Consider a spherical reactor of 5-cm diameter operating at steady conditions with a temperature variation that can be expressed in the form of T ( r ) = a − b r 2 , where a = 850 o C and b = 5 × 10 5 K/m 2 . The reactor is made of material with c = 200 J/kg . o C, k = 40 W/m .K, p = 9000 kg/m 3 . If the heat generation of the reactor is suddenly set to 9 MW/m 3 , determine the time rate of temperature change in the reactor. Is the heat generation of the reactor suddenly increased or decreased to 9 MW/m 3 from its steady operating condition?
Consider a spherical reactor of 5-cm diameter operating at steady conditions with a temperature variation that can be expressed in the form of T ( r ) = a − b r 2 , where a = 850 o C and b = 5 × 10 5 K/m 2 . The reactor is made of material with c = 200 J/kg . o C, k = 40 W/m .K, p = 9000 kg/m 3 . If the heat generation of the reactor is suddenly set to 9 MW/m 3 , determine the time rate of temperature change in the reactor. Is the heat generation of the reactor suddenly increased or decreased to 9 MW/m 3 from its steady operating condition?
Solution Summary: The author explains the time rate of temperature change in the reactor.
Consider a spherical reactor of 5-cm diameter operating at steady conditions with a temperature variation that can be expressed in the form of
T
(
r
)
=
a
−
b
r
2
,
where
a
=
850
o
C and b = 5
×
10
5
K/m
2
. The reactor is made of material with
c
=
200
J/kg
.
o
C, k = 40 W/m
.K, p = 9000 kg/m
3
. If the heat generation of the reactor is suddenly set to 9 MW/m3 , determine the time rate of temperature change in the reactor. Is the heat generation of the reactor suddenly increased or decreased to 9 MW/m3 from its steady operating condition?
Problem 5 (Optional, extra 6 points)
A 6-lb homogeneous disk of radius 3 in. spins as shown at the constant rate w₁ = 60 rad/s. The disk
is supported by the fork-ended rod AB, which is welded to the vertical shaft CBD. The system is
at rest when a couple Mo= (0.25ft-lb)j is applied to the shaft for 2 s and then removed. Determine
the dynamic reactions at C and D before and after the couple has been removed at 2 s.
4 in.
C
B
Mo
5 in
4 in.
Note: 2 rotating around CD induced by Mo is NOT
constant before Mo is removed.
and ₂ (two
unknowns) are related by the equation: ₂ =0+ w₂t
3 in.
Partial Answer (after Mo has been removed):
C-7.81+7.43k lb
D -7.81 7.43 lb
Problem 4.
A homogeneous disk with radius and mass m is mounted on an axle OG with length L and a
negligible mass. The axle is pivoted at the fixed-point O, and the disk is constrained to roll on a
horizontal surface. The disk rotates counterclockwise at the constant rate o₁ about the axle. (mg
must be included into your calculation)
(a). Calculate the linear velocity of G and indicate it on the figure.
(b). Calculate ₂ (constant), which is the angular velocity of the
axle OG around the vertical axis.
(c). Calculate the linear acceleration ā of G and indicate it on the
figure.
(d). Determine the force (assumed vertical) exerted by the floor on
the disk
(e). Determine the reaction at the pivot O.
1
Answers: N = mg +mr(r/L)² @² |j
mr w
IIG
C
R
L
i+
2L
=
Problem 2.
The homogeneous disk of weight W = 6 lb rotates at the constant rate co₁ = 16 rad/s with respect
to arm ABC, which is welded to a shaft DCE rotating at the constant rate 2 = 8 rad/s. Assume
the rod weight is negligible compared to the disk. Determine the dynamic reactions at D and E
(ignore mg).
Answers:
D=-7.12ĵ+4.47k lb
r-8 in.
9 in.
B
D
E=-1.822+4.47 lb
9 in.
E
12 in.
12 in.
x
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