Are these statement true? (Regarding heat flow)?

  1. Are these statements true? (Regarding heat flow)

    I'm revising for an exam, and this is what I think I know so far.
    It will really help if someone can confirm these statements for me.

    Consider an iron pipe with inner diameter 20mm and outer diameter 30mm. The inside surface of the pipe is maintained at 100 degrees celsius, and the outside surface is at 90 degrees celsius. Air is flowing past the outside of the pipe at temperature 20 degrees celsius.

    Are these statements true:

    1. The heat flow dQ/dt from the inside of the pipe to the outside of the pipe (through the iron) equals the temperature difference (between the inside and outside surface of the pipe), divided by the thermal resistance of the pipe
    2. The thermal resistance of the pipe (per meter) is given by (ln(r2/r1))/(2*pi*k) where k is the thermal conductivity of the pipe
    3. The heat flow dQ/dt from the outside of the pipe to the air equals the outside area of the pipe, multiplied by the temperature difference (between the air and the outside of the pipe), multiplied by the heat transfer coefficient between the air and the pipe
    4. The heat transfer coefficient between the air and the pipe can be found/estimated using the Reynolds number, the Prandtl number, and the thermal conductivity of the air
    5. The Reynolds number will depend on mainly the velocity of the air
    6. The Prandtl number will depend mainly on the specific heat capacity of the air
    7. If the heat flow from the inside to the outside of the pipe equals the heat flow from the outside of the pipe to the air, the temperature on the outside of the pipe will be constant
    8. If the heat flow from the inside to the outside of the pipe is greater than the heat flow from the outside of the pipe to the air, the temperature on the outside of the pipe will be increasing.

    Thank you!
    Last edited: Apr 18, 2013
  2. jcsd
  3. Simon Bridge

    Simon Bridge 15,067
    Science Advisor
    Homework Helper
    Gold Member

    The way to figure out if these statements are true is to work out the reasons for stating them in the first place - in terms of physics. If you make a statement simply because some authority says this is so, then you are relying on (a) the authority got it right, (b) you remembered it correctly, and (c) the statement does not apply to a similar but crucially different situation.

    eg. for #1
    heat flows from a hot place to a cold place
    the outside of the pipe is colder than the inside - therefore the direction of travel is correct
    heat does not flow instantly - it flows at a rate ... so there must be something resisting the flow... so the comment about "resistance" sounds good (it is actually the definition of thermal resistance). If the heat flow is a current, and the temperature difference is the voltage, then the expression is Ohms' Law.

    #2 the thicker the pipe, the more/less (pick one) resistance it should have.
    are the dimensions correct?

    ... etc.
    The idea is to think about what the statements would mean.
    i.e. 4,5 and 6 are saying that, how fast the air carries off heat depends mainly on how fast the air is going and how much heat each bit of air can hold. Does this sound reasonable? If the air is a bunch of trucks and the heat is dirt, and you want to know how fast the trucks can carry off the dirt, what would that depend on?

    Now you should be able to answer your own question.
  4. There is an inconsistency between items 1 and 2 with item 3. If statements 1 and 2 are correct, then dQ/dt represents the local heat flow per unit length of pipe. If item 3 is correct, then dQ/dt is the total heat flow over the entire length of the pipe.

    Item 5 is a little puzzling. The Reynolds number depends on the viscosity and density of the air, which are functions of the temperature. If the temperature of the outside of the pipe is thought to be variable, then the Re will be a function of the temperature in addition to the velocity. If the temperature is thought to be fixed, then the Re will depend mainly on the velocity.

    For item 6, the Prantdl number is equal to the heat capacity times the viscosity, over the thermal conductivity. Here again, the viscosity is a function of the temperature, as is the heat capacity. If the temperature is though of as being fixed, this then fixes the Prantdl number, which will then not depend on anything.
  5. I made up these statements to summarize what I have learnt when doing questions and past exams on the subject. I feel like I understand the physics enough to know the limitations of these statements, and I feel like they all sound reasonable and make sense. However after doing many questions I have found that sometimes things that I thought sounded reasonable were incorrect. That's why I've asked this question, just to check I have not missed something.
    I like the truck analogy, thats made me feel a little more confident with statements 5 and 6.

    Ah yes, statement 3 should say "The heat flow dQ/dt per meter from the outside of the pipe to the air equals...

    I didn't think of that. Thank you. In most my questions though it seems to be assumed that the outside air stays at a constant temperature, which makes the calculations a bit easier.
  6. AlephZero

    AlephZero 7,299
    Science Advisor
    Homework Helper

    I don't know how you are supposed to interpret 5 or 6.

    If you assume all the data given in the question is "fixed" (that includes the air temperature), then Re does mainly depend on the air velocity. On the other hand if anything can vary it also depends strongly on the pipe diameter.

    For 6, Pr for air is almost constant for quite a wide range of conditions, so you could argue it doesn't "depend" on anything relevant to the conditions in the question. But even a change in Pr from 0.700 to 0.701 depends on some things more than on others... :confused:
  7. Sorry I should have been more specific with the example situation I gave. The air flowing past this pipe is being continuously replaced by fresh air (with uniform temperature, viscosity, density etc..) ,and the only variable is the velocity of the air.
    In this case The Reynolds number will only change with a change in the velocity. Statement 6 was a bit useless to be honest, and I can see why it has caused confusion. The reason I said this was because I remember being taught about heat capacity rate, which equals the specific heat capacity multiplied by the mass flow rate. I got a little bit confused, and remembered that specific heat capacity was important, as so was mass flow rate, so I thought Re was like the mass flow rate and Pr was like the specific heat capacity. Looking over these equations again I can see it's much more than this though.
    I didn't really understand what it meant by dimensionless number. I had a revision lecture today and the tutor explained why we use dimensionless numbers, which has helped me understand this a lot more.
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