Thermocouple Effective Length under Sinusoidal Gas Temperature Condition

Abstract

When a thermocouple is used to measure gas temperature, the measured temperature is the thermocouple bead temperature, which is not equal to the gas temperature. The bead temperature results from its energy balance. Through the wire convection and conduction, the temperature of the bead is related to the gas temperature within a certain geometric range around it, and this range is quantified by the effective length. Under the sinusoidal incoming gas temperature condition, the analytical expression for the effective length is deduced, and its accuracy is validated by the one-dimensional numerical solution. The differences between the analytical and numerical effective lengths are less than 10.5% for the test cases. Similar to that under the uniform incoming gas temperature condition, the effective length under the sinusoidal gas temperature condition increases with the thermal conductivity and the diameter of the wire and decreases with the heat transfer coefficient of the wire. The influence of the amplitude, wavelength and phase of the gas temperature on the effective length are very weak, meaning that the theoretical expression under the uniform gas temperature can calculate the effective length under the non-uniform gas temperature with good accuracy.

1. Introduction

Gas temperature measurements are very common in many chemical and thermal processes. As the temperature-measurement tools, thermocouples are used extensively in industrial process applications and combustion applications such as furnaces, boilers, and engines. Accurate gas temperature measurement is crucial for safety, pollutant reduction, performance improvement, analysis and control of thermal and chemical processes. When the gas temperature is high, the readout of a thermocouple, which is the temperature of the thermocouple bead, can significantly deviate from the real gas temperature. The bead temperature is determined by the bead energy balance. The bead has conduction heat transfer with wires, has radiation and convection heat transfer with incoming gas and has radiation heat transfer with surrounding walls. The gas could have a catalytic reaction with the noble metal of the thermocouple, and part of the reaction heat release could be absorbed by the bead. The bead energy balance in a steady flow field is shown in Figure 1 and Equation (1). Similar energy conservation also applies for the thermocouple wires.

$$q_{cond}+q_{r,g}+q_{r,s}+q_c+q_{cat}=0$$

(1)

T_g is the gas temperature; T_∞ is the environmental temperature; q_cond is the conductive heat transfer rate between the bead and the wires; q_r,g is the absorbed gas radiation rate by the bead; q_r,s is the heat transfer rate with the environment through surface radiation; q_c is the convective heat transfer rate with the incoming gas; q_cat is the absorbed heat from the catalytic surface reaction by the bead. The catalytic reaction is negligible if the hot gas is in local equilibrium state [1].

The bead temperature is affected by the gas temperature contacting the bead, and it is also affected by the gas temperature contacting the thermocouple wires next to the bead. The surrounding gas temperature affects the temperature of the wires, which affects the conduction rate to the bead. As such, the bead temperature is influenced by the surrounding gas temperature in a certain geometric range through the wire conduction. This range is quantified by the effective length, outside of which the gas temperature does not influence the bead temperature [2].

Hindasageri et al. [3] studied the heat transfer of a B-type thermocouple in a gas flame numerically. They showed that the bead should penetrate into the flame at least 10mm to minimize the wire conduction loss and obtain a stable temperature reading. Shaddix [4,5,6] also stated that the thermocouple should be long enough to guarantee that the bead temperature is not influenced by the wire conduction. Liu et al. [2] derived an analytical expression for the effective length l_th of bare wire thermocouples:

$$l_{th}=-\frac{\ln (0.01)}{2}\sqrt{\frac{k_{}d}{h_w}}$$

(2)

k is the thermal conductivity of the wire with higher conductivity; h_w = h_c,w + h_r,w is the total heat transfer coefficient, which is the sum of the convection coefficient h_c,w and the radiation coefficient h_r,w of the wire with higher conductivity. Sun et al. [7] further studied the effective length of thermocouples with protection or insulation tubes. For the thermocouple with protection tubes, the existence of the tubes changes the heat transfer of the wire, which will influence the effective length when the wire projecting a length beyond the tube end is less than the effective length of the bare-wire thermocouple. The tube serves as a thermal resistance between the wire and the hot gas, and it reduces the heat transfer rate of the wire, which is equivalent to reducing the total heat transfer coefficient of the wire. This will increase the effective length.

The deduction of Equation (2) was based on the assumption of a constant incoming gas temperature. However, the gas temperature is normally not uniform, and the application accuracy of Equation (2) under non-uniform gas temperature condition is not certain. Since a non-uniform temperature profile can normally be expressed as a sum of sinusoidal functions through the Fourier transformation, the effective length of bare wire thermocouples with a sinusoidal incoming gas temperature profile is studied here. The analytical expression for the effective length is first deduced. Then, the result of the one-dimensional thermocouple heat transfer program is used to validate the accuracy of the analytical solution. Finally, the influences of the size, thermal conductivity and heat transfer coefficient of the thermocouple on the effective length are analyzed. The influences of the amplitude, wavelength and phase of the gas temperature profile on the effective length are also discussed.

2. 1D Numerical Simulations

For the bare wire thermocouple, the wire diameters are normally 0.5 mm or less, and the wire temperature gradients mainly exist in the axial direction. The bead diameter is also small, and the bead temperature is normally uniform. Therefore, the axial heat transfer assumption (one-dimensional) for the thermocouple is reasonable. Bradley et al. [8], Sato et al. [9] and Liu et al. [2] have developed their one-dimensional numerical codes for the bare wire thermocouples. Sun et al. [7] have developed the one-dimensional code for thermocouples with ceramic tubes. Li et al. [10] have validated the prediction accuracy of the one-dimensional programs in Liu et al. [2] and Sun et al. [7] with experimental data. For Hencken flames a the temperature of 1184–1994 K [10], the code predicts the bead temperature within the experimental uncertainty, i.e., ±35 K. The discretized energy conservation equations of the bead and the wires of bare wire thermocouples are briefed here for reference.

The discretized energy balance equation of the bead is shown in Equation (3). The discretization schematic is shown in Figure 2a. The energy balance equation of the wire is similar to that of the bead, and its discretization schematic and equation are shown Figure 2b and Equation (3) [2].

$$\begin{array}{l}{k}_{1.1}\frac{\pi d^2}{4}\left(\frac{T_{1.1}-T_b}{\Delta x}\right)+k_{2.1}\frac{\pi d^2}{4}\left(\frac{T_{2.1}-T_b}{\Delta x}\right)+\sigma \epsilon (\pi D^2-\pi d^2/2)\cdot \\ \left[(1-{\alpha }_g)T_{\infty }^4+{\epsilon }_g{T}_g^4-T_b^4\right]+h_{c,b}(\pi D^2-\pi d^2/2)(T_g-T_b)=0\end{array}$$

(3)

$$\begin{array}{l}\left(\frac{k_{w,i+1}+k_{w,i}}{2}\right)\frac{\pi d^2}{4}\left(\frac{T_{w,i+1}-T_{w,i}}{\Delta x}\right)+\left(\frac{k_{w,i-1}+k_{w,i}}{2}\right)\frac{\pi d^2}{4}\left(\frac{T_{w,i-1}-T_{w,i}}{\Delta x}\right)+\sigma \epsilon \pi d\Delta x\cdot \\ [(1-{\alpha }_g)T_{\infty }{}^4+{\epsilon }_g{T}_g{}^4-T_{w,i}{}^4]+h_{c,w}\pi d\Delta x(T_g-T_{w,i})=0\end{array}$$

(4)

k_1.1 is the linear average thermal conductivity between the bead and the first node of the wire 1; k_2.1 is the linear average thermal conductivity between the bead and the first node of the wire 2; k_w,i is the thermal conductivity of the wire node i; D is the bead diameter; d is the wire diameter; T_w,i is the wire temperature of the node i; ε is the surface emissivity of the thermocouple; ε_g is the volumetric emissivity of the gas, which is a function of the gas composition, the gas temperature and the size of the high-temperature gas zone [11]; α_g is the absorption coefficient of the environmental radiation, which is a function of the gas composition, the gas temperature, the high-temperature gas zone size and the environmental temperature [11]; h_c,b and h_c,w are the convection heat transfer coefficients of the bead and the wire, respectively, which can be obtained from the following correlations.

Bead in cross flow [12]:

$$N{u}_b=2+0.6{\mathrm{Re}}^{1/2}{\mathrm{Pr}}^{1/3}$$

(5)

Wires in cross flow [8]:

$$N{u}_w=0.42{\mathrm{Pr}}^{1/5}+0.57{\mathrm{Re}}^{1/2}{\mathrm{Pr}}^{1/3} 0.01 < \mathrm{Re} < 10,000$$

(6)

As shown in Figure 3, the sinusoidal incoming temperature T_g(x) is:

$$T_g\left(x\right)=T_h+T_m\sin \left(\frac{2\pi }{L_p}x+\phi \right)$$

(7)

T_h is the average temperature, and T_m, L_p and φ are the amplitude, the wavelength and the initial phase of the temperature wave, respectively.

The length of the incoming flow is infinity, whereas the wire length L_w of the thermocouple is finite. When L_w is increased from 0, the bead temperature varies with it; when L_w is more than a certain value, the bead temperature will not change. There is a numerical effective length l_sim,

$$L_w\ge l_{sim}, {q_{cond}|}_{L_w=l_{sim}}-{q_{cond}|}_{L_w=0}\ge 0.99\left({q_{cond}|}_{L_w\to \infty }-{q_{cond}|}_{L_w=0}\right)$$

(8)

l_sim is the effective length through the one-dimensional simulation, and q_cond is the conduction rate from the bead to the wire with higher conductivity. q_cond = 0W when L_w = 0m; in the real one-dimensional simulation, L_w = 0.04m is set to represent L_w→∞, which is long enough to guarantee that q_cond and T_b reach stable values. Equation (8) can be simplified as follows:

$$L_w\ge l_{sim}，{q_{cond}|}_{L_w=l_{sim}}\ge 0.99{q_{cond}|}_{L_w=0.04\mathrm{m}}$$

(9)

In the simulation, the ends of the thermocouple wires are set to be adiabatic. The one-dimensional simulation can predict the effective length accurately; however, it does not clearly indicate the relationship among the effective length, the incoming condition, and the thermocouple properties. Therefore, a theoretical solution is also researched based on the fin theory.

3. Theoretical Effective Length

Treating the bead as the base, the two wires can be considered as two fins. Figure 4 shows a schematic of the energy balance of the thermocouple wire. Ignoring the gas radiation, the differential equation of the fin is shown in Equation (10).

$$\begin{gathered} \frac{d^{2}T}{d{x}^2}+\frac{h_{c,w}P}{k{A}_c}\left(T_g-T\right)-\frac{h_{r,w}P}{k{A}_c}\left(T-T_{\infty }\right)=0; \\ h_{r,w}=\epsilon \sigma \left(T+T_{\infty }\right)\left(T^2+T_{\infty }^2\right)，A_c=\frac{\pi d^2}{4} \end{gathered}$$

(10)

P is the circumference of the wire cross section, and A_c is its area. Since a very short part of the wire will influence the bead temperature, h_c,w and h_r,w only vary slightly in this part of the wire, and they are assumed to be constant. k and h_r,w can be calculated approximately based on the bead temperature. h_c,w can be calculated using the film temperature (T_b+T_g)/2. However, in the one-dimensional simulation, the values of k, h_r,w and h_c,w are calculated according to the local condition, and they vary along the wire. This will cause some difference between the numerical and theoretical results.

Defining the excess temperature:

$$\theta \left(x\right)\equiv T\left(x\right)-T_{\infty }$$

(11)

The differential equation can be simplified:

$$\frac{d^2\theta }{d{x}^2}-\frac{h_{w}P}{k{A}_c}\theta +\frac{h_{c,w}P}{k{A}_c}{\theta }_g=0$$

(12)

where h_w = h_c,w + h_r,w, θ_g = T_g − T_∞.

The boundary conditions are:

$${\theta |}_{x=0}={\theta }_b=T_b-T_{\infty };{\frac{d\theta }{dx}|}_{x=L_w}=0$$

(13)

The heat transfer rate between the bead and the wire is:

$$q_{cond}={-k{A}_c\frac{d\theta }{dx}|}_{x=0}$$

(14)

Substituting the gas temperature Equation (7) into Equation (12) and combining the boundary condition of Equation (13), the following theoretical temperature profile can be obtained.

$$\begin{gathered} \theta =-M_t{}^2{T}_m\omega \cos \left(\omega L_w+\phi \right)\frac{\sinh mx}{m\cosh m{L}_w}+\left({\theta }_c-M_t{}^2{T}_{m}sin\phi \right)\frac{\cosh m\left(L_w-x\right)}{\cosh m{L}_w}+ \\ {M}_t{}^2{T}_m\sin \left(\omega x+\phi \right)+\frac{h_{c,w}}{h_w}{\theta }_h；M_t{}^2=\frac{m_c{}^2}{{\omega }^2+m^2},m_c=\sqrt{\frac{h_{c,w}P}{k{A}_c}}，m=\sqrt{\frac{h_{w}P}{k{A}_c}}， \\ \omega =\frac{2\pi }{L_p},{\theta }_c={\theta }_b-\frac{h_{c,w}}{h_w}{\theta }_h,{\theta }_h=T_h-T_{\infty } \end{gathered}$$

(15)

The heat transfer rate between the bead and the wire is as follows:

$$q_{cond}=k{A}_c\left(\left({\theta }_c-M_t{}^2{T}_m\sin \phi \right)m\tanh m{L}_w+\frac{M_t{}^2\omega T_m\cos \left(\omega L_w+\phi \right)}{\cosh m{L}_w}-M_t{}^2\omega T_m\cos \phi \right)$$

(16)

When L_w→∞,

$$q_{cond}=k{A}_c\left({\theta }_{c}m-M_t{}^2{T}_{m}m\sin \phi -M_t{}^2\omega T_m\cos \phi \right)$$

(17)

q_cond is not monotonic under the sinusoidal gas temperature condition, and there exists a maximum l_the satisfying:

$$L_w\ge l_{the}， \frac{\left|{q_{cond}|}_{L_w}-{q_{cond}|}_{L_w\to \infty }\right|}{\left|{q_{cond}|}_{L_w\to \infty }\right|}\le 0.01$$

(18)

In this condition, it can be said that the increase in L_w will not influence the bead temperature anymore. l_the is the theoretical effective length. Substituting Equation (16) into Equation (18), l_the is the maximum L_w satisfying the following equation:

$$\begin{gathered} \left|\left({\theta }_{c}m-M_t^2{T}_{m}m\sin \phi \right)\left(\tanh m{L}_w-1\right)+\frac{\omega M_t^2{T}_m\cos \left(\omega L_w+\phi \right)}{\cosh m{L}_w}\right|= \\ 0.01\cdot \left|\left({\theta }_{c}m-M_t^2{T}_{m}m\sin \phi \right)-\omega M_t^2{T}_m\cos \phi \right| \end{gathered}$$

(19)

The expression contains trigonometric and hyperbolic functions; there is no explicit expression for l_the. However, it can be obtained through simple numerical calculation. The effective length is related to the properties of the thermocouple wire, such as the wire’s diameter and conductivity, the heat transfer coefficient and the incoming temperature parameters, e.g., the amplitude, wavelength and initial phase. The quantitative relationship is discussed in the following section.

4. Results and Discussions

S type thermocouples are frequently used for temperature measurement when the gas temperature is more than 1500 K. Three S-type thermocouples with different sizes are used to study the effective length here. The diameters of the thermocouple wires are 0.05 mm, 0.125 mm and 0.250 mm for thermocouples 1–3, respectively; the corresponding bead diameters are 0.15 mm, 0.375 mm and 0.750 mm, respectively. For the S-type thermocouple, one wire is platinum (Pt, wire 1), and the other wire is platinum/10% rhodium (Pt/10%Rh, wire 2). Their thermal conductivities are given in reference [13].

$$\begin{gathered} k_{Pt}=0.0198T+64.141,T(K) \\ {k}_{Pt/10\%Rh}=0.006T+28.385,T(K) \end{gathered}$$

(20)

The thermal conductivity of Pt is about three times that of Pt/10%Rh. In the one-dimensional simulation, the conductivity of the bead uses the average value of the two wires. Equation (2) shows that the effective length is proportional to the square root of the wire conductivity, and the Pt wire has a larger effective length, whose value is used as the effective length of the thermocouple. The emissivity of the whole thermocouple is set to 0.55. Except when otherwise indicated, the incoming flow will be the air flow with the velocity 10 m/s. ∆x is set to 0.2 mm for all the simulations. The grid independence has been tested with ∆x = 0.1mm, 0.2 mm and 0.4 mm, and the solution difference with ∆x = 0.1 mm and 0.2 mm is negligible.

Figure 5 shows the one-dimensional simulated bead temperature variation with the thermocouple wire length L_w when the gas temperature profile is Equation (21) (unless stated otherwise, the incoming gas temperature profile is always this one). As L_w increases, the bead temperature varies slightly, then reaches a stable value for all the three thermocouples. The stable temperatures are all below 1800 K. As the thermocouple size goes up, the deviation in the bead temperature from 1800 K is larger. The thermocouple wire length for reaching a stable bead temperature also increases with the thermocouple size.

$$T_g\left(x\right)=1800+200\sin \frac{2\pi }{0.004}x，T_{\infty }=300K$$

(21)

Figure 6 shows the temperature profile of the Pt wire of thermocouple 3 (L_w = 0.04 m, the coordinate of the bead center is x = −D/2 = −0.000375 m, and only the result for x ≤ 0.02 m is demonstrated). Both the numerical and analytical solutions are demonstrated. The gas temperature profile is also shown for comparison. Because of the surface radiation loss, the wire temperature is much lower than the gas temperature. The numerical and analytical solutions are almost identical, and they are following the trend of the gas temperature. The wavelengths of the numerical and analytical solutions are almost the same as that of the incoming gas. The crest location difference between T_the and T_g exists only when x is very small, and it is very small (on the order of 0.001 m). When x increases from 0 m to 0.0075 m, the value of the term $-M_t{}^2{T}_m\omega \cos \left(\omega L_w+\phi \right)\sinh mx/(m\cosh m{L}_w)+\left({\theta }_c-M_t{}^2{T}_m\ast sin\phi \right)\cosh m(L_w-x)/\cosh m{L}_w$ in Equation (15) decreases monotonically from 1.79 to 0.9. When x is further increased, this term gradually approaches 0. Therefore, the solution is determined by the term $M_t{}^2{T}_m\sin \left(\omega x+\phi \right)+h_{c,w}{\theta }_h/h_w$, i.e., the wavelength of the wire temperature is the same as that of the gas temperature. The biggest wire temperature difference between T_sim and T_the is for thermocouple 1, which is 1.2%. The h_c,w difference causes this wire temperature deviation; the local h_c,w value is used in the one-dimensional simulation, whereas the constant value is used in the theoretical solution.

Figure 7 shows the numerical and theoretical q_cond variation with L_w for the three thermocouples. It can be seen that q_cond oscillates significantly with L_w, then reaches a stable value. The negative value of q_cond means that the wire conducts heat to the bead, i.e., the wire temperature next to the bead is larger than the bead temperature. This is consistent with the wire temperature profiles in Figure 6. Since the heat transfer rate is larger for the larger thermocouple, the larger the thermocouple size, the stronger the q_cond oscillation. The numerical and theoretical q_cond are very close in the stable region. Considering that the numerical solution is more accurate, the biggest error of the theoretical solution in the stable region is for thermocouple 1, and the relative error is −18.3%. L_w of the numerical and theoretical solutions are approximately the same when q_cond reaches the stable value. A comparison between Figure 5 and Figure 7 shows that L_w are very close for T_b and q_cond reaching stable values.

The theoretical (l_the) and numerical (l_sim) effective lengths of the three thermocouples are shown in

Table 1. Effective length of three thermocouples.

Thermocouple	1	2	3
d (mm)	0.050	0.125	0.250
Tb (K)	1741.4	1657.5	1574.3
lsim (mm)	3.80	7.80	12.40
lthe (mm)	4.20	8.40	12.60
lth (mm)	3.26	6.77	11.56

, including the theoretical effective length calculated with Equation (2) (l_th). The bead temperatures T_b are the stable values (L_w = 0.04 m), which are used to calculate the convection and radiation coefficients and the thermal conductivity of the wire in the theoretical solution.

It is clearly seen that the effective lengths go up with the thermocouple size. l_the is a little larger than l_sim, and the maximum error is about 10.5%, which shows the good accuracy of the theoretical solution. The theoretical length l_th by Equation (2) (the solution under the uniform incoming gas temperature T_h) is less than l_sim, and the maximum error is -14.2% (thermocouple 1). The accuracy of Equation (2) is not bad even for the sinusoidal incoming condition. This equation is explicit and simple, and it can be used to quickly estimate the effective length under a non-uniform incoming condition.

During the study of the effect of T_m, L_p, φ and thermocouple size, the following phenomenon is observed: when L_w is increased from 0.2 mm to l_sim, $\left({\theta }_{c}m-M_t^2{T}_{m}m\sin \phi \right)\left(\tanh m{L}_w-1\right)$ is very close to 0, and $e^{-m{L}_w}$ in $\cosh m{L}_w=0.5\left(e^{m{L}_w}+e^{-m{L}_w}\right)$ is also close to 0, so Equation (19) can be simplified.

$$\left|\frac{\omega M_t^2{T}_m\cos \left(\omega L_w+\phi \right)}{e^{m{L}_w}}\right|=0.005\cdot \left|\left({\theta }_{c}m-M_t^2{T}_{m}m\sin \phi \right)-\omega M_t^2{T}_m\cos \phi \right|$$

(22)

For thermocouple 1, if the thermal conductivity of the two wires is multiplied by 0.64, 0.81, 1.21 and 1.44, the simulated effect lengths l_sim are 8.4 mm, 10.0 mm, 12.4 mm, 14.2 mm and 16.0 mm, respectively. As expected, the effective length increases with the wire conductivity; however, the variation does not exactly follow the $\sqrt{k}$ rule under a uniform incoming condition. Instead, the variation is just slightly more than $\sqrt{k}$ times.

The cases with the incoming velocity 3 m/s, 5 m/s, 10 m/s, 20 m/s and 30 m/s for thermocouple 3 are simulated to study the influence of h_w on the effective length. The simulated h_w of the first Pt node are 642.8 W/m²/K, 750.8 W/m²/K, 946.2 W/m²/K, 1217.9 W/m²/K and 1424.2 W/m²/K, respectively, and the simulated effective lengths are 17.4 mm, 14.6 mm, 12.4 mm, 9.4 mm and 7.8 mm, respectively. As expected, the effective length decreases with the total heat transfer coefficient. Under the sinusoidal gas temperature condition, the variation is just slightly stronger than 1/$\sqrt{h_w}$ times.

The above analysis is focused on the conductivity, the wire diameter, and the heat transfer coefficient of the thermocouple. From Equation (22), we can see that the amplitude, wavelength and phase can also change the effective length. With the one-dimensional simulation, their effects are studied here.

When T_m are 100 K, 200 K, 400 K and 800 K, the stable bead temperatures (L_w = 0.04m) of thermocouple 3 are 1566.5 K, 1578.8 K, 1603.4 K and 1652.5 K, respectively. The effective lengths are 12.4 mm, 12.4 mm, 12.6 mm and 12.8 mm, respectively. The stable bead temperature increases monotonically and slightly with the gas temperature wave amplitude, and it increases 86 K when the amplitude increases from 100 K to 800 K. As T_m increases, the stable bead temperature increases, since the gas temperature next to the bead increases. However, T_m only has a very small influence on the effective length. Figure 8 shows the Pt wire temperature profiles. The first two wire temperature waves are not sinusoidal; after that, the wire temperature profile is very close to the sinusoidal shape, and the wavelength is consistent with that of the gas temperature. Without the thermocouple bead, the wire temperature profile should be sinusoidal. The conduction between the bead and the wire makes the first two wire temperature waves deviate from the sinusoidal shape.

When L_p are 0.04 m, 0.02 m, 0.01 m, 0.004 m and 0.002 m, the stable bead temperatures are 1574.3 K, 1586.3 K, 1591.1 K, 1578.8 K and 1563.1 K, respectively; the effective lengths are 12.0 mm, 12.2 mm, 12.2 mm, 12.4 mm and 12.4 mm, respectively. The influence of L_p on the stable bead temperature is small, and the bead temperature varies within 28 K as the wavelength changes 20 times. The influence of L_p on the effective length is also very small; the length only varies within 0.4 mm as the wavelength changes 20 times. Figure 9 shows the Pt wire temperature profiles. For the large gas temperature wavelengths, the wire temperature profiles are very close to the sinusoidal profile even for the first period. For the small gas temperature wavelengths, they need two or three periods for the wire temperature to reach the perfect sinusoidal profile.

When φ is 0, 0.25π, 0.50π, 1.25π and 1.50π, the stable bead temperatures is 1578.8 K, 1611.3 K, 1610.9 K, 1496.5 K and 1497.8 K, respectively. The effective lengths are 12.4 mm, 12.2 mm, 12.2 mm, 12.4 mm and 12.4 mm, respectively. The influence of φ on the stable bead temperature is relatively strong, since it significantly changes the gas temperature next to the bead, and the bead temperature varies within 114.8 K for the tested cases. Its influence on the effective length is negligible and only varies within 0.2 mm for the tested cases. Figure 10 shows the Pt wire temperature profiles. After the first two periods, the wire temperature almost reaches the perfect sinusoidal profile.

Overall, the amplitude, wavelength and phase of the gas temperature have little effect on the effective length. The effective length is mainly determined by the thermocouple parameters, such as the conductivity, the size and the heat transfer coefficient of the wire. As such, using the theoretical expression of the uniform incoming temperature should be result in calculating the effective length with good accuracy.

5. Conclusions

Using the fin theory, the analytical expressions for the wire temperature and the heat transfer rate under the sinusoidal incoming gas temperature condition are deduced. The accurate and simplified expressions for the effective length are given. Although the constant thermal conductivity and the constant convection and radiation heat transfer coefficients are assumed in the analytical deduction, the theoretical solution demonstrates very good accuracy compared with the numerical solution. Compared with the numerical effective length, the maximum error of the theoretical effective length is only 10.5% for the tested cases. The effective length under the sinusoidal gas temperature condition still increases with the conductivity and the diameter of the wire and decreases with the heat transfer coefficient of the wire. Compared with the theoretical result under the uniform incoming temperature condition, the dependence of the effective length on these parameters is approximately the same. The influence of the amplitude, wavelength and phase of the gas temperature on the effective length are very weak. Using the theoretical expression under a uniform incoming temperature can result in a calculation of the effective length under a non-uniform gas temperature quickly and easily with good accuracy.