Formula sheet

Mine Ventilation and Underground Safety Systems Formula Sheet

Mine ventilation formulas for airflow, pressure loss, fan power, leakage, dilution, heat load, uncertainty, reliability, limits, and release evidence.

Branch: Mining and Georesources Engineering
Content: Formula sheet
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

This formula sheet collects first-pass relationships used in underground mine ventilation and safety reviews. Use it to screen airflow quantities, pressure loss, fan duty, leakage, contaminant dilution, heat load, measurement margin, reliability, and operating release evidence.

These equations support engineering education, design review, commissioning, and troubleshooting. They do not replace the approved ventilation plan, legal exposure limits, mine-specific trigger-action response plan, competent-person review, calibrated surveys, fan curves, emergency procedures, or site standards.

Before calculating, state the operating mode: normal production, development heading, post-blast clearance, re-entry, maintenance, fan failure, emergency, or unoccupied standby. A result that is acceptable in one mode may be unsafe in another.

How to Use This Formula Sheet

Use this sheet as a release-oriented screening tool, not as a substitute for the mine ventilation plan. Start by defining the occupied area, mining state, applicable limit, airflow path and control objective. A calculation for normal production does not automatically support post-blast re-entry, fan-failure response, hot-work release, emergency egress or unoccupied standby.

Then separate four quantities that are often confused: fan-station flow, useful branch flow, face or workplace flow, and effective fresh-air quantity. The formulas only support a safety decision when the calculated quantity matches the hazard being controlled. Dilution, heat stress, smoke clearance, diesel exposure, battery charging, emergency refuge, and equipment cooling can each require a different controlling airflow.

Use pressure-loss, fan-law and system-curve formulas to understand whether a ventilation change is physically plausible. Use dilution, clearance and heat-load formulas to screen hazard control. Use uncertainty, reliability and interlock checks before accepting a result as release evidence. If the margin is smaller than the measurement uncertainty, the result should trigger adjustment or restriction rather than simple acceptance.

Symbols and basis

Use SI units unless a site standard states otherwise.

Symbol	Meaning	Common unit
$Q$	volumetric airflow	$\text{m}^3/\text{s}$
$\dot{m}$	mass airflow	$\text{kg/s}$
$A$	airway, duct, or station area	$\text{m}^2$
$v$	average air velocity	$\text{m/s}$
$\rho$	air density at ventilation condition	$\text{kg/m}^3$
$\Delta p$	pressure difference or fan pressure rise	$\text{Pa}$
$R$	ventilation branch resistance in square-law form	$\text{N s}^2/\text{m}^8$
$G$	contaminant source rate as volume flow at ventilation conditions	$\text{m}^3/\text{s}$
$C$	contaminant concentration as volume fraction	dimensionless
$\dot{Q}_h$	sensible heat load	$\text{W}$
$c_p$	specific heat of air	$\text{J/(kg K)}$
$\eta$	efficiency	dimensionless

Keep gauge pressure and absolute pressure separate. Gauge pressure is suitable for most ventilation pressure drops. Density calculations require absolute pressure and absolute temperature.

Basis and Validity Limits

The pressure-loss and network formulas in this sheet assume a defined airflow path, reasonably steady flow and a ventilation resistance that can be represented by a square-law relationship. Real mine networks are affected by regulators, doors, stoppings, leakage paths, advancing headings, auxiliary fans, duct damage, vehicles, obstructions and changing excavation geometry. A formula result is therefore a screen until it is reconciled with a survey or approved model.

Fan-law scaling is valid only for the same fan, comparable operating region and acceptable mechanical and electrical limits. It does not justify overspeeding a fan, operating near stall, ignoring vibration, exceeding motor or drive ratings, or bypassing protection. Fan power and specific fan power are useful efficiency indicators, but they do not prove that fresh air reaches the intended workplace.

Dilution and clearance formulas assume known source strength, representative mixing and reliable fresh-air delivery. They can be unsafe for stratified gases, explosive atmospheres, fires, diesel particulate, dust, recirculation, sensor placement errors or unknown transient releases. Heat-load formulas are sensible-heat screens; heat-stress control also depends on humidity, radiant heat, air speed, clothing, workload, exposure duration and work-rest practice.

Treat every formula as conditional on site rules, jurisdictional exposure limits and competent-person review. When the calculation supports entry, re-entry, restart or continued production, it must be backed by measured airflow, gas readings, fan status, interlock state, communication readiness, authority to release and a clear retest trigger.

Airflow, mass flow, and continuity

Volumetric airflow:

Q=Av

Mass airflow:

\dot{m}=\rho Q

Average velocity:

\displaystyle v=\frac{Q}{A}

For a circular duct:

\displaystyle A=\frac{\pi D^2}{4}

Steady continuity at a ventilation junction:

\displaystyle \sum Q_{in}=\sum Q_{out}+Q_{leak}

or, when leakage is treated as another branch:

\displaystyle \sum Q_{in}=\sum Q_{out}

The practical check is not only whether total intake equals total return. The useful question is whether enough fresh air reaches each occupied or hazard-controlled location.

Air density

Dry-air screening density:

\displaystyle \rho=\frac{p_{abs}}{R_{air}T}

where $p_{abs}$ is absolute pressure, $T$ is absolute temperature, and $R_{air}\approx287\ \text{J/(kg K)}$ .

Density ratio for approximate correction:

\displaystyle \frac{\rho_2}{\rho_1}=\frac{p_2T_1}{p_1T_2}

For humid, hot, deep, or high-altitude mines, use site psychrometric data rather than one constant density. Density affects pressure loss, fan duty, mass flow, heat capacity rate, and some fixed airflow measurements.

Dynamic pressure and Bernoulli screen

Velocity pressure:

\displaystyle q=\frac{1}{2}\rho v^2

Simple Bernoulli relation along a streamline:

\displaystyle p+\frac{1}{2}\rho v^2+\rho gz=\text{constant}-\text{losses}+\text{fan rise}

Use Bernoulli as an energy-accounting screen, not as a replacement for a calibrated mine ventilation network. Mine airways are rough, leaky, branching, and frequently disturbed by doors, regulators, duct damage, vehicles, and advancing headings.

Square-law pressure loss

For turbulent mine ventilation branches, a common screening relation is:

\Delta p=RQ^2

Estimate branch resistance from measured pressure drop and airflow:

\displaystyle R=\frac{\Delta p}{Q^2}

For branches in series carrying the same airflow:

R_{series}=\sum_i R_i

For parallel branches with the same pressure drop:

\displaystyle Q_i=\sqrt{\frac{\Delta p}{R_i}}

and:

Q_{total}=\sum_i Q_i

Do not add parallel resistances as if they were electrical resistors. In ventilation networks, each parallel branch flow depends on the same pressure difference and its own square-law resistance.

Fan power and specific fan power

Air power:

P_{air}=\Delta p_f Q

Electrical input power:

\displaystyle P_{in}=\frac{P_{air}}{\eta_{fan}\eta_{motor}\eta_{drive}}

If one combined efficiency is used:

\displaystyle P_{in}=\frac{\Delta p_f Q}{\eta_{overall}}

Installed overall efficiency:

\displaystyle \eta_{installed}=\frac{\Delta p_f Q}{P_{in}}

Specific fan power:

\displaystyle SFP=\frac{P_{in}}{Q}

where $SFP$ is often expressed as $\text{kW}/(\text{m}^3/\text{s})$ .

Fan power is a release check only when airflow distribution, fan curve position, motor current, protection settings, restart conditions, noise, vibration, and ventilation mode are also acceptable.

Fan laws

For the same fan and similar operating region:

\displaystyle \frac{Q_2}{Q_1}=\frac{N_2}{N_1}

\displaystyle \frac{\Delta p_2}{\Delta p_1}=\left(\frac{N_2}{N_1}\right)^2

\displaystyle \frac{P_2}{P_1}=\left(\frac{N_2}{N_1}\right)^3

where $N$ is rotational speed.

If density changes significantly:

\Delta p\propto \rho N^2

P\propto \rho N^3

The fan laws are not permission to overspeed a fan. Check blade stress, motor rating, variable-speed drive limits, stall margin, noise, vibration, and the installed fan curve.

System curve and operating point

For a simplified ventilation system:

\Delta p_{sys}=R_{sys}Q^2+\Delta p_{static}

The operating point is where the fan curve intersects the system curve:

\Delta p_{fan}(Q)=\Delta p_{sys}(Q)

Changing regulators, doors, ducts, leakage, or mined geometry changes $R_{sys}$ and therefore the operating point. A fan can be healthy while the ventilation system is not delivering safe branch quantities.

Leakage and delivery efficiency

Delivery loss:

Q_{loss}=Q_{fan}-Q_{delivered}

Leakage fraction:

\displaystyle L_f=\frac{Q_{loss}}{Q_{fan}}

Delivery efficiency:

\displaystyle \eta_Q=\frac{Q_{delivered}}{Q_{fan}}=1-L_f

For repeated similar duct sections with fractional loss $l$ per section:

Q_{out}=Q_{in}(1-l)^n

where $n$ is the number of sections.

This geometric estimate is only a screen. Real duct leakage depends on duct pressure, coupling quality, tears, bends, kinks, fan placement, brattice condition, and return-air short-circuiting.

Effective fresh-air quantity

If a measured face airflow contains a recirculated fraction $r$ :

Q_{fresh}=Q_{face}(1-r)

If intake air already contains contaminant concentration $C_{in}$ , use $Q_{fresh}$ and $C_{in}$ in dilution checks. Fan flow, duct outlet flow, and effective fresh-air flow are different quantities.

Dilution airflow

Steady well-mixed concentration screen:

\displaystyle C=C_{in}+\frac{G}{Q_{fresh}}

Required fresh airflow for a target concentration:

\displaystyle Q_{req}=\frac{G}{C_{lim}-C_{in}}

where $C_{lim}>C_{in}$ .

If the source is expressed as a mass emission rate $\dot{m}_g$ :

\dot{m}_g=\rho_g G

and:

\displaystyle G=\frac{\dot{m}_g}{\rho_g}

This calculation assumes mixing, steady source rate, known contaminant, and representative fresh-air delivery. It is not valid for stratified gases, explosive atmospheres, fires, transient releases, poorly placed sensors, or recirculated return air unless those effects are explicitly handled.

Clearance after a transient release

For a well-mixed volume $V$ with constant fresh airflow $Q$ and no continuing source, concentration decays as:

C(t)=C_0 e^{-Qt/V}

Time to reduce from $C_0$ to $C_{target}$ :

\displaystyle t=\frac{V}{Q}\ln\left(\frac{C_0}{C_{target}}\right)

With a continuing source $G$ :

C(t)=C_{ss}+(C_0-C_{ss})e^{-Qt/V}

where:

\displaystyle C_{ss}=C_{in}+\frac{G}{Q}

Post-blast or abnormal-condition re-entry should use measured gas evidence, airflow state, affected headings, and the approved re-entry procedure. A calculated decay time alone is not release evidence.

Heat load and temperature rise

Sensible temperature rise:

\displaystyle \Delta T=\frac{\dot{Q}_h}{\rho Q c_p}

Required airflow for a target sensible temperature rise:

\displaystyle Q_{req,h}=\frac{\dot{Q}_h}{\rho c_p \Delta T_{allow}}

Total sensible heat-load screen:

\dot{Q}_h=\sum_i P_i(1-\eta_i)+\dot{Q}_{geo}+\dot{Q}_{water}+\dot{Q}_{rock}+\dot{Q}_{people}+\dot{Q}_{other}

where the terms represent equipment losses, geothermal heat, hot water inflow, rock heat, metabolic heat, and other sources.

Heat-stress risk is not determined by dry-bulb temperature alone. Humidity, radiant heat, air speed, clothing, workload, acclimatization, exposure duration, and work-rest controls all matter.

Equipment-based airflow screen

Some sites define equipment airflow factors. If a site factor $k_i$ is assigned to equipment item $i$ :

Q_{fleet}=\sum_i k_i P_i U_i

where $P_i$ is rated or effective equipment power and $U_i$ is an operating or utilization factor.

The governing required quantity for a branch may be:

Q_{required}=\max(Q_{plan},Q_{dilution},Q_{heat},Q_{fleet},Q_{emergency})

Use only site-approved factors and limits. Do not import generic diesel, battery, or equipment factors without checking the jurisdiction, engine technology, aftertreatment, duty cycle, contaminant basis, heat load, and mine ventilation plan.

Venturi airflow measurement

For a simplified incompressible Venturi station:

\displaystyle Q=C_d A_t\sqrt{\frac{2\Delta p}{\rho(1-\beta^4)}}

where $C_d$ is discharge coefficient, $A_t$ is throat area, $\Delta p$ is differential pressure, $\rho$ is air density, and $\beta=d_t/d_1$ is diameter or equivalent-area ratio.

A fixed station should be checked against a traverse or other independent method. Dust buildup, tubing leaks, poor straight length, density assumptions, water in pressure lines, calibration drift, and electromagnetic interference can bias the reading.

Measurement uncertainty and guard margin

For independent relative standard uncertainties:

u_r=\sqrt{u_1^2+u_2^2+\cdots+u_n^2}

Absolute standard uncertainty in airflow:

u_Q=u_r Q

One-sided guard margin against a minimum required quantity:

M=Q_{meas}-Q_{min}-k u_Q

where $k$ is a coverage or guard factor chosen by the acceptance rule.

If $M<0$ , the measurement does not robustly demonstrate compliance. The engineering response can be to increase airflow, improve the measurement, repeat the survey, or restrict the operating state.

Reliability and availability

Availability for a repairable ventilation function:

\displaystyle A=\frac{MTBF}{MTBF+MTTR}

Series availability for functions that must all work:

A_{series}=\prod_i A_i

Parallel availability for redundant alternatives where any one can satisfy the function:

A_{parallel}=1-\prod_i(1-A_i)

Use these formulas only after defining the mission. A standby fan, door-status sensor, gas trip, communication link, and power supply do not have the same consequence if they fail in production, re-entry, emergency, or unoccupied mode.

Interlock and protection checks

A ventilation interlock should be checked as a function:

\text{hazard detected}\rightarrow \text{safe action}\rightarrow \text{confirmed state}

A simplified proof-test coverage screen is:

\displaystyle C_{test}=\frac{n_{detected}}{n_{credible}}

where $n_{credible}$ is the number of credible failure modes included in the test plan and $n_{detected}$ is the number that the proof test would reveal.

Electrical protective checks should confirm that circuit breakers, relays, starters, variable-speed drives, cable ratings, restart logic, and overcurrent protection support the ventilation safety function. A fan that passes an airflow test can still fail release if it trips on restart, loses communication, or cannot be safely isolated and restored.

Risk-priority screen

Risk priority number:

RPN=SOD

where $S$ is severity, $O$ is occurrence, and $D$ is detection ranking.

Use $RPN$ as a structured discussion aid, not as a substitute for mandatory controls. High-severity ventilation hazards often require stop-work, withdrawal, interlock repair, or competent signoff even when the numerical ranking appears improved.

Worked screening example

A production district is being reviewed before release after regulator adjustment. The survey gives:

Quantity	Value
Fan-station airflow	$Q_f=108\ \text{m}^3/\text{s}$
Fan pressure rise	$\Delta p_f=1.55\ \text{kPa}$
Electrical input power	$P_{in}=235\ \text{kW}$
Useful branch airflow sum	$Q_u=82\ \text{m}^3/\text{s}$
Controlled bypass and leakage	$Q_l=18\ \text{m}^3/\text{s}$
Face airflow in one heading	$Q_{face}=18.5\ \text{m}^3/\text{s}$
Recirculation fraction at face	$r=12\%$
Gas source screening rate	$G=0.018\ \text{m}^3/\text{s}$
Concentration limit above intake	$C_{lim}=0.15\%=0.0015$
District sensible heat load	$\dot{Q}_h=1.25\ \text{MW}$
Density and heat capacity	$\rho=1.18\ \text{kg/m}^3$ , $c_p=1010\ \text{J/(kg K)}$

Air power:

P_{air}=1550(108)=167{,}400\ \text{W}=167.4\ \text{kW}

Installed efficiency:

\displaystyle \eta_{installed}=\frac{167.4}{235}=0.712=71.2\%

This is plausible for a release screen, but only if the fan curve, motor current, controls, and branch quantities also pass.

Air-balance residual:

Q_{res}=Q_f-Q_u-Q_l=108-82-18=8\ \text{m}^3/\text{s}

Residual percentage:

\displaystyle \frac{8}{108}(100\%)=7.4\%

If the action limit is $8\%$ , the balance is acceptable but close enough to justify checking the largest leakage paths.

Effective fresh airflow at the face:

Q_{fresh}=18.5(1-0.12)=16.3\ \text{m}^3/\text{s}

Required dilution airflow:

\displaystyle Q_{req}=\frac{0.018}{0.0015}=12.0\ \text{m}^3/\text{s}

The dilution screen passes:

16.3>12.0

Sensible temperature rise for the district:

\displaystyle \Delta T=\frac{1.25\times10^6}{1.18(82)(1010)}=12.8\ \text{K}

This heat screen may control the release if the allowable temperature rise is lower than $12.8\ \text{K}$ or if humidity and workload make heat stress unacceptable. The same district can pass contaminant dilution and fail thermal control.

Now check uncertainty on a branch that measured $Q_{meas}=23.0\ \text{m}^3/\text{s}$ against a minimum $Q_{min}=22.0\ \text{m}^3/\text{s}$ . The relative standard uncertainty is estimated as $u_r=6\%$ , so:

u_Q=0.06(23.0)=1.38\ \text{m}^3/\text{s}

With a one-standard-uncertainty guard factor $k=1$ :

M=23.0-22.0-1(1.38)=-0.38\ \text{m}^3/\text{s}

The branch is numerically above its minimum but not robustly released under the guard rule. The practical decision is to adjust the regulator, improve measurement confidence, or restrict the operating state until there is a measurable margin.

Validation Evidence Package

Before using a ventilation calculation as release evidence, assemble a package that another engineer or mine official can audit. It should connect the formula result to the actual underground state, not only to a spreadsheet cell or survey note.

Include:

the operating mode and occupied locations are explicit;
measured fan flow is reconciled with branch quantities and leakage;
effective fresh-air quantity, not only total fan flow, controls exposure decisions;
contaminant, heat, and equipment airflow bases are separated;
density, pressure, and temperature assumptions match the measurement method;
fan duty is checked against curve, efficiency, motor, protection, and restart limits;
critical sensors, alarms, interlocks, and communication paths are proof-tested;
measurement uncertainty is smaller than the required safety margin;
bypasses, doors, regulators, stoppings, and ducts match the approved ventilation state;
the final release decision cites measured evidence, authority, residual limits, and retest triggers.

Also include the ventilation survey method, instrument calibration status, station locations, density or psychrometric basis, gas-monitor readings, fan curve or drive state, alarm/interlock proof-test evidence, communication status, isolation or restart constraints, deviations from the plan and the person or role authorized to accept the condition. For post-blast, abnormal gas, fire, fan outage or re-entry decisions, the evidence package should explicitly cite the governing procedure and the trigger-action response level.

Common Formula Mistakes

Common mistakes include treating fan flow as face airflow, ignoring recirculated return air, using one density without checking pressure and temperature, adding parallel branch resistances incorrectly, applying fan laws outside the valid operating region, and accepting a branch because it is slightly above a minimum while measurement uncertainty is larger than the margin.

Do not treat a dilution calculation as proof of safe entry when source strength, mixing or gas stratification is uncertain. A post-blast or abnormal-gas clearance time is only a planning estimate until measured gas levels, airflow state, affected headings, sensor status and the approved re-entry procedure agree.

Do not use a fan power or efficiency result as proof of safety. A fan can be electrically healthy and still deliver poor branch quantities because of doors, regulator settings, leakage, recirculation, damaged ducts, blocked stoppings or changed mine geometry. Conversely, a branch can show acceptable flow while a critical alarm, interlock, starter, communication link or restart logic remains unfit for release.

Another serious mistake is separating ventilation calculations from operations. Mining sequence, diesel or battery fleet, blasting, heat load, door condition, dewatering works, electrical trips, and emergency routes can change the required ventilation state. The formulas are useful only when they remain tied to the live mine configuration and validation evidence.

REF

Disciplines