One Dimensional Models for Conduction Heat Transfer in

One Dimensional Models for Conduction Heat Transfer in Manufacturing Processes P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Simplified Models for Complex Heat Transfer due to micro-molecular Movements!!!

Steady-State One-Dimensional Conduction • For conduction through a large wall the heat equation reduces to one dimensional Equation in Cartesian system. • Assume a homogeneous medium with invariant thermal conductivity ( k = constant) : One dimensional Transient conduction with heat generation.

Heat Transfer in Metal Cutting

Axi-symmetric Conduction Models for Manufacturing Processes No changes in T along and z directions: Steady state with no heat generation :

Axi-Symmetric Steady Conduction in Radial Systems Homogeneous and constant property material

At any radial location the surface are for heat conduction in a solid cylinder is: At any radial location the surface are for heat conduction in a solid sphere is: The GDE for cylinder: The GDE for sphere:

General Solution for Cylinder: General Solution for Sphere:

Boundary Conditions • No solution exists when r = 0. • Totally solid cylinder or Sphere have no physical relevance! • Inner wall at finite radius is essential for steady state conduction with no heat generation.

UNSTEADY-STATE HEAT CONDUCTION Applications where rate/duration of heating/cooling is a Design Parameter……

Welding Process : How to decide the Rate of Welding?

Injection Molding Process They need you to help them to the time required for a part in a mold to cool to an acceptable temperature for removal.

Resin Transfer Molding Process (1) Insert fiber preform (2) Close mold (3) Inject mixed resin/catalyst (4) Part solidifies via reaction (5) Open mold (6) Remove part The cycle time for step (4) is approximately the same 25 minutes for parts of all sizes made by Aero. Form. All other molding operations (1), (2), (3), (5) and (6) take a total of 5 minutes.

Selective LASER Sintering

General Conduction Equation • The general form of these equations in multi-dimensions for homogeneous material is: For Cartesian Geometry:

Cooling/Quenching of Hot Processed Products

Transient-conduction • If we have a ball with initial temperature of T 0 and it is left in fluid at Te. • Heat is transferred by convection at the surface. • As the surface temperature decreases, heat is transferred from the center of the ball to the surface, then to the fluid. • Temperature will vary with location within a system and with time. • Temperature and rate heat transfer variation of a system are dependent on its internal resistance and surface resistance.

First Law Analysis of Cooling /Quenching of Hot Object s Rate of Change in energy of hot object= Rate of Heat transfer Rate of Heat Transfer = Rate of Convection by fluid = Rate of Conduction transfer in the metal ball For spherical objects

T T 0 Rcond T 0 T At any instant: Rconv Total thermal resistance of the system: Rtot

• A Biot number is defined as:

Relationship between the Biot number and the temperature profile.

System with negligible internal resistance For this case Bi 1. 0. and the temperature profile within the body is quite uniform. The rate of change in internal energy of the body is equal to the rate of heat taken away from the surface by convection:

Rearranging: Integrating to any time interval:

Define, Thermal Time Constant

Thermal Time Constant The total energy transferred during time t

Hot Rolling of Steel Sheets