Study notes

Qwestrum Engineering360 · Chemical Engineering · Mechanical Operations

Filtration

Cake filtration is modelled by adding the growing cake resistance to a constant medium resistance; at constant pressure the filtrate volume follows a parabolic law, and plotting t/V against V linearises the data to extract specific cake resistance α and medium resistance R_m.

Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

Incompressible cake: α is constant; compressible: α rises with ΔP
Constant-rate filtration: ΔP rises as the cake thickens
Cake washing and deliquoring add to cycle time

Topic details

Introduction

This McCabe-Smith topic sizes filters and interprets filtration tests. You apply the constant-pressure filtration equation, linearise the data to obtain α and R_m from the slope and intercept, and account for cake compressibility and the additional time for washing and dewatering when estimating cycle time.

Key relations & formulas

\frac{V^{2}}{A^{2}} = \frac{(2 \Delta P t)}{(\mu \alpha c)} + \frac{(\Delta P t)}{(\mu R_{m})}

(constant-pressure cake filtration)

\frac{dV}{(A dt)} = \Delta P / (\mu(\alpha c \frac{V}{A} + R_{m}))

(Ruth / basic filtration rate)

Formulas (Indian textbook notation)

$\alpha = specific cake resistance; R_{m} = medium resistance$

Notation and sign conventions

Relation 1 —

\frac{V^{2}}{A^{2}} =

\frac{V^{2}}{A^{2}} = \frac{(2 \Delta P t)}{(\mu \alpha c)} + \frac{(\Delta P t)}{(\mu R_{m})}

(constant-pressure cake filtration)

Write this relation with symbols exactly as in Unit Operations of Chemical Engineering — McCabe, Smith & Harriott before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 2 —

dV/

\frac{dV}{(A dt)} = \Delta P / (\mu(\alpha c \frac{V}{A} + R_{m}))

(Ruth / basic filtration rate)

Relation 3 —

\alpha = specific cake resistance; R_{m} = medium resistance

Formulas (Indian textbook notation)

$\alpha = specific cake resistance; R_{m} = medium resistance$

Concept in depth

As filtrate passes through the medium it deposits a cake that itself becomes the dominant resistance; the flow rate therefore falls with time even at constant pressure. The specific cake resistance α reflects how tightly the cake packs — for a compressible cake it increases with pressure, so simply raising ΔP yields diminishing returns because the cake densifies. Because both cake and medium resistances add in series, early filtration is medium-controlled and later filtration cake-controlled. The linearised t/V versus V plot is the standard way to separate the two resistances from a single test.

Assumptions and validity limits

State assumptions explicitly before using any relation for filtration — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.

Wrong assumptions invalidate the entire solution even when the formula is correct. In Mechanical Operations viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Mechanical Operations papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to filtration.
4. Use equation 1:

\frac{V^{2}}{A^{2}} =

.
5. Use equation 2:

dV/

.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Filtration appears in solids processing industries. In Indian chemical curricula this topic is tested because it connects theory to size reduction, filtration, and fluidization.

GATE and semester exams often combine filtration with earlier units — revise prerequisites before attempting mixed problems.

Industry interview panels sometimes ask: "Where did you use filtration?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

Students assume α is constant for a clearly compressible cake, forget the medium resistance at the start of filtration, and mix up which of slope and intercept gives α versus R_m. Neglecting wash and dewatering time when the question asks for cycle time is common.

Quick revision checklist

Before attempting filtration problems, confirm you can:

1. Incompressible cake: α is constant; compressible: α rises with ΔP
2. Constant-rate filtration: ΔP rises as the cake thickens
3. Cake washing and deliquoring add to cycle time

Revise the solved examples in Unit Operations of Chemical Engineering — McCabe, Smith & Harriott and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Interpreting a filtration plot

Problem

A constant-pressure test plotted as t/V versus V gives a straight line. What do the slope and intercept represent?

Solution

The slope is proportional to the specific cake resistance α (cake-building term) and the intercept is proportional to the medium resistance R_m. Both scale inversely with ΔP, so higher pressure shifts the whole line down.

Conceptual check — Filtration

Problem

In a Mechanical Operations semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of filtration." What should a complete answer include?

Exams & GATE

McCabe-Smith Ch. 30 — linearise t/V vs V to get α and R_m from slope and intercept.

📖 Standard books (India)

Unit Operations of Chemical Engineering — McCabe, Smith & Harriott
Read: Syllabus unit
Momentum, heat, and mass transfer operations

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📚Key formulas & points