ME 314 Heat and Mass Transfer Laboratory
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EXPERIMENT # 5
 
LUMPED HEAT CAPACITY


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 solid-fluid 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)
 
where q is the heat transferred from the system to the surroundings, C is the specific heat of the metal, As is the total surface area, and m is the mass of the block. If we call the heat out the heat transferred by convection, we get,
 
(2)
 
This equation states that the rate of heat lost by convection from the block surfaces equals the negative of the time rate of change of energy stored in the block.

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)
 
The value, Lc, is the characteristic length which is defined as the solid's volume over the surface area, Lc = V/As. In order to use the Lumped Capacitance Method, the Biot number must be calculated and confirmed to be less than 0.1. Unless this requirement is satisfied, use of this method would create too much error.

Now, if the value of hAs/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/m2K. 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)
 
The individual heat rates can then be added and used in the same equation for the whole cube, and hoverall can be determined.
 
 
(5)
 
However, to find the individual heat rates, the convection coefficients for each face must be found. If we make an approximation by assuming that each face behaves as a vertical or horizontal flat plate, we can use some published free convection correlations for constant temperature (Bi << 0.1) flat plates. For the four vertical faces hvert can be found by using the Nusselt Number,
 
(6)
 
Where Lc is the characteristic length, not the same as defined earlier. Be extra careful when using equations that contain Lc; make sure that the Lc you use is correctly defined, and is is the corresponding Lc for that equation. The value of Nu for free convection on a vertical surface can be found using Equation (9.27) in your textbook.
 
(7)
 
where the Prandtl number can be found from the tables in the back of your textbook, and the Rayleigh number is defined as,
 
(8)
 
where n and a can be found from the tables in the back of your textbook, and for an ideal gas b = 1/Tfilm. The Film temperature is the average temperature between the surface and ambient. All properties of air are evaluated at the film temperature.

Lc for vertical surfaces is the length of the surface in the vertical direction. For horizontal plates, Lc is defined as the area/perimeter of the horizontal face.

For the bottom surface with free convection Equation (9.32) in your textbook:
 
(9)
 
 and for the upper surface Equation (9.30) in your textbook:
 
(10)
 
can be used. Make a table in your results section, like the example given, comparing these expected and experimental h values. Make sure when calculating the Nu and Ra, and h for each surface that you are using the correct Lc.

 
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:
 

  1. Bring a small container of water to a boil using an electric hot plate. The larger the mass of water, the longer it will take it to boil.

  2.  
  3. Set up the data acquisition system by the supplemental instructions (ask lab instructor).

  4.  
  5. Suspend the first block in the boiling water and obtain temperature data as a function of time as the block heats up.

  6.  
  7. Once the block has reached a steady temperature, remove it from the boiling water, quickly dry the remaining water from its surface and suspend it in the air.

  8.  
  9. Take temperature versus time data as the block cools while it is suspended in the air.

  10.  
  11. It will not be necessary to make several runs for this experiment; just one for each is sufficient. Make sure you always start with the same initial block temperature during these additional runs.

  12.  
  13. Repeat the above procedures for the second block.

  14.  
  15. Discuss whether assumptions made are valid (keep in mind all approximations used, especially in using the equations and finding slopes, etc). Comment on the sources of error. Keep in mind the correlations used are models of a slightly different physical situation. Also discuss errors, if any, within the equipment and methodology of the experiment.