one dimensional models for conduction heat transfer in manufacturing processes p m v subbarao...

One Dimensional Models for Conduction Heat Transfer in Manufacturing Processes

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Simplified Models for Complex Heat Transfer due to micro-molecular

Movements!!!

Steady-State One-Dimensional Conduction

• For conduction through a large wall the heat equation reduces to one dimensional Equation in Cartesian system .

• Assume a homogeneous medium with invariant thermal conductivity ( k = constant) :

),(2

2

txgx

Tk

t

TC p

One dimensional Transient conduction with heat generation.

21 CxCT

Heat Transfer in Metal Cutting

02

2

2

2

2

2

dz

Td

dy

Td

dx

Tdk

Axi-symmetric Conduction Models for Manufacturing Processes

),,,(1

2tzrg

z

Tk

z

Tk

rr

Tkr

rt

TC p

)(rgr

Tkr

rt

TC p

No changes in T along and z directions:

Steady state with no heat generation :

0

r

Tr

rk 0

r

TkA

rk

Axi-Symmetric Steady Conduction in Radial Systems

0

drdrdT

kAd

Homogeneous and constant property material

0

drdrdT

Ad

At any radial location the surface are for heat conduction in a solid cylinder is:

rlAcylinder 2

At any radial location the surface are for heat conduction in a solid sphere is:

24 rAsphere

The GDE for cylinder:0

drdrdT

rd

The GDE for sphere:0

2

drdrdT

rd

General Solution for Cylinder:

21 ln CrCrT

General Solution for Sphere:

r

CCrT 1

2

0dr

drdT

rd

Boundary Conditions

• No solution exists when r = 0. • Totally solid cylinder or Sphere have no physical

relevance!• Inner wall at finite radius is essential for steady state

conduction with no heat generation.

21 ln CrCrT r

CCrT 1

2

UNSTEADY-STATE HEAT CONDUCTION

Applications where rate/duration of heating/cooling is a Design Parameter……

Welding Process : How to decide the Rate of Welding?

Injection Molding Process

They need you to help them to 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.

Selective LASER Sintering

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 Cartesian Geometry:

• The general form of these equations in multi-dimensions for homogeneous material is:

Cooling/Quenching of Hot Processed Products

Transient-conduction

• 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.

•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.

First Law Analysis of Cooling /Quenching of Hot Object s

Rate of Change in energy of hot object= 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

cTdVd

dt

dU V

dt

drtrTrcd

dt

dU

R

0

2 ),(4For spherical objects

Total thermal resistance of the system: RtotT0

T

Rcond Rconv

TT0

At any instant:

econv

Rrcond

R

TtRThAdr

trdTkA

dt

drtrTrcd

),(

,),(4

0

2

•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

withinConductionby Object by Lost Heat

Object of Surface from removalheat Covection Bi

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 during time t

t

convdtQQ0

t

dtTThAQ0

t t

dteTThAQ c

0

0

t t

dteTThAQ c

0

0

Hot Rolling of Steel Sheets

1solid

sticcharacteri

k

hLBi