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  MODULE 6: Worked-out Problems Problem 1: For laminar free convection from a heated vertical surface, the local convection coefficient may be expressed as h x =Cx -1/4 , where h x  is the coefficient at a distance x from the leading edge of the surface and the quantity C, which depends on the fluid properties, is independent of x. Obtain an expression for the ratio , where is the average coefficient between the leading edge (x=0) and the x location. Sketch the variation of h x  and with x.  x x  hh /   x h   x h    Schematic: Boundary layer,h x =C x-1/4  where C is a constant T s x     Analysis:  It follows that average coefficient from 0 to x is given by x1/4xx01/4x0xx h34Cx34x3/4xC34hdxxxCdxhx1h        Hence 34hh xx    The variation with distance of the local and average convection coefficient is shown in the sketch.  x x  hh   x x  hh 34   4/1   Cxh  x   Comments:  note that x =4/3,   independent of x. hence the average coefficients for an entire plate of length L is =4/3L, where h L  is the local coefficient at x=L. note also that the average exceeds the local. Why?  hh  x /  L h    Problem 2: Experiments to determine the local convection heat transfer coefficient for uniform flow normal to heated circular disk have yielded a radial  Nusselt number distribution of the form         noD rra1k h(r)DNu  Where n and a are positive. The Nusselt number at the stagnation  point is correlated in terms of the Reynolds number (Re D =VD/  ) and Prandtl 0.361/2Do Pr0.814Rek 0)Dh(rNu    Obtain an expression for the average Nusselt number,, corresponding to heat transfer from an isothermal disk. Typically  boundary layer development from a stagnation point yields a decaying convection coefficient with increasing distance from the stagnation  point. Provide a plausible for why the opposite trend is observed for the disk.   k  Dh Nu  D /     Known:  Radial distribution of local convection coefficient for flow normal to a circular disk. Find: Expression for average Nusselt number. Schematic: Assumptions: Constant properties. Analysis: The average convection coefficient is    oos r0no2n23ono0r02oAss 2)r(nar2rrkNuoh]2 π 2 π r)a(r/r[1Nu Dk  π r1hhdAA1h     Where Nu o  is the Nusselt number at the stagnation point (r=0).hence,   0.361/2Doro2n2o Pr2)]0.841Re2a/(n[1NuD 2)]2a/(n[1NuNuD ror2)(na2r/ro2Nuk DhNuD o        Comments: The increase in h(r) with r may be explained in terms of the sharp turn, which the boundary layer flow must take around the edge of the disk. The boundary layer accelerates and its thickness decreases as it makes the turn, causing the local convection coefficient to increase.
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