劉秋曉,王發(fā)展,,張 院,陳海鵬
(1.西安建筑科技大學(xué)材料與礦資學(xué)院,西安710055;2.西安建筑科技大學(xué)機(jī)電與工程學(xué)院,西安710055)
Cu-Cr二元合金宏觀偏析與溫度場(chǎng)的數(shù)值模擬
劉秋曉1,王發(fā)展1,2,張 院2,陳海鵬2
(1.西安建筑科技大學(xué)材料與礦資學(xué)院,西安710055;2.西安建筑科技大學(xué)機(jī)電與工程學(xué)院,西安710055)
Cu-Cr材料是兩相復(fù)合材料,鉻粒子通常以嵌入在銅基體中的形式存在,凝固過(guò)程中存在著嚴(yán)重的偏析現(xiàn)象,進(jìn)而對(duì)凝固溫度場(chǎng)的分布產(chǎn)生影響.為分析Cr相偏析對(duì)溫度場(chǎng)分布的影響,基于Eulerian?Eulerian方法,建立了三維凝固偏析模型,利用Fluent模擬計(jì)算,偏析模型采用濃度梯度▽SCr和分布面積梯度▽SCr來(lái)表示,得到了Cu-6.5%Cr的凝固偏析分布和溫度場(chǎng)分布.結(jié)果表明:由于Cr的熔點(diǎn)比較高,當(dāng)t<ta時(shí),Cr先發(fā)生側(cè)向凝固,形成糊狀區(qū);同時(shí),Cr相的密度比Cu小,內(nèi)部Cr相會(huì)發(fā)生上浮,向合金頂部移動(dòng);當(dāng)t>ta時(shí),頂部的Cr會(huì)大量聚集,形成頂部偏析,兩側(cè)Cr相“困”于糊狀區(qū),形成壁面偏析;隨著凝固界面橫向推移,壁面偏析對(duì)壁面溫度場(chǎng)的分布產(chǎn)生影響,Cu的傳熱系數(shù)是Cr的3.85倍,隨著Cr相偏析度的增加,降低了基體壁面的傳熱效率,導(dǎo)致溫度梯度變大.研究工作將模擬結(jié)果與實(shí)驗(yàn)結(jié)果進(jìn)行了對(duì)比,證明了模型的準(zhǔn)確性.
凝固偏析;濃度梯度;分布面積梯度;溫度場(chǎng);傳熱效率
Cu-Cr合金是以純Cu為基體,加入Cr元素所構(gòu)成的一種觸頭材料,是一種具有良好的導(dǎo)電、導(dǎo)熱性能的合金,主要用于制作真空觸頭、接觸導(dǎo)線、電阻焊電極等零部件[1-4].當(dāng)Cr相彌散到基體中時(shí),合金的抗拉強(qiáng)度提高,塑性降低,但由于該合金凝固時(shí)存在液-固相變過(guò)程,液相Cr易于發(fā)生側(cè)相凝固及上浮,使得Cr在基體中的分布易產(chǎn)生偏析現(xiàn)象,嚴(yán)重限制了合金的使用,其制備和研究過(guò)程十分困難[5-7].
在Cu-Cr精煉金屬鑄造過(guò)程中,Cu和Cr的晶粒尺寸和比例各有不同,使得2種金屬無(wú)法熔融均勻,造成嚴(yán)重的偏析現(xiàn)象,從而最終影響產(chǎn)品質(zhì)量[8].因此,研究者以Cu-Cr合金為研究對(duì)象,建立了固、液兩相三維流動(dòng)凝固模型,對(duì)合金體系中的凝固過(guò)程進(jìn)行了數(shù)值模擬,揭示了Cu-Cr合金中的宏觀偏析與溫度梯度變化的密切聯(lián)系.實(shí)際上,XIA等[9]利用歐拉數(shù)值方法,建模分析了鑄件的應(yīng)力分布.岳強(qiáng)等[10]建立了數(shù)學(xué)模型,分析了鑄件流場(chǎng)、濃度場(chǎng)的變化情況.但是,到目前為止,對(duì)凝固過(guò)程中的溫度場(chǎng)與宏觀偏析聯(lián)系變化的研究比較少.
本文基于Eulerian?Eulerian方法,建立了三維凝固偏析模型,利用 fluent模擬計(jì)算,得到了Cu-6.5%Cr的凝固偏析分布和溫度場(chǎng)分布.
隨著凝固模擬技術(shù)的廣泛應(yīng)用,液、固兩相流動(dòng)對(duì)鑄件凝固的影響逐漸為人們所理解[11].Cu-Cr凝固偏析所應(yīng)用的數(shù)學(xué)模型引入如下假設(shè):1)凝固體系中銅、鉻的物理化學(xué)參數(shù)不同.2)銅和鉻在其液、固不同狀態(tài)的擴(kuò)散系數(shù)不同.3)糊狀區(qū)的流動(dòng)擴(kuò)散采用Blake?Kozeny假設(shè).4)Cr的熔點(diǎn)較高,不考慮汽相[12-15].
動(dòng)量守恒方程(Navier?Stokes方程)為
其中
κ-ε雙方程紊流模型:
雷諾在紊流方程中引入均化模型對(duì)N-S方程和連續(xù)性方程進(jìn)行一次平均,得到平均運(yùn)動(dòng)的雷諾方程為
由能量守恒原理可得傅里葉導(dǎo)熱微分方程:
連續(xù)方程:
其中
2.1 網(wǎng)格劃分及邊界條件
網(wǎng)格劃分:以鑄造模型為基礎(chǔ)建立三維基準(zhǔn)模型,將其導(dǎo)入Gambit軟件進(jìn)行網(wǎng)格劃分,劃分成三維網(wǎng)格模型,如圖1所示.
邊界條件設(shè)定:上底面為壓力出口;下底面為普通底面,h=30 W/(m2·K);側(cè)面為冷卻壁面,h=400 W/(m2·K),TEXT=298 K.
圖1 凝固模型Fig.1 Solidification model
2.2 模型計(jì)算
將劃分好的三維網(wǎng)格模型導(dǎo)入到Fluent中進(jìn)行數(shù)值模擬計(jì)算,使用SIMPLE算法對(duì)質(zhì)量、動(dòng)量、溶質(zhì)、熱焓和多相相變進(jìn)行耦合,時(shí)間步長(zhǎng)為0.01 s,為了使殘差更好地收斂,調(diào)節(jié)適合的松弛因子,每步最大迭代次數(shù)為40[16-17].導(dǎo)入到Fluent中的合金凝固模型所用的參數(shù)見表1.
表1 材料的物性參數(shù)Table 1 Physical property parameters of the material
3.1 偏析模型
Cr的熔點(diǎn)較高、密度較小,在凝固過(guò)程中主要以液相→固相的相變過(guò)程,Cr在基體內(nèi)以液相形式發(fā)生側(cè)向凝固和上浮.Cu-Cr合金凝固過(guò)程中,單位時(shí)間內(nèi)Cr在基體中的偏析模式用濃度梯度(▽SCr)和分布面積梯度(▽SCr)來(lái)表示.設(shè)Cr在合金基體中的濃度函數(shù)為c(a,t),分布函數(shù)為f(a,t),其中,a表示為x、y或z方向,因此,在凝固Δt時(shí)間后有
積分得
化簡(jiǎn)得
令a-a′=η,在三維方向上有
同理可得
式(11)和(12)揭示了偏析相隨時(shí)間、位置變化的規(guī)律[18].
合金中的凝固流動(dòng)有3種因素的作用:凝固初期,基體主要是合金液相,對(duì)溶質(zhì)產(chǎn)生浮力,導(dǎo)致向上流動(dòng);重力,導(dǎo)致向下流動(dòng);溶質(zhì)的擴(kuò)散對(duì)流作用,引起Cr相等軸晶的紊流,把周圍的熔體向上推動(dòng).合金凝固初期,2個(gè)向上的力量占主導(dǎo)地位.在基體內(nèi)部,下半部的Cr相等軸晶改變方向向內(nèi)移動(dòng),然后向頂部移動(dòng).同時(shí),由于重力作用,部分溶質(zhì)向下運(yùn)動(dòng),向上和向下流動(dòng)的相互作用,使上半部整個(gè)合金基體的湍流模式是高度不穩(wěn)定和無(wú)序的,開始產(chǎn)生宏觀偏析.
在凝固后期,從兩側(cè)向內(nèi)生長(zhǎng)的柱狀尖端在基體中心處相遇.然而在機(jī)體上部,等軸晶會(huì)大量聚集,并停止向柱狀晶尖端前沿的擴(kuò)散.在柱狀晶尖端,等軸晶擴(kuò)散受阻,此處為柱狀晶和等軸晶共存區(qū)域.此外,由于熔體相互作用的復(fù)雜性,氣流也會(huì)影響最終的Cr相偏析[19].
圖2為Cu-6.5%Cr合金凝固過(guò)程中Cr在基體中的擴(kuò)散分布面積隨時(shí)間和位置的變化規(guī)律.由于Cr的熔點(diǎn)比較高,但密度比Cu的要小,因此在凝固過(guò)程中會(huì)發(fā)生嚴(yán)重的偏析現(xiàn)象[18].ta為Cr上浮到基體頂部的臨界點(diǎn).從圖2可以看出:當(dāng)t<ta時(shí),Cr相比較均勻地分散在合金基體中,Cr相熔點(diǎn)較高,先發(fā)生側(cè)向凝固,形成糊狀區(qū);同時(shí),Cr相的密度比Cu小,內(nèi)部Cr相會(huì)發(fā)生上浮,向合金頂部移動(dòng).當(dāng)t>ta時(shí),頂部的Cr會(huì)大量聚集并發(fā)生偏析,兩側(cè)Cr相“困”于糊狀區(qū),無(wú)法自由移動(dòng),并伴隨著糊狀區(qū)前沿的推移,以柱狀晶形態(tài)持續(xù)在基體內(nèi)生長(zhǎng).由于目前的模型不包括熔化和后擴(kuò)散,固體樹突的溶質(zhì)濃度沒有討論.在不解決每個(gè)枝晶和枝晶的微觀細(xì)節(jié),能提供足夠的信息來(lái)‘重建’溶質(zhì)在整個(gè)合金的分布.
圖2 Cu-Cr6.5%合金凝固偏析云圖Fig.2 Cu-Cr6.5%segregation contours of alloy solidification
圖3給出了y=1 m處液相質(zhì)量分?jǐn)?shù)變化曲線.凝固初期(150 s),壁面附近剛開始凝固,內(nèi)部全是液相,Cr相向內(nèi)部“漂移”,中部Cr相的質(zhì)量分?jǐn)?shù)相對(duì)較高.隨著凝固的進(jìn)行(450 s),在x=0.1和0.4 m附近形成少量糊狀區(qū),Cr相開始凝聚成核,以柱狀晶形態(tài)生長(zhǎng),內(nèi)部液相Cr以等軸晶形態(tài)自由移動(dòng),在3種因素的綜合作用下向基體上部偏析.750 s時(shí)凝固繼續(xù)進(jìn)行,糊狀區(qū)擴(kuò)大,兩側(cè)偏析增大,內(nèi)部Cr相由于向頂部聚集,Cr相質(zhì)量分?jǐn)?shù)降低;1 050 s凝固后期兩側(cè)液相質(zhì)量分?jǐn)?shù)較少,固相質(zhì)量分?jǐn)?shù)占比增多,頂部Cr相大量聚集,兩側(cè)Cr相柱狀晶緩慢生長(zhǎng)[20],整個(gè)合金基體趨于完全凝固.
圖3 凝固過(guò)程中液相質(zhì)量分?jǐn)?shù)曲線Fig.3 Mass fraction of liquid phase during solidification
3.2 溫度場(chǎng)
圖4為Cu-6.5%Cr合金凝固過(guò)程中溫度場(chǎng)的變化趨勢(shì).凝固初期,溫度場(chǎng)均勻分布,由于此時(shí)Cr相分布均勻,整個(gè)機(jī)體的冷卻散熱也是均勻的,但隨著凝固的進(jìn)行,兩側(cè)壁面的水冷溫度一直保持298 K,導(dǎo)致壁面的冷卻加快,而機(jī)體底部的冷卻則會(huì)逐漸減慢.同時(shí),壁面不同區(qū)域出現(xiàn)溫度梯度差異,說(shuō)明合金基體的傳熱效率發(fā)生變化.凝固晶粒的生長(zhǎng)方向直接關(guān)系到溫度梯度和溶質(zhì)濃度.合金凝固初期,由于基體壁面的冷卻速率較快,主要形成柱狀晶;隨著凝固的進(jìn)行,凝固界面橫向推移,進(jìn)而 Cr相晶粒生長(zhǎng)也遵循橫向方向[21],與圖中溫度梯度方向基本一致.由此產(chǎn)生的溫度分布如圖4所示,可以看出,熱傳遞沿橫向垂直于重力方向.
圖4 Cu-6.5%Cr合金凝固溫度場(chǎng)云圖Fig.4 Cu-6.5%Cr temperature field of alloy solidification
圖5為t=450 s時(shí)Cr的等值線云圖和溫度的等值線云圖,可以看出,CCr與T有相類似的分布情況.
圖5 微觀的凝固偏析與溫度場(chǎng)云圖Fig.5 Microscopic solidification segregation and temperature field contours
觀察圖 5的實(shí)線區(qū)域(x軸方向 0.075~0.105 m),分析該區(qū)域的CCr、T的演變過(guò)程,得到圖6的宏觀偏析和溫度梯度隨凝固時(shí)間的變化曲線.從圖6(a)和6(b)中可以看出,450 s時(shí)y=1.24 m Zone1的溫度梯度約為 260 K,而 y=1.28 m Zone2溫度梯度約為140 K,這是由于Cr相偏析阻礙傳熱的進(jìn)行,使得熱量擴(kuò)散速率降低,從而導(dǎo)致溫度梯度增大.
從圖6(a)和6(b)可以看出,隨著凝固的進(jìn)行,150 s時(shí)y=1.24 m Zone1處Cr相質(zhì)量分?jǐn)?shù)逐漸增大,偏析度增大,并在x方向0.075~0.105 m處Cr相質(zhì)量分?jǐn)?shù)達(dá)到12.2%.由于Cu的導(dǎo)熱系數(shù)是Cr的3.85倍,隨著Cr相偏析的增大,對(duì)傳熱效率阻礙作用也在增加,使得溫度梯度增大.對(duì)比圖6(a)~6(d)可以看出,450 s時(shí)Zone1的Cr相質(zhì)量分?jǐn)?shù)差達(dá)到6.3%,Zone2的Cr相質(zhì)量分?jǐn)?shù)差只有4.1%,相對(duì)應(yīng)的Zone1溫度梯度差達(dá)到180 K,而Zone2的溫度梯度最大只有80 K,說(shuō)明同一時(shí)刻不同區(qū)域Cr的偏析度越大,溫度梯度越大.因此,偏析的形成對(duì)合金基體傳熱產(chǎn)生嚴(yán)重的影響,同樣,傳熱效率的改變也證明了偏析模型的準(zhǔn)確性.
圖6 溫度梯度與質(zhì)量分?jǐn)?shù)變化曲線Fig.6 Curves of temperature gradient and mass fraction change
為了驗(yàn)證體系模型的準(zhǔn)確性,實(shí)驗(yàn)選取Cu-Cr6.5%合金,質(zhì)量3 200 kg,實(shí)驗(yàn)參數(shù)與模擬參數(shù)一致,對(duì)比試驗(yàn)結(jié)果與模擬結(jié)果,但由于實(shí)際冶煉實(shí)驗(yàn)材料中含有雜質(zhì)及局部冷卻溫度出現(xiàn)差異,導(dǎo)致實(shí)驗(yàn)與模擬結(jié)果存在一定的偏差.實(shí)驗(yàn)將模型通過(guò)鍛造開坯、熱軋、冷拉、退火等工序拉成成品線材.選取凝固 4個(gè)截面位置(0.20 m,0.70 m,1.20 m,1.70 m),與模擬結(jié)果進(jìn)行對(duì)比,得到不同位置的Cr質(zhì)量分?jǐn)?shù)曲線,模擬與實(shí)驗(yàn)得到的Cr質(zhì)量分?jǐn)?shù)變化趨勢(shì)相似,如圖7所示.
圖7 實(shí)驗(yàn)結(jié)果與模擬結(jié)果Cr的質(zhì)量分?jǐn)?shù)變化曲線Fig.7 Experimental results and simulation results of Cr mass fraction change curve
1)依據(jù)Cu-6.5%Cr合金凝固過(guò)程中Cr在基體中的擴(kuò)散分布面積隨時(shí)間和位置的變化規(guī)律,當(dāng)t<ta時(shí),Cr相比較均勻地分散在合金基體中,Cr相熔點(diǎn)較高,先發(fā)生側(cè)向凝固,形成糊狀區(qū);同時(shí)Cr相的密度比Cu小,內(nèi)部Cr相會(huì)發(fā)生上浮,向合金頂部開始移動(dòng).當(dāng)t>ta時(shí),頂部的Cr會(huì)大量聚集,形成頂部偏析,兩側(cè)Cr相“困”于糊狀區(qū),形成壁面偏析.
2)隨著凝固界面橫向推移,Cr相晶粒也遵循此方向,進(jìn)而形成壁面偏析,對(duì)壁面溫度場(chǎng)的分布產(chǎn)生影響;Cu的傳熱系數(shù)是Cr的3.85倍,隨著Cr相偏析度的增加,降低了基體壁面的傳熱效率,導(dǎo)致溫度梯度變大;而且傳熱效率的改變也印證了模型的準(zhǔn)確性.
3)本文應(yīng)用計(jì)算機(jī)模擬了銅絡(luò)合金的凝固偏析過(guò)程,得到了直觀的宏觀偏析云圖和溫度場(chǎng)云圖;從理論上分析了宏觀偏析與溫度場(chǎng)分布的聯(lián)系機(jī)理,并進(jìn)行了實(shí)驗(yàn)驗(yàn)證,模擬結(jié)果與實(shí)驗(yàn)結(jié)果一致,證明模型的準(zhǔn)確性.
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(編輯 程利冬)
Numerical simulation of macrosegregation and temperature field of Cu-Cr two binary alloy
LIU Qiuxiao1,WANG Fazhan2,ZHANG Yuan2,CHEN Haipeng2
(1.College of Materials and Mineral Resources,Xi′an University of Architecture and Technology,Xi′an 71000,China;2.College of Mechanical and Electrical Engineering,Xi′an University of Architecture and Technology,Xi′an 71000,China)
There is a serious segregation phenomenon during solidification process because Cu-Cr material is a two?phase composite material and Cr usually emended in the copper.So the solidification temperature field distribution was effected seriously by segregation phenomenon.In order to analyze the effect of Cr phase segregation on the distribution of temperature field,a 3D segregation solidification model which adopted concentration gradient and distribution area gradient▽SCrwas built based on Eulerian?Eulerian and calculated by Fluent.The Cu-6.5%Cr segregation and the temperature field distribution were gained.The result shows that Cr frozen laterally at t<tabecause of the Cr higher melting point and formed mushy zone.While the internal Cr taken place to go up and moved to the alloy top because Cr phase density is smaller than Cu;Owing to the Cr was aggregated at the top at t<ta,the top segregation formed.The Cr phase in the both side was trapped in mushy zone,so the both side formed segregation.With transverse passage of solidification interface,the wall surface temperature field distribution was affected by surface segregation.The heat transfer coefficient of Cu is 3.85 times to Cr.With the increase of Cr phase segregation degree,the matrix wall surface heat transfer efficiency was reduced,and the temperature gradient was increased.Compared to the experimental results,the analytical method was proved to be correct.
segregation;concentration gradient;distribution area gradient;temperature field;transfer efficiency
TG111.4;TG146
A
1005-0299(2016)06-0073-06
2016-03-25.
十二五國(guó)家科技支撐計(jì)劃項(xiàng)目(2011BAE31B02);陜西省自然科學(xué)基金基礎(chǔ)研究計(jì)劃重點(diǎn)項(xiàng)目.
劉秋曉(1989—),男,碩士研究生.
王發(fā)展,E?mail:wangfz10_1@163.com.
10.11951/j.issn.1005-0299.20160613