Chapter 5

Time-Dependent Conduction

5.1 The Lumped Capacitance MethodThis method assumes spatially uniform solid temperature at any

instant during the transient process. It is valid if the temperature gradients within the solid are small.

Apply energy balance to the control volume of Fig. 5.1

If the energy exchange is through convection,

Letting θ = T-T∞, we obtain

(5.1)out stE E− =& &

−hAs (T − T∞ ) = ρVc dTdt

ρVchAs

dθdt = −θ

(5.2)

With the initial condition θ (0)= θi, it follows

The thermal time constant is defined as

where Rt is the resistance to convection heat transfer, Ct is the lumped thermal capacitance of the solid.

θθi

=T − T∞Ti − T∞

= exp −hAsρVc

⎛ ⎝ ⎜

⎞ ⎠ ⎟ t

⎡ ⎣ ⎢

⎤ ⎦ ⎥ (5.6)

τ t =ρVchAs

= ( 1hAs

)(ρVc) = RtCt

Q = qdt0t

∫ = hAs θdt0t

∫ = (ρVc)θi 1 − exp − tτt

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡ ⎣ ⎢

⎤ ⎦ ⎥

The total energy transfer Q can be obtained by

(5.8)

5.2 Validity of the Lumped Capacitance MethodTo find the criterion for the validity of the lumped capacitance

method, consider the steady-state energy balance (Fig. 5.3)

kAL (Ts ,1 − Ts,2 ) = hA(Ts,2 − T∞ )

Ts,1 − Ts,2Ts,2 − T∞

= (L / kA)(1 / hA) =

RcondRconv

= hLk ≡ Bi

k--conductivity of the solid

→

Criterion for the validity of the lumped capacitance method(Fig.5.4):

Lc= characteristic lengthBi =hLck < 0.1

(5.9)

(5.10)

Criterion for the validity of the lumped capacitance method (Fig.5.4): Bi =

hLck < 0.1 Lc= characteristic length

The dimensionless time, Fourier number, Fo = αtLc

2

Fo, with Bi, characterizes transient conduction problems.θθi

=T − T∞Ti − T∞

= exp(−Bi ⋅ Fo)EX 5.1

(5.12)

(5.13)

5.3 General Lumped Capacitance AnalysisA general situation including convection, radiation, an applied

surface heat flux, and internal energy generation (Fig.5.5) can be described as

" " ", conv rad ( , )( )s s h g s c r

dTq A E q q A Vcdt

ρ+ − + =&

" 4 4, sur ( , )or [ ( ) ( )]s s h g s c r

dTq A E h T T T T A Vcdt

εσ ρ∞+ − − + − =&

(5.14)

(5.15)

EXs 5.2, 5.3

5.4 Spatial EffectsConsider a representative 1-D heat transfer problem,DE:

IC:

BCs:

Eqs. 5.26-29 imply

However, after nondimensionalization with

∂ 2T∂x2 = 1

α∂T∂t

T(x, 0) = Ti

∂T∂x x=0

= 0 −k ∂T∂x x= L

= h[T(L,t) − T∞ ]

T = T(x, t,Ti,T∞ , L,k,α,h)

(5.27)

(5.26)

(5.28, 29)

(5.30)

θ∗ ≡ θθi

=T − T∞Ti − T∞

x∗ ≡ xL t∗ ≡ αt

L2 ≡ Fo

∂ 2θ∗

∂x*2 = ∂θ∗

∂Fo

θ∗(x∗, 0) = 1∂θ∗

∂x* x∗=0= 0 ∂θ∗

∂x* x∗=1= −Bi θ ∗(1, t∗ )

→

(5.31-33)

(5.35-37)

(5.34)

For a prescribed geometry, the transient temperature distribution is a universal function of x*, Fo, and Bi. That is, the dimensionless solution assumes a prescribed form that does not depend on the particular value of Ti, T∞, L, k, α, or h..The physical interpretation of Fo: Fo can not only viewed as the dimensionless time, it also provides a measure of the relative effectiveness for a material to conduct and store energy, as

θ∗ = f (x∗,Fo, Bi)

FoLtcLkttTcLLTkLEq ===ΔΔ 2232st //)//()/(~)/( αρρ&

(5.38)→

5.5 The Plane Wall with Convection

5.5.1 Exact SolutionThe problem depicted in Fig. 5.6a has the exact solution as

θ∗ = Cn exp(−ζn2Fo)cos(ζn x∗)

n=1

∞∑

Cn =4sin ζn

2ζn + sin(2ζn )

and the eigenvalues ζn are positive roots of the transcendental eq.

ζn tanζn = Bi

(5.39a)

(5.39b)

(5.39c)

where

5.5.2 Approximate SolutionFor Fo > 0.2, the exact infinite series solution can be approximated

by the first term of the series: (∵ζ1< ζ2 < ζ3 <…, cf. App. B.3)

where --temperature variation at midplane x* = 0.Eq. 5.40 implies that the time dependence of the temperature at any

location within the wall is identical.5.5.3 Total Energy Transfer

*1

1

[ ( , ) ] sin1 (1 *) ~ 1io

o i

T x t TQ dV dVQ T T V V

ζθ θζ∞

− −= = − −

−∫ ∫

θ∗ ≈ C1 exp(−ζ12Fo) cos(ζ1x

∗)

θ∗ = θ0∗ cos(ζ1x∗)

θ0∗ = C1 exp(−ζ1

2Fo)

or

(5.40a)

(5.40b)

[ ]o iQ cV T Tρ ∞= −[ ( , ) ] ;iQ c T x t T dVρ= − −∫

5.5.4 Additional ConsiderationsThe solution is also valid for the problem with insulation on one side (x* = 0) and experiences convective transport on the other side (x* = +1).The foregoing results may be used to determine the transient response of a plane wall to a sudden change in surface temperature (θ* = θi*, at x* = 1). The process is equivalent to having an infinite convection coefficient, in which case the Biot number is infinite (Bi= ∞) and the fluid temperature T∞ is replaced by the prescribed surface temperature Ts.

5.6 Radial Systems with ConvectionInfinite cylinder: (Fig. 5.6b)

with eigenvalues ζn being the positive roots of the equation

<Home Work> Solve for Eqs. 5.39 and 5.47 using the method of separation of variables.

EX 5.4

θ∗ = Cn exp(−ζn2Fo)J0(ζnr

∗)n=1

∞∑

Cn = 2ζn

J1(ζn )J0

2 (ζn ) + J12 (ζn)

ζnJ1(ζn )J0 (ζn) = Bi

where

(5.47a)

(5.47b)

(5.47c)

About Bessel EquationSimilar reasoning can be made for the Bessel Equation as for modified Bessel Equation, as both correspond to the cylindrical coordinate.

The Bessel Equation of order v is

The general solution of (3) is (B3a)

(B3)22

2 21 (1 ) 0vd dr drdr r

θ θ θ+ + − =

1 2( ) ( ) ( )r C J r C Y rν νθ = +

From www.-efunda.-com/-math

2.405 5.520 8.654

Compare , with the general solution

andwith the general solution

It can be seen that J0(mr) is similar to cosmx in the oscillating behavior, but J0 (mr) reflects damped amplitude in response to increased r. Their “periods”are also very similar except for the first half cycle. In fact, for large x

22

21 0d d mr drdr

θ θ θ+ + =

1 0 2 0( ) ( ) ( )r C J mr C Y mrθ = +2

22 0d m

dxθ θ+ =

1 2( ) cos sinx C mx C mxθ = +

2( ) ~ cos2 4n

nJ x xx

π ππ

⎛ ⎞− −⎜ ⎟⎝ ⎠

(B5a)

(B5)

(B4, B4a)

5.7 The Semi-Infinite SolidThree cases of surface conditions: (Fig. 5.7)

Using the similarity method with similarity variable η = x/(4αt)1/2, the PDE can be transformed into an ODE. Exact solutions can thus be obtained.Case 1 Constant surface temperature: T(0,t)=Ts

Case 2 Constant surface heat flux: →Eq. 5.59Case 3 Surface Convection: →Eq. 5.60

EX 5.6

′ ′ q s = ′ ′ q 0−k ∂T

∂x x=0= h[T∞ − T(0,t)]

"

( , )2

( )( )

s

i s

s is

T x t T xerfT T t

k T Tq tt

α

πα

− ⎛ ⎞= ⎜ ⎟− ⎝ ⎠−

=

(5.57)

(5.58)

A special case: interfacial contact between two semi-infinite solids at different initial temperatures

If contact resistance is negligible, there is no temperature jump at the interface. Also,

Since Ts does not change with time(no energy storage), from Eq 5.58

With Ts determined, the temperature profile can be described by Eq. 5.57.

(5.57)

(5.58)

" ", ,s A s Bq q=

, ,( ) ( )A s A i B s B i

A B

k T T k T Tt tπα πα

− −− =

, ,( ) ( )( ) ( )

A A i B B is

A B

k c T k c TT

k c k cρ ρ

ρ ρ+

→ =+

(5.62)

(5.63)