In this screencast we're going to use humidity charts to help us model a heating process.
Humidity charts are a graphical display of physical property for an air/water vapor mixture.
For those of you that are using Felberg and Rousseau as a resource we can give a humidity
chart in SI units for the figure 8-41. The physical property data that we'll use for
this problem include specific enthalpy and humid volume, both of these are given to us
on a per kilogram of dry air basis, so it's important to recognize that humidity charts
give us physical property for a humid air system. It's given to us in a basis of per
kilograms of dry air. We'll also look at relative humidity. In this problem we know that there's
100 cubic meters of humid air coming into the heater per minute. It's entering at 20
degrees C and a relative humidity of 60%. We want to calculate the heat that is required
to heat the air to 40 degrees C. That's shown in the flow chart below. On the inlet side
we know the volumetric flow rate, the temperature, and the relative humidity, and we know the
outlet is at 40 degrees C, and we want to find Q. Any time we're trying to calculate a heat
we need an energy balance, so for our open system our energy balance reduces to Q=deltaH.
The humidity charts provide specific enthalpy on a per kilogram of dry air basis, so with
that information we can rewrite our energy balance as Q is equal to the mass flow rate of dry air times
the change in specific enthalpy, which is given to us in a per kilogram of dry air basis.
So we need to calculate two parameters in this case, the mass flow rate of dry air as
well as the change in specific enthalpy, and we can look at the humidity charts to help
us with both. First we'll use the humidity charts to help us find the change in specific
enthalpy, which by definition is the specific enthalpy at the outlet minus that at the inlet.
To find a position on the humidity chart we need at least two pieces of information, and
we know two at the inlet. We know the air going into the heater is 20 degrees C, and
we can find the temperature along the x-axis of the humidity chart. Here it's labeled specifically
dry bulb temperature, which represents the temperature that we're used to thinking about,
the temperature that you can measure with a thermometer. So the inlet there is 20 degrees
C. We also know the relative humidity which can be found on the curve sweeping across
the humidity chart. We know that the relative humidity of the inlet air is 60%, in other
words the air is carrying 60% as much water as it could at saturation. If we follow the
relative humidity curve down the temperature scale to 20 degrees and find the intersection
of the 60% relative humidity curve with 20 degrees C. We know that our inlet air is represented
at this point on the humidity chart. The first step in calculating the specific enthalpy
is finding the enthalpy at saturation, which can be found on the scale in the upper left
region of the humidity chart. We get the enthalpy at saturation by following these diagonal
curves to the enthalpy scale. The value that I am going to use from the enthalpy scale
at the inlet, specifically at saturation, is 43 kilojoules per kilogram of dry air.
Keep in mind that this specifically represents the enthalpy at saturation. We know the inlet
air is not saturated, it's only 60% saturated. So we also need to account for the enthalpy
deviation, which are the curves that sweep more vertically across the humidity charts.
So our inlet condition is somewhere in between these two enthalpy deviation curves of -0.1
and -0.2 kilojoules per kilogram of dry air. For the enthalpy deviation I'll use the mid
point of these two curves, which is -0.15 kilojoules per kilogram of dry air, which
gives us an inlet enthalpy of 42.85 kilojoules per kilogram of dry air. We also need to find
the specific enthalpy at the outlet, but the only thing we know about the outlet is that it's
40 degrees C. Minimally we need two pieces of information to find where we are on the
humidity chart, but if we think about what we know about the process, it can help us
trace a path along the humidity chart to help us arrive at the outlet condition. We know
that there is no water being added or removed from the air, so the absolute humidity, or
the moisture content, which is found on the y-axis, is not changing, we're only heating
up the humid air. So we know that the moisture content has to stay constant, which implies
that although we don't know anything about the relative humidity at the outlet, we do
know that the moisture content stays the same and we can move horizontally across the humidity
chart until our final condition, or our final temperature, which we know is 40. So our outlet
condition is represented by this point on the humidity chart, and if we want to find
the specific enthalpy like we did at the inlet we can trace this diagonal curve up and left
to find the enthalpy at saturation. The enthalpy at saturation that I'll use in this case is
63 kilojoules per kilogram of dry air. Again, the air is not saturated at the outlet, in
fact the relative humidity is going down. As we increase the temperature the capacity
for air to carry water is going up, so the relative humidity goes down. We're approximately
20% relative humidity at the outlet. So again we need to find the enthalpy deviation for
air that is not saturated with water. The two enthalpy deviation curves that are closest
in this case are -0.6 and -0.8. It makes sense that the deviation is larger in this case
because we are further removed from saturation. I am going to use the mid point in this case,
which is -0.7, which gives us a specific enthalpy at the outlet of 62.3 kilojoules per kilogram
of dry air. If we plug these values in and calculate our change in specific enthalpy
from the inlet to the outlet we get a change of 19.45 kilojoules per kilogram of dry air.
Next we need to find the mass flow rate of dry air, which isn't changing in the process.
The mass flow rate of dry air into the heater is the same as the mass flow rate of the dry
air out of the heater. What we do know is the volumetric flow rate into the heater is
100 meters cubed of humid air per minute. A piece of information that's going to help
us on the humidity chart is the humid volume, which gives us the volume of humid air in
meters cubed per kilogram of dry air. If we look at the units of the volumetric flow rate
of the humid volume we can see that we can calculate the mass flow rate of dry air by
dividing the volumetric flow rate of humid air by the humid volume. Since we know the
volumetric flow rate at the inlet it's important that we find the humid volume at the inlet
as well. As the air is heated from the inlet to the outlet both the volumetric flow rate
and the humid volume will change, so we need to relate these two quantities at the same
point in the process. We know the inlet is 20 degrees C and 60% relative humidity, which
is represented by this point on the humidity chart, and we can get the humid volume at
this point by looking at the steeper of the set of diagonal curves, and there's only a
couple of them, so we're somewhere between 0.8 and 0.85 meter cubed per kilogram of dry
air, so we're pretty close to the 0.85 line, I'll use a value of 0.84 for the humid volume.
We can use that value to calculate our mass flow rate of dry air, so we have 100 meters
cubed of humid air per minute, and we're dividing that by our humid volume, which we are going
to use a value of 0.84 meters cubed of humid air per kilogram of dry air. So when we plug
these in we see that the units of humid air cancel out, and that gives us a value of 119
kilograms of dry air per minute. If we go back to the beginning of the problem we can
use these two numbers that we got from the humidity charts to solve our energy balance.
So again Q is equal to the mass flow rate of dry air times the change in specific enthalpy
given to us in a per kilogram of dry air basis. So using our number Q is equal to 119 kilograms
of dry air per minute, times our change in specific enthalpy, which is 19.45 kilojoules
per kilogram of dry air. So the heat required for this process is 2315 kilojoules per minute.