



OBJECTIVE:
The objective of this experiment is to study unsteady state heat transfer
by the lumped capacitance method and to determine and compare heat transfer
coefficients with expected values.
INTRODUCTION:
Up to this point, we have studied only steady state heat transfer where the heat transfer rate is constant or does not change with time. However, many situations arise where steady state is not prevalent, and the analysis becomes much more difficult. It is in these situations where unsteady heat flow causes temperature and other variables to change with time. We will see, however, that in some unsteady situations, for which a certain criterion is met, the use of the lumped capacitance theory greatly simplifies the analysis. The criterion as we will see is based on the assumption that temperature gradients within a solid are negligible compared to the temperature gradients between the solid and fluid. To verify whether this assumption is true or not depends on calculating the Biot number which we will define later.
For this experiment, we will use the lumped heat capacity approximation to determine the convection heat transfer coefficients of two specimens, through experimental measurements, and the through the use of free convection correlations. The experimental measured values can be compared to the values calculated by the equations.
THEORY:
To understand the lumped heat capacity theory we must start by considering
a hot metal block that is submerged in water. The essence of this theory
is that the temperature within the solid block is assumed to be spatially
uniform at any instant throughout the unsteady cooling process. This implies
that the temperature gradient within the solid is negligible compared to
that of the gradient across the solidfluid interface. The nice thing about
this assumption is that we no longer have to use the heat equation analysis,
which becomes very difficult. Instead we can use the energy balance method
on the solid. Writing the first law for unsteady heat flow with no work
we have,

(1)


(2)

Let us now define the Biot number. The Biot number, Bi, can be defined
as the ratio of temperature differences across the solid itself, and between
the solid and fluid. It can also be defined as the ratio of thermal resistance
of the solid by conduction and the fluid by convection.

(3)

Now, if the value of hA_{s}/Cm in Equation (2), was constant with respect to time, Equation (2) could be integrated to yield Equation 5.6 in the textbook. However, this should not be assumed and for this experiment a plot of T versus t should be made. Using this plot, we can determine dT/dt and T at a time value and then calculate h from Equation (2). By repeating this procedure at several different times, we obtain data to plot h versus block temperature for both blocks during the heating and cooling modes (4 plots). This is the method by which the experimental convection heat transfer coefficient (h) is found.
For this experiment we are to compare the h values with expected values. Note that we do not have boiling on the block surfaces so the h during heating should not be compared with that for a surface on which boiling is taking place. It is difficult to find expected h values for the heating process but past runs have yielded results between 1100  3400 W/m^{2}K. Just qualitatively compare your results to this range.
To find expected values of h for cooling at a particular time,
we must find the heat rate for each face of the block using the heat equation
for convection,

(4)


(5)


(6)


(7)


(8)

L_{c} for vertical surfaces is the length of the surface in the vertical direction. For horizontal plates, L_{c} is defined as the area/perimeter of the horizontal face.
For the bottom surface with free convection Equation (9.32) in your
textbook:

(9)


(10)

APPARATUS:
The apparatus for this experiment includes a hot plate, metal cylindrical container of water, timer, potentiometer, and two metal block specimens. The hot plate is used to heat the container of water to boiling where temperature becomes constant.
The data acquisition system, like the one used in previous experiments, is used to measure the temperature of each specimen and the time as the specimen heats up and cools down. The two specimens are made of copper and aluminum or steel respectfully and both have a thermocouple imbedded within them.
PROCEDURE: