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Homework Help: Convective heat transfer and steady state conduction problem

  1. Nov 12, 2014 #1
    • Homework problems require the use of the template and require some effort on the part of the student
    Hi all, I have a unique problem that I need help with

    A company's curing oven's exposed surface temperature is measured at 80°c when the surrounding air is 23°c. You think it should be insulated and wager to pay for the cost of this yourself if you can keep the savings incurred. Is this a smart move? the oven is 3.7m long and 2.4m diameter. The plant operates 16 hours/day 365 days/year. The insulation to be used is fiberglass (Kins = 0.08 W/m/°c, assumed constant) which costs $2.5/m2 per cm thickness plus $21.5/m2 for labour regardless of tthickness. Convective heat transfer coefficient on the outer surfaced is estimated to be h0= 20W/m2/K. The oven uses natural gas, whose unit cost is £7.1/GJ input and the oven's efficiency is 80%.
    Ignoring radiation heat loss from the outer surface, determine
    A) how much money you will make out of this venture, by installing one inch (1 inch = 2.54 cm) of insulation, if any, and
    B) the thickness of insulation (to nearest inches) that will maximise your earnings

    My thinking for part A is to:

    calculate the area of the oven exposed to the air,
    use q = hA (T2-T1) to obtain the heat transfer rate of the system with no insulation
    Use Rcond = ln(r2-r1)/(2*pi*L*K) to find the heat transfer rate *with* 1" insulation
    find Rconv ( 1/(h*A)
    use Rconv and Rcond to find Q
    calculate loss per year versus installation cost.

    This comes out to be a saving of $3432.67 for the year for me, but given the long answer nature of this question I've undoubtedly gone wrong somewhere and would appreciate other peoples' working if possible :)

    Any input would be greatly appreciated as I'm well and truly stuck
    Last edited: Nov 12, 2014
  2. jcsd
  3. Nov 12, 2014 #2
    Show us your detailed calculations of the heat load both with and without the insulation. From your description, it looks like you have the right idea.

  4. Nov 12, 2014 #3
    Area of the oven exposed to the air;
    (pi x d^2)/2 + (pi x d x L) = 32.42 m^2

    Heat transfer rate of the system with no insulation:
    q=20 x 32.42 x ( 80 - 23 )
    q=36.96 KW

    Heat transfer rate of the system with 1" insulation:
    Thermal circuits-
    Rcond= ln(r2-r1)/(2 x pi x L x K)
    Rcond= 5.661x10^-3 K/W

    Rconv=1/(h x A)
    Rconv=1.542x10^-3 K/W

    q=dT/(Rcond + Rconv)
    q= (80 - 23)/((5.661+1.542)x10^-3)
    q=7913 W

    Loss per year, without insulation;
    ( 36.96 x 10^-3/10^9 ) x 7.1 = $5516.7

    Loss per year, if fitted with 1" thick insulation
    ( 7913/10^9 ) x 7.1 = $1181.13

    Total cost of installation;
    assuming $2.5 x 2.54 = $6.35 /m^2 per inch thickness (not calculated by volume of a hollow cylinder)
    Cost by area= A x cost per inch = 32.42 x 6.35 = $205.867
    Cost by labour= 21.5 x 32.42 = $697.03
    Total cost of installation= $205.867 + $697.03 = $902.897

    Total loss with insulation (including installation) in a year= $902.897 + $1181.13 = $2084.03
    Total loss without insulation in a year= $5516.7
    Net savings= 5516.7 - 2084.03 = $3432.67

    Have yet to manage part B unfortunately
  5. Nov 12, 2014 #4
    Well, I didn't go through every last detail, but it looks pretty good. However, I have the following comments:
    1. Is that 7.1 British pounds per GJ, of 7.1 $ per GJ?
    2. What did you do about the 80%?
    3. Neglecting the 80% issue, I get the $ amounts for the natural gas that you got, even though the equations in your post are not complete, leaving out the sec/yr part.
    4. In part b, let T be the thickness of the insulation, and calculate the heat transfer and the economics as a function of T. Then determine the value of T that maximizes the return.

  6. Nov 13, 2014 #5

    Yes that's $/GJ and I'm unsure as to how to factor in the 80% efficiency of the oven. Thanks for pointing that out - I should definitely leave in the sec/yr calculation. This has been a big help!
  7. Nov 13, 2014 #6
    I believe I've ran into a snag with the very first calculation - surface area of the cylinder. Somehow I got 32.42m^2 instead of 36.95m^2 so here we go again!
  8. Nov 13, 2014 #7
    I used your 32.42, assuming that you got that right. So, just redo the calculation.

  9. Nov 13, 2014 #8
    yeah that's all good now, I'm still just a tad stumped as to where I throw the 80% efficiency into the mix, though :(
  10. Nov 13, 2014 #9
    Efficiency -- if a car is less efficient what does that mean? Same here 80% means what?

    Granted this is homework and that is the stated problem - but beyond the overall cost of running the of the oven, is the cost of removing the ~40KW heat in the factory. For example if the surrounding air is "maintained" at 23 C - then there is a cooling system that has to work harder to remove the heat- it also has an efficiency - and 40KW waste heat is significant.

    Also - had to laugh a little -- Question A = How much you will be paid to do this project? That is what you will "make".... Energy savings do not equal "money made" - unless you have a nice contract with your employer / customer... if you get my point.
  11. Nov 13, 2014 #10
    It isn't clear what they mean by this.
  12. Nov 13, 2014 #11
    I am quite sure ... 80% of the energy in is converted to "useful" heat. -- this inefficency comes mostly from how good the combustion is and how much heat goes out with the exhaust. So using "£7.1/GJ input and the oven's efficiency is 80%" when you use £7.1 of gas you get 1.0 GJ * 0.80 heat in the oven.

    A "perfect" oven would have 1 GJ of heat for every £7.1 spent.

    --- AND my apologies I overlooked the payment scheme - but the savings over how long...assuming 1 year? etc..
  13. Nov 13, 2014 #12
    Yeah assuming one year, sorry for slow replies this part B is taking it's toll on my sanity haha
  14. Nov 13, 2014 #13
    It shouldn't. You just do exactly the same thing you did in part a, but retain the insulation thickness as an algebraic variable. Or, just do part a over and over again for a bunch of different thicknesses, and plot a graph of the $$ return as a function of the thickness to find the minimum point.

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