1st Exam of Advanced Heat Transfer
Integrated Master in Mechanical Engineering
June 12, 2018
2nd semester
Remarks:
1- Duration of the exam: 3 h
2- Recommended time for the theoretical part: 45 minutes
3- The use of textbooks and calculator is forbidden during the theoretical part
The theoretical part must be solved in this paper sheet. Identify it:
Student number:
.
Integrated Master Course:
.
Name:
Theoretical Part (grade: 8/20)
Four statements are given for questions 1 to 8, only one of them being true. Indicate the
correct statement for every question in the following table. The grade is +0.5 for every
correct answer and -0.15 for every wrong answer. The overall grade of these 8 questions
will never be lower than 0.
Question
Answer
1
2
3
4
5
6
7
8
1. In the analytical solution for the condensation of a vapor on a vertical plate proposed
by Nusselt, it is considered that:
(a) The velocity profile inside the condensate liquid film varies quadratically with
the distance to the wall, and the temperature profile varies linearly.
(b) The velocity profile inside the condensate liquid film varies linearly with the
distance to the wall, and the temperature profile varies quadratically.
(c) The velocity and temperature profile inside the condensate liquid film vary
linearly with the distance to the wall.
(d) The velocity and temperature profile inside the condensate liquid film vary
quadratically with the distance to the wall.
2. Different combinations of fluid and type of surface material:
(a) influence the heat transfer coefficient for the nucleate pool boiling regime, but
do not influence the values of the critical heat flux and minimum heat flux (at
the Leidenfrost point).
(b) influence the values of the critical heat flux and minimum heat flux (at the
Leidenfrost point), but do not influence the heat transfer coefficient for the
nucleate pool boiling regime.
(c) do not influence the heat transfer coefficient for the nucleate pool boiling
regime, and the values of the critical heat flux and minimum heat flux (at the
Leidenfrost point).
(d) influence the heat transfer coefficient for the nucleate pool boiling regime, but
do not influence the values of the critical heat flux and minimum heat flux (at
the Leidenfrost point).
3. In the solution of a scalar transport equation through the finite volume method, the
most suitable option for linearizing the source term S! = 1 − ϕ#" ≈ S$ + S% ϕ"
(considering Su and Sp as functions of ϕ∗" known from the previous iteration) is the
following:
(a) S$ = 1 − ϕ∗" '
;
S% = 0
(b) S$ = 1 + ϕ∗" + ϕ∗" ( + ϕ∗" '
(c) S$ = −ϕ∗" #
;
;
S% = −1 − ϕ∗" − ϕ∗" ( − ϕ∗" '
S% = ϕ∗" )*
(d) none of the above options is suitable.
4. In the numerical solution of a scalar transport equation (without diffusion
phenomena), the convective terms should be discretized with:
(a) central differencing scheme because it does not induce false diffusion.
(b) upwind scheme because it always yields physically plausible solution.
(c) central differencing scheme whenever the local grid Péclet number is very
small.
(d) hybrid scheme because it yield the best compromise between upwind and
central differencing.
5. The hybrid differencing scheme used to approximate the steady-state onedimensional convection-diffusion equation:
(a) is bounded and produces physically realistic solutions.
(b) leads to false diffusion whenever the Peclet number is smaller than 2.
(c) is first order accurate in Dx, if the Peclet number is maintained always
smaller than 2.
(d) is less accurate than the upwind scheme for Peclet numbers greater than
2.
6. For estimating the average spectral absorption coefficient of an isothermal
polyatomic gas medium:
(a) wide band models are the most accurate option.
(b) information about the optical thickness of the medium is not required.
(c) Elsasser model is the most accurate option.
(d) none of the above options is true.
7. Regarding the radiative properties of particles, independent scattering can be
assumed whenever:
(a) the radiation wavelength and energy remain unchanged during the
scattering process.
(b) the clearance to wavelength ratio is very high.
(c) the volume fraction of particles in the medium is very high.
(d) the scattering by one particle is affected by the presence of surrounding
particles.
8. For solving a problem of radiative exchange in an enclosure containing a
participating gas medium:
(a) the zonal method is always the most practical choice.
(b) the zonal method does not request direct information about the gas
medium emissivity.
(c) the mean beam length method is typically accurate but time consuming.
(d) the presence of the medium can be neglected if its temperature is close to
zero Kelvin.
9.
In a clear and well-organized text with a number of words of less than 150, describe
the various regimes observed in the condensation of a vapor on a vertical plate, and
indicate the conditions of occurrence of the indicated regimes. (2 points)
10. Consider the following expression that relates the angle of incidence on a horizontal
surface with the latitude, solar declination and solar hour angles:
cos 𝜃 = sin 𝜙 ∙ sin 𝛿 + cos 𝜙 ∙ cos 𝛿 ∙ cos 𝜔
(a) At solar noon, for a place on earth located at the equator line, conclude about
the value of angle θ and its variation during the year. (1 point)
(b) For the place location given in question a), use the relation given above to
explain how the solar time of sunrise changes during the year. Quantitatively
justify your answer. (1 point)
1st Exam of Advanced Heat Transfer
Integrated Master in Mechanical Engineering
June 12, 2018
2nd semester
Remarks:
1- Duration of the exam: 3 h
2- Textbooks and lecture notes may be used in the practical part. The use of books of
solved problems is forbidden.
3- Each problem must be solved in a different paper sheet
Practical Part (grade: 12/20)
1. A long cylinder with a 5cm diameter, horizontally immersed in a water bath which
is at saturation temperature for atmospheric pressure, is subjected to a heat flux at its
external surface of 0.8 𝑀 W⁄m2 . The cylinder is made of copper, with an external
polished surface.
(a) Verify that the indicated heat flux is compatible with the nucleate pool boiling
regime, by calculating the critical heat flux. (1.5 points)
(b) Estimate the temperature of the external cylinder surface, assuming the heat
transfer regime is nucleate pool boiling. (2.5 points)
(c) What would be the differences introduced in the solution of the previous
questions, if the external diameter of the cylinder was 1 mm. Justify your
answer, presenting the numerical calculations required by a complete answer.
(1.0 point)
(continues in next page)
2. Consider a closed reactor with an inner volume of 4 m3 enclosed by walls with total
area of 12 m2. Inside the reactor, a gas mixture formed by CO2 diluted in air
(CO2/air volume ratio of 0.34) is maintained at 5 atm and 2000 K, while an internal
heat generation process occurs at a rate q''' [W/m3]. For simplification purposes
assume that the reactor walls are black and that air is composed by 21% O2 and 79%
N2 .
(a) Estimate the total emissivity ε and total absorptivity α of the gas mixture,
assuming that the reactor walls are kept at 500 ºC by means of external cooling.
Justify all hypothesis and approximations considered in your calculations.
(2 points)
(b) Under the conditions of question a), calculate the rate q''' of internal heat
generation in the reactor. (Note: if you did not solve question a), assume
ε = 0.14 and α = 0.26 ). (1 point)
(c) Instead of the original gas mixture, consider that the medium inside the reactor
consists of black particles of uniform size diluted in air (particle diameter of
1 mm and particle volume fraction of 0.02 %). Under this alternative scenario,
what are the total emissivity and total absorptivity of the medium? Justify all
hypothesis and approximations considered in your calculations. (Note: if you
did not solve question a), consider the mean beam length for radiation to be
1.2 m). (2 points)
(d) Under transient conditions, consider the following simplified model used to
estimate the evolution of the gas temperature T with time, as function of q'''
and reactor wall temperature Tw:
𝜌𝑐
𝑑𝑇
= 𝑞""" − 𝛾(𝑇 # − 𝑇$# )
𝑑𝑡
where all parameters, except T, can be treated as constants. Discretize and
linearize the above differential equation using implicit Euler method with time
step ∆t, and provide the resulting linear equation for the calculation of Tn+1 in
the form: an Tn+1 = Tn + b, where n is the time iteration index and coefficient
an can be a function of Tn. (2 points)