Heat Transfer and the Art of Warmth (98.4% Geeky, 1.6% Useful)Dec 11 '06 (Updated Dec 12 '06) Write an essay on this topic.
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The Bottom Line Stay warm. All the cool kids are doing it.
Most people don't like to be cold. Whether it's putting on a pair of gloves before playing in the snow or letting the water warm up before jumping in the shower, we go out of our way to stay warm and cozy. There is a definite science behind staying warm, and it is worth examining. As a disclaimer, I should state right up front that you'll probably find this a terrible waste of time if you're not a science geek. I think this is neat but don't expect to gain great insight into how to pick winter clothes. It's much more theoretical than prescriptive. The Basics What does it really mean to feel cold? Basically, cold is a feeling you get when your skin temperature drops below your point of comfort. This is important, because it isn't directly linked to the outside temperature. What matters is how that temperature affects your skin temperature, which is also a function of several other factors. Heat Energy is an atomic property. An increase in heat energy is really just an increase in atomic and subatomic movement. For example, molecules in hot air will travel more quickly, and electrons in a solid body will rotate around their nuclei more quickly when the temperature is higher. Temperature is what we use to measure this heat energy, but total heat energy can be different in two different bodies even if they share the same temperature. A kilogram of water at 100F will have less heat energy than a kilogram of iron filings at 100F. This is because the iron has a higher heat capacity, which is a measure of how much heat an amount of a substance has at different temperatures. Also, heat capacity is proportional to mass, so 2 kg of water at 100F will have twice as much heat energy as 1 kg of water at 100F. Heat naturally flows from higher temperature bodies to lower temperature bodies. Even though the kilogram of iron filings has more heat than the water, it won't transfer any of that heat to the water because they have the same temperature. In fact, if the iron were at 99F, it would still have more heat than the 100F water, but it could actually receive even more heat from the water until the temperature of the iron equaled that of the water. Heat Transfer As stated, heat transfer occurs naturally between two bodies of different temperature. If you want to stay warm on a cold day, you have to find a way to stop the heat from being transferred from your body to your surroundings. Obviously, most people are too warm when the air is 98.6F, so we do want some of our body heat to dissipate. The goal is to keep an unusually cold environment from taking enough heat to make us feel uncomfortable. Your skin temperature is determined by the rate at which heat is leaving your body. If you are losing a lot of heat, your skin will be much cooler than your core temperature. If you are losing no heat, your skill will be the exact same temperature as your core. Many things affect the rate at which heat leaves your body, with temperature playing a major but not solo role. Don't believe me? Stick a plastic bag and a spoon in the freezer overnight. By morning, they'll both be the same temperature as the freezer, but you'll find that the spoon feels much colder than the plastic bag. This is because the spoon can transfer heat from your body much faster. The reason why is discussed later. Heat transfer occurs at a rate that is proportional to the temperature difference (or temperature differential). This is because the temperature differential is the force that drives the transfer of heat, just like voltage is the force that drives the transfer of electrons (electricity) or like pressure is the force that drives the transfer of water in pipes. This driving force is divided by the resistance to change to get the rate of transfer. The more thermal resistance you can build up, the slower heat will be transferred. Just like adding resistors to an electrical circuit drops the amperage or adding a length of small tubing will slow the flow of water, adding materials that resist heat transfer will slow the transfer of heat energy, keeping your warmer. When you enter an environment, your skin temperature will change until you reach a sort of equilibrium. True thermal equilibrium only occurs when all the bodies are the same temperature, but this won't happen as long as you're producing heat (i.e. as long as your body is functioning). Instead, your skin temperature will decrease or increase until it stabilizes, and this balanced temperature is a function of the rate at which heat is being transferred to or from your body. If you step out of your house into cold morning air, the temperature of the skin on your nose will drop until it balances out. Hypothetically, the skin on your nose might be 80F in your home, but might drop to 60F if you stay outside for a while. So it makes sense that you stay warm by slowing heat transfer away from your body and not by keeping your environment at a certain temperature. After all, we can keep our hands warm with a good pair of gloves, even if it is really cold outside. To find out how to slow this heat transfer, you must understand the modes by which heat can be transferred. Conduction Conduction is the method of heat transfer that occurs when two objects are in direct contact with each other. If you were to put your hand on a hot pan, heat would be transferred from the pan to your hand by conduction. Similarly, sticking a spoon in boiling water will cause the end you're holding to heat up even if it is out of the water. Conduction makes sense when you consider what heat is. Imagine a pool table with balls scattered all over. You'd stand at one end and represent the source of heat. If you grab a ball and roll it across the table, it will probably hit another ball or even multiple balls. Keep rolling balls, and you'll find that the balls close to you move the fastest because you keep rolling them, but that balls at the other end are moving also because of the transferred motion, though they'll move more slowly. The same is true of heat. The atoms or electrons are excited near the source, but the increase of movement is transferred throughout the body, though the effect at the end opposite the source receives less of the boost. Convection Convection occurs when you are dealing with at least one fluid that is moving. Although not always taught this way, convection should be thought of as conduction with a bonus element. When you transfer heat from a source to a recipient by conduction, the temperature of the recipient increases while that of the source drops. As the heat is transferred, the temperature differential decreases, slowing the rate of heat transfer. Convection, however, allows one or both bodies to flow. If you poured cold water into a hot pan, the water would quickly heat up and the pan would cool slowly. However, if you poured cold water over a pan, the water would be constantly replaced with new water that hasn't already had its temperature increased by the pan. This maintains a greater temperature differential, and so provides a higher rate of heat transfer. The mechanism for the actual transfer is the same as for conduction, so the maintaining of the higher temperature differential is the only real difference. There are two forms of convection. The first is free convection which results when you have a lot of a fluid but no bulk motion. Say, for example, you add a lot of water to a large pot and put it on the stove. The water at the bottom of the pot will heat up, the extra heat will cause that water's density to drop a little, and the hot water will naturally drift upwards, being replaced by cooler water that was waiting at the top of the pot. This is better than if you put hot water over a cold source, since the cool water would tend to stay at the bottom and no replacement flow could occur. The other form of convection is forced convection. This is what you experience when you use a fan, put your hand under a running faucet, or drive with your hand out the window. In these cases, the fluid has net motion relative to the other body. Forced convection can transfer heat faster than free convection, because you can force the fluid to a high speed, replacing the fluid at a very fast rate. The faster you replace the fluid, the lesser the time the fluid is in contact with the other body, maintaining the higher differential. Convection explains wind chill. Standing in still air, you only experience free convection around your body. Once the wind starts blowing, you get forced convection, which makes it feel colder. Also, this explains goose bumps. Goose bumps are your body's attempt to trap air against your body. If your body can slow the air flowing over you by obstructing it with hair, you'll stay warmer because your body can warm that trapped air and the flowing air can't replace it as easily. Radiation Radiation is the only form of heat transfer that does not require a medium. Radiation is the mode by which heat energy is transferred through vacuums, which is important since space is a vacuum. All of the energy we get from the sun is via radiation. Basically, heat transfer is transmitted though electromagnetic waves. Beams of light, as well as invisible frequencies, help keep the surface of the earth inhabitable. The waves/particles (not getting into that today) interact with the molecules, adding energy directly. Despite the major role it plays in our environment, most heat on earth is transferred by conduction or convection, because these methods are more efficient. Reflective surfaces accept very little heat energy by radiation, while "black bodies" accept 100% of the heat and reflect none. Color is just our perception of waves reflected by a source. White contains all the colors and black is devoid of color. That is why a black shirt feels so hot on a sunny day. The black doesn't reflect as much of the heat it's receiving by radiation. Thermal Conductivity Conductivity is a property that should be understood. This is a property that is different for each material. The higher the thermal conductivity, the better it can conduct heat. This has to do with its atomic structure, which determines how readily it can transmit the heat energy. Metal has a high thermal conductivity, so holding a spoon that has been in the freezer will feel much colder than a plastic bag, since plastics generally have very low thermal conductivities. Insulation When you bundle up to go outside, you are attempting to isolate yourself from the cold environment. To put it another way, you are adding clothing in order to increase your thermal resistance. The colder the weather, the more thermal resistance is needed to maintain your comfort. You will also need additional clothing if it is windy or wet. Air has a very low thermal conductivity. You can test this yourself by noticing how much slower room temperature air cools down a hot object than room temperature water. Alternatively, you could just stick your hand in room temperature water and notice that it feels much colder than the air. Almost all home and personal insulation use this low conductivity to its advantage. Clothing does what your body is attempting to do with goose bumps; that is, trap air against you so it can be warmed up. Without the insulating clothing, air that you contact will drift away and new, cold air will drift in. When you add layers of clothing, the air can't escape. This allows your body to warm the nearby air, creating a sort of bubble around you. This air has a low enough thermal conductivity that the outside free air has a hard time accepting heat from it, thus raising your skin temperature back to a comfortable level. This also explains why you don't wear a coat in the summer. The trapped air can heat to a much warmer temperature because the higher ambient temperature leaves the trapped air disinclined to transfer much heat away. It is interesting to note that fiberglass, which is what makes up most home insulation, actually has a very high thermal conductivity. This would cause a problem, except that the insulation has a very low density. This fluffy design traps a great deal of air, and the final result is a higher resistance even though the main ingredient is thermally conductive. You could double this resistance by making your walls twice as thick and putting two layers. However, if you tried to press two layers into a wall designed for just one, you would double the amount of fiberglass in the area but reduce the amount of air, actually causing your house to be less insulated by adding more insulation! Winter clothes work much the same way. Many of them, though, use a material that is not thermally conductive to further boost total thermal resistance. A variety of natural and synthetic fibers are used to promote heat retention. The common adage for cold weather dressing is to do layers, and this is correct. The more layers you add, the more air can be trapped. There should be a disclaimer, though, about adding too many layers. If your clothing is very tight because of all the layers, you're actually reducing the amount of air you're trapping. If thick fabrics are squeezed to fit inside another layer, they can't trap the air they were intended. High performance insulation, such as might be used in spacecraft, takes advantage of the fact that both conduction and convection require a medium. This insulation reduces the air, creating a sort of vacuum around the object. This way, the only conduction that can occur is through the material that actually holds apart the walls. Only radiation can transfer heat through a vacuum, and this can further reduced by making the surfaces reflective. Moisture and Rain The fastest way to lose the insulative properties of your clothing is to get it wet. The reason is that water has a much higher thermal conductivity than air (about 25 times higher). When you get wet, the water creeps in and fills all the voids that the air was filling. Compared to air, water can wick heat away from your body much faster, hold more of it for its own heat, and transfer it to the surroundings much faster. Put a wet and dry sponge in the fridge and see which feels colder when you pull them out. Again, they'll be the same temperature, showing that feeling cold is not just a function of temperature. If you want to stay warm, keep as much of your insulation dry as possible. This means wearing a waterproof layer on the outside and keeping cracks around your waist and wrists as dry as possible. ------------------------------DO NOT CONTINUE--------------------- Below is the math. I wasn't originally going to put this in, but I have been convinced. Unless you're a math masochist (hmmh... mathochist?), abort now. This is even less interesting than the rest. q ------> Core . Body . . Skin Clothes . . . Air . . . Ambient . o-----/\/\/\/\/\-----o-----/\/\/\/\/\------/\/\/\/\/\--------o . . . . . . R_b . . . . . . . R_c . . . . . R_a If you are familiar with electric circuits, the above diagram will look somewhat familiar. Thermal Energy is transferred much like Electricity. For the heat to transfer from your core to the air, it must overcome resistances ("/\/\/\/\"). The first resistance is due to your body. A person with more fat will have a higher resistance in their body. This value is denoted by R_b. The more insulating your clothing, the higher the next resistance ("R_c") will be. Air in your clothes should be trapped, so all the heat transfer through your clothes should be by conduction. As you add layers and find other ways to trap more air (fluffier clothes, for example), you can cause R_c to increase. The final resistance is R_a, which is a measure of the resistance associated with the outside air trying to take heat from your clothes. Air really close to your clothes will be about the same temp as your clothes, while the air far away will have a temperature equal the atmospheric (or ambient) temperature. In the diagram, each "o" denotes a sort of node, where you might care about the temperature. The first one is your body temperature, which should be equal to 98.6F. The second is your skin temperature, which will determine whether you feel hot or cold. The last one is the outside temperature. At this point, it would be a good idea to explain steady-state. If you pour yourself a cup of coffee and forget it on your counter at home, it will slowly cool until it reaches room temperature. Once the temperature stops changing, you have reached steady-state. Similarly, if you turn on a heater, all the air nearby will start to heat up. This will cause a chain reaction, heating other nearby air. Now, if you just have this heater sitting outside, there's no way that you can provide enough heat to raise the overall outside temperature noticeably. However, as the heater runs, the nearby temperatures will stabilize, with air closest to the heater being warm and air furthest from the heater being cool. This, too, is steady-state. Basically, you reach steady-state when you are in conditions long enough that all the temperatures stabilize. In our case, pretend you step outside on a cold day. Steady-state will occur once your skin temperature has dropped as low as it is going to drop. q is the heat rate, which is the amount of heat energy being transferred per unit time. This could be measured in watts or kilowatts like most home electronics, but in our case, this is more properly measured in calories per second or calories per hour. Earlier I stated that the rate equals the temperature differential divided by the resistance. When you have your resistances in series like this (that is, when the heat travels from one layer to the next in succession), you can just add them up. We can say then, q = (Temp_Core - Temp_Ambient) / ( R_b + R_c + R_a ) Now, before steady-state, the temperature of your body, skin, and clothes are all changing. After steady-state, however, they are all leveled out. Your body is still producing energy to keep your core from cooling down. This heat has to go somewhere, and you know it isn't being stored in the body or clothes because their temperatures aren't increasing any more. You can deduce, then, that the rate at which your body is transferring heat to your skin has to equal the rate at which your skin is transferring heat to your clothes, which has to equal the rate at which your clothes are transferring heat to the air. And from an overall look, this all has to equal the rate that your body is transferring heat to the air. You can look at any piece or the overall "thermal circuit" and say that the heat entering has to equal the heat leaving. This is a major characteristic of steady-state. With this we can say, q = ( Temp_Core - Temp_Skin ) / R_b = ( Temp_Skin - Temp_Ambient ) / ( R_c + R_a ) These three equations for q will show us something interesting. There are others we could write, though. Everything has to balance. Let's look at what happens when we mess with R_c. Remember, R_c represents how much resistance to heat transfer your clothes provide. To change this in real life, we could add layers or replace clothes. In theory, your body would adjust it's production of heat to keep your Temp_Core at 98.6F. In this case, q can be found with the following, q = ( 98.6 - Temp_Ambient) / ( R_b + R_c + R_a ) = ( 98.6 - Temp_Skin ) / R_b = ( Temp_Skin - Temp_Ambient ) / ( R_c + R_a ) To find our skin temperature for reference, let's just rearrange some things and say the following, Temp_Skin = 98.6 - R_b * ( 98.6 - Temp_Ambient ) / ( R_b + R_c + R_a ) Things are starting to get hard to read because equations don't show up well in this sort of format. Just take a look at the previous equation. If you double R_c, the denominator of that second term increases, causing the second term to decrease. This would cause an increase in your skin temperature. If that didn't make sense, trying plugging in some numbers. If you pick Temp_Ambient of 50F and say all of your resistances are 5, you'll get Temp_Skin equal to 82.4F. If you double R_c to 10 and leave everything else the same, you'll get Temp_Skin equal to 86.45F. See? Double your clothes' thermal resistance, increase your warmth. I know this is kinda mathy, but I gave ample warning. How do you double R_c in real life? Just double your clothes (assuming you're not squeezing out air). If you're wearing a t-shirt, put on another. Understand that I made up those resistances completely, just to show that an increase in R_c will cause skin temperature to increase. Try this with different numbers to confirm it for yourself. You might also notice that if the ambient temperature is higher than your core (that is, Temp_Ambient > 98.6), increasing R_c will seem to decrease your skin temperature. This is a mistake, though, because all of this math is assuming that your body is adjusting its heat production to maintain 98.6F. If the outside temperature is greater than 98.6F, your body can't reverse heat production (suck heat in) to compensate! It has to use other means to keep your temperature under control, which I guess I should mention. I'll come back to that. If you just can't wait, scroll down to "Hot Bod" So, this mathematically shows how increasing clothing thermal resistance helps keep your skin warm. To go even deeper into this mess, let's look at how to calculate this resistance. Conduction is basically dependent on two factors: thermal conductivity (discussed previously) and thickness. Understand that the overall thermal conductivity of winter clothes will fall somewhere between the thermal conductivity of the pure material and the thermal conductivity of air. The more air in the clothes, the closer the conductivity will be to air. To get that actual thermal conductivity is rather involved and this is where I draw the line. Let's just say that you draw up another thermal circuit. This new one will have two paths instead of one to represent the resistance through your clothes, and these paths will be in parallel instead of in series. Basically, the heat can choose to go through the air in the clothes or through the material. You can do circuit analysis to find out what the overall result is. I think, though, you could fudge this by just doing a weighted average of the conductivities based on its volumetric ratio. If you have a down coat that's 5% down and 95% air, take the thermal conductivity of air times 0.95 and add it to the thermal conductivity of down multiplied by 0.05. This should get you in the ballpark. Heat transfer through a wall via conduction is calculated the following ways, q = Temp_Diff * k * A / t Where k is the thermal conductivity, A is the cross-sectional area, and t is the thickness. Using this to calculate the total resistance to heat transfer, we can derive the following: R = t / ( k * A ) q = Temp_Diff / R In some texts, you'll also see R calculated without the area term. Using this system, the heat rate would equal the temperature differential times the area divided by Resistance. For what we are doing here, though, it is fine to use a composite Resistance value that represents all material and geometric considerations. So like I stated earlier, heat rate (or heat flux) equals the temperature difference divided by the resistance. For you, this means that you can increase your thermal resistance, R, (and therefore increase your Temp_Skin), by either increasing t or decreasing k. To increase your thickness, simply add more layers up to the point where you aren't actually making the total clothes layer any thicker. To decrease k, find fabrics that have low thermal conductivities and/or trap as much air as possible. The reiterate, air will have a lower thermal conductivity than any fiber, so it is more important to find a material that catches air than to find one that is composed of fabric with a lower thermal conductivity. I'm going to stop now. I'm almost tempted (okay, so I am tempted), to walk through the mathematics for free and forced convection, but since you can't control wind speed or ambient temperature, the only point would be to show you mathematically what I already told you qualitatively: higher air speeds will increase heat rate, and forced convection is more potent than free convection. As I showed earlier, the effect of altering R_c is the same, regardless of what the other resistances are. R_a will be whatever it is, as most of us can't control the weather. R_b will be higher if you increase your fat, but be forewarned that the larger gut you're likely to haul around will increase your surface area. Convection, like Conduction, is proportional to cross-sectional area. Since you're transferring heat outward, this means that the more surface area you have, the more the air will be able to remove heat. The net effect will be that you stay warmer as you gain weight, but the increased surface area is definitely a consideration when looking at the numbers. Hot Bod So, back to what happens if the ambient temperature escalates to temps higher than your core temp. Your body wants to maintain 98.6F even with the hotter outside temp. Basically, it relies on the same mechanism most air conditioners use. Your body secretes water (sweat), and this water evaporates. Evaporation requires a lot of energy, so the water takes extra heat from your skin in order to vaporize. The energy you transfer to your evaporating sweat serves to decrease the overall temperature in your body, because your skin temperature dropping because of the sweat allows your core to transfer away its own heat, keeping your insides from, uh, cooking. Remember how forced convection maintains a higher temperature differential? When the air is cool, this increases the rate at which air wicks heat from your body. When things heat up, higher flows increase the rate at which air adds heat to your body. Wind Chill in reverse, basically.  |
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