Transient heat conduction in a cylindrical electrical heating element is modeled with the radial heat equation, internal Joule heat generation, and a convective boundary condition at the surface. For a long solid rod of radius (R), constant properties, and uniform electrical generation, the dimensionless temperature is represented as a steady radial profile plus an infinite series of decaying Bessel-function modes. The decay rates are determined by eigenvalues that depend on the surface Biot number.

The key idea is to separate the temperature into a generated-heat steady state and a transient correction. Cylindrical symmetry makes the spatial eigenfunctions (J_0(\lambda_n r)), while convection at the rod surface determines the eigenvalues through (\lambda_n J_1(\lambda_n)=\mathrm{Bi},J_0(\lambda_n)).

Defining the Cylindrical Heating Problem

Physical assumptions

A commonly used pilot-plant model assumes:

The heating element is a long cylinder, so axial and end effects are negligible.
Electrical heat generation is uniform within the solid.
Thermal conductivity (k), density (\rho), and heat capacity (C_p) are constant.
The initial temperature is uniform.
Heat leaves the outer surface by convection to the surrounding process fluid.

These assumptions are appropriate when the rod length is much greater than its radius and the electrical and material properties do not vary substantially with temperature.

Volumetric electrical generation

The internal heat-generation rate is written as (q'''), with units of (\mathrm{W/m^3}). For a resistive element, it can be related to electrical conditions through

[ q'''=\sigma E^2 ]

where (\sigma) is electrical resistivity and (E) is the electric field. Equivalent expressions can be written using current, voltage, resistance, and the element volume.

Governing energy equation

With cylindrical symmetry, the transient heat equation is

[ \rho C_p\frac{\partial T}{\partial t}

k\left[ \frac{1}{r}\frac{\partial}{\partial r} \left(r\frac{\partial T}{\partial r}\right) \right]+q''' ]

or, using thermal diffusivity (\alpha=k/(\rho C_p)),

[ \frac{\partial T}{\partial t}

\alpha\left[ \frac{1}{r}\frac{\partial}{\partial r} \left(r\frac{\partial T}{\partial r}\right) \right] +\frac{q'''}{\rho C_p}. ]

The radial coordinate satisfies (0\le r\le R).

Applying the Boundary and Initial Conditions

Centerline condition

The temperature must remain finite at the centerline. Symmetry also requires zero radial heat flux:

[ \left.\frac{\partial T}{\partial r}\right|_{r=0}=0. ]

This condition eliminates the second-kind Bessel function (Y_0), which is singular at (r=0).

Convective surface condition

At the outer surface, conduction within the rod balances convection to the surrounding fluid:

[ -k\left.\frac{\partial T}{\partial r}\right|_{r=R}

h,[T(R,t)-T_\infty], ]

where (h) is the convective heat-transfer coefficient and (T_\infty) is the bulk process-fluid temperature.

This is a Robin, or mixed, boundary condition because it relates both temperature and temperature gradient.

Initial condition

For a uniformly preheated or initially cold rod,

[ T(r,0)=T_i. ]

The initial condition determines how strongly each transient eigenmode contributes to the subsequent response.

Nondimensional Formulation

Dimensionless variables

Define the dimensionless radius, time, and temperature as

[ r^*=\frac{r}{R}, \qquad Fo=\frac{\alpha t}{R^2}, \qquad \Theta=\frac{k(T-T_\infty)}{q'''R^2}. ]

The Fourier number (Fo) measures the elapsed time relative to the characteristic internal conduction time.

The surface Biot number is

[ \mathrm{Bi}=\frac{hR}{k}. ]

It compares external convective resistance with internal conductive resistance.

Dimensionless equation

The governing equation becomes

[ \frac{\partial \Theta}{\partial Fo}

\frac{1}{r^} \frac{\partial}{\partial r^} \left( r^\frac{\partial \Theta}{\partial r^} \right)+1. ]

The dimensionless boundary conditions are

[ \left.\frac{\partial \Theta}{\partial r^}\right|_{r^=0}=0 ]

and

[ -\left.\frac{\partial \Theta}{\partial r^}\right|_{r^=1}

\mathrm{Bi},\Theta(1,Fo). ]

Constructing the Temperature Solution

Steady-state temperature profile

At steady state, the time derivative is zero:

[ \frac{1}{r^} \frac{d}{dr^} \left( r^\frac{d\Theta_{ss}}{dr^} \right)+1=0. ]

Applying the centerline and convective surface conditions gives

[ \Theta_{ss}(r^*)

\frac{1}{2\mathrm{Bi}} + \frac{1}{4}(1-r^{*2}). ]

The first term represents the temperature rise associated with convection from the surface to the process fluid. The second term represents the radial temperature rise caused by heat conduction from the center to the surface.

Transient correction

Write the full solution as

[ \Theta(r^*,Fo)

\Theta_{ss}(r^)+\theta(r^,Fo). ]

The transient correction satisfies the homogeneous diffusion equation

[ \frac{\partial \theta}{\partial Fo}

\frac{1}{r^} \frac{\partial}{\partial r^} \left( r^\frac{\partial \theta}{\partial r^} \right). ]

Using separation of variables gives

[ \theta(r^*,Fo)

\sum_{n=1}^{\infty} A_n J_0(\lambda_n r^*) e^{-\lambda_n^2 Fo}. ]

Therefore, the complete temperature field is

[ \boxed{ \Theta(r^*,Fo)

\frac{1}{2\mathrm{Bi}} + \frac{1}{4}(1-r^{2}) + \sum_{n=1}^{\infty} A_n J_0(\lambda_n r^) e^{-\lambda_n^2 Fo} } ]

where (J_0) is the first-kind, zero-order Bessel function.

Eigenvalue equation

Applying the convective boundary condition to a mode (J_0(\lambda r^*)) produces

[ \lambda_n J_1(\lambda_n)

\mathrm{Bi},J_0(\lambda_n), ]

or equivalently,

[ \lambda_n

\mathrm{Bi}, \frac{J_0(\lambda_n)}{J_1(\lambda_n)}. ]

The positive roots (\lambda_n) are found numerically. Each root defines one radial temperature mode.

Mode coefficients

The coefficients are obtained by expanding the initial departure from steady state in the Bessel-function basis:

[ A_n

\frac{ \displaystyle \int_0^1 r^* \left[\Theta(r^,0)-\Theta_{ss}(r^)\right] J_0(\lambda_n r^),dr^ }{ \displaystyle \int_0^1 r^J_0^2(\lambda_n r^),dr^* }. ]

The weighting factor (r^*) is required by the cylindrical geometry and the orthogonality relation for the eigenfunctions.

Interpreting the Model in a Pilot Plant

Predicting heating time

The slowest mode, associated with the smallest eigenvalue (\lambda_1), usually controls the late-time response:

[ \theta\sim A_1J_0(\lambda_1r^*)e^{-\lambda_1^2Fo}. ]

This allows engineers to estimate how long the element requires to approach its steady temperature before performing a full series calculation.

Estimating surface and center temperatures

The model provides temperature at any radial location. In particular:

The centerline temperature is obtained by setting (r^*=0), for which (J_0(0)=1).
The surface temperature is obtained by setting (r^*=1).
The largest temperature generally occurs at the center because heat must conduct outward through the rod.

These predictions are important when the heater is immersed in a process fluid, embedded in a wall, or used to supply heat to a jacketed or tubular unit operation.

Relating the model to fluid flow

The Biot number depends on the convection coefficient (h). In a pilot plant, (h) may change with fluid velocity, agitation, viscosity, and flow regime.

Those effects are commonly represented through correlations involving Reynolds and Nusselt numbers. Changing the process-fluid velocity therefore changes (\mathrm{Bi}), the eigenvalues, the surface temperature, and the overall heating time.

Evaluating the lumped-capacity approximation

When

[ \mathrm{Bi}<0.1, ]

internal temperature gradients are often small enough that the rod can be approximated as having a uniform temperature. The full Bessel-series solution becomes necessary when internal conduction resistance is significant and the center and surface temperatures differ appreciably.

Extending the Model to Real Equipment

Finite-length heating elements

A long-cylinder model neglects heat loss through the ends. For short elements, the temperature depends on both (r) and the axial coordinate (z), requiring a two-dimensional or three-dimensional formulation.

The solution then involves coupled radial Bessel functions and axial eigenfunctions.

Temperature-dependent properties

Electrical resistivity, thermal conductivity, and heat capacity may vary with temperature. In that case, (q'''), (k), and (\alpha) are no longer constant, and the analytical series solution may need to be replaced by numerical time integration.

Composite insulation and wall systems

If the heating element is surrounded by insulation, a metal sheath, or a process-vessel wall, each layer requires its own conduction equation. At every interface, the model must enforce

[ T_1=T_2 ]

and

[ -k_1\frac{\partial T_1}{\partial r}

-k_2\frac{\partial T_2}{\partial r}. ]

These conditions ensure continuity of temperature and heat flux across the composite structure.

Coupling to the process fluid

The constant-(T_\infty) boundary assumption is appropriate when the surrounding fluid is well mixed or behaves as a large thermal reservoir. If the fluid temperature changes significantly, the heater model must be coupled to a fluid energy balance involving mass, heat capacity, flow rate, heat-transfer area, and the overall heat-transfer coefficient.

Understanding the Trade-offs

Analytical solution versus numerical model

The Bessel-series solution is fast, transparent, and useful for design calculations and teaching experiments. It becomes less convenient when geometry, properties, generation rates, or boundary conditions vary with time or position.

Finite-difference, finite-volume, or finite-element methods handle those complications more naturally but require numerical discretization and verification.

Uniform generation versus nonuniform generation

The analytical profile above assumes uniform volumetric heat generation. Nonuniform electrical loading, temperature-dependent resistivity, or localized heating changes the source term and therefore changes both the steady profile and transient response.

Constant convection coefficient versus changing flow conditions

Using a constant (h) simplifies the calculation, but pilot-plant flow conditions may change during operation. Agitation speed, fluid viscosity, phase changes, and fouling can alter (h), so measured or correlation-based values should be checked against the operating range.

Center temperature versus average temperature

The center temperature is often the critical safety or material-quality variable, but it is not the same as the volume-average temperature. Process energy balances may require the average solid temperature, while material limits may depend on the hottest centerline temperature.

Making the Right Choice for Your Goal

The appropriate model depends on the required accuracy, available measurements, and operating conditions.

If your primary focus is estimating heater response time: Use the dimensionless Fourier and Biot numbers with the first few Bessel modes, checking whether the lumped-capacity approximation is valid.
If your primary focus is predicting the hottest point: Evaluate the full radial solution at the centerline, because the maximum temperature generally occurs at (r=0).
If your primary focus is determining convection performance: Use measured or correlation-based (h) values to calculate (\mathrm{Bi}), then compare predicted surface temperatures and heating rates with pilot-plant data.
If your primary focus is modeling a short, insulated, or composite element: Use a multidimensional or multilayer numerical model with temperature and heat-flux matching at interfaces.
If your primary focus is process control: Couple the cylindrical conduction model to the fluid or vessel energy balance instead of treating the surrounding fluid temperature as constant.

A cylindrical electrical heater is therefore modeled by combining internal Joule generation, radial conduction, convective surface loss, and Bessel-function transient modes to predict its temperature history throughout the pilot-plant operation.

Summary Table:

Model Element	Purpose	Key Relationship
Internal heat generation	Represents resistive electrical heating	(q'''=\sigma E^2)
Radial heat equation	Describes transient temperature within the cylinder	(\rho C_p\frac{\partial T}{\partial t}=k\nabla_r^2T+q''')
Surface convection	Models heat transfer to the process fluid	(-k\frac{\partial T}{\partial r}=h(T-T_\infty))
Fourier number	Measures progress of internal heat diffusion	(Fo=\alpha t/R^2)
Biot number	Indicates the importance of internal temperature gradients	(Bi=hR/k)
Bessel eigenfunctions	Represent cylindrical transient temperature modes	(J_0(\lambda_n r^*))
Eigenvalue equation	Determines modal decay rates	(\lambda_nJ_1(\lambda_n)=BiJ_0(\lambda_n))
Lumped-capacity check	Identifies when a uniform-temperature model may be adequate	Generally valid when (Bi<0.1)
Extended numerical model	Handles finite length, composite layers, changing properties, or variable fluid conditions	Use finite-difference, finite-volume, or finite-element methods

Design the Right Pilot-Plant Heating System

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How is transient heat conduction modeled in cylindrical electrical heating elements used in chemical engineering pilot plants?