ENGR 7901 - Heat Transfer II Convective Heat Transfer 1 Introduction In this portion of the course we will examine convection heat transfer principles. We are now interested in how to predict the value of the heat transfer coefficient h, which up until now has always been prescribed. We begin with Newton s law of cooling: Q = hat w T ) 1) which is used to relate the heat transfer Q, to the surface temperature and the fluid temperature by means of the coefficient h. Pedagogically, Newton s law of cooling is the starting point. However, we shall see it is not strictly required. It has found usefulness in the experimental determination of the heat transfer coefficient, but is still not required as the starting point for convective heat transfer analysis. It is useful, in cases where resistive networks are used and simple film resistances are required. We tend to work with dimensionless heat transfer coefficients, such that Eq. 1) can be written as: QL k f AT w T ) = hl = Nu L 2) k f The group on the right hand side is referred to as the Nusselt number. It can be interpreted as either the dimensionless heat transfer coefficient or the dimensionless heat transfer rate, if we consider the left hand side. We will now concentrate on what influences the value of h and how to predict its value for a given geometry, set of flow conditions, and fluid. Convection heat transfer occurs when heat is conducted into a moving fluid. Fluid motion occurs in one of two ways: Forced Flow, i.e. pressure driven flow, p Natural Flow, i.e. buoyancy induced flow, ρ Since flow of a fluid is an integral part of convection heat transfer, we must consider, in addition to a modified energy equation, the equation of continuity or conservation of mass), and the momentum transport equations or conservation of momentum). 1
We will consider several types of convective heat transfer problems. These include: External Forced Convection Internal Forced Convection External Natural Convection Internal Natural Convection Mixed Convection Combined Forced and Natural Convection) Phase Change, i.e. Boiling and Condensation Convection heat transfer is influenced by many factors. These include: Geometry, i.e. Internal Shape or External Shape Type of Flow, i.e. Laminar or Turbulent Surface Boundary Condition, i.e. Isothermal or Isoflux Position in Flow Stream Fluid Properties Single Phase or Phase Change For a given geometry, fluid, and flow type, the parameter which has a significant effect on the value of the heat transfer coefficient is position. At any point along the surface of a body, the heat transfer coefficient varies. This leads to two types of heat transfer coefficients, local and average. The local heat transfer coefficient h x as it is sometimes denoted, is used to calculate local heat transfer rates at points on the body. While the average heat transfer coefficient, h, as it is usually denoted, is used to calculate the total heat transfer rate. The average heat transfer coefficient for a body of constant and uniform temperature is given by the area averaged integral: h = 1 hda s 3) A s A s for an external flow, or the flow length averaged integral: h = 1 L L for internal flows and external flows over flat surfaces. 2 0 hdx 4)
The field of convective heat transfer is a combination of: Experimental Results Analytical Results Numerical Investigations Semi-Analytical Models In general, there are only a limited number of analytical results. However, many correlations and models have been developed which accurately predict the heat transfer coefficient. We will consider a number of these fundamental results. 2 Convective Transport Equations In order to fully understand the complexity of convective heat transfer, it is best to first review the derivation of the Navier-Stokes equations from first principles. Beginning first with conservation of mass and then proceeding to conservation of momentum. We will examine the equations first in terms of shear stresses and discuss the necessary constituitive relations for Newtonian fluids. 2.1 Conservation of Mass Starting with conservation of mass, we may derive the continuity equation by considering the flows in and out of an elementary control volume dv = dxdydz. The conservation law states that The mass flow into the control volume is: ṁ stored = ṁ in ṁ out 5) The mass flow out of the control volume is: ṁ in = ρudy dz) + ρvdx dz) + ρwdx dy) 6) ṁ out = ρudy dz) + ρu dy dz) x dx 3
The mass stored within the control volume is: ρv dx dz) +ρvdx dz) + dy ρw dx dy) +ρwdx dy) + dz 7) ṁ stored = ρ dx dy dz) t 8) Applying the conservation law yields the following equation in cartesian co-ordinates: ρ ρu) t = x + ρv) + ρw) ) 9) The above equation may be expanded and re-arranged to give the following alternate form of the continuity equation: ρ t + u ρ x + v ρ + w ρ u = ρ x + v + w ) 10) Finally, we may write the equation more compactly in the following form: Dρ Dt = ρ V 11) Here, the term Dρ/Dt is the substantial derivative or convective derivative of the fluid density. For incompressible flows ρ is constant and the continuity equation simplifies to: V = 0 12) or u x + v + w = 0 13) 2.2 Conservation of Momentum Next, we examine the momentum equations. The momentum equations may be easily derived if we consider the following conservation law for a control volume dv = dx dy dz, in each of the three flow directions: Ṁ stored = Ṁin Ṁout + ΣF external 14) 4
We will examine only the x-direction and then simply state the y and z components. The flow of momentum into the control volume in the x-direction is: Ṁ in = ρudy dz)u + ρvdx dz)u + ρwdx dy)u 15) The second and third terms arise from mass flow from the y and z directions. The flow of momentum out of the control volume in the x-direction is: ρu dy dz u) Ṁ out = ρu dy dz) u + dx x ρv dx dz u) +ρv dx dz) u + dy ρw dx dy u) +ρw dx dy) u + dz 16) Finally, the momentum stored within the control volume is: Ṁ stored = ρu) dx dy dz) 17) t The external forces consist of the tangential and normal stresses, which may be viewed as diffusion of momentum normal to the faces of the control volume, the pressure forces, and the body force due to gravity. The momentum diffusion into the control volume is: F stress,in = τ xx dy dz) + τ yx dx dz) + τ zx dx dy) 18) The momentum diffusion out of the control volume is: F stress,out = τ xx dy dz) + τ xx dy dz) dx x +τ yx dx dz) + τ yx dx dz) dy The pressure forces acting in the x-direction are given by: +τ zx dx dy) + τ zx dx dy) dz 19) F p,in = pdy dz) 20) 5
and F p,out = pdy dz) + and finally, the body force due to gravity is: p dy dz) dx 21) x F g = ρg x dx dy dz) 22) Combining all of these terms using the conservation law for the control volume results in: ρu) t ρuu) + + ρvu) x ρg x p x + ρwu) ) τxx x + τ yx + τ zx = ) 23) Similar balances for the y and z directions yield the following results: ρv) t ρw) t ρuv) + + ρvv) x ρg y p ρuw) + + ρvw) x ρg z p + ρwv) ) τxy x + τ yy + τ zy = ) + ρww) ) τxz x + τ yz + τ zz = ) 24) 25) We may further simplify the left hand side inertia terms) by expanding and using the equation of continuity. This yields the more familliar forms: u ρ t + u u x + v u + w u ) = ρg x p x τxx x + τ yx + τ ) zx 26) v ρ t + u v x + v v + w v ) = ρg y p τxy x + τ yy + τ ) zy 27) w ρ t + u w x + v w + w w ) = ρg z p τxz x + τ yz + τ ) zz 28) 6
It will be left as an excercise for the student to verify that the left hand side inertia terms) expand and simplify using the continuity equation. The above equations are valid for any type of fluid provided the appropriate constituitive relationships are used for the stresses. 2.3 Constituitive Relationships The equations derived above are valid for any type of fluid. All that is required are the correct constituitive relationships or stress-strain relations as they are sometimes referred. The derivation of the required stress-strain relationships is quite involved and beyond the scope of this course. Details of these relationships may be found in many of the referenced texts or any text on continuum mechanics. We are interested in the results for compressible and imcompressible Newtonian fluids. For Newtonian compressible fluids these are: τ xx = 2µ u x + 2 3 µ V 29) τ yy = 2µ v + 2 3 µ V 30) τ zz = 2µ w + 2 3 µ V 31) u τ xy = τ yx = µ + v ) x v τ yz = τ zy = µ + w ) w τ zx = τ xz = µ x + u ) 32) 33) 34) For incompressible fluids, we may substitute these relationships with V = 0 to obtain the following momentum equations: u ρ t + u u x + v u + w u ) = ρg x p ) 2 x + µ u x + 2 u 2 + 2 u 2 2 v ρ t + u v x + v v + w v ) = ρg y p ) 2 + µ v x + 2 v 2 + 2 v 2 2 35) 36) 7
w ρ t + u w x + v w + w w ) = ρg z p ) 2 + µ w x + 2 w 2 + 2 w 2 2 37) Finally, we may write the equation in vector form using the following compact notation: ρ D V Dt }{{} inertia = ρ g }{{} bodyforce p }{{} + µ 2 V }{{} pressure friction 38) 2.4 Conservation of Energy We may now derive the energy equation for a constant property, incompressible, Newtonian fluid with negligible viscous heating. Considering a control volume of dv = dx dy dz with uniformly distributed heat sources S and constant uniform thermal conductivity k, the following energy balance may be written for the control volume: Ė stored = Ėin Ėout + Ėgen 39) Each energy flow term is composed of the heat transported by convection and the heat transported by conduction. First, the heat flow into the control volume by means of conduction is Ė in,conduction = k dy dz T x k dx dz T k dx dy T 40) and by convection is Ė in,convection = ρu dy dz) h + ρv dx dz) h + ρw dx dy) h 41) Next, the heat flow out of the control volume by conduction is and by convection is Ė out,conduction = k dy dz T x k dy dz T x x k dx dz T k dx dy T 8 ) dx k dx dz T ) dy k dx dy T ) dz 42)
ρu dy dz h) Ė out,convection = ρu dy dz) h + dx x ρv dx dz h) +ρv dx dz) h + dy ρw dx dy h) +ρw dx dy) h + dz 43) Finally, the heat generated within the control volume due to sources is and that stored within the control volume is Ė gen = Sdx dy dz) 44) Ė stored = ρ C p dx dy dz) T t 45) Combining each of the heat transfer terms and dividing by the volume dv = dx dy dz, yields the following energy equation in cartesian co-ordinates: T u h) ρc p t + ρ v h) + + x ) ) w h) 2 T = k x + 2 T 2 + 2 T + S 2 46) 2 Using the thermodynamic relationship h = C p T, and expanding the differentials, yields the simpler expression when combined with the equation of continuity for an incompressible flow: T ρc p t + u T x + v T + w T ) ) 2 T = k x + 2 T 2 + 2 T + S 2 47) 2 which may be written in vector notation as: ρc p DT Dt }{{} convection = k 2 T }{{} conduction + S }{{} generation 48) If viscous dissipation is considered, the viscous dissipation function is included: where T ρc p t + u T x + v T + w T ) ) 2 T = k x + 2 T 2 + 2 T + µφ + 2 Ṡ 49) 2 9
[ u ) 2 ) 2 ) ] 2 v w Φ = 2 + + + x u + v ) 2 w + x + v ) 2 w + x + u ) 2 50) 3 Boundary Layer Concepts The concept of the boundary layer was very important in fluid dynamics for determining the viscous drag experienced by a body in a streaming flow. In convection heat transfer we also consider the boundary layer concept from a temperature perspective. Just as in viscous flow, where viscosity leads to a reduction in the fluid velocity as we move closer to the wall, the temperature field decreases as we move away from a heated surface. A thermal boundary layer may be defined in much the same manner, i.e. and δ H u U 0.99 51) ) Tw T δ T 0.99 52) T w T Theoretical and experimental analysis shows that the ratio of the hydrodynamic and thermal boundary thicknesses has the proportionality δ H ν ) 1/3 P r 1/3 δ T α where P r is the Prandtl number, a physical property of the flowing fluid. In general, fluids such as oils have very high Prandtl numbers P r >> 1, while fluids such as liquid metals have very low Prandtl numbers P r << 1, while gases have Prandtl numbers of order P r 1. Recall, in a viscous flow we also defined a friction coefficient related to the velocity gradient at the wall, i.e. 53) C f = τ w 1 ρu 2 2 = µ u y=0 1 2 ρu 2 54) We can also define the heat transfer rate in a similar manner using Fourier s Law and Newton s Law of cooling: 10
Q w = ka T or, by solving for the heat transfer coefficient k T y=0 h = = T w T = hat w T ) 55) y=0 q w T w T 56) where q w = Q w /A is the wall heat flux. The minus sign results from the fact the temperature decreases as we move away from the heated wall, while in the friction law the velocity increases as we move away from the wall. Since the thermal boundary layer thickness increases as we move further downstream in a particular flow, the resistance to heat transfer increases, i.e. h decreases. Two possibilities exist with Eq. 56), in that either the wall temperature remains constant or the heat flux remains constant. If the boundary condition at the wall is that of constant temperature, then k T y=0 hx) = = q wx) 57) T w T T w T or the local heat flux varies with position. On the other hand, if the heat flux is constant, then hx) = k T y=0 = T w T or the local wall temperature varies with position. q w T w x) T 58) These characteristics also hold for turbulent boundary layers, but the effect of thermal boundary condition is less pronounced for a turbulent flow than for a laminar flow. Thus, to summarize: If T w = Constant, then q w = q w x) - Isothermal Surface If q w = Constant, then T w = T w x) - Isoflux Surface 4 Non-Dimensionalization of the Energy Equation We may non-dimensionalize the energy equation by introducing the following dimensionless parameters: 11
and x = x L u = T = T T T w T 59) y = y L u v = v U U z = z L w = w U 60) Here L represents a characteristic length scale of the geometry under consideration. This leads to the following non-dimensional form of the energy equation: T w T )U u T ) T T + v + w = L x ) 2 T x + 2 T 2 + 2 T 2 2 αt w T ) L 2 + µu 2 ρc p L 2 Φ 61) This is traditionally written in a more compact manner using dimensionless groups: Re L P r u T ) T T + v + w = x ) 2 T x + 2 T 2 + 2 T + EcP rφ 62) 2 2 The dimensionless grouping Re L P r is also referred to as the Peclet number, P e = ReP r, i.e., P e L = UL ν ν α = UL α 63) It is a very important parameter in convective heat transfer. A number of special limits exist for these dimensionless groups. When Re L P r 0, we see that convection, the left hand side of the energy equation, becomes negligible as compared with the conduction terms. And when EcP r 0, the viscous dissipation or viscous generation terms become negligible relative to the conduction terms. In both cases the conduction terms remain, otherwise there would be no heat transfer problem. In general EcP r is only large for high speed flows Ma > 0.3 and/or very viscous fluids such as oils which have large Prandtl numbers. 55) may be non- Finally, the definition of the heat transfer coefficient given by Eq. dimensionalized to give: T y =0 = hl k = q w L kt w T ) = Nu L 64) 12
The dimensionless heat transfer coefficient is known as the Nusselt number Nu. It is often interpreted as the dimensionless heat transfer coefficient, but as can be seen by the above equation, it is the dimensionless heat transfer rate. The Nusselt number depends on several parameters. In forced convection it takes the form: if it is a local value, and if it is a mean value. Here Re denotes the Reynolds number. Nu L = fx, Re L, P r) 65) In natural convection flows the Nusselt number takes the form: if it is a local value, and Nu L = fre L, P r) 66) Nu L = fx, Ra L, P r) 67) Nu L = fra L, P r) 68) where Ra is the Rayleigh number, which we will define later. It plays a similar role as the Reynolds number, being the dominant dependent variable in natural convection flows. The subscript L, is used to denote the characteristic length scale used in non-dimensionalization. In most models or correlations, it represents either the flow length x or L, the diameter of a cylinder or sphere, D, the square root of surface area for an external flow over a three dimensional body, A s, or the square root of cross-sectional duct area, A c in internal flows. It is very important to know the length scale used in the dimensionless grouping. That is why we use it as a subscript in the symbol chosen for the dimensionless parameter. In most cases, the length scale is an intrinsic scale such as the diameter of a sphere or cylinder. But many more general models now exist, where the length scale is defined in a more generic manner such as the square root of a characteristic area. 5 Dimensionless Groups Before proceeding to the review of fluid dynamics and heat transfer models, a brief discussion on the use of dimensionless quantities is required. A number of important dimensionless quantities appear throughout the text. The student should familliarize himself or herself with these parameters and their use. Table 1 summarizes the most important groups that will be encountered during this course. 13
Table 1 Frequently Encountered Dimensionless Groups Group Biot Number Reynolds Number Prandtl Number Peclet Number Grashof Number Rayleigh Number Nusselt Number Definition Bi = hl k s Re = ρv L µ P r = ν α P e = V L α = ReP r gβ T L3 Gr = Ra = ν 2 gβ T L3 αν Nu = q/a)l k f T Stanton Number St = Nu ReP r Colburn Factor j = Nu ReP r 1/3 Friction Coefficient C f = τ 1 2 ρv 2 = GrP r = hl k f Fanning Friction Factor f = p/l)a/p ) 1 2 ρv 2 14
6 Laminar versus Turbulent Flow Due to mixing within the the hydrodynamic and thermal boundary layers, heat transfer coefficients are much higher in a turbulent flow than in a laminar flow. In forced flows, the Reynolds criterion is used to determine whether a flow is laminar or turbulent: Laminar External Flows, Re x < 500, 000 Turbulent External Flows, Re x > 500, 00 Laminar Internal Flows, Re D < 2300 Turbulent Internal Flows, Re D > 4000 It is very important in convective heat transfer to determine the type of flow, as is done in fluid mechanics for drag of friction coefficient modelling. The value of h is strongly influenced by Laminar-Transition-Turbulent fluid behaviour. 7 Film Temperature In convective heat transfer, we must use appropriate values of the fluid properties when computing the heat transfer coefficient. Thus, since temperature varies throughout the convecting fluid, we must use an average value that is indicative of the entire film. For this we define the mean film temperature as: T film = T w + T 2 69) The above definition will be used whenever we need thermal properties for a given system. 15