Page 17 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

14 CHAPTER 1 Introduction

Separating the variables in Eq. (1.19), integrating from x = 0 and rearranging
gives:

Q = kA ∆T (1.21)
Q˙=kA∆TL L

Thermal resistance circuits: For steady one-dimensional flow with no generation
heat conduction equation, Eq. (1.21) can be rearranged as:

Q = ∆T = ∆T (1.22)
/
Q˙=∆TL/KA=∆TRcond LKA R cond
Conduction thermal resistance (R cond ) is represented by:
L
R = (1.23)
Rcond=LKA cond KA
It is obvious that the thermal resistance R cond increases as wall thickness (L)
increases, area (A) and K decreases. The concept of a thermal resistance circuit can
be used for problems such as composite wall thickness.
The heat transfer rate for composite wall is given by:

Q = ∆T = ∆T (1.24)
Q˙=∆T∑Rcond=∆TR +R 2 ∑ R cond R 1 + R 2
1
1.8.2.2 Convective heat transfer
Convection heat transfer is due to the moving fluid. The fluid can be a gas or a liquid;
both have applications in bio and nano heat transfer. Convection is the energy trans-
fer between two mediums; typically, a surface and fluid that moves over the surface.
In convective heat transfer heat is transferred by diffusion (conduction) and by bulk
fluid motion (advection). An example of convection heat transfer is the flow of blood
inside the human vessels or air and water flow over the human skin. In convective
heat transfer it is important to examine the fluid motion near the surface. Close to
wall there exists a thin layer called “boundary layer” where fluid experience velocity
and temperature differences. Boundary layer thickness depends on flow Reynolds
number, structure of the wall surface, pressure gradient and Mach number. Outside
this layer, temperature and velocity are uniform and identical to free stream tempera-
ture and velocity.

The rate of convection heat transfer (Q) from/to the surface is given by Newton’s
Q˙
Law of Cooling as [51]:
(
Q˙=hA(Tw−T∞) Q = hA T w − T ) (1.25)
∞
2
The quantity h (W/m K) is called convective heat transfer coefficient and T
w
and T are surface and fluid temperature, respectively. For many situations of prac-
∞
tical interest, the quantity h is known mainly through experiments. Integrating