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Rates of Heat Transfer January 17, 2013 Coach Stephens

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Rate of Heat Transfer This topic is of great importance because of the frequent need to either increase or decrease the rate at which heat flows between two locations.

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Rates of Heat Transfer For instance, those of us who live in colder winter climates are in constant pursuit of methods of keeping our homes warm without spending too much money. Heat escapes from higher temperature homes to the lower temperature outdoors through walls, ceilings, windows and doors. We make efforts to reduce this heat loss by adding better insulation to walls and attics, caulking windows and doors, and buying high efficiency windows and doors.

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Temperature Difference In the graphs below, the slope of the line represents the rate at which the temperature of each individual sample of water is changing. The temperature is changing because of the heat transfer from the hot to the cold water. The hot water is losing energy, so its slope is negative. The cold water is gaining energy, so its slope is positive. The rate at which temperature changes is proportional to the rate at which heat is transferred.

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Temperature Difference The temperature of a sample changes more rapidly if heat is transferred at a high rate and less rapidly if heat is transferred at a low rate. When the two samples reach thermal equilibrium, there is no more heat transfer and the slope is zero. So we can think of the slopes as being a measure of the rate of heat transfer. Over the course of time, the rate of heat transfer is decreasing. Initially heat is being transferred at a high rate as reflected by the steeper slopes. And as time progresses, the slopes of the lines are becoming less steep and more gently sloped.

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Temperature Difference What variable contributes to this decrease in the heat transfer rate over the course of time? Answer: the difference in temperature between the two containers of water. Initially, when the rate of heat transfer is high, the hot water has a temperature of 70°C and the cold water has a temperature of 5°C. The two containers have a 65°C difference in temperature. As the hot water begins to cool and the cold water begins to warm, the difference in their temperatures decrease and the rate of heat transfer decreases. As thermal equilibrium is approached, their temperatures are approaching the same value. With the temperature difference approaching zero, the rate of heat transfer approaches zero. In conclusion, the rate of conductive heat transfer between two locations is affected by the temperature difference between the two locations.

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Materials The first variable that we have identified as affecting the rate of conductive heat transfer is the temperature difference between the two locations. The second variable of importance is the materials involved in the transfer. In the previous discussed scenario, a metal can containing high temperature water was placed within a Styrofoam cup containing low temperature water. The heat was transferred from water through the metal to water. The materials of importance were water, metal and water. What would happen if the heat were transferred from hot water through glass to cold water? What would happen if the heat were transferred from hot water through Styrofoam to cold water? Answer: the rate of heat transfer would be different. Replacing the inner metal can with a glass jar or a Styrofoam cup would change the rate of heat transfer. The rate of heat transfer depends on the material through which heat is transferred.

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Materials The effect of a material upon heat transfer rates is often expressed in terms of a number known as the heat transfer coefficient. Heat transfer coefficients are numerical values that are determined by experiment. The higher that the coefficient is for a particular material, the more rapidly that heat will be transferred through that material. Materials with relatively high heat transfer coefficients are referred to as thermal conductors. Materials with relatively low heat transfer coefficients are referred to as thermal insulators. The table below lists heat transfer coefficients (k) for a variety of materials, in units of W/m/°C.

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Heat Transfer Coefficients Material k k Aluminum 237 Sand0.06 Brass 110 Cellulose0.039 Copper 398 Glass wool0.040 Gold 315 Cotton wool0.029 Cast Iron 55 Sheep's wool0.038 Lead 35.2 Cellulose0.039 Silver 427 Expanded Polystyrene0.03 Zinc 113 Wood0.13 Polyethylene (HDPE) 0.5 Acetone0.16 Polyvinyl chloride (PVC) 0.19 Water0.58 Dense Brick 1.6 Air0.024 Concrete (Low Density) 0.2 Argon0.016 Concrete (High Density) 1.5 Helium0.142 Ice 2.18 Oxygen0.024 Porcelain 1.05 Nitrogen0.024

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Heat Transfer Coefficient As is apparent from the table, heat is generally transferred by conduction at considerably higher rates through solids (colored red) in comparison to liquids (blue) and gases (green). Heat transfer occurs at the highest rates for metals (first eight items in left-hand column) because the mechanism of conduction includes mobile electrons. Several of the solids in the right-hand column have very low heat transfer coefficients and are considered insulators.

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Insulators The structure of these solids is characterized by pockets of trapped air interspersed between fibers of the solid. Since air is a great insulator, the pockets of air interspersed between these solid fibers gives these solids low heat transfer coefficients. One of these solid insulators is expanded polystyrene, the material used in Styrofoam products. Such Styrofoam products are made by blowing an inert gas at high pressure into the polystyrene before being injected into the mold. The gas causes the polystyrene to expand, leaving air filled pockets that contribute to the insulating ability of the finished product. Styrofoam is used in coolers, pop can insulators, thermos jugs, and even foam boards for household insulation.

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Area Another variable that affects the rate of conductive heat transfer is the area through which heat is being transferred. For instance, heat transfer through windows of homes is dependent upon the size of the window. More heat will be lost from a home through a larger window than through a smaller window of the same composition and thickness. More heat will be lost from a home through a larger roof than through a smaller roof with the same insulation characteristics. Each individual particle on the surface of an object is involved in the heat conduction process. An object with a wider area has more surface particles working to conduct heat. As such, the rate of heat transfer is directly proportional to the surface area through which the heat is being conducted.

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Thickness or Distance A final variable that affects the rate of conductive heat transfer is the distance that the heat must be conducted. Heat escaping through a Styrofoam cup will escape more rapidly through a thin-walled cup than through a thick-walled cup. The rate of heat transfer is inversely proportional to the thickness of the cup. A similar statement can be made for heat being conducted through a layer of cellulose insulation in the wall of a home. The thicker that the insulation is, the lower the rate of heat transfer. Those of us who live in colder winter climates know this principle quite well. We are told to dress in layers before going outside. This increases the thickness of the materials through which heat is transferred, as well as trapping pockets of air (with high insulation ability) between the individual layers.

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A Mathematical Equation As is often the case in physics, the mathematical relationship between these variables and the rate of heat transfer can be expressed in the form of an equation. Let's consider the transfer of heat through a glass window from the inside of a home with a temperature of T 1 to the outside of a home with a temperature of T 2. The window has a surface area A and a thickness d. The coefficient of heat transfer of the window glass is k. The equation relating the heat transfer rate to these variables is: Rate = kA(T 1 - T 2 )/d

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A Mathematical Equation The units on the rate of heat transfer are Joule/second, also known as a Watt. This equation is applicable to any situation in which heat is transferred in the same direction across a flat rectangular wall. It applies to conduction through windows, flat walls, slopes roofs (without any curvature), etc. A slightly different equation applies to conduction through curved walls such as the walls of cans, cups, glasses and pipes. We will not discuss that equation here.

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Example Problem Let's calculate the rate of heat transfer on a cold day through a rectangular window that is 1.2 m wide and 1.8 m high, has a thickness of 6.2 mm, a coefficient of heat transfer of 0.27 W/m/°C. The temperature inside the home is 21°C and the temperature outside the home is -4°C. To solve this problem, we will need to know the surface area of the window. Being a rectangle, we can calculate the area as width height. Area = (1.2 m)(1.8 m) = 2.16 m 2.

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Example Continued We will also need to give attention to the unit on thickness (d). It is given in units of cm; we will need to convert to units of meters in order for the units to be consistent with that of k and A. d = 6.2 mm = 0.0062 m Now we are ready to calculate the rate of heat transfer by substitution of known values into the above equation. Rate = (0.27 W/m/°C)(2.16 m 2 )(21°C - -4°C)/(0.0062 m) Rate = 2400 W (rounded from 2352 W)

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Additional Information It is useful to note that the thermal conductivity of a house window is much lower than the thermal conductivity of glass itself. The thermal conductivity of glass is about 0.96 W/m/°C. Glass windows are constructed as double and triple pane windows with a low pressure inert gas layer between the panes. Furthermore, coatings are placed on the windows to improve efficiency. The result is that there are a series of substances through which heat must consecutively pass in order to be transferred out of (or into) the house. Like electrical resistors placed in series, a series of thermal insulators has an additive effect on the overall resistance offered to the flow of heat.electrical resistors placed in series The accumulative effect of the various layers of materials in a window leads to an overall conductivity that is much less than a single pane of uncoated glass.

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