Consider at 500 L rigid tank containing an air-water vapor...

Consider at $500 \mathrm{L}$ rigid tank containing an air-water vapor mixture at $100 \mathrm{kPa}$ and $35^{\circ} \mathrm{C}$ with a $70 \%$ relative humidity. The system is cooled until the water just begins to condense. Determine the final temperature in the tank and the heat transfer for the process.

Key Concepts

Energy Balance in Closed Systems

In a closed system, especially one with constant volume, the first law of thermodynamics applies where the change in internal energy is equal to the heat added to the system, since no work is done. This principle is essential for analyzing heat transfer during a cooling process, where lowering the system’s energy can lead to phase change conditions.

Phase Change and Dew Point

The phase change concept involves the transformation of a substance from one state (e.g., vapor) to another (e.g., liquid). The dew point is the temperature at which water vapor begins to condense into liquid water at a given pressure and humidity. Understanding these saturation conditions is fundamental when analyzing processes where condensation starts to occur.

Utilization of Thermodynamic Property Data

Thermodynamic property data, such as those found in tables or charts for water and air, provide necessary information including saturation pressures, temperatures, and internal energies. These data are crucial for performing accurate calculations in multi-component systems and for determining state changes during processes like cooling and condensation.

Constant Volume Process

A constant volume process occurs when the volume of the system remains fixed throughout the process. In such systems, no work is performed by boundary expansion or compression (since work is pressure times change in volume), so energy exchanges involve only heat transfer and changes in the internal energy of the components in the system.

Relative Humidity

Relative humidity is a measure of the amount of water vapor present in air compared to the maximum amount that air can hold at a specific temperature, expressed as a percentage. It indicates how close the air is to being saturated, which is critical in determining the conditions under which condensation will begin.

Transcript

00:01 Hi, everybody.

00:01 So for this one, we're asked, determine the final temperature in the tank and heat transfer for the process.

00:10 Okay.

00:11 And so since the system is cooled until the water just began to be dense, we have humidity 1 equals humidity 2.

00:21 And your 5 -2 is going to be 100 % humidity.

00:26 So we have your pv2 is going to be 100 % humidity.

00:27 So we have your pv2 is going to equal to your p gas two.

00:33 And since volume is constant and ideal gas, we have pv1, temperature 1 equals pv2 over temperature 2.

00:44 And we're just gonna go pv2 equals pv1 over temperature two divided by temperature one.

00:51 Okay? and so this also is obviously pv gas two as well.

00:59 And so we're gonna do three, 0 .9396 times temperature 2 equals 35 .2 plus 273 .15.

01:14 So we got to convert it into kelvin.

01:18 And so this is going to equal to 0 .01278 t2.

01:25 Okay.

01:26 And so now what we're going to do is assume that t2 equals.

01:32 Equals 30 degrees celsius.

01:35 So from that, we're going to spend on pv2 is going to equal to 0 .01278 times 303 .15.

01:48 And what we're doing is just adding this to this guy here.

01:58 So we're just adding that to this to make it to kelvin, okay? and so we get equals 3 .8742442k pascal's.

02:16 And now we're just going to assume that it is your t2 is 25 degrees celsius.

02:27 And from this, we have pv2 equals 0 .012...

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