q3

• Chris Nkinthorn

• 20190530

Questions

1. Describe the lumped approach for transient heat transfer problems.

   In general, heat transfer problems are dependent upon multiple variables: typically, spatial and temporal. The method of lumped analysis is to collect, or lump, the entirety of spatial domain into a single value, which represents the net quantity for the entire system. In which case, the problem can be simplified into an ordinary differential equation (ODE), where the temperature changes with time alone.

2. Describe the relevance of the Biot number when dealing with transient heat transfer problems.

   The Biot number represents a ratio between resistances or time constants :

   $$B i=\frac{R_{c o n d, x}}{R_{c o n v}}=\frac{L \overline{h} A_{s}}{k A_{c}}=\frac{\tau_{i n t}}{\tau_{e x t}}=\frac{L^{2}}{\alpha} \frac{1}{R_{c o m} C}.$$

   These methods are equivalent because they are both ratios for measures of energy loss; for these methods to be applicable they must be consistent. The first method of deriving the Biot number is by relative resistances. The other is a comparison of the relative rates at which energy can leave. Fundamentally, they are the same: simplification down into the same value happens as the Biot number represents the fractional amount of heat energy travel in a specific direction. When the value is low, then little heat energy moves that way and that specific dimension is can be neglected, such as a 3D problem being reduced into 2D.

3. Discuss the connection between time constant and lumped capacitance.

   Lumped capacitance $C$ is a measure of the entire amount of thermal energy of a system, independent of position. Another quantity, the thermal resistance $R$ is a dependent on the mode(s) of heat transfer. The time constant is the product of these two values, representing the rate at which energy leaves the system and is used as a way to scale exponential decay.

4. Describe what is meant by thermal depth of penetration and thermal wave.

   The thermal depth of penetration is linked to the concept of a thermal wave. In this case of a semi infinite domain, or half space, let’s assume the medium to be homogenous. For this example, let the plane of the half space be the $xy$ plane and that $z = 0 $ separates the conducting medium from everything else. We can apply what ever temperature to the surface instantaneously, similar to applying a heat source directly to the “floor”. Initially, only the surface will “feel” the temperature and for layers $z^-$, the temperature is still the initial temperature. However, as time propagates, the temperature increase propagates into the material as well, in an exponentially decaying manner. The distance into the half space with significant temperature rise from the initial temperature is the “thermal depth of penetration” and the form of temperature distribution can be though of as the “thermal wave” pushing into the half space.

5. Discuss what is meant by external and internal equilibration.

   Equilibration is a process toward an equilibrium distribution of temperature which is steady state and independent of time. Let us apply an arbitrary volume, being acted upon by externally applied boundary conditions, such as a prescribed temperature. Internally, the system wants to be isothermal so that there is no net motion of heat energy. The external equilibration process trends toward a steady state as well, but wants the applied boundary to win out so that the system is isothermal: with the boundary temperature throughout.

6. Describe the attributes associated with a self-similar solution.

   Self similarity allows for a method of reducing the governing partial differential equations (PDE’s), which are multivariate in nature, down into a single dependent variable using functional analysis by way of a similarity parameter, denoted as $\eta(x,t)$. This is not always possible, depending on the problem. It does serve as a method finding analytic results, when possible. The method of reduction is done by normalization of one of the two parameters, so that it can be scaled back into a multidimensional equation in post.

7. Discuss the transformations required when using the self-similar parameter solution process.

   Self similarity is a method of reduction in variable order, such that there is a function which casts a multidimensional problem into a scalar value. The region $(x,y)\rightarrow \eta[x,t]$ becomes mapped so that $T(x,y) = T(\eta)$, so that a PDE can be solved using ODE methods. This is because the PDE is a set of similar curves, which can be normalized and scaled back into a family of curves once the ODE is solved. Explanation in a physical sense, is that the thermal wave travels through the material, it takes the same amount of time when normalized by the thermal penetration distance.

8. Describe the role of all thermo-physical properties required to characterize the characteristic time.

   Thermophysical properties are those intrinsic to the solid and fluid materials. For the solid material, the properties are the linear thermal conductivity, $k$; density, $rho$; and specific heat capacity $c_p$. For fluids, the thermophysical property to consider is the heat transfer coefficient, $\bar{h}$. Together, they form the characteristic times for internal and external equilibration. For practical analysis, these may be time, space, or temperature dependent, let these be constant for the explanation. In the numerator, increase property values would accelerate equilibration while in the denominator there is an inverse relationship. For internal equilibration, $\tau = L^2/4\alpha, \alpha = k/\rho c_p$. The linear thermal conductivity $k$, serves as a succeptibility for the material to allow heat transfer while $\rho c_p$ together quantifies the total energy of the material. The heat transfer coefficient $\bar{h}$ relates the rate at which convection can remove heat from the material.

9. Discuss when does an infinite domain 1D HT problem becomes a bounded problem.

   The infinite domain or even half space is only useful for the smallest timescales, where depth of penetration is very small in comparison to the actual thickness of the material. Additionally, the infinite domain 1D HT problem has also been analytically solved. Heat transfer problems are those concerned with the influence of external variables such as geometry and phase changes. Symmetry can be used to apply an adiabatic boundary condition by “folding” the domain. The physics phase change force a boundary condition to exist as the thermophysical properties between the two phases are different.

10. For 1D transient diffusion HT problems, describe how Leibnitz formula could be used.

    Leibnitz’s rule is applied when the limits of integration are dependent on time. It is used to move the time derivative from inside (or outside) of the integral to the outside (or inside, depending on the application and the original problem posed). The requirement for use is the integral is bounded. A major application in HT, which is 1D and time dependent is when phase change occurs in a conducting rod. In this case, the depth of thermal penetration is time dependent and we would want to keep track of how far along the melt interface is along the length of the bar. This is similar to simplification of crystal boundary grain growth, where instead of measuring the crystal boundary as it moves into a melted liquid medium, we are watching from the other side.