**2. Mathematical models of the thermometers**

Generally, thermometers are considered as elements with lumped thermal capacity. It is assumed that the temperature of the thermometer is only a function of time and temperature differences inside it are neglected.

Based on these assumptions, the mathematical model of the thermometer is the differential equation describing the inertial system of the first order [14]:

$$
\tau \frac{\mathrm{d}T\left(t\right)}{\mathrm{d}t} + T\left(t\right) = T\_f\left(t\right) \tag{1}
$$

where *τ*—time constant of the thermometer in the first-order model in s, *T*(*t*)—measured temperature in °C and *Tf* (*t*)—fluid temperature in °C.

The time constant is expressed by formula:

Measurement of Transient Fluid Temperature in the Heat Exchangers http://dx.doi.org/10.5772/65686 177

$$
\pi = \frac{m\_\text{r}c\_\text{r}}{h\_\text{r}A\_\text{r}} \tag{2}
$$

where *mt* —thermocouple mass in kg, *ct* —average-specific heat of the thermocouple in J/(kg K), *ht* —heat transfer coefficient on the outer surface of the thermocouple in W/(m2 K) and *At* outer surface area of the thermocouple in m2 .

The ordinary differential Eq. (1) was solved for the initial condition:

$$T\begin{pmatrix} 0 \end{pmatrix} = T\_0 = 0 \tag{3}$$

where *T*0 signifies initial thermometer temperature in °C.

The initial problem, (Eqs. (1) and (3)), was solved using the Laplace transformation. The operator transmittance *G*(*s*) then assumes the following form:

$$G(s) = \frac{T(s)}{T\_f(s)} = \frac{1}{\pi s + 1} \tag{4}$$

where *T*(*s*)—Laplace transform of the thermometer temperature, *Tf* (*s*)—Laplace transform of the fluid temperature and *s*—complex variable.

For the step increase of the fluid temperature from *T*0 = 0°C to the constant value *Ts*, the Laplace transform of the fluid temperature assumes the form *Tf* (*s*) = *Ts*/*s* and the transmittance formula can be simplified to

$$\frac{T\{s\}}{T\_s} = \frac{1}{s\{\pi s + 1\}}\tag{5}$$

After writing Eq. (5) in the form:

differences occur because it takes time for heat to transfer through the heavy housing to the

Most of the scientific publications concerning the measurement of temperature mainly discuss the problem of temperature measurement at steady state [1–9]. Only the step response of thermometers is studied to estimate the dynamic error of the temperature measurement. Few studies refer to the measurement of the transient fluid temperature, despite the high practical

An example is the measurement of transient temperature steam or flue gases in power plants, which is very difficult. Measured temperature differs significantly from the real temperature of the fluid, which is caused by massive thermowells of the thermometers and their low heat transfer coefficients. Some thermometers may have a time constant of 3 min or more, which makes the implementation of a single temperature measurement requiring about 15 min [13].On the other side, some designs of thermometers need more than one time constant to describe the unsteady response of temperature sensor inserted into the thermowell. Measuring the fluid temperature in a controlled process may require the knowledge of two or three time

The problem of dynamic errors in temperature measurements becomes particularly important in superheated steam temperature control systems, which use injection coolers (spray attemperators). Due to the large inertia of the thermometer, the measurements of transient fluid temperature can be inaccurate, which causes the automatic control system of the

A similar problem occurs in measuring the exhaust gas temperature as the time constants of

Generally, thermometers are considered as elements with lumped thermal capacity. It is assumed that the temperature of the thermometer is only a function of time and temperature

Based on these assumptions, the mathematical model of the thermometer is the differential

*Tt T t*

where *τ*—time constant of the thermometer in the first-order model in s, *T*(*t*)—measured

+ = (1)

( ) () () <sup>d</sup> d *<sup>f</sup>*

*T t*

*t* t

(*t*)—fluid temperature in °C.

constants, which describe the transient response of the thermometer [14].

superheated steam temperature not to work properly.

**2. Mathematical models of the thermometers**

equation describing the inertial system of the first order [14]:

the thermometer and the time delay are large.

differences inside it are neglected.

temperature in °C and *Tf*

The time constant is expressed by formula:

temperature sensor.

significance of the problem [10–13].

176 Heat Exchangers– Design, Experiment and Simulation

$$\frac{T\left(s\right)}{T\_s} = \frac{1}{s} - \frac{1}{s + \frac{1}{\tau}}\tag{6}$$

it is easy to find the inverse Laplace transformation and determine the thermometer temperature as a function of time:

$$\mu\left(t\right) = \frac{T\left(t\right) - T\_0}{T\_s - T\_0} = 1 - \exp\left(-\frac{t}{\tau}\right) \tag{7}$$

where *u*(*t*) denotes unit‐step response of the thermometer.

For structurally complex thermometers that measure the temperature of fluid under high pressure, the accuracy of the first‐order model (1) is inadequate. The cross section of the temperature sensor placed in the massive housing is shown in **Figure 1**. This example will be analysed to show that the thermometers can be modelled as second‐order inertial systems [14].

**Figure 1.** Part of the cross section of the temperature sensor together with the housing [14].

Air gap can occur between the outer thermowell and the temperature sensor (**Figure 1**). The discussion assumes that the thermal capacity *cρ* is neglected because of its small value.

Introducing the overall heat transfer coefficient *uin* referenced to the inner surface of the housing

$$\frac{1}{\mu\_{\rm in}} = \frac{1}{h\_{\rm in}} + \frac{\left(1 + D\_{\rm in}/d\right)\left(D\_{\rm in} - d\right)}{4k\_{\rm w}} + \frac{D\_{\rm in}}{d} \cdot \frac{1}{h\_{\rm r}} \tag{8}$$

and accounting for the radiation heat transfer from the housing to the inner sensor, the heat balance equation for the sensor located within the housing assumes the form:

Measurement of Transient Fluid Temperature in the Heat Exchangers http://dx.doi.org/10.5772/65686 179

$$A\,\rho\text{c}\,\frac{\text{d}T}{\text{d}t} = P\_{in}u\_{in}\left(T\_w - T\right) + C\left(T\_w^4 - T^4\right) \tag{9}$$

where the symbol *C* denotes:

where *u*(*t*) denotes unit‐step response of the thermometer.

178 Heat Exchangers– Design, Experiment and Simulation

**Figure 1.** Part of the cross section of the temperature sensor together with the housing [14].

housing

Air gap can occur between the outer thermowell and the temperature sensor (**Figure 1**). The discussion assumes that the thermal capacity *cρ* is neglected because of its small value.

Introducing the overall heat transfer coefficient *uin* referenced to the inner surface of the

*in in in*

= + + × (8)

*DdD d D*

and accounting for the radiation heat transfer from the housing to the inner sensor, the heat

1 1 ( )( ) 1 1 4

*in in w T*

*u h k dh* + -

balance equation for the sensor located within the housing assumes the form:

For structurally complex thermometers that measure the temperature of fluid under high pressure, the accuracy of the first‐order model (1) is inadequate. The cross section of the temperature sensor placed in the massive housing is shown in **Figure 1**. This example will be analysed to show that the thermometers can be modelled as second‐order inertial systems [14].

$$C = \frac{\pi d \sigma}{\frac{1}{\mathcal{E}\_T} + \frac{d}{D\_{\text{in}}} \left(\frac{1}{\mathcal{E}\_w} - 1\right)}\tag{10}$$

In Eqs. (8) and (9), the symbols *hin* and *hT* represent heat transfer coefficient on the inner sur‐ face of the housing and the outer surface of the thermocouple in W/(m2 K), respectively, *Din* inner diameter of the housing in m, *d* outer diameter of the thermocouple in m, *k*o housing thermal conductivity in W/(m K), *A* surface area of the thermocouple cross section in m2 , *ρ* average density of the thermocouple in kg/m3 , *c* average‐specific heat of the thermocouple in J/(kg K), *Pin* perimeter of the internal surface of the housing in m, *Tw* housing temperature in °C, *σ* = 5.67 × 10−8 W/(m2 K4 ) Stefan‐Boltzmann constant and *εw* and *εT* emissivity of the inner housing and the thermocouple surface, respectively.

The convection and conduction of heat transfer between the fluid and the thermometer housing are characterised by the overall heat transfer coefficient *uout* referenced to the outer housing surface:

$$\frac{1}{\mu\_{\rm out}} = \frac{1}{h\_{\rm out}} + \frac{1 + D\_{\rm out}/D\_{\rm in}}{2} \cdot \frac{\mathcal{S}\_{\rm w}}{\mathcal{A}\_{\rm w}} \tag{11}$$

where *hout* is the heat transfer coefficient on the outer surface of the housing in W/(m2 K), *Dout* outer diameter of the housing in m, *δw* housing thickness in m and *kw* housing thermal conductivity in W/(m K).

The formulas (8) and (11) for the overall heat transfer coefficients were derived using the basic principles of heat transfer [2, 4, 8]. The heat transfer equation for the housing (thermowell) can be written in the following form:

$$A\_w \rho\_w \mathcal{L}\_w \frac{\mathbf{d}T\_w}{\mathbf{d}t} = P\_{au} \mu\_{au} \left(T\_f - T\_w\right) - P\_{in} \mu\_{in} \left(T\_w - T\right) - C \left(T\_w^4 - T^4\right) \tag{12}$$

where *Aw* is the surface area of the housing cross section in m2 , *ρw* average density of the housing in kg/m3 , *cw* average‐specific heat of the housing, J/(kg K) and *Pout* perimeter of the external surface of the housing in m.

Further analysis of the heat exchange between the housing of the thermometer and the temperature sensor omits the heat transfer by radiation [14]. This is possible when the gap between the thermowell and the temperature sensor is filled with a non‐transparent substance or if one of the two emissivities *εw* and *εT* is near to zero.

Transforming Eq. (9), we get:

$$T\_w = \frac{A\rho c}{P\_{\text{in}} u\_{\text{in}}} \frac{\text{d}T}{\text{d}t} + T \tag{13}$$

Substituting Eq. (13) into Eq. (12) yields:

$$\frac{\left(A\_{w}\rho\_{w}c\_{w}\right)\left(A\rho c\right)}{\left(P\_{n}u\_{in}\right)\left(P\_{out}u\_{out}\right)}\frac{\mathrm{d}^{2}T}{\mathrm{d}t^{2}} + \frac{A\_{w}\rho\_{w}c\_{w}}{P\_{out}u\_{out}}\left[\frac{1+\left(P\_{out}u\_{in}\right)\left(A\rho c\right)c\_{w}}{P\_{out}u\_{in}}\right]\frac{\mathrm{d}T}{\mathrm{d}t} + T = T\_{f}\tag{14}$$

Introducing the following coefficients:

$$a\_2 = \frac{\left(A\_u \rho\_u \mathcal{L}\_w\right)\left(A \rho \mathcal{c}\right)}{\left(P\_u \mu\_u\right)\left(P\_{uu} \mu\_{uu}\right)}, \qquad a\_1 = \frac{A\_u \rho\_u \mathcal{L}\_w}{P\_{uu} \mu\_{uu}} \left[\frac{1 + \frac{\left(P\_{uu} \mu\_{uu}\right)\left(A \rho \mathcal{L}\right)}{\left(P\_u \mu\_{ui}\right)\left(A\_u \rho \mathcal{L}\_w \mathcal{L}\_w\right)}}{P\_{uu} \mu\_{uu}}\right] \tag{15}$$

the ordinary differential equation of the second order (14) can be written in the form:

$$a\_2 \frac{\mathbf{d}^2 T}{\mathbf{d}t^2} + a\_1 \frac{\mathbf{d}T}{\mathbf{d}t} + T = T\_f \tag{16}$$

The initial conditions are:

$$T\left(0\right) = T\_0 = 0, \quad \left.\frac{\mathrm{d}T\left(t\right)}{\mathrm{d}t}\right|\_{t=0} = \nu\_\tau = 0\tag{17}$$

Equations (16) and (17) are solved using the Laplace transformation. The operator transmit‐ tance *G*(*s*) is as follows:

Measurement of Transient Fluid Temperature in the Heat Exchangers http://dx.doi.org/10.5772/65686 181

$$G(s) = \frac{T(s)}{T\_f(s)} = \frac{1}{\left(\tau\_1 s + 1\right)\left(\tau\_2 s + 1\right)}\tag{18}$$

The time constants *τ*1 and *τ*2 in Eq. (18) are:

Further analysis of the heat exchange between the housing of the thermometer and the temperature sensor omits the heat transfer by radiation [14]. This is possible when the gap between the thermowell and the temperature sensor is filled with a non‐transparent substance

d

( )( ) <sup>2</sup>

*w ww w ww in in w w w*

r

*A c Ac A c Pu A c a a Pu P u P u A c*

the ordinary differential equation of the second order (14) can be written in the form:

d d *<sup>f</sup> T T a a TT t t*

d 0 0, 0

*T t*

Equations (16) and (17) are solved using the Laplace transformation. The operator transmit‐

0

d *<sup>T</sup> t*

1 d d d d

*A c Ac TAc Pu A c <sup>T</sup> T T Pu P u t P u A c t*

+

*w ww w ww in in w w w*

r ( )( )

é ù ê ú + + + =

ë û

1

+

*Pu A c*

r

( )( ) ( )( )

é ù ê ú +

ë û

*Pu A c*

r

r

+ += (16)

== == (17)

*out out*

*w ww*

r

*A c*

r

r

*out out*

*w ww*

r

*A c*

r

= + (13)

*f*

(14)

(15)

d *<sup>w</sup> in in Ac T T T Pu t* r

or if one of the two emissivities *εw* and *εT* is near to zero.

2

*in in out out out out*

( )( ) ( )( )

r

2 1

 r

= =

,

*in in out out out out*

2 2 1 2 d d

( ) ( ) <sup>0</sup>

*T T <sup>v</sup> <sup>t</sup>* <sup>=</sup>

 r

Transforming Eq. (9), we get:

180 Heat Exchangers– Design, Experiment and Simulation

Substituting Eq. (13) into Eq. (12) yields:

( )( ) ( )( )

Introducing the following coefficients:

The initial conditions are:

tance *G*(*s*) is as follows:

r

$$
\pi\_{1,2} = \frac{2a\_2}{a\_1 \pm \sqrt{a\_1^2 - 4a\_2}} \tag{19}
$$

The differential Eq. (16) can be written in the following form:

$$
\tau\_1 \tau\_2 \frac{\mathbf{d}^2 T}{\mathbf{d}t^2} + \left(\tau\_1 + \tau\_2\right) \frac{\mathbf{d}T}{\mathbf{d}t} + T = T\_f \tag{20}
$$

Equation (20) is solved for a step change in fluid temperature from *T*0 = 0°C to the constant value *Tf* . Laplace transform of the constant fluid temperature *Tf* is *Tf* (*s*) = *Ts*/*s*, and the operator transmittance assumes the following form:

$$\frac{T\left(s\right)}{T\_s} = \frac{1}{s\left(\tau\_1 s + 1\right)\left(\tau\_2 s + 1\right)}\tag{21}$$

Eq. (21) can be written in another form:

$$\frac{T\left(s\right)}{T\_s} = \frac{1}{s} + \frac{\tau\_1}{\tau\_2 - \tau\_1} \cdot \frac{1}{\left(s + \frac{1}{\tau\_1}\right)} - \frac{\tau\_2}{\tau\_2 - \tau\_1} \cdot \frac{1}{\left(s + \frac{1}{\tau\_2}\right)}\tag{22}$$

Making inverse Laplace transformation of Eq. (22), the thermometer temperature as a function of time is obtained:

$$u\left(t\right) = \frac{T\left(t\right) - T\_0}{T\_f - T\_0} = 1 + \frac{\tau\_1}{\tau\_2 - \tau\_1} \exp\left(-\frac{t}{\tau\_1}\right) - \frac{\tau\_2}{\tau\_2 - \tau\_1} \exp\left(-\frac{t}{\tau\_2}\right) \tag{23}$$

If we assume in Eq. (23) with *τ*2 = 0, then we obtain Eq. (7) with *τ* = *τ*1.

A time delay in the time response of the first order inertial system does not appear. Measuring the temperature of the fluid at high pressure requires the use of thermometers with the massive housings. In such cases, there is a time delay between the temperature indicated by the sensor and the changing temperature of the fluid. The inertial system of second order is suitable to describe the response behaviour with a time delay [13].

#### **3. Identification of time constants**

The time constant *τ* in Eq. (7) or time constants *τ*1 and *τ*2 in Eq. (23) can be determined on the basis of experimental data. For this purpose, the method of least squares is proposed to use [13, 15]. Finding the minimum of the function S

$$S = \sum\_{i=1}^{N} \left[ u\_m \left( t\_i \right) - u \left( t\_i \right) \right]^2 = \min \tag{24}$$

allows to determine values of the time constants. In Eq. (24), *u*(*ti* ) denotes the approximating function given by Eq. (7), and *N* is the number of conducted measurements (*ti* , *um*(*ti* )). The sum of the squares of the differences of the measured values *um*(*ti* ) and the fitted values *u*(*ti* ) is minimised. When the time constant *τ* or time constants *τ*1 and *τ*2 are determined, their values can be substituted into Eq. (24) to calculate *S*min.

The uncertainties of the calculated time constant *τ* or time constants *τ*1 and *τ*2 are calculated using the mean square error [13, 16–18]:

$$
\Delta S\_N = \sqrt{\frac{S\_{\text{min}}}{N - m}} \tag{25}
$$

where *m* denotes the amount of time constants (i.e. *m* = 1 for Eq. (7) and *m* = 2 for Eq. (7) and *m* = 2 for Eq. (23)). Based on the determined mean square error *SN*, which is an approximation of the standard deviation, the uncertainties in the determined time constants can be calculated using the formulas given in the TableCurve 2D software [18].

#### **4. Determination of transient fluid temperature**

The fluid temperature can be determined on the basis of measured histories of the thermometer temperature *T*(*t*) and known time constant *τ*, using Eq. (1), or time constants *τ*1 and *τ*2, using Eq. (20). Temperature of the thermometer *T*(*t*) and its first- and second-order time derivatives can be smoothed using the formulas [3, 13]:

$$\begin{aligned} T\left(t\right) &= \frac{1}{693} \cdot \left[ -63f\left(t - 4 \cdot \Delta t\right) + 42f\left(t - 3 \cdot \Delta t\right) \right. \\ &+ 117f\left(t - 2 \cdot \Delta t\right) + 162f\left(t - \Delta t\right) + 177f\left(t\right) + 162f\left(t + \Delta t\right) \\ &+ 117f\left(t + 2 \cdot \Delta t\right) + 42f\left(t + 3 \cdot \Delta t\right) - 63f\left(t + 4 \cdot \Delta t\right) \right] \end{aligned} \tag{26}$$

A time delay in the time response of the first order inertial system does not appear. Measuring the temperature of the fluid at high pressure requires the use of thermometers with the massive housings. In such cases, there is a time delay between the temperature indicated by the sensor and the changing temperature of the fluid. The inertial system of second order is suitable to

The time constant *τ* in Eq. (7) or time constants *τ*1 and *τ*2 in Eq. (23) can be determined on the basis of experimental data. For this purpose, the method of least squares is proposed to

() () <sup>2</sup>

minimised. When the time constant *τ* or time constants *τ*1 and *τ*2 are determined, their values

The uncertainties of the calculated time constant *τ* or time constants *τ*1 and *τ*2 are calculated

min

where *m* denotes the amount of time constants (i.e. *m* = 1 for Eq. (7) and *m* = 2 for Eq. (7) and *m* = 2 for Eq. (23)). Based on the determined mean square error *SN*, which is an approximation of the standard deviation, the uncertainties in the determined time constants can be calculated

The fluid temperature can be determined on the basis of measured histories of the thermometer temperature *T*(*t*) and known time constant *τ*, using Eq. (1), or time constants *τ*1 and *τ*2, using Eq. (20). Temperature of the thermometer *T*(*t*) and its first- and second-order time derivatives

*mi i*

min

= é - ù= åë û (24)

*N m* <sup>=</sup> - (25)

) denotes the approximating

, *um*(*ti*

) and the fitted values *u*(*ti*

)). The sum

) is

1

*S u t ut* =

function given by Eq. (7), and *N* is the number of conducted measurements (*ti*

*N <sup>S</sup> <sup>S</sup>*

*i*

allows to determine values of the time constants. In Eq. (24), *u*(*ti*

of the squares of the differences of the measured values *um*(*ti*

using the formulas given in the TableCurve 2D software [18].

**4. Determination of transient fluid temperature**

can be smoothed using the formulas [3, 13]:

*N*

describe the response behaviour with a time delay [13].

use [13, 15]. Finding the minimum of the function S

can be substituted into Eq. (24) to calculate *S*min.

using the mean square error [13, 16–18]:

**3. Identification of time constants**

182 Heat Exchangers– Design, Experiment and Simulation

$$\begin{split} T'(t) &= \frac{\mathrm{d}T\left(t\right)}{\mathrm{d}t} = \frac{1}{1188\Delta t} \cdot \left[ 86f\left(t - 4 \cdot \Delta t\right) - 142f\left(t - 3 \cdot \Delta t\right) \right. \\ &- 193f\left(t - 2 \cdot \Delta t\right) - 126f\left(t - \Delta t\right) + 126f\left(t + \Delta t\right) \\ &+ 193f\left(t + 2 \cdot \Delta t\right) + 142f\left(t + 3 \cdot \Delta t\right) - 86f\left(t + 4 \cdot \Delta t\right) \right] \end{split} \tag{27}$$

$$\begin{split} T''(t) &= \frac{\mathbf{d}^2 T(t)}{\mathbf{d}t^2} = \frac{1}{462\left(\Delta t\right)^2} \cdot \left[ 28f\left(t - 4 \cdot \Delta t\right) + 7f\left(t - 3 \cdot \Delta t\right) \right. \\ &- 8f\left(t - 2 \cdot \Delta t\right) - 17f\left(t - \Delta t\right) - 20f\left(t\right) - 17f\left(t + \Delta t\right) \\ &- 8f\left(t + 2 \cdot \Delta t\right) + 7f\left(t + 3 \cdot \Delta t\right) + 28f\left(t + 4 \cdot \Delta t\right) \right] \end{split} \tag{28}$$

Symbol *f*(*t*) in Eqs. (26)–(28) denotes the temperature measured by the thermometer, and Δ*t* is a time step. Application of nine‐point digital filter allows eliminating the influence of random errors of the measured temperature *T*(*t*) on the determined temperature of the fluid *Tf* (*t*). If the measured temperature is not disturbed by significant errors, the first and second derivatives of the temperature can be estimated using the central difference method [15]:

$$T'(t) = \frac{f\left(t + \Delta t\right) - f\left(t - \Delta t\right)}{2\Delta t} \tag{29}$$

$$T''(t) = \frac{f\left(t + \Delta t\right) - 2f\left(t\right) + f\left(t - \Delta t\right)}{\left(\Delta t\right)^2} \tag{30}$$
