The dynamic mass balance is linearized by replacing the nonlinear vapor–liquid equilibrium relation with its local linear form and then collecting the composition terms. For stage (n), the liquid component balance is

[ H_n\frac{dx_n}{dt}

Lx_{n-1}-Lx_n+Vy_{n+1}-Vy_n. ]

If the equilibrium relation is approximated locally by (y_i=m_i x_i), with constant slope (m_i), the stage equation becomes a linear differential equation coupled to neighboring stages. This form is the basis for deriving transfer functions, studying disturbance propagation, and designing composition-control systems for absorption and distillation pilot plants.

Core takeaway: Linearization converts the stage balance into a first-order dynamic model by treating holdup, flow rates, and the local equilibrium slope as constant. The resulting model is generally a coupled set of first-order equations, not an independent equation for each stage, because each stage exchanges material with adjacent stages.

Starting With the Dynamic Stage Balance

Liquid component accumulation

For a constant liquid holdup (H_n), the accumulation of the absorbed or distilled component on stage (n) is

[ H_n\frac{dx_n}{dt}. ]

Here, (x_n) is the liquid-phase mole fraction of the component on stage (n). The units must be consistent: if (H_n) is molar holdup, (L) and (V) should be molar flow rates.

Flow contributions

The liquid stream entering stage (n) contributes (Lx_{n-1}), while the liquid stream leaving contributes (Lx_n).

The vapor stream entering from the adjacent stage contributes (Vy_{n+1}), and the vapor leaving stage (n) removes (Vy_n). Therefore,

[ H_n\frac{dx_n}{dt}

Lx_{n-1}-Lx_n+Vy_{n+1}-Vy_n. ]

This balance assumes constant (L), constant (V), constant holdup, no reaction, and negligible accumulation in the vapor phase.

Applying the Local Equilibrium Approximation

Linear equilibrium relation

The vapor and liquid compositions are related by the equilibrium function

[ y=f(x). ]

Around a selected operating point, this relation can be approximated as

[ y_i \approx y_{i,0}+m_i(x_i-x_{i,0}), ]

where

[ m_i=\left.\frac{dy}{dx}\right|{x{i,0}}. ]

For a simplified model with a constant equilibrium-line slope and compositions measured relative to the operating point, this reduces to

[ y_i=m x_i. ]

The slope (m) may represent a local volatility or Henry-law relationship, depending on the system.

Substitution into the stage balance

Using (y_n=mx_n) and (y_{n+1}=mx_{n+1}),

[ H_n\frac{dx_n}{dt}

Lx_{n-1}-Lx_n+mVx_{n+1}-mVx_n. ]

Collecting the terms involving (x_n) gives

[ \boxed{ H_n\frac{dx_n}{dt}

Lx_{n-1}

(L+mV)x_n + mVx_{n+1} } ]

or, after division by holdup,

[ \boxed{ \frac{dx_n}{dt}

\frac{L}{H_n}x_{n-1}

\frac{L+mV}{H_n}x_n + \frac{mV}{H_n}x_{n+1} }. ]

This is the standard linearized dynamic equation for an ideal equilibrium stage under constant-flow assumptions.

What “Linearized” Means in This Model

Linearizing the equilibrium relationship

The original equilibrium relation may be nonlinear, particularly for distillation systems with significant relative-volatility variation. Linearization replaces that nonlinear relation with its tangent near the selected steady operating point.

The approximation is reliable for relatively small composition changes around that point. Large disturbances may require the full nonlinear vapor–liquid equilibrium relationship.

Linearizing the material balance

The material balance itself is already linear in (x) when (H), (L), and (V) are constant. Additional nonlinearities arise when these quantities vary with composition, temperature, pressure, level, or total flow.

For a more general model, one writes

[ H_n(x_n)\frac{dx_n}{dt}

L(x_{n-1}-x_n) + V(y_{n+1}-y_n), ]

then expands the model about a steady state:

[ x_i=x_{i,0}+\delta x_i, \qquad y_i=y_{i,0}+\delta y_i. ]

The equilibrium perturbation is approximated by

[ \delta y_i=m_i\delta x_i. ]

The resulting perturbation model has the same structure as the boxed equation, but its coefficients are evaluated at the operating point.

Important Indexing Correction

The neighboring vapor composition must be retained

A frequent indexing mistake is to substitute (y_{n+1}=mx_{n-1}). Under the usual countercurrent stage numbering, the vapor entering stage (n) is (y_{n+1}), which is related to the liquid composition on stage (n+1):

[ y_{n+1}=mx_{n+1}. ]

Therefore, the correctly indexed equation is

[ H_n\frac{dx_n}{dt}

Lx_{n-1}

(L+mV)x_n + mVx_{n+1}. ]

The expression

[ H\frac{dx_n}{dt}

Lx_{n-1}-(L+mV)x_n+mVx_{n-1} ]

is only valid if the variable labeled (x_{n-1}) has been deliberately defined as the upstream liquid-equivalent composition of the entering vapor. It is not the usual stage-indexed equilibrium substitution.

Why the distinction matters

Retaining (x_{n+1}) produces a coupled column model. The composition on one stage responds to both neighboring stages, which is essential for representing disturbance propagation through a trayed column or discretized packed column.

Replacing the neighboring-stage term with the same upstream variable can produce a simpler single-stage model, but it changes the physical connectivity and may give incorrect dynamic predictions.

From Stage Equations to Transfer Functions

Laplace-domain form

Assuming zero initial perturbations and constant coefficients, taking the Laplace transform gives

[ H_n sX_n(s)

LX_{n-1}(s)

(L+mV)X_n(s) + mVX_{n+1}(s). ]

Rearranging,

[ \boxed{ \left[H_n s+(L+mV)\right]X_n(s)

LX_{n-1}(s)+mVX_{n+1}(s) }. ]

This equation is one row of the column’s dynamic transfer-function or state-space model.

Single-stage approximation

If the entering vapor composition is treated as a known input rather than as a state related to (x_{n+1}), define

[ u_n=Lx_{n-1}+Vy_{n+1}. ]

The balance becomes

[ H_n\frac{dx_n}{dt}

u_n-(L+mV)x_n. ]

The corresponding transfer function is

[ \boxed{ \frac{X_n(s)}{U_n(s)}

\frac{1}{H_n s+L+mV} } ]

with time constant

[ \boxed{ \tau_n=\frac{H_n}{L+mV} }. ]

If the input is only the upstream liquid composition and the vapor contribution is neglected or incorporated separately, then

[ \frac{X_n(s)}{X_{n-1}(s)}

\frac{L}{H_n s+L+mV}. ]

Application to Absorption and Distillation

Gas-liquid absorption

In absorption, the solute generally transfers from the gas phase into the liquid phase when the actual gas composition exceeds its equilibrium value:

[ y>y^*(x). ]

The equilibrium slope (m) describes how the equilibrium gas composition changes with liquid composition. A larger (mV) term increases the strength of the vapor-phase coupling and changes the stage response.

For packed absorption columns, the same concept is often written as a distributed differential model rather than as discrete tray balances. A finite-volume discretization produces stage-like equations with holdup and neighboring-cell coupling.

Distillation

In distillation, the equilibrium relation is commonly represented by

[ y_i=y^*(x_i), ]

where (y^*(x)) may be nonlinear. Around an operating composition,

[ \delta y_i \approx \left.\frac{dy^*}{dx}\right|{x{i,0}}\delta x_i. ]

The local slope can vary from stage to stage, so a distillation column usually has coefficients (m_n), (L_n), (V_n), and (H_n) rather than one universal (m), (L), and (V).

Boundary conditions

The internal stage equations are not sufficient by themselves. The condenser, reboiler, feed stage, reflux stream, distillate stream, bottoms stream, and column endpoints provide the boundary conditions and manipulated-variable relationships.

For example, reflux changes the liquid flow entering the upper stages, while boilup changes the vapor flow entering the lower stages. These boundary effects are often the dominant paths for control disturbances.

Extending the Model to Variable Holdup and Flow

Total mass balance

When liquid level or holdup varies, a total balance must be included. A general liquid holdup balance is

[ \frac{dH_n}{dt}

L_{n-1}-L_n+V_{n+1}-V_n, ]

with signs and stream definitions adjusted to the selected stage convention.

The component balance should then be written in terms of component holdup:

[ \frac{d(H_nx_n)}{dt}

\text{component in}

\text{component out}. ]

Expanding the derivative gives

[ H_n\frac{dx_n}{dt} + x_n\frac{dH_n}{dt}

\text{component in}

\text{component out}. ]

The additional (x_n,dH_n/dt) term is important when tray liquid levels or reboiler and condenser holdups change appreciably.

Energy and pressure effects

For accurate pilot-plant models, temperature affects both equilibrium and flow rates. Pressure affects vapor composition, saturation conditions, and volumetric flow.

A composition-only linearization is therefore appropriate mainly for modest disturbances around a fixed temperature and pressure operating point. Larger operating changes require coupled mass, energy, and pressure-balance equations.

Understanding the Trade-offs

Benefits of the linearized model

The linearized model is compact, computationally efficient, and suitable for transfer-function analysis. It provides direct insight into time constants, stage-to-stage coupling, disturbance attenuation, and controllability.

It is especially useful for controller tuning and educational pilot-plant experiments where the column operates near a stable nominal condition.

Limitations of constant equilibrium slope

A single value of (m) may be inadequate when relative volatility changes significantly with composition or temperature. The approximation can also fail near azeotropes, strongly nonideal mixtures, or wide operating ranges.

For these systems, use stage-specific local slopes or retain the nonlinear equilibrium function.

Limitations of constant flow and holdup

Assuming constant (L), (V), and (H) excludes level dynamics and hydraulic effects. This can lead to optimistic predictions of response speed and may hide important interactions between flow, pressure, and composition control.

Pilot plants should use measured or independently modeled holdups when tray flooding, weeping, foaming, or significant liquid-level variation is possible.

Confusing operating lines with dynamic balances

The operating line describes a steady material-balance relationship between phase compositions. It does not, by itself, describe how compositions evolve with time.

The dynamic model requires holdup terms such as (H,dx/dt). Removing those terms converts the model to a steady-state relation rather than a transient one.

Making the Right Choice for Your Goal

The appropriate model depends on whether the objective is control design, conceptual analysis, or high-fidelity pilot-plant prediction.

If your primary focus is controller design: Linearize (y^*(x)) around the intended operating point, retain neighboring-stage coupling, and assemble the equations in state-space form.
If your primary focus is a simple single-stage response: Treat the entering vapor composition as a known input and use the first-order transfer function (1/[Hs+(L+mV)]).
If your primary focus is column-wide disturbance propagation: Retain (x_{n-1}) and (x_{n+1}) in every stage equation and apply the correct condenser, reboiler, feed, reflux, and product boundary conditions.
If your primary focus is large pilot-plant transients: Use nonlinear equilibrium, variable holdup, changing flow rates, and coupled energy and pressure balances.

A sound linearized column model preserves the physical stage connectivity while simplifying the local equilibrium behavior around a clearly defined operating point.

Summary Table:

Key Element	Linearized Model Insight
Dynamic balance	`H_n dx_n/dt = Lx_{n-1} - Lx_n + Vy_{n+1} - Vy_n`
Equilibrium approximation	Replace `y=f(x)` with `delta y=m delta x` near a steady operating point
Coupled stage equation	`H_n dx_n/dt = Lx_{n-1} - (L+mV)x_n + mVx_{n+1}`
Main assumptions	Constant holdup, liquid and vapor flows, and negligible vapor-phase accumulation
Transfer-function form	`[H_n s + (L+mV)]X_n = LX_{n-1} + mVX_{n+1}`
Single-stage time constant	`tau_n = H_n/(L+mV)`
Critical indexing point	The entering vapor term is related to `x_{n+1}`, preserving neighboring-stage coupling
Best applications	Controller tuning, disturbance analysis, transfer functions, and state-space modeling
When nonlinear models are needed	Large disturbances, variable holdup or flow, strong nonideality, pressure changes, or energy effects

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