PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

11-9

11-20 The heating times of a sphere, a cube, and a rectangular prism with similar dimensions are to be determined. Assumptions 1 The thermal properties of the geometries are constant. 2 The heat transfer coefficient is constant and uniform over the entire surface. Properties The properties of silver are given to be k = 429 W/m⋅ºC, ρ = 10,500 kg/m3, and cp = 0.235 kJ/kg⋅ºC. Analysis For sphere, the characteristic length and the Biot number are

1.0 00023.0)C W/m.429(

)m 008333.0)(C. W/m12(

m 008333.06

m 05.06

6/

2

2

3

surface

<=°

°==

=====

khL

Bi

DD

DA

L

c

cπ

πV

Since 0.1< Bi , the lumped system analysis is applicable. Then the time period for the sphere temperature to reach to 25ºC is determined from

min 40.5==⎯→⎯=−−

⎯→⎯=−−

=°

°===

−−

∞

∞ s 24283303325)(

s 0005836.0m) 3C)(0.00833J/kg. 235)(kg/m 00(10,5

C. W/m12

)s 0005836.0(

1-3

2

1-tee

TTTtT

Lch

chAb

tbt

i

cpp ρρ V

Cube:

1.0 00023.0)C W/m.429(

)m 008333.0)(C. W/m12(

m 008333.06

m 05.066

2

2

3

surface

<=°

°==

=====

khL

Bi

LLL

AL

c

cV

min 40.5==⎯→⎯=−−

⎯→⎯=−−

=°

°===

−−

∞

∞ s 24283303325)(

s 0005836.0m) 3C)(0.00833J/kg. 235)(kg/m 00(10,5

C. W/m12

)s 0005836.0(

1-3

2

1-tee

TTTtT

Lch

chAb

tbt

i

cpp ρρ V

Rectangular prism:

1.0 00023.0)C W/m.429(

)m 008108.0)(C. W/m12(

m 008108.0m) 06.0(m) 05.0(2m) 06.0(m) 04.0(2m) 05.0(m) 04.0(2

m) 06.0(m) 05.0(m) 04.0(

2surface

<=°

°==

=++

==

khL

Bi

AL

c

cV

min 39.4==⎯→⎯=−−

⎯→⎯=−−

=°

°=

==

−−

∞

∞ s 23633303325)(

s 0005998.0m) 8C)(0.00810J/kg. 235)(kg/m 00(10,5

C. W/m12

)s 0005998.0(

1-3

2

1-tee

TTTtT

Lch

chAb

tbt

i

cpp ρρ V

The heating times are same for the sphere and cube while it is smaller in rectangular prism.

5 cm Air h, T∞

Air h, T∞ 5 cm

5 cm

5 cm

Air h, T∞ 5 cm

6 cm

4 cm