Betz revisited

Prof. Guy Houlsby Department of Engineering Science

Oxford Tidal Energy Workshopsponsored by the Oxford Martin School

30th March 2012

• Classical Lanchester-Betz theory– An oversight– An additional calculation– Resolving these problems

• Betz for an open channel

41

2

ut4 = a4u1u1

3

AT

ut2 = a2u1

41

2

ut4 = a4u1u1

3

AT

ut2 = a2u1

Equilib.Pressure

Bernouilli

Momentum

Bernouilli Bernouilli

41

2

ub4

ub4

ut4u1

3

AT Mixing

5

u5 = u1

A/B

41

2

ub4

ub4

ut4u1

3

AT Mixing

5

u5 = u1

A/B

Equilib.

Pressure

Momentum Momentum

Bernouilli

Bernouilli

Bernouilli

Continuity

242

42

1111

1

a

aa

BBB

42

24 1

1aa

a

BB

222

221 a

TTL

C

Au

TC

24231

1

21 aa

Au

PCP

aaa

aaa

4224

42242

31

111

21 B

B

Au

PC WPW

4Specify a

2424

212

1a

Au

TCT

Calculation

Results

PWP

PCCC

2

21 2

2

Au

TCTL

2716

21 3

1

Au

PCP

278

21 3

1

Au

PC WPW

98

21 2

1

Au

TCT

Optimal conditions as

32

0B

32

2 a 14 31

4 a

Mixing

41

2

5

u4b

h5u4b

u5u4t

h15

h1

u1

3

AT

h4

Mixing

41

2

5

u4b

h5u4b

u5u4t

h15

h1

u1

3

AT

h4

Hydrostatic

Continuity

HeadHead

Head

60

1

u6b

u6b

u6tu0

5

bT Mixing

7

u7

w

h0 h7

PLAN view of channel

60

1

u6b

u6b

u6tu0

5

bT Mixing

7

u7

w

h0 h7

Depth

Inner problem

Conclusions

• Anomalies in the original Betz calculation can be resolved by treating it as the limiting case of a confined flow

• Efficiency can be calculated

• An “inner and outer” calculation is proposed for flow in a channel