One Dimensional Non-Homogeneous Conduction Equation

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One Dimensional Non-Homogeneous Conduction
Equation

• P M V Subbarao
• Associate Professor
• Mechanical Engineering Department
• IIT Delhi

A truly non-homogeneous ODE... A Basis for
Generation of Tremendous Power.
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Homogeneous ODE

• How to obtain a non-homogeneous ODE for one
  dimensional Steady State Heat Conduction
  problems?
• Blending of Convection or radiation effects into
  Conduction model.

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Define
How to get strictly non-homogeneous Equation?
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Conduction with Thermal Energy Generation

• A truly non-homogenous ODE.
• Consider the effect of a process occurring within
  a medium such as thermal energy generation, qg ,
  e.g.,
• Conversion of electrical to thermal energy in an
  electric rod.
• Curing of concrete brides and dams.
• Nuclear fuel rod.
• Solid Propellant Rockets.

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Heat Transfer in Rocket Solid Propellent
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Fast Construction of Bridges RCC Technology
Convection Radiation
Convection Radiation
The term curing is used to include maintenance
of a favorable environment for the continuation
of chemical reactions, i.e. retention of
moisture within, or supplying moisture to the
concrete from an external source and protection
against extremes of temperature.
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Plane Wall with Thermal Energy Generation
For one-dimensional, steady-state conduction in
an isotropic medium properties
q
homogeneous medium with constant properties
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The temperature distribution is parabolic in x.
Applying Fouriers law
heat transfer rate
Thus, dT/dx is a function of x, and therefore
both the heat transfer rate and heat flux are
dependent on x for a medium with energy
generation.
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Boundary Conditions
Case 1 Simple Dirichlet Boundary Conditions

q
Maximum temperature occurs inside the slab close
the higher temperature surface.
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Boundary conditions
Case 1 Convection Boundary Conditions
At x -L
q
At x L
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heat transfer rate
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with
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Boundary Conditions
Case 3 Symmetric Dirichlet Boundary Conditions
q
Maximum temperature occurs at the center of the
slab. The left and right parts of the slab are
isolated at the axis.
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Modified Boundary Conditions
Case 3 Symmetric Dirichlet Boundary Conditions
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Boundary Conditions
Case 3 Symmetric Convection Boundary Conditions
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Radial Systems
For one-dimensional, steady-state conduction in
an isotropic homogeneous medium with
constant properties
q
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Cartridge Heaters
A Reliable heater should continue to provide
superior heat transfer, uniform temperatures
and resistance to oxidation and corrosion even
at high temperatures.
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Central Condition for a solid cylinder
One dimensional conduction is possible only if
there is axial Symmetry of temperature profile.
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Surface boundary condition
Dirichlet Boundary Condition
At r rO T(rO) TO
Maximum temperature occurs at center.
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Surface Convection Boundary Condition
All the heat generated in the cylinder of length
L, is transferred to ambient fluid by Convection
heat transfer.
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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/AC
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Hollow Cylinder with Heat Generation

• Hollow cylinder geometry has significant
  applications as a nuclear fuel rods.
• Nuclear fuel rods are made of hollow cylinder
  where the heat generated is carried away by a
  liquid metal coolant flowing either on the inside
  or outside the tubes.
• 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.

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Solid Propellant Rocket
Solid propellant rockets burn a solid block made
of fuel, oxidizer, and binder (plastic or
rubber). The block is called grain.
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Hybrid Rocket
Section A-A
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Sphere with Uniform Heat Generation
Bombs
GF- 3 US Baseball Grenade
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Solid Sphere with heat generation