Student Chapter
Meeting
Thursday, Sept. 3rd
7pm
ECJ 4.304
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Lecture Objectives:
• Review - Heat transfer
– Convection
– Conduction
– Radiation
Analysis of a practical problem
Example Problem –radiant
barrier in attic
Example Problem –heat transfer
in window construction
Convection
Convection coefficient – h [W/m2K]
Q  A  h  (Twall  Tair )  A  h  (Tw  T ) [W]
Heat flux
area
or
q  h  (Tw  T ) [W/m2]
Specific heat flux
Natural convection
Forced convection
T
T
Tw
Tw
L – characteristic length
Nusselt number:
hL
Nu 
k
Convection
Conduction
h – natural convection
k – air conduction
L- characteristic length
Which surface in this classroom
has the largest forced convection
A. Window
B. Ceiling
C. Walls
D. Floor
Which surface has the largest natural
convection
How to calculate h ?
What are the parametrs that affect h ?
What is the boundary layer ?
Laminar and Turbulent Flow
forced convection
Forced convection governing equations
1) Continuity
u v

 0
x y
2) Momentum
u
u
 2u
u v
v 2
x
y
y
u, v – velocities
n – air viscosity
Non-dimensionless momentum equation
Using
x*  x L ;
y*  y L ;
u*  u
U oo
v*  v
;
U oo
L = characteristic length and U0 = arbitrary reference velocity
*
*
2 *

u

u
1

u
*
*
u
v

 *2
*
*
U oo L y
x
y
n
ReL
Reynolds number
Forced convection
governing equations
Energy equation for boundary layer
2
T
T
 T
u
v
 
x
y
y 2

T –temperature,  – thermal diffusivity =k/rcp,
k-conductivity, r - density, cp –specific cap.
Non-dimensionless energy equations
T* 
T
1
T
* T
u
v

*
*
x
y
Re L . Pr . y *2
*
*
2
*
T  T
Tw  T
Air temperature outside
of boundary layer
*
Reynolds number
Re L 
UL
n
Inertial force
Viscous force
Wall temperature
Prandtl number
Pr 
n
a
Momentum diffusivity
Thermal diffusivity
Simplified Equation
for Forced convection
General equation
Nu  f (Re, Pr)
For laminar flow:
Nu  C L Re Pr 
For turbulent flow:
Nu  CT Re Pr 
1/ 3
Nu 
hL
k
Re L 
4/5
For air: Pr ≈ 0.7, n = viscosity is constant, k = conductivity is constant
Simplified equation:
h forced  f (U  , Ln )  C U m
m
Or:
h forced  C  ACH
m
ACH 
Volume flow rate
RoomVolume
UL
n
Natural convection
GOVERNING EQUATIONS
Natural convection
Continuity
u v

 0
x y
• Momentum which includes gravitational force
u
u
 2u
u v
 g T  T   v 2
x
y
y
• Energy
T
T
 2T
u
v
 
x
y
y 2
u, v – velocities , n – air viscosity , g – gravitation, ≈1/T - volumetric thermal expansion
T –temperature, T– air temperature out of boundary layer,  –temperature conductivity
Characteristic Number for Natural
Convection
Non-dimensionless governing equations
Using x*  x L ; y *  y L ; u *  u U ; v*  vU ;


T* 
T  T
Tw  T
L = characteristic length and U0 = arbitrary reference velocity Tw- wall temperature
The momentum equation become
*
u *
g Tw  T L *
1  2u *
* u
u
v

T 
 *2
*
*
2
x
y
U
Re L y
Gr
*
Multiplying by Re2 number Re=UL/n
g Tw  T L3
*
*
2 *

u

u

u
2
u * *  v* *  (Gr / Re L )  T *  (1 / Re L )  *2
x
y
y
n2
Grashof number
Characteristic Number for Natural Convection
Gr 
g Tw  T L3
n2
Buoyancy forces
Viscous forces
The Grashof number has a similar significance for natural convection
as the Reynolds number has for forced convection, i.e. it represents a
ratio of buoyancy to viscous forces.
General equation
Nu  f (Gr, Pr)
Natural convection
simplified equations
For laminar flow:
Nu  C L Gr  Pr 
For turbulent flow:
Nu  CT Gr  Pr
1/ 4
1/ 3
For air: Pr ≈ 0.7, n = constant, k= constant, = constant, g=constant
Simplified equation:
h forced  f ((Tw  T ) m , Ln )  f (T m , Ln )
Or:
Even more simple
h forced  C  T m
T∞ - air temperature outside of boundary layer, Ts - surface temperature
Forced and/or natural convection
In general,
Nu = f(Re, Pr, Gr)
GrL Re 2L  1  Nu  f (Re, Pr, Gr)
natural and forced convection
GrL Re 2L  1  Nu  f (Re, Pr)
forced convection
GrL Re 2L  1  Nu  f (Gr , Pr)
natural convection
Combined forced and natural
convention
Churchill and Usagi approach :
n
hcombined  (h1n  h2n )1/ n  (hnatural
 hnforced )1/ n
5
h
4
h
combined
3
n=2
n=3
2
h
n=6
1
1
h
2
T or ACH
0
0
1
2
3
4
This equation favors a dominant term (h1 or h2), and exponent coefficient ‘n’ determines
the value for hcombined when both terms have the same order of value
Example of general forced and
natural convection
Equation for convection at cooled ceiling surfaces
hnatural  (2.12  T 0.33 )3
h forced  1.19  ACH 0.8

hcombinbed  (2.12  T
n
)  (1.19  ACH )
0.33 3
0.8 3

1/ 3
What kind of flow is the most
common for indoor surfaces
A. Laminar
B. Turbulent
C. Transitional
D. Laminar, transitional, and turbulent
What about outdoor surfaces?
Conduction
Conductive heat transfer
k - conductivity
of material
• Steady-state
q  k / L(TS1  TS 2 )
• Unsteady-state
TS1
T
k  2T

 qsource
2

rc p x
L
• Boundary conditions
Tair
– Dirichlet
– Neumann
Tsurface = Tknown
T
 h(Tair  Tsurface)
x
TS2
h
Boundary conditions
Biot number
hL
Bi 
k solid
convention
conduction
Importance of analytical solution
1.0
T
  h
To
 h
  h
(T-Ts)/(To-Ts)
Ts
0
Analytical solution
Numerical -3 nodes, =60 min
Numerical -7 nodes, =60 min
Numerical -7 nodes, =12 min
0.8
0.7
T
-L/ 2
0.9
L/2
h omogenous
wall
L = 0.2 m
k = 0. 5 W/ mK
c p = 9 20 J/kgK
2
r = 120 0 k g/m
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
5
hour
6
7
8
9
10
What will be the daily temperature
distribution profile on internal surface
for styrofoam wall?
External temperature profile
T
time
A.
B.
What will be the daily temperature
distribution profile on internal surface
for tin glass?
External temperature profile
T
time
A.
B.
Conduction equation describes
accumulation
Important numbers
Nusselt number
Reynolds number
Prandtl number
Grashof number
Biot number
hL
Nu 
k
Re L 
Pr 
Gr 
Convection
Conduction
UoL
Inertial force
n
Viscous force
n
Momentum diffusivity
a
Thermal diffusivity
g Ts  T L3
hL
Bi 
k solid
n2
Buoyancy forces
Viscous forces
thermal internal resistance
surface film resistance
Reference book: Fundamentals of Heat and Mass Transfer, Incropera & DeWitt
Raiation
Radiation wavelength
Short-wave & long-wave radiation
• Short-wave – solar radiation
– <3mm
– Glass is transparent
– Does not depend on surface temperature
• Long-wave – surface or temperature radiation
– >3mm
– Glass is not transparent
– Depends on surface temperature
Radiation emission
The total energy emitted by a body,
regardless of the wavelengths, is given by:
Qemited  AT
4
Temperature always in K ! - absolute temperatures
 – emissivity of surface
– Stefan-Boltzmann constant
A - area
Surface properties
1
 r    

n
r


absorbed (α), transmitted (), and reflected (ρ) radiation
• Emission (   is same as Absorption (  ) for gray surfaces
• Gray surface: properties do not depend on wavelength
• Black surface:     
 Diffuse surface: emits and reflects in each direction equally
View (shape) factors
1
Fij 
Ai

cos  i  cos  j
Ai A j
l
2
dAi dA j
http://www.me.utexas.edu/~howell/
Ai Fij  A j F ji
For closed envelope – such as room
 Fij  1
j
n
Fi1  Fi 2  Fi 3  ...  Fin   Fij  1
j 1
i  1,2,..., n
View factor relations
A2
A3
A1
F11=0, F12=1/2
F22=0, F12=F21
F31=1/3, F13=1/3
A1=A2=A3
Radiative heat flux between two
surfaces
Q A, B 
 TA4  TB4 
1  A
1
1  B


 A AA AA FAB  B AB
Simplified equation for non-closed envelope

QA, B   A B FA, B AA TA4  TB4

Exact equations for closed envelope

Qi , j   i i , j Ai Ti 4  T j4

ψi,j - Radiative heat exchange factor
n
i, j  1,2,..., n
 i , j   j Fi , j   k , j 1   k Fi ,k
k 1
Summary
• Convection
– Boundary layer
– Laminar transient and turbulent flow
– Large number of equation for h for specific airflows
• Conduction
– Unsteady-state heat transfer
– Partial difference equation + boundary conditions
– Numerical methods for solving
• Radiation
–
–
–
–
Short-wave and long-wave
View factors
Simplified equation for external surfaces
System of equation for internal surfaces