Material balance fundamentals and degrees of freedom

The slurry tank at the front of the plant mixes ground corn meal with fresh water and with backset, a stream of thin stillage recycled from the back of the plant. The operators want mash at 32 percent solids by mass, and plant practice sets the backset at 30 percent of the liquid added. Theo is asked how much fresh water and backset the tank needs at the design meal rate. Before writing a single equation, Ingrid shows him how to tell whether the problem has enough information to be solved at all. That check, the degree-of-freedom analysis, is the core skill of this lesson.

A procedure for material balance problems

Experienced engineers solve balance problems in nearly the same order every time. The steps below look slow on easy problems, but they prevent nearly all of the errors that make hard problems go wrong, and they are the same steps a process simulator follows internally.

Labelling streams

There are two common ways to label a stream. One gives the total flow and the fractions of each component, such as \(\dot{m}_2\) kg/h with ethanol fraction \(w_{E,2}\). The other gives the flow of each component, such as \(\dot{m}_{E,2}\) and \(\dot{m}_{W,2}\). Both work. Total flow and fractions suit streams whose composition is specified; component flows suit streams whose composition is unknown and keep the equations linear. Choose for each stream whichever makes the known information easiest to write down, and never leave a quantity on the chart without either a number or a symbol.

When some fractions in a stream are known, write all but one of them and get the last by difference, since fractions sum to one. A stream with three components and two known fractions has its third fraction fixed, and counting it as a separate unknown would be an error in the degree-of-freedom analysis.

Independent equations

For a process with no reaction and \(N\) species, at most \(N\) independent material balances can be written for each system: one for each species, or one for total mass and \(N - 1\) for species. The total balance is the sum of the species balances, so writing all \(N + 1\) gives no new information. Any extra relations must come from elsewhere.

The other sources of equations are process specifications, physical relations and physical constraints. A specification is a stated fact about the process, such as a recovery, a split fraction or a ratio between streams: the backset rule in the slurry tank is one. Physical relations include equilibrium data, gas laws and density relations. The physical constraint that fractions in each stream sum to one is already used when you label all but one fraction, so it should not be counted twice.

Degree-of-freedom analysis

The number of degrees of freedom is the number of unknowns minus the number of independent equations relating them. If it is zero, the problem can be solved. If it is positive, it is underspecified: more information is needed, or some variables must be chosen. If it is negative, it is overspecified, and the extra data are either redundant or inconsistent. Design problems often start with positive degrees of freedom, and the designer picks values for some variables to optimize cost; plant data problems often have negative degrees of freedom, and the extra data reveal measurement errors.

Look at how the order of solution was chosen. The specification related only two unknowns, so it was used first to eliminate one; the total balance then expressed \(P\) in terms of \(K\); and the solids balance gave a single equation in \(K\). Looking for an equation with only one unknown, or a set of equations that can be solved together, is the main skill in hand solution. When the equations do not separate neatly, treat them as a linear system and solve them all at once.

Balances as linear systems

Material balances on nonreacting systems with known compositions are linear in the flow rates. Writing them in matrix form, \(\mathbf{A}\mathbf{x} = \mathbf{b}\), turns any balance problem into a standard calculation that a spreadsheet, a calculator or a few lines of code can solve. Process simulators build and solve systems with thousands of such equations. The program below solves the slurry tank and a beer column split with a small Gaussian elimination routine, so you can see exactly what the computer does.

# Solving material balance equations as a linear system (no libraries needed)

def solve(A, b):
    """Gaussian elimination with partial pivoting. A is a list of rows, b a list."""
    n = len(b)
    M = [row[:] + [bi] for row, bi in zip(A, b)]
    for c in range(n):
        p = max(range(c, n), key=lambda r: abs(M[r][c]))
        M[c], M[p] = M[p], M[c]
        for r in range(c + 1, n):
            f = M[r][c] / M[c][c]
            for k in range(c, n + 1):
                M[r][k] -= f * M[c][k]
    x = [0.0] * n
    for r in range(n - 1, -1, -1):
        x[r] = (M[r][n] - sum(M[r][k] * x[k] for k in range(r + 1, n))) / M[r][r]
    return x

# 1. Slurry tank: meal (88 % solids) + fresh water + backset (8 % solids) -> mash at 32 % solids
#    Unknowns: water W and backset K (kg/h); meal fixed at 120 000 kg/h; backset is 30 % of the liquid added.
meal, s_meal, s_back, s_mash = 120_000.0, 0.88, 0.08, 0.32
# total:  meal + W + K = P ; solids: s_meal*meal + s_back*K = s_mash*P ; backset rule: K = 0.30 (W + K)
# Unknowns ordered [W, K, P]
A = [[1, 1, -1],
     [0, s_back, -s_mash],
     [-0.30, 0.70, 0]]
b = [-meal, -s_meal * meal, 0]
W, K, P = solve(A, b)
print(f"Slurry tank: fresh water {W:9.0f} kg/h, backset {K:9.0f} kg/h, mash {P:9.0f} kg/h")

# 2. Beer column, two-component approximation: feed F at 13.5 % ethanol
F, zF = 280_000.0, 0.135
xD, xB = 0.58, 0.0005        # ethanol mass fractions in overhead vapour and bottoms
A = [[1, 1], [xD, xB]]       # unknowns [D, B]
D, B = solve(A, [F, zF * F])
rec = xD * D / (zF * F)
print(f"Beer column: overhead {D:9.0f} kg/h, bottoms {B:9.0f} kg/h, ethanol recovery {rec * 100:.2f} %")

Slurry tank: fresh water    158919 kg/h, backset     68108 kg/h, mash    347027 kg/h
Beer column: overhead     64987 kg/h, bottoms    215013 kg/h, ethanol recovery 99.72 %

Separation units and recovery

Separation units split one stream into two or more of different composition. For a two-component separation with a feed \(F\) of mass fraction \(z_F\), a top product \(D\) of fraction \(x_D\) and a bottom product \(B\) of fraction \(x_B\), the total and component balances give the product flows directly. The fractional recovery of a component is the fraction of its feed that leaves in the desired product.

Tie components and scaling

A tie component is a material that passes from one stream to another unchanged, such as an inert solid in a dryer or nitrogen in a combustion gas. A balance on a tie component often gives a ratio between two streams in one step. In a dryer, the dry solids leaving equal the dry solids entering, which links the wet feed and the dried product immediately.

Because nonreacting balances are linear and homogeneous in the flows, a solution on any basis can be scaled. If 100 kg of feed gives 23.2 kg of overhead, then 280 000 kg/h of feed gives 64 990 kg/h. Scaling multiplies every flow by the same factor but leaves every fraction and ratio unchanged. This is why the choice of basis is free, and why engineers often solve on 100 kg or 100 mol and scale at the end.

When the data do not agree

Plant problems are often overspecified. Suppose the meal flowmeter, the fresh water meter, the backset meter and the mash meter all report values, and the laboratory measures the solids in each stream. That is four measured flows for three unknowns, so the degrees of freedom are negative, and the measurements are very unlikely to satisfy every balance exactly. Theo finds that the measured mash flow is 2.1 percent higher than the sum of the three inlet meters. Ingrid does not average the numbers or pick the one she likes. She first checks which meters were calibrated recently, then uses the solids balance as an independent check: if the solids balance closes with the inlet meters but not with the mash meter, the mash meter is the likely culprit.

This habit, using the redundancy in overspecified data to find bad measurements, is one of the most practical uses of balances in operating plants. More formal methods, called data reconciliation, adjust every measurement by the smallest amount consistent with its accuracy so that all balances close, and they flag measurements that need large adjustments as probably faulty. Every major plant historian and simulator offers some form of it.

Choosing the system boundary

A balance can be written around any boundary: a single unit, a junction where streams meet, a group of units, or the whole process. Each boundary gives its own set of independent equations, but the sets overlap: for a process of several units, the balances on all the individual units together contain the same information as the balances on all but one unit plus the overall balance. A useful strategy is to start with the boundary that has zero degrees of freedom, solve it, and carry the results into the next boundary. Lesson 4 develops this for multiple-unit processes, where choosing the right order can turn a tangle of equations into a sequence of easy steps.