MECH 4822 (F09) Mid-Term Test Page 1 of 3

University of ManitobaDept. of Mechanical & Manufacturing Engineering

MECH 4822 Numerical Heat Transfer and Fluid Flow (F09)

Mid-Term Test 25 November 2009 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.• Ormiston, S.J., MECH 4822 Numerical Heat Transfer and Fluid Flow Supplementary Course

Notes V1.0, Department of Mechanical & Manufacturing Engineering, University of Mani-toba, August 2009.

• Mathematical reference tables.

Extra pages, complete 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.

The problems start on the next page in order to keep the first question on a single page.

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MECH 4822 (F09) Mid-Term Test Page 2 of 3

Values

1. The nodal temperatures in the domain shown in Figure 1 are to be determined using the finitevolume method presented in this course. The problem is steady-state two-dimensional steadyconduction with no energy generation. The left side and the bottom are insulated, the top isexposed to convection, and the right side has an applied heat flux.45

T1 T2

T3T4

TF,a

TF,b TF,c

TF,d

TF,e

TF,fTF,g

TF,h

h∞

h∞ = 500 [W/m2· K]

T∞

T∞ = 19 [◦C]

∆y = 0.10 [m]

∆x = 0.04 [m]

k = 25 [W/m· K]

Bi =h∞∆y

k= 2.0

q′′∆x

2k= 2.56 [K]

k ∆y

∆x= 62.5

k ∆x

∆y= 10.0

q′′

x

y

q′′ = 3, 200 [W/m2]

∆y

∆y

∆x ∆x

T4 =65.727 [◦C]

Figure 1: Two-dimensional domain for Question #1.

To reduce the solution to only three nodes (T1, T2, and T3), the value of T4 is given. The detailsof the geometry, boundary conditions, properties, and nodes used for the application of boundaryconditions (TF,a to TF,h) are also shown in the figure. Assume unit depth.

(a) Derive equations for nodes on the boundary: TF,a, TF,b, TF,c, TF,d, TF,e, and TF,f . Writeeach algebraic equation in terms of the boundary temperature and the appropriate nearestinterior nodal temperature. In your derivations, gather terms to use the quantities Bi andq′′ ∆x

2 kas appropriate.

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(b) Starting from the general finite volume algebraic equations, assemble equations of the form:

(aP )1T1 = (aW )

1TF,a + (aE)1 T2 + (aS)

1T4 + (aN)

1TF,b + (bP )

1(1)

(aP )2T2 = (aW )

2T1 + (aE)2 TF,d + (aS)

2T3 + (aN)

2TF,c + (bP )

2(2)

(aP )3T2 = (aW )

3T4 + (aE)3 TF,e + (aS)

3TF,f + (aN)

3T2 + (bP )

3(3)

for the T1, T2, and T3 control volumes. Show the equations in symbolic form first, thensubstitute the problem values given in Figure 1.

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(c) Substitute values into the boundary condition equations from part (a), absorb them asneeded into Equations (1) to (3), and use the value of T4 to obtain three equations in thethree unknown temperatures (Note: use degrees Celsius). Solve the set of equations usingthe TDMA. Show your work.

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(d) Using the temperature solution from part (c), determine TF,b.2

(e) Using overall energy conservation determine TF,c.4

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MECH 4822 (F09) Mid-Term Test Page 3 of 3

2. Figure 2 shows the cross-section of a long, solid bar with a thermal conductivity of k = 12 [W/m· K]and an internal generation rate of q̇G = 3000 [W/m3]. The top and bottom horizontal sur-faces of the bar are maintained at 60 [◦C] and 40 [◦C], respectively. The remaining surfacesare subjected to convection with ambient air at T∞ = 20 [◦C] with a heat transfer coefficient ofh∞ = 90 [W/m2

·K] . Assume a unit depth for the bar.55

T1 T2 T3 T4 T5

∆y

∆y

∆x∆x∆x∆x

h∞h∞

T∞T∞

x

y

Tt = 60 [◦C]

Tb = 40 [◦C]

∆x = 20 [cm]

∆y = 10 [cm]

k = 12 [W/m· K]h∞ = 90 [W/m2·K]

T∞ = 20 [◦C]

Figure 2: Nomenclature used in Question #2

(a) Using the energy balance approach of the finite difference method with ∆x = 20 [cm] and∆y = 10 [cm], determine T1, T2, T3, T4, and T5. Assume steady-state conditions and showyour work in the derivation and solution of your algebraic equations.

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(b) Determine the total energy transfer rate per unit depth from the bar by convection to theambient fluid.

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(c) Determine the heat transfer rates into each of horizontal isothermal surfaces.19

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