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AET432-CH6-page11

# AET432-CH6-page11 - Review problems 6-59 The Coutte ﬂow...

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Unformatted text preview: Review problems 6-59 The Coutte ﬂow of a ﬂuid between two parallel plates is considered. The temperature distribution is to be sketched and determined, and the maximum temperature of the ﬂuid, as well as the temperature of the fluid at the contact surfaces with the lower and upper plates are to be determined. Assumptions Steady operating conditions exist. Properties The viscosity and thermal conductivity of the ﬂuid are given to be p = 08 N-s/m2 and kf= 0.145 W/m-K. The thermal conductivity of lower plate is given to be kp = 15 W/m-K. Analysis: (a) Insulation ‘ v9 fee. 9". ova-’0 9‘2'7'9'0 9 7‘0. vvv .‘Q' 202020:¢I¢\$2€OI§OX¢I029202631020102033301020202.10‘ r, = 40°C k,, The sketch of temperature distribution is given in the ﬁgure. We observe from this ﬁgure that there are different slopes at the interface (y = 0) because of different conductivities (k,, > kf). The slope is zero at the upper plate 02 = L) because of adiabatic condition. (b) The general solution of the relevant differential equation is obtained as follows: u=lV———)iu—=lf— L dy L 2 2 _ 2 a; if? iLJLwCI dy kf L dy kf L2 —/-l V2 2 a T:————- +C +C Zkf L2 y 1y 2 Applying the boundary conditions: . . dT T(0)—T ””17“”? 7:7 y=0 y o p kp kal =T(C2 ‘TJ (1) 2 y=L.adiabatic dT =0————-)Cl:iZ—— dy L kf L k 2 FromEq. (1), C2 :b—fC] +Tv =b—g—V—+Tx kp ' kp L Substituting the coefﬁcients, the temperature distribution becomes ...
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