Page 17 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 3

As the interactions with the environment are part of the state of the system and its
evolution, it is important to recognize whether the boundary is crossed by mass flow
or permeable.An impermeable boundary defines a closed system. Open systems, or flow
systems, are those systems whose boundaries are permeable, that is, crossed by mass
fluxes.

1.3 First law analysis: energy, heat, and work interactions

The first law of thermodynamics introduces the energy of the system, E [J], an exten-
sive quantity, dependent on the amount of substance, as thermodynamic property, the
heat, Q [J], and the work, W [J], as interactions. It states that when the system evolves
from an initial state, ( ) 1 , to the final state, ( ) 2 , the change in the energy of the sys-
tem is a measure of the interactions that it undergoes

Q 1-2 2 W 1-2 5 E 2 2 E 1 : ð1:1Þ

The system here is a control volume and the signs suggest that the system receives
heat and executes work with respect to its environment. For a process that charac-
terizes the transition between two close states with small interactions, the first law
Eq. (1.1) is written as follows:
δQ 2 δW 5 dE; ð1:2Þ

where d( ) is the total, exact differential operator and δ( ) denotes a small variation.
Using Eq. (1.2) , the per-unit-time basis form of Eq. (1.3) becomes

dE
_
Q 2 _ W 5 ; ð1:3Þ
dt
_
_
where W [W] is power (work transfer rate), and Q [W] is heat transfer rate (thermal
power rate).
The change in energy, E 2 E 1 , distinguishes between the macroscopically discern-
ible forms of energy storage and a form of energy storage, that is, unidentifiable micro-
scopically, denoted by U [J], which is called for this reason as internal energy, and

E 2 2 E 1 5 U 2 2 U 1 1 1 2 2 1 mg z 2 2 z 1 Þ 1 E 2 2E 1 Þ : ð1:4Þ
mv 2 v
ð
ð
2 2 1 i
|fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl}
Energy Internal |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} Potential Other forms of
energy energy storage
Eq. (1.4) singles out several forms of energy: internal, kinetic, potential, and other
macroscopic forms of energy storage, E 2 2E 1 Þ , which may include electric energy,
ð
i
magnetic energy, chemical energy, and so on. Divided through the volume of the