ENG Heat Transfer II 1. 1 Forced Convection: External Flows Flow Over Flat Surfaces... 4

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1 ENG Heat Transfer II 1 Contents 1 Forced Convection: External Flows Flow Over Flat Surfaces Non-Dimensional form of the Equations of Motion Order of Magnitude Analysis for a Boundary Layer Flow Non-Dimensional form of the Energy Equation Order of Magnitude Analysis for a Thermal Boundary Layer Skin Friction and Heat Transfer Coefficients Solutions for Laminar Boundary Layer Flow Reynolds-Colburn (or Chilton-Colburn) Analogy Turbulent Boundary Layers Mixed Boundary Layer Conditions How to Evaluate Convection Heat Transfer Flow Across Cylinders and Spheres Flow Across Tube Banks Forced Convection: Internal Flow Hydrodynamic Fundamentals Thermodynamic Fundamentals Internal Flows: Correlations Fully Developed Laminar Flow Laminar Flow: Entry Region Turbulent Flow in Circular Tubes Turbulent Flow: Entry Region

2 ENG Heat Transfer II Flows in Noncircular Tubes Natural Convection Governing Equations Empirical Correlations Vertical Flat Plate (T s = const) Vertical Flat Plate (q s = const) Vertical Cylinders Horizontal Cylinders Horizontal Flat Surfaces (T s = const) Horizontal Flat Surfaces (q s = const) Inclined Surfaces Spheres Heat Exchangers Introduction Types of Heat Exchangers Overall Heat Transfer Coefficient Log Mean Temperature Difference Parallel Flow Counterflow Temperature distributions for Special Cases T lm for Other Exchanger Configurations Effectiveness - NTU Method

3 ENG Heat Transfer II 3 A Eulerian and Lagrangian Viewpoints 57 A.1 Lagrangian View A.2 Eulerian View B Conservation of Mass (Continuity Equation) 58 C Conservation of Momentum (Navier-Stokes Equations) 61

4 ENG Heat Transfer II 4 1 Forced Convection: External Flows 1.1 Flow Over Flat Surfaces Non-Dimensional form of the Equations of Motion Consider a two-dimensional, steady state, incompressible flow of a constant property, Newtonian fluid, with freestream velocity u, past a flat plate of length L. The governing equations for this flow are the continuity equation: u x + v y = 0 (1) and the Navier-Stokes equations: ρu u u + ρv x y = p ( 2 ) x + µ u x u y 2 ρu v ( v 2 ) + ρv x y = p y + µ v x v y 2 Define the following non-dimensional variables: x = x/l u = u/u p = p/ρu 2 y = y/l v = v/v (4) Substitution of these non-dimensional variables into Eq. (1) results in the following form of the continuity equation: u L (2) (3) u x + u v L y = 0 (5) which can be written in the following non-dimensional form: u x + v y = 0 (6)

5 ENG Heat Transfer II 5 Substitution of the non-dimensional variables into the x-component of the Navier- Stokes equations, Eq. (2), gives: ρu 2 u u L x + ρu2 L u v y = ρu2 p L x + µu ( 2 u L 2 x u ) y 2 (7) Multiplying this equation by L/ρu 2 gives: u u u + v x y = p x + µ ( 2 u ρu L x u ) y 2 (8) But Re L = ρu L/µ, therefore, the non-dimensional form of the x-component of the Navier-Stokes equations can be written as: u u u + v x y = p x + 1 ( 2 u Re L x u ) y 2 (9) Similarly, the non-dimensional form of the y-component of the Navier-Stokes equations is: u v v + v x y = p y + 1 ( 2 v Re L x v ) y 2 (10) Equations (6), (9), and (10) are the non-dimensional forms of the equations that govern the steady state, two-dimensional, incompressible flow of a constant property, Newtonian fluid, with freestream velocity u, past a flat plate of length L. Note: the only parameter in these equations is the Reynolds number, therefore, Re should appear as a parameter in the solutions for the hydrodynamic boundary layer on a flat plate.

6 ENG Heat Transfer II Order of Magnitude Analysis for a Boundary Layer Flow Consider the order of magnitude of the non-dimensional variables that appear in Eqs. (6), (9), and (10): O(x ) = 1 O(u ) = 1 O(v ) =? O(y ) = δ/l = δ 1 O(p ) =? O(Re L ) =? (11) Now, consider the order of magnitude of each term in Eqs. (6), (9), and (10): u x + v y = 0 (12) u u u + v x y = p x + 1 ( 2 u Re L x u ) y 2 (13) u v v + v x y = p y + 1 ( 2 v Re L x v ) y 2 (14) Note: 1. From the continuity equation O(v ) = δ 1 (since both terms must be of the same magnitude). 2. For the viscous terms to be the same order as the inertia terms in the x- component of the Navier-Stokes equations O(Re L ) = 1/δ For the pressure term to be the same order as the inertia terms in the x- component of the Navier-Stokes equations (to prevent infinite accelerations) O(p ) = 1.

7 ENG Heat Transfer II 7 The non-dimensional forms of the equations governing the flow in the boundary layer are: u x + v y = 0 (15) u u u + v x y = p x + 1 Re L ( 2 u ) y 2 (16) 0 = p y (17) Or in dimensional form: u x + v ρu u u + ρv x y = p x + µ y = 0 (18) ( 2 ) u y 2 (19) 0 = p y (20) One term has been eliminated from the x-component of the Navier-Stokes equations. The y-component of the Navier-Stokes equations has been reduced to a hydrostatic pressure distribution. By performing the order of magnitude analysis one equation has been simplified, and the number of equations that must be solved has been reduced by one Non-Dimensional form of the Energy Equation Consider a two-dimensional, steady state, incompressible flow of a constant property, Newtonian fluid at freestream temperature T and velocity u past a flat plate of length L, maintained at a constant temperature T s.

8 ENG Heat Transfer II 8 The energy equation may be written as: ρc p u T x + ρc pv T y = k ( 2 ) T x T y 2 + µφ (21) where Φ = 2 ( ) u 2 ( ) v 2 ( u x y y + v ) 2 (22) x Define the following non-dimensional variables x = x/l u = u/u θ = (T T s )/(T T s ) y = y/l v = v/u (23) Substituting these non-dimensional variables into Eqs. (21) and (22) gives: ρc p u (T T s ) L ( u θ ) θ + v x y = k(t T s ) L 2 ( 2 ) θ x θ y 2 + µu2 L 2 Φ (24) where ( u Φ ) 2 ( v ) 2 ( u ) 2 = 2 x + 2 y + y + v x (25) Simplifying Eq. (24): u θ θ + v x y = ( k 2 ) θ ρc p u L x θ y 2 + µu ρc p L(T T s ) Φ (26) But k ρc p u L = ( ) ( ) ( ) k µ k µ = = ρc p u L µ µc p ρu L 1 = 1 P rre L P e (27) µu ρc p L(T T s ) ( ) µu u = ρc p L(T T s u ( u 2 ) ( ) µ = c p (T T s ) ρu L ( u 2 ) ( ) µ = c p (T s T ) ρu L = Ec Re L (28)

9 ENG Heat Transfer II 9 Note: Reynolds # = Re L = ρu L µ inertia forces viscous forces Prandtl # = P r = µc p k = ν rate of diffusion of momentum α rate of diffusion of thermal energy Peclet # = P e = P rre L Eckert # = Ec = convective transport of thermal energy conductive transport of thermal energy u 2 kinetic energy/unit volume of flow c p (T s T ) thermal energy/unit volume of flow (29) (30) (31) (32) The non-dimensional form of the energy equation can be written as follows: u θ θ + v x y = 1 ( 2 ) θ P e x θ y 2 Ec Φ (33) Re L Note: the only parameters in the thermal problem are P r, Re L, and Ec Order of Magnitude Analysis for a Thermal Boundary Layer Similar to the analysis of the hydrodynamic boundary layer, consider the order of magnitude of each term in Eq. (33): O(x ) = 1 O(u ) = 1 O(Ec) =? O(y ) = δ T /L = δt 1 O(v ) = δ 1 O(P r) =? O(θ) = 1 O(Re L ) = 1/δ 2 O(P e) =? (34) Expanding all terms of the non-dimensional form of the energy equation: u θ θ +v x y = 1 ( 2 ) θ P e x θ y 2 Ec ( ( u ) 2 ( v ) 2 ( u ) ) 2 2 Re L x + 2 y + y + v x (35) Note: 1. For the conduction terms to be of the same order as the convection terms O(1/P e) = δt 2. This is sensible, since O(Re L) = 1/δ 2, and O(P r) 1 for most common fluids.

10 ENG Heat Transfer II For any of the viscous dissipation terms to be of the same order as the remainder of the equation, the only possibility is for O(Ec) 1, then the ( u / y ) 2 term will remain The non-dimensional form of the energy equation for the thermodynamic boundary layer on a flat plate (steady state, constant property, Newtonian fluid, incompressible flow) is: u θ θ + v x y = 1 2 ( θ Ec u ) 2 P e y 2 Re L y (36) But Ec 1 only at high velocities, e.g. for air, (c p 1000 J/kg oc, (T s T ) 100 o C u 316 m/s for Ec = u 2 /(c p (T s T )) = 1. The speed of sound at 300K is 347 m/s. therefore, at low velocities, viscous dissipation is negligible. Viscous dissipation is very important at high velocities, e.g. the space shuttle, and SR-71 Blackbird spy plane. Viscous dissipation gives rise to frictional heating. Neglecting viscous dissipation, the equation governing the thermal boundary layer on a flat plate for steady, two-dimensional, incompressible flow of a constant property, Newtonian fluid is: ρc p u T x + ρc pv T y = k 2 T y 2 (37) or u T x + v T y = α 2 T y 2 (38) Skin Friction and Heat Transfer Coefficients We would be interested in the frictional drag due to the hydrodynamic boundary layer. The frictional drag is due to the shear stress at the plate surface, i.e.: τ s = µ u y (39) y=0 The shear stress is often written in terms of a skin friction coefficient, C f : τ s = C f ρu 2 2 (40) Since the velocity gradient u/ y at y = 0 varies with x, the skin friction coefficient will also be a function of x. Further, the non-dimensional Navier-Stokes equations, Eqs. (9) and (10) illustrate that the only parameters that would influence the solution for the velocity gradient at the wall (i.e. y = 0) are Re and p/ x. But, the pressure

11 ENG Heat Transfer II 11 gradient is only a function of x, and it is determined by the geometry of the flow, therefore, for flows of different fluids past the same geometry, only the Reynolds number and position on the body should influence the skin friction coefficient: C f = C f (x, Re x ) (41) To solve for C f we need the velocity distribution in the boundary layer, therefore, we need to solve Eqs. (18) and (19). The heat flux at the surface of the plate exposed to the convection environment, q s, can be written as follows: q s = h(t s T ) = k T y (42) y=0 Heat is transferred from the wall to the fluid by conduction (since the molecules of fluid next to the wall have zero velocity relative to the plate). The heat transfer coefficient is defined as follows: h = k T y y=0 T s T (43) To determine h we need the temperature gradient at the wall, i.e. the temperature distribution, therefore, we need to solve the energy equation for the boundary layer, i.e. Eq. (37), which will require a prior solution for the hydrodynamic boundary layer. The non-dimensional form of the energy equation, Eq. (36), illustrates that the only non-dimensional parameters that should appear in the thermal boundary layer solution are x, y, P e (or Re and P r), and Ec. Instead of working with the heat transfer coefficent, it is common to use a nondimensional variable called the Nusselt number (N u): or locally: Nu L = hl k Actual heat transfer in the presence of flow Heat transfer if only conduction occurs Nu x = hx k Since only P r, Re, and Ec are the parameters of the flow: (44) (45) h = h(p r, Re, Ec) (46) Nu = Nu(P r, Re, Ec) (47) If Ec is small, h and Nu are only functions of P r and Re and: h = h(p r, Re) (48) Nu = Nu(P r, Re) (49)

12 ENG Heat Transfer II Solutions for Laminar Boundary Layer Flow Blausius (1908) developed an analytical solution for the hydrodynamic boundary layer (see Section 7.2.1, Incropera and DeWitt): δ = 5x Re 1/2 x (50) C f,x = 0.664Re 1/2 x (51) Polhausen (1921) developed an analytical solution for the thermal boundary layer: Nu x = h xx k = 0.332Re1/2 x P r 1/3 P r 0.6 (52) Note: 1. These correlations are for local values, i.e. they are functions of x. 2. The boundary layer thickens at the rate of x 1/2. For a given x, the thickness decreases with increasing Re (as the influence of viscous forces decreases). 3. The Nusselt number increases at the rate of x 1/2 but h x decreases at the rate of x 1/2 (i.e. the thermal boundary layer thickens with increasing x, decreasing the temperature gradient and the heat transfer rate at the surface of the plate). 4. The expected non-dimensional parameters have arisen in the analytical solutions. Often, we are not interested in the local skin friction, or heat transfer rate, but in the total frictional drag over a surface, or the total heat transfer rate from (or to) the surface. The total frictional drag force, D, would be defined as follows: Since ρu 2 /2 is constant: where D = L 0 τ s w dx = L 0 ρu 2 C f,x w dx (53) 2 D = C f ρu 2 2 A (54) C f = 1 L L 0 C f,x dx (55) Substituting Eq. (51) into Eq. (55) and performing the integral gives: C f = 1.328Re 1/2 L (56)

13 ENG Heat Transfer II 13 The total heat transfer rate from the plate, q, can be evaluated as follows: q = L 0 h x (T s T )w dx (57) but (T s T ) is constant for this problem, therefore, q = ha(t s T ) (58) where h = 1 L L Substituting h x from Eq. (52) into Eq. (59) gives: 0 h x dx (59) or an average Nusselt number can be defined as: h L = k L Re1/2 L P r1/3 P r 0.6 (60) Nu L = 0.664Re 1/2 L P r1/3 P r 0.6 (61) Note: C f, Nu L, and h L are twice the corresponding value at the position L from the leading edge (due to the exponent on Re x ). The above correlations for h and Nu have restrictions on the Prandtl number. Churchill and Ozoe experimentally obtained the following correlation for laminar flow over a flat isothermal plate which is valid for all Prandtl numbers: Nu x = Re 1/2 x P r 1/3 [ 1 + (0.0468/P r) 2/3 ] 1/4 P e x 100 (62) where Nu L = 2Nu L. The correlations developed thus far are based on the assumption that the fluid properties are constant and uniform. When there is a significant difference between the plate and freestream temperatures, the fluid properties are evaluated at a film temperature: T f = T s + T 2 (63)

14 ENG Heat Transfer II 14 When the plate is not heated over its entire length (e.g. heating starts at a location ξ from the leading edge of the plate) Eq. (52) is modified to give: Nu x = 0.332Re 1/2 x P r 1/3 [ ( ) ] ξ 3/4 1/3 1 (64) x Constant heat flux For a laminar boundary layer flow over a flat plate which has a constant and uniform wall heat flux (q s, W/m 2 ) the local Nusselt number is: Nu x = 0.453Re 1/2 x P r 1/3 P r 0.6 (65) Note: only the constant has changed between Eq. (52) and Eq. (65). If the heat flux is given, we would be interested in the local wall temperature, T s,x, or the mean temperature difference T s T : therefore So: q s = h x (T s,x T ) = Nu x k x (T s,x T ) (66) T s T = 1 L Using Eq. (65) to determine Nu x : and Note: T s,x T = T s T = L 0 q s x Nu x k q s x Nu x k (67) dx = q s L/k Nu L (68) q s L/k 0.680Re 1/2 L P r1/3 (69) q s = 3 2 h L(T s T ) (70)

15 ENG Heat Transfer II The mean Nusselt number for laminar flow over a flat plate with q s = const is: Nu L q s =const = 0.680Re 1/2 L P r1/3 (71) which is only 2% larger than the mean value for the constant T s boundary condition, Eq. (61), therefore, it is acceptable to use any of the Nu L correlations for constant T s to determine T s T for the constant q s boundary condition. 2. The correlation developed by Ozoe and Churchill may also be used for the constant heat flux boundary condition by replacing the constants and in Eq. (62) with and , respectively Reynolds-Colburn (or Chilton-Colburn) Analogy The skin friction coefficient for laminar boundary layer flow over a uniform temperature flat plate is given by Eq. (51). This equation can be rearranged as follows: C f,x 2 = 0.332Re 1/2 x (72) The expression for the local Nusselt number for a laminar boundary layer on a uniform temperature flat plate is Eq. (52). Dividing Eq. (52) by Re x P r: Nu x Re x P r = 0.332Re 1/2 x P r 2/3 (73) the left hand side of this equation is called the Stanton number, St x, therefore: Comparison of Eqs. (72) and (74) gives: or St x P r 2/3 = 0.332Re 1/2 x (74) St x P r 2/3 = C f,x 2 StP r 2/3 = C f 2 This is the Reynolds-Colburn analogy between fluid friction and heat transfer. For example, if experimental measurements are made of the frictional drag on a body C f St h heat transfer rate (or vice versa). The Reynolds-Colburn analogy applies for laminar and turbulent boundary layers on flat plates, and, in a modified form, for turbulent tube flow. (75) (76)

16 ENG Heat Transfer II Turbulent Boundary Layers Consider a steady, incompressible flow of a Newtonian fluid past a flat plate. The freestream conditions are constant at u and T, and the plate temperature is a uniform and constant T s. As the Reynolds number increases, the inertia forces begin to dominate the viscous forces and instabilities in the flow can no longer be damped out by viscous effects. The flow will go through a transition from a laminar boundary layer to a turbulent boundary layer. The turbulent portion of the boundary layer is thicker than the laminar portion, and instead of smooth lamina, it consists of eddies of varying size. The effect of these eddies is to give a mean velocity profile that is fuller than in the laminar portion of the boundary layer. This will result in higher velocity gradients at the surface of the plate higher wall shear stress (and C f,x ) higher frictional drag. Similarly, the mean temperature profile in the boundary layer becomes fuller, and the temperature gradient at the wall will increase. Since the temperature gradient at the wall increases, then the heat transfer rate will increase, therefore, by Eq. (43), the convection heat transfer coefficient will be higher than for laminar flow. In laminar flow, the shear stress is a function of the fluid properties (i.e. µ) and the velocity gradient. In turbulent flow, the shear stress is a function of µ, the velocity gradient, and the flow properties. This occurs, because the eddies will cause a transport of momentum through the boundary layer, and this transport is modelled as a shear stress: u τ t = ρɛ M (77) y where ɛ M is the eddy viscosity, which is due to the fluid motion.

17 ENG Heat Transfer II 17 An analytical solution cannot be found for turbulent flows, due to the dependence of the flow on the flow properties, therefore, experimental measurements are used to develop empirical correlations for C f and Nu in turbulent flows. The local skin friction coefficient is evaluated by the following equation: C f,x = Re 1/5 x Re x 10 7 (78) and this equation may be used for Re x < 10 8 to within 15% accuracy. Using the Reynolds-Colburn analogy: Nu x = Re 4/5 x P r 1/3 0.6 < P r < 60, Re x < 10 8 (79) For heating starting at a position ξ from the leading edge of the plate: Nu x = Re 4/5 x P r 1/3 [1 (ξ/x) 9/10 ] 1/9 0.6 < P r < 60, Re x < 10 8 (80) For uniform wall heat flux: Nu x = Re 4/5 x P r 1/3 0.6 P r 60 (81) i.e. 4% higher than for the constant wall temperature boundary condition Mixed Boundary Layer Conditions A laminar boundary layer will eventually go through a transition to a turbulent boundary layer. Define the plate length as L and the position where the critical Reynolds number, Re x,c, occurs as x c. When 0.95 < x c /L 1, then the mean laminar convection coefficient, Eq. (60), can be used to determine the total heat transfer rate from the plate. When x c /L 0.95 then the laminar and turbulent sections of the boundary layer should be accounted for when determining the total heat transfer rate from the plate: h L = 1 ( xc ) L h lam dx + h turb dx (82) L 0 x c where the transition is assumed to occur abruptly at x c. Substituting Eqs. (52) and (79) into Eq. (82) gives: where is dependent upon the critical Reynolds number. Nu L = (0.037Re 4/5 L A)P r1/3 (83) A = 0.037Re 4/5 x,c 0.664Re 1/2 x,c (84)

18 ENG Heat Transfer II 18 The critical Reynolds number for flow over a smooth flat plate is , therefore: and Nu L = (0.037Re 4/5 L 871)P r1/3 0.6 < P r < 60, < Re L 10 8 (85) C f = Re 1/ < Re L 10 8 (86) Re L L If the boundary layer is completely turbulent (e.g. it is tripped at the leading edge of the plate) A = 0: Nu L = 0.037Re 4/5 L P r1/3 (87) C f = 0.074Re 1/5 L (88) Note: these equations would also be appropriate when x c /L 1, since A 0.037Re 4/5 L. All of the foregoing correlations are to be used with properties evaluated at the film temperature. These correlations are acceptable for engineering calculations, but they may be up to 25% in error due to freestream turbulence, and surface roughness How to Evaluate Convection Heat Transfer The calculation of convection heat transfer rates uses the following procedure: 1. Identify the geometry of the flow. All of the convection heat transfer correlations are dependent upon the geometry involved. 2. Specify the reference temperature, and evaluate all fluid properties at this reference temperature. Usually the film temperature, or a mean bulk temperature is used, however, there are exceptions. 3. Calculate the Reynolds number. Is the flow laminar or turbulent, or both? Is there anything to cause the flow to be completely turbulent, e.g. a very rough surface, or freestream turbulence? 4. Determine the boundary condition on the surface, i.e. constant temperature or uniform flux. 5. Decide if local or mean values are required. 6. Pick an appropriate correlation based on the previous steps.

19 ENG Heat Transfer II Flow Across Cylinders and Spheres The total drag force, F D, acting on a cylinder in a cross flow is a function of a frictional component, due to the shear stress in the fluid at the surface of the cylinder, and a component due to the pressure differential acting on the cylinder due to the formation of the wake. These two drag components are called frictional and form (or pressure drag), respectively. For Re D < 2 the flow remains attached to the cylinder, and the drag is mainly due to friction. As the Reynolds number is increased boundary layer separation and wake formation become important, and form drag dominates frictional drag. As the freestream is brought to rest at the stagnation point on the cylinder, a maximum pressure is attained. As the flow expands about the cylinder, the pressure decreases, and the boundary layer develops in a favourable pressure gradient (dp/dx < 0). As the flow passes the maximum height of the cylinder, however, it will begin to decelerate, and consequently the pressure will increase, producing an adverse pressure gradient (dp/dx > 0). Since the pressure in a boundary layer is constant at any x location, this adverse pressure gradient will decelerate the flow within the boundary layer. If the flow has insufficient momentum to overcome the adverse pressure gradient it will separate from the surface of the cylinder and create a recirculation zone, or a wake. Thereafter, the pressure cannot increase, and this gives rise to a large pressure differential between the front and back of the cylinder high form drag.

20 ENG Heat Transfer II 20 A laminar boundary layer carries less momentum near the surface of the cylinder than a turbulent boundary layer, therefore, it will separate earlier (θ = 80 o ) than the turbulent boundary layer (θ = 140 o ) higher form drag. The transition Reynolds number for a cylinder is approximately

21 ENG Heat Transfer II 21 Figure 7.9, Incropera and DeWitt, above, shows the variation of Nu D as a function of angular position from the stagnation point on the cylinder, i.e. 0 o. The Nusselt number decreases from the stagnation point as the laminar boundary layer grows. Between 80 o and 100 o Nu increases rapidly, due to the transition from laminar to turbulent flow. The Nusselt number decreases as the turbulent boundary layer is established. The Nusselt number increases as 140 o is reached, due to boundary layer separation, and increased mixing in the wake region. Due to the large changes that occur in the flow over a cylinder, depending on position and Reynolds number, empirical correlations are used to determine overall heat transfer coefficients for cylinders in cross flow. The correlation developed by Hilpert: Nu D = hd k = CRem DP r 1/3 P r 0.6 (89) is widely used for gases, and liquids. The properties used in this equation are evaluated at the film temperature, and Nu D and Re D are based on the characteristic dimension of the cylinder, i.e. its diameter. The constants C and m are determined from Table 7.2, Incropera and Dewitt. Equation (89) may also be used for cylinders with noncircular cross-sections when Table 7.3, Incropera and DeWitt, is used to define C and m. Zhukauskas developed the following correlation: ( ) P r 1/4 Nu D = CRe m DP r n 0.7 < P r < 500, 1 < Re D < 10 6 (90) P r s

22 ENG Heat Transfer II 22 where C and m are obtained from Table 7.4, Incropera and DeWitt, and n = 0.37 for P r 10 and n = 0.36 for P r > 10. All properties are evaluated at T except for P r s. Churchill and Bernstein have developed the following correlation, which is valid for Re D P r > 0.2: Nu D = Re1/2 D P [ ( ) ] 5/8 4/5 [ r1/3 ReD 1 + (0.4/P r) 2/3 ] 1 + (91) 1/4 282, 000 where all properties are evaluated at the film temperature. Note: these correlations may be in error by as much as 20% in engineering calculations. The behaviour of fluid flow about a sphere is similar to that about a cylinder, however, the drag coefficient is reduced due to the three-dimensional nature of the flow. McAdams has proposed a correlation for the convection heat transfer coefficient on a sphere in a gas: Nu D = 0.37Re 0.6 D 17 < Re D < 70, 000 (92) where all properties are evaluated at the film temperature. Whitaker has developed the following correlation: Nu D = 2 + (0.4Re 1/2 D Re2/3 D )P r0.4 ( µ µ s ) 1/4 (93) which is valid for 0.71 < P r < 380, 3.5 < Re D < , and 1 < µ/µ s < 3.2. All properties in this correlation are evaluated at T except µ s which is evaluated at T s. Ranz and Marshall developed the following correlation for convection heat transfer from freely falling liquid drops: Nu D = Re 1/2 D P r1/3 (94) Note: Eqs. (93) and (94) reduce to Nu D = 2 when Re D 0, which is the value obtained for conduction from the surface of a sphere in a stationary infinite medium. 1.3 Flow Across Tube Banks Heat exchangers (e.g. boiler, air conditioner) often employ aligned or staggered arrays of tubes in a cross flow.

23 ENG Heat Transfer II 23 In these figures, D is the outer diameter of the tubes, N T and N L are the number of rows normal (transverse) and parallel (longitudinal) to the freestream flow, V, and S T and S L are the transverse and longitudinal spacings or pitches (center to center) of the tubes, respectively. The mean convection coefficient for a tube array can be found from the following relation: Nu D = 1.13C 1 Re m D,maxP r 1/3 (95) Where the constants C 1 and m are given as functions of S T /D and S L /D in Table 7.5, Incropera and DeWitt. This correlation is valid for N L 10, 2000 < Re D,max < 40, 000 and P r 0.7. The heat transfer coefficient is a function of the position of the tube in the tube bank. The first tube behaves like a cylinder in a cross flow. The coefficient on the other tubes is higher, because the first few rows of tubes behave as a turbulence grid, generating turbulence and increasing heat transfer rates. Usually the flow conditions stabilize after 10 rows, and there is little further change in the convection coefficient. For tube banks consisting of less than 10 rows parallel to the flow a correction factor C 2, based on the number of rows N L is applied to the Nusselt number. Where C 2 is obtained from the table below: Nu D NL <10 = C 2 Nu D NL 10 (96) N L Staggered Aligned

24 ENG Heat Transfer II 24 The Reynolds number, Re D,max, used in Eq. (95) is based on the maximum velocity in the tube bank. Re D,max = ρv maxd (97) µ For the aligned tube bank, the maximum velocity occurs between two tubes, i.e. A 1 : and, from mass conservation: V max = S T S T D V (98) For a staggered tube array, the maximum velocity may occur in one of two locations, i.e. A 1 or A 2, and V max is determined from Eq. (98), or: where V max = S D = Note: Eq. (99) should be used when 2(S D D) < (S T D). S T 2(S D D) V (99) [ SL 2 + S ] 1/2 T (100) 2 All of the properties used in the above relations are evaluated at the film temperature. Assuming the surface temperature of the tubes is uniform and higher than the external fluid temperature, the temperature of the external fluid will increase as it passes through the tube bank. Since q s = h T, q s will decrease as the fluid passes through the tube bank. Since the heat transfer rate to the fluid decreases, then the rate at which its temperature increases will decrease (i.e. the rate of increase is nonlinear). Using (T s T ) or (T s (T i + T o )/2) would lead to over and under estimations of

25 ENG Heat Transfer II 25 the heat transfer rate. An appropriate mean temperature difference to use is the log mean temperature difference (derived in Section 2.2): T lm = (T s T i ) (T s T o ) ln [(T s T i )/(T s T o )] (101) Note: Eq. (101) can be written in the following easy to remember form: T lm = T in T out ln [ T in / T out ] (102) The heat transfer rate to the fluid can then be written as: q = ha tot T lm = ṁc p (T o T i ) (103) where, A tot is the total surface area of the tubes exposed to convection: and ṁ is the mass flow rate of the freestream fluid: A tot = (N T N L )πdl (104) ṁ = ρ N T S T V L (105) The pressure drop through a tube bank can be evaluated using the following expression: ) p = χn L ( ρv 2 max 2 f (106) where the appropriate friction factor, f, and correction factor, χ, are obtained from Figs and 7.14, Incropera and DeWitt. These figures use P L S L /D and P T S T /D as independent variables. Note: 1. An alternate correlation has been proposed by Zhukauskas, see Eq. (7.57), Incropera and DeWitt. 2. Tube spacings of S T /S L < 0.7 are undesirable for inline tube arrangements, because they produce a preferred flow path between the rows of tubes, and much of the tube surface is not exposed to the main flow. See Fig. 7.12, Incropera and DeWitt.

26 ENG Heat Transfer II 26 2 Forced Convection: Internal Flow 2.1 Hydrodynamic Fundamentals Consider steady state flow in a circular tube of radius r o : The fully developed velocity profile is parabolic for laminar flow. The profile is flatter for turbulent flow. The friction and heat transfer rate are highest in the developing flow region, and asymptote to a constant value in the fully developed region. The length of the developing flow region (or hydrodynamic entry region), x fd,h, is a function of the Reynolds number for laminar flows: ( ) xfd,h 0.05Re D (107) D lam where Re D = ρu md (108) µ and u m is a mean velocity of the flow. The transition Reynolds number, Re D,c, is 2300 for tube flow. The hydrodynamic entry length for turbulent flow is independent of Reynolds number: ( ) xfd,h (109) D To derive an expression for the laminar velocity profile in a circular tube of radius r o, consider the steady, incompressible flow of a constant property Newtonian fluid in the tube. In the fully developed region v = 0, and u/ x = 0. The Navier-Stokes equations written in Polar-Cylindrical co-ordinates can be solved to give: u = 1 4µ ( dp dx ) r 2 o turb [ ( ) ] r 2 1 ro (110)

27 ENG Heat Transfer II 27 and the mean velocity is: u m = r2 o dp 8µ dx Defining the Moody (or Darcy) friction factor as: f = (dp/dx)d ρu 2 m/2 the pressure drop in a tube of length L can be written as: (111) (112) where L replaces dx. p = L 0 f ρu2 m 2D dx = f L D ρu2 m 2 (113) Substituting the expression for the mean velocity, Eq. (111), and the definition of the Reynolds number into Eq. (112) gives: f = 64 Re D (114) Experimental data must be used to determine the friction factor for turbulent flows in rough tubes, i.e. the Moody diagram, Fig. 8.3, Incropera and DeWitt. The following correlations exist for fully developed turbulent flow in smooth tubes: f = 0.316Re 1/4 D Re D (115) f = 0.184Re 1/5 D Re D (116) f = (0.79 ln Re D 1.64) Re D (117) 2.2 Thermodynamic Fundamentals Consider a fluid entering a tube with a uniform velocity and temperature.

28 ENG Heat Transfer II 28 The temperature profile in the fully developed region depends on the boundary condition applied at the tube wall (T s = const, or q s = const) The thermal entry length for laminar flow is: ( ) xfd,t = 0.05Re D P r (118) D lam i.e. if P r > 1, x fd,h < x fd,t and vice versa. The thermal entry length for turbulent flow is the same as the hydrodynamic entry length. A bulk (or mean) temperature is used in the calculation of heat transfer rates for internal flows. It is determined from the rate of transport of thermal energy through a cross-section, A c : Ė t = ṁc v T m = ρuc v T da c (119) A c i.e. The heat flux is defined as: T m = A c ρuc v T da c ṁc v (120) q s = h(t s T m ) (121) Note: T m is a function of x (T m increases if T s > T m and decreases if T s < T m ). It will be shown that h is constant in the fully developed region, therefore, if T s is constant then q s decreases with x, and if q s is constant then T s T m is a constant. In the thermally fully developed region T/ x 0, but: [ ] Ts (x) T (r, x) = 0 (122) x T s (x) T m (x) therefore: But r [ Ts T T s T m ] q s = k T r fd,t = T/ r r=r o f(x) (123) r=r o T s T m = h(t s T m ) (124) r=ro therefore h f(x) (125) k So in the thermally fully developed region the convection coefficient is constant. Since h is constant, the constant q s boundary condition gives: dt s dx = dt m fd,t dx (126) fd,t so (T s T m ) is constant.

29 ENG Heat Transfer II 29 Applying the 1st Law to an element of fluid flowing in a tube (i.e. an open system of constant volume fixed in space): Neglecting e k, e p, and all forms of work except for flow work: and if the fluid is assumed to have constant specific heats: dq conv = ṁdh (127) dq conv = ṁc p dt m (128) This equation can be integrated from the inlet to the exit of the tube to give: q conv = ṁc p (T m,o T m,i ) (129) Defining dq conv = q s P dx, Eq. (128) can be rearranged to give: dt m dx = q s P ṁc p = P ṁc p h(t s T m ) (130) For constant wall heat flux Eq. (130) can be integrated to give: T m (x) = T m,i + q s P ṁc p x (131) so the bulk temperature varies linearly with x. Note: T s T m increases until the fully developed region is reached (due to the higher h in the entrance region) and (T s T m ) becomes constant.

30 ENG Heat Transfer II 30 For the constant wall temperature boundary condition, Eq. (130) can be rewritten as: dt m dx = d( T ) = P h T (132) dx ṁc p where T = T s T m. Rearranging this equation, and integrating from inlet to outlet: or To d( T ) T i T = P L h dx (133) ṁc p 0 ln T o = P L ( ) 1 L h dx = P L h (134) T i ṁc p L 0 ṁc p which can be rearranged to give: T o = T ( s T m,o = exp P L ) h T i T s T m,i ṁc p So the temperature difference (T s T m ) decays exponentially. (135)

31 ENG Heat Transfer II 31 Using Eq. (134): or where q conv = ṁc p ((T s T m,o ) (T s T m,i )) = P Lh (T s T m,o ) (T s T m,i ) ln( T o / T i ) (136) q conv = ṁc p (T m,o T m,i ) = ha T lm (137) T lm = i.e. the log mean temperature difference. T o T i ln( T o / T i ) (138)

32 ENG Heat Transfer II Internal Flows: Correlations Fully Developed Laminar Flow An analytical solution can be obtained for fully developed laminar flow in a circular tube of diameter D: Nu D = hd k = 4.36 q s = const (139) = 3.66 T s = const (140) Table 8.1, Incropera and DeWitt, lists Nu D values for fully developed laminar flow in a variety of noncircular cross-sections Laminar Flow: Entry Region For a thermal entry region occurring in a fully developed velocity field, Hausen obtained the following relation for constant T s : Nu D = hd k = (D/L)Re D P r [(D/L)Re D P r] 2/3 (141) which gives a mean h over the entry region. Note: this value asymptotes to the fully developed value of Nu D = The correlation above is not generally applicable, since it assumes a fully developed velocity profile. For the combined entry region (i.e. hydrodynamic and thermal entry region) Seider and Tate obtained the following relation: Nu D = 1.86 ( ) ReD P r 1/3 ( ) µ 0.14 (142) L/D µs which is valid for T s = const, 0.48 < P r < 16, 700, < (µ/µ s ) < 9.75, and Re D P r(d/l) > 10. The properties in both of these correlations are evaluated at the mean bulk temperature of the fluid, except µ s Turbulent Flow in Circular Tubes The Fanning friction coefficient is defined as: C f = τ s ρu 2 m/2 (143)

33 ENG Heat Transfer II 33 and the shear stress at the wall of a circular tube is: ( ) du τ s = µ (144) dr r=r o Using the fully developed (laminar) velocity profile, and the mean velocity obtained from that profile: Using the Chilton-Colburn analogy: and the friction factor, Eq. (117): C f = f 4 (145) C f 2 = f 8 = StP r2/3 = Nu D Re D P r P r2/3 (146) Nu D = 0.023Re 4/5 D P r1/3 (147) The Dittus-Boelter correlation is the preferred form of the above correlation: Nu D = 0.023Re 4/5 D P rn (148) where n = 0.4 when T s > T m, and n = 0.3 when T s < T m. This correlation is valid for 0.7 P r 160, Re D 10, 000, and (L/D) 10. All properties should be evaluated at T m. This correlation should only be used for moderate T s T m. When the temperature difference between the wall and the bulk fluid conditions becomes large, there can be a large variation in properties in the fluid. For these conditions, Seider and Tate have developed the following correlation: Nu D = 0.027Re 4/5 D P r1/3 ( µ µ s ) 0.14 (149) which is valid for 0.7 P r 16, 700, Re D 10, 000, and (L/D) 10. All properties should be evaluated at T m except µ s. Both Eqs. (148) and (149) are valid for constant T s and q s. To reduce the errors (which may be as large as 25%) that may be induced by Eqs. (148) and (149), Petukhov developed the following correlation: Nu D = (f/8)re D P r (f/8) 1/2 (P r 2/3 1) (150) which can produce errors of up to 10%. This correlation is valid for 0.5 < P r < 2000, and 10 4 < Re D < , and for constant T s and q s boundary conditions. The friction factor may be obtained from a Moody diagram, or an appropriate smooth tube correlation. Fluid properties are evaluated at the fluid bulk temperature, except for µ s.

34 ENG Heat Transfer II 34 Gnielinski modified the Petukhov correlation for use at lower Reynolds numbers: Nu D = (f/8)(re D 1000)P r (f/8) 1/2 (P r 2/3 1) (151) This correlation is valid for 0.5 < P r < 2000, and 3000 < Re D < , and for constant T s and q s boundary conditions. The friction factor may be obtained from a Moody diagram, or an appropriate smooth tube correlation. Fluid properties are evaluated at the fluid bulk temperature, except for µ s Turbulent Flow: Entry Region The turbulent entry region is usually small 10 < (x fd,t /D) < 60, and it is often acceptable to assume that the fully-developed Nu D can also be used in the developing region. Nusselt developed the following correlation: Nu D = 0.036Re 4/5 D P r1/3 ( D L ) < L D < 400 (152) where the properties are evaluated at the mean bulk temperature of the fluid Flows in Noncircular Tubes The previously listed correlations may be used for tubes of noncircular cross-sections when the diameter D is replaced by the hydraulic diameter, D h : D h = 4A c P where A c is the cross-section of the tube, and P is the wetted perimeter. (153) Correlations for flow in a concentric tube annulus are given in Section 8.7, Incropera and DeWitt, however, the Dittus Boelter correlation, Eq. (148), may be used with the appropriate hydraulic diameter for fully developed turbulent flow in the annulus.

35 ENG Heat Transfer II 35 3 Natural Convection 3.1 Governing Equations Natural (or free) convection occurs when a body force acts on a fluid in which there are density gradients buoyancy forces. Fluid velocities in natural convection are much smaller than for forced convection, therefore, the heat transfer coefficients are much smaller. Natural convection often is the largest resistance in multi-mode heat transfer analyses. In general, fluid density decreases with increasing temperature ( ρ/ T < 0), therefore, fluids rise when heated. The presence of a density gradient, however, does not guarantee the presence of natural convection. If T 2 is sufficiently larger than T 1, the buoyancy forces become large enough to overcome the viscous forces, and an unstable fluid recirculation develops. When T 1 > T 2, the flow is stable as the lower density fluid is above the higher density fluid, and the flow is thermally stratified. If a vertical flat plate possessing a uniform temperature T s is placed in an infinite quiescent medium at temperature T, where T s > T, the fluid near the plate will be heated and begin to rise. This fluid motion will entrain fluid from the quiescent region, and lead to formation of a boundary layer.

36 ENG Heat Transfer II 36 The velocity of the fluid at the plate and at y = is zero. The boundary layer will initially be laminar, but instabilities in the flow will eventually overcome the damping effects of viscosity, and the flow will go through a transition to a turbulent boundary layer (with the expected increase in heat transfer rates). The equations governing the fluid motion are the continuity, Navier-Stokes and energy equations. The equations reduce to the same form as those used for a forced convection boundary layer on a flat plate, except for a modification in the momentum equation. Consider steady, two-dimensional natural convection of a constant property Newtonian fluid driven by a constant temperature (T s > T ) vertical flat plate. The x momentum equation is: u u x + v u y = 1 ρ p x g + ν 2 u y 2 (154) The flow is assumed incompressible, however, a variable density must be accounted for in a buoyancy force term (Boussinesq approximation). The flat plate boundary layer approximation illustrated that pressure is constant in the y direction, therefore, the x pressure gradient inside the boundary layer is the x pressure gradient (hydrostatic) in the quiescent portion of the fluid. Substituting Eq. (155) into Eq. (154) gives: p x = ρ g (155) u u x + v u y = g ρ (ρ ρ) + ν 2 u y 2 (156)

37 ENG Heat Transfer II 37 The first term on the RHS is the buoyancy force term. Defining the volume coefficient of expansion, β: β = 1 ( ) ρ (157) ρ T p which can be expressed in the following approximate form: β 1 ( ) ρ ρ (158) ρ T T then ρ ρ ρβ(t T ) (159) and the buoyancy force term can be replaced, to give the following form of the x momentum equation: u u x + v u y = gβ(t T ) + ν 2 u y 2 (160) The dependence of the buoyancy force on the temperature difference is now shown explicitly in the momentum equation. The equations that must be solved for the natural convection boundary layer are the following forms of the continuity, x momentum, and energy equations. u x + v y = 0 (161) u u x + v u y = gβ(t T ) + ν 2 u y 2 (162) u T x + v T y = α 2 T y 2 (163) Note: the energy and momentum equations must be solved simultaneously, due to the coupling through the buoyancy force term. Using the following nondimensional variables: x = x L y = y L (164) u = u u o v = v u o θ = T T T s T (165) where u o is an arbitrary reference velocity, the x momentum and energy equations can be written in the following nondimensional forms: u u u + v x y = gβ(t s T )L u 2 o θ + 1 Re L 2 u y 2 (166) u θ θ + v x y = 1 2 θ Re L P r y 2 (167)

38 ENG Heat Transfer II 38 The dimensionless parameter on the RHS of the momentum equation can be written in a more convenient form (without u o ), by multiplying it by Re 2 L. The result is the Grashof number, Gr L : Gr L = gβ(t s T )L u 2 o ( ) uo L 2 = gβ(t s T )L 3 ν ν 2 (168) The Grashof number is the ratio of buoyancy forces to viscous forces, and it plays a role similar to the Reynolds number in forced convection. From the nondimensional form of the governing equations we should expect: Nu L = f(gr L, Re L, P r) (169) This would be true when forced and free convection are of similar magnitude, i.e. Gr L /Re 2 L 1. When Gr L/Re 2 L 1 natural convection dominates and Nu L = f(gr L, P r). When Gr L /Re 2 L 1 forced convection dominates and Nu L = f(re L, P r). 3.2 Empirical Correlations The product Gr L P r appears frequently in Natural convection calculations, and it is defined as the Rayleigh number, Ra L : Ra L = Gr L P r = gβ(t s T )L 3 να (170) The critical Rayleigh number, Ra x,c, for transition to turbulent flow on a vertical flat plate is: Ra x,c = Gr x,c P r = gβ(t s T )x (171) να and as for forced convection, empirical correlations are required to model natural convection for turbulent flows. For most engineering calculations, the correlation for N u in natural convection heat transfer takes the following simple form: Nu L = hl k = CRan L (172) where the constants C and n depend on the flow geometry and the Rayleigh number. All properties in Eq. (172) are evaluated at the film temperature: T f = T + T s 2 (173)

39 ENG Heat Transfer II Vertical Flat Plate (T s = const) Equation (172) can be used to determine the Nusselt number for a uniform temperature vertical flat plate. The most commonly used constants are: Nu L = 0.59Ra 1/4 L Ra L < 10 9 (174) Nu L = 0.10Ra 1/3 L Ra L > 10 9 (175) The following, more accurate, correlations have been proposed by Churchill and Chu: Nu L = { Ra 1/6 } 2 L [1 + (0.492/P r) 9/16 ] 8/ Ra L (176) Nu L = Ra 1/4 L [1 + (0.492/P r) 9/16 ] 4/9 Ra L 10 9 (177) These correlations are valid for a wider range of Rayleigh numbers Vertical Flat Plate (q s = const) A modified Grashof number Gr, may be used in the analysis of constant flux boundary conditions for vertical flat plates. For laminar flow: and for turbulent flow: Grx = Gr x Nu x = gβq s x 4 kν 2 (178) Nu x = h xx k = 0.60(Gr xp r) 1/ < Gr x < (179) Nu x = 0.17(Gr xp r) 1/ < Gr xp r < (180) where all properties are evaluated at the film temperature. These correlations are valid for water and air. The mean heat transfer coefficient can be determined from these correlations using the following equation: h = 5 4 h x=l (181) Equation (177) may also be used for the constant wall heat flux boundary condition when Nu L and Re L are based on the temperature difference at the midpoint of the plate.

40 ENG Heat Transfer II Vertical Cylinders It has been determined experimentally that the correlations for free convection from vertical flat plates, Eqs. (174) through (177), may be used for vertical cylinders when: D L 35 Gr 1/4 L (182) Horizontal Cylinders The Nusselt number for natural convection from a long isothermal horizontal cylinder can be determined from: Nu D = hd k = CRan D (183) where the constants C and n are determined from Table 9.1, Incropera and DeWitt, and Nu D and Ra D are based on the diameter of the cylinder. A more accurate (and complicated) correlation valid for a wide range of Rayleigh numbers was developed by Churchill and Chu: Nu D = which is simplified to: for laminar flows. { Nu D = Ra 1/6 } 2 D [1 + (0.559/P r) 9/16 ] 8/27 Ra D < (184) 0.518Ra 1/4 D [1 + (0.559/P r) 9/16 ] 4/9 Ra D < 10 9 (185) Horizontal Flat Surfaces (T s = const) The heat transfer coefficient for free convection from horizontal surfaces is strongly dependent on the orientation of the surface and the sign of the temperature difference (T s T ).

41 ENG Heat Transfer II 41 When a heated surface faces upward, or a cooled surface faces downward, the fluid is free to move away from the surface, and this helps to promote heat transfer. If a heated surface faces downward, or a cooled surface faces upward, the fluid has to travel across the surface before it can rise, or fall, in its preferred direction. These orientations cause an extra resistance to the fluid flow, and act to decrease heat transfer. The characteristic length to be used in the Rayleigh number is defined by: L A s P where A s is the area for heat transfer, and P is the perimeter of the surface. (186) For the upper surface of a heated plate, and the lower surface of a cooled plate: Nu L = 0.54Ra 1/4 L 10 4 Ra L 10 7 (187) Nu L = 0.15Ra 1/3 L 10 7 Ra L (188) For the lower surface of a heated plate, and the upper surface of a cooled plate: Nu L = 0.27Ra 1/4 L 10 5 Ra L (189)

42 ENG Heat Transfer II Horizontal Flat Surfaces (q s = const) For the upper surface of a heated plate, and the lower surface of a cooled plate: Nu L = 0.13Ra 1/3 L Ra L < (190) Nu L = 0.16Ra 1/3 L Ra L (191) For the lower surface of a heated plate, and the upper surface of a cooled plate: The characteristic length is L = A s /P. Nu L = 0.58Ra 1/5 L 10 6 Ra L (192) The properties in these correlations are evaluated at: T e = T s (T s T )/4 (193) where T s is the mean surface temperature, related to the heat flux by: and the Nusselt number is defined as: h = q s T s T (194) Nu L = hl k = q s L (T s T )k (195)

43 ENG Heat Transfer II Inclined Surfaces Different fluid flows exist depending on which side of an inclined plate is considered. For a heated plate facing down, and a cooled plate facing up, it has been found that Eqs. (176) and (177) can be used to determine Nu L when g in the Rayleigh number is replaced by g cos θ. If θ > 60 o the horizontal plate correlations should be used. When a heated surface faces upward (or a cooled surface faces downward) parcels of fluid may rise (or fall) from the surface, depending on the Rayleigh number, and this results in three-dimensional flow, thinning of the boundary layer, promotion of heat transfer, and increase in the heat transfer coefficient. This is an area of continuing research.

44 ENG Heat Transfer II Spheres Churchill has obtained the following correlation for natural convection from isothermal spheres: 0.589Ra 1/4 D Nu D = 2 + [1 + (0.469/P r) 9/16 ] 4/9 Ra D (196) This correlation is valid for all fluids with P r 0.7, and the properties are evaluated at the film temperature.

45 ENG Heat Transfer II 45 4 Heat Exchangers 4.1 Introduction A heat exchanger is a device used to promote the exchange of heat between two fluids. Heat exchangers are in common use, e.g. car radiator, air conditioning systems, space heating, waste heat recovery, boilers, condensers, chemical processes, etc Types of Heat Exchangers Double pipe (or concentric tube) - parallel and counterflow The counterflow exchanger is a better design, because it will maintain a higher temperature difference between the hot and cold fluids. Cross flow - mixed and unmixed (e.g. car radiator) In an unfinned tube crossflow exchanger, the tube fluid is unmixed, and the external fluid is mixed. The temperature of the external fluid is relatively uniform over the outside of the tube due to this mixing.

46 ENG Heat Transfer II 46 In a finned tube crossflow exchanger, both fluids are unmixed, therefore, a temperature gradient can exist along the length of the tube, this will help to promote heat transfer, because a larger temperaure difference can be maintained between the internal and external fluids. Shell and tube exchanger (chemical processes, heating system, condenser) The tube fluid enters a header and is separated into tubes. These tubes make two or more passes inside a shell. The shell fluid is forced to flow across the tubes by baffles inside the shell (to promote heat transfer). By separating the flow into tubes a large area for heat transfer can be created. Plate heat exchanger (industrial processes) - fluids are forced to flow between plates, which have patterns stamped in them. Very large heat transfer areas can be created, but large pumping costs result, therefore, they are most often used for gases. Compact heat exchangers - exchangers with a very high surface area to volume ratio ( 700 m 2 /m 3 ). Typically, one fluid is a gas with a low heat transfer coefficient,

47 ENG Heat Transfer II 47 therefore, the large area is used to reduce the thermal resitance associated with the gas. The flow is usually laminar, to reduce the high pumping costs associated with large flow area. Compact heat exchangers are used in applications where space restrictions are more important than pumping costs. (e.g. home air/heat exchanger) All heat exhangers can be analysed using one of two methods: (1) the Log-Mean Temperature Difference (LMTD) method; and (2) the Effectiveness-Number of Transfer Units (ɛ-ntu) method Overall Heat Transfer Coefficient Consider a double pipe heat exchanger: The overall heat transfer coefficient, U, is defined as: therefore, the overall heat transfer coefficient is: q = T lm Rth = UA T lm (197) or UA = U i A i = U o A o = [ 1 + ln(d o/d i ) + 1 ] 1 (198) h i A i 2πkL h o A o [ 1 U i = + A i ln(d o /D i ) + A ] i 1 1 hi 2πkL A o h o (199) [ Ao 1 U o = + A o ln(d o /D i ) + 1 ] 1 A i h i 2πkL h o (200)

48 ENG Heat Transfer II 48 During operation of a heat exchanger, the heat transfer surfaces of the exchanger become coated with a film of deposits (chemical, biological, corrosion), and these deposits add a thermal resistance to the thermal circuit between the two fluids, i.e. a fouling resistance, R f. This fouling resistance may exist on both the inner and outer surfaces of the exchanger. The fouling resistance is defined in terms of a fouling factor, R f (m2 oc/w), which is dependent on the operating conditions, fluids, and heat exchanger materials. Examples of some fouling factors are given in Table 11.1, Incropera and DeWitt. Typical fouling factors are of order m 2 oc. Inclusion of internal and external fouling factors in UA, Eq. (198) gives: UA = U i A i = U o A o = [ 1 + R f,i + ln(d o/d i ) h i A i A i 2πkL + R f,o + 1 ] 1 (201) A o h o A o 4.2 Log Mean Temperature Difference Parallel Flow Assume: 1. The exchanger is insulated from its surroundings. 2. Axial conduction in the fluids is negligible.

49 ENG Heat Transfer II e k = e p = W = 0 4. Specific heats are constant. 5. U is constant. An energy balance on each fluid gives: Equation (202) can be integrated to give: dq = ṁ h c p,h dt h = C h dt h = ṁ c c p,c dt c = C c dt c (202) q = C h (T h,i T h,o ) = C c (T c,o T c,i ) (203) Also, the heat transfer rate between the two fluids can be written as: dq = U T da (204) where T = T h T c at any location x, and d( T ) = dt h dt c. Using Eq. (202), d( T ) can be written as follows: ( 1 d( T ) = dq + 1 ) (205) Ch Cc Substituting for dq using Eq. (204): d( T ) = U ( 1 Ch + 1 Cc ) T da (206) Rearranging Eq. (206) and integrating from the inlet to the outlet of the exchanger: o d( T ) i T ( ) To ln T i which can be rearranged to give: ( 1 = U + 1 ) o da (207) Ch Cc i ( 1 = UA + 1 ) (208) Ch Cc ( Th,i T h,o = UA + T ) c,o T c,i (209) q q or where q = UA (T h,o T c,o ) (T h,i T c,i ) ln [(T h,o T c,o )/(T h,i T c,i )] T lm,p F = (210) q = UA T lm,p F (211) T o T i ln( T o / T i ) = T i T o ln( T i / T o ) (212)

50 ENG Heat Transfer II Counterflow Assume: 1. The exchanger is insulated from its surroundings. 2. Axial conduction in the fluids is neglgible. 3. e k = e p = W = 0 4. Specific heats are constant. 5. U is constant. An energy balance on each fluid gives: Equation (213) can be integrated to give: dq = ṁ h c p,h dt h = C h dt h = ṁ c c p,c dt c = C c dt c (213) q = C h (T h,i T h,o ) = C c (T c,o T c,i ) (214) Also, the heat transfer rate between the two fluids can be written as: dq = U T da (215)

51 ENG Heat Transfer II 51 where T = T h T c at any location x, and d( T ) = dt h dt c. Using Eq. (213), d( T ) can be written as follows: Substituting for dq using Eq. (215): ( 1 d( T ) = dq 1 ) Ch Cc d( T ) = U (216) ( 1 Ch 1 Cc ) T da (217) Rearranging Eq. (217) and integrating from the inlet to the outlet of the exchanger (w.r.t. the hot side): o d( T ) i T ( ) To ln T i ( 1 = U 1 ) o da (218) Ch Cc i ( 1 = UA 1 ) (219) Ch Cc ( Th,i T h,o = UA T ) c,o T c,i (220) q q which can be rearranged to give: q = UA (T h,o T c,i ) (T h,i T c,o ) ln [(T h,o T c,i )/(T h,i T c,o )] (221) or where T lm,cf = q = UA T lm,cf (222) T o T i ln( T o / T i ) = T i T o ln( T i / T o ) (223) Temperature distributions for Special Cases For condensation, the temperature of the hot fluid is approximately constant, therefore, C h (a similar temperature distribution is obtained when C h >> C c ). For evaporation, the temperature of the cold fluid is approximately constant, therefore, C c (a similar temperature distribution is obtained when C c >> C h ).

52 ENG Heat Transfer II T lm for Other Exchanger Configurations For other heat exchanger configurations, the T lm,cf is used in conjunction with a correction factor, F, determined from figures in Section 11S.1 on the Incropera and DeWitt student website at Wiley.com, i.e.: and F = 1 for evaporation and condensation. q = UAF T lm,cf (224) The LMTD method is very useful in design calculations where the goal of the calculation is the size of the exchanger. For example, given the hot and cold mass flow rates, the inlet and outlet temperatures of the hot and cold fluids, and the diameters of the tubes used in a double pipe heat exchanger, one could evaluate U and then calculate the required area or length of the exchanger. In general, the LMTD method is used when all inlet and outlet temperatures are available, or may be easily evaluated. If all inlet and oulet temperatures are not available, the LMTD method becomes tedious, as an iterative solution is required. For example, a heat exchanger is to be used at off-design conditions, i.e. a mass flow rate is changed. In this case, the inlet temperatures, inlet mass flow rates, and geometry of the exchanger would be known. The desired properties would be the exit temperatures of the two fluids and the overall heat transfer rate. To determine the exit temperatures iterations would be required. 4.3 Effectiveness - NTU Method The Effectiveness - Number of Transfer Units (ɛ-nt U) method can be used for design (i.e. sizing) and off-design (i.e. performance) calculations, without iterations.