_{1}

A linear model of three layers plane wall exposed to oscillating temperatures with different amplitudes and frequencies was built by using a physical superposition. A physical superposition of two states was performed, one state is a wall which one surface is exposed to oscillating temperature and the other surface is exposed to zero relative temperature and a second state is a wall which one surface is exposed to relative zero temperature while the other surface is exposed to oscillating temperature with different amplitudes and frequencies. Temperature distributions were introduced for different amplitudes, frequencies and thermal conductivities. It was shown that increasing the frequency value decreases the temperature penetration length, high frequency value leads to extremum temperature values changes on the surface while low frequency value allows gradually temperature changes during the time period. Temperature distribution lines where there are at the same time heat flux entry and heat flux exit were not received for the same constraint frequencies.

Temperature oscillations on a plane wall have a high importance in some engineering areas. For example, the cylinder plane wall of a combustion engine is revealed to an oscillating temperature by the unsteady burning of the fuel into the engine cylinder space. Despite that the temperature oscillating amplitude may have a small value related to the temperature full value, a fatigue stress may appear causing a matter failure. Racopoulos et al. [^{−}^{6} m and the angular frequency of the signals is of order 10^{12} rad・s^{−}^{1}. Perez and Autrique [

An example of a system that could clearly describe the need for such a research is a chip cooling system. On the chip in the direction of the fins array, four layers are sometimes identified: an internal interface (TIM1-thermal interface material type 1), a lid or a cap, an external interface (TIM2) and the base of the fins array. The alternating current that is passing into the chip through its Ohm resistances causes fluctuations in the chip internal heat production leading to temperature fluctuations.

As reviewed above, works dealing with oscillating heat source or oscillating temperature were based upon experimental results combined with nonlinear models. Basically, the energy constraint was located on one surface only and heat was transferred along one homogeneous medium. This work purpose is to introduce temperature distributions of three layers plane wall exposed to temperatures oscillations with different amplitudes and frequencies. It was performed by using a physical superposition and complex numbers. Despite this physical model is based upon linear equations, it expand clearly the relevant knowledge, in principle, to unlimited different mediums while the energy source may be located around the system surfaces. The results received support previous works.

The current model is intend to solve the temperature distribution of three layers plane wall exposed to temperatures oscillations with different amplitudes and frequencies as is shown in

The main assumptions accompanied the physical model development are

unsteady heat transfer and one dimensional heat conduction without heat generation.

In order to solve the main problem two more simple problems were solved. After solving the more simple problems a physical superposition were performed in order to identify the main problem solution.

The structure problem is shown in

The characterized differential equation is:

∂ 2 T * ∂ x * 2 = 1 α * ∂ T * ∂ t * (1)

or:

∂ T * ∂ t * = α * ∂ 2 T * ∂ x * 2 (2)

All equations parameters are dimensionless, T * is a dimensionless temperature, x * is a dimensionless length, α * is a dimensionless thermal diffusivity and t * is a dimensionless time. The dimensionless length is defined as x / ( L 1 + L 2 + L 3 ) . If average values are defined: T a v = ( T 01 + T 02 ) / 2 ,

α a v = ( α 1 + α 2 + α 3 ) / 2 , average time period t a v = ( 2 π / ω 1 + 2 π / ω 2 ) / 2 = π ( 1 / ω 1 + 1 / ω 2 ) , average frequency ω a v = ( ω 1 + ω 2 ) / 2 then the dimensionless parameters may be defined: T * = T / T a v , α * = α / α a v , t * = t / t a v , ω * = ω / ω a v . Sub-indexes 1, 2 and 3 are related to every wall layer. The average parameters definition of T a v , t a v and ω a v is according to the constraint on the external system surfaces and is not related to a specific slab.

It is defined:

L 1 * = L 1 L 1 + L 2 + L 3 (3)

L 2 * = L 1 + L 2 L 1 + L 2 + L 3 (4)

The temperature distribution into every block layer is received as:

T 1 * = c 1 e ω 1 * 2 α 1 * x 1 * + i ( ω 1 * t * + ω 1 * 2 α 1 * x 1 * ) + c 2 e − ω 1 * 2 α 1 * x 1 * + i ( ω 1 * t * − ω 1 * 2 α 1 * x 1 * ) (5)

T 2 * = c 3 e ω 1 * 2 α 2 * x 2 * + i ( ω 1 * t * + ω 1 * 2 α 2 * x 2 * ) + c 4 e − ω 1 * 2 α 2 * x 2 * + i ( ω 1 * t * − ω 1 * 2 α 2 * x 2 * ) (6)

T 3 * = c 5 e ω 1 * 2 α 3 * x 3 * + i ( ω 1 * t * + ω 1 * 2 α 3 * x 3 * ) + c 6 e − ω 1 * 2 α 3 * x 3 * + i ( ω 1 * t * − ω 1 * 2 α 3 * x 3 * ) (7)

(A possible development solution to Equations (5)-(7), see the appendix)

where:

0 ≤ x 1 * ≤ L 1 * (8)

L 1 * ≤ x 2 * ≤ L 1 * + L 2 * (9)

L 1 * + L 2 * ≤ x 3 * ≤ 1 (10)

If average value are defined: k a v = ( k 1 + k 2 + k 3 ) / 2 then the dimensionless conductivities may be defined: k 1 * = k 1 / k a v , k 2 * = k 2 / k a v , k 3 * = k 3 / k a v .

Six boundary conditions have to be defined:

T 1 * ( x 1 * = 0 ) = T 01 * cos ( ω 1 * t * ) (11)

T 1 * ( x 1 * = L 1 * ) = T 2 * ( x 2 * = L 1 * ) (12)

k 1 * ∂ T 1 * ∂ x 1 * ( x 1 * = L 1 * ) = k 2 * ∂ T 2 * ∂ x 2 * ( x 2 * = L 1 * ) (13)

T 2 * ( x 2 * = L 2 * ) = T 3 * ( x 3 * = L 2 * ) (14)

k 2 * ∂ T 2 * ∂ x 2 * ( x 2 * = L 2 * ) = k 3 * ∂ T 3 * ∂ x 3 * ( x 3 * = L 2 * ) (15)

T 3 * ( x 3 * = 1 ) = 0 (16)

Boundary conditions 11 and 16 indicate the temperature constraints on the system external left surface and on the system right external surface. Boundary conditions 12 and 14 indicate the temperatures on the boundary surface between the first and the second slab and on the boundary surface between the second and the third slab, boundary conditions 13 and 15 indicate the heat flux through these surfaces.

Despite the problem is a time function, the time conditions are periodically steady and initial conditions is not needed.

By placing the temperatures distributions equations (Equations (5)-(7)) into the boundary conditions (Equations (11)-(16)) the following equations are received:

c 1 + c 2 = T 01 * (17)

c 1 e ω 1 * 2 α 1 * L 1 * + i ω 1 * 2 α 1 * L 1 * + c 2 e − ( ω 1 * 2 α 1 * L 1 * + i ω 1 * 2 α 1 * L 1 * ) − c 3 e ω 1 * 2 α 2 * L 1 * + i ω 1 * 2 α 2 * L 1 * − c 4 e − ( ω 1 * 2 α 2 * L 1 * + i ω 1 * 2 α 2 * L 1 * ) = 0 (18)

k 1 * c 1 ( ω 1 * 2 α 1 * + i ω 1 * 2 α 1 * ) e ω 1 * 2 α 1 * L 1 * + i ω 1 * 2 α 1 * L 1 * − k 1 * c 2 ( ω 1 * 2 α 1 * + i ω 1 * 2 α 1 * ) e − ( ω 1 * 2 α 1 * L 1 * + i ω 1 * 2 α 1 * L 1 * ) − k 2 * c 3 ( ω 1 * 2 α 2 * + i ω 1 * 2 α 2 * ) e ω 1 * 2 α 2 * L 1 * + i ω 1 * 2 α 2 * L 1 * + k 2 * c 4 ( ω 1 * 2 α 2 * + i ω 1 * 2 α 2 * ) e − ( ω 1 * 2 α 2 * L 1 * + i ω 1 * 2 α 2 * L 1 * ) = 0 (19)

c 3 e ω 1 * 2 α 2 * L 2 * + i ω 1 * 2 α 2 * L 2 * + c 4 e − ( ω 1 * 2 α 2 * L 2 * + i ω 1 * 2 α 2 * L 2 * ) − c 5 e ω 1 * 2 α 3 * L 2 * + i ω 1 * 2 α 3 * L 2 * − c 6 e − ( ω 1 * 2 α 3 * L 2 * + i ω 1 * 2 α 3 * L 2 * ) = 0 (20)

k 2 * c 3 ( ω 1 * 2 α 2 * + i ω 1 * 2 α 2 * ) e ω 1 * 2 α 2 * L 2 * + i ω 1 * 2 α 2 * L 2 * − k 2 * c 4 ( ω 1 * 2 α 2 * + i ω 1 * 2 α 2 * ) e − ( ω 1 * 2 α 2 * L 2 * + i ω 1 * 2 α 2 * L 2 * ) − k 3 * c 5 ( ω 1 * 2 α 3 * + i ω 1 * 2 α 3 * ) e ω 1 * 2 α 3 * L 2 * + i ω 2 * 2 α 3 * L 2 * + k 3 * c 6 ( ω 1 * 2 α 3 * + i ω 1 * 2 α 3 * ) e − ( ω 1 * 2 α 3 * L 2 * + i ω 1 * 2 α 3 * L 2 * ) = 0 (21)

c 5 e ω 1 * 2 α 3 * + i ω 1 * 2 α 3 * + c 6 e − ( ω 1 * 2 α 3 * + i ω 1 * 2 α 3 * ) = 0 (22)

By defining:

a 1 = e ω 1 * 2 α 1 * L 1 * + i ω 1 * 2 α 1 * L 1 * (23)

a 2 = e − ( ω 1 * 2 α 1 * L 1 * + i ω 1 * 2 α 1 * L 1 * ) (24)

a 3 = e ω 1 * 2 α 2 * L 1 * + i ω 1 * 2 α 2 * L 1 * (25)

a 4 = e − ( ω 1 * 2 α 2 * L 1 * + i ω 1 * 2 α 2 * L 1 * ) (26)

a 5 = k 1 * ( ω 1 * 2 α 1 * + i ω 1 * 2 α 1 * ) e ω 1 * 2 α 1 * L 1 * + i ω 1 * 2 α 1 * L 1 * (27)

a 6 = k 1 * ( ω 1 * 2 α 1 * + i ω 1 * 2 α 1 * ) e − ( ω 1 * 2 α 1 * L 1 * + i ω 1 * 2 α 1 * L 1 * ) (28)

a 7 = k 2 * ( ω 1 * 2 α 2 * + i ω 1 * 2 α 2 * ) e ω 1 * 2 α 2 * L 1 * + i ω 1 * 2 α 2 * L 1 * (29)

a 8 = k 2 * ( ω 1 * 2 α 2 * + i ω 1 * 2 α 2 * ) e − ( ω 1 * 2 α 2 * L 1 * + i ω 1 * 2 α 2 * L 1 * ) (30)

a 9 = e ω 1 * 2 α 2 * L 2 * + i ω 1 * 2 α 2 * L 2 * (31)

a 10 = e − ( ω 1 * 2 α 2 * L 2 * + i ω 1 * 2 α 2 * L 2 * ) (32)

a 11 = e ω 1 * 2 α 3 * L 2 * + i ω 1 * 2 α 3 * L 2 * (33)

a 12 = e − ( ω 1 * 2 α 3 * L 2 * + i ω 1 * 2 α 3 * L 2 * ) (34)

a 13 = k 2 * ( ω 1 * 2 α 2 * + i ω 1 * 2 α 2 * ) e ω 1 * 2 α 2 * L 2 * + i ω 1 * 2 α 2 * L 2 * (35)

a 14 = k 2 * ( ω 1 * 2 α 2 * + i ω 1 * 2 α 2 * ) e − ( ω 1 * 2 α 2 * L 2 * + i ω 1 * 2 α 2 * L 2 * ) (36)

a 15 = k 2 * ( ω 1 * 2 α 2 * + i ω 1 * 2 α 2 * ) e ω 1 * 2 α 2 * L 1 * + i ω 1 * 2 α 2 * L 1 * (37)

a 16 = k 3 * ( ω 1 * 2 α 3 * + i ω 1 * 2 α 3 * ) e − ( ω 1 * 2 α 3 * L 2 * + i ω 1 * 2 α 3 * L 2 * ) (38)

a 17 = e ω 1 * 2 α 3 * + i ω 1 * 2 α 3 * (39)

a 18 = e − ( ω 1 * 2 α 3 * + i ω 1 * 2 α 3 * ) (40)

The coefficients algebraic system equations are received as:

c 1 + c 2 = T 01 * (41)

a 1 c 1 + a 2 c 2 − a 3 c 3 − a 4 c 4 = 0 (42)

a 5 c 1 − a 6 c 2 − a 7 c 3 + a 8 c 4 = 0 (43)

a 9 c 3 + a 10 c 4 − a 11 c 5 − a 12 c 6 = 0 (44)

a 13 c 3 − a 14 c 4 − a 15 c 5 + a 16 c 6 = 0 (45)

a 17 c 5 + a 18 c 6 = 0 (46)

The relevant determinants are defined:

a 19 = [ 1 1 0 0 0 0 a 1 a 2 − a 3 − a 4 0 0 a 5 − a 6 − a 7 a 8 0 0 0 0 a 9 a 10 − a 11 − a 12 0 0 a 13 − a 14 − a 15 a 16 0 0 0 0 a 17 a 18 ] (47)

a 20 = [ T 01 * 1 0 0 0 0 0 a 2 − a 3 − a 4 0 0 0 − a 6 − a 7 a 8 0 0 0 0 a 9 a 10 − a 11 − a 12 0 0 a 13 − a 14 − a 15 a 16 0 0 0 0 a 17 a 18 ] (48)

a 21 = [ 1 T 01 * 0 0 0 0 a 1 0 − a 3 − a 4 0 0 a 5 0 − a 7 a 8 0 0 0 0 a 9 a 10 − a 11 − a 12 0 0 a 13 − a 14 − a 15 a 16 0 0 0 0 b 17 b 18 ] (49)

a 22 = [ 1 1 T 01 * 0 0 0 a 1 a 2 0 − a 4 0 0 a 5 − a 6 0 a 8 0 0 0 0 0 a 10 − a 11 − a 12 0 0 0 − a 14 − a 15 a 16 0 0 0 0 a 17 a 18 ] (50)

a 23 = [ 1 1 0 T 01 * 0 0 a 1 a 2 − a 3 0 0 0 b 5 − a 6 − a 7 0 0 0 0 0 a 9 0 − a 11 − a 12 0 0 a 13 0 − a 15 a 16 0 0 0 0 b 17 a 18 ] (51)

a 24 = [ 1 1 0 0 T 01 * 0 a 1 a 2 − a 3 − a 4 0 0 a 5 − a 6 − a 7 a 8 0 0 0 0 a 9 a 10 0 − a 12 0 0 a 13 − a 14 0 a 16 0 0 0 0 0 a 18 ] (52)

The differential equations coefficients are received as:

The structure problem is shown in

The temperature distribution into every block layer is received as:

where:

Six boundary conditions have to be defined:

According to what was written earlier, boundary conditions 66 and 71 indicate the temperature constraints on the system external left surface and on the system right external surface. Boundary conditions 67 and 69 indicate the temperatures on the boundary surface between the first and the second slab and on the boundary surface between the second and the third slab, boundary conditions 68 and 70 indicate the heat flux through these surfaces.

As explained in Section 2.1 initial condition is not needed.

By placing the temperatures distributions equations (Equations (60)-(62)) into the boundary conditions (Equations (66)-(71)) the following equations are received:

(76)

By defining:

The algebraic coefficients system equations are received as:

The relevant determinants are defined:

The differential equations coefficients are received as:

This stage is a simple superposition of the previous two stages solutions introduced in Sections 2.1 and 2.2.

A physical superposition may be defined as a superposition of structure and not as a superposition of parameters, for example, in this work a physical superposition is a combination of structure solutions introduced in Sections 2.1 and 2.2. A mathematical superposition is a solution combination related to the same parameter, for example a combination of different frequencies

The temperature value

Every line plot in Figures 4-6 introduce specific time. The time cycle

The temperatures boundary conditions introduced in the model are reference temperatures, boundary conditions on the left plane of the left board and boundary condition on the right plane of the right board, so, temperatures negative values are reference temperatures and not absolute values.

Exploring the blue line in the case presented by

along the left direction in case

Comparing

In

It may be stated that as the distance from the constraint surface is increasing (with the same conductivity) or as the thermal conductivity is becoming smaller the relevant surface has difficulty to follow the constraint surface temperature. It is actually shown in all figures. The tracking is seems to break away in some cases while the relevant surface has opposite temperature sign against the temperature constraint sign. It is shown for example in

There are cases where there are only heat flux entry into the wall, this is shown for example in

By looking for example in a

1) Increasing the frequency value decreases the temperature penetration length.

2) High frequency value causes steady extremum temperature values on the surface while low frequency value allows gradually temperature changes during the time period.

3) For high thermal conductivity, the temperature will follow the temperature constraint on the surface, for low thermal conductivity, the absolute temperature value will decrease, the constraint on the other surface may increase again the temperature value.

4) In systems that are working with high frequency and demand insulation, it is possible to save material thickness and insulation.

5) Temperature distribution lines where there are at the same time heat flux entry and heat flux exit were not received for the same constrained frequencies.

6) This work introduces actually a method that may be used for calculating any number of planes wall.

7) In order to build this physical model, a physical superposition had to be used.

8) This work problem is parallel to fluids layers between oscillating plates while the velocity vector is coming in place of the scalar temperature value and the kinematic viscosity substitutes the thermal diffusivity.

9) Using complex numbers is an appropriate tool to deal with this work and other parallel problems.

Sadik, S. (2018) A Three-Layers Plane Wall Exposed to Oscillating Temperatures with Different Amplitudes and Frequencies. Energy and Power Engineering, 10, 165-185. https://doi.org/10.4236/epe.2018.104012

We look for a solution to the equation:

try:

where X is a function of

The relevant derivatives are received as:

The basic equation will be received as:

or:

The last connection leads to two differential equations that has to be solved, the first equation is the time equation:

The solution of the time differential equation is:

where

The second equation is the location equation:

The solution of the location differential equation is:

The full solution is received as:

The full solution may be reduced to two coefficients only, the same notation will be used:

So, the full solution will be received as:

While using the equality:

the full solution will be received as: