Skip to main content

Advertisement

SpringerLink
Log in

Energy Conversion Performance and Optimization of Wearable Annular Thermoelectric Generators

  • Original Research Article
  • Published:
Journal of Electronic Materials Aims and scope Submit manuscript

Abstract

This paper presents a theoretical model for a human skin-wearable annular thermoelectric generator (WATEG) system and provides analytical solutions for its energy conversion performance. The Pennes equation is used to model the heat transfer of human skin, which is assumed to be a cylindrical multilayer structure composed of subcutis, dermis, and epidermis. The heat exchanges induced by blood perfusion and metabolic heat generation within the skin tissue are taken into account. It is found that the influence of skin effect and contact thermal resistance between the human skin and flexible substrate plays a significant role in the energy conversion performance of the WATEG and should be considered. The matched load resistance, optimal fill factor, and height of thermoelectric legs are determined through numerical analysis. The findings of this study can be applied to the practical design of WATEG devices and are expected to contribute to their optimization.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Abbreviations

A :

Area (m2)

c :

Specific heat (J/kg/K)

F :

Fill factor (%)

H :

Height (m) or microhardness (MPa)

h :

Convection coefficient or contact thermal conductance (W/m2/K)

I :

Current (A)

j :

Current density (A/m2)

K :

Thermal conductance (W/K)

P :

Power output (μW/cm2) or contact pressure (kPa)

Q :

Heat flux (W)

q :

Heat flux density (W/m2)

R :

Electric resistance (Ω)

r :

Radius (mm)

T :

Temperature (K)

α :

Seebeck coefficient (V/K)

Δa :

Asperity slope (rad)

\(\delta \) :

Thicknesses (mm)

\(\theta \) :

Angle (rad)

ε :

Surface roughness (μm)

ρ :

Density (kg/m3)

σ :

Electric conductivity (S/m)

λ :

Thermal conductivity (W/m/K)

ω :

Blood perfusion rate (mL/mL/s)

a:

Ambient environment

b:

Blood

c:

Contact interface

d:

Dermis

e:

Epidermis

eff:

Effective property

en:

Encapsulation layer

f:

Convection at heat sink

fm:

Fill material

i:

i-th layer

m:

Metabolism

n:

N-type thermoelectric leg

out:

Output

p:

P-type thermoelectric leg

s:

Body core temperature or subcutis

DPL:

Dual-phase-lag

TEG:

Thermoelectric generator

WATEG:

Wearable annular thermoelectric generator

WTEG:

Wearable thermoelectric generator

References

  1. X. Wang, Z. Liu, and T. Zhang, Flexible sensing electronics for wearable/attachable health monitoring. Small 13(25), 1602790 (2017).

    Article  Google Scholar 

  2. T.Q. Trung and N.-E. Lee, Flexible and stretchable physical sensor integrated platforms for wearable human-activity monitoring and personal healthcare. Adv. Mater. 28(22), 4338–4372 (2016).

    Article  CAS  Google Scholar 

  3. A.S. Dahiya, J. Thireau, J. Boudaden, S. Lal, U. Gulzar, Y. Zhang, T. Gil, N. Azemard, P. Ramm, T. Kiessling, C. O’Murchu, F. Sebelius, J. Tilly, C. Glynn, S. Geary, C. O’Dwyer, K.M. Razeeb, A. Lacampagne, B. Charlot, and A. Todri-Sanial, Review-energy autonomous wearable sensors for smart healthcare: a review. J. Electrochem. Soc. 167(3), 037516 (2019).

    Article  Google Scholar 

  4. W. Ren, Y. Sun, D. Zhao, A. Aili, S. Zhang, C. Shi, J. Zhang, H. Geng, J. Zhang, L. Zhang, J. Xiao, and R. Yang, High-performance wearable thermoelectric generator with self-healing, recycling, and Lego-like reconfiguring capabilities. Sci. Adv. 7(7), eabe0586 (2021).

    Article  CAS  Google Scholar 

  5. A. Nozariasbmarz, H. Collins, K. Dsouza, M.H. Polash, M. Hosseini, M. Hyland, J. Liu, A. Malhotra, F.M. Ortiz, F. Mohaddes, V.P. Ramesh, Y. Sargolzaeiaval, N. Snouwaert, M.C. Özturk, and D. Vashaee, Review of wearable thermoelectric energy harvesting: from body temperature to electronic systems. Appl. Energy 258, 114069 (2020).

    Article  Google Scholar 

  6. P. Kolarsick, M.A. Kolarsick, and C. Goodwin, Anatomy and Physiology of the Skin. J. Dermatol. Nurs. Assoc. 3(4), 203 (2011).

    Google Scholar 

  7. N. Djongyang, R. Tchinda, and D. Njomo, Thermal comfort: a review paper. Renew. Sust. Energy. Rev. 14(9), 2626–2640 (2010).

    Article  Google Scholar 

  8. G. Kelly, Body temperature variability (Part 1): a review of the history of body temperature and its variability due to site selection, biological rhythms, fitness, and aging. Altern. Med. Rev. 11(4), 278 (2006).

    Google Scholar 

  9. F. Suarez, A. Nozariasbmarz, D. Vashaee, and M.C. Öztürk, Designing thermoelectric generators for self-powered wearable electronics. Energy Environ. Sci. 9, 2099–2113 (2016).

    Article  CAS  Google Scholar 

  10. D. Wijethunge, D. Kim, and W. Kim, Simplified human thermoregulatory model for designing wearable thermoelectric devices. J. Phys. D Appl. Phys. 51, 055401 (2018).

    Article  Google Scholar 

  11. A.B. Zhang, G.Y. Li, B.L. Wang, and J. Wang, A theoretical model for wearable thermoelectric generators considering the effect of human skin. J. Electron. Mater. 50, 1514–1526 (2021).

    Article  CAS  Google Scholar 

  12. A.B. Zhang, D.D. Pang, B.L. Wang, and J. Wang, Dynamic responses of wearable thermoelectric generators used for skin waste heat harvesting. Energy 262, 125621 (2023).

    Article  CAS  Google Scholar 

  13. F. Suarez, D.P. Parekh, C. Ladd, D. Vashaee, M.D. Dickey, and M.C. Öztürk, Flexible thermoelectric generator using bulk legs and liquid metal interconnects for wearable electronics. Appl. Energy 202, 736–745 (2017).

    Article  Google Scholar 

  14. M.H. Jeong, K.C. Kim, J.S. Kim, and K.J. Choi, Operation of wearable thermoelectric generators using dual sources of heat and light. Adv. Sci. 9(12), 2104915 (2022).

    Article  CAS  Google Scholar 

  15. Z.G. Shen, S.Y. Wu, and L. Xiao, Theoretical analysis on the performance of annular thermoelectric couple. Energy Convers. Manag. 89, 244–250 (2015).

    Article  Google Scholar 

  16. S.C. Kaushik and S. Manikandan, The influence of Thomson effect in the energy and exergy efficiency of an annular thermoelectric generator. Energy Convers. Manag. 103, 200–207 (2015).

    Article  Google Scholar 

  17. A.B. Zhang, B.L. Wang, D.D. Pang, L.W. He, J. Lou, J. Wang, and J.K. Du, Effects of interface layers on the performance of annular thermoelectric generators. Energy 147, 612–620 (2018).

    Article  Google Scholar 

  18. A.B. Zhang, B.L. Wang, D.D. Pang, J.B. Chen, J. Wang, and J.K. Du, Influence of leg geometry configuration and contact resistance on the performance of annular thermoelectric generators. Energy Convers. Manag. 166, 337–342 (2018).

    Article  Google Scholar 

  19. Z.F. Wen, Y. Sun, A.B. Zhang, B.L. Wang, J. Wang, and J.K. Du, Performance analysis of a segmented annular thermoelectric generator. J. Electron. Mater. 49, 4830–4842 (2020).

    Article  CAS  Google Scholar 

  20. S.F. Fan and Y.W. Gao, Numerical analysis on the segmented annular thermoelectric generator for waste heat recovery. Energy 183, 35–47 (2019).

    Article  Google Scholar 

  21. H.H. Pennes, Analysis of tissue and arterial blood temperatures in the resting human forearm. J. Appl. Physiol. 1, 93–122 (1948).

    Article  CAS  Google Scholar 

  22. M. Bahrami, J.R. Culham, and M.M. Yovanavich, Modeling thermal contact resistance: a scale analysis approach. J. Heat Transf. 126, 896–905 (2004).

    Article  CAS  Google Scholar 

  23. F. Xu, T.J. Lu, and K.A. Seffen, Biothermomechanics of skin tissues. J. Mech. Phys. Solids 56, 1852–1884 (2008).

    Article  CAS  Google Scholar 

  24. Y.C. Wang, Y.G. Shi, D.Q. Mei, and Z.C. Chen, Wearable thermoelectric generator for harvesting heat on the curved human wrist. Appl. Energy 205, 710–719 (2017).

    Article  Google Scholar 

  25. H.N. Ho and L.A. Jones, Modeling the thermal responses of the skin surface during hand-object interactions. J. Biomech. Eng. T ASME 130, 021005 (2008).

    Article  Google Scholar 

  26. F.P. Incropera, D.P. DeWitt, T.L. Bergman, and A.S. Lavine, Fundamentals of Heat and Mass Transfer, 6th ed., (New York: Wiley, 1996).

    Google Scholar 

  27. M. Gao, New formulation of the theory of thermoelectric generators operating under constant heat flux. Energ. Environ. Sci. 15, 356–367 (2022).

    Article  Google Scholar 

  28. A.B. Zhang, J. Lou, B.L. Wang, and J. Wang, A Griffith crack model in a generalized nonhomogeneous interlayer of bonded dissimilar half-planes. J. Theort. Appl. Mech. 61, 495–507 (2023).

    Article  Google Scholar 

  29. A.B. Zhang and B.L. Wang, Temperature and electric potential fields of an interface crack in a layered thermoelectric or metal/thermoelectric material. Int. J. Therm. Sci. 104, 396–403 (2016).

    Article  CAS  Google Scholar 

Download references

Acknowledgments

The research was supported by the Natural Science Foundation of Zhejiang Province of China (LY21A020004), the Natural Science Foundation of Ningbo (2022J095), the Natural Science Youth Foundation of Henan Province (232300420336), the Key University Scientific Research Project of Henan Province (22A610007), and the Special Scientific Research Fund Project of Cultivating Master Graduates of Huangshan University (hsxyssd008).

Author information

Authors and Affiliations

  1. School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Chenchen Guo & Aibing Zhang

  2. Henan Province Key Laboratory of Water Pollution Control and Rehabilitation Technology, Henan University of Urban Construction, Pingdingshan, 467036, People’s Republic of China

    Dandan Pang

  3. School of Mechanical and Electrical Engineering, Huangshan University, Huangshan, 245021, People’s Republic of China

    Jianhua Cao

Authors
  1. Chenchen Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aibing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dandan Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jianhua Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Aibing Zhang or Dandan Pang.

Ethics declarations

Conflict of interest

The authors declare that they have no financial interests or personal relationships with other people or organizations that can inappropriately influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A

The constants \({B}_{i}\) and \({D}_{i}\) (\(i=s, d\) and \(e\)):

$$ \begin{array}{*{20}c} {B_{s} = - \frac{{\frac{{q_{m} }}{{\eta_{b} }}K_{0} \left( {\xi_{s} r_{d} } \right) + \left( {T_{d} - T_{s} - \frac{{q_{m} }}{{\eta_{b} }}} \right)K_{0} \left( {\xi_{s} r_{s} } \right)}}{{I_{0} \left( {\xi_{s} r_{s} } \right)K_{0} \left( {\xi_{s} r_{d} } \right) - I_{0} \left( {\xi_{s} r_{d} } \right)K_{0} \left( {\xi_{s} r_{s} } \right)}}} \\ \end{array} $$
(A1)
$$ \begin{array}{*{20}c} {D_{s} = \frac{{\frac{{q_{m} }}{{\eta_{b} }}I_{0} \left( {\xi_{s} r_{d} } \right) + \left( {T_{d} - T_{s} - \frac{{q_{m} }}{{\eta_{b} }}} \right)I_{0} \left( {\xi_{s} r_{s} } \right)}}{{I_{0} \left( {\xi_{s} r_{s} } \right)K_{0} \left( {\xi_{s} r_{d} } \right) - I_{0} \left( {\xi_{s} r_{d} } \right)K_{0} \left( {\xi_{s} r_{s} } \right)}}} \\ \end{array} $$
(A2)
$$ \begin{array}{*{20}c} {B_{d} = \frac{{T_{d} K_{0} \left( {\xi_{d} r_{e} } \right) - T_{e} K_{0} \left( {\xi_{d} r_{d} } \right) + \left( {T_{s} + \frac{{q_{m} }}{{\eta_{b} }}} \right)\left[ {K_{0} \left( {\xi_{d} r_{d} } \right) - K_{0} \left( {\xi_{d} r_{e} } \right)} \right]}}{{I_{0} \left( {\xi_{d} r_{d} } \right)K_{0} \left( {\xi_{d} r_{e} } \right) - I_{0} \left( {\xi_{d} r_{e} } \right)K_{0} \left( {\xi_{d} r_{d} } \right)}}} \\ \end{array} $$
(A3)
$$ \begin{array}{*{20}c} {D_{d} = \frac{{T_{d} I_{0} \left( {\xi_{d} r_{e} } \right) - T_{e} I_{0} \left( {\xi_{d} r_{d} } \right) + \left( {T_{s} + \frac{{q_{m} }}{{\eta_{b} }}} \right)\left[ {I_{0} \left( {\xi_{d} r_{d} } \right) - I_{0} \left( {\xi_{d} r_{e} } \right)} \right]}}{{I_{0} \left( {\xi_{d} r_{d} } \right)K_{0} \left( {\xi_{d} r_{e} } \right) - I_{0} \left( {\xi_{d} r_{e} } \right)K_{0} \left( {\xi_{d} r_{d} } \right)}}} \\ \end{array} $$
(A4)
$$ \begin{array}{*{20}c} {B_{e} = \frac{{T_{e} - T_{c} - \frac{{q_{m} }}{{4\lambda_{e} }}\left( {r_{c}^{2} - r_{e}^{2} } \right)}}{{\ln r_{e} - \ln r_{c} }}} \\ \end{array} $$
(A5)
$$ \begin{array}{*{20}c} {D_{e} = - \frac{{T_{e} \ln r_{c} - T_{c} \ln r_{e} + \frac{{q_{m} }}{{4\lambda_{e} }}\left( {r_{e}^{2} \ln r_{c} - r_{c}^{2} \ln r_{e} } \right)}}{{\ln r_{e} - \ln r_{c} }}} \\ \end{array} $$
(A6)

where \(K_{s} = \frac{{\lambda_{s} \theta \delta }}{{{\text{ln}}\left( {r_{d} /r_{s} } \right)}}, K_{d} = \frac{{\lambda_{d} \theta \delta }}{{{\text{ln}}\left( {r_{e} /r_{d} } \right)}}, K_{e} = \frac{{\lambda_{e} \theta \delta }}{{{\text{ln}}\left( {r_{c} /r_{e} } \right)}}\) are the thermal conductances of subcutis, dermis, and epidermis, respectively.

Appendix B

The temperatures \(T_{d}\), \(T_{e}\) and \(T_{c}\), and the constants \(\gamma_{1}\), \(\gamma_{2}\), \(\Omega_{c}\) and \(\Omega_{s}\):

$$ \begin{array}{*{20}c} {T_{d} = \frac{{\mu_{s1} T_{s} - \mu_{e1} T_{e} }}{{\mu_{d1} }}} \\ \end{array} $$
(B1)
$$ \begin{array}{*{20}c} {T_{e} = \frac{{\mu_{s2} T_{s} - \mu_{c1} T_{c} - \mu_{d2} T_{d} }}{{\mu_{e2} }}} \\ \end{array} $$
(B2)
$$ \begin{array}{*{20}c} {T_{c} = - \frac{{K_{c} }}{{K_{e} \Omega_{c} }}T_{1} + \frac{{\Omega_{s} }}{{\Omega_{c} }}T_{s} } \\ \end{array} $$
(B3)
$$ \begin{array}{*{20}c} {\gamma_{1} = \alpha I - K_{eff} - K_{a} } \\ \end{array} $$
(B4)
$$ \begin{array}{*{20}c} {\gamma_{2} = \alpha I + K_{{{\text{eff}}}} + K_{c} + \frac{{K_{c}^{2} }}{{K_{e} \Omega_{c} }}} \\ \end{array} $$
(B5)
$$ \begin{array}{*{20}c} {\Omega_{c} = \frac{{\mu_{d1} \mu_{c1} }}{{\mu_{e1} \mu_{d2} - \mu_{e2} \mu_{d1} }} - 1 - \frac{{K_{c} }}{{K_{e} }}} \\ \end{array} $$
(B6)
$$ \begin{array}{*{20}c} {\Omega_{s} = \frac{{\mu_{m} }}{{T_{s} }} - \frac{{\mu_{d2} \mu_{s1} - \mu_{d1} \mu_{s2} }}{{\mu_{e1} \mu_{d2} - \mu_{e2} \mu_{d1} }}} \\ \end{array} $$
(B7)

with

$$ \begin{array}{*{20}c} {\mu_{d1} = \frac{{I_{1} \left( {\xi_{s} r_{d} } \right)K_{0} \left( {\xi_{s} r_{s} } \right) + I_{0} \left( {\xi_{s} r_{s} } \right)K_{1} \left( {\xi_{s} r_{d} } \right)}}{{I_{0} \left( {\xi_{s} r_{s} } \right)K_{0} \left( {\xi_{s} r_{d} } \right) - I_{0} \left( {\xi_{s} r_{d} } \right)K_{0} \left( {\xi_{s} r_{s} } \right)}} + \mu_{e1} } \\ \end{array} $$
(B8)
$$ \begin{array}{*{20}c} {\mu_{e1} = \frac{{K_{d} \xi_{d} {\text{ln}}\frac{{r_{e} }}{{r_{d} }}}}{{K_{s} \xi_{s} {\text{ln}}\frac{{r_{d} }}{{r_{s} }}}}\frac{{I_{1} \left( {\xi_{d} r_{d} } \right)K_{0} \left( {\xi_{d} r_{d} } \right) + I_{0} \left( {\xi_{d} r_{d} } \right)K_{1} \left( {\xi_{d} r_{d} } \right)}}{{I_{0} \left( {\xi_{d} r_{d} } \right)K_{0} \left( {\xi_{d} r_{e} } \right) - I_{0} \left( {\xi_{d} r_{e} } \right)K_{0} \left( {\xi_{d} r_{d} } \right)}}} \\ \end{array} $$
(B9)
$$ \begin{array}{*{20}c} {\mu_{s1} = \frac{{\mu_{s11} \left( {1 + \frac{{q_{m} }}{{\eta_{b} T_{s} }}} \right) - \mu_{s12} \frac{{q_{m} }}{{\eta_{b} T_{s} }}}}{{\mu_{s14} }} - \frac{{K_{d} \xi_{d} {\text{ln}}\frac{{r_{e} }}{{r_{d} }}}}{{K_{s} \xi_{s} {\text{ln}}\frac{{r_{d} }}{{r_{s} }}}} \cdot \frac{{\mu_{s13} }}{{\mu_{s15} }}} \\ \end{array} $$
(B10)
$$ \begin{array}{*{20}c} {\mu_{s11} = I_{1} \left( {\xi_{s} r_{d} } \right)K_{0} \left( {\xi_{s} r_{s} } \right) + I_{0} \left( {\xi_{s} r_{s} } \right)K_{1} \left( {\xi_{s} r_{d} } \right)} \\ \end{array} $$
(B11)
$$ \begin{array}{*{20}c} {\mu_{s12} = I_{1} \left( {\xi_{s} r_{d} } \right)K_{0} \left( {\xi_{s} r_{d} } \right) + I_{0} \left( {\xi_{s} r_{d} } \right)K_{1} \left( {\xi_{s} r_{d} } \right)} \\ \end{array} $$
(B12)
$$ \mu_{s13} = I_{1} \left( {\xi_{d} r_{d} } \right)[K_{0} \left( {\xi_{d} r_{d} } \right) - K_{0} \left( {\xi_{d} r_{e} } \right)] + K_{1} \left( {\xi_{d} r_{d} } \right)\left[ {I_{0} \left( {\xi_{d} r_{d} } \right) - I_{0} \left( {\xi_{d} r_{e} } \right)} \right] $$
(B13)
$$ \begin{array}{*{20}c} {\mu_{s14} = I_{0} \left( {\xi_{s} r_{s} } \right)K_{0} \left( {\xi_{s} r_{d} } \right) - I_{0} \left( {\xi_{s} r_{d} } \right)K_{0} \left( {\xi_{s} r_{s} } \right)} \\ \end{array} $$
(B14)
$$ \begin{array}{*{20}c} {\mu_{s15} = I_{0} \left( {\xi_{d} r_{d} } \right)K_{0} \left( {\xi_{d} r_{e} } \right) - I_{0} \left( {\xi_{d} r_{e} } \right)K_{0} \left( {\xi_{d} r_{d} } \right)} \\ \end{array} $$
(B15)
$$ \begin{array}{*{20}c} {\mu_{d2} = \frac{{I_{1} \left( {\xi_{d} r_{e} } \right)K_{0} \left( {\xi_{d} r_{e} } \right) + I_{0} \left( {\xi_{d} r_{e} } \right)K_{1} \left( {\xi_{d} r_{e} } \right)}}{{I_{0} \left( {\xi_{d} r_{d} } \right)K_{0} \left( {\xi_{d} r_{e} } \right) - I_{0} \left( {\xi_{d} r_{e} } \right)K_{0} \left( {\xi_{d} r_{d} } \right)}}} \\ \end{array} $$
(B16)
$$ \begin{array}{*{20}c} {\mu_{e2} = - \frac{{I_{1} \left( {\xi_{d} r_{e} } \right)K_{0} \left( {\xi_{d} r_{d} } \right) + I_{0} \left( {\xi_{d} r_{e} } \right)K_{1} \left( {\xi_{d} r_{e} } \right)}}{{I_{0} \left( {\xi_{d} r_{d} } \right)K_{0} \left( {\xi_{d} r_{e} } \right) - I_{0} \left( {\xi_{d} r_{e} } \right)K_{0} \left( {\xi_{d} r_{d} } \right)}} + \mu_{c1} } \\ \end{array} $$
(B17)
$$ \begin{array}{*{20}c} {\mu_{c1} = \frac{{K_{e} }}{{K_{d} \xi_{d} r_{e} {\text{ln}}\frac{{r_{e} }}{{r_{d} }}}}} \\ \end{array} $$
(B18)
$$ \begin{array}{*{20}c} {\mu_{s2} = \frac{{K_{e} q_{m} r_{e} {\text{ln}}\frac{{r_{c} }}{{r_{e} }}}}{{2\lambda_{e} K_{d} \xi_{d} T_{s} {\text{ln}}\frac{{r_{e} }}{{r_{d} }}}} + \frac{{K_{e} q_{m} \left( {r_{c}^{2} - r_{e}^{2} } \right)}}{{4\lambda_{e} K_{d} \xi_{d} r_{e} T_{s} {\text{ln}}\frac{{r_{e} }}{{r_{d} }}}} - \left( {1 + \frac{{q_{m} }}{{\eta_{b} T_{s} }}} \right)\mu_{s21} } \\ \end{array} $$
(B19)
$$ \begin{array}{*{20}c} {\mu_{s21} = \frac{{I_{1} \left( {\xi_{d} r_{e} } \right)\left[ {K_{0} \left( {\xi_{d} r_{d} } \right) - K_{0} \left( {\xi_{d} r_{e} } \right) + K_{1} \left( {\xi_{d} r_{e} } \right)} \right]\left[ {I_{0} \left( {\xi_{d} r_{d} } \right) - I_{0} \left( {\xi_{d} r_{e} } \right)} \right]}}{{I_{0} \left( {\xi_{d} r_{d} } \right)K_{0} \left( {\xi_{d} r_{e} } \right) - I_{0} \left( {\xi_{d} r_{e} } \right)K_{0} \left( {\xi_{d} r_{d} } \right)}}} \\ \end{array} $$
(B20)
$$ \begin{array}{*{20}c} {\mu_{m} = \frac{{q_{m} }}{{4\lambda_{e} }}\left[ {r_{c}^{2} \left( {1 - 2{\text{ln}}\frac{{r_{c} }}{{r_{e} }}} \right) - r_{e}^{2} } \right]} \\ \end{array} $$
(B21)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, C., Zhang, A., Pang, D. et al. Energy Conversion Performance and Optimization of Wearable Annular Thermoelectric Generators. J. Electron. Mater. 52, 7325–7336 (2023). https://doi.org/10.1007/s11664-023-10636-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11664-023-10636-y

Keywords

Search

Navigation