CHAPTER 1
BASIC CONCEPTS
1.1 Convection Heat Transfer
x Examine thermal interaction between a surface and an adjacent moving fluid.
1.2 Important Factors in Convection Heat Transfer
x Three factors play major roles in convection heat transfer:
(1) Fluid motion
(2) Fluid nature
(3) Surface geometry.
1.3 Focal Point in Convection Heat Transfer
x The determination of temperature distribution in a moving fluid can be expressed as
T
T ( x, y , z , t )
(1.1)
1.4 The Continuum and Thermodynamic Equilibrium Concepts
x Continuum model:
x Material is composed of continuous matter.
x Behavior of individual molecules is ignored.
x Valid when molecular man free path O is small relative to the characteristic
dimension of the system (small Knudsen number, Kn 10 1 ).
x Knudson number: defined as
Kn
O
De
(1.2)
x Thermodynamic equilibrium:
x Depends molecular interaction with an adjacent surface.
x At thermodynamic equilibrium:
(1) Fluid and adjacent surface have the same velocity (no velocity slip).
(2) Fluid and adjacent surface have the same temperature (no temperature jump).
x Valid for
Kn 10 3
1.5 Fourier’s Law of Conduction
x Gives a relationship between heat transfer rate or heat flux and temperature:
(1.3b)
2
q cxc
k
wT
,
wx
q cyc
k
wT
,
wy
q czc
k
wT
wz
(1.8)
x Used in the formulation of the governing equation for temperature distribution.
1.6 Newton's Law of Cooling
x This law defines the heat transfer coefficient h in terms of heat flux and surface
temperature
q csc
h (Ts Tf )
(1.9)
x Used in the formulation of boundary conditions.
x Used to determine the rate of transfer or heat flux when temperature distribution is
difficult to obtain analytically.
1.7 The Heat Transfer Coefficient h
x Defined in (1.9).
x h is not a property. It depends on:
h f (geometry, fluid motion, fluid properties, 'T )
(1.10)
x It is determined analytically using (1.8) and (1.9). This requires the determination of
temperature distribution.
h
wT ( x,0, z )
wy
k
Ts Tf (1.12)
x It is determined experimentally using Newton’s law of cooling (1.9).
x Table 1.1 gives typical values for h. Use this table as a guide only.
1.8 Radiation: Stefan-Boltzmann Law
x Emissivity H : radiation surface property
x Absorptivity D: surface property defined as the fraction of radiation energy incident
on a surface which is absorbed by the surface
x q12 is the net heat exchanged by radiation between two bodies.
x Special case: a small surface which is completely enclosed by a much larger surface,
q12 is given by Stefan-Boltzmann radiation law
q12
H1V A1 (T14 T24 )
(1.14)
3
x Subscript 1 in (1.14) refers to the small surface. Temperature is in absolute
degrees.
1.9 Differential Formulation of Basic Laws
x Various formulation procedures:
x
x
x
x
Differential
Integral
Variational
Finite difference
x Differential formulation:
x Based on the assumption of continuum.
x The basic laws are applied to an infinitesimal element.
x The result is a partial differential equation which is valid at every point.
1.10 Mathematical Background
&
(a) Velocity Vector V
&
V
u i v j wk
(1.15a)
(b) Velocity Derivative
&
wV
wx
wu
wv
ww
j
i
k
wx
wx
wx
(1.15b)
(c) The Operator ’
x Cartesian:
’{
w
w
w
i
j k
wx
wy
wz
(1.16)
x Cylindrical:
w
w
1 w
iT iz
ir wr
r wT
wz
(1.17)
1 w
1
w
w
ir iT iI
wr
r wT
r sinT wI
(1.18)
’{
x Spherical:
’{
(d) Divergence of a Vector
4
&
&
div.V { ’ ˜ V
wu wv ww
wx wy wz
(1.19)
(e) Derivative of the Divergence
&
w
’ ˜V
wx
w § wu wv ww ·
¨ ¸
wx ¨© wx wy wz ¸¹
or
&
w
’ ˜V
wx
&
wV
’˜
wx
(1.20)
(1.21)
(f) Gradient of Scalar. The gradient of a scalar, such as temperature T, is a vector given by
wT
wT
wT
i
j
k
wx
wy
wz
Grad T { ’ ˜ T
(1.22)
(g) Total Differential and Total Derivative
x Consider the variable f:
f
x Total differential of f:
and t:
f ( x, y , z , t )
This is the total change in f resulting from changes in x, y, z
wf
wf
wf
wf
dx dy dz dt
wx
wy
wz
wt
df
x Total derivative
df
dt
Df
Dt
u
wf
wf
wf wf
v
w wx
wy
wz wt
(1.23)
where
u
x Example: f
wf
wf
wf
v
w
convective derivative
wx
wy
wz
wf
local derivative
wt
u in (1.23) gives
du
dt
Du
Dt
u
wu
wu wu
wu
v
w wx
wy
wz wt
(1.24)
x Cylindrical coordinates:
dv r
dt
Dv r
Dt
vr
wv r v T wv r vT2
wv
wv
vz r r
wr
r
wz
r wT
wt
(1.25a)
5
dv T
dt
Dv T
Dt
dv z
dt
vr
Dv z
Dt
wv T v T wv T v r v T
wv
wv
vz T T
wr
wz
r wT
r
wt
(1.25b)
wv z v T wv z
wv
wv
vz z z
wr
wz
r wT
wt
(1.25c)
vr
x Example: f = T
dT
dt
DT
Dt
u
wT
wT
wT wT
v
w
wx
wy
wz wt
(1.26)
1.11 Units
x Basic SI units:
Length (L): meter (m)
Time (t): second (s)
Mass (m): kilogram (kg)
Temperature (T): kelvin (K)
x Derived units:
Force (newton): N = kg-m /s2
Energy (joules): J = N u m = kg-m2 /s2
Power (watts): W = J/s = N u m/s = kg-m2 /s3
1.12 Problem Solving Format
Convection problems lend themselves to a systematic solution procedure. Use the following
format when solving problems. Read details of each step and take advantage of illustrative
examples and posted solutions to homework problems to develop skills in using this
problem solving methodology. The following is an outline of this method:
(1) Observations
(2) Problem Definition
(3) Solution Plan
(4) Plan Execution
(i) Assumptions
(ii) Analysis
(iii) Computations
(iv) Checking
(5) Comments