lecture #23: internal flows. 1 cell cellular sheet cellular bilayer bilayered canister ecto- derm...
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Lecture #23: Internal Flows
1 cell cellular sheetcellular bilayer
bilayered canister
ecto-derm
endo-derm
one way gut
mouthanus
cephalization
mesoderm
Body Plan Evolution
heart
lung/gill
body intestine
Basic circulatory circuit
convectionIn dedicatedplumbing
diffusionin dedicatedexchangers
Convection vs. Diffusionx
C1 C2S
x
CCDS
time
massJ 12
Fick’s Law:
C= concentration in mass/volumeD = diffusion coefficientUnits = L2/T
Basic strategy of circulatory systems:Pluming uses bulk flow (convection) to move fluids to capillary beds where diffusioncan take place over short distances.
Relative importance of bulk flow to diffusion given by Peclet number:
D
ulPe
Problems with gas exchange:
consider simple gas exchanger:
water
bloodDIFFUSION
x
CCDSJ 12
distance
equilibriumpartial
pressure(02)
drivingforce
convection
Problems with gas exchange:
consider countercurrent gas exchanger:
water
blood
x
CCDSJ 12
distance
partialpressure
(02)
drivingforce
What about lungs? x
CCDSJ 12
distance
partialpressure
(02)
blood
air
Birds have more efficient system
Birds have more efficient system
What determines flow in pipes?
x
r
aP1 P2
L
If Re < 2000 (i.e. laminar flow):
4)(
)(22 ra
L
Prux
• flow ~ pressure gradient• flow ~ 1 / viscosity• parabolic flow distribution
4)(
max2a
L
Pux
What is maximum flow velocity?
At center of pipe, r=0:
8)(
4
42
21
2 a
L
Pa
a
L
PQ
What determines flux through pipe?
Flux (Q) = velocity x area:
= Hagen-Poiseuille equation
Flux through a system:• proportional to pressure gradient• inversely proportional to viscosity• has fourth order dependence on diameter
distance
pre
ssu
reflo
w velo
city
heart lung intestine body
heartlung
intestinebody
Pressureis lost (drops)across networkof pipes.
10% of our totalmetabolic cost!
5% of our totalWeight in blood!
Problems with blood
Blood is not a ‘Newtonian’ fluid,Mostly because of red blood cells.
4
8
a
L
Q
P
From Hagen-Poiseuille Equation:
‘Resistance’
Blood is very viscous due to red blood cells
% hematocrit
visc
osity
carrying capacity
02 carried/unit cost
optimumat 58%
Thoughts about plumbing:
Consider simple branch point:
S0
S1
S1
If S1 = 2 S2 then velocity is same in all branches; flux is ½ the original value.
a0a2
If a0 = 2 a1 then 16 times thepressure is required in small pipe for same flux!
Consider change in diameter:
Circulatory systems cannot compensate with large trunks –Blood volume would become too large.
Murray’s Law: what is geometry of branching network?
1) Cost to pump = Q x pressure gradient, or
4
28
a
L
P
2) Cost to make new pipe2aM
Total cost 2
4
28aM
a
Q
3) Find optimum as a function of diameter:
)8
( 24
2
aMa
Qa da
dopt
6/13/1 )
16(M
Qaopt
3/1~Qaopt3kaQif then
a0
a1
a2
32
31
30 aaa
a.k.a. Murray’s Law
01
01
01
26.0
63.0
79.0
uu
SS
aa
For simple symmetrical branching case:
Mass flux ~ cube ofvessel diameter
But, by law of continuity,
Q0
Q1
Q2
210 QQQ thus
333
32
31
30 ... naaaaa
More generally……
How does a growing vascular network ‘know’ to follow Murray’s Law?
x
r
adu/dr
Shear stress at wall, = du/dr
It can be shown that:
3
4
r
Q
But by Murray’s Law:
3kaQ
So with r = a (at wall):
k4
Thus, shear stress at wall is constant in network obeying Murray’s Law.Algorithm could be: ‘Grow vessel until shear stress reaches certain value.’