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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

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 90ºC while the atmosphere is at 27ºC 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/ρCp is thermal diffusivity in m2/s, k is thermal conductivity in W/m-K, ρ is density in Kg/m3, Cp is specific heat J/Kg-K, T is temperature in Kelvin (K), t is time in seconds.

Table.1. Thermo-physical properties

 Material Specific heat (J/Kg-K) Density (Kg/m3) Thermal conductivity (W/m-K) Aluminium 903 2700 237 Copper 385 8933 401 Stainless steel 480 8055 15.1

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 90ºC while the other end which is open to the atmosphere is assumed to be equal to 27ºC.

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.

 Comparison for Aluminium for four thermocouples Thermocouple 1 Thermocouple 2 Thermocouple 3 Thermocouple 4 29.2822 25.136 24 24.062 24 24.031 24 24.032 33.42556 26.568 24.56942 24.186 24 24.032 24 24.001 36.73699 28.205 25.46273 24.448 24.06138 24.001 24 24.002 39.43078 29.867 26.52703 24.777 24.20583 24.032 24.00662 24.002 41.65852 31.414 27.66783 25.197 24.43458 24.095 24.02809 24.033 43.52893 32.868 28.82748 25.675 24.73931 24.157 24.07191 24.034 45.1211 34.231 29.97159 26.242 25.10807 24.219 24.14386 24.034 46.49333 35.503 31.08042 26.809 25.52841 24.358 24.2478 24.035 47.68923 36.654 32.14343 27.434 25.98887 24.477 24.38585 24.096 48.7419 37.717 33.1558 28.061 26.47952 24.658 24.55865 24.097 49.67674 38.722 34.11628 28.685 26.99216 24.867 24.76572 24.236 50.51357 39.577 35.02572 29.279 27.52013 25.105 25.00573 24.235 51.26802 39.136 35.88621 29.874 28.05818 25.315 25.27679 24.326 51.95256 40.464 36.7005 30.468 28.60221 25.555 25.57662 24.416 52.57731 41.876 37.47168 31.061 29.14904 25.853 25.90278 24.506 53.15049 42.434 38.20289 31.597 29.69628 26.122 26.25273 24.625 53.67891 35.682 38.89722 32.191 30.24207 26.421 26.62394 24.655 54.16826 43.259 39.55767 32.607 30.78506 26.749 27.01398 24.835 54.6233 43.639 40.187 33.081 31.32423 27.047 27.4205 24.985 55.04807 44.138 40.78783 33.527 31.85881 27.316 27.84132 25.135 55.44604 44.754 41.36253 33.942 32.38827 27.643 28.27441 25.344 55.82015 45.372 41.91331 34.417 32.91222 27.972 28.71788 25.583 56.17298 45.989 42.44215 34.831 33.43039 28.299 29.17003 25.792 56.50675 46.488 42.95087 35.275 33.94259 28.596 29.6293 25.971 56.8234 47.017 43.4411 35.689 34.44871 28.893 30.09427 26.21 57.12462 47.546 43.91433 36.192 34.94868 29.281 30.56368 26.479 57.41192 47.956 44.3719 36.576 35.44248 29.518 31.03638 26.688 57.6866 48.395 44.81503 37.019 35.93009 29.904 31.51136 26.956 57.94983 48.776 45.24479 37.432 36.41155 30.23 31.9877 27.224 58.20263 49.098 45.66217 37.787 36.88688 30.496 32.46459 27.463 58.44593 49.449 46.06805 38.199 37.35613 30.824 32.94131 27.731 58.68052 49.771 46.46324 38.584 37.81936 31.181 33.41723 27.999 58.90714 50.034 46.84846 38.938 38.27662 31.508 33.89178 28.267 59.12642 50.327 47.22434 39.293 38.72798 31.835 34.36447 28.564 59.33895 50.56 47.59148 39.648 39.1735 32.133 34.83485 28.832 59.54523 50.823 47.9504 39.973 39.61325 32.459 35.30256 29.129 59.74573 51.115 48.30157 40.356 40.0473 32.816 35.76725 29.456 59.94086 51.319 48.64544 40.651 40.47572 33.112 36.22864 29.694 60.13099 51.583 48.98239 40.975 40.89858 33.468 36.68648 30.02 60.31645 51.816 49.31278 41.271 41.31595 33.765 37.14054 30.317 60.49754 51.99 49.63692 41.536 41.7279 34.062 37.59065 30.583 60.67453 52.165 49.9551 41.859 42.1345 34.387 38.03665 30.85 60.84766 52.43 50.26759 42.125 42.53581 34.684 38.4784 31.208 61.01715 52.635 50.57464 42.449 42.93191 35.009 38.91578 31.506 61.1832 52.839 50.87646 42.743 43.32286 35.305 39.34872 31.803 61.34599 52.985 51.17325 42.948 43.70872 35.572 39.77713 32.07 61.50567 53.16 51.46519 43.242 44.08957 35.867 40.20096 32.337 61.66239 53.365 51.75246 43.477 44.46547 36.162 40.62015 32.635 61.8163 53.541 52.03522 43.771 44.83648 36.429 41.03468 32.961 61.9675 53.716 52.31359 44.007 45.20267 36.724 41.44451 33.228 62.11611 53.862 52.58773 44.241 45.5641 36.99 41.84964 33.525 62.26223 54.066 52.85774 44.506 45.92083 37.314 42.25006 33.821 62.40596 54.213 53.12375 44.74 46.27292 37.55 42.64577 34.118 62.54738 54.388 53.38585 44.975 46.62043 37.844 43.03678 34.384 62.68656 54.534 53.64415 45.208 46.96343 38.14 43.42309 34.651 62.82357 54.739 53.89874 45.444 47.30196 38.436 43.80474 34.948 62.9585 54.855 54.1497 45.649 47.63609 38.672 44.18174 35.184 63.09138 55.03 54.39712 45.914 47.96588 38.938 44.55412 35.538 63.22229 55.177 54.64108 46.121 48.29138 39.205 44.92191 35.746 63.35127 55.352 54.88163 46.327 48.61265 39.442 45.28514 36.012 63.47838 55.468 55.11886 46.532 48.92974 39.706 45.64384 36.277 63.60365 55.644 55.35283 46.768 49.24271 40.003 45.99806 36.604 63.72714 55.643 55.5836 46.944 49.55161 40.239 46.34784 36.839 63.84888 55.79 55.81122 47.15 49.85649 40.475 46.6932 37.106 63.96891 55.965 56.03576 47.415 50.15741 40.741 47.0342 37.371 64.08727 56.081 56.25727 47.56 50.45441 40.976 47.37089 37.634 64.20399 56.226 56.4758 47.766 50.74756 41.212 47.70329 37.901 64.3191 56.314 56.69139 47.942 51.03689 41.418 48.03146 38.108 64.43263 56.46 56.9041 48.177 51.32246 41.654 48.35545 38.374 64.54462 56.605 57.11398 48.353 51.60432 41.919 48.67529 38.64 64.65509 56.722 57.32106 48.558 51.88251 42.155 48.99104 38.907 64.76406 56.839 57.52539 48.735 52.15708 42.362 49.30274 39.144 64.87157 56.985 57.72702 48.911 52.42809 42.597 49.61044 39.38 64.97763 57.101 57.92597 49.116 52.69557 42.862 49.91418 39.617 65.08227 57.218 58.12231 49.292 52.95957 43.068 50.21402 39.852 65.18552 57.305 58.31605 49.437 53.22015 43.302 50.50999 40.088 65.28739 57.51 58.50724 49.702 53.47733 43.538 50.80214 40.384 65.38791 57.538 58.69591 49.819 53.73117 43.744 51.09053 40.59 65.4871 57.683 58.88211 49.994 53.98171 43.95 51.37519 40.796 65.58497 57.8 59.06587 50.141 54.22899 44.126 51.65617 41.033 65.68155 57.946 59.24721 50.376 54.47305 44.42 51.93352 41.27 65.77686 58.004 59.42618 50.463 54.71395 44.537 52.20728 41.446 65.87091 58.121 59.6028 50.669 54.95171 44.773 52.4775 41.711 65.96372 58.237 59.77712 50.844 55.18637 44.978 52.74422 41.947 66.05532 58.353 59.94915 51.019 55.41799 45.153 53.00748 42.153 66.14571 58.44 60.11894 51.166 55.6466 45.388 53.26734 42.359 66.23492 58.528 60.2865 51.311 55.87223 45.565 53.52382 42.594 66.32295 58.673 60.45188 51.487 56.09493 45.77 53.77698 42.8 66.40984 58.789 60.61511 51.633 56.31474 45.976 54.02686 43.006 66.49559 58.847 60.7762 51.749 56.53168 46.122 54.27349 43.211 66.58021 58.964 60.93519 51.925 56.74581 46.329 54.51692 43.417 66.66373 59.078 61.09211 52.099 56.95715 46.505 54.7572 43.653 66.74616 59.224 61.24698 52.275 57.16574 46.711 54.99435 43.829 66.82751 59.253 61.39983 52.391 57.37162 46.858 55.22843 44.004 66.9078 59.399 61.5507 52.537 57.57483 47.092 55.45946 44.24 66.98705 59.456 61.69959 52.653 57.77539 47.239 55.6875 44.415 67.06525 59.544 61.84655 52.799 57.97335 47.414 55.91257 44.591 67.14244 59.69 61.9916 52.975 58.16873 47.621 56.13473 44.827 67.21863 59.748 62.13475 53.063 58.36157 47.739 56.35399 44.971 67.29382 59.836 62.27604 53.268 58.5519 47.974 56.57041 45.178 67.36803 59.982 62.4155 53.415 58.73976 48.12 56.78401 45.355 67.44127 60.041 62.55314 53.532 58.92517 48.297 56.99484 45.561 67.51356 60.128 62.68898 53.679 59.10818 48.473 57.20293 45.738 67.58491 60.184 62.82306 53.764 59.2888 48.588 57.40832 45.882 67.65533 60.272 62.9554 53.882 59.46708 48.736 57.61103 46.03 67.72483 60.388 63.08601 54.028 59.64304 48.941 57.81111 46.266 67.79343 60.506 63.21492 54.204 59.81671 49.117 58.0086 46.442 67.86113 60.535 63.34216 54.292 59.98812 49.263 58.20351 46.589 67.92796 60.593 63.46774 54.409 60.15731 49.41 58.39589 46.765 67.99391 60.68 63.59169 54.555 60.32429 49.556 58.58577 46.941 68.05901 60.737 63.71402 54.641 60.4891 49.672 58.77318 47.087 68.12326 60.885 63.83476 54.789 60.65177 49.878 58.95816 47.324 68.18667 60.971 63.95394 54.817 60.81233 49.994 59.14073 47.411 68.24926 61.088 64.07156 54.993 60.9708 50.2 59.32093 47.646 68.31103 61.117 64.18765 55.168 61.1272 50.287 59.49879 47.764 68.372 61.204 64.30224 55.197 61.28158 50.462 59.67433 47.969 68.43218 61.321 64.41533 55.344 61.43394 50.639 59.84759 48.145 68.49158 61.408 64.52695 55.49 61.58433 50.786 60.0186 48.292 68.5502 61.495 64.63712 55.577 61.73276 50.902 60.18738 48.438 68.60806 61.524 64.74586 55.694 61.87926 51.048 60.35398 48.644 68.66516 61.611 64.85319 55.752 62.02385 51.164 60.5184 48.76 68.72153 61.698 64.95912 55.898 62.16657 51.311 60.68069 48.878 68.77716 61.758 65.06367 55.986 62.30743 51.457 60.84087 49.025 68.83207 61.874 65.16686 56.161 62.44645 51.604 60.99896 49.23 68.88626 61.873 65.26871 56.19 62.58367 51.691 61.155 49.347 68.93975 61.99 65.36923 56.366 62.71911 51.867 61.30901 49.494 68.99254 62.079 65.46845 56.454 62.85278 52.013 61.46102 49.67 69.04465 62.106 65.56638 56.541 62.98472 52.1 61.61105 49.787 69.09608 62.194 65.66303 56.629 63.11494 52.217 61.75913 49.904 69.14684 62.252 65.75843 56.745 63.24347 52.391 61.90529 50.079 69.19694 62.341 65.85259 56.862 63.37033 52.509 62.04955 50.226 69.24639 62.428 65.94552 56.979 63.49553 52.685 62.19193 50.373 69.29519 62.457 66.03725 57.096 63.61911 52.744 62.33245 50.49 69.34337 62.573 66.12778 57.183 63.74109 52.919 62.47116 50.665 69.39091 62.602 66.21713 57.242 63.86147 53.007 62.60806 50.783 69.43784 62.719 66.30533 57.388 63.9803 53.124 62.74317 50.9 69.48415 62.777 66.39237 57.474 64.09757 53.242 62.87654 51.017 69.52987 62.836 66.47829 57.592 64.21332 53.388 63.00816 51.163 69.57499 62.922 66.56309 57.65 64.32757 53.476 63.13808 51.25 69.61952 62.922 66.64678 57.736 64.44033 53.476 63.26631 51.397 69.66348 62.979 66.72939 57.824 64.55163 53.622 63.39287 51.513 69.70686 63.066 66.81092 57.911 64.66148 53.709 63.51778 51.63 69.74968 63.156 66.89139 58.058 64.7699 53.858 63.64107 51.778 69.79194 63.184 66.97082 58.116 64.87691 53.945 63.76276 51.865 69.83365 63.243 67.04921 58.204 64.98252 54.063 63.88286 51.983 69.87482 63.33 67.12658 58.263 65.08677 54.15 64.00141 52.128 69.91545 63.359 67.20295 58.379 65.18966 54.296 64.11841 52.245 69.95556 63.418 67.27833 58.437 65.29121 54.384 64.23389 52.361 69.99515 63.475 67.35272 58.554 65.39144 54.501 64.34787 52.478 70.03422 63.533 67.42615 58.611 65.49037 54.588 64.46037 52.595 70.07278 63.62 67.49862 58.699 65.58801 54.676 64.5714 52.683 70.11084 63.68 67.57015 58.815 65.68439 54.823 64.68099 52.8 70.1484 63.737 67.64075 58.902 65.77951 54.911 64.78916 52.947 70.18548 63.764 67.71044 58.931 65.87339 54.968 64.89592 53.004 70.22208 63.853 67.77921 59.049 65.96605 55.115 65.00129 53.152 70.2582 63.852 67.84709 59.106 66.05751 55.202 65.10529 53.239 70.29385 63.94 67.91409 59.194 66.14778 55.29 65.20794 53.357 70.32903 63.999 67.98022 59.281 66.23687 55.377 65.30925 53.474 70.36376 64.027 68.04549 59.368 66.32481 55.494 65.40925 53.561 70.39804 64.114 68.10991 59.456 66.4116 55.612 65.50795 53.679 70.43187 64.173 68.17349 59.514 66.49726 55.699 65.60536 53.795 70.46526 64.202 68.23625 59.63 66.58181 55.786 65.70151 53.883 70.49822 64.229 68.29819 59.63 66.66527 55.873 65.79641 54 70.53075 64.259 68.35932 59.716 66.74763 55.962 65.89007 54.059 70.56285 64.376 68.41966 59.774 66.82893 56.108 65.98251 54.234 70.59454 64.405 68.47922 59.862 66.90916 56.137 66.07376 54.293 70.62582 64.462 68.538 59.95 66.98836 56.254 66.16381 54.38 70.65669 64.491 68.59601 59.978 67.06652 56.311 66.2527 54.467 70.68716 64.55 68.65327 60.066 67.14367 56.399 66.34043 54.526 70.71723 64.608 68.70979 60.183 67.21982 56.516 66.42702 54.673 70.74691 64.638 68.76557 60.241 67.29497 56.604 66.51248 54.789 70.7762 64.694 68.82063 60.269 67.36915 56.661 66.59684 54.819 70.80512 64.782 68.87497 60.357 67.44237 56.779 66.68009 54.964 70.83366 64.753 68.92861 60.416 67.51463 56.836 66.76226 55.052 70.86182 64.839 68.98154 60.473 67.58595 56.895 66.84337 55.111 70.88963 64.868 69.03379 60.561 67.65634 56.982 66.92342 55.227 70.91707 64.897 69.08536 60.591 67.72582 57.099 67.00243 55.287 70.94415 64.983 69.13626 60.678 67.7944 57.157 67.08041 55.403 70.97088 65.013 69.1865 60.765 67.86209 57.244 67.15738 55.461 70.99726 65.071 69.23608 60.853 67.92889 57.303 67.23335 55.549

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.

Cite This Work

My Assignment Help. (2021). Transient Heat Conduction: Comparison Of Finite Difference And Semi-Infinite Methods. Retrieved from https://myassignmenthelp.com/free-samples/eng481-applied-heat-and-mass-transfter/journal-of-scientific-and-engineering.html.

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[Accessed 10 September 2024].

My Assignment Help. 'Transient Heat Conduction: Comparison Of Finite Difference And Semi-Infinite Methods' (My Assignment Help, 2021) <https://myassignmenthelp.com/free-samples/eng481-applied-heat-and-mass-transfter/journal-of-scientific-and-engineering.html> accessed 10 September 2024.

My Assignment Help. Transient Heat Conduction: Comparison Of Finite Difference And Semi-Infinite Methods [Internet]. My Assignment Help. 2021 [cited 10 September 2024]. Available from: https://myassignmenthelp.com/free-samples/eng481-applied-heat-and-mass-transfter/journal-of-scientific-and-engineering.html.

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