Enter Cylinder and Boundary Data
Use meters, watts, kelvin-per-watt terms, and one consistent temperature scale.
Example Data Table
| Mode | ri | ro | Length | k | Inner temperature | Outer temperature | Expected heat rate |
|---|---|---|---|---|---|---|---|
| Surface temperatures | 0.025 m | 0.050 m | 2.00 m | 0.040 W/m·K | 120 °C | 25 °C | 68.89 W outward |
| Fluid temperatures | 0.025 m | 0.050 m | 2.00 m | 0.040 W/m·K | 120 °C, hi 500 | 25 °C, ho 15 | Use both convection films |
Formula Used
The calculator uses steady, one-dimensional radial conduction without internal heat generation.
For fluid temperature mode, total resistance includes inner convection, conduction, optional added resistance, and outer convection.
How to Use This Calculator
- Choose surface temperatures when both wall temperatures are known.
- Choose fluid temperatures when convection films must be included.
- Enter inner radius, outer radius, active length, and conductivity in compatible SI units.
- Enter boundary temperatures using either Celsius or kelvin consistently.
- For convection mode, add both film coefficients and any known series resistance.
- Optionally enter a radius inside the wall to calculate its local temperature and gradient.
- Press Calculate Heat Conduction. Review direction, resistance, heat rate, fluxes, and surface temperatures.
Understanding Divergent Cylindrical Heat Conduction
Radial heat transfer changes shape inside a cylinder. Heat moves through curved layers. Each outward layer has a larger area. The total heat rate remains constant during steady operation. However, heat flux decreases as radius rises. This is the divergent part of the flow. It appears in pipes, sleeves, furnaces, and insulated ducts. A plane wall estimate cannot represent expanding area correctly. Cylindrical resistance therefore uses a logarithmic radius term. This gives realistic results for thick walls and insulation systems.
Why Radius Matters
The inner radius controls the smallest heat transfer area. The outer radius controls the largest area. A large radius ratio increases conduction resistance. It also changes heat flux near the inner surface. Thin walls may behave almost like flat layers. Thick insulation does not. The logarithmic radius term becomes essential as thickness grows. Enter every radius in the same unit. Meter inputs are convenient. The outer radius must always be larger than the inner radius for a valid wall model.
Boundary Conditions and Convection
Surface temperature mode suits known wall temperatures. It isolates conduction through the cylindrical layer. Fluid temperature mode includes convection on both sides. This suits pipes carrying hot or cold fluids. The inner convection coefficient represents the fluid film near the bore. The outer coefficient represents air, water, or another surrounding fluid. Small coefficients can dominate total resistance. The calculator adds active resistances in series. It then finds one heat rate and estimates both wall surface temperatures for the selected conditions.
Reading the Main Results
Heat rate is reported in watts. Its sign shows temperature direction. A positive result means heat moves outward from the inner boundary. A negative result means heat moves inward. Total resistance is reported in kelvin per watt. Larger resistance reduces heat rate for the same temperature difference. Inner and outer heat flux use different surface areas. Their magnitudes differ despite an unchanged heat rate. The local temperature gradient also changes with radius. It is normally steepest near the smaller radius.
Practical Design Checks
Use conductivity values at realistic operating temperatures. Conductivity can change as temperature changes. Insulation may absorb moisture or age during service. These effects can reduce actual resistance. Include additional resistance when joints, coatings, or gaps matter. Check the selected radial location before reading its temperature. It must lie inside the cylindrical wall. Confirm consistent temperature units. Celsius and kelvin differences are numerically equal. Do not mix millimeters with meters. Unit mistakes create very large and costly errors in thermal calculations.
Limits of This Steady Model
This tool assumes one dimensional radial conduction. It assumes constant conductivity and steady conditions. It also assumes no internal heat generation. Long cylinders often match these limits well. Short components can lose heat through their ends. Strong property changes require a detailed numerical model. Radiation can matter at high temperatures. Use equivalent resistance only when its assumptions fit the situation. Treat results as engineering estimates. Compare them with measurements when safety, cost, or reliability matters. Document boundary conditions and data.
Frequently Asked Questions
What does divergent heat conduction mean in a cylinder?
It describes radial heat flow through increasingly larger cylindrical areas. In steady conditions, the heat rate remains constant, but heat flux decreases as radius increases.
Why does the calculation use ln(ro / ri)?
The heat transfer area changes continuously with radius. The logarithmic term integrates that changing area across the cylindrical wall.
Can I use Celsius temperatures?
Yes. Celsius and kelvin temperature differences have equal numerical values. Use one scale consistently for both boundary temperatures.
When should I select fluid temperature mode?
Select it when temperatures are measured in the fluids rather than at the wall surfaces. Enter both convection coefficients for this mode.
Why are inner and outer heat flux values different?
The same heat rate crosses different cylindrical areas. The inner area is smaller, so its heat flux magnitude is larger.
What does a negative heat rate show?
It shows that the actual thermal direction is opposite the assumed inner-to-outer direction. Heat is moving inward from the outer boundary.
What is additional series resistance?
It represents known contact, coating, or interface resistance outside the explicitly modeled convection and cylindrical conduction terms.
Can the selected radius equal an interface radius?
Yes. Enter either wall radius to evaluate the corresponding wall temperature and flux. Values must remain within the cylindrical layer.
Does this model include heat generation?
No. It assumes no internal heat generation. Heated electrical conductors and reacting materials need a model that includes generation.
Is this suitable for very short cylinders?
Use caution. End losses can become important in short components. A two-dimensional or numerical model may provide better accuracy.
Which conductivity value should I enter?
Use the material conductivity near the expected operating temperature. Published room-temperature values may be inaccurate for hot or cold equipment.