Q.60 A feed stream (F1) containing components A and B is processed in a system
comprising of separation unit and a mixer as shown below in the schematic
diagram. The mole fractions of the components A and B are xA and
xB, respectively. If F1 + F2 =kg h-1, the degrees of freedom of the system
is ________.
Introduction
Chemical engineering problems frequently require determining the Degrees of Freedom (DOF) of a process. DOF tells us how many process variables must be specified to fully define a system based on mass balances and constraints.
This article analyzes a system containing a separator and a mixer, a common flowsheet in material balance problems.
Problem Statement
A feed stream F₁ containing components A and B is processed along with a recycle stream F₂.
A separator produces two streams that recombine in a mixer before exiting as stream F₆.
Given:
F₁ + F₂ = 100 kg/h
Each stream contains two components A and B with mole fractions x_A and x_B.
Question: What is the degrees of freedom (DOF) of the system?
Step-by-Step DOF Analysis
Variables Present
Six streams → F₁, F₂, F₃, F₄, F₅, F₆
Each stream has:
- One flow rate
- One independent composition (since x_A + x_B = 1)
Total variables: 6 flowrates + 6 compositions = 12
Independent Equations Available
For two-component system:
- Overall mass balance
- Component balances at:
- Mixer → 2 balances
- Separator → 2 balances
- Composition constraints: For 6 streams x_A + x_B = 1 gives 6 constraints
Total equations: 1 + 2 + 2 + 6 = 11
Given Constraint
F₁ + F₂ = 100 kg/h ⇒ 1 more equation
Total equations now: 11 + 1 = 12
Degrees of Freedom
DOF = Variables − Equations = 12 − 12 = 0
Correct Answer
0
The system is fully specified.
Explanation of All Possible Options
| Option | Meaning | Explanation |
|---|---|---|
| 0 (Correct) | System is fully determined | Number of unknowns equals number of equations. No additional specifications are required. |
| 1 | Under-specified by one variable | Incorrect because one extra constraint is already supplied: F₁ + F₂. |
| 2 | Under-specified by two variables | Wrong because mass balances and composition relations supply adequate equations. |
| >2 | Highly under-specified | Not possible—system has sufficient balances and constraints to define stream variables. |
Conclusion
The system contains exactly the right number of equations to solve all stream variables, meaning the flowsheet is completely defined.
📌 Final Answer: The degrees of freedom of the system is 0.
