Chapter 5: Transient Conduction

1. Chapter 5: Transient Conduction • In general, a numerical solution is needed. • The lumped capacitance method: • Temperature gradients within the solid are small: • Simple analysis • Energy balance Many problems are time dependent (unsteady, transient). Steady state --- an approximation

2. Consider a hot metal forging that is initially at a uniform temperature Ti and is quenched by immersing it in a liquid of lower temperature Lumped capacitance method Assuming the temperature of the solid is spatially uniform at any instant ---- lumped capacitance method

3. Applying the energy balance to the control volume Lumped capacitance method Separation of variables

4. The thermal time constant Lumped capacitance method the resistance to convection heat transfer the lumped thermal capacitance of the solid (the energy needed to raise the solid by 1 oC) solid to respond more slowly to changes in its thermal environment -- the characteristic length, the ratio of the solid’s volume to surface area)

5. Lumped capacitance method Fourier number, dimensionless time 0.1, the error associated with the lumped capacitance method is small In order for Bi to be small: Small Rcond – large k --- temperature uniform Large Rconv – small h --- heat transfer rate small Small Lc -- small size

6. General Lumped Capacitance Analysis

7. The above equation is a nonlinear, non-homogenous ordinary differential equation that cannot be integrated to obtain an exact solution. Special cases: No imposed heat flux, generation, or convection General Lumped Capacitance Analysis Separating variables and integrating:

8. General Lumped Capacitance Analysis No radiation, Equation (5.15) reduces to a linear first-order, non-homogenous differential equation Solution: