Phase-Shift Dependence of Traveling Electromagnetic Wave on Medium's Refractive Index: A Novel Concept for Simultaneous Sensing of Fluid Concentration/Composition, Temperature, and Pressure

1.1. Motivation

Measuring concentration/composition of fluids is very important in laboratories and industrial process pipelines for monitoring or controlling a certain process. The present work offers a new concept for the instantaneous sensing of composition/concentration in addition to temperature, pressure, and all other intensive fluid properties, if needed. These concepts utilize the principles of electromagnetics in addition to thermodynamic considerations. The unique advantage of using EM techniques over other sensing techniques lies in the speed of measurement owing to the speed of light. In addition, EM sensors need not be in direct contact with the fluid, can be clamped externally (e.g., on a pipe), can be used for small-scale (micro/nano) or large-scale (e.g., tank ), and can be used for moving or static fluids. Moreover, the sensitivity of the EM sensor depends on the working frequency, which can be adjusted as required. The EM sensor can cover the entire range of concentrations and compositions of binary or multifluid systems. The proposed EM technique can be applied to both fluids and solids; however, this study is limited to fluids only.

In previous studies, EM techniques have been successfully employed to sense the volumetric flow rate in pipes or in micro/nano fluidic conduits [1-3]. By applying the proposed concepts, all the flow process variables can be combined into one sensor for instantaneous measurement. Instantaneous measurement means a zero time constant for the sensor, which in turn adds a great advantage to process control for quick and more stable control, in addition to increasing simplicity in the design of complex control systems [4].

1.2. Literature review

Even though EM theory was developed approximately a century and a half ago [5] and the dependence of the phase shift on the refractive index is straightforward, a careful literature review reveals that this method was not used previously to measure concentration or composition of fluids. The proposed sensing concept equivalently uses two techniques which are close but not the same. These two techniques are refractometric sensor [6-8] and permittivity measurement by capacitive sensor [9-11]. The refractometric sensor measures the refractive index using an optical technique and then computes the concentration or composition from the calibrated data of concentration versus refractive index or using a preexisting concentration-refractive index model. The disadvantages of this method are that it requires a visible line of sight between the light-emitting source and the light-detecting sensor, the procedure is lengthy and can be time-consuming, and the components used are expensive and relatively large. Permittivity sensing by capacitive sensors depends on measuring the permittivity of the fluid. From this measurement the concentration can be computed using calibrated data or preexisting models for the permittivity versus concentration, or more practically, from the refractive index because the magnetic permeability is close to 1 for non-magnetic fluids [12], especially at high frequency, the usual excellent assumption is R = $$ \sqrt{{\epsilon }_r}$$. The disadvantage of capacitive sensors is that the capacitor needs to be in direct contact with the fluid. This method is used for, or at least recommended for, small-scale applications because a large distance between the plates of the capacitor results in very low sensitivity; therefore, it is difficult or impossible to detect the variation of permittivity for compositions using the current capability of electronics [13,14]. Because the refractive index is strongly temperature- and pressure-dependent for gases, in either case of refractive index measurement, temperature and pressure are required in conjunction with the refractive index.

The objective of this study is to develop a new EM sensor that is capable of instantaneous measurement of concentration/composition in addition to temperature, pressure, and all other intensive variables, if needed. The theory of the sensor is first introduced, and then the design of the sensor with the interfacing circuit is proposed; the concepts are validated experimentally, and conclusions are drawn.

2.1. Electromagnetic considerations

When an EM wave travels in a medium, its behavior is governed by Maxwell's four Eqs. [15]:

Considering one dimensional EM wave traveling with velocity u in charge free (ρ_v = 0) medium in the z-direction, the solution of Maxwell's equations in Cartesian coordinates is [16]:

and

where, α is the attenuation constant of the medium and β is the phase constant given by

η is the intrinsic impedance given by

for 0 ≤ θ_η ≤ 45° where

The EM wave traveling velocity is given by

For non-conducting fluid σ = 0 or for fluid with σ << ωε Eqs. (7) and (8) become $$ \alpha = 0, \beta = \omega \sqrt{\mu \epsilon }; \left|\eta \right|= \sqrt{\mu /\epsilon }\mathrm{a}\mathrm{n}\mathrm{d} \theta \eta = 0$$. Therefore, Eqs. (5) and (6) become

The refractive index, R, of a medium is defined as the ratio of speed of light in vacuum, $$ C=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}=299792458 \mathrm{m}/\mathrm{s}$$, to the speed of light in the medium, that is

Therefore, Eqs. (12) and (13) become

or

where $$ \theta =\omega \frac{R}{c}z$$ is the phase shift of the EM wave. Therefore, by sending EM wave through fluid at a fixed certain point and receiving it at a second fixed point, the phase shift of the wave at the second point will shift by $$ \omega \frac{R}{c}z$$ depending on the refractive index of the fluid. The refractive index then can be calculated from the measured phase shift θ as

In addition, the amplitude of the magnetic field also depends on the refractive index, $$ \left|H\right|=\frac{{\epsilon }^{4}c{E}_0}{R}$$. This concept can also be utilized to detect R; however, because the amplitude is proportional to ε⁴ and ε is in the order of 10^-12 F/m, it is impossible, or at least impossible, to accurately detect the amplitude. Therefore, the only practical choice for detecting R is phase shift. The sensitivity of the sensor from Eq. (18) can be given as

which is linear with frequency; therefore, the sensitivity can be increased as required by increasing the frequency. This is an excellent advantage in selecting the phase shift as a measure of the refractive index.

2.2. Thermodynamic considerations

This section discusses a mathematical model to obtain the concentration/composition, temperature, pressure, and all other properties, if needed, from the calibration data or a mathematical model of the dependence of the refractive index on the concentration/composition, temperature, pressure, and frequency. According to Gibbs phase rule F = 2 - p + n, where F is the number of degrees of freedom, p is the number of phases, and n is the number of chemical species present in the mixture. The refractive index can be specified by two independent intensive properties in addition to the composition x_i. In addition, it has been experimentally proven that the refractive index is a function of frequency [17-19] (Frequency is not an intensive property; therefore, the Gibbs rule is not violated). Thus, the general case for the refractive index can be selected as

Because $$ {\sum }_{i=1}^n{x}_i=1$$, only n-1 of mole or mass fraction needs to be specified. Differentiating both sides of Eq. (20) yields

where x_j, j ≠ i indicates that all mole fractions from j=1,...,n-1 except component i are held constant. Referring to Eq. (21), we obtain

From which the pressure can be obtained as:

or

From Eq. (18)

Substituting (25), (26), and (27) into (24) yields

Therefore, by sending EM waves at different frequencies and measuring the refractive index at these frequencies, $$ {\left(\frac{\partial \theta }{\partial \omega }\right)}_{T,x}$$ is measured by the sensor, and $$ {\left(\frac{\partial \theta }{\partial \omega }\right)}_{P,T,x}$$ and $$ {\left(\frac{\partial \theta }{\partial P}\right)}_{T,x,\omega }$$ are obtained from the calibration data/models. Hence, the pressure is obtained from Eq. (24) or (28). From Eq. (21), we obtain

In a similar step as above, the temperature can be calculated as

Also, from Eq. (21):

or

Therefore, the composition can be determined using Eq. (32). For example, for binary system Eq. (32) reduces to:

Then

By using Eq. (18) we obtain

and x₂ = 1 - x₁. T₀, P₀, x_1,0, ω₀ all are the values at the base point (center point of calibration). Once the temperature, pressure, and composition are known, the refractive index can be determined from the calibration data/model. All other intensive properties (density, viscosity, enthalpy, etc.), if required, can be computed based on the thermodynamic Gibbs phase rule. This can be achieved using equations of state and/or empirical correlations. Aspen HYSYS software [20] is an excellent tool for this purpose.

2.3. Design considerations

2.3.1. Proposed design of transmitting and receiving antennas

Although many antenna designs have been proposed to transmit and receive EM signals through a fluid to measure the refractive index [21-23], in this study, a simple and practical design is proposed. Fig. 1 shows the proposed design and the circuit used to transmit and receive signals. Circular antennas are selected because they are suitable for circular pipes, particularly for clamping-on configuration. Because EM waves travel through a fluid with the speed of light, which is much higher than the fluid velocity, it makes no difference whether the fluid is static or in motion when a sensor is used. Therefore, the sensor can be used for flow streams or samples collected in tubes for laboratory measurements.

Fig. 1.

Suggested interfacing circuit for each single transmitting/receiving antennas.

The resistance 10 KΩ is required to close the circuit for electricity to flow in the circuit, and simultaneously its value should be much higher than the radiation resistance of the antenna to force the electricity to transmit through the antenna and not through the resistance itself. The order of magnitude of the resistance was estimated by comparing the proposed antenna with a Hertzian dipole antenna. The radiation resistance of Hertzian dipole is in the order of 2 Ω [16]; therefore, the resistance should be much larger than 2 Ω (10 kΩ is believed to be satisfactory for the proposed design). To measure the composition/concentration, temperature, and pressure simultaneously, multitransmitting and corresponding receiving antennas (Fig. 1) must be used. The number of these antennas should be one for temperature, one for pressure, and n - 1 for composition, for a total of 1 + 1 + n - 1 = n + 1 transmitting/receiving antennas (array). Fig. 2 shows the final design of the EM array of the sensor.

Fig. 2.

The proposed sensor design. The number of transmitting/receiving antenna arrays must be (n + 1).

3.1. Experimental setup

Fig. 3 shows the equipment used to validate the proposed concepts and sensor design. Binary Helium (He) and carbon dioxide (CO₂) gases were used for validation. Two cylinders with relatively high pressure of He and CO₂ are connected to a glass cylinder (glass was selected for visibility). A thermometer and pressure gauge were placed inside the glass cylinder to measure the temperature and pressure, respectively. A stirrer was placed inside the glass cylinder to ensure a good mixing of the two gases. The temperature and pressure inside the glass cylinder were controlled using an electrical heater and hand valves, respectively. A sinusoidal signal was generated using a functional signal generator, and the phase shift was measured using an oscilloscope.

Fig. 3.

Equipment used for sensor validation: 1. Sensor, 2. Glass Cylinder, 3. Heater, 4. Electrical Motor, 5. Circumferential Bolts, 6. Pipes, 7. Pressure Gauges, 8. Stirrer, 9. Hand Valves, 10. He Cylinder, 11. CO2 Cylinder 12. Thermometer, 13. Signal Functional Generator, 14. Oscilloscope.

At the beginning of the experiment, the glass cylinder was flushed with He gas to expel air, and the He pressure was set to the required level. Subsequently, the He valve was closed so that its amount remained constant for the next course of the experiment. The specific molar volume of He inside the glass cylinder was calculated using the measured temperature and pressure. After the He valve was closed, the CO₂ valve was opened, and the amount of CO₂ inside the glass cylinder was increased to the required level. The temperature of the gas mixture inside the glass cylinder was controlled by turning on the heater and increasing or decreasing the input power to obtain the required temperature. At each point, the temperature and pressure were recorded, and the mole fraction of CO₂ was calculated as follows:

Assuming ideal gas we obtain

or

After estimating the mole fraction of CO₂ from Eq. (39), the CO₂ mole fraction was recalculated using the Peng-Robinson equation of state [25] with the help of the Aspen HYSYS software [20]. Owing to the initial He content, the mole fraction of CO₂ did not reach one. Therefore, to obtain a mole fraction of one for CO₂, the experiment was repeated; however, in this case, the experiment was started with CO₂ and He was subsequently added. Therefore, two sets of experiments were conducted: in the first set, He was initially present and CO₂ was then added to the He, whereas in the second set, CO₂ was initially present and He was added to the CO₂. Thus, all mole fractions from 0–1 were covered.

3.2. Validation based on calibration

The equipment shown in Fig. 3 was used to calibrate the sensor over the ranges listed in Table 1. The phase shift was measured at different temperatures, pressures, mole fractions, and frequencies, and Eqs. (28), (30), and (35) were used for calibration.

Table 1.

Calibration range of the sensor based on equipment capability.

3.3. Validation based on literature models of refractive index

If a reliable model for the refractive index is available, R = R(T,P,x_i,ω)then the sensor can be used without calibration. The proposed sensor was validated based on separate literature models of the refractive indices. For He, the following model of refractive index as a function of wavelength was selected [26]:

For CO₂, the dependence of the refractive index on the wavelength was selected as [27]:

where λ in Eqs. (40) and (41) is the vacuum wavelength (C = λω) and the conditions of the equations are: T = 0°C and P = 101325 Pa. The Lorentz–Lorenz equation [28] was selected to determine the dependence of the refractive index on density.

Assuming (κρ) = x_Hev(κρ)_He + x_CO2(κρ)_CO2, we obtain Lorentz–Lorenz mixing rule as

Because the Lorentz–Lorenz equation is dependent on density rather than temperature and pressure, the Peng-Robinson equation of state with the help of the Aspen HYSYS software was selected to relate the density with pressure and temperature. Eqs. (24), (29), and (34) are used in this case.

3.4. Results of validation

Fig. 4 shows the representative results of pressure versus CO₂ mole fraction at a fixed temperature for both sets of experiments, as discussed in Section 3.1. The figure shows excellent agreement between the measured values and the calibrated sensor measurement and good agreement for the sensor measurement without calibration when the literature models were used. The figure also shows that for the literature models, the sensor measurement is underpredicted. Fig. 5 shows a temperature versus pressure plot for a fixed CO₂ mole fraction. Again, this figure shows excellent agreement for the calibrated sensor and good agreement for the sensor measurements based on the literature models. In this case, the temperature was overpredicted.

Fig. 4.

Experimental results obtained for pressure versus CO2 mole fraction at 40.1oC.

Fig. 5.

Experimental results obtained for temperature as a function of pressure at CO2 mole fraction of 0.48.

Finally, Fig. 6 shows a bubble plot for random points of T, P, and mole fractions. Again, the figure shows excellent agreement for the calibrated sensor and good agreement for the model sensors reported in literature. The error in the sensor measurements was calculated as

Fig. 6.

Bubble plots of random points for P, T and CO2 mole fraction. The size of bubbles indicate the CO2 mole fraction.

Table 2 lists the errors based on all the validation points. The table shows negligible errors for the calibrated sensor and acceptable errors for the sensor when the literature models were used.

Table 2.

Error percentage of sensor measurement.