One-term approximation

Kutzelnigg W and Maeder F 1978 Natural states of interacting systems and their use for the calculation of intermolecular forces. III. One-term approximations of oscillator strength sums and dynamic polarizabilities Chem. Phys. 35 397... 

In the light of the discussions made in Chap. 4, the contribution of the frontier term in the formula of Sr might be more important than expected from the expression. Such a consideration has early been made and a one-term approximation of Sr (denoted by Sj-) has been proposed 6 ). Thus, Sr is approximated by the frontier term only ... 

Example 4 Rule of Thumb for Time Required to Diffuse a Distance R A general rule of thumb for estimating the time required to diffuse a distance R is obtained from the one-term approximations. Consider the equation for the temperature of a flat plate of thickness 2R in the limit as Bi —> From Table 5-2, the first eigenvalue is 8 = k/2, and from Table 5-1,... 

Assumptions 1 The egg is spherical in shape with a radius of r, = 2.5 cm. 2 Heat conduction in the egg is one dimensional because or 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. 

We mentioned earlier that the terms in the series solutions in Table 4-1 eon-verge rapidly with increasing time, and for t > 0.2, keeping the first letm and neglecting all the remaining terms in the series results in an error under 2 percent. We are usuaUy interested in the solution for times with t > 0.2, and thus it is very convenient to express the solution using this one-term approximation, given as... 

Coefficients used in the one-term approximate solution of transient one-dimensional heat conduction in plane walls, cylinders, and spheres (B = hUk for a plane wall of thickness ZL, and Bi = hrjkfor a cylinder or sphere of radius r )... 

Using the appropriate nonditnensional temperature relations based on the one-term approximation for the plane wall, cylinder, and sphere, and performing the indicatetl integrations, we obtain the following relations for the fraction of heat transfer in those geometries ... 

SOLUTION A long cylindrical shaft is allowed to cool slowly. The center temperature and the heat transfer per unit length are to be determined. Assumptions 1 Heat conduction in the shaft is one-dimensional since it is long and it has thermal symmetry about the centerline. 2 The thermal properties of the shaft and the heat transfer coefficient are constant. 3 The Fourier number is t > 0.2 so that the one-term approximate solutions are applicable. Properties The properties of stainless steel 304 at room temperature are k - 14,9 W/m °C, p = 7900 kg/m r. = 477 J/kg X, and a = 3.95 X 10 mVs (Table A-3). More accurate results can be oblained by using properties at average temperature. 

Using the one-term approximation, the solutions of onedimensional transient heat conduction problems are expressed analytically as... 

An experiment is to be conducted to determine heat transfer coefficient on the surfaces of tomatoes that are placed in cold water at 7°C, The tomatoes (k = 0,59 W/m °C, a = 0,141 X lQ- mVs, p = 999 kg/m , 3,99 kJ/kg °C) with an initial uniform temperature of 30°C are spherical in. shape willr a diameter of 8 cm. After a period of 2 hours, the temperatures at the center and the surface of the tomatoes arc measured to be 10.0°C and 7.1 C, respectively. Using analytical one-term approximation method (not the Hcisler charts), determine the heat transfer coefficient and the amount of heat transfer during this period if there are eight such tomatoes in water. 

We conclude with a few comments about the validity of the two-timing method. The rule of thumb is that the one-term approximation Xq will be within 0(e) of the true solution x for all times up to and including t O( /e), assuming that both x and Xq start from the same initial condition. If x is a periodic solution, the situation is even better Xq remains within 0(e) of x for all t. 

An approximate solution method follows the procedure outlined in Subsection D2. Carry out this procedure to determine a condition equivalent to (12-153). What result do you get from the one-term approximation corresponding to (12-154) ... 

Using Excel, construct a graph of the function / from the previous example in the region 0 < x < a and another graph of ci V " . Compare the graphs and comment on how well the partial sum with one term approximates the function. Let a = 1 for your graphs. Q... 

Equation 8.4 is a set on N linear algebraic equations for the Nconstants C . Suppose we decide that a one-term approximation is sufficient. We then set... 

---