lMaterialsForStudentsQuiz

8/18/2019 lMaterialsForStudentsQuiz

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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


4/8

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


5/8

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


6/8

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

219


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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.

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