lmaterialsforstudentsquiz
TRANSCRIPT
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1
I. EXTERNAL FLOW
Configuration N u Correlation Comment
Flat plate, laminar, const. T w N ux 0.332 Re1/2x P r1/3 0.6 P r 60, Re < R
Flat plate, laminar, const. T w N uL 0.664 Re1/2
L P r1/3
0.6 P r 60, Re < RFlat plate, laminar, const. q w N ux 0.453 Re
1/2x P r1/3 0.6 P r 60, Re < R
Flat plate, laminar, const. q w N uL 0.68 Re1/2L P r
1/3 0.6 P r 60, Re < R
Flat plate, turbulent, const. T w N ux 0.0296 Re4/5x P r1/3 0.6 P r 60, Re > R
Flat plate, turbulent, const. q w N ux 0.0308 Re4/5x P r1/3 0.6 P r 60, Re > R
Flat plate, turbulent N uL (0.037 Re4/5L − 871)P r
1/3 0.6 P r 60, Re > RRecr = 5× 105
Flat plate, laminar, const. T w N ux 0.564 Re1/2x P r1/2 P r 0.05, RexP r > 10
Flat plate, laminar, const. q w N ux 0.886 Re1/2x P r1/2 P r 0.05, RexP r > 10
Cylinder, cross flow: NuD = 0.3+ 0.62Re1/2D P r
1/3
[1 + (0.4/P r)2/3]1/4
1 +
ReD282000
5/84
/5
. (valid
when ReDP r > 0.2)
Sphere: NuD = 2 +
0.4Re1/2D + 0.06Re
2/3D
P r0.4
µ
µs
1/4
. (valid for 0.71 P r 380,
3.5 ReD 7.6 × 104, 1 µ/µs 3.2)
FIG. 1: NuD = CRemDPr
n
P r
Prs
1/4
, n = 0.37 if P r ≤ 10 and n = 0.36 if P r ≥ 10. All properties
except P rs are evluated at T ∞. P rs alone is evaluated at the surface temperature.
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714 Appendix A: Some thermophysical properties of selected materials
Table A.6 Thermophysical properties of gases at atmosphericpressure (101325 Pa)
T (K) ρ (kg/m3) cp (J/kg·K) µ (kg/m·s) ν (m2/s) k (W/m·K) α (m2/s) Pr
Air
100 3.605 1039 0.711×10−5 0.197×10−5 0.00941 0.251× 10−5 0.784
150 2.368 1012 1.035 0.437 0.01406 0.587 0.745
200 1.769 1007 1.333 0.754 0.01836 1.031 0.731
250 1.412 1006 1.606 1.137 0.02241 1.578 0.721
260 1.358 1006 1.649 1.214 0.02329 1.705 0.712
270 1.308 1006 1.699 1.299 0.02400 1.824 0.712
280 1.261 1006 1.747 1.385 0.02473 1.879 0.711290 1.217 1006 1.795 1.475 0.02544 2.078 0.710
300 1.177 1007 1.857 1.578 0.02623 2.213 0.713
310 1.139 1007 1.889 1.659 0.02684 2.340 0.709
320 1.103 1008 1.935 1.754 0.02753 2.476 0.708
330 1.070 1008 1.981 1.851 0.02821 2.616 0.708
340 1.038 1009 2.025 1.951 0.02888 2.821 0.707
350 1.008 1009 2.090 2.073 0.02984 2.931 0.707
400 0.8821 1014 2.310 2.619 0.03328 3.721 0.704
450 0.7840 1021 2.517 3.210 0.03656 4.567 0.703
500 0.7056 1030 2.713 3.845 0.03971 5.464 0.704
550 0.6414 1040 2.902 4.524 0.04277 6.412 0.706600 0.5880 1051 3.082 5.242 0.04573 7.400 0.708
650 0.5427 1063 3.257 6.001 0.04863 8.430 0.712
700 0.5040 1075 3.425 6.796 0.05146 9.498 0.715
750 0.4704 1087 3.588 7.623 0.05425 10.61 0.719
800 0.4410 1099 3.747 8.497 0.05699 11.76 0.723
850 0.4150 1110 3.901 9.400 0.05969 12.96 0.725
900 0.3920 1121 4.052 10.34 0.06237 14.19 0.728
950 0.3716 1131 4.199 11.30 0.06501 15.47 0.731
1000 0.3528 1142 4.343 12.31 0.06763 16.79 0.733
1100 0.3207 1159 4.622 14.41 0.07281 19.59 0.736
1200 0.2940 1175 4.891 16.64 0.07792 22.56 0.738
1300 0.2714 1189 5.151 18.98 0.08297 25.71 0.738
1400 0.2520 1201 5.403 21.44 0.08798 29.05 0.738
1500 0.2352 1211 5.648 23.99 0.09296 32.64 0.735
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Figure5.10 The heat removal from suddenly-cooled bodies as
a function of h and time. 213
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F
i g u r e
5 . 7
T h e t r a n s i e n t t e m p e r a t u r e d i s t r i b u t i o n i n a s l a b
a t s i x
p o s i t i o n s : x / L
=
0 i s t h e c e n t e r ,
x
/ L
=
1 i s o n e o u t s i d e b o u n d a r y .
20
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F
i g u r e
5 . 8
T h e t r a n s i e n t t e m p e r
a t u r e d i s t r i b u t i o n i n a l o n g c y l i n d e r
o f r a d i u s r o
a t s i x p o s i t i o n s :
r
/ r o
=
0 i s t h e c e n t e r l i n e ; r / r o
=
1 i s t h e o u t s i d e b o u n d a r y .
210
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F
i g u r e
5 . 9
T h e t r a n s i e n t t e m p e r a
t u r e d i s t r i b u t i o n i n a s p h e r e o f r a
d i u s r o
a t s i x p o s i t i o n s : r / r o
= 0
i
s t h e c e n t e r ; r / r o
=
1 i s t h e o u t s i
d e b o u n d a r y .
21
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Table 5.2 One-term coefficients for convective cooling [5.1].
Plate Cylinder SphereBi
λ̂1 A1 D1 λ̂1 A1 D1 λ̂1 A1 D1
0.01 0.09983 1.0017 1.0000 0.14124 1.0025 1.0000 0.17303 1.0030 1.0000
0.02 0.14095 1.0033 1.0000 0.19950 1.0050 1.0000 0.24446 1.0060 1.0000
0.05 0.22176 1.0082 0.9999 0.31426 1.0124 0.9999 0.38537 1.0150 1.0000
0.
10 0.31105 1.0161 0.9998 0.44168 1.0246 0.9998 0.54228 1.0298 0.99980.15 0.37788 1.0237 0.9995 0.53761 1.0365 0.9995 0.66086 1.0445 0.9996
0.20 0.43284 1.0311 0.9992 0.61697 1.0483 0.9992 0.75931 1.0592 0.9993
0.30 0.52179 1.0450 0.9983 0.74646 1.0712 0.9983 0.92079 1.0880 0.9985
0.40 0.59324 1.0580 0.9971 0.85158 1.0931 0.9970 1.05279 1.1164 0.9974
0.50 0.65327 1.0701 0.9956 0.94077 1.1143 0.9954 1.16556 1.1441 0.9960
0.60 0.70507 1.0814 0.9940 1.01844 1.1345 0.9936 1.26440 1.1713 0.9944
0.70 0.75056 1.0918 0.9922 1.08725 1.1539 0.9916 1.35252 1.1978 0.9925
0.80 0.79103 1.1016 0.9903 1.14897 1.1724 0.9893 1.43203 1.2236 0.9904
0.90 0.82740 1.1107 0.9882 1.20484 1.1902 0.9869 1.50442 1.2488 0.9880
1.00 0.86033 1.1191 0.9861 1.25578 1.2071 0.9843 1.57080 1.2732 0.9855
1.10 0.89035 1.1270 0.9839 1.30251 1.2232 0.9815 1.63199 1.2970 0.9828
1.20 0.91785 1.1344 0.9817 1.34558 1.2387 0.9787 1.68868 1.3201 0.9800
1.30 0.94316 1.1412 0.9794 1.38543 1.2533 0.9757 1.74140 1.3424 0.9770
1.40 0.96655 1.1477 0.9771 1.42246 1.2673 0.9727 1.79058 1.3640 0.9739
1.50 0.98824 1.1537 0.9748 1.45695 1.2807 0.9696 1.83660 1.3850 0.9707
1.60 1.00842 1.1593 0.9726 1.48917 1.2934 0.9665 1.87976 1.4052 0.9674
1.80 1.04486 1.1695 0.9680 1.54769 1.3170 0.9601 1.95857 1.4436 0.9605
2.00 1.07687 1.1785 0.9635 1.59945 1.3384 0.9537 2.02876 1.4793 0.9534
2.20 1.10524 1.1864 0.9592 1.64557 1.3578 0.9472 2.09166 1.5125 0.9462
2.40 1.13056 1.1934 0.9549 1.68691 1.3754 0.9408 2.14834 1.5433 0.9389
3.00 1.19246 1.2102 0.9431 1.78866 1.4191 0.9224 2.28893 1.6227 0.9171
4.00 1.26459 1.2287 0.9264 1.90808 1.4698 0.8950 2.45564 1.7202 0.8830
5.00 1.31384 1.2402 0.9130 1.98981 1.5029 0.8721 2.57043 1.7870 0.8533
6.00 1.34955 1.2479 0.9021 2.04901 1.5253 0.8532 2.65366 1.8338 0.8281
8.00 1.39782 1.2570 0.8858 2.12864 1.5526 0.8244 2.76536 1.8920 0.7889
10.00 1.42887 1.2620 0.8743 2.17950 1.5677 0.8039 2.83630 1.9249 0.7607
20.00 1.49613 1.2699 0.8464 2.28805 1.5919 0.7542 2.98572 1.9781 0.6922
50.00 1.54001 1.2727 0.8260 2.35724 1.6002 0.7183 3.07884 1.9962 0.6434
100.00 1.55525 1.2731 0.8185 2.38090 1.6015 0.7052 3.11019 1.9990 0.6259∞ 1.57080 1.2732 0.8106 2.40483 1.6020 0.6917 3.14159 2.0000 0.6079
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§5.3 Transient conduction in a one-dimensional slab 20
Table 5.1 Terms of series solutions for slabs, cylinders, and
spheres. J 0 and J 1 are Bessel functions of the first kind.
An f n Equation for λ̂n
Slab 2sin λ̂n
λ̂n + sin λ̂n cos λ̂ncos
λ̂n
x
L
cot λ̂n =
λ̂n
BiL
Cylinder 2J 1(λ̂n)
λ̂n
J 20(λ̂n)+ J
21(λ̂n)
J 0λ̂n
r
r o
λ̂n J 1(λ̂n) = Bir o J 0(λ̂n)
Sphere 2 sin λ̂n − λ̂n cos λ̂n
λ̂n − sin λ̂n cos λ̂n
r o
λ̂n r
sin
λ̂n r
r o
λ̂n cot λ̂n = 1− Bir o
The solution is somewhat harder to find than eqn. (5.33) was, but the
result is4
Θ =
∞n=1
exp−λ̂2n Fo
2sin λ̂n cos[λ̂n(ξ − 1)]λ̂n + sin λ̂n cos λ̂n
(5.34)
where the values of λ̂n are given as a function of n and Bi = hL/k by the
transcendental equation
cot λ̂n =λ̂n
Bi (5.35)
The successive positive roots of this equation, which are λ̂n = λ̂1, λ̂2,
λ̂3, . . . , depend upon Bi. Thus,Θ = fn(ξ,Fo,Bi), as we would expect. This
result, although more complicated than the result for b.c.’s of the first
kind, still reduces to a single term for Fo 0.2.
Similar series solutions can be constructed for cylinders and spheres
that are convectively cooled at their outer surface, r = r o. The solutions
for slab, cylinders, and spheres all have the form
Θ =T − T ∞
T i−T ∞
=
∞
n=1
An exp−λ̂2n Fof n (5.36)
where the coefficients An, the functions f n, and the equations for the
dimensionless eigenvalues λ̂n are given in Table 5.1.
4See, for example, [5.1, §2.3.4] or [5.2, §3.4.3] for details of this calculation.