Use a moving-boundary heat balance: the pseudosteady-state approximation treats heat conduction through the existing solid layer as instantaneous relative to the slower motion of the solid-liquid interface. The solid growth rate is then obtained by equating the latent heat released at the interface to the heat transferred through the solid, tube wall, and internal coolant film. As the layer thickens, its thermal resistance increases, so the growth rate decreases.

The practical calculation is a resistance-based Stefan balance: determine the heat rate from the phase-change temperature to the coolant bulk temperature, then divide that rate by the latent heat required to create additional solid.

What the Pseudosteady-State Approximation Assumes

Thermal profiles adjust faster than the interface moves

The temperature field in the solid is approximated as steady at each instant, even though the solid-liquid boundary is moving. This is reasonable when thermal diffusion through the solid is much faster than the time required for the layer thickness to change substantially.

The approximation does not mean that the process is truly steady. The solid thickness and heat-transfer resistance still vary with time.

Latent heat dominates the moving-boundary balance

For freezing or crystallization, the main energy requirement associated with interface motion is the latent heat of fusion or crystallization. Sensible heat accumulation in the solid and liquid is neglected or treated as a secondary correction.

The interfacial balance is therefore based on

[ \text{heat removed}=\text{latent heat released by new solid}. ]

The interface remains near the phase-change temperature

The solid-liquid interface is commonly assigned the equilibrium phase-change temperature, (T_m), or an effective crystallization temperature when concentration or pressure effects are important.

This provides the thermal driving force for removing heat through the solid and tube assembly.

Building the Thermal Resistance Model

Define the tube and solid geometry

Let

(R_i) be the inside radius of the tube,
(R_t) be the outside radius of the tube,
(R_s(t)) be the radius of the moving solid-liquid interface,
(k_w) be the tube-wall thermal conductivity,
(k_s) be the solid-layer thermal conductivity,
(h_i) be the internal coolant-side heat-transfer coefficient,
(T_c) be the bulk coolant temperature,
(\rho_s) be the solid density,
(L) be the latent heat of solidification or crystallization.

For a deposit growing outward from the tube surface, the solid occupies the region between (R_t) and (R_s), and its thickness is

[ \delta(t)=R_s(t)-R_t. ]

Express the resistance per unit tube length

For a cylindrical tube, the coolant-film, tube-wall, and solid-layer resistances per unit length are

[ \mathcal{R}_i'=\frac{1}{h_i,2\pi R_i}, ]

[ \mathcal{R}_w'=\frac{\ln(R_t/R_i)}{2\pi k_w}, ]

and

[ \mathcal{R}_s'=\frac{\ln(R_s/R_t)}{2\pi k_s}. ]

Additional resistances can be included when justified, such as a contact resistance at the tube-solid interface or a liquid-side resistance outside the growing layer.

The total resistance per unit length is therefore

[ \mathcal{R}_{\text{tot}}'

\frac{1}{h_i,2\pi R_i} + \frac{\ln(R_t/R_i)}{2\pi k_w} + \frac{\ln(R_s/R_t)}{2\pi k_s}. ]

Calculate the instantaneous heat-removal rate

Under the pseudosteady approximation, the heat removed per unit tube length is

[ q'(R_s)

\frac{T_m-T_c}{\mathcal{R}_{\text{tot}}'}. ]

This heat rate is evaluated using the current interface radius. Consequently, the calculation is quasi-steady but still captures the time dependence caused by layer growth.

Deriving the Solid Growth Rate

Write the latent-heat balance

When the interface moves outward by (dR_s), the newly formed solid volume per unit tube length is

[ dV'=2\pi R_s,dR_s. ]

The corresponding latent heat released is

[ dQ_{\text{latent}}'

\rho_s L,2\pi R_s,dR_s. ]

Dividing by time and equating this rate to the heat removed gives

[ q'(R_s)

2\pi R_s\rho_s L\frac{dR_s}{dt}. ]

Substituting the resistance expression produces the governing growth equation:

[ \boxed{ \frac{dR_s}{dt}

\frac{T_m-T_c} { 2\pi R_s\rho_s L \left[ \frac{1}{h_i,2\pi R_i} + \frac{\ln(R_t/R_i)}{2\pi k_w} + \frac{\ln(R_s/R_t)}{2\pi k_s} \right] } } ]

This equation gives the instantaneous radial growth rate of the solid layer.

Interpret the growth-rate decline

As (R_s) increases, the solid resistance

[ \frac{\ln(R_s/R_t)}{2\pi k_s} ]

also increases. The heat flux reaching the phase boundary therefore falls, and the interface advances more slowly.

The factor (2\pi R_s) in the latent-heat term also matters: a given radial increment creates more solid volume per unit tube length as the interface circumference grows.

Useful Simplifications for Engineering Estimates

Use a planar approximation for a thin layer

If the solid thickness is small compared with the tube radius, (\delta \ll R_t), the cylindrical resistance can be approximated by a planar resistance. The growth equation becomes

[ \frac{d\delta}{dt}

\frac{T_m-T_c} {\rho_s L \left[ \frac{1}{h_i} + \frac{t_w}{k_w} + \frac{\delta}{k_s} \right]}, ]

where (t_w) is the tube-wall thickness.

If the coolant-film and wall resistances are grouped into a constant resistance (R_0), then

[ \frac{d\delta}{dt}

\frac{T_m-T_c} {\rho_s L\left(R_0+\delta/k_s\right)}. ]

Integrate to estimate thickness versus time

For constant properties and temperature difference, integration gives

[ t

\frac{\rho_s L}{T_m-T_c} \left[ R_0\delta+\frac{\delta^2}{2k_s} \right]. ]

This result shows two limiting regimes.

At small thickness, the coolant-film and wall resistances dominate, so growth is approximately linear in time. At larger thickness, solid conduction dominates, and the familiar diffusion-like behavior appears:

[ \delta \propto \sqrt{t}. ]

Use the cylindrical form for substantial deposits

When the layer thickness is not small relative to the tube radius, the planar approximation can underestimate the changing geometry and resistance. The cylindrical equation should then be integrated numerically, or analytically after expanding the resistance terms.

For a tube-based pilot unit, this distinction is important because deposits can become thick enough that the local heat-transfer area and logarithmic conduction resistance both influence the result.

Applying the Model to a Tube-Based Unit Operation

Estimate the effective heat-transfer coefficient

The resistance model identifies which part of the system limits growth. The internal coolant film is represented by (1/h_i), while the growing solid contributes (\delta/k_s) in the planar limit.

An effective overall coefficient can be defined from the total resistance, subject to a consistent reference area:

[ U=\frac{1}{R_{\text{total}}}. ]

This allows measured growth rates to be compared with calculated values and can reveal whether the dominant limitation is coolant-side convection, tube-wall conduction, or solid-layer conduction.

Evaluate coolant operating conditions

The coolant temperature and flow rate affect the thermal driving force and the internal coefficient (h_i). Lowering (T_c) generally increases the driving force (T_m-T_c), while increasing coolant velocity can reduce the internal film resistance.

The model can therefore be used to predict how changes in coolant supply conditions affect the initial growth rate and the time required to reach a target layer thickness.

Account for temperature and composition effects

In crystallization systems, (T_m) may vary with concentration, pressure, or solute rejection at the interface. If the liquid becomes progressively concentrated, the phase-change temperature and available driving force may change during operation.

In that case, the balance should be updated as a function of position, time, or local composition rather than using a single constant (T_m).

Couple the model to axial tube behavior

A real heat-transfer unit may have a coolant temperature that changes along the tube. The local driving force is then

[ \Delta T(z,t)=T_m(z,t)-T_c(z,t), ]

and the growth equation must be solved along the tube length as well as in time.

This produces a local thickness (R_s(z,t)), which can identify regions where excessive deposition, blockage, or nonuniform crystallization is likely.

Understanding the Trade-offs

The approximation omits transient sensible heating

The pseudosteady model neglects the time required to cool newly formed solid and nearby liquid. This is most defensible when the Stefan number is small, meaning latent heat is large relative to the sensible heat associated with the temperature change.

When sensible effects are significant, a full transient moving-boundary model is more appropriate.

Constant properties may be unrealistic

Thermal conductivity, density, latent heat, viscosity, and heat-transfer coefficients can vary with temperature and composition. Treating them as constant simplifies the analysis but may bias predicted thickness and growth rates.

Property variation should be included when the operating range is broad or when experimental data show systematic deviations from the simple model.

Surface conditions can change the interface behavior

The model generally assumes a well-defined interface and one-dimensional radial heat flow. Roughness, fouling, nucleation behavior, natural convection, shear, and crystal morphology can violate these assumptions.

For crystallization, the deposit may also be porous or dendritic rather than a dense solid layer. Its effective conductivity and interfacial area may then differ substantially from ideal values.

Growth can reduce useful process capacity

A thicker solid layer increases thermal resistance and can reduce the effective heat-transfer duty of the unit. It may also constrict flow passages, increase pressure drop, or cause product-quality changes through excessive residence time or nonuniform crystal size.

Therefore, maximizing instantaneous growth rate is not always the same as optimizing the overall unit operation.

Making the Right Choice for Your Goal

Use the pseudosteady model as a first engineering tool, then increase model complexity only when the assumptions materially affect the result.

If your primary focus is estimating solid thickness over time: Use the cylindrical resistance balance when the deposit is substantial and the planar integrated equation for thin-layer estimates.
If your primary focus is identifying the heat-transfer limitation: Separate coolant-film, tube-wall, and solid-layer resistances to determine which operating variable will have the greatest effect.
If your primary focus is optimizing coolant operation: Evaluate coolant temperature and flow together, because driving force and internal convection both control the latent-heat removal rate.
If your primary focus is predicting nonuniform growth in a pilot unit: Couple the local growth equation to axial coolant temperature and composition balances.
If your primary focus is high-fidelity prediction: Include transient sensible heat, variable properties, external liquid resistance, and deposit morphology when experimental behavior cannot be explained by the pseudosteady model.

A pseudosteady-state resistance balance converts a moving solidification boundary into a practical heat-transfer calculation while making clear why growth slows as the solid layer becomes thicker.

Summary Table:

Key element	Engineering insight
Core assumption	Thermal profiles adjust faster than the solid-liquid interface moves.
Driving force	Phase-change temperature minus coolant bulk temperature, ΔT = Tm − Tc.
Heat-transfer path	Internal coolant film, tube wall, and growing solid layer.
Growth balance	Removed heat equals latent heat released by newly formed solid.
Cylindrical growth	Use the current interface radius and logarithmic solid resistance for thick deposits.
Thin-layer estimate	Growth is initially near-linear, then approaches δ ∝ √t as solid resistance increases.
Main controls	Coolant temperature, coolant velocity, thermal properties, latent heat, and deposit thickness.
Model limitations	Transient sensible heating, variable properties, composition effects, fouling, and porous deposits may require higher-fidelity modeling.

Predict and Improve Your Tube-Based Process

Need to control freezing, crystallization, or solid deposition in a pilot unit? Contact LABPARK to discuss your application with our team.

LABPARK provides Educational and Vocational Unit Operations Pilot Plants for universities, research institutes, and enterprises, including solutions for chemical engineering, bioprocess and biotechnology, and environmental and water treatment. We can help you evaluate coolant conditions, heat-transfer resistance, interface growth, axial temperature variation, and scale-up requirements so you can:

Estimate solid-layer thickness and growth time more reliably.
Identify whether coolant flow, tube-wall conduction, or deposit resistance limits performance.
Assess blockage, pressure-drop, and nonuniform-growth risks before pilot operation.
Select practical instrumentation and operating conditions for teaching, research, and production development.

Contact LABPARK to define your target material, tube geometry, coolant conditions, and process objective, and develop a pilot-plant approach suited to your needs.