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