Principals of heat transfer Frank Kreith, Raj M.

4.5 Dimensionless Boundary Layer Equations And Similarity Parameters 241 another geometrically similar system provided the similarity parameters have the same value in both. For example, if the Reynolds number is the same, the dimen- sionless velocity distributions for air, water, and glycerin flowing over a flat plate will be the same at given values of x*. Inspection of Eq. (4.9a) shows that v* is related to u*, y*, and x*: v* ϭ f1(u*, y*, x*) (4.10) and that from Eq. (4.9b) the solution for u*, accordingly, can be expressed in the form 0p* (4.11) u* = f2 a x*, y*, ReL, 0x* b The pressure distribution over the surface of a body is determined by its shape. From the y-momentum equation it can be shown that Ѩp*>Ѩy* ϭ 0 and p* is only a function of x*. Hence, dp*>dx* can be obtained independently. It represents the influence of the shape on the velocity distribution in the free stream just outside the boundary layer. 4.5.1 Friction Coefficient From Eq. (4.2), the surface shear stress ␶s is given by ts = 0u ` = mUq 0u* ` (4.12) m L 0y* y* = 0 y=0 0y Defining the local frictional drag coefficient Cf as Cfx = ts (4.13) rUq2 /2 and substituting Eq. (4.12) for ␶s gives Cfx = 2 0u* ` (4.14) ReL 0y* y* = 0 From Eq. (4.11), it is apparent that the dimensionless velocity gradient Ѩu*>Ѩy* at the surface (y* = 0) depends only on x*, ReL, and dp*>dx*. But since dp*>dx* is entirely determined by the geometric shape of a body, Eq. (4.14) reduces to the form Cfx = 2 f3(x*, ReL) (4.15) ReL for bodies of similar shape. The above relation implies that for flow over bodies of similar shape the local frictional drag coefficient is related to x* and ReL by a uni- versal function that is independent of the fluid or the free-stream velocity. Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

242 Chapter 4 Analysis of Convection Heat Transfer The average frictional drag over a body tq can be determined by integrating the local shear stress ␶ over the surface of the body. Hence, tq must be independent of x*, and the average friction coefficient Cqf depends only on the value of the Reynolds number for flow over geometrically similar bodies: Cqf = tq = 2 (4.16) rUq2 /2 ReL f4(ReL) EXAMPLE 4.2 For flow over a slightly curved surface, the local shear stress is given by the relation ts(x) = 0.3 a rm 0.5 xb U1q.5 From this dimensional equation, obtain nondimensional relations for the local and average friction coefficients. SOLUTION From Eq. (4.13), the local friction coefficient is Cfx = ts(x) = 0.6a rm 0.5 U1q.5 x rU2q 1 rUq2 b 2 = 0.6a m 0.5 0.6 0.6 b= Re0x.5 = (ReLx*)0.5 rUq x Integrating the local value and dividing by the area per unit width (L ϫ 1) gives the average shear tq: tq = 1 L a rm b 0.5 dx = 0.6 rm b 0.5 L x aL L0 0.3 U1q.5 Uq1.5 and the average friction coefficient is therefore Cqf = tq 1.2 rU2q/2 = ReL0.5 4.5.2 Nusselt Number In convection heat transfer, the key unknown is the heat transfer coefficient. From Eq. (4.1), we obtain the following equation in terms of the dimensionless parameters: hc = - kf c (Tq - Ts) d 0T* ` = kf 0T* ` (4.17) L (Ts - Tq) 0y* + 0y* y*=0 y*=0 L Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

4.6 Evaluation of Convection Heat Transfer Coefficients 243 Inspection of this equation suggests that the appropriate dimensionless form of the heat transfer coefficient is the so-called Nusselt number, Nu, defined by Nu = hcL K 0T* ` (4.18) kf 0y* y*=0 From Eqs. (4.9a) and (4.9c), it is apparent that for a prescribed geometry the local Nusselt number depends only on x*, ReL, and Pr: Nu ϭ f5(x*, ReL, Pr) (4.19) Once this functional relation is known, either from an analysis or from experiments with a particular fluid, it can be used to obtain the value of Nu for other fluids and for any values of Uϱ and L. Moreover, from the local value of Nu, we can first oh-cbtaanind the local value of hc and then an average value of the heat transfer coefficient an average Nusselt number NuL. Since the average heat transfer coefficient is obtained by integrating over the heat transfer surface of a body, it is independent of x*, and the average Nusselt number is a function of only ReL and Pr: NuL = hqcL = f6(ReL, Pr) (4.20) kf 4.6 Evaluation of Convection Heat Transfer Coefficients Five general methods are available for the evaluation of convection heat transfer coefficients: 1. Dimensional analysis combined with experiments 2. Exact mathematical solutions of the boundary layer equations 3. Approximate analyses of the boundary layer equations by integral methods 4. The analogy between heat and momentum transfer 5. Numerical analysis, or modeling with computational fluid dynamics (CFD) methods All five of these techniques have contributed to our understanding of convec- tion heat transfer. Yet no single method can solve all the problems, because each one has limitations that restrict its scope of application. Dimensional analysis is mathematically simple and has found a wide range of application [5, 6]. The chief limitation of this method is that the results obtained are incomplete and quite useless without experimental data. Dimensional analysis con- tributes little to our understanding of the transfer process but facilitates the interpre- tation and extends the range of experimental data by correlating them in terms of dimensionless groups. There are two different methods for determining dimensionless groups suitable for correlating experimental data. The first of these methods, discussed in the fol- lowing section, requires only listing of the variables pertinent to a phenomenon. This technique is simple to use, but if a pertinent variable is omitted, erroneous results ensue. In the second method, the dimensionless groups and similarity conditions are Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

244 Chapter 4 Analysis of Convection Heat Transfer deduced from the differential equations describing the phenomenon. This method is preferable when the phenomenon can be described mathematically, but the solution of the resulting equations is often too involved to be practical. This technique was presented in Section 4.5. Exact mathematical analyses require simultaneous solution of the equations describing the fluid motion and the transfer of energy in the moving fluid [7]. The method presupposes that the physical mechanisms are sufficiently well understood to be described in mathematical language. This preliminary requirement limits the scope of exact solutions because complete mathematical equations describing the fluid flow and the heat transfer mechanisms can be written only for laminar flow. Even for laminar flow, the equations are quite complicated, but solutions have been obtained for a number of simple systems such as flow over a flat plate, an airfoil, or a circular cylinder [7]. Exact solutions are important because the assumptions made in the course of the analysis can be specified accurately and their validity can be checked by experiment. They also serve as a basis of comparison and as a check on simpler approximate methods. Furthermore, the development of high-speed computers has increased the range of problems amenable to mathematical solution, and results of computations for different systems are continually being published in the literature. Approximate analysis of the boundary layer avoids the detailed mathematical description of the flow in the boundary layer. Instead, a plausible but simple equa- tion is used to describe the velocity and temperature distributions in the boundary layer. The problem is then analyzed on a macroscopic basis by applying the equa- tion of motion and the energy equation to the aggregate of the fluid particles con- tained within the boundary layer. This method is relatively simple; moreover, it yields solutions to problems that cannot be treated by an exact mathematical analy- sis. In instances where other solutions are available, they agree within engineering accuracy with the solutions obtained by this approximate method. The technique is not limited to laminar flow but also can be applied to turbulent flow. The analogy between heat and momentum transfer is a useful tool for analyzing turbulent transfer processes. Our knowledge of turbulent-exchange mechanisms is insufficient for us to write mathematical equations describing the temperature distri- bution directly, but the transfer mechanism can be described in terms of a simplified model. According to one such model that has been widely accepted, a mixing motion in a direction perpendicular to the mean flow accounts for the transfer of momentum as well as energy. The mixing motion can be described on a statistical basis by a method similar to that used to picture the motion of gas molecules in the kinetic the- ory. There is by no means general agreement that this model corresponds to condi- tions actually existing in nature, but for practical purposes, its use can be justified by the fact that experimental results are substantially in agreement with analytical pre- dictions based on the hypothetical model. Numerical methods can solve in an approximate form the exact equations of motion [8, 9]. The approximation results from the need to express the field vari- ables (temperature, velocity, and pressure) at discrete points in time and space rather than continuously. However, the solution can be made sufficiently accurate if care is taken in discretizing the exact equations. One of the most important advantages of numerical methods is that once the solution procedure has been Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

4.7 Dimensional Analysis 245 programmed, solutions for different boundary conditions, property variables, and so on can be easily computed. Generally, numerical methods can handle complex boundary conditions easily. Numerical methods for solving convection problems are discussed in [9] and are an extension of the methods presented in Chapter 3 for conduction problems. 4.7 Dimensional Analysis Dimensional analysis differs from other approaches in that it does not yield equa- tions that can be solved. Instead, it combines several variables into dimensionless groups, such as the Nusselt number, which facilitate the interpretation and extend the range of application of experimental data. In practice, convection heat transfer coef- ficients are generally calculated from empirical equations obtained by correlating experimental data with the aid of dimensional analysis. The most serious limitation of dimensional analysis is that it gives no infor- mation about the nature of a phenomenon. In fact, to apply dimensional analysis it is necessary to know beforehand what variables influence the phenomenon, and the success or failure of the method depends on the proper selection of these vari- ables. It is therefore important to have at least a preliminary theory or a thorough physical understanding of a phenomenon before a dimensional analysis can be performed. However, once the pertinent variables are known, dimensional analy- sis can be applied to most problems by a routine procedure that is outlined below.* 4.7.1 Primary Dimensions and Dimensional Formulas The first step is to select a system of primary dimensions. The choice of the primary dimensions is arbitrary, but the dimensional formulas of all pertinent variables must be expressible in terms of them. In the SI system, the primary dimensions of length L, time t, temperature T, and mass M are used. The dimensional formula of a physical quantity follows from definitions or physical laws. For instance, the dimensional formula for the length of a bar is [L] by definition.† The average velocity of a fluid particle is equal to a distance divided by the time interval taken to traverse it. The dimensional formula of velocity is there- fore [L>t] or [LtϪ1] (i.e., a distance or length divided by a time). The units of veloc- ity could be expressed in meters per second, feet per second, or miles per hour, since they all are a length divided by a time. The dimensional formulas and the symbols of physical quantities occurring frequently in heat transfer problems are given in Table 4.1. *The algebraic theory of dimensional analysis will not be developed here. For a rigorous and compre- hensive treatment of the mathematical background, Chapters 3 and 4 of Langhaar [5] are recommended. †Square brackets indicate that the quantity has the dimensional formula stated within the brackets. Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

246 Chapter 4 Analysis of Convection Heat Transfer TABLE 4.1 Important heat and mass transfer physical quantities and their dimensions Quantity Symbol Dimensions in MLtT System Length L, x L Time t Mass M t Force F Temperature T M Heat Q ML/t2 Velocity u, v, Uϱ Acceleration a, g T Work W ML2/t2 Pressure p Density ␳ L/t Internal energy e L/t2 Enthalpy i ML2/t2 Specific heat c M/t2L Absolute viscosity ␮ M/L3 Kinematic viscosity ␯ ϭ ␮/␳ L2/t2 Thermal conductivity k L2/t2 Thermal diffusivity ␣ L2/t2T Thermal resistance R Coefficient of expansion ␤ M/Lt Surface tension ␴ L2/t Shear stress ␶ ML/t3T Heat transfer coefficient h L2/t Mass flow rate m# Tt3/ML2 1/T M/t2 M/Lt2 M/t3T M/t 4.7.2 Buckingham ␲ Theorem To determine the number of independent dimensionless groups required to obtain a relation describing a physical phenomenon, the Buckingham ␲ theorem may be used.‡ According to this rule, the required number of independent dimensionless groups that can be formed by combining the physical variables pertinent to a prob- lem is equal to the total number of these physical quantities n (e.g., density, viscos- ity, heat transfer coefficient) minus the number of primary dimensions m required to express the dimensional formulas of the n physical quantities. If we call these groups ␲1, ␲2, and so forth, the equation expressing the relationship among the variables has a solution of the form F(␲1, ␲2, ␲3, . . .) ϭ 0 (4.21) ‡A more rigorous rule proposed by Van Driest [26] shows that the ␲ theorem holds as long as the set of simultaneous equations formed by equating the exponents of each primary dimension to zero is linearly independent. If one equation in the set is a linear combination of one or more of the other equations (i.e., if the equations are linearly dependent), the number of dimensionless groups is equal to the total number of variables n minus the number of independent equations. Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.