Advanced Cylindrical Conduction Calculator

Calculator Inputs

Use consistent units throughout. Leave the target field blank and select the variable you want the calculator to solve.

Solve For

Inner Radius, r₁ (m)

Must be positive.

Outer Radius, r₂ (m)

Must exceed inner radius unless solving for r₂.

Cylinder Length, L (m)

Thermal Conductivity, k (W/m·K)

Known Heat Rate, Q (W)

Required for modes solving conductivity, length, temperature, or radius.

Inner Temperature, T₁ (°C or K)

Outer Temperature, T₂ (°C or K)

Probe Radius for T(r) (m)

Optional. Must lie between r₁ and r₂.

Profile Points

Reset

Example Data Table

Case	r₁ (m)	r₂ (m)	L (m)	k (W/m·K)	T₁	T₂	Heat Rate (W)	Resistance (K/W)
Steel pipe wall	0.05	0.09	1.50	16.00	180	60	30,785.95	0.00389788
Insulated section	0.04	0.10	2.00	0.18	140	35	259.61	0.40444831

Formula Used

Radial heat transfer through a hollow cylinder:
Q = 2πkL(T₁ − T₂) / ln(r₂ / r₁)

Conduction thermal resistance:
R_cond = ln(r₂ / r₁) / 2πkL

Inner and outer surface heat flux:
q″_i = Q / 2πr₁L and q″_o = Q / 2πr₂L

Temperature at any radius r:
T(r) = T₁ − [Q ln(r / r₁)] / 2πkL

Temperature gradient:
dT/dr = −Q / 2πkLr

These equations assume steady, one-dimensional radial conduction, constant thermal conductivity, no internal heat generation, and negligible axial effects.

How to Use This Calculator

Select the quantity you want to solve from the dropdown.
Enter the known geometry, temperatures, conductivity, and heat rate where needed.
Use consistent units, such as meters, watts, and W/m·K.
Optionally enter a probe radius to estimate temperature inside the wall.
Choose the number of profile points for the Plotly graph.
Press the calculate button to see results above the form.
Download the generated summary and profile using the CSV or PDF buttons.

FAQs

1) What does this calculator measure?

It evaluates steady radial conduction through a hollow cylinder. You can compute heat rate, conductivity, length, one boundary temperature, or one radius, then review resistance, flux, gradients, and temperature variation.

2) Which units should I use?

Any consistent unit system works. A common choice is meters for radius and length, watts for heat rate, W/m·K for conductivity, and either Celsius or kelvin for temperatures.

3) Why must the outer radius exceed the inner radius?

The hollow cylinder model requires a positive wall thickness. If outer radius is not larger, the logarithmic radius ratio becomes invalid and the conduction equations no longer represent a real cylindrical wall.

4) Can I use Celsius or kelvin?

Yes. Temperature difference is what matters in these formulas. So Celsius and kelvin give identical heat-transfer results, as long as both boundary temperatures use the same scale.

5) What assumptions are built into the equations?

The model assumes steady one-dimensional radial conduction, constant conductivity, no internal heat generation, and negligible axial heat flow. For layered walls or variable properties, a more detailed model is needed.

6) Why does heat flux differ at the inner and outer surfaces?

The same total heat rate passes through different surface areas. Because the outer area is larger, outer surface heat flux is lower than inner surface heat flux for the same cylinder and heat rate.

7) What is the probe radius field for?

It estimates the wall temperature at a selected radial position between inner and outer surfaces. This helps check insulation performance, material limits, and safe operating temperatures inside the cylinder wall.

8) When should I avoid this calculator?

Avoid it when conductivity changes strongly with temperature, heat is generated within the wall, contact resistances matter, or axial and transient effects are important. Those cases need extended heat-transfer analysis.