Chapter 12 installation effects on temperature sensors
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“… the key to obtaining accurate gas temperature measurements from thermocouples is an understanding of the behavior of a bare wire thermocouple in response to its environment … radiation shields and stagnation tubes do not change a junction's sensitivity to its environment, they change its environment ...”

Robert J. Moffat (1961)

In this chapter we consider separately installations in fluids and installations in solids as they affect the temperature sensor. In fluid applications, the sensor faces a complex heat transfer effect [1]-[7].

Convective heat transfer between fluid and sensor is balanced against radiative heat transfer between fluid, its enclosing walls, and the sensor, and simultaneously against conductive heat transfer between sensor and its supports.

In surface applications, on the other hand, radiation and convection are assumed to affect sensor and surface roughly alike; thus conduction between sensor and its supports becomes the dominant mode of heat transfer to consider.

Only secondarily must the convection effects between sensor supports and the ambient fluid be considered.

12.1 Combined Heat Transfer Equations for Fluid Installations

For a physical model, consider a gas flowing in an enclosure into which is immersed a temperature sensor and its support (hereafter referred to simply as the sensor).

To be specific let Tgas > Tsensor > Twall, although in a numerical case consideration of algebraic signs will allow variations in this restriction.

The temperature sensor can receive heat by convection and radiation.

It can lose heat by conduction and radiation. The modes and quantities of heat transfer involved are affected by boundary conditions, gas properties, the thermodynamic state of the gas, and by the nature of the directed motion of the gas.

A model of the problem to be examined is represented schematically in Figure 12.1. For a differential element of the

sensor (see Figure 12.2), we can write a general heat balance expressing the conservation of thermal energy as



where the subscripts have the following meanings:

c = convection, r = radiation, and k = conduction.

Thus our first job is to determine reasonable expressions for the various terms in (12.1).


Heat will be transferred to the sensor from the moving gas by forced convection. This phenomenon has been described by Newton’s cooling equation as



where Ts is the static temperature of the gas, and Tx is the temperature of a differential element of the sensor.

If the gas moves with an appreciable velocity, however, the temperature even an adiabatic sensor attains will not be the static temperature of the gas (see Chapter 11).

Instead, the thermally isolated sensor will sense the adiabatic temperature of the gas, which can be defined in terms of (11.22) as


where cp must be considered constant over the temperature range (Tadi-Ts). Thus it is clear that the modified Newton cooling equation to use in the heat balance of (12.1) is


The temperature distribution through the gas surrounding the sensor call be visualized as in Figure 12.3. The two coefficients of (12.4), hc and R, are discussed at greater length in Section 12.3.


There will be an interchange of radiant energy between the sensor, the gas, and the enclosing walls. For a black body, this phenomenon is described by the Stefan-Boltzmann radiation equation


The sensor will radiate energy according to its absolute temperature, as indicated by (12.5), modified, however, by the emissivity of the sensor, which accounts for its non-black body characteristics. Some of this energy will be absorbed by the gas and the remainder by the enclosing walls; that is,



But this is not the complete story. There are reflections (radiant inter-changes) between the gas and the solid gray body (the sensor) that must be considered, Effects of these reflections are included by adjusting the sensor emissivity as recommended by McAdams [8] and by Grober [9]; thus


where αg,x signifies absorptivity of the gas, to be evaluated at the sensor temperature.

For the sensor emission absorbed by the enclosing walls we have


When (12.7) and (12.8) are combined according to (12.6), we have



Equation 12.9 represents the net rate of radiant heat transfer from the sensor.


The gas will radiate energy according to its absolute temperature and its emissivity. The sensor will absorb some fraction of this energy incident on its surface.

where єg,g signifies emissivity of the gas, to be evaluated at the gas temperature.

By Kirchhoff’s law for solid bodies, αx may be replaced by єx; and again considering the reflections between the fluid and the sensor, we replace єx with (єx +1)/2 to obtain




The enclosing walls will radiate according to their absolute temperature only, considering such enclosures to be black. However, the gas intercepts and absorbs some of the radiant energy emitted by the walls so that the net radiation received by the sensor from the walls is

whereαg,w signifies absorptivity of the gas, to be evaluated at the wall temperature



By combining (12.9), (12.11), and (12.12), we obtain the expression for the net emission from the sensor by radiation, as required by the heat balance of (12.1).


For convenience (12.13) also can be expressed in terms of a radiation coefficient as


patterned after Newton’s cooling law, where



The coefficients hr and є’ are discussed further in Section 12.3.


Heat will be transferred from the tip of the sensor (in the gas) to its base (at the wall) by means of conduction. This phenomenon is described by Fourier’s conduction equation


For an element of the sensor in the steady state (i.e., for the case of zero heat storage), the one-dimensional expression for the net conductive heat transfer, as required by the heat balance of equation (12.1), is


The Heat Balance

When (12.4), (12.14), and (12.18) are combined according to (12.1), we obtain


12.2 Solutions to Combined Heat Transfer Equation

Equation 12.19 is a second-order, first-degree, nonlinear differential equation, and as such has no known closed-form solution.

It is nonlinear because the coefficients a2 and a3 are both functions of the dependent variable Tx. The offender in both cases is the radiation coefficient hr. There are at least three approaches to a solution of (12.19).

Tip Solution

Here all conduction effects are neglected, and (12.19) reduces to


This solution leads to a sensor tip temperature that is usually too high, since any conduction tends to reduce Ttip.

Overall Linearization

Here hr, is based on an average Tx, justified by noting that (Tadi - Tw) << Tw in any practical problem. Thus for specifying the radiation coefficient only, we can approximate Tx, which is bounded by Tadi and Tw, by Tadi, Tw, or its average value


If, in addition, a right circular cylinder is assumed for the geometry of the sensor and support, the area dependence on x is removed. Under these conditions (12.19) is linearized, the coefficients a1, a2, and a3 are constants, and the closed solution is



This solution leads to quick, approximate answers for the case in which the gas can be considered transparent to radiation (i.e., for єg≈0), but, in general, overall linearization leads to unreliable results.

Stepwise Linearization

Here the solution is based on dividing the sensor and its support, lengthwise, into a number of elements, as indicated in Figure 12.4. The temperature Tx at the center of each element is taken to represent the temperature of that entire element.

The number of lengthwise divisions can be made as large as desired to enhance the finite difference approximation to the nonlinear equation (12.19).

Three heat balance equations will describe the heat transfer completely through the model of Figure 12.4.

TIP ELEMENT. The heat balance for element 1 is


From which we obtain

T2=T1-(Ac,0+Ac,1) Δx/kAk,1[hcTadi-(hc+hr,1)T1+Hr,1Tw]


The heat balance for any internal element is



=hr,xAr,x(Tx-Tw)+kAk,x(Tx-Tx+1)/ Δx

From which we obtain



BASE ELEMENT. The heat balance for element N is



From which we obtain

where T’w is the calculated value of the enclosing wall temperature.




For convection and radiation calculations, it is the surface area of the sensor that is required. Considering a right circular cylinder, we have

For conduction calculations it is the cross-sectional area of the sensor that is required. Again for a right circular cyclinder we have



An initial T1 is assumed (Tadi is a good first choice), and all other temperatures (T2, T3, ..., TN, and T’w) are obtained according to (12.24), (12.25) , and (12.26). The calculated wall temperature T’w is compared to the given wall temperature Tw, and adjustments in T1 are made for a second try.

Iteration schemes (such as Newton’s method) rapidly lead to a unique solution for the steady-state temperature distribution throughout the sensor-support combination [10].

Comparison of the Methods

The tip solution, in which conduction effects are neglected, leads to a sensor temperature that is always too high. The overall linearized solution is sometimes adequate (when єg is small), leads at times to sensor temperatures that are higher than the tip solution (whenєg≈0.5), and sometimes yields imaginary solutions (when єg≈1), in which case the ratio

exceeds 1.

Of course the stepwise linearized solution represents the most reliable solution of the three.

Trend Curves

Because of the complex relationships between the many variables involved in the heat transfer analysis, it is easy to lose track of the physical effects of the controlling variables.

In an effort to give a clearer picture of these effects, trend curves are presented in Figure 12.5 for a particular practical problem.

In Figure 12.5a, note the highly beneficial effect of ensuring a substantial convective film coefficient over the thermometer well.

In Figure 12.5b, the less dramatic effect of well thermal conductivity is seen.

In Figure 12.5c, the importance of fluid and well emissivity is shown;

Note the insulating effect the fluid emissivity has on the thermometer well, simplifying the temperature measuring problem in steam, for example, in Figure 12.5d, the importance of insulating the enclosing walls, at least in the vicinity of the thermometer well, is seen.

Graphical Solution

In Figure 12.6, a summary curve of many problems solved by the step linearized method (based on 20 steps) is given. The coordinates are the parameters of (12.22) and (12.23), that is, of the overall linearized solution.

Because the actual problem is nonlinear, no exact graphical solution is possible. Even under the extremes of all the variables, however, as noted in Figure 12.6, the limits of uncertainty that attend the summary curve are relatively narrow.

Thus some useful information concerning a given sensor installation can be obtained rapidly from Figure 12.6, as illustrated in several examples to follow.

12.3 Some Useful Heat Transfer Coefficients

Often we must estimate a heat transfer coefficient as an item of secondary importance in a particular job [11]. A case in point is the requirement of obtaining the coefficients hc, α, and hr, in the equations of Section 12.2.

There is such a confusing array of coefficients, exponents, and equations available in the various texts (see for example, Table 12.1) that it is believed best to present satisfactory values in graphical form for quick, ready estimates of these heat transfer coefficients.

Forced Convection Film Coefficients

The dimensional analysis that , for forced convection , yields an expression of the form


is well known, and advantages of the use of (12.29) in correlating large amounts of data for many different fluids are obvious [12]-[15].

The convective film coefficient hc, however, is often desired, and this requires an evaluation of the Reynolds number

and then an arithmetic operation to obtain hc from the Nusselt number and the Prandtl number

To simplify the procedure, the following is suggested. From (12.29)


Equation 12.30 can be rearranged to give


Thus through the term in parentheses the film coefficient is seen to be a function of the thermodynamic state of the fluid. If a particular reference state is chosen for a particular fluid (e.g., 500 psi, 500℉ steam), a unique family of curves can be plotted in terms of hc = f(G, D).

This has been done for appropriate values of a, b, and c as reported in the literature for

(a) forced convection inside cylinders (see Figure 12.7); and

(b) for forced convection across single cylinders (see Figures 12.5 and 12.9).

To obtain the actual film coefficient hc for any fluid at any state, it is necessary only to multiply the plotted reference value of h’c by the ratio


Correction curves representing this ratio for steam and air are given in Figures 12.10 through 12.12. These curves are required to account for variations in the thermodynamic state properties μ, k, cp.

Recovery Factor

In Section 11.7 the recovery factor was discussed at some length. Briefly, there are several recovery factors to choose from. The frictional recovery factor r, based on the local adiabatic temperature, remains constant around the periphery of a cylinder. This convenient-to-use recovery factor is identical to the flat-plate recovery factors of (11.19) and (11.21).

Another recovery factor is based on the undisturbed free stream velocity, the static temperature, and the mean adiabatic temperature.

This is the correct recovery factor to use, but unfortunately it varies with the Mach number, and no systematic correlation exists.

However, noting that its value for air varies from 0.55 to 0.82, which is similar to the Prl/2 evaluation, and considering the inherent difficulties in the use of this mean-free r, we recommend for analysis the constant valued local rs, as defined by (11.19) and (11.21).

This simplification tends to introduce a slightly greater rate of convective heat transfer to the sensor than actually exists and leads in turn to uncertainties that are on the optimistic side.

Radiation Coefficients

The coefficients hrand є’ of (12.15) and (12.16) also can be represented as the ratio hr/’ in convenient graphical form, as indicated in Figure 12.13.

In view of the scarcity of information on gaseous emissivity, it is suggested that advantage be taken of the simplification єg,x= єg,g = єg,w. This, together with the approximations that F of (12.9) equals l, leads to a more manageable form of (12.16), namely,


12.4 Fluid Applications and Examples

Several examples are given to illustrate the use of the graphical heat transfer coefficients in conjunction with the various combined heat transfer analyses toward solutions to typical fluid installations.

Example 1. Tip Solution (involving no Conduction).

Given that a spherical sensor of surface area = 0.8 in.2, disk area = 0.2 in.2, and emissivity = 1 is installed in a duct of radius 2 in. and length 6 ft, the sensor indicates 250℉ when the duct wall is at 200℉, and the convective heat transfer coefficient is 20 Btu/h-ft2-℉.

Find the gas temperature.

Solution (based on use of a radiation coefficient)




in agreement with (12.20).

Now from Figure 12.13,


From (12.16) є’ = 1 if the walls and sensor are considered to be black bodies (єw= єx = 1), and if the gas is assumed to be transparent to radiation (єg= 0 ). Thus


Example 2. Same as Example 1 except the Solution is now based on use of form and emissivity factors,



When the form factor FA is given according to Schenck [14] as

FA=4L/(L2+R2)1/2(Adisk/Asurface)sensor= 6/(36+1/36)1/2=1

and the emissivity factor is taken as unity, the result is


Note that it is the difference in fourth-power absolute temperatures that must be used in this form of the radiation heat transfer equation. Thus


We see that under the same conditions the two solutions agree within close limits.

Thermometer Well Solutions

Such problems resolve to the following. Given the installation details of a thermometer well, find whether the installation is satisfactory. The steps required are:

1. Remove area dependence on immersion length by approximating the given well by a right circular cylinder of the same surface area.

2. Determine the pertinent heat transfer coefficients.

3. Determine the value of


where L = length of immersed portion of thermometer well,

d = bore diameter of the well,

D = outside diameter of the well.

4. If X' ≤ 20, enter Figure 12.6 with


where a3 is defined under (12.19). Now compute either Ttip or Tadi, whichever is unknown, accounting to the following:



On the basis of the difference between Ttip and Tadi, the worth of the installation can be judged.

5. If X' > 20, it means that conduction effects are negligible. Thus (12.36) or (12.37) are solved for Ttip or Tadi by setting

Y = 0.

Example 3.

Air flows at a rate of 1 lb/sec at a pressure of 2 atmospheres in an uninsulated 6-in. pipe. The indicated well tip temperature is 200℉, and the indicated enclosing wall temperature is 180℉. The cylindrical well has a 3-in. immersion, 1/2in. OD, 1/8in. ID, k = 20 Btu/h-ft-℉,єwell = 0.9, єfluid = 0 (i.e., transparent). The installation is to yield fluid temperature to ±1℉.



from Figure 12.8 (low Reynolds number),

from Figure 12.10 (low Reynolds number),


from Figure 12.13,

by (12.16),


By (12.34)

Since X' is less than 20, form X according to (12.35).

and from Figure 12.6, Y = 0.070.

By (12.37)

This fluid temperature is not within requirements, and the installation must be changed. Insulating the wall in the well vicinity seems the simplest change.

One inch of 85% magnesia (k= 0.036 Btu/h-ft-℉) raises Twall to 196℉, although neither X nor Y changes appreciably.

Now, (12.37) yields

This is within specified bounds so the installation is now satisfactory. The generalized curve of Figure 12.6 does not always lead to reliable graphical solutions, since the ratio (Tx – A3)/(Tw - a3) does not reflect the radiation effect adequately, and the parameter (hr + hc)DL2[k(D2 – d2)] may mask the conduction effects when radiation is significant.

Whenever conduction effects can be neglected, however (i.e., whenever a very reliable approximation is easily obtained. This approach, based on the tip solution, as given by (12.20), is illustrated by the following example.

Example 4.

Steam flows at a rate of l.7lb/sec-ft2 at a pressure of 14.7psia in an uninsulated pipe. The indicated well tip temperature is 506.6℉, and the enclosing wall temperature is 350℉. The well has an 8-in. immersion, OD = 1/4in., ID ≈ 0, k = 10 Btu/h-ft-℉, єwell = 0.97, єfluid = 0.35. Would this installation be acceptable if fluid temperature to ±5℉ were required?


from Figure 12.8 (low Reynolds number),

from Figure 12.11,

from Figure 12.13,

for first try by assuming F = l, and

in (12.16).


which is much greater than the suggested 20; therefore, set Y = 0 and solve (12.37). There results

The estimate є’ is now corrected by using Tadi,1 to give




Further iteration will not change Tadi significantly. Thus we conclude that this installation is unsatisfactory, since

Tadi - Ttip = 18.7℉ which is greater than the five-degree requirement.

The information presented here should be used to correct the installation (if so required) until acceptable limits of uncertainty in the temperature measurement are obtained.

These solutions should never be used as corrections to a sensor indication.

12.5 Solid Applications

The main difficulty in sensing surface temperatures usually concerns the method of attachment of the sensor to the surface. That is, the sensor must attain and yet not upset the surface temperature.

In Chapter 8, Optical Pyrometry, a method was discussed that circumvents the above difficulty. A cavity is drilled into the surface whose temperature is required in an attempt to approach black body conditions.

By sighting on the cavity, one avoids the attachment problem and, hopefully, avoids the temperature perturbation problem also (recall that surface sightings require a knowledge of surface emissivity as a function of wavelength and temperature , and that such information is rarely available, or at best is highly uncertain).

Usually more mundane methods are employed in surface temperature measurement.

In particular, the thermocouple method is discussed briefly as most representative of the possible approaches, although RTDs are used in increasing numbers to perform this same function.

The following information is based on the work of A. J. Otter [17]. For treatment in greater detail, the excellent and comprehensive books by Baker, Ryder, and Baker should be consulted [18].

Some common methods of attachment of thermocouples to surfaces are shown in Figure 12.14.

The junction can be held to the surface by solder, braze, weld, insulating cement, peen, or simply by pressing.

The insulated thermocouple wires should be held in intimate contact with an isothermal portion of the surface for a length of at least 20 wire diameters to avoid steep temperature gradients in the vicinity of the thermocouple measuring junction.

The usual relation between indicated and true surface temperature is given by


where the desired temperature difference , Tsurface - Tsensor, is expressed as a fraction of the natural temperature difference, Tsurface - Tambient. The calibration factor z can be determined reliably only by experiment, although many theoretical approximations are to be found in the literature.

Typical calibration factors are given in Table 12.2, and the junction types for which these apply are shown in Figure 12.15.

Naturally the results are applicable only for the conditions given in Table 12.2.

It can be seen that the sensor size should be as small as possible and that, under the stated conditions, the junction types in order of preference are: separated, button, exposed, and (least desirable) grounded.

Certain general conclusions can be made as to minimizing surface temperature measurement errors.

1. Keep installation size as small as possible.

2. Keep sensor wires in an isothermal region for at least 20 wire diameters.

3. Locate sensor as close to surface as possible.

4. Disturb ambient conditions at the surface as little as possible by the sensor installation.

5. Reduce thermal resistance between sensor and surface to a minimum.

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