Water at 70^◦ F flows in a prototype circular pipe with a diameter of 6 ft and is fitted with an orifice meter with a diameter of 4.2 ft, as illustrated in Figure EP 11.15. A smaller model of the larger prototype is designed in order to study the flow characteristics of pipe flow in the flow

Question

Water at [latex] 70^{\circ } F [/latex] flows in a prototype circular pipe with a diameter of 6 ft and is fitted with an orifice meter with a diameter of 4.2 ft, as illustrated in Figure EP 11.15. A smaller model of the larger prototype is designed in order to study the flow characteristics of pipe flow in the flow-measuring device. The model fluid is crude oil at [latex] 68^{\circ } F [/latex] (s = 0.86,[latex] \mu = 150 imes 10^{-6} lb-sec/ft^{2} [/latex]), the ideal velocity (see Equation 8.140 [latex] v_{2i} = \sqrt{\frac{2(p_{1}-p_{2})}{ ho \left[1 - \left(\frac{A_{2}}{A_{1}} ight)^{2} ight] } } [/latex]) of flow of the oil in the smaller model pipe fitted with an orifice is 2ft/sec, and the model scale, λ is 0.25. (a) Determine the actual discharge in the model orifice meter. (b) Determine the ideal velocity of flow of the water in the prototype orifice meter in order to achieve dynamic similarity between the model and the prototype. (c) Determine the actual discharge in the prototype orifice meter in order to achieve dynamic similarity between the model and the prototype.

Accepted Answer

(a) In order to determine the actual discharge in the model orifice meter, the actual discharge equation, Equation 11.153 [latex] Q_{a} = \underbrace{\left[\underbrace{\left(\frac{v_{a}}{\sqrt{\frac{2\Delta p}{ ho } } } ight) }_{C_{v}} \underbrace{\left(\frac{A_{a}}{A_{i}} ight) }_{C_{c}} ight] }_{C_{d}} Q_{i} [/latex], is applied as follows:

[latex] Q_{a} = \underbrace{\left[\underbrace{\left(\frac{v_{a}}{\sqrt{\frac{2\Delta p}{ ho } } } ight) }_{C_{v}} \underbrace{\left(\frac{A_{a}}{A_{i}} ight) }_{C_{c}} ight] }_{C_{d}} Q_{i} [/latex]

where the [latex] C_{D} [/latex] or [latex] C_{d} [/latex] , or, in the case of an orifice meter, the orifice discharge coefficient, [latex] C_{o} [/latex], is used to model the flow resistance and is a function of both the geometry of the flow-measuring device, [latex] L_{i}/L [/latex] and R, as illustrated in Figure 8.29. Furthermore, in order to determine the geometry D and [latex]D_{o} [/latex] of the model pipe and orifice meter, the model scale, λ (inverse of the length ratio) is applied as follows:

[latex] D_{p}: = 6 ft [/latex] [latex] D_{op}: = 4.2 ft [/latex] [latex] \lambda : = 0.25 [/latex]

Guess value: [latex] D_{m}: = 1 ft [/latex] [latex] D_{om}: = 0.1 ft [/latex]

Given

[latex] \lambda = \frac{D_{m}}{D_{p}} [/latex] [latex] \lambda = \frac{D_{om}}{D_{op}} [/latex]
[latex] \left ( \begin{matrix} D_{m} \ D_{om} \end{matrix} ight ) : = Find (D_{m}, D_{om}) = \left ( \begin{matrix} 1.5 \ 1.05 \end{matrix} ight ) ft [/latex]
[latex] slug: = 1 lb \frac{sec^{2}}{ft} [/latex] [latex] s_{m} : = 0.86 [/latex] [latex] ho _{w} : = 1.94 \frac{slug}{ft^{3}} [/latex]
[latex] ho _{m} : = s_{m}. ho _{w} = 1.668 \frac{slug}{ft^{3}} [/latex] [latex] \mu _{m} : = 150 imes 10^{-6} lb \frac{sec}{ft^{2}} [/latex]

[latex]A_{om}: = \frac{\pi .D^{2}_{om}}{4} = 0.866 ft^{2} [/latex] [latex]A_{m}: = \frac{\pi .D^{2}_{m}}{4} = 1.767 ft^{2} [/latex] [latex] V_{im}: = 2 \frac{ft}{sec} [/latex]

[latex] R_{m}: = \frac{ ho _{m} .V_{m} .D_{m}}{\mu _{m}} = 3.337 imes 10^{4} [/latex] [latex] \frac{D_{om}}{D_{m}} = 0.7 [/latex] [latex] C_{om}: = 0.63 [/latex]

[latex] Q_{im}: = V_{im}. A_{om} = 1.732 \frac{ft^{3}}{sec} [/latex] [latex] Q_{am}: =C_{om}. Q_{im} = 1.091 \frac{ft^{3}}{sec} [/latex]

(b) To determine the ideal velocity of flow of the water in the prototype orifice meter in order to achieve dynamic similarity between the model and the prototype for pipe flow in the flow-measuring device, the geometry [latex] L_{i}/L [/latex] must remain a constant between the model and prototype as follows:

[latex] \left(\frac{L_{i}}{L} ight)_{p} = \left(\frac{L_{i}}{L} ight)_{m} [/latex]

where the geometry is modeled as follows:

[latex] \frac{D_{op}}{D_{p}} = 0.7 [/latex] [latex] \frac{D_{om}}{D_{m}} = 0.7 [/latex]

Furthermore, the R must remain a constant between the model and prototype as follows:

[latex] \underbrace{\left[\left(\frac{ ho vL}{\mu } ight)_{p} ight] }_{R_{p}} = \underbrace{\left[\left(\frac{ ho vL}{ \mu } ight)_{m} ight] }_{R_{m}} [/latex]

[latex] ho _{p} : = 1.936 \frac{slug}{ft^{3}} [/latex] [latex] \mu _{p} : = 20.50 imes 10^{-6} lb \frac{sec}{ft^{2}} [/latex]

Guess value: [latex] V_{ip}: = 10 \frac{ft}{sec} [/latex] [latex]R_{p} : = 1000 [/latex]

Given
[latex] R_{p} = \frac{ ho _{p} .V_{p} .D_{p}}{\mu _{p}} [/latex]
[latex] R_{p} = R_{m} [/latex]
[latex] \left ( \begin{matrix} V_{ip} \ R_{p} \end{matrix} ight ) : = Find (V_{ip}, R_{p}) [/latex]

[latex] V_{ip} = 0.059 \frac{ft}{s} [/latex] [latex]R_{p} = 3.337 imes 10^{4} [/latex]

(c) To determine the actual discharge in the prototype orifice meter in order to achieve dynamic similarity between the model and the prototype for pipe flow in the flow-measuring device, the orifice discharge coefficient, [latex] C_{o} [/latex] must remain a constant between the model and the prototype (which is a direct result of maintaining a constant R and a constant [latex] L_{i}/L [/latex] between the model and the prototype) as follows:

[latex] \underbrace{\left[\frac{ Q_{a}}{\left[\sqrt{\frac{2\Delta p}{ ho } } ight] \left[A_{i} ight] } ight]_{p} }_{C_{D_{p}}= C_{d_{p}}} = \underbrace{\left[\frac{ Q_{a}}{\left[\sqrt{\frac{2 \Delta p}{ ho } } ight] \left[A_{i} ight] } ight]_{m} }_{C_{D_{m}}= C_{d_{m}}} [/latex]

[latex]A_{op}: = \frac{\pi .D^{2}_{op}}{4} = 13.854 ft^{2} [/latex] [latex]A_{p}: = \frac{\pi .D^{2}_{p}}{4} = 28.274 ft^{2} [/latex]
Guess value: [latex] Q_{ap}: = 1 \frac{ft^{3}}{sec} [/latex] [latex] Q_{ip}: = 1 \frac{ft^{3}}{sec} [/latex] [latex] C_{op}: = 0.5 [/latex]

Given
[latex] C_{op} = \frac{Q_{ap}}{Q_{ip}} [/latex] [latex] Q_{ip} = V_{ip}.A_{op}[/latex]
[latex] C_{op} = C_{om} [/latex]
[latex] \left ( \begin{matrix} Q_{ap} \ Q_{ip} \ C_{op} \end{matrix} ight ) : = Find (Q_{ap}, Q_{ip}, C_{op} ) [/latex]

[latex] Q_{ap} = 0.514 \frac{ft^{3}}{sec} [/latex] [latex] Q_{ip} = 0.816 \frac{ft^{3}}{sec} [/latex] [latex] C_{op} = 0.63 [/latex]

Therefore, although the similarity requirements regarding the independent π term, [latex] L_{i}/L [/latex] ([latex] D_{op}/ D_{p}= D_{om}/ D_{m} =0.7 [/latex]) and the independent π term, R (“viscosity model”) ([latex] R_{p} = R_{m} = 3.337 imes 10^{4} [/latex]) are theoretically satisfied, the dependent π term (i.e., the orifice discharge coefficient,[latex] C_{o} [/latex] ) will actually/practically remain a constant between the model and its prototype ([latex] C_{op} = C_{om} = 0.63 [/latex]) only if it is practical to maintain/attain the model velocity, pressure, fluid, scale, and cost.

Question 11.15: Water at 70^◦ F flows in a prototype circular pipe with a di...

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