**5. Examples of application of the method for a step change in temperature of the fluid**

The first example demonstrates the use of the described method of transient fluid temperature measurement in the case of thin sheathed thermocouple application. During the experiment, the thermocouple with outer diameter 1.5 mm at the ambient temperature was suddenly immersed into hot water at saturation temperature. The results are presented in **Figure 2**. The time step Δ*t* of the measured temperature was 1.162 s. The measured history of the temperature was approximated by Eq. (7) in order to determine the time constant *τ* of the thermocouple. In the calculations, TableCurve 2D code [18] was used. The calculations of the time constant and the uncertainty at the confidence level of 95% gave the following results: *τ* = 1.54 ± 0.09 s.

**Figure 2.** Fluid and thermometer temperature changes determined from the first-order equation (1) for the sheathed thermocouple with outer diameter 1.5 mm.

Next, the transient fluid temperature *Tf* was determined using Eq. (1) together with Eqs. (26) and (27). The first-order time derivative d*T*/d*t* in Eq. (1) was also calculated using the central difference quotient of Eq. (29). The results obtained show that the use of a first-order model for thin thermometers is sufficient (**Figure 2**). The results also indicate that the central difference approximation of the time derivative in Eq. (1) leads to less accurate results, since it is more sensitive to random errors in the experimental data.

Another example shows the application of the method for measuring transient fluid temperature with an industrial thermometer with massive thermowell and its complex construction (**Figure 3**). As in the previous example, the thermometer at the ambient temperature was suddenly immersed into hot water at a temperature of about 100°C. To compare the two methods of determining the unknown fluid temperature (using the first-and second-order model) for this thermometer, Eqs. (7) and (23) were used as the functions approximating the transient response of the thermometer. The following values with the 95% confidence uncertainty were obtained: *τ* = 14.07 ± 0.39 s, *τ*1 = 3.0 ± 0.165 s and *τ*2 = 10.90 ± 0.2 s.

Next, the fluid temperature changes were determined from Eq. (1) for the first-order model and from Eq. (20) for the second-order model (**Figure 4**).

The analysis of the results presented in **Figure 4** indicates that the second-order model delivers more accurate results than the first-order model.

Measurement of Transient Fluid Temperature in the Heat Exchangers http://dx.doi.org/10.5772/65686 185

**Figure 3.** Industrial thermometer and its dimensions: *D* = 18 mm, *l* = 65 mm and *L* = 140 mm.

**Figure 2.** Fluid and thermometer temperature changes determined from the first-order equation (1) for the sheathed

and (27). The first-order time derivative d*T*/d*t* in Eq. (1) was also calculated using the central difference quotient of Eq. (29). The results obtained show that the use of a first-order model for thin thermometers is sufficient (**Figure 2**). The results also indicate that the central difference approximation of the time derivative in Eq. (1) leads to less accurate results, since

Another example shows the application of the method for measuring transient fluid temperature with an industrial thermometer with massive thermowell and its complex construction (**Figure 3**). As in the previous example, the thermometer at the ambient temperature was suddenly immersed into hot water at a temperature of about 100°C. To compare the two methods of determining the unknown fluid temperature (using the first-and second-order model) for this thermometer, Eqs. (7) and (23) were used as the functions approximating the transient response of the thermometer. The following values with the 95% confidence uncer-

Next, the fluid temperature changes were determined from Eq. (1) for the first-order model

The analysis of the results presented in **Figure 4** indicates that the second-order model delivers

tainty were obtained: *τ* = 14.07 ± 0.39 s, *τ*1 = 3.0 ± 0.165 s and *τ*2 = 10.90 ± 0.2 s.

was determined using Eq. (1) together with Eqs. (26)

thermocouple with outer diameter 1.5 mm.

184 Heat Exchangers– Design, Experiment and Simulation

Next, the transient fluid temperature *Tf*

it is more sensitive to random errors in the experimental data.

and from Eq. (20) for the second-order model (**Figure 4**).

more accurate results than the first-order model.

**Figure 4.** Fluid and industrial thermometer temperature changes determined from the first-order equation (1) and the second-order equation (20).
