# Heat Transfer for the Mechanical PE Exam

### Conduction

The three modes of heat transfer are (1) Conduction, (2) Convection and (3) Radiation. Following this discussion, this section will delve into the primary application of heat transfer concepts in the HVAC and Refrigeration field, which is Cooling and Heating Load Calculations. Another important application of heat transfer is heat exchangers, which will be discussed in this section and in the Mechanical Systems section. Finally, determining insulation requirements is discussed. Determining insulation requirements is an important and practical skill and can be used for determining the needed insulation for pipes, ducts, walls and roofs.

Conduction is a method of heat transfer through a medium or multiple mediums in physical contact due to a temperature difference. In the HVAC and Refrigeration field, heat transfer due to conduction is most commonly calculated during cooling load calculations for wall and roof heat loads. The formula for calculating heat transfer due to conduction through a material is as follows:

### Thermal Conductivity

The thermal conductivity for various materials can be found in ASHRAE Handbook – Fundamentals. Thermal conductivity is a measure of how well a material conducts and promotes heat transfer. For example, metals are excellent conductors and thus have a high conductivity. Aluminum has a thermal conductivity of 128 Btu/(hr*ft*℉) and iron has a conductivity of ~30 Btu/(hr*ft*℉). Poor conductors include materials like wood (Douglas fir –0.0833 Btu/(hr*ft*℉)) and insulation materials (Cellular Glass - 0.0275 Btu/(hr*ft*℉) ; Glass Fiber – 0.0221Btu/(hr*ft*℉)).

It is important to note that often times, thermal conductivity is given in units of(Btu*in)/(hr*〖ft〗^2*℉),. This value is basically a thermal conductivity value per inch thickness of materials. Insulation, masonry, plastering and wood materials often have thermal conductivity per inch of materials. For example, cellular glass has a unit thermal conductivity of 0.33(Btu*in)/(hr*〖ft〗^2*℉)), which means that for an inch in thickness of cellular glass material the thermal conductivity is 0.33.

Besides thermal conductivity, materials can also be classified by their R-Value of their U-Factors as shown below.

### U-Factor/R Value

U-Factor stands for the overall heat transfer coefficient and it is representative of a material’s ability to conduct heat. Similarly to thermal conductance, a higher U-factor value has a higher ability to conduct and transfer heat. U-factor is related to thermal conductance by the following formula.

This equation assumes that U does not vary based on temperature. For purposes of the exam, this is a safe assumption.

R-Value stands for thermal resistance and it is representative of a material’s ability to resist heat. This is opposite of the U-Factor and thermal conductance which are measures of a materials ability to conduct heat. The relationship between the R-Value, U-Factor and thermal conductance is shown in the following formula.

This equation assumes that R does not vary based on temperature. For purposes of the exam, this is a safe assumption.

R-values are typically used in the HVAC and Refrigeration field to describe building insulation and materials. For example, insulation manufacturers provide product data for their various products and the key value shown on the product data is the R-Value based on different thicknesses.

Notice that the unit R-Value is 5 for 1” of insulation. The corresponding R-values for various inches of thicknesses are found by simply multiplying the thickness in inches by the R-value for 1” of insulation, refer to the below equation.

A must have skill for the aspiring professional engineer is to be able to calculate the overall heat transfer coefficient, U-factor for a wall, roof, duct or pipe. The method in which the overall heat transfer coefficient will be described through this wall example.

It is important to be able to follow the flow of heat from the beginning to the end of this diagram. (1) The first method of heat transfer is convection, warm outdoor air moves across the outer surface of the concrete wall causing the outer surface of the wall to heat up. There would also be radiation loads acting upon the surface of the wall, but for simplicity it is assumed that there are no radiation loads. (2) Next the heat travels from the outer surface of the concrete wall to the inside surface, (3) then to the outer surface of the insulation and through the insulation,(4) then to the outer surface of the gypsum board and through the board. (5) Finally the outer surface of the gypsum board transmits heat both convectively and through radiation to the indoor air.

In order to find the overall heat transfer coefficient, all of the resistances must be summed. It is the opinion of the author, that each method of heat transfer should be converted to its equivalent R-Value in order to keep it simple.

### Convection

Convection is the second mode of heat transfer and is defined as the transfer of heat through the movement of fluids. In the HVAC and Refrigeration field, convective heat transfer can be found in heating and air conditioning systems, whenever a moving fluid passes over a surface at a different temperature.

One of the most common examples of convection is natural convection in a non-mechanically ventilation/air conditioned building. As people enter a building, the lights get turned on and the sun heats the building, the air in the building begins to get warmer. The warm air is less dense than the air around it and begins to rise up and out of the building. The empty space left by the warm air is then replaced by cooler outside air and the cycle continues. This convective heat transfer through the movement of air is called natural convection. It is referred to as natural because it does not rely on a mechanical source like a fan to move the air.

Convective heat transfer has a similar equation to conductive heat transfer, except the U-Factor or R-Value is replaced with the convective heat transfer coefficient. This convective heat transfer coefficient characterizes the moving fluid by taking into account its viscosity, thermal conductance, temperature, velocity and it also characterizes the surface that the fluid is moving upon.

The third and final mode of heat transfer is radiation. This heat transfer mode is very complicated and differs between theory and application in the HVAC and Refrigeration field. In theory, radiation heat gains for a typical building’s window must be calculated with a computer program like Trane Trace 700, Carrier HAP or any other load calculation program. Radiation heat gains are simplified in hand calculated application and it is the opinion of the writer that the simplified equations for radiation are what can be tested on during the PE exam. Thus only the simplified equations will be discussed in this section and the subsequent sections.

Radiation is the mode of heat transfer that requires no matter to transmit heat. All objects above absolute zero are said to radiate or project heat from its surface. For HVAC and Refrigeration the primary heat gain due to radiation is from solar radiation. Heat is radiated from the sun and transmitted to a building either by heating up the outer surface or transmitting through windows and skylights. These specific examples of solar radiation are described further in the Cooling Load Calculations part of this section.

### Cooling Load Calculations - Basics

Cooling load calculations are typically one of the first calculations completed by the HVAC and Refrigeration engineer. These calculations serve as the basis for determining air conditioning equipment sizes. In order to determine the mechanical equipment sizes, the engineer must first determine what heat is being transferred into the building and what heat is being transferred out of the building. The summation of the heat gained and lost by the building will determine the size of the air conditioning equipment.

The various heat gains and losses into a building can be characterized as either external or internal loads. External loads include the conduction and radiation heat loads transferred through roofs, walls, skylights and windows. In addition, outside air can be brought into a building through ventilation requirements or infiltration, which will cause a load upon the system. Internal loads include heat loads from people, both latent and sensible, loads from lighting and miscellaneous equipment like computers, televisions, motors, etc.

The various heat gains can also be organized into sensible and latent heat gains. Sensible heat gains are those characterized by only a change in temperature and no change in state. Latent heat gains are those characterized by moisture gains. It is important to note that in the table below, that ventilation, infiltration, people and miscellaneous equipment both have sensible and latent heat gains. These individual heat gains are discussed thoroughly in the following sections.

THERMAL MASS and TIME LAG FACTOR

When completing load calculations it is important to understand the time lag factor. When the sun shines upon a wall face early in the morning, although the wall does experience a heat load, the amount of heat load experienced IN the building at that time is minimal. This is due to the thermal mass of the wall. Thermal mass is also known as heat capacity and is defined as the ability of a material to absorb heat.

The use of thermal mass is shown in buildings that have high thermal mass walls that absorb heat during the day, store the heat during occupied periods and release the heat during the night when it is cool.

UNCERTAINTY

Calculating heat gains and determining cooling loads has very high uncertainty. This is because of the many assumptions that must be made like occupant loads, occupant, schedules, outdoor weather conditions, equipment schedules and heat gains, etc. The engineer should recognize that the following calculations are not the most accurate ways to calculate cooling load and are only shown to highlight concepts that could be tested on the professional engineering exam. There are multiple methods used to calculate cooling load calculations like the Radiant Time Series, Total Equivalent Time Difference and the CLTD/SCL/CLF methods. The CLTD/SCL/CLF method is shown in this section because it is the most practical method that can be tested without a computer and in a relatively short period of time (4-hours, 6 minutes per problem).

### Cooling Load Calculations - Roof/Wall

The loads from the roofs and walls are conductive loads. Heat from the outdoors is conducted through the roofing or wall materials as it enters the space. If the problem assumes no radiation loads or does not take into account time, then the only load is the conductive load from the temperature difference between the outdoors and indoors, which is as shown below.

However, the heat effect from the roofs and walls is not this simple. The radiation from the sun onto the building and the time it takes for the heat to transmit through the materials must be taken in to account. In order to calculate the total effect of the difference between the indoor and outdoor temperature, the effect of the solar radiation onto the walls and roofs and the time factor due to the heat storage of the roof/wall material, the engineer should use the Cooling Load Temperature Difference or CLTD. These values can be found in the ASHRAE Fundamentals book 1997 edition and older. These tables are organized by latitude, roof or wall type, month and wall facing orientation direction. In addition, the CLTD is organized by the hour of the day. It is not the opinion of the author that you will need to look-up these values in ASHRAE 1997 and that these values will be given to you as part of the problem. It is only important to understand what CLTD is and how to use it when given it in a problem.

It is also important to note that the CLTD is a simplified approach to determining the heat load due to roofs and walls. In actuality the heat load due to the roofs/walls will also be dependent on many other conditions like the indoor conditions and the heat radiated from the inner wall/roof to the indoor space.

### Cooling Load Calculations - Skylight/Window

The heat loads form the skylights and windows can be broken up into (2) types of loads, conductive and radiation loads. The conductive loads for skylights and windows use the same formula as that of the roofs and windows, shown below again.

The radiation loads or solar transmission is calculated by multiplying the area of the window or skylight by the shading coefficient and the solar cooling load factor.

The shading coefficient is the ratio of the specific window or skylight's solar transmission compared to 1/8" clear glass. The shading coefficient is typically specific to the glass manufacturer and can be found in the manufacturer's product data. During the exam, this value along with the solar cooling load factor should be given. The solar cooling load factor is given in the ASHRAE 1997 Fundamentals book and similarly to CLTD it serves as a simplified approach to calculating heat gain. In addition, SCL is organized similarly by skylight/window, orientation, month, latitude and hour.

In lieu of SC, the term Solar Heat Gain Coefficient (SHGC) is being used by window/skylight manufacturers. This term is simply found by dividing the SC by 1.15. A lower SHGC or SC means that the glass lets in less solar gain and a higher SHGC or SC means that the glass allows more solar gain through.

The National Fenestration Rating Council (NFRC) rates glass and certifies the SHGC and U-Factor. Additional values like Visible Transmittance, Air Leakage and Condensation Resistance are also tested and certified.

### Cooling Load Calculations - People

The heat loads from a person depend on the activity level of the person. ASHRAE has tabulated heat loads both sensible and latent heat gains from people based on their activity levels, refer to ASHRAE Fundamentals. The loads from people can be calculated using these heat gain values, the number of people and the cooling load factor, as shown in the equations below. The cooling load factor takes into account the time lag factor and if it is not given it should be assumed to be 1.0.

R-Value stands for thermal resistance and it is representative of a material’s ability to resist heat. This is opposite of the U-Factor and thermal conductance which are measures of a materials ability to conduct heat. The relationship between the R-Value, U-Factor and thermal conductance is shown in the following formula.

### Cooling Load Calculations - Lighting

The heat load from lighting in a building is found by summing up the number of lights of each type and wattage, then converting the watts to Btu/hr, multiplying this number by the usage factor and the special allowance factor, as shown in the below equation.

The wattage of the light is based on the manufacturer reported value for the lamps in the lighting fixture, without taking into account the ballast. The lighting use factor is the ratio of the time the lights will be in use. This factor is typically 1.0 for most applications like offices, classrooms, stores, hospitals, etc. The usage factor may vary for a movie theater or inactive storage space. The special allowance factor takes into account the heat from ballasts. This factor is typically 1.2 for fluorescent lights and 1.0 for incandescent lights due to the lack of ballasts in incandescent lights.

Finally, the space fraction is the fraction of the total heat from the lights that is transmitted to the space. Lights located at the ceiling may have a percentage of its heat transmitted into the plenum and not into the space. This means that the air conditioning system, if the return is ducted, will not see the percentage of the heat that is transmitted to the plenum. If the plenum is used as a return, then the air conditioning will see the total heat from the lighting. For example, the space fraction for a hung fluorescent light (non-ceiling) will be 1.0, because the light is completely into the space. On the other hand a ceiling recessed light could have a space fraction of 0.5, meaning that 50% of its heat is transmitted to the plenum and the other 50% is transmitted to the space.

### Cooling Load Calculations - Miscellaneous Equipment

The heat gains from miscellaneous equipment can be found by the following equations.

The first equation is used for motors, where P is equal to the nominal horsepower of the motor. Dividing the horsepower of the motor by the efficiency of the motor allows the heat gains due to the motor and the heat gains due to the inefficiency of the motor to be taken into account. If the motor is used continuously then the usage factor will be 1.0. Otherwise the usage factor will be the fraction of the time that it is used divided by the total time the space is occupied. The load factor of the motor takes into account the fact that motors rarely run at its nominally rated capacity. For example, if a 1 HP motor actually operates at 0.75 HP then the load factor will by 0.75.

The second equation describes heat gain from everyday appliances like microwaves, toasters, ranges, ovens and computers. The input energy is found by researching the manufacturer's product data or by referring to typical values reported in ASHRAE Fundamentals. ASHRAE Fundamentals also has typical usage factors and radiated heat fractions for typical equipment. Also shown in ASHRAE Fundamentals are the sensible heat gains for typical pieces of equipment, which bypasses the formula below.

### Cooling Load Calculations - Infiltration

Infiltration is described as outside air that leaks into a building structure. These leaks could be through the building construction or through entry doors. Infiltration heat gains are found by the following equations. These equations are discussed more in the Psychrometrics Section.

The first equation is the total heat gains using enthalpy. In this equation, the volumetric flow rate of the infiltration or ventilation air must be known. This value is converted and multiplied by the difference in enthalpy between the outdoor air conditions and the indoor air conditions.

The following two equation split the total heat gain into the sensible and latent heat loads.

Sensible Heat Gains are calculated by multiplying the CFM of the infiltrated air by the difference in the temperatures of the indoor and outdoor air.

Latent Heat Gains are calculated by multiplying the CFM of infiltrated air by the difference in the humidity ratio of the indoor air and the outdoor air.

It is important to note that these loads are not seen directly by the cooling coil. These are indirect loads that occur in each air conditioned space. Ventilation air is seen directly at the coil and thus this air must be cooled down to the supply air distribution temperature which is much lower than the room condition air.

### Heat Exchangers

Heat exchangers are mechanical devices designed to exchange or transfer heat from a hot fluid to a cold fluid. Heat exchangers are used heavily throughout the HVAC and Refrigeration field, for example a condenser or evaporator in a chiller is simply a heat exchanger. A cooling or heating coil is a heat exchanger that transfers heat from one fluid to another fluid. A chilled water air handling unit transfers heat from the hot air to the chilled water.

There are many different types of heat exchangers that will be briefly discussed, but first it is important to understand the two classifications of heat exchangers, counter-flow and parallel flow heat exchangers. These two classifications describe the relation of the direction of flow between the cold and hot fluid. First the parallel flow heat exchanger, this heat exchanger has both the cold and hot fluids entering at the same end of the heat exchanger. At the beginning of the heat exchanger there is a large difference between the cold and hot fluids and at the end of the heat exchange the difference between cold and hot is reduced, refer to the figure below.

The counter-flow heat exchanger is opposite of the parallel flow heat exchanger. The cold and hot fluids enter at opposite ends. The figure below shows the counter-flow heat exchanger, notice the change in directional arrows.

LOG MEAN TEMPERATURE DIFFERENCE (LMTD)

In heat exchangers that do not have a phase change, heat is transferred from the hot fluid to the cold fluid through the temperature difference between the cold and hot. However, in a heat exchanger as shown in the previous figures, the temperature difference between the cold and hot fluids is not always constant and depends on the location in the heat exchanger. Thus the log mean temperature difference is used. The LMTD describes the logarithmic average temperature difference between the cold and hot fluid through a generic heat exchanger (counter or parallel). LMTD cannot be used for heat exchangers with a phase change like a boiler or condenser. The equation for LMTD is shown below.

The LMTD is then used to calculate the total heat exchanged by the heat exchanger through the following equation. The U-value is the heat transfer coefficient of the heat exchanger which is given by the heat exchanger manufacturer. The Area value is the total area where heat exchange occurs, which is given by the heat exchanger manufacturer.

HEAT BALANCE

Often times in the HVAC and Refrigeration field, a heat balance is conducted on a heat exchanger to show that a balance of heat loss from the hot fluid is shown as a heat gain to the cold fluid. For example, cooling coils are heat exchangers that transfer heat from air to water. The heat balance governing this heat transfer would be as shown below.

If there is a phase change, then the following equation can be used. Heat balances are discussed further in the Refrigeration Section, Mechanical Systems and the Psychrometrics section. Basically, heat balances are integral to the HVAC and Refrigeration field, but luckily the equations governing a heat balance are fairly simple.