Psychrometrics Part 4 - Full Example

Introduction

If you’ve read the first 3 parts of this blog series you’ll now be familiar with reading a psychrometric chart, how to navigate a chart to analyze various processes, and how to make the necessary calculations using equations.

This post will walk through an example based on space load calculations, and then determining how to properly size a cooling coil to provide the necessary sensible and latent cooling.

Step 1: Space Assessment

In this example, we’ll assume our space has the following load calculation results:

• Sensible cooling: 45.5 MBH
• Latent cooling: 15.6 MBH
• Ventilation: 322 cfm
• Space condition: 75^oF, 50% RH
• Altitude: 1,158 ft
• Design day condition: 91^oF dry bulb, 74^oF wet bulb

We want to supply air in such a way that will address both the sensible and latent cooling needs of the space, as well as properly ventilate the space. We also want to limit our supply temperature to 55^oF - it’s okay to supply air at a warmer temperature, but any colder will result in aggressive drafts and stratification of the air.

Based on the altitude of this location, our $P/P_{o}$ value is:

$p/p_o=(1-6.8754\times 10^{-6}(1158))^{5.2559}=0.959$

This makes our $C_s$, $C_l$, and $C_t$ values:

• $C_s$=1.036
• $C_l$=4,642
• $C_t$=4.316

Step 2: Airflow

So, based on the information above, what airflow is necessary? We need to make sure that our supply airflow will address both sensible and latent loads. The higher our temperature difference between supply air and room air, the lower the necessary airflow. So, if we start with the assumption that we are using 55^oF air, our airflow can then be calculated as:

$Q=\frac{q_{sens}}{C_s\Delta T}=\frac{45,500}{1.036(75-55)}=2,196\space cfm$

We can then check if this airflow at this temperature will satisfy our latent load. We can assume that if there is no dehumidification taking place, that our air will be delivered at very close to the saturation line at 55^oF. In order to find the humidity ratio, W, at this state, we can use our psychrometric chart:

Even though this is a sea level chart, we can assume it will be accurate enough for our example. By plotting our 55^oF point just below the saturation line, and drawing a line to our humidity ratio axis, we get a humidity ratio of W=0.009. Notice how this line passes almost right through the point of 75^oF at 50% RH. This means that taking 75^oF air and cooling it to 55^oF will result in almost purely sensible cooling, not addressing the space latent load. In order to address the space latent load, we need to rearrange the latent cooling equation to solve for $W_{supply}$. By drawing a horizontal line on our chart for the room condition at 75^oF, 50% RH, we can see that our room humidity ratio is at $W_{space}=0.0092$.

$W_{supply}=W_{space}-\frac{q_{lat}}{C_lQ_{supply}}=0.0092-\frac{15,600}{(4,642)(2196)}=0.0077$

So, we now know that we will in fact need to dehumidify the air down to a humidity ratio of 0.0077. Checking our psychrometric chart, this corresponds to a dew point temperature of 50^oF, and then it will need to be reheated to a temperature of 55^oF to be in the acceptable range of supply temperatures.

It’s important to note here that although we started this with the assumption of supplying air at the coolest allowable temperature, we’ve confirmed that it is in fact the ideal supply temperature. If we supplied at a higher temperature, we would need to supply more air, and we would still need to dehumidify the air, which means we would be reheating more air to a higher temperature, wasting energy overall.

Step 3: Analyzing the Air Handling Unit

We now know the following specifications for the air handling unit:

• 2,196 cfm of supply air
• 322 cfm of outdoor air
• Supply air temperature of 55^oF
• Dehumidification down to 50^oF

The question we now need to answer is: how should we size our cooling coil and reheat coil?

The cooling coil will be sized to be able to cool the total mixed air (return air + outdoor air) from the mixed air point down to the dehumidification point of 50^oF at saturation (100% RH). In order to know the mixed air state, we need to use our mixing equations:

$h_{mixed}=\frac{h_1\dot m_1+h_2\dot m_2}{\dot m_1+\dot m_2}$

$W_{mixed}=\frac{W_1\dot m_1+W_2\dot m_2}{\dot m_1+\dot m_2}$

Where state 1 will be the return air and state 2 will be the outdoor air. In order to know the mass flow rates of each stream, we will also need to know the specific volume of each. The enthalpy of each can be determined from the humidity ratio and dry bulb temperature of the respective air streams. Both humidity ratio and specific volume can be determined from the psychrometric chart:

By plotting these points we know that:

• $W_1=0.0092$
• $W_2=0.0142$
• $v_1=13.68$
• $v_2=14.20$

The enthalpy values can now be calculated by our enthalpy equation:

$h_1=0.24T_{db,1}+W_1(1061+0.444T_{db,1})=0.24(75)+0.0092(1061+0.444(75))=28.07\space Btu/lb$

$h_2=0.24T_{db,2}+W_2(1061+0.444T_{db,2})=0.24(91)+0.0142(1061+0.444(91))=37.48\space Btu/lb$

The mass flow rate ($\dot m$) is equal to the volumetric flow rate divided by the specific volume ($Q/v$). Since mixing is measured on a volumetric flow basis, we can determine our relative flow rates as:

• $Q_{oa}=322\space cfm$
• $Q_{ra}=Q_{sa}-Q_{oa}=2,196-322=1,874\space cfm$
• $\dot m_1=Q_{ra}/v_{ra}=\frac{1,874}{13.68}=137\space lb/min$
• $\dot m_2=Q_{oa}/v_{oa}=\frac{322}{14.20}=22.7\space lb/min$

Plugging these values into the mixing equations we get:

$h_{mixed}=\frac{28.07(137)+37.48(22.7)}{137+22.7}=29.41\space Btu/lb$

$W_{mixed}=\frac{0.0092(137)+0.0142(22.7)}{137+22.7}=0.0099$

We can then calculate our dry bulb temperature at this mixed state as

$T_{db}=\frac{h_{mixed}-1061W_{mixed}}{0.24+0.444W_{mixed}}=\frac{29.41-1061(0.0099)}{0.24+0.444(0.0099)}=77.4^oF$

We now have everything we need to calculate the necessary sensible and latent cooling from our cooling coil. Remember, we are cooling the air down to 50^oF before reheating it. So, our sensible cooling from our coil is

$q_{sens}=C_sQ_{supply}\Delta T=1.036(2,196)(77.4-50)=62.34\space MBH$

$q_{lat}=C_lQ_{supply}\Delta W=4,642(2,196)(0.0099-0.0077)=22.43\space MBH$

$q_{total}=q_{sens}+q_{lat}=62.34+22.43=84.77\space MBH$

The final step is to determine our reheat capacity. Since reheating the air is a purely sensible process, we can just use our sensible heat equation to bring the air from 50^oF to 55^oF

$q_{reheat}=C_sQ_{supply}\Delta T=1.036(2,196)(55-50)=11.38\space MBH$

Conclusion

Our final air handling unit specifications are:

• 2,196 cfm supply flow
• 322 cfm outdoor air
• Cooling coil with
  • 62.34 MBH sensible cooling
  • 22.43 MBH latent cooling
• 50^oF leaving temperature off the cooling coil
• 11.38 MBH reheat capacity

Although this may seem like an overly lengthy procedure for determining the final AHU specifications, most of the above calculations can be automated by software or spreadsheets with just a few manual inputs from the user. By referring to your psychrometric chart and providing inputs to your calculation tool, you can run through the above calculations with multiple scenarios in just a few seconds.

Obtaining accurate psycrhometric calculations can be the difference between a comfortable indoor environment versus having condensation raining down from overhead pipes and ductwork.