Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 78

ð
V ð

2 ð

5 @
@

Þ 5 @
@

0 u gz Þρ dV

1 ð

u gz p νÞð 2 ρ vA ð u gz Þ m u gz p νÞð _ m Þ

1 1 1 1 1 1

t t

5 @
@ t ð

um gzm Þ 2 ð u gz p νÞð _ m Þ

1 1 1

5 @
@

Þ 1 @
@

u @

m
@

gz @

m
@

ν @

m
@

t ð

um

t ð

gzm Þ 2 p

t 2 t 2 t

m @

u

u @

m
@

m @

gz
@

gz @

m
@

u @

m
@

gz @

m
@

ν @

m
@

m @

u

ν @

m
@

5

t 1 t 1 t 1 t 2 t 2 t 2

p

t 5 t 2

p

@ @ t

m
p

@ u

@

m _

5

ν t

Because we are making the assumption that air will behave like an ideal gas, we can make
the following substitutions:

m

5 ρ

V

p ν 5 RT

@ u

C v dT
dt

t 5

@

to get

m
p

@ u

@ t 5 ρ RT C v dT

V

R 2 T 2 C v dT

pV

m _

5

dt 5 ν dt

3 : 98 atm 6L

m ð 717 Nm = kgk Þð 0 : 0001K = s Þ 5 3 : 24 E 2 4g = min

5

2 2

ð 287 Nm = kgk Þ ðð 37 273 Þ K Þ

1

This is consistent with normal breathing.

The second law of thermodynamics is a statement about the disorder of a system. It
states that the change of entropy of a system is greater than or equal to the amount of heat
added to the system at a particular temperature:

dQ
T

dS ð 3 : 49 Þ

$

The time rate of change of entropy can therefore be defined as

Q
T

dS
dt $

ð 3 : 50 Þ

for one specific volume of interest. When developing Equation 3.22 , we stated that for
energy conservation the system property was S and the inherent property was s (entropy