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

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.

pC

txgT

t

T

),(2

General Conduction Equation

pC

tzyxg

z

T

y

T

x

T

t

T

):,,(

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

dt

dU :enegyin change of Rate

dt

drtrTrcd

dt

cTdVd

dt

dU

R

V

0

2 ),(4

At any instant:

econv

Rrcond

R

TtRThAdr

trdTkA

dt

drtrTrcd

),(

,),(4

0

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

drtrdT

kA

TtRThABi

,),(

solid

sticcharacteri

sticcharacteri

cond

conv

cond

conv

k

hL

LkAhA

RTT

kA

TThABi

0

0

conv

cond

conv

cond

sticcharacteri

sticcharacteri

cond

conv

R

R

hA

kAL

LkAhA

Bi 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 )(

)(

TtThAdt

cVTdconv )(

TtThAdt

dTcV conv )(

Rearranging:

dtcV

hA

TtT

dT conv

)(

Integrating to any time interval:

00)(

dtcV

hA

TtT

dT convT

T

cV

hA

TT

TT

0

ln

cV

hA

eTT

TT

0

Define, Thermal Time ConstanthA

cVc

ceTT

TT

0

TT

TTT

0

*

cV

thAtt

c *

Thermal Time Constant CVhAc

1

ththc CR

The total energy transferred in time t

surfacesolidsolid

sticcharacteri

Ak

hV

k

hLBi

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

t

CL

kt

TCV

tL

TkA

F

Hot Rolling of Steel Sheets

1solid

sticcharacteri

k

hLBi