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Chapter 10: Problem 59

10-59 The Reynolds number for condensate flow is defined as \(\operatorname{Re}=4 \dot{m} / p \mu_{l}\), where \(p\) is the wetted perimeter. Obtain simplified relations for the Reynolds number by expressing \(p\) and \(\dot{m}\) by their equivalence for the following geometries: \((a)\) a vertical plate of height \(L\) and width \(w,(b)\) a tilted plate of height \(L\) and width \(W\) inclined at an angle \(u\) from the vertical, \((c)\) a vertical cylinder of length \(L\) and diameter \(D,(d)\) a horizontal cylinder of length \(L\) and diameter \(D\), and (e) a sphere of diameter \(D\).

Short Answer

Expert verified
Question: Write the simplified expressions for the Reynolds number for condensate flow in the following geometries: (a) vertical plate, (b) tilted plate, (c) vertical cylinder, (d) horizontal cylinder, and (e) sphere. Answer: (a) Vertical plate: $\operatorname{Re} = \frac{2 \rho L w v}{(L+w) \mu_l}$ (b) Tilted plate: $\operatorname{Re} = \frac{2 \rho L_p W v}{(L_p+W) \mu_l}$ (c) Vertical cylinder: $\operatorname{Re} = \frac{2 \rho D v}{\mu_l}$ (d) Horizontal cylinder: $\operatorname{Re} = \frac{2 \rho D v}{\mu_l (1 + 2L/\pi D)}$ (e) Sphere: $\operatorname{Re} = \frac{8 \rho D^2 v}{3 \mu_l}$

Step by step solution

01

(a) Vertical plate

For a vertical plate of height \(L\) and width \(w\), the wetted perimeter \(p\) is given as \(p=2(L+w)\). The mass flow rate \(\dot{m}\) is given by \(\rho L w v\), where \(\rho\) is the density of the fluid, and \(v\) is the velocity. Thus, the Reynolds number is: $$\operatorname{Re} = \frac{4 \rho L w v}{2(L+w) \mu_l} = \frac{2 \rho L w v}{(L+w) \mu_l}$$
02

(b) Tilted plate

For a tilted plate of height \(L\) and width \(W\) inclined at an angle \(u\) from the vertical, we first find the projected length \(L_p = L \cos u\). The wetted perimeter \(p\) is given as \(p=2(L_p+W)\). The mass flow rate \(\dot{m}\) is given by \(\rho L_p W v\). Thus, the Reynolds number is: $$\operatorname{Re} = \frac{4 \rho L_p W v}{2(L_p+W) \mu_l} = \frac{2 \rho L_p W v}{(L_p+W) \mu_l}$$
03

(c) Vertical cylinder

For a vertical cylinder of length \(L\) and diameter \(D\), the wetted perimeter \(p\) is given as \(p=\pi D\). The mass flow rate \(\dot{m}\) is given by \(\rho \pi (D/2)^2 v\). Thus, the Reynolds number is: $$\operatorname{Re} = \frac{4 \rho \pi (D/2)^2 v}{\pi D \mu_l} = \frac{2 \rho D v}{\mu_l}$$
04

(d) Horizontal cylinder

For a horizontal cylinder of length \(L\) and diameter \(D\), the wetted perimeter \(p\) is given as \(p=\pi D + 2L\). The mass flow rate \(\dot{m}\) is given by \(\rho \pi (D/2)^2 v\). Thus, the Reynolds number is: $$\operatorname{Re} = \frac{4 \rho \pi (D/2)^2 v}{(\pi D + 2L) \mu_l} = \frac{2 \rho D v}{\mu_l (1 + 2L/\pi D)}$$
05

(e) Sphere

For a sphere of diameter \(D\), the wetted perimeter \(p\) is given as \(p=\pi D\). The mass flow rate \(\dot{m}\) is given by \(\rho (4/3)\pi (D/2)^3 v\). Thus, the Reynolds number is: $$\operatorname{Re} = \frac{4 \rho (4/3)\pi (D/2)^3 v}{\pi D \mu_l} = \frac{8 \rho D^2 v}{3 \mu_l}$$

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Most popular questions from this chapter

A long cylindrical stainless steel rod \(\left(c_{p}=\right.\) \(450 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \rho=7900 \mathrm{~kg} / \mathrm{m}^{3}, \varepsilon=0.30\) ) with mechanically polished surface is being conveyed through a water bath to be quenched. The \(25-\mathrm{mm}\)-diameter stainless steel rod has a temperature of \(700^{\circ} \mathrm{C}\) as it enters the water bath. \(\mathrm{A}\) length of \(3 \mathrm{~m}\) of the rod is submerged in water as it is conveyed through the water bath during the quenching process. As the stainless steel rod enters the water bath, boiling would occur at \(1 \mathrm{~atm}\). In order to prevent thermal burn on people handling the rod, it must exit the water bath at a temperature below \(45^{\circ} \mathrm{C}\). Determine the speed of the rod being conveyed through the water bath so that it leaves the water bath without the risk of thermal burn hazard.

A \(10 \mathrm{~cm} \times 10 \mathrm{~cm}\) horizontal flat heater is used for vaporizing refrigerant-134a at \(350 \mathrm{kPa}\). The heater is supplied with \(0.35 \mathrm{MW} / \mathrm{m}^{2}\) of heat flux, and the surface temperature of the heater is \(25^{\circ} \mathrm{C}\). If the experimental constant in the Rohsenow correlation is \(n=1.7\), determine the value of the coefficient \(C_{s f}\). Discuss whether or not the Rohsenow correlation is applicable in this analysis.

What is the modified latent heat of vaporization? For what is it used? How does it differ from the ordinary latent heat of vaporization?

Saturated water vapor at atmospheric pressure condenses on the outer surface of a \(0.1\)-m-diameter vertical pipe. The pipe is \(1 \mathrm{~m}\) long and has a uniform surface temperature of \(80^{\circ} \mathrm{C}\). Determine the rate of condensation and the heat transfer rate by condensation. Discuss whether the pipe can be treated as a vertical plate. Assume wavy-laminar flow and that the tube diameter is large relative to the thickness of the liquid film at the bottom of the tube. Are these good assumptions?

Saturated steam at \(100^{\circ} \mathrm{C}\) condenses on a 2-m \(\times 2-\mathrm{m}\) plate that is tilted \(40^{\circ}\) from the vertical. The plate is maintained at \(80^{\circ} \mathrm{C}\) by cooling it from the other side. Determine (a) the average heat transfer coefficient over the entire plate and \((b)\) the rate at which the condensate drips off the plate at the bottom. Assume wavy-laminar flow. Is this a good assumption?
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