Transient Conduction:Spatial Effects

(when lumped capacitance does not apply)

CH EN 3453 – Heat Transfer

Reminders…

• Homework #5 due Friday

• Friday we start working on the project– Scientific reporting– Be there!!

• Midterm #1 coming up Wed. October 1– Covers chapters 1, 2, 3, 4, 5– Make sure you have read the material

• Career fair tomorrow!!

Review: The Biot Number

• If Bi < 0.1 then the lumped capacitance approach can be used– Eq. 5.5 to find time to reach a given T– Eq. 5.6 to find T after a given time– Eq. 5.8a to find total heat gain (loss) for given time

• L depends on geometry– General approach is L = V/As

• L/2 for wall with both sides exposed• ro/2 for long cylinder• ro/3 for sphere

– Conservative approach is to use the maximum length• L for wall• ro for cylinder or sphere (preferred to use this)

Bi = hL

k

Lumped Capacitance Equations

• Time to reach specifiedtemperature (5.5):

• Temperature after specified time (5.6):

• Thermal time constant (5.7):

• Heat transferred during heating (5.8a):

θθi

=T − T∞Ti − T∞

= exp −hAsρVc

⎛⎝⎜

⎞⎠⎟t

⎡

⎣⎢

⎤

⎦⎥

τ t =1hAs

⎛⎝⎜

⎞⎠⎟ρVc( ) = RtCt

Q = ρVc( )θi 1− exp −tτ t

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥

t = ρVchAs

lnθiθ

About that Pie…(Can we use a lumped analysis approach?)

5

30 cm

3 cm

ρ = 1000 kg/m3 c = 4184 J/kg·Kk = 0.68 W/m·K

Spatial Effects(When lumped analysis cannot be used)

Dimensionless Variables

Temperature: θ*  ≡  θθi = T − T∞

Ti − T∞

Position: x* ≡xL

Time: t* ≡ αtL2

=kt

ρcpL2

Exact Solution for a Plane Wall• Temperature distribution

where

and

Note: ζ is the eigenvalue (root) of the transcendental equation

Approximate Solution for a Plane Wall

• For plane wall with Fo > 0.2, temperature distribution

where the midpoint (x = 0) temperature θo* is

and C1 and ζ1 are found in a table.

• Total heat transfer is:

Table 5.1 – ζ1 and C1 vs. Bi

Midplane Temp for Plane Wall

Temp. Distribution for Plane Wall

Heat Transferred – Plane Wall

Spatial Effects

• Arises from inadequate solution using lumped capacitance method– Temperature gradients are no longer negligible in the

medium• Requires initial and boundary conditions• Exact solutions involve infinite series• Approximate solutions use only first term

– Use Table 5.1 to determine C1 and ζ1

– Can use the one-term approximation when Fo > 0.2– Equations for time, temperature, position, and fraction

of total energy transfer for walls, cylinders and spheres

Example – Book Problem 5.39The 150-mm-thick wall of a gas-fired furnace is constructed of brick (k = 1.5 W/m·K, ρ = 2600 kg/m3, cp = 1000 J/kg·K) and is well insulated at its outer surface. The wall is at an initial temperature of 20°C when the burners are fired and the inner surface is exposed to products of combustion for which T∞ = 950°C and h = 100 W/m2·K.

How long does it take for the outer surface of the wall to reach 750°C?

• Assume Fo > 0.2• Use the for plane wall:

• Solve for t via Fo:

Table 5.1 – ζ1 and C1 vs. Bi

Temperature Distribution over TimeProblem 5.39

Approximate Solutions for Cylinders and Spheres

• Similar approach as for plane wall– NOTE: For cylinders and spheres, use ro for

calculation of Bi and use that Bi to look up values in the table

• Cylinder:

with centerline T:

and total energy transfer:

...and for spheres

• Sphere:

with center T:

and total energy transfer: