# On the Reliability of Temperature Measurements in Natural Gas Pipelines

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory and Methods

^{3}; $P$ (${P}_{S}$) is the absolute pressure at operative (standard reference) conditions, bar; $T$ (${T}_{S}$) is the absolute temperature at operative (standard reference) conditions, K; $Z$ (${Z}_{S}$) is the compressibility factor at operative (standard reference) conditions, dimensionless.

- the presence of preheating systems to cope with the Joule–Thomson effect (i.e., the lowering of the temperature following a gas pressure reduction);
- the absence of a cabin protecting the measurement stretch from the external environment;
- the absence of insulation for the measurement stretch.

#### 2.1. The Experimental Campaign in the Laboratory

^{−1}) and in summer (${T}_{set}$ = 30, 40, 50 °C and $w$ = 0.5 m s

^{−1}). The above mentioned gas temperatures and velocities were chosen considering the typical values of the plant investigated in-field, i.e., pipe temperature never below 8 °C (due to winter climate that is not particularly cold) and not very high flow velocities due to the plant characteristics (medium pressure pipeline serving a distribution city network).

#### 2.2. Development of a Numerical Model for Estimating the Gas Temperature

^{−1}K

^{−1}) and emissivity ε (-).

^{−1}); (iii) $\delta x$ is the height of the one-dimensional element (m); (iv) ${T}_{pipe,int}$ is the internal pipe temperature (K); (v) $C$ is the external circumference of the pipe (m); (vi) ${T}_{char}$ is a characteristic temperature which in the case of small variations between ${T}_{pipe,int}$ and ${T}_{i}$ can be assumed to be equal to their average (K); (vii) ${h}_{c}$ is the convective heat transfer coefficient (W/m

^{2}); (viii) $\sigma $ is the proportionality factor in the Stefan–Boltzmann law (W m

^{−2}K

^{−4}). Finally, a Dirichlet and a Robin boundary condition were applied at node 1 and node 6, respectively, for the resolution of the numerical model.

^{−1}, the authors calculated ${h}_{c}$ by using the numerical correlations for forced convection [29] and neglecting, therefore, the free convection mechanism. Since the Richardson number, ${R}_{i}$, was lower than 0.1 in all the performed numerical analyses, this hypothesis was verified.

#### 2.3. The Experimental In-Field Campaign

## 3. Results

#### 3.1. In-Laboratory Campaign

^{−1}) to +11.60 °C (at ${T}_{set}=$ 50 °C and $w$ = 0.5 m s

^{−1}). In the winter regime, ${T}_{well}$ is always underestimated, with absolute errors ranging from −0.70 °C (at ${T}_{set}=$ 12 °C and $w$ = 7.0 m s

^{−1}) to −4.2 °C (at ${T}_{set}=$ 8 °C and $w$ = 0.5 m s

^{−1}). As expected, this latter effect is lower in respect to that in the summer regime and occurs to a greater extent at lower flow rates.

^{−1}. The results of the validation process are shown in Table 2 together with the estimation of ${T}_{well}$ and of the related absolute deviation ${\u2206}_{model}={T}_{well,model}-{T}_{well,meas}$ at different line pressures (i.e., 5, 24 and 30 bar, typical of the transmission network). It is worthy to note that, in Table 2, the average values of three runs for each of the 10 experimental points have been reported. The relative errors between numerical and experimental data have been found within 4% for all the investigated conditions, and therefore show good agreement.

^{−1}) to 5.87 °C (at ${T}_{set}$ = 50 °C and $w$ = 0.5 m s

^{−1}) in the summer regime. This leads the converted volume to be underestimated in the range between 0.05 and 1.95%. Conversely, the estimated error ranges between −0.11 °C (at ${T}_{set}$ = 12 °C and $w$ = 7 m s

^{−1}) and −2.72 °C (at ${T}_{set}$ = 8 °C and $w$ = 0.5 m s

^{−1}) in winter regime, leading to overestimation of the converted volume in the range 0.04−0.93%. Table 2 also shows that the estimated error decreases as the line pressure increases.

#### 3.2. In-Field Campaign

^{−1}) and in summer (i.e., August 2020, with measured average ${T}_{pipe}$ = 24.0 °C and $w$ = 0.5 m s

^{−1}). The significant correlation between ${T}_{pipe}$ and ${T}_{well}$ is evident in Figure 5a, which is a result of the combined effect of ambient temperature, solar radiation, and flow rates. In Figure 5b, the correlation between ${T}_{well}$ and the registered flow rate $Q$ is depicted. It is worth noticing that the correlation is negligible in the winter regime, whereas in the summer regime, a clear negative correlation is found (R = −0.87). This highlights the influence of the convective effect of the flow rate: the lower the flow rate, the higher the measured temperature.

## 4. Discussion

- to provide an effective insulation and/or shielding of the stretch and of the measurement sensors;
- to use suitable conductive coupling fluids in the well–probe contact;
- to adequate design the size (i.e., length and thickness) and inclination (e.g., oblique for small diameters) of the thermowells;
- to use shielded thermowells or finish them with low-emissivity surfaces;
- to adopt appropriate thermowell immersion lengths that conform to the applicable standards and manufacturer’s specifications;
- to reduce the distance between the sections of the pipe in which the gas flow rate and temperature are measured.

## 5. Conclusions

- in the laboratory at ambient pressure, the measured error varies: (i) from 1.88 °C (at ${T}_{set}$ = 30°C and w = 7 m s
^{−1}) to 11.60 °C (at ${T}_{set}$ = 50 °C and w = 0.5 m s^{−1}) in the summer regime, (ii) from −0.70 °C (at ${T}_{set}$ = 12 °C and w = 7 m s^{−1}) to −4.21 °C (at ${T}_{set}$ = 8 °C and w = 0.5 m s^{−1}) in the winter regime; - the measured error is greatly attenuated at higher line pressure (e.g., at 30 bar, the estimated error ranges between 0.16 °C and 5.87 °C and between −0.11 °C and −2.72 °C, for summer and winter regimes, respectively);
- the in-field campaign has shown relevant correlations in the summer between ${T}_{well}$ and: (i) the ambient temperature ${T}_{amb}$ (R = 0.84), (ii) the cumulated solar radiation $\sum {G}_{i}$ (R = 0.76), and (iii) the gas flow rate $Q$ (R = −0.87);
- an average error value equal to 0.54 °C (0.18%) has been found in the field in summer, whereas in winter, this was negligible (i.e., within 0.03 °C and 0.01%).
- the measured errors in-field were found to be consistent with the corresponding ones in the laboratory.

^{−1}).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Acronyms and Symbols

CFD | computational fluid dynamics |

DN | nominal diameter |

RTDs | resistance temperature sensors |

UAG | unaccounted for gas |

C | external circumference of the pipe, m |

${D}_{t}$ | thermowell diameter, m |

${E}_{meas}$ | absolute measured error, °C |

$E{\%}_{meas}\%$ | relative measured error, % |

${\Delta}_{model}$ | absolute deviation between model and measurement, °C |

$\Delta {\%}_{model}$ | relative deviation between model and measurement, °C |

${G}_{i}$ | solar radiation, W m^{−2} |

${h}_{c}$ | convective heat transfer coefficient, Wm^{−2} |

$Ktvo$ | coefficient for volumetric conversion, dimensionless |

${L}_{t}$ | thermowell length, m |

$P$ | absolute pressure at operative conditions, bar |

Ps | absolute pressure at standard reference conditions, bar |

${\dot{Q}}_{\infty ,i}$ | convective heat flux, W |

${\dot{Q}}_{\left(i+1\right)\left(i\right)}$ | conductive heat flux inside the duct from node i + 1, W |

${\dot{Q}}_{\left(i-1\right)\left(i\right)}$ | conductive heat flux inside the duct from node i − 1, W |

${\dot{Q}}_{r,i}$ | radiative heat flux, W |

R | correlation coefficient, dimensionless |

${R}_{\left(i+1\right)\left(i\right)}$ | conductive thermal resistance between i and i + 1, K W^{−1} |

${R}_{\left(i-1\right)\left(i\right)}$ | conductive thermal resistance between i and i − 1, K W^{−1} |

$T$ | absolute temperature at operative conditions, °C |

${T}_{char}$ | characteristic temperature, K |

${T}_{amb}$ | environmental ambient temperature, °C |

${T}_{flow}$ | flow temperature measured with a shielded sensor inside the pipe, °C |

${T}_{flow,in}$ | flow temperature measured at the pipe inlet, °C |

${T}_{flow,out}$ | flow temperature measured at the pipe outlet, °C |

${T}_{S}$ | absolute temperature at standard reference conditions, °C |

${T}_{set}$ | set temperature of the thermostatic bath, °C |

${T}_{pipe,ext}$ | temperature measured on the external surface of the pipe, °C |

${T}_{pipe,int}$ | temperature measured on the internal surface of the pipe, °C |

${T}_{well}$ | temperature measured in the thermowell, °C |

${T}_{well,ins}$ | temperature measured in the thermowell in an insulated portion of the pipe, °C |

${T}_{well,model}$ | temperature measured in the thermowell calculated through the model, °C |

${T}_{i}$ | nodal temperatures, K |

$V$ | volume at operative conditions, m^{3} |

${V}_{S}$ | volume at standard reference conditions, Sm^{3} |

$w$ | flow velocity, ms^{−1} |

$Z$ | compressibility factor at operative conditions, dimensionless |

${Z}_{S}$ | compressibility factor at standard reference conditions, dimensionless |

$\delta x$ | height of the one-dimensional element, m |

ε | thermowell emissivity, dimensionless |

${\lambda}_{t}$ | Thermowell conductivity, W m^{−1} K^{−1} |

$\mathsf{\sigma}$ | Stefan–Boltzmann constant, W m^{−2} K^{−4} |

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**Figure 4.**Experimental data trends of: (

**a**) daily average ambient and gas temperature and daily cumulated solar radiation (secondary axis); daily average ${T}_{well}$, ${T}_{pipe}$, ${T}_{amb}$, and daily cumulated solar radiation (secondary axis) in winter (

**b**) and summer (

**c**) regimes.

**Figure 5.**Correlation analysis of ${T}_{well}$ in winter and summer regimes with: (a) the average daily pipe temperature ${T}_{pipe}$, (b) the average daily flow rate $Q$, (

**c**) the average ambient temperature ${T}_{amb}$, (

**d**) the daily cumulated solar radiation $\sum {G}_{i}$.

Low flow (0.5 m s^{−1}) | High flow (7 m s^{−1}) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Winter Regime | Summer Regime | Winter Regime | Summer Regime | ||||||||

P = 1 bar | ${T}_{set}$ (°C) | 8 | 12 | 30 | 40 | 50 | 8 | 12 | 30 | 40 | 50 |

${T}_{flow}$ (°C) | 20.06 | 20.14 | 24.26 | 25.80 | 28.20 | 20.94 | 20.71 | 22.89 | 23.62 | 24.05 | |

${T}_{well,meas}$ (°C) | 15.85 | 17.52 | 27.56 | 33.48 | 39.80 | 19.45 | 20.01 | 24.77 | 27.53 | 30.49 | |

${E}_{meas}$ (°C) | −4.21 | −2.62 | 3.30 | 7.68 | 11.60 | −1.49 | −0.70 | 1.88 | 3.91 | 6.45 | |

$E{\%}_{meas}$ (%) | −1.44% | −0.89% | 1.11% | 2.57% | 3.85% | −0.51% | −0.24% | 0.64% | 1.32% | 2.17% | |

${T}_{pipe,ext}$ (°C) | 8.32 | 12.53 | 30.47 | 40.71 | 50.74 | 11.09 | 14.54 | 30.27 | 39.92 | 49.58 | |

${T}_{pipe,int}$ (°C) | 10.10 | 13.62 | 29.34 | 38.49 | 47.34 | 16.01 | 17.65 | 27.04 | 32.96 | 39.13 | |

${T}_{flow,in}$ (°C) | 20.87 | 20.93 | 23.02 | 22.51 | 22.97 | 21.11 | 20.88 | 22.84 | 23.44 | 23.68 | |

${T}_{flow,out}$ (°C) | 17.84 | 18.74 | 26.00 | 30.08 | 34.67 | 19.75 | 19.94 | 24.14 | 26.22 | 28.16 |

Low Flow (0.5 m s^{−1}) | High Flow (7 m s^{−1}) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Winter Regime | Summer Regime | Winter Regime | Summer Regime | ||||||||

${T}_{set}$ (°C) | 8 | 12 | 30 | 40 | 50 | 8 | 12 | 30 | 40 | 50 | |

P = 1 bar | ${T}_{flow}$ (°C) | 20.06 | 20.14 | 24.26 | 25.80 | 28.20 | 20.94 | 20.71 | 22.89 | 23.62 | 24.05 |

${T}_{well,meas}$ (°C) | 15.85 | 17.52 | 27.56 | 33.48 | 39.80 | 19.45 | 20.01 | 24.77 | 27.53 | 30.49 | |

${T}_{well,model}$ (°C) | 12.44 | 15.14 | 28.21 | 35.73 | 43.26 | 18.97 | 19.48 | 24.60 | 27.52 | 30.45 | |

${\u2206}_{model}$ (°C) | −3.41 | −2.38 | 0.65 | 2.25 | 3.46 | −0.48 | −0.53 | −0.17 | −0.01 | −0.04 | |

$\u2206{\%}_{model}$ (%) | −1.16% | −0.81% | 0.22% | 0.75% | 1.15% | −0.16% | −0.18% | −0.06% | 0.00% | −0.01% | |

${E}_{model}$ (°C) | −7.62 | −5.00 | 3.95 | 9.93 | 15.06 | −1.97 | −1.23 | 1.71 | 3.90 | 6.40 | |

$E{\%}_{model}$ (%) | −2.60% | −1.70% | 1.33% | 3.32% | 5.00% | −0.67% | −0.42% | 0.58% | 1.31% | 2.15% | |

P = 5 bar | ${T}_{well,model}$ (°C) | 14.53 | 16.50 | 27.17 | 33.18 | 39.46 | 20.10 | 20.19 | 23.63 | 25.32 | 26.88 |

${E}_{model}$ (°C) | −5.53 | −3.64 | 2.91 | 7.38 | 11.26 | −0.84 | −0.52 | 0.74 | 1.70 | 2.83 | |

$E{\%}_{model}$ (%) | −1.89% | −1.24% | 0.98% | 2.47% | 3.74% | −0.29% | −0.18% | 0.25% | 0.57% | 0.95% | |

P = 24 bar | ${T}_{well,model}$ (°C) | 17.01 | 18.12 | 25.91 | 30.03 | 34.72 | 20.72 | 20.57 | 23.09 | 24.09 | 24.84 |

${E}_{model}$ (°C) | −3.05 | −2.02 | 1.65 | 4.23 | 6.52 | −0.22 | −0.14 | 0.20 | 0.47 | 0.79 | |

$E{\%}_{model}$ (%) | −1.04% | −0.69% | 0.55% | 1.41% | 2.16% | −0.07% | −0.05% | 0.07% | 0.16% | 0.27% | |

P = 30 bar | ${T}_{well,model}$ (°C) | 17.34 | 18.33 | 25.74 | 29.60 | 34.07 | 20.77 | 20.60 | 23.05 | 23.99 | 24.68 |

${E}_{model}$ (°C) | −2.72 | −1.81 | 1.48 | 3.80 | 5.87 | −0.17 | −0.11 | 0.16 | 0.37 | 0.63 | |

$E{\%}_{model}$ (%) | −0.93% | −0.62% | 0.50% | 1.27% | 1.95% | −0.06% | −0.04% | 0.05% | 0.12% | 0.21% |

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**MDPI and ACS Style**

Ficco, G.; Cassano, M.; Cortellessa, G.; Zuena, F.; Dell’Isola, M.
On the Reliability of Temperature Measurements in Natural Gas Pipelines. *Sensors* **2023**, *23*, 3121.
https://doi.org/10.3390/s23063121

**AMA Style**

Ficco G, Cassano M, Cortellessa G, Zuena F, Dell’Isola M.
On the Reliability of Temperature Measurements in Natural Gas Pipelines. *Sensors*. 2023; 23(6):3121.
https://doi.org/10.3390/s23063121

**Chicago/Turabian Style**

Ficco, Giorgio, Marialuisa Cassano, Gino Cortellessa, Fabrizio Zuena, and Marco Dell’Isola.
2023. "On the Reliability of Temperature Measurements in Natural Gas Pipelines" *Sensors* 23, no. 6: 3121.
https://doi.org/10.3390/s23063121