A spherical shell of inner and outer radii r 1 and r o , respectively, is filled with a heat-generating material that provides for a uniform volumetric generation rate ( W/m 3 ) of q . . The outer surface of the shell is exposed to a fluid having a temperature T ∞ and a convection coefficient h. Obtain an expression for the steady-state temperature distribution T ( r ) in the shell, expressing your result in terms of r i , r o , q . , h , T ∞ and the thermal conductivity k of the shell material.
A spherical shell of inner and outer radii r 1 and r o , respectively, is filled with a heat-generating material that provides for a uniform volumetric generation rate ( W/m 3 ) of q . . The outer surface of the shell is exposed to a fluid having a temperature T ∞ and a convection coefficient h. Obtain an expression for the steady-state temperature distribution T ( r ) in the shell, expressing your result in terms of r i , r o , q . , h , T ∞ and the thermal conductivity k of the shell material.
Solution Summary: The author explains the one-dimension thermal differential equation of the sphere with the internal heat generation.
A spherical shell of inner and outer radii
r
1
and
r
o
,
respectively, is filled with a heat-generating material that provides for a uniform volumetric generation rate
(
W/m
3
)
of
q
.
.
The outer surface of the shell is exposed to a fluid having a temperature
T
∞
and a convection coefficient h. Obtain an expression for the steady-state temperature distribution
T
(
r
)
in the shell, expressing your result in terms of
r
i
,
r
o
,
q
.
,
h
,
T
∞
and the thermal conductivity k of the shell material.
A plane wall of thickness 8cm and thermal conductivity k=5W/mK experiences uniform volumetric heat generation, while convection heat transfer occurs at both of its surfaces (x= -L, x= + L), each of which is exposed to a fluid of temperature T∞ = 20˚C. The origin of the x-coordinate is at the midplane of the wall. Under steady-state conditions, the temperature distribution in the wall is of the form T(˚C) = a + bx - cx^2, where x is in meters, a =86˚C, b = -500˚C/m, and c=4459.
1) Heat Flux Entering the wall is ?
2) Temperature at the left face is /
Which formula is used to calculate the heat conduction in the AXIAL direction in a
vertically located pipe segment whose inner and outer surfaces are perfectly
insulated. Here r, is inner radius, r, outer radius, Tri pipe inner surface temperature,
Tro pipe outer surface temperature, L is the length of the pipe, T the temperature on
the lower surface, Ty the temperature on upper surface.
Tu
r;
Tro
r
You are asked to estimate the maximum human body temperature if the metabolic
heat produced in your body could escape only by tissue conduction and later on the surface by
convection. Simplify the human body as a cylinder of L=1.8 m in height and ro= 0.15 m in
radius. Further, simplify the heat transfer process inside the human body as a 1-D situation when
the temperature only depends on the radial coordinater from the centerline. The governing
dT
+q""=0
dr
equation is written as
1 d
k-
r dr
r = 0,
dT
dr
=0
dT
r=ro -k -=h(T-T)
dr
(k-0.5 W/m°C), ro is the radius of the cylinder (0.15 m), h is the convection coefficient at the
skin surface (15 W/m² °C), Tair is the air temperature (30°C). q" is the average volumetric heat
generation rate in the body (W/m³) and is defined as heat generated per unit volume per second.
The 1-D (radial) temperature distribution can be derived as:
T(r) =
q"¹'r² qr qr.
+
4k 2h
+
4k
+T
, where k is thermal conductivity of tissue
air
(A) q" can be calculated…
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