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

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 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

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 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.

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.

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