Mechanical Engineering, Heat Transfer, Homework Solutions

March 21 Homework Solutions 6-8 During air cooling of potatoes, the heat transfer coefficient for combined convection, radiation, and evaporation is determined experimentally to be as shown: Air Velocity m/s Heat Transfer Coefficient W/m2·oC 14.0 19.1 20.2 24.4 0.66 1.00 1.36 1.73 Consider an 8-cm-diameter potato initially at 20°C with a thermal conductivity of 0.49 W/m·oC. Potatoes are cooled by refrigerated air at 5°C at a velocity of 1 m/s. Determine the initial rate of heat transfer from a potato, and the initial value of the temperature gradient in the potato at the surface. This problem illustrates the fact that the convection heat transfer rate is equal to the conduction heat transfer rate. For an air velocity of 1 m/s, the heat transfer coefficient is found from the table to be 19.1 W/m2·oC. When the potato is at its initial temperature of 20oC, the convective heat flux is found as follows. q = h(T − T∞ ) = 286.5 W ( 20 C −5 C ) = m ⋅ C m 19.1 W 2 o o o 2 Since this is also the conduction heat flux leaving the surface of the potato we can find the temperature gradient there by the Fourier Law. dT q = −k dr 286.5 W ⇒ 2 dT q =− =− m 0.49 W dr k m⋅o C dT 585 oC = dr m If we model the potato as a sphere with an 8 cm diameter, its area is π(0.08 m)2 = 0.020106 m2 so the initial heat flow from the potato = (285.5 W/m2)( 0.020106 m2) = 6.9  = 5.76 W Q An average man has a body surface area of 1.8 m2 and a skin temperature of 33°C. The convection heat transfer coefficient for a clothed person walking in still air is expressed as h = 8.6V0.53 for 0.5 < V < 2 m/s, where V is the walking velocity in m/s. Assuming the average surface temperature of the clothed person to be 30°C, determine the rate of heat loss from an average man walking in still air at 10°C by convection at a walking velocity of (a) 0.5 m/s, (b) 1.0 m/s, (c) 1.5 m/s, and (d) 2.0 m/s. Although the units for h in this equation are not stated we will assume that they are W/m2·oC. (This means that the constant 8.6 has units of W/m2·oC divided by (m/s)0.53 = W·s0.53/m2.53·oC) The heat transfer is found by the usual convection formula hA(T – T∞). Combining this with the equation for the heat transfer coefficient gives the heat transfer as 8.6V0.53A(T – T∞). For example, when the walking velocity is 0.5 m/s, the heat transfer is 0.53 0.53 8.6 W ⋅ s 8.6 W ⋅ s  0.5 m  Q = V 0.53 A(T − T∞ ) =   2.53 o 2.53 o m ⋅ C m ⋅ C  s  0.53 (1.8 m )(33 C − 10 C ) = 214.4 W 2 o o Repeating the calculation for the other walking speeds specified gives the results in the following table. Walking speed (m/s) Heat transfer (W) 0.5 214.4 1 309.6 1.5 383.8 2 447 6-18C What is the physical significance of the Prandtl number? Does the value of the Prandtl number depend on the type of flow or the flow geometry? Does the Prandtl number of air change with pressure? Does it change with temperature? The Prandtl number can be written in either of the following forms Pr = µc p k = µ ρc p ν = α ρ k The thermal diffusivity, α, is a measure of the diffusion of heat in a body; we used this property in transient conduction. Similarly, the kinematic viscosity, ν, is a measure of the diffusivity of momentum. Thus, the Prandtl number is the ratio of the diffusivity of momentum to the diffusivity of heat. This is also a measure of the ratio of momentum boundary layer thickness to the thermal boundary layer thickness. The Prandtl number is a fluid property that does not depend on the type of flow or flow geometry. The values of µ, cp and k are known to change with temperature so the Prandtl number will change with temperature. The values of µ, cp and k are only very weak functions of pressure so the Prandtl number can be assumed to be independent of pressure. 6-37 Air at 15°C and 1 atm is flowing over a 0.3-m long plate at 65°C a velocity of 30 m/s. Using EES, Excel, or other software, plot the following on a combined graph for the range of x = 0.0 m to x = xcr: (a) The hydrodynamic boundary layer as a function of x; (b) the thermal boundary layer as a function of x. The equations for the momentum boundary layer thickness, δ, and the thermal boundary layer thickness, δt, are given, respectively, by equations 6-51 on page 378 and 6-63 on page 380. In these equations Rex = Vx/ν = ρVx/µ. Note that the ratio of the two boundary layer thicknesses is Pr1/3 showing that the Prandtl number is related to the ratio of momentum boundary layer thickness to thermal boundary layer thickness. 4.91 δ = x Re x δt 4.91 1 = x Re x Pr1 3 ⇒ δ = Pr1 3 δt To compute the Reynolds number we need properties for air at atmospheric pressure. These can be found from Table A-15 in the text, which gives the properties of air at atmospheric pressure as a function of temperature. For this problem where the surface temperature is 65oC and the air temperature is 15oC the average (film) temperature used to find the properties is 40oC. At this temperature we find the following properties for air: ρ = 1.127 kg/m3, µ = 1.918x10-5 kg/m·s, and Pr = 0.7255. We can compute ν = µ/ρ = (1.918x10-5 kg/m·s)/(1.127 kg/m3) = 1.702x10-5 m2/s. We then obtain the following computational formula for the Reynolds number 3m 1.763 x10 6 Vx s Re x = = = x ν 1.702 x10 − 5 m 2 m s The critical Reynolds number for flat plate flow is 500,000 (see page 366 of text) which corresponds to the following critical length for the flow in this problem. xcrit = Re x 500,000 = = 0.284 m V 1.763 x10 6 ν m We can also obtain a computational formula for the boundary layer thicknesses. 4.91 δ = x Re x ⇒ δ= ( 4.91x 4.91x1 2 4.91x1 2 = = = 3.698 x10 − 3 m1 2 6 Vx V 1.763 x10 ν ν m ) x From the dimensions of the prefactor we see that δ will be in meters when x is in meters. As noted above the thermal boundary layer thickness is a factor of Pr-1/3 times the momentum boundary layer thickness. δ 4.91 = x Re x δt 4.91 1 = x Re x Pr1 3 ⇒ δ = Pr1 3 δt ⇒ δt = δ Pr1 3 = 1.113 δ The formulas above were used in an Excel spreadsheet to compute the momentum and thermal boundary layer thicknesses. The results of the calculations and a plot of the boundary layer thicknesses as a function of downstream distance is shown below x 0 0.01 0.02 0.03 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.284 Momentum 0.000E+00 3.698E-04 5.230E-04 6.405E-04 7.396E-04 9.059E-04 1.046E-03 1.169E-03 1.281E-03 1.384E-03 1.479E-03 1.569E-03 1.654E-03 1.735E-03 1.812E-03 1.971E-03 Thermal 0.000E+00 4.116E-04 5.820E-04 7.128E-04 8.231E-04 1.008E-03 1.164E-03 1.301E-03 1.426E-03 1.540E-03 1.646E-03 1.746E-03 1.841E-03 1.930E-03 2.016E-03 2.193E-03 Air Flow 6-38 Liquid water at 15°C is flowing over a 0.3-m wide plate at 65°C a velocity of 30 m/s. Using EES, Excel, or other software, plot (a) The hydrodynamic boundary and (b) the thermal boundary layer as a function of x on the same graph for the range of x = 0.0 m to x = xcr. Use a critical Reynolds number of 500,000. The equations for the momentum boundary layer thickness, δ, and the thermal boundary layer thickness, δt, are given, respectively, by equations 6-51 on page 378 and 6-63 on page 380. In these equations Rex = Vx/ν = ρVx/µ. Note that the ratio of the two boundary layer thicknesses is Pr1/3 showing that the Prandtl number is related to the ratio of momentum boundary layer thickness to thermal boundary layer thickness. 4.91 δ = x Re x δt 4.91 1 = x Re x Pr1 3 δ = Pr1 3 δt ⇒ To compute the Reynolds number we need properties for water. These can be found from Table A-9 in the text, which gives the properties of water as a function of temperature. For this problem where the surface temperature is 65oC and the water temperature is 15oC the average (film) temperature used to find the properties is 40oC. At this temperature we find the following properties for water: ρ = 992.1 kg/m3, µ = 0.000653 kg/m·s, and Pr = 4.32. We can compute ν = µ/ρ = (0.000653 kg/m·s)/( 992.1 kg/m3) = 6.582x10-7 m2/s. We then obtain the following computational formula for the Reynolds number 30 m 4.558 x10 7 Vx s x Re x = = = m ν 6.582 x10 − 7 m 2 s We are given the critical Reynolds number for flat plate flow is 500,000; this corresponds to the following critical length for the flow in this problem. xcrit = Re x 500,000 = = 0.110 m V 4.558 x10 7 ν m We can also obtain a computational formula for the boundary layer thicknesses. δ 4.91 = x Re x ⇒ δ= 4.91x 4.91x1 2 = = Vx V ν ν 4.91x1 2 4.558 x10 7 m ( = 7.273 x10 − 4 m1 2 ) x From the dimensions of the prefactor we see that δ will be in meters when x is in meters. As noted above the thermal boundary layer thickness is a factor of Pr-1/3 times the momentum boundary layer thickness. δ 4.91 = x Re x δt 4.91 1 = x Re x Pr1 3 ⇒ δ = Pr1 3 δt ⇒ δt = δ Pr1 3 = 0.6140 δ The formulas above were used in an Excel spreadsheet to compute the momentum and thermal boundary layer thicknesses. The results of the calculations and a plot of the boundary layer thicknesses as a function of downstream distance is shown on the next page. Comparing the results of this problem and problem 6-37 for air, with the same flow velocity and temperature we see that the transition to turbulence occurs at a much shorter length for water and the boundary layer thicknesses are much thinner for water. In addition, because the Prandtl number is greater than one, the momentum boundary layer is thicker than the thermal boundary layer. x 0 0.0002 0.0004 0.0006 0.00085 0.0011 0.00135 0.00185 0.00235 0.00285 0.00335 0.00385 0.00435 0.007 0.009 0.011 6-51 Momentum 0.000E+00 1.029E-05 1.455E-05 1.781E-05 2.120E-05 2.412E-05 2.672E-05 3.128E-05 3.526E-05 3.883E-05 4.209E-05 4.513E-05 4.797E-05 6.085E-05 6.900E-05 7.628E-05 Thermal 0.000E+00 6.315E-06 8.931E-06 1.094E-05 1.302E-05 1.481E-05 1.641E-05 1.921E-05 2.165E-05 2.384E-05 2.585E-05 2.771E-05 2.945E-05 3.736E-05 4.236E-05 4.683E-05 A 4-m by 4-m flat plate maintained at a constant temperature of 80°C is subjected to parallel flow of air at 1 atm, 20°C, and 10 m/s. The total drag force acting on the upper surface of the plate is measured to be 2.4 N. Using momentum-heat transfer analogy, determine the average convection heat transfer coefficient, and the rate of heat transfer between the upper surface of the plate and the air. We can use the modified Reynolds analogy to find the heat transfer coefficient from the skin friction coefficient. Equation 6.83 on page 383 relates the local skin friction coefficient and the local heat transfer coefficient, however we can integrate the local variables over the length of the plate to get the average values of skin friction and heat transfer coefficients in the Reynolds analogy equation. C f ,x 2 hx Pr 2 3 = ρc pV L ⇒ C f ,L L L hx Pr 2 3 hL Pr 2 3 Pr 2 3 = C f , x dx = 2 dx = 2 hx dx = ρc pV ρc pV ρc pV ∫ 0 ∫ 0 ∫ 0 The total skin friction coefficient for the flat plate is based on the total drag force. The properties of air at 20oC are found from Table A-15: ρ = 1.204 kg/m3, cp = 1007 J/kg·oC, and Pr = 0.7309. Using these properties and the given data on velocity and plate dimensions we can compute the skin friction coefficient. C f ,L = 2.4 N 1 1 kg ⋅ m D 1 = = 0.002492 2 2 (4 m )(4 m ) 1 1.204 kg  10 m  N ⋅ s 2 A ρV   2 2 m3  s  We can use this value in the modified Reynolds analogy to find the heat transfer coefficient. hL = ρc pVC f , L 2 Pr 2 3 1.204 kg 1007 J 10 m (0.002492) 18.62 W m 3 kg ⋅o C s = = m 2 ⋅o C 23 1J 2(0.7309 ) W ⋅s 2 Finally, we can use this heat transfer coefficient to find the overall heat transfer. Q = hL A(T − T∞ ) = ρc pVC f , L 2 Pr 23 = kW ( = 17.9 kW 16 m )(80 C − 20 C ) 1000 W C 18.62 W 2 o m ⋅ 2 o o