Heisler charts

The calculations for the Heisler charts series solutions for the problems into a of the charts to values of the Fourier... 

The Heisler charts of Figs. 4-7 and 4-10 may be used for solution of this problem. We first calculate the center temperature of the plate, using Fig. 4-7, and then use Fig. 4-10 to calculate the temperature at the specified x position. From the conditions of the problem we have... 

The Heisler charts discussed above may be used to obtain the temperature distribution in the infinite plate of thickness 2L, in the long cylinder, or in the sphere. When a wall whose height and depth dimensions are not large compared with the thickness or a cylinder whose length is not large compared with its diameter is encountered, additional space coordinates are necessary to specify the temperature, the above charts no longer apply, and we are forced to seek another method of solution. Fortunately, it is possible to combine the solutions... 

To solve this problem we combine the solutions from the Heisler charts for an infinite cylinder and an infinite plate in accordance with the combination shown in Fig. 4-18/ For the infinite-plate problem... 

Check to see if an analytical solution is available with such aids as the Heisler charts and approximations. 

An infinite plate having a thickness of 2.S cm is initially at a temperature of 150°C, and the surface temperature is suddenly lowered to 30°C. The thermal diffusivity of the material is 1.8 x 10 6 m2/s. Calculate the center-plate temperature after 1 min by summing the first four nonzero terms of Eq. (4-3). Check the answer using the Heisler charts. 

A long steel bar 5 by 10 cm is initially maintained at a uniform temperature of 250°C. It is suddenly subjected to a change such that the environment temperature is lowered to 35°C. Assuming a heat-transfer coefficient of 23 W/m2 °C, use a numerical method to estimate the time required for the center temperature to reach 90°C. Check this result with a calculation, using the Heisler charts. 

Analytical solutions for the three transient cases covered in the Heisler charts Of Chap. 4 are given in Ref. 1 of Chap. 4. Heisler (Ref. 7 of Chap. 4) was able to show that for cutIL2 or ar/r02 > 0.2, the infinite series solutions for the center temperature (x = 0 or r = 0) could be approximated within 1 percent with a single term ... 

The transient temperature charts in Figs. 4-15, 4-16, and 4-17 for a large plane wall, long cylinder, and sphere were presented by M. P. Heisler in 1947 and are called Heisler charts. They were supplemented in 1961 with transient heal transfer charts by II. Grober. There are three charts associated with each geometry the first chart is to determine the temperature Tj at the center of the... 

Properties The properties of brass at room temperature are fc = 110 W/m °C, p = 8530 kg/m, Cp = 380 J/kg °C, and a = 33.9 X lO" m /s (Table A-3). More accurate results are obtained by using properties at average temperature. Analysis The temperature at a specified location at a given time can be determined from the Heisler charts or one-term solutions. Here we use the charts to demonstrate their use, Noting that the half thickness of the plate is (. = 0.02 m, from Fig, 4-15 we have... 

Discussion We notice that the Biot number In this case is Bi = 1/45.8 - 0.022, which is much less than 0.1. Therefore, we expect the lumped system analysis to be applicable. This is also evident from (T - TJ/tFo - TJ = 0.99, which indicates that the temperatures at the center and the surface of the plate relative to the surrounding temperature are within 1 percent of each other. Noting that the error involved in reading the Heisler charts is typically a few percent, the lumped system analysis in this case may yield just as accurate results with less effort. 

The diffusion coefficients in solids are typically very low (on the order of 10 to 10" mVs), and thus the diffusion process usually affects a thin layer at the surface. A solid can conveniently be treated as a semi-infinite medium during transient mass diffusion regardless of its size and shape when the penetration depth is small relative to the thickness of the solid. When this is not the case, solutions for one dimensional transient mass diffusion through a plane wall, cylinder, and sphere can be obtained from the solution.s of analogous heat conduction problems using the Heisler charts or one term solutions pieseiited in Chapter 4. 

For the transient region the reader is referred to any heat transfer text for Heisler charts. Alternately, the transient region can be solved numerically by first guessing the location of the interface and iterating the procedure until the heat loss between the cyclic equilibrium region equals that of the steady state region. The heat transfer coefficients include that of convection and radiation, which we will evaluate after treating radiative heat transfer. 

Heisler charts Aset of graphical plots used to determine the time taken for thermal penetration by heat conduction into a soUd body by heating or cooling at its surface. The plots are prepared for standard geometric shapes such as slabs, cylinders, and spheres with the Fourier number on the x-axis and a dimensionless temperature on the y-axis. The lines represent the reciprocal of the Biot number. They are named after M. P. Heisler, who com -piled them in 1947. 

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