The steady-state radial temperature profile is modeled with a cylindrical heat-conduction balance through the packed catalyst bed. Neglecting axial conduction, the effective radial conductivity of the packing, heat exchange with the bulk gas, and volumetric reaction heat release are combined in

[ k_e\frac{1}{r}\frac{d}{dr}\left(r\frac{dT}{dr}\right)-ha(T-T_G)+H_V=0 ]

Here, (k_e) is the effective thermal conductivity of the bed, (h) is the pellet-to-gas heat-transfer coefficient, (a) is catalyst surface area per unit volume, (T_G) is bulk-gas temperature, and (H_V) is the volumetric heat-release rate.

The profile results from a balance between radial conduction, heat transfer from the catalyst bed to the gas, and exothermic heat generation. A steep thermal boundary layer develops near the wall when radial conduction through the packing cannot redistribute heat effectively relative to the heat removed at the wall.

How the Radial Temperature Model Works

Radial conduction through the packing

The first term,

[ k_e\frac{1}{r}\frac{d}{dr}\left(r\frac{dT}{dr}\right), ]

represents radial conduction in cylindrical geometry. The (1/r) factor distinguishes this balance from a one-dimensional planar-wall equation.

The effective conductivity (k_e) accounts for heat transport through the combined solid-pellet and gas-filled structure of the packed bed.

Heat transfer to the bulk gas

The term

[ -ha(T-T_G) ]

represents heat exchanged between the catalyst-packing structure and the flowing gas. When the local bed temperature (T) exceeds (T_G), this term removes heat from the packed solid phase.

This formulation generally treats (T_G) as the local bulk-gas temperature at the axial position being analyzed. The radial model therefore describes the cross-sectional temperature field while axial temperature changes are handled separately or neglected for the local balance.

Heat generated by reaction

The source term (H_V) represents heat released by the exothermic catalytic reaction per unit reactor volume. A positive (H_V) increases the bed temperature unless the generated heat is removed by gas convection or wall cooling.

For a uniform heat-release rate and constant properties, the equation predicts a radial profile shaped by the competition between this source and the two heat-removal mechanisms.

Conditions Applied at the Reactor Boundaries

Symmetry at the centerline

At the reactor centerline, the temperature profile must be symmetric. The corresponding condition is

[ \left.\frac{dT}{dr}\right|_{r=0}=0 ]

This prevents a physically unrealistic temperature cusp or net radial heat flux through the centerline.

Heat transfer at the wall

At the reactor wall, the model requires a thermal boundary condition. A common form is a convective wall condition,

[ -k_e\left.\frac{dT}{dr}\right|_{r=R}

h_w\left[T(R)-T_w\right] ]

where (R) is the bed radius, (h_w) is the effective bed-to-wall heat-transfer coefficient, and (T_w) is the wall or cooling-jacket temperature.

The exact condition depends on the pilot plant configuration. The wall temperature may be prescribed, or heat transfer may be linked to a cooling-jacket energy balance.

Why the boundary condition matters

The interior heat generation determines how much heat must leave the bed, but the wall condition determines how readily that heat can be removed. A strongly cooled wall can produce a large center-to-wall temperature difference even when the gas temperature remains relatively uniform.

What Creates the Thermal Boundary Layer

High conductive resistance through the bed

A thermal boundary layer forms when radial heat conduction through the packing is relatively weak. In that case, the wall can remove heat efficiently from the outer region, but the resulting cooling penetrates only a limited distance into the bed.

The temperature gradient is then concentrated near the wall, while the central core remains closer to the temperature established by reaction heat generation and gas convection.

Strong wall cooling

A large difference between the internally generated bed temperature and the wall or jacket temperature increases the wall temperature gradient. Strong external cooling therefore makes the boundary layer more pronounced when the packing cannot conduct heat rapidly enough toward the wall.

Weak pellet-to-gas heat exchange

If (ha) is small, the gas removes less heat from the catalyst-packing structure. More reaction heat must then travel radially toward the wall, increasing the sensitivity of the profile to the bed's effective thermal conductivity.

Large heat generation

A high (H_V) raises the internal thermal load. If wall cooling and gas heat removal do not increase proportionally, the center becomes hotter and the wall region must support a larger heat flux.

This increases the possibility of a sharp near-wall temperature change and, in sufficiently reactive systems, thermal runaway.

The Characteristic Radial Thermal Scale

Conduction-to-gas-exchange length

The balance between radial conduction and pellet-to-gas heat exchange introduces a characteristic length,

[ \ell_T \sim \sqrt{\frac{k_e}{ha}} ]

This is the approximate distance over which temperature differences within the packed structure are smoothed by conduction and exchanged with the gas.

A smaller (\ell_T) means that temperature changes are confined to a thinner radial region. A larger (\ell_T) allows thermal effects to extend farther into the bed.

Interpretation of the resistance ratio

Relative to the reactor radius, the important comparison is between (\ell_T) and (R). When (\ell_T/R) is small, the wall influence is localized and the core can remain nearly uniform.

Equivalently, a large ratio such as

[ \frac{haR^2}{k_e} ]

indicates strong local solid-gas exchange relative to radial conduction across the reactor scale. Depending on the wall condition, this can sharpen the radial transition because the wall-affected region is short compared with the bed radius.

Profile shape

For constant coefficients, the homogeneous portion of the cylindrical equation has modified-Bessel-function behavior rather than the simple exponential form associated with planar geometry. The physically acceptable solution remains finite at (r=0), and its constants are determined by the centerline and wall boundary conditions.

The important engineering result is the length scale and gradient distribution: the core is relatively uniform when wall effects are confined, while the near-wall region carries most of the radial temperature change.

Understanding the Trade-offs

A uniform core is not proof of uniform temperature

Temperature measurements at one radial position can miss a sharp wall layer. A center thermocouple may indicate a stable bulk temperature while the wall region experiences substantially different temperatures.

Pilot plant instrumentation should therefore measure multiple radial positions when wall cooling or exothermic operation is important.

Steady-state models omit wall thermal storage

The stated equation is a steady-state bed balance. It does not describe the transient energy stored in the reactor wall, catalyst, or packing.

In a pilot plant, the wall heat capacity can be comparable to or greater than that of the packed bed. During startup or inlet-temperature changes, the wall can absorb or release heat and significantly slow the observed temperature response.

Axial effects can become important

Neglecting axial conduction is appropriate when radial behavior is the focus and axial conduction is small compared with convection and reaction transport. It can become inaccurate near the inlet, outlet, strong cooling transitions, or during transient operation.

Axial wall conduction can also redistribute heat independently of the packed bed, so it should be included when modeling dynamic pilot-plant data or extrapolating results to another reactor scale.

Effective properties are operating-dependent

The values of (k_e), (h), and (a) depend on packing structure, gas properties, flow conditions, and catalyst geometry. Treating them as constants is useful for a local model, but the resulting profile should be checked against measured temperatures.

Making the Right Choice for Your Goal

Use the radial equation as a local steady-state model, then select the level of detail according to the operating question.

If your primary focus is predicting the steady radial profile: Apply centerline symmetry and an appropriate wall heat-transfer condition, using measured or validated values of (k_e), (h), (a), (T_G), and (H_V).
If your primary focus is detecting a wall thermal boundary layer: Compare the characteristic scale (\ell_T=\sqrt{k_e/(ha)}) with the reactor radius and use several radial temperature measurements.
If your primary focus is preventing thermal runaway: Evaluate the combined effects of (H_V), wall cooling, gas heat removal, and the maximum predicted center temperature rather than relying only on the wall temperature.
If your primary focus is modeling startup or dynamic operation: Extend the steady-state balance to include bed and wall heat capacities, and consider axial wall conduction.

A sound radial model makes the balance between reaction heat generation, gas cooling, packing conduction, and wall removal visible enough to design safer and more effective pilot-plant operation.

Summary Table:

Key factor	Role in the radial temperature profile	Effect on the wall boundary layer
Effective conductivity, k_e	Describes radial heat transport through the packed bed	Lower k_e increases temperature gradients near the wall
Gas heat exchange, ha	Removes heat from the catalyst-packing structure to the bulk gas	Lower ha forces more heat toward the wall
Reaction heat, H_V	Represents volumetric heat released by the exothermic reaction	Higher H_V raises the core temperature and wall heat flux
Wall cooling	Sets the wall temperature or wall heat-transfer rate	Stronger cooling increases the center-to-wall temperature difference
Thermal scale, l_T = sqrt(k_e/(ha))	Estimates how far temperature effects penetrate radially	Smaller l_T/R produces a thinner, more localized wall layer
Boundary conditions	Centerline symmetry and wall heat transfer define the solution	Accurate conditions are essential for reliable profile predictions
Pilot-plant instrumentation	Radial thermocouples verify model predictions	Multiple measurement positions can detect layers missed by center probes

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