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Transient Heat Transfer in a Sphere with Convection Boundary Condition (Heisler Chart Derivation and One-Term Solution)
The purpose of this article is to derive the classic Heisler-chart solution form for transient convective heating of a solid sphere, so you can compute center, surface, and volume-average temperatures using Biot and Fourier numbers with a clear, reusable workflow.
1. Problem statement and why the “sphere Heisler chart” exists
Transient convective heating of a sphere describes situations where a solid sphere initially at a uniform temperature is suddenly exposed to a fluid at a different temperature, and heat transfer at the surface is controlled by convection while heat diffusion inside the solid is controlled by conduction.
The practical objective is to predict the temperature at the center, the surface, or the volume-average temperature as a function of time without running a full numerical simulation.
Heisler charts are a graphical representation of the analytical eigenfunction solution, commonly reduced to the first term of an infinite series for engineering accuracy over broad ranges of time.
2. Governing equation and boundary conditions for a solid sphere
2.1 Assumptions that define the classical solution
The standard “Heisler chart for a sphere” solution relies on constant thermal conductivity k, density ρ, and heat capacity cp, giving constant thermal diffusivity α = k/(ρcp).
The sphere is homogeneous and isotropic, internal heat generation is neglected, radiation is neglected, and the surrounding fluid temperature is spatially uniform and constant at T∞.
The convective heat transfer coefficient h is constant over the surface, and the initial solid temperature is uniform at Ti.
Note : If k or cp varies strongly with temperature, or if h changes significantly with time or position, the classical Heisler-chart form becomes an approximation and you should validate against a numerical model.
2.2 Differential equation in spherical coordinates
For radial symmetry, the transient heat conduction equation in a sphere reduces to the one-dimensional radial form.
∂T/∂t = α * (1/r^2) * ∂/∂r ( r^2 * ∂T/∂r ), 0 < r < R, t > 0.Initial condition is uniform temperature.
T(r,0) = Ti.Symmetry condition at the center requires a finite gradient, which is enforced by zero slope at r = 0.
∂T/∂r (0,t) = 0.Convective boundary condition at the surface r = R links conduction in the solid to convection into the fluid.
-k * ∂T/∂r (R,t) = h * ( T(R,t) - T∞ ).3. Nondimensionalization: Biot number and Fourier number for a sphere
Define the dimensionless temperature (also called the reduced temperature) using the fluid temperature as the reference.
θ(r,t) = ( T(r,t) - T∞ ) / ( Ti - T∞ ).Define dimensionless radius and time using the sphere radius R and diffusivity α.
x = r/R, Fo = α t / R^2.The dimensionless convection strength is the Biot number.
Bi = h R / k.In these variables the PDE becomes.
∂θ/∂Fo = (1/x^2) * ∂/∂x ( x^2 * ∂θ/∂x ), 0 < x < 1.Boundary conditions become.
∂θ/∂x (0,Fo) = 0. -∂θ/∂x (1,Fo) = Bi * θ(1,Fo).4. Separation of variables for the sphere and the eigenvalue equation
4.1 Standard separated form and spherical eigenfunctions
Assume a product solution θ(x,Fo) = X(x)G(Fo) and separate variables, yielding an eigenvalue problem for X(x).
The time function has the decaying exponential form G(Fo) = exp(-μ²Fo), where μ is an eigenvalue.
For the sphere, the finite-at-center radial eigenfunction is proportional to sin(μx)/(μx).
The series solution therefore takes the classical Heisler form for a sphere.
θ(x,Fo) = Σ[n=1..∞] A_n * ( sin(μ_n x) / (μ_n x) ) * exp( -μ_n^2 Fo ).4.2 Applying the convection boundary condition gives the sphere eigenvalue relation
Apply the surface condition -∂θ/∂x(1,Fo) = Bi θ(1,Fo) term-by-term, which leads to the transcendental equation defining μn.
1 - μ_n * cot(μ_n) = Bi.Each positive root μn generates one mode of the transient response, and the complete temperature field is a weighted sum of all modes.
Note : The roots must be computed numerically, and the first root μ1 typically dominates after a short initial time because higher modes decay faster as exp(-μn²Fo).
5. Coefficients for a uniform initial temperature in a sphere
The coefficients A_n are determined by expanding the uniform initial condition θ(x,0)=1 in the eigenfunctions sin(μx)/(μx) using orthogonality with the spherical weight x².
A convenient closed form for A_n is.
A_n = 4 * ( sin(μ_n) - μ_n cos(μ_n) ) / ( 2 μ_n - sin(2 μ_n) ).Using the eigenvalue equation, sin(μ_n) - μ_n cos(μ_n) = Bi * sin(μ_n), so an equivalent compact form is.
A_n = 4 * Bi * sin(μ_n) / ( 2 μ_n - sin(2 μ_n) ).These coefficients are specifically for a sphere that starts at a uniform temperature Ti and is suddenly exposed to a fluid at T∞ at Fo > 0.
6. Quantities used in Heisler charts for a sphere
6.1 Center temperature ratio
At the center x → 0, sin(μx)/(μx) → 1, so the center dimensionless temperature is.
θ_center(Fo) = θ(0,Fo) = Σ[n=1..∞] A_n * exp( -μ_n^2 Fo ).Heisler charts for a sphere often plot θ_center versus Fo for different Bi, which is exactly this series evaluated numerically and displayed graphically.
6.2 Surface temperature ratio
At the surface x = 1, the surface dimensionless temperature is.
θ_surface(Fo) = θ(1,Fo) = Σ[n=1..∞] A_n * ( sin(μ_n) / μ_n ) * exp( -μ_n^2 Fo ).6.3 Volume-average temperature ratio
The volume-average temperature is often used for energy accounting and for coupling to lumped external balances, and it is defined by averaging with the spherical volume element.
θ̄(Fo) = (1/V) ∫_V θ dV = 3 ∫[0..1] θ(x,Fo) x^2 dx.Substituting the series solution yields a modal series for θ̄(Fo) with coefficients that can be computed from μn and A_n.
In engineering practice, the same one-term strategy used for the center temperature is also used for the average temperature when Fo is not extremely small.
7. One-term approximation: the analytical backbone of Heisler-chart accuracy
The most-used engineering simplification is retaining only the first eigenvalue μ1 and coefficient A1, which gives compact formulas with controlled error for many heating and cooling scenarios.
7.1 One-term center temperature
θ_center(Fo) ≈ A_1 * exp( -μ_1^2 Fo ).7.2 One-term temperature at any radius
θ(x,Fo) ≈ A_1 * ( sin(μ_1 x) / (μ_1 x) ) * exp( -μ_1^2 Fo ).7.3 When the one-term form is typically appropriate
Because higher modes decay as exp(-μn²Fo), the first mode dominates once Fo is moderately large compared to 1/μ2², and this is the regime where Heisler charts are routinely applied.
Note : At very small times the surface layer responds rapidly and higher modes matter, so a multi-term evaluation or a numerical model is preferred if you need high accuracy near Fo ≪ 0.1.
8. Practical workflow: compute temperatures without looking up a Heisler chart
The chart is a visualization of the series, so you can reproduce it computationally using the same steps for any Bi and Fo.
| Step | What to compute | Formula | Output |
|---|---|---|---|
| 1 | Dimensionless groups | Bi = hR/k, Fo = αt/R² | Bi, Fo |
| 2 | Eigenvalue root(s) | Solve 1 − μ cot μ = Bi | μ1, μ2, … |
| 3 | Series coefficient(s) | A_n = 4( sin μ_n − μ_n cos μ_n ) / ( 2μ_n − sin 2μ_n ) | A1, A2, … |
| 4 | Center, surface, or field temperature | θ(x,Fo) = Σ A_n (sin(μ_n x)/(μ_n x)) exp(−μ_n²Fo) | θ values |
| 5 | Convert back to temperature | T = T∞ + θ (Ti − T∞) | T values |
8.1 Numerical root finding for μ1
The equation 1 − μ cot μ = Bi has one root in each interval (n−1)π < μ < nπ for n = 1,2,3,…, and μ1 lies between 0 and π.
A robust approach is bracketing and bisection, which only needs function evaluations and never requires derivatives.
Define f(μ) = 1 - μ * cos(μ)/sin(μ) - Bi. Find μ1 in (0, π) by bisection: 1) Choose a small ε, set a = ε, b = π - ε. 2) While (b - a) is larger than tolerance: m = (a + b)/2 if f(a)*f(m) < 0: b = m else: a = m 3) μ1 ≈ (a + b)/2.Note : Do not evaluate cot(μ) exactly at μ = nπ where sin(μ)=0, and always keep a small ε away from endpoints to avoid division by zero.
9. Interpretation of Biot number limits for a sphere
9.1 Small Biot number and the lumped-capacitance check
If Bi is very small, internal conduction is fast compared to surface convection, so the sphere temperature remains nearly uniform inside.
In that limit, a lumped-capacitance model can be used, which replaces the full PDE with a single exponential in time based on the thermal time constant of the sphere.
Note : A common engineering screening criterion is that the lumped model is reasonable when Bi is small enough that internal temperature gradients are negligible, and if gradients matter you should use the distributed Heisler-chart approach.
9.2 Large Biot number and strong surface control
As Bi increases, the surface is more strongly driven toward the fluid temperature, and the internal gradients increase, which is precisely the regime where the Heisler-chart solution is most valuable.
10. Worked example template you can reuse (symbolic workflow with placeholders)
This template shows the sequence of calculations without committing to any specific material, geometry, or process values.
Given: R, k, ρ, cp, h, Ti, T∞, and time t. 1) α = k / (ρ cp) 2) Bi = h R / k 3) Fo = α t / R^2 4) Solve 1 - μ cot μ = Bi for μ1 (and optionally μ2, μ3, ...) 5) Compute A1 = 4( sin μ1 - μ1 cos μ1 ) / ( 2 μ1 - sin 2 μ1 ) 6) Center temperature ratio: θ_center ≈ A1 exp( -μ1^2 Fo ) 7) Center temperature: T_center = T∞ + θ_center (Ti - T∞)If you need surface temperature, replace step 6 by θ_surface ≈ A1 (sin μ1 / μ1) exp( -μ1²Fo), and then convert back to temperature using the same linear mapping.
11. Common implementation pitfalls and how to avoid them
11.1 Mixing up R and characteristic length
For the classical sphere Heisler formulation, Bi is defined with R as Bi = hR/k and Fo is Fo = αt/R².
If you mix this with alternative definitions using Lc = V/A, you will not match the Heisler-chart normalization and you will not reproduce the chart curves.
11.2 Using the wrong eigenvalue equation
The sphere with a convection boundary uses 1 − μ cot μ = Bi, while a plane wall uses μ tan μ = Bi, and a cylinder uses a Bessel-function relation.
Using the wrong characteristic equation will produce incorrect μn and incorrect time constants.
11.3 Evaluating the center expression incorrectly
The factor sin(μx)/(μx) must be handled using the limit value 1 at x = 0 to avoid a division-by-zero numerical issue.
In code, you can treat x = 0 as a special case, or evaluate sin(μx)/(μx) using a stable “sinc” implementation.
FAQ
What is the physical meaning of the Heisler-chart curves for a sphere.
Each curve is the center temperature ratio θ_center plotted against Fo for a fixed Bi, and it represents the analytical eigenfunction series evaluated numerically and displayed graphically.
The curve shape is controlled by μ1 through the dominant decay exp(−μ1²Fo), while Bi controls μ1 through the eigenvalue equation and controls A1 through the coefficient formula.
How do I decide whether I should use a one-term approximation or multiple terms.
If you are interested in moderate to longer times where higher modes have decayed, the one-term approximation is often sufficient for engineering work because the second and higher terms decay much faster.
If you need high accuracy at early times, near the surface, or for strict error bounds, evaluate more terms of the series by computing additional roots μ2, μ3, and their coefficients.
Can the same derivation be used for cooling instead of heating.
Yes, because the governing equation is linear and the dimensionless temperature θ is defined relative to T∞, so the same formulas apply whether Ti is greater than or less than T∞.
The sign and direction of heat flow are handled automatically through the factor (Ti − T∞) in the mapping back to temperature.
What changes if there is internal heat generation inside the sphere.
Uniform volumetric heat generation adds a source term to the PDE and changes the steady reference state, so the eigenfunction structure can remain similar but the particular solution and the coefficients differ.
In that case, you typically solve for a new reduced temperature about the generation-influenced steady state, or you use superposition with a generation term if boundaries and properties remain constant.
Why does the eigenvalue equation have infinitely many roots.
The transient diffusion operator with boundary conditions forms a Sturm–Liouville problem, and its eigenfunctions form a complete basis for representing the initial condition.
Each eigenvalue corresponds to a spatial mode of the temperature field, and the time dependence of each mode decays exponentially with a rate proportional to μn².