A HEAT TRANSFER TEXTBOOK - THIRD EDITION Episode 2 Part 9 pdf

Figure 8.14 Fully developed film condensation heat transferon a helical reflux condenser [8.39].. Data are fora data correlation, to predict the heat transfer coefficient for steam densating

and, with h(x) from eqn (8.57),

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

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

Rotating horizontal disk10: In this case, g = ω2x, where x is the

distance from the center and ω is the speed of rotation The Nusselt

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 ].

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 metric figures One of the most complicated ones that have been dealtwith is the reflux condenser shown in Fig.8.14 In such a configuration,cooling water flows through a helically wound tube and vapor condenses

geo-on the outside, running downward algeo-ong the tube As the cgeo-ondensateflows, centripetal forces sling the liquid outward at a downward angle.This complicated flow was analyzed by Karimi [8.39], who found that

where B is a centripetal parameter:

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 asΓc (kg/m · s) Its calculation in eqn (8.50) involvedsubstituting eqn (8.48) in

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

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 ofΓc This is easy

to do, becauseΓc is equal to ρuavδ.

Rec = Γc

µ = ρ f (ρ f − ρ g )gδ

3

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

Rec 7, scallop-shaped ripples become visible on the condensate film.

When Rec 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 Figure8.15

shows these data in comparison with Nusselt’s theory, eqn (8.60) The

comparison is almost perfect up to Rec 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: Γc /µ and 4Γc /µ The

latter one, which is simply four times as large as the one we use, is more common in

the American literature.

Figure 8.15 Film condensation on vertical plates Data are for

a data correlation, to predict the heat transfer coefficient for steam densating at 1 atm But for other fluids with different Prandtl numbers,one should consult [8.41] or [8.42]

con-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 leaveseach one and flows directly onto the one below it without splashing,then

NuD for n tubes = NuD1 tube

This is a fairly optimistic extension of the theory, of course Inaddition, the effects of vapor shear stress on the condensate and ofpressure 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]

• 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 Π4 in the film condensation problem can properly

be interpreted as Pr Re2 Ja

8.2 A 20 cm high vertical plate is kept at 34◦C in a 20◦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 Nux 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 Nux = 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 = 24.0 W/m.]

8.6 The heat flux from a 3 m high electrically heated panel in a

wall is 75 W/m2in an 18◦C room What is the average

temper-ature of the panel? What is the tempertemper-ature at the top? at the

bottom?

Figure 8.16 Configuration for Problem8.4.

Figure 8.17 Configuration for

Problem8.5

8.7 Find pipe diameters and wall temperatures for which the film

condensation heat transfer coefficients given in Table 1.1arevalid

8.8 Consider Example8.6 What value of wall temperature (if any),

or what height of the plate, would result in a laminar-to-turbulenttransition 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

NuL as a result of film condensation

8.10 A laminar liquid film of temperature Tsatflows down a vertical

wall that is also at Tsat 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 perature everywhere below that position Call the line where

tem-the wall temperature changes x = 0 If the whole system is

Figure 8.18 Configuration for

Problem8.9

immersed in saturated vapor of the flowing liquid, calculate

δ(x), Nu x, and NuL , 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/m2K) = 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

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/m2K, excluding exterior

convec-tion It is a cold, still winter night with Toutside = −30 ◦C and

external convection laminar or turbulent?

8.14 Consider Example8.2 The sheets are mild steel, 2 m long and

6 mm thick The bath is basically water at 60◦C, and the sheets

are put in it at 18◦C (a) Plot the sheet temperature as a function

of time (b) Approximateh 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 A 77◦C vertical wall heats 27◦ C air Evaluate δtop/L, Ra L, and

L where the line in Fig.8.3ceases to be straight Comment on

the implications of your results [δtop/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 = 97.3 W/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◦C in 7◦C air What is the rate of heat removal from the

body? [Q = 3.38 W.]

8.21 Consider the plate dealt with in Example8.3 Plot h as a

func-tion of the angle of inclinafunc-tion of the plate as the hot side istilted 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 nopart of the wall should exceed 33◦C? How much heat will beadded to the room if the panel is 7 m in width?

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

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

Findh 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 hinside= 600 W/m2K

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 with a 1 in

layer of 85% magnesia insulation? [Qsaved 127 W/m.]

8.29 What is the maximum speed of air in the natural convection

b.l in Example8.1?

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.

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 satusatu-rated 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

Tsat, 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 oat the top The sun’srays pass through parallel glass plates (see Section 10.6) and

deposit q sW/m2in 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 theroom 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?

8.39 A 1 in cube of ice is suspended in 20◦C air Estimate the drip

rate in gm/min (Neglect∆T through the departing water film.

hsf= 333, 300 J/kg.)

8.40 A horizontal electrical resistance heater, 1 mm in diameter,

releases 100 W/m in water at 17 ◦C What is the wire

tempera-ture?

8.41 Solve Problem5.39using the correct formula for the heat

trans-fer coefficient

8.42 A red-hot vertical rod, 0.02 m in length and 0.005 m in

diame-ter, is used to shunt an electrical current in air at room

temper-ature How much power can it dissipate if it melts at 1200◦C?

Note all assumptions and corrections Include radiation using

Frod-room= 0.064.

8.43 A 0.25 mm diameter platinum wire, 0.2 m long, is to be held

horizontally at 1035◦C It is black How much electric power is

needed? Is it legitimate to treat it as a constant-wall-temperature

heater in calculating the convective part of the heat transfer?

The surroundings are at 20◦C and the surrounding room is

virtually black

8.44 A vertical plate, 11.6 m long, condenses saturated steam at

1 atm We want to be sure that the film stays laminar What is

the lowest allowable plate temperature, and what is q at this

temperature?

8.45 A straight horizontal fin exchanges heat by laminar natural

convection with the surrounding air

a Show that

d2θ

dξ2 = m2L2θ 5/4 where m is based on h o ≡ h(T = T o ).

b Develop an iterative numerical method to solve this

equa-tion for T (x = 0) = T o and an insulated tip (Hint : earize the right side by writing it as (m2L2θ 1/4 )θ, and

lin-evaluate the term in parenthesis at the previous iterationstep.)

c Solve the resulting difference equations for m2L2 valuesranging from 10−3 to 103 Use Gauss elimination or the

tridiagonal algorithm Express the results as η/η o where

η is the fin efficiency and η o is the efficiency that wouldresult ifh owere the uniform heat transfer coefficient overthe entire fin

8.46 A 2.5 cm black sphere ( F = 1) is in radiation-convection

equi-librium with air at 20◦C The surroundings are at 1000 K What

is the temperature of the sphere?

8.47 Develop expressions for h(D) and Nu D during condensation

on a vertical circular plate

8.48 A cold copper plate is surrounded by a 5 mm high ridge which

forms a shallow container It is surrounded by saturated watervapor at 100◦C Estimate the steady heat flux and the rate ofcondensation

a When the plate is perfectly horizontal and filled to

over-flowing with condensate

b When the plate is in the vertical position.

c Did you have to make any idealizations? Would they

re-sult in under- or over-estimation of the condensation?

8.49 A proposed design for a nuclear power plant uses molten lead

to remove heat from the reactor core The heated lead is thenused to boil water that drives a steam turbine Water at 5 atm

pressure (Tsat = 152 ◦C) enters a heated section of a pipe at

60◦C with a mass flow rate of ˙m = 2 kg/s The pipe is stainless steel (k s = 15 W/m·K) with a wall thickness of 12 mm and an

outside diameter of 6.2 cm The outside surface of the pipe

is surrounded by an almost-stationary pool of molten lead at

477◦C

a At point where the liquid water has a bulk temperature

of T b = 80 ◦C, estimate the inside and outside wall peratures of the pipe, T w i and T w o, to within about 5◦C.Neglect entry length and variable properties effects and

tem-take β ≈ 0.000118 K −1 for lead Hint: Guess an outside

wall temperature above 370◦ C when computing h for the

lead