NISTIR 7407

Critical Frequency Estimates for 

Thermowells Covered Under 

ASME PTC 19.3

David S. Bartran 

Dean C. Ripple

 
NISTIR 7407

Critical Frequency Estimates for 

Thermowells Covered Under 

ASME PTC 19.3

David S. Bartran 

Consultant 

Saint Louis, Missouri 

Dean C. Ripple 

Process Measurements Division 

Chemical Science and Technology Laboratory 

National Institute of Standards and Technology 

Gaithersburg, MD 20899-8363

March 2007

U.S. Department of Commerce 

Carlos M. Gutierrez, Secretary 

Technology Administration 

Robert Cresanti, Under Secretary of Commerce for Technology 

National Institute of Standards and Technology 

 William Jeffrey, Director 

 
Abstract 

The ASME Performance Test Code, PTC 19.3, establishes the velocity ratings of thermowells in terms of 

a simplified resonant frequency estimate. The theoretical basis for this is examined and compared with 

discrete element methods. It is found that the code calculation fails to guarantee conservative velocity 

ratings for shorter thermowells. The implications of this finding and possible remedies are discussed. 

Keywords: Thermowell design, velocity ratings, vortex shedding, critical frequency estimates. 

Introduction 

The traditional starting point for thermowell design and application is the ASME Performance Test Code, 

PTC 19.3 [1]. While it is the formal basis for the static pressure and velocity ratings of thermowells used 

in performance testing, it is at the same time, the de facto standard for thermowells used throughout 

industry. This rating is derived from a simplified analysis that describes the tendency for thermowell to 

experience flow-induced resonance when the vortex shedding rate coincides with the natural frequency of 

a transverse bending mode. In spite of the widespread acceptance of the code, there is a growing body of 

evidence that thermowells designed to its criteria are subject to increased risk of failure at considerably 

lower velocities. There are a number of reasons for this including use of an out-dated description of 

vortex shedding, limitations of the critical frequency estimate, and reliance on an incomplete stress 

analysis [3]. Similar conclusions are reached for thermowells designed to the code criteria although used 

in non-code applications [4]. A more complete discussion of flow-induced vibration as it relates to 

thermowells and intrusive pipe fittings generally may be found in the literature [5]-[9]. The present 

discussion is restricted to that aspect of the code dealing with resonant frequency calculations. 

The code takes a tabular approach to the critical frequency estimates. This is no longer justified given the 

range of computational tools now available, so it is beneficial to compare the theory used to develop these 

tables with the results of discrete element calculations. While the code estimate is found to be adequate 

for the longer thermowells, it is found to over-estimate the critical frequency (and thus velocity rating) of 

short thermowells commonly recommended for high velocity applications. It also lacks sufficient 

accuracy for use in more rigorous designs or in forensic studies. These deficiencies can only be remedied 

through the use of modern analysis methods for both the critical frequency and stress estimates. 

Murdock’s Approximation 

The PTC 19.3 critical frequency calculation [1,2] combines a single-degree-of-freedom (SDOF), discretemass model of the thermowell with a displacement variable taken as the tip deflection of a continuous 

beam. This highly unusual approach differs significantly from accepted methods for estimating the 

critical frequencies of vibrating beams or establishing the forced response, yet it appears to be 

conservative. The theoretical basis for this method is explained but not endorsed in the following. 

The traditional form of Rayleigh’s method is based on the fact that, in the absence of losses, the 

amplitudes of the kinetic and potential energies, in harmonically vibrating systems, are equal. Following 

the treatment in reference [10] for beams in which the shear and rotation of the neutral axis are ignored, 

the resonant frequency of the beam may be written as: 

2 2

2

2

0 2

d y EI dx

dx

y m dx

ω

⎡ ⎤

⎢ ⎥ ⎣ ⎦ =

∫

∫ . (1) 

The transverse deflection y(x), with x the spanwise coordinate for a beam, is characterized by an elastic 

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modulus, E, a moment of inertia, I, and lineal mass density, m. 

This relationship is exact where the deflection profile is a solution of the fourth-order Stürm-Liouville 

problem for the beam and forms the least upper bound for all estimates based on approximate mode 

shapes. One variation of this method uses the static deflection y(x) of the thermowell (as a horizontal 

beam) sagging under its own weight. Such a solution does satisfy the boundary conditions, but given that 

it is a particular integral for a weight-loaded beam, rather than an homogeneous mode shape, it can only 

provide a crude estimate. 

 

A discrete mass version of the latter results in a critical frequency estimate given by: 

2

0 2

i i

i i

g my

m y

ω = ∑

∑ , (2) 

where yi is the static displacement of the individual masses, mi, and g is the acceleration of gravity. This 

approach suffers from the same limitations as the continuous model but is less accurate and convergent 

only to the extent that the displacements represent actual dynamic modes of the system. 

The Rayleigh method by itself only produces a least upper bound for the resonant frequency. For design 

purposes, given the imprecise method used in the code, a lower bound is preferred to minimize the risk of 

exposing the thermowell to flow-induced resonance. Since the maximum deflection of a cantilever 

always occurs at its tip, a lower bound can be constructed by the expedient of replacing the SDOF 

displacement by the tip displacement of the continuous beam. The resonant frequency is now bounded 

by: 

0

tip m

g g

y y

≤ ≤ ω . (3) 

The lower bound constitutes the Murdock estimate used in PTC 19.3 while the upper bound is the SDOF 

Rayleigh estimate. The analytical expression for the tip deflection, ytip, as derived in reference [2] is only 

valid for linearly tapered thermowells in which shear can be ignored. 

The Code Method 

To simplify the calculations, PTC 19.3 relies on tabulated frequency factors with the resonant frequency 

of the thermowell expressed as: 

 0 2

K f E f L γ = (4) 

where fo (=ω0/2π) is the resonant frequency of the thermowell, Kf is a frequency factor according to the 

Murdock approximation, E is the modulus of elasticity, L is the unsupported length of the thermowell, 

and γ is the specific weight of the material. 

Recasting the Murdock critical frequency estimate in the form of Eq. (4) results in an analytical 

expression for the frequency factors tabulated in the code. They may also be determined by direct 

numerical integration of Eq. (11) of reference [2]. No evidence of a reported ill-condition [2] is observed. 

The frequency factors developed from the Murdock formula [2] are shown in Fig. (1) for the Design 

Classes currently recommended in the code. The thermowell dimensions for these classes in both 

customary US units and in metric equivalents are summarized in Table I, below. 

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Table I: ASME PTC 19.3 Thermowell Dimensions (As adopted 1974). Metric equivalents, in parentheses, are 

approximate. 

Size\Class I II III IV V 

Root 0.8125" 

(2.06 cm) 

0.9375" 

(2.38 cm) 

1.1250" 

(2.86 cm) 

1.2500" 

(3.18 cm) 

1.4375" 

(3.65 cm) 

Tip 0.6250" 

(1.59 cm) 

0.7500" 

(1.91 cm) 

0.9375" 

(2.38 cm) 

1.0625" 

(2.70 cm) 

1.2500" 

(3.18 cm) 

Bore 

(Nominal) 

0.2500" 

(0.64 cm) 

0.3750" 

(0.95 cm) 

0.5625" 

(1.43 cm) 

0.6875" 

(1.75 cm) 

0.8750" 

(2.22 cm) 

Note: Thermowell lengths used in the calculations and as suggested by the code are 2.5", 4.5", 7.5", 10.5", 16", and 

24". A metric conversion of: 1”=25.4 mm, applies. 

Fig. 1: PTC 19.3 frequency factors using the analytic expression in reference [2] for the design classes 

considered in the code (see Table I). 

For completeness, the equivalent Murdock estimate for a hollow cylinder is given by: 

2 2

2

( )

8 f

gD d K

π

+

≅ , (5) 

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where D is the outside diameter at the base (root) and d the nominal inside diameter (or bore) of the 

thermowell. This result is easily compared to the exact theory and suggests that in having ignored shear 

effects, the Murdock calculation under-estimates the actual resonant frequency of the uniform beam by 

some 25 %.

Discrete Beam Comparisons 

The frequency factors for the PTC 19.3 designs can be inferred from discrete beam calculations in similar 

fashion. Two beam models are considered to illustrate the importance of shear as the thermowell aspect 

ratio approaches unity. The discrete beam calculations are based on the Euler-Bernoulli beam elements 

that allows only simple bending and the Timoshenko beam elements that includes both shear and rotation 

of the neutral axis. Transfer matrix methods are used to establish the mode shapes and critical frequencies 

[10,11], with numerical error of the method held to less than 0.2 % in comparison with independent 

calculations [12]. The resulting frequency factors for PTC 19.3 designs are presented in Fig. 2. Rayleigh 

estimates for these designs [13] are shown as heavy dashed lines. 

Fig. 2: Comparison of the frequency factors developed from two discrete beam calculations one that includes 

simple bending only and the other that includes shear and rotation of the neutral axis. 

For aspect ratios L/D >5, the Murdock calculation under-estimates the critical frequency by some 15 % to 

20 % and therefore is sufficient for design. This difference is also in line with that found for hollow 

cylinders. When shear and rotation of the neutral axis are included in the beam calculations, it is 

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discovered that the code estimate is no longer conservative for small aspect ratios. While the lack of 

accuracy of the code method is a concern in forensic evaluations, arbitrary use of improved estimates 

should be approached cautiously to avoid a significant reduction in design margins [3]. In application 

critical, high velocity applications, a finite element based, dynamic analysis is recommended. It is noted 

that finite element methods that fail to include shear and rotation of the neutral axis, although an 

considerable improvement, suffer from the same defect as the Murdock calculation for small aspect ratios. 

 

Conclusions 

The analytical expression for critical frequency developed by Murdock duplicates the frequency factors 

contained in PTC 19.3, and is better suited for direct calculations. While it is generally conservative, 

discrete element calculations that include shear and rotation of the neutral axis are recommended where 

more accurate estimates are required or where severe service conditions are expected. Caution is advised, 

however, in simply combining these improved estimates within the frame work of the current code, 

without consideration of tensile failure in bending. 

References 

[1]. ASME Performance Test Code, PTC 19. 3: Temperature Measurement, 1974. 

[2]. Murdock, J.W. , "Power Test Code for thermometer wells," ASME J. of Eng. for Power, Oct. 1959, 

 pp. 403-416. 

[3]. Bartran, D.S., et al., "Static and Dynamic Stresses of Practical Thermowells," ISA Trans., 39, 2000, 

 pp. 133-142. 

[4]. Hamblin, M., "Flow Induced Fatigue of Intrusive Pipe Fittings Such as Thermowells," Failures 2000, 

 1999, pp. 1-20. 

[5]. Blevins, R. D., Flow Induced Vibration, Krieger Publishing, Malarbar Fla., 1994. 

[6]. Blevins, R.D., Tilden, B.W., Martens, D.H., "Vortex-induced Vibration and Damping of 

 Thermowells," ASME Trans., PVP-Vol. 328, 1996, pp. 465-484. 

[7]. Morishita, M., Dozaki, K., “History of Flow-Induced Vibration Incident Occurred in Monju,” ASME 

 Trans., PVP-Vol. 363, 1998, pp. 103-108. 

[8]. Weaver, D. S. et al, "Flow-Induced Vibrations in Power and Process Plant Components-Progress and 

 Prospects," ASME J. of Pressure Vessel Technology, 122, 2000, pp. 339-348. 

[9]. Haslinger, K.H., "Flow-Induced Vibration Testing of Replacement Thermowell Designs," J. of 

 Fluids and Structures, 18, 2003, pp. 425–440. 

[10]. Thompson, W.T., Dahleh, M.D., 1998, Theory of Vibration, Prentice-Hall, Chap. 12. 

[11]. Han, S.M, Benaroya, H., Wei, T., "Dynamics of Transversely Vibrating Beams Using Four 

 Engineering Theories," J. Sound and Vibration, 225 (5), 1999, pp. 935-988. 

[12]. Gaines, J.H., Volterra, E., "Transverse Vibrations of Cantilever Bars of Variable Cross-Section," J. 

 of the Acoustical Soc., 39 (4), 1966, pp. 674-679. 

[13]. John Jacq Dusting, Private communication. See web-site http://www.jacq.istos.com.au/. 

 

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