水平橢圓管外之層流膜狀凝結之熱力第二定律最佳化分析

行政院國家科學委員會專題研究計畫 成果報告

水平橢圓管外之層流膜狀凝結之熱力第二定律最佳化分析

研究成果報告(精簡版)

計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 95-2221-E-151-064- 執 行 期 間 ： 95 年 08 月 01 日至 96 年 10 月 31 日 執 行 單 位 ： 國立高雄應用科技大學模具工程系 計 畫 主 持 人 ： 楊勝安 計畫參與人員： 博士班研究生-兼任助理：康易隆 碩士班研究生-兼任助理：吳家瑞 報 告 附 件 ： 出席國際會議研究心得報告及發表論文 處 理 方 式 ： 本計畫可公開查詢

中 華 民 國 96 年 10 月 25 日

1.前言 1.1 研究目的 有效使用石油，天然氣以及電氣可改善工業產品的競爭力，同時可降低能源使用與其 產品相關的環境污染。可用能的保存取決於有效率的熱力熱傳過程的設計，亦即最小化熱 傳與黏滯等不可逆所產生的熵增率。 熱工程系統的熵增率會破壞系統可用能，同時減少其效率。事實上，挑選出最低值熵 增率設備，諸如凝結器與熱交換器，也算是吾人對減少全球氣候的變遷一份貢獻。雖然自 Nusselt後，已經陸續有許多膜狀凝結熱傳的研究，並將水平圓管上膜狀凝結應用在熱機工 程上，例如熱交換器與熱管。若能更多了解如何以及哪裡產生熵增，那麼設計一套減少總 熵增的工作參數得以保存有用能，將是可行的方法。截至目前為止，有許多研究探討單相 對流熱傳所產生的熵值，然而探討屬於相變化熱傳的膜狀凝結，其熵增分析的文獻卻相當 少。換句話說，橢圓管壁上膜狀凝結的熱力第二定律分析依然是個未解決的問題。 因此，本研究的重點在於橢圓管壁上膜狀凝結的熱力學第二定律分析，尋求系統整體 之最小熵產生率及其對應之最佳管徑，以及系統的總熵增量的最佳化設計。經由數值分析 可了解系統內各元件因不可逆性所造成能量損失之情形，而作為改善熱交換器系統與最佳 化分析之重要依據，如此我們可以針對熵增較大之元件進行分析與改善，以期能夠達到最 大的效果。 1.2文獻探討 公元1961年，Nusselt[1]首先研究均勻溫度分佈之垂直平板與水平圓管外的層流膜狀凝 結熱傳問題。其主要假設為：（一）壁面為等溫；（二）液膜內溫度分佈為線性；（三） 不考慮液體和蒸汽交界面之剪應力；（四）忽略液膜內慣性力與對流能之作用。其結果獲 得以凝結過程相關之參數，所直接表示的熱傳係數，雖然其理論預測值與實驗量測值相比 較，理論值稍微低。 其後根據Fujii 等人[2]實驗與研究結果顯示，當水平圓管表面發生凝結時，管周圍之表 面溫度分佈，並非如Nusselt 理論所假定，呈均勻定值。事實上，縱使管內能維持定溫，但 由於管壁局部對流熱傳係數之差異，造成表面溫度分佈之非均勻性。除非，冷媒與管壁局 部對流熱傳係數之差異，造成表面溫度分佈之非均勻性。且冷媒與管壁間具相當良好的熱 傳性能，方可接近均勻定溫分佈之假定。否則，當冷媒與管壁間熱阻愈大，非均勻溫度分 佈情況愈加明顯。此種非均勻性，在Shklover[3]等人的水平圓管之凝結熱傳研究當中，亦 可從所得溫度分佈發現此一類似點。另外，Lee 等人[4]，在其實驗量測指出，圓管表面溫 度，可以餘弦函數型態表示。Memory 與 Rose[5] 進而將此一分佈型態代入 Nusselt 理論求 解，獲知其平均凝結熱傳遞係數並不受此一非均勻性而影響。換句話說，在相同平均溫度 下，定溫之Nusselt 解與 Memory 等人考慮非均勻性之平均凝結熱傳係數值幾乎相等。 Jain 與Bankoff [6]於1964 年考慮垂直壁溫為等溫，並忽略液-汽介面上剪應力的影響， 但壁面具有壁吸流效應，以擾動參數與加速參數用應雙重擾動展開(Double perturbation expansion)方式求解垂直壁之薄膜冷凝熱傳問題，結果顯示冷凝熱傳與Pr數、次冷性參數、 與壁吸流速度有關，尤其當Pr數愈高愈利於熱傳遞。而Frankel與Bankoff[7]於1967年發表水 平圓管壁溫為等溫而且具有壁吸流效應的圓管外薄膜冷凝熱傳問題。

Yang[8]於 1970 年則考慮 Jain 與 Bankoff 之問題，但將統御方程式以級數展開法(Series expansion method)求解，其中乃將速度與溫度變數展開，結果顯示當 Pr =10～Pr =100 時當

塞數Nu 值對壁吸流速度參數值增加時愈趨敏感，但當次冷性參數 大於某一特定值時則Nu 值不減反增(當 Pr =10~Pr =100 時)，此乃由於當次冷性加大液膜厚度大增而使得熱阻抗減少 所致。

Bejan[9]率先研究熵增最佳化在單相對流熱傳領域上，在他的著作中，Bejan 在熱機工 程方面上提出具體的方法來分析熵增最佳化，他分析等熱通量之平板，與圓管之熵增變化。

爾後，Nag 與 Mukherzee[10], sahin[11]兩篇論文皆研究黏性流體在等溫圓管的熱力學第二定

律分析。

近來，Saouli與Aiboud-Saouli[12]研究傾斜平板的層流熱力學第二定律分析，而以上學 者的研究都屬單相流的熵增分析。至於凝結熱傳的熵增分析，首先由Adeyinka和Naterer[13]

則研究垂直平板上之熵增變化情形，並提供一個有用的參數用來設計雙相流系統。Lin et al.

[14]則是完成運用熱力第二定律於飽和蒸氣下水平管內凝結分析。本相變化熱傳研究室于公 元2005~2007年期間，由指導教授Yang與其研究生Dung[15,16], Tzeng[17,18], Lee[19,20], 與 Chen and Wu[21]分別針對不同之幾何外型作膜狀凝結熵增分析。

2. 分析 2.1 橢圓管為可變壁溫 考慮一具直立長軸“2a”之水平橢圓管如圖 1.所示，放置於大量靜止或由上往下緩慢流 動的純飽和蒸汽當中。且已知管壁表面溫壁(T_W)低於蒸汽飽和溫度(Tsat) 將於管壁上開始發 生膜狀凝結，經過一段特定時間後即可形成一層穩態層流膜狀凝結液，由於凝結液受到重 力影響與有效表面張力作用，開始往下流動。座標系統採用Yang[22]所建立橢圓極座標系， 建立沿橢圓周弧線方線之x 座標，以及法線方向之 y 座標，並且已知兩者對應凝結液流之 流速分量分別為u 與 v。並作以下之基本假設： (1)流體性質為均勻定值。 (2)凝結液流為穩態層流。 (3)液膜甚薄，可視液膜內溫度分佈為直線分佈。 (4)液-汽介面之黏滯力可忽略。 (5)忽略液膜內慣性力與對流兩項。 圖1. 橢圓管物理模型圖 基於上述假設，水平橢圓管壁上形成凝結薄膜液之邊界層統御方程式可表示如下：

質量連續方程式 0 = ∂ ∂ + ∂ ∂ y v x u (1) 動量方程式 )] ( [sin ) ( 2 2 φ φ ρ ρ μ g Bo y u v + − − = ∂ ∂ (2) 能量方程式 0 2 2 = ∂ ∂ y T k (3) 其邊界條件假定如下 0 = y , u =0, 在T =T_w 與 y=δ , =0 ∂ ∂ y u , 在T =T_sat. 再者，將動量方程式與能量方程式積分，並配合上式之邊界條件，則可求得速度分佈方 程式與溫度分佈方程 ] ) ( 2 1 )][ ( [sin ) ( 2 2 δ δ φ φ δ μ ρ ρ y y Bo g u= − v + − , (4) w T y T T =Δ ⋅ + δ . (5) 然後利用所獲得的速度分佈方程式(4)式，可推導出單位管長下之局部凝結液質量流率為：

[

sin ( )

]

3 ) ( ) ( 3 0 μ φ φ δ ρ ρ ρ ρ δ Bo g dy y u m& =

∫

= − _v + (6) 接著 x 與x+dx間凝結液取一能量平衡，可得： δ / ' T k dx m d h fg & = Δ (7) 其中， ' (1 0.68 / ) fg p fg fg h C T h h = + Δ 係Rohsenow[23]所建立次冷效應所修正之有效凝結潛 熱。將m& 代入(7)式，且利用變數變換dx=rdφ可得： δ φ φ δ μ ρ ρ ρ T k rd d g h v fg Δ = − ′ ) sin ( 3 ) ( 3 (8) 在(8)式中，只要上式右邊管壁溫度分佈已知，即可求出膜狀凝結液厚度分佈。至於管 壁溫度分佈可為均勻或變化值，依管內對流熱傳係數而定。亦可由實驗量測管周圍各點之 溫度值，再利用曲線湊配法表成某一型態分佈T_w(φ)，然後再將T_w(φ)代入下式，求出平均 壁溫為：

[

]

∫

− − = π φ φ φ π 0 3 2 2 2_{) (1 sin )} 1 ( ) ( 2 d e e T D a T _w e w , (9) 其次，再將凝結液膜內溫度差表為 ) ( ) ( ) ( _{sat w t} φ _t φ w sat T T T F TF T T = − = − =Δ Δ , (10) φ φ =

均勻溫度分佈，亦即為Nusselt 型之凝結理論。但是根據 Shklover 等人[3]與 Lee 等人[4]之 實驗，均一致發現水平圓管壁溫度分佈並非均勻，且後者更以餘弦函數表示為： ) cos( 1 ) (φ A φ Ft = − , (11) 式中，A 介於 0 與 1 之間，其大小則由管內與管外之對流熱傳係數比值而定。當管內對流 熱傳係數高於管外時，管壁溫度分佈傾向於均勻分佈，亦即A→0﹔反之，管壁溫度為變化 分佈。 再者，將(10)式代入(8)式右邊，再利用變數變換與分離，即可積分得到局部凝結液厚度為 4 1 0 2 2 3 2 4 1 0 2 3 2 2 3 1 2 3 1 * } ] ) sin 1 ( ) 1 ( [ 1 { } ) sin 1 ( )] ( [sin ) ( ) 1 ( 2 { )] ( [sin

∫

∫

− − − + − + = − π φ φ φ π φ φ φ φ φ φ φ δ d e e d e Bo F e Bo t (12) 另一方面，基於Bejan 所提出的熱力學第二定律理論，對流熱傳的局部熵增率定義為： 2 2 2 " _{( ) ( )} y u T y T T k S S S G _∂ ∂ + ∂ ∂ = μ (13) 將上列所算之速度與溫度分佈分程式分別代入，且積分並無因次化可得

(

)

_r

(

)

_{d H F} s Ra Ja I N N Br I Ja Ra N = + Θ + = 1/4 − 43/ / 3 2 / 2 1 , (14) 其中

( )

_φ

δ

π d F I t r =

∫

₀ ∗ 2 , 與 =

∫

π _δ∗ _φ + _{φ φ} 0 2 3_{[sin( ) ( )]} ) ( Bo d Id . (15) 為了解膜狀凝結造成的熵增變化是由熱傳遞效應或是由流動摩擦效應所主導，吾人將 引用不可逆率比參數ψ 來判別，而不可逆率比值定義如下： H F N N = ψ (16) 2.2 定值熱傳管壁與多孔隙介質管壁 考慮一具直立長軸“2a”之水平橢圓管，放置於大量靜止或由上往下緩慢流動的純飽和 蒸汽當中。若管壁為多孔隙性質材料，且已知管壁表面溫壁(T_W)低於蒸汽飽和溫度(T_sat) 將 於管壁上開始發生膜狀凝結，經過一段特定時間後即可形成一層穩態層流膜狀凝結液，由 於凝結液受到重力影響與有效表面張力作用，開始往下流動。由於管壁為多孔隙，部分凝 結液受到均勻速度(V_W)吸入管壁內。如圖一所示，至於座標系統採用 Yang[22]所建立橢圓 極座標系，建立沿橢圓周弧線方線之x 座標，以及法線方向之 y 座標，並且已知兩者對應 凝結液流之流速分量分別為u 與 v。並作以下之基本假設： (1)流體性質為均勻定值。 (2)凝結液流為穩態層流。 (3)液膜甚薄，可視液膜內溫度分佈為直線分佈。 (4)液-汽介面之黏滯力可忽略。 (5)忽略液膜內慣性力與對流兩項。 基於上述假設，在根據穩態穩流液流之方程式可由守恆原理建立如下：

連續方程式： 0 = ∂ ∂ + ∂ ∂ y v x u (17) 動量平衡方程式：

(

ρ ρ

)

[

φ

( )

φ

]

μ g Bo y u v + − − = ∂ ∂ _sin 2 2 (18) 能量平衡方程式： q udy dx d V h_{fg w} = ⎥⎦ ⎤ ⎢⎣ ⎡ _⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +

∫

δ

ρ

0 (19) 忽略液-汽介面之黏滯力下，建立邊界條件為： 0 = y , u =0, 在T =T_w 與 y=δ , =0 ∂ ∂ y u , 在T =T_sat 再者，利用動量方程式並配合上式之邊界條件，則可求得速度分佈方程式 ] ) ( 2 1 )][ ( [sin ) ( 2 2 δ δ φ φ δ μ ρ ρ y y Bo g u= − v + − (20) 將(19)式代入(18)式，再利用座標變換，並搭配無因次轉換可得到局部無因次凝結液厚度為： 3 1 3 1 2 * ) ( sin ) 1 ( 3 2 1 ) ( Re ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = − φ φ ρ ρ ρ μ δ δ Bo S s _w v (21) 式中，壁溪流參數Sw =ρVwhfg /q 接著採用Nusselt凝結理論，可將表面溫度表示為： * 3 1 Re Pr _δ ⎥⎦ ⎤ ⎢⎣ ⎡ = Δ Ra k qD T e ₍₂₂₎ 利用Yang[22]之積分公式，可計算出橢圓管表面之平均溫差如下 φ φ φ φ φ φ π π d I d I T T _f( )sin( ) _f( )sin( ) 0 0

∫

∫

Δ ⋅ = Δ (23) 根據Bejan[9]所提出的第二定律理論，對流熱傳的局部熵增率定義為： 2 2 2 " ) ( ) ( y u T y T T k S S S G ∂ ∂ + ∂ ∂ = μ (24) 將上列所算之速度與溫度分佈分程式分別代入，且積分並無因次化可得

(

)

r

(

)

d H F s Ra I N N Br I Ra N = + Θ + = 1/3 −1 Pr Re / 3 2 Pr Re / 2 1 (25) 其中

φ

δ

φ d I_r =

∫

c _∗ 0 1 與 =

∫

c ∗ + d Bo I_d φ δ φ φ φ 0 2 3_{[sin( ) ( )]} ) ( (26) 欲了解膜狀凝結造成的熵增變化是由熱傳遞效應或是由流動摩擦效應所主導，吾人將引用 不可逆率比參數ψ 來判別，而不可逆率比值定義如下：

H F N N = ψ (27) 結果與討論 3-1 可變溫之橢圓管壁

根據Lee[4]等人的實驗結果，將非均勻壁溫之變動大小 A=0 與 A=1 分別代入

θ θ) 1 cos ( A Ft = − 式中，並考慮非均勻壁溫下受到不同熱傳參數作用，即不同的Ra/Ja 值輸 入時，對系統熵增率的影響。圖2.則表示非均勻壁溫與均勻壁溫情況下對系統熵增率影響 的比較，結果顯示當系統在非均勻壁溫的條件下，整體的熵增率趨勢與均勻壁溫的狀況下 並沒有太大的變化，唯有系統的熵增率會整體的增大，其原因可歸咎於當考慮非均勻壁溫 下的條件下，會造成熱傳係數值提高，使得熵增率亦跟著增大。且橢圓離心率的變大，也 會造成熵增率的提升。而在表面張力參數 Bo 與無因次參數 Br/Θ 對熵增率的影響則顯示於 圖3.。圖中明顯可見，表面張力的作用對熵增率的影響不大，且較大的 Br/Θ 值會使熵增率 在低 Ra/Ja 下略為提升。 3-2 定值熱傳管壁與多孔隙介質管壁 為了增強其凝結熱傳特性，本節加入了壁吸流效應，加以探討此種增強凝結熱傳的可 行性。在此節中分別以 Sw 參數為 0 與 0.5 代表無壁吸流效應與有壁吸流效應的影響，並探 討其差異性。圖4.即為探討因有限溫差熱傳所造成的熵增率在不同壁吸流效應下的影響， 加入壁吸流的效應會使有限溫差熱傳造成的熵增率提高，亦即熱傳效果愈佳。且其值亦會 隨著 Ra/RePr 增加而變大。在圖 5.中亦可看出熵增率隨著 Ra/RePr 增加而變大。在圖 6.中 可看出橢圓離心率對熵增率的影響，在 e 大於 0.7 之後較為明顯。在不可逆率比部份，如圖 7 所示，具壁吸流效應下的不可逆率比皆較小於無壁吸流效應下的熵增。且隨著 Ra/RePr 變大而逐漸趨近於0。 4.結論 本文假設自然對流下，膜狀凝結情形，參考Bejan 學者所提出關於流體於物體外層流 動條件下，所提出熵增理論與分析模式，本文同時也是首次針對橢圓管膜狀凝結過程中熵 增變化方面研究，以可變壁溫、定值熱傳遞與多孔性介質材料等不同條件下，運用各種不 同無因次化參數所代表的熵增方程式，對於有限溫差熱傳與不可逆流體摩擦所帶來熵增影 響加以討論，提供熱交換器業者在高效率工程設計上的參考資料，減少地球能源浪費的窘 境。在此，將各分析之結果，綜合以下的結論： 1. 變溫管壁無因次平均熵增中，因有限溫差熱傳所影響的熵增值遠高於流體摩擦熵增值。 2. 比較等溫與變溫管壁平均熵增，若考慮橢圓管表面壁溫為非等溫條件下，熱項熵增值對 整體熵增影響更明顯。 3. 在離心率 e 方面，離心率愈大其熵增值愈高，且熱傳效益愈佳。 4. 表面張力參數在離心率 e 小於 0.7 時，幾乎可以忽略不計。 5. 在多孔性介質管壁研究上，壁吸效應明顯的影響熵增率，壁吸效果愈強時，其熵增值亦 隨之增加。

圖2. 不同溫度變動參數下 Ra/Ja 對熵增率的影響

圖3. 不同表面張力與 Br/Θ 下 Ra/Ja 對熵增的影響

圖5.壁吸效應下 Ra/RePr 對平均熵增的影響

圖6.壁吸效應下橢圓離心率對熵增率的影響

5.符號說明 A 非均勻管壁之溫度變動參數 a,b 橢圓的半長、短軸長 Bo Bond參數，一種重力與表面張力效應比值的量度

( )

φ Bo 表面張力作用函數，

[

(

2 2

) (

2

)

]

2 2 1 sin 1 ) 2 sin( 2 3 1 ) ( e e e Bo Boφ =± φ − φ − Br 布蘭克數，μu₀2 /kΔT p C 定壓比熱 e D 橢圓等效直徑，

[

φ

]

φ π π d e e a D_e =

∫

− − 0 3 2 2 2 _{(1 sin )} 1 2 Ft 非均勻溫度管壁表面之無因次溫度分佈函數 g 重力加速度 Id ,Ir 積分函數 Ja Jakob 參數，C_pΔT/h'fg k 熱傳導係數 m& 質量流率 F N 流體摩擦熵增 H N 有限溫差熵增 Nu 局部紐賽數， hD/k NS 平均熵增數 Pr Prandtl參數 q 局部熱傳遞參數 R 橢圓表面曲率半徑 Ra 瑞利數，ρ

(

ρ−ρv

)

gPrDe3 μ2 Re Reynold’s參數，4πD_eq μh_fg hfg 蒸發潛熱 h’fg 修正蒸發潛熱 s 無因次流線長 Sw 壁吸流效應之參數 G S ′′ 局部熵增率 sat T 蒸汽下之飽和溫度 w T 橢圓管壁溫度 T Δ 蒸汽介面與管壁間溫差 u,v x,y方向速度 希臘字母 ρ 凝結液體密度 v ρ 蒸汽密度 μ 動黏滯係數 θ 自橢圓直立長軸往下量起之角度 ψ 在對應θ 角度位置之表面切線與水平方位之夾角 δ 局部凝結液厚度 ∗ δ 無因次化局部凝結液厚度 Θ 無因次化溫差 ΔT Tsat ψ 平均熵增不可逆率比值 η 無因次化凝結液厚度

5.參考文獻

1. Nusselt, W., 1916, ''Die oberflachen kondensation des wasserdampers,''

Zeitsehriftdesvereines eutsher ingenieure, Vol.60, pp.541–546, 569-575.

2. Uehare, T. H. and Oda, K., 1972, ''Filmwise condensation on a surface with uniform heat flux and body force convection,'' Heat Transfer Japanese Res. 4, pp.76-83.

3. Shklover, G. G; Semyonov, V. P. and Usachyev, A. M., 1989,"Condensation on a horizontal tube with a spatially non-uniform temperature distribution,'' Heat Transfer-Soviet Research, Vol. 21, No1, pp.29-33.

4. Lee, W. C., Rahbar, S. and Rose, J. W., 1984,"Film condensation of refrigerant 113 and ethanediol on a horizontal tube- effect of vapor velocity," J. Heat Transfer, Vol.106, pp.524-530.

5. Memory, S. B. and Rose, J. W., 1991, "Free convection laminar film condensation on a horizontal tube with variable wall temperature," Int. J. Heat Mass Transfer, Vol.34, pp.2775-2778.

6. Jain, K. C. and Bankoff, S. G., 1964, "Laminar film condensation on a porous vertical wall with uniform suction velocity," Trans. ASME J. Heat Transfer, Vol. 86, pp.481-489.

7. Frankel, N. A. and Bankoff, S. G., 1967, "Laminar film condensation on a porous horizontal tube with uniform suction velocity," Trans. ASME J. Heat Transfer, Vol. 87, pp.95-102.

8. Yang, J. W. 1970, "Effect of uniform suction on laminar film condensation on a porous vertical wall," Trans. ASME J. Heat Transfer, Vol. 92, pp.252-256.

9. Bejan, A. “A study of Entropy generation in fundamental convective heat transfer,”

Transactions of the ASME, Vol.101, pp.718-725 (1979)

10. Nag, P. K., and Mukherjee, P., 1987,“Thermodynamic Optimization of Convective Heat Transfer Though a Duct with Constant Wall Temperature.” Int. J. Heat Mass Transfer, Vol.30, pp. 401-405.

11. Sahin, A. Z., 1998, “Second Law Analysis of Laminar Viscous Flow through a Duct Subjected to Constant Wall Temperature.” J. Heat Transfer, Vol.120, pp. 76-83.

12. Saouli, S., and Aiboud-Saouli, S., 2004,“Second Law Analysis of Laminar Falling Liquid Film along an Inclined Heated Plate.” Int. Comm. Heat Mass Transfer, Vol.31, pp. 879-886. 13. Adeyinka, O. B., and Naterer, G. F., 2004, ''Optimization Correlation for Entropy Production

and Energy Availability in Film Condensation.'' Int. Comm. Heat Mass Transfer, Vol.31, pp. 513-524.

14. Lin, W. W., and Lee, D. J., 2001, ''Second-Law Analysis of Vapor Condensation of FC-22 in Film Flows Within Horizontal Tubes.'' J. Chin. Inst. Chem. Engng. Vol.32, pp. 89-94

15. Dung, S. C., Tzeng, S. H., and Yang, S. A., 2006,"Entropy Generation of Free Convection Film Condensation from Downward Flowing Vapors onto a Cylinder or Sphere." Journal of

Mechanics

16. Dung, S. C., and Yang, S. A., 2006,''Second Law Based Optimization of Free Condensation Film-Wise Condensation on Horizontal Tube'', Int. Comm. in Heat and Mass Transfer. Vol. 33 , pp.636-644.

Film Condensation from Downward Flowing Vapors onto a Cylinder or Sphere." Journal of

Mechanics.

18. Tzeng, S. H., and Yang, S. A., 2007,"Second Law Analysis and Optimization for Film-wise Condensation from Downward Flowing Vapors onto a Sphere", Heat and Mass Transfer, Vol.43, pp365-369.

19. Li, G. C., and Yang, S. A., 2006, "Thermodynamic Analysis of Free Convection Film Condensation on an Elliptical Cylinder", J. of the Chinese Institute Engineers, Vol.29, No5, pp.903-908.

20. Li, G. C., and Yang, S. A., 2007,"Entropy Generation Minimization of Free Convection Film Condensation on an Elliptical Cylinder", Int. J. Thermal Sciences, Vol.46, pp.407-412.

21. Chen, B. C., Wu, J. R., Yang, S. A., 2007, ''Entropy Generation of Forced Convection Film Condensation from Downward Flowing Vapors onto a Horizontal Tube'', J. of The Chinese

Insititute Engineers (printed in 2008)

22. 楊勝安，1993，水平之平板、橢圓管與橢圓球膜狀凝結熱傳研究，成功大學，博士論

文。

23. Rohsenow, W. M., 1956, ''Heat transfer and temperature distribution in laminar film condensation,'' Trans ASME, Vol.78, pp.1645-1648.

6. 計畫成果自評 本計劃利用熱力學第二定律分析水平橢圓管凝結的熱傳效益，其研究成果與計畫所預 其之目標大致相符。且截至目前為止，關於熱力學第二定律應用在凝結熱傳方面的研究依 舊不多，因此在投稿至SCI國際期刊上皆獲得良好的評語，足以證實本研究價構之完善與在 學術價值上的重要性。唯缺少以實驗證實在實際上的數據是否誤差，這點在未來還需要投 入更多的人力與設備來增加本研究的全整性。且在目前散熱機制中，圓形管是屬於較普遍 與加工容易的散熱零件，而橢圓管僅是圓管的變形，有鑒於此，本研究針對如何有效的減 少圓管因熱傳現象所導致的熵增反應，尋找最佳熵增量分析，可作為日後業界研發參考所 用。

出席國際研討會心得報告 出國人員姓名

服務機關及職稱

楊勝安

國立高雄應用科技大學模具工程系教授兼系主任

會議時間地點 2007年7月27至29日, PRAGUE, CZECH REPUBLIC

會議名稱 International Conference on Fluid Mechanics, Heat

Transfer and Thermodynamics 發表論文題目

Entropy Generation Analysis of Free Convection Film Condensation on a Vertical Ellipsoid with Variable Wall Temperature

一、參加會議經過：

7 月 27 日:研討會由今天早上 7 點半開始報到註冊並領取資料。開幕茶會由主 席 Ardil C.教授主持，接著由 Kindler E.的教授做有關模擬和以電腦 程式設計的演講。緊接著即開始做分組口頭報告，排定的報告主題 分別為。

1. Applied Computer Science

2. Biosciences and Bioengineering

3. Modeling and Simulation

4. Computer Vision, Image and

Signal Processing

5. Embedded and Real-Time

Computing System

6. Urban ﹠Regional Planning

and Transportation

7. Electric Power and Energy

Systems

8. Ecosystem, Environment and

Sustainable Development

9. Fluid Mechanics, Heat Transfer

and Thermodynamics

10. Cybernetics, Informatics, and Systemics 總計約有 68 篇論文發表 每人發表和問題討論的時間以 20 分鐘為限。早上參加完開幕典禮 後，在會場與國際學者進行學術交流並將自己的名片與來自不同國 家的學者交換。這次參加的口頭報告時程被安排在第四梯次，也就 是今天下午的 4 點 15 分到下午 6 點 15 分。在發表過程中本人利用 約 15 分鐘發表使用熱力學第二定律分析在水平橢圓管外層流膜狀凝 結熱傳方面的研究及成效與經驗，尤其是本人所定義之橢圓座標 系，在大會上引起不少參與者的興趣。

報告人與學生在會場留影 報告人在大會發表論文之照片 7 月 28 日:今天早上及下午繼續是分組口頭報告的時程，由於研討會的內容涵 蓋各個領域，雖然每個專題發表的篇數並不多，但總量卻十分驚人。 7 月 29 日:今天早上是本次研討會的最後ㄧ組分組口頭報告，內容是關於流體 力學、熱傳學和熱力學方面之主題。在會中與許多熱流領域之國際 學者進行討論，了解國外研究之現況。總結座談時，主席 Ardil C. 教授將談論主題作綜合結論，並提出未來研究重點方向。下午自行 安排旅遊行程，參觀捷克最負盛名的布拉格古堡、小城區及查理大 橋。 二、 與會心得 參與國際學術研討會是一個學校國際交流重要指標之ㄧ，技職教育體系也開 始鼓勵學校同仁積極參與國際研討會，本人 2005 年第一次參加在東歐波蘭舉行 國際熱學研討會，2006 年雖然未獲得國科會補助出席研討會之費用，仍自行花 費前往日本富山參加國際輸送(熱流)研討會，以及今年(2007 年)於捷克布拉格 所舉辦的國際熱傳與流力研討會，整體而言，獲益良多；茲分述如下： (一) 國際研討會有較充裕的研討會與交流空間： 研討會中發表者大多數由教授親自報告，與會者也大抵為來自各國的 學有專精的教授或業界傑出研究學者。反觀國內研討會，大都由研究生做 論文發表，大部分是研究生之間的討論，教授們卻很少參與討論與指導， 研究生獲得激盪、切磋、啟發的水準進步較不明顯。 (二) 國際研討會可提升個人與學校的知名度： 參與國際研討會，可將自己的名片與國際學者交換，藉由與不同國家 的學者交流，可了解國外研究的狀況，感受國際上不管是學校或工業界目 前的研究活力，無形間促進了國際學者了解我們與學校的研究狀況；逐漸 地，個人知名度也跟著提升，被指為分組主持人甚至籌備大會的委員的機 率也會增加。個人在熱流領域雖然只發表 30 餘篇 SCI 級論文，但卻有幸被 指定為分組主持人，並推派成為下一屆 2008 HEAT 的籌備委員之ㄧ。

(三) 國際研討會可了解所屬研究領域目前與未來的研究重點方向： 一般而言，國際學術研討會，在最後一天總結座談時，均會將談論主 題作綜合結論，並提出未來研究重點方向，例如 2005 HEAT 研討會上曾明 確指出未來重點方向為微熱流或高效率熱傳，與目前國科會熱流學門所訂 定之方向雷同；因此，跨出去參與國際學術研討會，可了解世界各國熱門 研究的課題，不僅可讓自己的研究視野也更加寬廣，也可激發個人新的研 究點子。 （四） 國際研討會提升國內研究生國際交流能力： 本次捷克布拉格研討會，我的學生洪任儀(博士生)及何盈毅(碩士生) 皆有共同參與發表，經由這次的活動，我發現他們規劃、詢問等活動能力 相當好，由於他們敢於開口與外國學者交談，使英文溝通能力進步不少， 藉由這研討會，也增強了他們用英文發表研究的信心；相信明年在波蘭舉 行的 2008HEAT 研討會以在冰島舉行的 ISTP 研討會將會是我及學生一展身 手，學術交流提升研究水準的好機會。 三、 建議 感謝國科會研究計劃補助參加此次國際研討會議，遠赴國外參加研討會， 交通食宿費用是一筆極大的開銷，若無相當的補助費用贊助，研究生難以成 行，不利於參與國際學術交流以提升研究水準。另外建議類似的大型國際研討 會，如果能夠在台灣舉辦，無論是時間、金錢，都可以節省不少；另一方面也 可以解決國內研究生缺乏參與國際學術交流之窘境。建議學者參與際學術交流 時，可積極爭取在國內舉辦。當然若能由國科會形成政策及專案計畫輔助，這 或許是未來可以努力的方向。 四、 研討會之日程表與摘要

XXII. WASET International Conference Program

Time Paper Title Author

07:30 08:00

Welcome & Registration Opening Remarks on WASET 2007 Friday: July 27, 2007

08:00

09:00 Plenary Talk Conference Co-Chairman

09:00

10:00 Chair: Fikret Caliskan Session – I

10:00

10:15 Coffee Break

13:00 13:00 14:00

Lunch Break

14:00

16:00 Chair: Pavel K. Lopatin Session – III

16:00

16:15 Coffee Break

16:15

18:15 Chair: Nuri Yucel Session – IV

Saturday: July 28, 2007 08:00

11:00 Chair: Y. G. Saridakis Session – V

11:00

11:15 Coffee Break

11:15

13:15 Chair: Miguel Ángel Gómez-Nieto Session – VI

13:15

14:15 Lunch Break

14:15

16:15 Chair: Wudhichai Assawinchaichote Session – VII

Sunday: July 29, 2007

08:00

11:00 Chair: Wellington S. Mota Session – VIII

Abstract—This paper aims to perform the second law analysis of thermodynamics on the laminar film condensation of pure saturated vapor flowing in the direction of gravity on an ellipsoid with variable wall temperature. The analysis provides us understanding how the geometric parameter- ellipticity and non-isothermal wall temperature variation amplitude “A.” affect entropy generation during film-wise condensation heat transfer process. To understand of which irreversibility involved in this condensation process, we derived an expression for the entropy generation number in terms of ellipticity and A. The result indicates that entropy generation increases with ellipticity. Furthermore, the irreversibility due to finite temperature difference heat transfer dominates over that due to condensate film flow friction and the local entropy generation rate decreases with increasing A in the upper half of ellipsoid. Meanwhile, the local entropy generation rate enhances with A around the rear lower half of ellipsoid.

Keywords—Free convection; Non-isothermal; Thermodynamic second law; Entropy, Ellipsoid.

I. INTRODUCTION

HERE are two types of techniques to enhance condensation heat transfer process and thus to increase the performance of condensers. They are passive and active enhancement techniques. The passive techniques do not require the application of the external power, whereas the active techniques require activator or power supply to bring about the enhancement. When we attempt to enhance a rate of condensation, what we usually do is the increasing condensing area via fins or the formation of very thin condensation. The later is achieved by using objects having favorable surface tension due to surface curvature. One of objects having favorable surface tension is the object of elliptical cross section or vertical ellipsoids. As for this kind of passive enhancement

Sheng-An Yang is Professor and Chair of Department of Mold and Die Engineering, National Kaohsiung University of Applied Sciences. No. 415 Chien Kung Road, Kaohsiung 807 Taiwan, R.O.C. (phone: 886-7-3814526 ex5400 e-mail: [email protected]).

Ren-Yi Hung is with the Department of Mold and Die Engineering, National Kaohsiung University of Applied Sciences 415 Chien Kung Road, Kaohsiung 807 Taiwan, R.O.C. (e-mail: [email protected]).

Ying-Yi Ho. is with the Department of Mold and Die Engineering, National Kaohsiung University of Applied Sciences 415 Chien Kung Road, Kaohsiung 807 Taiwan, R.O.C. (e-mail: 1095316122@ cc.kuas.edu.tw).

of condensation heat transfer, several researches, such as Yang and Hsu [1] and Yang and Chen [2], Ali and McDonald [3], Karimi [4], and Memory et al. [5] confirmed that cylinders, fins, or extended surfaces of elliptical profiles with major axes aligned with gravity are superior to those of circular profiles.

All the above heat transfer analyses belong to the filed of energy analysis, i.e. first law analysis, but first law analysis does not account for the irreversibility or degradation of energy in the system. Second law analysis provides an effective technique for measuring and optimizing performance of a thermal system by accounting for the energy quality. Second law analysis of thermal systems is widely gaining acceptance over traditional energy methods in both industry and academia as it is developed into a set of standards for measuring the performance. Entropy generation is associated with thermodynamic irreversibility which is common in all types of heat transfer processes. Film condensation belongs to phase-change heat transfer, but little literature regarding its second-law analysis is investigated.

Adeyinka and Naterer [6] first investigated the physical significance of entropy generation in plate film condensation. Lin et al. [7] performed the second-law analysis on saturated vapor flowing through and condensing inside horizontal cooling tubes. They noted that in a tube case, an optimum Reynolds number exists at which the entropy generates at a minimum rate. Dung and Yang [8] presented the entropy generation minimization method to optimize a saturated vapor flowing slowly onto and condensing on an isothermal horizontal tube. They observed that entropy generation provides a useful parameter in the optimization of a two-phase system. More recently, we first conducted a study [9] on the local entropy generation rate of laminar free convection film condensation on an elliptical cylinder. That paper investigated how the geometric parameter-ellipticity affects local entropy-generation rate during film-wise condensation heat transfer process. The second law analysis of the film condensation outside ellipsoids still remains an unsettled question so far.

Since the current state of knowledge about second law analysis of free convection film condensation outside an ellipsoid is somewhat incomplete, this investigation into the entropy generation rate will thus help us achieve the complete thermodynamic analysis, including the first and second law

Entropy Generation Analysis of Free Convection

Film Condensation on a Vertical Ellipsoid with

Variable Wall Temperature

Sheng-An Yang, Ren-Yi Hung, and Ying-Yi Ho

T

analysis. We will derive an expression for the entropy generation number, which accounts for the combined action of the specified irreversibility. Basically, this study makes good engineering sense to focus on the irreversibility of film condensation heat transfer and try to understand the function of the entropy generation mechanism.

II. THERMAL ANALYSIS

Consider a vertical ellipsoid, with major axis “2a” in the direction of gravity and minor axis “2b”, situated in a quiescent pure vapor which is at its saturated temperatureT_sat. Moreover, the wall temperature T_w may be non-uniform and belowT_sat. Thus, condensation occurs on the wall and a continuous film of the liquid runs downward over the ellipsoid under the actions of the component of gravity, and of the surface tension forces.

Fig. 1 illustrates schematically a physical model and coordinate system where the curvilinear coordinates (x, y) are aligned along an ellipsoid surface and its normal. The following simplifications are made in the analysis:

1) The condensate film flow is laminar and steady-state. 2) The inertia effect is neglected.

3) The condensate film thickness is much smaller than the NOMENCLATURE

A the wall temperature variation amplitude Tsat saturation temperature of vapor

a semi-major axis of ellipse Tw wall temperature

b semi-minor axis of ellipse U reference velocity component in x direction

o

B Bond number, (ρ−ρv)ga2/σ u velocity component in x direction

Br Brinkman number, μu₀2/kΔT v velocity component in y direction

p

C specific heat of condensate at constant pressure Greek symbols

e

D equivalent diameter of ellipsoid δ thickness of condensate film

e ellipticity of ellipse δ∗ dimensionless thickness of condensate film

g acceleration due to gravity θ angle measured from top of the ellipsoid

h condensing heat transfer coefficient at angle φ μ absolute viscosity of condensate

fg

h′ latent heat of condensation corrected for condensate subcooling ρ density of condensate

Ja Jakob number, C_p(T_sat−T_w)/h'_fg ρv density of vapor

k thermal conductivity of condensate σ surface tension coefficient in the film

•

m condensate mass flow rate per unit length of ellipsoid φ

angle between the tangent to ellipsoid surface and the normal to direction of gravity

F

N ′′′ film flow friction irreversibility Ω dimensionless temperature difference, ΔT /T_s

H

N ′′′ heat transfer irreversibility Subscripts

S

N ′′′ the entropy generation number

Nu local Nusselt number, hD_e/k sat saturation

'' '

gen

S the volumetric entropy generation rate, V vapor

'' '

o

S characteristic transfer rate Superscripts

Ra Rayleigh number, (ρ−ρv)ρgPrDe3/μ2 * indicates dimensionless

Fig. 1 Physical model and coordinate system for condensate film flow on an ellipsoid

curvature diameter.

4) Viscous dissipation is ignored.

5) Compared with the normal conduction, the streamwise conduction is negligible.

Based on the above simplifications, the condensate film governing equations for conservations of mass, momentum, and energy are as follows:

0 = ∂ ∂ + ∂ ∂ y v x u (1) )] ( [sin ) ( 2 2 φ φ ρ ρ μ _υ g Bo y u _{=− − +} ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ (2) 0 2 2 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ y T k (3) , subject to the following boundary conditions:

0 = y ; u=0 ; T=T_w (4) δ = y ; =0 ∂ ∂ y u ; T=T_sat (5) On account of varying radius of surface curvature, the surface tension forces can be derived here, as expressed in Yang and Chen [2]: ) 2 sin( ) 1 sin 1 ( 2 3 1 ) ( ₂ 2 2 2 2 2 φ φ φ e e a e Bo Bo − − = (6)

Integrating (2) and (3) with the use of the boundary conditions gives, respectively. ) 2 1 )]( ( [sin ) ( (y) ₂ 2 2 δ δ φ φ δ μ ρ ρ _υ y y Bo g u = − + − (7) w T y T T ⎟+ ⎠ ⎞ ⎜ ⎝ ⎛Δ = δ (8)

Using (7), we can obtain the mass flow rate of condensate film.

θ π φ φ μ δ ρ ρ ρ θ π ρ υ δ sin 2 )] ( [sin 3 ) ( sin 2 3 0 r Bo g dy r u m + − = =

∫

 (9)

An energy balance at the condensate-vapor interface, as in the Nusselt-Rohsenow condensation theory, gives

θ π δ 2 rsin T k dx m d hfg Δ = ′  (10) where, h′_fg =h_fg

(

1+0.68C_pΔT h_fg

)

is the modified latent heat of condensation proposed by Rohsenow [10] to account for convection in the film. In order to derive the local film thickness δ at the circumferential arc length x in terms ofφ, we can substitute (9) into (10) and obtain

{

δ φ φ θ

}

_δ θ μ ρ ρ ρ υ _{g Bo r k} T dx d h r fg Δ = + ′ − sin )] ( [sin sin 3 ) ( ₃ (11) To solve the above equation, it is convenient at this point to express dx and rsinθ in terms of e andφ. the differential streamwise length can be written as proposed by Yang [11]

) cos(φ θ θ − = rd dx (12) Next, by using the geometric relationship for tangent to the surface ) 1 ( tan tan ₂ e − = θ φ (13) And in Yang [11]: 5 . 0 2 2 2 ] ) cos 1 ( ) 1 ( [ e e θ a r= − − (14) And with the help of (14), one may obtain the following expressions: 2 3 2 2 2 ) sin 1 )( 1 ( e e φ a dx= − − (15) Once the wall temperature distribution T_w(φ)is specified or fitted by experimental data, the mean wall temperature is really available as

( )

e πT φ φ e φ dφ D a T _w e w 2 1 ( )sin [(1 sin ) ] 2 2 2 0 2 2 _{⋅ −} − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − =

∫

(16)

and express the temperature difference across the film as ) ( ) ( ) ( _{sat w t} φ _t φ w sat T T T F TF T − = − =Δ (17) where ΔT=T_sat−T_w.

Representative numerical results for the common axisymmetric case that involves the cosine distribution of non-isothermal wall temperature variation are given as

) cos( 1 ) (φ A φ F_t = − (18) Here, the non-isothermally function is adopted from the experiment of Lee et al. [12] for circular tube. Note that

1

0≤ A≤ and the amplitude A depends largely on the ratio of the outside-to-inside heat transfer coefficients.

Inserting (12) through (18) into (11), and introducing the transformation method, we can obtain dimensionless local condensate liquid film thickness as follows.

{ } 4 1 6 13 2 2 3 4 3 1 2 6 1 2 2 3 1 4 1 ) sin 1 ( sin )] ( )[sin ( ) 1 ( 2 ) sin 1 ( )] ( [sin sin ) ( 2 ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − + − ⋅ − + = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ′ − Δ =

∫

− − − ∗ φ φ φ φ φ φ φ φ φ φ ρ ρ ρ μ δ δ υ d e Bo F e e Bo h g T ak t fg (19) As in Nusselt [13] theory, interpreting the result of model is straightforward by employing the usual idea of a local heat transfer coefficient as follows:

* 4 1 / ] [ δ Ja Ra k hD Nu= e = (20) where, 2 3 Pr ) ( μ ρ ρ ρ v g De Ra= − ' ) ( fg w sat p h T T C Ja= −

( ) 2 e v 0 D g 2 1 U μ ρ ρ− =

According to Bejan [14], together with the fifth item of above-mentioned assumptions, the entropy generation rate for convection heat transfer can be written as

2 2 2 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ = ′′′ y u T y T T k S s s gen μ (21) On the right-hand side of (21), the first term and the second term represents the entropy generation due to heat transfer and due to condensate film flow friction, respectively. Substituting (7) and (8) into (21), and assuming

sat sat sat T T T T T= + − ≈ =T_s yield 2 2 2 4 2 0 2 2 _D [sin ( )] (1 ) 4U δ φ φ δ μ δ y Bo T T T k S s s gen ⎟ + + − ⎠ ⎞ ⎜ ⎝ ⎛ Δ = ′′ (22)

Next, entropy generation number (N ′′S′ ) is the ratio of the

volumetric entropy generation rate (S'gen'' ) to a characteristics transfer rate (S_o'''). 0 S S NS gen_′′′ ′′′ = ′′′ ₍₂₃₎ where,

( )

2 s 2 e 2 0 T D T k S′′′= Δ (24) Further, by introducing the following dimensionless parameters T k Br Δ = μU20 ₍₂₅₎ s T T Δ = Ω and δ η = y (26) the entropy generation number can be expressed as:

( ) ( ) F H S N N Bo Ja Ra F Ja Ra N t ′′′ + ′′′ = − + Ω + = ′′′ ∗ ∗ 2 2 0.5 2 2 2 0.5 ) 1 ( )] ( [sin ) ( Br ) ( ) ( η φ φ δ δ φ (27)

Notably, N ′′H′ denotes the dimensionless entropy generation due to heat transfer irreversibility

( ) 2 2 0.5 ) ( ) ( ∗ = ′′′ δ φ t F Ja Ra N_H (28) F

N ′′′ denotes the dimensionless entropy generation due to fluid

friction irreversibility at the wall (y=0) and is evaluated as follows. ( )0.5 2 2 2 ) 1 ( )] ( [sin ) ( Br δ _{φ φ η} − + Ω = ′′′ ∗ Bo Ja Ra NF (29)

To understand which of the condensate flow fiction or heat transfer dominates, we introduce a criterion known as the

irreversibility distribution ratio in the following equation:

H F N N ′′′ ′′′ = ψ (30)

Fig. 2 The condensation film profile for varying A and ellipticity

Fig. 3 Irreversibility due to heat transfer for varying A and ellipticity

Fig. 4 Irreversibility due to film flow friction for varying A and ellipticity

III. RESULTS AND DISCUSSION

Fig. 2 shows the condensation film profile for varying

ellipticity and A. It demonstrates that the dimensionless condensate film thickness δ∗ decreases slightly as A increases.

Figs. 3, 4, and 5 indicate the variation of dimensionless entropy generation numbersN ′′_H′ ,N ′′_F′ , N ′′_S′ with φ π under

the surface tension effects for various ellipticities. Firstly, Fig.3 indicates that the dimensionless entropy number due to heat transfer declines with the film thickness. This may account for the fact that the finite temperature difference heat transfer via thinner film will cause the higher irreversibility. Secondly, Fig. 4 demonstrates that the dimensionless entropy number due to film flow friction varies significantly with the square of (sinφ+Bo(φ)). Note that if we ignore the effect of surface tension, Bo(φ), the maximum value of this entropy generation will occur at the mid of ellipsoid. Finally, it is clear that the local dimensionless entropy number N ′′_S′ is similar to N ′′_H′

because the entropy generation due to heat transfer dominates that due to film flow friction, as seen in Fig. 5. Besides, Fig. 5 also confirms the local entropy generation rate increases with ellipticity.

Fig. 6 shows entropy generation rate versus Bo1 and

Ja

Ra for e=0.7. The entropy generation number is markedly affected by the non-isothermal wall temperature variation. This may account for the larger temperature differences. From (27), one may clearly see that the higher value of Ra Ja produces more entropy generation because the heat transfer irreversibility varies as square root of Ra Ja. Additionally, the higher value of Bo1 yields more entropy generation because of film flow friction.

Fig. 7 shows entropy generation rate versus Br Φ and A for e=0.9. From (27), one may clearly see that the higher value of Br Φ produces more entropy generation. The local entropy generation increases slightly with the increase in Brinkman number. This can be explained as the fact that the condensate film flow friction plays an insignificant role in the entropy generation rate.

Finally, Fig. 8 indicates the dependence of the irreversibility distribution ratio with A. The irreversibility distribution ratio for the case A=0 is larger than that for the caseA=1. This may account for the more contribution to irreversibility caused by larger temperature difference heat transfer. Whenψ <1, heat transfer irreversibility dominates over the flow friction irreversibility and vice versa for ψ >1 . Increasing non-isothermal wall temperature variation amplitude will enhance the heat transfer irreversibility due to finite temperature difference heat transfer. Hence, the irreversibility distribution ratio for isothermal wall case is larger than that for the non-isothermal case.

Fig. 5 Local entropy generation for varying A and ellipticity

Fig. 6 Local entropy generation number for varying group parameters

1/Bo

Fig. 7 Local entropy generation number for varying group parameters

Φ /

Br

Fig. 8 The dependence of the irreversibility distribution ratio with A

IV. CONCLUDING REMARKS

This work performed the entropy generation analysis of the laminar film condensation on an ellipsoid under the effect of the non-isothermal wall temperature variation for various ellipticities. The conclusions from this study can be summarized as follows:

1.) The local entropy generation increases with the decreases in Bond number.

2.) The local entropy generation increases with Brinkman number.

3.) The local entropy generation rate reduces with increasing wall temperature variation amplitude.

4.) The local entropy generation rate enhances with the wall temperature variation amplitude around the rear lower half of ellipsoid perimeter.

5.) Compared to entropy generation due to film flow friction, entropy generation due to heat transfer is generally dominant in most cases.

6.) The entropy generation Increases with the ellipticity of ellipsoid.

ACKNOWLEDGMENT

Funding for this study is provided by the National Science Council of the Republic of China under contracts NSC 95-2221-E-151-064.

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