- Conduction Heat Transfer Schneider Pdf
- Conduction Heat Transfer Calculator
- Conduction Heat Transfer Schneider Model
S3.4 Modeling of Laser Machining Process (1)
The mathematical formulation for steady conduction heat transfer from a toroidal geometry into an infinite medium is presented. The torus is in thermal communication with the infinite medium through a heat transfer coefficient at the torus' external surface. The mathematical formulation is presented using the Toroidal coordinate system; a finite element grid is developed usign this coordinate. Being discharged. In a charging process, heat transfer from the hot gas increases thermal energy stored within the colder spheres; during discharge, the stored energy decreases as heat is transferred from the warmer 321 An experiment has been performed under conditions for which the wafer, initially at a uniform temperature.
Laser machining is a thermal process which removes material through phase changes (melting, vaporization). Before we introduce the readers to the models used in LMP, it is necessary to have a brief review of the related thermal processes. These include conduction, convection and radiation heat transfer. Knowledge of Fluid mechanics is also needed in reading the literatures of LMP. We will also talk about the energy involved in state changes of materials.
Review of heat transfer:
Heat transfer is energy in transit due to temperature differnece. Conduction, convection and radiation are the three basic heat transfer mechanisms.
Conduction Heat Transfer:
Conduction is the transfer of energy from a more energetic to the less energetic particles of substances due to interactions between the particles. In gas and liquids, heat conduction takes place through random molecular motions (difusions), in solid heat conduction is through lattice waves induced by atomic motions. In nonconductor, the energy transfer is exclusively through such lattice waves, in conductors, the motion of free electrons also contribute to heat conduction.
Fourier Law is the rate equation based on experimental evidences. This law states that heat flux at the surface normal is proportional to the temperature gradient.
Figure 3.16: 1D heat transfer
For 1D heat conduction in x-direction:
q'=-k dT/dx
Where k is thermal conductivity (W/m.K), T is temperature (K), q' is the heat flux in x direction.
General form of Fourier's Law is:
q' = - k Ñ T(x,y,z,t) = -k(i¶ T/¶ x + j¶ T/¶ y + k¶ T/¶ z)
Thermal conductivity k is a material property, it may vary with temperature and pressure. In laser machining which usually involves solids, the dependence on temperature should be considered. This makes k a function of temperature k(T), such function relations are found through interpolation of experimental results at various temperatures.
Using energy balance relations, ie., increase of stored energy in a volume of material = energy in + energy generation in this volume - energy out, we can deduce the general differential form of three dimensional heat conduction equation. In short:
Ein + Eg - Eout = D Est
General equation of Heat Conduction:
¶ (k¶ T/¶ x)/¶ x +¶ (k¶ T/¶ y)/¶ y +¶ (k¶ T/¶ z)/¶ z + dq/dt =r cp (¶ T/¶ t)
T= T(x,y,z,t). Steady state: (¶ T/¶ t)=0
If material is of constant property, we can pull k out and we get:
¶2T/¶ x2 + ¶2T/¶ y2 +¶2T/¶ z2 + (dq/dt)/k = (r cp/k) (¶ T/¶ t) =1/a (¶ T/¶ t)
a = k/(r cp) is the thermal difusivity.
In cylindrical coordinates we have:
Fourier Law: q' = - k Ñ T(r,f ,z,t) = -k(i¶ T/¶ r + j(1/r)¶ T/¶f + k¶ T/¶ z)
General equation of Heat Conduction:
(1/r)¶ (k¶ T/¶ r)/¶ r +(1/r2)¶ (k¶ T/¶f )/¶f +¶ (k¶ T/¶ z)/¶ z + dq/dt =r cp (¶ T/¶ t)
Where r is the radial direction, f is the circumferential direction, z is the axial direction.
When we assume axissymetric heat transfer, which happens frequently in laser drilling modeling, we mean that the variation of temperature in the f direction is zero. Then the above equation becomes:
(1/r)¶ (k¶ T/¶ r)/¶ r +¶ (k¶ T/¶ z)/¶ z + dq/dt =r cp (¶ T/¶ t)
Similarly in spherical coordinates:
Conduction Heat Transfer Schneider Pdf
q' = - k Ñ T(r,q ,f ,t) = -k(i¶ T/¶ r + j(1/r)¶ T/¶q + k(1/r sinq )¶ T/¶f )
(1/r2) ¶ (kr2¶T/¶ r)/¶ r + (1/r2sin2q) ¶ (k¶ T/¶f )/¶f + (1/r2 sinq ) ¶ (k sinq¶ T/¶q )/¶q + dq/dt =r cp (¶ T/¶ t)
These equations are complex, but they still state the conservation of energy.
Convection Heat Transfer:
Convection usually refers to the energy transfer between a solid surface and an adjacent moving gas or liquid. Convection heat transfer is a combination of diffusion or molecular motion within the fluid and the bulk or macroscopic motion of the fluid.
The rate of energy transfer from a system to the fluid is quantified by Newton's law of cooling:
q' = h(Tfluid - Ts)
where q' is the convective heat flux (W/m2), h is the convective heat transfer coefficient (W/m2K), Ts is the surface temperature (K) and
Tfluid is fluid temperature away from the surface (K).
The heat transfer coefficient, h, is an empirical parameter that encompasses the effects of the fluid flow near the surface, the fluid properties, and the surface geometry. The decision of h is the most important task in convection heat transfer analysis. Many emperical formulations are put forward, readers should refer to textbook of Heat Trasfer for detailed information. Convective heat transfer is coupled with the fluid dynamics of the system. The following figure show the boundary layer development of a uniform flow along a flat plate.
Figure 3.17: illustration of Boundary layer and convection heat transfer
Radiation Heat transfer:
Unlike conduction and convection, radiation does not require the presence of a medium to propagate. Actually, radiation transfers heat energy most efficiently in a vacuum. Radiation energy is transported by electromagnetic waves. Thermal radiation is energy emitted by matter at a finite temperature as a result of changes in the electron configurations of the atoms or molecules. Generally, radiation heat transfer analysis is focused on solid surfaces but emission may also occur from liquids and gases.
Stefan-Boltzmann Law states that emissive power E of a surface is proportional to T4 :
E=es Ts4,
Conduction Heat Transfer Calculator
Where e is surface emissivity; s =5.67*10-8W/m2.K4 is the Boltzmann constant.
The energy absorbed by the surface is:
Gabs=a G=as Tsur4,
where a is absorptivity; G is the irradiation from the surroundings.
So the net heat transfer of radiation energy from a finite surface to the infinite surrouding is:
q'rad=E- Gabs=es Ts4-as Tsur4
Where Ts and Tsur are the surface and surrounding temperatures.
Figure 3.18: radiation energy exchange between a surface and the surounding
The real analysis of radiation energy exchange is far more complex than what we see here. Radiation relates to direction, area, wavelength, surface condition, etc. We also see that radiation increases with 4th power of temperature, thus at high temperature the radiation heat transfer should be considered.
Review of Fluid Mechanics:
Detailed discussion of convective heat transfer is based on Fluid Mechanics. So it is necessary to give a brief review of some basic concepts and relations in fluid mechanics. These include:
1. Eulerian and Lagrangian Descriptions:
There are two basic coordinate systems which may be used in mechanics. In Eulerian framework the independent variables are the spatial coordianates x,y,z and time t. Attention is focused on the fluid that passes through a control volume which is fixed in space, the fluid inside the control volume at any instant in time will consist of different fluid particles from that which was there at some previous instan in time. In Lagrangian approach, attention is focused on a particular mass of fluid as it flows, in this frame, x,y,z,t are no longer independent variables, the independent varialbles are x0, y0, z0 at time t0 and time t. All positions of the mass we are considering can be calculated if the velocity field is known.
Let F be any field variable such as density or temperature, F=F(x,y,z,t) in Eulerian description or F= F(x0,y0,z0,t) in Lagrangian description. Then, the time derivative of F is DF/Dt, and we have:
, where u, v, w are velocities in the x, y, z directions.
2. Conservation of mass
Consider a specific mass of fluid whose volume V is arbitrarily chosen. This fluid mass is followed as it flows, its shape and size can change with time, but its mass will remain unchanged unless some mass generation/extinction mechanism exists (nuclear reaction for example). So the conservation of mass states that the Lagrangian derivative of the mass of the fluid inside the volume is zero. In mathematical form:
(Integration form)
The PDE form of conservation of mass is:
where xk and uk are coordinates and velocity in the k=1,2,3 directions.
If the fluid is incompressible, then we have:
Combine this relation with the general law of mass conservation, we have the well known Continuity Equation for incompressible flows:
3. Conservation of Momentum
The principle of conservation of momentum is an application of Newton's second law of motion to an element of the fluid. Considering a given mass of fluid in a Lagrangian frame, this law states that the rate at which the momentum of the fluid mass is changing is equal to the net external force acting on the mass. The external forces may be classified as body forces (such as gravitational or electromagnetic forces) and surface forces (such as pressure forces or viscous forces) . Thus we have:
After many manipulations, using continuity relation and Renolds' Transport Theorem, and relate the surface force to the stress tensor and surface normal ni, (i=1,2,3 , j=1,2,3). We got the Momentum equations:
Here we used summation convention. The above equation is in fact three equations. We don't have space to explain all the derivation details and the summation conventions. It's important to understand the physical meanings of these equations.
The constitutive relations relate the stress tensor with the pressure and fluid velocity field by introducing some material coefficients. Assuming Newtonian fluid, we have:
, where p is pressure, m is dynamic viscocity, l is another material coefficient.
Taking this relation into the Equation of momentum conservation, we derive the famous Navier-Stokes Equation:
Again this equation uses summation convention and represent three scalar equations.
Under the condition of incompressible flow and the dynamic viscosity to be constant, the Navier-Stokes equation becomes:
When viscous is neglected, we have the Euler equations:
4. Conservation of Energy
Apply the first law of thermodynamics to fluid element, we have the principle of conservation of energy in fluid mechanics. Consider any arbitrary mass of fluid of volume V and follow it in a lagrangian frame of reference as it flows. The total energy of this mass per unit volume is the combination of internal energy and kinetic energy, . The rate of change of the energy of the mass we are following is balanced by the works done by external forces minus the energy the mass give to the outside. In integral form, this relation is:
Applying the relation of continuity and momentum, we get the equation of conservation of thermal energy:
This equation is the balance of thermal energy only, it comes from the subtraction of mechanical energy from the original energy balance relation. But it is usually refereed to as the Energy Equation.
Taking stress relation into the above equation, we have:
where T is temperature, k is the thermal conductivity, is the dissipation function defined as:
For incompressible fluid in cartesian frame, .
5. State equations of materials
Two state equations are needed to make the above relations self-contained.
We know for perfect gas P=r RT, and internal energy e=CvT, with Cv be the constant volume specific heat.
Some times enthalpy is used in the energy equation, enthalpy is defined as:
From discussion of 2, 3, 4 and 5, we have shown seven scalar equations with 7 variables (assume k, l and m are known). For most general case, these relations are coupled. But usually we can simplify them, either for 1D or 2D flow, or we assume some specific conditions. Assume incompressible flow or neglecting viscous can simplify the equation. If we neglect the velocity in the energy equation, we can get the heat conduction equation. Assume irrotational flow results in the analysis of potential flow.
The model may becomes more complex when turbulence is considered. Turbulence surely takes effect in actual applications, but we don't have time to cover it here. Another complexity comes from shock-wave which is common for high energy laser machining. We will see some discussions in following sections.
Are you bored of these mess of math? I guess you need to figure them out in order to read the literatures in this area. I can not take the task of fluid mechanics, if you could not understand the meaning of these relations, you are suggested to turn for a book on Fluid Mechanics. If you feel comfortable for this section, then you will feel comfortable for the following sections.
Boundary layers and potential flows are discussed extensively in fluid mechanics. Let's show you some pictures in addition to the boring equations.
The flow pattern of a dipole
Flow around a cylinder
Flow around a airfoil
Figure 3.19 Some patterns of fluid field (Courtesy of UNIVERSITY OF GENOA, HYDRAULIC INSTITUTE)
- Open Access
- Total Downloads : 856
- Authors : B. K. Bharti, Dr. S. Singh, K. S. Deshwal
- Paper ID : IJERTV1IS8605
- Volume & Issue : Volume 01, Issue 08 (October 2012)
- Published (First Online): 29-10-2012
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License: This work is licensed under a Creative Commons Attribution 4.0 International License
Computational Study Of Steady State Conduction Heat Transfer With Different Cross-Section Using Altair® Hyperworks®
Computational Study Of Steady State Conduction Heat Transfer With Different Cross- Section Using
Altair® Hyperworks®
B. K. Bharti1 , Dr. S. Singp, K. S. Deshwal3
Conduction Heat Transfer Schneider Model
1M. Tech Scholar, 2Associate Professor, 3M. Tech Scholar
Department of Mechanical Engineering,
Bipin Tripathi Kumaon Institute of Technology, Dwarahat, Almora, Uttarakhand (India), 263653
In this study, the effect of temperature distribution with constant heat flux is considered. The study is considered on the Rectangular & Triangular profile block cross-section. This study is performed using Fourier Law of steady state heat conduction which depict the first law of thermodynamics. In the governing equation Fourier law is applied with the case of no internal heat generation to solve a one-dimensional heat conduction problem. The Problem is solved computationally using Altair Hyperworks Software. Analytical results were obtained for Rectangular & Triangular cross-section and these can be used to build up the Heat flow variation. The heat flux is maintained at constant value and temperature distribution within the section is obtained. Due to temperature difference heat will flow from higher temperature to lower temperature. The material of solid block cross-section provides conductive resistance. Hence, this is a conduction mode of heat transfer. The heat transfer takes place in one dimension only and properties are considered to be isotropic with two different Material Brass & Steel.
Keywords: Rectangular & Triangular Block cross-section temperature distribution.CAE
Software Altair Hyper work.
As we explore the propagation of energy, we must take into account the science of thermodynamics, which allows us to predict the trajectories of the processes, and the science of heat transfer for knowing the modes by which energy is
propagated from one system to other systems. We know that heat is not temperature because heat is energy in transit. Heat can exist in rotational, vibration and translational motions of the particles of a system, whereas temperature is the measurement of the average of the kinetic energy of the particles of a substance. The average of the molecular
kinetic energy depends on the translational motion of the particles of a system. The energy absorbed or stored by a substance causes an increase in the kinetic energy of the particles that form that substance. This kinetic energy or motion causes the particles to emit heat, which is transferred to other regions of that substance or towards other systems with a lower energy density.
To understand heat transfer we have to keep in mind that heat is not a substance, but energy that flows from one system toward other systems with lower density of energy. Heat is temperature difference and the surroundings. In most of the processes heat is either given up or absorbed, so there is transfer of heat. There are three fundamental types of heat transfer: conduction, convection and radiation. All three types may occur at the same time, and it is advisable to consider the heat transfer by each type in any particular case
Heat transfer generally takes place by three modes such as conduction, convection and radiation. Heat transmission, in majority of real situations, occurs as a result of combinations of these modes of heat transfer. Conduction is the transfer of thermal energy between neighbouring molecules in a substance due to a
temperature gradient. It always takes place from a region of higher temperature to a region of lower temperature, and acts to equalize temperature differences. Conduction needs matter and does not require any bulk motion of matter.
Conduction takes place in all forms of matter such as solids, liquids, gases and plasmas. In
Solids, it is due to the combination of vibrations of the molecules in a lattice and the energy
Transport by free electrons. In gases and liquids, conduction is due to the collisions and diffusion of the molecules during their random motion.
Convection occurs when a system becomes unstable and begins to mix by the movement of mass. A common observation of convection is of thermal convection in a pot of boiling water, in which the hot and less-dense water on the bottom layer moves upwards in plumes, and the cool and denser water near the top of the pot likewise sinks. Convection more likely occurs with a greater variation in density between the two fluids, a larger acceleration due to gravity that drives the convection through the convecting medium.
Radiation describes any process in which energy emitted by one body travels through a medium or through space
absorbed by another body. Radiation occurs in nuclear weapons, nuclear reactors, radioactive radio waves, infrared light, visible light, ultraviolet light, and X- rays substances.
Heat conduction is increasingly important in various areas, namely in the earth sciences, and in many other evolving areas of thermal analysis. A common example of heat conduction is heating an object in an oven or furnace. The material remains stationary throughout, neglecting thermal expansion, as the heat diffuses inward to increase its temperature. The importance of such conditions leads to analyze the temperature field by employing sophisticated mathematical and advanced numerical tools, Schneider (1957).
The section considers the various solution methodologies used to obtain the temperature field. The objective of conduction analysis is to determine the temperature field in a body and how the temperature within the portion of the body. The temperature field usually depends on boundary conditions, initial condition, material properties and geometry of the body, Teixeira and Rincon (2009).
Why one need to know temperature field. To compute the heat flux at any location, compute thermal stress, expansion,
deflection, design insulation thickness, heat treatment method, these all analysis leads to know the temperature field, Fried,(1957).
The solution of conduction problems involves the functional dependence of temperature on space and time coordinate. Obtaining a solution means determining a temperature distribution which is consistent with the conditions on the boundaries and also consistent with any specified constraints internal to the region. Keshavarz and Taheri (2007) have obtained this type of solution.
There are several methods for measuring thermal conductivity and diffusivity in the laboratory. The methods can basically be divided into steady-state and transient methods. A good summary of the most often used techniques is given by Beck (1988). Several techniques have been applied in measurements of thermal parameters in situ, such as 'passive' methods based on either temperature gradients in a borehole as indicators of lithologic (conductivity) variation Conaway and Beck(1977) annual temperature wave in the uppermost 15-30 m of bedrock for diffusivity determination Parasnis(1974); Tan and Ritchie(1997) or direct measurement of geothermal heat flow density and simultaneous temperature gradient in a drill hole which can be used
for in situ 18 conductivity estimation Oelsner and Rosler(1981); Jolivet and Vasseur(1982). Various active methods using either cylindrical, line or spherical sources for generating either a continuos heating signal or a heat pulse in the investigated medium have been developed for measurements in boreholes or soft sediments for terrestrial, marine and lunar studies (e.g. Beck et al(1971); Sass et al(1981); Mussman and Kessels, (1980); Langseth et al., (1972); Davis(1988).
The basic theory of heat conduction in a cylindrically symmetric geometry is developed by Carslaw and Jaeger (1959), Jaeger (1955, 1958,1959) and Blackwell
(1953, 1954, 1956). They discussed analytical solutions for an infinitely long conductive cylinder which produces heat dissipating to the surrounding medium. The molecules in a segment of a system at high temperature vibrate faster than the molecules in other regions of the same or anothers systems which are at lower temperatures. The molecules with higher motions strike the less energized molecules and transfer some of their energy to the molecules at the colder regions of the system. For example, heat is transferred by conduction from the cars bodywork to the materials inside the car which are in touch with the cars
The element having
Heat conduction rate into the element = Q(X)
Heat conduction rate out the element = Q(x+dX)
Net rate of heat conduction into the element Qnet = Q(X) Q(x+dX)
If the heat is generated within the element due to resistance heating, chemical or nuclear reaction etc. And the rate of volumetric heat generation is g (W/m3).
Then rate of energy generation, Qgen = g (A dX)
Due to unequal heat transfer to and from the element, its internal energy will change.
The rate of change of internal energy,
bodywork.
= =
(1)
Q(X+dX) = Q(X) + () dX + 2 () +
2
Where, T = F(X, t), temperature of element as function of time and direction,
0C,
g = G(X, t), the function of time and direction, W/m3,
k =K(X), the function of direction, W/m.K,
C =specific heat of the material (solid having only one specific heat), J/kg.K),
M = mass of the element = ( A dX), kg,
2 + 3() + 3 +…………………
3
2! 3!
If the control volume is considered small enough, then the higher powers of dX such as dX2, dX3, etc. are very small, therefore, neglected from above equation and it reduces to
Q(X+dX) = Q(X) + () dX (3)
Substituting this equation in eq(2)., we get
A = area of element normal to the heat
– () dX + gAdX = C AdX
(4)
transfer, m2,
= density of the material, kg/m3, t = time, s,
Substituting
Q(X) = -kA
dX = directional thickness of element, m.
Then, – {-kA }+ g = C
Making the energy balance on the element.
Net rate of heat gain by conduction + rate of energy generation = the net rate of change of internal energy.
If the conducting material is isotropic, its thermal conductivity is independent of direction; it is treated as constant quantity, then
1 {A }+ = C
(5)
Qnet + Qgen =
Or [Q(X) Q(X+dX)] + g A dX = C
Where = k/ C is known as thermal
AdX
According to Taylors series
(2)
diffusivity
The above eq.(5) is in general coordinate system. It is one dimensional time dependent differential equation for heat conduction with constant thermal
conductivity. It is known as unidirectional governing equation for heat conduction.
Substituting the value of constant in equation (7), we get
This above eq.(5) in particular coordinate system by introducing proper area A and directional thickness dX as described below.
If there is no internal heat generation within the material, the above equation reduces to:
T(x) = [ 2 1 ] x + T
1
q = – k
q = -k 2 1
Q = q * A
(8)
1 {
1
The heat conduction rate Q is given by
A } =
Q = -kA
2 1 (9)
It is known as unidirectional Fourier equation.
Therefore if the heat is not generated within the solid then it reduces to unidirectional Laplace equation.
In Cartesian coordinate
{ } = 0 (6)
Integrating the above , we get
= c
T(x) =Cx + C1 (7)
T (0) = T1 = C1
T (L) =T2 = C (L) + T1 C = (T2-T1)/L
Equation (1-9) is the solution for the rate of heat transfer through a one dimensional equation. The equation suggests that, under some limiting conditions, conduction of heat through a solid can be thought of as a flow that is driven by a temperature difference and resisted by a thermal resistance, in the same way that electrical current is driven by a voltage difference and resisted by an electrical resistance. Inspection of equation (1-7) suggests that the thermal resistance to conduction through a solid is given by:
Thermal resistance RTH =
Figure: Rectangular Block Section
Figure: Triangular Block section
ASSUMPTIONS:
Material is isotropic.
Heat flow is in one dimension only.
There is no heat generation.
Steady state condition
Figure: Brass Material
Figure: steel Material:
Figure: Brass Material
Figure: steel Material
Figure: Temperature grid in Brass Material
Figure: Temperature grid in Brass Material
Figure: Temperature grid in steel Material
Figure: Temperature grid in steel Material
REFERENCES
Blackwell, J.H., (1956).The axial- flow error in the thermal conductivity probe. Can. J.
Blackwell, J.H.,(1953) Radial-axial heat flow in regions bounded internally by circular cylinders. Can. J. Phys., 31,472-479.
Blackwell, J.H., (1954). A transient-flow method for determination of thermal constants of insulating materials in bulk part I -Theory. J. Appl. Phys., 25, 137-143.
Schneider P.J,(1957), Conduction Heat transfer, Addison Wesley,
Carslaw, H.S. and Jaeger, J.C., 1959. Conduction of heat in solids. Oxford University Press, Oxford, 510 pp.
Fried, E; (1970) Thermal conduction contribution to heat transfer at contacts 24-33
Parasnis, D., (1974). Estimates of in situ thermal diffusivity of the ore-bearing rocks in some drillholes in the Skellefte field (N. Sweden) using the annual temperature wave. J.Geophys. 40, 83-95.
Conaway, J.G. and Beck, A.E., (1977). Fine-scale correlation between temperature gradient logs
and lithology. Geophysics, 42, 1401-141
Oelsner, C. and Rosier, R., (1981). Eine Bohrlochsonde zur gleichzeitigen Messung von Warmestrom und
Warmeleitfahigkeit in situ. Neue Bergbautechnik, 11, 266-268.
Jolivet, J. and Vasseur, G., (1982). Sur un essai de mesure directe du flux geothermique in situ. Ann. Geophys., 38, 225-239.
Davis, E. E., 1988. Oceanic heat- flow density. In: R. Haenel, L. Rybach and L. Stegena ( eds.), Handbook of Terrestrial Heat- Flow Density Determination, Kluwer, Dordrecht,pp. 223-260.
Tan, Y. and Ritchie, A.I.M., (1997). In situ determination of thermal conductivity of waste rock dump material. Water, Air and Soil Pollution, 98, 345-359.
Keshavarz P and Taheri M, (2007), An improved lumped analysis for transient heat conduction by using the polynomial approximation method, Heat Mass Transfer, 43, 11511156