HW-3-key-2012

HW-3-key-2012 - ChE 306 HW-3 Solution(2012 PROBLEM 4.5...

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ChE 306 HW-3 Solution (2012) PROBLEM 4.5 KNOWN: Boundary conditions on four sides of a rectangular plate. FIND: Temperature distribution. SCHEMATIC: ANALYSIS: This problem differs from the one solved in Section 4.2 only in the boundary condition at the top surface. Defining θ = T – T , the differential equation and boundary conditions are g G ± g G ² ³ g G ± G µ u U¶u·¸¹ µ u U¶º·¸¹ µ u U¶»·u¹ µ u ¼ ½¾ ½¿ À ¿Á µ Ã Ä Å (1 a, b, c, d) The solution is identical to that in Section 4.2 through Equation (4.11), U µ Æ Ç È ÉÊË ÈÌÍ Î ÉÊËÏ ÈÌÐ Î Ñ ÈÁÒ (2) To determine C n , we now apply the top surface boundary condition, Equation (1d). Differentiating Equation (2) yields ÓU Ó¸ À ¿Á µ Ô Ç È ÕÖ × ÉÊË ÕÖ² × ØÙÉÏ ÕÖÚ × Ñ ÈÁÒ ¶Û¹ Substituting this into Equation (1d) results in Ã Ä Å Ü µ Ô Ý È ÉÊË ÕÖ² × Ñ ÈÁÒ where A n = Ç È ¶ÕÖÞ×¹ØÙÉ϶ÕÖÚ × ß ¹ . The principles expressed in Equations (4.13) through
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(4.16) still apply, but now with reference to Equation (4) and Equation (4.14), we should choose f(x) = g G u ±U ² ³ ´ µ¶· ¸ ¹º» ´¼½ ¾ ¿ ½ . Equation (4.16) then becomes À ´ ¸ g G u ± Á ¹º» ÂÃÄ Å ¿ ½ ¾ Æ Á ¹º» Ç ÂÃÄ Å ¾ Æ ¿ ½ ¸ g G u ±Ã ȵÉÊ· ´ËÌ Í Ê Â Thus Î ´ ¸ È Ï ÐÑ u Ò µÓÌ· ÔÕÖ ËÌ ´ × ¼ ØÙÚÛ × µ´¼Ü ¾ ² · (5) The solution is given by Equation (2) with C n defined by Equation (5). PROBLEM 4.39 KNOWN: Plane surface of two-dimensional system. FIND: The finite-difference equation for nodal point on this boundary when (a) insulated; compare result with Eq. 4.42, and when (b) subjected to a constant heat flux.
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