A2 2D Computational Heat Conduction

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Department of Mechanical Engineering

Indian Institute of Technology Bombay

ME415: Computational Fluid Dynamics & Heat Transfer Spring

!1"

#ssignment $ : 2D Computational Heat Conduction for Cartesian Geometry, on a

Uniform and Non-Uniform Grid, using Physical and Mathematical Approach

%eigtage: 5' (nstructor Prof! Atul "harma

Date Posted: 10th Feb. (Wednesday) Due Date: 17th Feb. (Wednesday, Early Morning !M)

"#$%#E &'M%&&%"# *+"'* M""D$E "#$- (#o late subission allo/ed): Create a

single zipped file consisting on (a) filled-in answer sheet of this doc file converted into a pdf file and

(b) all the computer programs. The name of the zipped file should be rollnuber!

1. D 2o3utational *eat 2ondution (2*2) on a uni5or grid, /ith e63liit ethod and 2

orres3onding to 2ase (a): Physial a33roah.

Consider ! conduction in a s"uare shaped ( L1=1m and L2=1m) long stainless-steel (densit# ρ: 770 8g9$

specific-heat c p: 00 ;9<g < $ thermal-conductivit# k : 1=. W9< ) plate. The plate is initiall# at a uniform

temperature of 30oC and is suddenl# sub%ected to a constant temperature of T wb = 100oC on the west

boundar#$ T sb = 200oC on the south boundar#$ T eb = 300oC on the east boundar#$ and T nb = 400oC on north

boundar#.

i. &sing the ph#sical approach (ph#sical law based '$ and flu* based solution methodolog#) of C'!

development$ develop a computer program for e*plicit method on a uniform -! Cartesian grid. &se the

stead# state stopping criterion for non-dimensional temperature$ given (slide + ,., ,.,/) as

{ }1

$ $

1

$ ma* for $

n n

i j i j cst c c nb wb

i j c i j

T T l l L T T T

T t

θ

ε

τ

+ −∂ = ≤ = ∆ = − ÷ ÷ ÷∂ ∆ ∆

ii. Present a C'! application of the code for a volumetric heat generation of 0 and 0 8W9. Consider

ma*imum number of grid points as imax ×jmax =12×12 and the stead# state convergence tolerance as

εst =10-4. Plot the stead# state temperature profiles with and without volumetric heat generation.

. D 2o3utational *eat 2ondution (2*2) on a nonuni5or grid, /ith i3liit ethod and

2 orres3onding to 2ase (b): Matheatial a33roah.

Consider ! conduction in a s"uare shaped ( L1=1m and L2=1m) long stainless-steel (densit# ρ: 770 8g9$

specific-heat c p: 00 ;9<g < $ thermal-conductivit# k : 1=. W9< ) plate. The plate is initiall# at a uniform



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temperature of 30oC and is suddenl# sub%ected to a constant temperature of T wb = 100oC on the west

boundar#$ insulated on the south boundar#$ constant incident heat flu* of q = 10 k!m2 on the east

boundar#$ and "=1000 !m2 #$ and T % = 30oC on north boundar#.

i. 0enerate a non-uniform ! Cartesian grid$ using an algebraic method (can be found in slide no. ,.1 to

,.). The method involves a transformation of a uniform grid in a 1! computational domain (2-coordinate) of unit length to a ph#sical domain (*- coordinate) of length 3$ using an algebraic e"uation

given ( Hoffmann and Chiang, 2000) as

( ) ( ) ( ) ( )

( )

( ) ( ) ( ){ }

4

4 x L

ξ

ξ

−

−

+ + − − − =+ + −

1

1

1 1 1 1

1 1 1

The above e"uation is also used in the #-direction to generate the ! grid. This e"uation results in a grid

which is finest near the two ends (east and west as well as north and south boundar#) of the domain and

graduall# become coarser at the middle of the domain. 5t is called as e"ual clustering of grids at both the

ends of the domain. Consider ma*imum number of grid points (for the temperature) as

imax ×jmax =12×12 and &=1#2.

ii. &sing the mathematical approach (P!6 based '$ and coefficient of 376s based solution

methodolog#) of C'! development$ develop the 0auss-8eidel method based computer program for

implicit method on a non-uniform -! Cartesian grid. &se the stopping criterion presented in the previous

problem$ with 'T c=T wb-T %.

iii. &sing the non-uniform grid and the program$ present a C'! application of the code for a volumetric heat

generation of 0 and (0 k!m3. Consider the convergence tolerance as εst =10-4 for the stead# state$ and

ε=10-4 for iterative solution b# the 0auss-8eidel method. Plot the stead# state temperature profiles with

and without volumetric heat generation.

est Wishes 5or your suess in the insight5ul 5ield o5 2o3utational Fluid Dynais

9eep Pla#ing with the codes in future also.



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Answer Sheet

Proble > 1: Physial !33roah, /ith e63liit ethod and uni5or grid:

Plot the stead# state temperature profiles with and without volumetric heat generation. ( figures).

Disussion on the Fig. .1: rite #our answer$ limited inside this te*t bo* onl#

(a) (b)

'ig. .1: C'! 7pplication of the e*plicit method based ! code on uniform Cartesian grid: 8tead# state

temperature contour for unstead# state heat conduction$ (a) without and (b) with heat generation.



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Proble > : Matheatial !33roah, /ith i3liit ethod and nonuni5or grid:

Plot the stead# state temperature profiles with and without volumetric heat generation. ( figures).

Disussion on the Fig. .: rite #our answer$ limited inside this te*t bo* onl#

(a) (b)

'ig. .: C'! 7pplication of the implicit method based ! code on non-uniform Cartesian grid: 8tead#

state temperature contour for unstead# state heat conduction$ (a) without and (b) with heat

A2 2D Computational Heat Conduction

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