Modelling

Lumped systems:

T = f(t)

but

T = const

throughout the

solid at any time

Temperature of a body, in general, varies with time and
position: T = f(x,y,z,t)

→

→

→

→ unsteady conditions

TRANSIENT HEAT CONDUCTION

T = f(x,y,z,t)

in:

• 1D systems – a large plane wall, a long cylinder, a sphere,

a semi-infinite medium

• multidimensional systems

LUMPED SYSTEM

const

z

y

x

T

t

f

T

=

=

)

,

,

(

)

(

h

– heat transfer coefficient

Assumption:
temperature of a medium

T

∞

∞

∞

∞

>

>

>

> T

i

During a differential time

dt

–

temperature rising by a
differential amount

dT

Energy balance of the solid for the time interval

dt

:

Heat transfer into the

body during

dt

The increase in the energy of

the body during

dt

=

or:

dT

mC

dt

T

T

hA

p

S

=

−

∞

)

(

Integrating from t = 0, at which T = T

i

, to any time t, at which

T = T(t), gives

t

VC

hA

T

T

T

t

T

p

S

i

ρ

−

=

−

−

∞

∞

)

(

ln

V

m

ρ

=

Since:

and

)

(

∞

−

=

T

T

d

dT

, because

const

T

=

∞

dt

VC

hA

T

T

T

T

d

p

S

ρ

−

=

−

−

∞

∞

)

(

we obtain:

Thus:

bt

i

e

T

T

T

t

T

−

∞

∞

=

−

−

)

(

where:

p

S

VC

hA

b

ρ

=

)

/

1

(

s

b

/

1

=

τ

- time constant

The

T

of a body approaches the

ambient temperature

T

∞

∞

∞

∞

exponentially:
- the larger

b

value, the higher

rate of

T

decay.

)

(t

T

temperature
of a body at
time

t

The

actual rate

of convection heat transfer between the body

and its environment – from Newton`s law of cooling:

]

)

(

[

)

(

∞

•

−

=

T

t

T

hA

t

S

Q

)

(W

The

total amount

of heat transfer between the body and the

surrounding medium over the time interval (0 – t) is the
change in the energy content of the body:

]

)

(

[

i

p

T

t

T

mC

Q

−

=

)

(kJ

Heat transfer to
or from a body
reaches its
maximum value
when the body
reaches the
environment
temperature

Lumped system

– great convenience in heat transfer analysis

Criterion for the applicability
→

→

→

→ definition of a

characteristic length

S

C

A

V

L

=

and a

Biot number

Λ

=

C

hL

Bi

Also:

body

the

within

Conduction

body

the

of

surface

the

at

Convection

T

T

L

h

Bi

C

_

_

_

_

_

_

_

_

_

/

=

∆

∆

Λ

=

or:

body

the

of

surface

the

at

ce

tan

resis

Convection

body

the

within

ce

tan

resis

Conduction

h

L

Bi

C

_

_

_

_

_

_

_

_

_

_

_

/

1

/

=

Λ

=

Meaning of Biot number

→

→

→

→

• Small Biot number represents small resistance to heat
conduction

→

→

→

→ small T gradients within the body

• Lumped system - a uniform T distribution throughout the body
→

→

→

→ conduction resistance = 0

Thus, lumped system analysis is exact when Bi = 0

and approximate when Bi

>

>

>

> 0.

Estimation of an accuracy of lumped system analysis

→

→

→

→ Typical uncertainty in the convection heat transfer coefficient

h

is about

20%

It is generally accepted that lumped system analysis is applicable

if

1

.

0

≤

Bi

The temperature variations with location within the body is
slight (

<

<

<

< 5%) →

→

→

→ small bodies with high thermal conductivity in

a medium that is a poor conductor of heat (air)

Analogy between heat transfer to a solid and
passenger traffic to an island

Lumped system – small island with plenty of fast buses
(no overcrowding at the harbour = accumulation of heat)

Example of non-lumped system

:

high value of the convection heat transfer coefficient

h

→

→

→

→ large T difference between the inner and outer regions

On the other hand:

The larger the thermal conductivity

Λ

Λ

Λ

Λ

,

the smaller the temperature gradient
within the body.

TRANSIENT SYSTEMS

The variation of the T profile with time in the plane etc.

where:

α

α

α

α

is the thermal diffusivity in (m

2

/s)

T

T

t

L

C

L

L

L

t

p

∆

∆

Λ

=

=

/

)

/

1

(

3

2

2

ρ

α

τ

The rate at which heat is conducted
across L of a body of volume L

3

The rate at which heat is stored
in a body of volume L

3

=

TRANSIENT HEAT CONDUCTION IN SEMI-INFINITE SOLIDS

For the special case of

h

→

→

→

→ ∞

∞

∞

∞

, the surface temperature

T

S

becomes equal to the fluid temperature

T

∞

∞

∞

∞

, and













=

−

−

t

x

erfc

T

T

T

t

x

T

i

S

i

α

2

)

,

(

du

e

erfc

u

∫

−

−

=

ξ

π

ξ

0

2

2

1

)

(

The complementary error function

t

x

α

ξ

2

=

Example

Minimum burial depth of water
pipes to avoid freezing

m

t

x

80

.

0

2

=

=

α

ξ

TRANSIENT HEAT CONDUCTION IN MULTIDIMENSIONAL

SYSTEMS

For 2D and 3D systems – a superposition approach called a
product solution (based on charts containing numerical
solutions for 1D systems)

Example

A solid bar of rectangular profile a

×

×

×

× b is the intersection of two

plane walls of thickness a and b.

Transient T distribution in the bar:

)

,

(

)

,

(

)

,

,

(

t

y

t

x

T

T

T

t

y

x

T

wall

wall

bar

i

θ

θ

=













−

−

∞

∞

where:

∞

∞

−

−

=

T

T

T

t

x

T

t

x

i

wall

)

,

(

)

,

(

θ

∞

∞

−

−

=

T

T

T

t

y

T

t

y

i

wall

)

,

(

)

,

(

θ