Heat Transfer Between Two Plates

Carl-Ola Danielsson & Erik BirgerssonFaxénLaboratoriet

March 18, 2002

Abstract

Heat transfer between two parallel plates is studied for a rarefied gas via DSMC simulations. The Knnumber is varied so that we cover the range from the continuum to the free molecular limit. The collisionmodel is based on the variable hard sphere model for the DSMC. Phenomena such as temperature jumpand non-symmetrical velocity distributions are observed.

1 IntroductionTwo plates are held at constant temperature T1 and T2. The gap between the plates is filled with rarefiedArgon gas. The heat exchange in a gas between the two plates is studied, using the simulation programDSMC1. Comparison with the solutions for the extreme values of the Knudsen number, Kn = λL ,where λis the mean free path, i.e. the continuum solution for low Kn and free molecular flow for high Knudsennumbers, are made.

T1 T2

xy

L

Figure 1. The geometry and coordinate system for the heat transfer considered in this paper.

2 Solution methods

2.1 Direct Simulation Monte Carlo (DSMC)

The simulations in this project were performed by using the code DSMC1.FOR. A good introduction to thiscode is given in the book by Bird [1].In the code there are a number of parameters that need to be set, such as boundary conditions and

number density of the gas. During our simulations we held the number density of the gas constant. TheKnutsen number was altered by changing the distance between the plates. Doing so we had to changethe number of real molecules represented by each computational molecule, in order to keep the number ofcomputational molecules constant.

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The gap between the plates was divided into 200 cells and each cell into 10 subcells. The time step waschoosen in such a way that the molecules needed at least 4 time steps to move through a cell and so thatat an average, more than 10 time steps elapsed between two collisions for one molecule. To minimize thestatistical error, a large number of samples were taken.

2.2 Boundary conditions

There are two types of boundary conditions for the Boltzman equation, the specular and diffusive reflection,see figure 2. In a specular reflection, the normal velocity is reversed, the tangential velocity is retained andthe energy preserved.

αα

a.) b.)

Figure 2. a.) Specular reflection b.) Diffusive reflection.

Diffusive reflection is a more realistic boundary condition since it allows for transfer of both energy andtangential momentum. It is only well defined for a large number of incoming molecules and the reflectedmolecules are assumed to leave the surface distributed as a half Maxwellian with the temperature taken asthe temperature of the wall.

2.3 Continuum limit

The Chapman-Enskog method provides a solution to the Boltzman equation, in which the distributionfunction is expressed in the form of a power series. From the solution of the Boltzmann equation using thismethod it is possible to derive expressions for the coefficient of viscosity µ [1] and heat conduction κ. For avariable hard sphere

µ ∝ Tω, κ = 3µCp2

, (1)

where T is the temperature, Cp is the heat capacity and ω = 1/2 + υ, where υ is an empirical parameter.The DSMC code uses ω = 0.81.In the continuum limit we solve

d

dx

µκdT

dx

¶= 0, (2)

with a finite volume method.

2.4 Free molecular flow

At high Knudsen numbers the gas acts as if no collisions takes place between the molecules. The gas can thenbe treated as being composed of two gases, each of them described by a half Maxwellian velocity distributiongiven by the diffusive boundary conditions. Conservation of mass then requires that

n1pT1 = n2

pT2, (3)

where n1 is the number density of the gas moving towards the surface with temperature T2 and vice versa.The total number density of the gas is given by n = n1 + n2, and the temperature by:

T =(n1T1 + n2T2)

n. (4)

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The expression for the heat flux is given in Gombosi [3] as

q = αnk

rk

2πm

γ + 1

γ − 1√T1T2√

T1 +√T2(T2 − T1) (5)

3 Results and Discussion• Temperature: The temperature profiles (∆T = 100) for different Kn numbers are depicted in figure 3.We see that the continuum solution1 agrees with the DMSC solution for Kn = 10−2. For Kn = 100,we are close to the free molecular flow.

0 20 40 60 80 100 120 140 160 180 200260

280

300

320

340

360

380

Cell [#]

T [K

]

Continuum solution

Kn = 10-2 Kn = 1

Kn = 10Kn = 100

Free molecular flow

Figure 3. The temperature profiles obtained from DSMC for Kn=10−2,1, 10 and 100.The continuum solution and the free molecular solution are depicted as well.

With larger Kn numbers the temperature jump at the walls increases. If we look at a layer, with athickness of O(λ) adjacent to the wall, the molecules do not collide, hence we will have an averagetemperature composed of the outgoing particles with the wall temperature and the incoming withanother temperature. The sum of these particles give rise to a local temperature which differs fromthe wall temperature at increasing mean free paths, i.e. higher Kn numbers.

• Heat flux : The corresponding heat fluxes (∆T = 100) for the different Kn numbers are shown in figure4.

0 20 40 60 80 100 120 140 160 180 200-40

-35

-30

-25

-20

-15

-10

-5

0

Cell [#]

q [W

m- 2

]

Kn = 10-2

Kn = 1

Kn = 10Kn = 100

0 20 40 60 80 100 120 140 160 180 200-41

-40.5

-40

-39.5

-39

-38.5

-38

Cell [#]

q [W

m- 2

]

Free molecular flow

Kn = 100 (DSMC)

0 20 40 60 80 100 120 140 160 180 200-2.6

-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

Continuum solution

Kn = 10-2 (DSMC)

Cell [#]

q [W

m- 2

]

Figure 4. The heat fluxes for the Kn numbers. The corresponding continuumand free molecular fluxes for Kn=10−2 and Kn=100.

1Note that a Lennard-Jones potential was used for estimating the heat conduction, whereas the DSMC is based on aVHS-model.

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The heat flux increases with the Kn number, since we decrease the distance between the plates toobtain higher Kn numbers. The heat flux from the continuum solution is a good approximation tothe DSMC computed mean heat flux. At the free molecular limit, we have good agreement (< 1%)between the mean heat flux from DSMC and the theoretical value.

• Velocity distributions: For high Kn number, see figure 5.b and d., the molecules do not collide witheach other. The velocity distribution will then be given by the sum of two half Maxwellians moving inopposite directions. This is clearly visible when the temperature difference between the plates becomeslarge.

-3000 -2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000

-2000

-1500

-1000

-500

0

500

1000

1500

2000

c´ y [m

s-1 ]

c´x [ms-1]

-1500 -1000 -500 0 500 1000 1500 2000-1500

-1000

-500

0

500

1000

1500

c´ y [m

s-1 ]

c´x [ms-1]

-1000 -500 0 500 1000

-1000

-800

-600

-400

-200

0

200

400

600

800

c´ y [m

s-1 ]

c´x [ms-1]

-1000 -500 0 500 1000

-800

-600

-400

-200

0

200

400

600

800

c´ y [m

s-1 ]

c´x [ms-1]

a.)

d.)c.)

b.)

Figure 5. a) Kn=10−2,∆T = 102. Distribution looks symmetrical. b) Kn=102,∆T = 102. Molecules moving withnegative Cx have slightly higer speed. c) Kn=10−2,∆T = 103. Again for low Kn number the distribution looks

rather symmetrical. d) Kn=102,∆T = 103. Two half Maxwellians clearly visible.

For low Kn numbers, see figure 5.a and c., the molecules collide with each other. This can be thoughtof as a large number of local equilibriums, each characterized by its own Maxwellian distribution.Summing all these together will result in a velocity distribution which is hard to distinguish from asymmetrical one, but the fact that we have heat transfer from the hotter to the colder surface, tells usthat the velocity distribution can not be perfectly symmetric.

References[1] G.A. Bird, Molecular Gas Dynamics and the Direct simulation of Gas and Flows, Oxford Science Publi-

cations, 1994

[2] R. B. Bird, Transport Phenomena, John Wiley & Sons, 2002

[3] T. I. Gombosi, Gaskinetic Theory, Cambridge University Press, 1994

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