Measurement of the Dynamic Response of a Contact Probe Thermosensor in Conductive Media

This paper describes a method for characterizing the step response of a thermistor probe embedded in a low-conductivity solid. We define the “step response” as the dynamic response of a finite-size thermosensor instantaneously plunged into an infinite homogeneous conductive solid. The final goal of this research is to evaluate and enhance the time-dependent response of contact-type thermosensors. We will use the step response as the parameter for optimizing the probe timedependent behavior. Although our research focuses on thermistors, the results could be applied to other contact-type sensors like thermocouples and RTD’s. Currently, there is no direct way for determining the step response of the probe in such a case, since the probe can not be instantaneously plunged into a solid. In this paper, we describe an indirect experimental method for determining the step response of the probe. It is achieved by self-heating the thermistor and analyzing its temperature response. The success of this approach results from the fact that the heat transfer processes controlling self-heating are the same as the processes controlling the step response. In this paper, we present an analytical expression for the step response of a spherical probe in a conductive solid. A relationship between the step response and the thermistor response to a step power self-heating is developed. Finally, a simple experimental method for determining the step response from the self-heating response is presented. INTRODUCTION The typical way to characterize the transient response of a temperature probe is the water-plunge test. In this test, the sensor, at a certain temperature, is plunged into water at another temperature flowing at a standard speed. Since this test involves the response of the sensor to a sudden change of the temperature surrounding the sensor, the plunge response is often called step-response. The water-plunge test, however, gives limited information concerning the behavior of the probe in the actual measurement situation, since the test conditions are generally different from the measurement conditions. The speed and the thermal properties of the fluid surrounding the probe in the measurement may be different from the speed and thermal properties of the fluid in the test. Differences are more significant when the measurement is performed by a probe embedded in a solid. In this case, the probe response strongly depends on the thermal properties of the medium in which the probe is located (Valvano and Yuan, 1992). Thus, the waterplunge test gives a response which is significantly different from the response of a probe embedded in a solid. The intuitive way to get the true response of a probe in this case would be to "plunge" the thermistor into a solid with similar thermal properties. Agar-gelled water can be used to simulate tissue. However, this test is not practical in most cases. A typical application of the conduction dominated environment is the thermal response of an oral thermometer. Other application include externally heated tissue using laser, ultrasound, or EM. It is clear from the previous discussion that a method to determine the actual time response of a temperature probe in tissue would be useful. A great deal of research and development concerning methods for the in situ measurement of the time response of temperature sensors has been performed by Kerlin et al. (1980, 1981, 1984, 1982(a), 1982(b)). However, Kerlin's methods were specific for sensors in convective media. NOMENCLATURE a = thermistor radius (cm) b = α α b m m b k k ⋅ c = 1− k k m b / cb = specific heat of the probe material kb = thermal conductivity of probe bead (W/cm K) km = thermal conductivity of medium (W/cm K) Q = heat deposited in the probe (J/cm3) r = radius coordinate (cm) t = time (s) T = temperature rise inside the bead (oC) To = reference initial temperature of medium (K) Tb = temperature rise of probe ( oC) Tm = temperature rise of medium ( oC) Tplunge = average temperature inside the bead ( oC) Tself = average temperature inside the bead during selfheating Tsuperp. = temperature due to superposition of plunge responses y = integration variable αb = thermal diffusivity of probe bead (cm 2/s) αm = thermal diffusivity of medium (cm 2/s) Γ = rate of volumetric heat generation (W/cm3) ∆Timp = increment in temperature due to unit impulse generation λ = integration variable for convolution operation ρb = density of the probe material THE STEP RESPONSE OF A SPHERICAL TEMPERATURE PROBE EMBEDDED IN TISSUE The probe is modeled by a sphere of radius a, thermal conductivity kb and thermal diffusivity αb. The medium is modeled as an infinite medium with thermal conductivity kb and thermal diffusivity αm. The temperature variables Tb(r,t) and Tm(r,t) are referred to an initial basal temperature in the probe (T0): T r t T r t T b b ( , ) ( , ) , = − 0 0 (1) T r t T r t T m m ( , ) ( , ) , = − 0 0 (2) The differential equation governing the system is the heat transfer equation in spherical coordinates applied to the thermistor and the medium: 1 1 2 2 r r r r T r t T r t t b b b ∂ ∂ ∂ ∂ α ∂ ∂ ( ( , )) ( , ) = for 0 < < r a (3) 1 1 2 2 r r r r T r t T r t t m m m ∂ ∂ ∂ ∂ α ∂ ∂ ( ( , )) ( , ) = for a r < < ∞ (4) The initial conditions are described by: T r V b ( , ) 0 = for 0 < < r a (5a) T r m( , ) 0 0 = for a r < < ∞ (5b) The boundary conditions are described by: T at T a t b m ( , ) ( , ) = for t > 0 (6.a) T t m ∞ = , 0 b g for t > 0 (6.b) k T r t r k T r t r b b