# Unsteady State Heat Transfer Transient Heat Conduction Characteristic Length Useful Tables Approximate Analytical Solutions

## 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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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 ones 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.

## Bowen-Harris, Eva Executive Producer

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