If the sudden enlargement and contraction in the system described in Example 13.10 are replaced with a conical diffuser and conical nozzle, respectively, and the included angle of each is 10°, by how much is the frictional pressure drop reduced?

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Accepted Answer

To compare the frictional pressure drops calculated for the sudden enlargement and contraction with those for a 10° included angle diffuser and nozzle, we will ﬁrst compare the loss coefficients for each component. In Example 13.10 we found K_E = 0.56, with a frictional pressure drop of ρK_E([latex]\bar V^2_1[/latex] /2) = 0.75 lb_f/ft² for the enlargement. For the diffuser with A₁/A₂ = 0.25 and θ = 10° we ﬁnd from Table 13.5 that K_D = 0.16. The ratio of the loss coefficients is K_D/K_E = 0.16/0.56 = 0.29, which can be interpreted as a 71% decrease in the loss coefficient realized by replacing the enlargement with a diffuser. Since the loss associated with either device is referenced to the same upstream velocity [latex]\bar V_{in}[/latex], the frictional pressure drop associated with the diffuser is simply 29% of that for the enlargement, or 0.22 lb_f/ft².

For the contraction we found K_C = 0.41, and ρK_C( [latex]\bar V^2_2[/latex] /2) = 0.55 lb_f/ft². From Table 13.5, we see that a nozzle with A₁/A₂ = 4.0 and θ = 10° has K_N = 0.80. It appears contradictory for K_C to be less than K_N but recall that the minor loss for the contraction is based on the higher downstream velocity [latex]\bar V_2[/latex], while for the nozzle the table values are based on the lower upstream velocity [latex]\bar V_{in}[/latex]. The loss for the nozzle is

[latex]ρK_N \frac{\bar V^2_1}{2}= (2.329×10^{−3} slug/\mathsf{ft} ^3)(0.8)\left(\frac{1}{2} \right)(509\ \mathsf{ft} /min)^2 \left(\frac{1\ min}{60\ s} \right)^2 [/latex] = 0.067 lb_f/ft²

which corresponds to a reduction in the frictional pressure drop of (0.55 lb_f/ft² − 0.067 lb_f/ft²)/0.55 lb_f/ft² × 100% = 88%.

The decrease in pressure drop due to the nozzle and diffuser can be translated into reduced operating cost by recalling that the power needed to move the ﬂuid, Q[latex]\Delta [/latex]p, is supplied by a fan typically driven by an electric motor. This savings is offset by increased initial construction costs. Another consideration in duct design is the amount of space an installation will require. In this example each gradual transition requires almost 6 ft of length.

TABLE 13.5 Loss Coefficients for Diffusers and Nozzles

Included Angle (°)
Area ratio, A₁/A₂	10	15	20	30	45	60	90	120	150	180
Diffuser loss coefficient, K_D
0.06	0.21	0.29	0.38	0.6	0.84	0.88	0.88	0.88	0.88	0.88
0.1	0.21	0.28	0.38	0.59	0.76	0.8	0.83	0.84	0.83	0.83
0.25	0.16	0.22	0.3	0.46	0.61	0.68	0.64	0.63	0.62	0.62
0.5	0.11	0.13	0.19	0.32	0.33	0.33	0.32	0.31	0.30	0.30
Nozzle loss coefficient, K_N
1	0	0	0	0	0	0	0	0	0	0
2	0.2	0.2	0.2	0.2	0.22	0.24	0.48	0.72	0.96	1
4	0.8	0.64	0.64	0.64	0.88	1.1	2.7	4.3	5.6	6.6
6	1.8	1.4	1.4	1.4	2	2.5	6.5	10	13	15
10	5	5	5	5	6.5	8	19	29	37	43

Question 13.11: If the sudden enlargement and contraction in the system desc...

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