Dual-Phase-Lag Model (Chap 7.1.2)

1. Yoon kichul Department of Mechanical Engineering Seoul National University Dual-Phase-Lag Model(Chap 7.1.2) Multi-scale Heat Conduction

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

3. 1. Introductory Explanation of the Chapter ∙The concept of this chapter - Temperature gradient  Heat flux Not instantaneously - Heat source  Temperature gradient Lagging behavior b/w heat flux and temperature gradient ∙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 lag in time (t)  phase lag (iωt)

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

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

6. 2. Heat Equation 2)

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

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

9. 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. 5. Parallel or Coupled Heat Diffusion Process Assumptions Solid-fluid heat exchanger Fluid is stationary, pipe is insulated from outside Rods are sufficiently thin  use average temp. in a cross section Cross-sectional area of the fluid is also sufficiently thin Heat transfer along x direction only Average convection coefficient h Constant Cs, Cf,κs, κf : 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

11. 5. Parallel or Coupled Heat Diffusion Process 1) 2) By combining eqn. 1) and 2) to eliminate Tf obtain differential equation for Ts 1) 1) 2) 3) relaxation time 2) 3)

12. 5. Parallel or Coupled Heat Diffusion Process Differential equation for Ts - Equation describes a parallel or coupled heat diffusion process - Solutions exhibit diffusion characteristics 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. 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 - f0 , f1 : equilibrium distributions Guyer and Krumhansl solved the BTE  derived following equation : average phonon speed When Same as Jeffrey’s equation

14. 6. Simplified BTE for Phonon System When Energy transfer by wave propagation The scattering rate for U process is usually very high At higher temperature N process contributes little to the heat conduction  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