Analysis: Numerical and Temperature Distribution

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  • Topic: Numerical analysis, Partial differential equation, Finite difference
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  • Published : April 24, 2013
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The University of Nottingham Malaysia Campus MM3CMT, Computer Modelling Techniques, spring 2013 Assignment: on a Finite Difference and Finite Volume Method A fluid enters a very long pipe by TA temperature and leaves the pipe by TB temperature. The length of pipe is L m with D mm diameter. Assume the thermal conductivity is constant and equal to κ W/mK, ρ=1 kg/m3, specific heat equal 1 and mass flow rate is 0.2 kg/s. calculate temperature distribution along the pipe.

TA

X=0

u Figure Q1.

TB X=L

Pre-processing: 1. Apply a suitable assumption for simplification to reduce the dimension of the problem. 2. Derive the governing equation by using general transport equation. (  )  .( U )  .( grad )  S . t 3. Generate uniform grids (Mesh) with 10 equal cells, find the cell size and draw schematic of computational domain and assign nodal points when the length of domain is L m. Solving: 4. Applied the finite difference method, FDM, to discretise the governing equation. Apply backward difference for the convection term and central difference for the diffusion term. Calculate the temperature distribution along the pipe. you may need to develop a code for this purpose using any software (Matlab®, Excell®, …) 5. FVM 5.1 Applied the finite volume method, FVM, to discretise the governing equation. 5.2 Write the neighbouring coefficient and their value for each nodal point; write the resulting set of equations as a matrix. You can use the Table Q1. 5.3 Develop a code using any language or software (Matlab®, Excell®, …) to solve the above matrix set by using iteration method to calculate temperature distribution. For the results, assume one digit after point as an acceptable accuracy (convergence criteria).

5.4 Now consider a uniform heat generation q kW/m3 , along the pipe. Repeat part 5.3 and show the results.

Node 1 2, 3, 4, … 10

aW

aE

Sp

SU

ap

Table Q1. Post processing: 6. Write a temperature distribution for each node. You...
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