heat transfer1.ppt
DESCRIPTION
LectureTRANSCRIPT
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HEAT TRANSFERChapter 9 to 12 Transport Phenomenaby Bird, Stewart and Lightfoot
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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)
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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
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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.
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HEAT CONDUCTION, contd. -2From our experience we know the following factsDirection of heat transferToT1Cross sectional areaz, thickness
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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
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HEAT CONDUCTION, contd. -4Heat flux as a vectorTemperature is a scalar, while heat flux is a vector
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HEAT CONDUCTION, contd. -4Components of the Heat Flux VectorFor isotropic materials,kx = ky = kz = kthe heat flux vector will become:WhereandGrad operator
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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.
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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
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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:
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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
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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
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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.
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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
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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
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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).
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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
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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
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Steady state heat Conduction in slab, contd. -6Step 8: Temperature distribution
Thus the temperature distribution across the slab is not linear. > 0 = 0 < 0
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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
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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
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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
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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.
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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
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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
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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.
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Heat conduction in slab: Gas film resistance, -6How about T(0) and T(L)?Recall the following equations:
You get
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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
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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).
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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
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Composite slab objectsLet deal with constant k, and N layers.
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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
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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
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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)
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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
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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
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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
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Heat conduction in cylinder, contd. -7Step 9: Heat fluxHeat flowwhich is independent of r.
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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
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The influence of shape on heat balanceSphereSlabCylinderThe general form for the three shapes is:S (the shape factor) = 0, 1, 2
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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
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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:
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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
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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
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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
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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
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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.
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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
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A typical variation in heat flux w.r.t time
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637079.81227975690.01252940030.98754876590.0123733938
638079.87490219090.01251957710.98755846670.0123638144
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643080.18728078690.01247080570.98760663250.0123162504
644080.2496105910.01246111970.98761619860.0123068036
645080.3118920210.01245145610.98762574250.0122973786
646080.37412518960.0124418150.98763526430.0122879753
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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:
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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.
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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
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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.
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THE ENERGY EQUATION, -3Term 1: Accummulation
Volume of elementInternal Energy per unit massK.E per unit volumeTerm 2: Net Energy Addition by Convection
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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:
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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:
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THE ENERGY EQUATION, -6ConvectionConductionPressure WorkViscous WorkGravity Work
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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
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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
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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
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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
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SPECIAL FORMS OF ENERGY EQUATION, -4For a Newtonian fluid with constant k:For ideal gases, sinceThermal Equation For ideal gases
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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:
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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
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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
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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:
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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
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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
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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
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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.
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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.
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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
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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.