Abstract
Being the second mode that traditionally is encountered in the study of heat transfer, with the analysis of convection we learn on the heat transfer process that is executed by a fluid stream. The stream purposefully acts as a heat carrier between two moving media in relative motion coming in contact. It is clear, then, that this topic is focussed on the effects of solid/fluid heat interaction. So, while studying convection, we exploit the notions learned with the fluid mechanics that focussed on the effects of solid/fluid dynamic interaction. As we recall that convective heat flux is ruled by the description of a heat transfer coefficient, we describe briefly some basic devices where convection is realized following the microscopic balance leading to join the distributions of the flow velocity vector and the temperature scalar. It is clear that, at this point, we should distinguish between the temperature of the solid \(T_\mathrm {s}\), the one studied in Chap. 2, and the temperature of the fluid \(T_\mathrm {f}\) that will be the subject of this chapter, but whichever temperature we consider, we really face with the same scalar variable. However, for the sake of clarity sometime we indulge in differentiating the nomenclature, but unless we will be engaged in two-phase (solid/fluid) modeling, we will hold that the subject at stake now is the fluid temperature \(T_\mathrm {f}\). Along the same lines that we exploit so far, then we derive and integrate the governing differential equations in various cases, and the concept of the thermal boundary layer is presented that carries analogies with the former fluid boundary layer. Finally, a numerical solution of the governing equations is completed, for the distribution of temperature and velocity, following the course cast so far.
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31 March 2019
In the original version of the book, the belated corrections from author.
Notes
- 1.
In this case, the distribution of temperature depends on the previously determined distribution of velocity or may be even a source of it. See Sect. 4.1.2.4 (p. 150). Of course, the first problem is solved once the distribution of the temperature is ascertained, as we will see in the course of the chapter.
- 2.
The coefficient \(\overline{h}\) was already introduced in Sect. 2.2.1 (p. 21) and employed in the development of conduction calculations, starting from Eq. (2.32). Here, to prepare the forthcoming discussion on the analogy between the heat transfer coefficient now at stake and a mass transfer coefficient, the subscript “T” is purposely added to refer to the former, while the subscript “M” will be used to refer to the later.
- 3.
As proposed, in a less explicit form, by the aforementioned scientist i. newton, in the early eighteenth century.
- 4.
Also called the primary or working fluid, and the secondary or service fluid.
- 5.
In this cases, the nomenclature refers to the secondary fluid being at constant temperature, when undergoes phase-change or corresponds to a heat sink. Thus, the secondary fluid direction is immaterial, and the primary fluid direction is irrelevant.
- 6.
Typically, several hundreds of square meters per cube meter.
- 7.
As an example, we would want to heat some liquid water from \(T_{\mathrm {C,}0}=30.0~^{\circ }\mathrm {C}\) up to \(T_{\mathrm {C,}L}=50.0~^{\circ }\mathrm {C}\) with some hot combustion exhaust, the available stream being at \(T_{\mathrm {H,}0}=180.0~^{\circ }\mathrm {C}\). The working fluid has \(\dot{m}_\mathrm {C}=5.0~\mathrm {kg/s}\) and \(c_\mathrm {C}=4.20~\mathrm {kJ/kgK}\), while the service fluid \(\dot{m}_\mathrm {H}=10.0~\mathrm {kg/s}\) and \(c_{p\mathrm {H}}=1.10~\mathrm {kJ/kgK}\). On the water side, Eq. (4.9) gives
$$\begin{aligned} \dot{Q}=\left( \dot{m}c\right) _\mathrm {C}\left( T_{\mathrm {C,}L}-T_\mathrm {C,0}\right) =10.0\times 4.20\times 20.0=840~\mathrm {kW} \end{aligned}$$Reapplying Eq. (4.9) on the exhaust side:
$$\begin{aligned} T_{\mathrm {H,}L}=T_\mathrm {H,0}-\frac{\dot{Q}}{\left( \dot{m}c_p\right) _\mathrm {H}}=180.0-\frac{840}{5.0\times 1.10}=27~^{\circ }\mathrm {C} \end{aligned}$$which represents a violation of the Second Law of Thermodynamics, the temperature of the working fluid being always higher (regardless the HEX type and stream arrangement).
- 8.
The algebraic sign of W is consistent with the sign convention adopted in engineering Thermodynamics textbooks.
- 9.
Since it is generally \(\mathrm {D}x^2=2x\mathrm {D}x\), last term of Eq. (4.35) can be written as
$$\begin{aligned} \frac{\rho _\mathrm {f}}{2}\frac{\mathrm {D}}{\mathrm {D}\theta }\left( u^2+v^2\right) =\rho _\mathrm {f}\left( u\frac{\mathrm {D}u}{\mathrm {D}\theta }+v\frac{\mathrm {D}v}{\mathrm {D}\theta }\right) \end{aligned}$$The two material derivatives on the right-hand side will include the temporal and spatial variations of the velocity components, so that Eqs. (3.63) can be employed to write:
$$\begin{aligned} \frac{\rho _\mathrm {f}}{2}\frac{\mathrm {D}}{\mathrm {D}\theta }\left( u^2+v^2\right) =u\frac{\partial \sigma _{xx}}{\partial x}+u\frac{\partial \tau _{yx}}{\partial y}\pm uS_x+v\frac{\partial \sigma _{yy}}{\partial y}+v\frac{\partial \tau _{xy}}{\partial x}\pm vS_y \end{aligned}$$With these terms devised, Eq. (4.35) is simplified:
$$\begin{aligned} \lambda _\mathrm {f}\nabla ^2T+\left( \sigma _{xx}\frac{\partial u}{\partial x}+\tau _{yx}\frac{\partial u}{\partial y}+\sigma _{yy}\frac{\partial v}{\partial y}+\tau _{xy}\frac{\partial v}{\partial x}\right) =\pm \dot{e}'''+\rho _\mathrm {f}\frac{\mathrm {D}e}{\mathrm {D}\theta } \end{aligned}$$Now, we apply Stokes’ viscosity law Eqs. (3.64) to resolve the normal and tangential stresses and come up with pressure and velocity components:
$$\begin{aligned} \lambda _\mathrm {f}\nabla ^2T+\left[ -p\frac{\partial u}{\partial x}+2\mu \left( \frac{\partial u}{\partial x}\right) ^2-p\frac{\partial v}{\partial y}+2\mu \left( \frac{\partial v}{\partial y}\right) ^2+\mu \left( \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right) ^2\right] =\pm \dot{e}'''+\rho _\mathrm {f}\frac{\mathrm {D}e}{\mathrm {D}\theta } \end{aligned}$$or
$$\begin{aligned} \lambda _\mathrm {f}\nabla ^2T+\left[ 2\mu \left( \frac{\partial u}{\partial x}\right) ^2+2\mu \left( \frac{\partial v}{\partial y}\right) ^2+\mu \left( \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right) ^2\right] =\pm \dot{e}'''+\rho _\mathrm {f}\frac{\mathrm {D}e}{\mathrm {D}\theta } \end{aligned}$$having exploited the continuity Eq. (3.59).
- 10.
This function \(\phi \) should be invoked when dealing with extremely viscous flows (thus with very strong velocity gradients) such as in lubrication or oil piping problems. Expressions other than for Cartesian 2-D space are available [1].
- 11.
As proposed by German physician and physicist j.r. von mayer , in mid-nineteenth century.
- 12.
It is worth to note the complete analogy with the definition of momentum eddy diffusivity \(\varepsilon _\nu \) Eq. (3.92).
- 13.
The Pr and Pe numbers were proposed in the early nineteenth century by the aforementioned scientists l. prandtl and j.c.e. péclet, respectively, while the Ec number was were proposed by Czech-German origin American engineer e.r.g. eckert, in mid-twentieth century.
- 14.
Recall Note 10 in Chap. 3, on kinematic viscosity and thermal diffusivity definitions.
- 15.
The Pe number is of particular interest here: with the Pe number increasing, the heat conduction portion decreases and the convective heat portion grows.
- 16.
This “average” adjective is purposely stressed-out here as this dimensionless Nu number depends on the definition of average heat transfer coefficient Eq. (4.6a).
- 17.
As proposed by German engineer e.k.w. nusselt in the early twentieth century.
- 18.
As proposed by the aforementioned scientist v.j. boussinesq at beginning twentieth century.
- 19.
As proposed by German engineer f. grashof, in mid-nineteenth century.
- 20.
See the discussion by Bejan, A.: Convection Heat Transfer. pp. 188–192. John Wiley & Sons, New York (1995). The topic is resumed later in Note 14 of Chap. 5.
- 21.
As proposed by British physicist Lord Rayleigh, j.w. strutt, in the late nineteenth century.
- 22.
Based on the definitions of average and local convective coefficients, in analogy with Eq. (4.76) it is \(\overline{h}_\mathrm {T}=\int _0^Lh_\mathrm {T}(x)\mathrm {d}x\).
- 23.
All of these cases pertain to the fluid side, only, that will be referred to later in Sect. 4.5.4 (p. 190) as “segregated solutions,” so they imply a conventionally positive Nusselt number, the direction of the heat flux to be determined based on general consideration for each configuration at stake.
- 24.
As proposed by German mathematician e. pohlhausen in the early twentieth century.
- 25.
As proposed by Churchill-Bernstein [1].
- 26.
As proposed by Whitaker [1].
- 27.
As proposed by Gnielinski. Colburn’s, Dittus-Boelter’s (accounting for either heating or cooling) and Sieder-Tate’s (accounting for temperature variation influence on properties) formulas have also been used with success [1].
- 28.
As proposed by Churchill-Chu [4].
- 29.
As proposed by Churchill-Chu [1].
References
Bejan, A.: Heat Transfer. Wiley, New York (1993)
Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena. Wiley, New York (2002)
Bergman, T.L., Incropera, F.P., Lavine, A.: Fundamentals of Heat and Mass Transfer. Wiley, New York (2011)
Jiji, L.M.: Heat Transfer Essentials: A Textbook. Begell House, New York (1993)
Mills, A.F.: Heat Transfer. Richard D. Irwin, Boston (1992)
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Ruocco, G. (2018). Heat Transfer by Convection. In: Introduction to Transport Phenomena Modeling. Springer, Cham. https://doi.org/10.1007/978-3-319-66822-2_4
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DOI: https://doi.org/10.1007/978-3-319-66822-2_4
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