This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Chapter 4 Transient Heat Conduction
4115E A plate, a long cylinder, and a sphere are exposed to cool air. The center temperature of each
geometry is to be determined.
Assumptions 1 Heat conduction in each geometry is onedimensional. 2 The thermal properties of the
bodies are constant. 3 The heat transfer coefficient is constant and uniform over the entire surface. 4 The
Fourier number is τ > 0.2 so that the oneterm approximate solutions (or the transient temperature charts)
are applicable (this assumption will be verified).
Properties The properties of bronze are given to be k = 15 Btu/h.ft.°F and α = 0.333 ft2/h.
Analysis After 5 minutes Plate: First the Biot number is calculated to be Bi = hL (7 Btu/h.ft 2 .°F)(0.5 / 12 ft )
= 0.01944
=
(15 Btu/h.ft.°F)
k 2 ro The constants λ 1 and A1 corresponding to this Biot number are, from Table 41, 2 ro λ1 = 0.1410 and A1 = 1.0033
The Fourier number is τ= αt
L2 = (0.333 ft 2 /h)(5 min/60 min/h)
(0.5 / 12 ft) 2 = 15.98 > 0.2 2L Then the center temperature of the plate becomes θ 0, wall = 2
2
T − 75
T0 − T∞
= A1 e −λ1 τ ⎯
⎯→ 0
= (1.0033)e − (0.1410) (15.98) = 0.730 ⎯ T0 = 312°F
⎯→
Ti − T∞
400 − 75 Cylinder: Bi = 0.02 ⎯Table 9⎯→ λ1 = 0.1995 and A1 = 1.0050
⎯ ⎯ −1 θ 0,cyl = 2
2
T − 75
T 0 − T∞
= A1 e − λ1 τ ⎯
⎯→ 0
= (1.0050)e − (0.1995) (15.98) = 0.532 ⎯ T0 = 248°F
⎯→
Ti − T∞
400 − 75 Sphere: Bi = 0.02 ⎯Table 9⎯→ λ1 = 0.2445 and A1 = 1.0060
⎯ ⎯ −1 θ 0, sph = 2
2
T − 75
T 0 − T∞
= A1 e − λ1 τ ⎯
⎯→ 0
= (1.0060)e − (0.2445) (15.98) = 0.387 ⎯ T0 = 201°F
⎯→
Ti − T∞
400 − 75 After 10 minutes τ= αt
L2 = (0.333 ft 2 /h)(10 min/60 min/h)
(0.5 / 12 ft) 2 = 31.97 > 0.2 Plate: θ 0, wall = 2
2
T − 75
T0 − T∞
= A1 e −λ1 τ ⎯
⎯→ 0
= (1.0033)e − (0.1410) (31.97 ) = 0.531 ⎯ T0 = 248°F
⎯→
Ti − T∞
400 − 75 492 ...
View
Full
Document
This note was uploaded on 01/19/2012 for the course PHY 4803 taught by Professor Dr.danielarenas during the Fall '10 term at UNF.
 Fall '10
 Dr.DanielArenas
 Thermodynamics, Mass, Heat

Click to edit the document details