Part 2: CM3110 Transport Processes and Unit Operations I. Professor Faith Morrison. CM2110/CM Review. Concerned now with rates of heat transfer
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1 CM30 anspot Pocesses and Unit Opeations I Pat : Pofesso Fait Moison Depatment of Cemical Engineeing Micigan ecnological Uniesity CM30 - Momentum and Heat anspot CM30 Heat and Mass anspot Fait. Moison, Micigan ec U. CM0/CM0 - Reiew Concened now wit ates of eat tansfe Fait. Moison, Micigan ec U.
2 Enegy alance on a Contol Volume ateof net net ateof enegy in in enegy accumulation conection conduction poduction Conectie and conduction tems - enegy tat passes toug oundaies e.g. cemical eaction, electical cuent (fist law of temo witten on a contol olume) conduction - Fouie s law conection - due to flow 3 Fait. Moison, Micigan ec U. Fait. Moison, Micigan ec U. 4
3 s was tue in momentum tansfe (fluid mecanics) soling polems wit sell alances on indiidual contol olumes is tedious and it is easy to mae eos. Instead, we use te geneal equation, deied fo all cicumstances: Geneal Enegy anspot Equation (micoscopic enegy alance) Fait. Moison, Micigan ec U. Recall Momentum alance on CV: ate of net momentum sum of foces accumulation flowing in acting on contol ol of momentum dp dt nˆ ds F CS on CV net momentum conected out (Reynolds anspot eoem) Fait. Moison, Micigan ec U. 3
4 Recall Micoscopic Momentum alance: Equation of Motion S ds V nˆ Micoscopic momentum alance witten on an aitaily saped olume, V, enclosed y a suface, S Gis notation: P g t geneal fluid Gis notation: P g t Naie-Stoes Equation Newtonian fluid 7 Fait. Moison, Micigan ec U. Geneal Enegy anspot Equation (micoscopic enegy alance) s fo te deiation of te micoscopic momentum l alance, te micoscopic i enegy alance l is deied don an aitay olume, V, enclosed y a suface, S. S ds nˆ ˆ Gis notation: C p t S V see andout fo component notation Fait. Moison, Micigan ec U. 4
5 Enegy alance on a Contol Volume ateof net net ateof enegy out in enegy accumulation conection conduction poduction t Cˆp S Fait. Moison, Micigan ec U. 9 Pat I: Momentum ansfe Momentum d tansfe: d momentum flu elocity gadient Pat II: Heat ansfe Heat tansfe: q temal conductiity d d eat flu tempeatue gadient 0 Fait. Moison, Micigan ec U. 5
6 Fouie s law of Heat Conduction: maes efeence to a coodinate system q d d q llows you to sole fo tempeatue pofiles Gis notation Heat flows down a tempeatue gadient Flu is popotional to tempeatue gadient Fait. Moison, Micigan ec U. Geneal Enegy anspot Equation (micoscopic enegy alance) ˆ C p t ate of cange conection S conduction (all diections) souce (enegy geneated pe unit olume pe time) elocity must satisfy equation of motion, equation of continuity see andout fo component notation Fait. Moison, Micigan ec U. 6
7 7 Equation of enegy fo Newtonian fluids of constant density,, and temal conductiity,, wit souce tem (souce could e iscous dissipation, electical enegy, cemical enegy, etc., wit units of enegy/(olume time)). CM30 Fall 999 Fait Moison Souce: R.. id, W. E. Stewat, and E. N. Ligtfoot, anspot Pocesses, Wiley, NY, 960, page 39. Gis notation (ecto notation) Note: tis andout is on te we p C p S C t ˆ ˆ Catesian (yz) coodinates: p p z y C S z y C z y t ˆ ˆ Cylindical (z) coodinates: y ( ) p p z C S z C z t ˆ ˆ Speical () coodinates: p C t sin sin sin ˆ sin Fait. Moison, Micigan ec U. Fouie s Epeiments: Simple One-dimensional Heat Conduction 4 Fait. Moison, Micigan ec U.
8 Eample : Heat flu in a ectangula solid ssumptions: wide, tall sla steady state Wat is te steady state tempeatue pofile in a ectangula sla if one side is eld at and te ote side is eld at? q > H HO SIDE COLD SIDE W 5 Fait. Moison, Micigan ec U. Eample : Heat flu in a sla Solution: q c c c Constant ounday conditions? 6 Fait. Moison, Micigan ec U. 8
9 Eample : Heat flu in a sla; tempeatue ounday conditions Solution: q Constant, and depends on Vaies linealy, and does not depend on 7 Fait. Moison, Micigan ec U. n Impotant ounday Condition in Heat ansfe: Newton s Law of Cooling ul fluid omogeneous solid wall wall Wat is te flu at te wall? 8 Fait. Moison, Micigan ec U. 9
10 e flu at te wall is gien y te empiical epession nown as Newton s Law of Cooling is epession sees as te definition of te eat tansfe coefficient. q ul wall depends on: geomety fluid elocity fluid popeties tempeatue diffeence 9 Fait. Moison, Micigan ec U. ul fluid solid wall () ul ul wall wall wall ( ) in solid e tempeatue diffeence at te fluid-wall inteface is caused y comple penomena tat ae lumped togete into te eat tansfe coefficient, 0 Fait. Moison, Micigan ec U. 0
11 How do we andle te asolute alue signs? q ul wall Heat flows fom ot to cold e coodinate system detemines if te flu is positie o negatie Fait. Moison, Micigan ec U. Eample : Heat flu in a ectangula solid ssumptions: wide, tall sla steady state and ae te eat tansfe coefficients of te left and igt walls Wat is te steady state tempeatue pofile in a ectangula sla if te fluid on one side is eld at and te fluid on te ote side is eld at? ul tempeatue on left H ul tempeatue on igt > W Newton s law of cooling ounday conditions Fait. Moison, Micigan ec U.
12 Polem-Soling Pocedue micoscopic eat-tansfe polems. setc system. coose coodinate system 3. pply te micoscopic enegy alance 4. sole te diffeential equation fo tempeatue pofile 5. apply ounday conditions 6. Calculate te flu fom Fouie s law q d d 3 Fait. Moison, Micigan ec U. Eample : Heat flu in a sla Solution: q c c c Constant ounday conditions? 4 Fait. Moison, Micigan ec U.
13 Rectangula sla wit Newton s law of cooling Cs is is te same as Eample, EXCEP tee ae diffeent ounday conditions. Wit Newton s law of cooling ounday condition, we now te flu at te ounday in tems of te eat tansfe coefficient, : q 0 0 e flu is positie ut, we do not (eat flows in te now tese temps +-diection) q w 0 w 5 Fait. Moison, Micigan ec U. How do we apply tese ounday conditions? Soln fom Eample : q c c c unnown constants to sole fo, c, c. w 0 w We can eliminate te wall temps fom te C y using te solution fo (). ten sole fo c, c. 6 Fait. Moison, Micigan ec U. 3
14 4 c fte some algea, Eample 4: Heat flu in a sla c 7 Fait. Moison, Micigan ec U. Sustituting ac into te solution, we otain te final esult. Solution: (temp pofile, flu) Eample : Heat flu in a sla wit Newton s law of cooling ounday conditions (eat tansfe coefficients, ) q empeatue pofile: 8 q Rectangula sla wit Newton s law of cooling Cs Flu: Fait. Moison, Micigan ec U.
15 Eample 4: Heat flu in a sla Eample: Wat is te tempeatue in te middle of a sla (ticness =, temal conductiity = =6 U/ ft o F) if te left side is eposed to a fluid of tempeatue 0 o F and te igt side is eposed to a fluid of tempeatue 50 o F? e eat tansfe coefficients at te two faces ae te same and ae equal to U/ ft o F 9 Fait. Moison, Micigan ec U. Eample: Eample 4: Heat flu in a sla Fo eat conduction in a sla wit Newton s law of cooling ounday conditions, we setced te solution as sown. If te eat tansfe coefficients ecame infinitely lage, ow would te setc cange? Wat ae te pedictions fo () and dte flu fo tis case? w w 0 30 Fait. Moison, Micigan ec U. 5
16 ounday Conditions on Heat-ansfe Polems wall tempeatue specified o 50 ounday C ounday wall flu specified -paticula alue gien -insulating ounday -Newton s law of cooling q q q ounday ounday ounday 5 W.40 m 0 wall tempeatue/flu continuity along ounday of two diffeent mateials q q ul ounday ounday ounday ounday 3 Fait. Moison, Micigan ec U. 6
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