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A criterion for determining the relative importance of the fluctuating component of a periodic heat source
Energy Conversion and Management 96 (2015) 12–17
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
A criterion for determining the relative importance of the fluctuating component of a periodic heat source Adam C. Malloy a,b,⇑, Ricardo F. Martinez-Botas a, Michael Lamperth b a b
Mechanical Engineering Department, Imperial College London, Exhibition Road, London SW7 2AZ, United Kingdom GKN-EVO eDrive Systems Ltd, Unit 14 Woking Business Park, Albert Drive, Woking GU21 5JY, United Kingdom
a r t i c l e
i n f o
Article history: Received 2 June 2014 Accepted 9 February 2015 Available online 7 March 2015 Keywords: Criterion Transient heat sources Numerical modeling Transient temperature response
a b s t r a c t Devices such as rotating electrical machines, transformers, and microprocessors experience thermal loading during operation. This is caused by device losses which manifest themselves as heat sources. Whether operated continuously or on a duty cycle these heat sources are often periodic in nature, exhibiting both mean and fluctuating components. This paper proposes a criterion which can be used to estimate the relative importance of the fluctuating component of a periodic heat source on the temperature response of a device, or a component within a device. It may be used by the heat transfer analyst to determine whether a periodic heat source can be modeled accurately by its mean value or whether it must be modeled as a function of time. During thermometric tests it enables the experimentalist to determine whether the measured temperature rise rate represents an instantaneous or a mean value of heat generation rate. The criterion is derived by considering a sinusoidal heat source acting on a thermal network element. A case study is presented where the criterion is used to estimate the relative importance of the fluctuating component of a range periodic heat sources present in a rotating electrical machine. Results are compared with numerical predictions and agreement is found to be fit for purpose. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Losses in devices such as rotating electrical machines, linear actuators, transformers, and microprocessors cause thermal loading which can age components, limit performance, and cause failure. Device losses manifest themselves as heat sources which are often periodic in nature [1–3]. For example Fig. 1 shows the computed instantaneous iron loss in a permanent magnet synchronous machine (PMSM). When thermometric experiments are performed on these devices, as in [4–6], the experimentalist must decide whether the measured temperature response represents a mean or an instantaneous value of heat generation rate. When numerically modeling the thermal performance of these devices the analyst must decide whether the heat source can be accurately represented by its equivalent mean value, or whether its variation over time must be modeled explicitly. It may be possible to make a computational saving when the source is represented by its equivalent mean ⇑ Corresponding author at: GKN-EVO eDrive Systems Ltd, Unit 14 Woking Business Park, Albert Drive, Woking GU21 5JY, United Kingdom. Tel.: +44 1483 745010. E-mail addresses: [email protected] (A.C. Malloy), [email protected]. uk (R.F. Martinez-Botas), [email protected] (M. Lamperth). http://dx.doi.org/10.1016/j.enconman.2015.02.032 0196-8904/Ó 2015 Elsevier Ltd. All rights reserved.
value as, when compared with the time varying heat source, a larger time step may be used during the simulation. Although thermal models of these devices are common [7–14] discussions regarding the level of detail to which the heat source must be modeled are rare. Often the experience of the analyst or empirical evidence is relied upon to justify modeling decisions a posteriori. Fig. 2, which is a similar to the work of [3], shows a typical application where a periodic heat source is present in an integrated circuit application. Presented with this data an experimentalist would have to make a judgement about whether the mean or instantaneous heat generation rate was being measured. It may also be unclear whether the measured response is due to an instantaneous heat generation rate, or whether it is noise in the temperature sensor measurement and can be filtered out without loss of fidelity. Similarly, given the heat source profile in Fig. 2 the analyst must decide whether to model the transient nature of the heat source, or whether to use its mean value and make a computational saving by increasing the simulation time step. In lieu of the above, this paper presents the derivation and application of a criterion which can be used to estimate the relative importance of the fluctuating component of a periodic heat source on the temperature response of a device or component. It may be used by the heat transfer analyst to determine whether a periodic heat source can be modeled accurately by its mean value, or
13
A.C. Malloy et al. / Energy Conversion and Management 96 (2015) 12–17
Fig. 1. Computed instantaneous iron loss in a PMSM.
Fig. 5. Criterion values plotted over a range of x⁄, each plot line represents a different value of q⁄.
Rth ¼
l 1 þ kA hA
ð1Þ
C th ¼ qC p V
Fig. 2. Temperature response of an integrated circuit to a periodic heat source.
ð2Þ
where k is the thermal conductivity (W/m K), A is the surface area (m2), h is the coefficient of convective heat transfer (W/m2 K), l is the length scale (m) V/A where V is the volume of the element, q is the density (kg/m3), and Cp is the heat capacity (J/kg K). A temperature difference h is defined as T–T1. A sinusoidally varying heat source acts as a forcing function on the element and is considered to be composed of mean and fluctuating components and q0 respectively. Fig. 4 shows the response of the thermal eleq ment to such a heat source. By assuming temperature independent material properties the temperature response can also be decomposed into mean and fluctuating components h and h0 respectively. The steady state mean component is given by Eq. (3).
h ¼ q Rth
ð3Þ
is the h is the mean value of the temperature rise (K), and q where mean value of the heat source (W). The steady state fluctuating component is given by Eq. (4). Fig. 3. Thermal network element used to derive the criterion.
whether it must be modeled as a function of time. It allows the experimentalist to determine whether a measured temperature rise over time corresponds to a mean or an instantaneous heat generation rate. It also enables the experimentalist to determine which frequencies can be filtered out of a transient temperature measurement without loss of fidelity. 2. Criterion derivation A thermal network element (Fig. 3) is used to derive the criterion. The thermal network consists of a thermal resistance Rth (Eq. (1)) and a thermal capacitance Cth (Eq. (2)).
C th
dh0 1 0 þ h ¼ q0 sin xt dt Rth
ð4Þ
where h0 is the fluctuating component of the temperature rise (K), t is the time (s), q0 is the fluctuating component of the heat source (W), and x is the angular velocity (rad/s). The solution of Eq. (4) is seen to be:
h0 ¼
q0 2
C th 1s þ x2
1
s
sin xt þ x cos xt
ð5Þ
where s = RthCth is the thermal time constant (s). Eq. (5) is at a maximum when:
d 1 sin xt þ x cos xt ¼ 0 dt s
Fig. 4. Definition of a periodic heat source and response of the thermal network element to the source.
ð6Þ
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A.C. Malloy et al. / Energy Conversion and Management 96 (2015) 12–17
Fig. 6. PMSM geometry.
Fig. 7. Finite element electromagnetic model of a PMSM.
Fig. 8. Finite volume thermal model of a PMSM.
Solving Eq. (6) for t and substituting back into Eq. (5) yields:
h0MAX ¼
0
q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 C th s þ x2
ð7Þ
where h0MAX is the maximum difference between the fluctuating and steady temperature components. The periodic heat source criterion m is then defined by Eq. (8) which can be seen to be the ratio between Eqs. (7) and (3).
q
ffi m ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ x2
ð8Þ
; and the characteristic where the heat source parameter q⁄ = q0 /q frequency x⁄ = xs. The criterion is simply the ratio between the maximum temperature rise above the mean, and the mean temperature rise above the outside temperature; for example m = 0.1 is interpreted as ‘the fluctuating temperature component is 10% of the magnitude of the steady component’ while m = 1 is interpreted as ‘the
fluctuating temperature component is the same magnitude as the steady component’. The criterion is seen to be similar to the equation for the gain of a low pass filter [15], with the addition of the heat source parameter q⁄ in the numerator. The introduction and definition of q⁄ is therefore a key contribution of the work, allowing the analogy to a low pass filter to be made. The case studies presented in Section 3 outline a procedure for determining q⁄ for non-sinusoidal heat sources. A comparison to a numerical prediction of m provides validation of the approach. In terms of existing non-dimensional parameters the characteristic frequency x⁄ is similar to the inverse of the Fourier number, see Eq. (9).
Fo ¼
kt
qC p l2
ð9Þ
In fact if the variable h in Rth was neglected they would be identical. It has been included here in order to allow the criterion to account for problems with different Biot numbers, where the Biot number is defined as in Eq. (10).
15
A.C. Malloy et al. / Energy Conversion and Management 96 (2015) 12–17 Table 1 Parameters defining the periodic heat source criterion for the magnet component. Parameter 2
A (m ) 3
V (m ) l (m) h (W/m2 K) k (W/m K) q (kg/m3) Cp (J/kg K) x (rad/s) 0
Value
Comment 2
6.46 10
4
2.06 10 4.86 103 274 6.16 7500 460 75,360
q (W)
398
(W) q Rth (K/W) Cth (J/K)
421 0.098 710
Table 2 Criterion values and numerically determined values of m in a PMSM. Component
Surface area of magnet in contact with heat transfer boundary Volume of magnet Length scale defined as V/A Heat transfer coefficient applied over surface area A Thermal conductivity of magnet material Density of magnet material Heat capacity of magnet material Characteristic angular velocity of q0 as determined from Fig. 9 Fluctuating component of the periodic heat source as determined from Fig. 9 Mean component of the periodic heat source Thermal resistance calculated using Eq. (1) Thermal resistance calculated using Eq. (2)
7
Magnet Core Winding
qm ¼
1
Z
r
Criterion m 1.79 10 2.62 107 2.05 107
Numerical m 7
5.09 10 1.02 107 1.66 107
Logarithmic error (%) 7.2 5.9 1.4
!
J dV
ð11Þ
V
where qm is the predicted magnet loss (W), r is the conductivity of !
the magnet material (S/m), and J is the current density (A/m3). Winding losses were predicted using Eq. (12). 2
qw ¼ 3i R
ð12Þ
tetrahedral elements. Ansoft uses the T X finite element formulation to solve 3D transient electromagnetic problems. A detailed description of this formulation is given in [2]. Magnet losses were calculated at each time step from Eq. (11). While core losses were calculated using the method described in [1].
where qw is the predicted winding loss (W), i is the phase current (A) and R is the phase resistance. The predicted transient losses from the FEA model were volume averaged in each PMSM component and used as source terms in the second component of the model; a 3D transient finite volume thermal model of the machine constructed using ‘Ansys CFX V13.0’. The mesh was generated using ‘Ansys Mesh’ and consisted of 418,743 first order finite volume elements. Details are given in Fig. 8 along with the predicted transient loss profiles. As can be seen, the loss profiles vary greatly around their mean values, though over very small time scales. The heat transfer coefficient boundary conditions on the rotor and core faces were determined using correlations taken from [16,17] respectively. As m is the ratio between the maximum temperature rise above the mean and mean temperature rise above ambient, values of m were calculated from volume averaged values of component temperatures predicted by the model. The proposed criterion was used to predict values of m for comparison with the numerical values. The parameters used to calculate the value of the criterion for the magnet component are given in Table 1. Fig. 9 shows how the angular velocity x and unsteady loss component q0 of the magnet were determined. Subsequently the characteristic frequency x⁄ was calculated as 5.26 106 and the heat source parameter q⁄ as 0.945, giving a criterion value of 1.79 107. It should be noted that heat transfer in this system is governed by a set of coupled partial differential equations. This is a good test for the criterion as it is ignorant of the coupling effect, and treats each component as a single isolated lump. Another challenge for the criterion is that each of the heat generating components has an anisotropic thermal conductivity. In this work the predominant direction of the heat flux was used to select which thermal conductivity component was most appropriate.
Fig. 9. Method used to determine the characteristic values of x and q0 from the FEA predicted, volume averaged, magnet losses.
Fig. 10. Duty cycle definition.
Bi ¼
hl k
ð10Þ
3. Results and discussion Fig. 5 shows m as a function of x⁄, where each plot line represents a different value of q⁄. It can be seen that when x⁄ < 1 the criterion value tends to the same value as q⁄, this indicates that the thermal capacitance of the element saturates within each heat source fluctuation and the temperature response will be ‘as unsteady as the heat source’. When x⁄ > 1 the criterion value is attenuated rapidly, indicating that the thermal capacitance is smoothing the temperature response to the heat source fluctuation. 3.1. Case study I A rotating electrical machine has been selected as a case study to demonstrate the value of the proposed criterion. The electrical machine is an axial flux PMSM and is shown in Fig. 6. A multiphysics numerical model of the machine was constructed in order to determine numerical values of m for comparison with values from the proposed criterion. The numerical model comprised of two components, ‘Ansoft Maxwell 14.0.20 was used to construct a 3D transient electromagnetic finite element model which predicted the losses generated in the machine’s magnet, core and winding components. Details of the finite element model are given in Fig. 7. The model comprised of 111,964 second order !
16
A.C. Malloy et al. / Energy Conversion and Management 96 (2015) 12–17
Fig. 11. Comparison between criterion values and numerically determined values of m in a PMSM operating on a duty cycle s Numerical D = 0.25, Numerical D = 0.50, Criterion D = 0.50, M Numerical D = 0.75, - - Criterion D = 0.75.
Table 2 presents the results from both the criterion and the numerical study and it can be seen that the criterion has predicted the amount of unsteadiness in the temperature response of each component reasonably well, despite the complex nature of the thermal system. The accuracy is certainly fit for purpose, it allows the analyst to say with confidence that only a small error in the temperature response of each component would be incurred if the heat sources were modeled using their mean values. It also allows the experimentalist to determine that the measured temperature rise in a heat generating component in this system represents the mean value of the heat source, rather than the instantaneous value. 3.2. Case study II To further demonstrate the value of the criterion it has been used to investigate the thermal response of the electrical machine when operated on a duty cycle. Fig. 10 presents the terminology used in this section when referring to duty cycles. The results of this analysis allow the analyst charged with modeling thermal performance over a range of duty cycles to identify which analyses must be transient, in order to determine peaks in temperature response, and which may be modeled accurately by a steady state analysis, as the peaks in temperature response are filtered out. In the ‘on’ state the machine is at the operating point previously investigated, and as shown the heat sources can be modeled by their mean values. In this case the selection of the fluctuating component of the heat source is made by considering the Fourier series representation of a duty cycle, see Eq. (13).
qðtÞ ¼
1 qB X 2q þ sin T n p n¼1
pnB T
cos
2pnt T
ð13Þ
where B is the ‘on’ time (s), and T is the time period (s). The magnitude of each harmonic of the series is seen to be proportional to 1/n, the most important component in the series is therefore the first one, which has the lowest frequency and greatest magnitude. It is the most likely to induce a transient temperature response, and was therefore used to define q0 and x. Using the same component values of s as before, the criterion has been calculated over a range of frequencies for three duty cycles: D = 0.25, 0.50, and 0.75. Fig. 11 compares the criterion values with numerical results from the finite volume model. Good agreement between the criterion and the numerical results can be seen for all components across the range investigated. The causes of the error in the criterion value can be identified by consideration of the assumptions made in the criterion’s derivation, and in the definition of the fluctuating component of the heat source. As previously stated the criterion neglects anisotropic material properties, coupling effects due to neighboring components and any 3D effects (the derivation assumed a lumped
Criterion D = 0.25, h
element). All of these effects were included in the finite volume model, along with an accurate representation of the heat source (i.e. it was modeled as a pulse train rather than by the first harmonic only). 4. Conclusions This paper has presented the derivation and application of a criterion which can be used to estimate the relative importance of the fluctuating component of a periodic heat source on the temperature response of a device or component. A case study was presented in which the criterion was used to investigate the range of heat sources present in an electrical machine. It was found that even though the heat sources varied greatly about their mean values only small fluctuations existed in the temperature response of each component. The investigation was extended to show what level of unsteadiness should be expected in the temperature response of each component when the electrical machine was operated on a duty cycle. The results obtained from the criterion were deemed fit for purpose when compared with the results of a numerical analysis which modeled the temperature response of the electrical machine explicitly. The criterion is useful to the analyst who must make a decision about whether to model a periodic heat source by its mean value or whether to model it as a function of time. It allows the experimentalist to determine whether a measured temperature response corresponds to a mean or an instantaneous heat generation rate. It also enables the experimentalist to determine which frequencies can be filtered out of a transient temperature measurement without loss of fidelity. References [1] Lin D, Zhou P, Fu W, Badics Z, Cendes Z. A dynamic core loss model for soft ferromagnetic and power ferrite materials in transient finite element analysis. IEEE Trans Magnet 2004;40(2):1318–21. http://dx.doi.org/10.1109/TMAG. 2004.825025. [2] Zhou P, Fu W, Lin D, Stanton S, Cendes Z. Numerical modeling of magnetic devices. IEEE Trans Magnet 2004;40(4):1803–9. http://dx.doi.org/10.1109/ TMAG.2004.830511. [3] Fisher T, Avedisian T, Krusius P. Transient thermal response due to periodic heating on a convectively cooled substrate. IEEE Trans Compon, Packag Manufact Technol 1996;19(1):255–62. http://dx.doi.org/10.1109/96.486510. [4] Malloy AC, Martinez-Botas RF, Jaensch M, Lamperth M. Measurement of heat generation rate in the permanent magnets of rotating electrical machines. In: Proc. of The 6th IET international conference on power electronics, machines and drives. Bristol; 2012, pp. 1–6. DOI: 10.1049/cp.2012.0287. [5] Gilbert A. A method of measuring loss distribution in electrical machines. Proc IEE-Part A Power Eng 1961;108(39):239–44. http://dx.doi.org/10.1049/pia.1961.0050. [6] Malloy AC, Martinez-Botas RF, Lamperth M. Measurement of magnet losses in a surface mounted permanent magnet synchronous machine. IEEE Trans Energy Convers 2014;PP(99):1–8. http://dx.doi.org/10.1109/TEC.2014.2353133. [7] Li H. Cooling of a permanent magnet electric motor with a centrifugal impeller. Int J Heat Mass Transfer 2010;53(4):797–810. http://dx.doi.org/10.1016/ j.ijheatmasstransfer.2009.09.022.
A.C. Malloy et al. / Energy Conversion and Management 96 (2015) 12–17 [8] Kim M-S, Lee K-S, Um S. Numerical investigation and optimization of the thermal performance of a brushless DC motor. Int J Heat Mass Transfer 2009;52(5–6):1589–99. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2008. 07.040. [9] Kolondzovski Z, Belahcen A, Arkkio A. Comparative thermal analysis of different rotor types for a high-speed permanent-magnet electrical machine. IET Electric Power Appl 2009;3(4):279–88. http://dx.doi.org/10.1049/ietepa.2008.0208. [10] Mellor P, Roberts D, Turner D. Lumped parameter thermal model for electrical machines of TEFC design. Proc IEE-Part B: Electric Power Appl 1991;138(5): 205–18. [11] Lim CH, Airoldi G, Bumby JR, Dominy RG, Ingram GI, Mahkamov K, et al. Experimental and CFD investigation of a lumped parameter thermal model of a single-sided, slotted axial flux generator. Int J Therm Sci 2010;49(9):1732–41. http://dx.doi.org/10.1016/j.ijthermalsci.2010.03.018. [12] Lim CH, Airoldi G, Dominy RG, Mahkamov K. Experimental validation of CFD modelling for heat transfer coefficient predictions in axial flux permanent
[13]
[14]
[15] [16]
[17]
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magnet generators. Int J Therm Sci 2011;50(12):2451–63. http://dx.doi.org/ 10.1016/j.ijthermalsci.2011.07.002. Tsili MA, Amoiralis EI, Kladas AG, Souflaris AT. Power transformer thermal analysis by using an advanced coupled 3D heat transfer and fluid flow model. Int J Therm Sci 2012;53:188–201. http://dx.doi.org/10.1016/j.ijthermalsci. 2011.10.010. Wakil NE, Chereches NC, Padet J. Numerical study of heat transfer and fluid flow in a power transformer. Int J Therm Sci 2006;45(6):615–26. http:// dx.doi.org/10.1016/j.ijthermalsci.2005.09.002. Fitzgerald AE, Higginbotham DE, Grabel A. Basic electrical engineering. McgrawHill College; 1981. Boutarfa R, Harmand S. Local convective heat transfer for laminar and turbulent flow in a rotor stator system. Exp Fluids 2005;38:209–21. http:// dx.doi.org/10.1007/s00348-004-0900-5. Howey D, Childs P, Holmes A. Air-gap convection in rotating electrical machines. IEEE Trans Ind Electron 2012;59(3):1367–75. http://dx.doi.org/ 10.1109/TIE.2010.2100337.
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
A criterion for determining the relative importance of the fluctuating component of a periodic heat source Adam C. Malloy a,b,⇑, Ricardo F. Martinez-Botas a, Michael Lamperth b a b
Mechanical Engineering Department, Imperial College London, Exhibition Road, London SW7 2AZ, United Kingdom GKN-EVO eDrive Systems Ltd, Unit 14 Woking Business Park, Albert Drive, Woking GU21 5JY, United Kingdom
a r t i c l e
i n f o
Article history: Received 2 June 2014 Accepted 9 February 2015 Available online 7 March 2015 Keywords: Criterion Transient heat sources Numerical modeling Transient temperature response
a b s t r a c t Devices such as rotating electrical machines, transformers, and microprocessors experience thermal loading during operation. This is caused by device losses which manifest themselves as heat sources. Whether operated continuously or on a duty cycle these heat sources are often periodic in nature, exhibiting both mean and fluctuating components. This paper proposes a criterion which can be used to estimate the relative importance of the fluctuating component of a periodic heat source on the temperature response of a device, or a component within a device. It may be used by the heat transfer analyst to determine whether a periodic heat source can be modeled accurately by its mean value or whether it must be modeled as a function of time. During thermometric tests it enables the experimentalist to determine whether the measured temperature rise rate represents an instantaneous or a mean value of heat generation rate. The criterion is derived by considering a sinusoidal heat source acting on a thermal network element. A case study is presented where the criterion is used to estimate the relative importance of the fluctuating component of a range periodic heat sources present in a rotating electrical machine. Results are compared with numerical predictions and agreement is found to be fit for purpose. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Losses in devices such as rotating electrical machines, linear actuators, transformers, and microprocessors cause thermal loading which can age components, limit performance, and cause failure. Device losses manifest themselves as heat sources which are often periodic in nature [1–3]. For example Fig. 1 shows the computed instantaneous iron loss in a permanent magnet synchronous machine (PMSM). When thermometric experiments are performed on these devices, as in [4–6], the experimentalist must decide whether the measured temperature response represents a mean or an instantaneous value of heat generation rate. When numerically modeling the thermal performance of these devices the analyst must decide whether the heat source can be accurately represented by its equivalent mean value, or whether its variation over time must be modeled explicitly. It may be possible to make a computational saving when the source is represented by its equivalent mean ⇑ Corresponding author at: GKN-EVO eDrive Systems Ltd, Unit 14 Woking Business Park, Albert Drive, Woking GU21 5JY, United Kingdom. Tel.: +44 1483 745010. E-mail addresses: [email protected] (A.C. Malloy), [email protected]. uk (R.F. Martinez-Botas), [email protected] (M. Lamperth). http://dx.doi.org/10.1016/j.enconman.2015.02.032 0196-8904/Ó 2015 Elsevier Ltd. All rights reserved.
value as, when compared with the time varying heat source, a larger time step may be used during the simulation. Although thermal models of these devices are common [7–14] discussions regarding the level of detail to which the heat source must be modeled are rare. Often the experience of the analyst or empirical evidence is relied upon to justify modeling decisions a posteriori. Fig. 2, which is a similar to the work of [3], shows a typical application where a periodic heat source is present in an integrated circuit application. Presented with this data an experimentalist would have to make a judgement about whether the mean or instantaneous heat generation rate was being measured. It may also be unclear whether the measured response is due to an instantaneous heat generation rate, or whether it is noise in the temperature sensor measurement and can be filtered out without loss of fidelity. Similarly, given the heat source profile in Fig. 2 the analyst must decide whether to model the transient nature of the heat source, or whether to use its mean value and make a computational saving by increasing the simulation time step. In lieu of the above, this paper presents the derivation and application of a criterion which can be used to estimate the relative importance of the fluctuating component of a periodic heat source on the temperature response of a device or component. It may be used by the heat transfer analyst to determine whether a periodic heat source can be modeled accurately by its mean value, or
13
A.C. Malloy et al. / Energy Conversion and Management 96 (2015) 12–17
Fig. 1. Computed instantaneous iron loss in a PMSM.
Fig. 5. Criterion values plotted over a range of x⁄, each plot line represents a different value of q⁄.
Rth ¼
l 1 þ kA hA
ð1Þ
C th ¼ qC p V
Fig. 2. Temperature response of an integrated circuit to a periodic heat source.
ð2Þ
where k is the thermal conductivity (W/m K), A is the surface area (m2), h is the coefficient of convective heat transfer (W/m2 K), l is the length scale (m) V/A where V is the volume of the element, q is the density (kg/m3), and Cp is the heat capacity (J/kg K). A temperature difference h is defined as T–T1. A sinusoidally varying heat source acts as a forcing function on the element and is considered to be composed of mean and fluctuating components and q0 respectively. Fig. 4 shows the response of the thermal eleq ment to such a heat source. By assuming temperature independent material properties the temperature response can also be decomposed into mean and fluctuating components h and h0 respectively. The steady state mean component is given by Eq. (3).
h ¼ q Rth
ð3Þ
is the h is the mean value of the temperature rise (K), and q where mean value of the heat source (W). The steady state fluctuating component is given by Eq. (4). Fig. 3. Thermal network element used to derive the criterion.
whether it must be modeled as a function of time. It allows the experimentalist to determine whether a measured temperature rise over time corresponds to a mean or an instantaneous heat generation rate. It also enables the experimentalist to determine which frequencies can be filtered out of a transient temperature measurement without loss of fidelity. 2. Criterion derivation A thermal network element (Fig. 3) is used to derive the criterion. The thermal network consists of a thermal resistance Rth (Eq. (1)) and a thermal capacitance Cth (Eq. (2)).
C th
dh0 1 0 þ h ¼ q0 sin xt dt Rth
ð4Þ
where h0 is the fluctuating component of the temperature rise (K), t is the time (s), q0 is the fluctuating component of the heat source (W), and x is the angular velocity (rad/s). The solution of Eq. (4) is seen to be:
h0 ¼
q0 2
C th 1s þ x2
1
s
sin xt þ x cos xt
ð5Þ
where s = RthCth is the thermal time constant (s). Eq. (5) is at a maximum when:
d 1 sin xt þ x cos xt ¼ 0 dt s
Fig. 4. Definition of a periodic heat source and response of the thermal network element to the source.
ð6Þ
14
A.C. Malloy et al. / Energy Conversion and Management 96 (2015) 12–17
Fig. 6. PMSM geometry.
Fig. 7. Finite element electromagnetic model of a PMSM.
Fig. 8. Finite volume thermal model of a PMSM.
Solving Eq. (6) for t and substituting back into Eq. (5) yields:
h0MAX ¼
0
q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 C th s þ x2
ð7Þ
where h0MAX is the maximum difference between the fluctuating and steady temperature components. The periodic heat source criterion m is then defined by Eq. (8) which can be seen to be the ratio between Eqs. (7) and (3).
q
ffi m ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ x2
ð8Þ
; and the characteristic where the heat source parameter q⁄ = q0 /q frequency x⁄ = xs. The criterion is simply the ratio between the maximum temperature rise above the mean, and the mean temperature rise above the outside temperature; for example m = 0.1 is interpreted as ‘the fluctuating temperature component is 10% of the magnitude of the steady component’ while m = 1 is interpreted as ‘the
fluctuating temperature component is the same magnitude as the steady component’. The criterion is seen to be similar to the equation for the gain of a low pass filter [15], with the addition of the heat source parameter q⁄ in the numerator. The introduction and definition of q⁄ is therefore a key contribution of the work, allowing the analogy to a low pass filter to be made. The case studies presented in Section 3 outline a procedure for determining q⁄ for non-sinusoidal heat sources. A comparison to a numerical prediction of m provides validation of the approach. In terms of existing non-dimensional parameters the characteristic frequency x⁄ is similar to the inverse of the Fourier number, see Eq. (9).
Fo ¼
kt
qC p l2
ð9Þ
In fact if the variable h in Rth was neglected they would be identical. It has been included here in order to allow the criterion to account for problems with different Biot numbers, where the Biot number is defined as in Eq. (10).
15
A.C. Malloy et al. / Energy Conversion and Management 96 (2015) 12–17 Table 1 Parameters defining the periodic heat source criterion for the magnet component. Parameter 2
A (m ) 3
V (m ) l (m) h (W/m2 K) k (W/m K) q (kg/m3) Cp (J/kg K) x (rad/s) 0
Value
Comment 2
6.46 10
4
2.06 10 4.86 103 274 6.16 7500 460 75,360
q (W)
398
(W) q Rth (K/W) Cth (J/K)
421 0.098 710
Table 2 Criterion values and numerically determined values of m in a PMSM. Component
Surface area of magnet in contact with heat transfer boundary Volume of magnet Length scale defined as V/A Heat transfer coefficient applied over surface area A Thermal conductivity of magnet material Density of magnet material Heat capacity of magnet material Characteristic angular velocity of q0 as determined from Fig. 9 Fluctuating component of the periodic heat source as determined from Fig. 9 Mean component of the periodic heat source Thermal resistance calculated using Eq. (1) Thermal resistance calculated using Eq. (2)
7
Magnet Core Winding
qm ¼
1
Z
r
Criterion m 1.79 10 2.62 107 2.05 107
Numerical m 7
5.09 10 1.02 107 1.66 107
Logarithmic error (%) 7.2 5.9 1.4
!
J dV
ð11Þ
V
where qm is the predicted magnet loss (W), r is the conductivity of !
the magnet material (S/m), and J is the current density (A/m3). Winding losses were predicted using Eq. (12). 2
qw ¼ 3i R
ð12Þ
tetrahedral elements. Ansoft uses the T X finite element formulation to solve 3D transient electromagnetic problems. A detailed description of this formulation is given in [2]. Magnet losses were calculated at each time step from Eq. (11). While core losses were calculated using the method described in [1].
where qw is the predicted winding loss (W), i is the phase current (A) and R is the phase resistance. The predicted transient losses from the FEA model were volume averaged in each PMSM component and used as source terms in the second component of the model; a 3D transient finite volume thermal model of the machine constructed using ‘Ansys CFX V13.0’. The mesh was generated using ‘Ansys Mesh’ and consisted of 418,743 first order finite volume elements. Details are given in Fig. 8 along with the predicted transient loss profiles. As can be seen, the loss profiles vary greatly around their mean values, though over very small time scales. The heat transfer coefficient boundary conditions on the rotor and core faces were determined using correlations taken from [16,17] respectively. As m is the ratio between the maximum temperature rise above the mean and mean temperature rise above ambient, values of m were calculated from volume averaged values of component temperatures predicted by the model. The proposed criterion was used to predict values of m for comparison with the numerical values. The parameters used to calculate the value of the criterion for the magnet component are given in Table 1. Fig. 9 shows how the angular velocity x and unsteady loss component q0 of the magnet were determined. Subsequently the characteristic frequency x⁄ was calculated as 5.26 106 and the heat source parameter q⁄ as 0.945, giving a criterion value of 1.79 107. It should be noted that heat transfer in this system is governed by a set of coupled partial differential equations. This is a good test for the criterion as it is ignorant of the coupling effect, and treats each component as a single isolated lump. Another challenge for the criterion is that each of the heat generating components has an anisotropic thermal conductivity. In this work the predominant direction of the heat flux was used to select which thermal conductivity component was most appropriate.
Fig. 9. Method used to determine the characteristic values of x and q0 from the FEA predicted, volume averaged, magnet losses.
Fig. 10. Duty cycle definition.
Bi ¼
hl k
ð10Þ
3. Results and discussion Fig. 5 shows m as a function of x⁄, where each plot line represents a different value of q⁄. It can be seen that when x⁄ < 1 the criterion value tends to the same value as q⁄, this indicates that the thermal capacitance of the element saturates within each heat source fluctuation and the temperature response will be ‘as unsteady as the heat source’. When x⁄ > 1 the criterion value is attenuated rapidly, indicating that the thermal capacitance is smoothing the temperature response to the heat source fluctuation. 3.1. Case study I A rotating electrical machine has been selected as a case study to demonstrate the value of the proposed criterion. The electrical machine is an axial flux PMSM and is shown in Fig. 6. A multiphysics numerical model of the machine was constructed in order to determine numerical values of m for comparison with values from the proposed criterion. The numerical model comprised of two components, ‘Ansoft Maxwell 14.0.20 was used to construct a 3D transient electromagnetic finite element model which predicted the losses generated in the machine’s magnet, core and winding components. Details of the finite element model are given in Fig. 7. The model comprised of 111,964 second order !
16
A.C. Malloy et al. / Energy Conversion and Management 96 (2015) 12–17
Fig. 11. Comparison between criterion values and numerically determined values of m in a PMSM operating on a duty cycle s Numerical D = 0.25, Numerical D = 0.50, Criterion D = 0.50, M Numerical D = 0.75, - - Criterion D = 0.75.
Table 2 presents the results from both the criterion and the numerical study and it can be seen that the criterion has predicted the amount of unsteadiness in the temperature response of each component reasonably well, despite the complex nature of the thermal system. The accuracy is certainly fit for purpose, it allows the analyst to say with confidence that only a small error in the temperature response of each component would be incurred if the heat sources were modeled using their mean values. It also allows the experimentalist to determine that the measured temperature rise in a heat generating component in this system represents the mean value of the heat source, rather than the instantaneous value. 3.2. Case study II To further demonstrate the value of the criterion it has been used to investigate the thermal response of the electrical machine when operated on a duty cycle. Fig. 10 presents the terminology used in this section when referring to duty cycles. The results of this analysis allow the analyst charged with modeling thermal performance over a range of duty cycles to identify which analyses must be transient, in order to determine peaks in temperature response, and which may be modeled accurately by a steady state analysis, as the peaks in temperature response are filtered out. In the ‘on’ state the machine is at the operating point previously investigated, and as shown the heat sources can be modeled by their mean values. In this case the selection of the fluctuating component of the heat source is made by considering the Fourier series representation of a duty cycle, see Eq. (13).
qðtÞ ¼
1 qB X 2q þ sin T n p n¼1
pnB T
cos
2pnt T
ð13Þ
where B is the ‘on’ time (s), and T is the time period (s). The magnitude of each harmonic of the series is seen to be proportional to 1/n, the most important component in the series is therefore the first one, which has the lowest frequency and greatest magnitude. It is the most likely to induce a transient temperature response, and was therefore used to define q0 and x. Using the same component values of s as before, the criterion has been calculated over a range of frequencies for three duty cycles: D = 0.25, 0.50, and 0.75. Fig. 11 compares the criterion values with numerical results from the finite volume model. Good agreement between the criterion and the numerical results can be seen for all components across the range investigated. The causes of the error in the criterion value can be identified by consideration of the assumptions made in the criterion’s derivation, and in the definition of the fluctuating component of the heat source. As previously stated the criterion neglects anisotropic material properties, coupling effects due to neighboring components and any 3D effects (the derivation assumed a lumped
Criterion D = 0.25, h
element). All of these effects were included in the finite volume model, along with an accurate representation of the heat source (i.e. it was modeled as a pulse train rather than by the first harmonic only). 4. Conclusions This paper has presented the derivation and application of a criterion which can be used to estimate the relative importance of the fluctuating component of a periodic heat source on the temperature response of a device or component. A case study was presented in which the criterion was used to investigate the range of heat sources present in an electrical machine. It was found that even though the heat sources varied greatly about their mean values only small fluctuations existed in the temperature response of each component. The investigation was extended to show what level of unsteadiness should be expected in the temperature response of each component when the electrical machine was operated on a duty cycle. The results obtained from the criterion were deemed fit for purpose when compared with the results of a numerical analysis which modeled the temperature response of the electrical machine explicitly. The criterion is useful to the analyst who must make a decision about whether to model a periodic heat source by its mean value or whether to model it as a function of time. It allows the experimentalist to determine whether a measured temperature response corresponds to a mean or an instantaneous heat generation rate. It also enables the experimentalist to determine which frequencies can be filtered out of a transient temperature measurement without loss of fidelity. References [1] Lin D, Zhou P, Fu W, Badics Z, Cendes Z. A dynamic core loss model for soft ferromagnetic and power ferrite materials in transient finite element analysis. IEEE Trans Magnet 2004;40(2):1318–21. http://dx.doi.org/10.1109/TMAG. 2004.825025. [2] Zhou P, Fu W, Lin D, Stanton S, Cendes Z. Numerical modeling of magnetic devices. IEEE Trans Magnet 2004;40(4):1803–9. http://dx.doi.org/10.1109/ TMAG.2004.830511. [3] Fisher T, Avedisian T, Krusius P. Transient thermal response due to periodic heating on a convectively cooled substrate. IEEE Trans Compon, Packag Manufact Technol 1996;19(1):255–62. http://dx.doi.org/10.1109/96.486510. [4] Malloy AC, Martinez-Botas RF, Jaensch M, Lamperth M. Measurement of heat generation rate in the permanent magnets of rotating electrical machines. In: Proc. of The 6th IET international conference on power electronics, machines and drives. Bristol; 2012, pp. 1–6. DOI: 10.1049/cp.2012.0287. [5] Gilbert A. A method of measuring loss distribution in electrical machines. Proc IEE-Part A Power Eng 1961;108(39):239–44. http://dx.doi.org/10.1049/pia.1961.0050. [6] Malloy AC, Martinez-Botas RF, Lamperth M. Measurement of magnet losses in a surface mounted permanent magnet synchronous machine. IEEE Trans Energy Convers 2014;PP(99):1–8. http://dx.doi.org/10.1109/TEC.2014.2353133. [7] Li H. Cooling of a permanent magnet electric motor with a centrifugal impeller. Int J Heat Mass Transfer 2010;53(4):797–810. http://dx.doi.org/10.1016/ j.ijheatmasstransfer.2009.09.022.
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