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Theory
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The purpose of this study was to characterize the differential equation
that governs the temperature of the tantalum foil plasma calorimeter. In particular, the exponential heating and cooling time decay dependence
on the power loss due to conduction were investigated in the limit of low
powers.
The
power incident on the calorimeter foil consists of four parts: the power from
the beam, power lost to radiation, power lost to conduction, power incident
on the foil from ambient temperature. Therefore, the temperature of the foil
as a function of time is governed by the following differential equation,
(1)
where
the temperature change is defined as
, Pinc is the incident
power, A is the area of the foil,
s is the Stefan Boltzmann constant, e is the emissivity of the
foil, k is a constant which characterizes
the power lost to thermal conduction, T is the temperature of the foil, and T0 is the temperature of the surroundings, or the ambient
(room) temperature.
This differential equation can be rewritten in terms of dimensionless
variables as
(2)
where
,
,
, and
.
Here y is a dimensionless variable which is proportional to DT, x is a dimensionless variable which is proportional to t, B a dimensionless variable which is proportional to the incident power, and C is a dimensionless variable which is proportional to the power loss to thermal conduction.
While the differential equation (1) can not be solved analytically, we can approximate equation (1) when the temperature change of the calorimeter foil is small compared to room temperature (DT << T0) as
. (3)
We can rewrite equation (3) in terms of the dimensionless variables as they are defined for equation (2) as
.
This equation can be solved analytically. The solution in the case where a beam of power Pinc is incident on the calorimeter and the initial temperature of the calorimeter is equal to the ambient temperature is:
and the solution in the case where the calorimeter has been heated to an equilibrium temperature and is allowed to cool off with Pinc = 0 is given by:
.
These solutions can be rewritten in the heating case as
(4)
and in the cooling case as
(5)
where kheating and kcooling are the dimensionless decay constants in the heating and cooling cases respectively.
Results
The computer algebra system Maple was used to generate an array of numerical solutions to equation (2) using Maple’s built-in numeric dsolve command. Solutions to equation (2) were generated in both the heating and cooling cases over a range of x values from 0.01 to 2 by increments of 0.01 and B values from 0.01 to 2 by increments of 0.02. This procedure was repeated for a range of C values from 0.1 to 2 by increments of 0.1 and the results of these calculations were saved.
The solutions to equation (2) for several different ranges of dimensionless power (B) as generated by Maple were imported into Excel. Excel’s solver function was used to fit equations (4) and (5) to the numeric solutions to equation (2) using a least squares fitting criterion. A typical example of the solution to equation (2) as generated by Maple is shown for C = 1 and B=1 in figure 1. The line through the data is the result of the least squares fit.
Figure 1: The solution curve to equation (2) for C = 1 and B = 1.
Excel was then used to fit a line to
the fitted values for kheating and kcooling versus B
(the dimensionless power) as shown in figure 2.
The red trend line in the plot is taken over a range of powers from
B = 1 to B = 3 while the green trend line is taken over a range from B = 0
to 2, and the black trend line is taken from B = 0 to 3. The trend lines indicate that as the incident power increases, the
rate of change in the dimensionless decay constants in both the heating decay
constants tends to decrease (ie:
). This means that the value of the
heating and cooling decay constants depends on the range of dimensionless
powers (B) taken and that the two parameters are not linearly related.
Figure 2: kheating and kcooling versus B (dimensionless power).
From figure 2:
Figure 3 is a plot of experimental results of the behavior predicted in figure 2 as a result of a numerical analysis of (eq 1). The theoretical and experimental results are in agreement in the following respects:
The k versus B data in both the theoretical
and experimental cases are roughly linear.
The slope of the heating and cooling
curves tends to decrease in both the heating and cooling case as the incident
power increases.
The heating decay constant (kheating)
> the cooling decay constant (kcooling).
The value of
is similar in the experimental and
theoretical cases.
Figure 3: Experimental results from calorimeter # 31.
Conclusion
The value of the heat loss due to conduction (C) cannot be determined given
a value for the incident power (B) from this analysis alone. This is because:
·
The governing equation (eq 1) is non linear and has no analytic solution.
·
The functional from of the relationship between B and k is not known.
·
The change in k is clearly not linear over all ranges of B and can
therefore not easily be approximated numerically.
The numerical analysis of the governing equation is however in close agreement
with the experimental results obtained for calorimeter # 31.