Mass and energy balance methodology: from control volume to reconciled stream table

Every equipment size, chemical consumption and utility load in a process design derives from the mass and energy balance. This note sets out a method that produces balances which close, can be audited and survive independent checking. It covers conservation laws, degrees of freedom, recycle, reaction, energy, reconciliation of plant data and propagation of uncertainty.

Technical note, RB Solutions Ltd, 22 October 2026

Conservation laws and the general balance equation

For any conserved or generated quantity and any defined control volume, the general balance applies [1][2]:

\frac{d M}{d t} = \sum_{in} {\dot{m}}_{j} - \sum_{out} {\dot{m}}_{j} + R_{gen} - R_{cons}

where M is the inventory of the quantity within the control volume (kg), t is time (s), ṁ_j are mass flows crossing the boundary (kg s⁻¹), and R_gen and R_cons are rates of generation and consumption by reaction (kg s⁻¹).

Total mass and the mass of each element are conserved in non-nuclear processes, so the reaction terms vanish for them. Chemical species are generally not conserved: a species balance requires the reaction terms. Moles are not conserved through reaction unless the stoichiometry is equimolar. Volume is not conserved at all, and volumetric "balances" are valid only as an approximation for dilute aqueous streams of near-constant density. Even then, the density basis has to be stated.

Control volumes

The control volume is chosen to make the unknown of interest appear in as few equations as possible. Typical choices are the overall plant boundary, which gives the external flows independent of internal recycles; a single unit; a mixing or splitting node; and a boundary drawn to cut a recycle stream. Each boundary is drawn on the block flow diagram before any calculation, and every stream crossing it is listed.

Steady-state and dynamic balances

At steady state dM/dt = 0, and the balance becomes algebraic. A dynamic balance is required where inventory changes matter to the design: batch and sequencing operations, equalisation and buffer tanks, start-up, and the response of pH or level to a disturbance. A steady-state balance on a batch process must be expressed over a complete cycle, and the cycle-average flows must then be distinguished from the instantaneous peak flows that size pumps and pipework.

Degrees of freedom analysis

Before solving, the problem is confirmed to be correctly specified [1]:

N_{DoF} = N_{unknowns} - N_{independent equations}

where N_DoF is the number of degrees of freedom (dimensionless count). The independent equations comprise material balances (one per component per unit, or total plus components less one), specified ratios, split fractions, equilibrium relations and performance specifications.

If N_DoF = 0 the problem is fully specified. If it is positive, design variables remain to be fixed. If it is negative, the problem is over-specified and the data are either redundant or inconsistent. The analysis is carried out per unit and for the whole flowsheet. A common error is counting a total balance and all component balances as independent equations when only C of the C + 1 are independent. Over-specification is not a fault in plant data analysis. It provides the redundancy that data reconciliation exploits.

Recycle, bypass and purge

Recycle streams couple units so that the flowsheet must be solved simultaneously or by iteration on a tear stream. The overall balance, drawn around the plant boundary and excluding the recycle, normally yields the fresh feed and product flows directly. Unit balances then resolve the recycle. A purge is required wherever a component enters with the feed but has no other exit, such as an inert in a gas loop or dissolved salts in a water recycle. Without one, that component accumulates without limit. At steady state with a purge fraction P of the recycle loop flow, the inert balance gives:

{\dot{n}}_{I, feed} = P x_{I, loop} {\dot{n}}_{loop} \Rightarrow x_{I, loop} = \frac{{\dot{n}}_{I, feed}}{P {\dot{n}}_{loop}}

where ṅ_I,feed is the inert entering with fresh feed (mol s⁻¹), P is the purge fraction of the loop flow (dimensionless), x_I,loop is the inert mole fraction in the loop (dimensionless) and ṅ_loop is the loop flow at the purge point (mol s⁻¹).

The same relationship governs concentration factors in water recycle systems such as filter backwash recovery and cooling circuits. Halving the purge doubles the loop concentration of every non-removed solute.

Water and solids balance closure

In water and effluent treatment, two balances must close at every separation step: the water balance and the dry solids balance. For a clarifier or thickener with feed F, overflow O and underflow U:

Q_{F} = Q_{O} + Q_{U}, Q_{F} X_{F} = Q_{O} X_{O} + Q_{U} X_{U}

where Q is volumetric flow (m³ h⁻¹) and X is suspended solids concentration (kg m⁻³); for concentrated underflows the balance is written on mass flow with slurry density stated.

Chemical additions such as coagulant, polymer and the precipitated hydroxide or carbonate solids they produce must enter the solids balance as generated mass. Omitting them is the most frequent cause of undersized sludge handling. A closure check is computed and reported for every node: (Σin − Σout − accumulation)/Σin. It should equal zero to the arithmetic precision of the calculation. A design balance that closes only to a few per cent contains an error, not an uncertainty.

Component balances with reaction

Where reaction occurs, the extent of reaction ξ provides a compact formulation [1][3]:

{\dot{n}}_{i, out} = {\dot{n}}_{i, in} + \sum_{r} ν_{i, r} {\dot{ξ}}_{r}

where ṅ_i is the molar flow of species i (mol s⁻¹), ν_i,r is its stoichiometric coefficient in reaction r (dimensionless, negative for reactants) and ξ̇_r is the rate of extent of reaction r (mol s⁻¹).

An element balance is the independent check on any reacting balance. Summing each element over all species in and out must close exactly, whatever the extents. In precipitation and neutralisation duties, a charge balance on each stream provides a further check that the ionic composition is physically possible.

Energy balance and heat of reaction

For an open steady-flow system with negligible kinetic and potential energy changes, the first law reduces to [1][2]:

\dot{Q} - {\dot{W}}_{s} = \sum_{out} {\dot{n}}_{i} {\hat{H}}_{i} - \sum_{in} {\dot{n}}_{i} {\hat{H}}_{i}

where Q̇ is heat added to the system (W), Ẇ_s is shaft work done by the system (W) and Ĥ_i is the molar enthalpy of species i relative to a stated reference state (J mol⁻¹).

Using elements in their standard states at 25 °C as the reference, the heat of reaction appears automatically through the enthalpies of formation. With reacting species as the reference, it appears explicitly as ξ̇ ΔH_r°. The standard heat of reaction follows from tabulated enthalpies of formation:

Δ H_{r}^{°} = \sum_{i} ν_{i} Δ H_{f, i}^{°}

where ΔH_r° is the standard enthalpy of reaction (J mol⁻¹ of extent) and ΔH_f,i° are standard enthalpies of formation (J mol⁻¹).

For neutralisation, H⁺(aq) + OH⁻(aq) → H₂O(l), the NBS tables give ΔH_f° = −285.83 kJ mol⁻¹ for liquid water, −229.99 kJ mol⁻¹ for OH⁻(aq) and zero for H⁺(aq) by convention [4]. Hence ΔH_r° = −285.83 − (−229.99) = −55.84 kJ mol⁻¹. Neutralising 0.05 mol L⁻¹ of hydroxide releases 2.79 kJ per litre. With a specific heat capacity of about 4.18 kJ kg⁻¹ K⁻¹ for dilute aqueous solution [2], this raises the temperature by roughly 0.67 K. That is negligible for dilute effluent, but significant for concentrated caustic or acid streams, where heat of dilution must also be included.

Data reconciliation by weighted least squares

Measured plant flows never satisfy the balance exactly. Where the system is over-determined, the measurements are adjusted by the minimum statistically weighted amount needed to satisfy the constraints [5]:

\min_{\hat{x}} {(\hat{x} - x)}^{T} V^{- 1} (\hat{x} - x) subject to A \hat{x} = 0

where x is the vector of measurements, x̂ the reconciled estimates, V the measurement covariance matrix (diagonal with σ_j² for independent errors, units of flow squared) and A the incidence (balance) matrix (dimensionless).

For linear constraints the solution is closed-form:

\hat{x} = x - V A^{T} {(A V A^{T})}^{- 1} A x

where Ax is the vector of balance residuals (units of flow) and AVA^T its covariance (units of flow squared).

The normalised residual, each element of Ax divided by the square root of the corresponding diagonal element of AVA^T, is approximately standard normal when only random error is present. A value outside ±1.96 indicates a gross error, such as a failed meter or an unmeasured stream, at the 5 % significance level [5]. A gross error must be located and removed before reconciliation, because least squares distributes it across all measurements.

Worked example: reconciling a three-stream split

Inputs. A splitter node has feed F₁ and two outlets F₂ and F₃. Meter readings and standard uncertainties, taken from calibration certificates, are: F₁ = 100.0 m³ h⁻¹ (σ = 2.0), F₂ = 62.0 m³ h⁻¹ (σ = 1.5) and F₃ = 41.0 m³ h⁻¹ (σ = 1.0). The constraint is F₁ − F₂ − F₃ = 0, so A = [1, −1, −1].

Route 1, closed form. The residual is Ax = 100.0 − 62.0 − 41.0 = −3.0 m³ h⁻¹. Its variance is AVA^T = 4.00 + 2.25 + 1.00 = 7.25 (m³ h⁻¹)², a standard deviation of 2.69 m³ h⁻¹. The normalised residual is 3.0/2.69 = 1.11, which is inside ±1.96, so no gross error is indicated. The adjustments are VA^T(−3.0/7.25):

F̂₁ = 100.0 − 4.00 × (−3.0)/7.25 = 101.66 m³ h⁻¹
F̂₂ = 62.0 − (−2.25) × (−3.0)/7.25 = 61.07 m³ h⁻¹
F̂₃ = 41.0 − (−1.00) × (−3.0)/7.25 = 40.59 m³ h⁻¹

The check 61.07 + 40.59 = 101.66 m³ h⁻¹ closes exactly. The least-certain meter, F₁, absorbs the largest adjustment.

Route 2, constrained substitution. Eliminating F₁ = F₂ + F₃ and minimising the weighted sum of squares over the two free variables, by solving the 2 × 2 normal equations, returns the same three values, 101.66, 61.07 and 40.59 m³ h⁻¹. The two routes agree to the precision reported.

Table 1. Measured and reconciled flows for the worked example
Stream	Measured (m³ h⁻¹)	σ (m³ h⁻¹)	Reconciled (m³ h⁻¹)	Adjustment / σ
F₁ feed	100.0	2.0	101.66	+0.83
F₂ outlet	62.0	1.5	61.07	−0.62
F₃ outlet	41.0	1.0	40.59	−0.41
Closure F₁ − F₂ − F₃	−3.0	2.69	0.00	normalised residual 1.11

Uncertainty propagation

A balance result inherits uncertainty from its inputs. For a function y = f(x₁, …, x_n) of uncorrelated inputs, the first-order law of propagation of uncertainty is [6]:

u_{c}^{2} (y) = \sum_{i = 1}^{n} {(\frac{\partial f}{\partial x_{i}})}^{2} u^{2} (x_{i})

where u_c(y) is the combined standard uncertainty of y and u(x_i) the standard uncertainty of each input, both in the units of the respective quantity.

Two consequences are important in design. First, a quantity obtained as a small difference of two large measured flows, such as a filter backwash loss or an unmetered leak, has an uncertainty that can exceed the quantity itself. It should be measured directly or bracketed, not inferred. Second, design inputs that are assumptions rather than measurements, such as a future influent load, are carried as bracketed cases (low, design, high), and the balance is solved at each bracket. A single nominal case is not sufficient.

Stream tables on PFDs

The balance is communicated through the stream table on the process flow diagram. Each stream number on the drawing corresponds to one column, listing at minimum total mass flow, volumetric flow at stated conditions, temperature, pressure, phase and the concentrations of design components. Normal, design and, where relevant, minimum and maximum cases are given. The table is generated from the calculation, not typed, so that drawing and calculation cannot diverge. It carries the closure check for each unit as a row or note. Graphical conventions for the PFD itself are covered in the companion note on PFD and P&ID standards.

Verification by an independent second route

A balance is not complete until it has been verified by a method that does not share the error modes of the first. Effective second routes include the following:

An overall balance around the plant boundary checked against the sum of unit balances.
An element or charge balance checked against a species balance.
A hand calculation of a governing stream checked against the flowsheet model.
Back-substitution of the solution into every original equation.
Comparison with an independent benchmark such as a published per-capita load, a vendor heat balance or historical plant data.

Agreement within a stated tolerance is recorded with the method used. Disagreement is resolved, never averaged.

Table 2. Balance types and their independent checks
Balance	Conserved?	Independent check
Total mass	Yes	Overall boundary versus sum of units
Element	Yes	Species balances with reaction extents
Chemical species	No, with reaction	Element and charge balance
Volume	No	Mass balance with stated densities
Energy	Yes	Heat of reaction via formation enthalpies and via measured temperature rise
Dry solids	Yes, including generated precipitate	Water balance and underflow concentration

References

Himmelblau, D.M. and Riggs, J.B. (2012) Basic Principles and Calculations in Chemical Engineering, 8th ed. Upper Saddle River: Prentice Hall.
Green, D.W. and Southard, M.Z. (eds) (2019) Perry's Chemical Engineers' Handbook, 9th ed. New York: McGraw-Hill.
Felder, R.M., Rousseau, R.W. and Bullard, L.G. (2016) Elementary Principles of Chemical Processes, 4th ed. Hoboken: Wiley.
Wagman, D.D. et al. (1982) The NBS tables of chemical thermodynamic properties. Journal of Physical and Chemical Reference Data, 11, Supplement 2.
Narasimhan, S. and Jordache, C. (2000) Data Reconciliation and Gross Error Detection. Houston: Gulf Publishing.
JCGM 100:2008 Evaluation of measurement data: Guide to the expression of uncertainty in measurement. Joint Committee for Guides in Metrology. bipm.org

Related technical notes

RB Solutions Ltd prepares mass and energy balances, stream tables and reconciliation of plant data for water, effluent and process plants, each verified by an independent second route.

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