Problems

5.6 Internal flows 167

a.Obtain an expression for the velocity boundary layer thickness in terms of local Reynolds number.

b.Derive an expression for local Nusselt number if the plate is subjected to a constant heat flux boundary condition and the Prandtl number (Pr) of the fluid is more than 1.

c.For a given local value of Reynolds number and Prandtl number, which boundary condition type (constant T or constant q) would exhibit a higher heat transfer rate? Comment.

5.2Use the integral form of the momentum and energy conservation for steady, laminar, incompressible, constant property two-dimensional flow and heat transfer over a flat plate given in Problem 5.1.

a.Assuming a quadratic profile for the velocity, obtain an expression for the velocity boundary layer thickness in terms of the local Reynolds number. Also derive an expression for the average skin friction coefficient c f .

b.Assuming a linear temperature profile and a liquid metal (Pr << 1) as the medium, derive expressions for the local and average Nusselt numbers.

c.If the velocity profile is assumed to be cubic, how will the expression for the Nusselt number change for (Pr << 1)?

5.3Consider a steady, turbulent boundary layer on an isothermal flat plate of length “L” at temperature Tw. The boundary layer is “tripped” at the leading edge

x= 0 by a fine wire such that the flow is turbulent right at the leading edge of the plate. Assume constant physical properties and velocity and temperature profiles of the form

u= y 1/7

u∞ δ

From experiments it is known that the wall shear stress is related to the boundary layer thickness by an expression of the form

(Note that in view of the nature of the velocity profile, this expression is required for evaluating the RHS of the momentum integral equation at the wall.) Additionally, heat flux at the wall is related to the wall temperature through the Newton’s law of cooling as Qw = hx (Tw − T∞ )

a.By employing the momentum integral equation, derive an expression for the boundary layer thickness as a function of local Reynolds number for the turbulent boundary layer.

168CHAPTER 5  Forced convection

b.Determine the average friction coefficient over the entire plate length, cf , L .

c.Using the energy integral equation, obtain an expression for local Nusselt number Nux and use this result to evaluate the average Nusselt number NuL .

5.4The local heat transfer coefficient for turbulent flow along a flat plate can be determined by

Stx Pr2/3 = 0.0296Rex−1/5

Assume that this relation is valid from the leading edge of the flat plate. Develop an expression for the average value of the heat transfer coefficient over the length L of the flat plate.

5.5Consider a thin horizontal flat plate of length 1 m that is maintained at a temperature of 373 K. Air at a temperature of 303 K and a velocity of 15 m/s flows over the top surface of the plate. The bottom surface of the plate is insulated. The plate is 1 m deep in the direction perpendicular to the plane of the paper. The flow is fully turbulent from the leading edge (i.e. x = 0). Properties of air at the film temperature of 338 K are given below for ready use, though you can actually get them from Table 5.2.

k = 0.03 W/mK, ν = 21 × 10−6 m2 /s, ρ = 1 kg/m3 and Pr = 0.7.

Determine the convective heat transfer rate from the plate in the region between x = 150 mm and x = 800 mm (x is the axial distance from the leading edge). Assume the flow and heat transfer to be two-dimensional, steady, and incompressible and that air has constant properties.

5.6Consider a laminar flow of a fluid over a flat plate maintained at a constant temperature. If the free stream velocity of the fluid is halved, determine

a.The change in the drag force on the plate.

b.The change in the rate of heat transfer between the fluid and the plate. Calculate these quantities if the velocities are tripled, assuming that the flow is still laminar.

5.7Consider a cylindrical pin fin, 10 mm in diameter and 80 mm long made of

aluminum (k = 205W/mK) The fin is placed in a cross flow of air at T∞ = 30 C with u∞ = 5 m/s The base of the fin is maintained at 100 °C, and the tip may be assumed to be insulated.

a.Determine the heat transfer coefficient from the fin.

b.Determine the heat transfer rate from the fin.

c.Determine the fin efficiency and fin effectiveness.

d.If the velocity of the air is increased by 50%, determine the quantities calculated in (b) and (c).

e.If the velocity of air is 5 m/s but the fin diameter is reduced to 6 mm, calculate the heat transfer rate, fin efficiency, and fin effectiveness.

5.6 Internal flows 169

f.Comment on the results obtained with a view to examining the effect of key parameters on the performance of the fin.

5.8Consider a 60 W bulb. The surface area of the bulb is known to be 33.9 ×10−3 m2 . Assume the bulb to be a sphere.

a.Determine its surface temperature when it is kept in surroundings with atmospheric air at 30 °C and air with a very mild velocity of 0.6 m/s flows over the bulb.

b.Is the bulb safe to touch?

c.If radiation is considered, will your estimate of the surface temperature be lower or higher?

d.Regardless of your answer to (c), determine the surface temperature when radiation is present, with the bulb assembly emissivity assumed to be 0.2. (Please note this is a crude assumption with the assembly consisting of a filament, glass envelope, fuse, lead wires, and filament support). Assume the temperature of the surroundings in so far as radiation is concerned to be 30 °C (The problem involves iterations.)

5.9Consider a bank of cylinders consisting of 16 rows (in the direction of the flow),

with each tube having a diameter of 50 mm in an array with ST = 8 mm and SH = 6 mm The cylinders are heated such that their surface temperatures are maintained at 150 °C. Air flows over the cylinders with a velocity of u∞ = 3 m/s at a temperature of 30 °C. Please use Table 5.2 to obtain the air properties. Determine the heat transfer coefficient for this configuration for

a.Inline arrangement

b.Staggered arrangement

5.10A fluid at a temperature of 30 °C and an average velocity of 2.8 m/s is in hydrodynamically and thermally fully developed flow through a circular tube of 5 mm diameter. The fluid then enters a 6 m long heated section.

The surface of the heated section is maintained at 120 °C. Determine the fluid outlet temperature and the total heat transfer rate. Properties of the fluid may be evaluated at an estimated mean temperature of 75 °C and are as follows.

5.11Water at 30 °C is flowing in a 12 mm diameter tube with a flow rate of 0.6 kg/s. The wall temperature of the tube is 55 °C.

a.Determine the entry length (hydrodynamic and thermal).

b.Determine the length of the tube such that the water is heated to 45 °C

c.Determine the additional length required for every 1 °C rise in the outlet temperature of the water.

Use water properties given at the end of the chapter (Table 5.3) at an appropriate bulk mean temperature.

170CHAPTER 5  Forced convection

5.12Hot water flowing in a thin-walled tube is cooled by blowing room air at 30 °C

in cross flow with a velocity of u∞ = 10 m/s. The inlet and outlet temperatures of water are 80 °C and 40 °C respectively. The mass flow rate of water is 0.18 kg/s and the diameter of the tube is 40 mm.

a.Determine the length of the tube required to accomplish the cooling.

b.Determine the outlet temperature of the water, if the length of the tube is doubled.

c.Determine the outlet temperature of the water for the tube length determined in (a) when the velocity of the air is doubled.

Table 5.3  Thermophysical properties of water at saturation pressure (Meyer et al., 1967; Harvey et al., 2000)