Theory

Andrew Cunningham
Solving the Equation of Temperature Change of the Tantalum Plasma Calorimeter.
11/26/99

Introduction

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. 

 

Theory

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,

 

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.