Question

To determine the effect of the temperature dependence of the thermal conductivity on the temperature distribution in a solid, consider a material for which this dependence may be represented by: k = k0 + a T, where “k0“ is a positive constant and “a” is a coefficient that may be positive or negative. Starting with a steady-state energy balance, derive a relationship for temperature (T) as a function of distance (x) from the lower temperature wall. You may assume that there is no heat generation within the wall.

Answer #1

The thermal conductivity of a sheet of rigid,extruded insulation
is k = 0.020 W/(m*K). The measured temperature difference across a
30-mm-thick sheet of the material is T1-T2=10 C.
a.) Assuming 1-D, steady state conditions withouth thermal
energy generation in the material, what is the heat flux through a
2 m x 3 m sheet of the insulation?
b.) What would be the effect on the rate of heat transfer
through the sheet if a material with a relatively higher thermal...

A nuclear fuel element consists of a solid cylindrical pin of
radius, r1 and thermal conductivity, kf. The fuel pin is in good
contact with a cladding material of radius, r2 and thermal
conductivity, kc. Consider steady state condition for which uniform
heat generation occurs within the fuel at a volumetric rate, g
(w/m3 ) and the outer surface of the cladding is exposed to a
coolant that is characterized by temperature, Tsur and convection
coefficient, h.
a) Obtain equations...

A
40-mm-thick plain wall of thermal conductivity 0.04 W/m-K and of
temperature 304 K is exposed to both cold fluid and surroundings
both at 298 K. The cold fluid and wall provide that the heat
transfer coefficient is 10.5 W/m2-K and the emissivity of the wall
is 0.83. Given that the area of the wall is 1.5m2 If the total heat
transferred is 140.6987W, find the temperature of the other side of
the plain wall

The steady-state temperature distribution inside a solid object
is described by the following expression, where x is the spatial
co-ordinate: T(x) =2x3- 3x2 + x +10
If the thermal conductivity and thickness of the solid are 10
W.m-1.K-1 and 0.8 m respectively, what will
be the form of heat flux expression?
At what point within the solid does the heat flux reach a
maximum (or minimum)?

Consider a cube of density, specific heat, and thermal
conductivity of 2700 kg/m3, 0.896 kJ/kg-K, and 204 W/m-K,
respectively. The cube is 5 cm in length, and is initially at a
temperature of 20 oC. For t>0, two of the boundary surfaces are
insulated, two are subjected to uniform heating at a rate of 10,000
W/m2, and two dissipate heat by convection to an ambient
temperature of 20 oC, with a heat transfer coefficient of 50
W/m2-K.
Assuming lumped capacitance...

Consider a large uranium plate of thickness 5 cm and thermal
conductivity k = 28 W/m K in which heat is generated uniformly at a
constant rate of q˙ = 6 × 10^5 W/m^3 . One side of the plate is
insulated while the other side is subjected to convection in an
environment at 30◦C with a heat transfer coefficient of h = 60 W/m2
K. Considering six equally spaced nodes with a nodal spacing of 1
cm,
(a) Sketch...

To determine the thermal conductivity of a long, 25.0 mm
diameter cylindrical rod, one half of the rod was inserted into a
furnace while the other half projected into air at 27.0 degrees C.
After steady state had been reached, the temperatures at two points
76.0 mm apart were measured and were found to be 126 degrees C. and
91.0 degrees C. The convective heat transfer coefficient of the
surface of the rod exposed to the air was 22.7 W/m^2...

The initial temperature distribution
across a wall with a thickness of 60 cm is described with the
following function:
T(x) = -250x+100x2+
20x3/ 2x
where T is temperature in
degrees Celsius and x is the coordinate in meters.
The wall has a thermal conductivity of
2.5 W cm-1 K-1.
(i) On a unit surface
area basis, determine the heat flux into the wall on the left
face.
(ii) On a unit surface area
basis, determine the heat flux out of...

Gas with a thermal conductivity of k = 0.04 W/mk and prandtl
number of 0.7 is flowing at anaverage velocity of 0.5 m/s through a
2 mm diameter tube at a Reynolds number of 50. The heat capacity
rate for the gas flowing through the tube is 0.001 kJ/s. The inlet
temperature of the gas is 20C. There is a constant heat flux of 200
W/m^2 transferred from the tube wall into the gas.
a) What is the Nu at...

71 mm thick horizontal plate made of an opaque material of
thermal conductivity k= 25 W/m-K. At the bottom, water flows at a
temperature of T∞,w = 25C, whereas air flows at the top of the
plate at T∞,a = 710 C having convection coefficient of ha= 71
W/m2-K. Assuming a diffused top of the plate that receives an
irradiated flux of 7100W/m2, of which 30% is reflected back. The
top and bottom surface temperatures are maintained at 43 C...

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