# Notes on Heisler Charts for GATE Mechanical Engineering

Heisler charts give a relation between temperature and time of an object under transient conditions. These charts are basically used to calculate the temperature or time of objects under unsteady state heat transfer which is a usual phenomenon in our daily life. These two charts included in geometry introduced in 1947 by M.P.Heisler which were supplemented by third chart per geometry in 1961 by H. Grober.

Q:-What is  TRANSIENT TEMPERATURE?

Whenever we encounter a phenomenon in which Temp of an body  changes with time that phenomenon is termed as transient temperature condition.

Transient temperature  Conditions are encountered in our daily life so it is important area of consideration and plays vital role in solving daily life situations.

Q:-Conditions for a body to be identified by a HEISLER chart?

1.Body must be at uniform temperature initially

2.Temperature of surrounding and convective heat transfer must remain constant.

3.There must be no heat generation from body itself.

Q:-What is HEISLER CHART?

This chart is basically a relation between TEMPERATURE and TIME of an object .If we need to find a the temperature of the body at any time or vice versa.We can use this graphical representation.There are various terms used under this chart such as FOURIER NUMBER and BIOT NUMBER to manipulate the temperature at any time or vice versa.

Here T is the Temperature at any  random time to be evaluated.in the above diagram , we can see a number of  lines in the temperature  Temperature time graph.These lines correspond to biot number. First the biot number is calculated, then the line corresponding to biot number is taken as refrence for calculating parameters Temperature and time of the body under transient condition.

Q:-What is FOURIER NUMBER?

The Fourier number (Fo) or Fourier modulus, named after Joseph Fourier, is a dimensionless number that characterizes transient heat conduction. Conceptually, it is the ratio of diffusive or conductive transport rate to the quantity storage rate, where the quantity may be either heat (thermal energy) or matter (particles). The number derives from non-dimensionalization of the heat equation (also known as Fourier's Law) or Fick's second law and is used along with the Biot number to analyze time dependent transport phenomena.

$F_{o}=\frac{\alpha&space;T}{L^{2}}=\frac{kt}{\rho&space;c_{p}L^{2}}$

Q:-What is BIOT NUMBER?

The Biot number (Bi) is a dimensionless quantity used in heat transfer calculations. It is named after the French physicist Jean-Baptiste Biot (1774–1862), and gives a simple index of the ratio of the heat transfer resistances inside of and at the surface of a body. This ratio determines whether or not the temperatures inside a body will vary significantly in space, while the body heats or cools over time, from a thermal gradient applied to its surface.

In general, problems involving small Biot numbers (much smaller than 1) are thermally simple, due to uniform temperature fields inside the body. Biot numbers much larger than 1 signal more difficult problems due to non-uniformity of temperature fields within the object. It should not be confused with Nusselt number, which employs the thermal conductivity of the fluid and hence is a comparative measure of conduction and convection, both in the fluid.

$Bi=\frac{hL}{k}$

EQUATION FOR MANIPULATING TEMPERATURE ON HEISLER CHART

$T=\frac{T(0,t)-T_{\infty}}{T_{i}-T_{\infty}}$

In this equation T(0,t) Temperature at any time t .The equation equals biot number which is marked from the HEISLER CHART.Hence we will be able to calculate the Temperature of the body under Transient conditions.

Similarly We can calculate the TIME at a particular temperature in the reverse manner.

First by manipulating biot number and then using temperature values , biot number to find fourier number.

Then from the fourier number we can calculate time as we already know the geometric parameters of the body and heat  transfer coefficients.

Use of Heisler chart can be illustrated in this   example for  calculating temperature at any time of an object under transient conditions.

NUMERICAL-

Boiling Eggs

An ordinary egg can be approximated as a 5-cm-diameter sphere (Fig.). The egg is initially at a uniform temperature of $5^{0}C$ and is dropped into boiling water at $95^{0}C$. Taking the convection heat transfer coefficient to be $h=1200W/m^{2}C$, determine how long it will take for the center of the egg to reach $70^{0}C$.

SOLUTION An egg is cooked in boiling water. The cooking time of the egg is to be determined.

Assumptions 1 The egg is spherical in shape with a radius of ro  2.5 cm. 2 Heat conduction in the egg is one-dimensional because of thermal symmetry about the midpoint. 3 The thermal properties of the egg and the heat transfer coefficient are constant. 4 The Fourier number is t 0.2 so that the one-term approximate solutions are applicable.

Analysis The temperature within the egg varies with radial distance as well as time, and the temperature at a specified location at a given time can be determined from the Heisler charts or the one-term solutions. Here we use the latter to demonstrate their use. The Biot number for this problem is

$Bi=\frac{hr_{o}}{k}=\frac&space;{1200\times&space;0.025}{0.627}=47.8$

which is much greater than 0.1, and thus the lumped system analysis is not applicable.

From tables, $\lambda_{1}=3.0754$ and $A_{1}=1.9958$

$\frac{T_{0}-T_{\infty}}{T_{i}-T_{\infty}}=A_{1}e^{\lambda^{2}_{1}\tau}\rightarrow&space;\tau&space;=&space;0.209$

which is greater than 0.2, and thus the one-term solution is applicable with an error of less than 2 percent. Then the cooking time is determined from the definition of the Fourier number to be t= 865 s= 4.4 min Therefore, it will take about 15 min for the center of the egg to be heated from $5^{o}C&space;to&space;70^{o}C$

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