Question 12.12: Analyzing a Steam-Spray Humidifier Moist air with a temperat...

Known Moist air enters a humidifier at a temperature of 22°C and a wet-bulb temperature of 9°C. The mass flow rate of the dry air is 90 kg/min. Saturated water vapor at 110°C is injected into the mixture at a rate of 52 kg/h.

Find Using the psychrometric chart, determine at the exit the humidity ratio and the temperature, in °C.

Schematic and Given Data:

Engineering Model

1. The control volume shown in the accompanying figure operates at steady state. Changes in kinetic and potential energy can be neglected and \dot{W}_{ cv }=0.

2. There is no heat transfer with the surroundings.

3. The pressure remains constant throughout at 1 bar. Figure A-9 remains valid at this pressure.

4. The moist air streams are regarded as ideal gas mixtures adhering to the Dalton model.

Analysis

a. The humidity ratio at the exit \omega_{2} can be found from mass rate balances on the dry air and water individually. Thus,

\dot{m}_{ a 1}=\dot{m}_{ a 2} (dry air)

\dot{m}_{ v 1}+\dot{m}_{ st }=\dot{m}_{ v 2} (water)

With \dot{m}_{ v 1}=\omega_{1} \dot{m}_{ a } \text {, and } \dot{m}_{ v 2}=\omega_{2} \dot{m}_{ a } \text {, where } \dot{m}_{ a } is the mass flow rate of the air, the second of these becomes

Using the inlet dry-bulb temperature, 22°C, and the inlet wet-bulb temperature, 9°C, the value of the humidity ratio \omega_{1} can be found by inspection of the psychrometric chart, Fig. A-9. The result is \omega_{1}=0.002 kg (vapor)/kg(dry air). This value should be verified as an exercise. Inserting values into the expression for \omega_{2}

b. The temperature at the exit can be determined using an energy rate balance. With assumptions 1 and 2, the steady-state form of the energy rate balance reduces to a special case of Eq. 12.55. Namely,

0=\dot{Q}_{ cv }+\dot{m}_{ a }\left[\underline{\left(h_{ a 1}-h_{ a 2}\right)}+\underline{\omega_{1} h_{ g 1}+\left(\omega_{2}-\omega_{1}\right) h_{ w }-\omega_{2} h_{ g 2}}\right] (12.55)

0=h_{ a 1}-h_{ a 2}+\omega_{1} h_{ g 1}+\left(\omega_{2}-\omega_{1}\right) h_{ g 3}-\omega_{2} h_{ g 2} (a)

In writing this, the specific enthalpies of the water vapor at 1 and 2 are evaluated as the respective saturated vapor values, and h_{ g 3} denotes the enthalpy of the saturated vapor injected into the moist air.

Equation (a) can be rearranged in the following form suitable for use with the psychrometric chart.

1 \left(h_{ a }+\omega h_{ g }\right)_{2}=\left(h_{ a }+\omega h_{ g }\right)_{1}+\left(\omega_{2}-\omega_{1}\right) h_{ g 3} (b)

As shown on the sketch of the psychrometric chart, Fig. E12.12b, the first term on the right of Eq. (b) can be obtained from Fig. A-9 at the inlet state, defined by the intersection of the inlet dry-bulb temperature, 22°C, and the inlet wet-bulb temperature, 9°C; the value is 27.2 kJ/kg(dry air). The second term on the right can be evaluated using the known humidity ratios \omega_{1} \text { and } \omega_{2} \text { and } h_{ g 3} from Table A-2: 2691.5 kJ/kg(vapor). The value of the second term of Eq. (b) is 25.8 kJ/kg (dry air). The state at the exit is then fixed by \omega_{2} \text { and }\left(h_{ a }+\omega h_{ g }\right)_{2}=53 kJ/kg (dry air), calculated from the two values just determined. Finally, the temperature at the exit can be read directly from the chart. The result is T_{2} \approx 23.5^{\circ} C.

Alternative IT Solution

2 The following program allows T_{2} to be determined using IT, where \dot{m}_{ a } is denoted as mdota, \dot{m}_{ st } is denoted as mdotst, w1 and w2 denote \omega_{1} \text { and } \omega_{2}, respectively, and so on.

Using the Solve button, the result is T_{2}=23.4^{\circ} C, which agrees closely with the values obtained above, as expected.

1 A solution of Eq. (b) using data from Tables A-2 and A-22 requires an iterative (trial) procedure. The result is T_{2}=24^{\circ} C, as can be verified.

2 Note the use of special Moist Air functions listed in the Properties menu of IT.

Skills Developed

Ability to…

• apply psychrometric terminology and principles.

• apply mass and energy balances for a spray humidification process in a control volume at steady state.

• retrieve necessary property data using the psychrometric chart.

• apply IT for psychrometric analysis.

Quick Quiz

Using the psychrometric chart, what is the relative humidity at the exit? Ans. ≈63%.