Exam 2 (4th July)
Group 1.1 (4/20) / Justify every answer, using equations if convenient
The figure represents a wall with heat generation, insulated on one side. The temperature at the fluid contact surface is 120ºC.
/ a) Compute the heat flux transferred to the fluid (per unit of area).
b) Compute the power (uniform) generated per unit of volume inside the wall.
c) Write the differential equation that describes the temperature inside the wall and the boundary conditions.
d) Compute the temperature at x=L.
Group 1.2 (6/20)
The figure represents temperature profiles in an wall isolated on one side, in several moments of time (initial situation, stationary situation and transition situations) for different boundary and heat generation conditions.
/ a) Is there any case with heat generation inside the wall? If so, for that case outline a chart with the heat flux through the wall in stationary regime.
b) Is there any case with heat removal inside the wall (equivalent to negative heat production)?
c) In two of the figures the temperature gradient at is null along the whole wall. This allows us to say something about the relationship between the wall and the fluid temperatures?
d) In figure (d) does the temperature gradient shown at seem acceptable?
e) In figures (a) and (c) the initial situation and the final are coincident, but developments over time are different. Indicate the plausible boundary conditions for each one of them.
f) In 2 figures the temperature is imposed on the right wall. Indicate a method to impose the temperature.
TEM – Transferência de Energia e Massa 2015/16 / Exame 2 – Parte 2, 4 de Julho
Grupo 2.1: About evolution equations / (2.5 valores/20)
The Dissolved Oxygen evolution equation for a generic 1D geometry in a eulerian reference is:
And in the Lagrangian reference is:
Where is the saturation concentration and is the concentration of Biological Oxygen Demand, i.e. the oxygen required for the respiration of the organic matter carried by the water.
a) Physically, what is the difference between partial derivative and total derivatives?
b) How to write the equation for a tank with uniform properties?
c) The value of the diffusivity , in a numerical model increases with the spatial step. Why?
Grupo 2.2: Resolução Numérica das equações / (2.5 valores/20)
a) Taylor’s series are used to compute differentials, transforming partial differential equations into algebraic equations, and providing information about truncation errors. Central differences for advection are second order accurate. What does it mean?
b) How does the numerical diffusion is related to the spatial step?
c) The implicit methods are more stable than the explicit. Why?
Grupo 2.3: Condições de Fronteira / (2.5 valores/20)
- The exchange of gases and heat through the free surface of a river are described by equations with similar forms. To what extent are related to wind speed?
- Why can we consider the bottom of a river as adiabatic?
Grupo 2.4: Resultados de um modelo 1D / (2.5 valores/20)
- For a 1D case and an instantaneous heat emission into a water body in equilibrium with the atmosphere, sketch as a function of space, (a) an initial distribution, and (b) the solution after some time in case of (b1) having only heat exchanges with the atmosphere (b2) having heat exchange with the atmosphere plus advection.
- Comment the statement "the importance of heat diffusion decreases when the heat exchange with the atmosphere increases".