One Dimensional Polar Geometries For Conduction with Heat Generation P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi A.

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Presentation transcript:

One Dimensional Polar Geometries For Conduction with Heat Generation P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi A Basis for Generation of Tremendous Power….

Radial Systems For one-dimensional, steady-state conduction in an isotropic homogeneous medium with constant properties: q’’’

Cartridge Heaters A Reliable heater should continue to provide superior heat transfer, uniform temperatures and resistance to oxidation and corrosion even at high temperatures.

Central Condition for a solid cylinder: One dimensional conduction is possible only if there is axial Symmetry of temperature profile. q’’’

Surface boundary condition: Dirichlet Boundary Condition:At r = r O T(r O ) = T O Maximum temperature occurs at center.

Surface Convection Boundary Condition: q’’’ All the heat generated in the cylinder of length L, is transferred to ambient fluid by Convection heat transfer.

Current Carrying Conductor An important practical application. Cooling of current carrying conductors enhances their current carrying capacity. Knowledge of temperature distribution is required to make sure that the conductor is not reaching its burn out condition. Uniform internal heat generation occurs due to Joule heating. Rate of heat generation per unit volume: Current Density : i/A C

Maximum Temperature at the axis of the cylinder:

Hollow Cylinder with Heat Generation Hollow cylinder geometry has significant applications as a nuclear fuel rods. Hollow electrical conductors of cylindrical shape are used for high current carrying applications, where the cooling is done by a fluid (Hydrogen) flowing on the inside. Annual reactors insulated from inside or outside are used in Chemical Processes.

Solid Propellant Rocket Solid propellant rockets burn a solid block made of fuel, oxidizer, and binder (plastic or rubber). The block is called grain.

Hybrid Rocket Section A-A

Positron Rocket Heat Generation is inside the matrix. The outer surface and core of the cooled by liquid hydrogen. The inner surface is treated as insulated. This is a problem of heat generation in a hallow cylinder with inner surface insulated.

In-Situ Thermal Processing of Shale Oil Heat Generation/absorption is inside the shale oil. The outer surface of the shale oils is surrounded by rock. The inner surface is electrical heated. This is a problem of heat generation in a hallow cylinder with outer surface insulated.

Boundary Conditions - I Solution : Inner surface insulated: Convection cooling of outer surface: riri rOrO

Boundary Conditions - II Solution : Heating of Inner surface : Insulated outer surface: riri rOrO

Sphere with Uniform Heat Generation Bombs GF- 3 US Baseball Grenade

Encapsulation of pharmaceutical products.

Solid Sphere with heat generation: Surface convection cooling: Surface radiative cooling: