§8.5 Film condensation 439 and, with h(x) from eqn. (8.57), h = 2 πD ⌠    ⌡ πD/2 0 1 √ 2 k x      ρ f  ρ f −ρ g  h  fg x 3 µk ( T sat −T w ) xg e (sin 2x/D) 4/3  x 0 ( sin 2x/D ) 1/3 dx      1/4 dx This integral can be evaulated in terms of gamma functions. The result, when it is put back in the form of a Nusselt number, is Nu D = 0.728   ρ f  ρ f −ρ g  g e h  fg D 3 µk ( T sat −T w )   1/4 (8.64) for a horizontal cylinder. (Nusselt got 0.725 for the lead constant, but he had to approximate the integral with a hand calculation.) Some other results of this calculation include the following cases. Sphere of diameter D: Nu D = 0.828   ρ f  ρ f −ρ g  g e h  fg D 3 µk ( T sat −T w )   1/4 (8.65) This result 9 has already been compared with the experimental data in Fig. 8.10. Vertical cone with the apex on top, the bottom insulated, and a cone angle of α ◦ : Nu x = 0.874 [ cos(α/2) ] 1/4   ρ f  ρ f −ρ g  g e h  fg x 3 µk ( T sat −T w )   1/4 (8.66) Rotating horizontal disk 10 : In this case, g = ω 2 x, where x is the distance from the center and ω is the speed of rotation. The Nusselt number, based on L = (µ/ρ f ω) 1/2 ,is Nu = 0.9034   µ  ρ f −ρ g  h  fg ρ f k ( T sat −T w )   1/4 = constant (8.67) 9 There is an error in [8.33]: the constant given there is 0.785. The value of 0.828 given here is correct. 10 This problem was originally solved by Sparrow and Gregg [8.38]. 440 Natural convection in single-phase fluids and during film condensation §8.5 This result might seem strange at first glance. It says that Nu ≠ fn(x or ω). The reason is that δ just happens to be independent of x in this config- uration. The Nusselt solution can thus be bent to fit many complicated geo- metric figures. One of the most complicated ones that have been dealt with is the reflux condenser shown in Fig. 8.14. In such a configuration, cooling water flows through a helically wound tube and vapor condenses on the outside, running downward along the tube. As the condensate flows, centripetal forces sling the liquid outward at a downward angle. This complicated flow was analyzed by Karimi [8.39], who found that Nu ≡ hd cos α k =    ρ f −ρ g  ρ f h  fg g(d cosα) 3 µk∆T   1/4 fn  d D ,B  (8.68) where B is a centripetal parameter: B ≡ ρ f −ρ g ρ f c p ∆T h  fg tan 2 α Pr and α is the helix angle (see Fig. 8.14). The function on the righthand side of eqn. (8.68) was a complicated one that must be evaluated numerically. Karimi’s result is plotted in Fig. 8.14. Laminar–turbulent transition The mass flow rate of condensate per unit width of film, ˙ m, is more com- monly designated as Γ c (kg/m · s). Its calculation in eqn. (8.50) involved substituting eqn. (8.48)in ˙ m or Γ c = ρ f  δ 0 udy Equation (8.48) gives u(y) independently of any geometric features. [The geometry is characterized by δ(x).] Thus, the resulting equation for the mass flow rate is still Γ c = ρ f  ρ f −ρ g  gδ 3 3µ (8.50a) This expression is valid for any location along any film, regardless of the geometry of the body. The configuration will lead to variations of g(x) and δ(x), but eqn. (8.50a) still applies. §8.5 Film condensation 441 Figure 8.14 Fully developed film condensation heat transfer on a helical reflux condenser [8.39]. It is useful to define a Reynolds number in terms of Γ c . This is easy to do, because Γ c is equal to ρu av δ. Re c = Γ c µ = ρ f (ρ f −ρ g )gδ 3 3µ 2 (8.69) It turns out that the Reynolds number dictates the onset of film insta- bility, just as it dictates the instability of a b.l. or of a pipe flow. 11 When Re c  7, scallop-shaped ripples become visible on the condensate film. When Re c reaches about 400, a full-scale laminar-to-turbulent transition occurs. Gregorig, Kern, and Turek [8.40] reviewed many data for the film condensation of water and added their own measurements. Figure 8.15 shows these data in comparison with Nusselt’s theory, eqn. (8.60). The comparison is almost perfect up to Re c  7. Then the data start yielding somewhat higher heat transfer rates than the prediction. This is because 11 Two Reynolds numbers are defined for film condensation: Γ c /µ and 4Γ c /µ. The latter one, which is simply four times as large as the one we use, is more common in the American literature. 442 Natural convection in single-phase fluids and during film condensation §8.5 Figure 8.15 Film condensation on vertical plates. Data are for water [8.40]. the ripples improve heat transfer—just a little at first and by about 20% when the full laminar-to-turbulent transition occurs at Re c = 400. Above Re c = 400, Nu L begins to rise with Re c . The Nusselt number begins to exhibit an increasingly strong dependence on the Prandtl num- ber in this turbulent regime. Therefore, one can use Fig. 8.15, directly as a data correlation, to predict the heat transfer coefficient for steam con- densating at 1 atm. But for other fluids with different Prandtl numbers, one should consult [8.41]or[8.42]. Two final issues in natural convection film condensation • Condensation in tube bundles. Nusselt showed that if n horizontal tubes are arrayed over one another, and if the condensate leaves each one and flows directly onto the one below it without splashing, then Nu D for n tubes = Nu D 1 tube n 1/4 (8.70) This is a fairly optimistic extension of the theory, of course. In addition, the effects of vapor shear stress on the condensate and of pressure losses on the saturation temperature are often important in tube bundles. These effects are discussed by Rose et al. [8.42] and Marto [8.41]. Problems 443 • Condensation in the presence of noncondensable gases. When the condensing vapor is mixed with noncondensable air, uncondensed air must constantly diffuse away from the condensing film and va- por must diffuse inward toward the film. This coupled diffusion process can considerably slow condensation. The resulting h can easily be cut by a factor of five if there is as little as 5% by mass of air mixed into the steam. This effect was first analyzed in detail by Sparrow and Lin [8.43]. More recent studies of this problem are reviewed in [8.41, 8.42]. Problems 8.1 Show that Π 4 in the film condensation problem can properly be interpreted as Pr Re 2  Ja. 8.2 A 20 cm high vertical plate is kept at 34 ◦ Cina20 ◦ C room. Plot (to scale) δ and h vs. height and the actual temperature and velocity vs. y at the top. 8.3 Redo the Squire-Eckert analysis, neglecting inertia, to get a high-Pr approximation to Nu x . Compare your result with the Squire-Eckert formula. 8.4 Assume a linear temperature profile and a simple triangular velocity profile, as shown in Fig. 8.16, for natural convection on a vertical isothermal plate. Derive Nu x = fn(Pr, Gr x ), com- pare your result with the Squire-Eckert result, and discuss the comparison. 8.5 A horizontal cylindrical duct of diamond-shaped cross section (Fig. 8.17) carries air at 35 ◦ C. Since almost all thermal resis- tance is in the natural convection b.l. on the outside, take T w to be approximately 35 ◦ C. T ∞ = 25 ◦ C. Estimate the heat loss per meter of duct if the duct is uninsulated. [Q = 24.0W/m.] 8.6 The heat flux from a 3 m high electrically heated panel in a wall is 75 W/m 2 in an 18 ◦ C room. What is the average temper- ature of the panel? What is the temperature at the top? at the bottom? 444 Chapter 8: Natural convection in single-phase fluids and during film condensation Figure 8.16 Configuration for Problem 8.4. Figure 8.17 Configuration for Problem 8.5. 8.7 Find pipe diameters and wall temperatures for which the film condensation heat transfer coefficients given in Table 1.1 are valid. 8.8 Consider Example 8.6. What value of wall temperature (if any), or what height of the plate, would result in a laminar-to-turbulent transition at the bottom in this example? 8.9 A plate spins, as shown in Fig. 8.18, in a vapor that rotates syn- chronously with it. Neglect earth-normal gravity and calculate Nu L as a result of film condensation. 8.10 A laminar liquid film of temperature T sat flows down a vertical wall that is also at T sat . Flow is fully developed and the film thickness is δ o . Along a particular horizontal line, the wall temperature has a lower value, T w , and it is kept at that tem- perature everywhere below that position. Call the line where the wall temperature changes x = 0. If the whole system is Problems 445 Figure 8.18 Configuration for Problem 8.9. immersed in saturated vapor of the flowing liquid, calculate δ(x),Nu x , and Nu L , where x = L is the bottom edge of the wall. (Neglect any transition behavior in the neighborhood of x = 0.) 8.11 Prepare a table of formulas of the form h(W/m 2 K) = C [ ∆T ◦ C/L m ] 1/4 for natural convection at normal gravity in air and in water at T ∞ = 27 ◦ C. Assume that T w is close to 27 ◦ C. Your table should include results for vertical plates, horizontal cylinders, spheres, and possibly additional geometries. Do not include your calculations. 8.12 For what value of Pr is the condition ∂ 2 u ∂y 2      y=0 = gβ(T w −T ∞ ) ν satisfied exactly in the Squire-Eckert b.l. solution? [Pr = 2.86.] 8.13 The overall heat transfer coefficient on the side of a particular house 10 m in height is 2.5 W/m 2 K, excluding exterior convec- tion. It is a cold, still winter night with T outside =−30 ◦ C and T inside air = 25 ◦ C. What is h on the outside of the house? Is external convection laminar or turbulent? 8.14 Consider Example 8.2. The sheets are mild steel, 2 m long and 6 mm thick. The bath is basically water at 60 ◦ C, and the sheets 446 Chapter 8: Natural convection in single-phase fluids and during film condensation are put in it at 18 ◦ C. (a) Plot the sheet temperature as a function of time. (b) Approximate h at ∆T = [ (60 +18)/2 −18 ] ◦ C and plot the conventional exponential response on the same graph. 8.15 A vertical heater 0.15 m in height is immersed in water at 7 ◦ C. Plot h against (T w − T ∞ ) 1/4 , where T w is the heater tempera- ture, in the range 0 <(T w − T ∞ )<100 ◦ C. Comment on the result. should the line be straight? 8.16 A77 ◦ C vertical wall heats 27 ◦ C air. Evaluate δ top /L, Ra L , and L where the line in Fig. 8.3 ceases to be straight. Comment on the implications of your results. [δ top /L  0.6.] 8.17 A horizontal 8 cm O.D. pipe carries steam at 150 ◦ C through a room at 17 ◦ C. The pipe has a 1.5 cm layer of 85% magnesia insulation on it. Evaluate the heat loss per meter of pipe. [Q = 97.3W/m.] 8.18 What heat rate (in W/m) must be supplied to a 0.01 mm hori- zontal wire to keep it 30 ◦ C above the 10 ◦ C water around it? 8.19 A vertical run of copper tubing, 5 mm in diameter and 20 cm long, carries condensation vapor at 60 ◦ C through 27 ◦ C air. What is the total heat loss? 8.20 A body consists of two cones joined at their bases. The di- ameter is 10 cm and the overall length of the joined cones is 25 cm. The axis of the body is vertical, and the body is kept at 27 ◦ Cin7 ◦ C air. What is the rate of heat removal from the body? [Q = 3.38 W.] 8.21 Consider the plate dealt with in Example 8.3. Plot h as a func- tion of the angle of inclination of the plate as the hot side is tilted both upward and downward. Note that you must make do with discontinuous formulas in different ranges of θ. 8.22 You have been asked to design a vertical wall panel heater, 1.5 m high, for a dwelling. What should the heat flux be if no part of the wall should exceed 33 ◦ C? How much heat will be added to the room if the panel is7minwidth? 8.23 A 14 cm high vertical surface is heated by condensing steam at 1 atm. If the wall is kept at 30 ◦ C, how would the average Problems 447 heat transfer coefficient change if ammonia, R22, methanol, or acetone were used instead of steam to heat it? How would the heat flux change? (Data for methanol and acetone must be obtained from sources outside this book.) 8.24 A 1 cm diameter tube extends 27 cm horizontally through a region of saturated steam at 1 atm. The outside of the tube can be maintained at any temperature between 50 ◦ C and 150 ◦ C. Plot the total heat transfer as a function of tube temperature. 8.25 A 2 m high vertical plate condenses steam at 1 atm. Below what temperature will Nusselt’s prediction of h be in error? Below what temperature will the condensing film be turbulent? 8.26 A reflux condenser is made of copper tubing 0.8 cm in diameter with a wall temperature of 30 ◦ C. It condenses steam at 1 atm. Find h if α = 18 ◦ and the coil diameter is 7 cm. 8.27 The coil diameter of a helical condenser is 5 cm and the tube diameter is 5 mm. The condenser carries water at 15 ◦ C and is in a bath of saturated steam at 1 atm. Specify the number of coils and a reasonable helix angle if 6 kg/hr of steam is to be condensed. h inside = 600 W/m 2 K. 8.28 A schedule 40 type 304 stainless steam pipe with a 4 in. nom- inal diameter carries saturated steam at 150 psia in a process- ing plant. Calculate the heat loss per unit length of pipe if it is bare and the surrounding air is still at 68 ◦ F. How much would this heat loss be reduced if the pipe were insulated witha1in. layer of 85% magnesia insulation? [Q saved  127 W/m.] 8.29 What is the maximum speed of air in the natural convection b.l. in Example 8.1? 8.30 All of the uniform-T w , natural convection formulas for Nu take the same form, within a constant, at high Pr and Ra. What is that form? (Exclude any equation that includes turbulence.) 8.31 A large industrial process requires that water be heated by a large horizontal cylinder using natural convection. The water is at 27 ◦ C. The diameter of the cylinder is 5 m, and it is kept at 67 ◦ C. First, find h. Then suppose that D is increased to 10 m. 448 Chapter 8: Natural convection in single-phase fluids and during film condensation What is the new h? Explain the similarity of these answers in the turbulent natural convection regime. 8.32 A vertical jet of liquid of diameter d and moving at velocity u ∞ impinges on a horizontal disk rotating ω rad/s. There is no heat transfer in the system. Develop an expression for δ(r ), where r is the radial coordinate on the disk. Contrast the r dependence of δ with that of a condensing film on a rotating disk and explain the difference qualitatively. 8.33 We have seen that if properties are constant, h ∝ ∆T 1/4 in natural convection. If we consider the variation of properties as T w is increased over T ∞ , will h depend more or less strongly on ∆T in air? in water? 8.34 A film of liquid falls along a vertical plate. It is initially satu- rated and it is surrounded by saturated vapor. The film thick- ness is δ o . If the wall temperature below a certain point on the wall (call it x = 0) is raised to a value of T w , slightly above T sat , derive expressions for δ(x),Nu x , and x f —the distance at which the plate becomes dry. Calculate x f if the fluid is water at 1 atm, if T w = 105 ◦ C and δ o = 0.1 mm. 8.35 In a particular solar collector, dyed water runs down a vertical plate in a laminar film with thickness δ o at the top. The sun’s rays pass through parallel glass plates (see Section 10.6) and deposit q s W/m 2 in the film. Assume the water to be saturated at the inlet and the plate behind it to be insulated. Develop an expression for δ(x) as the water evaporates. Develop an ex- pression for the maximum length of wetted plate, and provide a criterion for the laminar solution to be valid. 8.36 What heat removal flux can be achieved at the surface of a horizontal 0.01 mm diameter electrical resistance wire in still 27 ◦ C air if its melting point is 927 ◦ C? Neglect radiation. 8.37 A 0.03 m O.D. vertical pipe, 3 m in length, carries refrigerant through a 24 ◦ C room. How much heat does it absorb from the room if the pipe wall is at 10 ◦ C? 8.38 A 1 cm O.D. tube at 50 ◦ C runs horizontally in 20 ◦ C air. What is the critical radius of 85% magnesium insulation on the tube? 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