TimeTime--Dependent Conduction Dependent Conduction ::The Lumped Capacitance MethodThe Lumped Capacitance Method

Chapter FiveChapter FiveSections 5.1 thru 5.3Sections 5.1 thru 5.3

Transient Conduction• A heat transfer process for which the temperature varies with time,

as well as location within a solid.

• It is initiated whenever a system experiences a change in operating conditions and proceeds until a new steady state (thermal equilibrium) is achieved.

• It can be induced by changes in:– surface convection conditions ( ), ,h T∞

• Solution Techniques– The Lumped Capacitance Method– Exact Solutions– The Finite-Difference Method

– surface radiation conditions ( ),,r surh T– a surface temperature or heat flux, and/or

– internal energy generation.

5.1 The Lumped Capacitance Method• Based on the assumption of a spatially uniform temperature distribution

throughout the transient process.

• Why is the assumption never fully realized in practice?

• General Lumped Capacitance Analysis:

Consider a general case, which includes convection, radiation and/or an applied heat flux at specified surfaces as well as internal energy generation.

( ), , ,, , ,s c s r s hA A A

Hence, .( )( , ) T r t T t→

≈

First Law: stin out g

dE dTc E E Edt dt

ρ= ∀ = − +i i i

• Assuming energy outflow due to convection and radiation and withinflow due to an applied heat flux ,sq′′

( ) ( ), , , gs s h s c r s r surdTc q A hA T T h A T T Edt

ρ ∞′′∀ = − − − − +i

• Is this expression applicable in situations for which convectionand/or radiation provide for energy inflow?

• May h and hr be assumed to be constant throughout the transient process?

• Special Cases (Exact Solutions, ) ( )0 iT T≡

Negligible Radiation ( ), / :T T b aθ θ θ∞ ′≡ − ≡ −

, ,/ /gs c s s ha hA c b q A E cρ ρ⎛ ⎞′′≡ ∀ ≡ + ∀⎜ ⎟⎝ ⎠

i

The non-homogeneous differential equation is transformed into a homogeneous equation of the form:

d adtθ θ′

′= −

d a dtθθ′= −

′

Integrating from t=0 to any t and rearranging,

( ) ( )/, exp 1 expi i

T T b aor at atT T T T

∞

∞ ∞

−⎡ ⎤= − + − −⎣ ⎦− − (5.25)

To what does the foregoing equation reduce as steady state is approached?

How else may the steady-state solution be obtained?

1 1

2 2

2

ln( ) exp( ) exp( ) or / exp( )

at 0, ( ) /Hence, ( ) / [( ) / ] ex

i i

i

at C at CC at b a C at

t T T C T T b aT T b a T T b a

θ θθ θ

∞

∞ ∞

′ ′= − + ⇒ = − +′ = − − = −

= = ⇒ = − −− − = − −

初始條件:

p( )/ / = [1 ] exp( )

i i i

atT T b a b a atT T T T T T

∞

∞ ∞ ∞

−

−+ − −

− − −

Negligible Radiation and Source Terms , 0, 0) :gr s(h h E q ′′>> = =i

( ),s cdTc hA T Tdt

ρ ∞∀ = − − (5.2)

, 0is c

tc d hA

dtθ

θ

ρ θθ

∀= − ∫∫

,s c

i i t

hAT T texp t expT T c

θθ ρ τ

∞

∞

⎡ ⎤ ⎡ ⎤⎛ ⎞−= = − = −⎢ ⎥ ⎢ ⎥⎜ ⎟− ∀⎝ ⎠ ⎣ ⎦⎣ ⎦

The thermal time constant is defined as

( ),

1t

s c

chA

τ ρ⎛ ⎞

≡ ∀⎜ ⎟⎜ ⎟⎝ ⎠

(5.7)

ThermalResistance, Rt

Lumped ThermalCapacitance, Ct

The change in thermal energy storage due to the transient process is

0

toutstE Q E dt∆ ≡ − = −∫i

,0

t

s chA dtθ= − ∫ ( ) 1 expit

tcρ θτ

⎡ ⎤⎛ ⎞= − ∀ − −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦(5.8)

Negligible Convection and Source Terms , 0, 0 :gr sh h E q⎛ ⎞′′>> = =⎜ ⎟⎝ ⎠

i

Assuming radiation exchange with large surroundings,

( )4 4,s r sur

dTc A T Tdt

ρ ε σ∀ = − −

,

4 40 i

s r T

surT

tAc

dTT T

dtε σρ

=∀ −∫∫

3,

1n 1n4

sur sur i

s r sur sur sur i

T T T TctA T T T T Tρ

ε σ⎧ + +∀ ⎪= −⎨

− −⎪⎩ (5.18)

Result necessitates implicit evaluation of T(t).

1 12 tan tan i

sur sur

TTT T

− −⎫⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎪+ − ⎬⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎪⎣ ⎦⎭

5.2 Validity of The Lumped Capacitance Method

• The Biot Number: The first of many dimensionless parameters to be considered.

Definition:

/cBi Biot number = hL k≡

convection or radiation coefficienth →

thermal conductivity of t soe dh lik →

of the solid ( / or coordinate associated with maximum spa

characteristtial temperature difference)

ic leng th c sL A→ ∀

Physical Interpretation:/

/1/

c s cond solid

s conv solid fluid

L kA R TBihA R T

∆=

∆∼ ∼

Criterion for Applicability of Lumped Capacitance Method: 1Bi <<

Physical Interpretation: , ,2

,2

/ = 1/

s i s s cond

s s conv

T T L kA R hL BiT T hA R k∞

−= = =

−

,1 ,2 ,2kA: ( ) ( )L s s sT T hA T T∞− = −穩態下

0.1 ( Lumped Heat Capacitance )

/: ;

: / 2 ; : / 3 ( )

c

c s

c

c o c o c o

hLBik

L V APlate L LCylinder L r Sphere L r L r

≡ <

=

=

= = =

使用 方法之條件

保守考慮下:

[ ]

, ,

2

2

2

( )( )

/( )( ) ( )( ) ( )( )

s c s cc

i c

c c

c

c

c

hA A khLT T exp t = exp tT T c k cL

hL hLk c t exp t

tFo = Fourier numberL

exp exp Bi Fo k L k L

α

ρ ρ

ρ α

∞

∞

⎡ ⎤ ⎡ ⎤⎛ ⎞−= − −⎢ ⎥ ⎢ ⎥⎜ ⎟− ∀ ∀⎝ ⎠ ⎣ ⎦⎣ ⎦

⎡ ⎤ ⎡ ⎤= − = − ≡ −⎢ ⎥ ⎢

≡

⎥⎣ ⎦ ⎣ ⎦

(5.6)

暫態下:

Problem 5.12: Charging a thermal energy storage system consistingof a packed bed of aluminum spheres.

KNOWN: Diameter, density, specific heat and thermal conductivity of aluminum spheres used in packed bed thermal energy storage system. Convection coefficient and inlet gas temperature.

FIND: Time required for sphere at inlet to acquire 90% of maximum possible thermal energy and the corresponding center temperature.

Schematic:

ASSUMPTIONS: (1) Negligible heat transfer to or from a sphere by radiation or conduction due to contact with other spheres, (2) Constant properties.

ANALYSIS: To determine whether a lumped capacitance analysis can be used, first compute Bi = h(ro/3)/k = 75 W/m2⋅K (0.025m)/150 W/m⋅K = 0.013 <<1.

Hence, the lumped capacitance approximation may be made, and a uniform temperature may be assumed to exist in the sphere at any time.

From Eq. 5.8a, achievement of 90% of the maximum possible thermal energy storage corresponds to

( )stt

i

E0.90 1 exp t /

cVτ

ρ θ∆

− = = − −

( )tt ln 0.1 427s 2.30 984sτ= − = × =

3

t s 2

2700 kg / m 0.075m 950 J / kg KVc / hA Dc / 6h 427s.

6 75 W / m Kτ ρ ρ

× × ⋅= = = =

× ⋅

From Eq. (5.6), the corresponding temperature at any location in the sphere is

( ) ( ) ( )g,i i g,iT 984s T T T exp 6ht / Dcρ= + − −

( )( )2 3

T 984s

300 C 275 C exp 6 75 W / m K 984s / 2700 kg / m 0.075m 950 J / kg K= ° − ° − × ⋅ × × × ⋅

If the product of the density and specific heat of copper is (ρc)Cu ≈ 8900 kg/m3 × 400 J/kg⋅K = 3.56 × 106 J/m3⋅K, is there any advantage to using copper spheres of equivalent diameter in lieu of aluminum spheres?

Does the time required for a sphere to reach a prescribed state of thermal energy storage change with increasing distance from the bed inlet? If so, how and why?

( )T 984s 272.5 C= °