Psychrometrics Part 3 - Equations

Introduction

This blog post is a continuation of Part 2 in this series. Part 2 examined the different psychrometric processes and how they appear on a psyrchrometric chart. This post formalizes those visual representations into equations, making it easier to assess air states and processes with a spreadsheet.

From Part 1 of this series it was shown that with any two pieces of information, we can gather the remaining information about the psychrometric state of the air. While equations can fully replace the process of using a psychrometric chart to obtain this information, only some of those equations will be presented here, since some of them are excessively complicated. For all calculations necessary in psychrometric analysis, you can check out Chapter 1 of the ASHRAE Fundamentals Handbook.

Altitude Adjustment

Throughout the equations below, the values of $C_s$, $C_l$, and $C_t$ will appear in various spots. These are coefficients used for sensible, latent, and total cooling, and are dependent on air density, which varies with altitude.

Factor	Name	Sea Level Value
$C_s$	Sensible heat factor	1.08 $\frac{Btu/h}{^oF\cdot cfm}$
$C_l$	Latent heat factor	4840 $\frac{Btu/h}{cfm}$
$C_t$	Total heat factor	4.5 $\frac{Btu/h}{(Btu/lb)\cdot cfm}$

The sensible heat factor is equal to 1.08 at 0% humidity (dry air), and is sometimes increased to 1.10 for more accurate calculations, since 1.10 is the sea level value for air at 50% humidity.

These values each get adjusted for altitude by multiplying the numbers by $\frac{p}{p_o}$, where p is the pressure at the current altitude, and $p_o=14.696$ psi is the atmospheric pressure at sea level. The local pressure is calculated by the equation $p=p_o(1-6.8754\times 10^{-6}Z)^{5.2559}$ where Z is the altitude in feet.

Because of this change in heat factors with a change in altitude, a psychrometric chart is not perfectly accurate when your location is not at the altitude set for the particular chart you’re using. A standard psychrometric chart is set for sea level, but is considered fairly accurate up to a few thousand feet of elevation. There is a separate chart set for 5000 ft of elevation that should be used for locations that are particularly high in altitude.

Calculating Processes

When making calculations about processes, they can be split into the sensible and latent calculations. Adding these together will yield the total change in energy, which can also be encapsulated into one equation.

Sensible heating or cooling is calculated with the equation

$q_{sens}=C_s\times Q\times (t_f-t_i)$

Where $C_s$ is the sensible heat factor, Q is the total airflow [cfm] and $t_f$ and $t_i$ are the final and initial dry bulb temperatures of the air [^oF].

Latent heating or cooling (humidification or dehumidification) are calculated by the equation

$q_{lat}=C_l\times Q\times (W_f-W_i)$

Where $C_l$ is the latent heat factor, Q is still the airflow [cfm] and $W_f$ and $W_i$ are the final and initial humidity ratios of the air $[\frac{lb_{water}}{lb_{dry air}}]$.

Total heating or cooling is the result of adding each of these values together for a process that includes one or both of sensible/latent portions, or it can be calculated with the equation

$q_{total}=C_t\times Q\times (h_f-h_i)$

Where $C_t$ is the total heat factor, Q is the airflow [cfm], and $h_f$ and $h_i$ are the final and initial enthalpy values of the air.

The relationship between enthalpy, dry bulb temperature, and humidity ratio can be easily calculated with the equation

$h=0.24T_{db}+W(1061+0.444T_{db})$

With this equation, and basd on the equations above, making energy calculations about sensible, latent, and total cooling can easily be made. By combining these equations with a psychrometric chart, moving between air states and energy transfer can be performed much more quickly.

The last air process to be looked at is the mixing of two air streams. To calculate the final air state, first the enthalpy and humidity ratios of the new air stream can be calculated, and then the final temperature can be calculated be rearranging the above equation into

$T_{db,mixed}=\frac{h_{mixed}-1061W_{mixed}}{0.24+0.444W_{mixed}}$

The only thing remaining is to determine $h_{mixed}$ and $W_{mixed}$.

Recall from Part 2 of this series that the state of two mixed air streams is found by joining a straight line between the points on a psychrometric chart, and locating the new point on that line based on the relative quantities of airflow between the two streams. Mathematically this works out as a weighted average between the two air streams for both enthalpy and humidity ratio. The only complication here is that the weighted average is based on mass flow rate, whereas air flow is usually measured as a volumetric quantity. The equations for enthalpy and humidity ratio are defined as

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

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

Where $\dot m$ represents a mass flow rate. This can be combined with the relationship $\dot m=\frac{Q}{v}$ where Q is the volumetric flow rate and $v$ is the specific volume [$ft^3/lb$], which is a quantity that can be determined for each air state from the psychrometric chart. The equations then become

$h_{mixed}=\frac{h_1\frac{Q_1}{v_1}+h_2\frac{Q_2}{v_2}}{\frac{Q_1}{v_1}+\frac{Q_2}{v_2}}$

$W_{mixed}=\frac{W_1\frac{Q_1}{v_1}+W_2\frac{Q_2}{v_2}}{\frac{Q_1}{v_1}+\frac{Q_2}{v_2}}$

Since specific volume doesn’t change by a large amount with changing temperature and humidity, these equations can sometimes be approximated by removing the specific volume components, making them $h_{mixed}=\frac{h_1Q_1+h_2Q_2}{Q_1+Q_2}$ $W_{mixed}=\frac{W_1Q_1+W_2Q_2}{Q_1+Q_2}$

However, if the two air states are significantly different in temperature then these approximations may yield answers that are noticeably different than the correct values.

Air Conditioning

The above section is useful when analyzing coil loads and system loads. For example, air mixing calculations are used for mixing return air with ventilation air. A mixed air state is then known, and if there is a desired supply air temperature, then the sensible, latent, and total energy transfer equations can be applied to determine how much energy is required to bring the supply airflow from one state another. This will yield a sensible, latent, and total coil load, but one question remains: Where does the supply flow come from?

The supply airflow is determined based on the system requirements, which ultimately come from space load calculations. Space load calculations are complicated, and software such as HeatWise is used to determine those loads and provide a required airflow, as well as calculate the necessary system loads.

Once space load calculations are complete, the airflow is calculated by rearranging the sensible heat equation to solve for airflow

$Q=\frac{q_{sens}}{C_s(T_r-T_s)}$

Where $T_s$ is the supply temperature and $T_r$ is the room temperature. The above equation is for cooling, so for heating calculations $T_s$ and $T_r$ would need to be reversed.

Since cooling loads require latent cooling as well, but airflow is driven by the sensible load, you can determine the necessary humidity ratio required to meet the latent load by rearranging the latent heat equation

$W_s=W_r-\frac{q_{lat}}{C_lQ}$

Where $W_r$ is the humidity ratio of the room air. If $W_s$ needs to be lower than the saturation humidity ratio at $T_s$, then meeting the latent load can be accomplished by dehumidifying the air and reheating it to the desired supply air temperature. If this is not done, then a higher room humidity may need to be accepted.

Conclusion

Psychrometric processes can be analyzed using both a psychrometric chart and through equations. Using a chart cannot take into account unique altitudes, however it can provide useful information such as specific volume, or taking a dry bulb and wet bulb temperature state and equating it to a humidity ratio and enthalpy, which are much easier to use in calculations.

Calculation relative humidity and wet bulb temperature are much more complicated, but not impossible, and the ASHRAE Fundamentals Handbook (Chapter 1) gives detailed equations on how to calculate these values and more based on known inputs.

The fourth and final part of this series will go through an example of a particular system, step-by-step, to fully analyze the psycrhometric cycle of an air conditioning system.