1. Yoon kichul Department of Mechanical Engineering Seoul National University Multi-scale Heat Conduction

2. Seoul National University Contents 1. Introductory Explanation of the Chapter 2. Heat Flux Equation (Lagging Behavior) 4. Dual-Phase-Lag Model by Tzou 6. Simplified BTE for Phonon System 1) Gurtin and Pipkin 2) Joseph and Preziosi 5. Parallel or Coupled Heat Diffusion Process 3. Jeffrey Type Lagging Heat Equation

3. Seoul National University 1. Introductory Explanation of the Chapter ∙Title of this chapter “Dual-Phase-Lag Model” - “Dual” : Two different phenomena (heat  temp. gradient, and reverse) - “Phase-Lag” : Lag in time  phase lag by Fourier’s transform ∙The concept of this chapter - Temperature gradient  Heat flux - Heat source  Temperature gradient Not instantaneously Lagging behavior b/w heat flux and temperature gradient lag in time (t)  phase lag (iωt)

4. Seoul National University 2. Heat Equation 1) Gurtin and Pipkin : kernal function -  : Cattaneo equation : Fourier’s law

5. Seoul National University 2. Heat Equation 2) Joseph and Preziosi - : effective conductivity, : elastic conductivity 0 With assumption 1)2) 1) 2) 1) 2) By Leibniz integral rule

6. Seoul National University 2. Heat Equation 2)

7. Seoul National University 3. Jeffrey Type Lagging Heat Equation With eqn. (7.3) : Jeffrey’s eqn. : Fourier’s law : Cattaneo equation Generally,  Thermal process b/w Fourier’s law and Cattaneo equation By on both side,

8. Seoul National University 4. Dual-Phase-Lag Model by Tzou Extended from lagging concept With assumption Delay time : heat source  temperature gradient : temperature gradient  heat flux In certain cases such as short-purse laser heating, both and exist By Taylor expansion Only requirement : (does not require ) However, DPL model produces a negative conductivity component Generalized form of DPL needs to be considered same as heat equation by Joseph and Preziosi with

9. Seoul National University 4. Dual-Phase-Lag Model by Tzou Generalized form of DPL 1) 2) Here, is not defined by  can be theoretically allowed ∴ More general than Jeffrey’s equation  Can describe behavior of parallel heat conduction

10. Seoul National University 5. Parallel or Coupled Heat Diffusion Process : volumetric heat capacities d : rod diameter, N: number of rods, D : inner diameter of pipe : total surface area per unit length : total cross-sectional areas of rods and fluld Solid-fluid heat exchanger Assumptions Fluid is stationary, pipe is insulated from outside Rods are sufficiently thin  use average temp. in a cross section Heat transfer along x direction only Average convection coefficient h Cross-sectional area of the fluid is also sufficiently thin Constant C s, C f, κ s, κ f

11. Seoul National University 1)2) By combining eqn. 1) and 2) to eliminate T f  obtain differential equation for T s 1)2)3) 1) 2) 3) 5. Parallel or Coupled Heat Diffusion Process relaxation time

12. Seoul National University Differential equation for T s - Solutions exhibit diffusion characteristics 5. Parallel or Coupled Heat Diffusion Process - Equation describes a parallel or coupled heat diffusion process In this example, Dual-Phase-Lag model can still be applied Initial temperature difference b/w rods and fluid  local equilibrium X at the beginning

13. Seoul National University 6. Simplified BTE for Phonon System Callaway - No acceleration term, simplified scattering terms (two-relaxation time approx.) - : relaxation time for U process : Not-conserved total momentum after scattering - : relaxation time for N process : Conserved total momentum after scattering - f 0, f 1 : equilibrium distributions Guyer and Krumhansl solved the BTE  derived following equation : average phonon speed When Same as Jeffrey’s equation

14. Seoul National University 6. Simplified BTE for Phonon System When Energy transfer by wave propagation The scattering rate for U process is usually very high N process contributes little to the heat conduction At higher temperature  Heat transfer occurs by diffusion mechanism, rather than by wave-like motion Only at low temperature Mean free path of phonons in U process is longer than specimen size Scattering rate of N process is high enough to dominate other scatterings  Heat transfer occurs by wave-like motion called second sound