JP2004053272A - Method for calculating diffusion coefficient related to contaminant flowing out of solidified body - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method for more accurately calculating a diffusion coefficient used for an analysis related to the amount of contaminants flowing out of a solidified body obtained by performing the insolubilization treatment of the contaminants. <P>SOLUTION: A diffusion test is performed to dip the sample of the solidified body into the solvent for necessary time for measuring lapse time and the amount of change in contaminant concentration in the solvent. Measurement values of lapse time in the diffusion test and the amount of change in contaminant concentration are substituted for a relation expression among conditions listed below: (a) the shape of the sample is assumed to be a column having specific dimensions; (b) a specific initial contaminant concentration distribution is set to the contaminants in the sample; (c) the contaminant concentration near the boundary between the sample and the solvent is regarded to be always constant; (d) aging in the contaminant concentration distribution in the sample can be derived by modelling according to the Fick law, lapse time, the amount of change in contaminant concentration in the solvent, and the diffusion coefficient of contaminants in the solidified body. <P>COPYRIGHT: (C)2004,JPO

Description

【０００８】
続いて、各時間区間ｉ毎の拡散係数Ｄ_ｅ，ｉ（ｃｍ^２／ｓ）を、下式（数５）に基づき計算する。なお、時間区間ｉ終了時点における積算時間；ｔ_ｉ（ｓ）である。
【０００９】
【数５】

【００１０】
以上より、体積当たりの汚染物溶出可能量ρ・Ｕ_ｂｅｓ並びに拡散係数Ｄ_ｅ，ｉを得ることができ、経時的に固化体から溶出する汚染物質量を予測することが可能となる。ＣＥＮ案では、時刻ｔが経過したときに固化体から溶出する汚染物質量Ｍ_ｄｉｆ（ｔ）（ｇ）を、下式（数６）により求める。但し、有効拡散係数Ｄ_ｅ（ｃｍ^２／ｓ）は、各時間区間ｉ毎に算定した拡散係数Ｄ_ｅ，ｉの相乗平均値である。
【００１１】
【数６】

【００１２】
【発明が解決しようとする課題】
ところが、ＣＥＮ案に準拠して算定した固化体中の拡散係数の値を用いて汚染物溶出の予測を行い、現実の結果と比較してみると、誤差が存在することが分かる。検討したところ、ＣＥＮ案による拡散係数の算定値が真値よりも小さい値となっていると考えられ、このことが誤差を生じせしめている有力な要因であることが明らかとなった。
【００１３】
以上の問題に鑑みて、本発明は、拡散係数の算定値が真値より小さくなることの原因を解析し、より精確な拡散係数算出方法を提供しようとするものである。
【００１４】
【課題を解決するための手段】
上述した課題を解決すべく、本発明では、固化体の供試体並びに供試体中の汚染物の拡散を、以下に掲げる条件；
（ａ）供試体の形状を、所定の寸法を有する円柱と仮定
（ｂ）供試体中の汚染物について、所定の初期汚染物濃度分布を設定
（ｃ）供試体と溶媒との境界近傍における汚染物濃度は常に一定であると見なす
（ｄ）供試体中の汚染物濃度分布の経時変化はＦｉｃｋの法則に従う
でモデル化して経過時間並びに溶媒中の汚染物濃度の変化量と固化体中の汚染物の拡散係数との関係式を導き、該関係式に拡散試験における経過時間と汚染物濃度の変化量との各々の測定値を代入して拡散係数を算出することとした。
【００１５】
後に詳述するが、ＣＥＮ案における拡散係数の算出式（数５）は、供試体の形状を図３に示すような薄板状と仮定して導出されたものである。しかし、拡散試験に使用される供試体は、直方体状あるいは円柱状であることが通常である。つまり、薄板モデルより導かれた算出式に、非薄板形状をなす供試体に対する拡散試験の結果の値を代入することが、拡散係数の算定値に誤りをもたらす原因の一つとなっていた。本発明は、このような不合理に初めて着目してなされたものであって、供試体を所定の寸法を有する円柱としてモデル化して算出式を導出することにより、拡散係数の算出の精確化を図るものである。また、セメントあるいはセメント系固化材について配合試験等を行う際には、円柱状の供試体を作成することが多い。従って、円柱モデルに基づく拡散係数の算出式を提供することで、供試体の手配をより容易とし拡散試験のコストを低減できるという効用をも奏し得る。
【００１６】
さらに、ＣＥＮ案における拡散係数の算出式（数５）は、供試体表面の汚染物濃度を常に０と仮定して導出されている。しかしながら、供試体表面の汚染物濃度を常に０と見なすことができるのは、厳密には溶媒の体積並びに溶媒中の拡散係数がともに無限大の場合のみである。即ち、拡散試験の際に溶媒中に溶出する汚染物質量は、供試体表面の汚染物濃度を常に０と見なすことができる理想条件下での溶出汚染物質量よりも少ない。よって、拡散試験の結果得られる測定値を算出式（数５）に代入して算出した拡散係数の算定値は、真値よりも必ず小さくなる。そこで、本発明では、前記関係式を利用して得た拡散係数Ｄ_ｍ，ｊに、該拡散係数Ｄ_ｍ，ｊと溶媒中の汚染物の拡散係数Ｄ_ｆとを用いて定められる、拡散試験における溶媒の体積及び溶媒中の汚染物の拡散係数が有限であることに関する補正因子ｄ（Ｄ_ｍ，ｊ，Ｄ_ｆ）を乗じて、より真値に近づけた拡散係数Ｄ_{ｍ，ｊ＋１}を算出する。
【００１７】
【発明の実施の形態】
本発明の一実施形態を、図面を参照して説明する。まず、ＣＥＮ案における拡散係数の算出式（数５）の導出過程について説明する。汚染物を不溶化処理して得られる固化体の供試体からの汚染物の溶出を、供試体中の汚染物濃度と溶媒中の汚染物濃度との差によるものと考え、供試体中の汚染物濃度の変化量がその濃度勾配に比例する、即ち、供試体中の汚染物濃度分布の経時変化がＦｉｃｋの法則に従うものと仮定する。かつ、供試体中の汚染物について、時刻ｔ＝０において汚染物濃度Ｃ_０が供試体中に一様分布しているものとする。そして、供試体と溶媒との境界近傍における汚染物濃度を常に０であると見なして、供試体中の汚染物の外部への溶出に関する境界値問題を考える。
【００１８】
ここで特記しておくと、ＣＥＮ案における拡散係数の算出式の誘導は、実は、図３に示すような薄板モデルが起点となっている。固化体の供試体を薄板状をなすものと仮定した場合、該薄板中に含まれている汚染物の溶出はその大半が上面ないし下面を経由するものとなる。つまり、上面ないし下面からの溶出量が側面からの溶出量と比較して卓越するから、薄板中の汚染物濃度分布の経時変化を厚み方向の１次元問題として近似可能である。上面ないし下面での汚染物濃度を常に０とすれば、薄板中の汚染物濃度Ｃ_Ｓ（ｚ，ｔ）は下式（数７）で与えられる。
【００１９】
【数７】

【００２０】
なお、薄板の厚さを２Ｈ、薄板の上下面からの深度をｚとおいている。また、λ_ｎ＝（２ｎ＋１）π／２、Ｔ＝Ｄｔ／Ｈ^２、Ｄは拡散係数、Ｃ_Ｓ（ｚ，０）＝Ｃ_０である。即ち、図３に示すように、薄板中の汚染物濃度Ｃ_Ｓ（ｚ，ｔ）は厚さの中心面に対称に分布する。上式（数７）を空間パラメタｚで平均化すると、下式（数８）となる。
【００２１】
【数８】

【００２２】
しかして、下式（数９）で定義される平均溶出度Ｆ（ｔ）を導入する。
【００２３】
【数９】

【００２４】
平均溶出度Ｆ（ｔ）は、供試体を溶媒に浸して時間ｔが経過したときに該供試体中に当初含まれていた初期汚染物質量のうち何割が溶媒中に溶出したかを表し、Ｆ（０）＝０、Ｆ（∞）＝１である。Ｃ_ｄｉｆ（ｔ）は、時刻ｔまでに溶媒中に溶出した供試体体積当たりの汚染物質量を表し、Ｃ_ｄｉｆ（ｔ）＝Ｃ_０・Ｆ（ｔ）である。平均溶出度Ｆ（ｔ）と時間係数Ｔとの相関を、図４に示す。薄板の厚さ２Ｈ及び拡散係数Ｄを不変とすると、初期の平均溶出度Ｆ（ｔ）は√ｔにほぼ比例し、Ｆ（ｔ）の大きさが０．５程度までであれば下式（数１０）で近似できる。
【００２５】
【数１０】

【００２６】
上式（数１０）を拡散係数Ｄについての式に変形すると、
【００２７】
【数１１】

【００２８】
となる。（数１１）より、Ｃ_ｄｉｆ（ｔ）もまた√ｔに比例すると見なすことができる。Ｃ_ｄｉｆ（ｔ）〜√ｔの比例関係が成立する範囲内では、Ｃ_ｄｉｆ（ｔ）の増分についての下式（数１２）が成り立つ。なお、時刻ｔ_ｉ−１から時刻ｔ_ｉまでの供試体体積当たりの汚染物溶出質量；ΔＣ_{ｄｉｆ，ｉ}である。
【００２９】
【数１２】

【００３０】
式（数１２）を式（数１１）に代入すると、下式（数１３）が得られる。
【００３１】
【数１３】

【００３２】
時刻ｔ_ｉ−１から時刻ｔ_ｉまでの供試体表面積当たりの汚染物溶出質量Ｅ_{ｄｉｆ，ｉ}と、上記のΔＣ_{ｄｉｆ，ｉ}との間には、Ｅ_{ｄｉｆ，ｉ}＝ΔＣ_{ｄｉｆ，ｉ}・Ｖ_Ｓ／Ａなる関係が成り立つ。但し、供試体体積をＶ_Ｓ、供試体表面積をＡとおいている。既に述べたように、供試体が薄板状をなし、汚染物の溶出が専ら薄板の上下面を経由して起こっていると仮定しており、従ってＶ_Ｓ＝Ｂ・Ｌ・２Ｈ、Ａ＝２Ｂ・Ｌ＝Ｖ_Ｓ／Ｈ（即ち、薄板側面を無視）とすることができる。以上より、拡散係数Ｄに関する式、
【００３３】
【数１４】

【００３４】
を得ることができる。初期汚染物濃度Ｃ_０を、Ａｖａｉｌａｂｉｌｉｔｙ試験の結果得られる固化体の体積当たりの汚染物溶出可能量ρ・Ｕ_ｂｅｓとすれば、式（数１４）は式（数５）に一致する。
【００３５】
また、拡散試験では、溶媒中の汚染物濃度が測定される。各時間区間ｉにおいて供試体から溶出した汚染物質量は、溶媒中に存在する汚染物質量に等しいため、
【００３６】
【数１５】

【００３７】
が成立する。式（数１５）は、式（数４）に一致する。
【００３８】
因みに、式（数１０）より、時刻ｔまでに溶出する供試体体積当たりの汚染物質量Ｃ_ｄｉｆ（ｔ）を下式（数１６）のように表すことができる。
【００３９】
【数１６】

【００４０】
よって、時刻ｔまでに供試体から溶出する汚染物質量を、下式（数１７）を用いて求めることができる。
【００４１】
【数１７】

【００４２】
Ｃ_０をρ・Ｕ_ｂｅｓとし、ＤをＤ_ｅとすれば、式（数１７）は式（数６）に一致する。
【００４３】
このようにして、ＣＥＮ案における拡散係数の算出式（数５）は導出されている。しかしながら、該算出式を利用して得られる拡散係数の算定値は、拡散係数の真値と比べてやや小さくなる傾向があると思われる。以降では、ＣＥＮ案よりも精確に拡散係数を算出する方法について述べる。
【００４４】
ＣＥＮ案の方法が有している問題の一つは、拡散係数の算出式（数５）を導くために仮定した供試体のモデルにある。即ち、供試体を薄板状をなすものとしている点、及び、薄板の側面からの汚染物溶出を無視している点が、実際に拡散試験で使用される供試体にそぐわないことにある。そこで、供試体のモデルをより現実に即したものとして、拡散係数の算出式を導出することを試みる。
【００４５】
固化体の供試体の形状を、図５に示すような底面半径Ｒ、高さ２Ｈの寸法を有する円柱と仮定する。供試体中の汚染物について、所定の初期汚染物濃度分布、言い換えれば供試体体積当たりの汚染物質量の初期分布を予め設定する。議論を簡単にするため、ここでは、時刻ｔ＝０において汚染物濃度Ｃ_０が供試体中に一様分布しているものとする。そして、供試体と溶媒との境界近傍における汚染物濃度を常に一定であると見なし、供試体中の汚染物濃度分布の経時変化がＦｉｃｋの法則に従うものとして、供試体中の汚染物の外部への溶出に関する境界値問題を考え、拡散係数における経過時間並びに溶媒中の汚染物濃度の変化量と固化体中の汚染物の拡散係数との関係式を導く。
【００４６】
特に、供試体表面近傍での汚染物濃度を常に０と見なすとともに、円柱側面からのみ汚染物が溶出し上下底面からは溶出しないと仮定すれば、供試体中の汚染物濃度分布の経時変化を径方向の軸対称問題として近似可能であって、該円柱モデルに関する平均溶出度Ｆ（ｔ）を下式（数１８）で表すことができる。
【００４７】
【数１８】

【００４８】
なお、β_ｎはＢｅｓｓｅｌ関数Ｊ_０（ｘ）＝０の根で、時間係数Ｔ_Ｒ＝Ｄ_ｐ・ｔ／Ｒ^２、Ｄ_ｐは拡散係数である。上式（数１８）を基にＦ（ｔ）と√ｔとの相関をグラフにプロットすると、図６に示すようになる。図６では、Ｒ、Ｄ_ｐを変化させたときのＦ（ｔ）と√ｔとの相関を表示してある。図６によれば、Ｆ（ｔ）の大きさが０．３程度までであれば、Ｆ（ｔ）〜√ｔの比例関係を利用してＦ（ｔ）を線形近似できそうである。Ｒ、Ｄ_ｐを定めた場合にはＦ（ｔ）をＴ_Ｒにより一意に決定可能であるから、Ｆ（ｔ）〜√ｔの関係を線形近似する比例係数ｋに関して、Ｒ、Ｄ_ｐとの間に下式（数１９）に示す相似則を仮定できる。
【００４９】
【数１９】

【００５０】
上式（数１９）において、ｋ_０は、所定の基準供試体半径Ｒ_０並びに基準拡散係数Ｄ_０を設定した場合の、Ｆ（ｔ）〜√ｔの関係を線形近似する比例係数である。例えば、基準供試体半径Ｒ_０＝１、基準拡散係数Ｄ_０＝１とすると、図６よりｋ_０の値は略２．１となる。但し、ここでは、Ｆ（ｔ）の√ｔに対する初期接線勾配を、比例係数ｋ、ｋ_０として扱っている。上式（数１９）を拡散係数Ｄ_ｐについての式に変形すると、
【００５１】
【数２０】

【００５２】
となる。即ち、Ｄ_ｐ＝Ｎ_１・ｋ^２・Ｒ^２の形で拡散係数Ｄ_ｐを表すことができる。式（数２０）に、Ｒ_０＝１、Ｄ_０＝１並びにｋ_０＝２．１を代入することで、定数Ｎ_１≒０．２２７を得る。
【００５３】
平均溶出度Ｆ（ｔ）は、拡散試験により得られる測定値を下式（数２１）に代入することで計算できる。なお、拡散試験に係る時間区間ｉ終了時点での平均溶出度；Ｆ（ｔ_ｉ）、時間区間ｉ終了時点での積算時間；ｔ_ｉ、時間区間ｉ終了時点での溶媒中の汚染物濃度；Ｃ_ｆ，ｉ、供試体中の初期汚染物濃度；Ｃ_０、溶媒体積；Ｖ_ｆ、供試体体積；Ｖ_Ｓである。
【００５４】
【数２１】

【００５５】
よって、比例係数ｋの値を拡散試験の結果より決定することが可能である（例えば、ｋ＝Ｆ（ｔ）／√ｔとする）から、供試体の拡散係数Ｄ_ｐを式（数２０）により算定することができる。
【００５６】
しかしながら、式（数２０）は、円柱側面からのみ汚染物が溶出し上下底面からは溶出しないとの仮定の下で導かれたものである。このような仮定が成り立つには、供試体の上下底面を被覆してあるか、供試体の高さ２Ｈが底面半径Ｒと比べて著しく大きなものでなくてはならない。実際の拡散試験では供試体の上下底面を経由する汚染物溶出も生じ得るから、式（数２０）は拡散係数の算出式として必ずしも最適であるとは言えない。例えば、セメントあるいはセメント系固化材について配合試験等を行う場合に使用される供試体は、高さが直径の２倍程度となっており、底面からの汚染物溶出を無視することが妥当ではない。式（数２０）、（数２１）は、経過時間ｔ_ｉ並びに溶媒中の汚染物濃度の変化量Ｃ_ｆ，ｉと固化体中の汚染物の拡散係数Ｄ_ｐとの、いわば仮の関係式である。
【００５７】
そこで、補正因子を乗じることで拡散係数Ｄ_ｐを補正し、底面からの汚染物溶出を考慮に入れた拡散係数を求めることを考える。汚染物溶出の量と側面を経由する汚染物溶出の量との相関は、供試体の高さ２Ｈと底面半径Ｒとの関数として近似することが可能であると考えられる。よって、供試体の底面からの汚染物溶出を加味する補正因子を、供試体の高さ２Ｈと底面半径Ｒとを用いて近似的に定めることができると予想される。該補正因子ｃ（Ｈ，Ｒ）を定めるべく、本実施形態では、円柱側面及び底面より汚染物が溶出するとした場合の供試体中の汚染物濃度分布の経時変化を有限要素法を用いて解析した結果を援用することとする。
【００５８】
供試体の底面半径Ｒ、供試体中の拡散係数Ｄ_ｐの各々を所定値に定めるとともに、供試体側面及び底面近傍の汚染物濃度を常に０とする。このような前提条件の下で、有限要素法を用いて供試体中の汚染物濃度分布の経時変化を解析する。かつ、供試体中の汚染物濃度分布を空間パラメタで積分すれば、平均溶出度Ｆ（ｔ）を算出することができる。図７に示すものは、有限要素法を用いて解析した、Ｈ／Ｒ＝１／２、１、２、５の各場合におけるＦ（ｔ）と√ｔとの相関である。なお、底面半径Ｒ＝１、拡散係数Ｄ_ｐ＝１と定めている。この解析結果を利用して、比例係数比ｋ（Ｈ／Ｒ）／ｋ（∞）とＨ／Ｒとの相関をプロットすると、図８に示すようになる。ｋ（∞）は、供試体の高さ２Ｈが底面半径Ｒと比べて著しく大きい、即ち底面を経由した溶出を０としたときの比例係数である。図８に示すＨ／Ｒ−ｋ（Ｈ／Ｒ）／ｋ（∞）平面上のプロット点に近似曲線を当てはめると、ｋ（Ｈ／Ｒ）／ｋ（∞）とＨ／Ｒとの関係を表す近似式（数２２）を得る。
【００５９】
【数２２】

【００６０】
Ｎ_２は定数である。当該例では、Ｎ_２≒０．５４となる。ｋ（Ｈ／Ｒ）は、円柱の上下底面を経由した汚染物溶出を加味した比例係数であって、拡散試験の結果より決定されるものである。そして、上式（数２２）を利用すれば、拡散試験における条件と同等の底面半径Ｒ並びに拡散係数Ｄ_ｐの下でのｋ（∞）を、ｋ（Ｈ／Ｒ）より得ることが可能である。即ち、ｋ（∞）は、ｋ（Ｈ／Ｒ）から式（数２２）を除することで求められる。既に述べたように、ｋ（∞）は底面を経由した溶出を０としたときの比例係数であるから、ｋ＝ｋ（∞）を式（数２０）に代入することにより、好適に拡散係数Ｄ_ｐを算出することができる。式（数２０）によれば、拡散係数Ｄ_ｐはｋ^２に比例するため、底面からの汚染物溶出を考慮に入れた拡散係数について、
【００６１】
【数２３】

【００６２】
が成立する。比例係数ｋ（Ｈ／Ｒ）はＦ（ｔ）〜√ｔの線形勾配であるから、結果として、拡散係数Ｄ_ｐは下式（数２４）で表される。
【００６３】
【数２４】

【００６４】
以上より、経過時間ｔ_ｉ並びに溶媒中の汚染物濃度の変化量Ｃ_ｆ，ｉと固化体中の汚染物の拡散係数Ｄ_ｐとの関係式（数２１）、（数２０、あるいは数２４）を導出することができた。これら関係式に、拡散試験における測定値ｔ_ｉ、Ｃ_ｆ，ｉを代入すれば、拡散係数Ｄ_ｐを算出することができる。因みに、初期汚染物濃度Ｃ_０は、供用状態にある供試体より時刻ｔ＝∞までに溶出する供試体体積当たりの汚染物質量を意味する。言い換えるならば、供試体より溶媒中に溶出し得る汚染物の供試体体積当たりの最大質量を意味する。勿論、Ｃ_０に供試体体積Ｖ_Ｓを乗じたものが初期汚染物質量となる。供試体中の初期汚染物濃度Ｃ_０として、例えば、上述したＡｖａｉｌａｂｉｌｉｔｙ試験の結果得られる固化体の体積当たりの汚染物溶出可能量ρ・Ｕ_ｂｅｓの値を用いることができる。但し、Ａｖａｉｌａｂｉｌｉｔｙ試験以外の方法により初期汚染物濃度Ｃ_０を決定することを妨げない。
【００６５】
なお、通常、拡散試験は複数の時間区間に分けて行われる。拡散試験を５の時間区間に分けて行ったときには、５回の測定が実施され、その測定結果より５つのＦ（ｔ_ｉ）／√ｔ_ｉの値が得られる。即ち、５つの拡散係数Ｄ_ｐが算出される。よって、これらを相乗平均して、固化体の拡散係数の値とすることができる。このとき、Ｆ（ｔ_ｉ）と√ｔ_ｉとが略比例している時間区間ｉに係る拡散係数Ｄ_ｐのみを相乗平均して、固化体の拡散係数の値とすることが好ましい。例えば、Ｆ（ｔ_ｉ）＜０．５の条件を満たしている拡散係数Ｄ_ｐのみを相乗平均し、Ｆ（ｔ_ｉ）＜０．５の条件を満たさない拡散係数Ｄ_ｐの値は採用しないことが好ましい。これは、式（数２０）がＦ（ｔ）〜√ｔの比例関係を利用して導かれていることによる。
【００６６】
ＣＥＮ案による拡散係数算出方法が有している別の問題として、拡散係数の算出式（数５）の導出過程において供試体表面の汚染物濃度を常に０と仮定している点を挙げることができる。供試体表面の汚染物濃度を常に０と見なすことができるのは、厳密には溶媒の体積並びに溶媒中の拡散係数がともに無限大の場合のみである。しかし、現実には、溶媒の体積は無限ではあり得ず、供試体表面での汚染物濃度を常に０に保つことは不可能である。ＣＥＮ案に準拠して拡散試験を行うときには、溶媒の体積Ｖ_ｆを供試体の体積Ｖ_Ｓの４〜６倍に設定する。また、溶媒中の汚染物の拡散係数は、通常１０^−５ｃｍ^２／ｓ程度である。つまり、拡散試験の際に溶媒中に溶出する汚染物質量は、供試体表面の汚染物濃度を常に０と見なすことができる理想条件下での溶出汚染物質量よりも少ない。よって、拡散試験の結果得られる測定値を算出式（数５）に代入して算出した拡散係数の算定値は、真値よりも必ず小さくなる。このことは、本発明に係る算出方法により算出される拡散係数Ｄ_ｐに関しても言えることである。なぜならば、Ｄ_ｐの算出式（数２０、あるいは数２４）の導出過程においても、供試体と溶媒との境界近傍の汚染物濃度を常に０と仮定しているからである。
【００６７】
そこで、補正因子を乗じることで拡散係数を補正し、より真値に近づけることを考える。以下、拡散係数をＤ_ｍ，ｊと表記し、Ｄ_ｍ，ｊに補正因子を一回乗じて補正した値をＤ_{ｍ，ｊ＋１}とおくこととする。但し、算出式（数２４）より直接算出した拡散係数をＤ_ｍ，１とする（即ち、添字ｊ＝１）。溶媒体積Ｖ_ｆ、供試体体積Ｖ_Ｓを所定値に設定した場合、拡散係数の算出値Ｄ_ｍ，ｊを補正するための補正因子を、当該Ｄ_ｍ，ｊと、溶媒中の汚染物の拡散係数Ｄ_ｆとを用いて近似的に定めることができると予想される。該補正因子ｄ（Ｄ_ｍ，ｊ，Ｄ_ｆ）を定めるべく、本実施形態では、供試体中の汚染物濃度分布の経時変化を有限要素法を用いて解析した結果を援用することとする。
【００６８】
ここでは、固化体の供試体を直径５ｃｍ、高さ１０ｃｍの円柱形状のものとし、溶媒体積Ｖ_ｆを供試体体積Ｖ_Ｓの５倍に設定する。このような前提条件の下で、溶媒中の汚染物の拡散係数Ｄ_ｆ＝１０^−５ｃｍ^２／ｓと仮定し、有限要素法を用いて、溶媒中及び供試体中の汚染物濃度分布の経時変化を解析することにより、溶媒に関する物理的制約を考慮に入れた平均溶出度Ｇを計算する。他方、供試体を直径５ｃｍ、高さ１０ｃｍの円柱とし、溶媒体積Ｖ_ｆを無限大としたとき、即ち供試体と溶媒との境界面上の汚染物濃度を常に０としたときの平均溶出度Ｆは、式（数１８）を利用して得ることができる。そして、有限体積の溶媒中へ汚染物が拡散する場合を考慮した平均溶出度Ｇの、無限体積の溶媒中へ汚染物が拡散する場合における平均溶出度Ｆに対する比Ｇ／Ｆ＝αとおく。供試体中の汚染物の拡散係数を種々の値に変化させてαを計算し、αとＤ_ｍ，ｊ／Ｄ_ｆとの相関をプロットすると、図９に示すようになる。図９に示すＤ_ｍ，ｊ／Ｄ_ｆ−α平面上のプロット点に近似曲線（図９が片対数グラフであるため、図示例では近似直線）を当てはめると、α（Ｄ_ｍ，ｊ／Ｄ_ｆ）の近似式（数２５）を得る。
【００６９】
【数２５】

【００７０】
Ｎ_３、Ｎ_４は定数である。当該例では、Ｎ_３≒０．７１、Ｎ_４≒０．１２となる。勿論、供試体の寸法その他が異なる場合にあっても、上記と同様の方法により補正因子ｄ（Ｄ_ｍ，ｊ，Ｄ_ｆ）を導くことが可能である。式（数２４）によれば、拡散係数Ｄ_ｍ，ｊは［Ｆ（ｔ）／√ｔ］^２に比例する。よって、補正因子ｄ（Ｄ_ｍ，ｊ，Ｄ_ｆ）＝１／α^２とすることができる。従って、補正因子ｄ（Ｄ_ｍ，ｊ，Ｄ_ｆ）を１回乗じて補正した拡散係数Ｄ_{ｍ，ｊ＋１}は下式（数２６）で表すことができる。
【００７１】
【数２６】

【００７２】
さらに、所定の条件を満たすまで複数回の補正を行い、拡散係数Ｄ_ｍ，ｊをより真値に近づけていくことも好ましい。ｎ回の補正を行うことにより得られる拡散係数Ｄ_{ｍ，１＋ｎ}は、下式（数２７）で表される。
【００７３】
【数２７】

【００７４】
所定の条件としては、例えば、α≧１となった場合、｜α（Ｄ_{ｍ，ｊ＋１}／Ｄ_ｆ）−α（Ｄ_ｍ，ｊ／Ｄ_ｆ）｜が予め設定した値（例えば、０．００１）よりも小さくなった場合、あるいは、｜Ｄ_{ｍ，ｊ＋１}−Ｄ_ｍ，ｊ｜が所定値よりも小さくなった場合、等が考えられる。
【００７５】
以上に詳述した拡散係数算出の手順を総合すると、図１０、図１１に示すフローチャートのようになる。
【００７６】
因みに、図１０、図１１に示しているフローチャートに準拠し、コンピュータ１を、拡散試験を行うことにより得られる経過時間ｔ_ｉと溶媒中の汚染物濃度の変化量Ｃ_ｆ，ｉとの各々の測定値を格納する測定値格納手段、前記測定値格納手段に格納している測定値を関係式（数２１）、（数２０、あるいは数２４）に代入することで得られる拡散係数を算出する拡散係数算出手段、並びに、前記関係式を利用して得た拡散係数Ｄ_ｍ，ｊに補正因子ｄ（Ｄ_ｍ，ｊ，Ｄ_ｆ）を乗じて補正をした拡散係数Ｄ_{ｍ，ｊ＋１}を算出する補正手段として機能させるプログラムを作成することができる。あるいは、測定値格納部１１、拡散係数算出部１２並びに補正部１３を具備し、図１０、図１１に示しているフローチャートに準拠した処理を実行可能な拡散係数算出装置１を構成することができる。
【００７７】
本発明に係るプログラムを基に機能するコンピュータ１、若しくは本発明に係る拡散係数算出装置１は、例えば、図１２に示すようなハードウェア資源を具備してなる。即ち、コンピュータ若しくは拡散係数算出装置１は、プロセッサ１ａ、主記憶装置１ｂ、補助記憶装置１ｃ、外部とデータの授受を行うためのＩ／Ｏインタフェース１ｄ、等のハードウェア資源を具備する。通常、プロセッサ１ａによって実行されるべきプログラムが補助記憶装置１ｃに格納されており、プログラムの実行の際には補助記憶装置１ｃ主記憶装置１ｂに読み込まれ、プロセッサ１ａによって解読される。そして、当該プログラムに従って上記のハードウェア資源を作動し、少なくとも、図１３に示す測定値格納部１１、拡散係数算出部１２としての機能を発揮するようにしている。
【００７８】
測定値格納部１１は、拡散試験を行うことにより得られる、経過時間ｔ_ｉと、溶媒中の汚染物濃度の変化量Ｃ_ｆ，ｉとの各々の測定値を格納する。通常、測定値格納部１１は、主記憶装置１ｂまたは補助記憶装置１ｃの所要の領域を用いて構成される。
【００７９】
拡散係数算出部１２は、前記測定値格納部１１に格納している測定値を関係式（数２１）、（数２０、あるいは数２４）に代入することで得られる拡散係数を算出する。拡散係数算出部１２は、例えば、ソフトウェアを主体として構成される。
【００８０】
さらに、コンピュータ若しくは拡散係数算出装置１は、補正部１３としての機能をも発揮し得るものであることが望ましい。補正部１３は、前記拡散係数算出部１２によって算出される拡散係数Ｄ_ｍ，ｊに補正因子ｄ（Ｄ_ｍ，ｊ，Ｄ_ｆ）を乗じて、補正をした拡散係数Ｄ_{ｍ，ｊ＋１}を算出する。補正部１３もまた、例えば、ソフトウェアを主体として構成される。
【００８１】
当該コンピュータ若しくは拡散係数算出装置の１の動作に関して、図１０、図１１のフローチャートを参照して説明する。まず、拡散試験の結果得られる測定値ｔ_ｉ、Ｃ_ｆ，ｉを格納する（ステップＳ１）。ｉは時間区間を表示する添字で、拡散試験をＣＥＮ案と同様に行う場合にはｉ＝（１，２，…，５）となる。初期設定として、変数ｊ＝１、α_１＝１とする（ステップＳ２）。なお、α_ｊ＝α（Ｄ_ｍ，ｊ／Ｄ_ｆ）である。そして、各時間区間ｉ毎に、関係式（数２１）、（数２６）に測定値ｔ_ｉ、Ｃ_ｆ，ｉを代入した結果である拡散係数Ｄ_ｐを計算する（ステップＳ３、Ｓ４、Ｓ５、Ｓ６）。但し、ここで算出される拡散係数Ｄ_ｐは、補正因子ｃ及び補正因子ｄにより補正をしたものである。
【００８２】
全ての時間区間ｉについて拡散係数Ｄ_ｐを算出した後、Ｆ（ｔ_ｉ）／α_ｊ＜０．５を満足する時間区間ｉに係るＤ_ｐのみを選出してこれらを相乗平均する（ステップＳ７）。より精確な拡散係数の算定値を得るべく、Ｆ（ｔ_ｉ）と√ｔ_ｉとが略比例している期間内での測定値ｔ_ｉ、Ｃ_ｆ，ｉを用いて拡散係数の算出を行うことが好ましいためである。補正因子ｄを司るα_ｊは、供試体と溶媒との境界近傍における汚染物濃度が０よりも大きくなることに鑑みて平均溶出度Ｆを補正するものであり、通常、α_ｊ＜１（従って、Ｆ（ｔ_ｉ）＜Ｆ（ｔ_ｉ）／α_ｊ）となる。相乗平均した結果の値を、Ｄ_ｍ，ｊとおく。
【００８３】
得られたＤ_ｍ，ｊを用いて、式（数２５）を基にα_ｊ＋１＝α（Ｄ_{ｍ，ｊ＋１}／Ｄ_ｆ）を計算する（ステップＳ８）。所定の条件を満足する場合、例えば、α_ｊ＋１≧１となったとき（ステップＳ９）、｜α_ｊ＋１−α_ｊ｜が予め設定した値（例えば、０．００１）よりも小さくなったとき（ステップＳ１０）、Ｄ_ｍ，ｊを固化体中の汚染物の拡散係数の算定値として（ステップＳ１２）、処理を終了する。所定の条件を満足しない場合には、上記のステップＳ３に戻り、Ｄ_{ｍ，ｊ＋１}を算出する（ステップＳ１１）。
【００８４】
以降、実施例として、セメント系固化材で固化した模擬汚染土に関する拡散試験の結果を基に、ＣＥＮ案における拡散係数算出方法と本発明に係る拡散係数算出方法とを比較する。汚染物質は全クロムで、含有量の相異なる２つの試料Ｎｏ．１、Ｎｏ．２を用意した。図１４に示すものは、これら試料に関するＡｖａｉｌａｂｉｌｉｔｙ試験の結果である。Ａｖａｉｌａｂｉｌｉｔｙ試験の結果得られたρ・Ｕ_ｂｅｓの値を、供試体中の初期汚染物濃度Ｃ_０とする。
【００８５】
拡散試験に用いる供試体は、試料Ｎｏ．１、Ｎｏ．２とも、直径５ｃｍ、高さ１０ｃｍの円柱とする。拡散試験の結果得られる溶媒中の汚染物濃度の測定値Ｃ_ｆ，ｉ並びに該Ｃ_ｆ，ｉ及び積算時間ｔ_ｉを用いて算出した拡散係数Ｄ_ｅ（ＣＥＮ案法による）、Ｄ_ｍ（本発明による）を、図１５に示す。図１５中の「平均」の欄は、各時間区間について算出した拡散係数を相乗平均した値である。試料Ｎｏ．１、Ｎｏ．２ともに、ＣＥＮ案法による拡散係数Ｄ_ｅは本発明による拡散係数Ｄ_ｍより小さい。
【００８６】
さらに、本発明の方法を利用して算出した拡散係数Ｄ_ｍ，ｊに補正因子ｄ（Ｄ_ｍ，ｊ，Ｄ_ｆ）を乗じて補正をした拡散係数Ｄ_{ｍ，ｊ＋１}を、図１６に示す。特に、試料Ｎｏ．１の拡散係数は、４回の補正を実行した結果、補正前の値よりも約２６％大きくなっており、ＣＥＮ案法による拡散係数の値よりも約６０％も大きくなる。
【００８７】
本実施形態によれば、汚染物を不溶化処理して得られる固化体の供試体を溶媒に浸し所要の時間を経過させ、経過時間と、溶媒中の汚染物濃度の変化量とを測定する拡散試験を行い、
供試体及び該供試体中の汚染物の拡散を以下に掲げる条件；
（ａ）供試体の形状を、所定の寸法を有する円柱と仮定
（ｂ）供試体中の汚染物について、所定の初期汚染物濃度分布を設定
（ｃ）供試体と溶媒との境界近傍における汚染物濃度は常に一定であると見なす（ｄ）供試体中の汚染物濃度分布の経時変化はＦｉｃｋの法則に従う
でモデル化することにより導かれる、経過時間並びに溶媒中の汚染物濃度の変化量と固化体中の汚染物の拡散係数との関係式に、拡散試験における経過時間と汚染物濃度の変化量との各々の測定値を代入して、拡散係数を算出するものとしている。従って、供試体の実際の形状により即した算出式を導出して用いるものとなっており、より真値に近い拡散係数の値を得ることが可能となっている。
【００８８】
前記供試体を溶媒に浸して時間ｔが経過したときに溶媒中に溶出する汚染物質量の、該供試体中に当初含まれていた初期汚染物質量に対する比である平均溶出度が√ｔに比例すると仮定し、その比例係数を用いて前記関係式を導くことにより、拡散係数の近似式を簡潔に記述できる。
【００８９】
前記供試体の底面からの汚染物溶出を無視して前記平均溶出度の関数式Ｆ（ｔ）を導き、該関数式を基にＦ（ｔ）の√ｔに対する比例係数ｋを決定することで得られる、経過時間並びに溶媒中の汚染物濃度の変化量と固化体中の汚染物の拡散係数との仮の関係式に、供試体の高さ２Ｈと底面半径Ｒとを用いて定められる、供試体の底面からの汚染物溶出を加味する補正因子ｃ（Ｈ，Ｒ）を乗じて、前記関係式を導くことにより、拡散係数の算出式を径方向の１次元モデルに帰着して導出することができる。
【００９０】
さらに、前記関係式を利用して得た拡散係数Ｄ_ｍ，ｊに、該拡散係数Ｄ_ｍ，ｊと溶媒中の汚染物の拡散係数Ｄ_ｆとを用いて定められる、拡散試験における溶媒の体積及び溶媒中の汚染物の拡散係数が有限であることに関する補正因子ｄ（Ｄ_ｍ，ｊ，Ｄ_ｆ）を乗じて、補正をした拡散係数Ｄ_{ｍ，ｊ＋１}を算出することにより、溶媒の体積や溶媒中の拡散係数が有限であるという物理的制約を考慮に入れた、より真値に近い拡散係数の値を得ることが可能となる。
【００９１】
そして、コンピュータ１を、拡散試験を行うことにより得られる、経過時間と、溶媒中の汚染物濃度の変化量との各々の測定値を格納する測定値格納手段、及び、前記測定値格納手段に格納している測定値を前記関係式に代入することで得られる拡散係数を算出する拡散係数算出手段として機能させるプログラムを作成することができる。該プログラムを、さらに、コンピュータ１を、前記関係式を利用して得た拡散係数Ｄ_ｍ，ｊに、該拡散係数Ｄ_ｍ，ｊと溶媒中の汚染物の拡散係数Ｄ_ｆとを用いて定められる、拡散試験における溶媒の体積及び溶媒中の汚染物の拡散係数が有限であることに関する補正因子ｄ（Ｄ_ｍ，ｊ，Ｄ_ｆ）を乗じて、補正をした拡散係数Ｄ_{ｍ，ｊ＋１}を算出する補正手段として機能させるものとして構成してもよい。
【００９２】
加えて、上記の式（数１８）より、円柱状供試体より経時的に溶出する供試体体積当たりの汚染物質量Ｃ_ｄｉｆ（ｔ）を下式（数２８）のように表すことができる。
【００９３】
【数２８】

【００９４】
上式（数２８）における時間係数Ｔ_Ｒは供試体中の拡散係数により定まる。供試体中の拡散係数をＤとおくと、Ｔ_Ｒ＝Ｄ・ｔ／Ｒ^２となる。よって、拡散係数の算定値を得ることで、供試体より溶出する汚染物質量の予測を簡便に行うことが可能となる。しかも、ＣＥＮ案における予測の式（数６）ではＭ_ｄｉｆ（∞）＝∞となってしまうが、上式（数２８）ではＣ_ｄｉｆ（∞）＝Ｃ_０となる。即ち、式（数６）を用いるよりも式（数２８）を用いる方が、汚染物の溶出量に関して正しく予測できると言える。因みに、本発明に係る拡散係数算出方法を利用して得られる、補正因子ｃを乗じた算定値を式（数２８）に適用するときには、該算定値より補正因子ｃを除してから式（数２８）に代入してもよい。
【００９５】
なお、本発明は以上に詳述した実施形態に限られるものではない。各部の具体的構成は上記実施形態には限られず、本発明の趣旨を逸脱しない範囲で種々変形が可能である。
【００９６】
【発明の効果】
以上に詳述した本発明によれば、より精確に拡散係数を算出することが可能となる。
【図面の簡単な説明】
【図１】Ａｖａｉｌａｂｉｌｉｔｙ試験及び拡散試験の概要を説明する表
【図２】拡散試験における時間区間の区切りを説明する表
【図３】ＣＥＮ案法における薄板モデルを示す図
【図４】Ｆ（ｔ）と√ｔとの相関を示す図
【図５】本発明における円柱モデルを示す図
【図６】円柱底面からの汚染物溶出を無視した場合のＦ（ｔ）と√ｔとの相関を示す図
【図７】円柱底面からの汚染物溶出を加味した場合のＦ（ｔ）と√ｔとの相関を示す図
【図８】補正因子ｃ（Ｈ，Ｒ）を示す図
【図９】α（Ｄ_ｍ，ｊ／Ｄ_ｆ）を示す図
【図１０】本発明に係る拡散係数算出方法の手順を示すフローチャート
【図１１】本発明に係る拡散係数算出方法の手順を示すフローチャート
【図１２】コンピュータ若しくは拡散係数算出装置が具備するハードウェア資源の内容を示す図
【図１３】コンピュータ若しくは拡散係数算出装置の機能ブロック図
【図１４】本発明の一実施例に係る、Ａｖａｉｌａｂｉｌｉｔｙ試験結果を示す図
【図１５】同実施例に係る、拡散係数の算定値を示す図
【図１６】補正因子ｄを乗じて得た補正後の拡散係数の値を示す図
【符号の説明】
１…コンピュータ若しくは拡散係数算出装置
１１…測定値格納部
１２…拡散係数算出部
１３…補正部[0001]
TECHNICAL FIELD OF THE INVENTION
The present invention relates to quantitative evaluation of contaminant elution from a solid obtained by insolubilizing contaminants using cement, a solidifying material, or the like.
[0002]
[Prior art]
As a countermeasure to be applied to the ground containing harmful contaminants, a method of insolubilizing the contaminants using cement, solidifying material or the like is often used. This is because cement and solidifying material physically and chemically fix contaminants with hydrates formed by the reaction of soil and water with clay minerals, and at the same time, fill the hydrates into the soil gap, Is made impermeable to reduce or prevent contaminants from eluting outside the ground. Normally, the ground insolubilized with cement or solidifying material becomes a solidified body having significantly greater strength than the original state. There is also a treatment method in which soil and soil containing contaminants and contaminants themselves are solidified by insolubilizing using cement, a solidifying material, or the like, and then buried in soil.
[0003]
In Japan, regarding the safety related to soil contamination, an evaluation method based on the Ministry of the Environment Notification No. 46 (August 23, 1991) is used. When conducting the test according to the method specified in Notification No. 46 of the Ministry of the Environment, the target soil is air-dried, classified into 2 mm or less, immersed in a solvent (pure water), and eluted into the solvent after 6 hours. The concentration of the contaminated substance is compared with a reference value (however, cadmium and arsenic are also evaluated based on their contents). That is, if the above evaluation method is used for the solidified soil treated with cement, solidified material, or the like, the solidified soil is crushed and sieved. Despite the fact that the solidification treatment was performed to suppress the elution of contaminants, the test using a sample obtained by pulverizing the solidified body was performed to quantitatively evaluate the elution of contaminants from solidified soil. It is hard to say that it is appropriate.
[0004]
Contaminants contained in the solid are eluted to the outside over time, and the elution rate depends on the diffusion coefficient in the solid. Therefore, in order to quantitatively predict the effect due to the elution of the contaminant, the maximum amount of the contaminant eluted from the solidified body and the diffusion coefficient of the contaminant in the solidified body are required. Of course, in the method based on the Ministry of the Environment Notification No. 46 described above, only the value of the concentration of the contaminant eluted in the solvent after a predetermined time is obtained, and the quantitative estimation of the elution of the contaminant from the solidified body is performed. It is impossible to do. The Netherlands uses a diffusion test-based assessment of the safety of treating contaminants in concrete. The European Commission is also considering, with reference to the Netherlands, formulating standards for the safety assessment of concrete bodies containing contaminants. The following is a description of the European Commission proposal (revised on December 4, 1996, hereinafter referred to as the CEN proposal).
[0005]
According to the CEN proposal, an Availability test for determining the maximum elution amount of contaminants and a diffusion test for determining a diffusion coefficient in a solidified body are both performed. FIG. 1 shows an outline of the availability test and the diffusion test. In the Availability test, a sample pulverized to a particle size of 125 μm or less is immersed in two kinds of solvents of pH 7 and pH 4, and the amount of contaminants eluted in the solvent is measured, whereby the amount of contaminants per dry mass of the solidified body to be tested is measured. U _bes (G / dry-g). The dry density of the solidified product is ρ (dry-g / cm ³ ), The amount of contaminants that can be eluted per volume of the solidified product is ρ · U _bes (G / cm ³ )It can be expressed as. In the diffusion test, the concentration of contaminants in the solvent after the solid sample is immersed for a predetermined time is measured. The diffusion test is performed in five time intervals. FIG. 2 shows the length of each time section. At the end of each time interval, the concentration of contaminants eluted in the solvent is measured and the solvent is updated.
[0006]
Then, a diffusion coefficient is calculated for each time section. Contaminant concentration C in the solvent at the end of time interval i _{f, i} (G / cm ³ ) Is substituted into the following equation (Equation 4) to obtain the contaminant elution mass E per surface area of the specimen. _{dif, i} (G / cm ² ). The surface area of the specimen: A (cm ² ), Solvent volume; V _f (Cm ³ ).
[0007]
(Equation 4)

[0008]
Subsequently, the diffusion coefficient D for each time section i _{e, i} (Cm ² / S) is calculated based on the following equation (Equation 5). Note that the accumulated time at the end of time section i; t _i (S).
[0009]
(Equation 5)

[0010]
From the above, the contaminant elutable amount ρ · U per volume _bes And diffusion coefficient D _{e, i} And the amount of contaminants eluted from the solidified body over time can be predicted. According to the CEN proposal, the amount of contaminants M eluted from the solidified body after time t has elapsed _dif (T) and (g) are obtained by the following equation (Equation 6). However, the effective diffusion coefficient D _e (Cm ² / S) is the diffusion coefficient D calculated for each time interval i. _{e, i} Is the geometric mean of
[0011]
(Equation 6)

[0012]
[Problems to be solved by the invention]
However, when the elution of contaminants is predicted using the value of the diffusion coefficient in the solidified body calculated based on the CEN plan and compared with the actual result, it can be seen that there is an error. After an examination, it was considered that the calculated value of the diffusion coefficient according to the CEN proposal was smaller than the true value, and it was clarified that this was a major factor causing an error.
[0013]
In view of the above problems, the present invention seeks to analyze the cause of the calculated value of the diffusion coefficient being smaller than the true value and provide a more accurate diffusion coefficient calculation method.
[0014]
[Means for Solving the Problems]
In order to solve the above-mentioned problems, in the present invention, the solid specimen and the diffusion of contaminants in the specimen are set under the following conditions;
(A) Assuming that the shape of the specimen is a cylinder having the specified dimensions
(B) Predetermined initial contaminant concentration distribution for contaminants in the specimen
(C) The contaminant concentration near the boundary between the specimen and the solvent is always assumed to be constant
(D) The time-dependent change in the contaminant concentration distribution in the specimen follows Fick's law
The relationship between the elapsed time and the amount of change in the contaminant concentration in the solvent and the diffusion coefficient of the contaminant in the solidified body is derived by modeling with, and the elapsed time and the amount of change in the contaminant concentration in the diffusion test are derived from the relational expression. The diffusion coefficient was calculated by substituting the measured values of the above.
[0015]
As will be described in detail later, the formula for calculating the diffusion coefficient (Equation 5) in the CEN proposal is derived on the assumption that the specimen has a thin plate shape as shown in FIG. However, the specimen used for the diffusion test is usually a rectangular parallelepiped or a column. In other words, substituting the value of the result of the diffusion test for a specimen having a non-thin plate shape into the calculation formula derived from the thin plate model has been one of the causes of errors in the calculated value of the diffusion coefficient. The present invention has been made focusing on such unreasonableness for the first time, and by deriving a calculation formula by modeling a specimen as a cylinder having a predetermined dimension, it is possible to accurately calculate the diffusion coefficient. It is intended. In addition, when performing a mixing test or the like on cement or cement-based solidified material, a columnar specimen is often created. Therefore, by providing a calculation formula for the diffusion coefficient based on the cylindrical model, it is possible to obtain the effect that the arrangement of the specimen can be made easier and the cost of the diffusion test can be reduced.
[0016]
Further, the formula for calculating the diffusion coefficient (Equation 5) in the CEN plan is derived on the assumption that the contaminant concentration on the surface of the specimen is always 0. However, the contaminant concentration on the specimen surface can always be regarded as zero only when the volume of the solvent and the diffusion coefficient in the solvent are both infinite. That is, the amount of contaminants eluted in the solvent during the diffusion test is smaller than the amount of contaminants eluted under ideal conditions where the concentration of contaminants on the surface of the specimen can always be regarded as zero. Therefore, the calculated value of the diffusion coefficient calculated by substituting the measured value obtained as a result of the diffusion test into the calculation formula (Equation 5) is always smaller than the true value. Therefore, in the present invention, the diffusion coefficient D obtained by using the relational expression _{m, j} The diffusion coefficient D _{m, j} And diffusion coefficient D of contaminants in solvent _f And the correction factor d (D D) for the finite diffusion volume of the solvent and contaminants in the solvent in the diffusion test, determined using _{m, j} , D _f ) To make the diffusion coefficient D closer to the true value. _{m, j + 1} Is calculated.
[0017]
BEST MODE FOR CARRYING OUT THE INVENTION
An embodiment of the present invention will be described with reference to the drawings. First, the process of deriving the diffusion coefficient calculation formula (Equation 5) in the CEN plan will be described. Elution of contaminants from the specimen of solidified solid obtained by insolubilizing contaminants is considered to be due to the difference between the concentration of contaminants in the specimen and the concentration of contaminants in the solvent. It is assumed that the amount of change in concentration is proportional to the concentration gradient, that is, the change over time of the contaminant concentration distribution in the specimen follows Fick's law. Further, with respect to the contaminant in the specimen, the contaminant concentration C at time t = 0. ₀ Are uniformly distributed in the specimen. Then, assuming that the concentration of the contaminant in the vicinity of the boundary between the specimen and the solvent is always 0, a boundary value problem concerning the elution of the contaminant in the specimen to the outside is considered.
[0018]
Here, it should be noted that the derivation of the diffusion coefficient calculation formula in the CEN plan actually starts from a thin plate model as shown in FIG. If it is assumed that the solidified specimen is in the form of a thin plate, most of the contaminants contained in the thin plate pass through the upper surface or the lower surface. That is, since the amount of elution from the upper surface or the lower surface is superior to the amount of elution from the side surface, it is possible to approximate the temporal change of the contaminant concentration distribution in the thin plate as a one-dimensional problem in the thickness direction. If the contaminant concentration on the upper surface or the lower surface is always 0, the contaminant concentration C in the thin plate _S (Z, t) is given by the following equation (Equation 7).
[0019]
(Equation 7)

[0020]
Note that the thickness of the thin plate is 2H, and the depth from the upper and lower surfaces of the thin plate is z. Also, λ _n = (2n + 1) π / 2, T = Dt / H ² , D is the diffusion coefficient, C _S (Z, 0) = C ₀ It is. That is, as shown in FIG. _S (Z, t) is distributed symmetrically in the center plane of the thickness. When the above equation (Equation 7) is averaged by the spatial parameter z, the following equation (Equation 8) is obtained.
[0021]
(Equation 8)

[0022]
Thus, an average elution degree F (t) defined by the following equation (Equation 9) is introduced.
[0023]
(Equation 9)

[0024]
The average dissolution degree F (t) represents the percentage of the initial contaminants initially contained in the specimen eluted into the solvent when the specimen was immersed in the solvent and the time t had elapsed. , F (0) = 0 and F (∞) = 1. C _dif (T) represents the amount of contaminants per sample volume eluted in the solvent up to time t, _dif (T) = C ₀ F (t). FIG. 4 shows the correlation between the average dissolution degree F (t) and the time coefficient T. Assuming that the thickness 2H of the thin plate and the diffusion coefficient D are unchanged, the initial average elution degree F (t) is almost proportional to Δt. Equation 10) can be approximated.
[0025]
(Equation 10)

[0026]
When the above equation (Equation 10) is transformed into an equation for the diffusion coefficient D,
[0027]
[Equation 11]

[0028]
It becomes. From (Equation 11), C _dif (T) can also be considered proportional to Δt. C _dif Within the range where the proportional relationship between (t) and Δt is established, C _dif The following equation (Equation 12) holds for the increment of (t). Note that time t _i-1 To time t _i Elution mass per specimen volume up to _{dif, i} It is.
[0029]
(Equation 12)

[0030]
By substituting Equation (Equation 12) into Equation (Equation 11), the following Equation (Equation 13) is obtained.
[0031]
(Equation 13)

[0032]
Time t _i-1 To time t _i Eluted mass of contaminant per specimen surface area up to _{dif, i} And the above ΔC _{dif, i} Between E _{dif, i} = ΔC _{dif, i} ・ V _S / A holds. However, the specimen volume is V _S , And the surface area of the specimen is denoted by A. As described above, it is assumed that the specimen is in the form of a thin plate, and the elution of contaminants occurs exclusively through the upper and lower surfaces of the thin plate. _S = B · L · 2H, A = 2B · L = V _S / H (that is, the side of the thin plate is ignored). From the above, the equation for the diffusion coefficient D,
[0033]
[Equation 14]

[0034]
Can be obtained. Initial contaminant concentration C ₀ Is determined by the amount of contaminant elutable ρ · U per volume of the solidified body obtained as a result of the Availability test. _bes Then, Expression (Equation 14) matches Expression (Equation 5).
[0035]
In the diffusion test, the contaminant concentration in the solvent is measured. Since the amount of contaminants eluted from the specimen in each time interval i is equal to the amount of contaminants present in the solvent,
[0036]
[Equation 15]

[0037]
Holds. Equation (Equation 15) matches Equation (Equation 4).
[0038]
Incidentally, from the equation (Equation 10), the contaminant amount C per sample volume eluted by the time t is obtained. _dif (T) can be expressed as the following equation (Equation 16).
[0039]
(Equation 16)

[0040]
Therefore, the amount of contaminants eluted from the specimen by time t can be obtained using the following equation (Equation 17).
[0041]
[Equation 17]

[0042]
C ₀ Is ρ · U _bes And D is D _e Then, Equation (Equation 17) matches Equation (Equation 6).
[0043]
In this way, the formula for calculating the diffusion coefficient (Equation 5) in the CEN plan is derived. However, it is considered that the calculated value of the diffusion coefficient obtained by using the calculation formula tends to be slightly smaller than the true value of the diffusion coefficient. Hereinafter, a method of calculating the diffusion coefficient more accurately than the CEN plan will be described.
[0044]
One of the problems of the method of the CEN proposal lies in the model of the specimen assumed to derive the equation (5) for calculating the diffusion coefficient. That is, the point that the specimen is formed in a thin plate shape and the point that the elution of contaminants from the side surface of the thin plate is neglected are incompatible with the specimen actually used in the diffusion test. Therefore, we try to derive the diffusion coefficient calculation formula with the model of the specimen being more realistic.
[0045]
It is assumed that the shape of the solid specimen is a cylinder having a bottom radius R and a height 2H as shown in FIG. For the contaminants in the specimen, a predetermined initial contaminant concentration distribution, in other words, an initial distribution of the amount of the contaminants per volume of the specimen is set in advance. For simplicity of discussion, here, at time t = 0, the contaminant concentration C ₀ Are uniformly distributed in the specimen. The concentration of contaminants in the vicinity of the boundary between the specimen and the solvent is considered to be always constant, and the time-dependent change in the concentration distribution of the contaminants in the specimen conforms to Fick's law. Considering the boundary value problem concerning the elution of water, a relational expression between the elapsed time in the diffusion coefficient, the amount of change in the concentration of the contaminant in the solvent and the diffusion coefficient of the contaminant in the solidified body is derived.
[0046]
In particular, assuming that the contaminant concentration near the surface of the specimen is always 0 and that the contaminant elutes only from the side of the cylinder and does not elute from the upper and lower bottom surfaces, the change over time in the concentration distribution of the contaminant in the specimen This can be approximated as a radially axisymmetric problem, and the average elution degree F (t) for the cylinder model can be expressed by the following equation (Equation 18).
[0047]
(Equation 18)

[0048]
Note that β _n Is the Bessel function J ₀ (X) = 0 and the time coefficient T _R = D _p ・ T / R ² , D _p Is the diffusion coefficient. FIG. 6 is a graph plotting the correlation between F (t) and Δt based on the above equation (Equation 18). In FIG. 6, R, D _p Are displayed, the correlation between F (t) and Δt when Δ is changed. According to FIG. 6, if the magnitude of F (t) is up to about 0.3, it is likely that F (t) can be linearly approximated using the proportional relationship between F (t) and Δt. R, D _p , F (t) is replaced by T _R Can be uniquely determined by the following equation. Therefore, for a proportional coefficient k that linearly approximates the relationship between F (t) to Δt, R, D _p And a similarity rule shown in the following equation (Equation 19) can be assumed.
[0049]
[Equation 19]

[0050]
In the above equation (Equation 19), k ₀ Is a predetermined reference specimen radius R ₀ And reference diffusion coefficient D ₀ Is a proportional coefficient that linearly approximates the relationship between F (t) and Δt. For example, the reference specimen radius R ₀ = 1, reference diffusion coefficient D ₀ = 1, k ₀ Is approximately 2.1. However, here, the initial tangent gradient of F (t) with respect to Δt is represented by a proportional coefficient k, k ₀ Is treated as. The above equation (Equation 19) is calculated using the diffusion coefficient D _p Transforming into the formula for,
[0051]
(Equation 20)

[0052]
It becomes. That is, D _p = N ₁ ・ K ² ・ R ² The diffusion coefficient D in the form _p Can be represented. In equation (20), R ₀ = 1, D ₀ = 1 and k ₀ = 2.1, the constant N ₁ ≒ 0.227 is obtained.
[0053]
The average dissolution F (t) can be calculated by substituting the measured value obtained by the diffusion test into the following equation (Equation 21). The average dissolution degree at the end of time section i relating to the diffusion test; F (t _i ), Integrated time at the end of time section i; t _i Contaminant concentration in the solvent at the end of time interval i; C _{f, i} , Initial contaminant concentration in the specimen; C ₀ , Solvent volume; V _f , Specimen volume; V _S It is.
[0054]
(Equation 21)

[0055]
Therefore, the value of the proportionality coefficient k can be determined from the result of the diffusion test (for example, k = F (t) / Δt), so that the diffusion coefficient D of the specimen is obtained. _p Can be calculated by the equation (Equation 20).
[0056]
However, the equation (Equation 20) is derived on the assumption that the contaminant elutes only from the side surfaces of the cylinder and does not elute from the upper and lower bottom surfaces. In order to hold such an assumption, the upper and lower bottom surfaces of the specimen must be covered, or the height 2H of the specimen must be significantly larger than the bottom radius R. In an actual diffusion test, elution of contaminants via the upper and lower bottom surfaces of the test specimen may occur, and therefore, Expression (20) is not always optimal as a calculation expression of the diffusion coefficient. For example, the specimen used when performing a compounding test or the like on cement or cement-based solidified material has a height of about twice the diameter, and it is not appropriate to ignore elution of contaminants from the bottom surface. . Equations (20) and (21) are equivalent to the elapsed time t _i And the amount of change C in the concentration of contaminants in the solvent _{f, i} And diffusion coefficient D of contaminants in solidified material _p Is a temporary relational expression.
[0057]
Therefore, the diffusion coefficient D is multiplied by the correction factor. _p Is considered, and the diffusion coefficient taking into account the elution of contaminants from the bottom surface is considered. It is believed that the correlation between the amount of contaminant elution and the amount of contaminant elution via the side surface can be approximated as a function of the specimen height 2H and bottom radius R. Therefore, it is expected that a correction factor that takes into account the elution of contaminants from the bottom surface of the specimen can be approximately determined using the height 2H and the bottom radius R of the specimen. In order to determine the correction factor c (H, R), in the present embodiment, the time-dependent change of the contaminant concentration distribution in the specimen when the contaminant elutes from the side and bottom surfaces of the cylinder is analyzed using the finite element method. The results obtained will be referred to.
[0058]
Bottom radius R of specimen, Diffusion coefficient D in specimen _p Are set to predetermined values, and the contaminant concentration near the side and bottom surfaces of the specimen is always set to zero. Under such preconditions, the change over time of the contaminant concentration distribution in the specimen is analyzed using the finite element method. In addition, by integrating the contaminant concentration distribution in the specimen with a spatial parameter, the average elution degree F (t) can be calculated. FIG. 7 shows the correlation between F (t) and Δt in each case of H / R = １ ／, 1, 2, 5, analyzed using the finite element method. Note that the bottom radius R = 1 and the diffusion coefficient D _p = 1. Using this analysis result, plotting the correlation between the proportional coefficient ratio k (H / R) / k (Ｒ) and H / R is as shown in FIG. k (∞) is a proportional coefficient when the height 2H of the specimen is significantly larger than the radius R of the bottom surface, that is, when the elution through the bottom surface is set to 0. When an approximate curve is applied to plot points on the H / R-k (H / R) / k (∞) plane shown in FIG. 8, the relationship between k (H / R) / k (∞) and H / R is obtained. An approximate expression (Equation 22) is obtained.
[0059]
(Equation 22)

[0060]
N ₂ Is a constant. In this example, N ₂ ≒ 0.54. k (H / R) is a proportional coefficient taking into account the elution of contaminants through the upper and lower bottom surfaces of the cylinder, and is determined from the result of the diffusion test. Then, using the above equation (Equation 22), the base radius R and the diffusion coefficient D are equivalent to the conditions in the diffusion test. _p It is possible to obtain k (∞) under k (H / R). That is, k (∞) is obtained by dividing the equation (Equation 22) from k (H / R). As described above, k (∞) is a proportionality coefficient when the elution via the bottom surface is set to 0. Therefore, by substituting k = k (∞) into the equation (Equation 20), it is possible to suitably obtain the diffusion coefficient. D _p Can be calculated. According to equation (20), the diffusion coefficient D _p Is k ² Because the diffusion coefficient takes into account the elution of contaminants from the bottom,
[0061]
(Equation 23)

[0062]
Holds. Since the proportional coefficient k (H / R) is a linear gradient from F (t) to Δt, the diffusion coefficient D _p Is represented by the following equation (Equation 24).
[0063]
[Equation 24]

[0064]
From the above, the elapsed time t _i And the amount of change C in the concentration of contaminants in the solvent _{f, i} And diffusion coefficient D of contaminants in solidified material _p (Equation 21) and (Equation 20 or 24) could be derived. These relational expressions include the measured value t in the diffusion test. _i , C _{f, i} , The diffusion coefficient D _p Can be calculated. Incidentally, the initial pollutant concentration C ₀ Means the amount of contaminants per specimen volume eluted from the specimen in service by time t = ∞. In other words, it means the maximum mass of a contaminant that can be eluted into the solvent from the specimen per volume of the specimen. Of course, C ₀ Specimen volume V _S Is the initial contaminant amount. Initial contaminant concentration C in the specimen ₀ For example, the contaminant elutable amount ρ · U per volume of the solidified body obtained as a result of the above-described Availability test _bes Can be used. However, the initial contaminant concentration C was determined by a method other than the Availability test. ₀ Does not prevent you from deciding.
[0065]
Note that the diffusion test is usually performed in a plurality of time sections. When the diffusion test is performed in five time intervals, five measurements are performed, and five F (t _i ) / √t _i Is obtained. That is, five diffusion coefficients D _p Is calculated. Therefore, these can be geometrically averaged to obtain the value of the diffusion coefficient of the solidified body. At this time, F (t _i ) And Δt _i Is substantially proportional to the diffusion coefficient D for the time interval i. _p It is preferable to calculate the diffusion coefficient of the solidified body by geometrically averaging only the values. For example, F (t _i ) <Diffusion coefficient D satisfying condition <0.5 _p Only the geometric mean, and F (t _i ) <0.5 diffusion coefficient not satisfying the condition _p Is not preferably adopted. This is because the equation (Equation 20) is derived using the proportional relationship between F (t) and Δt.
[0066]
Another problem of the diffusion coefficient calculation method according to the CEN proposal is that the concentration of contaminants on the surface of the specimen is always assumed to be 0 in the process of deriving the diffusion coefficient calculation formula (Equation 5). it can. Strictly speaking, the contaminant concentration on the specimen surface can always be regarded as zero only when the volume of the solvent and the diffusion coefficient in the solvent are both infinite. However, in reality, the volume of the solvent cannot be infinite, and it is impossible to keep the concentration of the contaminant on the specimen surface always zero. When performing a diffusion test in accordance with the CEN proposal, the volume of the solvent V _f Is the volume V of the specimen _S Is set to 4 to 6 times. The diffusion coefficient of contaminants in the solvent is usually 10 ^-5 cm ² / S. That is, the amount of contaminants eluted in the solvent during the diffusion test is smaller than the amount of eluted contaminants under ideal conditions where the concentration of contaminants on the surface of the specimen can always be regarded as zero. Therefore, the calculated value of the diffusion coefficient calculated by substituting the measured value obtained as a result of the diffusion test into the calculation formula (Equation 5) is always smaller than the true value. This means that the diffusion coefficient D calculated by the calculation method according to the present invention is _p The same can be said for. Because D _p This is because the contaminant concentration near the boundary between the specimen and the solvent is always assumed to be 0 in the process of deriving the calculation formula (Equation 20 or Eq. 24).
[0067]
Therefore, it is considered that the diffusion coefficient is corrected by multiplying the correction factor to make the diffusion value closer to the true value. Hereinafter, the diffusion coefficient is D _{m, j} Notation and D _{m, j} Is multiplied by a correction factor once and the corrected value is D _{m, j + 1} I will keep it. However, the diffusion coefficient directly calculated from the calculation formula (Equation 24) is D _{m, 1} (That is, the subscript j = 1). Solvent volume V _f , Specimen volume V _S Is set to a predetermined value, the calculated value D of the diffusion coefficient _{m, j} The correction factor for correcting _{m, j} And the diffusion coefficient D of the contaminant in the solvent _f It is expected that it can be approximately determined using The correction factor d (D _{m, j} , D _f In this embodiment, the result of analyzing the change over time in the concentration distribution of contaminants in the specimen using the finite element method is used to determine).
[0068]
Here, the specimen of the solidified body was a cylinder having a diameter of 5 cm and a height of 10 cm, and the solvent volume V _f Is the specimen volume V _S Set to 5 times. Under these preconditions, the diffusion coefficient D of the contaminant in the solvent is _f = 10 ^-5 cm ² / S using the finite element method to analyze the change over time in the concentration distribution of contaminants in the solvent and the specimen to calculate the average dissolution G taking into account the physical constraints on the solvent. . On the other hand, the specimen was a cylinder having a diameter of 5 cm and a height of 10 cm, and the solvent volume V _f Is set to infinity, that is, when the concentration of the contaminant on the interface between the specimen and the solvent is always set to 0, the average elution degree F can be obtained by using the equation (Equation 18). Then, a ratio G / F = α of the average elution degree G in consideration of the case where the contaminant diffuses into the finite volume of the solvent with respect to the average elution degree F when the contaminant diffuses into the infinite volume of the solvent is set. Calculate α by changing the diffusion coefficient of contaminants in the specimen to various values, and calculate α and D _{m, j} / D _f FIG. 9 shows a plot of the correlation with. D shown in FIG. _{m, j} / D _f When an approximation curve (an approximation straight line in the example shown in FIG. 9 is applied to plot points on the α-plane, since FIG. 9 is a semilogarithmic graph), α (D _{m, j} / D _f ) Is obtained.
[0069]
(Equation 25)

[0070]
N ₃ , N ₄ Is a constant. In this example, N ₃ ≒ 0.71, N ₄ ≒ 0.12. Of course, even when the dimensions and the like of the specimen are different, the correction factor d (D _{m, j} , D _f ) Can be derived. According to equation (24), the diffusion coefficient D _{m, j} Is [F (t) / √t] ² Is proportional to Therefore, the correction factor d (D _{m, j} , D _f ) = 1 / α ² It can be. Therefore, the correction factor d (D _{m, j} , D _f ) Is multiplied once to correct the diffusion coefficient D _{m, j + 1} Can be expressed by the following equation (Equation 26).
[0071]
(Equation 26)

[0072]
Further, a plurality of corrections are performed until a predetermined condition is satisfied, and a diffusion coefficient D _{m, j} Is also preferably brought closer to the true value. Diffusion coefficient D obtained by performing n corrections _{m, 1 + n} Is represented by the following equation (Equation 27).
[0073]
[Equation 27]

[0074]
As the predetermined condition, for example, when α ≧ 1, | α (D _{m, j + 1} / D _f ) -Α (D _{m, j} / D _f ) | Is smaller than a preset value (for example, 0.001), or | D _{m, j + 1} -D _{m, j} Becomes smaller than a predetermined value.
[0075]
When the procedure of the diffusion coefficient calculation described in detail above is integrated, the flowchart shown in FIGS. 10 and 11 is obtained.
[0076]
Incidentally, the elapsed time t obtained by performing the diffusion test on the computer 1 in accordance with the flowcharts shown in FIGS. _i And the amount of change C in the concentration of contaminants in the solvent _{f, i} And a diffusion value obtained by substituting the measured values stored in the measured value storage means into a relational expression (Equation 21) or (Equation 20 or 24). Diffusion coefficient calculating means for calculating a coefficient, and a diffusion coefficient D obtained by using the relational expression. _{m, j} To the correction factor d (D _{m, j} , D _f )) And the diffusion coefficient D corrected _{m, j + 1} A program can be created that functions as a correction unit that calculates. Alternatively, it is possible to configure the diffusion coefficient calculation device 1 including the measurement value storage unit 11, the diffusion coefficient calculation unit 12, and the correction unit 13 and capable of executing the processing based on the flowcharts illustrated in FIGS. .
[0077]
The computer 1 that functions based on the program according to the present invention, or the diffusion coefficient calculation device 1 according to the present invention includes, for example, hardware resources as shown in FIG. That is, the computer or the diffusion coefficient calculation device 1 includes hardware resources such as a processor 1a, a main storage device 1b, an auxiliary storage device 1c, and an I / O interface 1d for exchanging data with the outside. Usually, a program to be executed by the processor 1a is stored in the auxiliary storage device 1c. When the program is executed, the program is read into the main storage device 1b of the auxiliary storage device 1c and decoded by the processor 1a. Then, the above-mentioned hardware resources are operated according to the program, and at least the functions as the measured value storage unit 11 and the diffusion coefficient calculation unit 12 shown in FIG.
[0078]
The measured value storage unit 11 stores an elapsed time t obtained by performing a diffusion test. _i And the change amount C of the contaminant concentration in the solvent _{f, i} Are stored. Normally, the measurement value storage unit 11 is configured using a required area of the main storage device 1b or the auxiliary storage device 1c.
[0079]
The diffusion coefficient calculation unit 12 calculates a diffusion coefficient obtained by substituting the measured value stored in the measured value storage unit 11 into the relational expression (Equation 21) or (Equation 20 or 24). The diffusion coefficient calculation unit 12 is configured mainly by software, for example.
[0080]
Further, it is desirable that the computer or the diffusion coefficient calculation device 1 can also function as the correction unit 13. The correction unit 13 calculates the diffusion coefficient D calculated by the diffusion coefficient calculation unit 12. _{m, j} To the correction factor d (D _{m, j} , D _f ) And corrected diffusion coefficient D _{m, j + 1} Is calculated. The correction unit 13 is also configured mainly by software, for example.
[0081]
The operation of the computer or the diffusion coefficient calculation device 1 will be described with reference to the flowcharts of FIGS. First, the measured value t obtained as a result of the diffusion test _i , C _{f, i} Is stored (step S1). i is a subscript indicating a time section, and i = (1, 2,..., 5) when the diffusion test is performed in the same manner as the CEN plan. Initially, variables j = 1, α ₁ = 1 (step S2). Note that α _j = Α (D _{m, j} / D _f ). Then, for each time section i, the measured values t are expressed by the relational expressions (Equation 21) and (Equation 26). _i , C _{f, i} Is the diffusion coefficient D obtained by substituting _p Is calculated (steps S3, S4, S5, S6). However, the diffusion coefficient D calculated here _p Is corrected by the correction factor c and the correction factor d.
[0082]
Diffusion coefficient D for all time intervals i _p Is calculated, then F (t _i ) / Α _j D for time interval i satisfying <0.5 _p Are selected and geometrically averaged (step S7). To obtain a more accurate diffusion coefficient calculation value, F (t _i ) And Δt _i Is a measured value t within a period in which _i , C _{f, i} This is because it is preferable to calculate the diffusion coefficient by using. Α controlling correction factor d _j Is to correct the average elution degree F in view of the fact that the contaminant concentration near the boundary between the test sample and the solvent becomes larger than 0, and usually, α _j <1 (therefore, F (t _i ) <F (t _i ) / Α _j ). The value of the result of the geometric mean is expressed as D _{m, j} far.
[0083]
D obtained _{m, j} , And based on the equation (Equation 25), α _{j + 1} = Α (D _{m, j + 1} / D _f ) Is calculated (step S8). When a predetermined condition is satisfied, for example, α _{j + 1} When ≧ 1 (step S9), | α _{j + 1} −α _j Is smaller than a preset value (for example, 0.001) (step S10), D _{m, j} Is set as the calculated value of the diffusion coefficient of the contaminant in the solidified body (step S12), and the process ends. If the predetermined condition is not satisfied, the flow returns to step S3, and D _{m, j + 1} Is calculated (step S11).
[0084]
Hereinafter, as an example, the diffusion coefficient calculation method in the CEN plan and the diffusion coefficient calculation method according to the present invention will be compared based on the results of a diffusion test on a simulated contaminated soil solidified with a cement-based solidification material. The contaminant is total chromium and two sample Nos. 1, No. 2 was prepared. FIG. 14 shows the results of the Availability test on these samples. Ρ · U obtained as a result of Availability test _bes Is the initial contaminant concentration C in the specimen. ₀ And
[0085]
Specimens used for the diffusion test were sample Nos. 1, No. Each of them is a column having a diameter of 5 cm and a height of 10 cm. Measured value C of contaminant concentration in solvent obtained as a result of diffusion test _{f, i} And the C _{f, i} And integration time t _i Diffusion coefficient D calculated using _e (According to the CEN proposal), D _m (According to the invention) is shown in FIG. The column of “average” in FIG. 15 is a value obtained by geometrically averaging the diffusion coefficients calculated for each time section. Sample No. 1, No. 2 Both have a diffusion coefficient D according to the CEN proposal method. _e Is the diffusion coefficient D according to the present invention. _m Less than.
[0086]
Further, the diffusion coefficient D calculated using the method of the present invention is _{m, j} To the correction factor d (D _{m, j} , D _f )) And the diffusion coefficient D corrected _{m, j + 1} Is shown in FIG. In particular, the sample No. As a result of performing the correction four times, the diffusion coefficient of 1 is about 26% larger than the value before correction, and about 60% larger than the value of the diffusion coefficient according to the CEN proposal method.
[0087]
According to the present embodiment, a sample of a solidified body obtained by insolubilizing contaminants is immersed in a solvent, and a required time is passed. Test,
Conditions for the specimen and the diffusion of contaminants in the specimen as follows:
(A) Assuming that the shape of the specimen is a cylinder having the specified dimensions
(B) Predetermined initial contaminant concentration distribution for contaminants in the specimen
(C) The contaminant concentration near the boundary between the specimen and the solvent is assumed to be always constant. (D) The temporal change in the contaminant concentration distribution in the specimen follows Fick's law.
The relationship between the elapsed time and the amount of change in the contaminant concentration in the solvent and the diffusion coefficient of the contaminant in the solidified body, which is derived by modeling with The diffusion coefficient is calculated by substituting the respective measured values of. Therefore, a calculation formula suitable for the actual shape of the specimen is derived and used, and it is possible to obtain a value of the diffusion coefficient closer to the true value.
[0088]
The average elution degree, which is the ratio of the amount of contaminants eluted in the solvent when the time t has elapsed after immersing the specimen in the solvent to the amount of the initial contaminants initially contained in the specimen, is Δt. By assuming proportionality and using the proportionality coefficient to derive the relational expression, an approximate expression for the diffusion coefficient can be simply described.
[0089]
By neglecting the elution of contaminants from the bottom surface of the specimen and deriving a function formula F (t) of the average elution degree, a proportional coefficient k of F (t) to Δt is determined based on the function formula. The obtained elapsed time and the provisional relation between the amount of change in the concentration of the contaminant in the solvent and the diffusion coefficient of the contaminant in the solidified body are determined using the height 2H and the bottom radius R of the specimen. By multiplying by the correction factor c (H, R) taking into account the elution of contaminants from the bottom surface of the specimen and deriving the above-mentioned relational expression, the formula for calculating the diffusion coefficient is derived as a radial one-dimensional model. be able to.
[0090]
Further, the diffusion coefficient D obtained using the above relational expression _{m, j} The diffusion coefficient D _{m, j} And diffusion coefficient D of contaminants in solvent _f And the correction factor d (D D) for the finite diffusion volume of the solvent and contaminants in the solvent in the diffusion test, determined using _{m, j} , D _f ) And corrected diffusion coefficient D _{m, j + 1} By calculating, it is possible to obtain a value of the diffusion coefficient closer to the true value in consideration of the physical constraint that the volume of the solvent and the diffusion coefficient in the solvent are finite.
[0091]
Then, the computer 1 stores the measured values of the elapsed time and the amount of change in the concentration of the contaminant in the solvent, which are obtained by performing the diffusion test, on the measured value storing means, and on the measured value storing means. It is possible to create a program that functions as a diffusion coefficient calculation unit that calculates a diffusion coefficient obtained by substituting the stored measurement value into the relational expression. The program and the computer 1 are used to obtain a diffusion coefficient D obtained by using the above relational expression. _{m, j} The diffusion coefficient D _{m, j} And diffusion coefficient D of contaminants in solvent _f And the correction factor d (D D) for the finite diffusion volume of the solvent and contaminants in the solvent in the diffusion test, determined using _{m, j} , D _f ) And corrected diffusion coefficient D _{m, j + 1} May be configured to function as a correction unit that calculates.
[0092]
In addition, according to the above equation (Equation 18), the amount of contaminants C per volume of the specimen eluted with time from the cylindrical specimen is expressed as C _dif (T) can be expressed as in the following equation (Equation 28).
[0093]
[Equation 28]

[0094]
The time coefficient T in the above equation (Equation 28) _R Is determined by the diffusion coefficient in the specimen. If the diffusion coefficient in the specimen is D, then T _R = D · t / R ² It becomes. Therefore, by obtaining the calculated value of the diffusion coefficient, it becomes possible to easily predict the amount of contaminants eluted from the specimen. Moreover, the prediction formula (Equation 6) in the CEN proposal is M _dif (∞) = ∞, but in the above equation (Equation 28), C _dif (∞) = C ₀ It becomes. That is, it can be said that the use of the equation (Equation 28) can more correctly predict the elution amount of the contaminant than the use of the equation (Equation 6). Incidentally, when the calculated value obtained by using the diffusion coefficient calculating method according to the present invention and multiplied by the correction factor c is applied to the equation (Equation 28), the correction factor c is divided from the calculated value, and then the equation ( Expression 28) may be substituted.
[0095]
Note that the present invention is not limited to the embodiment described in detail above. The specific configuration of each unit is not limited to the above embodiment, and various modifications can be made without departing from the spirit of the present invention.
[0096]
【The invention's effect】
According to the present invention described in detail above, it is possible to calculate the diffusion coefficient more accurately.
[Brief description of the drawings]
BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a table outlining the availability test and the diffusion test.
FIG. 2 is a table illustrating time section breaks in a diffusion test.
FIG. 3 is a diagram showing a thin plate model in the CEN proposal method.
FIG. 4 is a diagram showing a correlation between F (t) and Δt.
FIG. 5 is a diagram showing a cylinder model according to the present invention.
FIG. 6 is a diagram showing the correlation between F (t) and Δt when the elution of contaminants from the bottom of the cylinder is ignored.
FIG. 7 is a diagram showing a correlation between F (t) and Δt when contaminant elution from the bottom of the cylinder is taken into account.
FIG. 8 is a diagram showing a correction factor c (H, R).
FIG. 9: α (D _{m, j} / D _f Figure showing
FIG. 10 is a flowchart showing a procedure of a diffusion coefficient calculation method according to the present invention.
FIG. 11 is a flowchart showing a procedure of a diffusion coefficient calculation method according to the present invention.
FIG. 12 is a diagram showing contents of hardware resources provided in a computer or a diffusion coefficient calculation device.
FIG. 13 is a functional block diagram of a computer or a diffusion coefficient calculation device.
FIG. 14 is a diagram showing the results of an Availability test according to one example of the present invention.
FIG. 15 is a diagram showing calculated values of a diffusion coefficient according to the embodiment.
FIG. 16 is a diagram showing a value of a diffusion coefficient after correction obtained by multiplying by a correction factor d.
[Explanation of symbols]
1. Computer or diffusion coefficient calculation device
11 ... Measured value storage
12 ... Diffusion coefficient calculator
13 Correction unit

Claims (10)

A method of calculating a diffusion coefficient used to analyze the amount of contaminants eluted from a solid obtained by insolubilizing contaminants,
The specimen of the solidified body is immersed in a solvent to allow a required time to elapse, and the elapsed time and a diffusion test for measuring the amount of change in the contaminant concentration in the solvent are performed.
Conditions for the specimen and the diffusion of contaminants in the specimen as follows:
(A) The shape of the specimen is assumed to be a cylinder having a prescribed dimension. (B) The prescribed initial contaminant concentration distribution is set for the contaminants in the specimen. (C) Contamination near the boundary between the specimen and the solvent. (D) The time-dependent change in the contaminant concentration distribution in the specimen is derived by modeling according to Fick's law, and the elapsed time and the amount of change in the contaminant concentration in the solvent A diffusion coefficient calculation method for calculating a diffusion coefficient by substituting each measured value of an elapsed time and a change amount of a contaminant concentration in a diffusion test into a relational expression of a diffusion coefficient of a contaminant in a solidified body. The average elution degree, which is the ratio of the amount of contaminants eluted in the solvent when the sample is immersed in the solvent and the time t has elapsed to the amount of the initial contaminants initially contained in the sample, is Δt. 2. A method according to claim 1, wherein the proportionality is assumed to be proportional, and the relational expression is derived using the proportionality coefficient. By neglecting the elution of contaminants from the bottom surface of the specimen and deriving a function formula F (t) of the average elution degree, a proportional coefficient k of F (t) to Δt is determined based on the function formula. The obtained elapsed time and the provisional relation between the amount of change in the concentration of the contaminant in the solvent and the diffusion coefficient of the contaminant in the solidified body are determined using the height 2H and the bottom radius R of the specimen. 3. The diffusion coefficient calculation method according to claim 2, wherein the relational expression is derived by multiplying a correction factor c (H, R) taking into account the elution of contaminants from the bottom surface of the specimen. 4. The method according to claim 3, wherein the relational expression is represented by the following expression (Equation 1).

Furthermore, the volume of the solvent in the diffusion test is determined by using the diffusion coefficient D _{m, j} obtained by using the above relational expression and the diffusion coefficient D _{m, j} and the diffusion coefficient D _f of the contaminant in the solvent. And a correction factor d (D _{m, j} , D _f ) relating to the fact that the diffusion coefficient of the contaminant in the solvent is finite, and calculating a corrected diffusion coefficient D _{m, j + 1.} The diffusion coefficient calculation method according to 1, 2, 3, or 4. The diffusion coefficient D _{m, 1} calculated by substituting the measured values of the elapsed time and the amount of change in the contaminant concentration into the relational expression is corrected based on the following expression (Equation 2), and the resulting value D 6. The diffusion coefficient calculation method according to claim 5 _{, wherein m, 1 + n} is a diffusion coefficient of a contaminant in the solidified body.

7. The diffusion coefficient calculation method according to claim 5, wherein the correction factor d is represented by the following equation (Equation 3).

It is used for carrying out the method of calculating a diffusion coefficient according to claim 1, 2, 3, 4, 5, 6, or 7.
Obtained by performing a diffusion test, elapsed time, measured value storage means for storing each measured value of the amount of change in the concentration of contaminants in the solvent, and,
A program that functions as a diffusion coefficient calculation unit that calculates a diffusion coefficient obtained by substituting a measured value stored in the measured value storage unit into the relational expression. Further, a computer is used to determine a diffusion coefficient D _{m, j} obtained by using the above relational expression by using the diffusion coefficient D _{m, j} and a diffusion coefficient D _f of a contaminant in a solvent. Functions as correction means for calculating a corrected diffusion coefficient D _{m, j + 1} by multiplying by a correction factor d (D _{m, j} , D _f ) relating to the finite volume of the solvent and the diffusion coefficient of contaminants in the solvent. 9. The program according to claim 8, wherein the program is executed. It is used for carrying out the diffusion coefficient calculation method according to claim 1, 2, 3, 4, 5, 6, or 7.
Obtained by performing a diffusion test, elapsed time, a measured value storage unit that stores each measured value of the amount of change in the concentration of contaminants in the solvent,
A diffusion coefficient calculating unit configured to calculate a result obtained by substituting the measured values stored in the measured value storage unit into the relational expression;

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