ENGR Heat Transfer II

Transcription

1 ENGR Heat Transfer II Internal Flows 1 Introduction In this chapter we examine laminar and turbulent internal forced convection flows. We will consider two fundamental geometries, the plane channel and the circular tube. We will also give some consideration to non-circular ducts. We begin by reviewing laminar flows and considering several important types of flow. 2 Internal Forced Convection If we focus our attention to the tube, we may recall from fluid mechanics that there exists in an internal flow, two distinct flow regions: the developing region and the fully developed region. In short tubes developing flow prevails over most of the flow length, where as in a fully developed flow the flow field remains unchanged over most of the duct length. In laminar flow we may define the extent of the developing region through the definition of the entrance length: L h 0.05DRe D (1) Eq. (1) is based on finding the point along the duct axis where u 0.99u max. From the above equation, we see that for laminar flow, the entrance length increases with increasing Reynolds number, which in turn is affected by duct size, flow speed, and viscosity. Thus a small diameter duct containing a very viscous fluid will very nearly be fully developed, whereas a larger diameter duct with low viscosity fluid will have considerable flow development occurring. In general, we may define a flow according to the value of the entrance length in relation to its actual length: L >> L h, Fully Developed Flow L << L h, Developing Flow It should be clear that a duct of L h L, is not simply one of fully developed flow, as most of the duct will be undergoing flow development and pressure drop will be larger owing to 1

2 the acceleration of the fluid in the region outside the boundary layers. However, that is not to say that the above definition of L h, is not useful. As we shall see it will become important when considering thermal boundary layers. 2.1 Mean Velocity and Bulk Temperature Two important parameters in internal forced convection are the mean flow velocity u and the bulk or mixed mean fluid temperature T m (z). The mass flow rate is defined as: where ṁ = ρua c (2) u = 1 A c A c uda c (3) while the bulk or mixed mean temperature is defined as: T m (z) = A c ρuc p T da c = 1 ua c A c ut da c (4) Since temperature is increasing as the fluid flows downstream, we must use alternate approaches to formulate the Nusselt number and heat transfer rate. 2.2 Thermal Boundary Layers The thermal boundary grows in relation to the hydrodynamic boundary layer. In practice, the thermal boundary layer scales with the hydrodynamic boundary layer through the Prandtl number according to: L t 0.05DRe D P r (5) Eq. (5) gives the approximate length of thermal boundary development which is defined on the basis of Nu D,z 1.05Nu D,fd, i.e. where the local Nusselt number is five percent greater than the fully developed flow value. There are several fundamental problems in laminar internal flow that can be considered. Unfortunately, the course text does not address all of these adequately, but we shall. The following problems arise as a result of considering the thermal entrance length in proportion to the hydrodynamic entrance length: L >> L h, L >> L t, i.e. thermally and hydrodynamically fully developed flow. This rarely occurs in practice, but it affords many theoretical solutions. 2

3 L >> L h, L << L t, i.e. hydrodynamically fully developed, but thermally developing flow, sometimes called the thermal entrance problem. This type of flow is characteristic of high Prandtl number fluids P r, e.g. oils. L << L h, L << L t, i.e. hydrodynamically and thermally developing flow, sometimes called the combined entrance problem L << L h, L >> L t, i.e. hydrodynamically developing flow and thermally fully developed. This type of flow occurs with low Prandtl number fluids P r 0, e.g. liquid metals. Under the special case of P r 0 we experience a type of flow called plug flow or slug flow, for which there is no flow field, but merely a uniform velocity at every point in the duct cross-section. In such cases, the heat transfer coefficient is higher, than more traditional fully developed flows. Plug flows also occur inside porous ducts. In general, heat transfer is always higher in developing flows, since the thermal resistance of the boundary layer is lower. In the thermal entrance region, heat is being transferred from a warmer wall temperature (in the case of heating) to the lowest temperature which is the inlet fluid temperature. However, when the thermal boundary layers merge, there ceases to be a constant sink temperature and the bulk fluid temperature rises quickly. The local heat transfer rate is: We also often define a Nusselt number as: Nu D = Q z = h z A(T w T m (z)) (6) Q z /AD k f (T w T m (z)) = h zd (7) k f One must now be careful, as mean Nusselt numbers are difficult to obtain from the above equation. As we shall see shortly, we use an integrated mean temperature difference from inlet to exit to calculate the heat transfer rate. 2.3 Energy Balances in Internal Flow We desire to relate the heat transfer rate, the duct geometry, and the flow for specific boundary conditions, i.e. isothermal or isoflux. We begin by examining a control volume for a section of duct which yields the following expressions for heat transfer: dq conv = (T m + dt m T m ) = dt m (8) 3

4 or Q conv = (T m,o T m,i ) (9) We may also write: dq conv = q w P dz = h z (T w T m )P dz (10) which may be combined with the result given earlier to yield: dt m dz = q wp = h zp (T w T m ) (11) Now, there exists two possible solution paths, one for isothermal walls, T w = Constant and one for isoflux walls, q w = Constant Constant Wall Flux, q w In the case of constant wall flux, we have: which gives: dt m dz = q wp = Constant (12) or a linear variation in bulk temperature. T m (z) = T m,i + q wp z (13) With constant heat flux, we are most often interested in how the wall temperature varies. We see from above result that bulk temperature varies linearly. This also implies that wall temperature must vary linearly in fully developed flow, since the heat transfer coefficient is constant. In the entry region where a boundary layer exists, the wall to fluid temperature difference is not constant. In problems involving isoflux boundaries, we are only concerned with local wall conditions, and hence no mean heat transfer coefficient exists. We may define a local heat transfer coefficient (or Nusselt number), but in general it is of little use Constant Wall Temperature, T w In the case of constant wall temperature, we have: dt m dz = h zp (T w T m ) (14) 4

5 which may be written as: or dt m dz = h zp (T m T w ) (15) d( T ) T = h zp dz (16) Integrating from inlet to exit gives: which gives: To d( T ) T i T L h z P = dz (17) 0 ( ) To ln T i = P L 1 L L 0 h z dz = hp L (18) We may now substitute for the T s and take the exponential of each side: ( T w T m,o = exp P hl ) T w T m,i (19) The above result illustrates the exponential behaviour of the bulk fluid for constant wall temperature. It may also be written as: T w T (z) T w T m,i to get the local variation in bulk temperature. ( = exp P hz ) We are interested in relating the wall temperature, the inlet and exit temperatures, and the heat transfer in one single expression. To do this we write: (20) Q conv = ((T w T m,i ) (T w T m,o )) = ( T i T o ) (21) or Combining with Eq. (18) gives: = Q conv ( T i T o ) (22) or ( ) To ln T i = hp L Q conv ( T i T o ) (23) 5

6 Q conv = ha ( T o T i ) ( ) = ha T LMT D (24) To ln T i where A = P L and T LMT D = ( T o T i ) ( ) (25) To ln T i which is the Log Mean Temperature Difference. The above expression requires knowledge of the exit temperature, which is only known if the heat transfer rate is known. An alternate equation can be derived which eliminates the outlet temperature. Using Eq. (19) we may write: or by substituting Eq. (9), we get ( (T w T m,i + T m,i T m,o ) = (T w T m,i ) exp P hl ) T w T m,i + Q conv ( = (T w T m,i ) exp P hl ) (26) (27) Finally, after re-arranging: [ ( Q conv = (T w T m,i ) 1 exp P hl )] (28) In some problems involving a single fluid, it is more convenient to consider thermal resistance. We can define a thermal resistance from the above equation as: R t = T w T m,i Q conv = 1 [1 exp ( P hl )] (29) 2.4 Film Temperature and Bulk Temperature In external flows we used the concept of the film temperature to evaluate the fluid properties. Recall that the film temperature was the average of the wall temperature and the stream temperature. We have already seen that the bulk fluid (or stream) temperature varies with axial position. Therefore, we cannot define a constant film temperature. We can however, define a mean bulk temperature which is the average of the inlet and exit temperatures: T bulk = T m,i + T m,o 2 (30) 6

7 In many problems we do not initially know the exit temperature, but we most often know the wall temperature and the inlet temperature. In these instances, we can estimate the mean bulk temperature by assuming that the exit temperature is close to the wall temperature. If required, we can resolve the problem after the exit temperature is obtained from the heat transfer rate and the enthalpy balance, i.e. Eq. (9), if the mean bulk temperature changes appreciably. 3 Some Fundamental Results in Laminar Flow In this section we review many useful models for predicting the heat transfer coefficient for the isothermal wall condition. We will begin with the simplest and most conservative estimates of the heat transfer coefficient, thermally fully developed flow and proceed to the most general case, the combined thermal entrance problem, for which we expect the highest heat transfer coefficients. Further, in this section laminar flow is assumed, that is Re D < Hydrodynamic and Thermally Fully Developed Flows In a thermally fully developed flow, the temperature profile does not change its relative shape and hence the heat transfer rate at the wall remains constant. This generally occurs in long tubes. The results for a tube and channel lend easily themselves to analysis, but the solution procedures are still cumbersome, thus we will only summarize the solutions. In a circular duct, the solutions for the heat transfer coefficient for constant wall temperature is: Nu D = In the case of the constant flux condition, we obtain: (Q/A)D = hd = 3.66 (31) k f T LMT D k f T m (z) = T w (z) 11 q w D (32) 48 k f We may use Newton s law of cooling to express the above equation as a Nusselt number which gives: Nu D = q w D k f (T w (z) T m (z)) = hd = 4.36 (33) k f 7

8 However, caution is advised when using the Nusselt form of the equation, as we must now know the correct mean temperature difference, which in this case is the constant wall to fluid temperature difference. It is important that the student understand the differences between Eqs. (31) and Eqs.(33). The above equation in practice is of little use as we usually know the heat flux and desire the local wall temperature, and thus Eq. (32) is more direct. In general, most engineering thermal systems are not designed to allow fully developed flow to occur, as it is not efficient use of surface area. In practice, under the constraints of constant pressure drop (or pumping power) or fixed volume, the optimal flow length that is achieved in practice is of the order of the thermal entrance length. 8

9 In a plane channels of spacing b, the solutions for the heat transfer coefficient for constant wall temperature is: In the case of the constant flux condition, we obtain: Nu 2b = (Q/A)(2b) k f T LMT D = h(2b) k f = 7.54 (34) Nu 2b = q w (2b) k f (T w (z) T m (z)) = h(2b) = 8.23 (35) k f For other shapes we use the hydraulic diameter D h = 4A c /P in the definition of the Nusselt number. Results are also constants for fully developed flows, see the Table given above. 3.2 Combined Hydrodynamic and Thermally Developing Flows For simultaneous development of boundary layers in the entrance region of tubes and channels expressions for 0.1 < P r < can be found in handbooks. These models are general and will predict the fully developed flow limit if the duct is long. These are recommended for use in most analyses which fall under laminar flow Circular Duct For laminar developing flow in circular tubes with constant wall temperature, one may use the following model due to Stephan. This correlation is valid for all values of the dimensionless duct length L = L/(DRe D ) and for 0.1 < P r < : Nu D = Nu(P r ) tanh(2.432p r 1/6 (L ) 1/6 ) (36) where Nu(P r ) = tanh(2.264(l ) 1/ (L ) 2/3 ) L tanh(l ) (37) and L = L DRe D P r A simpler model due to Hausen given in the text for the P r case is: 9

10 0.0668/L Nu D = (38) /(L ) 2/3 The above results are shown graphically in the figure below Plane Channel For laminar developing flow in plane channels with constant wall temperature, one may use the following model, also due to Stephan, which is valid for all L = L/(2bRe 2b ) and for 0.1 < P r < 1000: Nu 2b = A simpler model for the special case when P r is: Nu 2b = (L ) P r 0.17 (L ) 0.64 (39) 0.03/L /(L ) 2/3 (40) The Sieder-Tate Model Sieder and Tate developed a very simple model for supposed combined entry length problems. Their model is based on experimental data and takes the form: ( ) 1/3 ( ) µ Nu D = 1.86 (41) L µ w 10

11 for L = L DRe D P r 0.05 ( ) µ The above model is valid for laminar flows, 0.6 < P r < 16, 700 and < µ w < The figure above shows this model is only approximately valid in the Prandtl number range, as it represents a best fit. There has been much historical debate in the many heat transfer texts regarding this model and its range of application. The concensus is that it is not strictly applicable to the combined entrance region problem, despite what your text says. Clearly from the above figure, it is at best, only an approximation. Further, in light of the fact that it was developed in the late thirties, it is wiser to use newer models that are strictly applicable to the combined entrance problem such as thos given abaove. 4 Turbulent Flows in Tubes, Channels and Ducts Turbulent flows are much easier to predict, owing to shorter entrance lengths, so short, as to not being important in most applications. Recall from fluids, that the turbulent entrance length is: L h 4.4DRe 1/6 D (42) We can also reasonable expect that the thermal entrance length will scale with Prandtl 11

12 number, such that L t L h P r. For a tube of similar diameter, we see that the ratio of turbulent to laminar entrance length is: L h,tur L h,lam 88DRe 5/6 D (43) Thus at Re D = 4000, we see that it is only 8.7 percent of the laminar entrance length. Further, as the the Reynolds number increases, this reduces to 4.1 percent at Re D = 10, 000, and 0.6 percent at Re D = 100, 000. The simplest model for predicting the heat transfer coefficient in a tube is the Dittus-Boelter equation: Nu D = 0.023Re 4/5 D P rn (44) where n = 0.4 for heating of the fluid and n = 0.3 for cooling of the fluid. The well known Colburn equation uses n = 1/3 and does not consider the heating/cooling effect. The above equation is valid for Re D > 10, 000, 0.7 < P r < 160 and for tubes of length greater than ten diameters, i.e. L/D > 10. More refined models include the Petukov and Gnielinski correlations. Both models require a friction factor to be calculated as a secondary parameter. The Petukov model takes the form: where or Nu D = (f/8)re D P r (f/8) 1/2 (P r 2/3 1) (45) f = (1.82 log 10 Re D 1.64) 2 (46) f = (0.79 ln Re D 1.64) 2 (47) The above model is valid for 0.5 < P r < 2000 and 10 4 < Re D < To obtain agreement at lower Reynolds numbers, Gnielinski modified the above correlation and proposes: Nu D = (f/8)(re D 1000)P r (f/8) 1/2 (P r 2/3 1) (48) The above model is valid for 0.5 < P r < 2000 and 2300 < Re D <

13 The Dittus-Boelter, Petukov, and Gnielinski models all apply to the case of isothermal or isoflux conditions, as there is little difference in boundary condition in turbulent flows. Further, the thermal properties for each should be evaluated at the mean bulk temperature. Finally, for non-circular ducts, all of the equations are valid provided the hydraulic diameter is used, D D h = 4A c /P as a length scale. 13