Skin Temperature Measurement Model

Chris Boljkovac | March 13, 2025

Background

Sometimes, it’s necessary to measure a process temperature at a location without a thermowell. One option is to install a skin thermocouple mounted under the insulation on the outer surface of the pipe or equipment. As expected, this will lead to a less-accurate measurement. To improve the accuracy, a measurement model can be developed that relates the fluid temperature, skin temperature, and heat transfer in the system.

A common approach to estimate the fluid temperature from the skin temperature is,

$T_f=T_s+C_t(T_s-T_a) (1)$

where,

$T_f$ is the fluid temperature estimate
$T_s$ is the measured skin temperature
$T_a$ is the ambient temperature
$C_t$ is the temperature correction factor

A correction factor of $C_t=0.06$ is often used but it would be expected to vary with the fluid properties and flowrate, pipe and insulation thickness and materials, ambient environment, etc.

Heat Transfer Model

A simple heat transfer model can be developed to determine the factors that affect the temperature correction factor $C_t$ . See Figure 1 for an example of fluid flowing through an insulated pipe.

Figure 1: Pipe Skin Temperature Measurement

Heat transfer from the fluid through the pipe wall can be expressed as,

$\tilde{Q}_{s}=U_{o,s}(T_f-T_s) (2)$

where,

$\tilde{Q}_{s}$ is the heat flux through the pipe wall
$U_{o,s}$ is the overall fluid-pipe heat transfer coefficient

Heat transfer from the outside pipe wall through the insulation can be expressed as,

$\tilde{Q}_{a}=U_{o,a}(T_s-T_a) (3)$

where,

$\tilde{Q}_{a}$ is the heat flux through the pipe wall to the ambient atmosphere
$U_{o,a}$ is the overall pipe-ambient heat transfer coefficient

Note: $U_{o,s}$ and $U_{o,a}$ are based on outside pipe area

At steady-state, these heat fluxes are equal, and therefore,

$U_{o,s}(T_f-T_s)=U_{o,a}(T_s-T_a) (4)$

and, re-arranging for $T_f$ ,

$T_f=\dfrac{U_{o,a}}{U_{o,s}}(T_s-T_a) (5)$

Therefore, by comparing equations (1) and (5), the temperature correction factor $C_t$ is,

$C_t=\dfrac{U_{o,a}}{U_{o,s}} (6)$

the ratio of the overall heat transfer coefficients.

The overall heat transfer coefficients for flat surfaces (and, large diameter pipes),

$U_{o,a}= \dfrac{x_a}{k_a}+\dfrac{1}{h_a} (7)$

$U_{o,s}=\dfrac{1}{h_i}+\dfrac{x_w}{k_w} (8)$

where,

$x_a$ is the thickness of the insulation
$x_w$ is the thickness of the pipe wall
$k_a$ is the thermal conductivity of the insulation
$k_i$ is the thermal conductivity of the pipe wall
$h_i$ is the fluid-pipe convective heat transfer coefficient
$h_a$ is the insulation-ambient convective heat transfer coefficient

and, for circular cross sections (as in our pipe example),

$U_{o,a}= \dfrac{D_o\ln(D_o/D_a)}{2k_a}+\dfrac{(D_o/D_a)}{h_a} (9)$

$U_{o,s}=\dfrac{(D_o/D_i)}{h_i}+\dfrac{D_o\ln(D_o/D_i)}{2k_w} (10)$

where,

$D_i$ is the inside pipe diameter
$D_o$ is the outside pipe diameter
$D_a$ is the outside insulation diameter

Therefore, from equations (9) and (10), the temperature correction factor $C_t$ can be expected to vary with,

pipe wall and insulation thickness,
pipe wall and insulation thermal conductivity, and,
fluid-pipe and insulation-ambient convective heat transfer coefficients.

For item (3) above, the convective heat transfer coefficients are strongly affected by fluid velocity but are also functions of fluid properties (e.g. density, heat capacity, viscosity, and thermal conductivity).

Recommendation

As shown above, the temperature correction factor $C_t$ is affected by many system-specific factors. Using the nominal value $C_t=0.06$ in equation (1) is not expected to result in accurate fluid temperature estimates and is not recommended. When using skin temperatures, the heat transfer models should first be considered.

Chris Boljkovac