Microsystems for Bioelectronics

2.5 SUMMARY

In this chapter upper bounds for energy and power of nanomorphic implementation of different energy

sources were derived based on fundamental physics of operation and assuming ideal conditions.

Results of the derivations in this chapter are summarized in Table 2.5.1. It can be seen that the

electrochemical galvanic cell provides the best combination of stored energy and power delivery

among the options considered. In subsequent chapters the galvanic cell power and energy estimates

will be used as a reference for available energy and power.

APPENDIX: A KINETIC MODEL TO ASSESS THE LIMITS OF HEAT REMOVAL

A simple approach to represent heat transfer in a body is to consider the transfer of energy that occurs

when two masses collide. In Figure A1, energy is transferred from moving Ball 1 with mass m

1

to

stationary Ball 2 with mass m

2

.

Table 2.5.1 Energy and power of nanomorphic implementation of different energy sources

Stored energy, J Power, W

Galvanic cell 10

–5

10

–6

Supercapacitor 10

–7

1

Radioisotopes 10

–5

10

–14

(Bio) Fuel cell Sustainable 10

–8

Solar Sustainable 10

–7

Laser Sustainable 10

–7

RF Sustainable 10

–8

/10

–12

Vibration Sustainable 10

–12

Thermal Sustainable 10

–8

0.0

0.5

1.0

1.5

2.0

2.5

0 1020304050

Q, W/cm

2

u

0

,

m/s

FIGURE 2.4.8

Maximum heat removal rate for forced air cooling (T

a

¼ 300 K) of a Si surface (T

h

¼ 305 K)

Appendix: A kinetic model to assess the limits of heat removal 43

The energy transfer in this system can be calculated from the momentum and energy conservation:

m

1

v

!

þ m

2

u

!

¼ m

1

V

!

þ m

2

U

!

m

1

v

!

2

2

þ

m

2

u

!

2

2

¼

m

1

V

!

2

2

þ

m

2

U

!

2

2

(A1)

where v and u are the corresponding velocities of Balls 1 and 2 before collision, while V and U are the

corresponding velocities aft er collision.

Solution of (A1) for V and U is:

V ¼ v

m

1

 m

2

m

1

þ m

2

þ u

2m

2

m

1

þ m

2

U ¼ v

2m

1

m

1

þ m

2

þ u

m

2

 m

1

m

2

þ m

1

(A2)

Let the energy of Ball 1 before the collision be E

1b

and after the collision E

1a

. Similarly, the energy of

Ball 2 before and after collision is correspondingly E

2b

and E

2a

.

Energy change in Ball 1 as a result of the collision is:

DE

1

¼ E

1b

 E

1a

¼

m

1

2

ðv

2

 V

2

Þ¼

m

1

2



v

2

4m

1

m

2

ðm

1

þ m

2

Þ

2

 u

2

4m

2

2

ðm

1

þ m

2

Þ

2

 uv

4m

2

ðm

1

 m

2

Þ

ðm

1

þ m

2

Þ

2



¼

4m

1

m

2

ðm

1

þ m

2

Þ

2



v

2

m

1

2

 u

2

m

2

2



 uv

2m

1

m

2

ðm

1

 m

2

Þ

ðm

1

þ m

2

Þ

2

(A3)

or

DE ¼

4m

1

m

2

ðm

1

þ m

2

Þ

2

ðE

1b

 E

2b

Þuv

2m

1

m

2

ðm

1

 m

2

Þ

ðm

1

þ m

2

Þ

2

(A4)

m

1

m

2

Before:

After :

2

2

1

1

v m

E

v

b

2

2

2

2

U m

E

U

a

2

2

1

1

V m

E

V

a

2

2

2

2

u m

E

u

b

FIGURE A1

Two colliding balls of arbitrary masses and initial velocities

44 CHAPTER 2 Energy in the small: Integrated micro-scale energy sources