Computational Fluid Mechanics and Heat Transfer Third Edition_10 docx

7. Forced convection in a variety of configurations The bed was soft enough to suit me But I soon found that there came such a draught of cold air over me from the sill of the window that this plan would never do at all, especially as another current from the rickety door met the one from the window and both together formed a series of small whirlwinds in the immediate vicinity of the spot where I had thought to spend the night. Moby Dick, H. Melville, 1851 7.1 Introduction Consider for a moment the fluid flow pattern within a shell-and-tube heat exchanger, such as that shown in Fig. 3.5. The shell-pass flow moves up and down across the tube bundle from one baffle to the next. The flow around each pipe is determined by the complexities of the one before it, and the direction of the mean flow relative to each pipe can vary. Yet the problem of determining the heat transfer in this situation, however difficult it appears to be, is a task that must be undertaken. The flow within the tubes of the exchanger is somewhat more tractable, but it, too, brings with it several problems that do not arise in the flow of fluids over a flat surface. Heat exchangers thus present a kind of micro- cosm of internal and external forced convection problems. Other such problems arise everywhere that energy is delivered, controlled, utilized, or produced. They arise in the complex flow of water through nuclear heating elements or in the liquid heating tubes of a solar collector—in the flow of a cryogenic liquid coolant in certain digital computers or in the circulation of refrigerant in the spacesuit of a lunar astronaut. We dealt with the simple configuration of flow over a flat surface in 341 342 Forced convection in a variety of configurations §7.2 Chapter 6. This situation has considerable importance in its own right, and it also reveals a number of analytical methods that apply to other configurations. Now we wish to undertake a sequence of progressively harder problems of forced convection heat transfer in more complicated flow configurations. Incompressible forced convection heat transfer problems normally admit an extremely important simplification: the fluid flow problem can be solved without reference to the temperature distribution in the fluid. Thus, we can first find the velocity distribution and then put it in the energy equation as known information and solve for the temperature distribution. Two things can impede this procedure, however: • If the fluid properties (especially µ and ρ) vary significantly with temperature, we cannot predict the velocity without knowing the temperature, and vice versa. The problems of predicting velocity and temperature become intertwined and harder to solve. We en- counter such a situation later in the study of natural convection, where the fluid is driven by thermally induced density changes. • Either the fluid flow solution or the temperature solution can, itself, become prohibitively hard to find. When that happens, we resort to the correlation of experimental data with the help of dimensional analysis. Our aim in this chapter is to present the analysis of a few simple problems and to show the progression toward increasingly empirical so- lutions as the problems become progressively more unwieldy. We begin this undertaking with one of the simplest problems: that of predicting laminar convection in a pipe. 7.2 Heat transfer to and from laminar flows in pipes Not many industrial pipe flows are laminar, but laminar heating and cool- ing does occur in an increasing variety of modern instruments and equip- ment: micro-electro-mechanical systems (MEMS), laser coolant lines, and many compact heat exchangers, for example. As in any forced convection problem, we first describe the flow field. This description will include a number of ideas that apply to turbulent as well as laminar flow. §7.2 Heat transfer to and from laminar flows in pipes 343 Figure 7.1 The development of a laminar velocity profile in a pipe. Development of a laminar flow Figure 7.1 shows the evolution of a laminar velocity profile from the en- trance of a pipe. Throughout the length of the pipe, the mass flow rate, ˙ m (kg/s), is constant, of course, and the average,orbulk, velocity u av is also constant: ˙ m =  A c ρudA c = ρu av A c (7.1) where A c is the cross-sectional area of the pipe. The velocity profile, on the other hand, changes greatly near the inlet to the pipe. A b.l. builds up from the front, generally accelerating the otherwise undisturbed core. The b.l. eventually occupies the entire flow area and defines a velocity pro- file that changes very little thereafter. We call such a flow fully developed. A flow is fully developed from the hydrodynamic standpoint when ∂u ∂x = 0orv = 0 (7.2) at each radial location in the cross section. An attribute of a dynamically fully developed flow is that the streamlines are all parallel to one another. The concept of a fully developed flow, from the thermal standpoint, is a little more complicated. We must first understand the notion of the mixing-cup,orbulk, enthalpy and temperature, ˆ h b and T b . The enthalpy is of interest because we use it in writing the First Law of Thermodynam- ics when calculating the inflow of thermal energy and flow work to open control volumes. The bulk enthalpy is an average enthalpy for the fluid 344 Forced convection in a variety of configurations §7.2 flowing through a cross section of the pipe: ˙ m ˆ h b ≡  A c ρu ˆ hdA c (7.3) If we assume that fluid pressure variations in the pipe are too small to affect the thermodynamic state much (see Sect. 6.3) and if we assume a constant value of c p , then ˆ h = c p (T −T ref ) and ˙ mc p ( T b −T ref ) =  A c ρc p u ( T −T ref ) dA c (7.4) or simply T b =  A c ρc p uT dA c ˙ mc p (7.5) In words, then, T b ≡ rate of flow of enthalpy through a cross section rate of flow of heat capacity through a cross section Thus, if the pipe were broken at any x-station and allowed to discharge into a mixing cup, the enthalpy of the mixed fluid in the cup would equal the average enthalpy of the fluid flowing through the cross section, and the temperature of the fluid in the cup would be T b . This definition of T b is perfectly general and applies to either laminar or turbulent flow. For a circular pipe, with dA c = 2πr dr, eqn. (7.5) becomes T b =  R 0 ρc p uT 2πr dr  R 0 ρc p u 2πr dr (7.6) A fully developed flow, from the thermal standpoint, is one for which the relative shape of the temperature profile does not change with x.We state this mathematically as ∂ ∂x  T w −T T w −T b  = 0 (7.7) where T generally depends on x and r . This means that the profile can be scaled up or down with T w − T b . Of course, a flow must be hydrody- namically developed if it is to be thermally developed. §7.2 Heat transfer to and from laminar flows in pipes 345 Figure 7.2 The thermal development of flows in tubes with a uniform wall heat flux and with a uniform wall temperature (the entrance region). Figures 7.2 and 7.3 show the development of two flows and their sub- sequent behavior. The two flows are subjected to either a uniform wall heat flux or a uniform wall temperature. In Fig. 7.2 we see each flow de- velop until its temperature profile achieves a shape which, except for a linear stretching, it will retain thereafter. If we consider a small length of pipe, dx long with perimeter P, then its surface area is Pdx(e.g., 2πRdx for a circular pipe) and an energy balance on it is 1 dQ = q w Pdx = ˙ md ˆ h b (7.8) = ˙ mc p dT b (7.9) so that dT b dx = q w P ˙ mc p (7.10) 1 Here we make the same approximations as were made in deriving the energy equa- tion in Sect. 6.3. 346 Forced convection in a variety of configurations §7.2 Figure 7.3 The thermal behavior of flows in tubes with a uni- form wall heat flux and with a uniform temperature (the ther- mally developed region). This result is also valid for the bulk temperature in a turbulent flow. In Fig. 7.3 we see the fully developed variation of the temperature profile. If the flow is fully developed, the boundary layers are no longer growing thicker, and we expect that h will become constant. When q w is constant, then T w − T b will be constant in fully developed flow, so that the temperature profile will retain the same shape while the temperature rises at a constant rate at all values of r . Thus, at any radial position, ∂T ∂x = dT b dx = q w P ˙ mc p = constant (7.11) In the uniform wall temperature case, the temperature profile keeps the same shape, but its amplitude decreases with x, as does q w . The lower right-hand corner of Fig. 7.3 has been drawn to conform with this requirement, as expressed in eqn. (7.7). §7.2 Heat transfer to and from laminar flows in pipes 347 The velocity profile in laminar tube flows The Buckingham pi-theorem tells us that if the hydrodynamic entry length, x e , required to establish a fully developed velocity profile depends on u av , µ, ρ, and D in three dimensions (kg, m, and s), then we expect to find two pi-groups: x e D = fn ( Re D ) where Re D ≡ u av D/ν. The matter of entry length is discussed by White [7.1, Chap. 4], who quotes x e D  0.03 Re D (7.12) The constant, 0.03, guarantees that the laminar shear stress on the pipe wall will be within 5% of the value for fully developed flow when x> x e . The number 0.05 can be used, instead, if a deviation of just 1.4% is desired. The thermal entry length, x e t , turns out to be different from x e . We deal with it shortly. The hydrodynamic entry length for a pipe carrying fluid at speeds near the transitional Reynolds number (2100) will extend beyond 100 di- ameters. Since heat transfer in pipes shorter than this is very often im- portant, we will eventually have to deal with the entry region. The velocity profile for a fully developed laminar incompressible pipe flow can be derived from the momentum equation for an axisymmetric flow. It turns out that the b.l. assumptions all happen to be valid for a fully developed pipe flow: • The pressure is constant across any section. • ∂ 2 u  ∂x 2 is exactly zero. • The radial velocity is not just small, but it is zero. • The term ∂u  ∂x is not just small, but it is zero. The boundary layer equation for cylindrically symmetrical flows is quite similar to that for a flat surface, eqn. (6.13): u ∂u ∂x +v ∂u ∂r =− 1 ρ dp dx + ν r ∂ ∂r  r ∂u ∂r  (7.13) 348 Forced convection in a variety of configurations §7.2 For fully developed flows, we go beyond the b.l. assumptions and set v and ∂u/∂x equal to zero as well, so eqn. (7.13) becomes 1 r d dr  r du dr  = 1 µ dp dx We integrate this twice and get u =  1 4µ dp dx  r 2 +C 1 ln r + C 2 The two b.c.’s on u express the no-slip (or zero-velocity) condition at the wall and the fact that u must be symmetrical in r : u(r = R) = 0 and du dr     r =0 = 0 They give C 1 = 0 and C 2 = (−dp/dx)R 2 /4µ,so u = R 2 4µ  − dp dx   1 −  r R  2  (7.14) This is the familiar Hagen-Poiseuille 2 parabolic velocity profile. We can identify the lead constant (−dp/dx)R 2  4µ as the maximum centerline velocity, u max . In accordance with the conservation of mass (see Prob- lem 7.1), 2u av = u max ,so u u av = 2  1 −  r R  2  (7.15) Thermal behavior of a flow with a uniform heat flux at the wall The b.l. energy equation for a fully developed laminar incompressible flow, eqn. (6.40), takes the following simple form in a pipe flow where the radial velocity is equal to zero: u ∂T ∂x = α 1 r ∂ ∂r  r ∂T ∂r  (7.16) 2 The German scientist G. Hagen showed experimentally how u varied with r , dp/dx, µ, and R, in 1839. J. Poiseuille (pronounced Pwa-zói or, more precisely, Pwä-z´ e ¯e) did the same thing, almost simultaneously (1840), in France. Poiseuille was a physician interested in blood flow, and we find today that if medical students know nothing else about fluid flow, they know “Poiseuille’s law.” §7.2 Heat transfer to and from laminar flows in pipes 349 For a fully developed flow with q w =constant, T w and T b increase linearly with x. In particular, by integrating eqn. (7.10), we find T b (x) − T b in =  x 0 q w P ˙ mc p dx = q w Px ˙ mc p (7.17) Then, from eqns. (7.11) and (7.1), we get ∂T ∂x = dT b dx = q w P ˙ mc p = q w (2πR) ρc p u av (πR 2 ) = 2q w α u av Rk Using this result and eqn. (7.15) in eqn. (7.16), we obtain 4  1 −  r R  2  q w Rk = 1 r d dr  r dT dr  (7.18) This ordinary d.e. in r can be integrated twice to obtain T = 4q w Rk  r 2 4 − r 4 16R 2  +C 1 ln r + C 2 (7.19) The first b.c. on this equation is the symmetry condition, ∂T /∂r = 0 at r = 0, and it gives C 1 = 0. The second b.c. is the definition of the mixing-cup temperature, eqn. (7.6). Substituting eqn. (7.19) with C 1 = 0 into eqn. (7.6) and carrying out the indicated integrations, we get C 2 = T b − 7 24 q w R k so T −T b = q w R k   r R  2 − 1 4  r R  4 − 7 24  (7.20) and at r = R, eqn. (7.20) gives T w −T b = 11 24 q w R k = 11 48 q w D k (7.21) so the local Nu D for fully developed flow, based on h(x) = q w  [T w (x) − T b (x)],is Nu D ≡ q w D (T w −T b )k = 48 11 = 4.364 (7.22) [...]... eqn (7.12) and its context.] For other situations, the coefficient changes With qw = constant, it is 0.043 at a 5% error level; when the velocity and temperature profiles develop simultaneously, the coefficient ranges between about 0.028 and 0.053 depending upon the Prandtl number and the wall boundary condition [7.4, 7.5] §7.2 Heat transfer to and from laminar flows in pipes Figure 7.4 Local and average... suggested C1 = 4.8 and C2 = 0.025.) For uniform wall heat flux, many more data are available, Heat transfer surface viewed as a heat exchanger §7.4 and Lyon [7.21] recommends the following equation, shown in Fig 7.8: NuD = 7 + 0.025 Pe0.8 D (7.54) In both these equations, properties should be evaluated at the average of the inlet and outlet bulk temperatures and the pipe flow should have L/D > 60 and PeD > 100... predict the heat transfer coefficient on the outer tube to within ±10%, irrespective of the heating configuration The heat transfer coefficient on the inner surface, however, is sensitive to both the diameter ratio and the heating configuration For that surface, the hydraulic diameter approach is not very accurate, Do ; other methods have been developed to accurately especially if Di predict heat transfer in... resistance around its perimeter This might be true for a copper duct heated at a fixed rate in watts per meter of duct length Laminar entry length formulæ for noncircular ducts are also given by Shah and London [7.5] 7.6 Heat transfer during cross flow over cylinders Fluid flow pattern It will help us to understand the complexity of heat transfer from bodies in a cross flow if we first look in detail at the... resistance is in the sublayer on the wall, this ratio determines the heat transfer coefficient to within about ±20% across a broad range of duct shapes In fully-developed laminar flow, where the thermal resistance extends into the core of the duct, the heat transfer coefficient depends on the details of the duct shape, and Dh alone cannot define the heat transfer coefficient Nevertheless, the hydraulic diameter provides... considered Heat travels first from the air at T∞ through the outside heat transfer coefficient to the duct wall, and then through the inside heat transfer coefficient to the flowing air — effectively through two resistances in series from the fixed temperature T∞ to the rising temperature Tb We have seen in Section 2.4 that an overall heat transfer coefficient may be used to describe such series resistances... reasonably accurate They gave maximum errors of +25% and −40% in the range 0.67 Pr < 100 and usually were considerably more accurate than this However, subsequent research has provided far more data, and better theoretical and physical understanding of how to represent them accurately Modern formulations During the 1950s and 1960s, B S Petukhov and his co-workers at the Moscow Institute for High Temperature... Turbulent pipe flow §7.3 357 Figure 7.5 Heat transfer to air flowing in a 1 in I.D., 60 in long pipe (after Kreith [7.9]) in the laminar range The rate of rise of NuD with ReD becomes very great in the transitional range, which lies between ReD = 2100 and about 5000 5000, the flow is turbulent and it turns out in this case Above ReD that NuD Re0.8 D The Reynolds analogy and heat transfer A form of the Reynolds... pipe is a heat exchanger whose overall heat transfer coefficient, U (between the wall and the bulk), is just h Thus, if we wish to know how much pipe surface area is needed to raise the bulk temperature from Tbin to Tbout , we can calculate it as follows: ˙ Q = (mcp)b Tbout − Tbin = hA(LMTD) or A= ˙ (mcp)b Tbout − Tbin h ln Tbout − Tw Tbin − Tw Tbout − Tw − Tbin − Tw (7.56) By the same token, heat transfer. .. than heat flux, is changed Only liquid metals give fairly long thermal entrance lengths, and, for these fluids, xet depends on both Re and Pr in a complicated way Since liquid metals have very high thermal conductivities, the heat transfer rate is also more strongly affected by the temperature distribution in the center of the pipe We discusss liquid metals in more detail at the end of this section When heat . between about 0.028 and 0.053 depending upon the Prandtl number and the wall boundary con- dition [7.4, 7.5]. §7.2 Heat transfer to and from laminar flows in pipes 353 Figure 7.4 Local and average Nusselt. α = 1.541 × 10 −7 , and ν = 0.556 10 −6 m 2 /s. Therefore, Re D = (0.001 m)(0.2m/s)/0.556× 10 −6 m 2 /s = 360, and the flow is laminar. Then, noting that T is greatest at the wall and setting. of 100 µm, resulting in heat transfer coefficients in the range of 10 4 W/m 2 K for water [7.2]. If, instead, liquid sodium (k ≈ 80 W/m·K) is used as the working fluid, the laminar flow heat transfer