Transfer function methods

Figure 11P.1 can be used to determine the dimensionless dispersion parmeter ( l/uL) for a system of interest. Use the transfer function method to evaluate the mean residence time and QjJuL) for a system subjected to the arbitrary input shown in the figure. Note that the output response has been shifted 62.5 sec to the left. Response values for the input and output streams were as follows. 

Another passive method is the transference function method (TFM) introduced by Muramatsu [6]. The method consists of an oscillator that drives a crystal through a known measuring impedance and a radiofrequency voltmeter which measures the transference modulus of the system. Muramatsu [6] neglected the effect of the parasitic capacitance and his expression for the quartz impedance resulted in a nonlinear relationship between the measured resistance R with the ac voltage divider and the value of R measured by an impedance analyser. Calvo and Etchenique [74] improved the method and introduced an analytical expression to fit the entire transfer function around resonance in order to obtain the same values of R, L and C as measured by a frequency response analyser. 

A flow sheet schematics of the real time transfer function method is depicted in Fig. 12.5. The BVD equivalent electrical circuit elements R, L and C are obtained by nonlinear fit of each transfer function spectra,... 

Fig. 12.5. Flowchart describing the voltage divider transfer function method (TFM) Real time measurement and nonlinear fit to the BVD equivalent electrical circuit.

The methods described in this chapter and this book apply to electrochemical impedance spectroscopy. Impedance spectroscopy should be viewed as being a specialized case of a transfer-function analysis. The principles apply to a wide variety of frequency-domain measurements, including non-electrochemical measurements. The application to generalized transfer-function methods is described briefly with an introduction to other sections of the text where these methods are described in greater detail. Local impedance spectroscopy, a relatively new and powerful electrochemical approach, is described in detail. 

A parallel development has taken place for related transfer-fimction methods. For electrochemical systems, impedance spectroscopy, which relies on measurement of current and potential, provides the general system response. As described in Chapters 14 and 15, transfer-function methods allow the experimentalist to isolate the portion of the response associated with specific inputs or outputs. 

M. J. Hopkins, A. J. Sheppard, and P. Eisenklam [Chem. Eng. Sci., 24, 1131 (1969)] have indicated that the data in Figure PI 1.24 can be used to determine the dimensionless dispersion parmeter, TiJuL. Use the transfer function method to... 

Equation (7.3) is the reflection coefficient for the two microphone transfer function method. When the specimen is backed by a rigid back waU, there wiU be no transmitted waves and thus by conservation of energy, all the incident waves are reflected and absorbed. Thus the absorption coefficient ( i), can be expressed as in... 

One of the drawbacks of the two-microphone transfer function method is that the absorption coefficient determined may not be a true representation of the material s characteristic. In the case of a porous material, such as silica aerogels, the reflected wave from the rigid wall could contribute to a rise in the absorbed energy by the material. To account for this uncertainty, the four-microphone impedance tube setup is usually used to determine the transmission loss (TL) and absorption coefficient (Feng 2013). In the absence of additional microphones downstream of the specimen, a sound meter could be used instead to measure the TL of the specimen under test. However, the sound meter picks up discrete transmitted signals at periodic interval, which could result in a mismatch with the generated signals from the source. 

The question is, how does one explain such a low dependence of rate on medium basicity in this system The transfer function method has shed some light on this problem. 

There are many acoustical methods proposed for measuring flow resistivity (Delany and Bazley, 1971 Smith and Parott, 1983). A method that uses a standard impedance tube directly to measure the static flow resistivity without any additional requirements to tube modification or sensor location change is described in ISO Standard, 10534-2 (1998) and by Tao et al. (2015). In the method, the specific acoustic impedance on the front surface of the test specimen is measured first by using the traditional transfer function method with the test specimen being placed against and with a known interval to the rigid termination, and then the characteristic impedance, the propagation constant, and the static flow resistivity are calculated based on the obtained impedance transfer functions. 

ISO 10534-2, 1998. Aconstics— Determination of sound absorption coefficient and impedance in impedance tabes— Transfer function method. International Organization for Standardization. 

---