HEAT TRANSFERChapter 9 to 12 Transport Phenomenaby Bird, Stewart and Lightfoot

CONTENTS OF HEAT TRANSFER Heat conduction: Fourier law Thermal conductivity of homogeneous media and heterogeneous mediaSteady state heat conduction in slab, cylinder and sphere (with special emphasis on temperature distribution in Nuclear Reactor Fuel Elements)Transient heat conduction in semi-infinite objectsTransient heat conduction in finite objects with symmetry conditions and asymmetry conditionsTransient heat conduction of objects in finite environmentForced convection & heat conduction (with special emphasis on Nuclear Reactor Heat Transfer)Free convectionEquations of changeHeat conduction with phase change (if time permitted)

MODES OF HEAT TRANSFER Three modes of heat transfer ConductionConvection RadiationForcedNaturalHeat is transferred due to temperature gradient.Heat is transferred by the motion of the fluid.If the fluid is forced, forced convection.If the motion of the fluid is induced by the temperature, natural convectionOnly occur at high temperature

HEAT CONDUCTION Points to RememberTransfer occurs due to the pure random motion of the molecules at molecular level onlySimplest among the three modes of heat transfer. It occurs in all three phases of matter (gas, liquid and solid). In solids, it is the only mechanism of heat transfer.The driving force for the heat conduction is the temperature gradient, that is whenever a gradient in temperature exists there exists a conductive heat flow. The heat flow is measured as the amount of energy transferred through any given plane per unit area per unit time. It is called heat flux.

HEAT CONDUCTION, contd. -2From our experience we know the following factsDirection of heat transferToT1Cross sectional areaz, thickness

HEAT CONDUCTION, contd. -3Experience shows that Conductive heat fluxheat flux: energy/area/timeProperty of material through which the heat is conducted:Thermal conductivityTemperature gradientPoint wise:Negative sign:Heat is moving in the direction of negative temperature gradientFouriers lawThe fundamental equation in heat conduction

HEAT CONDUCTION, contd. -4Heat flux as a vectorTemperature is a scalar, while heat flux is a vector

HEAT CONDUCTION, contd. -4Components of the Heat Flux VectorFor isotropic materials,kx = ky = kz = kthe heat flux vector will become:WhereandGrad operator

MOMENTUM vs HEAT ?Heat / momentum fluxDriving ForceProperty of MaterialREMEMBERThere are three components of the HEAT FLUX VECTOR, while there are nine components of the stress tensor / MOMENTUM FLUX VECTOR.

THERMAL CONDUCTIVITYIn general, ksolid > kliquid > kgasProperty of material through which the heat is conductedUnits: W/m/KInsulating materials have low k

MaterialsThermal conductivityHydrogen @ 100K0.067Water @ 293K0.600Liquid sodium @373K83.70Aluminum206.0

MaterialsThermal conductivityGlass @ 298K1.00Brick @ 293K0.6 1.00Glass wool @298K0.07

THERMAL CONDUCTIVITY, contd. -2Temperature dependence:GasLiquidSolidGas: k increases with temperature Liquid: no general rule; e.g. k of alcohols, benzene, toluene decrease with T, while k of glycerin, ethylene glycol increase with T Solid: also no general rule; e.g. k of polyethylene, polypropylene, PTFE decrease with T, while those of insulation materials increase with TFairly independent of pressurePressure dependence:

PROCEDURE OF TRANSPORT PHENOMENA ANALYSIS1. Draw a physical diagram as carefully as possible.2. Identify all possible energy transport mechanisms3. Set a frame of coordinates and draw the direction of all transport processes identified in step 2.4. Draw a shell in such a way that its surfaces are perpendicular to the transport direction.5. Carry out the energy shell balance as below:

This should give a first order ODE in terms of heat fluxSHELL BALANCE BY FIRST PRINCIPLES

Shell Balance By First Principles, contd. -2Procedure Of Transport Phenomena Analysis, contdApply the Fourier law: This should give a second order ODE in terms of temperature.Impose physical constraints on the boundary of the physical system.This gives rise to BCs. Note that the number of boundary conditions must match the order of the differential equation.Solve the equation for the temperature distribution.Obtain the mean temperature, heat flux.Energy production:Chemical reactionElectrical heatViscous dissipation (degradation of mechanical energy)Nuclear Reactions

BOUNDARY CONDITIONSTemperature is specified. This is called boundary condition of the first kind (Dirichlet BC)At the bounding surface, the heat flux equals to the flux supplied. Boundary condition of the second kind (Neumann BC)At the bounding surface, the heat flux entering the domain equals the heat flux across the thin film surrounding the object. BC of the third kind (Robin BC).Concept of heat transfer coefficientAt the interface between the two domains, the temperatures and the heat fluxes of those domains are continuous. BC of the fourth kind.

Steady state heat Conduction in slabenergy inenergy outconstantcross sectionarea, Athin shellzz+zSteps 1 to 4:Steps 5 and 6: Energy balance equationrate of energy in;units: Joule/secrate of energy out;units: Joule/secno heatproductionSteady state

Steady state heat Conduction in slab, contd. -2First order ODE with respect to heat fluxFouriers lawSecond-order ODE with respect to Tk is constantk is temperaturedependent

Steady state heat Conduction in slab, contd. -3Step 7: Physical constraint on the boundaryBoundary conditions of the first kind.Step 8: Temperature distributionTemperature distribution is linear (only valid for constant k and slab geometry).

Steady state heat Conduction in slab, contd. -4Step 9: Heat fluxApply the Fouriers law;Be Very Careful and Rememberthat the heat flux in general is a function of zSo to calculate the heat flux, you first decide where do we evaluate the heat flux at? At z=0 or z=L ?Fourier law

Steady state heat Conduction in slab, contd. -5Would the heat flux be proportional to temperature difference and temperature distribution linear if k=k(T)Look at the case when k takes the following functional form:Step 6: Heat balance equationStep 7: Boundary conditionssame as before, that is:z = 0; T = T0z = L; T = TL

Steady state heat Conduction in slab, contd. -6Step 8: Temperature distribution

Thus the temperature distribution across the slab is not linear. > 0 = 0 < 0

Steady state heat Conduction in slab, contd. -7Can you explain the shape of the temperature distribution?z = 0z = LT0 > 0Lower k, highertemperature gradientHigher k, lowertemperature gradient < 0Lower k, highertemperature gradientHigher k, lowertemperature gradientTL

Steady state heat Conduction in slab, contd. -8Step 9: Heat flux

Apply the Fouriers lawkindependent of z,as one wouldexpect physicallyWould this minus sign cause a concern to you as it could give zero flux?Answer: Of course not, see the constraint on

Steady state heat Conduction in slab, contd. -9Note: If we define an average thermal conductivity across the slab as:The heat flux equation is:So far so goodBut are there other things that need to be addressed?1. Existence of gas film surrounding the slab faces.2. Slab is made of many different materials.LETS CONSIDER THESE ONE BY ONE.which is an interesting equation

Heat conduction in slab: Gas film resistanceWhat has been done so far in steps 1 to 6 is still valid, that is the heat balance equation is:

Physical Definition of the ProblemHeat Balance EquationLets study the constant k first, and then deal with temperature dependent k.

Heat conduction in slab: Gas film resistance, -2Step 7: Physical constraintsWith the existence of the gas film, we talk of BC of the third kind.

Local Heat Transfer Coefficients = f(Re, Pr, Gr)Step 8: Temperature distributionNote: T(0) and T(L) are yet known, but we proceed anyway.Step 9: Heat fluxthis is the fluxthrough the slab

Heat conduction in slab: Gas film resistance, -3For steady state operation

Rearrange the above equation as:

Final heat flux equation

written in terms of known parameters andoperating conditions.

Heat conduction in slab: Gas film resistance, -4write the heat flux equation as:Where Bi0 and BiL are dimensionless and are defined as:which are called the Biot numbersPhysical significance of Biot number1. Bi >>1: Heat transfer through the object is limiting.2. Bi

Heat conduction in slab: Gas film resistance, -5Avoid the confusion Bi and Nu

Their definitions:Nu is seen in many correlations of heat transfer coefficients; hence it involves only film properties.Bi is seen in the analysis of heat transfer to or from an object.

Heat conduction in slab: Gas film resistance, -6How about T(0) and T(L)?Recall the following equations:

You get

Heat conduction in slab: Gas film & k(T)Steps 1 to 6:The heat balance equation is:Step 7: Boundary conditionswith T(0) being the solid object temperature at z=0, and T(L) being that at z=L.Step 8 and 9: Temperature distribution & heat flux

Heat conduction in slab: Gas film & k(T), -2This flux must be the same as fluxes through the film at both sides of the solid object:Thus, we getTwo nonlinear algebraic equations in terms of T(0) and T(L) as kavg is afunction of T(0) and T(L).

So farHeat conductionThermal conductivity, kFirst principles of solving heat conduction problemsHeat conduction in slab object:Heat conduction with gas film in slab objectBC of the first kindBC of the third kindConstant thermal conductivityT-dependent thermal conductivityNow let us turn to composite objectsobjects of different geometries than slab, Cylindersphere

Composite slab objectsLet deal with constant k, and N layers.

Composite slab objectsUtilizing results obtained earlier for a single slab object, one can write:In the ratio form;The interfacial temperatures can be found by equating the above two equationsU-1

Heat conduction in cylinderArea is changing along the heat flow direction.Steps 1 to 4Step 5:Heat balance around the annulus shellIncoming /outgoing areasrate of energy in/out;units: Joule/secno heatproductionSteady state

Heat conduction in cylinder, contd. -2Divide by 2Lr and take limit r0The final heat balance equation valid at a point is:First-order ODE w.r.t. qStep 6:Apply the Fourier law of heat conductionk is constantk =k(T)

Heat conduction in cylinder, contd. -3Step 7: Physical constraintsAt r= R1: T = T1At r= R2: T = T2BC of first kindCase of constant thermal conductivityStep 8: Temperature distributionalways associated withcylindrical geometry

Heat conduction in cylinder, contd. -4Step 9: Heat fluxUnlike the slab case, this heat flux is not a constant, but rather decreases as the heat moves away from the center.This is not entirely unexpected.

rheat fluxareaHeat flow =flux areasmalllargesmallthe samelargesmalllargethe same

Heat conduction in cylinder, contd. -5Although the heat flux changes with distance r, the heat flow (energy/time) must be a constant, a requirement of steady state.Total rate of heat flow out of the cylinder:constant as expectedCompare this heat flow with that for the slab geometry.Irrespective of the geometry, the heat flow is proportional to1. thermal conductivity2. temperature differenceOnly difference is geometrical factor

Heat conduction in cylinder, contd. -6When the annulus is very thin, that is no curvatureR2 - R1

Heat conduction in cylinder, contd. -7Step 9: Heat fluxHeat flowwhich is independent of r.

Heat conduction in composite cylinders wallsExactly the same analysis as that of composite wall slabsHeat flowSimilarly the analysis of composite cylindrical walls with fluidsat the two surfaces, leads to

The influence of shape on heat balanceSphereSlabCylinderThe general form for the three shapes is:S (the shape factor) = 0, 1, 2

Transient Heat ConductionEngine OilAirQuery:You will feel the heat first with which system?Answer:Air system will transport heat quicker than the oil system.Does this seem odd as the thermal conductivity of oil (0.14 W/m/K) is about 5 times larger than that of air(0.03 W/m/K)? Answer: No

Transient Heat Conduction, -2The heat transfer in this system is transient; that is heat travels from the left surface to the right surfacePortion of which is conducted through and portion is absorbed by the material to increase its latent heat.Since oil has a much higher heat capacity per unit volume, most of the heat is retained by the oil Hence it retards the movement of heat front to the right surface.Proper parameter characterises the transient situation isTHE THERMAL DIFFUSIVITYIt is defined as:

Transient Heat Conduction, -3The larger is this parameter, the faster is the heat transport.We shall see that this parameter will evolve naturally during the analysis.Now, coming back to our example.

Oil System Air System 848 1.1 Cp 2130 928 k 0.140.03

7.7510-8 m2/s 2.9410-5 m2/s

Heat conduction in a Semi-infinite slab objectCase 1Constant temperature at x=0 surfaceHeat balance over the shell:rate of energy in;units: Joule/secrate of energy out;units: Joule/secno heatproductionaccumulationof energydivide by Ax and take the limit when x 0ORthermal diffusivity

Heat conduction in a Semi-infinite slab object, -2Initial and boundary conditions

At t= 0: T = T0At x= 0: T = TsAt x : T = T0far away from the surface, the temperature is not yet disturbed by the change in surface temperatureThis condition is maintained for t > 0+condition of the object before the surface temperature is changedTemperature distribution

Heat conduction in a Semi-infinite slab object, -3The heat penetration front can be defined as the front at which the temperature is such that:From the table of error function,erf(2)=0.995; The penetration front is proportional to1 the square root of time,2. the square root of k3. the inverse of the square root of Cp

Heat conduction in a Semi-infinite slab object, -4Front travels fast high conductivitylow volumetric heat capacityFront travels slowLow conductivityhigh volumetric heat capacityThe square root dependence of the heat front with respect to time is the characteristics of conduction (diffusion) type problems.Heat flux: Obtained from the Fourier lawWhat we can observe from this are:1. Heat flux decays rapidly with x for a given time2. Heat flux increases to a maximum and then decays w.r.t time.

Heat conduction in a Semi-infinite slab object, -5Of interest to engineers is the heat flux entering the object at x=0:This heat flux is infinite at t=0! Is this acceptable physically?Answer:The infinite flux at t=0 is due to the sudden exposure of the surface to a new temperature, that is infinite temperature gradient at t=0.If we are interested in the amount of heat has entered the object up to time t, we find:The amount of heat transfer is proportional to the square root of1. The thermal conductivity2. The volumetric heat capacity

A typical variation in heat flux w.r.t time

Chart1

0.1338567456

0.2389863071

0.2941235222

0.3253045482

0.3438189831

0.3550198282

0.3617177247

0.3655097907

0.3673602534

0.3678794412

0.3674684997

0.3663992447

0.3648605141

0.3629862323

0.3608730174

0.3585915687

0.3561942274

0.3537201112

0.3511986739

0.3486522153

0.34609768

0.3435479644

0.3410128787

0.3384998624

0.3360145226

0.3335610421

0.3311424923

0.3287610732

0.3264183011

0.3241151538

0.3218521855

0.3196296174

0.3174474085

0.3153053135

0.3132029274

0.3111397224

0.3091150768

0.3071282982

0.3051786427

0.3032653299

0.3013875547

0.2995444977

0.2977353324

0.2959592316

0.2942153728

0.2925029415

0.2908211346

0.2891691628

0.2875462523

0.2859516462

0.2843846054

0.2828444095

0.2813303568

0.2798417652

0.2783779716

0.2769383319

0.2755222213

0.2741290336

0.272758181

0.2714090938

0.2700812199

0.2687740244

0.267486989

0.2662196118

0.2649714065

0.2637419024

0.2625306432

0.2613371872

0.2601611065

0.2590019865

0.2578594256

0.2567330348

0.2556224369

0.2545272665

0.2534471695

0.2523818025

0.2513308326

0.250293937

0.2492708025

0.2482611255

0.2472646112

0.2462809738

0.2453099356

0.2443512273

0.2434045873

0.2424697615

0.2415465032

0.2406345727

0.2397337372

0.2388437702

0.2379644518

0.237095568

0.2362369108

0.2353882781

0.2345494729

0.2337203038

0.2329005845

0.2320901337

0.2312887748

Sheet1

tSQRT tQ

00000

0.10.3162277663.16227766020.04232921960.1338567456

0.20.44721359552.23606797750.10687792570.2389863071

0.30.54772255751.82574185840.16109808780.2941235222

0.40.6324555321.58113883010.20574066110.3253045482

0.50.70710678121.41421356240.24311673440.3438189831

0.60.77459666921.29099444870.27499717650.3550198282

0.70.83666002651.19522860930.30263476110.3617177247

0.80.8944271911.11803398870.32692189540.3655097907

0.90.94868329811.05409255340.34850853670.3673602534

1110.36787944120.3678794412

1.11.04880884820.95346258920.38540421390.3674684997

1.21.0954451150.91287092920.40137026280.3663992447

1.31.14017542510.87705801930.41600499180.3648605141

1.41.18321595660.84515425470.42949110210.3629862323

1.51.22474487140.81649658090.44197737720.3608730174

1.61.26491106410.7905694150.45358644280.3585915687

1.71.3038404810.76696498880.46442045280.3561942274

1.81.34164078650.74535599250.47456532820.3537201112

1.91.37840487520.72547625010.48409396420.3511986739

21.41421356240.70710678120.49306869140.3486522153

2.11.44913767460.69006555930.50154318710.34609768

2.21.48323969740.67419986250.50956397880.3435479644

2.31.51657508880.65938047340.51717163690.3410128787

2.41.54919333850.64549722440.5244017320.3384998624

2.51.58113883010.6324555320.53128560910.3360145226

2.61.61245154970.62017367290.53785101930.3335610421

2.71.64316767250.60858061950.54412263830.3311424923

2.81.67332005310.59761430470.55012249650.3287610732

2.91.70293863660.58722021950.55587033660.3264183011

31.73205080760.57735026920.56138391380.3241151538

3.11.76068168620.56796183420.56667924870.3218521855

3.21.7888543820.55901699440.57177084160.3196296174

3.31.81659021250.55048188260.57667185530.3174474085

3.41.84390889150.54232614450.58139427110.3153053135

3.51.87082869340.53452248380.58594902350.3132029274

3.61.89736659610.52704627670.59034611610.3111397224

3.71.92353840620.51987524490.59459472210.3091150768

3.81.9493588690.5129891760.59870327190.3071282982

3.91.97484176580.50636968350.60267952960.3051786427

420.50.60653065970.3032653299

4.12.02484567310.49386479830.61026328610.3013875547

4.22.04939015320.48795003650.61388354410.2995444977

4.32.07364413530.48224282220.61739712590.2977353324

4.42.09761769630.47673129460.62080932170.2959592316

4.52.12132034360.47140452080.62412505580.2942153728

4.62.1447610590.46625240410.62734891860.2925029415

4.72.16794833890.4612656040.63048519580.2908211346

4.82.190890230.45643546460.63353789370.2891691628

4.92.21359436210.45175395150.6365107630.2875462523

52.23606797750.44721359550.63940731920.2859516462

5.12.25831795810.44280744280.64223086140.2843846054

5.22.28035085020.43852900970.64498448960.2828444095

5.32.30217288660.43437224280.64767111970.2813303568

5.42.32379000770.43033148290.65029349780.2798417652

5.52.34520787990.42640143270.65285421250.2783779716

5.62.36643191320.42257712740.65535570650.2769383319

5.72.38746727730.41885390830.65780028740.2755222213

5.82.40831891580.41522739930.66019013690.2741290336

5.92.42899156030.41169348480.66252731960.272758181

62.44948974280.40824829050.66481379140.2714090938

6.12.4698178070.40488816510.66705140630.2700812199

6.22.48997991960.40160966450.66924192360.2687740244

6.32.50998007960.39840953640.67138701390.267486989

6.42.52982212810.39528470750.67348826480.2662196118

6.52.54950975680.39223227030.67554718620.2649714065

6.62.56904651570.38924947210.67756521530.2637419024

6.72.58843582110.38633370460.67954372090.2625306432

6.82.60768096210.38348249440.68148400770.2613371872

6.92.62678510730.38069349380.683387320.2601611065

72.64575131110.3779644730.68525484530.2590019865

7.12.66458251890.37529331250.68708771780.2578594256

7.22.6832815730.37267799620.68888702130.2567330348

7.32.70185121720.37011660510.69065379220.2556224369

7.42.72029410170.3676073110.69238902180.2545272665

7.52.73861278750.36514837170.69409365940.2534471695

7.62.75680975040.36273812510.6957686140.2523818025

7.72.77488738510.36037498510.69741475690.2513308326

7.82.79284800880.3580574370.69903292350.250293937

7.92.81069386450.35578403350.70062391520.2492708025

82.82842712470.35355339060.70218850130.2482611255

8.12.84604989420.35136418450.70372742060.2472646112

8.22.86356421270.34921514790.70524138280.2462809738

8.32.88097205820.34710506730.70673107010.2453099356

8.42.89827534920.34503277970.70819713860.2443512273

8.52.91547594740.34299717030.70964021960.2434045873

8.62.93257565970.34099716970.71106092070.2424697615

8.72.94957624080.33903175180.71245982680.2415465032

8.82.96647939480.33709993120.71383750170.2406345727

8.92.9832867780.33520076160.71519448840.2397337372

930.33333333330.71653131060.2388437702

9.13.01662062580.33149677210.71784847340.2379644518

9.23.03315017760.32969023670.71914646410.237095568

9.33.04959013640.32791291790.72042575320.2362369108

9.43.06594194340.32616403650.72168679470.2353882781

9.53.08220700150.32444284230.72293002750.2345494729

9.63.0983866770.32274861220.72415587550.2337203038

9.73.11448230050.32108064950.72536474830.2329005845

9.83.13049516850.31943828250.72655704230.2320901337

9.93.14642654450.31782086310.72773314060.2312887748

103.16227766020.3162277660.72889341410.230496336

204.4721359550.22360679770.79962948870.1788025893

305.47722557510.18257418580.83312283570.1521067234

406.32455532030.1581138830.85375254850.1349901306

507.07106781190.14142135620.86812344540.122771195

607.74596669240.12909944490.87888656040.1134637671

708.36660026530.11952286090.88734372210.1060578603

808.944271910.11180339890.89422004490.0999768404

909.48683298050.10540925530.89995613540.0948637061

100100.10.9048374180.0904837418

11010.48808848170.09534625890.90905811050.08667529

12010.95445115010.09128709290.91275562760.0833228078

13011.4017542510.08770580190.91603033140.0803411748

14011.83215956620.08451542550.91895747990.0776660824

15012.24744871390.08164965810.92159477540.0752478983

16012.64911064070.07905694150.92398730970.0730476107

17013.03840481040.07669649890.92617090490.0710340658

18013.4164078650.07453559920.92817443120.0691820374

19013.78404875210.0725476250.93002145330.0674708476

20014.14213562370.07071067810.93173142340.0658833608

21014.49137674620.06900655590.93332056130.0644052375

22014.83239697420.06741998620.93480251470.0630243727

23015.16575088810.06593804730.93618886190.0617304655

24015.49193338480.06454972240.93748949880.0605146869

25015.81138830080.06324555320.93871294140.0593694193

26016.12451549660.06201736730.93986656370.0582880499

27016.43167672520.06085806190.94095678790.0572648065

28016.73320053070.05976143050.94198923670.0562946243

29017.02938636590.0587220220.94296885740.0553730379

30017.32050807570.05773502690.94390002240.0544960932

31017.60681686170.05679618340.94478661290.0536602738

32017.888543820.05590169940.94563208750.0528624407

33018.16590212460.05504818830.94643953960.0520997819

34018.43908891460.05423261450.94721174570.0513697694

35018.70828693390.05345224840.94795120610.0506701233

36018.9736659610.05270462770.94866017910.0499987815

37019.23538406170.05198752450.94934071030.0493538734

38019.49358868960.05129891760.9499946580.0487336977

39019.74841765810.05063696840.95062371440.0481367029

400200.050.95122942450.0475614712

41020.24845673130.04938647980.9518132020.0470067035

42020.49390153190.04879500360.95237634340.0464712071

43020.73644135330.04822428220.95292004010.0459538849

44020.97617696340.04767312950.95344538940.0454537255

45021.21320343560.04714045210.95395340340.0449697947

46021.44761058950.04662524040.9544450180.0445012284

47021.67948338870.04612656040.95492109930.0440472258

48021.90890230020.04564354650.9553824510.0436070433

49022.13594362120.04517539510.95582981920.0431799898

50022.3606797750.04472135950.95626389850.0427654216

51022.58317958130.04428074430.95668533580.0423627387

52022.8035085020.0438529010.95709473480.0419713806

53023.02172886640.04343722430.95749265950.0415908234

54023.23790007720.04303314830.95787963750.0412205765

55023.45207879910.04264014330.9582561630.0408601801

56023.66431913240.04225771270.95862269950.0405092027

57023.87467277260.04188539080.95897968210.0401672388

58024.08318915760.04152273990.95932752010.0398339071

59024.2899156030.04116934850.95966659810.0395088486

60024.49489742780.0408248290.95999727890.0391917248

61024.69817807050.04048881650.96031990420.0388822164

62024.8997991960.04016096640.96063479670.0385800218

63025.0998007960.03984095360.96094226140.0382848561

64025.29822128130.03952847080.96124258630.0379964495

65025.4950975680.0392232270.96153604440.0377145466

66025.69046515730.03892494720.96182289390.0374389054

67025.88435821110.03863337050.962103380.0371692963

68026.07680962080.03834824940.9623777350.0369055014

69026.26785107310.03806934940.96264617970.0366473137

70026.45751311060.03779644730.96290892370.0363945364

71026.64582518890.03752933130.96316616640.0361469821

72026.832815730.03726779960.96341809780.0359044726

73027.01851217220.03701166050.96366489850.0356668381

74027.20294101750.03676073110.96390674070.0354339165

75027.38612787530.03651483720.96414378860.0352055534

76027.56809750420.03627381250.96437619910.0349816014

77027.7488738510.03603749850.96460412160.0347619196

78027.92848008750.03580574370.96482769910.0345463733

79028.10693864510.03557840330.96504706830.0343348338

80028.28427124750.03535533910.96526235990.034127178

81028.46049894150.03513641840.96547369890.0339232879

82028.63564212660.03492151480.9656812050.0337230505

83028.80972058180.03471050670.9658849930.0335263575

84028.98275349240.0345032780.96608517290.0333331053

85029.15475947420.0342997170.96628185010.033143194

86029.32575659720.0340997170.96647512590.0329565283

87029.49576240750.03390317520.96666509730.0327730161

88029.66479394840.03370999310.96685185770.0325925695

89029.83286778040.03352007620.96703549670.0324151035

900300.03333333330.96721610050.0322405367

91030.1662062580.03314967720.9673937520.0320687906

92030.33150177620.03296902370.96756853080.0318997898

93030.4959013640.03279129180.96774051390.0317334616

94030.65941943350.03261640370.9679097750.0315697359

95030.82207001480.03244428420.96807638550.0314085454

96030.98386676970.03227486120.96824041380.031249825

97031.14482300480.0321080650.96840192610.0310935119

98031.3049516850.03194382820.96856098630.0309395458

99031.46426544510.03178208630.96871765590.0307878681

100031.62277660170.03162277660.96887199430.0306384226

101031.78049716410.03146583880.96902405890.0304911548

102031.93743884530.03131121460.96917390510.0303460121

103032.09361307180.03115884760.96932158640.0302029436

104032.24903099320.03100868360.96946715450.0300619003

105032.4037034920.030860670.96961065950.0299228346

106032.55764119220.03071475580.96975214980.0297857005

107032.71085446760.0305708920.9698916720.0296504536

108032.86335345030.0304290310.97002927170.0295170508

109033.01514803840.03028912660.97016499240.0293854503

110033.16624790360.03015113450.97029887690.0292556119

111033.31666249790.03001501130.9704309660.0291274964

112033.46640106140.02988071520.97056129980.0290010658

113033.61547262790.02974820590.97068991680.0288762835

114033.76388603230.02961744390.97081685440.0287531137

115033.91164991560.02948839120.9709421490.028631522

116034.05877273190.0293610110.97106583580.0285114747

117034.2052627530.02923526730.97118794880.0283929393

118034.35112807460.02911112550.97130852130.0282758843

119034.49637662130.02898855180.97142758550.0281602789

120034.64101615140.02886751350.97154517260.0280460933

121034.78505426190.02874797870.97166131290.0279332988

122034.92849839310.02862991670.9717760360.027821867

123035.07135583350.02851329740.97188937040.0277117707

124035.21363372330.02839809170.97200134410.0276029833

125035.35533905930.02828427120.9721119840.027495479

126035.49647869860.02817180850.97222131660.0273892327

127035.63705936240.02806067670.97232936730.02728422

128035.777087640.02795084970.97243616110.027180417

129035.91656999210.02784230230.97254172230.0270778007

130036.05551275460.02773500980.97264607430.0269763484

131036.19392214170.02762894820.97274924020.0268760384

132036.33180424920.02752409410.97285124230.0267768492

133036.46916505760.02742042490.97295210230.02667876

134036.60601043540.02731791820.97305184140.0265817506

135036.74234614170.02721655270.97315048030.0264858013

136036.87817782920.02711630720.97324803920.0263908928

137037.01351104660.02701716130.97334453750.0262970064

138037.1483512420.02691909510.97343999440.0262041238

139037.28270376460.0268220890.97353442860.0261122271

140037.41657386770.02672612420.97362785810.0260212991

141037.5499667110.02663118210.97372030080.0259313226

142037.68288736280.02653724460.97381177390.0258422813

143037.81534080240.02644429430.97390229430.0257541589

144037.9473319220.02635231380.97399187830.0256669397

145038.07886552930.02626128660.97408054220.0255806083

146038.20994634910.02617119610.97416830150.0254951497

147038.34057902540.02608202650.97425517150.0254105492

148038.47076812330.02599376220.97434116730.0253267926

149038.60051813120.0259063880.97442630330.0252438659

150038.72983346210.0258198890.97451059390.0251617553

151038.85871845550.02573425060.9745940530.0250804476

152038.98717737920.02564945880.97467669410.0249999297

153039.11521443120.02556549960.97475853060.0249201889

154039.24283374070.02548235960.97483957540.0248412126

155039.37003937010.02540002540.97491984130.0247629887

156039.49683531630.02531848420.97499934070.0246855054

157039.62322551230.02523772330.97507808570.0246087509

158039.74921382870.02515773030.97515608830.0245327138

159039.87480407480.02507849310.97523335990.0244573831

1600400.0250.9753099120.0243827478

161040.12480529550.02492223930.97538575570.0243087972

162040.2492235950.02484519970.97546090190.024235521

163040.37325847640.02476887020.97553536120.0241629088

164040.49691346260.02469323990.97560914410.0240909507

165040.62019202320.02461829820.97568226060.0240196368

166040.74309757490.02454403470.97575472090.0239489577

167040.86563348340.02447043920.97582653470.0238789039

168040.98780306380.02439750180.97589771150.0238094662

169041.10960958220.02432521280.97596826080.0237406356

170041.23105625620.02425356250.97603819170.0236724033

171041.35214625630.02418254170.97610751320.0236047606

172041.47288270670.02411214110.97617623410.0235376991

173041.59326868620.02404235180.97624436310.0234712105

174041.71330722920.02397316510.97631190870.0234052866

175041.83300132670.02390457220.9763788790.0233399194

176041.95235392680.02383656470.97644528230.0232751012

177042.07136793590.02376913440.97651112650.0232108242

178042.19004621950.02370227320.97657641950.0231470811

179042.30839160260.0236359730.97664116880.0230838643

180042.42640687120.0235702260.97670538210.0230211666

181042.54409477240.02350502470.97676906660.0229589811

182042.66145801540.02344036150.97683222970.0228973006

183042.77849927240.02337622910.97689487830.0228361185

184042.89522117910.02331262020.97695701950.022775428

185043.01162633520.02324952770.97701866010.0227152224

186043.12771730570.02318694480.97707980670.0226554955

187043.24349662090.02312486450.9771404660.0225962409

188043.35896677740.02306328020.97720064440.0225374523

189043.47413023860.02300218530.97726034820.0224791236

190043.58898943540.02294157340.97731958360.0224212489

191043.70354676680.02288143810.97737835670.0223638224

192043.81780460040.02282177320.97743667360.0223068381

193043.9317652730.02276257270.97749454010.0222502905

194044.04543109110.02270383050.9775519620.022194174

195044.15880433160.02264554070.9776089450.0221384831

196044.27188724240.02258769760.97766549450.0220832125

197044.38468204230.02253029550.97772161620.0220283569

198044.49719092260.02247332870.97777731540.0219739111

199044.60941604640.0224167920.97783259730.0219198699

200044.721359550.02236067980.97788746720.0218662285

201044.83302354290.02230498680.97794193010.0218129819

202044.94441010850.0222497080.97799599120.0217601252

203045.05552130430.02219483810.97804965520.0217076537

204045.16635916250.02214037210.9781029270.0216555628

205045.27692569070.02208630520.97815581150.0216038478

206045.38722287160.02203263250.97820831320.0215525042

207045.49725266430.02197934910.97826043680.0215015277

208045.6070170040.02192645050.97831218680.0214509137

209045.71651780260.0218739320.97836356770.0214006581

210045.82575694960.0218217890.97841458370.0213507566

211045.93473631140.02177001720.97846523930.0213012051

212046.04345773290.02171861210.97851553870.0212519995

213046.15192303690.02166756950.9785654860.0212031357

214046.26013402490.02161688510.97861508530.0211546098

215046.36809247750.02156655460.97866434060.021106418

216046.47580015450.02151657410.9787132560.0210585563

217046.58325879540.02146693950.97876183530.0210110211

218046.69047011970.02141764680.97881008220.0209638087

219046.79743582720.02136869220.97885800080.0209169153

220046.90415759820.02132007160.97890559450.0208703374

221047.01063709420.02127178150.97895286710.0208240715

222047.11687595760.0212238180.97899982230.020778114

223047.22287581250.02117617750.97904646340.0207324617

224047.32863826480.02112885640.97909279410.020687111

225047.43416490250.02108185110.97913881770.0206420587

226047.5394572960.02103515810.97918453770.0205973016

227047.64451699830.0209887740.97922995730.0205528363

228047.74934554530.02094269540.97927507990.0205086597

229047.8539444560.0208969190.97931990870.0204647688

230047.95831523310.02085144140.97936444680.0204211604

231048.06245936280.02080625950.97940869740.0203778315

232048.16637831520.020761370.97945266350.0203347791

233048.27007354460.02071676980.97949634830.0202920003

234048.37354648980.02067245580.97953975460.0202494922

235048.47679857420.02062842490.97958288550.020207252

236048.5798312060.02058467420.97962574390.0201652768

237048.68264577860.02054120080.97966833260.0201235639

238048.78524367060.02049800150.97971065440.0200821105

239048.88762624630.02045507370.97975271210.020040914

240048.98979485570.02041241450.97979450850.0199999717

241049.09175083450.02037002110.97983604620.0199592809

242049.1934955050.02032789070.9798773280.0199188392

243049.29503017550.02028602060.97991835630.019878644

244049.39635614090.02024440830.97995913390.0198386928

245049.49747468310.02020305090.97999966330.0197989831

246049.59838707050.0201619460.98003994690.0197595125

247049.69909455920.02012109090.98007998730.0197202785

248049.7995983920.02008048320.98011978690.0196812789

249049.89989979950.02004012040.98015934810.0196425113

2500500.020.98019867330.0196039735

251050.09990019950.01996011960.98023776480.019565663

252050.1996015920.01992047680.98027662490.0195275778

253050.29910535980.01988106930.98031525590.0194897156

254050.39841267340.01984189480.98035366010.0194520741

255050.49752469180.01980295090.98039183960.0194146514

256050.59644256270.01976423540.98042979670.0193774453

257050.69516742250.01972574610.98046753350.0193404536

258050.79370039680.01968748080.98050505210.0193036744

259050.89204260.01964943730.98054235460.0192671055

260050.99019513590.01961161350.98057944320.0192307451

261051.08815909780.01957400730.98061631970.019194591

262051.18593556830.01953661660.98065298630.0191586414

263051.28352561980.01949943940.9806894450.0191228944

264051.38093031470.01946247360.98072569760.019087348

265051.47815070490.01942571720.98076174620.0190520004

266051.57518783290.01938916840.98079759260.0190168496

267051.67204273110.0193528250.98083323870.018981894

268051.76871642220.01931668520.98086868640.0189471317

269051.86520991960.01928074720.98090393760.0189125608

270051.96152422710.0192450090.98093899390.0188781797

271052.05766033930.01920946880.98097385730.0188439867

272052.15361924160.01917412470.98100852950.0188099799

273052.24940191050.01913897510.98104301230.0187761577

274052.34500931320.0191040180.98107730720.0187425185

275052.44044240850.01906925180.98111141620.0187090606

276052.53570214630.01903467470.98114534070.0186757824

277052.63078946780.0190002850.98117908260.0186426822

278052.72570530590.0189660810.98121264340.0186097585

279052.8204505850.01893206110.98124602470.0185770097

280052.91502622130.01889822370.98127922820.0185444343

281053.00943312280.01886456690.98131225540.0185120307

282053.10367218940.01883108940.98134510780.0184797975

283053.19774431310.01879778950.98137778710.0184477331

284053.29165037790.01876466560.98141029460.018415836

285053.38539126020.01873171620.98144263210.0183841049

286053.47896782850.01869893980.98147480080.0183525382

287053.57238094390.01866633480.98150680230.0183211346

288053.665631460.01863389980.9815386380.0182898926

289053.75872022290.01860163330.98157030930.0182588109

290053.85164807130.01856953380.98160181770.0182278881

291053.9444158370.01853759990.98163316450.0181971229

292054.03702434440.01850583030.98166435120.0181665139

293054.12947441090.01847422330.98169537910.0181360597

294054.22176684690.01844277780.98172624950.0181057591

295054.3139024560.01841149240.98175696370.0180756108

296054.40588203490.01838036560.98178752320.0180456136

297054.49770637380.01834939610.98181792910.0180157661

298054.58937625580.01831858260.98184818270.0179860671

299054.6808924580.01828792390.98187828540.0179565154

300054.77225575050.01825741860.98190823840.0179271097

301054.86346689740.01822706540.98193804290.0178978489

302054.95452665610.01819686310.98196770010.0178687318

303055.04543577810.01816681050.98199721130.0178397572

304055.13619500840.01813690630.98202657760.017810924

305055.22680508590.01810714920.98205580020.0177822309

306055.31726674380.01807753820.98208488040.0177536769

307055.40758070880.01804807190.98211381920.0177252608

308055.4977477020.01801874930.98214261770.0176969816

309055.58776843870.01798956910.98217127730.017668838

310055.67764362830.01796053020.98219979880.0176408292

311055.76737397440.01793163150.98222818350.0176129538

312055.85696017510.01790287190.98225643250.017585211

313055.94640292280.01787425010.98228454670.0175575997

314056.03570290450.01784576530.98231252740.0175301188

315056.12486080160.01781741610.98234037550.0175027672

316056.21387729020.01778920170.98236809210.0174755441

317056.3027530410.01776112080.98239567820.0174484483

318056.39148871950.01773317260.98242313480.017421479

319056.48008498580.01770535580.98245046310.017394635

320056.56854249490.01767766950.98247766380.0173679155

321056.65686189690.01765011270.98250473810.0173413194

322056.74504383640.01762268440.98253168690.0173148459

323056.83308895350.01759538360.98255851120.0172884939

324056.9209978830.01756820920.9825852120.0172622626

325057.0087712550.01754116040.98261179020.017236151

326057.09640969450.01751423610.98263824660.0172101583

327057.1839138220.01748743540.98266458240.0171842834

328057.27128425310.01746075740.98269079830.0171585256

329057.35852159880.01743420110.98271689520.017132884

330057.44562646540.01740776560.98274287420.0171073576

331057.53259945460.017381450.9827687360.0170819456

332057.61944116360.01735525340.98279448160.0170566472

333057.7061521850.01732917480.98282011170.0170314615

334057.79273310720.01730321350.98284562740.0170063877

335057.8791845140.01727736850.98287102930.016981425

336057.96550698480.0172516390.98289631850.0169565724

337058.05170109480.01722602410.98292149560.0169318293

338058.1377674150.01720052290.98294656160.0169071948

339058.2237065120.01717513470.98297151720.0168826682

340058.30951894850.01714985850.98299636320.0168582485

341058.39520528260.01712469360.98302110050.0168339352

342058.48076606890.01709963920.98304572990.0168097273

343058.56620185740.01707469440.9830702520.0167856241

344058.65151319450.01704985850.98309466780.016761625

345058.73670062240.01702513060.98311897790.016737729

346058.82176467940.017000510.98314318320.0167139355

347058.90670590010.01697599590.98316728440.0166902438

348058.9915248150.01695158760.98319128210.0166666531

349059.0762219510.01692728420.98321517730.0166431628

350059.1607978310.01690308510.98323897050.016619772

351059.24525297440.01687898950.98326266260.0165964801

352059.32958789680.01685499660.98328625420.0165732864

353059.41380311010.01683110570.9833097460.0165501903

354059.49789912260.01680731610.98333313880.0165271909

355059.58187643910.01678362720.98335643320.0165042877

356059.66573556070.01676003810.983379630.01648148

357059.74947698520.01673654820.98340272980.0164587672

358059.83310120660.01671315680.98342573320.0164361484

359059.91660871580.01668986320.9834486410.0164136232

3600600.01666666670.98347145380.0163911909

361060.08327554320.01664356660.98349417230.0163688508

362060.1664358260.01662056240.98351679710.0163466023

363060.24948132560.01659765330.98353932890.0163244447

364060.3324125160.01657483860.98356176820.0163023776

365060.4152298680.01655211780.98358411580.0162804001

366060.4979338490.01652949010.98360637230.0162585118

367060.58052492340.0165069550.98362853820.016236712

368060.66300355240.01648451180.98365061420.0162150002

369060.74537019390.01646215990.98367260090.0161933757

370060.8276253030.01643989870.98369449890.0161718379

371060.90976933140.01641772760.98371630880.0161503864

372060.99180272790.01639564590.98373803110.0161290204

373061.07372593840.01637365310.98375966660.0161077395

374061.15553940570.01635174850.98378121560.016086543

375061.23724356960.01632993160.98380267890.0160654305

376061.3188388670.01630820180.9838240570.0160444013

377061.4003257320.01628655850.98384535040.0160234549

378061.48170459580.01626500120.98386655970.0160025908

379061.56297588650.01624352930.98388768540.0159818084

380061.64414002970.01622214210.98390872820.0159611072

381061.7251974480.01620083920.98392968850.0159404867

382061.80614856140.016179620.98395056690.0159199463

383061.88699378710.0161584840.9839713640.0158994856

384061.96773353930.01613743060.98399208010.0158791039

385062.048368230.01611645930.9840127160.0158588009

386062.1288982680.01609556950.9840332720.0158385759

387062.209324060.01607476070.98405374870.0158184286

388062.28964600960.01605403250.98407414670.0157983583

389062.36986451810.01603338420.98409446630.0157783647

390062.4499799840.01601281540.98411470820.0157584471

391062.52999280350.01599232550.98413487270.0157386052

392062.609903370.01597191410.98415496050.0157188385

393062.68971207460.01595158070.9841749720.0156991465

394062.76941930590.01593132470.98419490760.0156795286

395062.84902544990.01591114570.98421476790.0156599846

396062.92853089020.01589104320.98423455330.0156405138

397063.00793600810.01587101660.98425426430.0156211158

398063.08724118240.01585106560.98427390140.0156017902

399063.16644678940.01583118970.98429346490.0155825365

400063.24555320340.01581138830.98431295550.0155633543

401063.3245607960.0157916610.98433237350.0155442432

402063.40346993660.01577200740.98435171930.0155252026

403063.48228099240.0157524270.98437099350.0155062323

404063.56099432830.01573291940.98439019650.0154873316

405063.63961030680.0157134840.98440932870.0154685003

406063.71812928830.01569412050.98442839050.0154497378

407063.79655163090.01567482840.98444738230.0154310438

408063.87487769070.01565560730.98446630470.0154124178

409063.95310782130.01563645670.9844851580.0153938595

410064.03124237430.01561737620.98450394260.0153753684

411064.10928169930.01559836540.9845226590.0153569442

412064.18722614350.01557942380.98454130760.0153385863

413064.26507605220.01556055110.98455988880.0153202945

414064.34283176860.01554174680.98457840290.0153020682

415064.42049363360.01552301050.98459685040.0152839073

416064.49806198640.01550434180.98461523170.0152658111

417064.57553716390.01548574030.98463354720.0152477794

418064.6529195010.01546720560.98465179730.0152298118

419064.73020933070.01544873730.98466998230.0152119079

420064.80740698410.0154303350.98468810270.0151940673

421064.884512790.01541199830.98470615880.0151762896

422064.96152707560.01539372680.9847241510.0151585745

423065.0384501660.01537552010.98474207970.0151409217

424065.11528238440.01535737790.98475994520.0151233306

425065.1920240520.01533929980.9847777480.0151058011

426065.26867548830.01532128530.98479548840.0150883327

427065.34523701080.01530333420.98481316680.015070925

428065.42170893520.0152854460.98483078350.0150535778

429065.49809157530.01526762040.98484833880.0150362906

430065.5743852430.0152498570.98486583320.0150190632

431065.65059024870.01523215550.9848832670.0150018951

432065.72670690060.01521451550.98490064050.014984786

433065.80273550540.01519693660.98491795410.0149677357

434065.8786763680.01517941850.98493520810.0149507437

435065.95452979140.01516196090.98495240290.0149338098

436066.03029607690.01514456330.98496953880.0149169335

437066.10597552420.01512722550.98498661620.0149001147

438066.18156843110.01510994710.98500363530.0148833529

439066.25707509390.01509272780.98502059660.0148666478

440066.33249580710.01507556720.98503750020.0148499991

441066.40783086350.0150584650.98505434670.0148334065

442066.48308055440.01504142090.98507113620.0148168696

443066.55824516920.01502443460.98508786910.0148003882

444066.63332499580.01500750560.98510454570.014783962

445066.70832032060.01499063380.98512116640.0147675906

446066.78323142830.01497381870.98513773140.0147512738

447066.85805860180.01495706010.98515424110.0147350112

448066.93280212270.01494035760.98517069580.0147188025

449067.0074622710.0149237110.98518709570.0147026475

450067.0820393250.01490711980.98520344120.0146865458

451067.15653356150.01489058390.98521973260.0146704971

452067.23094525590.01487410290.98523597010.0146545012

453067.30527468190.01485767650.98525215410.0146385578

454067.37952211170.01484130440.98526828490.0146226666

455067.45368781620.01482498630.98528436270.0146068272

456067.52777206450.01480872190.98530038790.0145910395

457067.60177512460.0147925110.98531636070.0145753031

458067.67569726280.01477635310.98533228150.0145596177

459067.74953874380.01476024810.98534815040.0145439832

460067.82329983130.01474419560.98536396780.0145283991

461067.89698078710.01472819540.9853797340.0145128653

462067.97058187190.01471224720.98539544920.0144973814

463068.04410334480.01469635060.98541111370.0144819472

464068.11754546370.01468050550.98542672770.0144665625

465068.19090848490.01466471150.98544229170.0144512269

466068.26419266350.01464896840.98545780570.0144359402

467068.33739825310.01463327590.98547327020.0144207022

468068.41052550590.01461763370.98548868530.0144055126

469068.48357467310.01460204150.98550405130.0143903711

470068.5565460040.01458649910.98551936850.0143752774

471068.62943974710.01457100630.98553463710.0143602314

472068.70225614930.01455556270.98554985740.0143452328

473068.77499545620.01454016820.98556502960.0143302813

474068.84765791220.01452482230.9855801540.0143153766

475068.92024376050.0145095250.98559523090.0143005186

476068.99275324260.01449427590.98561026050.014285707

477069.06518659930.01447907480.9856252430.0142709416

478069.13754406980.01446392140.98564017870.014256222

479069.2098258920.01444881540.98565506780.0142415481

480069.28203230280.01443375670.98566991060.0142269197

481069.35416353760.0144187450.98568470730.0142123365

482069.42621983080.014403780.98569945810.0141977982

483069.49820141560.01438886160.98571416340.0141833047

484069.57010852370.01437398940.98572882320.0141688556

485069.64194138590.01435916320.98574343790.0141544509

486069.71370023170.01434438280.98575800770.0141400902

487069.78538528950.01432964790.98577253280.0141257733

488069.85699678630.01431495840.98578701350.0141115

489069.92853494820.01430031390.98580144990.0140972702

4900700.01428571430.98581584240.0140830835

491070.07139216540.01427115930.9858301910.0140689397

492070.1427116670.01425664870.98584449610.0140548387

493070.21395872620.01424218230.98585875780.0140407801

494070.28513356320.01422775980.98587297640.0140267639

495070.35623639740.01421338110.98588715210.0140127898

496070.42726744660.01419904590.98590128520.0139988576

497070.49822692810.01418475390.98591537570.013984967

498070.56911505750.0141705050.9859294240.0139711179

499070.63993204980.0141562990.98594343020.01395731

500070.71067811870.01414213560.98595739460.0139435432

501070.78135347670.01412801470.98597131740.0139298172

502070.85195833570.01411393590.98598519870.0139161319

503070.9224929060.01409989920.98599903880.013902487

504070.99295739720.01408590420.98601283790.0138888824

505071.06335201780.01407195090.98602659620.0138753178

506071.13367697510.01405803890.98604031390.0138617931

507071.20393247570.01404416810.98605399110.013848308

508071.27411872480.01403033830.98606762820.0138348624

509071.34423592690.01401654930.98608122520.0138214561

510071.41428428540.01400280080.98609478240.0138080888

511071.48426400260.01398909280.98610829990.0137947605

512071.554175280.01397542490.9861217780.0137814708

513071.62401831790.01396179690.98613521690.0137682197

514071.69379331570.01394820880.98614861670.0137550068

515071.7635004720.01393466030.98616197770.0137418321

516071.83313998430.01392115120.98617530.0137286954

517071.9027120490.01390768120.98618858380.0137155965

518071.97221686180.01389425040.98620182920.0137025351

519072.04165461730.01388085830.98621503660.0136895112

520072.11102550930.01386750490.9862282060.0136765245

521072.18032973050.013854190.98624133770.0136635748

522072.24956747280.01384091330.98625443170.0136506621

523072.31873892710.01382767470.98626748840.013637786

524072.38784428340.01381447410.98628050790.0136249465

525072.45688373090.01380131120.98629349030.0136121434

526072.52585745790.01378818580.98630643580.0135993764

527072.59476565150.01377509780.98631934470.0135866455

528072.66360849830.01376204710.9863322170.0135739504

529072.73238618390.01374903330.9863450530.013561291

530072.80109889280.01373605640.98635785280.0135486671

531072.86974680890.01372311620.98637061650.0135360785

532072.93833011520.01371021240.98638334450.0135235252

533073.00684899380.0136973450.98639603680.0135110068

534073.07530362580.01368451380.98640869350.0134985234

535073.14369419160.01367171850.98642131490.0134860746

536073.21202087090.01365895910.98643390120.0134736603

537073.28028384220.01364623540.98644645240.0134612805

538073.34848328360.01363354710.98645896880.0134489348

539073.41661937190.01362089410.98647145050.0134366232

540073.48469228350.01360827630.98648389770.0134243455

541073.55270219370.01359569360.98649631050.0134121015

542073.62064927720.01358314560.9865086890.0133998912

543073.68853370780.01357063240.98652103350.0133877143

544073.75635565830.01355815360.98653334420.0133755706

545073.82411530120.01354570920.9865456210.0133634601

546073.89181280760.0135332990.98655786430.0133513826

547073.95944834840.01352092290.98657007420.0133393379

548074.02702209330.01350858070.98658225070.0133273259

549074.09453421140.01349627220.98659439420.0133153465

550074.1619848710.01348399720.98660650460.0133033994

551074.22937423960.01347175580.98661858220.0132914846

552074.2967024840.01345954760.98663062710.0132796018

553074.36396977030.01344737250.98664263950.0132677511

554074.43117626370.01343523040.98665461950.0132559321

555074.49832212880.01342312110.98666656720.0132441448

556074.56540752920.01341104450.98667848290.0132323891

557074.63243262820.01339900050.98669036650.0132206647

558074.69939758790.01338698880.98670221840.0132089716

559074.76630257010.01337500940.98671403860.0131973095

560074.83314773550.01336306210.98672582720.0131856785

561074.89993324430.01335114670.98673758450.0131740783

562074.9666592560.01333926320.98674931050.0131625088

563075.03332592920.01332741140.98676100540.0131509698

564075.09993342210.0133155910.98677266930.0131394613

565075.16648189190.01330380210.98678430230.0131279831

566075.23297149520.01329204440.98679590470.013116535

567075.29940238810.01328031790.98680747650.013105117

568075.36577472570.01326862230.98681901780.0130937288

569075.43208866260.01325695760.98683052880.0130823705

570075.49834435270.01324532360.98684200970.0130710417

571075.56454194930.01323372010.98685346050.0130597425

572075.63068160480.01322214710.98686488150.0130484727

573075.69676347110.01321060440.98687627260.0130372321

574075.76278769950.01319909190.98688763410.0130260206

575075.82875444050.01318760950.98689896610.0130148381

576075.8946638440.01317615690.98691026860.0130036846

577075.96051605930.01316473420.98692154190.0129925597

578076.0263112350.0131533410.98693278610.0129814635

579076.0920495190.01314197750.98694400130.0129703958

580076.15773105860.01313064330.98695518750.0129593565

581076.22335600060.01311933840.9869663450.0129483455

582076.2889244910.01310806260.98697747390.0129373625

583076.35443667530.01309681590.98698857420.0129264076

584076.41989269820.01308559810.98699964610.0129154807

585076.48529270390.0130744090.98701068980.0129045815

586076.5506368360.01306324860.98702170530.0128937099

587076.61592523750.01305211670.98703269280.0128828659

588076.68115805070.01304101330.98704365230.0128720494

589076.74633541740.01302993810.9870545840.0128612601

590076.81145747870.01301889110.98706548810.0128504981

591076.87652437510.01300787210.98707636460.0128397631

592076.94153624670.01299688110.98708721360.0128290552

593077.00649323270.01298591790.98709803530.0128183741

594077.0713954720.01297498240.98710882980.0128077197

595077.13624310270.01296407450.98711959720.012797092

596077.20103626250.0129531940.98713033760.0127864908

597077.26577508830.01294234090.98714105110.012775916

598077.33045971670.0129315150.98715173780.0127653675

599077.39509028360.01292071620.98716239790.0127548452

600077.45966692410.01290994450.98717303140.012744349

601077.52418977330.01289919960.98718363850.0127338788

602077.58865896510.01288848160.98719421920.0127234345

603077.65307463330.01287779020.98720477380.0127130159

604077.71743691090.01286712530.98721530220.012702623

605077.78174593050.01285648690.98722580470.0126922557

606077.84600182410.01284587490.98723628120.0126819138

607077.91020472310.01283528910.98724673190.0126715972

608077.97435475850.01282472940.9872571570.0126613059

609078.03845206050.01281419570.98726755650.0126510397

610078.10249675910.0128036880.98727793050.0126407986

611078.16648898350.01279320610.98728827920.0126305824

612078.23042886240.01278274980.98729860250.012620391

613078.29431652430.01277231920.98730890070.0126102244

614078.35815209660.0127619140.98731917390.0126000824

615078.42193570680.01275153430.98732942210.012589965

616078.48566748140.01274117980.98733964540.0125798719

617078.54934754660.01273085050.9873498440.0125698032

618078.61297602810.01272054630.98736001790.0125597588

619078.67655305110.01271026710.98737016730.0125497385

620078.74007874010.01270001270.98738029210.0125397422

621078.80355321940.01268978310.98739039270.0125297699

622078.86697661250.01267957820.98740046890.0125198215

623078.93034904270.01266939790.9874105210.0125098968

624078.99367063250.01265924210.98742054910.0124999958

625079.05694150420.01264911060.98743055310.0124901183

626079.12016177940.01263900350.98744053330.0124802643

627079.18333157930.01262892050.98745048970.0124704337

628079.24645102460.01261886160.98746042240.0124606264

629079.30952023560.01260882670.98747033150.0124508423

630079.37253933190.01259881580.98748021710.0124410813

631079.43550843290.01258882860.98749007920.0124313433

632079.49842765740.01257886510.98749991810.0124216283

633079.56129712370.01256892530.98750973370.0124119361

634079.62411694960.0125590090.98751952630.0124022666

635079.68688725250.01254911610.98752929570.0123926198

636079.74960814950.01253924660.98753904220.0123829955

637079.81227975690.01252940030.98754876590.0123733938

638079.87490219090.01251957710.98755846670.0123638144

639079.93747556680.01250977710.98756814490.0123542573

6400800.01250.98757780050.0123447225

641080.0624756050.01249024580.98758743360.0123352098

642080.1249024960.01248051440.98759704420.0123257191

643080.18728078690.01247080570.98760663250.0123162504

644080.2496105910.01246111970.98761619860.0123068036

645080.3118920210.01245145610.98762574250.0122973786

646080.37412518960.0124418150.98763526430.0122879753

647080.43631020880.01243219630.98764476420.0122785936

648080.498447190.01242259990.98765424210.0122692335

6490

6500

6510

6520

6530

6540

6550

6560

6570

6580

6590

6600

6610

6620

6630

6640

6650

6660

6670

6680

6690

6700

6710

6720

6730

6740

6750

6760

6770

6780

6790

6800

6810

6820

6830

6840

6850

6860

6870

6880

6890

6900

6910

6920

6930

6940

6950

6960

6970

6980

6990

7000

7010

7020

7030

7040

7050

7060

7070

7080

7090

7100

7110

7120

7130

7140

7150

7160

7170

7180

7190

7200

7210

7220

7230

7240

7250

7260

7270

7280

7290

7300

7310

7320

7330

7340

7350

7360

7370

7380

7390

7400

7410

7420

7430

7440

7450

7460

7470

7480

7490

7500

7510

7520

7530

7540

7550

7560

7570

7580

7590

7600

7610

7620

7630

7640

7650

7660

7670

7680

7690

7700

7710

7720

7730

7740

7750

7760

7770

7780

7790

7800

7810

7820

7830

7840

7850

7860

7870

7880

7890

7900

7910

7920

7930

7940

7950

7960

7970

7980

7990

8000

8010

8020

8030

8040

8050

8060

8070

8080

8090

8100

8110

8120

8130

8140

8150

8160

8170

8180

8190

8200

8210

8220

8230

8240

8250

8260

8270

8280

8290

8300

8310

8320

8330

8340

8350

8360

8370

8380

8390

8400

8410

8420

8430

8440

8450

8460

8470

8480

8490

8500

8510

8520

8530

8540

8550

8560

8570

8580

8590

8600

8610

8620

8630

8640

8650

8660

8670

8680

8690

8700

8710

8720

8730

8740

8750

8760

8770

8780

8790

8800

8810

8820

8830

8840

8850

8860

8870

8880

8890

8900

8910

8920

8930

8940

8950

8960

8970

8980

8990

9000

9010

9020

9030

9040

9050

9060

9070

9080

9090

9100

9110

9120

9130

9140

9150

9160

9170

9180

9190

9200

9210

9220

9230

9240

9250

9260

9270

9280

9290

9300

9310

9320

9330

9340

9350

9360

9370

9380

9390

9400

9410

9420

9430

9440

9450

9460

9470

9480

9490

9500

9510

9520

9530

9540

9550

9560

9570

9580

9590

9600

9610

9620

9630

9640

9650

9660

9670

9680

9690

9700

9710

9720

9730

9740

9750

9760

9770

9780

9790

9800

9810

9820

9830

9840

9850

9860

9870

9880

9890

9900

9910

9920

9930

9940

9950

9960

9970

9980

9990

10000

Sheet1

Sheet2

Sheet3

Heat conduction in a Semi-infinite slab object, -6If there is a film heat transfer resistance at x=0, the solution for the temperature distribution isWhen heat transfer coefficient h is infinite (that is no film resistance), the second term in the RHS is zero, as erf()=1.Case 2: Constant heat flux at x=0The heat balance equation is still the same as before as the equation describes point wise heat balance within the domain (i.e. object) and it is not affected by the way how we impose conditions on the system.How we impose the system is through the boundary condition.The boundary condition for this case is:

Heat conduction in a Semi-infinite slab object, -7The solution for the heat flux is:The temperature distribution is:The quantity of interest is the rise in temperature at the x=0 surface:Due to the constant supply of heat flux, the x=0 temperature increases with time and follows the square root dependence.

THE ENERGY EQUATIONThe equation of energy, which describes energy transport in homogeneous fluid or solid.An energy balance without any source term can be written as

THE ENERGY EQUATION, -2Some comments about the terms used in Energy EquationKinetic energy: is energy associated with the fluid bulk/observable motion, i.e. v2 on a unit volume basis.Internal energy: is energy associated with the random translational and internal motions of the molecules plus the energy of interaction between the molecules.Potential energy: does not appear in Eq. It is taken into account as the work done on the system by gravity.The Source Terms: Not included in the energy equation except the viscous dissipation term which appears automatically. Rest can be added as and when needed

Let us do this term-by-term.

THE ENERGY EQUATION, -3Term 1: Accummulation

Volume of elementInternal Energy per unit massK.E per unit volumeTerm 2: Net Energy Addition by Convection

THE ENERGY EQUATION, -4Term 3: Net Energy Addition by Conductionqx, qy, qz are components of the heat flux vector qTerm 4: Work done on the system surfaces by fluid.Work done by pressure forcesWork done by viscous forcesThe work done by the pressure forces:

THE ENERGY EQUATION, -5The work by the viscous forcesThe rate of work done against the three components of the volume forces (gravity)Term 4: Work done on the systems volume by external forcesPutting all those terms into the shell energy balance equation, then dividing by xyz, we get the following energy balance equation per unit volume:

THE ENERGY EQUATION, -6ConvectionConductionPressure WorkViscous WorkGravity Work

THE ENERGY EQUATION, -7Such a massive energy equation can be written in a compact vector-tensor:rate of gain of energy per unit volumerate of energy input per unit volume by convectionrate of energy input per unit volume by conductionrate of work done on fluid per unit volume by pressure forcesrate of work done on fluid per unit volume by viscous forcesrate of work done on fluid per unit volume by gravitational forces

SPECIAL FORMS OF ENERGY EQUATIONUsing Equation of continuity one can transform the energy equation in Substantial Derivative formTake scalar product with VThis is called the Mechanical Energy form of Equation of MotionSubtract the Mechanical Energy equation from the Energy Equation

SPECIAL FORMS OF ENERGY EQUATION, -2rate of gain of internal energy per unit volumerate of energy input per unit volume by conductionReversible rate of internal energy increase per unit volume by volume changeIrreversible rate of internal energy increase per unit volume by viscous dissipationInternal Energy is not a convenient energy to be measured, needs to be replaced by more convenient terms

SPECIAL FORMS OF ENERGY EQUATION, -3Equations written in terms of internal energy (often difficult to measure in an experiment) are not directly too useful.One prefers equations written in terms of temperature and heat capacities (usually easier to measure in an experiment.From thermodynamics:Thus thermal energy equation becomesExpress q in terms of temperature gradient and stress tensors in terms of velocity gradient, the above equation will be in terms of temperature & velocity

SPECIAL FORMS OF ENERGY EQUATION, -4For a Newtonian fluid with constant k:For ideal gases, sinceThermal Equation For ideal gases

SPECIAL FORMS OF ENERGY EQUATION, -5For solid, Liquid and gases if pressure is constant and viscous dissipation is neglectedThermal Equation for such cases is:In particular for solids the Thermal Equation is:

SOURCE TERMS (NUCLEAR)Nuclear Source term in particular and other source terms in general are evaluated on volumetric basisHeat generated per unit volume per unit time.If we know the following then this term can be calculatedNumber of nuclear reactions/fissions occurring per unit volume per unit timeEnergy released in each fission reaction To calculate the volumetric rate of fission reactions we needNumber of atoms available for fissionThe number of atoms among these participating in fissionNumber of neutrons available for these fission reactions

LETS DISCUSS EACH TERM AND QUANTIFY

SOURCE TERMS (NUCLEAR), -2Number of Atoms per unit volume of fuel available NffFuel available is generally in the form of a compoundIf available in pure metallic/compound form then

Where,Av = Avogadros Numberff = density of fissionable fueli = number of fissionable atoms in one molecule of fuelIf the fuel is not pure it has some enrichment ratio rWhere,r =enrichment or mass ratio of fissionable fuel to total fuelf = mass fraction of fuel in fuel material and can be calculated as

SOURCE TERMS (NUCLEAR), -3Where,Mnf = Molecular Mass of non-fissionable fuel material such as U238MO2= Molecular Mass of non fuel elements that make up the the balance of the fuel material such as O2 in UO2If Mnf Mff then above equation simplifies to

Thus finally the number density of fissionable atoms is:

SOURCE TERMS (NUCLEAR), -4Number of neutrons available for these fission reactionsDepend on the neutron flux available in the reactorComes from the Physics of the reactor and can be calculatedFlux Distribution isSinusoidal in the axial directionBessel function of radius in the radial directionConstant according to the above functions for homogeneous coreEffected by the different non-fissile materials present in the core, such as control rods, cladding and moderator etc for heterogeneous coresDepends on the temperature of the fuel and moderatorThe accuracy of flux distribution determination is extremely important from thermal hydraulics point of view For our purpose we will assume that the flux distribution is known to us for the time being

SOURCE TERMS (NUCLEAR), -5Energy released in each fission reaction200 Mev is liberated90% of this can be recoveredRest is lostIf Gf is the energy released in one fission reaction then energy available for Heat transfer is 0.9Gf

SOURCE TERMS (NUCLEAR), -6How many fission reactions will take place for a known neutron flux and Number density NffThis depends on the probability of the reaction to occurThe microscopic neutron crossection f is neededThis again comes from the Physics of nuclear materials

THUS THE FINAL SOURCE TERM RELATION BECOMESVolumetric source strengthMev/cm3sec by 1.5477E-8 to convert to BTU/hr ft3Neutron flux, No/cm2 secEffective Neutron fission microscopic crossection, cm2Energy released MeVNumber density, No/cm3

Problem-1An electric wire with radius ro of 0.50 mm is made of copper [electrical conductivity = 5.1 x 107 ohm-1 m-1 and thermal conductivity = 380 W/(m K)]. It is insulated (see figure) to an outer radius r1 of 1.50 mm with plastic [thermal conductivity = 0.350 W/(m K)]. The ambient air is at 38.0oC and the heat transfer coefficient from the outer insulated surface to the surrounding air is 8.500 W/(m2 K). Determine the maximum current in amperes that can flow at steady-state in the wire without any portion of the insulation getting heated above its maximum allowable temperature of 93.0oC.

Problem-2A solid is formed from the conical section of a sphere of radius R as shown in the figure. The spherical surface at r = R is insulated, while the two conical surfaces at q = q1 and q = q2 are held at temperatures T1 and T2, respectively. The thermal conductivity k of the solid material may be assumed constant.a) Establish an expression for the temperature T (q) in the solid object at steady state. b) Find the total rate of heat flow across each of the conical surfaces.

Problem-3A heated sphere of diameter D is placed in a large amount of stagnant fluid. Consider the heat conduction in the fluid surrounding the sphere in the absence of convection. The thermal conductivity k of the fluid may be considered constant. The temperature at the sphere surface is TR and the temperature far away from the sphere is Ta. Establish an expression for the temperature T in the surrounding fluid as a function of r, the distance from the center of the sphere.Also develop an expression for the Nusselt number

Problem-4An oil (of viscosity m and thermal conductivity k) acts as a lubricant between two coaxial cylinders. The inner cylinder is stationary and the outer cylinder of radius R rotates at an angular velocity W. The clearance between the cylinders is b, which is small compared to the radii of the cylinders; so, curvature effects can be neglected and the cylindrical system can be approximated by a plane narrow slit (to be solved in Cartesian coordinates) as shown in the figure. Derive an expression for the maximum temperature in the lubricant if both cylinders are at temperature T0. Neglect the temperature dependence of m and k, but explicitly take into account the heat generated by viscous dissipation.