ME415: CFD Theory - Quiz 2
(01.10.2021)
Total Marks: 60 marks
Duration: 60 min
Section A: MCQ
[2 marks each]
1. Consider an unsteady state heat conduction in a 2D rectangular plate where
the right/east wall is subjected to the convective boundary condition. Select the
appropriate first-order FDM based algebraic formulation for temperature, at
the boundary grid points, on the east wall.
(T∞: Ambient temperature , h : convection heat transfer coefficient , Δx: Width
of CV in x-direction, k: thermal conductivity of slab )
*+,-./0123,4 56∆%-8 9
A. 𝑇"#$%,' =
(+,56∆%)
*,-./0123,4 56∆%-8 9
B. 𝑇"#$%,' =
(+,56∆%)
*+,-./0123,4 56∆%-8 9
C. 𝑇"#$%,' =
(+,<6∆%)
*+,-./0123,4 <6∆%-8 9
D. 𝑇"#$%,' =
(+,<6∆%)
2. Which of the following statement(s) is(are) true of Implicit time stepping
solution methodology ?
A. Time step size should be lower than certain value to obtain stable solutions
B. For 1D problem, implicit time stepping involves 3 temperatures at current
time step and 1 temperature at previous time step. (FOR INTERIOR GRID
POINTS)
C. For 1D problem, implicit time stepping involves 1 temperature at current
time step and 3 temperatures at previous time step. (FOR INTERIOR GRID
POINTS)
D. Iterative schemes for a particular time step are NOT required in implicit
time stepping
3. Which of the following statement(s) is(are) acceptable for the Coefficient of
LAEs form of FVM based algebraic formulation for a 2D unsteady conduction
problem ?
A. All the coefficients for temperature in the linear algebraic equations are
negative
B. Certain coefficients for temperature in the linear algebraic equations are
positive
C. Certain coefficients for temperature in the linear algebraic equations are
negative
D. All the coefficients for temperature in the linear algebraic equations are
positive
4. For a 2D unsteady state heat conduction in a 2D rectangular slab with width
of control volume as twice of it's height, select the appropriate time step size(s)
to be considered for an explicit time stepping scheme.
(Δx – width of CV, Δy – height of CV, Given Δx = 2 Δy)
+∆% ?
A. ∆𝑡 ≤ @A
+∆B ?
B. ∆𝑡 ≤ @A
∆B ?
C. ∆𝑡 ≤ CDA
∆% ?
D. ∆𝑡 ≤ CDA
5. Which of the following data-structure(s) is suited to store width of control
volume for the solution of 2D heat conduction on a non-uniform Cartesian grid?
A. single-valued variable
B. 1D row-vector (matrix with one column)
C. 2D matrix
D. 1D column-vector (matrix with one row)
6. Which is an unconditionally stable solution method
a. Explicit
b. Implicit
c. Crank-Nicolson
d. All the above
7. For FDM based discretization of temperature gradient at the boundary grid
point on west face, which of the following method is used
a. Forward difference
b. Backward difference
c. Central difference
d. All the above
e. Depending on the accuracy required any of the above could be selected
Section B: Short Answer Questions
[8 marks each]
8. State physical situations for each of the boundary conditions (Dirichlet,
Neumann and Robin) associated with temperature?
9. Derive the grid Fourier number criterion from the LAE of cell-center temperature
corresponding with only unsteady and diffusion term. Assume the grid to be uniform
and the LAE is solved explicitly?
Section C: Long Answer Questions
9. Derive Linear algebraic equation(LAE) for a general internal to solve 1D
unsteady heat conduction without heat generation with implicit time marching
scheme. (assume aE, aW and aP to be coefficients of TE, TW and TP respectively).
(Derive from energy conservation law neglecting advection and heat generation
term)
[12 marks]
10. Consider an electronic device which needs to be cooled through the use of a
cylindrical extended surface. Assume temperature variation across radial
direction to be negligible. The base of surface is kept at constant temperature
100 C while end of pin is exposed to ambient air at 30C.
Air 30 0C
Base
100 0C
12cm
(Assume following properties à h=25 W/m2K, k = 16.2 W/mK, ρ = 1200 kg/m3,
Cp =1.4 )
(i) Formulate the conduction equation assuming negligible advection and
indicate the boundary conditions necessary for the study.
(ii) Discretise using first order FVM with 5 points along axial direction and write
the linear algebraic equations assuming that 1D STEADY state solution is
required. Solve the algebraic equations using suitable method.
(iii) Using the temperature values found above, find the heat transfer rate on
right end of pin.
[18 marks]
END OF PAPER
