Eigenvalue problems for rectangular pilot-plant models are formulated by separating the spatial transport operator from time. For constant properties and no internal generation, transient temperature or concentration satisfies a diffusion equation such as (\partial \theta/\partial t = D\nabla^2\theta). In a rectangular domain, separation of variables reduces each spatial direction to an eigenvalue problem of the form (\mathcal{L}_x\Phi=-\lambda^2\Phi), with sine-cosine basis functions selected to satisfy the equipment’s boundary conditions.

The geometry determines the spatial operator, but the boundary conditions determine the admissible eigenvalues and eigenfunctions. Once those functions are chosen to satisfy the homogeneous boundary conditions, the transient field is represented as a sum of exponentially decaying spatial modes.

Start With the Governing Diffusion Model

Heat conduction and mass diffusion share the same structure

For a rectangular solid, slab, membrane, or packed region with constant properties, the transient heat-conduction equation is

[ \frac{\partial \theta}{\partial t}

D_H \left( \frac{\partial^2\theta}{\partial x^2} + \frac{\partial^2\theta}{\partial y^2} + \frac{\partial^2\theta}{\partial z^2} \right), ]

where (\theta) is temperature and

[ D_H=\frac{k}{\rho C_p} ]

is thermal diffusivity.

Mass diffusion has the analogous form,

[ \frac{\partial c}{\partial t}

D \left( \frac{\partial^2c}{\partial x^2} + \frac{\partial^2c}{\partial y^2} + \frac{\partial^2c}{\partial z^2} \right), ]

where (c) is concentration and (D) is the mass diffusivity.

The rectangular domain supplies the coordinate structure

For a rectangular region with dimensions (L_x), (L_y), and (L_z),

[ 0<x<L_x,\qquad 0<y<L_y,\qquad 0<z<L_z. ]

The model may represent a drying slab, catalyst bed, thermal-storage block, membrane section, or rectangular pilot-plant channel, provided the properties and geometry support the constant-coefficient approximation.

Separate Time and Space

The separated solution defines the basis functions

A separated mode can be written as

[ \theta(x,y,z,t)

X(x)Y(y)Z(z)T(t). ]

Substitution into the three-dimensional diffusion equation gives

[ \frac{1}{D_H}\frac{1}{T}\frac{dT}{dt}

\frac{1}{X}\frac{d^2X}{dx^2} + \frac{1}{Y}\frac{d^2Y}{dy^2} + \frac{1}{Z}\frac{d^2Z}{dz^2}. ]

Because each term depends on a different independent variable, each spatial term is assigned a separation constant.

Each coordinate becomes a one-dimensional eigenvalue problem

For the (x)-direction, the spatial equation is

[ \frac{d^2X}{dx^2}+\lambda_x^2X=0, ]

or, using an operator notation,

[ \mathcal{L}_xX=-\lambda_x^2X, \qquad \mathcal{L}_x=\frac{d^2}{dx^2}. ]

The general solution is

[ X(x)=A_x\sin(\lambda_x x)+B_x\cos(\lambda_x x). ]

The same formulation applies in the (y)- and (z)-directions, with eigenvalues (\lambda_y) and (\lambda_z).

The temporal factor decays by the total spatial wavenumber

For a mode with three spatial eigenvalues, the time equation is

[ \frac{dT}{dt} + D_H \left( \lambda_x^2+\lambda_y^2+\lambda_z^2 \right)T =0. ]

Therefore,

[ T(t)

\exp\left[ -D_H \left( \lambda_x^2+\lambda_y^2+\lambda_z^2 \right)t \right]. ]

Higher spatial modes decay more rapidly, so the long-time response is dominated by the lowest admissible eigenvalues.

Impose the Physical Boundary Conditions

Dirichlet boundaries produce sine-type restrictions

A prescribed temperature or concentration becomes homogeneous after subtracting a suitable reference or steady-state solution. The resulting condition is typically

[ X(0)=0 \quad\text{or}\quad X(L_x)=0. ]

For example, imposing

[ X(0)=0 ]

eliminates the cosine term:

[ X(x)=A_x\sin(\lambda_x x). ]

If the opposite boundary also satisfies

[ X(L_x)=0, ]

then

[ \sin(\lambda_x L_x)=0, ]

so the eigenvalues are

[ \lambda_{x,n}=\frac{n\pi}{L_x}, \qquad n=1,2,3,\ldots ]

The corresponding basis functions are sine functions.

Neumann boundaries produce cosine-type restrictions

An insulated wall or zero-flux boundary is represented by

[ \frac{dX}{dx}=0. ]

At (x=0),

[ X'(0)=0 ]

eliminates the sine term and gives

[ X(x)=B_x\cos(\lambda_x x). ]

If the other wall is also insulated,

[ X'(L_x)=0, ]

the eigenvalues satisfy

[ \sin(\lambda_xL_x)=0. ]

This gives

[ \lambda_{x,n}=\frac{n\pi}{L_x}, \qquad n=0,1,2,\ldots ]

The (n=0) mode is the constant spatial mode, which is physically important for insulated systems because the total average temperature or concentration can remain nonzero.

Robin boundaries represent surface transfer resistance

A convective heat-transfer or mass-transfer boundary is commonly written as

[ -k\frac{\partial \theta}{\partial x}

h(\theta-\theta_\infty), ]

or, after nondimensionalization and homogenization,

[ X'(L_x)+A X(L_x)=0. ]

Here, (A) contains the relevant transfer coefficient and conductivity or diffusivity scale. The same mathematical form describes convective heat transfer and interfacial mass transfer.

Applying this condition to

[ X(x)=A_x\sin(\lambda_xx)+B_x\cos(\lambda_xx) ]

produces a characteristic equation involving (\lambda_x). Unlike the pure Dirichlet or Neumann cases, the eigenvalues are generally roots of a transcendental equation and must be computed numerically.

Boundary conditions must be homogenized before selecting the basis

A nonzero prescribed surface temperature or concentration is not itself represented by a basis function that vanishes at the boundary. The usual procedure is to write

[ \theta=\theta_{\mathrm{ss}}+\theta', ]

where (\theta_{\mathrm{ss}}) satisfies the nonhomogeneous boundary conditions and (\theta') satisfies homogeneous ones.

The eigenfunctions are constructed for (\theta'), because this allows the boundary terms arising during integration by parts to vanish or reduce to the appropriate Robin form.

Build the Rectangular Basis

The multidimensional eigenfunctions are products

Once the one-dimensional eigenfunctions have been determined, a three-dimensional rectangular eigenfunction is formed as

[ \Phi_{mnp}(x,y,z)

X_m(x)Y_n(y)Z_p(z). ]

Its total eigenvalue is

[ \Lambda_{mnp}^2

\lambda_{x,m}^2 + \lambda_{y,n}^2 + \lambda_{z,p}^2. ]

The complete transient solution can then be written as

[ \theta'(x,y,z,t)

\sum_{m,n,p} C_{mnp} \Phi_{mnp}(x,y,z) \exp\left(-D_H\Lambda_{mnp}^2t\right). ]

For mass diffusion, replace (D_H) by the mass diffusivity (D).

Two-dimensional models use the same construction

If the pilot-plant component is modeled as a rectangular slab or channel with negligible variation in one direction, the solution reduces to

[ \theta'(x,y,t)

\sum_{m,n} C_{mn} X_m(x)Y_n(y) \exp\left[ -D_H \left( \lambda_{x,m}^2+\lambda_{y,n}^2 \right)t \right]. ]

A one-dimensional slab model is the corresponding single-coordinate version.

Coefficients come from inner products

The coefficients are determined by projecting the initial temperature or concentration field onto the eigenfunctions. In one dimension,

[ C_n

\frac{\displaystyle\int_0^{L_x} f(x)X_n(x),dx} {\displaystyle\int_0^{L_x} X_n^2(x),dx}, ]

where (f(x)) is the initial disturbance after any steady-state component has been removed.

For a rectangular three-dimensional domain, the projection uses the multidimensional inner product

[ C_{mnp}

\frac{\iiint f(x,y,z) \Phi_{mnp}(x,y,z),dV} {\iiint \Phi_{mnp}^2(x,y,z),dV}. ]

This orthogonality is what makes the spectral expansion computationally practical.

Understand the Role of Eigenvalues

Eigenvalues determine response time

Each mode has a characteristic decay time approximately equal to

[ \tau_{mnp}

\frac{1}{D_H\Lambda_{mnp}^2}. ]

The smallest nonzero eigenvalue controls the dominant long-time response and therefore strongly influences heating, cooling, drying, or diffusion residence-time estimates.

Dimensionless groups connect the model to equipment operation

For convective boundaries, the eigenvalue equations depend on dimensionless groups such as the Biot number,

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

The Biot number compares internal conduction resistance with external convective resistance. It therefore changes the eigenvalues and determines whether the rectangular component behaves approximately as a spatially uniform body or develops substantial internal gradients.

Basis functions can support analytical and numerical workflows

The eigenfunction expansion provides analytical benchmark solutions for process models and experimental verification. It can also be used to check finite-difference, finite-element, or CFD predictions for rectangular pilot equipment under controlled boundary conditions.

Understanding the Trade-offs

Analytical solutions require restrictive assumptions

The separated rectangular formulation is simplest when the geometry is rectangular, properties are constant, and the diffusion coefficient does not vary with temperature or concentration. Strong property variation, nonlinear radiation, moving boundaries, phase change, or internal generation can invalidate the simple sine-cosine formulation.

Robin conditions usually require numerical root finding

For mixed boundaries, eigenvalues are not generally available as simple multiples of (\pi/L). The characteristic equation must be solved numerically, and care is needed to identify all relevant roots without missing closely spaced or repeated modes.

Ideal geometry may not represent the entire pilot plant

A real pilot unit may include corners, supports, baffles, contact resistances, nonuniform flow, or spatially varying transfer coefficients. A rectangular eigenfunction model remains useful, but its predictions should be compared with measurements or a higher-fidelity numerical model when those effects are significant.

Incorrect basis functions produce incorrect boundary behavior

Using sine functions for an insulated wall, or cosine functions for a fixed-value wall, imposes the wrong physics. In transform methods, the mismatch also leaves boundary terms unresolved during integration by parts, so the resulting transformed equations do not correctly represent the pilot equipment.

Other geometries require other eigenfunctions

The sine-cosine basis is specific to Cartesian rectangular geometry. Cylindrical systems generally require Bessel functions, while spherical systems use spherical Bessel functions and, where angular variation exists, Legendre polynomials.

How to Apply This to Your Project

The correct formulation depends on the geometry, the measured initial field, and the actual thermal or mass-transfer boundary conditions.

If your primary focus is a rectangular slab or block: Formulate independent one-dimensional eigenvalue problems in each Cartesian direction and combine their sine-cosine eigenfunctions as products.
If your primary focus is fixed surface temperature or concentration: Homogenize the boundary condition and use sine-type eigenfunctions satisfying zero values at the relevant boundaries.
If your primary focus is insulated surfaces: Use Neumann eigenvalue problems and include the constant zero-eigenvalue mode when the physics permits it.
If your primary focus is convective heat or mass transfer: Apply Robin conditions, derive the transcendental characteristic equation, and calculate its roots numerically.
If your primary focus is predicting transient response time: Identify the lowest nonzero combined eigenvalue, because its exponential decay mode governs the long-time behavior.
If your primary focus is validating a pilot-plant simulation: Use the eigenfunction series as a benchmark, while checking whether property variation, internal generation, or geometric details require a numerical model.

A reliable rectangular transport model is built by matching the operator, eigenfunctions, eigenvalues, and boundary conditions to the actual pilot-plant physics.

Summary Table:

Key Aspect	Formulation or Insight
Governing model	Diffusion equation for temperature or concentration with constant properties and no internal generation
Spatial domain	Rectangular coordinates with dimensions (L_x), (L_y), and (L_z)
Separation of variables	Express the field as (\theta=X(x)Y(y)Z(z)T(t))
Eigenvalue problem	Each coordinate satisfies (\mathcal{L}X=-\lambda^2X), commonly (X''+\lambda^2X=0)
Dirichlet boundary	Fixed temperature or concentration produces sine-type eigenfunctions
Neumann boundary	Insulated or zero-flux surfaces produce cosine-type eigenfunctions and may include a constant mode
Robin boundary	Convective heat or mass transfer produces transcendental eigenvalue equations requiring numerical roots
Multidimensional basis	Product eigenfunctions: (\Phi_{mnp}=X_mY_nZ_p)
Transient decay	Each mode decays as (\exp[-D\Lambda_{mnp}^2t])
Dominant response	The lowest nonzero eigenvalue controls long-time heating, cooling, drying, or diffusion behavior
Coefficients	Determined by projecting the initial field onto the eigenfunctions using inner products
Model limitations	Property variation, phase change, internal generation, complex geometry, and nonlinear transfer may require numerical methods

Improve Your Pilot-Plant Transport Model

Need reliable educational or industrial equipment for studying transient heat conduction, mass diffusion, or coupled transport processes? Contact LABPARK to discuss your project.

LABPARK provides Educational and Vocational Unit Operations Pilot Plants for universities, research institutes, and enterprises, including systems for chemical engineering, bioprocess and biotechnology, and environmental and water treatment. Our pilot plants help you validate eigenfunction models, investigate boundary-condition effects, measure response times, and connect theoretical transport equations with practical operating data.

Tell us your application, process conditions, geometry, and training or research objectives. LABPARK can help identify a suitable pilot-plant configuration for teaching, laboratory research, process development, and industrial demonstration.