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Unformatted text preview: PROBLEM 4.72
KNOWN: Straight fin of uniform cross section with insulated end. FIND: (:1) Temperature distribution using finiteedifference method and validity of assuming one—
dimensional heat transfer, (b) Fin heat transfer rate and comparison with analytical solution, Eq. 3.76, (c)
Effect of convection coefﬁcient on ﬁn temperature distribution and heat rate. SCHEMATIC:
Tm: OC
_: h = 500 Wimz K
Tb = 100 00 —o 2 {
LR = 50 WimK I W
x L = 48 mm ASSUMPTIONS: (l) Steadystate COnditions, (2) Onedimensional conduction in fin, (3) Constant
properties, (4) Uniform ﬁlm coefﬁcient. ANALYSIS: (3) From the analysis of Problem 4.45. the finitedifference equations for the nodal
arrangement can be directly written. For the nodal spacing Ax = 4 mm, there will be 12 nodes. With E
3) w representing the distance normal to the page, 2
LP .szo h” szzﬂAx2=———500W/m KX23 (4X10_3mm)=0.0533
kAo kf'w kw 50W/mKx6x10_ m Node I: 100+T2 + 0.0533X30(2+0.0533)T1= 0 or 2.053T. + '1‘; = 101.6
Node in: Tn+1+Tn_1+1.60A2.0533Tn : 0 or Tn_1o2.053Tn +T,,_] = 71.60
Node I2: T1 1 + (0.0533/2)30—(0.0533/2+ 1)le = 0 or T11 —l.0267T;2 = —0.800 Using matrix notation, Eq. 4.52, where {A} [T] = [C]. the Amatrix is ttidiagonal and only the nonzero
terms are shown below. A matrix inversion routine was used to obtain [T]. Tridiagonal Matrix A Cofumn Matrices
Nonzero Terms Values Node C T a” 21].; 4.053 1 l —lOl.6 85.8 all an an 1 2.053 1 2 l .6 74.5 3.3.; 33.3 33.4 1 2.053 1 3 l a“ 34,4 34,5 1 2.053 1 4 l.6 58.6 314 355 35,5 1 l 5 ads 3“ am 1 2.053 1 6 —l .6 48.8 37.5 31'"; 3.7.3 1 l 7  1 213‘? 33,3 3,3,9 1 l 8 ] 39,3 ago all") 1 l 9 l.6 41.2 310.9 3mm am.“ 1 2.053 I 10 l .6 39.9
an‘go all,“ am; I I 312.11 312,12 al2.13 I  l .027 1 12 0.8 38.9 The assumption of onedimensional heat conduction is justiﬁed when Bi 5 h(wf2)fk < 0.1. Hence. with
Bi = 500 wxm3.K(3 x 103 m)f50 WlmK = 0.03, the assumption is reasonable. PROBLEM 4.72 (Cont) (b) The ﬁn heat rate can be most easily found from an energy balance on the control volume about Node
0. I f r T _T AX
(if =QI+Qconv _k'“' D 1+h[2 ](T0 T00) Air 2
o —3
1 —85. c o
q} 2SOW/mK(6x10_3m)M+500W/m2K 2324“ (100730) c
4x10‘ in 2
q} =(1065+140)w/m =1205 w/m. < From Eq. 3.76. the fin heat rate is
q = (thAC)UZ 9btanh mL. Substituting numerical values with P = 2(w + t): 2 e and A, = w t , m = (hP/m)1’2 = 57.74 m'I and M
= (hpmcim = 17.321? W/K. Hence. with at, = 70°C. q’ = 17.32 w/Kx70thanh(57.44x0.043) : 1203 win
and the ﬁnitedifference result agrees very well with the exact (analytical) solution. (c) Using the IHT F infreDiﬁerence Equations Tao! Pad for ID, SS conditions. the fin temperature distribution and heat rate were computed for h = 10. 100, 500 and 1000 WlmzK. Results are plotted as
follows. 100 1300
90
G 1500
g 80
m— n
g 7° ,5 1200
g 60 E
E 50 500
a 6!
'— 40 t E
<6 600
30 g
300
+ h: 10 waer 0 . ,
—e— n = 100 wirmnaK o 2 4
+ h: 500 WImAZK 00 00 500 300 1000
E' h = 1000 WW3" Convection coefficient, htwtmnaK) The temperature distributions were obtained by ﬁrst creating 3 Lookup Table consisting of 4 rows of nodal temperatures corresponding to the 4 values of h and then using the LO0KUPVAL2 interpolating
function with the Explore feature of the IHT menu. Speciﬁcally, the function T_EVAL = LOOKUPVAL2(tO467. h, x) was entered into the workspace, where t0467 is the ﬁle name given to the
Lookup Table. For each value of h, Explore was used to compute T(x), thereby generating 4 data sets which were placed in the Brewer and used to generate the plots. The variation of q’ with h was simply generated by using the Explore feature to solve the ﬁnitedifference model equations for values of h
incremented by 10 from 10 to 1000 wmﬁs. Although q’f increases with increasing h, the effect of changes in h becomes less pronounced. This trend is a consequence of the reduction in ﬁn temperatures, and hence the fin efficiency, with increasing h. For 10 S h S 1000 WimZK. 0.95 2 1"; 0.24. Note the nearly isothermal ﬁn for h = 10 WlmzK and
the pronounced temperature decay for h = 1000 W/mEK. ...
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 Fall '08
 Rothamer
 Thermodynamics, Heat, Heat Transfer, Thermal conductivity, heat rate

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