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Heat Transfer for Engineering Students

Here are the key steps to solve this problem: 1. The problem describes a semi-infinite solid (asphalt) with an initial uniform temperature of 50°C 2. At time t=0, the surface temperature is suddenly changed to 20°C 3. We want to calculate the total energy transferred from the asphalt over 30 minutes with the surface at 20°C 4. Use the semi-infinite solid solution for constant surface temperature case 5. Identify parameters: k=1 W/m-K (typical for asphalt), T0=50°C, T∞=20°C, t=30 minutes 6. Plug into the semi-

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460 views8 pages

Heat Transfer for Engineering Students

Here are the key steps to solve this problem: 1. The problem describes a semi-infinite solid (asphalt) with an initial uniform temperature of 50°C 2. At time t=0, the surface temperature is suddenly changed to 20°C 3. We want to calculate the total energy transferred from the asphalt over 30 minutes with the surface at 20°C 4. Use the semi-infinite solid solution for constant surface temperature case 5. Identify parameters: k=1 W/m-K (typical for asphalt), T0=50°C, T∞=20°C, t=30 minutes 6. Plug into the semi-

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CH EN 3453 – Heat Transfer

Transient Conduction:
Semi-Infinite Solids

Reminders…
• Homework #5 due Friday
– For #5, make it simpler by using h = 198 W/m2·K
– For #4, temperature is 550°C
– Help session this afternoon at 4:30 in MEB 2325
– Returned Monday

• Friday we start working on the project

• Midterm #1 coming up Wed. October 1


– Covers chapters (1, 2) 3, 4, 5
– Won’t cover section 5.10 (FDA of transient conduction)

• Midterm Review Session Monday, Sept. 29

• Survey...
Calendar...

Review of 5.1 to 5.6


• Lumped Capacitance Method and Validity
– Assumes solid is spatially uniform,
– Resistance to conduction within the solid is small
compared to the resistance to
heat transfer between the solid and
its surroundings
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 5.39


The 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 Time


Problem 5.39
Approximate Solutions for Cylinders
and Spheres
• Similar approach as for plane wall
– NOTE: 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:


5.7 The Semi-Infinite Solid
• An idealized geometry to analytically solve heat
transfer problems
• Method for transient conduction problems
• Analytical derivation is found on page 284-286
(6th ed.) or 311-314 (7th ed.)
• Three cases
– Constant surface temperature
– Constant surface heat flux
– Surface convection
Three%Cases

Gaussian Error Function


• Appendix B.2 (page 961 in 6th ed.)
Slabs%of%Different%Temperature
• Temperature%slopes%are%not%linear,%indica9ve%
of%transient%behavior

Example – Book Problem 5.68


Asphalt pavement may achieve temperatures as
high as 50°C on a hot summer day. Assume that
such a temperature exists throughout the
pavement, when suddenly a rainstorm reduces the
surface temperature to 20°C. Calculate the total
amount of energy (J/m2) that will be transferred
from the asphalt over a 30-minute period in which
the surface is maintained at 20°C.

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