Question 13.3: An existing heat exchanger has a cross section as shown in F......

An energy-equation analysis using equation (6-10) will follow the same steps as in Example 2 in this chapter, yielding, as the governing equation,

${\frac{\delta Q}{d t}}-{\frac{\delta W_{\mathrm{s}}}{d t}}=\iint_{\mathrm{c.s.}}\left(e+{\frac{P}{\rho}}\right)\rho(\mathbf{v}\cdot\mathbf{n})\,d A+{\frac{\partial}{\partial t}}\iiint_{\mathrm{c.v.}}e\rho\,d V+{\frac{\delta W_{\mu}}{d t}}$    (6-10)

$0=\frac{P_{2}-P_{1}}{\rho}+g h_{L}$

The equivalent diameter for the shell is evaluated as follows:

${\mathrm{Flow~area}}={\frac{\pi}{4}}(25-9)=4\pi\operatorname{in}_{\cdot}^{2}$

${\mathrm{Wetted\ perimeter}}=\pi(5+9)=14\pi\,\mathrm{in}.$

thus

$D_{\mathrm{eq}}=4\frac{4\pi}{14\pi}=1.142\,\mathrm{in}.$

Substituting the proper numerical values into the energy equation for this problem reduces it to

$0=-\frac{3\,\mathrm{lb}_{f}/\mathrm{in}^{2}(\,144\,\mathrm{in.}^{2}/\mathrm{ft}^{2})}{1.94\,\mathrm{slugs/ft}^{3}}+2f_{f}v_{\mathrm{avg}}^{2}\,\mathrm{ft}^{2}/\mathrm{s}^{2}\frac{5\,\mathrm{ft}}{(1.142/12)\,\mathrm{ft}}\frac{g}{g}$

or

$0=-223+105f_{f}v_{\mathrm{avg}}^{2}$

As $f_{f}$ cannot be determined without a value of Re, which is a function of $υ_{avg}$, a simple trial-and-error procedure such as the following might be employed:

1. Assume a value for $f_{f}$.

2. Calculate $υ_{avg}$ from the above expression.

3. Evaluate Re from this value of $υ_{avg}$.

4. Check the assumed value of $f_{f}$ using equation (13-15a).

${\frac{1}{\sqrt{f_{f}}}}=-3.6\log_{10}\left[{\frac{6.9}{\mathrm{Re}}}+\left({\frac{e}{3.7D}}\right)^{10/9}\right]$    (13-15a)

5. If the assumed and calculated values for $f_{f}$ do not agree, repeat this procedure until they do.

Employing this method, we find the velocity to be 23.6 fps, giving a flow rate for this problem of $2.06\;\mathrm{ft}^{3}/\mathrm{min}(0.058\;{\mathrm{m}}^{3}/{\mathrm{s}}).$

Notice that in each of the last two examples in which a trial-and-error approach was used, the assumption of $f_{f}$ was made initially. This was not, of course, the only way to approach these problems; however, in both cases a value for $f_{f}$ could be assumed within a much closer range than either D or $υ_{avg}$.