PowerPoint Presentation

Forms of Energy Generation:Degradation of electrical energy to heatHeat from nuclear source (by fission)Heat from viscous dissipationOverall Shell Energy Balance

Energy GenerationLet S = rate of heat production per unit volume (W/m3)(Se)(Sn)(Sv)S is the notation from BSL. He denotes Se, Sn, Sv, and Sc for each heat source.1Electrical Heat SourceConsider an electrical wire (solid cylinder):

Shell Heat Balance:Rate of Heat IN:Rate of Heat OUT:Generation:We can explore many other geometries. However, the most practical is a solid cylinder because this is the shape of electrical wires.2Electrical Heat SourceRate of Heat IN:Rate of Heat OUT:Generation:

The Shell:Rate of Heat INArea perpendicular to qr at r = rYou may or may not explain this. This will help them understand shell heat balances so they dont complain about derivations in the exam. (Spoon-feeding purposes)3Electrical Heat SourceRate of Heat IN:Rate of Heat OUT:Generation:

The Shell:Rate of Heat OUTArea perpendicular to qr at r = r + drYou may or may not explain this. This will help them understand shell heat balances so they dont complain about derivations in the exam. (Spoon-feeding purposes)4Electrical Heat SourceRate of Heat IN:Rate of Heat OUT:Generation:

The Shell:Generation = Volume X SeToo smallYou may or may not explain this. This will help them understand shell heat balances so they dont complain about derivations in the exam. (Spoon-feeding purposes)5Electrical Heat SourceConsider an electrical wire (solid cylinder):

Shell Heat Balance:The answer to the question in red: You cannot cancel r from the (rq) evaluated at (r + dr) because (rq) is actually (r + dr)*q.6Electrical Heat SourceConsider an electrical wire (solid cylinder):We now have:Q: Is this correct?NO!

Spoon-feeding purposes7Electrical Heat SourceConsider an electrical wire (solid cylinder):We now have:We must adhere to the definition of the derivative:

Spoon-feeding purposes8Electrical Heat SourceConsider an electrical wire (solid cylinder):We now have:Boundary conditions:Note: The problem statement will tell you hints about what boundary conditions to use.Integrating:

Let the students verify the integration.When there is heat generation, we should avoid ODE solutions that yield an infinite temperature at r = 0.That is why we need the boundary condition: at r = 0, q is finite.9Electrical Heat SourceConsider an electrical wire (solid cylinder):We now have:Applying B.C. 1:

Because q has to be finite at r = 0, all the terms with radius, r, below the denominator must vanish. Therefore:

The heat flux is found to be linear against radius. Q: Does it make sense that the q profile does not contain the variable L?A: Yes, because q is heat flux. If you want the heat flow rate, then multiply q by the transfer area (which includes L).10Electrical Heat SourceConsider an electrical wire (solid cylinder):We now have:

Substituting Fouriers Law:11Electrical Heat SourceConsider an electrical wire (solid cylinder):We now have:

Applying B.C. 2:This is it! But, we rewrite it into a nicer form12Electrical Heat SourceConsider an electrical wire (solid cylinder):

Temperature Profile:

Important assumptions:Temperature rise is not large so that k and Se are constant & uniform.The surface of the wire is maintained at T0.Heat flux is finite at the center.13Electrical Heat SourceOther important notesLet:These imply the following :We dont often see the value of Se. On the other hand, we usually report values of E, ke, and I. Therefore, we can use these information to get Se.14Electrical Heat SourceTemperature Profile:

The stress profile versus the temperature profile:Heat flux profile:15Electrical Heat SourceQuantities that might be asked for:Maximum Temperature

Average Temperature Rise

Heat Outflow Rate at the SurfaceSubstituting r = 0 to the profile T(r):In getting the maximum temp., you dont always just substitute r = R in the temperature profile. The more appropriate way is to get the derivative of the temperature profile, and equate to 0.

Be reminded that the dA element in the average temperature rise integral, dA = r dr d(theta).

16Electrical Heat SourceExamples for Review:Example 10.2-1 and Example 10.2-2Bird, Stewart, and Lightfoot, Transport Phenomena, 2nd Ed., p. 29517Nuclear Heat SourceConsider a spherical nuclear fuel assembly (solid sphere):Before doing a balance, let:

Actually, for the solid cylinder geometry with constant S term, the same profile can be derived for a nuclear heat source. However, we will learn in this particular example how to:Handle a case where the S term, Sn, is dependent on radius, r.Handle the derivation for a spherical geometry made up of 2 different materials (the cladding and the fissionable material); and,Handle the boundary conditions between the 2 differing solid materials in the sphere.18Nuclear Heat SourceConsider a spherical nuclear fuel assembly (solid sphere):Before doing a balance, let:

These are just notations. You can adhere to any notation you like.Take note that the generation term only appears in the fissionable material, and not in the cladding.19Nuclear Heat SourceConsider a spherical nuclear fuel assembly (solid sphere):

For the fissionable material:Rate of Heat IN:Rate of Heat OUT:Generation:This slide focuses on the shell balance at the fissionable material only.20

Electrical Heat SourceRate of Heat IN:Rate of Heat OUT:Generation:Generation = Volume X SnToo smallYou may or may not explain this. This will help them understand shell heat balances so they dont complain about derivations in the exam. (Spoon-feeding purposes)21

Nuclear Heat SourceFor the fissionable material:For the Al cladding:No generationhere!In case you get lost, Al is aluminum.22

Nuclear Heat SourceFor the fissionable material:For the Al cladding:No generationhere!23

Nuclear Heat SourceFor the fissionable material:For the Al cladding:No generationhere!24

Nuclear Heat SourceFor the fissionable material:For the Al cladding:No generationhere!Integrating:Integrating:

25Nuclear Heat SourceIntegrating:Integrating:

Boundary Conditions:Boundary Conditions:For the fissionable materialFor the Al claddingThe B.C. at the fissionable material is summoned because of symmetry of the sphere around r = 0.The B.C. at the Al cladding is summoned because the heat flux is constant across a series of resistances.26Nuclear Heat Source

Inserting Fouriers Law:Inserting Fouriers Law:

For the fissionable materialFor the Al claddingSome observations:In the fissionable material, heat flux increases with radius (cubic). This is expected because of the parabolic increase of the generation term Sn as the radius increases. However, this stops at R(F).In the Al cladding, the heat flux decreases with radius. The cladding is there to prevent heat loss.27Nuclear Heat Source

For the fissionable materialFor the Al claddingBoundary Conditions:Boundary Conditions:At r = R(F),

T(F) = T(C)R(F)R(C)At r = R(C),

T(C) = T0The B.C. at the fissionable material is referred to as continuity of temperature. This B.C. is summoned at solid-solid interfaces.

28Nuclear Heat SourceFor the fissionable materialFor the Al cladding

Again, we learned in this particular example how to:Handle a case where the S term, Sn, is dependent on radius, r.Handle the derivation for a spherical geometry made up of 2 different materials (the cladding and the fissionable material); and,Handle the boundary conditions between the 2 differing solid materials in the sphere.

Final note: To get Tmax, we just substitute r = 0 in the T(F) profile. Tmax can indicate thermal deterioration.

29Recall the Overall Shell Energy Balance:Overall Shell Energy Balance

Q by Convective TransportQ by Molecular TransportW by Molecular TransportW by External ForcesEnergy GenerationSteady-State!We need to account for work by molecular motion when dealing with viscous dissipation. This slide is just a recall from a previous lecture.30Overall Shell Energy Balance

Q by Convective TransportQ by Molecular TransportW by Molecular TransportHow can we account for all these terms at once?We need all these terms for viscous dissipation:31Combined Energy Flux VectorConvective Energy FluxHeat Rate from Molecular MotionWork Rate from Molecular MotionCombined Energy Flux Vector:We introduce something new to replace q:The combined energy flux vector is designed to integrate the energy balance terms into a single variable, e, which works just like q in itself.

However, we can simplify this further!32Combined Energy Flux VectorCombined Energy Flux Vector:We introduce something new to replace q:Recall the molecular stress tensor:When dotted with v:Substituting into e:The molecular stress tensor is the momentum molecular transport tensor.It consists of the pressure force and the stress tensor itself. Delta is just a unit vector like i-hat or j-hat.33Combined Energy Flux VectorCombined Energy Flux Vector:We introduce something new to replace q:Simplifying the boxed expression:Finally:We look into the boxed expression. In the end, we were able to reduce the combined energy flux vector into a form involving tau and enthalpy. This is the form to be used in the viscous dissipation part.34Viscous Dissipation SourceConsider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders:

Explain that viscous dissipation occurs as heat is released by the friction caused by the rubbing cylindrical shells of the fluid.The inner cylinder is stationary while the outer cylinder is moving. The distance between the inner and outer cylinder is b.The temperature at the outer cylinder is Tb while the temperature at the inner cylinder is T0.

We will only make a balance in the zoomed in portion of the flow. Because b is relatively small, the curvature is neglected, and the coordinate system adopted is simply Cartesian. Yey!! It is already found that the velocity only varies linearly with x, as well as temperature. Note that x is the vertical direction.35Viscous Dissipation SourceConsider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders:

We now make a shell balance shown in red on the left.Rate of Energy IN:Rate of Energy OUT:When the combined energy flux vector is used, the generation term will automatically appear from e.We now make a shell balance. We used the combined energy flux vector because the fluid is in motion.36Viscous Dissipation SourceConsider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders:We now make a shell balance shown in red on the left.Rate of Energy IN:Rate of Energy OUT:When the combined energy flux vector is used, the generation term will automatically appear from e.At this point, we cant invoke a boundary condition because we dont know what ex is!So we use the equation of the combined energy flux tensor next.37Viscous Dissipation SourceConsider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders:When the combined energy flux vector is used, the generation term will automatically appear from e.Fouriers Law:Newtons Law:The convection term is cancelled because the velocity component in the x-direction is zero. Remember that we only need ex.We also invoke both Fouriers and Newtons Law.38Viscous Dissipation SourceConsider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders:When the combined energy flux vector is used, the generation term will automatically appear from e.Substituting the velocity profile:Integrating:Recall the velocity profile and substitute in the vz term. The last integration is due to dT/dx.39Viscous Dissipation SourceConsider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders:When the combined energy flux vector is used, the generation term will automatically appear from e.Boundary Conditions:

After applying the B.C.:

This is the temperature profile!! This equation can only be applied when Tb is not equal to T0.40Viscous Dissipation SourceConsider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders:When the combined energy flux vector is used, the generation term will automatically appear from e.Q: So where is Sv?After applying the B.C.:

Check if the units of Sv are indeed W/m3.41Viscous Dissipation SourceConsider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders:Temperature Profile:New Dimensionless Number:The Brinkman number measures how important is viscous heating to the temperature profile. The higher the value of Br, the larger the temperature rise.42Viscous Dissipation SourceScenarios when viscous heating is significant:

Flow of lubricant between rapidly moving parts.Flow of molten polymers through dies in high-speed extrusion.Flow of highly viscous fluids in high-speed viscometers.Flow of air in the boundary layer near an earth satellite or rocket during reentry into the earths atmosphere.

43