Chapter 2 Flashcards
Is heat transfer a scalar or a vector quantity? Explain. Answer the same question for temperature.
Heat transfer is a vector quantity since it has direction as well as magnitude. Therefore,
we must specify both direction and magnitude in order to describe heat transfer
completely at a point. Temperature, on the other hand, is a scalar quantity.
How does transient heat transfer differ from steady heat transfer? How does one-dimensional heat transfer differ from two-dimensional heat transfer?
The term steady implies no change with time at any point within the medium while transient implies variation with time or time dependence. Therefore, the temperature or
heat flux remains unchanged with time during steady heat transfer through a medium at any location although both quantities may vary from one location to another. During
transient heat transfer, the temperature and heat flux may vary with time as well as location. Heat transfer is one-dimensional if it occurs primarily in one direction. It is twodimensional if heat transfer in the third dimension is negligible
Consider a third order linear and homogeneous differential equation. How many arbitrary
constants will its general solution involve?
The general solution of a 3rd order linear and homogeneous differential equation will involve 3 arbitrary constants.
How is the order of a differential equation determined?
The order of a differential equation is the order of the highest order derivative in the equation.
What is the geometrical interpretation of a derivative? What is the difference between
partial derivatives and ordinary derivatives?
Geometrically, the derivative of a function y(x) at a point represents the slope of the tangent line to the graph of the function at that point. The derivative of a function that
depends on two or more independent variables with respect to one variable while holding the other variables constant is called the partial derivative. Ordinary and partial derivatives are equivalent for functions that depend on a single independent variable.
Consider a hot dog being cooked in boiling water in a pan. Would you model the heat transfer to the hot dog as one-, two-, or three-dimensional? Would the heat transfer be steady or transient? Also, which coordinate system would you use to solve this problem, and where would you place the origin? Explain.
Heat transfer to a hot dog can be modelled as two-dimensional since temperature differences (and thus heat transfer) will exist in the radial and axial directions (but there will be symmetry about the centre line and no heat transfer in the azimuthal direction. This would be a transient heat transfer process since the temperature at any point within the hot dog will change with time during cooking. Also, we would use the cylindrical coordinate system to solve this problem since a cylinder is best described in cylindrical coordinates. Also, we would place the origin somewhere on the centre line, possibly at the centre of the hot dog. Heat transfer in a very long hot dog could be considered to be one-dimensional in preliminary calculations.
From a heat transfer point of view, what is the difference between isotropic and anisotropic materials?
Most engineering materials are isotropic in nature, and thus they have the same properties in all directions. For such materials we do not need to be con- cerned about the variation of properties with direction. But in anisotropic ma- terials such as the fibrous or composite materials, the properties may change with direction. For example, some of the properties of wood along the grain are different than those in the direction normal to the grain.
What is heat generation in a solid? Give examples.
Heat generation in a solid is the conversion of mechanical, electrical, nuclear or chemical energy into heat.
In order to size the compressor of a new refrigerator, it is desired to determine the rate of heat transfer from the kitchen air into the refrigerated space through the walls, door, and the top and bottom section of the refrigerator. In your analysis, would you treat this as a transient or steady-state heat transfer problem? Also, would you consider the heat transfer to be one-dimensional or multidimensional? Explain.
The heat transfer process from the kitchen air to the refrigerated space is transient in nature since the thermal conditions in the kitchen and the refrigerator, in general, change with time. However, we would analyze this problem as a steady heat transfer problem under the worst anticipated conditions such as the lowest thermostat setting for the refrigerated space, and the anticipated highest temperature in the kitchen (the so-called design conditions). If the compressor is large
enough to keep the refrigerated space at the desired temperature setting under the presumed worst conditions, then it is large enough to do so
under all conditions by cycling on and off. Heat transfer into the refrigerated space is three-dimensional in nature since heat will be entering through all six sides of the refrigerator. However, heat transfer through any wall or floor takes place in the direction normal to the surface, and thus it can be analyzed as being one-dimensional. Therefore, this problem can be simplified greatly by considering the heat transfer to be one-dimensional at each of the four sides as well as the top and bottom sections, and then by adding the calculated values of heat transfer at each surface.
What is a boundary condition? How many boundary conditions do we need to specify for a two-dimensional heat transfer problem?
The mathematical expressions of the thermal conditions at the boundaries are called the boundary conditions. To describe a heat transfer problem completely, two boundary conditions must be given for each direction of the coordinate system along which heat transfer is significant. Therefore, we need to specify two boundary conditions for one-dimensional problems, four boundary conditions for two-dimensional problems.
What is an initial condition? How many initial conditions do we need to specify for a two-dimensional heat transfer problem?
An initial condition is a mathematical expression for the temperature distribution of the medium initially. Note that we need only one initial condition for a heat conduction problem regardless of the dimension since the conduction equation is first order in time.
What is a thermal symmetry boundary condition? How is it expressed mathematically?
Some heat transfer problems possess thermal symmetry as a result of the symmetry in imposed thermal conditions. For example, the two surfaces of a large hot plate of thickness L suspended vertically in the air will be subjected to the same thermal conditions, and thus the temperature distribution in one half of the plate will be the same as that in the other half.
How is the boundary condition on an insulated surface expressed mathematically?
on an insulated surface, the first derivative of temperature with respect to the space variable (the temperature gradient) in the direction normal to the insulated surface is zero
It is claimed that the temperature profile in a medium must be perpendicular to an insulated surface. Is this a valid claim? Explain.
This also means that the temperature function must be perpendicular to an insulated surface since the slope of temperature at the surface must be zero.
Why do we try to avoid the radiation boundary conditions in heat transfer analysis?
The radiation boundary condition involves the fourth power of temperature, and thus it is a nonlinear condition. As a result, the application of this boundary condition results in powers of the unknown coefficients, which makes it difficult to determine them. Therefore, it is tempting to ignore radiation exchange at a surface during a heat transfer analysis in order to avoid the complications associated with nonlinearity. This is especially the case when heat transfer at the surface is dominated by convection, and the role of radiation is minor.
