Simultaneous prediction of the performance coefficients in a compact absorption heat transformer using new neural network configurations

Abstract

The calculation of the performance of absorption heat transformers (AHTs) depends on multiple variables. In this work, artificial neural network (ANN) models with new configurations were developed to simultaneously estimate the coefficient of performance (COP) and Carnot coefficient of performance (COP_Carnot) of an AHT prototype. The variables used to train the models were: the inlet and outlet temperatures corresponding to the main components of the AHT. The output parameters to simulate were the COP and COP_carnot, which are important values to determine the performance and real efficiency based on the Carnot cycle, respectively. To find the appropriate model, it was necessary to explore learning algorithms, activation functions, and multilayers. The results show a good estimation of the output parameters through three configurations of the ANN model. However, based on the number of coefficients obtained during learning and the simultaneous simulation of two output parameters, a multilayer ANN model was proposed as the best configuration. Therefore, an architecture of four neurons in the first hidden layer and four neurons in the second hidden layer (08:04:04:02) was sufficient to reproduce the output parameters, achieving a value of R² of 0.9265, 0.9573 and with a mean absolute percentage error of 2.41, 1.14% for COP and COP_Carnot, respectively. In the three configurations, the use of hyperbolic tangent sigmoid activation function (TANSIG) in the hidden layers and the adjustment of the coefficients with the Levenberg–Marquardt learning algorithm obtained the best results. The influence of each of the variables selected for the ANN model was analyzed through a correlation matrix and a sensitivity analysis. Other experimental variables were added in the training of the ANN model to consult the impact caused during the simultaneous prediction of the performance coefficients.

Appendices

Appendix 1

Weighting and bias coefficients produced from the ANN model for the prediction of the COP.

Number of neurons (s)	${\text{Wi}}$								${\text{Wo}}$	Bias
	Variable								Output layer (l = 1)
	$T_{1}$ (k = 1)	$T_{2}$ (k = 2)	$T_{3}$ (k = 3)	$T_{4}$ (k = 4)	$T_{5}$ (k = 5)	$T_{6}$ (k = 6)	$T_{7}$ (k = 7)	$T_{8}$ (k = 8)	$Wo_{1,s}$	$b1_{\left( s \right)}$	$b2_{\left( l \right)}$
1	− 2.1331	− 0.4083	− 0.7737	0.7099	2.3708	2.1206	0.7128	1.8999	0.0181	0.6232	0.5970
2	0.2441	− 1.3668	− 2.0000	2.0153	− 0.9345	− 1.8978	− 0.4147	2.3316	0.0196	− 0.6881
3	− 0.9422	0.0293	0.3891	0.4074	− 2.9774	1.0543	2.4282	1.5277	− 0.0635	− 0.9256
4	− 1.7725	− 2.3908	0.7713	1.4164	2.0856	− 1.3312	1.3290	0.1880	− 0.2312	− 0.3950
5	1.1707	0.5308	2.0106	1.5114	1.0232	0.6669	− 0.6753	− 3.2822	0.0064	− 1.3759
6	0.1900	− 2.7516	1.1800	0.9036	− 1.7018	0.8779	0.9966	− 1.9703	0.0056	2.8516
7	− 1.6872	1.7982	1.1574	1.9072	− 0.2321	− 0.4701	0.9764	2.1916	0.0681	− 5.6599

Weighting and bias coefficients produced from the ANN model for the prediction of the COP_Carnot.

${\text{Wi}}$									${\text{Wo}}$	Bias
Number of neurons (s)	Variable								Output layer (l = 1)
	$T_{1}$ (k = 1)	$T_{2}$ (k = 2)	$T_{3}$ (k = 3)	$T_{4}$ (k = 4)	$T_{5}$ (k = 5)	$T_{6}$ (k = 6)	$T_{7}$ (k = 7)	$T_{8}$ (k = 8)	$Wo_{1,s}$	$b1_{\left( s \right)}$	$b2_{\left( l \right)}$
1	1.2221	0.9570	− 2.1169	− 1.1600	1.2064	2.6072	− 0.8197	− 0.7379	− 0.1288	− 2.2468	− 0.1578
2	− 1.0986	2.3541	1.9413	− 1.0455	1.7354	− 0.9605	− 0.5717	2.1373	− 0.2415	− 0.6347
3	− 2.2385	− 1.0154	0.2286	− 2.0534	0.9983	2.0025	− 1.0321	− 1.7013	0.0497	3.4225
4	− 0.3122	0.0743	2.4414	1.1720	− 1.8615	− 0.6122	− 0.8637	2.5789	0.0107	− 1.2252
5	− 2.3927	1.7122	1.3911	1.7826	− 0.5359	− 1.7792	− 1.7396	0.7528	− 0.6157	− 1.3462
6	− 1.7680	− 1.6591	1.5389	− 0.5904	1.2875	− 2.0185	0.1877	2.2252	− 0.3378	− 0.0802
7	− 0.1275	1.5380	− 2.9774	− 2.2634	0.2984	1.4155	1.9892	0.2671	0.1017	− 2.2307

Appendix 2

Weighting and bias coefficients produced from the ANN model for the simultaneous prediction of COP and COP_Carnot.

${\text{Wi}}$										Wo
Number of neurons (s)	Variable									Output layer, ($l = 1$)		Output layer, ($l = 2$)
	$T_{1}$ (k = 1)	$T_{2}$ (k = 2)	$T_{3}$ (k = 3)	$T_{4}$ (k = 4)	$T_{5}$ (k = 5)	$T_{6}$ (k = 6)	$T_{7}$ (k = 7)	$T_{8}$ (k = 8)	$b_{1\left( s \right)}$	${\text{Wo}}_{1,s}$	$b_{{2\left( {1,{\text{s}}} \right)}}$	${\text{Wo}}_{2,s}$	$b_{{2\left( {2,s} \right)}}$
1	1.6747	− 0.5274	− 1.2592	2.0081	1.3907	− 2.5409	− 1.7215	− 2.2429	− 0.7503	0.8681	− 0.1336	0.2726	− 0.3537
2	− 1.9968	1.3322	0.3864	0.9460	0.6943	− 2.8574	− 0.0128	2.9998	1.6963	0.5568		0.3913
3	1.0649	1.8916	0.5598	1.1567	− 2.6749	0.7416	− 0.9553	− 3.1411	− 0.8227	− 0.0196		0.2331
4	− 2.9780	0.0266	− 1.4290	− 0.3328	− 1.3324	− 1.6011	1.3466	− 2.1245	5.9750	0.0486		− 0.0470
5	3.1788	− 0.7218	− 1.8047	0.4607	0.2179	− 2.1842	− 1.2756	− 2.3456	1.5401	− 0.1880		0.2781
6	1.7075	0.4295	2.1427	− 2.0244	0.5295	1.1132	2.6168	2.4397	− 4.9063	− 0.0085		0.1543
7	− 1.2740	− 2.8032	0.3272	1.2777	− 1.1816	− 1.7473	− 2.1240	2.4022	2.4572	− 0.0075		− 0.0177
8	2.5366	− 1.4103	1.1171	2.7357	− 1.5617	− 1.1458	− 0.7763	− 1.2546	− 0.8566	− 0.1683		− 0.0054
9	2.3108	0.3890	2.5240	− 1.3345	0.0620	− 2.5060	1.8813	0.4995	− 1.3727	− 0.3583		− 0.1582
10	− 1.2346	1.5734	2.6753	− 2.4446	0.2297	1.2647	1.8880	− 0.8900	− 2.6726	0.3199		0.0300
11	1.7013	− 2.2208	− 0.4385	− 2.1839	0.2297	− 1.4996	− 2.2048	0.3056	3.1038	1.1650		0.1081
12	− 1.0138	2.5953	− 2.7922	1.6915	− 0.9334	− 1.9297	− 1.3175	− 1.0216	0.6049	− 0.0304		− 0.4375
13	− 1.5880	− 0.6965	− 0.7669	3.1960	− 2.2016	1.6127	− 1.7808	0.5850	− 0.7899	0.4847		− 0.1436
14	0.4511	2.7091	− 0.2609	− 0.3156	2.2022	− 2.4428	− 1.6971	1.1723	0.0231	0.6328		0.1326
15	− 1.1389	2.1478	2.4430	0.6585	− 2.5051	0.9171	− 0.9272	2.4435	− 3.0446	1.0608		0.0156
16	1.8798	− 2.5119	2.3131	− 2.5288	1.2460	− 0.1742	0.3855	1.1100	1.4270	0.8586		0.6079

Appendix 3

Weighting and bias coefficients produced from the multilayer ANN model for the simultaneous prediction of COP and COP_Carnot.

(s1)	${\text{Wi}}$									${\text{Wo}}$						LW
	First hidden layer								Bias	(s2)	Second hidden layer				Bias	Output layers		Bias
	$T_{1}$ (k = 1)	$T_{2}$(k = 2)	$T_{3}$ (k = 3)	$T_{4}$ (k = 4)	$T_{5}$ (k = 5)	$T_{6}$ (k = 6)	$T_{7}$ (k = 7)	$T_{8}$ (k = 8)	$b_{{1\left( {\text{s}} \right)}}$		$s_{1} = 1$	$s_{1} = 2$	$s_{1} = 3$	$s_{1} = 4$	$b_{{2\left( {{\text{s}}2} \right)}}$	${\text{LW}}_{s3,1}$	${\text{LW}}_{s3,2}$	$b_{{3\left( {{\text{s}}3,1} \right)}}$	$b_{{3\left( {{\text{s}}3,2} \right)}}$
1	1.9887	0.7834	1.0414	− 1.7030	1.0367	1.2334	− 1.5372	1.9950	− 3.5116	1	1.5136	1.2618	0.4001	− 0.1371	− 2.2623	− 0.0821	− 0.0838	− 0.7702	− 0.1314
2	− 1.0213	0.2249	0.4104	0.7932	1.3645	1.9303	2.4361	1.6075	− 4.5641	2	− 1.0002	1.0356	0.6990	− 1.0029	0.8045	− 0.0059	− 0.1786
3	0.0483	− 2.0885	− 1.3767	0.5722	− 2.0218	− 1.8054	− 1.5816	0.7019	4.2296	3	0.4918	− 0.2132	0.6851	− 1.1641	1.2014	− 0.0076	0.1381
4	2.0201	− 0.0463	− 2.3453	0.6314	0.2979	1.6527	− 2.6486	− 0.5006	0.3462	4	0.6300	1.0686	0.9701	1.3983	2.2163	1.1926	0.7475

Appendix 4

Weighting and bias coefficients produced from the ANN model contemplating twelve input variables for the simultaneous prediction of COP and COP_Carnot.

Number of neurons (s)	${\text{Wi}}$													Wo
	Variable													Output layer, ($l = 1$)		Output layer, ($l = 2$)
	$T_{1}$ (k = 1)	$T_{2}$ (k = 2)	$T_{3}$ (k = 3)	$T_{4}$ (k = 4)	$T_{5}$ (k = 5)	$T_{6}$ (k = 6)	$T_{7}$ (k = 7)	$T_{8}$ (k = 8)	$P_{1}$ (k = 9)	$P_{2}$ (k = 10)	$X_{1}$ (k = 11)	$X_{2}$ (k = 12)	$b_{1\left( s \right)}$	${\text{Wo}}_{1,s}$	$b_{{2\left( {1,{\text{s}}} \right)}}$	${\text{Wo}}_{2,s}$	$b_{{2\left( {2,s} \right)}}$
1	− 1.6138	− 2.1650	0.7990	− 1.4483	0.9342	− 1.6123	− 0.5266	− 0.4265	− 1.6099	− 0.8184	1.9543	0.4207	4.6934	0.1277	− 0.6248	− 0.0595	0.0530
2	1.3654	− 1.2615	− 1.9379	− 1.3412	1.8591	− 0.6898	0.8984	− 2.1796	− 0.1650	0.3952	− 0.3812	1.3437	0.8860	0.1213		0.0473
3	0.2644	1.2766	1.5811	1.1820	0.3581	0.0344	− 0.5857	− 1.8320	− 0.4457	2.6837	1.0152	0.9165	− 2.8019	0.2323		− 0.0504
4	− 0.3920	− 1.7987	0.8456	− 0.7068	0.7527	− 0.5965	− 1.3426	0.2267	2.4616	0.3208	1.1875	0.5110	0.6268	0.1933		− 0.3198
5	− 1.5465	− 1.8394	0.5179	1.6199	0.9502	− 0.3752	1.6878	− 0.0382	− 1.0893	− 0.9167	1.6557	− 1.0934	− 1.7476	− 0.9754		− 0.8633