Biomedical Engineering Reference
In-Depth Information
interactions the system experiences with its environment. This
relationship is expressed as the first law of thermodynamics, the
conservation of energy:
Although there are a large number of energy storage mech-
anisms in various materials, those that are likely to be most
relevant to processes encountered in biomedical applications
include:
mechanical
, related to velocity (kinetic), relative position
in the gravity field (potential), and elastic stress;
sensible
, related
to a change in temperature; and
latent
, related to a change in
phase or molecular reconfiguration such as denaturation. Thus,
dE
dt
∑
∑
(
)
=
QW
−
+
mh
−
hQ
+
(1.1)
in
out
gen
where
E
is the energy of the system;
Σ
Q
is the sum of all heat
flows, taken as positive into the system;
W
is the rate at which
work is performed on the environment;
E
=
KE
+
PE
+
SE
+
U
+
L
(1. 2)
where substitution of a constitutive relation for each term
yields
Σ−
in out
is the sum
of all mass flows crossing the system boundary, with each having
a defined enthalpy,
h
, as it enters or leaves the system; and
Q
gen
is
the rate at which energy generation and dissipation occur on the
interior of the system. These terms are illustrated in Figure 1.1,
depicting how the energy interactions with the environment
affect the system energy. In this case the system is represented on
a macroscopic scale, but there are alternative situations in which
it is of advantage to define the boundary as having microscopic
differential scale dimensions.
For the special case of a steady state process, all properties
of the system are constant in time, including the energy, and
the time derivative on the left side of Equation 1.1 is zero. For
these conditions, the net effects of all boundary interactions are
balanced.
Each term in Equation 1.1 may be expressed in terms of a spe-
cific constitutive relation, which describes the particular energy
flow as a function of the system temperature, difference between
the system and environmental temperatures, and/or spatial tem-
perature gradients associated with the process as well as many
thermal properties of the system and environment. When the
constitutive relations are substituted for the individual terms
in the conservation of energy (Equation 1.1), the result is a par-
tial differential equation that can be solved for the temperature
within the system during a heat transfer process as a function of
position and time. There are well-known solutions for many of
the classical problems of heat transfer (Carslaw and Jaeger 1959),
but numerous biomedical problems involve nonlinearities that
require a numerical solution method.
Development of the specific equations for the various consti-
tutive relations constitutes a major component of heat transfer
analysis. We will review these relations briefly in the following
sections. The one constitutive equation we will discuss here is
that for system energy storage.
mh
(
h
)
1
2
1
2
2
,
PE
=
mgz
,
=κ
2
,
Umc
p
=
KE
mV
SE
x
,
L
=
m
Λ (1. 3)
with properties defined as:
KE
is kinetic energy;
V
is velocity;
PE
is potential energy;
g
is the acceleration of gravity;
z
is position
along the gravity field;
SE
is the elastic energy; κ is the spring
constant;
x
is the elastic deformation;
U
is the internal energy;
c
p
is the specific heat;
T
is the temperature;
L
is the latent energy;
and Λ is the latent heat. The most commonly encountered mode
of energy storage is via temperature change.
1.2.2 Conduction Heat transfer
Energy can be transmitted through materials via conduc-
tion under the action of an internal temperature gradient.
Conduction occurs in all phases of material: solid, liquid, and
gas, although the effectiveness of the different phases in trans-
mitting thermal energy can vary dramatically as a function of
the freedom of their molecules to interact with nearest neigh-
bors. The conductivity and temperature of a material are key
parameters used to describe the process by which a material may
be engaged in heat conduction.
The fundamental constitutive expression that describes the
conduction of heat is called Fourier's law:
=
QkA
dT
dr
(1.4)
cond
where
r
is a coordinate along which a temperature gradi-
ent exists, and
A
is the area normal to the gradient and the
cross section through which the heat flows. The negative sign
accounts for the fact that heat must flow along a negative gradi-
ent from a higher to a lower temperature. This phenomenon is
described by the second law of thermodynamics and is illus-
trated in Figure 1.2.
For a process in which the only mechanism of heat transfer is
via conduction, a microscopic system may be defined as shown
in Figure 1.3. Equation 1.4 may be applied to the conservation
of energy (Equation 1.1) to obtain a partial differential equa-
tion for the temporal and spatial variations in temperature.
A microscopic system of dimensions
dx
,
dy
, and
dz
is defined
in the interior of the tissue as shown. The various properties
and boundary flows illustrated represent the individual terms
W
E
(
t
)
m
out
m
in
Q
gen
Q
conv
Q
conv
Q
rad
FIGURE 1.1
A thermodynamic system that interacts with its envi-
ronment across its boundary by flows of heat, mass, and work that con-
tribute to altering the stored internal energy.
