Changeset 262

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Ignore:
Timestamp:
06/19/2006 12:15:24 PM (4 years ago)
Author:
powell
Message:

Addition to the tundish problem: drag force calculation.

Location:
trunk/matml/transport/problems
Files:
13 modified

Legend:

Unmodified
Added
Removed

  • trunk/matml/transport/problems/alpha.html

    r261 r262  
    266266    (<a href="tundish-solution.pdf">solution</a>, 
    267267    <a href="tundish.zip">source</a> <a href="tundish/">directory</a>, 
    268     <a href="tundish.meta">metadata</a>), last modified May 9, 2005.</li> 
     268    <a href="tundish.meta">metadata</a>), last modified June 19, 2006.</li> 
    269269  <li><a href="fen.ps">Nitriding of an iron thin film</a>, by Adam 
    270270    Powell (<a href="fen-solution.ps">solution</a>, 

  • trunk/matml/transport/problems/boundary_layers.html

    r257 r262  
    7373    (<a href="tundish-solution.pdf">solution</a>, 
    7474    <a href="tundish.zip">source</a> <a href="tundish/">directory</a>, 
    75     <a href="tundish.meta">metadata</a>), last modified May 9, 2005.</li> 
     75    <a href="tundish.meta">metadata</a>), last modified June 19, 2006.</li> 
    7676  <li><a href="drainup.ps">Tank drainage through a rising tube</a>, by Adam 
    7777    Powell (<a href="drainup-solution.pdf">solution</a>, 

  • trunk/matml/transport/problems/chrono.html

    r260 r262  
    66of most recent modification: 
    77<ul> 
     8  <li><a href="tundish.ps">New tundish design</a>, by Adam Powell 
     9    (<a href="tundish-solution.pdf">solution</a>, 
     10    <a href="tundish.zip">source</a> <a href="tundish/">directory</a>, 
     11    <a href="tundish.meta">metadata</a>), last modified June 19, 2006.</li> 
    812  <li><a href="electroporation.pdf">Water disinfection by electroporation</a>,  
    913    by Paulo Jacob Silva, Thomas Schilling, Jonathan Tejada and Joy Yuan 
    … …  
    157161    <a href="torsionviscometer.meta">metadata</a>), last modified May 9, 
    158162    2005.</li> 
    159   <li><a href="tundish.ps">New tundish design</a>, by Adam Powell 
    160     (<a href="tundish-solution.pdf">solution</a>, 
    161     <a href="tundish.zip">source</a> <a href="tundish/">directory</a>, 
    162     <a href="tundish.meta">metadata</a>), last modified May 9, 2005.</li> 
    163163  <li><a href="contflow.pdf">Batch and continuous flow reactors</a>, by 
    164164Adam 

  • trunk/matml/transport/problems/drag_force.html

    r204 r262  
    4444    (<a href="tundish-solution.pdf">solution</a>, 
    4545    <a href="tundish.zip">source</a> <a href="tundish/">directory</a>, 
    46     <a href="tundish.meta">metadata</a>), last modified May 9, 2005.</li> 
     46    <a href="tundish.meta">metadata</a>), last modified June 19, 2006.</li> 
    4747  <li><a href="particles.pdf">Settling of magnesium hydroxide particles in 
    4848    water</a>, by Adam Powell (<a href="particles-solution.pdf">solution</a>, 

  • trunk/matml/transport/problems/metals.html

    r259 r262  
    9494    (<a href="tundish-solution.pdf">solution</a>, 
    9595    <a href="tundish.zip">source</a> <a href="tundish/">directory</a>, 
    96     <a href="tundish.meta">metadata</a>), last modified May 9, 2005.</li> 
     96    <a href="tundish.meta">metadata</a>), last modified June 19, 2006.</li> 
    9797  <li><a href="fen.ps">Nitriding of an iron thin film</a>, by Adam 
    9898    Powell (<a href="fen-solution.ps">solution</a>, 

  • trunk/matml/transport/problems/reynolds_number.html

    r260 r262  
    5555    (<a href="tundish-solution.pdf">solution</a>, 
    5656    <a href="tundish.zip">source</a> <a href="tundish/">directory</a>, 
    57     <a href="tundish.meta">metadata</a>), last modified May 9, 2005.</li> 
     57    <a href="tundish.meta">metadata</a>), last modified June 19, 2006.</li> 
    5858  <li><a href="glassfloat.pdf">Plate glass casting</a>, by Adam Powell 
    5959    (<a href="glassfloat-solution.ps">solution</a>, 

  • trunk/matml/transport/problems/tundish.meta

    r169 r262  
    44Title: New tundish design 
    55Author: Adam C. Powell, IV 
    6 Date: January 5, 2003 
    7 Copyright: 1994, 2003 Adam C. Powell, IV 
     6Date: April 26, 2006 
     7Copyright: 1994, 2003, 2006 Adam C. Powell, IV 
    88License: MIT Open CourseWare, http://ocw.mit.edu/OcwWeb/Global/terms-of-use.htm 
    99Description: Characterize the average and maximum velocity, particle size for complete removal, and entrance length of a new tundish design (the "Hyers Tundish") with vertical plates, and comment on its prospects for success. 
    10 Source file: tundish.zip (md5sum 49477f47c31242288423b6302c84b6ad) 
     10Source file: tundish.zip (md5sum ede2f7146fb4b217a095f30e1b6c8db6) 
    1111Source directory: tundish 
    1212Problem source format: LaTeX 
    13 Problem source file: tundish/tundish.tex (md5sum b685b24b12da54cf9fbd6a268d11e934) 
     13Problem source file: tundish/tundish.tex (md5sum cfca9e724a1fda9ad6d04e4a63438ab0) 
    1414Solution source format: (PDF)LaTeX 
    15 Solution source file: tundish/tundish-solution.tex (md5sum 918bad54318cd83b11f219078e884025) 
     15Solution source file: tundish/tundish-solution.tex (md5sum 3369ef9e2115f1421fce7036bf68e7c4) 
    1616Problem printable format: Postscript 
    17 Problem printable file: tundish.ps (md5sum e03c6b0fd08930b06bc8b101fa1c07d0) 
     17Problem printable file: tundish.ps (md5sum 9d48986b017ca04e26bce54043d019a4) 
    1818Solution printable format: PDF 
    19 Solution printable file: tundish-solution.pdf (md5sum b32ce70abd07bdb227133a35be4f26de) 
     19Solution printable file: tundish-solution.pdf (md5sum 63206355ad8d50e0554620d41e95a942) 
    2020Difficulty level: Junior 
    21 Solution time: 2 hours 
     21Solution time: 2.5 hours 
    2222Keywords: Viscous flow, Reynolds number, Drag force, Boundary layers, Metal 
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  • trunk/matml/transport/problems/tundish.ps

    r169 r262  
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    3373 b Fi(=)23 b(1m)307 2906 y Fg(\017)41 b Fi(Flo)n(w)27 
    3374 b(rate:)36 b Ff(Q)23 b Fi(=)g(2)1026 2873 y Fh(liters)p 
    3375 1025 2887 146 4 v 1055 2935 a(sec)1208 2906 y Fi(\(=0.002)1506 
    3376 2873 y Fh(m)1561 2848 y Fe(3)p 1504 2887 87 4 v 1534 
    3377 2935 a Fh(s)1601 2906 y Fi(\))243 3071 y(\(a\))41 b(Using)31 
     3320b Ff(L)22 b Fi(=)h(1)p Ff(:)p Fi(5m,)k Ff(W)1988 2881 
     3321y Fc(c)2045 2869 y Fi(=)c(0)p Ff(:)p Fi(15m,)j Ff(W)35 
     3322b Fi(=)23 b(1m)307 3012 y Fg(\017)41 b Fi(Flo)n(w)27 
     3323b(rate:)36 b Ff(Q)23 b Fi(=)g(2)1026 2979 y Fh(liters)p 
     33241025 2993 146 4 v 1055 3040 a(sec)1208 3012 y Fi(\(=0.002)1506 
     33252979 y Fh(m)1561 2954 y Fe(3)p 1504 2993 87 4 v 1534 
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    33793328(assuming)f(that)h(\015o)n(w)f(is)g(distributed)h(equally)g(to)f(eac)n 
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    3382 (through)f(those)g(c)n(hannels.)238 3292 y(\(b\))42 b(Assuming)24 
     3331(through)f(those)g(c)n(hannels.)238 3427 y(\(b\))42 b(Assuming)24 
    33833332b(\015o)n(w)g(is)g(fully-dev)n(elop)r(ed,)h(what)f(is)h(the)f(Reynolds) 
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    3385 (hannels?)390 3391 y(What)31 b(w)n(ould)f(the)h(Reynolds)f(n)n(um)n(b)r 
     3334(hannels?)390 3526 y(What)31 b(w)n(ould)f(the)h(Reynolds)f(n)n(um)n(b)r 
    33863335(er)g(b)r(e)h(without)g(the)g(v)n(ertical)e(plates)h(b)r(et)n(w)n(een)h 
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     3336(the)f(c)n(hannels)g(\(that)h(is,)390 3626 y(one)21 b(big)g(op)r(en)g 
    33883337(b)r(o)n(x\)?)35 b(Will)22 b(the)f(\015o)n(w)g(lik)n(ely)f(b)r(e)i 
    33893338(laminar)e(in)h(eac)n(h)g(situation)g(\(with)h(and)f(without)g(the)h 
    3390 (plates\)?)247 3612 y(\(c\))42 b(T)-7 b(reating)39 b(a)g(c)n(hannel)g 
     3339(plates\)?)247 3759 y(\(c\))42 b(T)-7 b(reating)39 b(a)g(c)n(hannel)g 
    33913340(as)f(a)h(pair)g(of)h(parallel)e(plates)h(with)h(fully-dev)n(elop)r(ed) 
    3392 f(steady-state)f(laminar)h(\015o)n(w)390 3712 y(b)r(et)n(w)n(een)28 
     3341f(steady-state)f(laminar)h(\015o)n(w)390 3858 y(b)r(et)n(w)n(een)28 
    33933342b(them,)g(calculate)f(the)h(maxim)n(um)f(v)n(elo)r(cit)n(y)g(of)h 
    3394 (molten)f(metal)h(in)g(a)f(c)n(hannel.)238 3833 y(\(d\))42 
     3343(molten)f(metal)h(in)g(a)f(c)n(hannel.)238 3991 y(\(d\))42 
    33953344b(Consider)32 b(a)g(ceramic)g(particle)g(whic)n(h)h(en)n(ters)f(a)g(c)n 
    33963345(hannel)g(at)h(the)g(b)r(ottom)g(cen)n(ter)f(and)h(rises)f(through)g 
    3397 (the)390 3932 y(molten)f(steel)f(as)g(it)h(is)f(b)r(eing)h(carried)e 
     3346(the)390 4091 y(molten)f(steel)f(as)g(it)h(is)f(b)r(eing)h(carried)e 
    33983347(forw)n(ard)f(b)n(y)i(the)h(\015o)n(w)f(of)g(metal)h(in)f(the)h(cen)n 
    3399 (ter)f(of)g(the)h(c)n(hannel.)45 b(Its)390 4032 y Ff(y)s 
     3348(ter)f(of)g(the)h(c)n(hannel.)45 b(Its)390 4190 y Ff(y)s 
    34003349Fi(-direction)26 b(v)n(elo)r(cit)n(y)f(is)i(the)g(maxim)n(um)f(v)n(elo) 
    34013350r(cit)n(y)g(of)g(the)h(\015uid)g(\(since)g(it)g(is)f(in)h(the)g(cen)n 
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     3351(ter)f(of)g(the)h(c)n(hannel\),)390 4290 y(and)22 b(its)f 
    34033352Ff(z)t Fi(-direction)f(v)n(elo)r(cit)n(y)h(is)h(its)f(up)n(w)n(ard)g 
    34043353(terminal)g(v)n(elo)r(cit)n(y)-7 b(.)34 b(What)22 b(is)g(the)g(minim)n 
    3405 (um)g(up)n(w)n(ard)e(terminal)390 4231 y(v)n(elo)r(cit)n(y)26 
     3354(um)g(up)n(w)n(ard)e(terminal)390 4390 y(v)n(elo)r(cit)n(y)26 
    34063355b(this)h(particle)f(can)h(ha)n(v)n(e)e(and)i(still)g(reac)n(h)e(the)i 
    34073356(top)g(b)r(efore)f(it)h(gets)g(to)f(the)h(end)g(of)g(the)g(c)n(hannel?) 
    3408 36 b(\(If)390 4331 y(y)n(ou)26 b(didn't)g(get)g(an)g(answ)n(er)f(to)h 
     335736 b(\(If)390 4489 y(y)n(ou)26 b(didn't)g(get)g(an)g(answ)n(er)f(to)h 
    34093358(part)f(1c,)h(use)g(y)n(our)f(v)-5 b(alue)26 b(from)g(part)f(1a)g(as)h 
    34103359(the)g(molten)g(steel)h(\015o)n(w)e(v)n(elo)r(cit)n(y)390 
    3411 4431 y(in)j(the)g(cen)n(ter.\))247 4552 y(\(e\))42 b(What)26 
     33604589 y(in)j(the)g(cen)n(ter.\))247 4722 y(\(e\))42 b(What)26 
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    3414 (Stok)n(es)390 4651 y(\015o)n(w)i(around)g(the)h(particle,)f(b)r(e)h 
    3415 (sure)f(to)g(c)n(hec)n(k)g(it!\))252 4772 y(\(f)6 b(\))43 
    3416 b(Calculate)21 b(the)g(en)n(trance)f(length)i(for)e(\015o)n(w)h(in)g 
    3417 (these)g(c)n(hannels.)34 b(What)22 b(do)r(es)e(this)i(tell)f(y)n(ou)g 
    3418 (ab)r(out)g(the)g(v)-5 b(alidit)n(y)390 4872 y(of)28 
    3419 b(our)f(assumptions)f(ab)r(o)n(v)n(e?)p 0 4995 1560 4 
    3420 v 92 5049 a Fb(1)127 5072 y Fa(This)f(problem)f(w)n(as)h(inspired)g(b)n 
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    3422 (the)f(Univ)n(ersit)n(y)g(of)g(Massac)n(h)n(usetts)i(at)e(Amherst\),)g 
    3423 (and)h(the)0 5151 y(design)e(sho)n(wn)g(here)g(is)f(informally)f(kno)n 
    3424 (wn)j(as)e(the)i(\\Hy)n(ers)f(T)-6 b(undish".)1929 5400 
    3425 y Fi(1)p eop end 
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     3370Fi(1)p eop end 
     3371%%Page: 2 2 
     3372TeXDict begin 2 1 bop 252 83 a Fi(\(f)6 b(\))43 b(Calculate)21 
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     3377(the)g(plates)f(\(not)h(the)g(ends\),)i(and)e(assuming)f(uniform)g(v)n 
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     3379(hannels,)f(estimate)h(the)g(drag)e(force)h(on)g(a)g(single)g(plate.) 
     33801929 5400 y(2)p eop end 
    34263381%%Trailer 
    34273382 

  • trunk/matml/transport/problems/tundish/tundish-solution.tex

    r137 r262  
    66 
    77  \begin{enumerate} 
    8   \item \label{uav} Average velocity is flow rate divided by total cross 
    9     section area of the four channels: 
     8  \item Average velocity is flow rate divided by total cross section area of 
     9    the four channels: 
    1010    $$u_{av} = \frac{Q}{A} = \frac{\rm 0.002\frac{m^3}{s}}{\rm 
    1111      4\times0.15m\times1m} = 0.0033\frac{\rm m}{\rm s}.$$ 
    … …  
    2424    more likely to be turbulent without the channels than with them. 
    2525 
    26   \item \label{maxu} In well-behaved channel flow between parallel plates 
    27     (laminar, fully-developed, steady-state, no edge effects, etc.), average 
    28     velocity is two-thirds of maximum velocity, so maximum velocity is 
     26  \item In well-behaved channel flow between parallel plates (laminar, 
     27    fully-developed, steady-state, no edge effects, etc.), average velocity is 
     28    two-thirds of maximum velocity, so maximum velocity is 
    2929    $$u_{max} = \frac{3}{2}u_{av} = 0.005\frac{\rm m}{\rm s}.$$ 
    3030 
    … …  
    3737    \frac{\rm 1m \cdot 0.005\frac{m}{s}}{\rm 1.5m} = 
    3838    0.00333\frac{\rm m}{\rm s}$$ 
     39 
     40  \item With a Reynolds number of 673, the entrance length $L_e$ goes as: 
     41    $$\frac{L_e}{H}=\frac{{\rm Re}_H}{100},$$ 
     42    so the entrance length is about 6.7 times the 0.15m channel width, or about 
     43    a meter.  Flow is therefore not likely to be fully-developed, since this is 
     44    a large fraction of the 1.5m channel length. 
     45 
     46    The real bad news of this is that uneven flow characteristics at the 
     47    entrance will persist throughout much or all of the channels, and 
     48    high-velocity regions will flow through quickly and ``short circuit'' the 
     49    attempt to float out the ceramic particles.  It's not clear at this point 
     50    whether this difficulty can be overcome so the tundish design will succeed. 
    3951 
    4052  \item For Stokes flow, the relationship was derived in Problem Set 8: 
    … …  
    5567    diameter of about 100 microns, which is quite good! 
    5668 
    57   \item With a Reynolds number of 673, the entrance length $L_e$ goes as: 
    58     $$\frac{L_e}{H}=\frac{{\rm Re}_H}{100},$$ 
    59     so the entrance length is about 6.7 times the 0.15m channel width, or about 
    60     a meter.  Flow is therefore not likely to be fully-developed, since this is 
    61     a large fraction of the 1.5m channel length. 
     69  \item This drag force needs to be calculated in two parts: the entrance 
     70    region in the first meter, and the fully-developed region from there to the 
     71    end. 
    6272 
    63     The real bad news of this is that uneven flow characteristics at the 
    64     entrance will persist throughout much or all of the channels, and 
    65     high-velocity regions will flow through quickly and ``short circuit'' the 
    66     attempt to float out the ceramic particles.  It's not clear at this point 
    67     whether this difficulty can be overcome so the tundish design will succeed. 
     73    In the entrance region, drag on a flat plate provides a means of 
     74    calculating the force.  Using 1 meter as the length and the average 
     75    velocity as the far stream velocity, the global/average Reynolds number is: 
     76    $${\rm Re}_L = \frac{\rho U_\infty L}{\mu} = 
     77    \frac{\rm7000\frac{kg}{m^3}0.0033\frac{m}{s}\cdot1m} 
     78    {\rm5.2\times10^{-3}\frac{kg}{m\cdot s}} = 4483$$ 
     79    Because this is much less than 10$^5$, the laminar correlation works: 
     80    $$f_L = \frac{1.328}{\sqrt{{\rm Re}_L}} = 0.198;\  
     81    F_d = f_lKA = {\rm 0.198 \cdot \frac{1}{2}7000\frac{kg}{m^3} 
     82      (0.0033\frac{m}{s})^2 \cdot 1m\times1m = 0.0015 N}$$ 
     83    In the fully-developed region, flow is laminar and parabolic, so measuring 
     84    $x$ from the center of a channel, $y$-velocity is given by: 
     85    $$u_y = u_{max}\left(1-\frac{x^2}{\left(\frac{W_c}{2}\right)^2}\right),$$ 
     86    and the shear stress $\tau_{xy}$, which is uniform over the plate surface 
     87    in this region, is calculated using the Newtonian fluid definition: 
     88    $$\tau_{xy} = 
     89    -\mu\left.\frac{\partial u_y}{\partial x}\right|_{x=\frac{W_c}{2}} = 
     90    -\frac{\mu u_{max}}{\left(\frac{W_c}{2}\right)^2}[-2x]_{x=\frac{W_c}{2}} = 
     91    \frac{\mu u_{max}}{4W_c} = 
     92    \frac{\rm5.2\times10^{-3}\frac{kg}{m\cdot s}\cdot0.005\frac{m}{s}} 
     93    {\rm4\cdot0.15m} = 4.3\times10^{-6}\frac{\rm N}{\rm m^2}.$$ 
     94    This over a half square meter adds a negligible amount of force, which 
     95    shows that most of the force is applied near the entrance where the 
     96    boundary layer is thinest. 
     97     
     98    For the two-sided plate we double this, and the force is a puny 3 
     99    milliNewtons!  This is nothing one need worry about. 
    68100  \end{enumerate} 
    69101\end{enumerate} 
    70102\end{document} 
     103%%% Local Variables: 
     104%%% TeX-PDF-mode: t 
     105%%% End: 

  • trunk/matml/transport/problems/tundish/tundish.tex

    r169 r262  
    5959  \item Calculate the entrance length for flow in these channels.  What does 
    6060    this tell you about the validity of our assumptions above? 
     61 
     62  \item Considering only the faces of the plates (not the ends), and assuming 
     63    uniform velocity at the entrances of the channels, estimate the drag force 
     64    on a single plate. 
    6165  \end{enumerate} 
    6266\end{enumerate} 

  • trunk/matml/transport/problems/viscous_flow.html

    r260 r262  
    4242    (<a href="tundish-solution.pdf">solution</a>, 
    4343    <a href="tundish.zip">source</a> <a href="tundish/">directory</a>, 
    44     <a href="tundish.meta">metadata</a>), last modified May 9, 2005.</li> 
     44    <a href="tundish.meta">metadata</a>), last modified June 19, 2006.</li> 
    4545  <li><a href="polyflow.pdf">Non-Newtonian polymer flow in a channel</a>, by 
    4646    Adam Powell (<a href="polyflow-solution.ps">solution</a>,