ANALYSIS OF EXPERIMENTAL DATA (MECHANICAL/AEROSPACE ENGINEERING) Engineering design variables are often dependent on several independent variables. Often this functional dependence is best characterized by multivariate power equations. As discussed in Sec. 17.3, a multiple linear regression of

Question

ANALYSIS OF EXPERIMENTAL DATA (MECHANICAL/AEROSPACE ENGINEERING)

Engineering design variables are often dependent on several independent variables. Often this functional dependence is best characterized by multivariate power equations. As discussed in Sec. 17.3, a multiple linear regression of log-transformed data provides a means to evaluate such relationships.

For example, a mechanical engineering study indicates that fluid flow through a pipe is related to pipe diameter and slope (Table 20.4). Use multiple linear regression to analyze these data. Then use the resulting model to predict the fl ow for a pipe with a diameter of 2.5 ft and a slope of 0.025 ft/ft.

Accepted Answer

The power equation to be evaluated is

[latex]Q=a_{0} D^{a_{1}} S^{a_{2}}[/latex] (20.6)

TABLE 20.4 Experimental data for diameter, slope, and flow of concrete circular pipes.

Experiment	Diameter, ft	Slope, ft/ft	Flow,[latex]f t^{3} / s[/latex]
1	1	0.001	1.4
2	2	0.001	8.3
3	3	0.0001	24.2
4	1	0.01	4.7
5	2	0.01	28.9
6	3	0.01	84·0
7	1	0.05	11.1
8	2	0.05	69·0
9	3	0.05	200·0

where Q = flow ([latex]ft^3/s[/latex]), S = slope (ft/ft), D = pipe diameter (ft), and [latex]a_0, a_1[/latex], and [latex]a_2[/latex] = coefficients. Taking the logarithm of this equation yields

[latex]\log Q=\log a_{0}+a_{1} \log D+a_{2} \log S[/latex]

In this form, the equation is suited for multiple linear regression because log Q is a linear function of log S and log D. Using the logarithm (base 10) of the data in Table 20.4, we can generate the following normal equations expressed in matrix form [recall Eq. (17.22)]:

[latex]\left[\begin{array}{ccc}n & \sum x_{1 i} & \sum x_{2 i} \\sum x_{1 i} & \sum x_{1 i}^{2} & \sum x_{1 i} x_{2 i} \\sum x_{2 i} & \sum x_{1 i} x_{2 i} & \sum x_{2 i}^{2}\end{array} ight]=\left\{\begin{array}{l}a_{0} \a_{1} \a_{2}\end{array} ight\}=\left\{\begin{array}{c}\sum y_{i} \\sum x_{1 i} y_{i} \\sum x_{2 i} y_{i}\end{array} ight\}[/latex] (17.22)

[latex]\left[\begin{array}{crr}9 & 2.334 & -18.903 \2.334 & 0.954& -4.903 \-18.903 & -4.903 & 44.079\end{array} ight]\left\{\begin{array}{c}\log a_{0} \a_{1} \a_{2}\end{array} ight\}=\left\{\begin{array}{r}11.691 \3.945 \-22.207\end{array} ight\}[/latex]

This system can be solved using Gauss elimination for

[latex]\log a_{0}=1.7475[/latex]

[latex]a_1 = 2.62[/latex]

[latex]a_2 = 0.54[/latex]

Since [latex]\log a_{0}=1.7475,[/latex] then [latex]a_{0}=10^{1.7475}=55.9[/latex] and Eq. (20.6) is

[latex]Q=55.9 D^{2.62} S^{0.54}[/latex] (20.7)

Eq. (20.7) can be used to predict fl ow for the case of D = 2.5 ft and S = 0.025 ft/ft, as in

[latex]Q=55.9(2.5)^{2.62}(0.025) S^{0.54}=84.1 ft ^{3} / s[/latex]

It should be noted that Eq. (20.7) can be used for other purposes besides computing flow. For example, the slope is related to head loss [latex]h_L[/latex] and pipe length L by [latex]S = h_L/L[/latex]. If this relationship is substituted into Eq. (20.7) and the resulting formula solved for [latex]h_L[/latex], the following equation can be developed:

[latex]h_{L}=\frac{L}{1721} Q^{1.85} D^{4.85}[/latex]

This relationship is called the Hazen-Williams equation.

Question 20.4: ANALYSIS OF EXPERIMENTAL DATA (MECHANICAL/AEROSPACE ENGINEER...

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