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ECH140:

Đối với từng vấn đề trên đây trả lời các câu sau đây:

(I) các biến phụ thuộc là gì: các biến phụ thuộc là gì

(Ii) Là vấn đề được đồng nhất hoặc không đồng nhất

(Iii) là vấn đề được tuyến tính phi tuyến

(Iv) Có vấn đề đã có hệ số biến hay hệ số liên tục

(V) Có vấn đề có một giải pháp ổn định nhà nước usHxL = 0.

±²³´ µ²´¶·¸¹´º» ³¼ ½´²´²¾ ¿ÀÁ ¾·³Â³¾» ±ÁÿÀÄÅÁ ÆÇÈÉÇ¿Ê ËÀ ÌÍÅÊËÄ¿Î ·ÀÉËÀÅÅÇËÀÉ ECH140: Mathematical Methods for Chemical Engineering ¼ËÀ¿Î ÆÇ¿ÄÏËÄÅ ·Ð¿Ê ½¿ÇÄÍ ÑÒÓÔ ±²³´µ¶ ·³³¸ ¹º¶ º³»µ´ º³ ¼µ²² ½¾³ºµ´¿ º³ ÀÁµ´»Â³º´ ¹´¸µ¶ ³Ã ¹º´ÄµÃµ¶Å ±²³´ µ¶· ¸¹º»¼ Total Mass Balance for fixed Control volume ρ ⅆ ± + ρ · ± ² = ³ Species Mass Balance ρ ± + ρ · ± ² = ´ ±µ ² = ¶µ ·µ µ ¸ Table of Laplace Transforms ±²³´µ¶ · ¸¹º »²¼½¾¿À Consider the following PDE ± = ² ³ + ´ ± ³ µ¶ · ( ³¸ ¹ ) = º + » ¼½¾ ( ¿ π ³ / À ) Á¶Â · ( ¹¸ ± ) = α Á¶Ã · ( À¸ ± ) = ¹ (i) What is the steady state solution U s ( x ) (10 pts) (ii) Determine the solution u ( x , t ) that satisfies the BCs and initial condition. Note you do not need to evaluate integrals. Just show how any coefficients in the solution are determined. (15 pts) (iii) Now if Q 0 = 0, and α = 0, can you determine a simple formula for the solution found in (ii) (5 pts) You may assume the following orthogonality conditions for part (ii) 0 L Sin ( n π x / L ) Sin ( m π x / L ) x = 0 if n m L 2 if n = m , where n , m are integers ±²³´µ¶ ¹ ¸¹º »²¼½¾¿À Consider the following PDE
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± = - ² + ³ ´µ ¶ ( ³· ¸ ) = ¹ ( ³ ) ºµ» ¶ ³ ( ¸· ± ) = ¸ ºµ¼ ¶ ( ½· ± ) = ¾ ± (i) Determine the steady state solution(s) for a > 0. (ii) Determine the transient solution. ±²³´µ¶ · ¸¹º »¼½¾ You are required to analyze the temperature distribution in a spherical shell with a heat source. The mathematical statement of the problem is given by ¿ ± = α ¿ À + ¼ α À ¿ À + Á À · À ± < À < À ² subject to the following IC and BCs ´µ ¶ ¿ ( À· ¸ ) = ¿ ± ºµ» ( ± ± ) = ² ºµ¼ ¶ ¿ ( À ² ) = ¿ ± The units of of the various quantities that appear in the above equation are: ¿ ²à Äűà [ Æ ] · α ²à ÄÅ±Ã Ç Ã Â²Ã Äűà [ È / ( Ç · Æ )] Á ²à ÄÅ±Ã Æ   Ç · à   ²à ÄÅ±Ã È   Ç · Æ  À ²à Äűà [ Ç ] ± ²à Äűà [ à ] (i) Use the following dimensionless variables to make the above system dimensionless: θ = ¿ - ¿ ² ¿ ± - ¿ ² · ξ = À À ± · τ = α ± À ± · ϵ = À ² À ± Where appropriate identify any other dimensionless variables that result when using the above vari- ables. Note: your final equation and BCs must be dimensionally consistent. (ii) Show that the steady state solution θ ³ ( ξ ) is given by θ [ ξ ] = É ϵ ´ ( ξ - » ) + É ξ ( ξ µ - » ) + ϵ ( É + ¼¸ ( ξ - » ) - É ξ ´ ) ¼¸ ( ϵ - » ) ξ (iii) The transient solution is given by 2 ± = - ² + ³ ´µ ¶ ( ³· ¸ ) = ¹ ( ³ ) ºµ» ¶ ³ ( ¸· ± ) = ¸ ºµ¼ ¶ ( ½· ± ) = ¾ ± (i) Determine the steady state solution(s) for a > 0. (ii) Determine the transient solution. ±²³´µ¶ · ¸¹º »¼½¾ You are required to analyze the temperature distribution in a spherical shell with a heat source. The mathematical statement of the problem is given by ¿ ± = α ¿ À + ¼ α À ¿ À + Á À · À ± < À < À ² subject to the following IC and BCs ´µ ¶ ¿ ( À· ¸ ) = ¿ ± ºµ» ( ± ± ) = ² ºµ¼ ¶ ¿ ( À ² ) = ¿ ± The units of of the various quantities that appear in the above equation are: ¿ ²à Äűà [ Æ ] · α ²à ÄÅ±Ã Ç Ã Â²Ã Äűà [ È / ( Ç · Æ )] Á ²à ÄÅ±Ã Æ   Ç · à   ²à ÄÅ±Ã È   Ç · Æ  À ²à Äűà [ Ç ] ± ²à Äűà [ à ] (i) Use the following dimensionless variables to make the above system dimensionless: θ = ¿ - ¿ ² ¿ ± - ¿ ² · ξ = À À ± · τ = α ± À ± · ϵ = À ² À ± Where appropriate identify any other dimensionless variables that result when using the above vari- ables. Note: your final equation and BCs must be dimensionally consistent. (ii) Show that the steady state solution θ ³ ( ξ ) is given by θ [ ξ ] = É ϵ ´ ( ξ - » ) + É ξ ( ξ µ - » ) + ϵ ( É + ¼¸ ( ξ - » ) - É ξ ´ ) ¼¸ ( ϵ - » ) ξ (iii) The transient solution is given by 2 ECH140FinalPracticeExam_2016.nb
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