**6. Experimental determination of time constant as a function of fluid velocity**

The method presented in this chapter can also be used to determine the temperature of the flowing fluid. In this case, the time constant of the thermometer *τ* should be determined as a function of fluid velocity *w*. After specifying function *τ*(*w*), it should be substituted into Eq. (1).

Research has shown that for thin sheathed thermocouples with a diameter ranging from 0.5 to 6 mm, the equation of time constant as a function of fluid (air) velocity is as follows [15]:

$$
\pi = \frac{1}{a + b\sqrt{w}}\tag{31}
$$

where *τ* is the time constant of the thermometer in s, *a* constant in 1/s, *b* constant in (m·s)−1/2 and *w* fluid velocity in m/s.

As an example, the constants *a* and *b* were determined for K‐type sheathed thermocouples with grounded hot junction with the outer diameter of 0.5, 1.0, 1.5 and 3.0 mm.

**Figure 5.** Wind tunnel used for determining thermocouple time constant—overall view [19]: (1) test chamber with an opening for the thermometer, (2) differential pressure measurement, (3) data acquisition system and (4) opening for the thermometer insertion.

The thermocouple time constant *τ* for various air velocities *w* was determined in an open benchtop wind tunnel (**Figure 5**). The WT4401‐S benchtop wind tunnel is designed to give uniform flow rate over a 100 mm × 100 mm cross section [19].

The calculated time constants *τ* of the sheathed thermocouples with the outer diameter of 0.5, 1.0, 1.5 and 3.0 mm for different velocities of the air are shown in **Figure 6**. The experimental data collected for all thermocouples, as presented in **Figure 6**, were approximated by the least squares method. The best estimates for the constants *a* and *b*, with uncertainty at the 95% confidence level, for the tested thermocouples are [15]:

**•** *dt* = 0.5 mm

**6. Experimental determination of time constant as a function of fluid**

The method presented in this chapter can also be used to determine the temperature of the flowing fluid. In this case, the time constant of the thermometer *τ* should be determined as a function of fluid velocity *w*. After specifying function *τ*(*w*), it should be substituted into

Research has shown that for thin sheathed thermocouples with a diameter ranging from 0.5 to 6 mm, the equation of time constant as a function of fluid (air) velocity is as follows [15]:

> 1 *a bw*

where *τ* is the time constant of the thermometer in s, *a* constant in 1/s, *b* constant in (m·s)−1/2

As an example, the constants *a* and *b* were determined for K‐type sheathed thermocouples

**Figure 5.** Wind tunnel used for determining thermocouple time constant—overall view [19]: (1) test chamber with an opening for the thermometer, (2) differential pressure measurement, (3) data acquisition system and (4) opening for the

The thermocouple time constant *τ* for various air velocities *w* was determined in an open benchtop wind tunnel (**Figure 5**). The WT4401‐S benchtop wind tunnel is designed to give

The calculated time constants *τ* of the sheathed thermocouples with the outer diameter of 0.5, 1.0, 1.5 and 3.0 mm for different velocities of the air are shown in **Figure 6**. The experimental data collected for all thermocouples, as presented in **Figure 6**, were approximated by the least

uniform flow rate over a 100 mm × 100 mm cross section [19].

<sup>=</sup> <sup>+</sup> (31)

t

with grounded hot junction with the outer diameter of 0.5, 1.0, 1.5 and 3.0 mm.

**velocity**

186 Heat Exchangers– Design, Experiment and Simulation

Eq. (1).

and *w* fluid velocity in m/s.

thermometer insertion.

*a* = 0.004337 ± 0.000622 1/s and *b* = 0.022239 ± 0.001103 (m·s)−1/2

**•** *dt* = 1.0 mm

*a* = 0.020974 ± 0.006372 1/s and *b* = 0.103870 ± 0.011240 (m·s)−1/2

**•** *dt* = 1.5 mm

```
a = 0.040425 ± 0.003301 1/s and b = 0.056850 ± 0.004479 (m·s)−1/2
```
**•** *dt* = 3.0 mm

*a* = 0.128220 ± 0.035716 1/s and *b* = 0.220641 ± 0.051122 (m·s)−1/2

The time constant of the thermocouple *τ* strongly depends on the heat transfer coefficient *ht* on the outer thermometer surface, which results from Eq. (2). The heat transfer coefficient is a function of Nusselt number, and this, in turn, is a function of fluid (air) velocity [20].

**Figure 6.** Time constants *τ* of sheathed thermocouple with outer diameters of 0.5, 1.0, 1.5 and 3.0 mm as a function of air velocity *w* with 95% confidence intervals.

When the velocity of the fluid and its temperature varies over time, time constant as a function of the velocity formulated by Eq. (31) can be used in Eq. (1) to determine the transient fluid temperature based on measurements made using a sheathed thermocouple.

To show how the described method can improve the temperature readings, the experimental measurements were presented [21]. The temperature of the flowing air in an open wind tunnel (**Figure 7**) was measured by K‐type sheathed thermocouples with outer diameters of 0.5, 1.0 and 1.5 mm. During measurements, the temperature and velocity of air flowing through the tunnel were changed. The thermocouples were placed in the tunnel behind the heat exchanger and very close to each other (i.e. they measured the same temperature, but did not influence each other). The air velocity was measured by the vane anemometer FV A915 S220. Both the temperature and velocity data were collected using the Ahlborn ALMEMO 5990‐0 data acquisition system.

**Figure 7.** Diagram of an open wind tunnel [22]: (A) heat exchanger, (B) fan, (C) chamber, (D) air channel, (E) water outlet pipe and (F) hot water feeding pipe.

The comparison of the computed temperatures with the measured temperatures, when the time constants of the thermocouples are known, shows that the above method provides decent results (**Figures 8** and **9**).

The time histories of temperature obtained from calculations are very similar, especially for thermocouples with the outer diameter of 0.5 and 1.0 mm. In the small degree, the temperature calculated using the measurements with the thermocouple with the outer diameter of 1.5 mm deviates from them. This difference is due to the large time constant of the sheathed thermo‐ couple with the outer diameter of 1.5 mm.

When the velocity of the fluid and its temperature varies over time, time constant as a function of the velocity formulated by Eq. (31) can be used in Eq. (1) to determine the transient fluid

To show how the described method can improve the temperature readings, the experimental measurements were presented [21]. The temperature of the flowing air in an open wind tunnel (**Figure 7**) was measured by K‐type sheathed thermocouples with outer diameters of 0.5, 1.0 and 1.5 mm. During measurements, the temperature and velocity of air flowing through the tunnel were changed. The thermocouples were placed in the tunnel behind the heat exchanger and very close to each other (i.e. they measured the same temperature, but did not influence each other). The air velocity was measured by the vane anemometer FV A915 S220. Both the temperature and velocity data were collected using the Ahlborn ALMEMO 5990‐0 data

**Figure 7.** Diagram of an open wind tunnel [22]: (A) heat exchanger, (B) fan, (C) chamber, (D) air channel, (E) water

The comparison of the computed temperatures with the measured temperatures, when the time constants of the thermocouples are known, shows that the above method provides decent

The time histories of temperature obtained from calculations are very similar, especially for thermocouples with the outer diameter of 0.5 and 1.0 mm. In the small degree, the temperature calculated using the measurements with the thermocouple with the outer diameter of 1.5 mm deviates from them. This difference is due to the large time constant of the sheathed thermo‐

temperature based on measurements made using a sheathed thermocouple.

acquisition system.

188 Heat Exchangers– Design, Experiment and Simulation

outlet pipe and (F) hot water feeding pipe.

couple with the outer diameter of 1.5 mm.

results (**Figures 8** and **9**).

**Figure 8.** Temperature of the air measured by the thermocouples with outer diameters of 0.5, 1.0 and 1.5 mm and tem‐ perature calculated by Eq. (1) when the velocity of the air was constant.

**Figure 9.** Temperature of the air measured by the thermocouples with outer diameters of 0.5, 1.0 and 1.5 mm and tem‐ perature calculated by Eq. (1) when the velocity of the air was changed.
