Cummins CollegeTransient heat conduction with spatial variations

Transient heat conduction problems for small bodies can be solves using lumped heat

capacity analysis. Lumped system analysis assumes a uniform temperature distribution

throughout the body, which is the case only when the thermal resistance of the body to heat

conduction (the conduction resistance) is zero. Thus, lumped system analysis is exact when

Bi = 0 and approximate when Bi > 0. The smaller the Biot number, the more accurate the

lumped system analysis. It is generally accepted that lumped system analysis is applicable if

Bi ≤ 0.1 Thus, when Bi ≤ 0.1 , the variation of temperature with location within the body

is slight and can reasonably be approximated as being uniform. Relatively small bodies of

highly conductive materials approximate this behavior.

In general, however, the temperature within a body changes from point to point as well as

with time. When geometries are large and spacial variations cannot be ignored lumped heat

capacity analysis approach for the solution cannot be used. These are the typical situation

where

Bi ≥ 0.1

Consider a plane wall of thickness 2L, initially at a uniform temperature Ti, At time t

= 0, is placed in a large medium that is at a constant temperature T∞ and kept in that

medium for t > 0. Heat transfer takes place between these bodies and their environments

by convection with a uniform and constant heat transfer coefficient h. The variation of the

temperature profile with time in the plane wall is illustrated in fig. 1

The above problem can be expresses in form of differential equations as

∂2T

∂x2=

1

α

∂T

∂t

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Figure 1: Transient temperature profiles in a plane wall exposed to convection

Boundary Conditions

∂T (0, t)

∂x= 0 and− k∂T (L, t)

∂x= h[T (L, t)− T∞]

The analytical solution obtained above for one-dimensional transient heat conduction in

a plane wall involves infinite series and implicit equations, which are difficult to evaluate.

Therefore it is required to present the solutions in tabular or graphical form using simple

relations. one-term approximation solutions of above equation are presented in graphical

form, known as the transient temperature charts.

The transient temperature charts for a large plane wall, long cylinder, and sphere were

presented by M. P. Heisler in 1947 and are called Heisler charts. They were supplemented

in 1961 with transient heat transfer charts by H. Grober. There are three charts associated

with each geometry: the first chart is to determine the temperature T0 at the center of the

geometry at a given time t. The second chart is to determine the temperature at other

locations at the same time in terms of T0. The third chart is to determine the total amount

of heat transfer up to the time t.

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The first chart for the plane wall is plotted using 3 different variables. Plotted along the

vertical axis of the chart is dimensionless temperature at the midplane. Fourier Number is

plotted along the horizontal axis.The curves within the graph are a selection of values for

the inverse of the Biot Number. The charts for the cylinder and sphere are also plotted on

the similar lines.

The values for the Biot number and Fourier number, as used in the Heisler charts are

evaluated on the basis of characteristics parameter, which is, semithickness for plate, and

surface radius for cylinders and spheres

Although Heisler-Grober Charts are a faster and simpler alternative to the exact solu-

tions of these problems, there are some limitations. First, the body must be at uniform

temperature initially. Additionally, the temperature of the surroundings and the convective

heat transfer coefficient must remain constant and uniform. Also, there must be no heat

generation from the body itself.

Exercise

1. A slab 10 cm thick at initially at 500 ◦C is immersed in liquid at 100 ◦C resulting in

heat transfer coefficient of 1200 W/m2K. Determine the temperature at the centerline

and at the surface 1 min after immersion. Properties for the slab are; α = 8.4 ×

10−5 m2/s, k = 215 W/mK, cp = 0.9 kJ/kgK, ρ = 2700 kg/m3

2. A long steel cylinder 12 cm in diameter and initially at 20 ◦C is placed into a furnace

at 820 ◦C with local heat transfer coefficient of 140 W/m2K. Calculate the time

required for the axis temperature to reach 800 ◦C. Also calculate the corresponding

temperature at radius of 5.4 cm. α = 6.11× 10−6 m2/s, k = 21 W/mK

3. A metallic sphere of radius 10 mm is initially at uniform temperature of 335 ◦C. It is

then quenched in water bath at 20 ◦C with h = 6000 W/m2K until the center of sphere

cools to 50 ◦C. Compute the time required for cooling of the sphere. Properties of

sphere are; α = 6.66× 10−6 m2/s, k = 20 W/mK, cp = 1000 kJ/kgK, ρ = 3000 kg/m3

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Fig

ure

2:T

emp

erat

ure

dis

trib

uti

onin

pla

ne

wal

lat

Mid

-Pla

ne

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Figure 3: Offset temperature distribution for wall

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Fig

ure

4:T

emp

erat

ure

dis

trib

uti

onin

acy

linder

atce

nte

rline

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Figure 5: Offset temperature distribution for cylinder

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Fig

ure

6:T

emp

erat

ure

dis

trib

uti

onin

spher

eat

cente

r

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Figure 7: Offset temperature distribution for sphere

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