Transient Heat Conduction: Comparison Of Finite Difference And Semi-Infinite Methods

The objective of this report is to know heat transfer through one dimensional transient heat conduction experiment. For achieve the above objective, we already had experimental results of the three different metal strips tested in the lab and now we will compare the results with the theoretical methods known as Finite difference method, semi-infinite solid method. Thereby, we will be discussing the potential reasons influenced the different theoretical methods in order to define the most accurate method of approximating the results with reference to the experimental results.

In the real world most of the heat transfer problems are related to time. For example, temperature of an engine cylinder rises after the combustion phase in an engine. During this process we can encounter, the temperature of the engine will be raised with respect to time from the start of an engine to certain point of

time. By the theoretical methods of transient heat conduction we would be able to figure out the time taken for the engine to reach certain temperature from initial conditions. Transient heat conduction is defined as change in temperature according to the time through conduction method of heat transfer. Transient is referred to a non-equilibrium state of change of temperature with time. So, this report is more related to transient conduction heat transfer.

After comparing the plots in excel sheet, the finite difference explicit theoretical method seems to be very close to the experimental results. Since the properties of the metals remains same in both experimental and theoretical methods. Although, some minor factors are neglected. The other methods results are not quite close to the experimental results.

The comparison between the semi-infinite solid method and finite difference method may vary upon number of factors:

1) In semi-infinite solid method erf function plays a major role, as Fourier number in the explicit number

2) Within 300 sec the surface temperature is reached in the explicit method, whereas it still takes more time in semi-infinite method to reach the surface temperature

3) Properties of the materials remains same in both methods

4) Due to same material properties, in both methods we can observe the same trend in the variation of temperature across the four nodes.

The Factors affecting these both methods are:

1) Fourier number

2) Stability criteria

3) Gaussian error function

4) Material properties

The Fourier number is mainly affected by the stability criterion, if we do not control the stability criterion as per the one dimensional transient conduction, otherwise we can see the rapid fluctuations in the temperaturewithin minute time. The Gaussian error function cannot be controlled in the excel software due to the default entry, while manual entry of error function during calculations might be affected.

Primary objective of the present work is to analyse the transient heat conduction equation for one-dimension. Different methods like finite difference explicit and semi-infinite method have been adopted to solve the 1-D transient heat conduction. Three different metal strips have been considered made of copper, aluminium and stainless steel. All the three materials have different thermal properties like, thermal conductivity, density and specific heat. Four thermocouples have been to measure the temperature. Experiments have also been conducted for the same set of model developed. At the end comparisons have been done for different cases considered, and then the advantage and disadvantage of the methods have been discussed.

Heat transfer is phenomenon which occurs in most of the real life applications. Like heat transfer from automobile vehicles in the engines, heat transfer from human body, heat transfer from electronic equipment etc. Transfer of heat occurs due to the change in temperature. There are basically three modes of heat transfer, conduction, convection and radiation. Conduction mostly occurs in the solid while convection occurs in the liquids. Heat transfer due to radiation occurs when there is high temperature difference or in the electromagnetic waves.

Actually, the purpose of this lab it to analyze the conductive heat transfer between metal strips and water so that we can familiar with the conduction way of these two materials.  We have three kinds of metal trips which are respectively copper, aluminum, stainless steel. Their lengths are 12cm. One end of these metal trips immerses in boiling water and the other end and sides insulated. There are four thermocouples which are placed along each metal strip 30mm apart. One of thermocouples is used to measure water temperature. All temperatures of these thermocouples are recorded by a data logger. The data logger records these temperatures of thermocouples from 0~300 seconds. Figure 1 shows the schematic of the experimental setup.

As present problem deals with the conduction problem where three metal strips are dipped in the water. The water is boiling at the temperature 90^ºC while the atmosphere is at 27^ºC temperatures. Two modes of heat transfer, convection and radiation can be neglected inside the metal strips, as it is well known fact that inside the solid body heat transfer by conduction dominates over convection and radiation. So the one-dimensional heat conduction equation govern the physical problem and can be written as

A simplified form of the above governing equation can also be written as

Where:  α=k/ρC_p is thermal diffusivity in m²/s, k is thermal conductivity in W/m-K, ρ is density in Kg/m³, C_p is specific heat J/Kg-K, T is temperature in Kelvin (K), t is time in seconds.

Table.1. Thermo-physical properties

At the end of the metal strips temperature is assumed to be equal to the point at which water is boiling which is equal to 90^ºC while the other end which is open to the atmosphere is assumed to be equal to 27^ºC.

The above equation is partial differential equation and it is difficult to solve the equation directly. To solve this we will convert this partial differential equation into algebraic form using the expansion of Taylor series. Taylor series can be used for either forward derivative terms or for backward derivative term. One can also use the central differencing scheme which is more accurate compared to the backward and forward differencing scheme. Taylor series expansion can be represented as,

Description

While writing the above equation we have neglected the higher order terms. Using the above equation we can find the second order partial differential term as,

Now the temporal derivative term is first order partial term, which can be discretized either by explicit scheme or by implicit scheme [1-3],

Above equation represents the explicit and implicit discretization of 1-D heat conduction equation. In explicit scheme only one term is at the next time level (n+1) which is an unknown quantity while all other terms are at same time level (n) which are known, while in implicit scheme spatial derivative terms are also at the next time level (n+1) which makes it difficult to handle in computer code. Explicit scheme is simple to code but requires a condition to be satisfy which is known as stability condition and leaves a limit on the time step for a particular grid selection while implicit scheme is free of this stability condition represented in eq.

First we consider the explicit scheme for simplicity of the scheme, as we know that term at time ‘n+1’ is unknown while terms at time ‘n’ are known. So from above equation it can be observed that all the terms on right hand side are at time ‘n’ (known) while only one term at time ‘n+1’ is there in left hand side which can be easily calculated as,

The below equation shows the implicit scheme, it can be observed from the scheme that all the terms are at next time level (n+1), which makes this scheme difficult to code.

For this method, we have two situations, one is the temperature of node 0, and the second one is the temperature of node1-4. So these equations are,

The equation for node 0:

The equation for node 1-4:

From the code it has been found that results for implicit scheme are diverging.

After comparing the plots, the finite difference explicit theoretical method seems to be very close to the experimental results. Since the properties of the metals remains same in both experimental and theoretical methods. Although, some minor factors are neglected. The other methods results are not quite close to the experimental results.

Finite difference explicit method and semi-infinite solid method are both numerical approach to solve the partial differential equation. As their approach is different they are giving different results. The difference between the two methods can be due to the number of variables.

In finite difference method, initially the temperature increases at faster rate while in semi-infinite solid method temperature increases gradually. This may be due to the fact that in semi-infinite solid approach uses ‘erf’ function which plays the vital in calculation of the temperatures.

As in finite difference approach there is one stability criteria which one have to satisfy while writing the code in the MATLAB, while for semi-infinite solid approach there is no condition which have to satisfied for the convergence of the problem.

Trent of the temperature increment is same this is due the fact that we have consider same material for both the cases to solve the problem.

Different variables or conditions which affect the two methods are,

Properties of the material which one has to use while deciding the stability condition

The stability condition is also termed as Fourier number.

‘erf’ function is also a parameter which separates the two methods

To solve a problem using the numerical approach one has to consider the stability criteria which help in the convergence of the problem. If one does not consider the stability criteria in his/her numerical approach their code can diverge and can give improper results. In heat conduction equation  is the stability criteria which are termed as Fourier number. Value of the above number should be less than 0.5, else the code will diverge. This divergence of the code is also termed as divergence. The Gaussian error function cannot be controlled in the excel software due to the default entry, while manual entry of error function during calculations might be affected.

Below results represents the comparison of the temperature for four thermocouples for all of the materials considered. From the figure it can be seen that for aluminium and copper temperature increases while for stainless steel it does not increases this is due to the lower thermal conductivity of the stainless steel compared to the copper and aluminium. Figure 8 shows the figures for theoretical calculations while figure 9 is for the experimental conducted.

Table below shows the comparison of the results between the numerical method and the experimental method. Table for copper and stainless steel have not been included here. Only their results have been presented in the figures.

Conclusion

• Temperature of copper and aluminum metal strips increases with increment in the time. This is due to the larger thermal conductivity of the aluminum and copper compared to the base material water.
• Extreme node (End) shows less increment in the temperature compared to the other nodes this is due to the fact that, it is furthest from the boiling water.
• Figure also shows that initially the temperature rises suddenly up to 50 seconds time, and then it increases gradually.
• Copper has shown highest increment in the temperature compared to aluminum as it has larger thermal conductivity compared to it.

References

Namiki, T., 1999, A New FDTD Algorithm Based on Alternating-Direction Implicit Method, IEEE Transaction on Microwave Theory and Techniques, 47(10), 2003-2007.

Liu, Y. & Sen, M. K., 2009, An Implicit Staggered-Grid Finite-Difference Method for Seismic Modelling, Geophysical Journal International, 179(1), 459-474.

Tamsir, M. & Srivastava V. K., 2011, A Semi-Implicit Finite-Difference Approach for Two-Dimensional Coupled Burgers’ Equation, International Journal of Scientific & Engineering Research, 2(6), 1-6.