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分析熔于一冷雙組分金屬粉末層與恒熱流美國(guó),MO65211,哥倫比亞州,哥倫比亞大學(xué),機(jī)械與航天工程系 陳鐵兵,張鈺2005年2月1日收到,2005年7月18日接受,2005年10月11日在網(wǎng)上發(fā)表摘要:熔煉一冷雙組分金屬粉末層進(jìn)行調(diào)查分析. 粉末床考慮的是一個(gè)混合兩種金屬粉末的顯著不同的熔點(diǎn). 收縮所致熔化,是考慮到在物理模型. 溫度分布在液體及固體階段得到利用精確解與積分近似解, 分別. 影響孔隙率,斯特凡數(shù)目,而冷的表面溫度和固液界面也進(jìn)行了研究. 目前的工作提供了強(qiáng)有力的基礎(chǔ),對(duì)復(fù)雜的立體選擇性激光燒結(jié)( SLS )過(guò)程可以 基礎(chǔ). 2005 Elsevier公司有限公司保留所有權(quán)利.關(guān)鍵詞:熔化; 金屬; 粉層 1. 說(shuō)明直接有選擇性的激光 焊接(SLS) 是涌現(xiàn)堅(jiān)實(shí)自由格式制造技術(shù)(SFF) 通過(guò)哪3-D 分開(kāi)被修造從基于金屬的粉末床以CAD 數(shù)據(jù) 1 。被制造的層數(shù)被創(chuàng)造有選擇性地熔化粉末的薄層以掃描激光束。在焊接層數(shù)以后, 新層數(shù)粉末被放置得相似和3-D 部份可能被建立在層數(shù)由層數(shù)過(guò)程。一張混雜的金屬粉末床, 包含二型金屬粉末擁有顯著不同熔點(diǎn), 廣泛地被使用在金屬直接SLS 粉末 2,3 。高熔點(diǎn)粉末從未融解在焊接過(guò)程和戲劇中一個(gè)重大角色作為支持結(jié)構(gòu)必要避免煮沸的 現(xiàn)象, 哪些是球形的形成與近似激光束的直徑。材料分析特殊物質(zhì)物產(chǎn)和方法基于金屬的粉末系統(tǒng)為SLS 應(yīng)用由Storch 等 4 并且Tolochko 等演講。 5 。根本問(wèn)題在直接SLS 周到地被回顧由Lu 等 6 。在近的充分的密度的制造對(duì)象從金屬粉末, 直接SLS 體會(huì)通過(guò)熔化和resolidification 被被指揮的激光 導(dǎo)致射線。這是一個(gè)好起點(diǎn)調(diào)查被簡(jiǎn)化 1-D 模型得到更好的理解對(duì)熔化處理在直接SLS 在一更加復(fù)雜之前 3-D 模型被調(diào)查?;救刍湍桃褟V泛調(diào)查和詳細(xì)的評(píng)語(yǔ)可參。 7.8。熔化燒結(jié)的金屬粉末,是明顯不同于正常熔煉由于體積分?jǐn)?shù)的 天然氣在粉末明顯降低后熔化. 因此,有相當(dāng)密度的變化粉末床伴隨熔化過(guò)程.熔化和凝固一維半無(wú)限體密度變化下的邊界條件,對(duì)第一類(lèi)已由zckert和德雷克9 ,Crank10 ,并卡斯勞Legates的11和charach和zarmi 12. 命名:cp 比熱 (J kg1 K1)hsl 潛熱融化或凝固(J kg1)k 導(dǎo)熱 (W m1 K1)Kg 因次氣體導(dǎo)熱Ks 因次有效導(dǎo)熱燒粉q00 熱流 (W m2)s 固液界面位置 (m)S 量綱固液界面位置s0 位置液面 (m)S0 因次液面位置Sc 冷參數(shù)Ste 斯特凡人數(shù)t 時(shí)間(s)T 溫度 (K)w 速度液相 (m s1)W 量綱流速的液相z 坐標(biāo) (m)Z 因次坐標(biāo)希臘符號(hào)熱擴(kuò)散(m2 s1) 量綱熱擴(kuò)散 參數(shù)區(qū)分兩種情況下熔化 熱穿透深度 (m) 因次熱穿透深度 體積氣體(安) (孔隙燒粉)因次溫度密度(kg m3) 因次時(shí)間 體積低熔點(diǎn)粉末粉末混合物 標(biāo)g 燃?xì)鈏 初次l 液相m 熔點(diǎn)p 制件s 燒固(混合兩種固體粉末)應(yīng)當(dāng)指出,在熔化補(bǔ)充下發(fā)生的邊界條件指定熱流而不指明 溫度. 古德曼和Shea 13研究熔化和凝固的有限板在指定的熱流用 熱平衡積分法. 張等. 14調(diào)查熔化問(wèn)題,一冷了半地區(qū)遭受恒熱流加熱. 張等. 15解決熔煉有限板的邊界條件中的第二類(lèi)用一個(gè)半確切方法. 收縮形成的,由于密度變化,在凝固過(guò)程二維腔數(shù)值金泳三和RO 16,他的結(jié)論是密度變化發(fā)揮著越來(lái)越重要的作用比對(duì)流的凝固過(guò)程. 張和Faghri17求解了潰壩熔化問(wèn)題,在一個(gè)半無(wú)限雙組份金屬粉末床受到一 不斷加熱熱流. 影響孔隙率的固體階段,初步阻力參數(shù)和量綱導(dǎo)熱氣體的影響. 由于補(bǔ)充了金屬粉末其實(shí)是一個(gè)逐層過(guò)程中, 因此,有必要對(duì)熔融混合金屬粉末床的厚度有限,在補(bǔ)充的過(guò)程. 本文 熔化的混合粉末床有限厚度遭受不斷加熱熱流將予以追究. 2. 物理模型物理模型的熔化問(wèn)題是列圖. 1 . 粉末床有限厚度含有兩種金屬粉末的顯著不同的熔點(diǎn). 起始溫度粉末床下面,我的熔點(diǎn)低熔點(diǎn)粉末. 在時(shí)間t = 0 ,一恒熱流, q00 , 突然適用于頂面粉床 和底部表面的粉末床假定為絕熱. 由于起始溫度粉末床低于熔點(diǎn)的低熔點(diǎn)粉末 其熔化不同時(shí)開(kāi)始,加上熱供暖.只有經(jīng)過(guò)一定時(shí)間的預(yù)熱, 在它的表面溫度粉末達(dá)到熔點(diǎn)低熔點(diǎn)粉末 將熔化的開(kāi)始. 粉末與高熔點(diǎn)永遠(yuǎn)融化在整個(gè)過(guò)程中. 因此,這個(gè)問(wèn)題可以分為兩個(gè)問(wèn)題:一個(gè)是熱傳導(dǎo)預(yù)熱期間和其他被熔化. 物理模型被視為一個(gè)傳導(dǎo)控制問(wèn)題. 在自然對(duì)流效應(yīng)液體地區(qū)由于溫差不考慮由于溫度 最高的是在液體表面并隨宜 2.1.時(shí)間預(yù)熱預(yù)熱期間,純傳導(dǎo)傳熱發(fā)生在粉末混合物. 理事方程以及相應(yīng)的初始和邊界條件的預(yù)熱問(wèn)題2.2. 熔化熔化后開(kāi)始,在液相方面的理事方程:其中W是速度液體表面所誘發(fā)的收縮. 因?yàn)橐后w是不可壓縮的收縮速度西經(jīng)Eq. (5) 是受到以下邊界條件:理事方程式為固相,其相應(yīng)的邊界條件溫度在固液界面滿意能量平衡,在固液界面基于質(zhì)量守恒定律,在固液界面上的收縮速度, W時(shí),固液界面速度,副/藥物療法, 有以下關(guān)系17 : 2.3 . 非維管方程確定了以下無(wú)量綱變量:非維管方程以及相應(yīng)的初始和邊界條件的預(yù)熱問(wèn)題,成為熔化,非維方程和相應(yīng)的邊界條件3. 近似解當(dāng)頂面的混合金屬粉末床受到恒定磁場(chǎng)加熱, 熱流將穿透的頂面,并進(jìn)行向下的底部表面. 深度上的熱流滲透到瞬間的時(shí)間定義為熱穿透深度, 以后,便再?zèng)]有熱傳導(dǎo). 古德曼和Shea 13引入一個(gè)參數(shù), , 分類(lèi)兩宗熔煉有限板. 當(dāng)是大于1 頂部表面溫度達(dá)到熔點(diǎn)在較短的時(shí)間比熱穿透深度到達(dá)底部 表面上顯示一個(gè)較短的預(yù)熱時(shí)間. 如果小于1 , 表面溫度仍低于熔點(diǎn)時(shí),熱穿透深度已達(dá)底部表面. 預(yù)熱持續(xù)到頂部表面溫度達(dá)到熔點(diǎn)低熔點(diǎn)粉末. 參數(shù)也可以用表示因次參數(shù)定義的情商. ( 13 ) ,即 由此可以看到,價(jià)值是由四個(gè)基本無(wú)量綱參數(shù):斯特凡數(shù)Ste, 冷資深參數(shù)Sc,有效導(dǎo)熱系數(shù)的固相KS和體積分?jǐn)?shù)氣體ES在固相. 預(yù)熱和熔煉兩個(gè)二 1和 1將得到討論FIG.3 . 影響孔隙的液相表面溫度(專(zhuān)題= 0.02 )3.1.1 預(yù)熱3.1.1. 1當(dāng)是大于1, 熔化開(kāi)始在之前滲透深度到達(dá)底部和因此, 預(yù)熱時(shí)間, sm, 對(duì)應(yīng)的熱量滲透深度、Dm, 和溫度發(fā)行在時(shí)間sm 是 17 那里Eq 。(38) 是表面溫度在上面粉末床。熔化的解答溫度發(fā)行在液體 熔化開(kāi)始當(dāng)表面溫度粉末床到達(dá)低熔化的熔點(diǎn)點(diǎn)粉末。液體層數(shù)被形成作為結(jié)果熔化, 溫度發(fā)行不取決于 的價(jià)值。它可能是由Eqs 的一種確切的解答獲得。(19)-(21) 并且 (24) 17 , 即。那里S0 是液體表面的無(wú)維的地點(diǎn)。3.2.2. 溫度發(fā)行在固體( 1)熔化開(kāi)始在熱漲潮到達(dá)底部之前粉末床, 如此問(wèn)題熔化半無(wú)限二組分粉末床。解答為熔化一張無(wú)限粉末床包含a 二粒金屬粉末混合物由張 17 獲得了。溫度發(fā)行在液體階段由Eq 測(cè)量。(38) 。度發(fā)行在堅(jiān)實(shí)區(qū)域被獲得由 17 , 堅(jiān)實(shí)液體接口的地點(diǎn)并且被獲得由 17 , 熱量滲透深度滿足等式在那時(shí)候熱量滲透深度到達(dá)底下表面, 即, D = 1, 溫度發(fā)行在固體是圖5. 多孔性的作用在液體階段在液體表面和液體堅(jiān)實(shí)接口的地點(diǎn)(Ste = 0.02) 。熱量滲透深度到達(dá)對(duì)的時(shí)間底部, =1, 被獲得從當(dāng) =1, 問(wèn)題成為熔化在有限平板。溫度發(fā)行在固體, hs(Z, s), 并且液體堅(jiān)實(shí)接口地點(diǎn), S 可能被獲得由解決的Eqs 。(22)-(24) 使用積分式近似方法相同與 事例; 1.4.結(jié)果和討論分析解答的檢驗(yàn)是由舉辦結(jié)果與數(shù)字比較結(jié)果被獲得從陳和張 20 , 調(diào)查二維熔化和resolidification 二組分金屬粉末層數(shù)在SLS 過(guò)程中服從對(duì)移動(dòng)的激光束。為了使用二維代碼在Ref 。 20 解決熔化在粉末層數(shù)服從了對(duì)恒定的熱漲潮, 高斯激光束由恒定的熱化替換了熱漲潮在整個(gè)粉末床的上面和 激光 掃描速度調(diào)整到零數(shù)字解答。參量被使用在本論文是轉(zhuǎn)換成對(duì)應(yīng)的參量在Ref 。 20 為代碼檢驗(yàn)的目的。比較液體表面和液體的瞬間地點(diǎn)堅(jiān)實(shí)接口由分析和數(shù)字獲得解答被顯示在圖2 。它能被看見(jiàn)預(yù)熱時(shí)間由分析獲得和數(shù)字解答幾乎是相同。地點(diǎn)液體表面和液體堅(jiān)實(shí)接口被獲得分析和數(shù)字解答行動(dòng)在非常相似趨向。它采取完全地熔化整個(gè)的時(shí)間粉末層數(shù)被獲得從分析解答是大約4% 長(zhǎng)比那被獲得從數(shù)字解答。多孔性, 冷資深 的作用, 無(wú)維導(dǎo)熱性和Stefan 編號(hào)在表面液體表面的溫度、地點(diǎn), 和地點(diǎn)粉末床的固液 接口將被調(diào)查。圖3 展示怎么表面溫度被多孔性影響在液體階段為Ste = 0.02 和幾個(gè)不同的冷資深 的參量。收縮的作用由固定隔絕冷資深 參量, 固相的多孔性, 和無(wú)維的導(dǎo)熱性。它能被看見(jiàn)那表面溫度增加當(dāng)多孔性在液體階段增量。這是因?yàn)橛行У纳仙鱾鲗?dǎo)性減少隨著容量分?jǐn)?shù)的增加氣體。當(dāng)Sc = 0.1, 預(yù)熱時(shí)間是更短的與比較當(dāng)Sc = 3.0 。作用收縮在表面溫度為Ste = 0.15 被顯示在圖4 。如同我們能看, 多孔性增量液體階段導(dǎo)致更高的表面溫度并且更加高級(jí)的Sc 要求更長(zhǎng)的預(yù)熱時(shí)間。當(dāng)Sc = 3.0, 你可能觀察期間熔化的過(guò)程顯著被變短當(dāng) Ste 增量從0.02 到0.15 。圖5 顯示地點(diǎn) 固液 接口和液體表面對(duì)應(yīng)對(duì)圖3 的條件。堅(jiān)實(shí)液體接口快速地行動(dòng)當(dāng)更多氣體被駕駛從液體。因而斷定對(duì)應(yīng)的地點(diǎn)液體表面移動(dòng)向下重大由于混雜的金屬粉末床的收縮。 固液 接口和液體的地點(diǎn)浮出水面對(duì)應(yīng)于圖4 的條件被顯示在圖6 。多孔性減退在液體階段并且加速固液 接口的行動(dòng)和液體表面向下。圖7 顯示最初冷資深 的作用表面溫度為Ste = 0.02 。它能被看見(jiàn)預(yù)熱時(shí)間增加當(dāng)冷資深 參量, Sc, 被增加從0.1 到0.5 。同樣趨向被觀察當(dāng)Sc 增加從1.0 到3.0 。最初冷資深 的作用在表面溫度為Ste = 0.15 被顯示在圖8 。比較對(duì)事例Ste = 0.02, 預(yù)熱時(shí)期為 Ste = 0.15 顯著被變短。同時(shí), 預(yù)熱時(shí)間為Ste = 0.15 增量當(dāng)Sc 被增加從1.0 到3.0 。無(wú)花果。7(a) 和8(a) 表明更低的液體表面溫度可能被獲得如果更大的最初的冷資深 的價(jià)值被使用; 但是, 這些變動(dòng)不是明顯的從無(wú)花果。7(b) 和 8(b) 。圖9 顯示固液 接口的地點(diǎn)并且液體表面對(duì)應(yīng)于條件圖7 。它能被看見(jiàn)在圖9(a), 存在更加了不起的最初冷資深 減少移動(dòng)的速度 固液 接口堅(jiān)固。在之前熱量滲透深度到達(dá)底部, 固液 接口寧可慢慢地行動(dòng), 液體堅(jiān)實(shí)接口快速地行動(dòng)在上升暖流以后滲透深度到達(dá)了底部。在a 更高的冷資深 的參量, 熔化發(fā)生在之后熱量滲透深度到達(dá)了底部依照被顯示在圖9(b) 。這些現(xiàn)象的原因是那預(yù)熱帶來(lái)平均溫度整個(gè)粉末床非常緊挨低熔化的粉末的熔點(diǎn)以便熔化的過(guò)程能非常迅速進(jìn)行。關(guān)系在之間 固液 接口和液體表面, 然而, 只是同樣為所有冷資深 的參量從它依靠氣體的容量分?jǐn)?shù)在固體并且液體階段(參見(jiàn)Eq 。(26) 。地點(diǎn) 固液 接口和液體表面對(duì)應(yīng)對(duì)圖的條件8 并且被密謀在圖10 。一個(gè)相似的趨向可能并且被觀察在圖9 。為了防止燒結(jié)部分氧化的空氣,蘇等. 21使用氬氣作為保護(hù)氣體的粉床. 相比空氣,有因次導(dǎo)熱Kg=3.7 氬具有更低因次導(dǎo)熱對(duì)Kg=2.5 . 表面溫度不同量綱導(dǎo)熱系數(shù)在不同的訪問(wèn)是顯示圖. 11 . 可以看出,預(yù)熱時(shí)間隨導(dǎo)熱性的氣體,是減少兩 訪問(wèn)值分別為0.02和0.15 . 但是,兩者的表面溫度不同的氣體是微不足道. 效果因次導(dǎo)熱性的氣體對(duì)所在地的固液界面和液體 面及相應(yīng)條件的無(wú)花果. 11列圖. 12 . 當(dāng)訪問(wèn) Ste= 0.02速度的固液界面增加時(shí),氬氣作為保護(hù)氣體. 當(dāng)Ste= 0.15 , 速度的固液界面更快Kg =2.5 前的熱穿透深度達(dá)到筆 他底部的粉層. 相比于Kg=3.7 所花費(fèi)的時(shí)間的熱穿透深度達(dá)到底部的粉層較長(zhǎng)Kg =2.5 . 一旦熱穿透深度到達(dá)底部的粉層, 熔化過(guò)程既Kg=3.7 ,Kg=2.5 全速自更 熱能可以用來(lái)供應(yīng)潛伏的熔化.圖6. 多孔性的作用在液體階段在液體表面和液體堅(jiān)實(shí)接口的地點(diǎn)(Ste = 0.15) 。 圖7. 冷資深 的作用在表面溫度(Ste = 0.02) 。圖8. 冷資深 的作用在表面溫度(Ste = 0.15) 。圖9. 冷資深 的作用在液體表面和液體堅(jiān)實(shí)接口的地點(diǎn)(Ste = 0.02) 。圖10. 冷資深 的作用在液體表面和液體堅(jiān)實(shí)接口的地點(diǎn)(Ste = 0.15) 。圖11. 氣體無(wú)維的導(dǎo)熱性的作用在表面溫度。5 . 結(jié)論 熔煉一冷雙組分金屬粉末層受到一恒定熱流加熱調(diào)查分析. 收縮所致熔煉,也會(huì)一并考慮. 增加的斯特凡數(shù)量和貨值冷人數(shù)加速熔化過(guò)程. 熔融的粉層充滿氬較低因次導(dǎo)熱也是調(diào)查,以避免 氧化. 熔煉有限粉末床不同于熔化半無(wú)限板自固液界面熔煉 在有限粉末床動(dòng)作快,在一個(gè)半無(wú)限板. 物理模型與這個(gè)調(diào)查結(jié)果提供了強(qiáng)有力的基礎(chǔ),而進(jìn)一步調(diào)查的復(fù)雜三維 選擇性激光燒結(jié)( SLS )過(guò)程中,可以根據(jù). 鳴謝 支持這項(xiàng)工作由辦公室和海軍研究公司( ONR )的資助下,一些n00014 - 04 - 1 - 0303非常認(rèn)同. 參考 1 J.G. 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Hub中國(guó),上海200030 , 上海交通大學(xué), 中國(guó)B校機(jī)械與動(dòng)力工程; 太原030024 ,太原科技大學(xué), 學(xué)校材料科學(xué)與工程摘要:近年來(lái),快速原型制造技術(shù)( RPM )中已逐漸走向成熟, 已被廣泛地用于制造功能性和實(shí)用金屬和陶瓷零件. 不過(guò),研究選擇性激光燒結(jié)( SLS )的石英砂則非常有限. 實(shí)驗(yàn)中的快速原型制造的石英砂進(jìn)行了基于SLS . 微觀形貌的燒結(jié)模式,在不同的工作條件下觀察三維光學(xué)顯微鏡(庵) . 影響工藝參數(shù)如激光功率,掃描速度,重疊率, 激光光束直徑與粉末混合比對(duì)尺寸精度和燒結(jié)質(zhì)量調(diào)查系統(tǒng). 它表示, 走出效應(yīng) ,變形的燒結(jié)樣品減少通過(guò)降低切片厚度和 優(yōu)化工藝參數(shù). 最后,有條件的選擇性激光燒結(jié)硅砂的方式獲得. 2006 Elsevier公司訴乙版權(quán)所有. 關(guān)鍵詞:快速原型制造; 補(bǔ)充; 硅砂模式; 燒結(jié)質(zhì)量 1. 序言 選擇性激光燒結(jié)( SLS )的一個(gè)重要分支,快速原型制造( RPM ) ,是結(jié)合計(jì)算機(jī)工程, 數(shù)控技術(shù),激光技術(shù)和材料加工技術(shù). 它也是世界上最重要的突破,在最近20年. 在加工前, 復(fù)合添加劑制造方法,可以用來(lái)制作三維任意形狀部件的CAD模式,而不涉及的 專(zhuān)用工具和模具. 因此,生產(chǎn)的靈活性和處理速度可以大大提高,導(dǎo)致時(shí)間和總成本可以降低. 相比其他的RPM技術(shù),廣泛的材料,如有機(jī)聚合物,蠟, 金屬和陶瓷,可用于前,后處理簡(jiǎn)單,它是那么費(fèi)時(shí). 由于 它一直高度集中,自發(fā)明在八十年代末已迅速發(fā)展 1-5 . 目前, 炎熱的實(shí)地研究的領(lǐng)域選擇性激光燒結(jié)主要集中在金屬加工, 陶瓷材料,取得了顯著成就,已經(jīng)取得的成果. 然而,作為一種豐富而廉價(jià)的原料, 研究補(bǔ)充硅砂還很有限 6-8 . 在這個(gè)文件中,快速原型制造硅砂模式基于選擇性激光燒結(jié)技術(shù)進(jìn)行了研究. 噴涂工藝參數(shù)對(duì)成形質(zhì)量的影響結(jié)合實(shí)驗(yàn)研究和微觀分析方法, 合格硅砂模式取得. 2 . 實(shí)驗(yàn)選擇性激光燒結(jié) 2.1 . 實(shí)驗(yàn)系統(tǒng)和材料 實(shí)驗(yàn)系統(tǒng)包括一個(gè)3kW分橫流CO2激光機(jī),自行設(shè)計(jì)和建造粉末壓裝置 一套計(jì)算機(jī)模擬軟件和西門(mén)子數(shù)控系統(tǒng). 精確的數(shù)值控制系統(tǒng)為0.1毫米. 原則的選擇性激光燒結(jié)系統(tǒng)顯示圖. 1 . 在SLS過(guò)程中, 粉末進(jìn)行了掃描沿預(yù)定軌道由激光束根據(jù)CAD模型的組件. 經(jīng)過(guò)掃描一層,活塞被降下來(lái)了距離一層厚度. 然后粉末預(yù)填在上燒結(jié)層粉末壓輥, 最后整個(gè)硅砂模式可以燒結(jié)掃描逐層. 實(shí)驗(yàn)材料所用的是自制的石英砂-酚醛樹(shù)脂(酚醛樹(shù)脂)的化合物. 主要成分的石英砂中SiO2:99%, Al2O3:0.22%,而微TiO2含量,熔點(diǎn)為1750C號(hào) 粘接劑酚醛樹(shù)脂粒度200網(wǎng)及軟化點(diǎn)105-115C號(hào) 固化劑8-12成胺. 2.2 . 實(shí)驗(yàn)方法為了考察了工藝參數(shù)對(duì)加工質(zhì)量及尺寸精度的燒結(jié)樣品, 長(zhǎng)度,寬度及高度的多軌激光燒結(jié)樣品分別標(biāo)示為L(zhǎng)號(hào) 鎢和H的影響激光功率P ,掃描速度六,重疊, 激光光束直徑D和粉末配比對(duì)燒結(jié)質(zhì)量和準(zhǔn)確性分別研究的情況下, 其他參數(shù)不變,在實(shí)驗(yàn)結(jié)果的平均值多種實(shí)驗(yàn). 在焊接以后樣品, 表面的微形態(tài)學(xué)在另外工作之下情況被觀察了在KEYENCE 三維顯微鏡下。3 . 結(jié)果與討論 3.1 . 準(zhǔn)分子激光功率對(duì)燒結(jié)質(zhì)量 形成機(jī)制的選擇性激光燒結(jié)石英砂在于吸收激光能量. 在加熱作用激光束,粘接劑軟化,熔融后固化, 石英砂顆粒被插入的粘合,構(gòu)成了固態(tài)連接骨架. 因?yàn)檐浕c(diǎn)硅砂相當(dāng)?shù)?大約只有110C號(hào) 和優(yōu)化加熱淬火溫度約為250三,所需的功率較低. 在此同時(shí),因?yàn)闄?quán)力的激光切割機(jī)跳躍一系列的瓦特?cái)?shù), 這導(dǎo)致一個(gè)很明顯的錯(cuò)誤,同時(shí)分析了影響激光功率. 在另一方面,由于電流相對(duì)穩(wěn)定, 它可以用來(lái)作為參考值的激光功率. 據(jù)觀察,從實(shí)驗(yàn)中,在一次電流為0.8 , 粘接劑酚醛樹(shù)脂尚未完全融化; 在電流為1.0 , 燒結(jié)效果最佳; 而電力增加值1.2 , 樣品表面被碳化,并敬獻(xiàn)了黑顏色. 因此,優(yōu)化電流在實(shí)驗(yàn)條件下為1.0甲.圖2 顯示不同的激光束 力量的作用在微被焊接的樣品的形態(tài)學(xué)。圖2(a) 顯示形態(tài)學(xué)原始的硅土沙子。它表示, 五谷原物硅土沙子是基本上一致規(guī)則半透明水晶提出在牛奶的顏色白色和蒼白黃色。在混合和按以后, 硅土的外部沙子由粘合劑, 硅土的空白包裝了沙粒用bonders 被填裝了。圖2(b) 微形態(tài)學(xué)在更低的力量, 它之下能被看見(jiàn)表面樣品提出淡黃的下面低力量, 零件的顏色黏合劑不充分地被熔化, 并且接合力量是不是足夠。在中等力量之下, 樣品的表面是在一種深褐色的顏色, 硅土沙粒是mosaicked 在堅(jiān)實(shí)黏合劑和形式一個(gè)半透明的保稅的身體與理想的被焊接的作用, 依照被顯示在圖2(c)。圖2(d) 顯示被焊接的表面下面更高的激光束 力量, 因?yàn)槟芰渴翘? 地方區(qū)域在表面當(dāng)前黑棕色顏色并且黏合劑被碳化了。圖1. SLS 系統(tǒng)的原則。3.2. 掃描速度F 的作用在維度被焊接的樣品圖3 表明掃描速度的影響對(duì)維度被焊接的樣品在激光束直徑3 毫米, 重疊的寬度0.5 毫米、激光束 力量12W, 和粉末比率硅土沙子和PF 樹(shù)脂14:1 。以增加掃描加速, 樣品減退的長(zhǎng)度、寬度和高度逐漸。這個(gè)結(jié)果原因是那以增加掃描速度, 激光束的停留時(shí)間在掃描斑點(diǎn)相應(yīng)地變短了, 當(dāng)激光束 力量是恒定的, 實(shí)際輸入能量在單位時(shí)間減少, 和熱受影響的區(qū)域并且被減少了, 導(dǎo)致a 維度變小樣品。3.3. 重疊的作用, 激光束直徑D 和粉末混合物比率在樣品的高度 在實(shí)驗(yàn), 激光束直徑D 被設(shè)置了到3mm 調(diào)整defocusing 的數(shù)額, 重疊? 毗鄰掃描軌道變化了在0.5-2.0 毫米, 激光束力量的范圍 P 12W 是, 粉末比率嗎? 是11:1, 和掃描速度F 是650 mm/min 。用增加重疊, 焊接深度增加了。當(dāng)重疊被增加了, 激光束 能量的吸收增加了, 更多黏合劑的數(shù)量被熔化了, 并且變?nèi)岷偷酿ず蟿┑纳疃仍黾恿?。在情況下, 其它實(shí)驗(yàn)性參量是固定的, 激光束能量密度減少了以增加激光束 依照被顯示放光直徑, 和被焊接的深度迅速地被減少, 在圖4 。圖2. 激光束 力量的作用在被焊接的樣品的微形態(tài)學(xué): (a) 原始的硅土沙子微粒(b) 低力量, (c) 中等力量, 和(d) 大功率。圖4 并且顯示關(guān)系在粉末混合物之間比率? 并且被焊接的維度在激光束 的情況下察覺(jué)3 毫米, 激光束 力量12W, 重疊1.1mm 和掃描加速650 mm/min 。它能被看見(jiàn), 以增加?, 樣品的長(zhǎng)度、寬度和厚度輕微地變化, 但不非常極大。那是以增加粘合劑, 被焊接的樣品的維度輕微地增加。由于粘合劑內(nèi)容的減少, 連接橋梁在硅土沙子粉末之中減少了, 變?nèi)岷筒⑶胰刍瘏^(qū)域和混雜的粉末的深度減少了相應(yīng)地。所以, 耐壓強(qiáng)度被焊接樣品可能被提高以更高的黏合劑內(nèi)容, 但許多粘合劑導(dǎo)致很多變形并且收縮, 導(dǎo)致波動(dòng)維度。PF 樹(shù)脂的固體化收縮收效一個(gè)區(qū)別在實(shí)際維度和被設(shè)計(jì)的部分之間被焊接的樣品, 但那里仍然有一些規(guī)則被跟隨。隨后實(shí)驗(yàn)的粉末混合物比率被選擇作為11:1 。3.4. 崗位處理被焊接的樣品和質(zhì)量分析在崗位之前處理, 有殘余的未熔化的接合粉末在有選擇性的激光束 被焊接的樣品, 和粘合劑的發(fā)行不是同類(lèi)的, 力量樣品是降低, 并且熱量藏品過(guò)程必需。沙子樣式可能只被使用在熔鑄在硬化的過(guò)程以后, 在中接合的聚集的現(xiàn)象代理可能被消滅, 水并且gasifiable 事態(tài)可能是揮發(fā), 和黏合劑可能一致地被分布當(dāng)被焊接的樣品舉行在大約250C 30 分鐘。在有圖3. 掃描速度的作用在被焊接的樣品的維度。有條理改進(jìn)被焊接的樣品的力量沒(méi)有碳化, 崗位處理溫度不應(yīng)該超過(guò) 300?C 。由于有選擇性的激光束 焊接是被碾壓的材料疊加性制造過(guò)程、切的厚度和過(guò)程參量有一個(gè)重大作用在質(zhì)量適當(dāng)?shù)貨](méi)控制和階段效應(yīng) 將出現(xiàn)。樣品的維度準(zhǔn)確性和表面結(jié)束將很大地影響。同時(shí), 在處理SLS, 由于不均勻的熱化和收縮, 變形是有義務(wù)發(fā)生。圖4. 重疊, 斑點(diǎn)大小和粉末比率的聯(lián)系對(duì)維度被焊接的樣品。避免這些瑕疵和改進(jìn)表面完成, 切的厚度應(yīng)該是足夠稀薄, 和合理處理參量應(yīng)該被選擇。層數(shù)厚度并且有一個(gè)作用在被焊接的組分的物產(chǎn)。稀薄的切的層數(shù)導(dǎo)致一更高尺寸準(zhǔn)確性和機(jī)械力量, 但制造業(yè)時(shí)間將被延長(zhǎng)。如果層數(shù)是太稀薄的, 均一和 粉末層的compactibility 將變得困難。避免這些瑕疵和改進(jìn)表面而且, 激光束 功率密度由激光束 力量和斑點(diǎn)大小決定, 并且熱化粉末的溫度和期間依靠在激光束 功率密度和掃描速度。在條件下低功率密度和迅速掃圖5. 沙子樣式焊接了由有選擇性的激光束 焊接。描速度, 粉末的部份不要有足夠時(shí)間完全地熔化, 和力量樣品是更低。但是, 粉末溫度過(guò)份地是更高, 蟲(chóng)膠黏合劑將被燒焦和將被氣化, 并且被焊接的表面將成為概略, 結(jié)合的物產(chǎn)在層數(shù)和焊接之間質(zhì)量將被減少。根據(jù)上述分析, 它結(jié)束更好焊接作用可能被獲得以媒介激光束 力量和降低掃描速度。優(yōu)化處理參量是如下: 電流1.0 A, 掃描速度650 mm/min, 激光束直徑3 毫米, 重疊0.5mm 和粉末混合物比率11:1 。沙子樣式焊接了以早先參量被顯示在圖5 。4. 結(jié)論有選擇性激光束 焊接硅土沙子有特征高靈活性, 更短的前置時(shí)間, 低成本, 做法集中化并且形成沒(méi)有模子。它是特別適當(dāng)?shù)臑閺?fù)雜形狀的鑄件和生產(chǎn)的發(fā)展和小全部片斷。處理參量有重要作用在物產(chǎn)并且被焊接的樣品的準(zhǔn)確性。以適當(dāng)?shù)妮斎?激光束 力量, 粉末比率和重疊, 表面準(zhǔn)確性并且維度精確度可能被改進(jìn), 夾層接合力量和整體組分的機(jī)械力量罐頭并且被改進(jìn)。粘合劑可能一致地被分布和力量樣品可能被提高以崗位處理和舉行, 而是藏品溫度無(wú)法超過(guò)300C 。形成樣品的精確度可能被并且改進(jìn)減少切的厚度和過(guò)程的優(yōu)化參量。鳴謝這工作由全國(guó)自然科學(xué)支持了中國(guó)的基礎(chǔ)在被授予的數(shù)量的50375096 之下。1 Liu Hongjun, Fan Zitian, Huang Naiyu, A note on rapid manufacturingprocess of metallic parts based on SLS plastic prototype, J. Mater. Process.Technol. 142 (2003) 710713.2 Deng. Qilin, Hu. Dejin, Pei. Jingyu, An Experimental study on selective 激光束sintering of ceramic powders, Aviat. Precision Manuf. Technol. 37 (2001)2831 (in Chinese).3 D. King, T. Tansey, Rapid tooling: selective 激光束 sintering injection tooling,J. Mater. Process. Technol. 132 (2003) 4248.4 Y.P. Kathuria, Microstructuring by selective 激光束 sintering of metallic powder,Surf. Coat. Technol. 116119 (1999) 643647.5 J.P. Kruth, M.C. Leu, T. Nakagawa, Progress in additive manufacturing andrapid prototyping, Keynote papers, CIRP Annals (1998) 525540.6 Y. Tang, J.Y.H. Fuh, H.T. Loh, Direct 激光束 sintering of silica sand, Mater.Des. 24 (2003) 623629.7 Li Jianguo, Xu Mingheng, Pattern-less rapid precision casting technologybased on SLS prototyping, Mach. Build. Autom. 5 (2003) 1213, 16 (inChinese).8 Fan Zitian, Huang naiyu, Li Yan, Investigation on the pre-coated castingmolds (or cores) made by the selected 激光束 sintering, J. Nanchang Univ. 22(2000) 2429 (in Chinese)subcooled Yuwen 11 is investigated points solid approximate solution respectively The e ects of porosity Stefan number and subcooling on the surface temperature and solid liquid interface are also investigated The present work provides a strong foundation upon which the investigation of complex selectively fusing a thin layer of the powders with scan imate diameter of the laser beam The particular melting and resolidification induced by a directed laser sity change of the powder bed accompanies the melting process Melting and solidification in 1 D semi infinite body with density change under the boundary condition of the first kind have been investigated by Zckert and Drake 9 Crank 10 Carslaw and Jaeger 11 and Corresponding author Tel 1 573 884 6939 fax 1 573 884 5090 E mail address zhangyu missouri edu Y Zhang Applied Thermal Engineering 26 1359 4311 see front matter C211 2005 Elsevier Ltd All rights reserved ning laser beam After sintering of a layer a new layer of the powder is deposited in the same manner and a 3 D part can be built in a layer by layer process A mixed metal powder bed which contains two types of the metal powders possessing significantly di erent melting points is used extensively in direct SLS of metal powders 2 3 The high melting point powder never melt in the sintering process and plays a significant role as the support structure necessary to avoid boiling phenom enon which is the formation of spheres with the approx beam It is a good starting point to investigate a simpli fied 1 D model to get a better understanding of the melting process in direct SLS before a much more com plicated 3 D model is investigated Fundamentals of melting and solidification have been investigated extensively and detailed reviews are avail able in Refs 7 8 Melting in SLS of the metal powders is significantly di erent from the normal melting since the volume fraction of the gas in the powders decreases significantly after melting Therefore a significant den three dimensional selective laser sintering SLS process can be based C211 2005 Elsevier Ltd All rights reserved Keywords Melting Metal Powder layer 1 Introduction Direct Selective Laser Sintering SLS is an emerging technology of Solid Freeform Fabrication SFF via which 3 D parts are built from the metal based powder bed with CAD data 1 A fabricated layer is created by material properties and methods of material analysis of the metal based powder system for SLS applications are addressed by Storch et al 4 and Tolochko et al 5 Fundamental issues on direct SLS are thoroughly re viewed by Lu et al 6 In fabrication of near full density objects from metal powder direct SLS is realized via Analysis of melting in a metal powder layer with Tiebing Chen Department of Mechanical and Aerospace Engineering University Received 1 February 2005 Available online Abstract Melting of a subcooled two component metal powder layer mixture of two metal powders with significantly di erent melting physical model The temperature distributions in the liquid and doi 10 1016 j applthermaleng 2005 07 034 two component constant heat flux Zhang of Missouri Columbia Columbia MO 65211 United States accepted 18 July 2005 October 2005 analytically The powder bed considered consists of a Shrinkage induced by melting is taken into account in the phases are obtained using an exact solution and an integral 2006 751 765 Nomenclature c p specific heat J kg C01 K C01 h sl latent heat of melting or solidification J kg C01 k thermal conductivity W m C01 K C01 K g dimensionless thermal conductivity of gas K s dimensionless e ective thermal conductivity of unsintered powder q 00 heat flux W m C02 s solid liquid interface location m S dimensionless solid liquid interface location s 0 location of liquid surface m S 0 dimensionless location of liquid surface Sc subcooling parameter Ste Stefan number t time s T temperature K w velocity of liquid phase m s C01 W dimensionless velocity of the liquid phase z coordinate m Z dimensionless coordinate Greek symbols a thermal di usivity m 2 s C01 752 T Chen Y Zhang Applied Thermal Charach and Zarmi 12 It should be noted that melting during SLS occurs under the boundary condition of specified heat flux instead of specified temperature Goodman and Shea 13 studied melting and solidifica tion in the finite slab under a specified heat flux by using the heat balance integral method Zhang et al 14 investigated the melting problem in a subcooled semi infinite region subjected to constant heat flux heating Zhang et al 15 solved melting in a finite slab with the boundary condition of the second kind by using a semi exact method Shrinkage formation due to density change during the solidification process in 2 D cavity was investigated numerically by Kim and Ro 16 who concluded that the density change played a more impor tant role than convection in the solidification process Zhang and Faghri 17 analytically solved a one dimensional melting problem in a semi infinite two component metal powder bed subjected to a constant heating heat flux E ects of the porosity of the solid phase initial subcooling parameter and dimensionless thermal conductivity of the gas were investigated Since SLS of the metal powder is actually a layer by layer pro cess it is necessary to investigate melting in a mixed me tal powder bed with the finite thickness during the SLS process In this paper melting of the mixed powder bed with finite thickness subjected to constant heating heat flux will be investigated C22a dimensionless thermal di usivity b parameter to distinguish between two melting cases d thermal penetration depth m D dimensionless thermal penetration depth e volume fraction of gas es porosity for unsintered powder h dimensionless temperature q density kg m C03 s dimensionless time volume fraction of the low melting point powder in the powder mixture Subscripts g gas i initial l liquid phase m melting point p sintered part s unsintered solid mixture of two solid pow ders Engineering 26 2006 751 765 2 Physical model The physical model of the melting problem is shown in Fig 1 A powder bed with finite thickness contains two metal powders with significantly di erent melting points The initial temperature of the powder bed is below the melting point of the low melting point pow der At time t 0 a constant heat flux q 00 is suddenly applied to the top surface of the powder bed and the bottom surface of the powder bed is assumed to be adiabatic Since the initial temperature of the powder bed is below the melting point of the low melting point powder its melting does not start simultaneously with the addition of heat heating Only after a finite period of time of preheating in which the surface tem perature of the powder reaches the melting point of the low melting point powder will the melting start The powder with the high melting point will never melt during the entire process Therefore the problem can be subdivided into two problems one being heat conduction during preheating and the other being melting The physical model is considered as a conduc tion controlled problem The e ect of natural convec tion in the liquid region due to the temperature di erence is not considered since the temperature is highest at the liquid surface and decreases with increas ing z T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 753 2 1 Duration of preheating During preheating pure conduction heat transfer oc curs in the powder mixture The governing equation and the corresponding initial and boundary conditions for the preheating problem are z Fig 1 Physical q s s 0 H 0 a s o 2 T s oz 2 oT s ot 0 z H 0 t t m 1 T T i 0 z H 0 t 0 2 C0 k s oT s oz q 00 z 0 t t m 3 oT s oz 0 z H 0 t t m 4 2 2 Melting After melting starts the governing equation in the liquid phase is a l o 2 T l oz 2 oT l ot w oT l oz s 0 z t m 5 where w is the velocity of liquid surface induced by the shrinkage Since the liquid is incompressible the shrink age velocity w is w ds 0 dt s 0 z t m 6 Eq 5 is subjected to the following boundary condition C0k l oT l oz q 00 z 0 t t m 7 The governing equation for the solid phase and its cor responding boundary conditions are a s o 2 T s oz 2 oT s ot s t z t m 8 oT s 0 z H 0 t t m 9 Liquid solid interface Low melting point powder High melting point powder model Original surface Liquid surface oz The temperature at the solid liquid interface satisfies T l z t T s z t T m z s t t t m 10 The energy balance at the solid liquid interface is k s oT s oz C0 k l oT l oz 1 C0e s q l h sl ds dt z s t t t m 11 Based on the conservation of mass at the solid liquid interface the shrinkage velocity w and the solid liquid interface velocity ds dt have the following relationship 17 w e s C0e l 1 C0e l ds dt 12 2 3 Non dimensional governing equations By defining the following dimensionless variables h l qc p p T l C0 T m Uq l h sl h s qc p p T s C0 T m Uq l h sl Sc qc p p T m C0 T i Uq l h sl s a p t H 2 Z z H S s H S 0 s 0 H D d H W w C1 H a p K s k s k p 1 C0e s K g k g k p C22a s a s a p Ste q 00 H Uq l h sl a p 13 The non dimensional governing equation and the corre sponding initial and boundary conditions for the pre heating problem become o 2 h s oZ 2 1 C22a s C1 oh s os 0 Z 1 s s m 14 h C0Sc 0 Z 1 s 0 15 oh s oZ C0 Ste K s 1 C0e s Z 0 s s m 16 h s C0Sc Z D s s m 23 h l Z s h s Z s 0 Z S s s s m 24 K s oh s oZ C0 1 C0e l 1 C0e s oh l oZ dS ds Z S s s s m 25 W e s C0e l 1 C0e l dS ds S 0 Z t m 26 3 Approximate solutions When the top surface of the mixed metal powder bed is subjected to constant flux heating the heat flux will penetrate through the top surface and conduct down ward the bottom surface The depth to which the heat flux penetrates at an instant in time is defined as the thermal penetration depth beyond which there is no heat conduction Goodman and Shea 13 introduced a 00 754 T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 oh s oZ 0 Z D s s m 18 For melting the non dimensional equation and corre sponding boundary conditions are o 2 h l oZ 2 oh l os W oh l oZ S 0 Z s m 19 W dS 0 ds S 0 Z t m 20 oh l oZ C0 Ste 1 C0e l Z S 0 s s m 21 o 2 h s oZ 2 1 C22a s C1 oh s os S s Z s m 22 Fig 2 Validation of analytical parameter b q H 2k s T m C0 T i to classify two cases of melting in a finite slab When b is greater than 1 the top surface temperature reaches the melting point in a shorter time than the thermal penetration depth reaches the bottom surface indicating that a shorter preheating time is needed If b is less than 1 the surface tempera ture is still below the melting point when the thermal penetration depth has reached the bottom surface Pre heating continues until the top surface temperature reaches the melting point of low melting point powder The parameter b can also be expressed using non dimensional parameters defined in Eq 13 i e b Ste 2K s Sc 1 C0 e s It can be seen that the value of b is determined by four basic non dimensional parame ters Stefan number Ste subcooling parameter Sc e ec tive thermal conductivity of the solid phase K s and solutions T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 755 volume fraction of gas e s in the solid phase Preheating and melting for both b 1 will be discussed 3 1 Preheating 3 1 1 b 1 The heat balance integral method 18 19 is employed here Integrating the heat conduction Eq 14 with re spect to Z from 0 to D the integral equation is obtained oh s oZ D s C0 oh s oZ 0 s C20C21 1 C22a s d ds H ScD 27 where H R D 0 h s Z s dZ h s Z s is assumed to be a second degree polynomial function which satisfies boundary conditions specified by Eqs 16 18 Then h s Z s can be determined b Fig 3 E ect of porosity in the liquid phase l l l l h s Z s C0Sc Q 2K s D 1 C0e s DC0 Z 2 28 The Eqs 16 18 and 28 can be substituted into Eq 27 and then an ordinary di erential equation for the thermal penetration depth D is obtained which can be solved easily D 6 C1 C22a s C1s 1 2 29 When the thermal penetration depth reaches the bottom surface i e D 1 the temperature distribution in the powder bed is h s Z s C0Sc Ste 2K s 1 C0e s 1 C0 Z 2 0 Z 1 s s D 1 s m 30 a l l l l on surface temperature Ste 0 02 756 T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 which becomes the initial condition of the next stage of preheating After the thermal penetration depth reaches the bottom the problem becomes a conduction problem in a finite slab In a manner analogous to that described previously the temperature of the powder is h s Z s C0Sc Ste 2K s 1 C0e s 1 C0 Z 2 Ste C1 C22a s K s 1 C0e s C2 sC0s D 1 0 Z 1 s D 1 s 1 When b is greater than 1 melting starts before the penetration depth reaches the bottom and therefore the preheating time s m corresponding thermal penetra tion depth D m and temperature distribution at time s m are 17 s m 2 3 1 C0e s 2 K 2 s Sc 2 C1 1 Ste 2 C22a s 34 D m 2 1 C0e s K s Sc C1 1 Ste 35 a l l l l on surface temperature Ste 0 15 T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 757 h s Z s Sc 1 C0 Z D m C18C19 2 C0 1 Z 0 s s m 36 h s 0 s C0Sc Ste C1 6a s s p 2 C1 K s C1 1 C0e s C138 Z 0 0 s s m 37 where Eq 38 is the surface temperature on the top of the powder bed 3 2 Solution of melting 3 2 1 Temperature distribution in the liquid Melting starts when the surface temperature of the powder bed reaches the melting point of the low melting point powder A liquid layer is formed as the result of melting the temperature distribution of which a b K Fig 5 E ect of porosity in the liquid phase on the location of the l l l l l l l does not depend on the value of b It can be obtained by an exact solution of Eqs 19 21 and 24 17 i e h l Z s 2Ste sC0s m p 1C0e l ierfc ZC0S 0 2 sC0s m p C18C19 C0ierfc SC0S 0 2 sC0s m p C18C19C20C21 38 where S 0 is dimensionless location of liquid surface 3 2 2 Temperature distribution in the solid b 1 Melting begins before the heat flux reaches the bot tom of the powder bed thus the problem is melting in semi infinite two component powder bed The solution for melting of an infinite powder bed containing a mixture of two metal powders has been obtained by a l l l l l l l l l liquid surface and the liquid solid interface Ste 0 15 T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 759 Zhang 17 The temperature distribution in the liquid phase is given by Eq 38 The temperature distribution in the solid region ob tained by 17 h s Z s Sc DC0 Z DC0 S C18C19 2 C0 1 43 The location of solid liquid interface is also obtained by 17 dS ds Ste 1 C0e s erfc 1 C0e s S 2 1 C0e l sC0s m p C18C19 C0 2K s Sc DC0 S 44 The thermal penetration depth satisfies the equation a Fig 7 E ect of subcooling on surface l dD ds 6K s DC0 S 1 2 3 Sc C18C19 C0 2Ste 1 C0e s erfc 1 C0e s S 2 1 C0e l sC0s m p C18C19 45 At the time that the thermal penetration depth reaches the bottom surface i e D 1 the temperature distribu tion in the solid is h s Z s D 1 Sc 1 C0 Z 1 C0 S C18C19 2 C0 1 46 The time that thermal penetration depth reaches to the bottom s D 1 is obtained from D s D 1 1 47 b l temperature Ste 0 02 760 T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 When s s D 1 the problem becomes melting in a finite slab The temperature distribution in the solid h s Z s and the liquid solid interface location S can be ob tained by solving Eqs 22 24 using the integral approximate method identical to the case of b 1 4 Results and discussion The validation of the analytical solution was conducted by comparing the results with the numerical results obtained from Chen and Zhang 20 who inves tigated the two dimensional melting and resolidification of a two component metal powder layer in SLS process subjected to a moving laser beam In order to use the two dimensional code in Ref 20 to solve melting in a b Fig 8 E ect of subcooling on surface l a powder layer subjected to constant heat flux the Gaussian laser beam was replaced by a constant heat ing heat flux on the top of the entire powder bed and the laser scanning velocity was set to zero in numerical solution The parameters used in the present paper were converted into corresponding parameters in Ref 20 for purpose of code validation The comparisons of instantaneous locations of liquid surface and liquid solid interface obtained by analytical and numerical solutions are shown in Fig 2 It can be seen that the preheating time obtained by the analytical and numer ical solutions are almost the same The locations of liquid surface and liquid solid interface obtained by analytical and numerical solutions move at very similar trends The time it takes to completely melt the entire powder layer obtained from analytical solution is about l temperature Ste 0 15 T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 761 l 4 longer than that obtained from the numerical solution The e ects of porosity subcooling dimensionless thermal conductivity and Stefan number on the surface temperature location of the liquid surface and the loca tion of the solid liquid interface of the powder bed will be investigated Fig 3 shows how the surface tempera ture is influenced by the porosity in the liquid phase for Ste 0 02 and several di erent subcooling parame ters The e ect of shrinkage is isolated by fixing the sub cooling parameter porosity of the solid phase and the dimensionless thermal conductivity It can be seen that the surface temperature increases as porosity in the liquid phase increases This is because the e ective ther mal conductivity decreases with increasing volume frac tion of the gas When Sc 0 1 the preheating time is a b Fig 9 E ect of subcooling on the location of the liquid much shorter compared to when Sc 3 0 The e ect of shrinkage on the surface temperature for Ste 0 15 is shown in Fig 4 As we can see the increase of poros ity in the liquid phase results in higher surface tempera tures and that a higher Sc requires a longer preheating time When Sc 3 0 one can observe that the duration of the melting process is shortened significantly when Ste increases from 0 02 to 0 15 Fig 5 shows the loca tions of solid liquid interface and liquid surface corre sponding to the conditions of Fig 3 The solid liquid interface moves faster when more gas is driven out from the liquid It follows that the corresponding location of the liquid surface moves downward significantly due to the shrinkage of the mixed metal powder bed The locations of solid liquid interface and liquid surface corresponding to the conditions of Fig 4 are shown in l surface and the liquid solid interface Ste 0 02 762 T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 l Fig 6 The decrease of porosity in the liquid phase also expedites the motion of the solid liquid interface and liquid surface downward Fig 7 shows the e ect of the initial subcooling on the surface temperature for Ste 0 02 It can be seen that the preheating time increases when the subcooling parameter Sc is increased from 0 1 to 0 5 The same trend is observed when Sc increases from 1 0 to 3 0 The e ect of the initial subcooling on the surface tem perature for Ste 0 15 is shown in Fig 8 Compared to the case of Ste 0 02 the preheating time for Ste 0 15 is significantly shortened Meanwhile the preheating time for Ste 0 15 increases when Sc is in creased from 1 0 to 3 0 Figs 7 a and 8 a indicate that a lower liquid surface temperature can be obtained if a larger initial subcooling value is used however a b l Fig 10 E ect of subcooling on the location of the liquid these changes are not apparent from Figs 7 b and 8 b Fig 9 shows the location of the solid liquid interface and the liquid surface corresponding to the conditions of Fig 7 It can be seen in Fig 9 a the existence of greater initial subcooling reduces the moving velocity of the solid liquid interface substantially Before the thermal penetration depth reaches the bottom the solid liquid interface moves rather slowly however the liquid solid interface moves much faster after the ther mal penetration depth has reached the bottom
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