MECH 4820 (F08) Test #1 Page 1 of 4

University of ManitobaDept. of Mechanical & Manufacturing Engineering

MECH 4820 Computational Methods for Thermofluids (F08)

Term Test #1 22 October 2008 Duration: 110 minutes

1. You are permitted to use the following reference material during this test:

• Incropera, F.P., and Dewitt, D.P., Bergman, T.L., and Lavine, A.S., Fundamentals of

Heat and Mass Transfer, 6th Ed. John Wiley and Sons, New York, 2007. (5th editionis also acceptable)

• Ormiston, S.J., MECH 4820 Computational Thermofluids Supplementary Course Notes

V10.0, Department of Mechanical & Manufacturing Engineering, University of Mani-toba, July 2008.

• Mathematical reference tables.

Extra pages, problem solutions, and class notes are not permitted.2. Ask for clarification if any problem statement is unclear to you.3. Clear solutions are required. Marks will not be assigned for answers that require unreasonable

effort for the instructor to decipher.4. The weight of each problem is indicated. The test will be marked out of 100. You may solve

the test problems in any order.

Values

1. The governing equation for steady, one-dimensional heat conduction in the pin fin (constantarea, square cross-section fin) shown in Figure 1(a) is:52

kAc

d2T

dx2− h∞ P (T − T∞) = 0 (1)

where P is the perimeter of the fin and Ac is the cross-sectional area. The pin fin has a sidedimension, s, uniform thermal conductivity, k, and is exposed to a fluid with an ambienttemperature of T∞ except where it is supported by the insulating wall and where the left faceof the fin is exposed to an incident heat flux of q′′. The convection heat transfer coefficientbetween the fin surface and the fluid is h∞.

(b) typical control volume nomenclature

fin

(a) fin geometry nomenclature

h∞

h∞

T∞

T∞

q′′

TW TP TE

L

w

s

skx

x

(δx)w

(δx)p

(δx)e

Ac

AcP

Ac = s2

P = 4 s

Side ViewEnd View

insulating wall

Figure 1: Nomenclature used in Question #1

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MECH 4820 (F08) Test #1 Page 2 of 4

For the nomenclature given in Figure 1(b), the finite volume method described in this courseproduces the following algebraic equation for the typical control volume:

aP TP = aW TW + aETE + bP

where

aP = aE + aW + h∞P (δx)p aE =kAc

(δx)e

aW =kAc

(δx)w

bP = h∞P (δx)pT∞

(a) If the heat transfer coefficient, h∞, is set to zero, what do the coefficients become? Towhat conduction application do these coefficients correspond?

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(b) Using the grid and problem specifications shown in Figure 2, determine the algebraicequations for nodal temperatures T1, T2, and T5 (do not determine the coefficients forT3, T4). When doing this, clearly derive and simplify the boundary condition equationsfor TL and TR and substitute them into the equations for T1 and T5, respectively. Useall temperature values in degrees Celsius.32

Show that your equations for T1, T2, and T5 match the appropriate entries in the set ofequations shown in Equation (2) below.

32 −32 0 0 0−32 44.8 −12.8 0 00 −12.8 28.8 −8 00 0 −8 24 −80 0 0 −8 17.455

T1

T2

T3

T4

T5

=

1600

160160

189.091

(2)

(a) grid

(b) problem specifications

x

x = 0 x = w x = L + w

k = 125 [W/m·K] L = 0.30 [m]

h∞ = 250 [W/m2·K] T∞ = 20 [◦C] s = 0.08 [m]

w = 0.05 [m]q′′ = 25000 [W/m2]

TR

T1 T2

T3 T4 T5

TL

wall

wall

Figure 2: Grid and problem specifications for Question #1

(c) Given that T4 = 24.898 [◦C] and T5 = 22.245 [◦C], solve the equation set in part (b) forT1 to T3 using the TDMA (Tri-Diagonal Matrix Algorithm). Show your work.13

(d) Using the numerical solution, calculate the heat transfer rate though the fin. Is this theexpected value? Explain why or why not.

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MECH 4820 (F08) Test #1 Page 3 of 4

2. Consider a fin with an insulated tip condition. Referring to Incropera et al., the dimensionlesstemperature distribution is given by:20

T (x) − T∞

Tb − T∞

=θ

θb

=cosh (m(L − x))

cosh (mL)

where

cosh(u) =1

2

(

e−u + eu)

The conditions for the problem are given as: Tb = 125 [◦C], T∞ = 25 [◦C], m = 10 [m−1],L = 0.2 [m]. Note that Appendix B.1 of Incropera et al. has tabulated values of hyperbolicfunctions.

(a) For the given conditions, derive an equation in terms of x only that would enable youto solve for the value of x at which the fin temperature is 75 [◦C]. Cast the resultingequation in a form suitable for using a root search for its solution (but do not solve it).

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(b) Solve the equation from part (a) for x using a Newton-Raphson root search. Use aninitial guess of x = 0.1 [m] and perform three iterations. Show your work.

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3. Re-derive the algebraic equation for T (tn+1) for the lumped capacitance model problem inChapter 1 of the Supplementary Course Notes using the Fully Implicit time-weightingapproach. Show your work in the derivation. Also, show the limit of the new equation forT (tn+1) as the time step goes to infinity. How does this compare to the same limit for theoriginal equation (in the notes)?12

4. This question requires that you write the lines of MATLAB source code that will performa particular calculation described below. The MATLAB code that sets up some variables isgiven; you do not need to re-write that code in your solution.16

The calculation to be performed is that of the dimensionless temperature profile in a planewall at a given dimensionless time. This calculation uses the four term approximation of theanalytical solution for transient conduction in a plane wall:

θ∗ =4

∑

n=1

Cn exp(

− ζ2

n Fo)

cos (ζnx∗) (3)

where ζn are the eigenvalues of the transcendental equation and Cn are the coefficients of theseries. The values of the four eigenvalues and coefficients (which correspond to a particularBiot number) and the dimensionless time are given in the set up code.

The MATLAB code should calculate θ∗ at 21 evenly spaced location for x∗ from 0 to 1. Thex∗ and θ∗ values should be stored in two separate vectors (one-dimensional arrays) namedxsval(i) and tsval(i), respectively. The first index (i=1) should correspond to x∗ = 0and the last index (i=21) should correspond to x∗ = 1. Note that each evaluation of θ∗ at agiven x∗ requires the summation of the four terms as given by Equation (3).

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MECH 4820 (F08) Test #1 Page 4 of 4

Table 1 shows the correspondence between the variables in the set up code shown in Figure 3,and the notation used in Equation (3). Note that the built-in functions in MATLAB forexponential and cosine are exp and cos, respectively. Note also that exponentiation usesthe “∧” symbol.

(a) Write the MATLAB source code that goes in the section labelled “New code section togo in here” in Figure 3. You do not need to do any input or output.14

(b) List all the new variables that you created in part (a) and give their meaning.2

Table 1: Table of variables for setup code in Figure 3Variable Meaning

C(n) Cn

Z(n) ζn

Fo Fo

% C_n coefficients

C(1) = 1.1347;

C(2) = -0.17468;

C(3) = 0.055986;

C(4) = -0.026344;

% Zeta_n values

Z(1) = 0.9190;

Z(2) = 3.4791;

Z(3) = 6.4705;

Z(4) = 9.5526;

% set Fourier number

Fo = 0.0201;

%%%<new_code_section_begins>%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

%--New code section to go in here

%

%%%<new_code_section_ends>%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Figure 3: Set up code in MATLAB for Question #4

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