Transient Conduction:Transient Conduction:Spatial Effects and the Role ofSpatial Effects and the Role of

Analytical SolutionsAnalytical Solutions

Solution to the Heat Equation for a Plane Wall withSymmetrical Convection Conditions

• If the lumped capacitance approximation can not be made, consideration mustbe given to spatial, as well as temporal, variations in temperature during thetransient process.

• For a plane wall with symmetrical convectionconditions and constant properties, the heatequation and initial/boundary conditions are:

2

2

1T Tx tα

∂ ∂=

∂ ∂ (5.26)

( ),0 iT x T= (5.27)

0

0x

Tx =

∂=

∂(5.28)

( ),x L

Tk h T L t Tx ∞=

∂⎡ ⎤− = −⎣ ⎦∂

(5.29)

• Existence of seven independent variables:( ), , , , , ,iT T x t T T k hα∞= (5.30)

How may the functional dependence be simplified?

• Non-dimensionalization of Heat Equation and Initial/Boundary Conditions:

Dimensionless temperature difference: *i i

T TT T

θθθ

∞

∞

−≡ =

−* xx

L≡Dimensionless coordinate:

Dimensionless time: *2

tt FoLα

≡ ≡

Fourierthe NumberFo →

The Biot Number:solid

hLBik

≡

( )* *, ,f x Fo Biθ =• Exact Solution:

( ) ( )* 2 *1

exp cosn n nn

C Fo xθ ζ ζ∞

== −∑ (5.39a)

( )4sin tan

2 sin 2n

n n nn n

C Biζ ζ ζζ ζ

= =+

(5.39b,c)

See Appendix B.3 for first four roots (eigenvalues ) of Eq. (5.39c)1 4,...,ζ ζ

• The One-Term Approximation :( )0.2Fo >Variation of midplane temperature (x*= 0) with time : ( )Fo

( )( ) ( )

* 21 1exp

oo

i

T TC Fo

T Tθ ζ∞

∞

−≡ ≈ −

−(5.41)

1 1Table 5.1 and as a function of C Biζ→

Variation of temperature with location (x*) and time :( )Fo( )* * *1coso xθ θ ζ= (5.40b)

Change in thermal energy storage with time:

stE QΔ = − (5.43a)

1 *

1

sin1o oQ Qζ θ

ζ⎛ ⎞

= −⎜ ⎟⎝ ⎠

(5.46)

( )o iQ c T Tρ ∞= ∀ − (5.44)

Can the foregoing results be used for a plane wall that is well insulated on oneside and convectively heated or cooled on the other?

Can the foregoing results be used if an isothermal condition is instantaneously imposed on both surfaces of a plane wall or on one surface ofa wall whose other surface is well insulated?

( )s iT T≠

Graphical Representation of the One-Term ApproximationThe Heisler Charts

• Midplane Temperature:

• Temperature Distribution:

• Change in Thermal Energy Storage:

Radial Systems• Long Rods or Spheres Heated or Cooled by Convection.

2

//

o

o

Bi hr kFo t rα

=

=

• One-Term Approximations:Long Rod: Eqs. (5.49) and (5.51)

Sphere: Eqs. (5.50) and (5.52)

1 1, Table 5.1C ζ →

• Graphical Representations:Long Rod: Figs. D.4 – D.6

Sphere: Figs. D.7 – D.9

The Semi-Infinite Solid• A solid that is initially of uniform temperature Ti and is assumed to extend

to infinity from a surface at which thermal conditions are altered.

• Special Cases:Case 1: Change in Surface Temperature (Ts)

( ) ( )0, ,0s iT t T T x T= ≠ =

( ), xerf2 t

s

i s

T x t TT T α

− ⎛ ⎞= ⎜ ⎟− ⎝ ⎠ (5.57)

( )s is

k T Tq

tπα−

′′ =(5.58)

( ) ( )1

2 22 /, exp

4

erfc2

oi

o

q t xT x t Tk t

q x xk t

α πα

α

′′ ⎛ ⎞− = −⎜ ⎟⎝ ⎠

′′ ⎛ ⎞− ⎜ ⎟⎝ ⎠ (5.59)

Case 2: Uniform Heat Flux ( )s oq q′′ ′′=

( )0

0,x

Tk h T T tx ∞=

∂⎡ ⎤− = −⎣ ⎦∂

( )

2

2

,2

2

i

i

T x t T xerfcT T t

hx h t x h texp erfck k kt

α

α αα

∞

− ⎛ ⎞= ⎜ ⎟− ⎝ ⎠⎡ ⎤⎛ ⎞⎡ ⎤⎛− + +⎢ ⎥⎜ ⎟⎜⎢ ⎥ ⎜⎝⎣ ⎦ ⎢ ⎥⎠⎝⎣ ⎦ (5.60)

Case 3: Convection Heat Transfer ( ),h T∞

Multidimensional Effects• Solutions for multidimensional transient conduction can often be expressed

as a product of related one-dimensional solutions for a plane wall, P(x,t),an infinite cylinder, C(r,t), and/or a semi-infinite solid, S(x,t). See Equations (5.64) to (5.66) and Fig. 5.11.

• Consider superposition of solutions for two-dimensional conduction in ashort cylinder:

( ) ( ) ( )

( ) ( )

, ,, ,

,i

Plane Infinitei iWall Cylinder

T r x t TP x t x C r t

T T

T x t T T r,t Tx

T T T T

∞

∞

∞ ∞

∞ ∞

−=

−

− −=

− −

Problem 5.66: Charging a thermal energy storage system consisting ofa packed bed of Pyrex spheres.

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

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

SCHEMATIC:

ASSUMPTIONS: (1) One-dimensional radial conduction in sphere, (2) Negligible heat transfer to or from a sphere by radiation or conduction due to contact with adjoining spheres, (3) Constant properties.

ANALYSIS: With Bi ≡ h(ro/3)/k = 75 W/m2⋅K (0.0125m)/1.4 W/m⋅K = 0.67, the lumped capacitance method is inappropriate and the approximate (one-term) solution for one-dimensional transient conduction in a sphere is used to obtain the desired results.

To obtain the required time, the specified charging requirement ( )/ 0.9oQ Q = must first be used to obtain the dimensionless center temperature,

*.oθ

From Eq. (5.52),

( ) ( )

31

oo1 1 1

Q1Q3 sin cos

ζθζ ζ ζ

∗ ⎛ ⎞= −⎜ ⎟−⎡ ⎤ ⎝ ⎠⎣ ⎦

With Bi ≡ hro/k = 2.01, 1 2.03ζ ≈ and C1 ≈ 1.48 from Table 5.1. Hence,

( )

( )

3

o0.1 2.03 0.837 0.155

5.3863 0.896 2.03 0.443θ ∗ = = =

− −⎡ ⎤⎣ ⎦

From Eq. (5.50c), the corresponding time is

2o o

211

rt ln

Cθ

αζ

∗⎛ ⎞= − ⎜ ⎟⎜ ⎟

⎝ ⎠

( )3 7 2k / c 1.4 W / m K / 2225 kg / m 835 J / kg K 7.54 10 m / s,α ρ −= = ⋅ × ⋅ = × ( ) ( )

( )

2

27 20.0375m ln 0.155/1.48

t 1,020s7.54 10 m /s 2.03−

= − =×

From the definition of *,oθ the center temperature is ( )o g,i i g,iT T 0.155 T T 300 C 42.7 C 257.3 C= + − = ° − ° = °

The surface temperature at the time of interest may be obtained from Eq. (5.50b) with r 1,∗ =

( ) ( )o 1s g,i i g,i1

sin 0.155 0.896T T T T 300 C 275 C 280.9 C2.03

θ ζζ

∗ ×⎛ ⎞= + − = ° − ° = °⎜ ⎟⎝ ⎠

Is use of the one-term approximation appropriate?

Problem: 5.82: Use of radiation heat transfer from high intensity lampsfor a prescribed duration (t=30 min) to assess

ability of firewall to meet safety standards corresponding tomaximum allowable temperatures at the heated (front) andunheated (back) surfaces.

( )4 210 W/msq′′ =

KNOWN: Thickness, initial temperature and thermophysical properties of concrete firewall. Incident radiant flux and duration of radiant heating. Maximum allowable surface temperatures at the end of heating.

FIND: If maximum allowable temperatures are exceeded.

SCHEMATIC:

ASSUMPTIONS: (1) One-dimensional conduction in wall, (2) Validity of semi-infinite medium approximation, (3) Negligible convection and radiative exchange with the surroundings at the irradiated surface, (4) Negligible heat transfer from the back surface, (5) Constant properties.

ANALYSIS: The thermal response of the wall is described by Eq. (5.59)

( ) ( )1/ 2 2

o oi

2 q t / q xx xT x, t T exp erfck 4 t k 2 t

α πα α

⎛ ⎞′′ ′′− ⎛ ⎞= + −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

where, 7 2pk / c 6.92 10 m / sα ρ−= = × and for

( )1 / 2ot 30 min 1800s, 2q t / / k 284.5 K.α π′′= = = Hence, at x = 0,

( )T 0,30 min 25 C 284.5 C 309.5 C 325 C= ° + ° = ° < °

At ( ) ( )1 / 22 ox 0.25m, x / 4 t 12.54, q x / k 1, 786K, and x / 2 t 3.54.α α′′= − = − = =Hence,

( ) ( ) ( )6T 0.25m, 30 min 25 C 284.5 C 3.58 10 1786 C ~ 0 25 C−= ° + ° × − ° × ≈ °

Both requirements are met.

Is the assumption of a semi-infinite solid for a plane wall of finite thickness appropriate under the foregoing conditions?

COMMENTS: The foregoing analysis may or may not be conservative, since heat transfer at the irradiated surface due to convection and net radiation exchange with the environment has been neglected. If the emissivity of the surface and the temperature of the surroundings are assumed to be ε = 1 and Tsur = 298K, radiation exchange at Ts = 309.5°C would be ( )4 4 2rad s surq T T 6,080 W / m K,εσ′′ = − = ⋅ which is significant (~ 60% of the prescribed radiation). However, under actual conditions, the wall would likely be exposed to combustion gases and adjoining walls at elevated temperatures.

Transient Conduction:�Spatial Effects and the Role of�Analytical Solutions