First principles before FEA

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One exchanger tube. Three different questions.

A straight tube stands in for one member of a tube-bundle heat exchanger. First ask what pressure does to it. Then ask how it expands when heated. Finally ask what happens if the tube sheets prevent that axial expansion. These are separate verification cases, not a combined pressure-and-temperature design analysis.

No actual alloy strength, temperature-dependent properties, fatigue limit, or allowable stress has been supplied. Consequently hardware safety is NOT_ESTABLISHED, even when the equation checks pass.

1. Pressure proof: equilibrium before a contour plot.

Tension is positive. A long, rotationally symmetric elastic cylinder must satisfy radial equilibrium. The Lamé solution simultaneously satisfies that equation and the inner/outer pressure tractions.

\epsilon_r=u_r',\quad \epsilon_\theta=\frac{u_r}{r},\quad \epsilon_z=\mathrm{const},\qquad u_r''+\frac{u_r'}{r}-\frac{u_r}{r^2}=0\Rightarrow u_r=C_1r+\frac{C_2}{r}

\frac{d\sigma_r}{dr}+\frac{\sigma_r-\sigma_\theta}{r}=0,\qquad \sigma_r=A-\frac{B}{r^2},\quad\sigma_\theta=A+\frac{B}{r^2}

A=\frac{p_i a^2-p_o b^2}{b^2-a^2},\qquad B=\frac{a^2b^2(p_i-p_o)}{b^2-a^2}

The check is explicit: differentiation gives 2B/r³; the hoop-minus-radial term supplies −2B/r³. Their sum is zero. At r = a, radial stress is −pᵢ; at r = b, it is −pₒ. The script tests both tractions rather than accepting a plausible-looking stress contour.

\sigma_z\pi(b^2-a^2)=\pi(p_i a^2-p_o b^2),\qquad \sigma_z=A

u_r(r)=\frac{(1-2\nu)Ar+(1+\nu)B/r}{E},\qquad \epsilon_z=\frac{(1-2\nu)A}{E}

The thin-wall estimate pᵢa/t gives 20 MPa, versus 21.048 MPa at the bore in the thick-wall calculation. Geometry determines whether that approximation is adequate. These stresses are not applicable at ports, tube-sheet joints, welds, or end-cap transitions.

2. Thermal proof: the boundary condition is the physics.

For the thermal-only axial model, the transverse surfaces are traction-free. Axial elastic strain is total axial strain minus thermal strain. A sliding end and two axially fixed ends solve different engineering problems.

\sigma_z=E\left(\frac{du}{dx}-\alpha\Delta T\right)

\sigma_z=0\quad\Rightarrow\quad u(L)=\alpha\Delta T L=1.280\ \mathrm{mm}

u(0)=u(L)=0\quad\Rightarrow\quad \sigma_z=-E\alpha\Delta T=-243.200\ \mathrm{MPa}

S=\pi(b^2-a^2),\qquad |R|=S E\alpha\Delta T=16.045\ \mathrm{kN}

Negative stress means compression. It does not mean the real tube can sustain it: yielding, buckling, support compliance, gradients, and tube-sheet behavior are omitted. Nor is −EαΔT the formula for fully triaxial restraint. As COMSOL's constraint example explains, adding restraints can produce a converged but physically irrelevant result.

3. Actual small FE calculations, compared with the derivations.

The downloadable Python implementation assembles and solves stiffness matrices. This is an educational radial FE / axial-bar calculation, not an ANSYS or CalculiX run. It does not replace Aero's existing conventional-solver qualification infrastructure.

\mathbf K=\sum_e\int_e\mathbf B^T\mathbf D\mathbf B\,2\pi rL\,dr,\qquad\mathbf K\mathbf q=\mathbf f

The pressure model uses the full isotropic radial/hoop/axial constitutive matrix, three-point Gauss integration, and the closed-end pressure force. Linear radial elements are refined through five meshes. Stress error is evaluated at quadrature points against the independent analytical field—not against the next mesh alone.

\mathbf k_e=\frac{ES}{\ell}\begin{bmatrix}1&-1\\-1&1\end{bmatrix},\qquad\mathbf f_{th,e}=ES\alpha\Delta T\begin{bmatrix}-1\\1\end{bmatrix}

Both thermal cases were solved with 2, 4, 8, and 16 elements. The linear free-expansion field and zero-displacement restrained field are represented exactly by these elements; stresses and reactions agree to floating-point precision. That exactness is specific to this simple problem, not proof that a real exchanger is mesh-independent.

Try the first-principles equations.

Change synthetic inputs locally in your browser. This recalculates the closed-form equations only; it does not run FEA or change the frozen verification record above. No prompt, geometry, or input is sent to a server.

ν = 0.30; outside gauge pressure = 0; uniform constant properties. No allowable stress, buckling check, material certification, combined-load model, or engineering PASS is inferred from these numbers.

What would make this an exchanger FEA qualification?

1. Freeze the real question, geometry, material data, pressure loads, temperatures, restraints, outputs, and acceptance criteria before solving.
2. Use the relevant conventional solver. Retain native fields, mesh, element formulation, reactions, warnings, and input decks.
3. Check pressure and reaction equilibrium, thermoelastic assumptions, mesh sensitivity, and relevant local stresses away from artificial singularities.
4. Investigate tube-sheet contact, joints, thermal gradients, yielding, buckling, fatigue, and mapped-fluid loads where physically applicable.
5. Separate numerical verification, physical validation, code compliance, and the engineer's release authority. Never promote an LLM explanation into observed solver evidence.

Existing retained conventional examples remain available in Aero's pressure-tube case, free thermal case, and restrained thermal case. They have their own inputs and dispositions; these new numbers do not rescore them.

The local LaTeX / 3090 verification: what actually worked?

This new public page is a static proof-and-calculation notebook, not a public GPU endpoint or a repair of the private LaTeX app. Its math rendering and deterministic examples need no GPU. The failed model calculation is why numerical checks—not model confidence—provide the evidence here. Read the sanitized runtime receipt.

Equations and proof you can keep.

Reproduce with Python 3 and NumPy: download verify.py and inputs.json into the same folder, then run python verify.py. The public record pins NumPy 2.3.5, input and implementation hashes. The PDF retains the derivations and scope limitations.

Primary-source context

• NPTEL: thick-cylinder elasticity, boundary-value formulation (indexed reference; full page retrieval timed out during this publication).
• COMSOL: thermal expansion constraints and physically irrelevant converged solutions. The derivations and new numerical results above are our own synthetic calculations.

Notebook Rev 1.0 · October 12, 2026. This is a new public educational artifact, not a new Aero software release, training result, or revision of historical qualification evidence.