welcome to our next heat transfer lecture this one is going to focus on the three modes of heat transfer and their rate equations and actually this lecture is specifically about the mode of heat transfer called conduction so the three different modes of heat transfer are shown here we have conduction which is heat transfer within a solid it could also be a stationary fluid but it basically just means that the material the atoms and molecules are moving um but they're kind of bound there's not a bulk of motion they're removing vibrating translating but they're basically staying in place conduction is going to use a rate equation which is the equation that we use to quantify the amount of heat flowing by conduction which is comes from fourier's law and this is something you'll hear repeatedly so fourier's law is the rate equation associated with conduction for convection convection is heat transfer from a surface to a moving fluid so unlike conduction convection has this bulk fluid motion and the rate equation associated with convection is called newton's law of cooling finally for radiation which is typically from a surface to another surface although there are exceptions to that that uses a rate equation called the stefan boltzmann law so today we're going to focus on conduction in this video before we get to that point i want to ask a quick review question so just think to yourself what is the driving force for heat transfer and if you remember from previous lectures the driving force is a temperature difference so whenever one material has a higher temperature than surrounding materials there's going to be a natural inclination for heat to flow from the higher temperature material to the lower temperature material and the greater the difference between the two the more the flow of heat will happen so if there's a bigger temperature difference there will be a bigger driving force and more heat transfer will occur conduction as a reminder it's heat transfer through a solid or a stationary fluid that is caused by random motion of its constituent atoms molecules or electrons so here on the left if we have this solid if the left hand side of the solid has a higher temperature than the right hand side we're going to naturally see a flow of heat going from left to right this particular diagram shows the heat flux which is denoted by q double prime to help you get your head around how conduction works we're going to do this thought experiment so let's just imagine that we have a metal rod which is just kind of suspended and initially it is all it's isothermal it's at a uniform temperature throughout the rod but let's say we introduced a change into our system and we added a heat source something like a bunsen burner on the far left hand side of this rod so what would happen is as you add energy from this bunsen burner into the left hand side of the rod you're going to that energy is going to go into those atoms and molecules and they're going to start vibrating and moving around faster than they normally would but remember because it's a solid those atoms and molecules are bound so they might be vibrating or moving in place but they're not moving very far distances so as those atoms and molecules become energized and start to move around that energy or that amount of molecular activity could be measured by a temperature so if you had a surface or a skin thermocouple there you'd actually see the temperature rise but on a micro scale what that means is there's just more molecular activity right there so those molecules they are moving and bouncing around so naturally if we thought of this in like chunks or sections naturally the molecules from this chunk or section are going to want to convey that energy to the next chunk of this material so those molecules are going to bump into their adjacent molecules which is going to cause the adjacent molecules to become more energized they'll start moving around more the temperature here will rate will rise and again so on and so on that heat will gradually propagate its way down the length of this rod and that is how conduction occurs a similar visual from the book although the coordinate system is tilted you see that if you have a higher temperature on one side of a solid material represented by these darker red atoms those atoms are going to be more energized they're going to impact the adjacent atoms because they're these atoms are at a higher energy state than these so they're naturally going to transfer some of that energy here those will transfer it here and so on and so on you can see this plot here where we have the higher temperature energy is going to flow downhill from high temperature to low with the flux going against the temperature gradient so that takes us to fourier's law so fourier's law is a way of putting math to what i just described so we see that our heat flux is proportional to the negative of k k is the thermal conductivity of a material so it's a measure of how well that material come that conducts heat multiplied by the temperature gradient and notice the units of the temperature gradient are going to be in degrees celsius per meter or kelvin per meter which because it's relative those would be equivalent units so our heat flux is equal to minus k times the gradient of temperature so what that tells us is that heat flows against the temperature gradient that's what this negative sign does for us so you'll you're going to be solving a lot of problems with fourier's law so it's really important to if you're confused about it to work through that confusion to keep on repeating problems and just get better and better at it and naturally over time it'll start to make more sense to you so as an attempt to do that let's i have this picture of a solid material and if we plotted if it were if we had x going in this direction from left to right and if we were to plot our temperature as a function of x just for the sake of simplicity let's assume this temperature gradient is well this temperature profile is linear so this is called a temperature profile it's a plot or a graph of or data representing our temperature as a function of some spatial coordinate or coordinates so if our temperature gradient is going this way notice that it's a negative slope so the temperature gradient is negative so if our temperature gradient is negative our overall rate of heat transfer is going to be positive because the thermal conductivity is also always positive which means if our if our heat flux is positive that just means it's going with our coordinate system so if we plotted q in the x direction we would see if this is zero heat flux we would see because our slope here our gradient is constant and our thermal conductivity is constant we would see this constant positive flow of heat so heat is going to be flowing this way and it's going to be the same flow of heat all the way through that solid so that's how fourier's law works if we have a constant slope then if the slope is constant through a material then you you can express the fourier's law differently in terms of an absolute temperature difference so first of all if our system is one dimensional meaning our temperature only varies in one dimension in this case x meaning that it does not vary in the y direction or in the z direction but only in the x direction then our gradient becomes a little bit simpler we just have to worry about the derivative of temperature with respect to one spatial coordinate so fourier's law takes on this one dimensional form which we'll work quite a bit with especially in chapter three we have our heat flux in the x direction is just equal to minus k remember the additive inverse of our thermal conductivity multiplied by the derivative of temperature with respect to x but when that derivative of temperature with respect to x is constant because we have a constant slope temperature profile then you can just do rise over the run you could take the temperature here and the temperature here take the difference divide that by the total width of our wall and get this form of fourier's law remember this only applies when we know that dt dx is constant and under this particular set of circumstances a one-dimensional system where the it's steady which means it's not changing with time this particular type of system is called a plane wall and it has constant thermal conductivity only under those conditions where we know dtdx is constant can we extrapolate and use the simpler algebraic form of fourier's law and just with a little bit of algebraic manipulation you can switch the positions of those temperatures and get this form so if we know that these conditions apply we can express fourier's law algebraically to get our heat flux just by measuring those two end point temperatures and putting them into our equation so heat flux if you remember has units of watts per meter squared so that tells us how much heat is flowing per unit area but more specifically the area that we're talking about is the cross-sectional area so it's the area that is normal or perpendicular to the flow of heat so it would be this area of heat's flowing this way it'd be the area that intercepts that flow of heat by being perpendicular to it so shown on this diagram if we were to take a slice out of this wall that area would be the width of the wall multiplied by the height of the wall would be the appropriate area so if we use fourier's law to quantify the heat flux by measuring the temperature and plugging in the wall thickness and thermal conductivity if instead of wanting to know flux in terms of heat loss per square meter if we wanted the total rate of heat loss we would just multiply our flux by the total cross-sectional area of the wall and that would give us the total rate of heat loss so you could do this in your own home if you knew that these conditions applied to the walls of your home you could calculate the flux so how much heat is your home losing or gaining by conductive heat transfer if you apply this form of fourier's law you get the flux but then if you wanted to know something a little bit more practical like how much energy is my house losing in total then you would just multiply that by the cross area of all the walls in your home assuming that it's all uniform throughout which would be a big assumption but then you can get the total rate of heat loss in your house in terms of watts and this is something that would actually impact your heating bill or your air conditioning bill is how many watt hours or rather kilowatt hours is required to heat or cool your home so heat transfer has some really practical applications but it's really important to get very used to going back and forth between these two different forms so the heat flux in terms of watts per meter squared or the total rate of heat loss in terms of watts and to go in between the two for this particular conduction problem you just multiply by the total cross-sectional area so we're going to do an example problem and i'm going to show you how you could apply fourier's law to get a heat flux from a known temperature profile so if we took some solid rod of material like that shown here so this solid rod has a temperature profile that can be described by the equation t as a function of x is equal to five x plus one hundred again a temperature profile is just an equation or a collection of data or a plot that shows you how temperature changes as a function of of your spatial coordinates in your system so this has this particular temperature profile and a thermal conductivity of 10 that's measured in watts per meter per kelvin then what is the heat flux so you can pause right now and you can take a moment to try and figure this out yourself so go ahead and do that if you'd like to so if you've paused i'm going to walk us through how to do this problem now so if you notice this temperature this this rod has actually the reverse temperature profile of the example i showed previously where now we have cold on the left hand side and hot on the right hand side so our if we plotted temperature as a function of x that would reflect that where it's starting off colder and getting hotter the deeper you go into the solid so that's how our temperature profile looks like so this is going to have a positive temperature gradient with respect to x so think about that temperature is increasing as x increases so that should tell us that the flux is going to be going the opposite way of our temperature gradient so let's just go ahead and think through that so fourier's law tells us that our flux in the x direction is equal to minus k dt dx so dt dx is positive our temperature is increasing with x our thermal conductivity is always a positive number a finite and positive number and then we have this minus sign so that minus sign is pretty critical because that's telling us which direction heat is flowing so we have a positive temperature gradient and a positive thermal conductivity so that's going to tell us that our our flux is negative and what does it mean to have a negative heat flux well that is really just a function of how we've defined our coordinate system so if we find our coordinate system with x being positive from left to right and this tells us that our flux is actually going the opposite way of our coordinate system it's going against x we could redefine our coordinate system to have x start over here and that would give us a positive flux so really energy and flux are really um positive terms you can't really have negative energy but you could have heat flowing against your coordinate system so if we were to plot how our flux looks like we have if 0 or up here the slope of this line is constant which means our heat flow is going to be constant and the slope of this line is positive and because of this negative sign here our flow of heat is going to be constant and negative which again that negative flux just means that the heat is flowing against our coordinate system all right let's put some actual numbers to this now so our first step would be take the derivative of our temperature profile so that derivative the derivative of 5x plus 100 is just five that's going to have units of the change in temperature is kelvin and a change in distance is meters so our our temperature gradient is five kelvin per meter which means every meter we're going we're increasing our temperature by five so if we plugged everything in here now we would get that our flux in the x direction is equal to minus 10 and i'll just put the units here on the bottom minus 10 watts per meter per kelvin multiplied by 5 kelvin per meter which gives us minus 50 and we're going to have watts per meter squared so that's a flux term it's as i've mentioned in previous lectures it's really important to have a conceptual understanding of what's going on and we've done just that we've thought about how our temperature profile changes we've thought about which direction we're expecting heat to flow so we're in really good shape and then we only do unit conversions at the end we plug in units just to make sure we've done the math correctly and that tells us we're getting the units that we want and that way you expect for a flux so that just confirms that we've done that problem correctly so i always recommend writing out your equation algebraically and then applying units at the end only to check your math don't try and randomly plug in numbers until you get the units to line up because that's going to be really confusing for you and inevitably you're gonna something will get screwed up and most importantly that conveys that you don't really have a fundamental understanding conceptually of what's happening in your problem so in this particular problem we get q in the x direction the q double prime index direction is minus 50 watts per meter squared and again that negative flow of heat just means it's going against our coordinate system some things to remember for conduction so the driving force is a temperature gradient so that's dt dx and it's that same driving force that shows up in our rate equation where we have q double prime i'll get rid of the x here i'll put this in more general terms q double prime is equal to minus the gradient of temperature so this particular form is what shows up in fourier's law and actually i forgot to put the thermal conductivity in there so the rate equation for conduction is called fourier's law you'll want to commit that to memory we'll be using it over and over again and you'll know whenever you need to quantify the rate of heat transfer by conduction you need to use fourier's law that'll apply ubiquitously for conduction problems when we're solving conduction problems often trying to solve for the temperature profile so we'd be doing the reverse of what we just did before we applied foray's law to a known temperature profile to get the rate of heat transfer often we will we'll work backwards from that we'll apply fouriers a lot to get the rate and then we'll do some integration to get the temperature profile but generally with conduction we're looking for solving for this profile what is our equation as a function of x or as a function of r or as a function of x y and z and this could also be dynamic so we'll use conduction in a solid you have these temperature gradients and you can solve and figure out what your temperature profile is and we typically won't be looking for just a single temperature but that concludes this lecture thank you for listening