How Is The Thermal Resistance Of A Thick-Walled Hollow Cylinder Calculated? Key Equations

The thermal resistance of a thick-walled hollow cylinder under steady-state radial conduction is the key geometric adjustment that separates cylindrical heat transfer from flat-plate calculations. In an educational pilot plant experiment, you compute it using the formula (R = \frac{\ln(r_o / r_i)}{2\pi k L}), where (r_i) and (r_o) are the inner and outer radii, (k) is the material's thermal conductivity, and (L) is the cylinder's length. This single expression captures how the decreasing heat flux area forces a logarithmic temperature profile through the wall, allowing students to directly calculate the conductive heat flow from a measured temperature difference.

The central insight for any pilot-plant experiment is that a thick cylinder does not behave like a plane wall. Its thermal resistance depends on the logarithmic ratio of the outer to inner radii, not simply on wall thickness. Understanding this distinction is the foundation for correctly analyzing both pure conduction setups and the tube-wall contribution in shell-and-tube heat exchangers.

Why the Logarithmic Term Defines Cylindrical Conduction

The need to calculate (R) correctly goes beyond plugging numbers into a formula. In an educational context, the deep goal is to internalize why radial geometry distorts the temperature field and how that distortion feeds into every measured heat balance.

The Physical Origin of the Logarithm

Heat flows radially outward, always moving from a smaller inner surface to a larger outer surface. As the radius increases, the same amount of energy spreads over a proportionally larger area.

This means the thermal flux is not constant—it weakens as you move outward. The conduction equation, when integrated over this changing area, naturally produces a log term.

Comparing the Thick-Wall Resistance to a Flat Plate

If you treated the cylinder as a flat plate of thickness (x = r_o - r_i) and area (A), you would write (R_{\text{flat}} = x/(kA)). For a thick-walled tube, this overestimates the actual resistance and leads to significant error.

The reason is that a flat plate formula ignores the expanding area. The logarithmic expression automatically accounts for this by substituting the logarithmic mean area, which is embedded inside the term (\ln(r_o/r_i)). In fact, you can rewrite the thick-wall resistance as (R = \frac{r_o - r_i}{k , A_{\text{lm}}}), where (A_{\text{lm}} = 2\pi L , r_{\text{lm}}) and (r_{\text{lm}} = \frac{r_o - r_i}{\ln(r_o/r_i)}).

A Quick Derivation Touchpoint for the Lab

For instructional clarity, it helps to see that under steady-state, Fourier’s law in cylindrical coordinates is (Q = -k (2\pi r L) \frac{dT}{dr}). Separating variables and integrating from (r_i) to (r_o) at constant (Q) immediately yields:

(T_i - T_o = \frac{Q}{2\pi k L} \ln\left(\frac{r_o}{r_i}\right)).

Rearranging this into (Q = \frac{\Delta T}{R}) directly defines the cylindrical conduction resistance. This is often the “aha” moment in a pilot plant exercise, because it shows that the resistance is not a fundamental property but a convenient definition derived from the integrated heat equation.

The Role of the Cylinder Resistance in a Complete Heat Transfer Loop

In most pilot-plant experiments, the thick-walled cylinder does not stand alone. It is immersed in fluids that impose their own resistances. The conducted heat must cross a hot-side convective film and a cold-side convective film.

Building the Three-Resistor Network

The overall thermal circuit becomes a series network:

Hot fluid convective resistance: (R_i = 1/(h_i A_i))
Cylinder conduction resistance: (R_{\text{cyl}} = \ln(r_o/r_i)/(2\pi k L))
Cold fluid convective resistance: (R_o = 1/(h_o A_o))

The total resistance is simply (R_{\text{total}} = R_i + R_{\text{cyl}} + R_o). The steady-state heat flow is then (Q = (T_{\text{hot}} - T_{\text{cold}})/R_{\text{total}}), which should match experimental calorimetric measurements.

Where Many Educational Texts Simplify—And Why It Matters

You will often see the tube wall resistance written as (x_m/(k_m A_m)), where (x_m) is the wall thickness and (A_m) is the log mean area. This is not an approximation—for a hollow cylinder, it is mathematically identical to the logarithmic form. However, if the manual substitutes the arithmetic mean area, the error becomes significant for thick walls.

As an educator or a discerning student, you must check which area definition the pilot plant manual uses. The thick-wall formula (\ln(r_o/r_i)/(2\pi k L)) is the safest, most transparent starting point because it removes any ambiguity about the area.

Understanding the Trade-offs

An objective advisor must highlight that the beautiful theoretical formula comes with practical caveats. Ignoring these leads to mismatches between calculated and observed heat duties.

Assumption of purely radial conduction: The formula assumes no end losses and no circumferential temperature variation. In a short educational cylinder, axial conduction through flanges and end caps can be a measurable heat leak that the simple formula ignores.
Uniform thermal conductivity: The value (k) is treated as constant. For materials with significant temperature dependence (e.g., stainless steel over a wide range), an average (k) must be chosen carefully. In an experiment, a 5°C uncertainty in (k) can shift (R_{\text{cyl}}) enough to confuse data analysis.
Ideal contact at interfaces: The resistance formula applies only to the solid wall. In a real apparatus, thermal contact resistance between a temperature sensor and the wall, or between layers, adds an unaccounted term. Students should be taught to identify these as systematic errors.

When the Thick-Wall Formula Becomes Overkill

For a tube with a radius ratio (r_o/r_i) less than about 1.5, the arithmetic mean area introduces an error of less than 1.5% in the resistance. Many shell-and-tube heat exchangers fall into this category. Here, using (R \approx (r_o - r_i)/(k \cdot 2\pi \cdot L \cdot r_{\text{mean}})) is acceptable and pedagogically simpler. The key is to know the boundary where “thick” treatment becomes non-negotiable.

Making the Right Choice for Your Goal

Your approach to the thick-walled cylinder calculation should shift depending on the learning objective of the pilot plant exercise. Here is how to frame the analysis:

If your primary focus is fundamental conduction physics: Use (R = \ln(r_o/r_i)/(2\pi k L)) explicitly. Derive it from Fourier’s law, plot the logarithmic temperature profile, and compare it directly with a false flat-plate prediction to cement the physical reason for the shape.
If your primary focus is overall heat exchanger performance: Treat the cylinder resistance as one component of a series network. You can legitimately present it in the equivalent form (R_{\text{cyl}} = x_w/(k A_{\text{lm}})) to connect seamlessly with convective resistance terms written in terms of areas. However, always note the origin of (A_{\text{lm}}) so the concept isn’t lost.
If your primary focus is experimental error analysis: Use the thick-wall formula as a baseline, then deliberately perturb it by assuming a flat-plate geometry or arithmetic mean area. The resulting difference in calculated (Q) versus the measured hot-fluid energy balance becomes a powerful lesson in the significance of geometric assumptions.

When you treat the thick-walled cylinder’s resistance as more than a formula—when you understand it as the consequence of a radial heat flow that must cross an ever-expanding area—you move from simply collecting data to genuinely interpreting heat transfer behavior.

Summary Table:

Parameter/Method	Formula / Definition	Key Application & Context
Thick-Walled Cylinder	$R = \frac{\ln(r_o / r_i)}{2\pi k L}$	Precise radial conduction analysis; required when $r_o/r_i > 1.5$.
Log Mean Area ($A_{\text{lm}}$)	$2\pi L \frac{r_o - r_i}{\ln(r_o/r_i)}$	Used to convert cylindrical geometry to an equivalent flat plate form.
Flat Plate Equation	$R = \frac{x}{k A}$	Simplified calculation; introduces high error for thick walls if $r_o/r_i$ is large.

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