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….

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.

Define: How to get strictly non-homogeneous Equation?

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, q g, e.g., Conversion of electrical to thermal energy in an electric rod. Curing of concrete brides and dams. Nuclear fuel rod. Solid Propellant Rockets.

Heat Transfer in Rocket Solid Propellent

Fast Construction of Bridges : RCC Technology 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.

Plane Wall with Thermal Energy Generation For one-dimensional, steady-state conduction in an isotropic medium properties: homogeneous medium with constant properties: q’’’

The temperature distribution is parabolic in x. Applying Fourier’s 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.

Boundary Conditions Case 1: Simple Dirichlet Boundary Conditions: q’’’ Solution : & Maximum temperature occurs inside the slab close the higher temperature surface.

Boundary conditions Case 1: Convection Boundary Conditions: At x = -L : At x = L : Solution : q’’’

heat transfer rate:

with

Boundary Conditions Case 3: Symmetric Dirichlet Boundary Conditions: q’’’ Solution : Maximum temperature occurs at the center of the slab. The left and right parts of the slab are isolated at the axis.

Modified Boundary Conditions Case 3: Symmetric Dirichlet Boundary Conditions: Solution : ```

Boundary Conditions Case 3: Symmetric Convection Boundary Conditions: Solution : At x = L :

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

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.

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

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

Solid Sphere with heat generation: