Simulation and Study of Heat Transfer

Keerthana P. G.

Why should we study heat transfer?

Insulation and radiant barriers Heat exchangers

used in refrigeration, air conditioning, space heating, power generation, and chemical processing.

Heat sinks also help to cool electronic and optoelectronic devices such as CPUs, higher-power lasers, and light-emitting diodes (LEDs).

In order to understand these materials and their uses, it is necessary to understand their mechanical properties such as stress coefficients, thermal conductivity etc.

Our project is an attempt to verify several theoretical predictions made regarding Hard Sphere models in 1D system

What this is all about.

To simulate heat conduction in a 1-dimensional chain with different atoms to verify theoretical predictions

To study one dimensional heat conduction and verify its properties. The long term objective is to study heat transfer and investigate the

influence of the working fluid on a finite time Carnot’s Engine.

Isolated 3d system

FCC lattice of 4*8*8*8 argon atoms Defining the lattice Initialising positions and momenta of all atoms. Positions by forced initialisation Velocities using Marsaglia Bray method.

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Algorithm

Forces calculation from lennard jones potential

Periodic Boundary Condition with minimum image condition Velocity verlet algorithm to update forces and positions

Stabilization

Heat Conduction in 1D Chain of Atoms N particles in a 1D box of length L Forced position initialisation. Velocity from Maxwell-Boltzmann distribution in compliance with the

initial system temperature. The left and right hand walls are kept at two different temperatures,

in our case, 8K and 2K respectively. Atoms have a small but finite radius.

Interactions:1. Particle-particle interactions by hard collisions. We assume that all interactions are elastic.

2. Interaction with the wall: The atoms at the end, upon hitting the wall bound back with a velocity sampled from Rayleigh distribution.

Here, TL is the temperature of the wall from which the atom is bounding back.

Local temperatures are calculated according the expression described.

Temperature profile is obtained at steady state. BBGKY(Boboligov-Born-Green-Kirkwood-Yvon) Equations to model the

system.

Theoretical Prediction

Surprisingly, the temperature profiles in the case of equal masses and the one with arbitrarily small mass differences completely different.

But energy density is constant in space at steady state. Seemingly contradicting.

Temperature profile doesn’t change under m(i)-->c*m(i) Also, from the boundary conditions, it can be verified that T(cT1, cT2,

x)=cT(T1, T2, x)

Explanation: Temperature also depends on n(x), i.e. the local number density.T(x,t)= 2 ε(x,t)/n(x,t)

1D system(with 1 particle type)

N particles of same mass in a 1D box of length L Collisions are assumed to be elastic. Initialisation is done at a certain temperature,the left and right hand

walls are kept at 3K and 2K respectively. Temperature is calculated for every individual atom and a

temperature profile is obtained.

Temperature profile

Theoritical prediction says that the temperature profile will be flat with themperature .Here is the simulated temperature profile

Energy profile

1D system with Particles of Alternating Mass N particles in a tube of length L. Alternate particles have alternate

masses. The ratio of masses (m1/m2) is varied to get different temperature

profiles. Once again we assume elastic collisions.

Temperature profile

Expected profile

Observations:

Temperature has a smooth and continuously varying profile with jumps at the boundaries that tend to smoothen with increase in system size.

For small =()/ and large N , the temperature profile depends on and only by a scaling factor of

Energy Profile

The energy density profile here is supposed to be the same as the profile for a same mass system

The Way Forward

Study of the effects of working fluid on the performance of a finite time Carnot’s cycle.

Acknowlegments:

Physics Review Letter, 2001, Heat Conduction in a One-Dimensional Gas of Elastically Colliding Particles of Unequal Mass, Abhishek Dhar, Raman Research Institute, Bangalore

Research Article, 2008, Heat Transfer in Low Dimensional systems, Abhishek Dhar, Raman Research Institute, Bangalore

Thank You