Heat Transfer (13): Transient heat conduction, lumped heat capacity model and examples

00:42:51
https://www.youtube.com/watch?v=XKPLjuo4Yxw

Sintesi

TLDRThis lecture covers Chapter 5 on transient heat conduction, introducing the lumped heat capacity (LHC) model. This model simplifies heat conduction analysis by assuming temperature varies solely with time. A specific example involving a copper sphere cooling in air is discussed, calculating critical parameters like the Biot number, energy loss, and temperature change over time. Other geometrical scenarios are outlined (plane wall, cylinder, sphere) to illustrate application of the Biot number, highlighting when to use the LHC model for heat transfer computations. Finally, the importance of initial conditions and conservation of energy principles are emphasized for transient heat conduction analyses.

Punti di forza

  • 📘 Introduced Chapter 5 on Transient Conduction.
  • 📏 Explained the heat diffusion equation in 2D.
  • 🧊 Discussed the Lumped Heat Capacity model for simplification.
  • 🌡️ Analyzed the cooling of a copper sphere in air.
  • 🔍 Calculated the Biot number to determine model applicability.
  • ⏳ Introduced the time constant tau for cooling analysis.
  • 📉 Presented temperature changes as exponential decay.
  • 🔄 Discussed energy loss due to convection.
  • 📝 Reviewed geometries for applying the LHC model.
  • ❓ Covered practical example-solving for real-world scenarios.

Linea temporale

  • 00:00:00 - 00:05:00

    The lecture introduces Chapter 5 on transient conduction, focusing on the heat diffusion equation relevant for heat transfer over time. It starts by reminding students of the fundamental equation from Chapter 2 and introduces the lumped heat capacity (LHC) model to simplify analysis when temperature variation is only with time and not spatially inside the body.

  • 00:05:00 - 00:10:00

    The LHC model assumes a body starts at a high initial temperature, T_i, and is cooled by a fluid at temperature T_infinity, promoting the condition where we can write a basic energy balance based on convection and change in energy storage, leading to the formation of a differential equation to derive temperature as a function of time.

  • 00:10:00 - 00:15:00

    A simplification involves defining a dimensionless temperature difference, theta, leading to a solution that can be expressed in exponential form with time constant tau. The response of the body to cooling depends on convection coefficient, specific heat, and geometry, which affects the cool rate. A graphical representation of cooling rates illustrates the influence of airflow on heat loss.

  • 00:15:00 - 00:20:00

    Using a practical example, the cooling of a copper sphere is described. The initial temperature and ambient conditions are specified. It demonstrates the application of the LHC model by calculating the Biot number and determining whether the lumped model is appropriate based on dimensionless parameters, allowing for simplified analysis.

  • 00:20:00 - 00:25:00

    The lecture explores how the cooling happens exponentially when the Biot number condition is satisfied. It discusses the relationship between the time constant tau and the cooling efficiency, illustrating how higher convection rates lead to faster cooling with different conditions.

  • 00:25:00 - 00:30:00

    The discussion moves to clarify the equations governing energy loss due to convection. Two key equations are presented, emphasizing how integral forms can be derived to find the total energy lost over a specified time period due to cooling.

  • 00:30:00 - 00:35:00

    An example concerning the energy lost from the copper sphere is worked through, leading to the conclusion that the internal energy change correlates directly with the energy lost to convection, reinforcing the energy balance concept.

  • 00:35:00 - 00:42:51

    The chapter ends with a preview of what's to come, specifically tackling more complex scenarios where the Biot number exceeds the critical value, indicating that more sophisticated methods will be necessary for future lessons.

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Video Domande e Risposte

  • What is the lumped heat capacity model?

    The lumped heat capacity model simplifies the study of transient heat conduction by assuming the temperature of an object only varies with time and not position within the object.

  • When can the lumped heat capacity model be used?

    It can be used when the Biot number (hlc/K) is less than one-tenth.

  • What does the Biot number indicate?

    The Biot number is a dimensionless parameter that indicates the ratio of thermal resistance within a body to thermal resistance at its surface, helping determine the applicability of the lumped heat capacity model.

  • What is the significance of the time constant tau?

    The time constant tau describes how quickly the temperature of an object decreases over time when subjected to cooling.

  • How does the temperature of the copper sphere change over time?

    The temperature of the copper sphere decreases exponentially over time when cooled by air.

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Scorrimento automatico:
  • 00:00:15
    >> Heads up of where we are.
  • 00:00:18
    We're starting a new chapter this morning,
  • 00:00:21
    Chapter 5, which is the
  • 00:00:24
    last of the four chapters on heat conduction.
  • 00:00:27
    We are going to finish about half
  • 00:00:30
    of that this morning and the rest
  • 00:00:31
    of it on Monday after our first mid-term.
  • 00:00:35
    This chapter is titled Transient Conduction.
  • 00:00:41
    So now time becomes
  • 00:00:44
    one of the variables in our solution.
  • 00:00:47
    We go back to Chapter 2 in general,
  • 00:00:52
    to get the heat diffusion equation
  • 00:00:54
    in two dimensions.
  • 00:00:55
    We write it here for
  • 00:00:57
    constant properties with no generation.
  • 00:01:00
    So this is the heat diffusion equation from
  • 00:01:02
    Chapter 2 with
  • 00:01:03
    constant property with no generation.
  • 00:01:06
    If we could solve this with
  • 00:01:09
    correct initial and boundary conditions,
  • 00:01:12
    we would get the temperature as
  • 00:01:14
    a function of x, y, and time.
  • 00:01:16
    Now it can be rather difficult, but luckily,
  • 00:01:23
    there is a possibility we can use
  • 00:01:26
    a simpler model to get results.
  • 00:01:30
    This model is called the
  • 00:01:32
    lumped heat capacity model, abbreviated LHC.
  • 00:01:36
    Lumped Heat Capacity.
  • 00:01:39
    Here's what the model says,
  • 00:01:41
    take a body initially at a temperature T_i,
  • 00:01:45
    initial temperature, at time zero, T_i.
  • 00:01:49
    Let's say it starts out hot,
  • 00:01:51
    just to talk about something.
  • 00:01:52
    It starts out hot.
  • 00:01:54
    We blow a cool fluid
  • 00:01:57
    over it at T_infinity with
  • 00:01:59
    a convection coefficient on the surface
  • 00:02:01
    of H. With time,
  • 00:02:05
    the body starts to cool.
  • 00:02:06
    Starts out hot, blow cool fluid over it,
  • 00:02:10
    temperature of the body starts to decline.
  • 00:02:12
    We're going to assume with
  • 00:02:15
    the simple model that
  • 00:02:16
    the temperature does not
  • 00:02:17
    vary inside the body with x and y,
  • 00:02:20
    but it only varies with time.
  • 00:02:23
    So now time is not a function,
  • 00:02:27
    or temperature is not
  • 00:02:28
    a function of x, y and time.
  • 00:02:30
    Now temperature is only a function of time.
  • 00:02:36
    We're not going to solve
  • 00:02:39
    the differential equation now.
  • 00:02:41
    Instead, we're going to write
  • 00:02:44
    an energy balance on this object.
  • 00:02:48
    So go back to the end of Chapter 1,
  • 00:02:54
    energy balance on a control volume.
  • 00:02:56
    E dot_in minus E dot_out
  • 00:02:59
    minus E dot_g equal E dot_storage.
  • 00:03:02
    Problem said no generation.
  • 00:03:05
    E dot_g, zero.
  • 00:03:08
    I'm looking at the model
  • 00:03:10
    where the object starts
  • 00:03:11
    out hot is going to be
  • 00:03:13
    cooled by a colder fluid.
  • 00:03:14
    So nothing comes in,
  • 00:03:17
    but energy does go up by convection.
  • 00:03:21
    There is a convection leaving.
  • 00:03:23
    What happens in?
  • 00:03:25
    The body has a change in stored energy.
  • 00:03:28
    We're taking energy out of the body
  • 00:03:30
    by the cool fluid blowing over it.
  • 00:03:34
    So we can then write down
  • 00:03:37
    what the convection is
  • 00:03:39
    and the change in stored energy.
  • 00:03:42
    So we have our convection minus
  • 00:03:45
    hA_s and T minus T_infinity.
  • 00:03:53
    Just to review Chapter 1,
  • 00:03:56
    convection heat transfer,
  • 00:03:57
    Newton's Law of Cooling.
  • 00:03:59
    Equal the time rate
  • 00:04:02
    of change of stored energy,
  • 00:04:04
    Rho cVdT, dtime.
  • 00:04:12
    Capital V is the volume.
  • 00:04:16
    Rho times the volume is the mass.
  • 00:04:19
    The mass times c gives me
  • 00:04:22
    a measure of how much energy
  • 00:04:24
    the object can store.
  • 00:04:25
    Mass times c, specific heat.
  • 00:04:29
    Units of the equation there
  • 00:04:32
    are in Watts, both sides Watts.
  • 00:04:35
    To simplify matters, we let
  • 00:04:39
    Theta equal T minus T_infinity.
  • 00:04:45
    Put that in the above equation then.
  • 00:04:48
    Well, what happens, of course,
  • 00:04:49
    is d Theta dt is equal to dt, dt.
  • 00:04:57
    Because a differential of a
  • 00:04:59
    constant, T_infinity is zero.
  • 00:05:02
    Put that up there.
  • 00:05:07
    Write it down, d Theta over t.
  • 00:05:24
    There's our little differential equation.
  • 00:05:28
    The initial condition is
  • 00:05:36
    when t equals zero, T equals T_i.
  • 00:05:41
    So Theta is equal to T_i minus T_infinity,
  • 00:05:47
    and we call that Theta_i.
  • 00:05:49
    So when time equals zero,
  • 00:05:52
    Theta equal Theta_i, a constant.
  • 00:05:55
    Solve then Theta over Theta_i,
  • 00:06:26
    and we get that solution for
  • 00:06:29
    temperature of the object
  • 00:06:30
    as a function of time.
  • 00:06:32
    Now, just to remind you
  • 00:06:34
    what Theta over Theta_i is.
  • 00:06:46
    So Theta over Theta_i
  • 00:06:49
    is a measure of the temperature.
  • 00:06:59
    If we want then we could plot
  • 00:07:01
    a curve of Theta
  • 00:07:05
    over Theta_i from zero to one.
  • 00:07:13
    This is time.
  • 00:07:15
    Start at one.
  • 00:07:18
    There's different curves, parametric values.
  • 00:07:23
    Each particular curve is
  • 00:07:26
    a different value of the time constant called
  • 00:07:31
    Tau, is in seconds.
  • 00:07:49
    It's Rho cV over hAs.
  • 00:07:55
    So if you want,
  • 00:07:57
    I'll just add this to it.
  • 00:07:59
    You might want to express this as exp,
  • 00:08:03
    it's the simplified form of this
  • 00:08:05
    then minus t over Tau.
  • 00:08:21
    >> We engineers love the dimensionless stuff.
  • 00:08:25
    So this is
  • 00:08:27
    a dimensionless temperature difference ratio.
  • 00:08:30
    This is dimensionless,
  • 00:08:33
    T over tau is dimensionless.
  • 00:08:35
    T is in seconds,
  • 00:08:36
    tau is and seconds, dimensionless,
  • 00:08:38
    so we like to express things very
  • 00:08:39
    simply and lot of
  • 00:08:41
    times that's in dimensionless form.
  • 00:08:44
    This is called the time constant tau.
  • 00:08:50
    Now, let's just for
  • 00:08:56
    sake of talking about something.
  • 00:08:57
    Let's just take a copper sphere
  • 00:09:00
    six centimeters in diameter.
  • 00:09:01
    I don't know if that size is a baseball size,
  • 00:09:04
    six centimeters in diameter,
  • 00:09:07
    initially, at 300 degrees Fahrenheit,
  • 00:09:10
    that's Ti 300, I'm
  • 00:09:15
    going to blow air over
  • 00:09:17
    it at 30 degrees Celsius not Fahrenheit.
  • 00:09:20
    let's just say 70 degrees Fahrenheit,
  • 00:09:22
    make it English units.
  • 00:09:25
    So 300 degrees Fahrenheit initial,
  • 00:09:28
    and now T infinity, 30 degrees Fahrenheit.
  • 00:09:31
    I blow air over it. Of course,
  • 00:09:32
    the temperature's going to drop with time.
  • 00:09:34
    How? Exponentially. Yeah, drop
  • 00:09:38
    with time exponentially.
  • 00:09:39
    But these three curves,
  • 00:09:41
    let's say you take that copper sphere
  • 00:09:43
    and put it in this room on a stand
  • 00:09:45
    at 300 Fahrenheit and
  • 00:09:49
    the air newsroom is really still.
  • 00:09:51
    Nobody's air's blowing,
  • 00:09:53
    air in the room is really still.
  • 00:09:56
    H is going to be really low,
  • 00:09:59
    time constant may be really high.
  • 00:10:03
    This arrow means
  • 00:10:05
    high time constants, low time constants.
  • 00:10:07
    The direction of the arrow tells me.
  • 00:10:09
    Increasing time constant curves this way.
  • 00:10:11
    So copper sphere in this room,
  • 00:10:14
    it cools real slow.
  • 00:10:18
    Now, I put a floor fan in front
  • 00:10:21
    of it, turn the floor fan on.
  • 00:10:23
    The air blows by really fast.
  • 00:10:25
    H becomes really big: H,
  • 00:10:29
    really big, time constant, really small.
  • 00:10:32
    Boy, the temperature drops
  • 00:10:33
    really fast to that copper sphere.
  • 00:10:35
    So that's the story
  • 00:10:37
    this graph is trying to tell you.
  • 00:10:38
    You can say, "What if the density is bigger?"
  • 00:10:41
    "What if the surface area is smaller?"
  • 00:10:44
    You can play the same game
  • 00:10:45
    with all these variables. So there
  • 00:10:50
    is the temperature variation with time,
  • 00:10:54
    very simple, because it's
  • 00:10:57
    the simplest possible model that
  • 00:10:59
    we can use in transient studies.
  • 00:11:02
    But now we say, "Yeah,
  • 00:11:03
    but when can we use it?"
  • 00:11:06
    Well, when we can use
  • 00:11:08
    it is when something special happens.
  • 00:11:11
    This lumped heat capacity model
  • 00:11:18
    can be used when
  • 00:11:22
    a dimensionless parameter
  • 00:11:25
    called the Biot number,
  • 00:11:31
    defined as hlc over K is less than one-tenth.
  • 00:11:43
    If that's satisfied, we can take the easy way
  • 00:11:47
    out and model the problem
  • 00:11:49
    as a lumped heat capacity model.
  • 00:11:52
    What is l sub c?
  • 00:11:55
    L sub c is called a characteristic length.
  • 00:12:06
    It's the volume of
  • 00:12:09
    the object divided by its surface area,
  • 00:12:13
    the area touching the
  • 00:12:14
    fluid cooling the object.
  • 00:12:21
    So there's some simple geometries.
  • 00:12:25
    We'll put those geometries
  • 00:12:26
    up on the board here.
  • 00:12:28
    The first one is a plane wall.
  • 00:12:59
    This particular plane wall
  • 00:13:02
    has convection on both sides.
  • 00:13:18
    The wall is 2L thick.
  • 00:13:22
    X is measured from the
  • 00:13:24
    centerline of the wall.
  • 00:13:25
    On both sides of the wall,
  • 00:13:28
    we have convection to a fluid at T infinity,
  • 00:13:37
    with a convection coefficient of
  • 00:13:40
    h. L sub c
  • 00:13:50
    is equal to L.
  • 00:13:55
    It's called the half thickness.
  • 00:14:07
    Another common geometry is
  • 00:14:10
    a cylinder whose radius
  • 00:14:20
    is r note.
  • 00:14:22
    Again, it has a fluid blowing over it at
  • 00:14:26
    T infinity with a convection coefficient
  • 00:14:29
    of h on the surface.
  • 00:14:35
    For this case, L sub C
  • 00:14:39
    equal the radius divided by 2.
  • 00:14:42
    Now, we have a sphere.
  • 00:14:53
    Again, we have a fluid blowing over
  • 00:14:56
    the sphere at temperature T infinity,
  • 00:14:59
    with a convection coefficient h
  • 00:15:01
    on the surface of the sphere.
  • 00:15:04
    L sub c equal or not over 3.
  • 00:15:14
    So those are three common geometries
  • 00:15:19
    that we engineers study.
  • 00:15:21
    I'll tell you later on to give you
  • 00:15:23
    some idea of what they really couldn't be,
  • 00:15:26
    but for right now, plane wall,
  • 00:15:29
    a cylinder, and a sphere.
  • 00:15:37
    Biot number, hlc over K. Here's LC.
  • 00:15:42
    I don't care what the object is,
  • 00:15:45
    you can find LC part.
  • 00:15:48
    For instance, what if I had
  • 00:15:52
    a cube of aluminum,
  • 00:16:01
    each side a, L sub c,
  • 00:16:09
    volume over surface area.
  • 00:16:12
    Volume a cubed. How many sides?
  • 00:16:16
    Six. What's one side? A squared.
  • 00:16:22
    L sub c, length of one side divided by six.
  • 00:16:26
    That's what goes in the Biot number.
  • 00:16:29
    So for any object,
  • 00:16:30
    it's the volume divided
  • 00:16:32
    by the surface area touching the fluid.
  • 00:16:37
    Well, that's half the story.
  • 00:16:42
    We know the full story.
  • 00:16:43
    The full story is,
  • 00:16:44
    once we get the temperature of an object,
  • 00:16:46
    whether it's of temperatures of function of
  • 00:16:48
    the geometry or of time,
  • 00:16:51
    the second thing is find out something about
  • 00:16:53
    how much heat transfer
  • 00:16:54
    has occurred by convection,
  • 00:16:56
    or how much energy the body
  • 00:16:58
    has lost by convection.
  • 00:17:02
    So now, we want to
  • 00:17:04
    find the energy lost by the body.
  • 00:17:29
    How is that energy lost by the body?
  • 00:17:31
    Well, convection to the fluid takes it out.
  • 00:17:34
    Now, it could be the reverse.
  • 00:17:36
    Hot gas, initial temperature low.
  • 00:17:38
    You heat the body.
  • 00:17:39
    Doesn't matter either way.
  • 00:17:43
    So Q equal integral zero to time t,
  • 00:17:51
    Rho hAs, and then we have our Theta d time.
  • 00:18:04
    Convection, by convection,
  • 00:18:07
    here's convection, Newton's law,
  • 00:18:09
    h A Delta t.
  • 00:18:11
    Theta is Delta t. T minus t infinity.
  • 00:18:18
    Dt, time integrate with respect to time.
  • 00:18:22
    Where you see Theta in here,
  • 00:18:25
    put the boxed equation right there,
  • 00:18:28
    put that right there, into there,
  • 00:18:31
    and integrate the exponential function.
  • 00:18:36
    Just so you know, the Newton's law
  • 00:18:39
    h A Delta t is in watts.
  • 00:18:42
    A watt is a joule per second.
  • 00:18:46
    What's this guy here?
  • 00:18:48
    Seconds. When I multiply joules
  • 00:18:50
    per second times seconds, I get joules.
  • 00:18:54
    Capital Q is joules,
  • 00:19:00
    little q is watts.
  • 00:19:04
    Q in joules is energy,
  • 00:19:07
    and of course, watts is
  • 00:19:09
    the heat transfer, the power watts.
  • 00:19:12
    So now, we've got two Qs,
  • 00:19:14
    a capital Q and a lowercase q.
  • 00:19:17
    You talk about energy,
  • 00:19:19
    you talk about capital Q.
  • 00:19:20
    You talk about heat transfer,
  • 00:19:22
    you talk about little q.
  • 00:19:24
    Put those guys in there,
  • 00:19:26
    do the integration, I get q.
  • 00:19:55
    >> So there are the two equations that
  • 00:19:59
    we need to solve.
  • 00:20:09
    So let's see what I got here.
  • 00:20:14
    Any questions before we go on now?
  • 00:20:21
    Wait, let me think here. Here it is.
  • 00:20:26
    So we're going to then take an example.
  • 00:20:29
    The example that I'm going to
  • 00:20:31
    work is a copper sphere.
  • 00:20:34
    Let's put that on the middle here.
  • 00:20:36
    So we're going to look at
  • 00:20:38
    a copper sphere being cooled by air.
  • 00:20:46
    I might need him. I'll leave him up here.
  • 00:21:13
    Diameter is 15 centimeters.
  • 00:21:22
    Initial temperature of 300.
  • 00:21:30
    Air temperature, T_infinity, 30 degrees
  • 00:21:34
    C. Properties at T_average.
  • 00:21:56
    You don't want to take the properties
  • 00:21:58
    at their starting temperature.
  • 00:22:00
    You don't know what the final temperature is.
  • 00:22:04
    So probably your best guess is to take
  • 00:22:07
    the properties at the average temperature,
  • 00:22:10
    starting temperature plus air temperature.
  • 00:22:12
    Once you've solved the problem,
  • 00:22:14
    go back and put
  • 00:22:16
    the right final temperature in there.
  • 00:22:20
    I want to know how
  • 00:22:21
    the temperature varies with time.
  • 00:22:23
    So find T as a function of time.
  • 00:22:29
    I'll put the properties down for copper.
  • 00:22:55
    Step 1, Chapter 5, check Biot.
  • 00:23:01
    Maybe you get lucky and the Biot's less than
  • 00:23:04
    one-tenth and you can
  • 00:23:06
    use this model, the simple model.
  • 00:23:08
    Biot equal hl_c over k. H, 25;
  • 00:23:19
    k, 393; l_c from the table geometry a sphere,
  • 00:23:26
    l_c, one-third the radius, the radius,
  • 00:23:30
    seven and a half, 0.20 is 159,
  • 00:23:43
    which is a lot less than one-tenth.
  • 00:23:46
    So we can use lumped heat capacity.
  • 00:23:54
    First [inaudible] done. I get
  • 00:23:56
    the easy about this time.
  • 00:23:59
    Here's the equation in the box.
  • 00:24:01
    There's the equation.
  • 00:24:28
    If I want to find the time constant Tau,
  • 00:24:36
    time constant Rho c
  • 00:24:42
    over h r_naught over three l_c.
  • 00:24:51
    V Over As is l_c.
  • 00:24:56
    Put those numbers in there,
  • 00:24:59
    I get 3546 seconds.
  • 00:25:05
    Time constant's almost an hour.
  • 00:25:08
    It's cooling really slow.
  • 00:25:12
    So then I know everything here.
  • 00:25:16
    The properties, I know h,
  • 00:25:17
    I know T_infinity, I know T_i.
  • 00:25:19
    So T as a function
  • 00:25:21
    of t then when you put the numbers in,
  • 00:25:36
    put T in seconds,
  • 00:25:39
    and the time will come out to be degrees
  • 00:25:41
    C. So that's my solution then.
  • 00:25:57
    Now, if somebody asked me,
  • 00:26:00
    by the way, in the first five minutes,
  • 00:26:09
    how much energy is lost by the copper sphere?
  • 00:26:12
    I say, okay, here's over here.
  • 00:26:15
    The first five minutes,
  • 00:26:17
    60 seconds per minute times 5,
  • 00:26:21
    300 for T. I know that,
  • 00:26:24
    I know that. I know that.
  • 00:26:25
    I know that. I know Tau.
  • 00:26:27
    I just found Tau. Put it in here.
  • 00:26:29
    So many joules of energy is
  • 00:26:32
    how much energy the sphere
  • 00:26:34
    has lost to the air.
  • 00:26:36
    Now you say, well,
  • 00:26:37
    how much heat transfer has occurred?
  • 00:26:41
    How much energy has going out by convection?
  • 00:26:45
    It's the same answer.
  • 00:26:47
    The change in internal energy
  • 00:26:49
    is the same as the amount
  • 00:26:50
    of energy lost by
  • 00:26:52
    convection. Says it right here.
  • 00:26:54
    Change in storage is the same as
  • 00:26:57
    how much energy goes out
  • 00:26:58
    of the body by convection.
  • 00:27:04
    Any questions on that then?
  • 00:27:07
    So now of course the obvious question is,
  • 00:27:11
    what if the Biot is greater than one-tenth?
  • 00:27:14
    Now we got
  • 00:27:14
    much more difficulty mathematically.
  • 00:27:17
    That's going to be Monday.
  • 00:27:19
    We'll save that one for Monday.
  • 00:27:21
    So I want to go over
  • 00:27:24
    some things about
  • 00:27:25
    the midterm on Friday with you.
  • 00:27:26
    So let's do that right now.
  • 00:27:28
    Any questions before I erase
  • 00:27:29
    any of this? Yes.
  • 00:27:32
    >> Where did we get the two [inaudible].
  • 00:27:34
    >> Which one now, here?
  • 00:27:36
    It's this guy here, 300 minus 30.
  • 00:27:43
    Let's take a look then,
  • 00:27:46
    a little bit of review.
  • 00:27:48
    It's going to be not in great detail,
  • 00:27:51
    but just to jog your memory of what
  • 00:27:54
    could occur on the first midterm.
  • 00:27:57
    So here we have Chapter 1. Is Elizabeth here?
  • 00:28:08
    In front here when were done,
  • 00:28:10
    a paper for you.
  • 00:28:12
    Let's take a look and see Chapter 1.
  • 00:28:17
    Fourier's Law, Newton's law, convection.
  • 00:28:29
    Fourier's law, KA over
  • 00:28:34
    L Delta T. Newton's law,
  • 00:28:36
    HAS Delta T. Radiation.
  • 00:28:45
    Small object in a large surroundings.
  • 00:28:50
    Epsilon Sigma AT object
  • 00:28:55
    to the fourth minus
  • 00:28:56
    t surroundings to the fourth.
  • 00:28:59
    That's for a black body, Epsilon is 1.
  • 00:29:03
    Alpha, that's the emissivity, is 1.
  • 00:29:07
    The Epsilon activity Alpha,
  • 00:29:09
    if you multiply the solar input on the street
  • 00:29:12
    out there times Alpha of black asphalt,
  • 00:29:15
    then that's how much is absorbed
  • 00:29:17
    by the black asphalt.
  • 00:29:19
    Alpha is the absorptivity.
  • 00:29:21
    Epsilon is the emissivity.
  • 00:29:27
    Then we had, like over here,
  • 00:29:31
    energy balance on the control volume.
  • 00:29:40
    Then we had a surface energy balance.
  • 00:29:49
    That gets us to Chapter 2.
  • 00:29:56
    Chapter 2, real short.
  • 00:29:59
    But Chapter 2 starts out with the tables in
  • 00:30:03
    the back just to
  • 00:30:11
    become familiar with what's there for you.
  • 00:30:16
    What was the solid?
  • 00:30:18
    What was the liquid?
  • 00:30:20
    What was the gas? At what temperature?
  • 00:30:23
    All that stuff is in the tables.
  • 00:30:26
    Then heat diffusion equation.
  • 00:30:37
    Different versions, the rectangular version.
  • 00:30:41
    There's one of them right there.
  • 00:30:43
    That's the heat diffusion equation
  • 00:30:45
    for constant properties no generation.
  • 00:30:48
    So obviously that should
  • 00:30:50
    be on your equation sheets.
  • 00:30:51
    Rectangular, cylindrical, spherical, three equations.
  • 00:31:02
    Know how to specify
  • 00:31:04
    initial conditions and boundary conditions
  • 00:31:07
    to apply to the partial
  • 00:31:10
    or ordinary differential equation.
  • 00:31:25
    >> Okay. That's Chapter 2,
  • 00:31:27
    let's go to Chapter 3.
  • 00:31:30
    Chapter 3 is our resistances.
  • 00:31:33
    Yeah. We have conduction resistances.
  • 00:31:53
    Again, we have plane
  • 00:32:00
    wall, cylindrical, spherical,
  • 00:32:10
    and we have contact resistance.
  • 00:32:17
    Then we have convection resistance.
  • 00:32:26
    Then we put these in
  • 00:32:29
    series; parallel, series parallel, or others.
  • 00:32:51
    We had a problem where the heat input from
  • 00:32:54
    a chip came in to
  • 00:32:56
    a certain point and is split,
  • 00:32:58
    and some went down, and some went up.
  • 00:33:00
    So that's neither of these three.
  • 00:33:02
    So there's other geometries
  • 00:33:04
    that you construct these circuits.
  • 00:33:12
    Okay. Fins; possibly use Table 3.4.
  • 00:33:22
    Fins of a uniform cross section area.
  • 00:33:25
    They give you qf,
  • 00:33:27
    and that gives you the temperature
  • 00:33:29
    of the fin as a function of x.
  • 00:33:37
    Maybe you can use Table 3.5.
  • 00:33:43
    Table 3.5 gives you
  • 00:33:48
    fin efficiency and then use that to get qf.
  • 00:34:08
    But maybe you can use Figure
  • 00:34:11
    3.19 or maybe Figure 3.20.
  • 00:34:18
    They give you a to f,
  • 00:34:21
    and put it in that equation to get qf.
  • 00:34:27
    Then maybe you have
  • 00:34:29
    multiple fins on a surface.
  • 00:34:39
    We had problems like that.
  • 00:34:51
    Now we have Chapter 4 then.
  • 00:34:57
    Chapter 4, draw a flux plot,
  • 00:35:06
    show the adiabats and the isotherms.
  • 00:35:20
    Find S, conduction shape factor.
  • 00:35:35
    Use it to solve for heat transfer.
  • 00:35:48
    Okay. If you want a resistance,
  • 00:35:58
    the resistance is 1 over SK.
  • 00:36:22
    You can combine that resistance
  • 00:36:24
    with other resistances,
  • 00:36:26
    like the convection resistance, 1 over HA_s.
  • 00:36:30
    By the way, this
  • 00:36:32
    conduction shape factor then,
  • 00:36:33
    you use q equal SK Delta
  • 00:36:39
    T. That's how you put S in there to get q.
  • 00:36:47
    Now, the last half of Chapter 4,
  • 00:36:50
    numerical methods will not be on the midterm.
  • 00:36:52
    So stop there after
  • 00:36:55
    the conduction shape vector, S.
  • 00:37:09
    Now, let's talk about
  • 00:37:13
    maybe what you should look at as far
  • 00:37:16
    as example problems in the textbook.
  • 00:37:23
    Of course, I told
  • 00:37:25
    you before and I'll tell you again you,
  • 00:37:26
    anything I box in the lecture probably should
  • 00:37:29
    be on your equation sheet
  • 00:37:31
    because I think it's an important equation.
  • 00:37:33
    But beyond that, there may be
  • 00:37:35
    equations or whatever you
  • 00:37:36
    use to solve homework.
  • 00:37:38
    If that's important, you
  • 00:37:40
    put that on your equation sheet.
  • 00:37:41
    Not just what I box in class,
  • 00:37:43
    but what you use to solve homework problems.
  • 00:37:46
    So here are example problems in the text.
  • 00:38:02
    It depends what textbook you're looking at.
  • 00:38:05
    If it's the Heat Transfer textbook,
  • 00:38:07
    Introduction to Heat Transfer,
  • 00:38:09
    it's going to be the left-hand side.
  • 00:38:13
    If it's the Heat and Mass Transfer textbook,
  • 00:38:17
    it might be slightly different.
  • 00:38:19
    So these are the example problems
  • 00:38:22
    worked in the chapters in
  • 00:38:24
    the textbook that I think probably are
  • 00:38:25
    important for you to look
  • 00:38:26
    at before the first midterm;
  • 00:38:29
    Chapter 1, Chapter 2, Chapter 3,
  • 00:38:54
    Chapter 4.
  • 00:38:56
    Now, these same problems
  • 00:38:59
    are in the Heat and Mass Transfer textbook,
  • 00:39:01
    but sometimes the numbers change
  • 00:39:03
    because the author
  • 00:39:04
    changed a textbook slightly.
  • 00:39:05
    So these are the ones that are
  • 00:39:07
    similar to the left hand column,
  • 00:39:09
    but they're in the Heat
  • 00:39:10
    and Mass Transfer textbook,
  • 00:39:12
    and it depends which one you've got in
  • 00:39:13
    your hands or on your laptop.
  • 00:39:19
    Most are the same numbers,
  • 00:39:21
    but there's a couple
  • 00:39:22
    of those slightly different.
  • 00:39:38
    Got it.
  • 00:39:53
    Okay. So we've got 1,
  • 00:39:56
    2, 3, 4, 5, 6, 7,
  • 00:39:58
    8, 9, 10, 11, 12, 13,
  • 00:40:00
    14; 2, 4, 7,
  • 00:40:03
    9, 1, 3, 4.
  • 00:40:07
    Right. 1, 2, 3, 6.
  • 00:40:12
    Six, so it should be okay.
  • 00:40:13
    I thought I missed one; 6, 9, 10,
  • 00:40:16
    4.1, 3.9, 3.10, 4.1.
  • 00:40:26
    Okay. Fifteen in-chapter worked examples.
  • 00:40:42
    You've worked 18 homework problems.
  • 00:40:52
    I've worked 21 problems worked in class.
  • 00:41:07
    You've got six previous exam problems.
  • 00:41:23
    So you've got a library now of
  • 00:41:26
    60 problems you can
  • 00:41:28
    review for the first midterm.
  • 00:41:31
    If you want to look at stuff,
  • 00:41:36
    that's what you look at right there.
  • 00:41:39
    What's most important?
  • 00:41:42
    Well, obviously, what I asked for
  • 00:41:43
    the last two years, that's that one.
  • 00:41:46
    When I assign homework,
  • 00:41:49
    why do I assign certain problems?
  • 00:41:51
    Because I think they tell
  • 00:41:52
    a really good story.
  • 00:41:53
    That's the next important one.
  • 00:41:55
    Why I work in class?
  • 00:41:56
    Because I want to go over
  • 00:41:57
    that with you to see if you can
  • 00:41:58
    understand what I'm doing up here
  • 00:41:59
    in the board in class.
  • 00:42:01
    This one here, this is worked class,
  • 00:42:05
    this in Chapter 1, that's what
  • 00:42:07
    the textbook author thinks are important.
  • 00:42:09
    That's all right, but that's not
  • 00:42:11
    number 1 on the list.
  • 00:42:12
    This is number 1, this
  • 00:42:14
    is number 2, this is number 3,
  • 00:42:16
    and that's number 4, if you've got
  • 00:42:18
    prioritize your time for looking at stuff,
  • 00:42:20
    that's what I would suggest.
  • 00:42:24
    Any questions on that then?
  • 00:42:27
    You know what you can bring to
  • 00:42:28
    the test on Friday,
  • 00:42:29
    what I'm going to give you one Friday,
  • 00:42:31
    so we'll see you on Friday then.
  • 00:42:33
    If you have any questions,
  • 00:42:34
    hang on after class,
  • 00:42:35
    I'll talk to you about
  • 00:42:36
    any questions you have.
Tag
  • transient conduction
  • lumped heat capacity
  • heat diffusion
  • Biot number
  • energy balance
  • temperature change
  • convection
  • characteristic length
  • copper sphere
  • cooling process