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\title{Finned Electric Motor with Prescribed Heat Fluxand Influence of the Internal and External Heat Convection Coefficients on the Temperature of the Core}
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             \author[1]{Marcus Vinicius Ferreira  Soares}

             \author[2]{Élcio  Nogueira}

             \affil[1]{  Rio de Janeiro State University}

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\date{\small \em Received: 6 December 2018 Accepted: 31 December 2018 Published: 15 January 2019}

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\begin{abstract}
        


One of a major objective is to analyze the effect of internal and external convection coefficients in the heat transfer generated by electric motors and characterize intervals of feasible values, in practical terms, for these coefficients. Another objective is to determine the motor core temperature, considering the environmental and operational conditions of the motor installation and the heat flux Q0 (fixed) in the fins. To achieve the objectives were developed analytical solutions for determining the temperature variations in the fins, performance and electric motor efficiency, considering heat flow constant at the base of the fins and the possible variations in temperature of surround media. The heat flux at the base of the fins, in this case, is the minimum necessary for satisfactory electric motor performance and the core temperature is within the safety range stipulated by the manufacturer. The obtained results characterize a range of possible values for the inner and outer heat transfer coefficients.

\end{abstract}


\keywords{finned electric motor; constant heat flow; efficiency; efficacy.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
Factors that allow electric motors to lose up to 4\% of their performance over their lifetime are improper installations, lack of regular maintenance, cleanliness and quality lubrication \hyperref[b2]{(Cardoso et. Al., 2009)}. Therefore, overheating the electric motor and consequently burning it is a problem for maintenance personnel, since heat produced by the electric motor must be dissipated efficiently, avoiding overheating and consequent burning. The high temperatures of the installation environment and the inefficient heat dissipation generated by the difference in net power supplied by the motor and power absorbed in the line is the main cause of overheating (Santos, Rafael Simões, 2011).\par
Fins or the extended surfaces are extensively used in engineering applications to increase the heat transfer efficiency of surfaces, and are of vital importance in the design of heat exchange devices in different fields of applications in order to provide an enhanced heat transfer effect through an increase in the total heat exchange area (Campo, A.; Kundu, B.,2017), \hyperref[b5]{(Incropera, Frank P. Et Al., 2008)}, (Santos, T. A. M., 2017). Once the temperature distribution through the fin is known, the heat transfer rate and the efficiency can be readily determined.\par
In electric motors, it is common to use extended surfaces, which increase the exchange area as a mechanism for heat transfer optimization. In fact, an important industrial application of fins occurs in electric motors and are of vital importance in the industry as they are used in machines of all types, including, for example, computer ventilation and other electronic equipment. A well-dimensioned finned ventilation system can contribute to energy savings.\par
The heat dissipation is directly associated with efficient ventilation, the temperature difference between the engine housing surface and the medium (in this case, the air surrounding the engine), the total heat exchange area of the engine housing and of extended surfaces (fins). Important feature is the geometry of the fins, as they have a great influence on the area available for exchange and, in order to heat transfer as good as possible, the construction material of these fins must have high thermal conductivity, such as aluminum and copper (Duarte, Denise Freire; Novais, Ariane Silva; Nogueira, Élcio, 2012), (Novais, Ariane; Chagas, R. D. F.; Nogueira, Élcio, 2014), (Marcus Vinicius Ferreira \hyperref[b6]{Soares, 2015)},  {\ref (Voigdlener, Thiago, 2004)}.\par
Ventilation depends directly on the internal and external convection heat transfer to which the motor is subjected. This dependence is because the heat exchange occurs between a moving fluid (air) and the motor surface (housing), which are at different temperatures. The air moves due to the internal and external convection process causing a random molecular displacement on the surface. According to  {\ref Ordenes (2008)}, the convection coefficient is a proportionality constant that considers several factors, including surface geometry and the nature of the flow.  {\ref Novais and Comitra (2014)} showed that one of the factors that contribute to burning is the thermal resistance that exists between the housing and the motor core. According to  {\ref Moreira (2012)}, besides the convection acting in the region outside the motor housing, and its respective convection coefficient, there is also the convection in the internal region and the resistance associated with the conduction process that occurs in the air layer, between the core and the motor housing.\par
The contact interface between the motor wall and its insulation is promoted by a mechanical union that generates a thermal resistance due to imperfect contact, resulting from small roughness and undulation. In this region, conduction occurs at the contact points and conduction through the trapped fluids in the interstices of roughness and undulation. However, it is important to note that some motors do not use this contact interface between the motor wall and its insulation, being air the only element that serves as thermal resistance between the motor core and the housing.\par
Thus, we have to determine two convection coefficients, one of them referring to the external region and the other to the internal part of the motor housing, the latter is also dependent on the microscopic imperfections on the surfaces and the contact pressure of the materials, the fluid contained in the interstices, of the applied oxide film and the metallic shim involved. This latter convection coefficient is undoubtedly the most complex of the two processes and the most difficult to obtain experimentally.\par
Although the internal resistance is crucial for the correct motor thermal sizing, the usual theoretical models still consider the temperature at the fin base attached to the motor housing equal to the motor core temperature.\par
In Table  {\ref 1} below, some physical and geometrical characteristics of the engine.\par
Table  {\ref 1}: Engine Data II. 
\section[{Theoretical Analysis}]{Theoretical Analysis}\par
Analytical solutions for temperature profiles were developed considering constant heat flux in the fins, and constant thermo physical properties. Variations were also simulated between the motor housing construction materials: cast iron and aluminum.\par
In this context we have the Ordinary Differential Equation (E.D.O.) of the rectangular profile fin, described by the equation below:?? 2 ??(??) ???? 2 = ?. ?? ?? ??. ?? [??(??) ? ?? ? ]\par
at where: From the first boundary condition (r = 0), we must:?? ?? = ?? ?? . ?????? = ?? ?? . ?? ?? 2 ??(??) ???? 2 = ?. ?? ?? ??. ?? ?? [??(??) ? ?? ? ] ??(??) = ??(??) ? ?? ? ?? 2 ??(??) ???? 2 = ?.???. [?? 1 . ??. ???????(??. 0) + ?? 2 .\par
??. cosh(??. 0)] = ?? 0 as ???????(0) = 0 e ???????(0) = 1, we have:???. [?? 2 . ??] = ?? 0 ?? 2 = ? ?? 0 ?? . ??\textbf{13}\par
From the 2nd boundary condition (r = L), we must: ???. [?? 1 . ??. ???????(????) + ?? 2 . ??. ??????? (????) = ? 2 . [?? 1 . ???????(????) + ?? 2 . ???????(????) ] ? ?? ? 2 . ?? 1 . ??. ???????(????) ? ?? ? 2 ?? 2 . ??. ???????(????) = ?? 1 . ???????(????) + ?? 2 .?? 0 = 0,25 . ? ? 750 32 ? ?? ?\par
Q 0 is the flow prescribed on the fins and A is the area of the fin. Note that 750 W is the engine power. The calculated value of Q 0 for the motor studied is 7709.7 W / m².\par
To determine the motor core temperature, we have: ???= ?? ?? ? ?? ?? 1 ? 1 ?? + ln ? D 2 D 1 ? ? 2?????? 20 ?? = ?? . ?? 1 . ?? e ?? 1 = ?? 2 ? 2 . ?? 0 ?? 0 = ?????????? ???????? ????????????????? © 2019 Global Journals???? = ?? ?? ? ?? ?? 1 ? 1 + A. ln ? ?? 2 ?? 1 ? ? 2?????? 21 ???? = ?? 0 = ?? ?? ? ?? ?? 1 ? 1 + ?? 1 . ln ? ?? 2 ?? 1 ? ? 2?? 22 ?? 0 . ? 1 ? 1 + ?? 1 . ln ? ?? 2 ?? 1 ? ? 2?? ? = ?? ?? ? ?? ??\textbf{23}?? ?? = ?? ?? + ?? 0 . ? 1 ? 1 + ?? 1 .ln ? ?? 2 ?? 1 ? ? 2?? ? 24\par
where T b is the temperature at the base of the fin, which depends on the operating conditions, i.e. external temperature T ? and external convection coefficient h 2 .\par
III. 
\section[{RESULTS AND DISCUSSION}]{RESULTS AND DISCUSSION}\par
The Table \hyperref[tab_2]{2} shows the numerical values for motor core temperature with h 2 = 1000 W/ (m 2 .ºC) and h 1 e T ? as variables.\par
We analyzed the temperatures ambient strictly below 45 °C for cast iron and aluminum materials. The data in Table \hyperref[tab_2]{2} and Figure  {\ref 1}, related to the internal and external heat transfer coefficients, show that under the best possible condition [h2 = 1000 W / (m². ° C)], there is no considerable difference between the analyzed materials.  For analysis purposes, a maximum engine core working temperature of 98 ° C is assumed. The manufacturer sets the maximum operating temperature equal to 120 ° C.\par
Analyzing Figure  {\ref 2}, for the external temperature of 40 ° C, we have that, for Cast Iron the motor to work in the acceptable temperature range, the value of h 2 must be greater than or equal to 200 W/ (m 2 . ºC).\par
For the value of h 1 = 200 W / (m². ° C), shown in Figure  {\ref 3}, the motor works at the limit established in the research and we can conclude that the motor works properly for values of h 1 above 200 W / (m². ° C).\par
Analyzing Figure  {\ref 4}, for the external temperature of 40 ° C, we arrive at the following conclusions: for h 1 = 100 W / (m². ° C) the motor temperature is equal to 128.61 ° C, which goes beyond the limit value. From research of 98 ° C. In the same Figure  {\ref 4, for h}  Figure  {\ref 6}, is the graph constructed from the data generated for engine efficiency. Note how aluminum results are far superior to cast iron.?? ?????????? = ?? 0 ? 2 . [?? ?? + (?? ?????????? ? 32 . ?? ?? )] . (?? ?? ? ?? ? ) 25\par
Another parameter to be analyzed is the effectiveness of the engine, which is the ratio between fin and non-fin motor heat exchange. The efficacy formulagiven by:?? ?????????? = ?? 0 ? 2 . ?? ?????????? . (?? ?? ? ?? ? )\textbf{26}\par
Again, Figure  {\ref 7}, demonstrates that the results for aluminum are much higher than those obtained by cast iron.\par
Looking at Figure  {\ref 7}, we can conclude that a reasonable effectiveness value for a fin of 2.5 is achieved when h 2 is near 400 W / (m². ° C) for cast iron.\par
Based on the results observed above, and in the graphs of efficiency and effectiveness, we conclude that for any values of h 1 ? 200 W / (m². ° C) and h 2 ? 200 W / (m². ° C) the engine temperature will have a satisfactory value. However, the values of h 1 and h 2 are physically limited. For h 1 to assume values greater than 400 W / (m². ° C), forced ventilation must be used with the aid of a fan inside the engine or by drilling holes in the housing to increase natural ventilation. In addition, for h 2 to assume values greater than 400 W / (m². ° C), ventilation must be increased in the environment where the engine is installed, which would generate more costs for the company.\par
The above results are consistent with the results presented and experimentally confirmed by Micallef (2006), Figure  {\ref 8} above, in his doctoral dissertation entitled "End Winding Cooling in Electric Machines". In his studies, using experimental and numerical method CFD -Computation Fluid Dynamic, the author applied three distinct turbulence models, and the results showed that the closer to the motor core, the higher the heat transfer coefficient by convection acting on the internal surfaces.\par
The measured internal coefficient assumed values close to300 W / (m 2 .K), a value that is included in the working range proposed by this study.\par
IV. 
\section[{CONCLUSION}]{CONCLUSION}\par
The motor core temperature was computationally simulated by varying the internal (h 1 ) and external (h 2 ) convection heat transfer coefficients and the ambient temperature. The results demonstrated two things: the importance of the internal heat transfer coefficients and their physical limitations.\par
Regarding the heat transfer coefficients, we can conclude that the values of h 1 and h 2 cannot be treated separately. For very low h 1 and h 2 values the fins tend to be at the same temperature as the outside temperature, so there is no efficient heat exchange, overheating the engine. For very high h 1 values, which is physically impossible to obtain, we have that the fin base tends to stay with the motor core temperature, with is the usually conditions generally used by the one-dimension and two-dimension fin models.\par
We conclude that for any values of h 1 ? 200 W / (m². ° C) and h 2 ? 200 W / (m². ° C) the motor temperature will have a satisfactory value. However, as already mentioned, the values of h 1 and h 2 are physically limited. For h 1 to assume values greater than 400 W / (m². ° C), which is already a critical value to obtain for mechanical reasons, it is necessary to use forced ventilation inside the engine or drill holes in the housing to increase nature ventilation. The same reasoning can be used for the external coefficient h 2 , which would entail a higher cost to the company, as it would be necessary to increase ventilation at the engine installation site.\par
Thus, the ranges, 200 ? h 1 ? 300 and 200 ? h 2 ? 400, are the limit values that guarantee the lowest value for the heat transfer rate required for efficient heat removal. Any values above these will increase the transfer rate and consequently decrease the motor core temperature, however, considering the difficulties of achieving these values in practical terms.\begin{figure}[htbp]
\noindent\textbf{2019}\includegraphics[]{image-2.png}
\caption{\label{fig_0}29 Year 2019 Finned}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Figure1:}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{24}\includegraphics[]{image-4.png}
\caption{\label{fig_2}Figure 2 :Figure 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-5.png}
\caption{\label{figure5}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-6.png}
\caption{\label{figure6}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-7.png}
\caption{\label{figure7}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-8.png}
\caption{\label{figure8}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-9.png}
\caption{\label{figure9}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-10.png}
\caption{\label{figure10}}\end{figure}
  \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{P{0.0631324004305705\textwidth}P{0.10430570505920345\textwidth}P{0.09149623250807319\textwidth}P{0.10430570505920345\textwidth}P{0.09149623250807319\textwidth}P{0.10522066738428416\textwidth}P{0.09241119483315394\textwidth}P{0.10522066738428416\textwidth}P{0.09241119483315394\textwidth}}
h1 W/(m²°C)\tabcellsep ?? ? = 15 Iron Cast\tabcellsep °C Al\tabcellsep ?? ? = 20 Iron Cast\tabcellsep °C Al\tabcellsep ?? ? = 30 Iron Cast\tabcellsep °C Al\tabcellsep ?? ? = 40 Iron Cast\tabcellsep °C Al\\
100\tabcellsep 93,86\tabcellsep 93,40\tabcellsep 98,86\tabcellsep 98,40\tabcellsep 108,86\tabcellsep 108,40\tabcellsep 118,86\tabcellsep 118,40\\
150\tabcellsep 68,16\tabcellsep 67,70\tabcellsep 73,16\tabcellsep 72,70\tabcellsep 83,16\tabcellsep 82,70\tabcellsep 93,16\tabcellsep 92,70\\
200\tabcellsep 55,31\tabcellsep 54,85\tabcellsep 60,31\tabcellsep 59,85\tabcellsep 70,31\tabcellsep 69,85\tabcellsep 80,31\tabcellsep 79,85\\
250\tabcellsep 47,60\tabcellsep 47,14\tabcellsep 52,60\tabcellsep 52,14\tabcellsep 62,60\tabcellsep 62,14\tabcellsep 72,60\tabcellsep 72,14\\
300\tabcellsep 42,46\tabcellsep 42,00\tabcellsep 47,46\tabcellsep 47,00\tabcellsep 57,46\tabcellsep 57,00\tabcellsep 67,46\tabcellsep 67,00\\
350\tabcellsep 38,79\tabcellsep 38,33\tabcellsep 43,79\tabcellsep 43,33\tabcellsep 53,79\tabcellsep 53,33\tabcellsep 63,79\tabcellsep 63,33\\
400\tabcellsep 36,04\tabcellsep 35,58\tabcellsep 41,04\tabcellsep 40,58\tabcellsep 51,04\tabcellsep 50,58\tabcellsep 61,04\tabcellsep 60,58\\
450\tabcellsep 33,90\tabcellsep 33,43\tabcellsep 38,89\tabcellsep 38,43\tabcellsep 48,89\tabcellsep 48,43\tabcellsep 58,89\tabcellsep 58,43\\
500\tabcellsep 32,18\tabcellsep 31,72\tabcellsep 37,18\tabcellsep 36,72\tabcellsep 47,18\tabcellsep 46,72\tabcellsep 57,18\tabcellsep 56,72\\
550\tabcellsep 30,78\tabcellsep 30,32\tabcellsep 35,78\tabcellsep 35,32\tabcellsep 45,78\tabcellsep 45,32\tabcellsep 55,78\tabcellsep 55,32\\
600\tabcellsep 29,61\tabcellsep 29,15\tabcellsep 34,61\tabcellsep 34,15\tabcellsep 44,61\tabcellsep 44,15\tabcellsep 54,61\tabcellsep 54,15\\
650\tabcellsep 28,62\tabcellsep 28,16\tabcellsep 33,62\tabcellsep 33,16\tabcellsep 43,62\tabcellsep 43,16\tabcellsep 53,62\tabcellsep 53,16\\
700\tabcellsep 27,78\tabcellsep 27,32\tabcellsep 32,78\tabcellsep 32,32\tabcellsep 42,78\tabcellsep 42,32\tabcellsep 52,78\tabcellsep 52,32\\
750\tabcellsep 27,04\tabcellsep 26,58\tabcellsep 32,04\tabcellsep 31,58\tabcellsep 42,04\tabcellsep 41,58\tabcellsep 52,04\tabcellsep 51,58\\
800\tabcellsep 26,40\tabcellsep 25,94\tabcellsep 31,40\tabcellsep 30,94\tabcellsep 41,40\tabcellsep 40,94\tabcellsep 51,40\tabcellsep 50,94\\
850\tabcellsep 25,83\tabcellsep 25,37\tabcellsep 30,83\tabcellsep 30,37\tabcellsep 40,83\tabcellsep 40,37\tabcellsep 50,83\tabcellsep 50,37\\
900\tabcellsep 25,33\tabcellsep 24,87\tabcellsep 30,33\tabcellsep 29,87\tabcellsep 40,33\tabcellsep 39,87\tabcellsep 50,33\tabcellsep 49,87\\
950\tabcellsep 24,88\tabcellsep 24,42\tabcellsep 29,88\tabcellsep 29,42\tabcellsep 39,88\tabcellsep 39,42\tabcellsep 49,88\tabcellsep 49,42\\
1000\tabcellsep 24,47\tabcellsep 24,01\tabcellsep 29,47\tabcellsep 29,01\tabcellsep 39,47\tabcellsep 39,01\tabcellsep 49,47\tabcellsep 49,01\end{longtable} \par
 
\caption{\label{tab_2}Table 2 :}\end{figure}
 			\footnote{© 2019 Global Journals} 		 		\backmatter  			 
\subsection[{Authors}]{Authors}			 			  				\begin{bibitemlist}{1}
\bibitem[Duarte and Freire]{b3}\label{b3} 	 		\textit{},  		 			Denise Duarte 		,  		 			Freire 		.  		Novais, Ariane Silva.  	 
\bibitem[Novais and Silva]{b8}\label{b8} 	 		\textit{},  		 			Ariane Novais 		,  		 			Silva 		.  		 	 
\bibitem[Soares ()]{b6}\label{b6} 	 		‘Analysis of the Influence of Convection Heat Transfer Coefficient on the Temperature of the Finned Electric Motor Core with Heat Flow Prescribed at the Base’.  		 			Marcus Vinicius Ferreira Soares 		.  	 	 		\textit{Oswaldo Aranha Foundation. Advisor: Élcio Nogueira}  		2015.  		 			Volta Redonda University Center 		 	 	 (Course Conclusion Paper. (Degree in Mechanical Engineering. in Portuguese) 
\bibitem[Beng and Micallef (2006)]{b0}\label{b0} 	 		\textit{End Winding Cooling in Electric Machines},  		 			Christopher Beng 		,  		 			Micallef 		.  		September 2006.  		 			University of Nottingham 		 	 
\bibitem[Cardoso and Balbino ()]{b2}\label{b2} 	 		‘Energy Savings Assessment Attributed to Energy Labeling Actions in Induction Motors In Brazil’.  		 			Rafael Cardoso 		,  		 			Balbino 		.  	 	 		\textit{BrazilianJournal of Energy}  		2009. 15 p. .  	 	 (N. 1.1 Sem.) 
\bibitem[Campo and Kundu ()]{b1}\label{b1} 	 		‘Exact Analytic Heat Transfer from an Annular Fin with Stepped Rectangular Profile’.  		 			A Campo 		,  		 			B Kundu 		.  		 \xref{http://dx.doi.org/10.7726/ajhmt.2017.1013}{10.7726/ajhmt.2017.1013}.  	 	 		\textit{American Journal of Heat and Mass Transfer}  		2017. 4  (4)  p. .  	 
\bibitem[Incropera ()]{b5}\label{b5} 	 		\textit{Heat and Mass Transfer Fundamentals},  		 			Frank P Incropera 		.  		2008. Brazil, LTC. p. 644.  	 	 (6th ed. Riode Janeiro) 
\bibitem[Moreira]{b7}\label{b7} 	 		‘Processos de Transferência de Calor’.  		 			José R Moreira 		.  		 set/2012.  	 	 		\textit{USP}  		 	 
\bibitem[Nogueira ()]{b4}\label{b4} 	 		\textit{Rectangular Profile Fin Analytical Solution: Comparison of Thermal Performance Between Aluminum and Cast Iron in Electric Motors},  		 			Élcio Nogueira 		.  		2012. Uni FOA, Volta Redonda, Brazil.  	 
\bibitem[Ordenes et al.]{b11}\label{b11} 	 		\textit{Surrounding Heat Transfer in Building},  		 			M Ordenes 		,  		 			R Lambert 		,  		 			S Güths 		.  		 	 	 (Departamento de) 
\bibitem[Novais et al. ()]{b10}\label{b10} 	 		‘Theoretical Analysis of Thermal Performance of Finned Electric Induction Motors’.  		 			Ariane ; Novais 		,  		 			R D F Chagas 		,  		 			Élcio Nogueira 		.  	 	 		\textit{Cadernos UniFOA (Online), v. IX}  		2014. p. .  	 	 (in Portuguese) 
\bibitem[Comitre and Chagas ()]{b9}\label{b9} 	 		‘Thermal Performance of Heat-Flowed Electric Motors Prescript On The Rectangular Flip Bases’.  		 			Roberta Duarte Ferreira Comitre 		,  		 			Chagas 		.  	 	 		\textit{Advisor: Élcio Nogueira},  				2014.  		 			Degree in Mechanical Engineering) -Volta Redonda University Center 		 	 	 (Oswaldo Aranha Foundation. in Portuguese) 
\end{bibitemlist}
 			 		 	 
\end{document}
