第八講傳遞性質(zhì)的理論與計(jì)算演示文稿當(dāng)前1頁(yè)，總共57頁(yè)。（優(yōu)選）第八講傳遞性質(zhì)的理論與計(jì)算當(dāng)前2頁(yè)，總共57頁(yè)。微觀尺度—本構(gòu)方程在微觀尺度，我們用物理量場(chǎng)描述動(dòng)量、能量和質(zhì)量的密度在時(shí)空中的分布，并通過(guò)本構(gòu)方程給出了動(dòng)量、能量和質(zhì)量的傳遞通量與速度、溫度和濃度的梯度之間關(guān)系的數(shù)學(xué)描述。在本構(gòu)方程中出現(xiàn)的表征不同物質(zhì)特性的系數(shù)稱為物質(zhì)的傳遞性質(zhì)，分別命名為粘度系數(shù)、導(dǎo)熱系數(shù)、擴(kuò)散系數(shù)。傳遞性質(zhì)隨溫度、壓力、化學(xué)成分而變化，其變化的原因和規(guī)律并不能在微觀尺度下予以解釋，需要在分子尺度下進(jìn)行探討。當(dāng)前3頁(yè)，總共57頁(yè)。分子尺度—傳遞現(xiàn)象機(jī)理我們?cè)谖⒂^層次描述的動(dòng)量、內(nèi)能和組分質(zhì)量通量，從分子層次描述，是單個(gè)分子的速度、動(dòng)能和空間位置變化的統(tǒng)計(jì)平均值。根據(jù)現(xiàn)代物理學(xué)的觀點(diǎn)，只要溫度不等于絕對(duì)零度，分子就在空間中不斷地隨機(jī)運(yùn)動(dòng)，我們稱之為分子的熱運(yùn)動(dòng)。當(dāng)前4頁(yè)，總共57頁(yè)。分子尺度—傳遞現(xiàn)象機(jī)理分子的運(yùn)動(dòng)可分解為平動(dòng)、轉(zhuǎn)動(dòng)和振動(dòng)。動(dòng)量傳遞與分子攜帶其平動(dòng)量(平動(dòng)速度與質(zhì)量的乘積)進(jìn)行平動(dòng)有關(guān)；能量傳遞包含了分子平動(dòng)、轉(zhuǎn)動(dòng)和振動(dòng)的動(dòng)能在空間的變化；而質(zhì)量傳遞則是不同分子平動(dòng)所引起的不同分子的空間數(shù)密度的變化。當(dāng)前5頁(yè)，總共57頁(yè)。傳遞性質(zhì)傳遞性質(zhì)的值可以隨化學(xué)組成、溫度和壓力而變化。并不存在普適性的通用方法可以預(yù)測(cè)各種情況下的傳遞性質(zhì)。相對(duì)而言普適性較好的估算傳遞性質(zhì)的經(jīng)驗(yàn)方法是所謂對(duì)應(yīng)狀態(tài)關(guān)聯(lián)法（參見(jiàn)

§1.3,§9.2,§17.2）。該法將對(duì)比傳遞性質(zhì)關(guān)聯(lián)為對(duì)比溫度和對(duì)比壓力的函數(shù)：當(dāng)前6頁(yè)，總共57頁(yè)。傳遞性質(zhì)這些關(guān)聯(lián)式是對(duì)應(yīng)狀態(tài)原理的直接推論：所有物質(zhì)在相同的對(duì)比熱力學(xué)狀態(tài)下具有相同的對(duì)比性質(zhì)。Allsubstanceshavethesamereducedpropertiesatthesamereducedthermodynamicstate.這項(xiàng)原理是相似理論的應(yīng)用，其可靠性取決于所選的物理量是否涵蓋了決定欲預(yù)測(cè)性質(zhì)所需的全部物理量。當(dāng)前7頁(yè)，總共57頁(yè)。動(dòng)量傳遞—粘度的理論與計(jì)算動(dòng)量傳遞是只存在于流體中的現(xiàn)象，流體的分子之間的相互約束較為寬松，分子能夠在空間中進(jìn)行平動(dòng)。分子平動(dòng)時(shí)自然攜帶其平動(dòng)量實(shí)現(xiàn)空間遷移，而遷移的總動(dòng)量，則與分子平動(dòng)的平均速度，分子平動(dòng)速度的分布，以及分子的數(shù)密度都有關(guān)。當(dāng)流體的相態(tài)不同時(shí)，分子間約束的差異導(dǎo)致上述參數(shù)有很大差別，因而流體的粘度以及粘度與溫度和壓力的關(guān)系就有顯著區(qū)別，相應(yīng)的理論模型與計(jì)算公式也就有所不同。當(dāng)前8頁(yè)，總共57頁(yè)。粘度的理論與計(jì)算—低壓氣體低壓氣體的特點(diǎn)是分子間距很大，分子間約束最為寬松，每一個(gè)分子在分子間作用迫使其明顯改變平動(dòng)速度(碰撞)前能夠在空間中運(yùn)動(dòng)遠(yuǎn)大于其直徑的距離(自由行程)，影響動(dòng)量傳遞的因素是分子的平均平動(dòng)速度分布和分子間的相互作用。對(duì)這兩個(gè)因素提出不同的物理模型，就可以得到不同的粘度理論和計(jì)算公式。當(dāng)前9頁(yè)，總共57頁(yè)。低壓氣體—簡(jiǎn)單氣體分子運(yùn)動(dòng)論物理模型1）氣體分子是直徑d質(zhì)量m的剛性球體。2）分子質(zhì)心間距離大于2d時(shí)，分子間沒(méi)有作用力；分子質(zhì)心間距離等于2d時(shí)，發(fā)生剛性碰撞。3）分子的運(yùn)動(dòng)行為可用平衡態(tài)統(tǒng)計(jì)規(guī)律描述。當(dāng)前10頁(yè)，總共57頁(yè)。簡(jiǎn)單氣體分子運(yùn)動(dòng)論

(2)數(shù)學(xué)模型平衡態(tài)下氣體分子的平動(dòng)速度服從Maxwell-Boltzmann分布，平均平動(dòng)速度為:(1.4-1)氣體分子對(duì)單位面積表面的撞擊頻率為:(1.4-2)氣體分子在兩次碰撞間的平均自由行程為:(1.4-3)當(dāng)前11頁(yè)，總共57頁(yè)。簡(jiǎn)單氣體分子運(yùn)動(dòng)論

(3)到達(dá)y=y0平面的分子的最后一次碰撞到y(tǒng)0平面的平均距離為(參見(jiàn)右下圖):(1.4-4)以上結(jié)果都是在平均平動(dòng)速度處處為零的條件下獲得的。假定其對(duì)存在平均速度梯度的情況仍然可用。當(dāng)前12頁(yè)，總共57頁(yè)。簡(jiǎn)單氣體分子運(yùn)動(dòng)論

(4)考慮y0平面上的單位面積S，在單位時(shí)間內(nèi)從y>y0區(qū)間穿過(guò)S的分子有Z個(gè)，這些分子進(jìn)行最后一次碰撞的平均位置在y0+a平面，其在x方向的平均速度分量為vx|y+a。在同一時(shí)間內(nèi)必然也有Z個(gè)分子從y<y0區(qū)間穿過(guò)S,其在x方向的平均速度分量為vx|y-a。于是單位時(shí)間內(nèi)通過(guò)單位面積S傳遞的x方向動(dòng)量為(1.4-5)當(dāng)前13頁(yè)，總共57頁(yè)。簡(jiǎn)單氣體分子運(yùn)動(dòng)論

(5)(1.4-7)作為一級(jí)近似，我們有(1.4-6)把Z和vx的表達(dá)式代入式(1.4-5)，得與牛頓粘性定律比較，我們有(1.4-8)當(dāng)前14頁(yè)，總共57頁(yè)。簡(jiǎn)單氣體分子運(yùn)動(dòng)論

(6)(1.4-9)把和

的表達(dá)式代入式(1.4-8)，得式中稱為碰撞截面積(CollisionCrossSection),表征分子發(fā)生相互作用的范圍(參見(jiàn)右圖)。當(dāng)前15頁(yè)，總共57頁(yè)。簡(jiǎn)單氣體分子運(yùn)動(dòng)論

(7)將式(1.4-8)與實(shí)驗(yàn)數(shù)據(jù)比較，有以下結(jié)果：式(1.4-8)表明粘度與壓強(qiáng)無(wú)關(guān)，當(dāng)壓強(qiáng)小于1MPa時(shí)與實(shí)驗(yàn)結(jié)果符合良好。當(dāng)壓強(qiáng)大于1MPa時(shí)，因理想氣體近似已不適用，粘度隨壓強(qiáng)增大。式(1.4-8)表明粘度正比于溫度的平方根，實(shí)驗(yàn)數(shù)據(jù)則顯示粘度隨溫度增大的冪指數(shù)大于0.6，增長(zhǎng)更快。當(dāng)前16頁(yè)，總共57頁(yè)。Chapman—Enskog理論

(1)Chapman和Enskog分析了式(1.4-8)所依據(jù)的簡(jiǎn)單分子運(yùn)動(dòng)論的物理模型，針對(duì)該模型最主要的兩個(gè)誤差來(lái)源：采用了平衡態(tài)的分子速度分布，而傳遞過(guò)程是非平衡態(tài)過(guò)程；采用了剛球分子的完全彈性碰撞模型，而實(shí)際分子間的作用是遠(yuǎn)吸近斥的“柔性碰撞”過(guò)程。當(dāng)前17頁(yè)，總共57頁(yè)。Chapman—Enskog理論

(2)提出了相應(yīng)的改進(jìn)措施：采用線性校正的Maxwell-Boltzmann分布(近平衡態(tài)分子速度分布)代替平衡態(tài)分布：(D.4-1)式中校正因子是傳遞推動(dòng)力的線性齊次函數(shù)，可表示為當(dāng)前18頁(yè)，總共57頁(yè)。Chapman—Enskog理論

(3)(D.4-2)式中d稱為“廣義擴(kuò)散推動(dòng)力”，表達(dá)式為(D.4-3)當(dāng)前19頁(yè)，總共57頁(yè)。Chapman—Enskog理論

(4)2)采用Lennard-Jones勢(shì)函數(shù)表征分子間的相互作用勢(shì)能場(chǎng)：式中為是分子的特征直徑，為特征能量。右圖為L(zhǎng)ennard-Jones勢(shì)函數(shù)的圖像。當(dāng)前20頁(yè)，總共57頁(yè)。Chapman—Enskog理論

(5)Chapman-Enskog得到的低壓氣體粘度公式為式中稱為粘度的碰撞積分因子，表征分子間作用勢(shì)能場(chǎng)對(duì)“碰撞”過(guò)程的影響，其值隨溫度升高而緩慢減小(附錄E)，亦可用下式計(jì)算：(1.4-14)當(dāng)前21頁(yè)，總共57頁(yè)。粘度的理論與計(jì)算—小分子液體小分子液體的特點(diǎn)是分子間距較小，分子間有較強(qiáng)的相互作用，大多數(shù)分子都不能在空間中自由平動(dòng)，而時(shí)時(shí)處于“約束”狀態(tài)。分子間的相對(duì)距離不易發(fā)生變化，但相對(duì)位置則可以發(fā)生變化。液體的分子動(dòng)力學(xué)理論相當(dāng)復(fù)雜，且其結(jié)果亦不易于應(yīng)用。教材上介紹了基于簡(jiǎn)單物理模型的Eyring理論，可用于液體粘度的粗略估算。當(dāng)前22頁(yè)，總共57頁(yè)。小分子液體—Eyring理論物理模型1）液體分子在空間中形成可變形的擬晶格結(jié)構(gòu)，但結(jié)構(gòu)中存在大量的空穴。分子間相互作用勢(shì)能形成約束分子平動(dòng)的勢(shì)壘，使分子處于勢(shì)阱中，大多數(shù)分子只能在平衡位置鄰域里振動(dòng)。2）小部分動(dòng)能較高的分子能夠沖破勢(shì)壘的約束，在擬晶格分子的間隙中平動(dòng)。3）剪切應(yīng)力使特定方向上的勢(shì)壘高度發(fā)生畸變，從而改變分子沖破勢(shì)壘的概率。當(dāng)前23頁(yè)，總共57頁(yè)。Eyring理論

(2)Eyring的液體擬晶格模型及分子間能量勢(shì)壘在剪切應(yīng)力下的畸變。當(dāng)前24頁(yè)，總共57頁(yè)。Eyring理論

(3)數(shù)學(xué)模型靜止液體中分子沖破勢(shì)壘在晶格間平動(dòng)的頻率可用速率方程描述:(1.5-1)當(dāng)液體中存在剪應(yīng)力場(chǎng)時(shí)，分子順著力的方向運(yùn)動(dòng)時(shí)剪應(yīng)力做的功將增強(qiáng)分子沖破勢(shì)壘的能力，等價(jià)于該方向的勢(shì)壘被減弱；反之在逆著剪應(yīng)力的方向上的勢(shì)壘被增強(qiáng)。該現(xiàn)象的數(shù)學(xué)描述為：當(dāng)前25頁(yè)，總共57頁(yè)。Eyring理論

(4)(1.5-2)故分子在順(+)/逆(-)剪應(yīng)力方向的平動(dòng)頻率為：(1.5-3)于是大量分子在A、B兩層位置之間沿剪應(yīng)力方向的平均平動(dòng)速度差等于順、逆方向的平動(dòng)頻率差乘以每次平動(dòng)的距離a：當(dāng)前26頁(yè)，總共57頁(yè)。Eyring理論

(5)(1.5-4)因?yàn)锳、B層間距離很小，故速度梯度可表為：(1.5-5)將式(1.5-3)代入式(1.5-5)，得(1.5-6)該式表明液體的剪切應(yīng)力與速度梯度之間并不是線性齊次函數(shù)關(guān)系，不遵循牛頓粘性定律。當(dāng)前27頁(yè)，總共57頁(yè)。Eyring理論

(6)故式(1.5-6)中的雙曲正弦函數(shù)可展開(kāi)為泰勒級(jí)數(shù)并只保留第一項(xiàng)：此結(jié)論顯然與大多數(shù)實(shí)驗(yàn)事實(shí)不符。考慮在通常的情況下有將其代入式(1.5-6)，得：當(dāng)前28頁(yè)，總共57頁(yè)。Eyring理論

(7)整理得(1.5-7)與牛頓粘性定律比較，有當(dāng)前29頁(yè)，總共57頁(yè)。Eyring理論

(8)式(1.5-7)中的參數(shù)a、和既難以理論計(jì)算又難以直接實(shí)驗(yàn)測(cè)定，因此需要與可以測(cè)定的物性參數(shù)相關(guān)聯(lián)。a/應(yīng)該在1的量級(jí)，不妨取其等于1。應(yīng)該僅占液體的蒸發(fā)內(nèi)能增量的一部分，故有經(jīng)驗(yàn)公式(1.5-10)(1.5-8)再引入蒸發(fā)內(nèi)能增量與蒸發(fā)焓和正常沸點(diǎn)之間的Trouton規(guī)則：當(dāng)前30頁(yè)，總共57頁(yè)。Eyring理論

(9)我們得到液體粘度的估算式(1.5-11)此式與早已得到成功應(yīng)用的Underwood液體粘度經(jīng)驗(yàn)關(guān)聯(lián)式具有相同的函數(shù)形式。這一形式吻合對(duì)Eyring理論和Underwood關(guān)聯(lián)式的合理性都提供了積極的支持。當(dāng)前31頁(yè)，總共57頁(yè)?！?.4ThermalConductivityofLiquids

Bridgman’Theory(1)PhysicalPictureTheliquidmoleculesareassumedbeingarrangedinacubiclattice,whichisborrowedfromthetheoryofsolidstructure,withoscillationabouttheirbalanceposition,rotationandvibrationabouttheirmasscenters.當(dāng)前32頁(yè)，總共57頁(yè)。§9.4ThermalConductivityofLiquids

Bridgman’Theory(2)Amoleculewithstrongeroscillation,rotationandvibrationmayinfluenceitsneighborsbyenhancingtheirsimilarmotions,whichmeansitmaytransportitskineticenergytotheneighbors.Bythisway,theenergyofchaoticthermalmotionofmoleculesistransportedinthemolecularlattice.Meanwhile,ifanorderedoscillationofmoleculesissuperposedonthechaoticthermalmotion,theenergycorrespondingtotheorderedmotionshouldbetransportedinthesameway.Thesoundwaveissuchamotion.當(dāng)前33頁(yè)，總共57頁(yè)。§9.4ThermalConductivityofLiquids

Bridgman’Theory(3)MathematicModelIntheabovephysicalpicture,thekineticenergyofamoleculewillbetransportedadistance,a,ineachtimeofinteractionwithaneighbor,whichissimilartothatoccursintherigid-spheregastheory.Bridgmanthenborrowedtheexpressionofthermalconductivityfromthesimplekinetictheoryofgaseswithamodificationthatthemeanvelocityinarbitrarydirection,,wasreplacedbythemeanvelocityiny-direction,

,becausethemotioninalatticeisnotarbitrary.

Thatis當(dāng)前34頁(yè)，總共57頁(yè)。§9.4ThermalConductivityofLiquids

Bridgman’Theory(4)Theheatcapacityatconstantvolumeofaliquidmoleculeisaboutthesameasforastructureparticleofsolidathightemperature,

.Thelatticelengthisapproximately

.Themeanvelocityshouldbeequaltothesonicvelocityvs.InsertingtheseintoEq.(9.4-1),weget(9.4-1)(9.4-2)Comparisonofthisformulawithexperimentaldataforpolyatomicliquidsshowsagoodagreementifthecoefficient3ischangedto2.8.當(dāng)前35頁(yè)，總共57頁(yè)。Thisequationislimitedtodensitieswellabovethecriticaldensity,becausetheassumptionofquasi-latticestructureisvalidonlyforstronglyconfinedmolecules.Thesuccessofthisequationforpolyatomicfluidsseemstoimplythattheenergytransferininteractionsofpolyatomicmoleculesisincomplete,sincetheheatcapacityusedhere,,

islessthantheheatcapacitiesofpolyatomicliquids.§9.4ThermalConductivityofLiquids

Bridgman’Theory(5)Thepracticallyapplicableresultsthenis(9.4-3)當(dāng)前36頁(yè)，總共57頁(yè)。Thisdeductionmaybecomprehendedrationallybecausetheinfluenceofrotationofamoleculeonitsneighborsshouldbeweakerthanthatofoscillation.Furtherweakeristheinfluenceofvibrationofmolecularsize,especiallythevibrationsresultingfrommultiplefreedomsofapolyatomicmoleculewhichmaycounteractwitheachother.Thevelocityoflow-frequencysoundisgivenby§9.4ThermalConductivityofLiquids

Bridgman’Theory(6)(9.4-4)當(dāng)前37頁(yè)，總共57頁(yè)。§9.5Thermalconductivitiesofsolids

Introduction(1)Insolids,moleculesaremorestronglyconfinedbyinteractionpotentialsthanthatinliquids,suchthattheycouldonlyoscillateabouttheirbalancepositionsinlimitedamplitudes.Becausetherearevarietiesofstructuresofsolids,whicharefromsimpletoverycomplicated,thequantitativedependenceofmoleculartransportpropertiesontemperatureandothermaterialpropertiescouldnotbeestimatedwithoneortwotheoreticalmodels.Thus,thermalconductivitiesofsolidshavetobemeasuredexperimentally.當(dāng)前38頁(yè)，總共57頁(yè)?！?.5Thermalconductivitiesofsolids

Introduction(2)Formostsolidstheprincipalmechanismsforenergytransportaretheoscillationandrotation,perhapsvibration,ofstructuralparticles.Anexceptionisthatfreeelectronsarethemajorheatcarriersinpuremetals.Ingeneral,Metalsarebetterheatconductorsthannonmetalsbecausefreeelectronscanmovefastthroughthelatticeandtransferenergybetweenstructuralparticles.Crystalline

materials,withanorderedlatticeresultinginstronginteractionsbetweenstructuralparticles,conductheatmorereadilythanamorphousmaterials.當(dāng)前39頁(yè)，總共57頁(yè)。§9.5Thermalconductivitiesofsolids

Introduction(3)Inporoussolids,thethermalconductivityisstronglydependentontheporestructure,andthefluidcontainedinthepores.Dryporoussolidsareverypoorheatconductorsbecauseofweakconductivityforthedrygascontainedintheporesandarethereforeexcellentforthermalinsulation.Theconductivitiesofnonmetalsincreasewithincreasingtemperaturebecauseoftheenhancementofthermalmotionsofmoleculesandotherstructuralparticles,whereastheconductivitiesofmostpuremetalsdecreasethoughthermalmotionsofatomsenhancetoo.當(dāng)前40頁(yè)，總共57頁(yè)。Wiedemann-Franz-LorenzEquation

ThermalconductivitiesofpuremetalsInpuremetals,therearelotsoffreeelectronsmovingveryfastintheintervalofatomswhichareconfinedtoalatticestructure.Thefreeelectrons,astheymovepastanatom,mustinteractwiththeatomsothattheenergystatesbothofelectronsandatomwillbeinfluencedwitheachother.Suchthat,asanelectronmovespasttwoatomsinseries,theelectronalsotransfersthethermalenergybetweenthetwoatomswhileittransferselectricchargeinspace.當(dāng)前41頁(yè)，總共57頁(yè)。Wiedemann-Franz-LorenzEquation

ThermalconductivitiesofpuremetalsWhentemperaturerises,theenhancementofchaoticmotionsofatomsyieldsstrongerenergybarriersinthelatticethatthefastmotionoffreeelectronsthroughthelatticeisrestrictedmorethanatlowertemperature.Therefore,bothratesoftheheattransferandtheelectricitytransferreduce.Thetotalrateofheattransferincludethecontributionsofthermalmotionofatomsandfastmotionoffreeelectrons.Thereductionofheatconductivitywithtemperatureincreasingmeansthatthecontributionoffreeelectronsisdominantunderusualconditions.當(dāng)前42頁(yè)，總共57頁(yè)。Wiedemann-Franz-LorenzEquation

ThermalconductivitiesofpuremetalsThethermalconductivitykandtheelectricalconductivityke

ofpuremetalsarerelatedapproximatelybyso-calledWiedemann-Franz-Lorenzequation.(9.5-1)(9A.6-1)ThevalueofLorenznumbercanbederivedbyuseofkinetictheoryto“electrongas”as當(dāng)前43頁(yè)，總共57頁(yè)?！?.6

EffectiveThermalConductivityofCompositeSolids（1）Wehavediscussedhomogeneousmaterials.Nowweturnourattentionbrieflytothethermalconductivityoftwo-phasesolids——onesolidphasedispersedinasecondsolidphase,orsolidscontainingpores.Forsteadyconductionthesematerialscanberegardedashomogeneousmaterialswithaneffectivethermalconductivitykeff,andthetemperatureandheatfluxcomponentsarereinterpretedastheanalogousquantitiesaveragedoveravolumethatislargewithrespecttothescaleoftheheterogeneitybutsmallwithrespecttotheoveralldimensionsoftheheatconductionsystem.當(dāng)前44頁(yè)，總共57頁(yè)。§9.6

EffectiveThermalConductivityofCompositeSolids（2）

Maxwellfirstconsideredamaterialmadeofspheresofthermalconductivityk,embeddedinacontinuoussolidphasewiththermalconductivityko.Thevolumefraction

ofembeddedspheresistakentobesufficientlysmallthatthespheresdonot"interact"thermally;thatis,oneneedstoconsideronlythethermalconductioninalargemediumcontainingonlyoneembeddedsphere.Forsmallvolumefraction

,theconductivityofheterogeneoussolidswasestimatedby(9.6-1)當(dāng)前45頁(yè)，總共57頁(yè)?！?.6

EffectiveThermalConductivityofCompositeSolids（3）Forlargevolumefraction

,Rayleighshowedthat,ifthespheresarelocatedattheintersectionsofacubiclattice,thethermalconductivityofthecompositeisgivenby(9.6-2)ComparisonofthisresultwithEq.(9.6-1)showsthattheinteractionbetweenthespheresissmall,evenat=/6,themaximumpossiblevalueofforthecubiclatticearrangement.ThereforethesimplerresultofMaxwellisoftenused,andtheeffectsofnonuniformspheredistributionareusuallyneglected.當(dāng)前46頁(yè)，總共57頁(yè)?！?.6

EffectiveThermalConductivityofCompositeSolids（4）Fornonsphericalinclusions,however,Eq.(9.6-1)doesrequiremodification.Thusforsquarearraysoflongcylindersparalleltothezaxis,Rayleighshowedthethermalconductivitytensoris(9.6-3)(9.6-4)當(dāng)前47頁(yè)，總共57頁(yè)?！?.6

EffectiveThermalConductivityofCompositeSolids（5）Thecompositesolidcontainingalignedembeddedcylindersisanisotropic.TheeffectivethermalconductivitytensorhasbeencomputeduptoO(2)foramediumcontainingspheroidalinclusions.Forcomplexnonsphericalinclusions,oftenencounteredinpractice,noexacttreatmentispossible,butsomeapproximaterelationsareavailable.Forsimpleunconsolidatedgranularbedsthefollowingexpressionhasprovensuccessful:(9.6-5)當(dāng)前48頁(yè)，總共57頁(yè)?！?.6

EffectiveThermalConductivityofCompositeSolids（6）inwhich(9.6-6)Thegkareshapefactorsforthegranulesofthemedium,andtheymustsatisfy:g1+g2+g3=1Forspheres:g1+g2+g3=1/3Forunconsolidatedsolids:g1+g2=1/8,g3=3/4Thestructureofconsolidatedporousbeds-forexample,sandstones——isconsiderablymorecomplex.Somesuccessisclaimedforpredictingtheeffectiveconductivityofsuchsubstances,butthegeneralityofthemethodsisnotyetknown.當(dāng)前49頁(yè)，總共57頁(yè)?！?.6

EffectiveThermalConductivityofCompositeSolids（7）Forsolidscontaininggaspocket,thermalradiationmaybeimportant.Thespecialcaseofparallelplanarfissuresperpendiculartothedirectionofheatconductionisparticularlyimportantforhigh-temperatureinsulation.Forsuchsystemsitmaybeshownthat(9.6-7)whereistheStefan-Boltzmannconstant,k1isthethermalconductivityofthegas,andListhetotalthicknessofthematerialinthedirectionoftheheatconduction.當(dāng)前50頁(yè)，總共57頁(yè)?！?.6

EffectiveThermalConductivityofCompositeSolids（8）

Forgas-filledgranularbeds,sincethethermalconductivitiesofgasesaremuchlowerthanthoseofsolids,mostofthegas-phaseheatconductionisconcentratednearthepointsofcontactofadjacentsolidparticles.Asaresult,thedistancesoverwhichtheheatisconductedthroughthegasmayapproachthemeanfreepathofthegasmolecules.Whenthisistrue,theconditionsforthedevelopmentsof§9.3areviolated,andthethermalconductivityofthegasdecreases.Veryeffectiveinsulatorscanthusbepreparedfrompartiallyevacuatedbedsoffinepowders.當(dāng)前51頁(yè)，總共57頁(yè)?！?.6

EffectiveThermalConductivityofCompositeSolids（9）Forcylindricalductsfilledwithgranularmaterialsthroughwhichafluidisflowing,theeffectivethermalconductivitiesintheradialandaxialdirectionsarequitedifferentandaredesignatedbykeff,rrandkeff,zz.Conduction,convection,andradiationallcontributetotheflowofheatthroughtheporousmedium.Forhighlyturbulentflow,theenergyistransportedprimarilybythetortuousflowofthefluidintheintersticesofthegranularmaterial;thisgivesrisetoahighlyanisotropicthermalconductivity.Forabedofuniformspheres,theradialandaxialcomponentsareapproximately(9.6-8,9)v0isthesuperficialvelocityofthesphericalparticles;Dpisthediameterofthesphericalparticles.當(dāng)前52頁(yè)，總共57頁(yè)。§17.5

TheoryofDiffusioninColloidalSuspensions（1）Nextweturntothemovementofsmallcolloidalparticlesinaliquid.Specificallyweconsiderafinelydivided,dilutesuspensionofsphericalparticlesofmaterialAinastationaryliquidB.WhenthespheresofAaresufficientlysmall,butstilllargewithrespecttothemoleculesofthesuspendingmedium,thecollisionsbetweenthespheresandthemoleculesofBwillresultinanerraticmotionofthespheres.ThisrandommotionisreferredtoasBrownianmotion.Themovementofeachspherecanbedescribedbyanequationofmotion,calledtheLangevinequation:當(dāng)前53頁(yè)，總共57頁(yè)?！?7.5

TheoryofDiffusioninColloidalSuspensions——Langevinequation(1)inwhichuAistheinstantaneousvelocityofthesphereofmassm.Theterm-uAgivestheStokes'lawdragforce,=6BRAbeingthe"frictioncoefficient."FinallyF(t)istherapidlyoscillating,irregularBrownianmotionforce.Eq.(17.5-1)cannotbe"solved"intheusualsense,sinceitcontainstherandomlyfluctuatingforceF(t).EquationssuchasEq.(17.5-1)a