, for the purpose of stabilizing the solution, it is assumed that the heat flux value at each time step is unchanged until a certain future time. The time step varies depending on the thermophysical properties, shape, and the like of the target material. In Equation 8, q indicates the heat flux, and the heat flux q in the m time step_mFrom the heat flux q in the future time m + r-1 time step_{m + r-1}Is assumed to be constant.
[0039]
(Equation 8)

[0040]
Then, when the minimization of Expression 7 is expanded using the assumption of Expression 8, it can be expanded in a matrix form as shown in Expression 9.
[0041]
(Equation 9)

[0042]
X of Equation 9^TX is derived from the first term on the right side of Equation 4, and X^TThe second term following X (α₀H₀ ^TH₀+ Α₁H₁ ^TH₁) Is derived from the second term on the right-hand side of Equation 4 (the superscript T represents a transposed matrix). The structure of X is as follows:_{j, i, k}As shown. Here, i indicating the number of divisions in the time direction changes up to a maximum M time steps, j indicating the number of thermocouples changes up to J, and k indicating the number of divisions of the heat flux distribution is maximum. p. Note that the superscript * in equation 9 indicates a reference value in the iterative convergence calculation, and T^*Is the temperature reference value, q^*Is the heat flux reference value. In the one-dimensional case, since the boundary conditions at both ends are estimated, the division number k of the heat flux distribution is p = 2 at the maximum.
[0043]
(Equation 10)

[0044]
Equation 9 is a simultaneous equation for estimating a change in the heat flux when a temperature change occurs. In each time step, the heat flux q at both ends is obtained using the equation 9. First, the calculated temperature at the thermocouple position in the previous time step is set to the initial T^*Then, q is obtained by Expression 9. This q is given as a boundary condition of the forward problem heat conduction equation model calculated in parallel, and the temperature distribution is calculated. The calculated temperature value obtained here is used as the next temperature reference value T.^*Is re-corrected (substituting into Equation 9 to obtain q again). This operation is repeated until q is equal to or smaller than the fixed residual (convergence) by q and T.^*Is repeated, and the heat flux (final q) at both ends in each time step is obtained. By repeating this calculation procedure, two changes in the heat flux q at both ends can be simultaneously estimated.
[0045]
Equation 10 represents a kind of sensitivity matrix, and in short, indicates the ratio of the magnitude of the change of the calculated temperature T at the thermocouple position to the unit change of the heat flux q at the boundary end point. In Equation 10, the value per unit time step can be calculated in each time step by the forward problem calculation performed at the same time as the inverse analysis.
[0046]
Hereinafter, a more desirable solution will be described by taking a one-dimensional inverse problem analysis as an example. As described above, even if a one-dimensional inverse problem in which the heat flux of the two end faces (the inner surface and the outer surface of the reaction vessel) is an unknown boundary condition is constructed (formulated), in principle, a solution can be obtained. it can.
[0047]
However, there may be multiple solutions depending on the number of thermocouples, thermophysical conditions of materials, and the like, and calculation may be unstable. One of the reasons is that if a combination of "heat flux difference between unknown end faces" can be appropriately selected, there may be a myriad of combinations of heat flux that express temperature changes at discrete temperature measurement points. Because there is. In particular, in the case of a substance with low thermal conductivity, even when estimating the boundary condition that the surface temperature becomes extremely large or small, it is necessary to reproduce only the temperature change at the discrete measurement points. However, it may happen that the object is recognized as one solution. This is not only impossible as a real phenomenon, but also makes the inverse problem calculation very unstable.
[0048]
Also, as a practical problem, the temperature of the thermocouple at the start of the inverse problem analysis (the temperature at the discrete measurement points) is given as a known value, but the initial conditions of the temperature distribution in other analysis regions are unknown. General. For this reason, the calculation is started from the tentative initial temperature distribution arbitrarily given, and while proceeding with the calculation steps, the actual temperature distribution is searched and estimated, and the calculation is performed stably while gradually correcting it to an appropriate temperature distribution. It is required that the calculation logic be able to proceed (the temperature distribution referred to here is, for example, calculated in parallel in the calculation procedure of the inverse problem analysis to correct the solution of the above equation 9). Is the calculated value of the forward problem heat conduction equation model). Thus, the uncertainty of the initial temperature distribution is also one of the major factors that makes the inverse problem calculation unstable.
[0049]
The above can be said to indicate that in order to stabilize the inverse problem, it is necessary to give some measure of the surface temperature (restriction condition) in the process of the inverse problem analysis. Based on this concept, a method of appropriately giving the constraint condition will be described with reference to the flowchart of FIG.
[0050]
First, a temporary heat flux q is given as a heat flux on one of the inner surface and the outer surface of the reaction vessel, here, on the outer surface. The provision of the temporary heat flux q includes the heat transfer coefficient h and the reference temperature T._bAnd using
q = h (T_surf−T_b)
(Step S201).
[0051]
T_surfIndicates the temperature at the unknown boundary, here the outer surface of the reaction vessel. This surface temperature T_surfIn order to correct the value of the heat flux in the process of the inverse problem analysis, usually the forward problem analysis is also performed at the same time, but this corresponds to the surface temperature obtained in the forward problem analysis.
[0052]
Also, the reference temperature T_bIs the temperature outside the inside and outside and inside surfaces of the reaction vessel. In the present embodiment, the cooling condition of the reaction vessel is determined based on, for example, the water temperature in the case of water cooling.
[0053]
As a result, the heat flux q on the left side of the above equation can be given as if it were known heat flux information. By providing the temporary heat flux information in this manner, the heat transfer coefficient h and the reference temperature T_bThus, physical validity can be ensured as compared with the case where an arbitrary heat flux is given, and it is possible to prevent an extreme temperature distribution from occurring.
[0054]
Next, the provisional heat flux q (= h (T (T_surf−T_b)), The heat flux at the inner surface of the reaction vessel where the sum of the squares of the differences between the temperatures T and Y shown in Equation 2 or 5 is minimized, and the temporary heat flow at the inner surface of the reaction vessel It is calculated as a bundle (step S202). This step is the main calculation procedure of the inverse problem analysis, and the formulation and calculation procedures shown in Equations 4 to 9 can be directly applied as one of specific solutions. In this case, when solving Equation 9, the temporary heat flux q (= h (T (T_surf−T_b)) Is given as known and means solving the temporary heat flux on the inner surface of the reaction vessel as unknown.
[0055]
Here, the heat flux information on one side (outer surface of the reaction vessel) is given as described above, and the heat flux on the other side (inner surface of the reaction vessel) obtained by inverse problem analysis indicates the possibility of one solution. It is just showing. In addition, the heat transfer coefficient h and the reference temperature T assumed to be known_bIs also an approximate value, which is originally unknown.
[0056]
Therefore, the heat transfer coefficient h and the external reference temperature T_bIs changed by several points, that is, the value of the temporary heat flux q on the outer surface of the reaction vessel is changed by several points (point K), and the temporary heat flux q on the outer surface of the reaction vessel is When each provisional heat flux information q is given, K combinations with the heat flux on the inner surface of the reaction vessel that minimizes the sum of the square of the difference between the temperatures T and Y are obtained (step S203).
[0057]
Then, as shown in the following Expression 11, K heat fluxes of the temporary heat flux q on the outer surface of the reaction vessel and the heat fluxes on the inner surface of the reaction vessel obtained corresponding to the respective temporary heat flux information q Of the combinations, the combination that minimizes the value of the square of the difference between the temperatures T and Y is selected, and the combination is set as the heat flux on the inner surface and the outer surface of the reaction vessel (step S204).
[0058]
(Equation 11)

[0059]
The brackets in the above equation show the result of one case in which the heat flux on one side is known and the inverse problem is analyzed, and means that the result of the least-square difference is further selected from the K-case calculation. I do.
[0060]
By repeating this procedure at each time step, it is possible to sequentially and simultaneously calculate the temporal change of the heat flux on the inner surface and the outer surface of the reaction vessel.
[0061]
As described above, the one-dimensional inverse problem analysis for simultaneously obtaining the heat flux changes on the inner surface and the outer surface of the reaction vessel can be stably executed. If the temperature change and the heat flux change on the inner surface and the outer surface of the reaction vessel can be simultaneously estimated, for example, whether the temperature fluctuation at a certain temperature measurement point is due to the heat flux change on the inner surface of the reaction vessel, It is possible to discriminate whether the change is caused by a change in heat flux on the outer surface of the reaction vessel, which is caused by poor contact of a cooling device installed in the reaction vessel.
[0062]
The above method is simple when applied to one-dimensional inverse problem analysis, and is often effective as an actual problem. The reason is that, generally, the upper and lower ends of the reaction vessel are often set under adiabatic conditions (symmetric), and there is no practical problem.
[0063]
Therefore, the inverse problem analysis can be performed assuming one dimension in the thickness direction within the range demarcated by the broken line in FIG. 2, and the results can be combined into two dimensions by combining the results in the vertical direction.
[0064]
In order to more strictly consider the heat flow in the vertical direction in FIG. 2, a two-dimensional inverse problem analysis is required. Such a two-dimensional analysis is equivalent to refining the heat flux divisions at the left and right ends in FIG. 2 upward and obtaining the minimum heat flux distribution in a least square manner at the temperatures at these thermocouple positions. That is, the present invention may be applied according to the same method as the inverse problem formulation disclosed in Japanese Patent Application Laid-Open No. 2002-206958.
[0065]
In this case, the heat flux at the upper and lower ends in FIG. 2 may be unknown or known, but in consideration of the stability of calculation, an appropriate heat flux (for example, heat insulation, etc.) is considered from physical considerations. It is desirable to give it and make it known.
[0066]
Extension to three-dimensional analysis can be easily performed based on the same idea.
[0067]
Here, for example, in order to estimate the “current” change of the heat flux change on the surface of the reaction vessel by the inverse problem analysis described above, it is necessary to estimate a situation in the reaction vessel as close as possible in the past.
[0068]
For this purpose, it is conceivable to shorten the unit time step of the inverse problem calculation. If the unit time step of the inverse problem calculation is long, a change that goes far back in the past is estimated at least for that time step. In addition, since the inverse problem calculation is a calculation result obtained by performing time averaging in the calculation time step, the restored heat flux change also captures a dull change, and a rapid change below the time step cannot be captured.
[0069]
However, shortening the unit time step of the inverse problem calculation means that during that short time, the small temperature transmitted from the heat flux change position (for example, the inner surface of the blast furnace in contact with the hot metal) to the thermocouple position It means that change must be captured. In particular, when the position of the thermocouple is far from the position where the heat flux changes, and the thermal conductivity of the substance between them is small, the movement of the temperature in a short time becomes very small.
[0070]
From such a small temperature change, a temperature change due to a heat flux change is theoretically extracted, and the heat flux change is restored (a short-time inverse problem analysis). Ingenuity is required.
[0071]
Therefore, the present inventors have conducted intensive studies so as to be able to estimate a past heat flux distribution as close as possible in a short time step while utilizing temperature data measured by a thermocouple.
[0072]
In the operation of a blast furnace or the like, since only the change in temperature is observed, when observing the global temperature flow, it is general to evaluate the value obtained by rounding down the decimal temperature data.
[0073]
In addition, even if you look closely at the current temperature change over time, the manufacturer's guaranteed range of the temperature measurement device that converts the thermocouple contact voltage change to temperature may be up to one decimal place in most cases. Temperature data is often used. That is, the collected and recorded temperature data is coarse, such as a decimal point or a single digit.
[0074]
However, in order to estimate the boundary condition change (heat flux change) at a distant position that causes a very subtle temperature change (inverse problem analysis), the behavior of the temperature change after the decimal point is estimated. That is very important.
[0075]
That is, in the inverse problem analysis, the time transition of the temperature and the flow of the temperature change are more important than the accuracy of the temperature itself. If the temperature changes in a stepwise manner in pursuit of only the accuracy of the temperature itself, the inverse problem analysis becomes unstable. In particular, when the inverse problem analysis is performed in a short time step, a stepwise temperature change that cannot be considered in the time step may occur. Therefore, even if an inverse problem analysis is performed using this stepwise temperature change to obtain a heat flux solution, there is a possibility that a physically impossible solution can be obtained. In fact, in most cases, the solution diverges, making it impossible to continue the inverse problem analysis.
[0076]
Generating temperature data with an artificially increased number of decimal places from temperature data with a small number of decimal places in a rational manner is an effective method for smoothing step-like time changes. It is.
[0077]
As one of the methods, the arithmetic unit 102 collects zero- or one-digit decimal temperature data with a sampling time shorter than the time step in the inverse problem analysis, averages them over time, and calculates them in the inverse problem analysis. This is a representative value of the temperature data used in the time step. For example, small digit temperatures sampled at 5-minute intervals shorter than the time step (1 hour) in the inverse problem analysis are simply averaged over a 1-hour range, and used as representative values of 1-hour step temperature data. This makes it possible to increase the number of decimal points of the one-hour temperature data apparently. By using this method, the temperature data used for the inverse problem calculation is based on the temperature data (measured with the thermocouple) of 0 decimal place and 1 decimal place where the accuracy measured by the thermocouple is guaranteed. It is possible to increase the number of digits after the decimal point.
[0078]
Further, the number of digits after the decimal point of the temperature data used for the inverse problem calculation may be increased by another method. For example, in a thermocouple, even if the assurance range of the temperature data itself is one decimal place, it is actually calculated or calibrated based on a conversion formula for converting the voltage at the thermocouple contact into temperature. A method of converting the temperature from the voltage-temperature conversion table is generally used.
[0079]
In the former method, a multi-digit temperature calculation value is rounded to one digit after the decimal point, and it is actually possible to obtain the number of decimal digits according to the capacity of the computer. For example, as a conversion formula for converting a voltage of a K-type thermocouple into a temperature, there is a conversion formula shown below.
T = a + b.X + c.X²+ DX³+ EX⁴+ FX⁵+ GX⁶+ HX⁷+ IX⁸
T: Temperature [° C]
X: Contact voltage [V]
a = 0.265584602
b = 24152.10900
c = 672333.4248
d = 2213040.682
e = -8609639.9
f = 4.83506 × 10¹⁰
g = −1.184452 × 10¹²
h = 1.38690 × 10^Thirteen
i = −6.333708 × 10^Thirteen
By directly converting the contact voltage to the temperature using the above conversion formula, it is possible to easily obtain the decimal part of the temperature data. It is needless to say that the contact voltage here is a true contact voltage that has undergone cold junction temperature compensation.
[0080]
(Second embodiment)
In addition to increasing the number of digits after the decimal point as described in the first embodiment, a filter 101a is provided in the input unit 101 as shown in FIG. It is mentioned. That is, the temperature data measured by the thermocouple is corrected using the filter 101a, and the corrected temperature data is used for the inverse problem analysis. If a filter is used, even if the original temperature data has a small number of digits after the decimal point, it is possible to apparently generate digits after the decimal point in the temperature data. Calculation of problem analysis is very stable.
[0081]
What seems to be the most effective in theory is to use a low-pass filter. The low-pass filter corrects the measured data in order to remove the influence of fluctuation due to high-frequency noise mixed in the measured data sampled at equal time intervals.
[0082]
The reason why it is effective to use a low-pass filter to correct the temperature data in the inverse problem analysis is that the change in the heat flux far away from the thermocouple slows down while it is transmitted through the refractory by the heat diffusion phenomenon. Then, when the temperature reaches the thermocouple position, the temperature change is considered to be extremely reduced (low frequency change is transmitted).
[0083]
This means that the temperature change at the thermocouple position caused by the change inside the furnace far away is a low-frequency change in time. However, high-frequency noise due to other causes is superimposed on the actual change in the contact voltage of the thermocouple (the original signal before the temperature conversion). Therefore, using temperature data that has been subjected to a process of removing high-frequency noise from an actual measured temperature change in an inverse problem analysis is a physically very meaningful method. Therefore, by this method, not only the calculation stability of the inverse problem analysis can be improved, but also the estimation accuracy of the heat flux can be improved.
[0084]
The low-pass filter processing includes not only the temperature data obtained by increasing the number of digits by taking the time average described above or using a conversion formula, but also the temperature data obtained by increasing the number of digits after the decimal point by some other method, or the original data. It is very effective to add the measured temperature data with a small number of digits after the decimal point. That is, increasing the number of digits after the decimal point described in the first embodiment and performing low-pass filter processing may be used in combination, whereby the temperature change can be synergistically smoothed.
[0085]
A wide variety of filters can be considered. W. Hamming's Digital filters, Dover publications Inc. 1998 and the like. There are many other reference materials.
[0086]
As a classic method of the low-pass filter, a method of correcting the temperature at the intermediate point using the measured temperature data Y sampled at equal time intervals is known. Expressions (1) to (3) shown in the following Expression 12 are derived as correction expressions for the correction. The subscript i indicates the number of times of sampling, which means that the current time i is corrected by the measured temperatures of the preceding and following times.
[0087]
(Equation 12)

[0088]
Equations (1) to (3) show equations of the three-term method, the five-term method, and the seven-term method, respectively, and the performance as a low-pass filter improves as the number of measured temperature data used for correction increases. That is, only lower frequency signals can be extracted. Basically, since it can be expressed by the coefficient of each term, if it is expressed by Windows display, it will be as each of the equations (4) to (6).
[0089]
Generally, as the number of terms increases, the performance as a low-pass filter improves, and the degree of freedom in filter design increases. In other words, since the high-frequency disturbance can be removed in a wide range by the polynomial filter, the calculation of the inverse problem analysis is extremely stable. On the other hand, when the number of terms increases, the measured temperature at several time points before and after the time point must be used to obtain the corrected temperature value at a certain time point, which is substantially retroactive to the past. This means that “present” itself in the case of “estimating the current heat flux by inverse problem analysis” is pulled back by the low-pass filter. Therefore, there is a trade-off between ensuring the stability of the inverse problem analysis using a low-pass filter and making the "current" estimation time as close as possible to the present, so it is necessary to select an appropriate combination while considering the relationship between the two. is there.
[0090]
From this point of view, the method described in the first embodiment uses the temperature data with as many decimal places as possible, suppresses the stepwise temperature change, and reduces the number of terms in the low-pass filter. Is effective in estimating the “current” heat flux.
[0091]
(Example)
Fig. 5 schematically shows an example of an inverse problem analysis assuming a one-dimensional transient heat conduction model using a thermocouple embedded in a carbon brick installed on the bottom of the blast furnace bottom. Is shown. Two thermocouples are embedded so as to be biased toward the cooling surface, TC1 indicates a high-temperature-side thermocouple, and TC2 indicates a low-temperature-side thermocouple. From the temperature changes measured by the thermocouples TC1 and TC2, the unsteady heat flux q1 on the high-temperature heat flux surface and the unsteady heat flux q2 on the cooling surface are simultaneously estimated by the inverse problem analysis described above. The high-temperature heat flux surface position is a fixed point 4.0 m behind the cooling surface. The assumed thermal properties of the carbon brick are as follows: specific heat Cp = 712 J / (kg · K), density ρ = 2300 kg / m³, And thermal conductivity k = 21.2 W / (m · K).
[0092]
FIG. 6A shows the estimation result of the high-temperature-side heat flux q1 obtained in the present embodiment. The horizontal axis in FIG. 6 represents the date, and shows the results from September 1 to October 1. Actually, the calculation is started before September 1 and the calculation ends on October 1. The time step of the inverse problem analysis is an eight-hour step.
[0093]
Case 1 is a calculation result in a case where a value of one digit after the decimal point of the temperature data measured by the thermocouple is rounded off, and a filter process is performed on data having a decimal point of 0 digit. Case 2 is a calculation result in the case where the numerical value of the two decimal places of the temperature data is rounded off, and the filter processing is performed on the data having one decimal place.
[0094]
FIGS. 6B and 6C show graphs in which temperature data (temperature data used for inverse problem analysis) of the high-temperature-side thermocouple TC1 and the low-temperature-side thermocouple TC2 after filtering are plotted. FIG. 6D shows a temperature difference between Case 1 and Case 2 at each position of the thermocouples TC1 and TC2 (Case 1-Case 2).
[0095]
As for the filter, in order to stabilize the calculation by applying a low-pass filter with as many terms as possible to past temperature data, a maximum of 21 terms of the Spencer equation (Equation (10)) is used. We use a technique to reduce the number. Specifically, expressions (4) to (10) are used in the notation of the Windows format. That is, the number of terms decreases as October 1 approaches, and finally equation (4) is used, and no filtering is performed on the last temperature data.
[0096]
In both Case 1 and Case 2, the past analysis is very stabilized by the filter effect, but becomes gradually unstable as it approaches the present, and the final estimated heat flux is realistically Is generating an incredibly large vibration. This is considered to indicate that the effect of the low-pass filter gradually decreases, and the calculation becomes unstable.
[0097]
Comparing Case 1 and Case 2, it can be seen that Case 2 is able to reduce the magnitude of the final vibration. This indicates that relatively stable calculation can be performed with temperature data having a large number of digits after the decimal point, even with a low-pass filter having a small number of terms.
[0098]
Furthermore, in any case, since vibration occurs in the final estimation result, when comparing the same digit number of temperature data, it is possible to increase the number of terms of the low-pass filter and smooth the temperature data as much as possible. It also shows that this is an effective means for obtaining a stable solution. In particular, when a low-pass filter in the form of the equations (4) to (10) is used, even if the equation (4) is used, since the last data is not filtered, there are many values after the decimal point of the original temperature data. In this case, the solution to the inverse problem is more stable because it is apparently easier to smooth.
[0099]
In case 1 and case 2, the inverse problem analysis was attempted without filtering the temperature data at all, but the solution diverged, and the calculation could not proceed.
[0100]
FIG. 7 shows a schematic diagram similar to FIG. 5 for cases where the installation depth of the thermocouple is different. FIG. 8A shows the estimation results of the heat flux q1 on the high-temperature side when the calculation time step of the inverse problem analysis is set to 8 hours and when the calculation time step is set to 6 hours. These calculation results are extracted and shown from September 1 to October 1 among the calculation results for a longer period.
[0101]
The temperature data used for the calculation is one digit after the decimal point, and smoothing is performed using the equation (10) as a filter. FIGS. 8B and 8C show graphs in which the temperature data of the high-temperature thermocouple TC1 and the low-temperature thermocouple TC2 after the smoothing are plotted. FIG. 8D substantially shows the difference (TC1−TC2) between the high temperature side thermocouple temperature and the low temperature side thermocouple temperature corresponding to the steady heat flux.
[0102]
When comparing the transition of the pseudo steady heat flux in FIG. 8D and the correspondence between the peaks of the unsteady heat flux estimated by the inverse problem analysis in FIG. 8A, the inverse problem analysis is intuitive. Compensates for the delay time, and it can be seen that the peak of the unsteady heat flux appears earlier (shown by the broken line in FIG. 8A).
[0103]
The pseudo steady heat flux in the graph of FIG. 8D can be interpreted as an average heat flux passing around the thermocouple position, and the value of the unsteady heat flux estimated at a high temperature surface at a depth of 4 m and In comparison, peaks do not always correspond to each other, but in this case, a peak lag of about 2 to 3 days is observed. In other words, the unsteady heat flux based on the inverse problem analysis detects the movement in the furnace approximately two to three days earlier than the conventional steady method.
[0104]
In addition, comparing the 8-hour step calculation and the 6-hour step calculation in the graph of FIG. 8A, the 6-hour step calculation shows a movement with a clearer contour. If the time step is lengthened, only the motion averaged within that time can be captured. Therefore, if the time step is shortened as much as possible, there is a high possibility that the actual internal motion can be expressed.
[0105]
(Other embodiments)
The inverse problem analysis device described above is configured by a computer CPU or MPU, RAM, ROM, RAM, or the like, and is realized by the operation of the program stored in the RAM, ROM, or the like as described above. .
[0106]
Therefore, the program itself realizes the functions of the above-described embodiment, and constitutes the present invention. As a transmission medium of the program, a communication medium (a wired line or a wireless line such as an optical fiber) in a computer network (a WAN such as a LAN or the Internet, a wireless communication network or the like) system for transmitting and supplying the program information as a carrier wave Can be used.
[0107]
Furthermore, means for supplying the program to a computer, for example, a storage medium storing the program constitutes the present invention. As such a storage medium, for example, a flexible disk, a hard disk, an optical disk, a magneto-optical disk, a CD-ROM, a magnetic tape, a nonvolatile memory card, a ROM, and the like can be used.
[0108]
It should be noted that the shapes and structures of the respective parts shown in the above-described embodiments are merely examples of the specific embodiments for carrying out the present invention, and these limit the technical scope of the present invention. It must not be interpreted. That is, the present invention can be embodied in various forms without departing from its spirit or its main features. For example, some programs may be executed on another computer in order to use the present invention in a network environment.
[0109]
【The invention's effect】
As described above, according to the present invention, the inverse problem analysis using an unsteady heat conduction equation is performed based on the temperature data measured at the temperature measurement points arranged inside the wall of the reaction vessel. When calculating the temperature or heat flux at the surface of the reaction vessel, for example, even if the time step in the inverse problem analysis is shortened in order to estimate a change in the temperature or heat flux as close as possible in the past, the decimal point of the temperature data is By increasing the number of digits of or by performing a filtering process, a small temperature change can be captured, and the inverse problem analysis can be stabilized.
[Brief description of the drawings]
FIG. 1 is a diagram illustrating a configuration of an inverse problem analysis apparatus according to a first embodiment.
FIG. 2 is a diagram showing a two-dimensional cross section near a furnace wall of a reaction vessel (blast furnace) in which a plurality of thermocouples are embedded.
FIG. 3 is a flowchart for explaining processing in inverse problem analysis.
FIG. 4 is a diagram illustrating a configuration of an inverse problem analysis apparatus according to a second embodiment.
FIG. 5 is a diagram showing an arrangement relationship between thermocouples TC1 and TC2 in the embodiment.
FIG. 6 is a diagram for describing an analysis result in the example.
FIG. 7 is a diagram showing an arrangement relationship of thermocouples TC1 and TC2 in another embodiment.
FIG. 8 is a diagram for describing an analysis result in another embodiment.
[Explanation of symbols]
101 Input unit
101a filter
102 arithmetic unit
103 Inverse Problem Analysis Unit
104 Output unit

Claims (14)

Based on the temperature data measured at the temperature measurement points arranged inside the wall of the reaction vessel accompanied by the temperature change reaction, the inverse problem analysis using the unsteady heat conduction equation is performed to obtain the temperature at the surface of the reaction vessel. Or it is an inverse problem analysis method to find the heat flux,
Having a procedure to increase the number of decimal places of the temperature data measured at the temperature measurement point,
An inverse problem analysis method, wherein the temperature data with the number of decimal places increased is used for the inverse problem analysis. In the procedure for increasing the number of digits after the decimal point of the temperature data, temperature data is sampled with a sampling time shorter than the calculation time step in the inverse problem analysis, and they are averaged over time to calculate the calculation time step in the inverse problem analysis. 2. The inverse problem analysis method according to claim 1, wherein a representative value of the temperature data used in step (a) is used. Based on the temperature data measured at the temperature measurement points arranged inside the wall of the reaction vessel accompanied by the temperature change reaction, the inverse problem analysis using the unsteady heat conduction equation is performed to obtain the temperature at the surface of the reaction vessel. Or it is an inverse problem analysis method to find the heat flux,
Having a procedure for filtering the temperature data measured at the temperature measurement point,
An inverse problem analysis method, wherein the temperature data after the filter processing is used for the inverse problem analysis. Based on the temperature data measured at the temperature measurement points arranged inside the wall of the reaction vessel accompanied by the temperature change reaction, the inverse problem analysis using the unsteady heat conduction equation is performed to obtain the temperature at the surface of the reaction vessel. Or it is an inverse problem analysis method to find the heat flux,
A procedure for increasing the number of decimal places of the temperature data measured at the temperature measurement point,
Performing a filtering process on the temperature data with the number of digits after the decimal point increased,
An inverse problem analysis method, wherein the temperature data after the filter processing is used for the inverse problem analysis. 5. The method according to claim 3, wherein the filtering is low-pass filtering. A plurality of temperature measurement points are arranged at least in the thickness direction inside the wall of the reaction vessel, and the temperature or heat flux on the inner surface and the outer surface of the reaction vessel are obtained by the inverse problem analysis. Item 6. The inverse problem analysis method according to any one of Items 1 to 5. The difference between the temperature measured at each of the temperature measurement points and the temperature at each of the temperature measurement point positions calculated from the assumed values of the temperature or heat flux on the inner surface and outer surface of the reaction vessel by an unsteady heat conduction equation. 7. The inverse problem analysis method according to claim 6, wherein the assumed value that minimizes the sum of the squares of the differences is obtained as a temperature or a heat flux on the inner surface and the outer surface of the reaction vessel. Based on the temperature data measured at the temperature measurement points arranged inside the wall of the reaction vessel accompanied by the temperature change reaction, the inverse problem analysis using the unsteady heat conduction equation is performed to obtain the temperature at the surface of the reaction vessel. Or it is an inverse problem analyzer for finding the heat flux,
Means for increasing the number of digits after the decimal point of the temperature data measured at the temperature measurement point,
An inverse problem analysis apparatus, wherein the temperature data with the number of digits after the decimal point increased is used for the inverse problem analysis. Based on the temperature data measured at the temperature measurement points arranged inside the wall of the reaction vessel accompanied by the temperature change reaction, the inverse problem analysis using the unsteady heat conduction equation is performed to obtain the temperature at the surface of the reaction vessel. Or it is an inverse problem analyzer for finding the heat flux,
Filter means for filtering the temperature data measured at the temperature measurement point,
An inverse problem analysis device, wherein the temperature data after the filter processing is used for the inverse problem analysis. Based on the temperature data measured at the temperature measurement points arranged inside the wall of the reaction vessel accompanied by the temperature change reaction, the inverse problem analysis using the unsteady heat conduction equation is performed to obtain the temperature at the surface of the reaction vessel. Or it is an inverse problem analyzer for finding the heat flux,
A means for increasing the number of digits after the decimal point of the temperature data measured at the temperature measurement point, and a filter means for performing a filtering process on the temperature data with the number of digits after the decimal point increased,
An inverse problem analysis device, wherein the temperature data after the filter processing is used for the inverse problem analysis. Based on the temperature data measured at the temperature measurement points arranged inside the wall of the reaction vessel accompanied by the temperature change reaction, the inverse problem analysis using the unsteady heat conduction equation is performed to obtain the temperature at the surface of the reaction vessel. Alternatively, a computer program for causing a computer to execute a process for determining a heat flux,
By performing the process of increasing the number of decimal places of the temperature data measured at the temperature measurement point,
A computer program characterized by using the temperature data in which the number of digits after the decimal point is increased in the inverse problem analysis. Based on the temperature data measured at the temperature measurement points arranged inside the wall of the reaction vessel accompanied by the temperature change reaction, the inverse problem analysis using the unsteady heat conduction equation is performed to obtain the temperature at the surface of the reaction vessel. Alternatively, a computer program for causing a computer to execute a process for determining a heat flux,
By performing a process of performing a filter process on the temperature data measured at the temperature measurement point,
A computer program characterized by using the temperature data after the filter processing for the inverse problem analysis. Based on the temperature data measured at the temperature measurement points arranged inside the wall of the reaction vessel accompanied by the temperature change reaction, the inverse problem analysis using the unsteady heat conduction equation is performed to obtain the temperature at the surface of the reaction vessel. Alternatively, a computer program for causing a computer to execute a process for determining a heat flux,
A process of increasing the number of digits after the decimal point of the temperature data measured at the temperature measurement point and a process of performing a filtering process on the temperature data with the number of digits after the decimal point increased,
A computer program characterized by using the temperature data after the filter processing for the inverse problem analysis. A computer-readable storage medium storing the computer program according to any one of claims 11 to 13.

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