unsteady-state heat conduction
DESCRIPTION
UNSTEADY-STATE HEAT CONDUCTION. P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi. Applications where rate/duration of heating/cooling is a Design Parameter……. Measurement of Gas temperature. T 1. T 2 > T 1. T 2. When to record the Gas temperature?. - PowerPoint PPT PresentationTRANSCRIPT
UNSTEADY-STATE HEAT CONDUCTION
P M V SubbaraoAssociate Professor
Mechanical Engineering DepartmentIIT Delhi
Applications where rate/duration of heating/cooling is a Design Parameter……
Measurement of Gas temperature
T1
T2
T2 > T1
When to record the Gas temperature?
Welding Process : How to decide the Rate of Welding?
Injection Molding Process
They need you to help them estimate the time required for a part in a mold to cool to an acceptable temperature for removal.
Resin Transfer Molding Process
(1) Insert fiber preform (2) Close mold(3) Inject mixed resin/catalyst(4) Part solidifies via reaction(5) Open mold(6) Remove part
The cycle time for step (4) is approximately the same 25 minutes for parts of all sizes made by AeroForm. All other molding operations (1), (2), (3), (5) and (6) take a total of 5 minutes.
pCtxgT
tT
),(2
General Conduction Equation
pCtzyxg
zT
yT
xT
tT
):,,(
2
2
2
2
2
2
For Rectangular Geometry:
• The general form of these equations in multidimensions is:
Transient-conduction• Temperature will vary with location
within a system and with time.• Temperature and rate heat transfer
variation of a system are dependent on its internal resistance and surface resistance.
• If we have a ball with initial temperature of T0 and it is left in fluid at Te.
• Heat is transferred by convection at the surface.
• As the surface temperature decreases, heat is transferred from the center of the ball to the surface, then to the fluid.
Thermal Analysis of Cooling of Ball
Rate of Change in energy of metal ball = Rate of Heat transfer
Rate of Heat Transfer = Rate of Convection by fluid =Rate of Conduction transfer in the metal ball
dtdU :enegyin change of Rate
dt
drtrTrcd
dt
cTdVd
dtdU
R
V
0
2 ),(4
At any instant:
econv
Rrcond
R
TtRThAdr
trdTkAdt
drtrTrcd
),(,
),(40
2
•Now, if the system itself is copper or the volume is small, the temperature response within the slab is considerably different from that if it is glass or the volume is large.
•The response has to do with what is called the internal resistance of the material. •Further, if the convection coefficient is very high, then the surface temperature almost becomes identical to the fluid temperature quickly. •Alternatively, for a low convection coefficient a large temperature difference exists between the surface and the fluid. •The value of the convection coefficient controls what is known as the surface resistance to heat transfer.•Thus, the temperature variation within the system is dependent on the internal and surface resistances. •The larger internal resistance or the smaller surface resistance, the larger temperature variation within the system, and vice versa.
Total thermal resistance of the system: Rtot
T0 T
Rcond Rconv
TT0
•A Biot number is defined as:
Rrcond
conv
drtrdTkA
TtRThABi
,),(
solid
sticcharacteri
sticcharacteri
cond
conv
cond
conv
khL
LkAhA
RTTkA
TThABi
0
0
conv
cond
conv
cond
sticcharacteri
sticcharacteri
cond
conv
RR
hA
kAL
LkAhABi 1
Relationship between the Biot number and the temperature profile.
System with negligible internal resistance
For this case Bi 1.0.and the temperature profile within the body is quite uniform.
The rate of change in internal energy of the body is equal to the rate of heat taken away from the surface by convection:
TtThAdt
dVtcTd
convV )(
)(
TtThAdtcVTd
conv )(
TtThAdtdTcV conv )(
Rearranging:
dtcV
hATtT
dT conv
)(
Integrating to any time interval:
00)(
dtcV
hATtT
dT convT
T
cVhA
TTTT
0
ln
cVhA
eTTTT
0
Define, Thermal Time Constant hAcV
c
ceTTTT
0
TTTTT
0
*
cVthAtt
c *
Thermal Time Constant CVhAc
1
ththc CR
The total energy transferred in time t
surfacesolidsolid
sticcharacteri
AkhV
khLBi
The Fourier number (Fo) or Fourier modulus, named after Joseph Fourier, is a dimensionless number that characterizes transient heat conduction. Conceptually, it is the ratio of the heat conduction rate to the rate of thermal energy storage. It is defined as:
22sticcharacteristiccharacteri
sticcharacterio L
tCL
ktTCV
tL
TkA
F
Hot Rolling of Steel Sheets
1solid
sticcharacteri
khLBi