Burt Rutan's White Knight and SpaceShip One, Photo Courtesy of Scaled Composites
Thermodynamics and Propulsion

18.3 Transient Heat Transfer (Convective Cooling or Heating)

All the heat transfer problems we have examined have been steady state, but there are often circumstances in which the 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 $ T_1 - T_w$ between two locations in the object, as shown in Figure 18.6.

Figure 18.6: Temperature variation in an object cooled by a flowing fluid
Image fig10CooledObject_web

If the heat transfer within the body and from the body to the fluid are of the same magnitude,

$\displaystyle h(T_w - T_\infty) \approx \frac{k}{L}(T_1-T_w),$

where $ L$ is a relevant length scale, say half the thickness of the object. The ratio of the temperature difference is

$\displaystyle \frac{T_1-T_w}{T_w-T_\infty} \approx \frac{hL}{k}.$ (18..14)

If the Biot number is small the ratio of temperature differences described in Equation (18.14) is also $ (T_1-T_w)/(T_w-T_\infty) \ll 1$ . We can thus say $ (T_1-T_w) \ll T_w
- T_\infty$ 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, $ T
= T(t)$ . The first law applied to the object is (using the fact that for solids $ c_p = c_v = c$ ),

$\displaystyle \dot{Q}_\textrm{in} = \rho V c \frac{dT}{dt},$ (18..15)

where $ \rho$ is the density of the object and $ V$ is its volume. In terms of heat transferred to the fluid, $ \dot{Q}_\textrm{out} =
-\rho VcdT/dt$ . The rate of heat transfer to the fluid is $ Ah(T -
T_\infty)$ , so the expression for the time evolution of the temperature is

$\displaystyle Ah(T -T_\infty)=-\rho Vc \frac{dT}{dt}.$ (18..16)

The initial temperature, $ T(0)$ , is equal to some known value, which we can call $ T_i$ . Using this, Equation (18.16) can be written in terms of a non-dimensional temperature difference $ (T-T_\infty)/(T_i-T_\infty)$ ,

$\displaystyle \frac{d}{dt}\left(\frac{T-T_\infty}{T_i-T_\infty}\right) + \frac{hA}{\rho Vc}\left(\frac{T-T_\infty}{T_i-T_\infty}\right) = 0.$ (18..17)

At time $ t=0$ , this non-dimensional quantity is equal to one. Equation (18.17) is an equation you have seen before, $ \left(\frac{dx}{dt}+\frac{x}{\tau}=0\right)$ which has the solution $ x = ae^{-t / \tau}$ . For the present problem the form is

$\displaystyle \frac{T-T_\infty}{T_i-T_\infty} =ae^{-hAt/\rho Vc}.$ (18..18)

The constant $ a$ 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 $ \tau =
\rho V c/hA$ .

The time constant, $ \tau$ , 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

$\displaystyle \frac{dE}{dt}+\frac{E}{RC} = 0.$

There are, in fact, a number of physical processes which have (or can be modeled as having) this type of exponentially decaying behavior.

Figure 18.7: Voltage change in an R-C circuit
Image fig10CircuitWithExponentialVoltageDecay_web

Muddy Points

In equation $ \dot{Q}_\textrm{in} =\rho V c
dT/dt$  (18.15), what is $ c$ ? (MP 18.5)

In the lumped parameter transient heat transfer problem, does a high density ``slow down'' heat transfer? (MP 18.6)