added mathjax, switched to kelvins

# Engineering Science Building

Smart control of HVAC systems using data-driven methods.

Documentation at https://hazrmard.github.io/EngineeringScienceBuilding

This repository is a mirror of https://git.isis.vanderbilt.edu/SmartBuildings/EngineeringScienceBuilding

---

title: Thermodynamics concepts

order: 0

hasequations: true

---

# Thermodynamics concepts

## Heat engine

![https://commons.wikimedia.org/wiki/File:Heat_engine.png](img/0-heat-engine.png)

A system that converts thermal energy into mechanical energy. This is achieved by reducing the temperature of the working medium. The thermal energy of the medium at high temperature in the hot reservoir (`Q_H`) is converted to useful work (`W`) and the rest (`Q_L`) is left with the medium as it reaches a cooler temperature in the cold reservoir.

A system that converts thermal energy into mechanical energy. This is achieved by reducing the temperature of the working medium. The thermal energy of the medium at high temperature in the hot reservoir $Q_H$ is converted to useful work $W$ and the rest $Q_L$ is left with the medium as it reaches a cooler temperature in the cold reservoir.

```

$$

+Q_H -W -Q_L = 0

```

$$

*Note: By convention, all variables indicate energy supplied **to** and work done **on** the system*.

Efficiency is the ratio of useful work done *by* the system to energy provided *to* the system. For a heat engine:

```

efficiency = -W / Q_H

```

$$

\eta = \frac{-W}{Q_H}

$$

An efficiency of 0.25 (1/4) means that one unit of work is done for 4 units of supplied energy.

## Carnot's theorem

The theorem puts the theoretical maximum limit on the efficiency of a *heat engine*. The greater the temperature difference between the hot (`T_H`) and cold (`T_L`) reservoirs, the higher the achievable efficiency. The maximum achievable efficiency is in a Carnot engine:

The theorem puts the theoretical maximum limit on the efficiency of a *heat engine*. The greater the temperature difference between the hot $T_H$ and cold $T_L$ reservoirs, the higher the achievable efficiency. The maximum achievable efficiency is in a Carnot engine:

```

efficiency <= 1 - T_L / T_H

```

$$

\eta <= 1 - \frac{T_L}{T_H}

$$

*Note*: Temperature should be in absolute units like Kelvin.

More information on [the theorem at Wikipedia][2].

A heat pump is a heat engine in reverse. Work is done *on* the system to take energy supplied by the cold reservoir *to* the system and expel it *from* the system to the hot reservoir.

```

$$

+Q_L +W -Q_H = 0

```

$$

More information at [hyperphysics][8].

```

COP = Useful heat removed (or supplied) / Work done by the system

```

COP (cooling) = Q_L / -W

<div>$$

\begin{align*}

COP_{cooling} &= \frac{Q_L}{-W} \\

COP (heating) = -Q_H / -W

```

COP_{heating) &= \frac{-Q_H}{-W}    \\

\end{align*}

$$</div>

The theoretical maximum COP is for a Carnot cycle:

```

COP (max cooling): T_L / (T_H - T_L)

<div>$$

\begin{align*}

COP_{max cooling} &= \frac{T_L}{T_H - T_L}

COP (max heating): T_H / (T_H - T_L)

```

COP_{mac heating) &= \frac{T_H}{T_H - T_L}

\end{align*}

$$</div>

*Note*: Temperature should be in absolute units like Kelvin.

A `COP=4` for cooling means that a heat pump cools the cold reservoir by 4 units of heat energy using 1 unit of work to do so.

A $COP=4$ for cooling means that a heat pump cools the cold reservoir by 4 units of heat energy using 1 unit of work to do so.

## Wet-bulb temperature

The ratio of partial pressure of water vapor to the equilibrium pressure of water vapor.

```

relative humidity = partial pressure / equilibrium pressure

```

$$

\text{relative humidity} = \frac{\text{partial pressure}}{\text{equilibrium pressure}}

$$

A 100% humidity (dewpoint) means that no net evaporation can occur because the air is saturated with water vapor.

order: 1

---

# Industry terms

## HVAC

[*HVAC*][1] stands for Heating, Ventilation, and Air Conditioning.