Heat Transfer & Thermal Analysis - Lecture Notes, Dispense di Ingegneria Aerospaziale

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Heat transfer &

thermal

analysis

Prof. Manfredo gherardo

guilizzoni

A.Y. 2019/

Lecture Notes by Andrea Pizzetti

HTTA Page 1

Heat

Heat is not an energy, but it's defined as the flow of thermal energy, so the thermal energy passing an unit surface in a unit time. Thermal energy is related to microscopic motions: the heat flow happens due to temperature differences. We can see an analogy with the voltage of the electric field: in this case the temperature is the voltage, therefore the potential. The term Thermal Analysis refers to two possible definitions: The set of experimental techniques which exploit thermal properties to characterize the materials or to detect defects/failures.

• The model, analysis, design and control of the systems from an heat transfer point of view. The second one will be the focus of our study. However, the full phenomena it's highly complex. Therefore, we'll try to have a compromise between accuracy and tractability: the introduced simplifications in fact have to be consistent with the problem.

Medium

Since we can't follow each one of the particles, we have to approach differently according to the type of medium:

• Vacuum (transfer only by radiation).
• Rarefied media (statistical mechanics).
• Dense media (continuum mechanics). We'll not consider in our study the rarefied media case. For the majority of the course indeed we'll use continuum mechanics, while at a certain point we'll study also radiation. We can consider the material domain as a continuum whenever the Knudsen Number Kn is very small, much smaller than 1. Knudsen number is defined as the ratio between the Mean Free Path and the characteristic dimension of the problem. The elements in a continuum domain are densely packed, hence the mean free path and therefore the Knudsen number is very small.

Heat Transfer Model

The model is based on four fundamentals: Balance Equations (or Conservative Equations when possible), such as the conservation of energy or the entropy balance.

• Constitutive Relations, typically empirical such as the ideal gas law. Jump Conditions , such as the Laplace equations, whenever there are discontinuities or interface surfaces between two different fields.

• BCs or ICs. As starting points we have mass, momentum, energy conservations and the 2nd law of thermodynamics, which states that the heat spontaneously flows from regions at higher thermal interaction potential (temperature) to regions at lower thermal interaction potential. A simple trick to get a preliminary result is to see the heat flow as a fluid, a lazy fluid, which always choose the easiest path. The general balance equation for an extensive quantity Y in a fixed volume system V with surface contour C is: The dot means divided by time (not derived!), two superscripts is divided by area, three superscripts is divided by volume. The specific quantity is y because it's multiplied per which is the infinitesimal mass. The variation in time of y is given by the surface integral of the contribution of y transferred by velocity w (carried Preliminary concepts lunedì 9 marzo 2020 10:

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Heat Conduction Equation

Conduction happens in solids or fluids in absence of macroscopic mass transport. Therefore, we can consider a closed system, in which there is no mass exchange but only energy one. Moreover, incompressibility assumption is used, since conduction affects more solids than fluids. We can infer that mechanical work Lm is not present since volume is constant (constant mass and density). If a device in the system introduces electrical work Le , this work is accounted into the thermal energy due to Joule effect, in particular in the heat source U. The conservation of energy then becomes: In integral terms: Heat flux Q has a direction in space because it's a variation, while U characterizes the unit volume, so it's scalar. As constitutive relation, we use the Fourier Law which respects the 2nd Law of thermodynamics: The Conductivity is a tensor, function of position p for non-homogenous materials, direction d for anisotropic materials and temperature (local property). However, for our study in the most of the cases it will be assumed constant and written as λ. If that is the case, for fixed heat flux, the higher is the conductivity, the more conductive is the material and the lowest will be the temperature difference between two interfaces, because the heat is more free to move through the material. Typical values of conductivity are:

• Gases around 0.01-0.
• Liquids around 0.1- 1
• Rubber or plastic 1- 10
• Stainless steel 15
• Iron 80
• Aluminum 200
• Copper 400
• Diamond 1000 When phenomena of de-localized electrons or lattice vibrations happen, conductivity gets larger. Employing perfect fluid/perfect solid assumption ( ) and the Fourier law together with the conservation of energy, the Heat Conduction Equation can be derived: When heat source is not present, often this form is used: Where α is called Thermal Diffusivity , which is a measure of how quickly a material can carry heat away from an heat source, since each material does not just transmit heat ( , but must be warmed Conduction martedì 10 marzo 2020 12:

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away from an heat source, since each material does not just transmit heat ( , but must be warmed as well ( ). The Heat conduction equation can't be solved analytically, unless we consider simplified cases. Due to the complexity of the solution, we'll use numerical methods such as CFD to obtain solutions in all of the other cases.

Steady-state 1D

If temperature variations are relevant along only one direction, we can make use of the Poisson equation in which conductivity is a constant scalar and the transient term disappears: If no heat source is present, furthermore, the Laplace equation is obtained: We obtained the same equations of electromagnetism, so we'll have also the same solutions and we can exploit the same techniques, following the electrical analogy. In this way the temperature profile is computed for cartesian, cylindrical and spherical coordinate systems. The heat flux is obtained through the Fourier law, so deriving by the principal dimension the temperature profile and multiplying for the conductivity, which is assumed constant. In cartesian coordinates, the heat flux is constant. In case of cylindrical and spherical coordinates to obtain again constant results we use the Heat Transfer Rate instead of the heat flux, which would depend on the radius. The heat transfer rate is the amount of thermal energy passing in the unit time. To obtain the heat flux, we just divide the heat transfer rate for the cross sectional area. In presence of an heat source, we have to integrate the equations, obtaining temperature and heat flux profiles as function of radius for all the three reference frames:

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Temperature gradient, therefore slope, has an inverse proportionality with the conductivity. Considering the red curve, at high temperature values, near T left, the conductivity is high since there is a low slope, while when decreasing the temperature the slope increases and therefore the conductivity gets lower. So the material linked with the red curve has higher conductivity at higher temperatures. For the blue curve it's vice versa.

Steady-state 2D/3D and Transient 1D

In some cases it's possible to develop an analytical solution for these problems. For steady-state 2D/3D we can use the Shape factor S method which models the conduction between two surfaces at fixed temperature across a passive domain, using the electric analogy. If we have a heat source, however, this method cannot be applied. For transient 1D case, instead: Heisier charts are used for plates, cylinders, spheres with or without heat sources and different BCs.

Semi-Infinite Medium method is used whenever a boundary is solicitated and all the other boundaries are very far from the part of the domain we're interested in.

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With numerical methods typical integral-differential problems of many engineering applications are converted into algebraic problems, which can be solved with the support of computers. Different methods are employed depending on the phenomenon. Conduction: Finite Difference Method (FDM), Finite Volume Method (FVM) or Finite Element Method (FEM).

• Convection: Lattice Boltzmann methods (LB).
• Radiation: Monte Carlo methods (MC). A numerical method is composed by two parts: System of equations, in general made by conservation equations, constitutive relations and models. This is the case for FDM, FVM, FEM, LB methods.

○ Algorithm, which describes what is happening. This is the case for MC method.

• Solving method, which can be based on:
• Discretization method , which often gives the name to the numerical method.

Discretization method

The discretization can be made on:

• Space, generating volume elements or surface faces.
• Time , generating time steps. Properties , considering approximations of their variations (such as linear or constant piecewise) or binning them into specific ranges (such as in the case of wavelengths for radiation).

In general we follow some steps: Choice of domain dimensions and analysis of possible symmetries (symmetry planes or axis- symmetries).

2. Choice of coordinates system (cartesian, cylindrical, spherical, curvilinear).
3. Modelling of the geometry.
4. Creation of the mesh. Meshing is the most important part, which is the division of the domain in small subparts. There are different types of meshes: Structured or Grids , which are always regular and can be uniform or with grading along certain directions.

Block-Structured, with conservation enforced at interfaces, but without matching of them (for instance when dealing with rotating parts).

• Unstructured , usually with refined cells at the boundaries. Chimera Grids , with some cells fixed and others moving, or both moving with different behaviours.

Meshes can employ different shapes for the elements. In case of 3D domains, the most common are tetrahedra and hexahedra (any polyedrum with 6 faces, so for instance a cube). In 2D domains, triangles or quadrilaterals are employed. In regions where the input quantity shows large gradients usually the mesh is refined. The Self-Adapting Mesh can change the mesh in time, allowing to account for the transient phase. With Snapping instead we means the movement and superposition of the nodes from the mesh to the actual surface of the body, often mapped with tomography. Numerical Methods martedì 17 marzo 2020 12:

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This phenomenon can be studied in three cases:

• Single-phase , when dealing with liquid or gas.
• Two-phase , in case of boiling/condensation or multicomponent mixtures.
• Marangoni , when dealing with thermocapillary convection due to surface tension gradients. Example of a Marangoni case can be seen in the following figure: There is also a classification based on the boundaries of the problem:
• Internal Convection. When a solid surface completely encloses the fluid. External Convection. When at least one side of the domain is occupied by the fluid up to an infinite distance from the wall.

A fluid flowing in a hot tube is an example of internal convection: Last classification line divides the type of motion in three types: Forced , when fluid motion is not forced by temperature gradients, but it's generated by an external source such as a pump or a fan.

• Free , when fluid motion is forced only by temperature gradients.
• Mixed , when there is a superposition of the two above. The following picture depicts a case of free convection for a plate: Classification martedì 31 marzo 2020 11:

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• 2. Convection Page

law. To compute the heat flux given to or removed from a solid wall we'll use the Newton Law: The complexity of the phenomenon is hidden inside the Convection Coefficient h , which is a difficult property to find.

Correlations

Under a series of assumptions, for very simple cases there are semi-analytical solutions from the boundary layer theory which allow to retrieve h. Its value depends on fluid properties, fluid velocity and area of the solid-fluid interface. Instead of using these variables, Correlations are in general developed employing dimensionless numbers. To retrieve them, an adimensionalization is needed: The characteristic velocity used to non-dimensionalize is different depending on the case. The system of equations is rewritten accordingly to the adimensionalization, obtaining equations in which dimensionless coefficients called Numbers are present. The BCs are obtained from the link between solid and fluid layer. In fact, till now we saw equations in which there is no reference to the interface. To account for this, we can set the conductive heat flux equal to the convective heat flux at the interface: It should be noticed, however, that the vectors of the heat fluxes have to be parallel to be able to write this boundary condition. To proceed, we can retrieve the derivative from the dimensionless numbers, in this manner we can express the missing convective coefficient as function of them:

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Each one of the dimensionless numbers has a physical meaning: Nusselt. For high Nu numbers, convection is prevalent, indeed convective coefficient is much higher than conductivity.

○ Liquid metals: 0.01/0. ○ Air: 0. ○ Water: 5/ ○ Oils: 20 Prandtl. For Pr numbers close to 1, the fluid is equally good in transporting momentum and thermal energy, therefore fluid dynamics boundary layer and thermal boundary layer have approximately the same shape. At low values of Pr, the fluid dynamics b.l. is lower than the thermal b.l.: the fluid transports better heat than momentum. At high values of Pr, it's vice versa.

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An heterogenous system is composed of different phases, which are parts with different physical properties. Phases are separated by Interfaces so jump conditions must be provided in case of multi- phase systems. Interfaces are characterized by an higher energy density, called Excess Energy, because of the different attitude towards bonding between the atoms of the two phases. The lower the compatibility, the higher is the excess energy. This can be understood by thinking of each atom as characterized by a certain amount of energy, part of it used to bond to all the atoms around. At the interface there are less atoms to bond, so this excess energy makes the interface bonds stronger:

Two-phase

We can express the excess energy from an energy viewpoint or from a force viewpoint: As an elastic membrane, an interface cannot resist normal forces, but it can resists to tangential ones. If an interface is curved, a pressure difference exists between the two sides, which depends on the curvature K. This pressure difference, between internal and external, is called Laplace-Young Overpressure: The proof for a spherical drop can be obtained accounting the fact that at equilibrium the pressure difference between the interface which tends to enlarge the drop is equal to the surface tension which keeps the interface connected and the drop as small as possible. Actually, the shape of the droplets are rarely spherical because of the presence of other mechanical or thermal phenomena. To account for these effects, usually a local surface tension is employed. For a cylinder the overpressure is just σ/R because the second radius is infinite.

Three-phases

We could have a drop of liquid on a solid surface, surrounded by a vapour. We can identify the Static Contact Angle as the one between the liquid-vapour interface and the solid-liquid interface, measured in the point of contact of the three interfaces: Capillarity lunedì 11 maggio 2020 09:

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measured in the point of contact of the three interfaces: For an ideal situation, with a homogenous isotropic flat and smooth solid surface, in absence of any external field, the Young Equation expresses the relation between the interface energies and the contact angle: Energies of the interfaces. The energy can vary among them if solid is not homogenous or if there are temperature differences.

• Gravitational field. It rules the flattening.
• Morphology of the surface. The superficial roughness can change a lot the situation. In a real situation, the shape of interfaces and the contact angle at each point are the result of different factors: A smooth surface can have different Wettability , which is defined according to the value of In case of a rough surface, characterized by cavities, the wetting state has to be described by two models depending on whether the liquid is able to fill the cavities or not. If it's able, the Wenzel Model is employed, which account for the larger wetted area with the r parameter:

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The convection phenomena linked to a phase change in a fluid are Vaporization , from liquid to vapor, or Condensation, from vapor to liquid. They can occur within the fluid or at an interface with a solid. The two most important characteristics are: Constant temperature. During the phase change, the heat transfer occurs at constant temperature.

Very large heat transfer coefficient. Thanks to the combined effect of latent heat and buoyancy-driven flow due to large density difference between liquid and vapour, the heat transfer coefficient and therefore the heat transfer rate is on the scale of thousands.

The phenomena are very complex because we have moving and changing area interfaces and dynamic interactions between the phases such as bubbles. Moreover, the phase change is a non- equilibrium phenomenon which should be combined with the heat transfer and fluid dynamics concepts. The problem therefore is usually faced only experimental using correlations to determine a Two- phase Convective Coefficient. The Newton equation is written using the Excess Temperature, difference between the wall temperature and the Saturation Temperature (temperature at which the phase change occurs) : The saturation temperature is a function of pressure. So for each saturation temperature there is also a Saturation Pressure. The applications of the phase change phenomenon are several:

• Power generation with Rankine cycles.
• Chemical processes.
• Thermal control of compact devices which require heat dissipation.

Vaporization

In general we speak of Vaporization when there is change from liquid to vapour phase. If the phenomenon is restricted to the gas-liquid interface we have Evaporation, otherwise we have Boiling. In the first case the vaporization happens even if the temperature is under the saturation one. For instance, in a pool after the rain, the water vaporizes even if the temperature is much lower than 100°. In this case, the vapour saturation pressure exerted by the vaporized molecules is lower than the total ambient pressure. Homogenous. Nucleation happens by statistical clustering of high-energy molecules within the bulk liquid.

Heterogenous. Nucleation happens at the micro cavities of the fluid solid interface (such as in the walls of a pot). This is the case in which capillary phenomena plays an important role.

• Pool Boiling. Initially still liquid pool, so free convection.
• Flow Boiling. Liquid stream in forced motion, so forced convection. During boiling, instead, the vapour saturation pressure has reached the total ambient one and vapour bubbles start to form within the bulk substance. We can have different cases of boiling:

Pool Boiling

If we place a liquid in a tank and we heat it up using a wire or a duct in which an heating fluid is moving, we will notice three different type of nucleations: Phase Change martedì 12 maggio 2020 11:

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In the Nukiyama Curve we can see the amount of heat flux dissipated within the liquid for different excess temperature values: The red curve depicts a mapping of the possible steady-state heat flux with different fixed heater temperature (Twall). We can notice that we have a transition region in which heat flux is lower with respect to cases at lower temperatures. This because the jet of vapor has completely surrounded the heater and since in general gases are less convective than liquids, the bulk fluid cannot absorb too much energy. In the film boiling zone, instead, the radiation begins to become a dominant effect since it scales with the fourth power of temperature, so the heat flux return to be high even if the vapor cushion is still present. In the blue curve the heat flux given by the heater is set constant. However, the heat flux which can be absorbed by the fluid is not always equal to it. When the latter is lower than the former, the temperature of the heater will rise. In transition region, due to the generation of the vapor cushion, a small increase in the heat flux absorbed by the bulk fluid will result in a suddenly increase in temperature difference because of the low convection coefficient. This jump can cause the melting or Burnout of the wire, so the design point should be not too close to the maximum transfer rate point.

Bubble Nucleation

Let's see what happens in the bubble nucleation region. On the surface of the heater crevices act as nucleation site. They are always present due to the roughness or maybe because they are engineered in that way. Indeed the zone of bubbles nucleation is good in terms of heat exchange. The crevices are so small that can't be filled with liquid due to the surface tension. Instead, inside them a gas is present. So we'll have a gas-liquid interface between the crevice and the liquid and therefore evaporation. When the saturation temperature is reached, the bubble begins to create. However, since it's very small, the Laplacian-Young overpressure it's high (low radius). The saturation temperature inside the bubble is much higher due to the higher pressure and the bubble cannot survive in gas phase, so it must condense unless we provide a Superheating. In this way, we can enlarge the bubble because

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