Calculations rounding off

Numerical conditioning. By mapping our data from the native coordinate system into an appropriate factor space, we can eliminate problems caused by highly colinear data such as a set of very similar spectra. This can reduce calculation round-off errors and make it possible... 

Experimental errors in the / v / data or calculation round-off errors usually result in mole ratios that are not whole numbers. 

To check these hand calculations, run the CD program TPF. The input screen is displayed in Fig. 6.4 and the output answers in Fig. 6.5. Please note that all input data in Fig. 6.4 is the same as that used in the eight two-phase steps of Example 6.1. All answers in Fig. 6.5 agree with the eight-step two-phase flow hand calculations. The small differences are negligible and are due to the hand-calculation round-offs. 

Complete the following calculations. Round off the answers to the correct number of significant figures. 

When you have finished your calculations, round off the answer to the correct number of significant figures. In the examples in this book, only the final answer... 

Check The amount of O2 seems correct 900 g/(30 g/mol) = 30 mol. To check the approximate size of the final calculation, round off the values, including that for R ... 

It is better to round off only the final answer and to carry through extra digits in intermediate calculations. If there are enough steps to the calculation, rounding off in each step may lead to a cumulative error in the final answer. 

The heat capacity can therefore be obtained by keeping a running count of and E during the simulation, from which their expectation values (E ) and (E) can be calculated at the enc of the calculation. Alternatively, if the energies are stored during the simulation then the value of ((E — (E)) ) can be calculated once the simulation has finished. This seconc approach may be more accurate due to round-off errors (E ) and (E) are usually botf large numbers and so there may be a large uncertainty in their difference. 

All numerical computations inevitably involve round-off errors. This error increases as the number of calculations in the solution procedure is increased. Therefore, in practice, successive mesh refinements that increase the number of finite element calculations do not necessarily lead to more accurate solutions. However, one may assume a theoretical situation where the rounding error is eliminated. In this case successive reduction in size of elements in the mesh should improve the accuracy of the finite element solution. Therefore, using a P C" element with sufficient orders of interpolation and continuity, at the limit (i.e. when element dimensions tend to zero), an exact solution should be obtaiiied. This has been shown to be true for linear elliptic problems (Strang and Fix, 1973) where an optimal convergence is achieved if the following conditions are satisfied ... 

Usually rounded off from JANAF Thermochemical Tables, NSRDS-NBS-37, 1971 (1141 ppj- Equilibrium constants can he calculated hy combining AhJ values from Table 2-221, h-r— ho is from Table 2-222, and 5 values from the above, using the formula In kp = —AG/(RT), where AG = AhJ + hj — /i29s) T . 

Step 3. Refer to Figure 4-7 and select an efficiency at a pressure ratio of 3 and a volume ratio, r of 2.44. The adiabatic efficiency is 74%. Now, from Figure 4-5, select a value of efficiency ratio using the tip speed ratio just calculated. Because the value is. 99+, round off to an even 1.0. With a multiplier of 1.0, the final adiabatic efficiency is the value read directly off the curve or T a = 74. The molecular weight correction for efficiency, per rule of thumb, is 0.6 for a final efficiency of 73.4. 

If the pressure coefficient is now, or was in an earlier step, 5% uiutei the 0.29 value, calculate a new mean blade velocity using the rounded off number of stages and the original pressure coefficient, 0.29. Use the calculated blade velocity in the subsequent step for compressor speed Calculate the speed. 

The temperature rises given in the preceding table are based upon a reference ambient temperature of 40°C. Motors intended for use in higher ambient temperatures should have temperature rises not exceeding the value calculated from the appropriate formula, rounded off to the nearest 5 degrees. 

Rounding off In calculations, round up if the last digit is above 5 and round down if it is below 5. 

For numbers ending in 5, always round to the nearest even number. For example, 2.35 rounds to 2.4 and 2.65 rounds to 2.6. In a calculation with multiple steps, round off only in the final step if possible, carry all digits in the memory of the calculator until that stage. 

We have seen that we must sometimes reduce the number of digits in our calculated result to indicate the accuracy of the measurements that were made. To reduce the number, we round off numbers other than integer digits using the following rules. 

The performance statistics, the SEE and the correlation coefficient show that including the square term in the fitting function for Anscombe s nonlinear data set gives, as we noted above, essentially a perfect fit. It is clear that the values of the coefficients obtained are the ones he used to generate the data in the first place. The very large /-values of the coefficients are indicative of the fact that we are near to having only computer round-off error as operative in the difference between the data he provided and the values calculated from the polynomial that included the second-degree term. 

Note When doing these calculations, do not round off your answers until the end. The answers here are rounded after each step to illustrate the concept of significant figures. 

Note When doing an extensive calculation, keep all the numbers in your calculator. The answers here are rounded off to the appropriate number of significant figures after each step only to illustrate the concept of... 

Note To find e-95 66, take the inverse In of-95.66 on your calculator, inv In of-95.66 = 2.85 x 10 42. Keep one more significant figure and round off to three significant figures at the end, particularly when working with logarithms. 

For simplicity, we will assume that depth below the sea bottom varies linearly with time. The first step in the calculation consists in removing the long-term drift over the whole period by fitting the data with a parabola and determining a periodogram out of the residuals from this fit. Depth has also been scaled to 1 in order to minimize round-off errors. Applying the method shown above, the best-fit parabola is obtained for... 

Carry as many numbers as possible throughout the calculation and only round off the final answer. 

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