3 Engineering Example 2
3.1 Heat flow in an insulated metal plate
Introduction
Thermal insulation is important in many domestic (e.g. central heating) and industrial (e.g cooling and heating) situations. Although many real situations involve heat flow in more than one dimension, we consider only a one dimensional case here. The flow of heat is determined by temperature and thermal conductivity. It is possible to model the amount of heat (J) crossing point in one dimension (the heat flow in the direction) from temperature (K) to temperature (K) (in which ) in time s by
,
where is the thermal conductivity in .
Problem in words
Suppose that the upper and lower sides of a metal plate connecting two containers are insulated and one end is maintained at a temperature (K) (see Figure 7).
The plate is assumed to be infinite in the direction perpendicular to the sheet of paper.
Figure 7:
- Find a formula for .
- If ( ), (K), ), (m), calculate the value of required to achieve a heat flow of .
Mathematical statement of the problem
- Given express as the subject of the formula.
-
In the formula found in part
- substitute , and to find .
Mathematical analysis
-
Divide both sides by
Multiply both sides by
Add to both sides
which is equivalent to
-
Substitute
and
to find
:
So the temperature in container 2 is to 3 sig.fig.
Interpretation
The formula can be used to find a value for that would achieve any desired heat flow. In the example given would need to be about K ( ) to produce a heat flow of .
Exercises
- The formula for the volume of a cylinder is . Find when cm and cm.
-
If
, find
when
- ,
- .
-
For the following formulae, find
at the given values of
.
- , , , .
- , , , .
- , , , , , .
- If find if and .
- If find if and .
- Evaluate when and .
-
To convert a length measured in metres to one measured in centimetres, the length in metres is multiplied by 100. Convert the following lengths to cm.
- 5 m,
- 0.5 m,
- 56.2 m.
-
To convert an area measured in
to one measured in
, the area in
is multiplied by
. Convert the following areas to
.
- ,
- ,
- .
-
To convert a volume measured in
to one measured in
, the volume in
is multiplied by
. Convert the following volumes to
.
- ,
- ,
- .
- If evaluate when , , and .
-
The moment of inertia of an object is a measure of its resistance to rotation. It depends upon both the mass of the object and the distribution of mass about the axis of rotation. It can be shown that the moment of inertia,
, of a solid disc rotating about an axis through its centre and perpendicular to the plane of the disc, is given by the formula
where is the mass of the disc and is its radius. Find the moment of inertia of a disc of mass 12 kg and diameter 10 m. The SI unit of moment of inertia is .
-
Transpose the given formulae to make the given variable the subject.
- , for ,
- , for ,
- for ,
- for .
-
Transpose the formula
for
- ,
- ,
- ,
- .
-
Transpose
,
- for ,
- for .
- Transpose for each of , and .
-
When a ball is dropped from rest onto a horizontal surface it will bounce before eventually coming to rest after a time
where
where is the speed immediately after the first impact, and is a constant called the acceleration due to gravity. Transpose this formula to make , the coefficient of restitution, the subject.
- Transpose for .
-
Make
the subject of
- ,
- .
-
In the design of orifice plate flowmeters, the volumetric flowrate, Q (
), is given by
where is a dimensionless discharge coefficient, (m) is the head difference across the orifice plate and ( ) is the area of the orifice and ( ) is the area of the pipe.
- Rearrange the equation to solve for the area of the orifice, , in terms of the other variables.
-
A volumetric flowrate of
passes through a
cm inside diameter pipe. Assuming a discharge coefficient of
, calculate the required orifice diameter, so that the head difference across the orifice plate is
mm.
[Hint: be very careful with the units!]
-
- ,
-
- ,
- ,
- ,
-
- ,
- ,
- .
-
- ,
- ,
- .
-
- ,
- ,
- .
- .
-
- ,
- ,
- ,
-
- ,
- ,
- ,
-
- ,
- , ,
-
- ,
-
-
m
Substituting in answer
- gives so diameter