ME421Lec3

7/28/2019 ME421Lec3

http://slidepdf.com/reader/full/me421lec3 1/17

ME421

HeatExchanger andSteamGenerator Design

Lecture 3

Heat Transfer Correlations

7/28/2019 ME421Lec3


Task

• We need to find average heat transfer coefficient (h) touse in U calculation in place of hi or ho. (Q = UAF∆ Tlm)

• Average Nusselt number (Nu=hL/k) can be obtained from

an appropriate correlation, with general formNu = f(Re, Pr)

• Chapter 3 of our textbook deals with forced convection

correlations for single phase flow only.• Chapters 6-8 of Incropera & DeWitt are good references

as well.

• Both sides or only one side of a heat exchanger may besingle phase, the correlations here are for the singlephase side(s).

• We will cover phase change in Chapter 7 of our textbook.

7/28/2019 ME421Lec3


Correlations

• Most of the correlations are empirical (obtained fromexperimental results)

• For the sake of generality they are given in terms of non-dimensional groups (Re, Pr, Nu etc.)

• They are only valid for the given conditions or ranges of variables. (Re range, temperatures for propertyevaluation, etc.)

• Check for the ranges and conditions and select the rightcorrelation.

• We need to determine some properties and plug theminto the correlation. These properties are generally eitherevaluated at mean (bulk) fluid temperature or at wall

temperature. Each correlation should also specify this.

7/28/2019 ME421Lec3


Selecting the right correlation

• Calculate Re and check the flow regime (laminar orturbulent)

• Calculate hydrodynamic entrance length (xfd,h or Lhe) tosee whether the flow is hydrodynamically fullydeveloped. (fully developed flow vs. developing)

• Calculate thermal entrance length (xfd,t

or Lte) to

determine whether the flow is thermally fully developed

• A heat transfer problem may involve combinations of hydrodynamic and thermal development; design

correlations must be selected accordingly.

7/28/2019 ME421Lec3


Dimensionless Groups

• Re=UL/ν where U and L are characteristic velocity andlength, and ν is kinematic viscosity.

(ratio of inertial to viscous forces)

• Pr=ν/α where α is thermal diffusivity.

(ratio of momentum and thermal diffusivities)

• Local Nux=hxL/k where hx is the local heat transfercoefficient and k is thermal conductivity of fluid.

(dimensionless temperature gradient)

• Mean Nu=hL/k is average Nu over a given length.

• Peb=PrbRe is Pecletnumber based on bulk fluidproperties.

(dimensionless independent heat transfer parameter)

7/28/2019 ME421Lec3


Characteristic length

• In case of internal flow: L=Dh hydraulic diameter or L=De

equivalent diameter

• Dh=4Ac/Pw where Ac is cross sectional area and Pw iswetted perimeter

• De=4A

c/P

hwhere P

his heated perimeter.

• For hydrodynamical considerations use Dh (Re, etc.)

• For thermal considerations use De (Nu, etc.)

• Circular tube => L=D (diameter of the tube)

7/28/2019 ME421Lec3


Characteristic velocity

• For internal flow, characteristic velocity is the mean fluidvelocity um.

• Related to mean fluid velocity

– Mass flow rate:

– Mass velocity:

Note that usually mass flow rate is known.

m cm u A ρ =&

mG u ρ =

7/28/2019 ME421Lec3


Concentric annular duct• Concentric annular ducts are quite common in heat

exchanger designs (e.g. double pipe heat exchanger)

• Ac=π(Di

2

-do

2

)/4, Pw=π(Di+do), Ph=πdo• Dh=(Di

2-do2)/(Di+do)=Di-do

7/28/2019 ME421Lec3


Variable properties

• Wall temperature Ts or Tw

• Fluid temperature Tb (fluid mean bulk temperature)

• For small changes Ti or To may also be used

• For example there may be a large radial temperaturegradient in circular duct. At what temperature properties are

evaluated matters.

• There may be a need for temperature correction incorrelations.

• Indices cp and vp correspond to constant and variableproperties.

7/28/2019 ME421Lec3


• Some properties are strong functions of temperature.

Convention for correction:– For liquids lump all property variations to µ (dynamic

viscosity). Sometimes variations are lumped to Pr.

– For gases use temperature dependence directly(everything depends on T)

• Fluids:

• Gases:

where n and m depends on the case.

,

n m

b b

cp w cp w

Nu f Nu f

µ µ

µ µ ⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

,

n m

w w

cp b cp b

T TNu f Nu T f T

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

7/28/2019 ME421Lec3


Correlations in the textbook

• Tables 3.1-3.5 are for circular ducts, both laminar andturbulent flow

• Tables 3.6 and 3.7 are correlations specificallydeveloped for variable properties.

• Table 3.8 shows laminar flow correlations for ducts of various cross sections

• For turbulent flow, Tables 3.3, 3.4, 3.6, 3.7 can also beused for noncircular ducts by considering Dh

• Section 3.8 deals with tube bundles (in External Flow

chapter of Incropera & DeWitt, it is a bit better)• Section 3.9 is for helical coils and spirals

• Section 3.10 is for bends (for example U bend in U tube

shell-and-tube heat exchanger)

7/28/2019 ME421Lec3


Tables in the textbook

• Table 3.1: Laminar forced convection correlations insmooth straight circular ducts

• Table 3.2: Exponents n and m for variable physical

properties associated with Eqns. (3.21) and (3.22) forlaminar forced convection through circular ducts

• Table 3.3: Turbulent forced convection correlations

through a circular duct with constant properties• Table 3.4: Turbulent flow isothermal Fanning friction

factor correlations for smooth circular ducts

• Table 3.5: Exponentsn

andm

for variable physicalproperties associated with Eqns. (3.21) and (3.22) forturbulent forced convection through circular ducts

7/28/2019 ME421Lec3


Tables in the textbook (continued)

• Table 3.6: Turbulent forced convection correlations incircular ducts for liquids with variable properties

• Table 3.7: Turbulent forced convection correlations in

circular ducts for gases with variable properties• Table 3.8: Nu and f for hydrodynamically and thermally

developed laminar flow in ducts of various cross sections

7/28/2019 ME421Lec3


Table 3.3

7/28/2019 ME421Lec3


Table 3.4

7/28/2019 ME421Lec3


Which correlation is better?

• Usually newer correlations are more accurate thanolder ones.

• More general ones are less accurate compared to

more specific ones.• Usually textbook correlations are as general as

possible. Search journal databases for morespecialized geometries/materials/fluids etc.

• You can obtain a first approximation using a generalcorrelation then search for more specific ones forbetter accuracy.

• Cited reference search may lead to newercorrelations (Scopus, Web of Science, etc.)• Study the examples in the book.

7/28/2019 ME421Lec3


Problem 3.1

A fluid flows steadily with a velocity of 6 m/s through acommercial iron rectangular duct whose sides are 1 in. by 2 in.and the length of the duct is 6 m. The average temperature of the fluid is 60oC. The fluid completely fills the duct. Calculatethe surface heat transfer coefficient if the fluid is

(a) Water;

(b) Air at atmospheric pressure;(c) Engine oil (ρ=864 kg/m3, cp=2047 J /kgK, υ=0.0839×10-3

m2/s, Pr=1050, k=0.140 W/mK).

ME421Lec3

Documents

Transcript of ME421Lec3