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This 0 page Class Notes was uploaded by Pietro Jones on Monday November 2, 2015. The Class Notes belongs to PTFE 3210 at Georgia Institute of Technology - Main Campus taught by Wallace Carr in Fall. Since its upload, it has received 25 views. For similar materials see /class/233938/ptfe-3210-georgia-institute-of-technology-main-campus in Polymer Science at Georgia Institute of Technology - Main Campus.

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MSEPTFE 3210 Course Review for Final Exam Final Exam Period 3 Monday December 9 2013 250 to 540 Textbook Theodore Bergman Adrienne Lavine Frank Incropera and David DeWitt Fundamentals of Heat and Mass Transfer Sixth or Seventh edition John Wiley amp Sons 2011 ISBN 13 978047050197 9 Book is required 1209 M Final Exam Period 2 0pm 2 540pm Grading Final grade will be the average of the 7 highest grades each worth maximum of 100 points but homework grade will not be dropped Homework One Grade Quiz 1 One Grade Quiz 2 One Grade Quiz 3 One Grade Quiz 4 One Grades Final Exam Three Separate Grades Each grade counts 100 points Grading Scheme Grading will be no tougher than this A 90 100 B 80 89 C 7079 D 6069 Chapter 1 Introduction heat transfer heat conduction Fourier s law thermal conductivity relative values convection forced convection free convection also referred to as natural convection Newton s law or Newton s law of cooling heat transfer coefficient relative values Chapter 1 Introduction thermal radiation equation for thermal emission of a blackbody equation for thermal emission of concentric bodies blackbody graybody emissivity steady state transient ux heat ux mass ux mass transfer rate pVA Conservation of energy First Law of Thermodyamics Summa of Important Equations Conduction 6T 2 194 q 8x Fourier s Law Convection q hATs Too Radiation q TAT4 q 0847 T004 Fm 9 1 S 9 8 00 9 blackbody Thermal Conductivity Relative values of thermal conductivity for various states of mater at normal temperatures and pressure l 1 1 0 1 DD 1000 Thermal conductivity WImK FIGURE 24 Range of thermal conductivity for vaxious states of matter at normal temperatures and pressure Table 11 Typical values of the convection heat transfer coef cient h PROCESS Wm2 39 K Free convection Gases 2 25 Liquids 501000 Forced convection Gases 75 250 Liquids 50 20000 Convection with phase change Boiling or condensation 2500 100000 132 The Surface Energv Balance q md Surroundings T5111 gt q mm Fluid 1100 T00 T00 Control surfaces Fig 9 The energy balance for conservation of energy at the surface of a medium heat transfer rate rate of energy generation per unit volume heat transfer rate per unit length heat ux Wm3 Wm Wm2 Chapter 2 Introduction to Conduction Review 21 The Conduction Rate Equation 22 The Thermal Properties of Matter 221 Thermal Conductivity 222 Other Relevant Properties thermal diffusivity 23 The Heat Diffusion Equation 24 Boundary and Initial Conditions 25 Summary TERMINOLOGY 39HME Steady State Unsteady Transient SPACE One dimensional Twodimensional Threedimensional Heat diffusion or conduction equation Includes effects of Heat conduction Heat storage Heat generation 2K8T2 Kai 2K5Tjqpcal ax ax ay ay 82 82 at This equation is referred to as the general form in Cartesian coordinates of the HEAT DIFFUSION EQUATION or Heat Conduction Equation Figure Differential control volume dr rd dz for 12 r 6r k conduction analysis in cylindrical coordinates r I z raljii 39 6T a r26 63 62 62 q pcpat 224 Kyquot qrdr i gram 5 I luby I 1 739 V 8 Figure Differential control volume dr rsin6d rd6 for conduction analysis in spherical coordinates r I 9 1 a 2M 1 6 6T kr k r2 or or rzsm26 6415 6415 1 i ksin6 q pc rzsin666 66 p at 227 If K constant then 82T Lial E 219 8x2 832 622 K aar qquot K where or called Thermal Diffusivity Governs Conduction heat flow in solids having uniform physical properties Significance of or K lt Proportional to heat conduction or OC lt Proportional to heat storage Ability of body to conduct thermal energy relative to its ability to store thermal energy High or gt Responds quickly to changes in thermal environment Body will quickly reach equilibrium after perturbation Low or gt Responds more slowly taking longer to reach a new equilibrium condition Typical Boundary Conditions 80 The variation of temperature in a body depends on the conditions at the boundaries of the body If there are variations of T with time then Txt will depend on the conditions existing in the body at some initial time The number of BC and initial conditions depend on the order of the differential equation governing the problem being considered Example 161 aZT a at 6x2 Boundary conditions Initial condition Need 2 boundary conditions and 1 initial condition TABl 121 Bmmdar conditions for the heat dif lsiuu Equation at the surfacc x 0 l Conmm surface temperature rm n 7 T 224 2 Consmm suxmcc heat ux at Finns heAl nx k M u 7 7 i i q a 3x 1 u bl Adulme or lmulmcd surface 226 3 Com action HH UCC condmun 227 hl39lx 7 T0r u k 0x Boundary Condition when Radiation is Neglected qcondw qcondf qconv Velocity of the fluid layer adjacent to the wall is zero Heat transfer per unit area between surface and the fluid layer must be by conduction alone 6T 6T qw qcondf y0 qconv T00 w y0 Boundary Condition when Radiation and Convection are Important x0 qgond s qgonv qr ad Fluid TOO Solid x0 I I I CIcond S CIconv CIrad Chapter 3 OneDimensional SteadyState Conduction Review 31 The Plane Wall 32 Skipped 33 Radial Systems 34 Summary of OneDimensional Conduction Result 35 Conduction with Thermal Generation 36 Heat Transfer from Extended Surfaces 37 The Bioheat Equation Skip Rest of Chapter Steady State Heat Flow Through a Plane Wall without internal heat generation Assumptions 1 A Constant 2 K Constant 3 1dimensional in x 4 Steady state 5 No generation Find 1 An expression for q TxL2 3 Tx N 23 Governing Equation 62T 62T aZT q 1 6T 5X2 332 022 k a at aZT 5x2 0 1 at xo TTH 6T 2 at xL TTL C1 ax TClxC2 Apply BC 1 THC10C2 e C2TH 239 TLC1LC2C1LTH 9 C1 THTLL 24 Temperature distribution T TH TH TL 6 Linear relationship TH TL 6X L q KA KAMKA 6X L L TX L2 TH TH TL 2 TH M Q L 2 TXL2TH q q 2amp4 WhCI CATZTH TL AT E Where RT LKA LKA R T 25 Electrical Circuit R V iR 9 l VR i current flow V Potential R Resistance Thermal Circuit RT q ATR q Heat transfer rate q AT Potential AT R Thermal resistance 26 i R1 R2 R3 E1 39 E2 39 E3 E EE1E2 E3 EiRliR2iR2 iR1R2R2 EiR eq Req R1 R2 R2 In general for a series of resistances Req R1R2R2Rn 27 Parallel circuit iT i1i2 iT ER1ER2 E1R11R2 iT EReq Thus E 1Req 1R11R2 In general L E J LZLLLL x W L Req R1 R2 R3 Rn Req 28 Thermal Resistance for Convection Assumptions 1 Steady state 2 h is known q 3 A is known Too AT AT qhAltTs Twgt 1 hA R AT Ts TOO 1 TTs R hA Thermal resistance for convection 29 Thermal Circuits Should be able to use thermals circuits to solve steady state problems involving composite structures with convection at surfaces with components in parallel andor series for the following geometries Plane Wall Cylindrical Wall Spherical Wall 30 Overall Heat Transfer Coefficient See pages 100101 Also called overall heat conductance Some times referred to as transmittance Has same units as h 31 q AT TWA TOO2 UAAT iR R1R2R3R4R5 Called overall heat transfer coefficient sometimes 1 1 referred to as overall heat transfer conductance ZKiRi i Area on which U is based is entirely arbitrary U When a value of U is quoted make sure that you find out which area that it is based on 32 Steady State Problems with Uniform Internal Heat Generation Be able to solve steady state problems with uniform internal heat generation in Cartesian cylindrical and spherical coordinates Important Steps 1Write governing equation heat diffusion equation 2Write the Boundary Conditions 3State assumptions and use them to simplify the governing equa on 4Solve the governing equation 5Apply boundary conditions to determine arbitrary constants and obtain the temperature distribution 6Use Fourier s law with the temperature distribution to find heat ux Example of Problem involving Uniform Internal Heat Generation in Cartesian Coordinates T Flat plate that generates engrgy internally and uniformly at a constant rate Assume Steady state 1dimensional in x K Constant Tx0 To Tx2L To ho ho h0 constant for both faces TOO environmental temperature TX 239 gt Knowns TOO To ho K Determine A The temperature distribution B clNFL C ToToo Another Problem involving Uniform lnternal Heat Generation in Cylindrical Coordinates Given Long cylinder Uniform internal generation q 1dimensional in r K Constant Steady state conditions T T0 at r r0 Find Tr Steady State Problems with Uniform Internal Heat Generation A problem of this type will be on the exam Chapter 5 Transient Conduction Review 51 The Lumped Capacitance Method 52 Validity of the Lumped Capacitance Method 53 General Lumped Capacitance Analysis 54 Spatial Effects 55 The Plane Wall with Convection 56 Radial Systems with Convection 57 SemiInfinite Solids Transient Conduction Lumped Capacitance Method Bi lt 01 Exact Solutions oneterm solution for F0 gt 02 for Plane wall Infinite Cylinder Sphere Rcond th Rconv K Then internal resistance is small compared to the external or fluid resistance If Bi ltlt10 In general L represents a characteristic length of the solid L will change as heat flow from solids with different geometrics are considered For irregularly shaped bodies volume Surfacearea ln bodies that resemble plates cylinders or spheres the error introduced by the assumption that the temperature at any instant is uniform will be less than 5 when internal resistance is less than 10 of the external surface resistance ie Bi lt 01 Energy B alance for Lump ed Cap acit ance Ap p roach Rate of Rate of Rate of Rate of heat owing heat 2 heat owing change in in generated out internal energy 2 x 2 E2 Thus EST Eout Eout CIconv dT V hA T T La 62T Q uselll atzmTz 6QZ Q Solve governing equation and applyIC t 6eexp ltIz Twgtexp where T p VC S What is L for the lump ed cap acitance method L 1 AS How is it different from that us ed in the exact solution approach What is Bi for the two ap proaches Transient Temperature Response of Lumped Capacitance Solids Corresponding to Different Thermal Time Constant If LVr 1PMA R1 9 X i i H m s 3 0368 i i i i 71 Ti 2 U3 HA I Be able to solve Problem 516 Given K 60WmK Tso 22223 p 7850kgm2 c 430JkgK TOO 1300K Ti 300K 1553 f fij iii395 A How long will it take for 2 33113 h 25Wm K ffffff Ts i 9 L a B What will Tso equal when TSJ 1200K R LKA R RA LK R 001m2KW Transient Temperature Distribution for Different Biot Numbers in a Plane Wall Symmetrically Cooled by Convection Infinite Flat Plate T Txt h gtX 92L x0 x L surface Infinite Flat Plate 62T62T62T pc 6X2 ay2 622 H3 at T T No generation 614 6X2 05 at Boundaryconditions 1 i 0 dueto symmetry 6X H 2 hTS Tw K A 6X S TXLt Tw 6X X23 K Noteas h gtoo TS gtTw Bi Roond gt00 R COHV Assumgtions 1 1dimensional in x 2 q 0 3 Constant properties So if have 30 TX L t gt 0 2 TS Can use our solution by letting Bi gt oo Then the solution applies LCtQ TQJO JR HzT Q mE TQ wan 6139 7i Tvoo X X E 532 L ail 2170 533 PO is the Fourier Number See p 261 When we applythe BC39s and LO we endup with the followingsolution T T00 00 TFTOO 2 E On exp CnZFOcos nX 6 where F0 2 FourierNum berzft 2 We can not determine the values oan and 5n until Biisspeoi ed because the boundary conditions lead to the following equation 4 quot tanquot 23139 or cot n 2 539c 1 Solution of the equation leads to 1 2 3 in y2mllL n I 086Bi 2 343Bi 3 644Bi 4 For each C C m WVquot 53 2 sm2 n If F0gt02 the rst term is much greater then other terms so they can be neglected For this case l 00 Value of C1 and 1 for various Bi is tabulated in Table 51 TABLF Iovfficicnls used in he mHemJ approximation 10 the herieh solnlinns for U39ansieut quotnerdimensinnal nonrlllt lion Plane Wall Cylinder Sphere Bi rad C C 001 0 0998 10017 030 0 02 U 1410 10033 10061 0 07 732 10049 10090 04 0 1987 10066 1 120 005 0 2217 10 10149 0 06 02425 0098 l 0179 0 07 02615 1 01 H 10209 0 08 02791 1 0110 10239 0 09 02956 1 0145 10268 010 01111 10160 10298 0 15 03779 1 0237 1 0445 0 20 0432 1 0311 10592 0 25 04801 1 03512 10737 0 0 0 52 M 10450 10880 04 0 5932 0580 11164 0 5 065313 1 0701 11441 06 07051 10814 11713 07 07506 I 0919 1 1978 08 07910 1 1016 1 2236 0 9 0 11274 1 1107 12188 10 08603 1 1191 12732 20 1 0769 1 1795 14793 30 1925 3102 16227 4 0 2616 1 2287 17201 5 0 1138 12402 17870 6 0 5496 1 2479 18358 70 1 5766 1 2532 18674 8 0 1 39711 1 2570 18921 90 14149 12598 19100 10 0 1428 1 211211 9249 20 0 14961 269 19781 30 0 l 5202 2717 19898 400 1 5325 1 2723 19942 50 0 15400 1 2727 19962 1000 1 5552 1 2731 l 11990 1 5707 1 2731 20000 quotBI VLLk for the plane NIH and hr k for rhe m uitc c311ndcr and sphch See ngure 3 a 4n 39 0 391 1 quot L I I To egz39m 59929 TH b FICI RH 5 One dimensional systems with an initial uniform temperature subjorted to sudden convention con ditions 0 Plant wall b Infinite 1r ylindcr or sphcrv Infinite Long Solid Cylinder TTrt h k h Circular Cylinder 1aaT162T aZTq laT r 2 2 2 rar 6r ra 62 K aat Ergo T l dim 1araT1aT F5 5 EE BC 1 0 arr0 12 q ShTs Tw K A 6r rr0 aT EH KTr r0t Tm C t 0 Trt 0 To The solution for the infinite cylinder problem is 6 2C explt F0gtJolt nrgt 54720 n1 6 2 C1 em 12F0Jo lr C2 eXp 22F0 0 2r n 3 547b n J0 Cn i 547c Once Bi is specified for a given problem I can be found using either Table 51 or Appendix B4 6 r r r0 at t I E 2 F0 Next consider a sphere a Tlx O T J r Tlr O 39139 r39 To quot 3a quot h 1 m b FIGI RE 50 One dimensional systems with an initial uniform tmnperature subjcrted to sudden commLion con ditions a Plane wall b Infinite cylinder or sphcrv Sphere 72 ar ar 66 66 2 1 6 r r 0 lidim r26r E 35 6T 3 l r26T16T 0 r0 BC 1 2 hTS Tw K A 6r rr0 aT h EH0 ETrr0t Tw C t o Trt 0 To Sphere co 1 6 C ex 2F0 sn r 548a n n nric 4 r r r0 IfF0202 8 E C1 r 1 Note As r gt0 atcenter 1 sm r mm 1 1 sinx sinltrgt lama 1 3915 X where x 1r sinx sin0 0 I39m gt undefined a0 X 0 0 dgnx si z AXZ39 21 X 1X 1 Section 57 The SemiInfinite Solid SemiInfinite Solid Assume constant properties onedimensional in xdirection a d no generation K 62T82T82T 6T 8x2 832 622 30 06 at T O No generation 62T i 8T 8x2 a at Initial Condition T Ti at t 0 Need two Boundary Conditions BC 1 The interior boundary condition is To a oo 2 Ti 556 SemiInfmite Solid The SemiIn nite Solid A solid that is initially of uniform temperature I and is assumed to extend to in nity from a surface at which thermal conditions are altered Case 1 Tx O TI 0 Three poss1ble cases for second boundary condltlon no TV Case 1 Change in Surface Temperature TS TX T0z TS Tx0 2T1 lint Case 2 m 0 Ti Case 2 Constant Heat Flux CI q 7 k amde 4 Case 3 Surface Convection 8T ase ka x x0 hToo TOgtt n9 O3T A 7k analIA 0 mer no n T I A A Introduce a similarity variable n which can transform our partial differential equation into a ordinary differential equation 7 E x4CEIUZ Case 1 Constant Surface Temperature T0 t T T T quot7 5 2773912f exp 7142 du E erfn 560 T 7 Ts o 1 Where the Gaussian error function erf 17 is a standard mathematical function that is tabulated in Appendix Bi Note that erf17 asymptotically approaches unity as 17 becomes in nite Thus at any nonzero time temperatures everywhere are predicted to have changed from T become closer to The in nite speed at Which boundary condition information propagates into the semiin nite solid is physically unrealistic but this limitation of Fonrier s law is not important except at extremely small time scales as dismissed in Section 23 The surface heat ux may be obtained by applying Fourier s laW at x 0 in which case derf 17 9 17 quotIa 7 at k km m MM 995 10 113 CTs TiXZ n39lkali 776016107 77 kTs Ti 56 MIN2 7 Is Case 2 Constant Surface Heat Flux q q 2112at7T 2 7x2 42x x T t 7T 7 rf 562 m k exp W k e c N gt The complementary errorfunction erfc w is de ned as erfc w E 1 erfw Case 3 Surface Convection fizz 7 hTcc 7 T0 t 10 Tx I 7 Ti erfc lt x gt T m e T 2 7 hx him x hxE exp gterfc 2 k 563 The complementary error function erfc w is de ned as erfc w E 1 7 erfw Appendix B l Mathematical Relations and Funvtians 132 Gaussian Error Function1 1015 w erf w w erf w w erfw 000 000000 036 038933 104 085865 002 002256 038 040901 108 087333 004 004511 040 042839 112 088679 006 006762 044 046622 116 089910 008 009008 048 050275 120 091031 010 011246 0 52 053790 130 093401 012 013476 0 56 057162 140 095228 014 015695 0 60 060386 150 096611 016 017901 0 64 063459 160 097635 018 020094 0 68 066378 170 098379 020 022270 0 72 069143 180 098909 022 024430 0 76 071754 190 099279 0 24 026570 0 80 074210 200 099532 0 26 028690 0 84 076514 220 099814 0 28 030788 0 88 078669 240 099931 0 30 032863 0 92 080677 260 099976 032 034913 0 96 082542 280 099992 034 036936 100 084270 300 099998 1quot le Gaussian error function is de ned as 2 W 0 erfw 7 W L 3 dz The complementary error function is de ned as erfcwE 1 7 erfw What is the thermal penetration depth 6 It is defined on page 314 of the seventh edition of your text as the depth to which significant temperature effects propagate within a medium Note that penetration depth is used in various fields to indicate depth measured from the surface at which significant effects propagate within the medium What are signi cant effects If 6 is smaller than the thickness of the plane wall our textbook indicates that the semiinfinite slab solution can be used for the plane wall case S WAQJWM 739 7 O 0 0 Note that when n 0 the ratio is zerogt 7st 13 306 72st 5 90 Sf wWm z m 5 quot 23932 VZ EZ 7 Note that penetration depth is higher for materials with larger thermal diffusivity and increases with t 2 5 1 0 Ewe2 F 5553 Note Heat Transfer Relationships in Chapter 5 may also be applicable to Mass Transfer Problems when symbols are appropriately changed Details were covered in Chapter 14 Also analogies between convective heat transfer and convective mass mass transfer will be introduced during our discussions of Chapters 6 and 7 Chapter 6 Introduction to Convection Review 61 The Convection Boundary Layer 62 Local and Average Convection Coefficients 63 Laminar and Turbulent 64 The Boundary Layer Equations 65 Boundary Layer Similarity The Normalized Boundary Layer Equations 66 Physical Significance of the Dimensionless Parameters 67 Boundary Layer Analogies Convection Mechanisms involved in Convection Random motion of molecules Conduction or Diffusion Bulk fluid motion Advection Summary of Important Convective Transfer Relationsh ips q hTsTOO qhAst71 Too NghmC C as aoo NA hmAsCAs CA00 quot1 hm IOLS pAoo nA hmAs pAs I0Aoo For mass transfer calculations we need to know either CAs or pAs Assumption that is almost always made Thermodynamics equilibrium exists at the interface between the gas and the liquid or solid interface For heat transfer TS temperature at the surface is equal to the temperature of the vapor at the interface For mass transfer At the surface the vapor is in a saturated state If we know the temperature at the surface we can find the pressure of the vapor from saturation tables in our thermodynamics books The Convective Boundary Layers Velocity BL Thermal BL Concentration BL Boundary Layer Thickness u Prandthumber Pr 0 L MomentumDiffusiVity y 05 Thermal Diffusivity 9 Pr ltl Thermal BL is bigger than Momentum boundary lay er Pr 1 5TH 5 Pr gtl MomentumBL is bigger than Thermal boundary lay er 1 a and D have the same units gt m2 s Sc 2 Schmidt Number 2 DAB a Le 2 Lewis Number 2 AB Laminar and turbulent flow Surface friction and convective transfer rates depend strongly on whether the boundary layer is lam inar or turbulent Laminar motion is highly ordered and it is possible to identify a streamline along which particles flow Turbulent highly irregular and characterized by velocity fluctuation These enhance transfer of momentum energy and species NJ 1 r 1 m l n I 5V I 1 AL 7 I gt T IEI39I I LT m inn of vulntity bound Jam lhickuess am the local he ll lv dnifcr roefficimll I I39nr l l K 4 mmar 74lt4immmgm 39 T 10w um an iamlmmal Hm I a 9 lt7 La remitth Local Reynolds Number Wax y The critical Reynolds Number is the value of Rex Rex Reynolds Number 2 for which transition begins For a at plate the value varies from about 1gtlt105 to approximately 3 x 106 A representative value of 5 gtlt105 is often assumed Rexc 225x105 for a at plate y 37quot 39 39 6 so 393 yquot t V 39 This pnmmetcr is equal to the dimenhtttnlcm temperature gradient at the sure fuutx and it provide a measure til the convection hcztl transfer occurring in tin surluce From Equation 070 it l39ullnws that for a prescribed gt39rmtelr Nu f4 Rt39L PM 6811 The Nussclt number is to the thermal buundur layer what the friction cocl39r l39icicnt Ls m the veluclty boundary Jnyet Eqmttion 6 31 implies that in it given gettmelry the Nusselt number must be some unva l 39 39 7 1n P39 It39 thin function were knnwn it could he used to compute the value ul Ntt or ii 39ercnt fluids and for dil l39et39ent Ltlues of Vttnu L From knowledge nl39 NlL the Inuit convection cce 39teicnt It may be found and the um heal ux ma then e mntpuletl from Equation 6 Mnrcuver nee the average hcut tratts39 39cuel39 L39ient is obtained by integrtttt g twer the surface nl39 the hut it must he indce pendent uftltc spatial variable Hence the functional dependence of the Lll t lquot trey Nusselt number i a 5 6 32 Sherwood Number DAB 6 This paramater is equal to the dimensionless concentration gmdiint ill the bun Face and it pruvidcs a measure of the convection muss transfer occurring at the surface From Equation 683 it follows thatfm I prescribed genmem Sh f7t REL St 685 The Sherwood number is to the concemmtiun boundary layer what the Nusselt number is to the thermal boundary luyen and Equation 685 implies that it must be a universal function ol39x R3 and 5439 As for Lhe Nusselt numA her it is also possible to work with an uvm39age Sherwood number that depends on only ReL and SC The Reynolds Analogy 673 IfPrSc1anddpdx0 then ReL Cf NuSh seeEq666 on page 384 Extension of Reynold analogy It has been shown that the analogy can be applied over a range of Pr and Sc if certain corrections are added Modified Reynolds analogy or ChiltonColburn analogy has the form C j StPrm 1H 06 lt Pr lt 60 C 7 SzmSc23 M 06 lt Sc lt 3000 Approximations valid for laminar ow only for dpdx z 0 jH and jM 3 C olburn j factors for heat and mass transfer respectively Approximations valid in turbulent ow even when dpdx 5 O Chapter 7 External Review 71 The Empirical Method 72 The Flat Plate in Parallel Flow 73 Methodology for a Convection Calculation Be aware that the other sections of this chapter contain correlations for other geometries such as a cylinder ln cross flow Experimental or empirical approach Perform heat or mass transfer measurements under controlled laboratory conditions Correlate the data in terms of appropriate dimensionless parameters For heat transfer the correlations is often of the form t 2 CRef Pr 71 C m and n depend on nature of surface geometry and the type of flow Similarly for mass transfer the correlation is often of the firm ShL CReZ Sc 73 And the Friction Coefficient is of the form CReT Methodology Convective Calculation Section 73 0quot Determine the flow geometry Determine reference temperature then evaluate pertinent fluid properties In mass transfer the pertinent fluid properties are these of the bulk fluid Calculate Re Determine what h hm etc is required Select appropriate correlation At what temperature should the fluid properties be calculated since the fluid properties vary across the boundary layer The two approaches are usually used are 1 Evaluate all fluid properties at mean boundary layer temperature Tf Ts TOO 2 2 Evaluate all properties at T00 and multiply the correlation by an additional parameter to account for property variation Common forms of the parameter are Pr r r or amp Pt x Tf Note that both methods are used must read information on the correlation of interest Also note that n is sometimes evaluated at the surface temperature Mass Transfer Molecular Diffusion or Molecular Mass Transfer is defined as The transport of one constituent of a mixture from a region of higher chemical potential to a region of lower chemical potential We considered only point to point concentration difference or ordinary diffusion Modes of Mass Transfer Diffusion analogous to heat conduction Convective Mass Transfer analogous to convective heat transfer i Free ii Forced a Molecular transport b Storage of diffusing species c Motion of globes of mass Note b and 0 together are referred to as advection in your textbook Fick s Law L pDAvaA 1412 jAgtllt CDABVXA AquotpDABVmAmA Aquot Bquot 1420 NAquot 420va XAIVAquotIVBquot 1431 Be able to write all four forms of Fick s Law Diffusion in a Stationary Medium 1441 Stationary media with specified surface concentrations HA pDAvaa mA AuJF BH Assumptions A1 molar or mass average velocity of the mixture in zero Then Haquot jA pDAvaA 143945 NAquot jA CDABVXa 1446 These equations are good approximations for nonstationary media for maltlt1 and nB z0 or xaltlt1 and NB zO Examples 1 dilute gas mixture 2 liquid solutions in which motion of solvent in negligible and 3 species diffusing in a solid Mass Balance anquotA a HA anquotA 510A n A X y Z 6x 8y 62 at anquot anquot anquot n3 BX By B28 0E3 8X 8y 82 at anquot anquot anquot 8p 8x ay 82 at O Mass Diffusion Equations aZpA a2pA a2pA hA 1 810A 8x2 ay2 522 DAB DAB at 393 520A 520A 520A NA 1 50A 1438b 8x2 ay2 822 DAB DAB at Mass balance combined with Fick s law For stationary medium Vk ro HAHZJA Evaporation and Sublimation Gas phase in contact with a liquid or solid interface diffusion into the gas phase is of interest Raoult s Law PA0 XA0 pAsat 1460 pA Partial pressure in the gas phase at interface xAO Mole fraction of species A in the liquid or solid at the interface Note that for a pure liquid xAO 1 pAsalt Saturation pressure of species A at the surface temperature Henry s Law mass transfer within the liquid phase is of interest XA0 pAH 1461 Gassolid interface gassolid interface XAO IA Gas phase LiqUiq including SpeCleS B specnes A SPBC ES D If species A is only weakly soluble in a liquid Henry s law may be used xAO Mole fraction atA in the liquid at the interface pA Partial pressure of gas at interface H Henry s constant See Table A9 If Gas A dissolves in Species B CAO S pA 1462 Gassolid Equation 1462 can be used if species A interface dissolves in solid and the gas and solid can be treated as a solution CAO IA Gas phase Solid including speCies A SEE B l CAO Molar concentration ofA in the solid at the interface pA Partial pressure of gas at interface 8 Solubility See Table A10 Evaporation in a Column Find A xAfx B XBgX C NAquot in terms of knowns Assumptions x 1 Steady state 2 1dimensional in x 3 No mass generation 4 Constant P T C and DAB 5 Species B does not dissolve BFVaporalion nfliqludAiulu alnnary gas in the Boundary conditions x0 9 XAXA0 and XBXBY0 xL 9 XAXA L and xBxB L Transient Diffusion of a Dilute SpeciesA in a Stationary Medium Example Dyeing Process Three Steps in Dyeing Process 1Transport of dye through the dye bath to the fiber surface 2Adsorption of dye at the fiber surface 3 Diffusion and adsorption of dye within the fiber Stationary medium CAEQ concentration CsEQ MW CsEQ gt X xO xL C KPartition coefficient 2 E SEQ Assume at any time during dyeing that K E CAX L 2 Concentration in substance at wall CS1W Concentration in dye liquor at wall CAEQgt dye concen a on CSOO C CS OO dunngch ng 39 39 V xO xL Initial condition tO CACAi Boundary conditions 1 xO acAaxo 2 xL NAquotDaCAaxxLhMCSWCsw Recall that K M so that the second boundary condition can be written CS39W D 6X K hmim m hMCAX L KCW Compare this equation and boundary conditions with these for transient heat transfer in a plane wall a2CA laCA 621 1 6T 5X2 D 5quot WQE At t0 CACAi and TTi BC1 atx09 amp0 and 50 8X 8X BC 2 at xL9 495 2CAXL KCSOO and X xL aT Kfa hTx L Tw xL The forms are identical We can obtain the mass transfer relationships from the heat transfer equations by making the following substitutions TCA ocD TiCAi hhMK KfD TOO KCSOO Note The relationship in Table 142 ignores the existence of the partition coefficient The solution for T is T T 2 EC exp 3F0cosg nx 5393 6 7i Too n1 C KC 00 ew C e 2F0 cos x 7 CM KCSM Xp 4 M 4 Dt FOM F Fourier Number for mass transfer 4sin n Z 24 sinlt24ngt h KL h L t B M M BiM is Biot Number for mass transfer Special case The case when the surface concentration is suddenly changed is equivalent to letting hM 9 00 The second boundary condition can be written acA ax When hM 900 CfxL 9KIOCOOC s concentration of substrate at surface h cfx L Kpr hML Since BiM K then BiM 9 oo Mass Transfer in SemiInfinite Solid Assume constant properties onedimensional in xdirection a d no generation K 82T 82T 627 pc 6T q 8x2 832 822 0 8t T O No generation 62T 1 8T 2 For mass transfer the heat transfer arameters 9 czar p must be replaced with the appropriate Initial Condition Tnauo mass transfer parameters This will be discussed in class Need two Boundary Conditions BC 1 The interior boundary condition is T0 gt oo 2 Ti 556 SemiInfmite Solid The Second Boundary Condition A solid that is initially of uniform temperature T and is assumed to extend to in nity from a surface at which thermal conditions are altered Case 1 Th 0 T 0 Three poss1ble cases for second boundary condltlon no 2 TV Case 1 Change in Surface Temperature TS TN T0z TS Tx0 2T1 Case 2 Constant Heat Flux CI q 7 Case 3 Surface Convection 3T C 3 k a hTw T0r mfg 2 Ti x x0 7k analIA 0 hm A no n For mass transfer the heat transfer parameters T h A A must be replaced with the appropriate mass transfer parameters Introduce a similarity variable n which can transform our partial differential equation into a ordinary differential equation 7 E x4CEIUZ For mass transfer 1 x4Dt12 Case 1 Constant Surface Temperature T0 t TS T Ts 12 77 2 n T T 27739 exp in du erfn 5 60 o 1 5 You should be able to write the analogous equation for mass transfer Where the Gaussian error function erf 17 is a standard mathematical function that is tabulated in Appendix Bi Note that erf17 asymptotically approaches unity as 17 becomes in nite Thus at any nonzero time temperatures everywhere are predicted to have changed from T become closer to The in nite speed at Which boundary condition information propagates into the semiin nite solid is physically unrealistic but this limitation of Fourier s law is not important except at extremely small time scales as discussed in Section 23 The surface heat ux may be obtained by applying Foulier s laW at x 0 in which case ni iemm 995 10 derf 17 9 17 d1 9x 770 113 kTs TiXZ nlkalii776010107 77 MRE Gm For mass transfer 5 770mm y DA 5 4 0 Sr HAN 0 m 0 m t V 7quotquot 5 May need to use the partition coefficient solubility or Henry s constant Case 2 Constant Surface Heat Flux q q 2112at7T 2 7x2 42x x T t 7T 7 rf 562 m k exp W k e c N gt The complementary error function erfc w is de ned as erfc w E 1 erfw For mass transfer the heat transfer parameters must be replaced with the appropriate mass transfer parameters Case 3 Surface Convection fizz f hTcc 7 T0 0 10 TXfT1 erfclt x gt Tm Ti ZVat 7 exp gterfc Tia 563 The complementary error function erfc w is de ned as erfc w E 1 7 erfwl For mass transfer the heat transfer parameters must be replaced with the appropriate mass transfer parameters May need to use the partition coefficient solubility or Henry s constant Compare this equation and boundary conditions with these for transient heat transfer in a plane wall 52CA 2150A 62T16T 6x2 Dat 35 At t0 CACAi and TTi BC 1 at xoo 9 CACAi and TTi BC 2 at xL9 49 z bazm qu and 0x H K mfg hTx L TOO xL The forms are identical We can obtain the mass transfer relationships from the heat transfer equations by making the following substitutions TCA ocD TiCAi hhMK KfD T KC 8

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