|Thermodynamics and Propulsion|
transient response to heat transfer is critical. An example
is the heating up of gas turbine compressors as they are brought up
to speed during take-off. The disks that hold the blades are large
and take a long time to come to temperature, while the casing is
thin and in the path of high velocity compressor flow and thus comes
to temperature rapidly. The result is that the case expands away
from the blade tips, sometimes enough to cause serious difficulties
with aerodynamic performance.
To introduce the topic as well as to increase familiarity with modeling of heat transfer problems, we examine a lumped parameter analysis of an object cooled by a stream. This will allow us to see what the relevant non-dimensional parameters are and, at least in a qualitative fashion, how more complex heat transfer objects will behave. We want to view the object as a ``lump'' described by a single parameter. We need to determine when this type of analysis would be appropriate. To address this, consider the temperature difference between two locations in the object, as shown in Figure 18.6.
If the heat transfer within the body and from the body to the fluid are of the same magnitude,
where is a relevant length scale, say half the thickness of the object. The ratio of the temperature difference is
If the Biot number is small the ratio of temperature differences described in Equation (18.14) is also . We can thus say and neglect the temperature non-uniformity within the object.
The approximation made is to view the object as having a spatially uniform temperature that is a function of time only. Explicitly, . The first law applied to the object is (using the fact that for solids ),
where is the density of the object and is its volume. In terms of heat transferred to the fluid, . The rate of heat transfer to the fluid is , so the expression for the time evolution of the temperature is
The initial temperature, , is equal to some known value, which we can call . Using this, Equation (18.16) can be written in terms of a non-dimensional temperature difference ,
At time , this non-dimensional quantity is equal to one. Equation (18.17) is an equation you have seen before, which has the solution . For the present problem the form is
The constant can be seen to be equal to unity to satisfy the initial condition. This form of equation implies that the solution has a heat transfer ``time constant'' given by .
The time constant, , is in accord with our intuition, or experience; high density, large volume, or high specific heat all tend to increase the time constant, while high heat transfer coefficient and large area will tend to decrease the time constant. This is the same form of equation and the same behavior you have seen for the R-C circuit, as shown schematically in Figure 18.7. The time dependence of the voltage in the R-C circuit when the switch is opened suddenly is given by the equation
There are, in fact, a number of physical processes which have (or can be modeled as having) this type of exponentially decaying behavior.
In the lumped parameter transient heat transfer problem, does a high density ``slow down'' heat transfer? (MP 18.6)