Intro to PLUS by Leta Moser and Kristen Cetin • PLUS accreditation• Peer-Led Undergraduate Studying (PLUS)
– assists students enrolled by offering class-specific, weekly study groups.
– Students can attend any study group at any point in the semester to review for an exam, discuss confusing concepts, or work through practice problems.
– http://www.utexas.edu/ugs/slc/support/plus
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]
khLNu
[W] )()( TThATThAQ wairwall
Conduction
Convection
Natural convection Forced convection
wT
T
L – characteristic length
wT
T
[W/m2] )( TThq w
h – natural convectionk – air conductionL- characteristic length
or
Nusselt number:
area Specific heat fluxHeat flux
Which surface in this classroom has the largest forced convection
A. Window B. CeilingC. WallsD. 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 Flowforced convection
Forced convection governing equations
v 2
2
yuv
yu
xuu
0 v
yxu
1) Continuity
2) Momentumu, v – velocities – air viscosity
oooo UUuuLyyLxx v v;; ; ****
2*
*2
*
**
*
** 1 v
yu
LUyu
xuu
oo
Non-dimensionless momentum equation
Using
L = characteristic length and U0 = arbitrary reference velocity
ReL Reynolds number
Forced convection governing equations
TTTTT
w
*
2
2 v
yT
yT
xTu
Energy equation for boundary layer
Non-dimensionless energy equations
2*
*2
*
**
*
**
.Pr.Re1 v
yT
yT
xTu
L
T –temperature, – thermal diffusivity =k/cp,
k-conductivity, - density, cp –specific cap.
Wall temperature
Air temperature outside of boundary layer
LU
LRe Inertial force
Viscous force a
Pr Momentum diffusivity
Thermal diffusivity
Reynolds number Prandtl number
Simplified Equation for Forced convection
Pr) (Re, fNu
LU
LRe 3/1PrRe LCNu
5/4PrRe TCNu
For laminar flow:
For turbulent flow:
For air: Pr ≈ 0.7, = viscosity is constant, k = conductivity is constant
khLNu
General equation
mnmforced UCLUfh ),(
Simplified equation:
mforced ACHCh
Or:
RoomVolumeACH rate flow Volume
Natural convection
GOVERNING EQUATIONSNatural convection
Continuity
• Momentum which includes gravitational force
• Energy
v 2
2
yuvTTg
yu
xuu
0 v
yxu
2
2 v
y
TyT
xT
u
u, v – velocities , – air viscosity , g – gravitation, ≈1/T - volumetric thermal expansion T –temperature, – air temperature out of boundary layer, –temperature conductivity T
Characteristic Number for Natural Convection
TTTTTUU
uuLyyLxxw
***** ;v v;; ;
2*
*2*
2*
**
*
**
Re1 v
yuT
ULTTwg
yu
xuu
L
Non-dimensionless governing equations
Using
L = characteristic length and U0 = arbitrary reference velocity Tw- wall temperature
The momentum equation become
2
3
LTTg w
Multiplying by Re2 number Re=UL/
Gr
2*
*2*2
*
**
*
** )Re/1()Re/( v
yuTGr
yu
xuu LL
Grashof number Characteristic Number for Natural Convection
2
3
LTTwgGr
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.
Buoyancy forces
Viscous forces
Pr) ,( GrfNu
General equation
Even more simple
Natural convection simplified equations
4/1Pr GrCNu L
3/1Pr GrCNu T
For laminar flow:
For turbulent flow:
For air: Pr ≈ 0.7, = constant, k= constant, = constant, g=constant
),(),)(( nmnmforced LTfLTTwfh
Simplified equation:
mforced TCh
Or:
T∞ - air temperature outside of boundary layer, Ts - surface temperature
Forced and/or natural convection
Gr) Pr, (Re, 1Re2 fNuGr LL
Pr) (Re, 1Re2 fNuGr LL
Pr) ,( 1Re2 GrfNuGr LL
In general, Nu = f(Re, Pr, Gr)
natural and forced convection
forced convection
natural convection
Combined forced and natural convention
nnforced
nnatural
nnncombined hhhhh /1/1
21 )()(
0 1 2 3 40
1
2
3
4
5h
T or ACH
n=2
n=3
n=6
h2
h1
hcombined
Churchill and Usagi approach :
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
8.019.1 ACHh forced
3/138.0333.0 )19.1()12.2( ACHThcombinbed
333.0 )12.2( Thnatural
Equation for convection at cooled ceiling surfaces
n
What kind of flow is the most common for indoor surfaces
A. Laminar B. TurbulentC. TransitionalD. Laminar, transitional, and turbulent
What about outdoor surfaces?
Conduction
Conductive heat transfer• Steady-state
• Unsteady-state
• Boundary conditions
– Dirichlet Tsurface = Tknown
– Neumann
)(/ 21 SS TTLkq
sourcep
qxT
ckT
2
2
)( surfaceair TThxT
L
Tair
k - conductivity of material
TS1 TS2
h
Boundary conditions
Biot number
solidkhLBi
convention
conduction
0 1 2 3 4 5 6 7 8 9 100.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Analytical solution Numerical -3 nodes, =60 min Numerical -7 nodes, =60 min Numerical -7 nodes, =12 min
(T-T
s)/(T
o-Ts
)
hour
Ts
0
T
-L / 2 L/2
h
h
h
T o
T
h omogenous wa ll
L = 0.2 mk = 0 . 5 W/ m Kc = 9 20 J/kgK
= 120 0 k g/mp
2
Importance of analytical solution
What will be the daily temperature distribution profile on internal surface
for styrofoam wall? A.
B.
External temperature profile
T
time
What will be the daily temperature distribution profile on internal surface
for tin glass? A.
B.
External temperature profile
T
time
Conduction equation describes accumulation
Important numbers
LUo
L Re Inertial force
Viscous forceReynolds number
a
Pr Momentum diffusivity
Thermal diffusivityPrandtl number
2
3
LTTsgGr
Buoyancy forcesViscous forces
khLNu
Conduction
Convection Nusselt number
solidkhLBi thermal internal resistance
surface film resistance
Grashof number
Biot number
Reference book: Fundamentals of Heat and Mass Transfer, Incropera & DeWitt