ETHT

Conduction Montu Faldu 140080125005Haresh Gajera 140080125006Vishal Gajjar 140080125007Sarthak Gokani 140080125008

Guided by : Dr. Manish Maheta

Many heat transfer problems require the understanding of the complete time history of the temperature variation. For example, in metallurgy, the heat treating process can be controlled to directly affect the characteristics of the processed materials. Annealing (slow cool) can soften metals and improve ductility. On the other hand, quenching (rapid cool) can harden the strain boundary and increase strength. In order to characterize this transient behavior, the full unsteady equation is needed:

2 21, or

kwhere = is the thermal diffusivityc

T Tc k T Tt t

Transient heat transfer with no internal resistance: Lumped Parameter Analysis

Solid

Valid for Bi<0.1

Total Resistance= Rexternal + Rinternal

GE: dTdt

hAmcp

T T BC: T t 0 Ti

Solution: let T T, therefore

ddt

hA

mcp

Lumped Parameter Analysis

ln

hA

mct

i

tmchA

i

pi

ii

p

p

eTTTT

e

tmchA

TT

Note: Temperature function only of time and not of space!

- To determine the temperature at a given time, or- To determine the time required for the temperature to reach a specified value.

)exp(T0

tcVhA

TTTT

tL

BitLLc

kk

hLtcVhA

ccc

c2

11

ck

Thermal diffusivity: (m² s-1)

Lumped Parameter Analysis

Lumped Parameter Analysis

tL

Foc

2

khLBi C

T = exp(-Bi*Fo)

Define Fo as the Fourier number (dimensionless time)

and Biot number

The temperature variation can be expressed as

thickness2La with wallaplane is solid the when) thickness(half cL

sphere is solid the whenradius) third-one(3cL

cylinder.a is solid the whenradius)- (half2or

cL, examplefor

problem thein invloved solid theof size the torealte:scale length sticcharacteria is c Lwhere

L

or

Graphical Representation of the One-Term Approximation:The Heisler ChartsMidplane Temperature:

Change in Thermal Energy Storage

Temperature Distribution

Assumptions in using Heisler charts:•Constant Ti and thermal properties over the body•Constant boundary fluid T by step change•Simple geometry: slab, cylinder or sphere

),(''. trgqt

H

),(.. trgTktTC p

Incorporation of the constitutive equation into the energy equation above yields:

Dividing both sides by Cp and introducing the thermal diffusivity of the material given by

smm

sm

Ck

p

2

Thermal DiffusivityThermal diffusivity includes the effects of properties like mass

density, thermal conductivity and specific heat capacity. Thermal diffusivity, which is involved in all unsteady heat-

conduction problems, is a property of the solid object. The time rate of change of temperature depends on its numerical

value. The physical significance of thermal diffusivity is associated with

the diffusion of heat into the medium during changes of temperature with time.

The higher thermal diffusivity coefficient signifies the faster penetration of the heat into the medium and the less time required to remove the heat from the solid.

pp CtrgT

Ck

tT

),(..

This is often called the heat equation.

pCtrgT

tT

),(..

For a homogeneous material:

pCtxgT

tT

),(2

This is a general form of heat conduction equation.

Valid for all geometries.

Selection of geometry depends on nature of application.

xqxxq

yyq

yqzzq

zq

),(. txgTktTC p

For an isotropic and homogeneous material:

),(2 txgTktTC p

):,,(2

2

2

2

2

2

tzyxgzT

yT

xTk

tTC p

):,,(12

2

2

2

2 tzrgzTT

rrTr

rk

tTC p

):,,(sin1sin

sin11

2

2

2222

2 trgTr

Trr

Trrr

ktTC p

X

Y

zyxkk ,,

),(. txgTktTC p

),,,( tzyxgz

zTk

yyTk

xxTk

tTC p

),,,(2

2

2

2

2

2

tzyxgzTk

zT

zk

yTk

yT

yk

xTk

xT

xk

tTC p

More service to humankind than heat transfer rate calculations

Steady-State One-Dimensional Conduction

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

• For conduction through a large wall the heat equation reduces to:

),,,(2

2

tzyxgxTk

xT

xk

tTC p

),,,(2

2

tzyxgxTk

tTC p

One dimensional Transient conduction with heat generation.

02

2

dx

TdA

0),,,(2

2

tzyxg

xTk

No heat generation

211 CxCTCdxdT

Apply boundary conditions to solve for constants: T(0)=Ts1 ; T(L)=Ts2

211 CxCTCdxdT

The resulting temperature distribution is:

and varies linearly with x.

Applying Fourier’s law:

heat transfer rate:

heat flux:

Therefore, both the heat transfer rate and heat flux are independent of x.

Wall Surfaces with Convection

2112

2

0 CxCTCdxdT

dxTdA

Boundary conditions:

110

)0(

TThdxdTk

x

22 )(

TLThdxdTk

Lx

Wall with isothermal Surface and Convection Wall

2112

2

0 CxCTCdxdT

dxTdA

Boundary conditions:

1)0( TxT

22 )(

TLThdxdTk

Lx

Electrical Circuit Theory of Heat TransferThermal ResistanceA resistance can be defined as the ratio of a

driving potential to a corresponding transfer rate.

iVR

Analogy:Electrical resistance is to conduction of electricity as thermal resistance is to conduction of heat.The analog of Q is current, and the analog of the temperature difference, T1 - T2, is voltage difference. From this perspective the slab is a pure resistance to heat transfer and we can define

qTR

RTq thth

WKmW

KmmkAL

LTTkA

TTq

TRss

ss

condth /1.

212

21

WKmW

KmhATThA

TTq

TRs

s

convth /1.1

2

2

WKmW

KmAhTTAh

TTq

TRrsurrsr

surrs

radth /1.1

2

2

The composite Wall The concept of a thermal

resistance circuit allows ready analysis of problems such as a composite slab (composite planar heat transfer surface).

In the composite slab, the heat flux is constant with x.

The resistances are in series and sum to Rth = Rth1 + Rth2.

If TL is the temperature at the left, and TR is the temperature at the right, the heat transfer rate is given by

21 thth

RL

th RRTT

RTq

Wall Surfaces with Convection

2112

2

0 CxCTCdxdT

dxTdA

Boundary conditions:

110

)0(

TThdxdTk

x

22 )(

TLThdxdTk

LxRconv,1 Rcond Rconv,2

T1 T2

Heat transfer for a wall with dissimilar materials

For this situation, the total heat flux Q is made up of the heat flux in the two parallel paths:

Q = Q1+ Q2 with the total resistance given by:

Composite Walls

The overall thermal resistance is given by