From: Snapshot-Content-Location: https://learning.oreilly.com/library/view/heat-and-mass/9789332515680/xhtml/chapter007.xhtml Subject: 7. Condensation - Heat and Mass Transfer, 2nd Edition Date: Sat, 30 Nov 2019 09:40:45 -0000 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----MultipartBoundary--se38IPiZtiI4mLA8tHmQydhT6WEmWKrpeEuXeFmY6F----" ------MultipartBoundary--se38IPiZtiI4mLA8tHmQydhT6WEmWKrpeEuXeFmY6F---- Content-Type: text/html Content-ID: Content-Transfer-Encoding: quoted-printable Content-Location: https://learning.oreilly.com/library/view/heat-and-mass/9789332515680/xhtml/chapter007.xhtml 7. Condensation - Heat and Mass= Transfer, 2nd Edition =20 =20 =20
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7

Condensation

If vapor comes in contacts wi= th a surface maintained at a temperature below the saturation temperature o= f the vapor, then condensation occurs. The liquid condensate will flow down= on the surface under the influence of gravity. When the liquid wets the su= rface, spreads out and forms a film and this process is known as filmwise c= ondensation. If the surface is not wetted by the liquid, then droplets form= and run down the surface. This process is known as dropwise condensation. = Heat transfer rate in dropwise condensation is nearly 10 times higher than = in the filmwise condensation process. But dropwise condensation is very dif= ficult to achieve in practical cases. All practical design methods are base= d on filmwise condensation. The heat transfer coefficient for filmwise cond= ensation on a flat plate and a circular tube were found by Nusselt in 1916.=

= 7.1 CONDENSATION ON VERTICAL SURFACES (NUSSELT THEORY)

An analytical model for filmw= ise condensation for vapor on a vertical wall is shown in Fig. 7.1 The film thickness (=CE=B4) is seen to be = zero at the top of the vertical wall and to increase with distance in the <= em>x direction. The following assumptions were first made by Nusselt.<= /p>

 =

3D"Filmwise

 

Figure 7.1 Filmwise Condensation on a Vertical Surface

 

  1. The vertical wall is kept at an u= niform temperature (Tw) which is lower = than the saturation temperature (Tv) of= the vapor.
  2. The drag force of vapor on the motion = of the condensate is very small.
  3. The flow of condensate is under gravit= y force.
  4. The flow acceleration in the condensat= e layer is very small.
  5. Fluid properties are constant and = heat transfer across the liquid film is by pure conduction.

The velocity distribution (<= em>u) at any location (x) across the liquid film is determine= d by equating the shear force acting upward to the net gravity force acting= downward. Assuming unit thickness

 

3D"image"

 

Integrating the equation, we= get

 

3D"Image"

 

=CE=BCl =E2=80=93 viscosity of the liquid

=CE=B4 =E2=80=93 th= ickness of liquid film at =E2=80=98x=E2=80=99

=CF=81l =E2=80=93 density of liquid

kl<= /sub> =E2=80=93 thermal conductivity of liquid

g =E2=80=93 acceler= ation due to gravity

Tv<= /sub> =E2=80=93 temperature of vapor

Tw<= /sub> =E2=80=93 temperature of the wall

Using the boundary condition = u =3D 0 at y =3D 0, the value of C is zero.

 

3D"image"

 

The average velocity at x= is

 

3D"image"

 

The m= ass flow rate (m) of the liquid at distance =E2=80=98x=E2= =80=99 per unit width of the plate is written as

 

3D"image"

 

The rate of heat transfer (d= Q) over the element is

 

dQ =3D hfg =C2=B7 dm     (7.6)<= /div>

 

where hfg is the latent heat of condensation and dm is the = rate of condensation of vapor over distance =E2=80=98dx=E2=80=99.<= /p>

As per the assumption, heat = conduction from the vapor to wall is

 

3D"image"

 

From Equation (7.5),

 

3D"image"

 

From Equations (7.7) and (7= .8),

 

3D"image"

 

Integrating Equation (7.9) and using the boundary conditions =CE=B4 =3D 0 at x =3D 0, the thickness of the water layer b= ecomes

 

3D"image"

 

The local heat transfer coef= ficient can be obtained by

 

3D"image"

 

where = 3D"Image"

Then average heat transfer c= oefficient,

 

3D"image"

 

where =E2=80=98L=E2= =80=99 is the length of the plate

 

3D"image"

 

The value of 3D"image" is obtained by substituting the value of =E2=80=98C= =E2=80=99 as

 

3D"image"

 

The physical properties must= be taken at the film temperature. The above equation is also applicable to= vertical tubes. In the above equation, substitute g =3D g sin =CE=B8 for inclined surfaces, where =E2=80=98=CE=B8= =E2=80=99 is the angle to the horizontal.

= 7.2 CONDENSATION ON HORIZONTAL SURFACE

For a horizontal tube with di= ameter (D), the Nusselt's equation is

 

3D"image"

 

Then the ratio of the conden= sation on a vertical tube of length L to that on a horizontal tube= of diameter D, is given by

 

3D"image"

 

Equation (7.14) gives two conditions:

  1. If L =3D 2.87 D= , then 3D"image" for the same temperature difference.
  2. If L =3D 50 D, t= hen 3D"image" for the same temperature difference.

If horizontal tubes are arra= nged in vertical tiers, the condensate from one tube drains out the tube ju= st below. The heat transfer coefficient is given by

 =

3D"image"

 

where N is the numbe= r of tubes.

= 7.3 REYNOLDS NUMBER FOR CONDENSATE FLOW

The Reynolds number (Re) is defined as

 

3D"image"

 

where U is the avera= ge velocity of the condensate liquid film, =CE=BCl= is the dynamic viscosity of liquid, and =CF=81l is the density of liquid.

 

3D"image"

 

3D"image"

 

where m is the mass = flow rate =3D =CF=81lUA.

There is a transition from l= aminar to turbulent condensation when the Reynolds number exceeds 1800.

= 7.4 CONDENSATION HEAT TRANSFER COEFFICIENTS FROM EXPERIMENTS

McAdams found by experiments = that the average heat transfer for vertical surfaces is about 20% higher th= an that found from Nusselt theory. McAdams recommended the following equati= ons:

For a vertical surface

 

3D"image"

 

The above equation can be wr= itten in terms of Reynolds number (Re < 1800) as

 

3D"image"

 

The other empirical correlat= ion for film condensation on a vertical plate for turbulence flow (Re > 1800) is

 

3D"Image"
= 7.5 DROPWISE CONDENSATION

If the condensing surface is = very smooth and if some oil substance is applied on it, then vapor will con= dense and form droplets. This type of dropwise condensation gives much high= er rate of heat transfer, 10 to 20 times higher than filmwise condensation.= Substances such as oleic acid, benzyl mercaptan, and some fats and waxes w= hen applied make the surface very smooth to promote dropwise condensation. = A comparison of heat transfer coefficients for filmwise and dropwise conden= sation is shown in Table 7.1.

 

Ta= ble 7.1 Approximate condensation heat transfer coefficients

Temperature difference (=C2=B0C) Filmwise condensation h (W/m2K) Dropwise condensation h (W/m2K)
2
20,900
1,10,300
4
13,930
92,890
6
11,610
75,470
8
9,290
65,020
10
6,965
55,730
12
5,800
46,445
14
4,645
40,640
= 7.6 INFLUENCE OF NONCONDENSABLE GASES

If the condensing gas has non= condensable gases in it, then the surface heat transfer coefficient will de= crease significantly. The presence of even one percent by volume of non-con= densable gases will reduce the heat transfer coefficient by 30=E2=80=9350%.=

= 7.7 CONCLUSIONS

The condensation is the phase= change process from vapor to liquid when the vapor comes in contact with a= surface whose temperature is lower than the saturation temperature corresp= onding to the vapor pressure. During condensation, the latent heat is liber= ated and flow of heat to the surface. The modes of condensation are (i) fil= mwise condensation, (ii) dropwise condensation, (iii) homogenous condensati= on, and (iv) direct contact condensation.

During the condensation proc= ess, the condensate tends to wet the surface and thereby form a liquid film= ; then it is known as film condensation. The thin liquid film reduces the h= eat transfer from the vapor to surface.

During the condensation proc= ess, the vapor condenses into small liquid droplets of various sizes, which= fall down along the surface in random fashion. This is known as dropwise c= ondensation. In this, the heat transfer rate is 5 to 10 times higher than f= ilm condensation.

The Jacob number is defined = as the ratio between maximum sensible heat of liquid to latent heat of cond= ensation. The rate of condensation over the horizontal tube is higher than = the vertical tube. Turbulent flow is hardly ever reached on a horizontal tu= be but it is established over the lower portion of the vertical surface.

The surface coatings used to= stimulate dropwise condensation are silicon, teflon, waxes, and fatty acid= s. The presence of noncondensable gases reduces the rate of heat transfer b= y adding thermal resistance and also exhibits the transfer by offering diff= usional resistance.

KEY CONCEPTS

  • Condensation is a convection heat trans= fer process associated with change in phase of the fluid from vapour to liq= uid.

  • Due to fluid movement this process is classified as a convection he= at transfer process.

  • Larger heat transfer is possible with condensation mode with smalle= r temperture difference.

  • Combined latent heat and bouyancy induced flows make the heat trans= fer coefficient larger.

  • Condensation of vapour is common in power plant and refrigeration c= ondensers. Condensation starts when the vapour temperture reaches its satur= ation temperature.

  • When vapour condenses out as droplets suspended in a gas phase and = forms fog, the processes is known as homogeneous condensation.

  • Direct contact condensation occurs when the vapour comes in contact= with cold liquid.

  • A clean and uncontaminated surface will produce film condensation.<= /p>

  • Coated surfaces can produce dropwise condensation.

  • Film condensation has both laminar and turbulent flows.

  • Boundary layer conditions are applicable in film condensation

= EXAMPLE PROBLEMS

7.1 A tube of oute= r diameter 3 cm (Do) and length 4 m (L) is maint= ained at an uniform temperature of 35 =C2=B0C (Tw) by the flow of cooling water through tube. It is used to condense steam = at 65 =C2=B0C (Tv). Assumingfilm condensation, d= etermine the average heat transfer coefficient over the entire length of th= e tube and rate of condensate flow for (i) vertical position based on McAda= ms results and (ii) horizontal positions of the tube based on Nusselt's the= ory. Assume flow as laminar flow.

  1. Vertical tube

    Average heat transfer co= efficient,

     

    3D"image"

     

    The properties of the co= ndensate at film temperature, Tf =3D (6= 5 + 35)/2 =3D 50=C2=B0C are given by,

     

    3D"image"

     

    Condensate h= eat =3D Heat transfer at the surface of the tube

     

    3D"image"

     

    Mass flow rate,

     

    3D"image"

     

    Check for laminar flow

     

    3D"image"

     

  2. Horizontal tube

     

    3D"image"

     

    Mass flow rate,

     

    3D"image"

     

    The condensate flow rate with= the horizontal tube is 2.18 times higher than with the vertical tube.

7.2 Saturated steam = at a temperature Tv of 85=C2=B0C condenses on th= e outer surface of 256 horizontal tubes (N) each of outer diameter 1.3 cm (= D0) and arranged in a 16 =C3=97 16 array. Tube s= urfaces are maintained at an uniform temperature (Tw) of 75=C2=B0C. Determine the total condensate rate based on Nussel t= =E2=80=99 s theory.

Average heat transfer coeffi= cient,

 

3D"image"

 

The properties of condensate= at Tf =3D 85 + 75/2 =3D 80=C2= =B0C are

 

3D"image"

 

Therefore,

3D"image"

 

Heat flow rate in the array = of tubes per metre length, Q =3D hm(N=CF=80D= 0)(Tv = =E2=88=92 Tw)

 

Q =3D 7977.6=C3=97256=C3= =97=CF=80=C3=970.03=C3=97(85=E2=88=9275) =3D 1924.78 kW/m

 

m =3D Q/hfg

 

Mass of condensate per metre= length,

=20 3D"image"

 

7.3 Find Nusselt's h= eat transfer coefficient and the mass of condensate per hour if a vertical = square plate of side 40 cm is exposed to steam at atmospheric pressure. The= plate is maintained at an uniform temperature of 96 =C2=B0C (Tw).

Average heat transfer co= efficient,

 

3D"image"

 

The properties of water at fi= lm temperature, Tf =3D (100 + = 96)/2 =3D 98=C2=B0C, are as follows:

 

3D"image"

 

Therefore,

3D"image"

 

Condensate heat =3D Heat transfer a= t the surface

 

3D"image"

 

Therefore,

3D"image"

 

Check for laminar flow

 

3D"image"
= REVIEW QUESTIONS
  1. Define condensation process.
  2. Differentiate between filmwise and dro= pwise condensation.
  3. Derive an expression for condensation = heat transfer coefficient.
  4. Sketch the condensation process in a v= ertical surface.
  5. Discuss the difference in heat tra= nsfer between Nusselt's theory and experimental correlations.
OBJECTIVE QUESTIONS
  1. Condensation occurs on a surface = when the
    1. surface temperature is above the = saturation temperature of the vapour
    2. surface temperature is below the satur= ation temperature of the vapour
    3. surface temperature is above or below = the saturation temperature of the vapour
    4. vapour temperature is below the sa= turation temperature of the vapour
  2. The condensation process is known as = =EF=AC=81lmwise condensation, if
    1. the liquid wets the surface
    2. the liquid forms a =EF=AC=81lm on the = surface
    3. the surface is not wetted by the liqui= d
    4. both (a) and (b)
  3. The condensation process is known as d= ropwise condensation, if
    1. the liquid wets the surface
    2. the liquid forms a =EF=AC=81lm on the = surface
    3. the surface is not wetted by the liqui= d and droplets form on the surface
    4. both (a) and (b)
  4. The heat transfer rate in dropwise con= densation is than that in =EF=AC=81lmwise condensation
    1. greater
    2. lower
    3. very much lower
    4. none of the above
  5. The assumptions in Nusselts's theory o= n condensation on a vertical surface are
    1. the surface temperature of the pl= ate is uniform and less than the saturation temperature of the vapour
    2. the film of condensate is subjected to= gravity
    3. the fluid properties are constant
    4. all of the above
  6. The equivalent diameter for the estima= tion of Reynolds number in condensate =EF=AC=82ow is equal to
    1. 4 =C3=97 the cross-sectional area= for condensate flow/ wetted perimeter, that is, 4 A/P
    2. 4 / the cross-sectional area for conde= nsate flow/ wetted perimeter, that is, 4 P/A
    3. 4 =C3=97 the cross-sectional area for = condensate flow =C3=97 wetted perimeter, that is, 4 P/A
    4. 4 / the cross-sectional area for c= ondensate flow =C3=97 wetted perimeter, that is, 4/P A
  7. McAdams found by experiment that the a= verage condensation heat transfer for vertical surface is about higher than= that of Nusselt's theory
    1. 10%
    2. 20%
    3. 25%
    4. 30%
  8. For the same temperature difference, t= he condensation heat transfer coef=EF=AC=81cients for vertical and horizont= al tubes are equal if
    1. the vertical tube length is 2.87 = times the horizontal tube diameter
    2. the vertical tube length is 8.72 times= the horizontal tube diameter
    3. the vertical tube length is 8.27 times= the horizontal tube diameter
    4. the vertical tube length is 2.78 t= imes the horizontal tube diameter
  9. For the same temperature difference, t= he condensation heat transfer coef=EF=AC=81cient on horizontal tubes is 2.0= 5 times that of the vertical surface if the vertical tube is times the hori= zontal tube diameter
    1. 20
    2. 30
    3. 40
    4. 50
  10. Laminar to turbulent condensation take= s place when Reynolds number is greater than
    1. 180
    2. 280
    3. 1800
    4. 580

Answers:

  1. b
  2. b
  3. c
  4. a
  5. d
  6. a
  7. b
  8. a
  9. d
  10. c
EXERCISE PROBLEMS
  1. A vertical= plate of height 1 m is used to condense saturated steam existing at atmosp= heric pressure. If the surface temperature of the plate is maintained at 35= 2.9 K, determine whether the condensate =EF=AC=82ow is laminar or turbulent= .(Ans. Laminar, Re =3D 465)
  2. Find the local Nusselt heat transfer c= oefficient for Problem 1 at the midpoint of the vertical plate. (An= s. 4760 W/m2K)
  3. Calculate the average Nusselt heat tra= nsfer coefficient for Problem 1 for the entire length of the plate. (Ans. 5515 W/m2K)
  4. Determine the rate of condensation for= Problem 1 based on Nusselt's theory. (Ans. 2.93 kg/min)
  5. Find the rate of heat transfer between= the vertical surface of Problem 1 and steam based on Nusselt's theory. (Ans. 110 kW)
  6. A vertical tube of outer diameter 1.99= 5 cm and length 0.299 m is used to condense saturated steam existing at atm= ospheric pressure. If the tube surface temperature is 370 K, find the avera= ge Nusselt's heat transfer coefficient and total heat transfer rate. (Ans. 9635 W/m2K, 1.2 kW)
  7. A condenser uses a tube 0.1195 m in di= ameter and 1.995 m long in a vertical position to cond-ense steam at 75=C2= =B0C. If the surface temperature is maintained at 49.5=C2=B0C, and the rate= of condensation of the steam using Nusselt's theory. (Ans. 1.6 kg/ min)
  8. Find the percentage increase in heat t= ransfer for Problem 7 if the tube is kept in the horizontal instead of vert= ical position in the condenser. (Ans. 50%)
  9. A square vertical plate of side 0.599 = m maintained at 287.5 K is used to condense steam with saturation temperatu= re of 311 K at 0.068 bar. Find the average Nusselt's heat transfer coef.cie= nt, heat transfer rate and rate of condensation. (Ans. 4.4= kW/m2K, 36 kW, 0.9 kg/ min)
  10. A vertical square plate of side 0.299 = m is maintained at 370.5 K to condense steam at atmospheric pressure. Estim= ate the Nusselt heat transfer coefficient and rate of condensation. (Ans. 1.32 kW/m2K, 3.75 kg/hr)
  11. A condenser has 100 horizontal tubes, = each having a diameter of 1.195 cm, which are arranged in square array and = used to condense steam available at atmospheric pressure. If the tubes are = maintained at 370.5 K, find Nusselt's heat transfer coefficient and the rat= e of condensation. (Ans. 12.5 kW/m2K, 16 kg/hr)
  12. Water flows through a vertical tube of= diameter 7.99 cm and length 99.95 cm and maintains its surface temperature= at 325 K. The tube is kept in an environment of saturated steam at atmosph= eric pressure. Find the heat transfer coefficient on the surface assuming l= aminar film condensation. Also find the heat transfer rate to water and rat= e of condensation of steam. (Ans. 4.1 kW/m2K, 51 kW, 21 gs/s)
  13. Saturated steam at 0.16 bar is con= densed in a condenser that consists of a square array of 400 tubes. If the = diameter of each tube is 0.598 cm and length 1 m, find the heat transfer co= efficient and rate of steam condensation. Assume laminar film condensation<= a id=3D"page_246">.

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