The gas volumetric balance is determined by conserving a non-condensable dry-gas component, not by assuming that total gas volume is constant. First calculate the dry-gas molar flow at the inlet, then use the unchanged dry-gas flow to determine the total outlet molar flow after water vapor has been added. Finally, apply the ideal-gas law at the actual inlet and outlet temperatures and pressures to obtain the corresponding volumetric flow rates.

Core takeaway: In evaporation and humidification pilot plants, water addition changes the total gas moles and composition, while temperature and pressure may also change the gas volume. A dry, non-reacting carrier gas—such as nitrogen or the dry-air fraction—is the tie component used to close the balance.

Why Total Gas Volume Is Not Conserved

Water vapor increases the total gas flow

When liquid water evaporates into a passing gas stream, the molar flow of the dry carrier gas remains essentially unchanged, but the molar flow of water vapor increases.

Therefore,

[ \dot n_{\text{total,out}}

\dot n_{\text{total,in}} ]

when evaporation occurs and no other gas component is removed.

The volumetric flow may increase even further if the outlet gas is warmer or maintained at a lower absolute pressure.

The dry carrier gas is the tie component

The balance is based on a component that does not react, condense, dissolve significantly, or otherwise leave the gas phase.

For an air–water system, this may be treated as dry air. In a controlled laboratory system, nitrogen is often used as the conserved component.

The fundamental balance is:

[ \dot n_{d,\text{in}}=\dot n_{d,\text{out}} ]

where (\dot n_d) is the molar flow rate of dry gas.

Step 1: Determine the Inlet Dry-Gas Flow

Convert inlet volume to total molar flow

For a measured inlet gas volumetric flow rate, use the ideal-gas equation:

[ \dot n_{\text{in}}

\frac{P_{\text{in}}\dot V_{\text{in}}} {R T_{\text{in}}} ]

Use absolute pressure and absolute temperature.

If the inlet gas composition is known, the dry-gas molar flow is:

[ \dot n_{d,\text{in}}

y_{d,\text{in}}\dot n_{\text{in}} ]

where (y_{d,\text{in}}) is the inlet mole fraction of dry gas.

If the inlet flow is already reported on a dry-gas basis, then (\dot n_{\text{in}}) may directly represent (\dot n_d).

Use humidity or water-vapor composition consistently

The water-vapor mole fraction is related to the dry-gas mole ratio by:

[ Y=\frac{\dot n_w}{\dot n_d}

\frac{y_w}{1-y_w} ]

where:

(Y) is the molal humidity, in mol water/mol dry gas;
(y_w) is the mole fraction of water vapor;
(\dot n_w) is the water-vapor molar flow.

The humidity ratio must be defined consistently at both inlet and outlet.

Step 2: Determine Water Evaporation

Apply the water component balance

At steady state, the water balance is:

[ \dot n_{w,\text{out}}

\dot n_{w,\text{in}} + \dot n_{\text{evap}} ]

Using the dry-gas basis:

[ \dot n_{\text{evap}}

\dot n_d\left(Y_{\text{out}}-Y_{\text{in}}\right) ]

This gives the molar rate of water evaporated.

The corresponding mass rate is:

[ \dot m_{\text{evap}}

\dot n_{\text{evap}}M_w ]

where (M_w) is the molecular weight of water.

For a pilot plant with a known liquid-water feed, the liquid feed rate should first be converted to a molar rate:

[ \dot n_{w,\text{liquid}}

\frac{\dot m_{w,\text{liquid}}}{M_w} ]

Only the portion that actually enters the gas phase belongs in (\dot n_{\text{evap}}).

Calculate the outlet total molar flow

Because the dry-gas molar flow is conserved:

[ \dot n_{\text{total,out}}

\dot n_d+\dot n_{w,\text{out}} ]

Alternatively, if the outlet dry-gas mole fraction is known:

[ \dot n_{\text{total,out}}

\frac{\dot n_d}{y_{d,\text{out}}} ]

This is often the most convenient approach when outlet humidity is measured directly.

Step 3: Convert Molar Flow to Volumetric Flow

Calculate the inlet volume

The inlet volumetric flow is:

[ \dot V_{\text{in}}

\frac{\dot n_{\text{total,in}}RT_{\text{in}}} {P_{\text{in}}} ]

Calculate the outlet volume

The outlet volumetric flow is:

[ \dot V_{\text{out}}

\frac{\dot n_{\text{total,out}}RT_{\text{out}}} {P_{\text{out}}} ]

Combining the dry-gas balance with the gas law gives:

[ \dot V_{\text{out}}

\frac{\dot n_d(1+Y_{\text{out}})RT_{\text{out}}} {P_{\text{out}}} ]

Similarly, the inlet flow can be written as:

[ \dot V_{\text{in}}

\frac{\dot n_d(1+Y_{\text{in}})RT_{\text{in}}} {P_{\text{in}}} ]

The volumetric expansion ratio is therefore:

[ \frac{\dot V_{\text{out}}}{\dot V_{\text{in}}}

\frac{1+Y_{\text{out}}}{1+Y_{\text{in}}} \frac{T_{\text{out}}}{T_{\text{in}}} \frac{P_{\text{in}}}{P_{\text{out}}} ]

This expression shows the three contributors to volume change:

Water addition, through (Y_{\text{out}}/Y_{\text{in}});
Temperature change, through (T_{\text{out}}/T_{\text{in}});
Pressure change, through (P_{\text{in}}/P_{\text{out}}).

Applying the Balance to a Humidification Pilot Plant

Countercurrent humidification

In a humidification column, unsaturated gas contacts circulating liquid water. The gas gains water vapor while the dry carrier gas remains approximately constant.

The required material-balance equations are:

[ \dot n_{d,\text{in}}=\dot n_{d,\text{out}} ]

and

[ \dot n_{d} \left(Y_{\text{out}}-Y_{\text{in}}\right)

\dot n_{\text{liquid evaporated}} ]

The outlet volumetric flow is then calculated from the outlet humidity, temperature, and pressure.

Adiabatic operation

In an insulated humidification column, evaporation draws latent heat from the gas and liquid phases. The volumetric balance remains a material balance, but the outlet temperature must be obtained from a separate energy balance.

For adiabatic humidification, the relationship may be expressed as:

[ (\theta-\theta_s)s

(H_s-H)\lambda ]

where (s) is the humid heat, (H) is humidity, (\theta_s) is the adiabatic saturation temperature, and (\lambda) is the latent heat of vaporization.

The humid heat is commonly represented as:

[ s=C_a+\mathscr{H}C_w ]

For an air–water system, a frequently used approximation is:

[ s\approx 1.00+1.9\mathscr{H} \quad \text{kJ/(kg·K)} ]

Once the outlet temperature is determined from the energy balance, it is inserted into the ideal-gas equation for the outlet volumetric flow.

Practical Measurement Procedure

Measure flow at defined conditions

A volumetric flow reading is meaningful only when its temperature, pressure, and moisture basis are known.

Record:

Inlet and outlet temperature;
Inlet and outlet absolute pressure;
Volumetric flow rate;
Gas composition or humidity;
Liquid feed and discharge rates.

Do not compare inlet and outlet volumes directly unless they have been corrected to the same reference conditions.

Correct gas-displacement measurements

If a gas burette or leveling vessel is used, equalize the liquid levels before reading. This makes the gas pressure inside the measuring tube equal to the external pressure.

Allow the apparatus to reach thermal equilibrium before recording volume.

If the gas is collected over a liquid, subtract the liquid’s vapor pressure:

[ P_{\text{dry gas}}

P_{\text{total}}-P_{\text{vapor}} ]

The displacement liquid must also be selected so that it does not significantly dissolve or react with the gas being measured.

Understanding the Trade-offs

Ideal-gas assumptions

The ideal-gas equation is normally adequate for air–water and nitrogen–water pilot-plant conditions at moderate pressure. At high pressure or when strongly nonideal gases are present, a compressibility factor should be included:

[ \dot V

\frac{Z\dot nRT}{P} ]

Humidity measurement errors

Humidity sensors may be affected by temperature gradients, condensation, response time, and poor gas mixing.

A small error in outlet humidity can produce a significant error in calculated water evaporation because the dry-gas flow is multiplied by the humidity difference.

Pressure-drop effects

Packed columns, spray chambers, and humidification towers may have measurable pressure drops.

Using the inlet pressure for the outlet calculation can overestimate or underestimate the outlet volume, depending on the actual operating conditions.

Ignoring liquid carryover

The gas outlet may contain entrained liquid droplets in addition to water vapor. Those droplets do not belong in the gas-phase molar balance and should be removed with a demister or accounted for separately.

Confusing mass and molar humidity

Humidity may be reported as kilograms of water per kilogram of dry gas, moles of water per mole of dry gas, or water-vapor mole fraction.

These quantities are not interchangeable without conversion. The balance must use one consistent definition throughout.

How to Apply This to Your Pilot Plant

Use the following workflow to close the gas volumetric balance:

If your primary focus is outlet gas volume: Measure or calculate outlet humidity, temperature, and absolute pressure, then use (\dot V_{\text{out}}=\dot n_{\text{total,out}}RT_{\text{out}}/P_{\text{out}}).
If your primary focus is evaporation rate: Conserve the dry-gas flow and calculate (\dot n_{\text{evap}}=\dot n_d(Y_{\text{out}}-Y_{\text{in}})).
If your primary focus is humidification performance: Compare inlet and outlet humidity on a dry-gas basis while checking that the dry carrier-gas balance closes.
If your primary focus is experimental validation: Correct all flow measurements to their actual temperature, pressure, and moisture conditions, and separately check water and energy balances.

Once the dry-gas tie component and the actual stream conditions are known, the evaporation or humidification volumetric balance follows directly from component balances and the ideal-gas law.

Summary Table:

Key element	Calculation or principle
Conserved component	Dry, non-reacting gas such as nitrogen or dry air
Dry-gas balance	n_d,in = n_d,out
Water evaporation	n_evap = n_d(Y_out - Y_in)
Total outlet flow	n_total,out = n_d + n_w,out
Outlet gas volume	V_out = n_total,outRT_out/P_out
Volume-change factors	Humidity, temperature, and absolute pressure
Measurement controls	Record flow, temperature, pressure, humidity, and liquid rates
Important checks	Account for pressure drop, condensation, liquid carryover, and non-ideal gas behavior

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