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...

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...

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...

A solid cylinder of radius R is well insulated at both
ends, and its exterior surface at r R is held at a fixed
temperature, TR. Heat is generated in the solid at a rate per unit
volume given by q = r(1-r/R), where「= constant. The thermal
conductivity of the solid may be assumed constant. Use the
conduction equation together with arn appropriate set of boundary
conditions to derive an expression for the steady- state
temperature profile, T(r), in the...

The fuel element of a nuclear reactor is in the shape of a plane
wall of thickness L = 20 mm. It is being maintained at a constant
temperature of 250ºC on both of its surfaces. At normal operating
power, heat is generated uniformly within the element at a
volumetric rate of q = 107 W/m3. A departure from the steady-state
conditions associated with normal operation will occur if there is
a change in the generation rate. Consider a sudden...

Air at freestream velocity U∞=16
ms, and free steam temperature
T∞=50℃ is
flowing over a plate surface that is at temperature
Ts=100℃. The velocity and thermal
boundary layers developing on the surface have been shown in the
figure. Also shown are the tangents to the velocity and temperature
profiles at the surface y=0. If the density of air is
ρ∞=1.1
kgm3, viscosity
μ=1.963×10-5kgm∙s, and
thermal conductivity k=0.0274Wm∙K then,
calculate
(a) the wall shear
stress, τw
(b) coefficient of skin
friction,...

A copper bus bar measuring 5cm by 10cm by 2.5m long is in a room
in which the air is maintained at 300K. The bus bar is supported by
two plastic pedestals to which it is attached by an adhesive. The
pedestals are square in cross section, measuring 8 cm on a side.
The pedestals are mounted on a wall whose temperature is 300K. If
1kW of energy is dissipated in the copper bar. What will be its
equilibrium temperature?...

A thin metallic wall may be constructed using copper (properties
are given in the table). The piping is required
to have a radius r = 0.008 m and carries steam at 385 K. The wall
is inside a room surrounded by air at a temperature
of 298 K. The wall is insulated with a material (properties are
given in the table).
Properties
Copper
Insulation material
Thermal Conductivity (W/m per ˚C)
385
0.071
Density (kg/m3)
8940
453
i. If the external...

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