Unsteady State Heat Transfer Transient Heat Conduction Characteristic Length Useful Tables Approximate Analytical Solutions
Bowen-Harris, Eva, Executive Producer has reference to this Academic Journal, PHwiki organized this Journal Unsteady State Heat TransferHT3: Experimental Studies of Thermal Diffusivities in addition to Heat Transfer Coefficients Transient Heat ConductionMany heat conduction problems encountered in engineering applicationsinvolve time as in independent variable. The goal of analysis is to determine the variation of the temperature as a function of time in addition to position T (x, t) within the heat conducting body. In general, we deal with conducting bodies in a three dimensional Euclidean space in a suitable set of coordinates (x R3) in addition to the goal is to predict the evolution of the temperature field as long as future times (t > 0).Here we investigate solutions to selected special cases of the following as long as m of the heat equationSolutions to the above equation must be obtained that also satisfy suitable initial in addition to boundary conditions.Example: Point Thermal Explosion
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One-Dimensional ProblemsImposed Boundary Temperature in Cartesian Coordinates: A simple but important conduction heattransfer problem consists of determining the temperature history inside a solid flat wall which is quenched from a high temperature. More specifically, consider the homogeneous problem of finding the one-dimensional temperature distribution inside a slab of thickness L in addition to thermal diffusivity a, initially at some specified temperature T (x, 0) = f(x) in addition to exposed to heat extraction at its boundaries x = 0 in addition to x = L such that T (0, t) = T (L, t) = 0 (Dirichlet homogeneous conditions), as long as t > 0. The thermal properties are assumed constant.Convection at the Boundary in Cartesian Coordinates: Same geometry BUT the boundary conditionsspecify values of the normal derivative of the temperature or when linear combinations of the normal derivative in addition to the temperature itself are used. Consider the homogeneous problem of transient heat conduction in a slab initially at a temperature T = f(x) in addition to subject to convection losses into a medium at T = 0 at x = 0 in addition to x = L. Convection heat transfer coefficients at x = 0 in addition to x = L are, respectively h1 in addition to h2. Assume the thermal conductivity of the slab k is constant.Imposed Boundary Temperature in addition to Convection at the Boundary in Cylindrical Coordinates:a long cylinder (radius r = b) initially at T = f(r) whose surface temperature is made equal to zero as long as t > 0. A long cylinder (radius r = b) initially at T = f(r) is exposed to a cooling medium which extracts heat uni as long as mly from its surface.Imposed Boundary Temperature in addition to Convection at the Boundary in Spherical Coordinates:quenching problem where a sphere (radius r = b) initially at T = f(r) whose surface temperature is made equal to zero as long as t > 0.Consider a sphere with initial temperature T (r, 0) = f(r) in addition to dissipating heat by convection into a medium at its surface r = b.Characteristic LengthFirst Problem: Slab/Convection The first problem is the 1D transient homogeneous heat conduction in a plate of span L froman initial temperature Ti in addition to with one boundary insulated in addition to the other subjected to a convective heat flux condition into a surrounding environment at T. This problem is equivalent to the quenching of a slab of span 2L with identical heat convection at the external boundaries x = L in addition to x = L).The mathematical as long as mulation of the problem is to find T (x, t) such that:Boundary conditions: Initial conditions: as long as all x when t = 0, Introduction of the following non dimensional parameters simplifies the mathematical as long as mulation of the problem. The dimensionless distance (X), time (t) in addition to temperature (q): in addition to a Biot number: The Biot number (Bi) is a dimensionless number used in heat transfer calculations. It is named after the French physicist Jean-Baptiste Biot (17741862), in addition to gives a simple index of the ratio of the heat transfer resistances inside (1/k) of in addition to at the surface of a body (1/hL).Biot number smaller than 0.1 imply that the heat conduction inside the body is much faster than the heat convection away from its surface, in addition to temperature gradients are negligible inside of it.
First Problem Formulation: Slab/Convection Boundary conditions: Initial conditions: as long as all x when t = 0, Introduction of the following non dimensional parameters simplifies the mathematical as long as mulation of the problem. The dimensionless distance (X), time (t) in addition to temperature (q) in addition to a Biot number:With the new variables, the mathematical as long as mulation of the heat conduction problem becomes: in addition to Second Problem Formulation: Solid Cylinder/Convection Boundary conditions: Introduction of the following non dimensional parameters simplifies the mathematical as long as mulation of the problem. The dimensionless distance (X), time (t) in addition to temperature (q) in addition to a Biot number:With the new variables, the mathematical as long as mulation of the heat conduction problem becomes:1D transient homogeneous heat conduction in a solid cylinder of radius b from an initial temperature Ti in addition to with one boundary insulated in addition to the other subjected to a convective heat flux condition into a surrounding environment at T.Third Problem Formulation: Sphere/Convection Introduction of the following non dimensional parameters simplifies the mathematical as long as mulation of the problem. The dimensionless distance (X), time (t) in addition to temperature (q) in addition to a Biot number:With the new variables, the mathematical as long as mulation of the heat conduction problem becomes:Cooling of a sphere (0 r b) initially at a uni as long as m temperature Ti in addition to subjected to a uni as long as m convective heat flux at its surface into a medium at T with heat transfer coefficient h.In terms of the new variable U(r, t) = rT (r, t) the mathematical as long as mulation of the problem is:
Biot numberFourier numberIn physics in addition to engineering, the Fourier number (Fo) or Fourier modulus, named after Joseph Fourier, is a dimensionless number that characterizes heat conduction. Conceptually, it is the ratio of diffusive/conductive transport rate by the quantity storage rate in addition to arises from non-dimensionalization of the heat equation. The general Fourier number is defined as:Fo = Diffusive transport rate (a/L2)/storage rate (1/t) The thermal Fourier number is defined by the conduction rate to the rate of thermal energy storage:Compare with non-dimensionless time parameter:So Fo=tTo underst in addition to the physical significance of the Fourier number t, we may express it as There as long as e, again the Fourier number is a measure of heat conducted through a body relative to heat stored. Thus, a large value of the Fourier number indicates faster propagation of heat through a body.Lumped System AnalysisIn heat transfer analysis, some bodies are observed to behave like a lump whose interior temperature remains essentially uni as long as m at all times during a heat transfer process. The temperature of such bodies can be taken to be a function of time only, T(t). Heat transfer analysis that utilizes this idealization is known as lumped system analysis , which provides great simplification in certain classes of heat transfer problems without much sacrifice from accuracy.if the internal temperature of a body remains relatively constant with respect to distance can be treated as a lumped system analysis heat transfer is a function of time only, T = T (t) Typical criteria as long as lumped system analysis is Bi< 0.1Consider a small hot copper ball coming out of an oven. Mea-surements indicate that the temperature of the copper ball changes with time, but it does not change much with position at any given time. Thus the temperature of the ball remains nearly uni as long as m at all times, in addition to we can talk about the temperature of the ball with no reference to a specific location. Lumped System AnalysisConsider a body of arbitrary shape of mass m, volume V, surface area As, density r, in addition to specific heat cp initially at a uni as long as m temperature Ti . At time t = 0, the body is placed into a medium at temperature T, in addition to heat transfer takes place between the body in addition to its environment, with a heat transfer coefficient h. For the sake of discussion, we assume that T >Ti, but the analysis is equally valid as long as the opposite case. We assume lumped system analysis to be applicable, so that the temperature remains uni as long as m within the body at all times in addition to changes with time only, T= T(t).During a differential time interval dt, the temperature of the body rises by a differential amount dT. An energy balance of the solid as long as the time interval dt can be expressed as:If Bi< 0.1orIntegrating from t=0, at which T =Ti, to any time t, at which T=T(t), givesorThe reciprocal of b has time unit, in addition to is called the time constant.The temperature of a body approaches the ambient temperature Texponentially. The temperature of the body changes rapidly at thebeginning, but rather slowly later on. A large value of b indicates that thebody approaches the environment temperature in a short time. The largerthe value of the exponent b, the higher the rate of decay in temperature.Note that b is proportional to the surface area, but inversely proportionalto the mass in addition to the specific heat of the body. This is not surprising sinceit takes longer to heat or cool a larger mass, especially when it has alarge specific heat.orLumped System AnalysisIf Bi< 0.1Once the temperature T(t) at time t is available, the rate of convection heat transfer between the body in addition to its environment at that time can bedetermined from Newtons law of cooling as:The total amount of heat transfer between the body in addition to the surrounding medium over the time interval t 0 to t is simply the change in the energy content of the body:The amount of heat transfer reaches its upper limit when the body reaches the surrounding temperature T. There as long as e, the maximum heat transfer between the body in addition to its surroundings is(A)(B)Exact Solution of One-Dimensional Transient Conduction ProblemThe non-dimensionalized partial differential equation as long as mulated above together with its boundary in addition to initial conditions can be solved using several analytical in addition to numerical techniques, including the Laplace or other trans as long as m methods, the method of separation of variables, the finite difference method, in addition to the finite-element method. Here we discuss the method of separation of variables developed by J. Fourier in 1820s in addition to is based onexp in addition to ing an arbitrary function (including a constant) in terms of Fourier series. The method is appliedby assuming the dependent variable to be a product of a number of functions, each being a function of a single independent variable. This reduces the partial differential equation to a system of ordinary differential equations, each being a function of a single independent variable. In the case of transient conduction in a plain wall, as long as example, the dependent variable is the solutionfunction u(X, t), which is expressed as u(X, t)= F(X)G(t), in addition to the application of the method results in two ordinary differential equation, one in X in addition to the other in t.The method is applicable if (1) the geometry is simple in addition to finite (such as a rectangular block, a cylinder, or asphere) so that the boundary surfaces can be described by simple mathematical functions, in addition to (2) the differential equation in addition to the boundary in addition to initial conditions in their most simplified as long as m are linear (no terms that involve products of the dependent variable or its derivatives) in addition to involve only one nonhomogeneous term (a term without the dependent variable or its derivatives). Separation of VariablesLet us consider the Slab/Convection experiment. Recall that in this case we have:The heat conduction equation in cylindrical or spherical coordinates can be nondimensionalized in a similar way. Note that nondimensionalization reduces the number of independent variables in addition to parameters from 8 to 3from x, L, t, k, a, h, Ti, in addition to T to X, Bi, in addition to Fo. That is,This makes it very practical to conduct parametric studies in addition to to present results in graphical as long as m. Recall that in the case of lumped system analysis, we had u f(Bi, Fo) with no space variable.Separation of VariablesFirst, we express the dimensionless temperature function u(X, t) as a product of a function of X only in addition to a function of t only as:Substituting to: all the terms that depend on X are on the left-h in addition to side of the equation in addition to all the terms that depend on t are on the r the terms that are function of different variables are separated ( in addition to thus the name separation of variables). Considering that both X in addition to t can be varied independently, the equality in Eq. C can hold as long as any value of X in addition to t only if it is equal to a constant. Further, it must be a negative constant that we will indicate by -l2 since a positive constant will cause the function G(t) to increase indefinitely with time (to be infinite), which is unphysical, in addition to a value of zero as long as the constant means no time dependence, which is again inconsistent with the physical problem. Setting Eq. C equal to -l2 gives:whose general solutions are: in addition to we have (C)Separation of VariablesThen it follows that there are an infinite number of solutions of the as long as m , in addition to the solution of this linear heat conduction problem is a linear combination of them,The constants An are determined from the initial conditionThis is a Fourier series expansion that expresses a constant in terms of an infinite series of cosine functions. Now we multiply both sides of last eq. by cos(lmX), in addition to integrate from X=0 to X=1. The right-h in addition to side involves an infinite number of integrals of the as long as m: It can be shown that all of these integrals vanish except when n m, in addition to the coefficient An becomes: Separation of VariablesThis completes the analysis as long as the solution of one-dimensional transient heat conduction problem in a plane wall. Solutions in other geometries such as a long cylinder in addition to a sphere can be determined using the same approach. The results as long as all three geometries are summarized in Table. Note that the solution as long as the plane wall is also applicable as long as a plane wall of thickness L whose left surface at x =0 is insulated in addition to the right surface at x=L is subjected to convection since this is precisely the mathematical problem we solved.Approximate Analytical SolutionsThe analytical solutions of transient conduction problems typically involve infinite series, in addition to thus the evaluation of an infinite number of terms to determine the temperature at a specified location in addition to time. However, as demonstrated in Table, the terms in the summation decline rapidly as n in addition to thus ln increases because of the exponential decay function . This is especially the case when the dimensionless time t is large. There as long as e, the evaluation of the first few terms of the infinite series (in this case just the first term) is usually adequate to determine the dimensionless temperature q. For example, as long as t> 0.2, keeping the first term in addition to neglecting all the remaining terms in the series results in an error under 2 percent.We are usually interested in the solution as long as times with t> 0.2, in addition to thus it is very convenient to express the solution using this one-term approximation, given as:where the constants A1 in addition to l1 are functions of the Bi number only, in addition to their values are listed in Table (see next slide) against the Bi number as long as all three geometries. The function J0 is the zeroth-order Bessel function of the first kind (see next slide).Useful Tables
Approximate Analytical SolutionsNoting that cos (0)= J0(0)= 1 in addition to the limit of (sin x)/x is also 1, these relations simplify to the next ones at the center of a plane wall, cylinder, or sphere:Comparing the sets of equations above with approximate solution we notice that the dimensionless temperatures anywhere in a plane wall, cylinder, in addition to sphere are related to the center temperature bywhich shows that time dependence of dimensionless temperature within a given geometry is the same throughout. That is, if the dimensionless center temperature q0 drops by 20 percent at a specified time, so does the dimensionless temperature q0 anywhere else in the medium at the same time. Once the Bi number is known, these relations can be used to determine thetemperature anywhere in the medium.Graphical Solutions : Heisler ChartsThe solutions obtained as long as 1D non homogeneous problems with Neumann boundary conditions in Cartesian coordinate systems using the method of separation of variables have been collected in addition to assembled in the as long as m of transient temperature nomographs by Heisler. The given charts are a very useful baseline against, which to validate ones own analytical or numerical computations.Indeed, the determination of the constants A1 in addition to l1 usually requires interpolation. For those who prefer reading charts to interpolating, these relations are plotted in addition to the one-term approximation solutions are presented in graphical as long as m, known as the transient temperature charts. The transient temperature charts shown in next slides as long as a large plane wall, long cylinder, in addition to sphere were presented by M. P. Heisler in 1947 in addition to are called Heisler charts.There are three charts associated with each geometry: the first chart is to determine the temperature T0 at the center of the geometry at a given time t. The second chart is to determine the temperature at other locations at the same time in terms of T0. The third chart is to determine the total amount of heat transfer up to the time t. These plots are valid as long as t > 0.2.Transient temperature in addition to heat transfer charts as long as a plane wall of thickness 2L initially at a uni as long as m temperature Ti subjected to convection from all sides to an environment at temperature T with a convection coefficient of h.
Transient temperature in addition to heat transfer charts as long as a long cylinder of radius ro initially at a uni as long as m temperature Ti subjected to convection from all sides to an environment at temperature T with a convection coefficient of h.Transient temperature in addition to heat transfer charts as long as a sphere of radius ro initially at a uni as long as m temperature Ti subjected to convection from all sides to an environment at temperature T with a convection coefficient of h.Useful RelationshipAgain the temperature of the body changes from the initial temperature Ti to the temperature of the surroundings T at the end of the transient heat conduction process in addition to the maximum amount of heat that a body can gain (or lose) is simply the change in the energy content of the body:The amount of heat transfer Q at a finite time t isAssuming constant properties, the ratio of Q/Qmax becomesUsing the appropriate non-dimensional temperature relations based on the one termapproximation as long as the plane wall, cylinder, in addition to sphere, in addition to per as long as ming the indicated integrations, we obtain the following relations as long as the fraction of heat transfer in those geometries:These Q/Qmax ratio relations based on the one-term approximation are also plotted in Heisler charts, against the variables Bi in addition to h2at/k2 as long as the large plane wall, long cylinder, in addition to sphere, respectively. Note that once the fraction of heat transfer Q/Qmax has been determined from these charts or equations as long as the given t, the actual amount of heat transfer by that time can be evaluated by multiplying this fraction by Qmax.
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