A Three-Layers Plane Wall Exposed to Oscillating Temperatures with Different Amplitudes and Frequencies

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.


Introduction
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 S. Sadik DOI: 10.4236/epe.2018.104012166 Energy and Power Engineering appear causing a matter failure.Racopoulos et al. [1] [2] investigated the temperature oscillations in the combustion chamber walls of a diesel engine.Based on theoretical model and experimental measurements, the penetration of the oscillating temperature part of the temperature full value into the inner cylinder wall was shown for few matters.It was observed that when increasing the degree of insulation, the wall temperature swings highly increase, while at the same time the corresponding depth inside which they disappear, decreases.Investigation of the efficiency of a thermoelectric power generator (TPG) was performed by Yan and Malen [3].Thermoelectric power generator converts heat directly into electricity without moving parts.It was shown that the use of a periodic heat source can increase the efficiency of a TPG.Temperature oscillations may be also formed by a periodic heating.Two methods of periodic heating implementation are an induction heating and a direct resistance heating.Surface hardening, welding and melting are some applications of induction heating.Direct resistance heating has major metal working applications as heating prior to forming and heat treating.Major nonmetals application is a glass melting.Sahin and Yilbas [4] analyzed the temperature rise in an insulated slab which is subjected to a direct resistance heating and to a conduction heating.It was found that for both heating methods, the thermal penetration depth for high frequencies is small.
Additional area related to temperature oscillations is the controlling temperature in buildings.Ma and Wang [5] investigated the dynamic heat transfer performance of an exterior Planar Thermal Mass (ePTM) subject to sinusoidal heating and cooling.Analytical solution was performed when the mean value of outdoor air temperature was equal to indoor air temperature.The analytical solution showed that the time lag and the decrement factor are independent of environment temperatures.Rojas-Trigos et al. [6] analyzed a sample that is uniformly heated on one of its surfaces by a power density modulated by a periodical square wave.The solution comprises a transient term and an oscillatory term superposed to it.Comparison between experimental results and the theoretical transient response showed a good adequacy.Non-Fourier behavior or a non-Fourier model is a modification of Fourier law in order to get a finite velocity of the heat wave.Cossali [7] investigated the effects of non-Fourier characteristics of the material on oscillating thermal fields.A general solution in terms of temperature Fourier transform was shown for a 1-D slab with convective boundary conditions.Al-Sanea and Zedan [8] investigated numerically the effect of the insulation layer in a composite wall on the rate of heat transfer under steady periodic conditions for summer and winter conditions in Riyadh Saudi Arabia.
The numerical model was validated against a semi-analytic procedure based on the Laplace transform technique developed for a three-layered composite wall.
The wall was subjected to a sinusoidal ambient temperature variation and a periodic solar insolation on one side and constant room temperature on the other side.The thermal performance with an insulation layer placed on the inside of a wall structure was compared to that when the insulation layer was placed on the 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
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 Figure 1.
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.

A Three Layers Plane Wall Exposed at One Surface (the Left
Surface) to Temperature Oscillations Where the Other Surface (the Right Surface) Is Maintained at a Relative Zero Temperature The structure problem is shown in Figure 2.
The characterized differential equation is: or: 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 L L + + .If average values are defined: ( ) ω is according to the constraint on the external system surfaces and is not related to a specific slab.
It is defined: Figure 2. The first building problem, a three layers plane wall exposed at one surface (the left surface) to temperature oscillations while the other surface (the right surface) is maintained at a relative zero temperature.
The temperature distribution into every block layer is received as: e e (A possible development solution to Equations ( 5)- (7), see the appendix) where: If average value are defined: ( ) Six boundary conditions have to be defined: 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: e e e e 0 e e e e 0 By defining: The coefficients algebraic system equations are received as: The relevant determinants are defined: 17 18 The differential equations coefficients are received as: The structure problem is shown in Figure 3.
The temperature distribution into every block layer is received as:  T c c 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: e e e e 0  e e e e 0 By defining: The algebraic coefficients system equations are received as: The relevant determinants are defined: The differential equations coefficients are received as:   ω related to the first structure introduced in Section 2.1.
The temperature value * 1 T is the sum of the temperature value introduced in Equation ( 5) and Equation (60), the temperature value * 2 T is the sum of the temperature value introduced in Equation ( 6) and Equation ( 61), the temperature value * 3 T is the sum of the temperature value introduced in Equation (7)   and Equation (62).The coefficients values of Equations ( 5)-( 7),  By looking for example in a Figure 6(d) it is seen that the center of the distribution temperature lines are extremum points.At those points surfaces the temperature is not changing and no heat flux transfer the surfaces.This is due to the symmetry of this case, there are the same boundary conditions on the external planes of the system, this causes to temperature symmetry plots and to extremum points.A plane along extremum points is adiabatic, so, we got here a solution of two cases, symmetry and adiabatic.

Conclusions
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.

Appendix
We look for a solution to the equation: ( ) ( ) where X is a function of * x only and i T is a function just of * t only.
The relevant derivatives are received as: ( ) ( ) The basic equation will be received as: ( ) ( ) ( ) ( ) The last connection leads to two differential equations that has to be solved, the first equation is the time equation: While using the equality: the full solution will be received as: outside.The result of the model application showed that the location of the insulation layer has a minimal effect on mean daily heat transfer rates.While dealing with improved heat transfer models in order to cover finite speed of heat propagation Al-Nimr et al.[9] investigated the thermal behavior of a two-layered thin slab carrying periodic signals under the effect of the dual-phase-lag heat conduction model.Two types of periodic signals were considered, a periodic heating source and a periodic imposed temperature at the boundary.The deviation among the prediction of three models (the classical Fourier diffusion model, the thermal wave model and the dual-phase-lag model) were estimated.It was found that the deviation among the three models increases as the frequency of the signal increases and as the thickness of the plate decreases.It was shown that the use of the dual-phase-lag heat conduction model is necessary when the metal film thickness is of order and the angular frequency of the signals is of order 10 −6 m and the angular frequency of the signals is of order 10 12 rad•s −1 .Perez and Autrique [10] investigated an experimental technique dedicated to thermal diffusivity or thermal conductivity identification in isotropic and orthotropic materials.The method is based on analysis of thermal waves induced by periodic excitation in planar samples.It was shown that phase-lag spatial fluctuations as a single frequency are more informative than a frequency sweep at a single point in space in orthotropic specific configurations.

Figure 1 .
Figure 1.The main problem structure.

Figure 3 .
Figure 3.The second building problem, a three layers plane wall exposed on one surface to temperature oscillations (the right surface) while the other surface (the left surface) is maintained at a relative zero temperature.

7 ) 8 ) 9 )
In order to build this physical model, a physical superposition had to be used.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 S. Sadik DOI: 10.4236/epe.2018.104012183 Energy and Power Engineering the kinematic viscosity substitutes the thermal diffusivity.Using complex numbers is an appropriate tool to deal with this work and other parallel problems.

1 c 2 c
is a constant coefficient.The second equation is the location equation: and 3 c are constant coefficients.The full solution may be reduced to two coefficients only, the same notation 75)

2.3. A Three Layers Plane Wall Exposed at One Surface to Temperature Oscillations and the Other Surface Is Exposed to Different Temperature Oscillation
* 3