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Heat Transfer
Radial Conduction Through a Sphere Simulator
Visualise steady-state heat conduction flowing radially through a spherical shell — for example the insulation around a cryogenic Dewar or an LNG tank. Change the inner and outer radii, conductivity and surface temperatures to see the thermal resistance, heat flow, heat flux and temperature profile in real time.
Parameters
Inner radius r₁
m
Radius of the inner (hot) surface of the shell
Outer radius r₂
m
Radius of the outer (cold) surface. Must be greater than r₁
Thermal conductivity k
W/(m·K)
~0.03–0.05 for insulation foam, ~1 for glass, ~400 for copper
Cross-section through the centre of the shell. Heat flows radially outward from the hot inner surface to the cooler outer surface. Because the same Q crosses a larger surface as r grows, the arrow density thins out toward the outside.
Steady radial conduction through a spherical shell. Q: heat flow [W]; R_th: thermal resistance [K/W]; T₁, T₂: inner and outer surface temperatures; r₁, r₂: inner and outer radii; k: conductivity. The heat flow is uniform along the path, while the heat flux q(r) = Q/(4πr²) falls with 1/r².
Temperature distribution in the shell. T is a linear function of 1/r, so the drop concentrates near the small inner radius.
What is Radial Conduction Through a Sphere?
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"Radial conduction through a sphere" — where does this actually show up? I learned the flat-plate version in class, but the spherical case never clicked.
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Everyday examples are a vacuum flask of coffee, a giant Moss-type spherical LNG tank, or a blood-storage Dewar. Whenever you want to keep an interior temperature isolated from the outside, heat ends up flowing radially through the spherical wall. Pressure vessels and spacecraft fuel tanks often use spheres too — partly for stress reasons, but the by-product is "minimum outside area for a given volume", which is also a sweet spot for insulation.
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Less outside area means heat escapes more slowly, right? So is the sphere just unconditionally better at insulating?
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That is the subtle bit: "geometrically nice" is not the same as "the wall conducts less". Wall for wall, a sphere actually conducts more heat than a flat plate of the same thickness, because the area grows as r² outward. Slide the inner radius r₁ smaller on the chart below and you will see Q go up, not down. One way to read it: the small inner surface has to carry the entire temperature drop, while the big outer surface only has to "let it out" gently — so the inner side is the bottleneck.
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Oh, it really does — Q rises as r₁ shrinks. So if I wrap a thicker insulation jacket around a small sphere, it might actually cool faster?
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Exactly the "critical radius" effect, famous from cylindrical wire insulation. Adding insulation increases the thermal resistance, but it also grows the outer surface area for convection. Wrap a small sphere with a low-k insulation and at first the heat loss rises with each extra layer. The critical radius for a sphere is r_c = 2k/h, a bit more forgiving than the cylinder's r_c = k/h, but it is a real effect — engineers really do end up scratching their heads when "more insulation made it cool faster".
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Interesting — thicker is not always better. Also, the T(r) curve is not a straight line; it drops steeply near the inside. Is that the sphere thing as well?
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Yes — that is the geometric signature of the sphere. In steady state the same Q has to cross 4πr², a surface that grows fast, so dT/dr ∝ 1/r² shrinks fast outward. Integrate it and T is linear in 1/r, dropping sharply near the inner radius and softly near the outer. For a 20 cm-diameter hot sphere covered with a 1 cm shell, most of the temperature drop lives in the first few millimetres on the inside. That tells you the inner-surface contact and surface quality of the insulation matter much more than the outside finish.
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So I can use this analytical solution as a quick sanity check before running a CAE mesh, right?
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That is one of the main uses. If your FEM run on a sphere gives a heat flow that is an order of magnitude off the value here, the problem is almost always a wrong boundary condition or a unit mix-up. If it agrees, this same number becomes the baseline you compare against once you add convection or radiation. In the cryogenic and pressure-vessel worlds, the formula on this page is one of the first things people write down on a notepad.
Frequently Asked Questions
For steady radial conduction through a spherical shell with inner radius r₁, outer radius r₂ and conductivity k, the thermal resistance is R_th = (r₂ − r₁) / (4π k r₁ r₂). Like R = L/(kA) for a plate or R = ln(r₂/r₁)/(2π k L) for a cylinder, it satisfies the thermal Ohm's law ΔT = Q · R_th. The product r₁·r₂ in the denominator is the geometric mean of the inner and outer areas, which averages the area that spreads as the heat flows outward.
For the same wall thickness, no — a sphere's area grows as r² outward, so it tends to pass more heat than a flat plate or a cylindrical shell. The reason cryogenic Dewars and LNG spherical tanks are still chosen is geometric: a sphere has the smallest outer surface area per unit enclosed volume, so the total surface available for heat loss is minimised. Low heat loss in practice comes from the combination of spherical shape AND a thick, low-conductivity insulation layer, not from the spherical wall alone.
No — this is the well-known critical insulation radius effect. When you wrap a small sphere with a low-conductivity insulation, the outer surface area grows, making convective heat loss to the surroundings easier; below the critical radius, adding insulation can actually increase the total heat loss. The critical radius is r_c = k/h for cylinders and r_c = 2k/h for spheres, where h is the outside heat-transfer coefficient. This simulator uses surface temperatures as boundary conditions and so does not include outside convection — the rule of thumb is to design above the critical radius.
Because in steady state the same heat flow Q must pass through successively larger spherical surfaces 4πr², the temperature gradient dT/dr scales as 1/r² and drops rapidly outward. Integrating gives T as a linear function of 1/r, not of r. So T(r) drops steeply near the small inner radius and only gently near the outer one — the T(r) chart on the right shows exactly this. Compared with a flat plate (linear) or a cylinder (logarithmic), a sphere concentrates the temperature drop most strongly on the inner side.
Real-World Applications
Cryogenic storage and LNG spherical tanks: A liquid-hydrogen or liquid-nitrogen Dewar, or a Moss-type LNG sphere holding −162 °C cargo, is exactly this spherical-shell problem. The inner wall sits at the cryogen temperature, the outer wall at ambient, with multi-layer vacuum insulation (MLI) or perlite-filled annulus between them. Designers measure performance as "boil-off rate per day" and work backward from Q = ΔT / R_th to choose the insulation thickness and effective k. The Q-vs-r₁ sensitivity chart here shows how the per-volume heat loss falls when you scale the tank up.
Pressure vessels and reactor containment: Spheres give the smallest outside area per volume and the most uniform stress under internal pressure, which is why they are used for high-pressure gas tanks and for some reactor containment vessels. In accident-condition heat-transfer analyses, the same form of resistance network appears: hot, high-pressure fluid inside, ambient air or cooling water outside, with multiple layers of steel, insulation and concrete adding their R_th values in series.
Spherical insulation, catalyst pellets, electronic components: Catalyst pellets, spherical phase-change-material capsules, solder balls inside a semiconductor package, even a scoop of ice cream — wherever a roughly spherical body is heated or cooled, this steady solution gives the first estimate. Combined with the thermal time constant τ ≈ ρ c r² / k, it tells you whether the response is "seconds" or "hours" before you build a transient model.
Earth and planetary science: Heat transport inside the Earth, the cooling of planets, the temperature of subsurface oceans in icy moons — all start from radial conduction through a sphere. Real cases of course add convection, radiation, latent heat and internal heating, but you first compute "how many watts would conduct out if it were pure conduction" and use that as a yardstick. The line scientists scribble first in a spreadsheet is exactly R_th = (r₂ − r₁) / (4π k r₁ r₂).
Common Misconceptions and Pitfalls
The biggest pitfall is mixing up the spherical resistance formula with the plate or cylinder version. R = (r₂ − r₁) / (4π k r₁ r₂) is specific to a sphere; the plate uses R = L/(kA) and the cylinder uses R = ln(r₂/r₁)/(2π k L). Even at the same thickness, the sphere has the r₁·r₂ product in its denominator, so its resistance depends strongly on the absolute radii — a thin layer on a very small sphere has a much smaller resistance than the same layer on a large one. If you wire up a multi-layer network using the flat-plate formula everywhere "with an average area", you will simultaneously underestimate the thin inner layer and overestimate the thick outer layer.
Next, using inner and outer surface temperatures as the inputs hides the real convective and radiative resistances. The tool above asks you for surface temperatures; in a real tank you usually know the fluid temperature inside and the ambient temperature outside, but the wall surface temperatures are themselves unknown and are set by the convection coefficients h_in and h_out. To predict the real heat flow you need to add 1/(h_in · A_in) and 1/(h_out · A_out) in series with this conduction R_th. Without those, this tool will overestimate the real heat flow, because it assumes the temperature drop happens entirely inside the wall.
Finally, treating the conductivity k as a single constant. Most insulating materials are strongly temperature-dependent: polyurethane foams and multi-layer vacuum insulation change quickly at cryogenic temperatures, while at high temperatures the radiative contribution becomes important and the apparent k can rise as the cube of temperature. The industry standard for cryogenic equipment is to use the "effective k integrated over the operating temperature range", not just a single mid-point value. The simulator here is fine for first-cut estimates, but for design that has to hit a number, always check the manufacturer's k(T) curve.
How to Use
Enter inner radius (mm) and outer radius (mm) to define the spherical shell geometry—typical range 10–500 mm for industrial applications.
Set thermal conductivity (W/m·K) of the shell material: use 0.04 for mineral wool insulation, 50 for stainless steel, or 200 for aluminum.
Specify inner surface temperature (°C) and ambient/outer surface temperature; the simulator calculates steady-state radial heat flow and thermal resistance using R_th = (r_o − r_i)/(4πk·r_i·r_o).
Review thermal resistance, heat flow rate Q (W), surface heat flux values (W/m²), and mid-radius temperature to validate insulation performance or heat transfer design.
Worked Example
Cryogenic tank with inner radius r_i = 150 mm, outer radius r_o = 200 mm, filled with aerogel insulation (k = 0.018 W/m·K). Inner wall at 77 K (−196°C), outer wall at 298 K (25°C). Temperature difference ΔT = 221 K. Thermal resistance R_th = (0.2 − 0.15)/(4π × 0.018 × 0.15 × 0.2) = 29.5 K/W. Heat inleak Q = 221/29.5 = 7.49 W. Inner surface flux = 7.49/(4π × 0.15²) = 2.65 W/m². At mid-radius r = 0.175 m: T_mid ≈ 77 + 221 × (0.175 − 0.15)/(0.2 − 0.15) = 188 K (−85°C).
Practical Notes
Spherical geometry reduces thermal resistance compared to planar walls of identical thickness; use this simulator to compare spherical vs. cylindrical pipe insulation (e.g., LNG pipelines, 300 mm diameter).
Heat flux is non-uniform on spherical surfaces—outer surface flux is always lower than inner due to larger area; critical for corrosion or condensation risk assessment.
At extreme temperature gradients (>500 K difference), validate whether thermal conductivity remains constant; most insulants exhibit k-value variation with mean temperature.
Classification output flags inadequate insulation when R_th < 1 K/W for cryogenic duty; verify against industrial standards like ASME B36.19M for thermal design margins.