External Forced Convection : Flow over Bluff Objects (Cylinders, Spheres, Packed Beds) and Impinging Jets Chapter 7 Sections 7.4 through 7.8
7.4 The Cylinder in Cross Flow Conditions depend on special features of boundary layer development, including onset at a stagnation point and separation, as well as transition to turbulence. Stagnation point: Location of zero velocity ( u = 0) and maximum pressure. Followed by boundary layer development under a favorable pressure gradient ( dp / dx < 0) and hence acceleration of the free stream flow ( du / dx 0). > As the rear of the cylinder is approached, the pressure must begin to increase. Hence, there is a minimum in the pressure distribution, p(x), after which boundary layer development occurs under the influence of an adverse pressure gradient dp / dx > 0, du / dx < 0. ( )
Separation occurs when the velocity gradient du / dy y = 0 reduces to zero and is accompanied by flow reversal and a downstream wake. Location of separation depends on boundary layer transition. Re ρ VD VD D = µ ν
What features differentiate boundary development for the flat plate in parallel flow from that for flow over a cylinder? Force imposed by the flow is due to the combination of friction and form drag. The dimensionless form of the drag force is C D FD = Figure 7.8 A f 2 ( ρv /2)
Heat Transfer Considerations The Local Nusselt Number: 5 How does the local Nusselt number vary with for Re 2 x 10? What conditions are associated with maxima and minima in the variation? 5 How does the local Nusselt number vary with θ for Re > 2 10? What conditions are associated with maxima and minima in the variation? The Average Nusselt Number ( Nu D hd k ) θ / : D D < Flow over a circular cylinder [Churchill and Bernstein Correlation]: Nu D 1/2 1/3 5/8 0.62Re Pr D ReD = 0.3 + 1 2/3 1/4 + 1 282,000 + ( 0.4/Pr) 4/5 for ( Re Pr) 0.2 D
Cylinders of Noncircular Cross Section: Nu D = hd m 1/3 = C ReD Pr k for Pr 0.7
Circular Cylinders [Zukauskas]: Nu D Pr = C m n 1/4 ReD Pr ( ) 0.7 Pr 500 Prs 1 Re 10 D 6 If Pr 10, n = 0.37 ; If Pr 10, n = 0.36.
7.5 The Sphere See Eqs.. (7.47) - (7.49) Nu sph hd = k µ 12 23 0.4 = 2 + 0.4 Re + 0.06 Re Pr µ s 14 (7.48)
7.6 Flow Across Tube Banks A common geometry for two-fluid heat exchangers. Aligned and Staggered Arrays: Aligned: V max = S T S T V D S T max S D D T T S V max = T V S 2 D D S T D ( S D D) Staggered: V = V if 2( S D) ( S D) or, if 2( ) ( )
Nu = C Re N 10 m D 1 D,max L 2000 Re 40, 000 Pr = 0.7 D,max where Re D,max ρvmaxd µ more general form: m 1/3 D = 1.13 1 ReD,max Pr L 10 Nu C N 2000 Re 40,000 Pr 0.7 D,max If N L < 10, Nu = C Nu D ( 10) 2 D N < ( N 10) L L
Flow Conditions: How do convection coefficients vary from row-to-row in an array? How do flow conditions differ between the two configurations? Why should an aligned array not be used for S T /S L < 0.7? Average Nusselt Number for an Isothermal Array [Zukauskas]: Nu C C ( ) 1/4 m 0.36 D = 2 ReD,maxPr Pr/ Prs Cm, Table 7.7 ; C Table 7.8 All properties are evaluated at ( T + T )/2 except for Pr s. i 2 o
Fluid Outlet Temperature (T o ) : Ts T o πdnh = exp T T ρvn S c s i T T p N = N x N T L What may be said about T o as N? Total Heat Rate: = A = N( π DL) q ha T T = m s m ( T T ) ( T T ) s i s o T n T s s T T i o s Pressure Drop: 2 ρvmax p = NLχ f 2 χ, f Figures 7.13 and 7.14
7.8 The Sphere and Packed Beds Flow over a sphere Boundary layer development is similar to that for flow over a cylinder, involving transition and separation. ( 1/ 2 2/3) 0.4 ( ) 1/ 4 Nu = 2 + 0.4Re + 0.06Re Pr µ / µ D D D s What are the limiting values of the Nusselt number and the convection coefficient for slow flows over small spheres. CD Figure 7.8 Gas Flow through a Packed Bed Flow is characterized by tortuous paths through a bed of fixed particles. Large surface area per unit volume renders configuration desirable for the transfer and storage of thermal energy.
For a packed bed of spheres: ε 0.575 j H = 2.06 Re D ε void fraction (0.3 < ε < 0.5) A q ha T = p, t m, total surface area of particles pt Ts T ha o pt, = exp Ts T i ρvac, bc p Acb, cross-sectional area of bed
7.7 Impinging Jet Characterized by large convection coefficients and used for cooling and heating in numerous manufacturing, electronic and aeronautic applications. Flow and Heat Transfer for a Circular or Rectangular Jet: Significant Features: Mixing and velocity profile development in the free jet. Stagnation point and zone. Velocity profile development in the wall jet.
Local Nusselt number distribution: Average Nusselt number: hd k VD e Re = ν ( ) h Nu = = f Re, Pr, A r,h / Dh h, A r from Fig. 7.17 Correlations Section 7.7.2 Jet Arrays Slot Jets What is the nature and effect of jet interactions and discharge conditions? Nusselt number correlations for arrays of circular and slot jets Section 7.7.2.
Problem: Extrusion Process Problem 7.63: Cooling of extruded copper wire by convection and radiation. KNOWN: Velocity, diameter, initial temperature and properties of extruded wire. Temperature and velocity of air. Temperature of surroundings. FIND: (a) Differential equation for temperature distribution T(x) along the wire, (b) Exact solution for negligible radiation and corresponding value of temperature at prescribed length ( x = L = 5m) of wire, (c) Effect of radiation on temperature of wire at prescribed length. Effect of wire velocity and emissivity on temperature distribution.
Problem: 7.78 Measurement of combustion gas temperature with a spherical thermocouple junction. KNOWN: Velocity and temperature of combustion gases. Diameter and emissivity of thermocouple junction. Combustor temperature. FIND: (a) Time to achieve 98% of maximum thermocouple temperature rise for negligible radiation, (b) Steady-state thermocouple temperature, (c) Effect of gas velocity and thermocouple emissivity on measurement error. SCHEMATIC:
ASSUMPTIONS: (1) Validity of lumped capacitance analysis, (2) Constant properties, (3) Negligible conduction through lead wires, (4) Radiation exchange between small surface and a large enclosure (parts b and c). PROPERTIES: Thermocouple: 0.1 ε 1.0, k = 100 W/m K, c = 385 J/kg K, ρ = 8920 kg/m 3 ; Gases: k = 0.05 W/m K, ν = 50 10-6 m 2 /s, Pr = 0.69. ANALYSIS: (a) If the lumped capacitance analysis may be used, it follows from Equation 5.5 that ρvc Ti T Dρc t = ln = ln 50 has T T 6h ( ). Neglecting the viscosity ratio correlation for variable property effects, use of V = 5 m/s with the Whitaker correlation yields ( ) ( 1/2 2/3) 0.4 VD 5 m s( 0.001m) NuD = hd k = 2 + 0.4 ReD + 0.06 ReD Pr ReD = = = 100 ν 6 2 50 10 m s ( ( ) ( ) )( ) 0.05 W m K h = 2 + 0.4 100 1/2 + 0.06 100 2/3 0.69 0.4 = 328 W m 2 K 0.001m Since Bi = h( ro 3) k = 5.5 10-4, the lumped capacitance method may be used. ( 3 ) 0.001m 8920 kg m 385J kg K t = ln( 50) = 6.83s 6 328W m 2 K
(b) Performing an energy balance on the junction, q conv = q rad. Hence, evaluating radiation exchange from Equation 1.7 and with ε = 0.5, ( ) ε σ ( 4 4) = has T T As T Tc 0.5 5.67 10 8 W m 2 K 4 4 4 1000 T K T 400 K 4 328 W m 2 K ( ) = ( ) T = 936 K Parametric calculations to determine the effects of V and ε yield the following results: 1000 990 Temperature, T(K) 950 Temperature, T(K) 970 950 930 910 900 0 5 10 15 20 25 Velocity, V(m/s) Emissivity, epsilon = 0.5 890 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Emissivity Velocity, V = 5 m/s
Since the temperature recorded by the thermocouple junction increases with increasing V and decreasing ε, the measurement error, T - T, decreases with increasing V and decreasing ε. The error is due to net radiative transfer from the junction (which depresses T) and hence should decrease with decreasing ε. For a prescribed heat loss, the temperature difference ( T - T) decreases with decreasing convection resistance, and hence with increasing h(v). COMMENTS: To infer the actual gas temperature (1000 K) from the measured result (936 K), correction would have to be made for radiation exchange with the cold surroundings. What measures may be taken to reduce the error associated with radiation effects?