MSG 327 Mathematical Modelling [Pemodelan Matematik]

Transcription

1 UNIVRSITI SAINS MALAYSIA First Semester amination Academic Session 15/16 December 15/Janary 16 MSG 37 Mathematical Modelling [Pemodelan Matematik] Dration : 3 hors [Masa : 3 jam] Please check that this eamination paper consists of SVN pages of printed materials before yo begin the eamination. [Sila pastikan bahawa kertas peperiksaan ini mengandngi TUJUH mka srat yang bercetak sebelm anda memlakan peperiksaan ini.] Instrctions: Answer THR (3) qestions. [Arahan: Jawab TIGA (3) soalan.] In the event of any discrepancies, the nglish version shall be sed. [Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan, versi Bahasa Inggeris hendaklah digna pakai]. /-

2 - - [MSG 37] 1. Consider the following advection-dispersion-daecay eqation (1) at steady state c c k c (1) with the bondary conditions c = kg/m 3 when = and = +. Sppose a factory discharges a chemical into the river at = m at the rate W kg/s and mied niformly. Therefore, at this discharge point, = m, the chemical concentration will be a maimm ( c ). Assme that the dispersion coefficient m /s, the velocity m/s and the chemical decay rate k s -1 are constant. Show that the analytical soltion to (1) is c ce ce 4k with 1 1. W Show that c kg/m 3 with = A m 3 /s in which A m is the cross sectional area of the river. (c) Sppose W =.1 kg/s, =.3 m/s, = 1 m /s, k = 1-4 s -1 and A = 1 m. Find the chemical concentration c() in the river at = 5 m, m, m, 5 m and 1 m. Sketch the graph of c() for (, ). [ 1 marks ]...3/-

3 - 3 - [MSG 37] 1. Pertimbangkan persamaan adveksi-sebaran-pereptan (1) berikt pada keadaan mantap c c k c (1) dengan syarat-syarat sempadan c = kg/m 3 apabila = and = +. Andaikan sat kilang melepaskan sat bahan kimia ke dalam sngai pada = m pada kadar W kg/s dan dicampr seragam. Maka, pada titik pelepasan ini, = m, kepekatan bahan kimia tersebt akan menjadi maksimm ( c ). Andaikan bahawa pekali sebaran m /s, halaj m/s dan kadar pereptan bahan kimia k s -1 adalah malar. Tnjkkan bahawa penyelesaian analitikal ntk (1) ialah c ce ce 4k dengan 1 1. W Tnjkkan bahawa c kg/m 3 dengan = A m 3 /s di mana A ialah las keratan rentas sngai. (c) Andaikan W =.1 kg/s, =.3 m/s, = 1 m /s, k = 1-4 s -1 dan A = 1 m. Cari kepekatan bahan kimia c() dalam sngai pada = 5 m, m,, 5 m dan 1 m. Lakarkan graf c() ntk (, ). [ 1 markah ]...4/-

4 - 4 - [MSG 37]. qation (1) can be solved nmerically by sing finite segment method to obtain the chemical concentration in a niform river. Sppose that a niform river has cross-sectional area A = 1 m, length L = 15 m, velocity =.3 ms -1 and dispersion coefficient = 1 m s -1. This river is divided into three niform segments with = 5 m for each segment as shown in Figre 1. A chemical is released into segment 1 of the river at the rate of W =.1 kg s -1. This chemical decays at the rate of k = 1-4 s -1. The water at the pstream (c ) and downstream (c d ) segments is assmed to have a chemical concentration of. kg m -3. W =.1 kg s c c 15 m Figre 1. A niform river divided into three segments (c 1, c, c 3 ) with pstream segment (c ) and downstream segment (c d ) Find the concentration for every segment c 1, c, c 3 after steady state is achieved. Sppose that a lake represented by segment 4 is added to the system as shown in Figre. W =.1 kg s -1 c c c 15 m Figre. A niform river divided into three segments (c 1, c, c 3 ) with pstream and downstream segments (c and c d ) and lake segment (c 4 ) (i) Find the concentration c 4 at Segment 4 after steady state is achieved. (ii) Find and discss the concentration at Segment 4 when 4. Here, 4 represents the dispersion between Segment and Segment 4. (iii) Find and discss the concentration at Segment 4 when 4. [ 1 marks ]...5/-

5 - 5 - [MSG 37]. Persamaan (1) boleh diselesaikan secara berangka dengan menggnakan kaedah segmen terhingga ntk mendapatkan kepekatan kimia di dalam sngai yang seragam. Andaikan sat sngai seragam mempnyai las keratan rentas A = 1 m, panjang L = 15 m, halaj =.3 ms -1 dan pekali sebaran = 1 m s -1. Sngai ini dibahagikan kepada tiga segmen seragam dengan = 5 m bagi setiap segmen seperti yang ditnjkkan dalam Rajah 1. Sat bahan kimia dilepaskan ke dalam segmen 1 sngai tersebt pada kadar W =.1 kg s -1. Bahan kimia ini merept pada kadar k = 1-4 s -1. Air di segmen hl (c ) dan segmen hilir (c d ) dianggap mempnyai kepekatan kimia. kg m -3. W =.1 kg s c c 15 m Rajah 1. Sngai seragam dibahagikan kepada tiga segmen (, c, ) dengan segmen hl (c ) dan segmen hilir (c d ) Cari kepekatan ntk setiap segmen c 1, c, c 3 selepas keadaan mantap dicapai. Andaikan sat tasik yang diwakili oleh Segmen 4 ditambah ke sistem tersebt seperti yang ditnjkkan dalam Rajah. 4 W =.1 kg s c 4 c c 15 m Rajah. Sngai seragam dibahagikan kepada tiga segmen (c 1, c, c 3 ) dengan segmen hl dan hilir (c, c d ) dan segmen tasik (c 4 ) (i) Cari kepekatan c 4 di segmen 4 selepas keadaan mantap dicapai. (ii) Cari dan bincangkan kepekatan pada Segmen 4 apabila 4. Di sini, 4 mewakili sebaran antara Segmen dan Segmen 4. (iii) Cari dan bincangkan kepekatan pada Segmen 4 apabila 4. [1 markah]...6/-

6 - 6 - [MSG 37] 3. Let BOD concentration in a fish tank at time t day be denoted by t mg/l and DO concentration in the fish tank at time t be denoted by ct mg/l. The initial concentrations of BOD and DO in that tank are denoted by mg/l and c mg/l, respectively. BOD decay rate, day -1, DO reaeration rate, day -1 and DO satration concentration, cs mg/l are assmed constants. Any additional symbols sed mst be defined, with proper dimension and nit. Write the differential eqations to describe the changes of BOD and DO concentrations with respect to time in a niformly-mied closed tank. Then find the soltions of the differential eqations. Sketch the time series for BOD and DO concentrations in the closed tank when: (i) c (ii) c (iii) c Now consider the case with the tank opened so that air can enter the tank to reaerate the water. Write the differential eqations to describe the changes of BOD and DO concentrations with respect to time in this open tank. Then find the soltions of the differential eqations. Sketch the time series for BOD and DO concentrations in the open tank when: (i) c (ii) c (c) Sppose the open tank has pre water (cin = 8 mg/l) flowing into it at the rate of m 3 /day. If the BOD concentration in the open tank were to remain constant at c mg/l and the volme of water in the tank were to remain constant at V m 3, derive the steady state DO concentration in the tank. Let c 6 mg/l, c mg/l, =.3 day -1, =.6 day -1, cs 7 mg/l and V = 1 m 3. Calclate the reqired flow m 3 /day to maintain a constant DO concentration of 5 mg/l in the tank. [ 1 marks ]...7/-

7 - 7 - [MSG 37] 3. Biarkan kepekatan BOD dalam sat tangki ikan pada masa t hari diwakili oleh t mg/l dan kepekatan DO dalam tangki ikan tersebt pada masa t diwakili oleh ct mg/l. Kepekatan awal BOD dan DO dalam tangki tersebt masingmasing diwakili oleh mg/l dan c mg/l. Kadar pereptan BOD, hari -1, kadar pengdaraan semla DO, hari -1 dan kepekatan tep DO, cs mg/l dianggap malar. Simbol tambahan yang dignakan mestilah didefinisikan, dengan dimensi dan nit yang betl. Tlis persamaan pembezaan ntk menggambarkan perbahan kepekatan BOD dan DO terhadap masa dalam sat tangki terttp yang bercampr seragam. Kemdian cari penyelesaian bagi persamaan pembezaan tersebt. Lakarkan siri masa bagi kepekatan BOD dan DO dalam tangki terttp tersebt apabila: (i) c (ii) c (iii) c Sekarang pertimbangkan kes dengan tangki terbka spaya dara dapat mask tangki ntk pengdaraan semla. Tlis persamaan pembezaan ntk menggambarkan perbahan kepekatan BOD dan DO terhadap masa dalam tangki terbka ini. Kemdian cari penyelesaian bagi persamaan pembezaan tersebt. Lakarkan siri masa bagi kepekatan BOD dan DO dalam tangki terbka tersebt apabila: (i) c (ii) c (c) Andaikan tangki terbka tersebt mempnyai air bersih (cin = 8 mg/l) mengalir ke dalamnya pada kadar m 3 /hari. Jika kepekatan BOD dalam tangki tersebt dikekal malar pada c mg/l dan isipad air dalam tangki tersebt dikekal malar pada V m 3, terbitkan kepekatan DO dalam tangki tersebt pada keadaan mantap. Biar c 6 mg/l, c mg/l, =.3 hari -1, =.6 hari -1, cs 7 mg/l dan V = 1 m 3. Kira aliran m 3 /hari yang diperlkan ntk mengekalkan kepekatan DO dalam tangki pada 5 mg/l. [ 1 markah ] - ooo O ooo -