Heat and mass transfer are quantitative in nature, i.e ...

1 Chapter 1: Introduction1.4 Fundamentals of Momentum, Heat and Mass Transfer

Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Heat and mass transfer are quantitative in nature, i.e., The amount of energy that can be transferred by a given heat

pipe design in order to determine its suitability for a particular application.

The restriction of the dimensions of a certain flat-plate heat exchanger in order to maintain a stable flow in a system.

Two highly-recommended analytical tools can organize and filter information so the design process can be efficiently and effectively executed. Dimensional analysis Scale analysis Applicable to both single- and multiphase systems.

1.4.4 Dimensional Analysis



Dimensional analysis is used to interpolate the experimental laboratory results (prototype models) to full scale system.

Two criteria must be fulfilled to perform such an objective: Dimensional similarity, in which all dimensions of the prototype

to full scale system must be in the same ratio. Dynamic similarity, in which relevant dimensionless groups are

the same between prototype model and full scale system. The convective heat transfer coefficient is a function of

the thermal properties of the fluid, the geometric configuration, flow velocities, and driving forces.




Consider forced convection in a circular tube with a length L and a diameter D. The flow is assumed to be incompressible and natural convection is negligible compared with forced convection. The heat transfer coefficient can be expressed as

(1.161)where k is the thermal conductivity of the fluid, is viscosity, is specific heat, is density, U is velocity, and is the temperature difference between the fluid and tube wall.

Equation (1.161) can also be rewritten as

(1.162)


( , , , , , , , )ph h k c U T D Lµ ρ= ∆

( , , , , , , , , ) 0pF h k c U T D Lµ ρ ∆ =

µpc ρ T∆



It can be seen from eq. (1.162) that eight dimensional parameters are required to describe the convection problem.

The theory of dimensional analysis shows that it is possible to use fewer dimensionless variables to describe the convection problem.

Buckingham’s Π theorem (Buckingham, 1914) # dimensionless variables required to describe problem = #

dimensional variables - # primary dimensions required to describe problem

Dimensionless variables can be identified using Buckingham’s Π theorem and are formed from products of powers of certain original dimensional variables.

Any of such dimensionless groups can be written as(1.163)


( )a b c d e f g h iph k c U T D Lµ ρΠ = ∆



Substituting dimensions (units) of all variables into eq. (1.163) yields

(1.164) For to be dimensionless, the components of each primary

dimension must be summed to zero, i.e.,

(1.165)

which gives a set of five equations with nine unknowns.


2 3

2 3M Q

a b c d e fg h i

c d e a b c e f h i c d f a b d g a b d

Q Q M Qt M L T L LL T LT Lt MT L tL t T− + − − − − + + + − + − − − − + + +

Π = =

02 3 0

00

0

c d ea b c e f h ic d fa b d ga b d

− + =− − − − + + + =− + − =− − − + =

+ + =

Π



According to linear algebra, the number of distinctive solutions of eq.(1.165) is four (9-5=4), which coincides with the Π theorem.

In order to obtain the four distinctive solutions, we have free choices on four of the nine components.

If we select , the solutions of eq. (1.164) become , which give us the first nondimensional variable

(1.166)

which is the Reynolds number ( ).


0 and 1a d i f= = = =0, 1, 1, 0,b c e g= = − = = and 1h =

1UDρµ

Π =

1 ReΠ =



Similarly, we can set and get the solutions of eq. (1.165) as . The second nondimensional variable becomes

(1.167)which is the Nusselt number ( ).

Following a similar procedure, we can get two other dimensionless variables:

(1.168)

(1.169)

which are the Prandtl number ( ) and the aspect ratio of the tube.


1, c 0 and 0a f i= = = =1, 0, 0, 0,b d e g= − = = =

2hDk

Π =

2 NuΠ =

3pck

µΠ =

4LD

Π =

3 Pr /ν αΠ = =

and 1h =



Equation (1.162) can be rewritten as (1.170)

or (1.171)

If the flow and heat transfer in the tube are fully developed, no change of flow and heat transfer in the axial direction, eq. (1.171) can be simplified further:

(1.172) The number of nondimensional variables is three, as

opposed to the nine dimensional variables in eq. (1.161).


1 2 3 4( , , , ) 0F Π Π Π Π =

Nu (Re,Pr, / )f L D=

Nu (Re,Pr)f=



Example 1.3:For boiling and condensation processes, obtain the appropriate

dimensionless parameters using a similar procedure to the one above.

Solution:The heat transfer coefficient for boiling or condensation depends on the

properties of the fluid, a characteristic length L, the temperature difference , buoyancy force , latent heat of vaporization , and surface tension, :

(1.173)


T∆ ( )v gρ ρ−lvhl σ

[ , , , , , , ( ) , , ]p v vh h k c L T g hµ ρ ρ ρ σ= ∆ −l l



1.4.4 Dimensional Analysis There are 10 dimensional variables in eq. (1.173) and there are five

primary dimensions in the boiling and condensation problem. Therefore, it will be necessary to use (10 – 5) = 5 dimensionless variables to describe the liquid-vapor phase change process, i.e.:

(1.174)where the Grashof number is defined as:

(1.175) The new dimensionless parameters introduced in eq. (1.174) are

the Jakob number, Ja, and the Bond number, Bo, which are defined as

(1.176)

(1.177)

( )Nu Gr,Ja,Pr,Bof=

3

2

( )Gr vg Lρ ρ ρµ

−= l

Ja p

v

c Th

∆=

l2( )Bo v gLρ ρ

σ−

= l



Table 1.8 provides a summary of the definitions, physical interpretations, and areas of significance of the important dimensionless numbers for transport phenomena in multiphase systems.

Table 1.9 summarizes the existing correlations in literature for various heat transfer modes for both single-phase and two-phase systems in different geometric configurations.




Table 1.8 Summary of dimensionless numbers for transport phenomena




Table 1.8 Summary of dimensionless numbers for transport phenomena cont’d




Table 1.9 Correlations for convective heat transfer for various modes and geometries




1.4.4 Dimensional AnalysisTable 1.9 Correlations for convective heat transfer for various modes and geometries cont’d




Example 1.4:Dry air at 35 °C and velocity of 1.5 m/s flows over a flat plate of 2 m × 2 mcovered with thin layer of water (see Fig. 1.13) . The plate surface temperature is kept at 30 °C. Calculate the heat transfer due to convection and evaporation from the plate to air. The radiation heat transfer from the liquid surface and the heat conduction in the liquid can be neglected.

Water

2m

2m

Air

qconvqeva

Figure 1.13 Dry air flows over a flat plate covered with water

o

1.5m/s

35 C

uT

∞

∞

=

=o30CwT =




Solution: The properties of the air at average temperature of liquid water and air (32.5°C) are

The diffusivity of water vapor in the air is (see Appendix B)

Reynolds number is

According to Table 1.9, the Nusselt number is

3 6 21.1431kg/m , 16.44 10 m /s,ρ ν −= = ×326.7 10 W/m-K,k −= × Pr=0.706.

2

2.072-10 5 2

H O air 1.87 10 2.64 10 m /sTDp

−= × = ×

56

1.5 2Re 1.825 1016.44 10L

u Lν∞

−×= = = ××

0.5 1/3Nu 0.664Re Pr 252.57L= =




The heat transfer coefficient is

The heat transfer from the plate to the air due to convection is

The Schmidt number is

The Sherwood number is

3 2/ 252.57 26.7 10 / 2 3.37W/m -Kh Nuk L −= = × × =

( ) 3.37 2 2 5 67.44Ws wq hA T T∞= − = − × × × = −

6

516.44 10Sc 0.6232.64 10D

ν −

−×= = =

×

0.33 0.33Sc 0.623Sh Nu 252.57 242.26Pr 0.706

= = × =




The mass transfer coefficient is

The concentration of the water vapor at the surface of flat plate is

The concentration of the water vapor at the incoming air is since the air is dry.

The latent heat for evaporation is taken as that at 30°C, i.e., .

The heat transfer due to evaporation is therefore

3, , @30 0.03420kg/mv w v sat Cρ ρ= =o

, 0vρ ∞ =

2430.2 kJ/kgvh =l

, ,( )eva m v w v vq h A hρ ρ ∞= − l

512Sh / 242.26 2.64 10 / 2 0.0032m/smh D L −= = × × =

0.0032 2 2 (0.03420 0) 2430.2 1.06W= × × × − × =

Heat and mass transfer are quantitative in nature, i.e ...

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