kuliah-6-internal-convection.ppt education

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INTERNAL FORCED INTERNAL FORCED
CONVECTIONCONVECTION
Nazaruddin Sinaga
Laboratorium Efisiensi dan Konservasi Energi

2. Internal FlowInternal Flow
The development of the boundary layer for
laminar flow in a circular tube is
represented in Fig. 17.11. Because of
viscous effects, the uniform velocity profile
at the entrance will gradually change to a
parabolic distribution as the boundary
layer begins to fill the tube in the entrance
region.
2

3. 3

4. Beyond the hydrodynamic entrance length,
the velocity profile no longer changes, and
we speak of the flow as hydrodynamically
fully developed. The extent of the entrance
region, as well as the shape of the velocity
profile, depends upon Reynolds number,
which for internal flow has the form
4

5. where where uu
mm is the mean (average) velocity; D, is the mean (average) velocity; D,
the tube diameter, is the characteristic the tube diameter, is the characteristic
length; length; and is the mass flow rate. In a and is the mass flow rate. In a
fully developed flow, the fully developed flow, the critical Reynolds critical Reynolds
number corresponding number corresponding to the onset of to the onset of
turbulence isturbulence is
ReRe
DD 2300
≈
2300
≈

6. although much larger Reynolds numbers although much larger Reynolds numbers
(Re(Re
DD ≈≈ 10,000) are needed to achieve fully 10,000) are needed to achieve fully
turbulent conditions. For turbulent conditions. For laminar flow laminar flow (Re(Re
DD
== 2300), the 2300), the hydrodynamic entry length hydrodynamic entry length has has
the formthe form
(X(X
LL/D )/D )
lamlam  0.05 Re 0.05 Re
DD

7. while for while for turbulent flow, turbulent flow, the entry length is the entry length is
approximately independent of Reynolds number approximately independent of Reynolds number
and that, as a first approximationand that, as a first approximation
For the purposes of this text, we shall assume For the purposes of this text, we shall assume
fully developed turbulent flow for (x/D) > 10.fully developed turbulent flow for (x/D) > 10.
17.4017.40

8. If fluid enters the tube at If fluid enters the tube at x =x = 0 with a 0 with a
uniform temperature Tuniform temperature T(r,0) (r,0) that is less than that is less than
the the constant constant tube surface temperature, tube surface temperature, TT
ss , ,
convection heat transfer occurs, and a convection heat transfer occurs, and a
thermal boundary layer begins to develop. thermal boundary layer begins to develop.

10. In the In the thermal entrance regionthermal entrance region, , the the
temperature of the central portion of temperature of the central portion of
the flow outside the the flow outside the thermal boundary thermal boundary
layer, layer, δδtt, remains unchanged, but in the , remains unchanged, but in the
boundary layer, the temperature boundary layer, the temperature
increases sharply to that of the tube increases sharply to that of the tube
surface. surface.

11. At the At the thermal entrance lengththermal entrance length, , xx
fdfd,,tt, the , the
thermal boundary layer has filled the thermal boundary layer has filled the
tube, the fluid at the centerline begins tube, the fluid at the centerline begins
to experience heating, and the to experience heating, and the thermally thermally
fully developed flow fully developed flow condition has been condition has been
reached. For reached. For laminar flow, laminar flow, the thermal the thermal
entry length may be expressed asentry length may be expressed as

12. From this relation and by comparison of From this relation and by comparison of
the hydrodynamic and thermal boundary the hydrodynamic and thermal boundary
layers of Fig. 17.11layers of Fig. 17.11a a and 17.11and 17.11bb, it is , it is
evident that we have represented a fluid evident that we have represented a fluid
with a Pr < 1 (gas), as the hydrodynamic with a Pr < 1 (gas), as the hydrodynamic
boundary layer boundary layer has developed more has developed more
slowlyslowly than the thermal boundary layer than the thermal boundary layer
(x(x
fd,hfd,h>x>x
fd,tfd,t). For liquids having Pr > 1, the ). For liquids having Pr > 1, the
inverse situation would occur.inverse situation would occur.
α
υ
Pr

13. For For turbulent flow, turbulent flow, conditions are nearly conditions are nearly
independent of Prandtl number, and to a independent of Prandtl number, and to a
first approximation the first approximation the thermal entrance thermal entrance
length length isis

14. The Mean Temperature. The Mean Temperature.
TThe temperature and velocity profiles at a e temperature and velocity profiles at a
particular particular location in the flow direction location in the flow direction x x
each depend on radius, each depend on radius, rr. The . The mean mean
temperature of the fluid, also referred to temperature of the fluid, also referred to
as the average or bulk temperature, as the average or bulk temperature,
shown on the figure as shown on the figure as TT
mm((xx), is defined in ), is defined in
terms of the energy transported by the terms of the energy transported by the
fluid as it moves past location fluid as it moves past location xx. .

15. For incompressible flow, with constant For incompressible flow, with constant
specific heat specific heat cc
pp, the , the mean temperature mean temperature is is
found fromfound from

16. where where uu
mm is the mean velocity. For a is the mean velocity. For a
circular tube, circular tube, dAdA
cc = 2 = 2ππrdr, and it follows rdr, and it follows
that :that :
17.4417.44

17. The The mean temperaturemean temperature is the fluid reference is the fluid reference
temperature used for determining the convection temperature used for determining the convection
heat rate with Newton’s law of cooling and the heat rate with Newton’s law of cooling and the
overall energy balance.overall energy balance.
Newton’s Law of CoolingNewton’s Law of Cooling. . To determine the To determine the
convective heat flux at the tube surface, Newton’s convective heat flux at the tube surface, Newton’s
law of cooling, also referred to as the law of cooling, also referred to as the convection convection
rate equation, rate equation, is expressed asis expressed as

18. where where h h is the is the local local convection coefficient. convection coefficient.
Depending upon the method of surface Depending upon the method of surface
heating (cooling), heating (cooling), TT
ss can be a constant or can be a constant or
can vary, but the mean temperature will can vary, but the mean temperature will
always change in the flow direction. Still, always change in the flow direction. Still,
the the convection coefficient is a constant for convection coefficient is a constant for
the fully developed conditions the fully developed conditions we examine we examine
next.next.

19. Fully Developed ConditionsFully Developed Conditions. . The temperature The temperature
profile can be conveniently represented as the profile can be conveniently represented as the
dimensionless ratio (dimensionless ratio (Ts -Ts - T T )/()/(Ts -Ts - TmTm). While the ). While the
temperature profile temperature profile TT((rr) continues to change with ) continues to change with
xx, the , the relative shape relative shape of the profile given by this of the profile given by this
temperature ratio temperature ratio is independent of is independent of x x for fully for fully
developed conditions. The requirement for such developed conditions. The requirement for such
a condition is mathematically stated asa condition is mathematically stated as
17.4617.46

20. where Twhere T
ss is the tube surface temperature, is the tube surface temperature,
T T is the local fluid temperature, and is the local fluid temperature, and TT
mm is is
the mean temperature. Since the the mean temperature. Since the
temperature ratio is independent of temperature ratio is independent of xx, the , the
derivative of this ratio with respect to derivative of this ratio with respect to r r
must also be independent of must also be independent of xx. .

22. Evaluating this derivative at the tube surface Evaluating this derivative at the tube surface
(note that (note that Ts Ts and and Tm Tm are constants in sofar as are constants in sofar as
differentiation with respect to differentiation with respect to r r is concerned), is concerned),
we obtainwe obtain

23. Substituting for from Fourier’s law, is
of the form

24. and for and for q”q”
ss from Newton’s law of from Newton’s law of
cooling, we obtaincooling, we obtain

25. Hence, Hence, in the thermally fully developed flow in the thermally fully developed flow
of a fluid with constant properties, the local of a fluid with constant properties, the local
convection coefficient is a constant, convection coefficient is a constant,
independent of xindependent of x. The last equation is not . The last equation is not
satisfied in the entrance region where satisfied in the entrance region where h h
varies with varies with xx..

26. Because the thermal boundary layer Because the thermal boundary layer
thickness is zero at the tube entrance, thickness is zero at the tube entrance,
the coefficient is extremely large near the coefficient is extremely large near x x
== 0, and decreases markedly as the 0, and decreases markedly as the
boundary layer develops, until the boundary layer develops, until the
constant value associated with the fully constant value associated with the fully
developed conditions is reached.developed conditions is reached.

27. Hydrodynamically fully developed:
Thermally fully developed:

29. Energy Balances and Methods Energy Balances and Methods
of Heatingof Heating
Because the flow in a tube is completely Because the flow in a tube is completely
enclosed, an energy balance may be enclosed, an energy balance may be
applied to determine the convection heat applied to determine the convection heat
transfer rate, transfer rate, qq
convconv, in terms of the , in terms of the
difference in temperatures at the tube difference in temperatures at the tube
inlet and outlet. inlet and outlet.

30. From an energy balance applied to a From an energy balance applied to a
differential control volume in the tube, differential control volume in the tube,
we will determine how the mean we will determine how the mean
temperature temperature TT
mm((xx) varies in the flow ) varies in the flow
direction with position along the tube direction with position along the tube
for two for two surface thermal conditions surface thermal conditions
(methods of heating/cooling).(methods of heating/cooling).

31. Overall Tube Energy BalanceOverall Tube Energy Balance. .
Fluid moves at a constant flow rate and Fluid moves at a constant flow rate and
convection heat transfer occurs along the convection heat transfer occurs along the
wall surface. Assuming that fluid kinetic and wall surface. Assuming that fluid kinetic and
potential energy changes are negligible, potential energy changes are negligible,
there is no shaft work, and regarding there is no shaft work, and regarding cc
pp as as
constant, the energy rate balance reduces constant, the energy rate balance reduces
to giveto give

32. where Twhere T
mm denotes the mean fluid denotes the mean fluid
temperature and the subscripts temperature and the subscripts i i and and o o
denote inlet and outlet conditions, denote inlet and outlet conditions,
respectively. It is important to recognize respectively. It is important to recognize
that this that this overall energy balance is a general overall energy balance is a general
expression that applies irrespective of the expression that applies irrespective of the
nature of the surface thermal or tube flow nature of the surface thermal or tube flow
conditionsconditions..

34. Energy Balance on a Differential Control VolumeEnergy Balance on a Differential Control Volume. .
We can apply the same analysis to a differential We can apply the same analysis to a differential
control volume within the tube as shown in Fig. control volume within the tube as shown in Fig.
17.1417.14b b by writing Eq. 17.48 in differential formby writing Eq. 17.48 in differential form
17.4917.49

35. We can express the rate of convection heat We can express the rate of convection heat
transfer to the differential element in terms of transfer to the differential element in terms of
the surface heat flux asthe surface heat flux as
where where P P is the surface perimeter. Combining is the surface perimeter. Combining
Eqs. 17.49 and 17.50, it follows thatEqs. 17.49 and 17.50, it follows that
17.5017.50

36. By rearranging this result, we obtain an By rearranging this result, we obtain an
expression for the axial variation of Texpression for the axial variation of T
m m in terms in terms
of the of the surface heat fluxsurface heat flux
17.5117.51

37. or, using Newton’s law of cooling, Eq. 17.45, with or, using Newton’s law of cooling, Eq. 17.45, with
q”q”
ss=h(T=h(T
ss - T - T
m), m), in terms of the tube wall in terms of the tube wall surface surface
temperaturetemperature
17.5217.52

38. Thermal Condition: Constant Surface Heat Thermal Condition: Constant Surface Heat
Flux, Flux, q”q”
ss
For For constant surface heat flux constant surface heat flux thermal condition thermal condition
(Fig. 17.15), we first note that it is a simple matter (Fig. 17.15), we first note that it is a simple matter
to determine the total heat transfer rate, to determine the total heat transfer rate, qq
convconv. .
Since Since q”q”
ss is independent of is independent of xx, it follows that, it follows that
17.5317.53

39. This expression can be used with the overall This expression can be used with the overall
energy balance, Eq. 17.48, to determine the fluid energy balance, Eq. 17.48, to determine the fluid
temperature change, temperature change, TT
mm,,oo --TT
mm,,ii..
For constant For constant q”q”
ss it also follows that the right-hand it also follows that the right-hand
side of Eq. 17.51 is a constant independent of side of Eq. 17.51 is a constant independent of xx. .
HenceHence

41. Integrating from Integrating from x =x = 0 to some axial position 0 to some axial position xx, ,
we obtain the we obtain the mean temperature distribution, Tmean temperature distribution, T
mm((xx))
17.5417.54

42. Thermal Condition: Constant Surface Thermal Condition: Constant Surface
Temperature, Temperature, TT
ss

43. Results for the total heat transfer rate and the Results for the total heat transfer rate and the
axial distribution of the mean temperature are axial distribution of the mean temperature are
entirely different for the entirely different for the constant surface constant surface
temperature temperature condition (Fig. 17.17). Defining condition (Fig. 17.17). Defining ΔΔT T as as
((Ts -Ts - TmTm), Eq. 17.52 may be expressed as), Eq. 17.52 may be expressed as

44. With constant, separate variables and With constant, separate variables and
integrate from the tube inlet to the outletintegrate from the tube inlet to the outlet

45. From the definition of the average convection From the definition of the average convection
heat transfer coefficient, Eq. 17.8, it follows thatheat transfer coefficient, Eq. 17.8, it follows that
where or simply is the average value of where or simply is the average value of h h
for the entire tube. Alternatively, taking the for the entire tube. Alternatively, taking the
exponent of both sides of the equationexponent of both sides of the equation
17.55a17.55a
17.55b17.55b

46. If we had integrated from If we had integrated from x =x = 0 to some axial 0 to some axial
position, we obtain the position, we obtain the mean temperature mean temperature
distribution, Tdistribution, T
mm((xx))
17.5617.56

47. where is now the average value of where is now the average value of h h from the from the
tube inlet to tube inlet to xx. This result tells us that the . This result tells us that the
temperature difference (temperature difference ( TT
ss-T-T
mm) ) decreases decreases
exponentially exponentially with distance along the tube axis. with distance along the tube axis.
The axial surface and mean temperature The axial surface and mean temperature
distributions are therefore as shown in Fig. 17.18.distributions are therefore as shown in Fig. 17.18.

48. Determination of an expression for the total heat Determination of an expression for the total heat
transfer rate qtransfer rate q
convconv is complicated by the is complicated by the
exponential nature of the temperature decrease. exponential nature of the temperature decrease.
Expressing Eq. 17.48 in the formExpressing Eq. 17.48 in the form
and substituting for from Eq. 17.55a, we and substituting for from Eq. 17.55a, we
obtain the obtain the convection rate equationconvection rate equation
17.5717.57

49. where where AA
ss is the tube surface area (Ais the tube surface area (A
ss = P.L ) and = P.L ) and
ΔΔTT
lmlm is the is the log mean temperature difference log mean temperature difference
((LMTDLMTD))
17.5817.58

50. Convection Correlations for Tubes: Fully Convection Correlations for Tubes: Fully
Developed RegionDeveloped Region
To use many of the foregoing results for To use many of the foregoing results for
internal flow, the convection coefficients internal flow, the convection coefficients
must be known. In this section we must be known. In this section we
present correlations for estimating the present correlations for estimating the
coefficients for coefficients for fully developed laminar fully developed laminar
and and turbulent turbulent flows in flows in circular circular and and
noncircular tubesnoncircular tubes. .

51. Laminar FlowLaminar Flow
The problem of laminar flow (ReThe problem of laminar flow (Re
DD < < 2300) in tubes 2300) in tubes
has been treated theoretically, and the results has been treated theoretically, and the results
can be used to determine the convection can be used to determine the convection
coefficients. For flow in a coefficients. For flow in a circular circular tube tube
characterized by characterized by uniform surface heat flux uniform surface heat flux and and
laminar, fully developed conditions, laminar, fully developed conditions, the the Nusselt Nusselt
number is a constant, number is a constant, independent of Reindependent of Re
DD, Pr, and , Pr, and
axial locationaxial location
17.6117.61

52. When the thermal surface condition is When the thermal surface condition is
characterized by a characterized by a constant surface temperature, constant surface temperature,
the results are of similar form, but with a the results are of similar form, but with a
smaller value for the Nusselt numbersmaller value for the Nusselt number
17.6217.62

54. Guide for Selection of Internal Guide for Selection of Internal
Flow CorrelationsFlow Correlations

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