## Thermal Systems

Introduction of Thermal Systems

Thermal systems are those that involve the storage and transfer of heat.  Heat stored in a material object is manifested as a higher temperature.  For example, a hot block of metal has more heat stored in it than an equivalent cool block. Heat flows between objects by one of three mechanisms: conduction, convection (or mass transfer), and radiation.  Conductive heat transfer occurs when a temperature difference exists across an object.

An example is the flow of heat that occurs through the wall of a building if the temperature inside is higher (or lower) than the temperatures outside.  Convective heat transfer involves the flow of heat in a liquid or gas, as when a fan blows cool air across a hot object; the air carries away some of the heat of the object.   Radiative heat transfer, like conductive transfer, is caused by a temperature difference between objects, does not require a physical medium for heat flow (i.e., radiative heat can flow through a vacuum).  It is exemplified by the heat transfer from sun to earth, but it is highly nonlinear (it depends on the fourth power of the temperature difference) and will not be discussed here.  Our discussion will also be limited in several other ways listed, and briefly discussed, here.

A list of the fundamental units of interest is listed below. The next tab (system elements) gives a description of the building blocks of these system (thermal resistance, capacitance and fluid flow). This is followed by a description of methods to go from a drawing of a system to a mathematical model of a system in the form of differential equations.  Methods for solving the equation are given elsewhere.  The last section discusses topics relevant to energy storage and dissipation in these systems.

There are two fundamental physical elements that make up thermal systems, thermal resistances and thermal capacitance.  There are also three sources of heat, a power source, a temperature source, and fluid flow.  All five of these are described below, along with the important mathematical relationships used to describe each one.

### A Note on Temperature

In practice temperature when we discuss temperature, we will use degrees Celsius (°C), while SI unit for temperature is to use Kelvins (0°K = -273.15°C).  However, we will generally be interested in temperature differences, not absolute temperatures (much as electrical circuits deal with voltage differences).  Therefore, we will generally take the ambient temperature (which we will label θa) to be our reference temperature, and measure all temperatures relative to this ambient temperature.  We will also assume that the ambient temperature is constant.  Thus, if the ambient temperature is =25°C, and the temperature of interest is θi=32°C, we will say that θi=7° above ambient.   Note: this is consistent with electrical systems in which we assign one voltage to be ground (and assume that it is constant) and assign it the value of zero volts.  We then measure all voltages relative to ground.

### Thermal Energy

Thermal energy refers to several distinct thermodynamic quantities, such as the internal energy of a system; heat or sensible heat, which are defined as types of energy transfer (as is work); or for the characteristic energy of a degree of freedom in a thermal system kT, where T is temperature and k is the Boltzmann constant.

### Relation to Heat and Internal Energy

In thermodynamics, heat is energy in transfer to or from a thermodynamic system, by mechanisms other than thermodynamic work or transfer of matter. Heat refers to a quantity transferred between systems, not to a property of any one system, or ‘contained’ within it. On the other hand, internal energy is a property of a single system. Heat and work depend on the way in which an energy transfer occurred, whereas internal energy is a property of the state of a system and can thus be understood without knowing how the energy got there.

In a statistical mechanical account of an ideal gas, in which the molecules move independently between instantaneous collisions, the internal energy is the sum total of the gas’s independent particles’ kinetic energies, and it is this kinetic motion that is the source and the effect of the transfer of heat across a system’s boundary. For such a gas, the term ‘thermal energy’ is effectively synonymous with ‘internal energy’. In many statistical physics texts, “thermal energy” refers to kT, the product of Boltzmann’s constant and the absolute temperature, also written as kBT. In a material, especially in condensed matter, such as a liquid or a solid, in which the constituent particles, such as molecules or ions, interact strongly with one another, the energies of such interactions the contribute strongly to the internal energy of the body. The term ‘thermal energy’ is also applied to the energy carried by a heat flow, although this can also simply be called heat or quantity of heat.

### Historical Context

In an 1847 lecture titled “On Matter, Living Force, and Heat”, James Prescott Joule characterized various terms that are closely related to thermal energy and heat. He identified the terms latent heat and sensible heat as forms of heat each affecting distinct physical phenomena, namely the potential and kinetic energy of particles, respectively. He described latent energy as the energy of interaction in a given configuration of particles, i.e. a form of potential energy, and the sensible heat as an energy affecting temperature measured by the thermometer due to the thermal energy, which he called the living force.

### Useless Thermal Energy

If the minimum temperature of a system’s environment is Te and the system’s entropy is S, then a part of the system’s internal energy amounting to S. Te cannot be converted into useful work. This is the difference between the internal energy and the Helmholtz free energy.

### Thermal Resistance

Though heat transfer through via conduction and heat transfer via convection occur as a result of very different mechanisms, the resulting mathematical relationship is identical. Therefore, we will cover both mechanisms here.

#### – Notation for Thermal Systems, and Depictions of Thermal Systems

In thermal systems we use θ to represent temperature. The subscript on θ denotes the specific temperature being measured (For example θa might represent the ambient temperature). Thermal resistances always exist between two distinct temperatures.  We use “R” to represent the resistance, and two subscripts to denote the two temperatures. We can use two representations for thermal systems.  In the representation (above) on the left, the thermal resistance between interior and ambient is shown as a shaded rectangle.  The hash marks above and below the box indicate insulators through which no heat can flow. The only path for heat to flow is from θi to θa.

In the representation on the right, temperatures are shown as voltages (with ambient temperature shown as ground since this temperature doesn’t change) Thermal resistances (between two temperatures) are shown as electrical resistance, and heat flow is shown as a current. Each depiction has its own strengths.  The depiction at left does a better job of physically describing the system.  The description at right is easier to understand and analyze if you have experience with electrical circuits. As systems get more complicated, the circuit is often a more compact and useful representation of a thermal system.

### Thermal Capacitance

In addition to thermal resistance, objects can also have thermal capacitance (also called thermal mass).   The thermal capacitance of an object is a measure of how much heat it can store.  If an object has thermal capacitance its temperature will rise as heat flows into the object, and the temperature will lower as heat flows out.  To understand this, envision a rock in the sun. During the day heat goes in to the rock from the sunlight, and the temperature of the rock increases as energy is stored in the rock as an increased temperature. At night energy is released, and the rock cools down. We represent a thermal capacitance in isolation in diagrams (and equations) as shown below (in the drawing at the left the coil represents a power source, see below, and the stippled object is the thermal capacitance). In the thermal analog, one end of the capacitor is always connected to the constant ambient temperature.

$\displaystyle {{C}_{1}}\frac{{d{{\theta }_{1}}}}{{dt}}={{q}_{1}}$

Note: the electrical model will always have one side of the capacitance connected to ground, or ambient.  Also, we could write the equation as $\displaystyle {{C}_{1}}\frac{{d\left( {{{\theta }_{1}}-{{\theta }_{a}}} \right)}}{{dt}}={{q}_{1}}$ but since θa is constant, it can be removed from the derivative.

The thermal capacitance of an object is determined by its mass and specific heat

$\displaystyle C=m.{{c}_{p}}$

Where “C” is the thermal capacitance, “m” is the mass in kilograms, and “cp” is the specific heat in J/(kg.K).  It is always assumed that the capacitor is at a single uniform temperature, though this is obviously a simplification in many cases.

### Power Source (or Heat Source)

A common part of a thermal model is a controlled power source that generates a predetermined amount of power, or heat, in a system.  This power can either be constant or a function of time.  In the electrical analog, the power source is represented by a current source.  An example of a power source is the quantity “qi” in the diagrams for the thermal capacitance, above. In practice a power source is often an electrical heating element comprised of a coil of wire that is heated by a current flowing through it. Therefore, we use a diagram of a coil of wire to represent the power source. An ideal power source generates power that is independent of temperature. Examples of power sources you might be familiar with are electric hair dryers (typically on the order of 1000 W) and space heaters for individual rooms (typically several hundred Watts).

### Temperature Source

Another common source used in thermal systems is a controlled temperature source that maintains a constant temperature. An ideal temperature source maintains a given temperature independent of the amount of power required. A refrigerator is an example of such a source.  Another such source is the ambient surroundings. We will assume that the temperature of the ambient surroundings is constant regardless of the heat flow in or out (we will also take ambient temperature to be our reference temperature, i.e., θa=0).

### Mass Transfer (Fluid Flow)

If fluid with specific heat “cp” J/(kg.K) flows into a system with a flow rate of “G” kg/sec and a temperature of “θin” °C above ambient, and flows out at a temperature of  “θout” °C below ambient then the rate of heat flow into the system is given by

We can cancel the K and °C since a temperature difference (θin – θout) is the same in Kelvin or Celsius.

If you carefully observe this equation, it makes sense intuitively. Heat into a system goes up with mass flow rate into the system (increased mass flow, yields increased heat flow). Heat into a system also goes up with the specific heat of the mass (higher specific heat indicates increased capacity to store heat).  Finally, heat into a system increases with an increased inflow temperature, or a decreased outflow temperature (if the temperature difference between inflow and outflow increases, more heat is being taken from the fluid). Note: the mass flow rate at the input and output must be equal or the mass (and thermal capacitance) of the system would be changing. This is not allowed for the systems being studied.

### Mathematical Modeling

While the previous page (System Elements) introduced the fundamental elements of thermal systems, as well as their mathematical models, no systems were discussed. This page discusses how the system elements can be included in larger systems, and how a system model can be developed.

#### – The Energy Balance

To develop a mathematical model of a thermal system we use the concept of an energy balance. The energy balance equation simply states that at any given location, or node, in a system, the heat into that node is equal to the heat out of the node plus any heat that is stored (heat is stored as increased temperature in thermal capacitances).

Heat in = Heat out + Heat stored

To better understand how this works in practice it is useful to consider several examples.

#### – Examples Involving only Thermal Resistance and Capacitance

##### * Example: Two thermal resistances in series

Consider a situation in which we have an internal temperature, θi, and an ambient temperature, θa with two resistances between them.  An example of such a situation is your body.  There is a (nearly) constant internal temperature, there is a thermal resistance between your core and your skin (at θs), and there is a thermal resistance between the skin and ambient. We will call the resistance between the internal temperature and the skin temperature Ris, and the temperature between skin and ambient Rsa.

1. Draw a thermal model of the system showing all relevant quantities.
2. Draw an electrical equivalent
3. Develop a mathematical model (i.e., an energy balance).
4. Solve for the temperature of the skin if θi, = 37 °C, θa = 9 °C, Ris= 0.75 °C/W; for a patch of skin and Rsa= 2.25 °C/W for that same patch.

Solution:

a) In this case there are no thermal capacitances or heat sources, just two know temperatures (θi, and θa), one unknown temperature (θs), and two resistances ( Ris and Rsa).

b) Temperatures are drawn as voltage sources. Ambient temperature is taken to be zero (i.e., a ground temperature), all other temperatures are measured with respect to this temperature.

c) There is only one unknown temperature (at θs), so we need only one energy balance (and, since there is no capacitance, we do not need the heat stored term).

Heat in = Heat out + Heat stored; (Heat stored=0)

$\displaystyle \frac{{{{\theta }_{i}}-{{\theta }_{s}}}}{{{{R}_{{is}}}}}=\frac{{{{\theta }_{s}}-{{\theta }_{a}}}}{{{{R}_{{sa}}}}}=\frac{{{{\theta }_{s}}}}{{{{R}_{{sa}}}}}$

Note: The first equation included θa, but the second does not, since θa is our reference temperature and is taken to be zero.

$\displaystyle {{\theta }_{s}}={{\theta }_{i}}\frac{{{{R}_{{sa}}}}}{{{{R}_{{is}}}+{{R}_{{sa}}}}}$

Note: You may recognize this result as the voltage divider equation from electrical circuits.

We can now solve numerically (we use 28 °C for the internal temperature since it is 28 °C above ambient (37°-9°=28°):

$\displaystyle {{\theta }_{s}}={{\theta }_{i}}\frac{{{{R}_{{sa}}}}}{{{{R}_{{is}}}+{{R}_{{sa}}}}}=28{}^\circ \frac{{2.25}}{{0.75+2.25}}=28(0.75)=21{}^\circ$

This says that θs is 21 °C above ambient. Since the ambient temperature is 9 °C, the actual skin temperature is 30 °C.

Note: If Rsa is lowered, for example by the wind blowing, the skin gets cooler, and it feels like it is colder.  This is the mechanism responsible for the “wind chill” effect.

* Example: Heating a Building with One Room

Consider a building with a single room. The resistance of the walls between the room and the ambient is Rra, and the thermal capacitance of the room is Cr, the heat into the room is qi, the temperature of the room is θr, and the external temperature is a constant, θa.

1. Draw a thermal model of the system showing all relevant quantities.
2. Draw an electrical equivalent
3. Develop a mathematical model (i.e., a differential equation).

Solution:

a) We draw a thermal capacitance to represent the room (and note its temperature). We also draw a resistance between the capacitance and ambient.

b) To draw the electrical system, we need a circuit with a node for the ambient temperature, and a node for the temperature of the room. Heat (a current source) goes into the room. Energy is stored (as an increased temperature) in the thermal capacitance, and heat flows from the room to ambient through the resistor.

c) We only need to develop a single energy balance equation, and that is for the temperature of the thermal capacitance (since there is only one unknown temperature). The heat into the room is qi, heat leaves the room through a resistor and energy is stored (as increased temperature) in the capacitor.

Heat in = Heat out + Heat stored

$\displaystyle {{q}_{i}}=\frac{{{{\theta }_{r}}-{{\theta }_{a}}}}{{{{R}_{{ra}}}}}+C\frac{{d{{\theta }_{r}}}}{{dt}}$

by convention we take the ambient temperature to be zero, so we end up with a first order differential equation for this system.

$\displaystyle {{q}_{i}}=\frac{{{{\theta }_{r}}}}{{{{R}_{{ra}}}}}+C\frac{{d{{\theta }_{r}}}}{{dt}}$

* Example: Heating a Building with One Room, but with Variable External Temperature

Consider the room from the previous example. Repeat parts a, b, and c if the temperature outside is no longer constant but varies. Call the external temperature θe(t) (this will be the temperature relative to the ambient temperature). We will also change the name of the resistance of the walls to Rre to denote the fact that the external temperature is no longer the ambient temperature.

Solution:

The solution is much like that for the previous example. Exceptions are noted below.

a) The image is as before with the external temperature replaced by θe(t).

b) To draw the electrical system, we need a circuit with a node for the external temperature and a node for the temperature of the room. Though perhaps not obvious at first, we still need a node for the ambient temperature since all of our temperatures are measured relative to this, and our capacitors must always have one node connected to this reference temperature. Heat flows from the room to the external temperature through the resistor.

c) We still only need to develop a single energy balance equation, and that is for the temperature of the thermal capacitance (since there is only one unknown temperature). The heat into the room is qi, heat leaves the room through a resistor and energy is stored (as increased temperature) in the capacitor.

Heat in = Heat out + Heat stored

$\displaystyle {{q}_{i}}=\frac{{{{\theta }_{r}}-{{\theta }_{e}}}}{{{{R}_{{ra}}}}}+C\frac{{d{{\theta }_{r}}}}{{dt}}$

(the ambient temperature is taken to be zero in this equation). In this case we end up with a system with two inputs (qi and θe).

* Example: Heating a Building with Two Rooms

Consider a building that consists of two adjacent rooms, labeled 1 and 2.  The resistance of the walls room 1 and ambient is R1a, between room 2 and ambient is R2a and between room 1 and room 2 is R12. The capacitance of rooms 1 and 2 are C1 and C2, with temperatures θ1 and θ2, respectively. A heater in room 1 generates a heat qin. The temperaturexternal temperature is a constant, θa.

a) Draw a thermal model of the system showing all relevant quantities.

b) Draw an electrical equivalent

c) Develop a mathematical model (i.e., a differential equation).

In this case there are two unknown temperatures, θ1 and θ2, so we need two energy balance equations.  In both cases we will take θa to be zero, so it will not arise in the equations.

The two first order energy balance equations (for room 1 and room 2) could be combined into a single second order differential equation and solved. Details about developing the second order equation are here.

#### – Examples Involving Fluid Flow

So far, we have not considered fluid flow in any of the examples; let us do so now.

* Example: Cooling a Block of Metal in a Tank with Fluid Flow.

Consider a block of metal (capacitance=Cm, temperature=θm). It is placed in a well-mixed tank (at termperature θt, with capacitance Ct). Fluid flows into the tank at temperature θin with mass flow rate Gin, and specific heat cp. The fluid flows out at the same rate. There is a thermal resistance to between the metal block and the fluid of the tank, Rmt, and between the tank and the ambient Rta. Write an energy balance for this system.

Note: the resistance between the tank and the metal block, Rmt, is not explicitly shown.

Solution:

Since there are two unknown temperatures, we need two energy balance equations.

Aside: Modeling a Fluid Flow with and Electrical Analog

To model this system with an electrical analog, we can represent the fluid flow as a voltage source at θin, with a resistance equal to 1/(Gin·cp). If you sum currents at the nodes θt and θm you can show that this circuit is equivalent to the thermal system above.

To model this system with an electrical analog, we can represent the fluid flow as a voltage source at θin, with a resistance equal to 1/(Gin·cp). If you sum currents at the nodes θt and θm you can show that this circuit is equivalent to the thermal system above.

#### – Solving the Model

Thus far we have only developed the differential equations that represent a system.  To solve the system, the model must be put into a more useful mathematical representation such as transfer function or state space. Details about developing the mathematical representation are here.

### Basic Concept of Cooling Systems

Thermal energy (heat) flows naturally from high temperatures to low temperatures, and moving heat from low temperatures to higher temperatures requires work. There are a number of ways to accomplish this, but the most common method is the vapor-compression refrigeration cycle. The vapor-compression cycle is used in liquid-chilling equipment, direct expansion (DX) cooling systems, refrigeration equipment of all types, and in heat pumps. Insulation is an important component in many of these applications to either increase efficiency, control condensation, or in some cases, to control noise.

The commercial development of the vapor-compression refrigeration cycle is a fairly recent event. The first practical system was patented by James Harrison (a Scottish-born Australian) in 1856. By the 1890s, mechanical refrigeration was being utilized in breweries and meat-packing plants. In 1902, Willis Carrier (known as the “Father of Air Conditioning”) developed a system to control temperature and humidity in the Sackett-Wilhelms printing and publishing plant in Brooklyn, New York. The subsequent adoption of refrigeration and air conditioning was a significant contributor to our current standard of living and to the growth and development of our country. Refrigerated rail cars and trucks enabled the shipment of foods over long distances. Air conditioning greatly increased worker comfort and productivity during summer months, particularly in Southern cities. The importance of this technology was recognized by the National Academy of Engineering when they cited the development of air conditioning and refrigeration as one of the top 20 engineering achievements of the twentieth century.

The basic vapor-compression refrigeration cycle is used in a wide variety of systems and equipment. This ranges from small home refrigerators to large and complex commercial refrigeration installations totaling thousands of tons of cooling capacity. Numerous home appliances utilize this technology, including refrigerators, freezers, air conditioners (home and automobile), dehumidifiers, and heat-pump water heaters. Larger-scale applications include cold-storage warehouses, food- and beverage-processing plants of all types, ice making, transportation systems (truck, rail, and marine), chemical and petrochemical processing, and natural-gas processing.

The basic vapor-compression cycle requires 4 components: a compressor, condenser, expansion valve, and evaporator. The compressor (usually powered by an electric motor providing the work, W) compresses the refrigerant to a high pressure and high temperature. The refrigerant then flows to the condenser, which is a heat exchanger where heat (QH) is removed (by air or water cooling) and the refrigerant is condensed to a liquid. The refrigerant then passes through the expansion valve where the refrigerant expands to a low pressure and a low temperature. The cold refrigerant then flows to another heat exchanger called the evaporator, where it absorbs heat (QL) and boils back into a vapor on its way back to the compressor. During its circuit around the system, the refrigerant picks up heat at the evaporator and rejects that heat (along with the energy added by the compressor) at the condenser.

Note that where cooling is desired (for refrigeration or air conditioning) the evaporator is located either within the cooled space or in duct work/piping systems serving that space. When heating is desired, the situation is reversed and the condenser is located in the heated space or in duct work/piping serving that space. In the latter situation, the device is called a heat pump or reverse-cycle refrigeration system. Many systems utilize a reversing value that can reverse the direction of refrigerant flow and allow the system to supply either heating or cooling to the same space.

The working fluid in this cycle is called the refrigerant. A wide variety of refrigerants are used in vapor-compression systems. Refrigerant selection depends on the application and involves many competing requirements. A primary consideration is the fluid’s thermodynamic properties (boiling point, latent heat of vaporization, specific heat, specific volume). Other important considerations include chemical stability, safety, environmental consequences, cost, availability, compatibility with compressor lubricants and equipment materials, and current regulations. Early industrial systems used many different working fluids–mainly ammonia, carbon dioxide, sulfur dioxide, or methyl chloride. In the 1930s, Freon® was developed by the DuPont Corporation specifically for use as a refrigerant and became widely used under the designations R-11, R-12, and R-22. These chlorofluorocarbon (CFC) and hydrochlorofluorocarbon (HCFC) refrigerants have been largely phased out due to concerns over atmospheric ozone depletion and global-warming potential. Alternative replacement refrigerants are now commonly used (R-134a, R-410A), and new refrigerants are under development.

The vapor-compression cycle uses mechanical energy (at the compressor shaft) to move thermal energy from a low temperature to a higher temperature. The efficiency of any conversion process is expressed as the ratio of the output (the energy sought) to the input (the energy that costs). For the vapor-compression cycle, the efficiency is often expressed as the coefficient of performance (COP). For a cooling system, the objective (energy sought) is the energy absorbed at the evaporator (designated as QL in Figure 1). The energy that costs is the work done at the compressor (W in Figure 1). The coefficient of performance is therefore QL/W (expressed in consistent units). For a heat pump, the energy sought is the heat rejected at the condenser (QH), so the coefficient of performance is QH/W. The coefficient of performance for modern cooling systems is usually greater than 1.0. In other words, the amount of heat transferred to the evaporator is greater than the amount of work required. Large water-cooled centrifugal chillers used for commercial HVAC systems can have full-load COPs of around 6.0. In other words, the energy absorbed at the evaporator is 6 times the amount of mechanical work required. Expressed as a percentage, this is a conversion efficiency of 600%.

There are a number of other terms used to express the efficiency of cooling systems. For smaller unitary equipment, efficiency is often expressed as the energy efficiency ratio (EER). Units on the EER are BTUH/watt. For example, a 3-ton unit (36,000 BTUH) that draws 3 kW (3,000 watts) would have an EER of 36,000/3,000 or 12.0 BTUH/watt. EER can be converted to COP by dividing by 3.412 BTUH/watt (so an EER of 12.0 equals a COP of 12.0/3.412 or 3.52). A variation on the EER is the seasonal energy efficiency ratio (SEER). This term is intended to reflect the fact that at part-load conditions (i.e., cooler weather), the efficiency improves. The SEER measures the average performance of a cooling system over a cooling season.

For larger central systems, performance is sometimes expressed as the total electrical power drawn by the system divided by the cooling effect achieved, or kW/ton. This term is a little confusing since lower kW/ton values have better performance. kW/ton can be converted to COP by inverting and multiplying by 3.517 (i.e., COP = 3.517/(kW/ton).

The use of insulation on vapor-compression cooling systems will vary greatly depending on the application. Small installations using packaged equipment will require little or no insulation. Large custom-built industrial refrigeration applications (where loads are remote from mechanical rooms) may require miles of piping requiring insulation.

As a general rule, the suction lines (which transport cold vapor back to the compressor) require insulation to control condensation and/or ice formation and to limit parasitic heat gains. Any heat gain to these suction lines will increase the amount of work required from the compressor. Hot gas lines on cooling systems generally do not require insulation since the object here is to reject heat to the environment. The obvious exception is where the hot gas lines are accessible by personnel and are warm enough for burns. Another exception would be where hot gas lines run through conditioned space where heat loss would add to the cooling load. Hot liquid lines typically do not require insulation since again the objective here is to reject heat to the environment.

Source:
Websites: 1. Wikipedia 2. Swarthmore 3. Insulation
Books and Articles: 1. Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics Press, New York. / 2. Born, M. (1949). Natural Philosophy of Cause and Chance, Oxford University Press, London, p. 31. / 3. Robert F. Speyer (2012). Thermal Analysis of Materials. Materials Engineering. Marcel Dekker, Inc.

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