ME 144: Heat Transfer Convection Relations for External Flows. J. M. Meyers

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1 ME 144: Heat Transfer Convection Relations for External Flows

2 Empirical Correlations Generally, convection correlations for external flows are determined experimentally using controlled lab conditions Correlations for the average Nusselt number are sought of the form: Here material properties are evaluated at the film temperature: Nu Re Pr 2 2

3 Flat Plate: Laminar Boundary Layer Flow Velocity Boundary Layer Mass: -Momentum: Energy: kinematic viscosity + Recall our assumptions/simplifications: 0 Related to variations in $ due to changes in geometry think of Bernoulli-like effects Note: # " 0for a flat plate! Neglect viscous dissipation which is an adequate assumption for low speed incompressible flows 3

4 Flat Plate: Laminar Boundary Layer Flow Velocity Boundary Layer Mass: -Momentum: Energy: kinematic viscosity + These are the governing equations for an isothermal flat plate ( ) with a uniform flow$, Solution Approach: Since there is no intrinsic length scale in the problem we seek the solution in terms of a similarity variable. 4

5 Flat Plate: Laminar Boundary Layer Flow Velocity Boundary Layer The solution procedure for this problem was first obtained by Blasius(1908) This was done by first introducing the stream function as defined by (such that continuity is still satisfied): & & ( & & " + 0 The momentum equation in terms of& then becomes: & ( & "( & " & (( & ((( + Base on the dimensional analysis we performed in last class we saw that: )~ Re " This allowed Blasius to reason that the similarity variable' would be given by: ' Re " $ 5

6 Flat Plate: Laminar Boundary Layer Flow Velocity Boundary Layer Further, he reasoned that the stream function could be defined in terms of'with proper scaling: &, ~ $ ), ' $, ' Substitution of the stream function into the momentum equation leads to the nonlinear ODE: 2, --- +,, -- 0 This equation is subject to boundary conditions: & ",0 & (,0 0 At the wall lim ( & (, $ At the BL edge 6

7 Flat Plate: Laminar Boundary Layer Flow Velocity Boundary Layer The numerical solution of the Blasius equation yields, by defining the height at which: $ 0.99 ) 5 Re " And the wall friction coefficient: $ Re " 7

8 Flat Plate: Laminar Boundary Layer Flow Knowledge of the velocity field then permits the direct calculation of the thermal field Defining again our dimensionless temperature: Assumingthat;;(')onecanderiveanotherODEforthetemperaturefield: Which is subject to: ; '0 0 ; ' 1 Evidently this solution depends on Pr Thermal Boundary Layer ; ; -- + Pr 2,;- 0 8

9 Flat Plate: Laminar Boundary Layer Flow Thermal Boundary Layer Numerical results for? 0.6 (typical for gases) shows that: ; B 0.332Pr G/I CDE In terms of the convection coefficient(or Nusselt number): h 1 K L (DE hk $ ; (0) Local solution (Eq 7.23): Average solution (Eq. 7.30): Nu " h " K 0.33Re " G/ Pr G/I Nu " h O " K 0.664Re " G/ Pr G/I Pr 0.6 Pr 0.6 It also follows from the ODE solution that: ) ) M Pr G/I 9

10 Flat Plate: Turbulent Boundary Layer The results for turbulent boundary layers are essentially obtained from experiments Thekeyfindingsfortheplateare: Re PQRM ~5 10 T ) 0.37Re " VG/T Recall Nu " Re " W/T Pr G/I 10

11 Convection for a Cylinder in a Cross Flow X Y X Z 8[ /2 11

12 Convection for a Cylinder in a Cross Flow 12

13 Convection for a Cylinder in a Cross Flow As engineers we are generally interested in average conditions Hilpertcorrelation (Eq. 7.52): Values of and _can be found in Table 7.2 Nu X Re X Pr G/I Pr 0.7 for various Re X all properties are evaluated at the film temperature + 2 Zukauskas correlation (Eq. 7.53): Nu X Re X Pr Pr Pr 0.7 Pr Re X 10 ] G/W All properties evaluated at except for Pr which is evaluated at Values of C and m can be found in Table 7.4 for various Re X If Pr 10,^0.37; if Pr 10,^0.36 Churchill and Bernstein correlation (Eq. 7.54): Nu X Re X G/ Pr G/I /Pr /I G/W 1+ Re X 282,000 T/a W/T all properties are evaluated at the film temperature: Re X Pr

14 Convection for a Cylinder in a Cross Flow Example 7.4 Experiments have been conducted on a metallic cylinder (b12.7 mm,c94 mm). The cylinder is heated internally by an electrical heater and is subjected to a cross flow of air in a low-speed wind tunnel. Under a specific set of operating conditions for which the upstream air velocity and temperature were maintainedconstant($ 10m/sand 26.2C). The heater power dissipation was measured to be 46 W, while the average cylinder surface temperature was determined to be C. It is estimated that 15% of the power dissipation is lost through the cumulative effect of surface radiation and conduction through the end pieces. 14

15 Convection for a Cylinder in a Cross Flow Example 7.4 Find 1. Determine the convection heat transfer coefficient from the experimental observations. 2. Compare the experimental result with the convection coefficient computed from an appropriate correlation. Assume 1. Steady-state, incompressible flow conditions. 2. Uniform cylinder surface temperature. Care must be taken when using these (or any empirically-based) correlations! 15

16 Convection for a Cylinder in a Cross Flow Example 7.4 Determine flow properties and non-dimensional parameters from Table A.4 for air: d 300 d: V]_ f K VI g _ d Pr0.707 Re X $ b d 350 d: V]_ f K30 10 VI g _ d Pr0.700 Re X $ b d 400 d: Pr

17 Convection for a Cylinder in a Cross Flow Example 7.4: Analysis Newton s law of cooling The convection heat transfer coefficient may be determined from the data by using: ho i Z As 15% of the power is lost through the end pieces, thus: i0.85 And the area is: Zjbc ho W j m m C ho102 W m K 17

18 Convection for a Cylinder in a Cross Flow Example 7.4 Hilpert correlation (Eq. 7.52): Nu X Re X Pr G/I Pr 0.7 All properties evaluated at the film temperature: Hence, from Table 7.2, 0.193and _0.618 Pr0.700 Re X $ b 6071 Nu X E.]Ga G/I 37.3 honu X K b W/m K m ho 88 W m K 18

19 Convection for a Cylinder in a Cross Flow Example 7.4 Zukauskas correlation (Eq. 7.53): Nu X Re X Pr Pr Pr G/W 0.7 Pr Re X 10 ] Pr0.707 Re X $ b 7992 Pr Evaluated at Evaluated at f From Table 7.4, 0.26and _0.6and since Pr<10, ^0.37 Nu X E.] ho Nu X K b E.Io W/m K m G/W 50.5 ho105 W m K 19

20 Convection for a Cylinder in a Cross Flow Example 7.4 Churchill and Bernstein correlation (Eq. 7.54): Nu X Re X G/ Pr G/I /Pr /I G/W 1+ Re X 282,000 All properties evaluated at the film temperature: T/a W/T Re X Pr 0.2 Pr0.700 Re X $ b 6071 No table look-up for this correlation Nu X G/ 0.70 G/I /0.70 /I G/W ,000 T/a W/T 40.6 honu X K b W/m K m ho96.0 W m K 20

21 Convection for a Cylinder in a Cross Flow Example 7.4 Comments Uncertainties associated with measuring the air velocity, estimating the heat loss from cylinder ends, and averaging the cylinder surface temperature, which varies axially and circumferentially, render the experimental result accurate to no better than 15%. Accordingly, calculations based on each of the three correlations are within the experimental uncertainty of the measured result. Recognize the importance of using the proper temperature when evaluating fluid properties. 21

22 Convection for a Sphere in a Cross Flow Boundary layer effects associated with flow over a sphere are much like those for the circular cylinder, with transition and separation playing prominent roles. 1) Laminar separation 2) Turbulent BL transition 3) Turbulent separation

23 Convection for a Sphere in a Cross Flow Numerous heat transfer correlations have been proposed The Whitaker correlation is an expression of the form: Nu X Re X G/ +0.06Re X /I Pr E.W p p G/W (Eq. 7.52) All properties evaluated at 0.71 Pr Re X W 1.0 p # p

24 Convection for Tube Banks in a Cross Flow Heat transfer to or from a bank (or bundle) of tubes in cross flow is relevant to numerous industrialapplications,suchassteamgenerationinaboileroraircoolinginthecoilofanair conditioner. Typically, one fluid moves over the tubes, while a second fluid at a different temperature passes through the tubes. In this section we are specifically interested in the convection heat transfer associated with cross flow over the tubes. 24

25 Convection for Tube Banks in a Cross Flow The tube rows of a bank can be either aligned or staggered Flow around the tubes in the first row of a tube bank is similar to that for a single (isolated) cylinder in cross flow For downstream rows, flow conditions depend strongly on the tube bank arrangement ho Nu X K b Aligned Staggered 25

26 Convection for Tube Banks in a Cross Flow Typically, we wish to know the average heat transfer coefficient for the entire tube bank. Zukauskas has proposed a correlation of the form: Nu X G Re X,q" Pr E.I] Pr Pr r Re X,q" 2 10 ] 0.7 Pr 500 G/W (Eq. 7.58) All properties evaluated at the average of bank inlet ( ) and outlet temperatures ( E ) r represents number of tube bank rows Empirically derived constants G and _can be found from Table 7.5 ho Nu X K b 26

27 Convection for Tube Banks in a Cross Flow Nu X G Re X,q" Pr E.I] Pr Pr honu X K b G/W Staggering will increase heat flux BUT at a penalty of flow losses (headloss!!!) 27

28 Convection for Tube Banks in a Cross Flow The previous correlation is generally good for tube banks of 20 or more rows If less than 20 rows are present then a simple correction may be applied: Nu X B (st ue) Nu X B (st ve) (Eq. 7.59) The coefficient can be found in Table

29 Convection for Tube Banks in a Cross Flow Again, staggering will increase heat flux BUT at a penalty of flow losses there is generally as much interest in the pressure drop associated with flow across a tube bank as in the overall heat transfer rate The power required to move the fluid across the bank is often a major operating expense and is directly proportional to the pressure drop, which may be expressed as: xr y 8[ q" 2, (Eq. 7.65) Here, xis the head loss of the bank, yis the correction factor, and, is the friction factor Both yand, can be found in Tables nd 7.15 for aligned and staggered banks, respectively 29

30 Convection for Tube Banks in a Cross Flow xr y 8[ q" 2, 30

31 Convection for Tube Banks in a Cross Flow xr y 8[ q" 2, 31

32 Convection for Tube Banks in a Cross Flow For flows through tube banks, the total heat transfer is often sufficient to change the temperature of the free stream low by a significant amount. For overall convection calculations, the `log mean temperature difference' z is used to characterize the mean temperature difference between the surface and free stream: 32

33 Convection for Tube Banks in a Cross Flow Example (Problem 7.90) A preheater involves the use of condensing steam at 100 on the inside of a bank of tubes to heat air that enters at 1 atm and 25. The air moves at 5 m/s in cross flow over the tubes. Each tube is 1 m long and has an outside diameter of 10 mm. The bank consists of 196tubesinasquare,alignedarrayforwhich } 15 mm. Whatisthetotalrateofheattransfertotheair? What is the pressure drop associated with the airflow? 33

34 Convection for Impinging Jets 34

35 Convection for Impinging Jets 35

36 Convection for Impinging Jets 36

37 Convection for Impinging Jets 37

38 Convection for Impinging Jets Example (Problem 7.100) A circular transistor of 10-mm diameter is cooled by impingement of an air jet exiting a 2-mm-diameter round nozzle with a velocity of 20 m/s and a temperature of 15C. The jet exit and the exposed surface of the transistor are separated by a distance of10mm. Ifthetransistoriswellinsulatedatallbutitsexposed surface and the surface temperature is not to exceed 85C, what is the transistor s maximum allowable operating power? 38

39 Convection for Impinging Jets Example (Problem 7.100) A circular transistor of 10-mm diameter is cooled by impingement of an air jet exiting a 2-mm-diameter round nozzle with a velocity of 20 m/s and a temperature of 15C. The jet exit and the exposed surface of the transistor are separated by a distance of10mm. Ifthetransistoriswellinsulatedatallbutitsexposed surface and the surface temperature is not to exceed 85C, what is the transistor s maximum allowable operating power? 39

40 Approaching These External Flow Problems Application of a convection correlation for any flow situation are facilitated by following a few simple rules. 1. Become immediately cognizant of the flow geometry. Flat plate? Cylinder? Sphere? Tube bank? Impinging jets? 2. Specify the appropriate reference temperature and evaluate the pertinent fluid properties at that temperature. May need to refer to material properties table(table A.4) 3. Calculate the Reynolds number. Boundary layer conditions are strongly influenced by this parameter determine whether the flow is laminar or turbulent. 4. Decide whether a local or surface average coefficient is required. 5. Select the appropriate correlation. 40

41 References Bergman, Lavine, Incropera, and Dewitt, Fundamentals of Heat and Mass Transfer, 7 th Ed., Wiley, 2011 D. E. Hitt, External Convection Relations, ME 144 Lecture Notes, University of Vermont, Spring 2008 Chapman, Heat Transfer, 3 rd Ed., MacMillan, 1974 Y. A. Çengeland A. J. Ghajar, Heat and Mass Transfer, 5 th Ed., Wiley,

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