MECH 4820 Computational Methods for Thermofluids (F07)home.cc.umanitoba.ca/~engsjo/teaching/MECH-4822/... · MECH 4820(F07) Test #2 Page 3 of 3 2. Figure 3 shows part of a computational

MECH 4820 (F07) Test #2 Page 1 of 3

University of ManitobaDept. of Mechanical & Manufacturing Engineering

MECH 4820 Computational Methods for Thermofluids (F07)

Term Test #2 14 November 2007 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 ofHeat 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 NotesV9.0, Department of Mechanical & Manufacturing Engineering, University of Manitoba,July 2007.

• Mathematical reference tables.

Extra pages, problem solutions, and class notes are not permitted.2. Clear solutions are required. Marks will not be assigned for answers that require unreasonable

effort for the instructor to decipher.3. Ask for clarification if any problem statement is unclear to you.4. The weight of each problem is indicated. The test will be marked out of 100.5. You may solve the test problems in any order.

Values

1. This question uses the finite difference energy balance method to predict the temperaturein the fin heat transfer application shown in Figure 1. In the application, two fluid regionsare separated by a wall of insulation with thickness Lw. Heat is transferred steadily fromthe hotter fluid (fluid 1) to the colder fluid (fluid 2) via the circular rod that is shown incross-section in the figure. The rod has a thermal conductivity, k, diameter d. cross-sectionalarea, Ac, and perimeter, P . The portion of the rod that is exposed to fluid 1 has a lengthL1. The convection heat transfer coefficient and ambient fluid temperature for this side areh1 and T∞,1, respectively. The portion of the rod that is exposed to fluid 2 has a length L2.The conditions on this side of the wall have convection heat transfer coefficient and ambientfluid temperature h2 and T∞,2, respectively. The values of these parameters are given inFigure 1.

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The computational mesh to be used in the numerical analysis is shown in Figure 2. In thismesh, all nodes are on the grid lines, and in each of the three regions (“exposed to fluid 1”,“in the wall”, “exposed to fluid 2”) the grid lines are equally spaced. Overall, however, thespacing is not uniform.

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MECH 4820 (F07) Test #2 Page 2 of 3

d

L1 L2Lw

T∞,1h1T∞,2h2

circular rod

insulating wall

fluid 1 fluid 2

Lw = 5.0 [cm]

k = 160.0 [W/m·K]

L1 = 40.0 [cm] L2 = 20.0 [cm]

h1 = 230.4 [W/m2·K]

T∞,1 = 100 [◦C]

h2 = 80.0 [W/m2·K]

T∞,2 = 20 [◦C]

d = 4.0 [cm]

Figure 1: Schematic of fin application in Question #1

T1

T2 T3T4 T5

T6

T7fluid 1 fluid 2

wall

L1 L2

Lw∆x1 ∆x2

∆x3 ∆x4

∆x5 ∆x6

∆x1 = 0.200 [m]∆x2 = 0.200 [m]

∆x3 = 0.025 [m]∆x4 = 0.025 [m]

∆x5 = 0.100 [m]∆x6 = 0.100 [m]

Ac = 0.0012566 [m2] P = 0.12566 [m]

Figure 2: Grid used in Question #1

(a) Using the finite difference energy balance method on control volumes for T1 to T5, theproblem parameters from Figure 1, and the grid from Figure 2, determine the algebraicequations for nodal temperatures T1 to T5. When doing this for each node, clearlyshow the control volume you are using and the equations for the energy flowsyou use to derive the algebraic equation. Use all temperature values in degrees Celsius.

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(b) Given that T5 = 67.200 [◦C], T6 = 47.271 [

◦C], and T7 = 40.978 [◦C], solve the equation

set from part (a) for T1 to T4. Show your work. No credit will be given for only thetemperature values (with no work shown).

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(c) Using the numerical solution, calculate the heat transfer rate though the fin.5

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MECH 4820 (F07) Test #2 Page 3 of 3

2. Figure 3 shows part of a computational mesh and a corner node (node 1) in a domainconsisting of two materials. A transient 2D conduction analysis is to be performed on thedomain using the finite difference method with the Explicit approach and the mesh shown.There is no internal energy generation.

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material A

material B

T1 T2

T3

T4

∆y

∆y

∆x

h∞T∞

xy

Insulation

∆x = 0.020 [m]

∆y = 0.040 [m]

kA = 20 [W/m · K]

ρA = 7800 [kg/m3]

CpA = 460 [J/kgK̇]

kB = 1.2 [W/m · K]

ρB = 1600 [kg/m3]

CpB = 850 [J/kg · K]

h∞ = 40 [W/m2· K]

T∞ = 30 [◦C]

Figure 3: Nomenclature used in Question #2

(a) Using the energy balance method, derive a transient finite difference equation for T p+11in symbolic form. Sketch the control volume used and indicate on the sketch the energyflows. Note that for a control volume made up of more than one material, the specificheat (Cp) is calculated using a mass-weighting of the component specific heat values.

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(b) Determine the time step restriction, in seconds, for the T p+11 equation for the conditionsgiven in Figure 3.

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