Thick Wall Cylinder Temperature: From Log Formula to FEM Validation
In a homogeneous thick-walled cylinder with constant conductivity and one-dimensional radial heat flow, the steady-state temperature follows a logarithmic profile in r. Sudden heating or thermal shock needs time-dependent methods, because transient stresses often exceed steady-state values. Temperature gradients drive mainly hoop stress, and the radius ratio and temperature-dependent material properties change the magnitudes materially.
TL;DR:
When the inner surface temperature is unknown, solve a heat balance using fluid temperature and convection coefficient before applying the radial profile.
Use Fourier number to judge transients: low values mean gradients are still developing, while Biot number shows whether surface convection or wall conduction controls resistance.
A hot bore puts the inner layer in compression and the cooler exterior in tension; a cold bore reverses this, adding to pressure stress.
For hot cyclic loads, temperature dependent models matched barrel surface temperatures with maximum error under about 3%, compared with about 10% for constant property models.
Table of Contents
1. How does temperature vary across the wall at steady state?
4. Should you include temperature-dependent material properties?
5. How do you validate with FEM and avoid common mismatches?
6. Worked example: steady-state profile and approximate hoop stress
7. How do you model thermal contact resistance in a thick-wall cylinder?
8. How does heat generation change the temperature distribution?
9. What boundary conditions do real installations actually have?
10. How do analytical results compare with commercial simulation software?
11. How do you choose heat transfer coefficients for inner and outer surfaces?
13. How Jewlz Engineering Toolkit speeds up these calculations
1. How does temperature vary across the wall at steady state?
For steady radial conduction with constant properties, the heat equation in cylindrical coordinates reduces to T = C1 ln r + C2, as textbook derivations for thick-wall cylinders show. Fixing the two surface temperatures gives the closed form, which Clarkson course notes on cylindrical conduction present as:
T® = T1 + (T2 − T1) · ln(r/r1) / ln(r2/r1)
Here T1 is the temperature at the inner radius r1 and T2 the temperature at the outer radius r2. The assumptions matter as much as the formula:
Heat flows only in the radial direction, so you are away from the end caps.
Thermal conductivity is constant. If k varies with temperature, the profile deviates from the logarithm.
There is no internal heat generation and the state is steady.
Both surface temperatures are known.
Often you do not know the inner surface temperature, only a convection coefficient h and a fluid temperature. Then write a heat balance: the convective flux at the inner wall, h(Tfluid − T1), equals the conductive flux through the wall, which per unit length is 2πk(T1 − T2)/ln(r2/r1) when divided across the area 2πr1. Solving that single equation gives T1, and the logarithmic formula takes over from there.
The radius ratio r2/r1 controls how steep the gradient is. Because flux density falls as 1/r, the temperature drop is concentrated on the inner surface in thick walls, and the profile flattens toward the outside. The maximum temperature sits at whichever surface is hotter, so for inner heating it is the bore.
2. When does a transient solution become necessary?
Steady state is only the end point. Under thermal shock the temperature field depends on time, and thermal-shock studies typically obtain it with Laplace transforms. A Dirac delta in the boundary condition can represent a sudden inner-surface heating event, and the resulting transient stresses often exceed the steady-state ones, which matters for fatigue life. A classic thick-wall thermal-shock paper derives the time-dependent temperature by Laplace transform and then solves the stress response, showing how the radius ratio shapes the transient stresses.
Several analytical routes exist:
Laplace transform. Transform time away, solve the radial ordinary differential equation with Bessel-type functions, then invert. This is the standard route for step or impulse inner-surface loading.
Eigenfunction expansion and separation of variables. Series solutions in Bessel eigenfunctions suit finite-time responses. Exact unsteady solutions for functionally graded cylinders were built this way and validated against FEM.
Green’s functions. These give exact expressions for several boundary conditions, and steady-periodic heating of a cylinder shows the approach used in thermal-property measurement.
Short-time Laplace asymptotics. For a sudden inner-surface temperature rise, thermoelastic studies of graded cylinders obtain compact short-time solutions.
Numerical transient FEM. Use it when properties vary with temperature or the geometry departs from an ideal cylinder.
Three dimensionless groups tell you which regime you are in. Thermal diffusivity α = k/(ρc) sets how fast a disturbance diffuses. The Biot number Bi = hL/k compares surface convection to internal conduction: a low Bi means the wall equilibrates internally and the surface resistance dominates, while a high Bi means steep internal gradients. The Fourier number Fo = αt/L² measures elapsed diffusion time against wall thickness. Transient effects dominate when Fo is small and gradients are still developing; our note on Bi and Fo thresholds for transient conduction gives practical cutoffs.
The rule of thumb: if loading is slow compared with the wall’s diffusion time, steady state is fine. For thermal shock, rapid cycling or startup transients, solve the time-dependent problem. For pressure-vessel cases, see our write-up on thermal shock screening and fracture modeling.
3. How does a temperature field turn into stress?
For an axisymmetric cylinder in plane strain, the temperature field enters through thermal strain αΔT®. Equilibrium and compatibility then give radial, hoop and axial stresses as integrals of T® across the wall, as developed in textbook plane-strain derivations, which treat locations away from the end caps. Using the elastic modulus E, Poisson’s ratio ν and expansion coefficient α, the standard results take this shape:
Radial stress is zero at both free surfaces and reaches its extreme inside the wall.
Hoop stress carries the largest thermal component, because hotter inner material is restrained by cooler outer material.
Axial stress follows from the plane-strain constraint and tracks the hoop distribution in sign.
The sign follows the gradient. With a hot bore, the hot inner layer wants to expand but is held back, so it goes into compression and the cooler outer layer goes into tension. Reverse the heat flow, as in a cold quench of the bore, and the bore goes into tension, which is why thermal shock cracks tend to start at the cooled surface.
Superposition makes combined loading practical. Compute Lamé pressure stresses and thermal stresses separately, then add them component by component, as applied thermoelastic FEM studies do for combined pressure and temperature loads. The effect can help or hurt:
Thermal hoop compression at a hot bore offsets the tensile Lamé hoop stress from internal pressure, which is beneficial in steady operation.
During a cold-bore transient, thermal tension adds to pressure tension at the same location, which is the harmful case.
Shutdown and restart cycles can swap which surface is critical, so check both.
Our guides on pressure vessel stress formulas and FEA cover the pressure side of that superposition in detail.
4. Should you include temperature-dependent material properties?
Many real components run hot enough that conductivity, modulus and expansion all drift. Coupled thermoelastic modeling with temperature-dependent material properties (TDMP) is often essential, since temperature changes alter the properties and the thermo-mechanical response in turn.
The evidence favors TDMP where accuracy matters. In a coupled transient analysis under cyclic loads, TIMP (temperature-independent) models tended to overestimate residual temperature and stress compared with TDMP, and in the cited machine-gun barrel experiment TDMP matched measured outer-surface temperatures with a maximum error under about 3%, against about 10% for TIMP.
Functionally graded materials (FGMs) add another layer. In unsteady FGM cylinder solutions, time and material heterogeneity significantly altered temperature, displacement and stress distributions, so the grading index shifts both the temperature profile and where stress peaks. Finite-length graded cylinders also couple heat and moisture, and series-method studies show the grading index affects the results.
Closed-form models stop being reliable in several situations: near end caps where the plane-strain assumption fails, under non-axisymmetric loading, and once plasticity sets in. Move to coupled numerical models in those cases.
Pro Tip: Run the TIMP version first as a cheap bound, then repeat with TDMP. If the two disagree by more than your safety margin, TDMP is not optional.
5. How do you validate with FEM and avoid common mismatches?
FEM is the standard cross-check for analytical work, and studies of asymmetric, time-dependent flux in hollow cylinders note that analytical results can diverge slightly from numerical ones. Special functions such as Kelvin functions and numerical precision issues are the cause, so validate safety-critical analytical results against a numerical model.
Practical validation steps:
Use an axisymmetric model for a long cylinder; use a three-dimensional model only if end effects, ports or non-axisymmetric loads matter.
Refine the mesh through the wall, with elements concentrated near the heated or cooled surface where the gradient is steepest, and repeat until peak stress stops changing.
For transients, choose time steps small enough to resolve the early, steep part of the response, then halve them and compare.
Compare surface temperatures first, since a wrong temperature field guarantees wrong stresses.
Check units and do an energy balance: heat in at the bore should match heat out at the outer surface at steady state.
Run a TDMP sensitivity case to see how much property variation moves the answer.
If your numerical result disagrees with the logarithmic profile at steady state, suspect the boundary conditions or a unit error before suspecting the physics. Our practical FE workflow for pressure vessel thermal analysis walks through these checks.
6. Worked example: steady-state profile and approximate hoop stress
The inputs below are illustrative, not market data. Take a steel cylinder with r1 = 0.10 m and r2 = 0.20 m, an inner wall at T1 = 300 °C and an outer wall at T2 = 100 °C. Assume E = 200 GPa, ν = 0.3 and α = 12 × 10⁻⁶ per °C.
Find the profile. ln(r2/r1) = ln 2 = 0.693. At r = 0.15 m, ln(1.5) = 0.405, so T = 300 + (100 − 300)(0.405/0.693) = 300 − 117 = 183 °C.
Check the gradient. At r = 0.10 m the temperature is 300 °C, and the first half of the wall radius already carries more than half of the total drop, which shows the inner-surface concentration.
Estimate hoop stress at the bore. Use the thermoelastic integral for a hot-bore cylinder: σθ(r1) = αE/(1 − ν) · [ (1/r²)·∫T r dr − T(r1) ] in the standard thin-to-thick form, with the integral evaluated from r1 to r. Evaluating it numerically for this profile gives a compressive hoop stress at the bore of order a few hundred megapascals, with the outer surface in tension. Treat this as an order-of-magnitude estimate, not a design value.
Decide whether steady state is enough. If the bore reached 300 °C over hours, the steady result is acceptable. If it was a sudden step, the transient peak can be higher, so run a time-dependent model.
Pro Tip: Using TDMP would typically lower conductivity at the hot side and shift the profile, so rerun with temperature-dependent k and E before trusting the number.
7. How do you model thermal contact resistance in a thick-wall cylinder?
Real cylinders are often built up: a liner shrunk into a sleeve, a bushing in a housing, or stacked shells. At each interface the surfaces touch only at microscopic peaks, so a temperature jump appears across the junction even in steady state.
The standard treatment adds a contact resistance in series. Define a contact conductance hc and a per-length interface resistance Rc = 1/(hc · 2π ri), where ri is the interface radius. The total resistance is then the sum of each layer’s logarithmic resistance, ln(rj+1/rj)/(2πkj), plus Rc at each junction. The steady heat flow per unit length is the overall temperature difference divided by that sum, and each layer’s temperatures follow from its own logarithmic profile.
Practical consequences:
The jump at the interface is flux times resistance, so it grows with heat flow.
Contact pressure raises hc, so a tight shrink fit conducts better than a loose one.
Thermal expansion changes contact pressure as the part heats, which couples the thermal and mechanical problems.
For stress work, treat each layer separately and enforce displacement and traction continuity at the interface.
When hc is not known, bracket it with a perfect-contact case and a poor-contact case. If the answer barely moves, you can stop worrying about it.
8. How does heat generation change the temperature distribution?
Everything above assumed no source term. When heat is generated inside the wall, for example by electrical resistance, induction or friction, or absorbed at a surface, the governing equation gains a source and the profile is no longer a pure logarithm.
For uniform volumetric generation q‴ in a steady, constant-property cylinder, the solution adds a term proportional to r² to the logarithmic part: T® = −q‴ r²/(4k) + C1 ln r + C2. The constants are fixed by your boundary conditions. Two cases show how much the source matters:
Insulated outer surface. All generated heat leaves through the bore, so the temperature peaks at the outer surface, the reverse of the no-source case.
Cooled outer surface with insulated bore. Heat flows outward, the peak sits at the bore and the profile is steeper near it.
Surface heating, such as radiation or a flame on one boundary, is handled as a flux boundary condition instead of a source term. External heating of the outer wall flips the hot side and therefore flips the sign of the thermal hoop stress.
Generation raises both the peak temperature and the gradient, so thermal stress rises with it. Keep the signs straight: a hot core with a cooler skin puts the core in compression and the skin in tension, which is the opposite of what a cold-bore quench does.
9. What boundary conditions do real installations actually have?
The textbook case of two fixed surface temperatures is rare. Real walls see convection, radiation, insulation and changing environments, and the boundary condition you choose often matters more than the formula you apply.
Common practical situations:
Convective inner and adiabatic outer. A good model for a well-insulated pipe or barrel carrying hot fluid. It gives a uniform wall temperature at steady state, so thermal stress vanishes, and all the action is in the transient.
Convective on both sides. The normal case for heat exchanger shells. Use the heat-balance method from earlier to reduce it to surface temperatures.
Imposed flux. Typical for electric heating or concentrated sources; the surface temperature floats with the wall’s response.
Radiation to surroundings. Becomes significant at high temperature and makes the problem nonlinear, so iterate.
Several habits prevent mistakes. Measure or bound the fluid temperature at the wall, not the bulk value, since films and stratification shift it. Keep in mind that h itself varies with flow rate, so a shutdown can change the boundary condition abruptly. Where a boundary condition is uncertain, run both extremes and design for the worse one. For readers who want external definitions of thermal classes and cooling concepts, the datacenter thermal glossary on average waste heat temperature is a useful quick reference.
10. How do analytical results compare with commercial simulation software?
Analytical solutions and commercial FEM or CFD packages answer different questions, and the best practice is to use both. Applied thermoelastic work routinely uses ABAQUS or ANSYS for validation of stress and displacement under combined pressure and temperature loads.
Where the two agree, you gain confidence. Where they disagree, the cause is usually one of a short list:
Different assumptions. The closed form assumes plane strain, constant properties and a perfect cylinder; the software model may include end effects or variable properties. Match the assumptions before comparing.
Numerical approximation. Evaluating special functions can introduce small differences, as noted in the asymmetric heat-flux study.
Mesh and time step. An under-resolved gradient will underpredict peak stress.
Boundary condition mismatch. A film coefficient entered in the wrong units is the classic culprit.
Analytical results are fast, transparent and good for sensitivity studies and sanity checks. Commercial software handles geometry, nonlinearity, contact and plasticity that closed forms cannot. A sound workflow runs the closed form first, builds the axisymmetric model, confirms it reproduces the closed form on a simplified case, and only then adds real-world complexity. For safety-critical parts, keep the independent check.
11. How do you choose heat transfer coefficients for inner and outer surfaces?
The convection coefficient h links the wall to the fluid, and it is usually the least certain input in the whole calculation. It depends on fluid properties, velocity, geometry and whether the flow is natural or forced.
Typical approach:
Inner surface, forced flow. Use a pipe-flow correlation based on Reynolds and Prandtl numbers to get the Nusselt number, then h = Nu · k_fluid / D. Laminar and turbulent flow differ greatly, so check the Reynolds number first.
Outer surface, natural convection. Use a free-convection correlation based on the Rayleigh number. Values are low, which is why uninsulated outer walls are often close to adiabatic in a short transient.
Outer surface, forced flow. Use a cross-flow correlation for a cylinder if air or water moves across it.
Combined radiation. At elevated temperature add a radiative coefficient to h, or model radiation separately.
Remember how h enters the physics. It sets the Biot number, so a larger h pushes the surface temperature toward the fluid temperature and steepens internal gradients, raising thermal stress. A small h keeps the wall nearly uniform and the stress small.
Because correlations carry real uncertainty, run a sensitivity sweep: vary h up and down by a wide margin and see whether your conclusion changes. If it does, measure h rather than estimating it.
12. Joel’s rules of thumb for working engineers
Here is how I work through these problems. First, check Bi and Fo before anything else. A low Biot number indicates that the wall is nearly uniform and the radial formula may be a useful first pass; check Fo to see whether you are still in the transient window. Second, scale effort to risk: start with the closed form, move to a quick axisymmetric FEM check, and use a full three-dimensional model only if end effects matter. Third, never trust a single run; vary h and the properties. The free toolkit features make these quick checks cheap.
— Joel
13. How Jewlz Engineering Toolkit speeds up these calculations
We built our toolkit so you can get from formula to verified result without a license barrier. It combines conduction, convection and radiation models in one place, supports variable material properties across a wide temperature range, and includes a built-in property database. That covers the workflows in this article: steady and transient thermal analysis, temperature-dependent properties, and pressure-plus-thermal superposition for cylinders.
The toolkit can help with these calculations. Students and early-career engineers can use the free basic tools for checks, while experienced designers can use the paid features for deeper work. You can also see our thermal analysis tools for the steady and transient modules in detail.
Ready to run your own cylinder? Open the Jewlz Engineering Toolkit and try the free tools today.
FAQ
What is a thick-walled cylinder?
A thick-walled cylinder is one whose wall is large enough compared with its radius that stress and temperature vary significantly across the wall. Thin-wall approximations, which assume uniform values, no longer hold, so you use the radial formulas shown above.
What is the hoop stress for a thick-walled cylinder?
For pressure loading, hoop stress comes from the Lamé equations and varies with radius, peaking at the bore. Thermal gradients add a further hoop component computed from the temperature field, and the two are superposed, as covered in the pressure vessel stress guide.
How do you calculate thermal expansion of a cylinder?
For a uniform temperature change, each dimension grows by its original length times α times ΔT. For a thick wall with a gradient, the unrestrained expansion is not uniform, so you integrate αΔT® across the wall to find displacements and stresses.
Why can thermal shock be worse than steady heating?
During a sudden temperature change the steep gradient exists before the wall has equilibrated, and transient stresses often exceed steady-state values. That makes time-dependent analysis essential for startup, quench and rapid-cycling cases.
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