Consider the microchannel cooling arrangement of 8.107.

Fully developed conditions are known to exist for water flowing through a 25-mm-diameter tube at 0.01 kg/s and 27 C. What is the maximum velocity of the water in the tube? What is the pressure gradient associated with the flow?

What is the pressure drop associated with water at 27 C flowing with a mean velocity of 0.2 m/s through a 600-m-long cast iron pipe of 0.15-m inside diameter?

Water at 27 C flows with a mean velocity of 1 m/s through a 1-km-long pipe of 0.25-m inside diameter. (a) Determine the pressure drop over the pipe length and the corresponding pump power requirement, if the pipe surface is smooth. (b) If the pipe is made of cast iron and its surface is clean, determine the pressure drop and pump power requirement. (c) For the smooth pipe condition, generate a plot of pressure drop and pump power requirement for mean velocities in the range from 0.05 to 1.5 m/s

An engine oil cooler consists of a bundle of 25 smooth tubes, each of length L 2.5 m and diameter D 10 mm. (a) If oil at 300 K and a total flow rate of 24 kg/s is in fully developed flow through the tubes, what is the pressure drop and the pump power requirement? (b) Compute and plot the pressure drop and pump power requirement as a function of flow rate for 10   30 kg/s.

For fully developed laminar flow through a parallelplate channel, the x-momentum equation has the form The purpose of this problem is to develop expressions for the velocity distribution and pressure gradient analogous to those for the circular tube in Section 8.1. (a) Show that the velocity profile, u(y), is parabolic and of the form where is the mean velocity um  a2 12 dp dx um u(y) 3 2 um 1  y2 (a/2) 2 d and dp/dx p/L, where p is the pressure drop across the channel of length L. (b) Write an expression defining the friction factor, f, using the hydraulic diameter Dh as the characteristic length. What is the hydraulic diameter for the parallel-plate channel? (c) The friction factor is estimated from the expression , where C depends upon the flow cross section, as shown in Table 8.1. What is the coeffi- cient C for the parallel-plate channel? (d) Airflow in a parallel-plate channel with a separation of 5 mm and a length of 200 mm experiences a pressure drop of p 3.75 N/m2 . Calculate the mean velocity and the Reynolds number for air at atmospheric pressure and 300 K. Is the assumption of fully developed flow reasonable for this application? If not, what is the effect on the estimate for um?

Consider pressurized water, engine oil (unused), and NaK (22%/78%) flowing in a 20-mm-diameter tube. (a) Determine the mean velocity, the hydrodynamic entry length, and the thermal entry length for each of the fluids when the fluid temperature is 366 K and the flow rate is 0.01 kg/s. (b) Determine the mass flow rate, the hydrodynamic entry length, and the thermal entry length for water and engine oil at 300 and 400 K and a mean velocity of 0.02 m/s.

Velocity and temperature profiles for laminar flow in a tube of radius ro 10 mm have the form T(r) 344.8  75.0(r/ro) 2  18.8(r/ro) 4 u(r) 0.1[1  (r/ro) 2 ] f with units of m/s and K, respectively. Determine the corresponding value of the mean (or bulk) temperature, Tm, at this axial position

At a particular axial station, velocity and temperature profiles for laminar flow in a parallel plate channel have the form with units of m/s and C, respectively. Determine corresponding values of the mean velocity, um, and mean (or bulk) temperature, Tm. Plot the velocity and temperature distributions. Do your values of um and Tm appear reasonable?

In Chapter 1, it was stated that for incompressible liquids, flow work could usually be neglected in the steady-flow energy equation (Equation 1.12d). In the trans-Alaska pipeline, the high viscosity of the oil and long distances cause significant pressure drops, and it is reasonable to question whether flow work would be significant. Consider an L 100 km length of pipe of diameter D 1.2 m, with oil flow rate 500 kg/s. The oil properties are
900 kg/m3 , cp 2000 J/kg
K, 0.765 N
s/m2 . Calculate the pressure drop, the flow work, and the temperature rise caused by the flow work. 8.10

When viscous dissipation is included, Equation 8.48 (multiplied by
cp) becomes This problem explores the importance of viscous dissipation. The conditions under consideration are laminar, fully developed flow in a circular pipe, with u given by Equation 8.15. (a) By integrating the left-hand side over a section of a pipe of length L and radius ro, show that this term yields the right-hand side of Equation 8.34. (b) Integrate the viscous dissipation term over the same volume. (c) Find the temperature rise caused by viscous dissipation by equating the two terms calculated above. Use the same conditions as in Problem 8.9

Consider a circular tube of diameter D and length L, with a mass flow rate of . (a) For constant heat flux conditions, derive an expression for the ratio of the temperature difference between the tube wall at the tube exit and the inlet temperature, Ts(x L)  Tm,i, to the total heat transfer rate to the fluid q. Express your result in terms of , L, the local Nusselt number at the tube exit NuD(x L), and relevant fluid properties. (b) Repeat part (a) for constant surface temperature conditions. Express your result in terms of , L, the average Nusselt number from the tube inlet to the tube exit , and relevant fluid properties.

Water enters a tube at 27 C with a flow rate of 450 kg/h. The heat transfer from the tube wall to the fluid is given as q s(W/m) ax, where the coefficient a is 20 W/m2 and x (m) is the axial distance from the tube entrance. (a) Beginning with a properly defined differential control volume in the tube, derive an expression for the temperature distribution Tm(x) of the water. (b) What is the outlet temperature of the water for a heated section 30 m long? (c) Sketch the mean fluid temperature, Tm(x), and the tube wall temperature, Ts(x), as a function of distance along the tube for fully developed and developing flow conditions. (d) What value of a uniform wall heat flux, q s (instead of q s ax), would provide the same fluid outlet temperature as that determined in part (b)? For this type of heating, sketch the temperature distributions requested in part (c).

Consider flow in a circular tube. Within the test section length (between 1 and 2) a constant heat flux q s is maintained. (a) For the following two cases, sketch the surface temperature Ts(x) and the fluid mean temperature Tm(x) as a function of distance along the test section x. In case A, flow is hydrodynamically and thermally fully developed. In case B, flow is not developed. (b) Assuming that the surface flux q s and the inlet mean temperature Tm,1 are identical for both cases, will the exit mean temperature Tm,2 for case A be greater than, equal to, or less than Tm,2 for case B? Briefly explain why

Consider a cylindrical nuclear fuel rod of length L and diameter D that is encased in a concentric tube. Pressurized water flows through the annular region between the rod and the tube at a rate , and the outer surface of the tube is well insulated. Heat generation occurs within the fuel rod, and the volumetric generation rate is known to vary sinusoidally with distance along the rod. That is, sin(x/L), where (W/m3 ) is a constant. A uniform convection coefficient h may be assumed to exist between the surface of the rod and the water. (a) Obtain expressions for the local heat flux q (x) and the total heat transfer q from the fuel rod to the water. (b) Obtain an expression for the variation of the mean temperature Tm(x) of the water with distance x along the tube. (c) Obtain an expression for the variation of the rod surface temperature Ts(x) with distance x along the tube. Develop an expression for the x-location at which this temperature is maximized.

Consider the laminar thermal boundary layer development near the entrance of the tube shown in Figure 8.4. When the hydrodynamic boundary layer is thin relative to the tube diameter, the inviscid flow region has a uniform velocity that is approximately equal to the mean velocity um. Hence the boundary layer development is similar to what would occur for a flat plate. (a) Beginning with Equation 7.21, derive an expression for the local Nusselt number NuD, as a function of the Prandtl number Pr and the inverse Graetz number GzD 1 . Plot the expression using the coordinates shown in Figure 8.10a for Pr 0.7. (b) Beginning with Equation 7.25, derive an expression for the average Nusselt number , as a function of the Prandtl number Pr and the inverse Graetz number GzD 1 . Compare your results with the Nusselt number for the combined entrance length in the limit of small x.

In a particular application involving fluid flow at a rate m through a circular tube of length L and diameter D, the surface heat flux is known to have a sinusoidal variation with x, which is of the form q s(x) q s,m sin(x/L). The maximum flux, q s,m, is a known constant, and the fluid enters the tube at a known temperature, Tm,i . Assuming the convection coefficient to be constant, how do the mean temperature of the fluid and the surface temperature vary with x?

A flat-plate solar collector is used to heat atmospheric air flowing through a rectangular channel. The bottom surface of the channel is well insulated, while the top surface is subjected to a uniform heat flux q o, which is due to the net effect of solar radiation absorption and heat exchange between the absorber and cover plates. (a) Beginning with an appropriate differential control volume, obtain an equation that could be used to determine the mean air temperature Tm(x) as a function of distance along the channel. Solve this equation to obtain an expression for the mean temperature of the air leaving the collector. (b) With air inlet conditions of 0.1 kg/s and Tm,i 40 C, what is the air outlet temperature if L 3 m, w 1 m, and q o 700 W/m2 ? The specific heat of air is cp 1008 J/kg
K. 8.1

Atmospheric air enters the heated section of a circular tube at a flow rate of 0.005 kg/s and a temperature of 20 C. The tube is of diameter D 50 mm, and fully developed conditions with h 25 W/m2
K exist over the entire length of L 3 m. (a) For the case of uniform surface heat flux at q s 1000 W/m2 , determine the total heat transfer rate q and the mean temperature of the air leaving the tube Tm,o. What is the value of the surface temperature at the tube inlet Ts,i and outlet Ts,o? Sketch the axial variation of Ts and Tm. On the same figure, also sketch (qualitatively) the axial variation of Ts and Tm for the more realistic case in which the local convection coefficient varies with x. (b) If the surface heat flux varies linearly with x, such that q s (W/m2 ) 500x (m), what are the values m of q, Tm,o, Ts,i , and Ts,o? Sketch the axial variation of Ts and Tm. On the same figure, also sketch (qualitatively) the axial variation of Ts and Tm for the more realistic case in which the local convection coeffi- cient varies with x. (c) For the two heating conditions of parts (a) and (b), plot the mean fluid and surface temperatures, Tm(x) and Ts(x), respectively, as functions of distance along the tube. What effect will a fourfold increase in the convection coefficient have on the temperature distributions? (d) For each type of heating process, what heat fluxes are required to achieve an air outlet temperature of 125 C? Plot the temperature distributions

Fluid enters a tube with a flow rate of 0.015 kg/s and an inlet temperature of 20 C. The tube, which has a length of 6 m and diameter of 15 mm, has a surface temperature of 30 C. (a) Determine the heat transfer rate to the fluid if it is water. (b) Determine the heat transfer rate for the nanofluid of Example 2.2.

Water at 300 K and a flow rate of 5 kg/s enters a black, thin-walled tube, which passes through a large furnace whose walls and air are at a temperature of 700 K. The diameter and length of the tube are 0.25 m and 8 m, respectively. Convection coefficients associated with water flow through the tube and airflow over the tube are 300 W/m2
K and 50 W/m2
K, respectively. (a) Write an expression for the linearized radiation coefficient corresponding to radiation exchange between the outer surface of the pipe and the furnace walls. Explain how to calculate this coefficient if the surface temperature of the tube is represented by the arithmetic mean of its inlet and outlet values. (b) Determine the outlet temperature of the water, Tm,o.

Slug flow is an idealized tube flow condition for which the velocity is assumed to be uniform over the entire tube cross section. For the case of laminar slug flow with a uniform surface heat flux, determine the form of the fully developed temperature distribution T(r) and the Nusselt number NuD

Superimposing a control volume that is differential in x on the tube flow conditions of Figure 8.8, derive Equation 8.45a.

An experimental nuclear core simulation apparatus consists of a long thin-walled metallic tube of diameter D and length L, which is electrically heated to produce the sinusoidal heat flux distribution where x is the distance measured from the tube inlet. Fluid at an inlet temperature Tm,i flows through the tube at a rate of . Assuming the flow is turbulent and fully developed over the entire length of the tube, develop expressions for: (a) the total rate of heat transfer, q, from the tube to the fluid; (b) the fluid outlet temperature, Tm,o; (c) the axial distribution of the wall temperature, Ts(x); and (d) the magnitude and position of the highest wall temperature. (e) Consider a 40-mm-diameter tube of 4-m length with a sinusoidal heat flux distribution for which q o 10,000 W/m2 . Fluid passing through the tube has a flow rate of 0.025 kg/s, a specific heat of 4180 J/kg
K, an entrance temperature of 25 C, and a convection coefficient of 1000 W/m2
K. Plot the mean fluid and surface temperatures as a function of distance along the tube. Identify important features of the distributions. Explore the effect of 25% changes in the convection coefficient and the heat flux on the distributions.

Water at 20 C and a flow rate of 0.1 kg/s enters a heated, thin-walled tube with a diameter of 15 mm and length of 2 m. The wall heat flux provided by the heating elements depends on the wall temperature according to the relation where q s,o 104 W/m2 ,  0.2 K1 , Tref 20 C, and Ts is the wall temperature in C. Assume fully developed flow and thermal conditions with a convection coefficient of 3000 W/m2
K. (a) Beginning with a properly defined differential control volume in the tube, derive expressions for the variation of the water, Tm(x), and the wall, Ts(x), q temperatures as a function of distance from the tube inlet. (b) Using a numerical integration scheme, calculate and plot the temperature distributions, Tm(x) and Ts(x), on the same graph. Identify and comment on the main features of the distributions. Hint: The IHT integral function DER(Tm, x) can be used to perform the integration along the length of the tube. (c) Calculate the total rate of heat transfer to the water.

Engine oil is heated by flowing through a circular tube of diameter D 50 mm and length L 25 m and whose surface is maintained at 150 C. (a) If the flow rate and inlet temperature of the oil are 0.5 kg/s and 20 C, what is the outlet temperature Tm,o? What is the total heat transfer rate q for the tube? (b) For flow rates in the range 0.5  , compute and plot the variations of Tm,o and q with . For what flow rate(s) are q and Tm,o maximized? Explain your results.

Engine oil flows through a 25-mm-diameter tube at a rate of 0.5 kg/s. The oil enters the tube at a temperature of 25 C, while the tube surface temperature is maintained at 100 C. (a) Determine the oil outlet temperature for a 5-m and for a 100-m long tube. For each case, compare the log mean temperature difference to the arithmetic mean temperature difference. (b) For 5  L  100 m, compute and plot the average Nusselt number and the oil outlet temperature as a function of L.

In the final stages of production, a pharmaceutical is sterilized by heating it from 25 to 75 C as it moves at 0.2 m/s through a straight thin-walled stainless steel tube of 12.7-mm diameter. A uniform heat flux is maintained by an electric resistance heater wrapped around the outer surface of the tube. If the tube is 10 m long, what is the required heat flux? If fluid enters the tube with a fully developed velocity profile and a uniform temperature profile, what is the surface temperature at the tube exit and at a distance of 0.5 m from the entrance? Fluid properties may be approximated as
1000 kg/m3 , cp 4000 J/kg
K, m 2  103 kg/s
m, k 0.8 W/m
K, and Pr 10. 8.2

An oil preheater consists of a single tube of 10-mm diameter and 5-m length, with its surface maintained at 175 C by swirling combustion gases. The engine oil (new) enters at 75 C. What flow rate must be supplied to maintain an oil outlet temperature of 100 C? What is the corresponding heat transfer rate?

Engine oil flows at a rate of 1 kg/s through a 5-mmdiameter straight tube. The oil has an inlet temperature of 45 C and it is desired to heat the oil to a mean temperature of 80 C at the exit of the tube. The surface of the tube is maintained at 150 C. Determine the required length of the tube. Hint: Calculate the Reynolds numbers at the entrance and exit of the tube before proceeding with your analysis

Air at p 1 atm enters a thin-walled (D 5-mm diameter) long tube (L 2 m) at an inlet temperature of Tm,i 100 C. A constant heat flux is applied to the air from the tube surface. The air mass flow rate is m . 135  106 kg/s. (a) If the tube surface temperature at the exit is Ts,o 160 C, determine the heat rate entering the tube. Evaluate properties at T 400 K. (b) If the tube length of part (a) were reduced to L 0.2 m, how would flow conditions at the tube exit be affected? Would the value of the heat transfer coefficient at the tube exit be greater than, equal to, or smaller than the heat transfer coefficient for part (a)? (c) If the flow rate of part (a) were increased by a factor of 10, would there be a difference in flow conditions at the tube exit? Would the value of the heat transfer coefficient at the tube exit be greater than, equal to, or smaller than the heat transfer coefficient for part (a)? 8.31 T

To cool a summer home without using a vaporcompression refrigeration cycle, air is routed through a plastic pipe (k 0.15 W/m
K, Di 0.15 m, Do 0.17 m) that is submerged in an adjoining body of water. The water temperature is nominally at T
17 C, and a convection coefficient of ho
1500 W/m2
K is maintained at the outer surface of the pipe. If air from the home enters the pipe at a temperature of Tm,i 29 C and a volumetric flow rate of i 0.025 m3  /s, Pl what pipe length L is needed to provide a discharge temperature of Tm,o 21 C? What is the fan power required to move the air through this length of pipe if its inner surface is smooth?

Batch processes are often used in chemical and pharmaceutical operations to achieve a desired chemical composition for the final product. Related heat transfer processes are typically transient, involving a liquid of fixed volume that may be heated from room temperature to a desired process temperature, or cooled from the process temperature to room temperature. Consider a batch process for which a pharmaceutical (the cold fluid, c) is poured into an insulated, highly agitated vessel (a stirred reactor) and heated by passing a hot fluid (h) through a submerged heat exchanger coil of thinwalled tubing and surface area As. The flow rate, , mean inlet temperature, Th,i, and specific heat, cp,h, of the hot fluid are known, as are the initial temperature, Tc,i Th,i, the volume, Vc, mass density,
c, and specific heat, cv,c, of the pharmaceutical. Heat transfer from the hot fluid to the pharmaceutical is governed by an overall heat transfer coefficient U. (a) Starting from basic principles, derive expressions that can be used to determine the variation of Tc and Th,o with time during the heating process. Hint: Two equations may be written for the rate of heat transfer, q(t), to the pharmaceutical, one based on the logmean temperature difference and the other on an energy balance for flow of the hot fluid through the tube. Equate these expressions to determine Th,o(t) as a function of Tc(t) and prescribed parameters. Use the expression for Th,o(t) and the energy balance for flow through the tube with an energy balance for a control volume containing the pharmaceutical to obtain an expression for Tc(t). (b) Consider a pharmaceutical of volume Vc 1 m3 , density
c 1100 kg/m3 , specific heat cv,c 2000 J/kg
K, and an initial temperature of Tc,i 25 C. A coiled Co tube of length L 40 m, diameter D 50 mm, and coil diameter C 500 mm is submerged in the vessel, and hot fluid enters the tubing at Th,i 200 C and m . h 2.4 kg/s. The convection coefficient at the outer surface of the tubing may be approximated as ho 1000 W/m2
K, and the fluid properties are cp,h 2500 J/kg
K, h 0.002 N
s/m2 , kh 0.260 W/m
K, and Prh 20. For the foregoing conditions, compute and plot the pharmaceutical temperature Tc and the outlet temperature Th,o as a function of time over the range 0  t  3600 s. How long does it take to reach a batch temperature of Tc 160 C? The process operator may control the heating time by varying m . h. For 1  m . h  5 kg/s, explore the effect of the flow rate on the time tc required to reach a value of Tc 160 C. 8.33 The ev

The evaporator section of a heat pump is installed in a large tank of water, which is used as a heat source during the winter. As energy is extracted from the water, it begins to freeze, creating an ice/water bath at 0 C, which may be used for air conditioning during the summer. Consider summer cooling conditions for which air is passed through an array of copper tubes, each of inside diameter D 50 mm, submerged in the bath. (a) If air enters each tube at a mean temperature of Tm,i 24 C and a flow rate of m . 0.01 kg/s, what tube length L is needed to provide an exit temperature of Tm,o 14 C? With 10 tubes passing through a tank of total volume V 10 m3 , which initially contains 80% ice by volume, how long would it take to completely melt the ice? The density and latent heat of fusion of ice are 920 kg/m3 and 3.34  105 J/kg, respectively. (b) The air outlet temperature may be regulated by adjusting the tube mass flow rate. For the tube length determined in part (a), compute and plot Tm,o as a function of m . for 0.005  m .  0.05 kg/s. If the dwelling cooled by this system requires approximately 0.05 kg/s of air at 16 C, what design and operating conditions should be prescribed for the system? 8.3

A liquid food product is processed in a continuous- flow sterilizer. The liquid enters the sterilizer at a temperature and flow rate of Tm,i,h 20 C, m . 1 kg/s, respectively. A time-at-temperature constraint requires that the product be held at a mean temperature of Tm 90 C for 10 s to kill bacteria, while a second constraint is that the local product temperature cannot exceed Tmax 230 C in order to preserve a pleasing taste. The sterilizer consists of an upstream, Lh 5 m heating section characterized by a uniform heat flux, CH0 an intermediate insulated sterilizing section, and a downstream cooling section of length Lc 10 m. The cooling section is composed of an uninsulated tube exposed to a quiescent environment at T
20 C. The thin-walled tubing is of diameter D 40 mm. Food properties are similar to those of liquid water at T 330 K. (a) What heat flux is required in the heating section to ensure a maximum mean product temperature of Tm 90 C? (b) Determine the location and value of the maximum local product temperature. Is the second constraint satisfied? (c) Determine the minimum length of the sterilizing section needed to satisfy the time-at-temperature constraint. (d) Sketch the axial distribution of the mean, surface, and centerline temperatures from the inlet of the heating section to the outlet of the cooling section. 8.

Water flowing at 2 kg/s through a 40-mm-diameter tube is to be heated from 25 to 75 C by maintaining the tube surface temperature at 100 C. (a) What is the required tube length for these conditions? (b) To design a water heating system, we wish to consider using tube diameters in the range from 30 to 50 mm. What are the required tube lengths for water flow rates of 1, 2, and 3 kg/s? Represent this design information graphically. (c) Plot the pressure gradient as a function of tube diameter for the three flow rates. Assume the tube wall is smooth.

Consider the conditions associated with the hot water pipe of Problem 7.56, but now account for the convection resistance associated with water flow at a mean velocity of um 0.5 m/s in the pipe. What is the corresponding daily cost of heat loss per meter of the uninsulated pipe?

A thick-walled, stainless steel (AISI 316) pipe of inside and outside diameters Di 20 mm and Do 40 mm is heated electrically to provide a uniform heat generation rate of . The outer surface of the pipe is insulated, while water flows through the pipe at a rate of . m 0.1 kg/s (a) If the water inlet temperature is Tm,i 20 C and the desired outlet temperature is Tm,o 40 C, what is the required pipe length? (b) What are the location and value of the maximum pipe temperature?

An air heater for an industrial application consists of an insulated, concentric tube annulus, for which air flows through a thin-walled inner tube. Saturated steam flows through the outer annulus, and condensation of the steam maintains a uniform temperature Ts on the tube surface. Consider conditions for which air enters a 50-mmdiameter tube at a pressure of 5 atm, a temperature of Tm,i 17 C, and a flow rate of , while saturated steam at 2.455 bars condenses on the outer surface of the tube. If the length of the annulus is L 5 m, what are the outlet temperature Tm,o and pressure po of the air? What is the mass rate at which condensate leaves the annulus?

Consider fully developed conditions in a circular tube with constant surface temperature Ts Tm. Determine whether a small- or large-diameter tube is more effective in minimizing heat loss from the flowing fluid characterized by a mass flow rate of . Consider both laminar and turbulent conditions

Consider the encased pipe of Problem 4.29, but now allow for the difference between the mean temperature of the fluid, which changes along the pipe length, and that of the pipe. (a) For the prescribed values of k, D, w, h, and T
and a pipe of length L 100 m, what is the outlet temperature Tm,o of water that enters the pipe at a temperature of Tm,i 90 C and a flow rate of ? (b) What is the pressure drop of the water and the corresponding pump power requirement? (c) Subject to the constraint that the width of the duct is fixed at w 0.30 m, explore the effects of the flow rate and the pipe diameter on the outlet temperature. m

Water flows through a thick-walled tube with an inner diameter of 12 mm and a length of 8 m. The tube is immersed in a well-stirred, hot reaction tank maintained at 85 C, and the conduction resistance of the tube wall (based on the inner surface area) is R cd 0.002 m2
K/W. The inlet temperature of the process fluid is Tm,i 20 C, and the flow rate is 33 kg/h. (a) Estimate the outlet temperature of the process fluid, Tm,o. Assume, and then justify, fully developed flow and thermal conditions within the tube. (b) Do you expect Tm,o to increase or decrease if combined thermal and hydrodynamic entry conditions exist within the tube? Estimate the outlet temperature of the water for this condition. 8

Atmospheric air enters a 10-m-long, 150-mm-diameter uninsulated heating duct at 60 C and 0.04 kg/s. The duct surface temperature is approximately constant at Ts 15 C. (a) What are the outlet air temperature, the heat rate q, and pressure drop p for these conditions? (b) To illustrate the tradeoff between heat transfer rate and pressure drop considerations, calculate q and p for diameters in the range from 0.1 to 0.2 m. In your analysis, maintain the total surface area, As DL, at the value computed for part (a). Plot q, p, and L as a function of the duct diameter.

NaK (45%/55%), which is an alloy of sodium and potassium, is used to cool fast neutron nuclear reactors. The NaK flows at a rate of m . 1 kg/s through a D 50-mmdiameter tube that has a surface temperature of Ts 450 K. The NaK enters the tube at Tm,i 332 K and exits at an outlet temperature of Tm,o 400 K. Determine the tube length L and the local convective heat flux at the tube exit. 8.4

The products of combustion from a burner are routed to an industrial application through a thin-walled metallic duct of diameter Di 1 m and length L 100 m. The gas enters the duct at atmospheric pressure and a mean temperature and velocity of Tm,i 1600 K and um,i 10 m/s, respectively. It must exit the duct at a temperature that is no less than Tm,o 1400 K. What is the minimum thickness of an alumina-silica insulation (kins 0.125 W/m
K) needed to meet the outlet requirement under worst case conditions for which the duct is exposed to ambient air at T
250 K and a cross-flow velocity of V 15 m/s? The properties of the gas may be approximated as those of air, and as a first estimate, the effect of the insulation thickness on the convection coefficient and thermal resistance associated with the cross flow may be neglected. 8.45 L

Liquid mercury at 0.5 kg/s is to be heated from 300 to 400 K by passing it through a 50-mm-diameter tube whose surface is maintained at 450 K. Calculate the required tube length by using an appropriate liquid metal convection heat transfer correlation. Compare your result with that which would have been obtained by using a correlation appropriate for Pr 0.7.

The surface of a 50-mm-diameter, thin-walled tube is maintained at 100 C. In one case air is in cross flow over the tube with a temperature of 25 C and a velocity of 30 m/s. In another case air is in fully developed flow through the tube with a temperature of 25 C and a mean velocity of 30 m/s. Compare the heat flux from the tube to the air for the two cases

Consider a horizontal, thin-walled circular tube of diameter D 0.025 m submerged in a container of noctadecane (paraffin), which is used to store thermal energy. As hot water flows through the tube, heat is transferred to the paraffin, converting it from the solid to liquid state at the phase change temperature of T
27.4 C. The latent heat of fusion and density of paraffin are hsf 244 kJ/kg and
770 kg/m3 , respectively, and thermophysical properties of the water may be taken as cp 4.185 kJ/kg
K, k 0.653 W/m
K, 467  106 kg/s
m, and Pr 2.99. (a) Ass (a) Assuming the tube surface to have a uniform temperature corresponding to that of the phase change, determine the water outlet temperature and total heat transfer rate for a water flow rate of 0.1 kg/s and an inlet temperature of 60 C. If H W 0.25 m, how long would it take to completely liquefy the paraffin, from an initial state for which all the paraffin is solid and at 27.4 C? (b) The liquefaction process can be accelerated by increasing the flow rate of the water. Compute and plot the heat rate and outlet temperature as a function of flow rate for 0.1   0.5 kg/s. How long would it take to melt the paraffin for m 0.5 kg/s? m

Consider pressurized liquid water flowing at m . 0.1 kg/s in a circular tube of diameter D 0.1 m and length L 6 m. (a) If the water enters at Tm,i 500 K and the surface temperature of the tube is Ts 510 K, determine the water outlet temperature Tm,o. (b) If the water enters at Tm,i 300 K and the surface temperature of the tube is Ts 310 K, determine the water outlet temperature Tm,o. (c) If the water enters at Tm,i 300 K and the surface temperature of the tube is Ts 647 K, discuss whether the flow is laminar or turbulent. 8.49 C

Cooling water flows through the 25.4-mm-diameter thin-walled tubes of a steam condenser at 1 m/s, and a surface temperature of 350 K is maintained by the condensing steam. The water inlet temperature is 290 K, and the tubes are 5 m long. (a) What is the water outlet temperature? Evaluate water properties at an assumed average mean temperature, . (b) Was the assumed value for reasonable? If not, repeat the calculation using properties evaluated at a more appropriate temperature. (c) A range of tube lengths from 4 to 7 m is available to the engineer designing this condenser. Generate a plot to show what coolant mean velocities are possible if the water outlet temperature is to remain at the value found for part (b). All other conditions remain the same

The air passage for cooling a gas turbine vane can be approximated as a tube of 3-mm diameter and 75-mm length. The operating temperature of the vane is 650 C, and air enters the tube at 427 C. (a) For an airflow rate of 0.18 kg/h, calculate the air outlet temperature and the heat removed from the vane. (b) Generate a plot of the air outlet temperature as a function of flow rate for 0.1   0.6 kg/h. Compare this result with those for vanes having 2- and 4-mm-diameter tubes, with all other conditions remaining the same

The core of a high-temperature, gas-cooled nuclear reactor has coolant tubes of 20-mm diameter and 780-mm length. Helium enters at 600 K and exits at 1000 K when the flow rate is 8  103 kg/s per tube. (a) Determine the uniform tube wall surface temperature for these conditions (b) If the coolant gas is air, determine the required flow rate if the heat removal rate and tube wall surface temperature remain the same. What is the outlet temperature of the air?

Air at 200 kPa enters a 2-m-long, thin-walled tube of 25-mm diameter at 150 C and 6 m/s. Steam at 20 bars condenses on the outer surface. (a) Determine the outlet temperature and pressure drop of the air, as well as the rate of heat transfer to the air. (b) Calculate the parameters of part (a) if the pressure of the air is doubled.

Heated air required for a food-drying process is generated by passing ambient air at 20 C through long, circular tubes (D 50 mm, L 5 m) housed in a steam condenser. Saturated steam at atmospheric pressure condenses on the outer surface of the tubes, maintaining a uniform surface temperature of 100 C. (a) If an airflow rate of 0.01 kg/s is maintained in each tube, determine the air outlet temperature Tm,o and the total heat rate q for the tube. (b) The air outlet temperature may be controlled by adjusting the tube mass flow rate. Compute and plot Tm,o as a function of for 0.005   0.050 kg/s. If a particular drying process requires approximately 1 kg/s of air at 75 C, what design and operating conditions should be prescribed for the air heater, subject to the constraint that the tube diameter and length be fixed at 50 mm and 5 m, respectively?

Consider laminar flow of a fluid with Pr 4 that undergoes a combined entrance process within a constant surface temperature tube of length L xfd,t with a flow rate of . An engineer suggests that the total heat transfer rate can be improved if the tube is divided into N shorter tubes, each of length LN L/N with a flow rate of . Determine an expression for the ratio of the heat transfer coefficient averaged over the N tubes, each experiencing a combined entrance process, to the heat transfer coefficient averaged over the single tube, . 8

A common procedure for cooling a high-performance computer chip involves joining the chip to a heat sink within which circular microchannels are machined. During operation, the chip produces a uniform heat flux at its interface with the heat sink, while a liquid coolant (water) is routed through the channels. Consider a square chip and heat sink, each L on a side, with microchannels of diameter D and pitch S C1D, where the constant C1 is greater than unity. Water is supplied at an inlet temperature Tm,i and a total mass flow rate (for the entire heat sink). (a) Assuming that is dispersed in the heat sink such that a uniform heat flux is maintained at the surface of each channel, obtain expressions for the longitudinal distributions of the mean fluid, Tm(x), and surface, Ts(x), temperatures in each channel. Assume laminar, fully developed flow throughout each channel, and express your results in terms of m . , q c, C1, D, and/or L, as well as appropriate thermophysical properties. (b) For L 12 mm, D 1 mm, C1 2, q c 20 W/cm2 , m . 0.010 kg/s, and Tm,i 290 K, compute and plot the temperature distributions Tm(x) and Ts(x). (c) A common objective in designing such heat sinks is to maximize while maintaining the heat sink at an acceptable temperature. Subject to prescribed values of L 12 mm and Tm,i 290 K and the constraint that Ts,max  50 C, explore the effect on of variations in heat sink design and operating conditions. 8.56

One way to cool chips mounted on the circuit boards of a computer is to encapsulate the boards in metal frames that provide efficient pathways for conduction to supporting cold plates. Heat generated by the chips is then dissipated by transfer to water flowing through passages drilled in the plates. Because the plates are made from a metal of large thermal conductivity (typically aluminium or copper), they may be assumed to be at a temperature, Ts,cp (a) Consider circuit boards attached to cold plates of height H 750 mm and width L 600 mm, each with N 10 holes of diameter D 10 mm. If operating conditions maintain plate temperatures of Ts,cp 32 C with water flow at per passage and Tm,i 7 C, how much heat may be dissipated by the circuit boards? (b) To enhance cooling, thereby allowing increased power generation without an attendant increase in system temperatures, a hybrid cooling scheme may be used. The scheme involves forced airflow over the encapsulated circuit boards, as well as water flow through the cold plates. Consider conditions for which Ncb 10 circuit boards of width W 350 mm are attached to the cold plates and their average surface temperature is Ts,cb 47 C when Ts,cp 32 C. If air is in parallel flow over the plates with u
10 m/s and T
7 C, how much of the heat generated by the circuit boards is transferred to the air? 8.57 Refri

Refrigerant-134a is being transported at 0.1 kg/s through a Teflon tube of inside diameter Di 25 mm and outside diameter Do 28 mm, while atmospheric air at V 25 m/s and 300 K is in cross flow over the tube. What is the heat transfer per unit length of tube to Refrigerant-134a at 240 K? 8

Oil at 150 C flows slowly through a long, thin-walled pipe of 30-mm inner diameter. The pipe is suspended in a room for which the air temperature is 20 C and the convection coefficient at the outer tube surface is 11 W/m2
K. Estimate the heat loss per unit length of tube

Exhaust gases from a wire processing oven are discharged into a tall stack, and the gas and stack surface temperatures at the outlet of the stack must be estimated. Knowledge of the outlet gas temperature Tm,o is useful for predicting the dispersion of effluents in the thermal plume, while knowledge of the outlet stack surface temperature Ts,o indicates whether condensation of the gas products will occur. The thin-walled, cylindrical stack is 0.5 m in diameter and 6.0 m high. The exhaust gas flow rate is 0.5 kg/s, and the inlet temperature is 600 C (a) Consider conditions for which the ambient air temperature and wind velocity are 4 C and 5 m/s, respectively. Approximating the thermophysical properties of the gas as those of atmospheric air, estimate the outlet gas and stack surface temperatures for the given conditions. (b) The gas outlet temperature is sensitive to variations in the ambient air temperature and wind velocity. For T
25 C, 5 C, and 35 C, compute and plot the gas outlet temperature as a function of wind velocity for 2  V  10 m/s.

A hot fluid passes through a thin-walled tube of 10-mm diameter and 1-m length, and a coolant at T
25 C is in cross flow over the tube. When the flow rate is m
18 kg/h and the inlet temperature is Tm,i 85 C, the outlet temperature is Tm,o 78 C. m Assuming fully developed flow and thermal conditions in the tube, determine the outlet temperature, Tm,o, if the flow rate is increased by a factor of 2. That is, m
36 kg/h, with all other conditions the same. The thermophysical properties of the hot fluid are
1079 kg/m3 , cp 2637 J/kg
K, 0.0034 N
s/m2 , and k 0.261 W/m
K. 8.6

Consider a thin-walled tube of 10-mm diameter and 2-m length. Water enters the tube from a large reservoir at m
0.2 kg/s and Tm,i 47 C. (a) If the tube surface is maintained at a uniform temperature of 27 C, what is the outlet temperature of the water, Tm,o? To obtain the properties of water, assume an average mean temperature of T _ m 300 K. (b) What is the exit temperature of the water if it is heated by passing air at T
100 C and V 10 m/s in cross flow over the tube? The properties of air may be evaluated at an assumed film temperature of Tf 350 K. (c) In the foregoing calculations, were the assumed values of T _ m and Tf appropriate? If not, use properly evaluated properties and recompute Tm,o for the conditions of part (b). 8.62

Water at a flow rate of m
0.215 kg/s is cooled from 70 C to 30 C by passing it through a thin-walled tube of diameter D 50 mm and maintaining a coolant at T
15 C in cross flow over the tube. (a) What is the required tube length if the coolant is air and its velocity is V 20 m/s? (b) What is the tube length if the coolant is water and V 2 m/s? 8.6

The problem of heat losses from a fluid moving through a buried pipeline has received considerable attention. Practical applications include the trans-Alaska pipeline, as well as power plant steam and water distribution lines. Consider a steel pipe of diameter D that is used to transport oil flowing at a rate m
o through a cold region. The pipe is covered with a layer of insulation of thickness t and thermal conductivity ki and is buried in soil to a depth z (distance from the soil surface to the pipe centerline). Each section of pipe is of length L and extends between pumping stations in which the oil is heated to ensure low viscosity and hence low pump power requirements. The temperature of the oil entering the pipe from a pumping station and the temperature of the ground above the pipe are designated as Tm,i and Ts, respectively, and are known. Consider conditions for which the oil (o) properties may be approximated as
o 900 kg/m3 , cp,o 2000 J/kg
K, o 8.5  104 m2 /s, ko 0.140 W/m
K, 548 Pro 104 ; the oil flow rate is m
o 500 kg/s; and the pipe diameter is 1.2 m. (a) Expressing your results in terms of D, L, z, t, m
o, Tm,i , and Ts, as well as the appropriate oil (o), insulation (i), and soil (s) properties, obtain all the expressions needed to estimate the temperature Tm,o of the oil leaving the pipe. (b) If Ts 40 C, Tm,i 120 C, t 0.15 m, ki 0.05 W/m
K, ks 0.5 W/m
K, z 3 m, and L 100 km, what is the value of Tm,o? What is the total rate of heat transfer q from a section of the pipeline? (c) The operations manager wants to know the tradeoff between the burial depth of the pipe and insulation thickness on the heat loss from the pipe. Develop a graphical representation of this design information. 8.64 T

To maintain pump power requirements per unit flow rate below an acceptable level, operation of the oil pipeline of Problem 8.63 is subject to the constraint that the oil exit temperature Tm,o exceed 110 C. For the values of Tm,i , Ts, D, ti , z, L, and ki prescribed in Problem 8.63, operating parameters that are variable and affect Tm,o are the thermal conductivity of the soil and the flow rate of the oil. Depending on soil composition and moisture and the demand for oil, representative variations are 0.25  ks  1.0 W/m
K and 250  m
o  500 kg/s. Using the properties prescribed in Problem 8.63, determine the effect of the foregoing variations on Tm,o and the total heat rate q. What is the worst case operating condition? If necessary, what adjustments could be made to ensure that Tm,o  110 C for the worst case conditions?

Consider a thin-walled, metallic tube of length L 1 m and inside diameter Di 3 mm. Water enters the tube at m
0.015 kg/s and Tm,i 97 C. (a) What is the outlet temperature of the water if the tube surface temperature is maintained at 27 C? (b) If a 0.5-mm-thick layer of insulation of k 0.05 W/m
K is applied to the tube and its outer surface is maintained at 27 C, what is the outlet temperature of the water? (c) If the outer surface of the insulation is no longer maintained at 27 C but is allowed to exchange heat by free convection with ambient air at 27 C, what is the outlet temperature of the water? The free convection heat transfer coefficient is 5 W/m2
K. 8.6

A circular tube of diameter D 0.2 mm and length L 100 mm imposes a constant heat flux of q 20  103 W/m2 on a fluid with a mass flow rate of m
0.1 g/s. For an inlet temperature of Tm,i 29 C, determine the tube wall temperature at x L for pure water. Evaluate flui fluid properties at T _ 300 K. For the same conditions, determine the tube wall temperature at x L for the nanofluid of Example 2.2.

Repeat Problem 8.66 for a circular tube of diameter D 2 mm, an applied heat flux of q 200,000 W/m2 , and a mass flow rate of m
10 g/s. 8

Heat is to be removed from a reaction vessel operating at 75 C by supplying water at 27 C and 0.12 kg/s through a thin-walled tube of 15-mm diameter. The convection coefficient between the tube outer surface and the fluid in the vessel is 3000 W/m2
K. (a) If the outlet water temperature cannot exceed 47 C, what is the maximum rate of heat transfer from the vessel? (b) What tube length is required to accomplish the heat transfer rate of part (a)?

A heating contractor must heat 0.2 kg/s of water from 15 C to 35 C using hot gases in cross flow over a thinwalled tube. Your assignment is to develop a series of design graphs that can be used to demonstrate acceptable combinations of tube dimensions (D and L) and of hot gas conditions (T
and V) that satisfy this requirement. In your analysis, consider the following parameter ranges: D 20, 30, or 40 mm; L 3, 4, or 6 m; T
250, 375, or 500 C; and 20  V  40 m/s.

A thin-walled tube with a diameter of 6 mm and length of 20 m is used to carry exhaust gas from a smoke stack to the laboratory in a nearby building for analysis. The gas enters the tube at 200 C and with a mass flow rate of 0.003 kg/s. Autumn winds at a temperature of 15 C blow directly across the tube at a velocity of 5 m/s. Assume the thermophysical properties of the exhaust gas are those of air. (a) Estimate the average heat transfer coefficient for the exhaust gas flowing inside the tube. (b) Estimate the heat transfer coefficient for the air flowing across the outside of the tube. (c) Estimate the overall heat transfer coefficient U and the temperature of the exhaust gas when it reaches the laboratory

A 50-mm-diameter, thin-walled metal pipe covered with a 25-mm-thick layer of insulation (0.085 W/m
K) and carrying superheated steam at atmospheric pressure is suspended from the ceiling of a large room. The steam temperature entering the pipe is 120 C, and the air temperature is 20 C. The convection heat transfer coefficient on the outer surface of the covered pipe is 10 W/m2
K. If the velocity of the steam is 10 m/s, at what point along the pipe will the steam begin condensing?

A 50-mm-diameter, thin-walled metal pipe covered with a 25-mm-thick layer of insulation (0.085 W/m
K) and carrying superheated steam at atmospheric pressure is suspended from the ceiling of a large room. The steam temperature entering the pipe is 120 C, and the air temperature is 20 C. The convection heat transfer coefficient on the outer surface of the covered pipe is 10 W/m2
K. If the velocity of the steam is 10 m/s, at what point along the pipe will the steam begin condensing?

Pressurized water at Tm,i 200 C is pumped at m
2 kg/s from a power plant to a nearby industrial user through a thin-walled, round pipe of inside diameter D 1 m. The pipe is covered with a layer of insulation of thickness t 0.15 m and thermal conductivity k 0.05 W/m
K. The pipe, which is of length L 500 m, is exposed to a cross flow of air at T
10 C and V 4 m/s. Obtain a differential equation that could be used to solve for the variation of the mixed mean temperature of the water Tm(x) with the axial coordinate. As a first approximation, the internal flow may be assumed to be fully developed throughout the pipe. Express your results in terms of m
, V, T
, D, t, k, and appropriate water (w) and air (a) properties. Evaluate the heat loss per unit length of the pipe at the inlet. What is the mean temperature of the water at the outlet? 8.74 W

Water at 290 K and 0.2 kg/s flows through a Teflon tube (k 0.35 W/m
K) of inner and outer radii equal to 10 and 13 mm, respectively. A thin electrical heating tape wrapped around the outer surface of the tube delivers a uniform surface heat flux of 2000 W/m2 , while a convection coefficient of 25 W/m2
K is maintained on the outer surface of the tape by ambient air at 300 K. What is the fraction of the power dissipated by the tape, which is transferred to the water? What is the outer surface temperature of the Teflon tube?

The temperature of flue gases flowing through the large stack of a boiler is measured by means of a thermocouple enclosed within a cylindrical tube as shown. The tube axis is oriented normal to the gas flow, and the thermocouple senses a temperature Tt corresponding to that of the tube surface. The gas flow rate and temperature are designated as m
g and Tg, respectively, and the gas flow may be assumed to be fully developed. The stack is fabricated from sheet metal that is at a uniform temperature Ts and is exposed to ambient air at T
and large surroundings at Tsur. The convection coefficient associated with the outer surface of the duct is designated as ho, while those associated with the inner surface of the duct and the tube surface are designated as hi and ht , respectively. The tube and duct surface emissivities are designated as t and s, respectively. (a) Neglecting conduction losses along the thermocouple tube, develop an analysis that could be used to predict the error (Tg  Tt ) in the temperature measurement. (b) Assuming the flue gas to have the properties of atmospheric air, evaluate the error for Tt 300 C, Ds 0.6 m, Dt 10 mm, m
g 1 kg/s, T
Tsur 27 C, t s 0.8, and ho 25 W/m2
K. 8.76 In

In a biomedical supplies manufacturing process, a requirement exists for a large platen that is to be maintained at 45  0.25 C. The proposed design features the attachment of heating tubes to the platen at a relative spacing S. The thick-walled, copper tubes have an inner diameter of Di 8 mm and are attached to the platen with a high thermal conductivity solder, which provides a contact width of 2Di. The heating fluid (ethylene glycol) flows through each tube at a fixed rate of m
0.06 kg/s. The platen has a thickness of w 25 mm and is fabricated from a stainless steel with a thermal conductivity of 15 W/m
K. Considering the two-dimensional cross section of the platen shown in the inset, perform an analysis to determine the heating fluid temperature Tm and the tube spacing S required to maintain the surface temperature of the platen, T(x, w), at 45  0.25 C, when the ambient temperature is 25 C and the convection coefficient is 100 W/m2
K.

Consider the ground source heat pump of Problem 5.100 under winter conditions for which the liquid is discharged from the heat pump into high-density polyethylene tubing of thickness t 8 mm and thermal conductivity k 0.47 W/m
K. The tubing is routed through soil that maintains a uniform temperature of approximately 10 C at the tube outer surface. The properties of the fluid may be approximated as those of water. (a) For a tube inner diameter and flow rate of Di 25 mm and m
0.03 kg/s and a fluid inlet temperature of Tm,i 0 C, determine the tube outlet temperature (heat pump inlet temperature), Tm,o, as a function of the tube length L for 10  L  50 m. (b) Recommend an appropriate length for the system. How would your recommendation be affected by variations in the liquid flow rate? Pol

For a sharp-edged inlet and a combined entry region, the average Nusselt number may be computed from Equation 8.63, with C 24 ReD 0.23 and m 0.815  2.08  106 ReD [23]. Determine at x/D 10 and 60 for ReD 104 and 105 . 8.7

Fluid enters a thin-walled tube of 5-mm diameter and 2-m length with a flow rate of 0.04 kg/s and a temperature of Tm,i 85 C. The tube surface is maintained at a temperature of Ts 25 C, and for this operating condition, the outlet temperature is Tm,o 31.1 C. What is the outlet temperature if the flow rate is doubled? Fully developed, turbulent flow may be assumed to exist in both cases, and the fluid properties may be assumed to be independent of temperature.

Air at 3  104 kg/s and 27 C enters a rectangular duct that is 1 m long and 4 mm  16 mm on a side. A uniform heat flux of 600 W/m2 is imposed on the duct surface. What is the temperature of the air and of the duct surface at the outlet?

Air at 25 C flows at 30  106 kg/s within 100-mmlong channels used to cool a high thermal conductivity metal mold. Assume the flow is hydrodynamically and thermally fully developed. (a) Determine the heat transferred to the air for a circular channel (D 10 mm) when the mold temperature is 50 C (case A). (b) Using new manufacturing methods (see Problem 8.105), channels of complex cross section can be readily fabricated within metal objects, such as molds. Consider air flowing under the same conditions as in case A, except now the channel is segmented into six smaller triangular sections. The flow area of case A is equal to the total flow area of case B. Determine the heat transferred to the air for the segmented channel. (c) Compare the pressure drops for cases A and B.

A cold plate is an active cooling device that is attached to a heat-generating system in order to dissipate the heat while maintaining the system at an acceptable temperature. It is typically fabricated from a material of high thermal conductivity, kcp, within which channels are machined and a coolant is passed. Consider a copper cold plate of height H and width W on a side, within which water passes through square channels of width w h. The transverse spacing between channels  is twice the spacing between the sidewall of an outer channel and the sidewall of the cold plate. Consider conditions for which equivalent heat-generating systems are attached to the top and bottom of the cold plate, maintaining the corresponding surfaces at the same temperature Ts. The mean velocity and inlet temperature of the coolant are um and Tm,i , respectively. (a) Assuming fully developed turbulent flow throughout each channel, obtain a system of equations that may be used to evaluate the total rate of heat transfer to the cold plate, q, and the outlet temperature of the water, Tm,o, in terms of the specified parameters. (b) Consider a cold plate of width W 100 mm and height H 10 mm, with 10 square channels of width w 6 mm and a spacing of  4 mm between channels. Water enters the channels at a temperature of Tm,i 300 K and a velocity of um 2 m/s. If the top and bottom cold plate surfaces are at Ts 360 K, what is the outlet water temperature and the total rate of heat transfer to the cold plate? The thermal conductivity of the copper is 400 W/m
K, while average properties of the water may be taken to be
984 kg/m3 , cp 4184 J/kg
K, 489  106 N
s/m2 , k 0.65 W/m
K, and Pr 3.15. Is this a good cold plate design? How could its performance be improved? 8.83 The col

The cold plate design of Problem 8.82 has not been optimized with respect to selection of the channel width, and we wish to explore conditions for which the rate of heat transfer may be enhanced. Assume that the width and height of the copper cold plate are fixed at W 100 mmand H 10 mm, while the channel height and spacing between channels are fixed at h 6 mm and  4 mm. The mean velocity and inlet temperature of the water are maintained at um 2 m/s and Tm,i 300 K, while equivalent heat-generating systems attached to the top and bottom of the cold plate maintain the corresponding surfaces at 360 K. Evaluate the effect of changing the channel width, and hence the number of channels, on the rate of heat transfer to the cold plate. Include consideration of the limiting case for which w 96 mm (one channel). 8.8

A device that recovers heat from high-temperature combustion products involves passing the combustion gas between parallel plates, each of which is maintained at 350 K by water flow on the opposite surface. The plate separation is 40 mm, and the gas flow is fully developed. The gas may be assumed to have the properties of atmospheric air, and its mean temperature and velocity are 1000 K and 60 m/s, respectively. (a) What is the heat flux at the plate surface? (b) If a third plate, 20 mm thick, is suspended midway between the original plates, what is the surface heat flux for the original plates? Assume the temperature and flow rate of the gas to be unchanged and radiation effects to be negligible.

Air at 1 atm and 285 K enters a 2-m-long rectangular duct with cross section 75 mm  150 mm. The duct is maintained at a constant surface temperature of 400 K, and the air mass flow rate is 0.10 kg/s. Determine the heat transfer rate from the duct to the air and the air outlet temperature

A double-wall heat exchanger is used to transfer heat between liquids flowing through semicircular copper tubes. Each tube has a wall thickness of t 3 mm and an inner radius of ri 20 mm, and good contact is maintained at the plane surfaces by tightly wound straps. The tube outer surfaces are well insulated. (a) If hot and cold water at mean temperatures of Th,m 330 K and Tc,m 290 K flow through the St adjoining tubes at m
h m
c 0.2 kg/s, what is the rate of heat transfer per unit length of tube? The wall contact resistance is 105 m2
K/W. Approximate the properties of both the hot and cold water as 800  106 kg/s
m, k 0.625 W/m
K, and Pr 5.35. Hint: Heat transfer is enhanced by conduction through the semicircular portions of the tube walls, and each portion may be subdivided into two straight fins with adiabatic tips. (b) Using the thermal model developed for part (a), determine the heat transfer rate per unit length when the fluids are ethylene glycol. Also, what effect will fabricating the exchanger from an aluminum alloy have on the heat rate? Will increasing the thickness of the tube walls have a beneficial effect? 8.87

Consider laminar, fully developed flow in a channel of constant surface temperature Ts. For a given mass flow rate and channel length, determine which rectangular channel, b/a 1.0, 1.43, or 2.0, will provide the highest heat transfer rate. Is this heat transfer rate greater than, equal to, or less than the heat transfer rate associated with a circular tube?

You have been asked to perform a feasibility study on the design of a blood warmer to be used during the transfusion of blood to a patient. This exchanger is to heat blood taken from the bank at 10 C to 37 C at a flow rate of 200 ml/min. The blood passes through a rectangular cross-section tube, 6.4 mm  1.6 mm, which is sandwiched between two plates held at a constant temperature of 40 C. (a) Compute the length of the tubing required to achieve the desired outlet conditions at the speci- fied flow rate. Assume the flow is fully developed and the blood has the same properties as water. (b) Assess your assumptions and indicate whether your analysis over- or underestimates the necessary length.

A coolant flows through a rectangular channel (gallery) within the body of a mold used to form metal injection parts. The gallery dimensions are a 90 mm and b 9.5 mm, and the fluid flow rate is 1.3  103 m3 /s. The coolant temperature is 15 C, and the mold wall is at an approximately uniform temperature of 140 C. To minimize corrosion damage to the expensive mold, it is customary to use a heat transfer fluid such as ethylene glycol, rather than process water. Compare the convection coefficients of water and ethylene glycol for this application. What is the tradeoff between thermal performance and minimizing corrosion?

An electronic circuit board dissipating 50 W is sandwiched between two ducted, forced-air-cooled heat sinks. The sinks are 150 mm in length and have 20 rectangular passages 6 mm  25 mm. Atmospheric air at a volumetric flow rate of 0.060 m3 /s and 27 C is drawn through the sinks by a blower. Estimate the operating temperature of the board and the pressure drop across the sinks

To slow down large prime movers like locomotives, a process termed dynamic electric braking is used to switch the traction motor to a generator mode in which mechanical power from the drive wheels is absorbed and used to generate electrical current. As shown in the schematic, the electric power is passed through a resistor grid (a), which consists of an array of metallic blades electrically connected in series (b). The blade material is a high-temperature, high electrical resistivity alloy, and the electrical power is dissipated as heat by internal volumetric generation. To cool the blades, a motor-fan moves high-velocity air through the grid. (a) Treating the space between the blades as a rectangular channel of 220-mm  4-mm cross section and 70-mm length, estimate the heat removal rate per blade if the airstream has an inlet temperature and velocity of 25 C and 50 m/s, respectively, while the blade has an operating temperature of 600 C. (b) On a locomotive pulling a 10-car train, there may be 2000 of these blades. Based on your result from part (a), how long will it take to slow a train whose total mass is 106 kg from a speed of 120 km/h to 50 km/h using dynamic electric braking?

A printed circuit board (PCB) is cooled by laminar, fully developed airflow in adjoining, parallel-plate channels of length L and separation distance a. The channels may be assumed to be of infinite extent in the transverse direction, and the upper and lower surfaces are insulated. The temperature Ts of the PCB board is uniform, and airflow with an inlet temperature of Tm,i is driven by a pressure difference p. Calculate the average heat removal rate per unit area (W/m2 ) from the PCB

Water at m
0.02 kg/s and Tm,i 20 C enters an annular region formed by an inner tube of diameter Di 25 mm and an outer tube of diameter Do 100 mm. Saturated steam flows through the inner tube, maintaining its surface at a uniform temperature of Ts,i 100 C, while the outer surface of the outer tube is well insulated. If fully developed conditions may be assumed throughout the annulus, how long must the system be to provide an outlet water temperature of 75 C? What is the heat flux from the inner tube at the outlet? 8.9

For the conditions of Problem 8.93, how long must the annulus be if the water flow rate is 0.30 kg/s instead of 0.02 kg/s?

Referring to Figure 8.11, consider conditions in an annulus having an outer surface that is insulated (q o 0) and a uniform heat flux q i at the inner surface. Fully developed, laminar flow may be assumed to exist. (a) Determine the velocity profile u(r) in the annular region. (b) Determine the temperature profile T(r) and obtain an expression for the Nusselt number Nui associated with the inner surface.

Consider the air heater of Problem 8.38, but now with airflow through the annulus and steam flow through the inner tube. For the prescribed conditions and an outer tube diameter of Do 65 mm, determine the outlet temperature and pressure of the air, as well as the mass rate of steam condensation.

Consider a concentric tube annulus for which the inner and outer diameters are 25 and 50 mm. Water enters the annular region at 0.04 kg/s and 25 C. If the inner tube wall is heated electrically at a rate (per unit length) of q 4000 W/m, while the outer tube wall is insulated, how long must the tubes be for the water to achieve an outlet temperature of 85 C? What is the inner tube surface temperature at the outlet, where fully developed conditions may be assumed?

It is common practice to recover waste heat from an oilor gas-fired furnace by using the exhaust gases to preheat the combustion air. A device commonly used for this purpose consists of a concentric pipe arrangement for which the exhaust gases are passed through the inner pipe, while the cooler combustion air flows through an annular passage around the pipe. Consider conditions for which there is a uniform heat transfer rate per unit length, qi 1.25  105 W/m, from the exhaust gases to the pipe inner surface, while air flows through the annular passage at a rate of m
a 2.1 kg/s. The thin-walled inner pipe is of diameter Di 2 m, while the outer pipe, which is well insulated from the surroundings, is of diameter Do 2.05 m. The air properties may be taken to be cp 1030 J/kg
K, 270  107 N
s/m2 , k 0.041 W/m
K, and Pr 0.68. (a) If air enters at Ta,1 300 K and L 7 m, what is the air outlet temperature Ta,2? (b) If the airflow is fully developed throughout the annular region, what is the temperature of the inner pipe at the inlet (Ts,i,1) and outlet (Ts,i,2) sections of the device? What is the outer surface temperature Ts,o,1 at the inlet? 8.99 A co

A concentric tube arrangement, for which the inner and outer diameters are 80 mm and 100 mm, respectively, is used to remove heat from a biochemical reaction occurring in a 1-m-long settling tank. Heat is generated uniformly within the tank at a rate of 105 W/m3 , and water is supplied to the annular region at a rate of 0.2 kg/s. (a) Determine the inlet temperature of the supply water that will maintain an average tank surface temperature of 37 C. Assume fully developed flow and thermal conditions. Is this assumption reasonable? (b) It is desired to have a slight, axial temperature gradient on the tank surface, since the rate of the biochemical reaction is highly temperature dependent. Sketch the axial variation of the water and surface temperatures along the flow direction for the following two cases: (i) the fully developed conditions of part (a), and (ii) conditions for which entrance effects are important. Comment on features of the temperature distributions. What change to the system or operating conditions would you make to reduce the surface temperature gradient?

Consider the air cooling system and conditions of Problem 8.31, but with a prescribed pipe length of L 15 m. (a) What is the air outlet temperature, Tm,o? What is the fan power requirement? (b) The convection coefficient associated with airflow in the pipe may be increased twofold by inserting a coiled spring along the length of the pipe to disrupt flow conditions near the inner surface. If such a heat transfer enhancement scheme is adopted, what is the attendant value of Tm,o? Use of the insert would not come without a corresponding increase in the fan power requirement. What is the power requirement if the friction factor is increased by 50%? (c) After extended exposure to the water, a thin coating of organic matter forms on the outer surface of the pipe, and its thermal resistance (for a unit area of the outer surface) is R t,o 0.050 m2
K/W. What is the corresponding value of Tm,o without the insert of part (b)?

Consider sterilization of the pharmaceutical product of Problem 8.27. To avoid any possibility of heating the product to an unacceptably high temperature, atmospheric steam is condensed on the exterior of the tube instead of using the resistance heater, providing a uniform surface temperature, Ts 100 C. (a) For the conditions of Problem 8.27, determine the required length of straight tube, Ls, that would be needed to increase the mean temperature of the pharmaceutical product from 25 C to 75 C. (b) Consider replacing the straight tube with a coiled tube characterized by a coil diameter C 100 mm and a coil pitch S 25 mm. Determine the overall C length of the coiled tube, Lcl (i.e., the product of the tube pitch and the number of coils), necessary to increase the mean temperature of the pharmaceutical to the desired value. (c) Calculate the pressure drop through the straight tube and through the coiled tube. (d) Calculate the steam condensation rate

An engineer proposes to insert a solid rod of diameter Di into a circular tube of diameter Do to enhance heat transfer from the flowing fluid of temperature Tm to the outer tube wall of temperature Ts,o. Assuming laminar flow, calculate the ratio of the heat flux from the fluid to the outer tube wall with the rod to the heat flux without the rod, q o/q o,wo, for Di/Do 0, 0.10, 0.25 and 0.50. The rod is placed concentrically within the tube.

An electrical power transformer of diameter 230 mm and height 500 mm dissipates 1000 W. It is desired to maintain its surface temperature at 47 C by supplying ethylene glycol at 24 C through thin-walled tubing of 20-mm diameter welded to the lateral surface of the transformer. All the heat dissipated by the transformer is assumed to be transferred to the ethylene glycol. Assuming the maximum allowable temperature rise of the coolant to be 6 C, determine the required coolant flow rate, the total length of tubing, and the coil pitch S between turns of the tubing.

A bayonet cooler is used to reduce the temperature of a pharmaceutical fluid. The pharmaceutical fluid flows through the cooler, which is fabricated of 10-mmdiameter, thin-walled tubing with two 250-mm-long straight sections and a coil with six and a half turns and a coil diameter of 75 mm. A coolant flows outside the cooler, with a convection coefficient at the outside surface of ho 500 W/m2
K and a coolant temperature of 20 C. Consider the situation where the pharmaceutical fluid enters at 90 C with a mass flow rate of 0.005 kg/s. The pharmaceutical has the following properties:
1200 kg/m3 , 4  103 N
s/m2 , cp 2000 J/kg
K, and k 0.5 W/m
K. Cool (a) Determine the outlet temperature of the pharmaceutical fluid. (b) It is desired to further reduce the outlet temperature of the pharmaceutical. However, because the cooling process is just one part of an intricate processing operation, flow rates cannot be changed. A young engineer suggests that the outlet temperature might be reduced by inserting stainless steel coiled springs into the straight sections of the cooler with the notion that the springs will disturb the flow adjacent to the inner tube wall and, in turn, increase the heat transfer coefficient at the inner tube wall. A senior engineer asserts that insertion of the springs should double the heat transfer coefficient at the straight inner tube walls. Determine the outlet temperature of the pharmaceutical fluid with the springs inserted into the tubes, assuming the senior engineer is correct in his assertion. (c) Would you expect the outlet temperature of the pharmaceutical to depend on whether the springs have a left-hand or right-hand spiral? Why?

The mold used in an injection molding process consists of a top half and a bottom half. Each half is 60 mm  60 mm  20 mm and is constructed of metal (
7800 kg/m3 , c 450 J/kg
K). The cold mold (100 C) is to be heated to 200 C with pressurized water (available at 275 C and a total flow rate of 0.02 kg/s) prior to injecting the thermoplastic material. The injection takes only a fraction of a second, and the hot mold (200 C) is subsequently cooled with cold water (available at 25 C and a total flow rate of 0.02 kg/s) prior to ejecting the molded part. After part ejection, which also takes a fraction of a second, the process is repeated. (a) In conventional mold design, straight cooling (heating) passages are bored through the mold in a location where the passages will not interfere with the molded part. Determine the initial heating rate and the initial cooling rate of the mold when five 5-mm-diameter, 60-mm-long passages are bored in each half of the mold (10 passages total). The velocity distribution of the water is fully developed at the entrance of each passage in the hot (or cold) mold. (b) New additive manufacturing processes, known as selective freeform fabrication, or SFF, are used to construct molds that are configured with conformal cooling passages. Consider the same mold as before, but now a 5-mm-diameter, coiled, conformal cooling passage is designed within each half of the SFF-manufactured mold. Each of the two coiled passages has N 2 turns. The coiled passage does not interfere with the molded part. The conformal channels have a coil diameter C 50 mm. The total water flow remains the same as in part (a) (0.01 kg/s per coil). Determine the initial heating rate and the initial cooling rate of the mold. (c) Compare the surface areas of the conventional and conformal cooling passages. Compare the rate at which the mold temperature changes for molds con- figured with the conventional and conformal heating and cooling passages. Which cooling passage, conventional or conformal, will enable production of more parts per day? Neglect the presence of the thermoplastic material.

Consider the pharmaceutical product of Problem 8.27. Prior to finalizing the manufacturing process, test trials are performed to experimentally determine the dependence of the shelf life of the drug as a function of the sterilization temperature. Hence, the sterilization temperature must be carefully controlled in the trials. To promote good mixing of the pharmaceutical and, in turn, relatively uniform outlet temperatures across the exit tube area, experiments are performed using a device that is constructed of two interwoven coiled tubes, each of 10-mm diameter. The thin-walled tubing is welded to a solid high thermal conductivity rod of diameter Dr 40 mm. One tube carries the pharmaceutical product at a mean velocity of up 0.1 m/s and inlet temperature of 25 C, while the second tube carries pressurized liquid water at uw 0.12 m/s with an inlet temperature of 127 C. The tubes do not contact each other but are each welded to the solid metal rod, with each tube making 20 turns around the rod. The exterior of the apparatus is well insulated. ( (a) Determine the outlet temperature of the pharmaceutical product. Evaluate the liquid water properties at 380 K. (b) Investigate the sensitivity of the pharmaceuticals outlet temperature to the velocity of the pressurized water over the range 0.10 uw 0.25 m/s.

An extremely effective method of cooling high-powerdensity silicon chips involves etching microchannels in the back (noncircuit) surface of the chip. The channels are covered with a silicon cap, and cooling is maintained by passing water through the channels. Consider a chip that is 10 mm  10 mm on a side and in which fifty 10-mm-long rectangular microchannels, each of width W 50 m and height H 200 m, have been etched. Consider operating conditions for which water enters each microchannel at a temperature of 290 K and a flow rate of 104 kg/s, while the chip and cap are at a uniform temperature of 350 K. Assuming fully developed flow in the channel and that all the heat dissipated by the circuits is transferred to the water, determine the water outlet temperature and the chip power dissipation. Water properties may be evaluated at 300 K. 8.

An ideal gas flows within a small diameter tube. Derive an expression for the transition density of the gas
c below which microscale effects must be accounted for. Express your result in terms of the gas molecule diameter, universal gas constant, Boltzmanns constant, and the tube diameter. Evaluate the transition density for a D 10-m-diameter tube for hydrogen, air, and carbon dioxide. Compare the calculated transition densities with the gas density at atmospheric pressure and T 23 C. 8

Consider the microchannel cooling arrangement of Problem 8.107. However, instead of assuming the entire chip and cap to be at a uniform temperature, adopt a more conservative (and realistic) approach that prescribes a temperature of Ts 350 K at the base of the channels (x 0) and allows for a decrease in temperature with increasing x along the side walls of each channel. (a) For the operating conditions prescribed in Problem 8.107 and a chip thermal conductivity of kch 140 W/m
K, determine the water outlet temperature and the chip power dissipation. Heat transfer from the sides of the chip to the surroundings and from the side walls of a channel to the cap may be neglected. Note that the spacing between channels,  S  W, is twice the spacing between the side wall of an outer channel and the outer surface of the chip. The channel pitch is S L/N, where L 10 mm is the chip width and N 50 is the number of channels. (b) The channel geometry prescribed in Problem 8.107 and considered in part (a) is not optimized, and larger heat rates may be dissipated by adjusting related dimensions. Consider the effect of reducing the pitch to a value of S 100 m, while retaining a width of W 50 m and a flow rate per channel of m
1 104 kg/s. 8.110 The

The onset of turbulence in a gas flowing within a circular tube occurs at ReD,c
2300, while a transition from incompressible to compressible flow occurs at a critical Mach number of Mac
0.3. Determine the critical tube diameter Dc, below which incompressible turbulent flow and heat transfer cannot exist for (i) air, (ii) CO2, (iii) He. Evaluate properties at atmospheric pressure and a temperature of T 300 K.

Due to its comparatively large thermal conductivity, water is a preferred fluid for convection cooling. However, in applications involving electronic devices, water must not come into contact with the devices, which would therefore have to be hermetically sealed. To circumvent related design and operational complexities and to ensure that the devices are not rendered inoperable by contact with the coolant, a dielectric fluid is commonly used in lieu of water. Many gases have excellent dielectric characteristics, and despite its poor heat transfer properties, air is the common choice for electronic cooling. However, there is an alternative, which involves a class of perfluorinated liquids that are excellent dielectrics and have heat transfer properties superior to those of gases. Consider the microchannel chip cooling application of Problem 8.109 but now for a perfluorinated liquid with properties of cp 1050 J/kg
K, k 0.065 W/m
K, 0.0012 N
s/m2 , and Pr 15. (a) For channel dimensions of H 200 m, W 50 m, and S 20 m, a chip thermal conductivity of kch 140 W/m
K and width L 10 mm, a channel base temperature (x 0) of Ts 350 K, a channel inlet temperature of Tm,i 290 K, and a flow rate of m
1 104 kg/s per channel, determine the outlet temperature and the chip power dissipation for the dielectric liquid. (b) Consider the foregoing conditions, but with air at a flow rate of m
1 106 kg/s used as the coolant. Using properties of cp 1007 J/kg
K, k 0.0263 W/m
K, and 185  107 N
s/m2 , determine the air outlet temperature and the chip power dissipation. 8.112 Many of the so

Many of the solid surfaces for which values of the thermal and momentum accommodation coefficients have been measured are quite different from those used in micro- and nanodevices. Plot the Nusselt number NuD associated with fully developed laminar flow with constant surface heat flux versus tube diameter for 1 m  D  1 mm and (i) t 1, p 1, (ii) t 0.1, p 0.1, (iii) t 1, p 0.1, and (iv) t 0.1, p 1. For tubes of what diameter do the accommodation coefficients begin to influence convection heat transfer? For which combination of t and p does the Nusselt number exhibit the least sensitivity to changes in the diameter of the tube? Which combination results in Nusselt numbers greater than the conventional fully developed laminar value for constant heat flux conditions, NuD 4.36? Which combination is associated with the smallest Nusselt numbers? What can you say about the ability to predict convection heat transfer coefficients in a small-scale device if the 558 Chap accommodation coefficients are not known for material from which the device is fabricated? Use properties of air at atmospheric pressure and T 300 K.

A novel scheme for dissipating heat from the chips of a multichip array involves machining coolant channels in the ceramic substrate to which the chips are attached. The square chips (Lc 5 mm) are aligned above each of the channels, with longitudinal and transverse pitches of SL ST 20 mm. Water flows through the square cross section (W 5 mm) of each channel with a mean velocity of um 1 m/s, and its properties may be approximated as
1000 kg/m3 , cp 4180 J/kg
K, 855  106 kg/s
m, k 0.610 W/m
K, and Pr 5.8. Symmetry in the transverse direction dictates the existence of equivalent conditions for each substrate section of length Ls and width ST. (a) Consider a substrate whose length in the flow direction is Ls 200 mm, thereby providing a total of NL 10 chips attached in-line above each flow channel. To a good approximation, all the heat dissipated by the chips above a channel may be assumed to be transferred to the water flowing through the channel. If each chip dissipates 5 W, what is the temperature rise of the water passing through the channel? (b) The chip-substrate contact resistance is R t,c 0.5  104 m2
K/W, and the three-dimensional conduction resistance for the Ls  ST substrate section is Rcond 0.120 K/W. If water enters the substrate at 25 C and is in fully developed flow, estimate the temperature Tc of the chips and the temperature Ts of the substrate channel surface. 8.114 Conside

Consider air flowing in a small-diameter steel tube. Graph the Nusselt number associated with fully developed laminar flow with constant surface heat flux for tube diameters ranging from 1 m  D  1 mm. Evaluate air properties at T 350 K and atmospheric pressure. The thermal and momentum accommodation coefficients are t 0.92 and p 0.87, respectively. Compare the Nusselt number you calculate to the value provided in Equation 8.53, NuD 4.36.

An experiment is designed to study microscale forced convection. Water at Tm,i 300 K is to be heated in a straight, circular glass tube with a 50-m inner diameter and a wall thickness of 1 mm. Warm water at T
350 K, V 2 m/s is in cross flow over the exterior tube surface. The experiment is to be designed to cover the operating range 1  ReD  2000, where ReD is the Reynolds number associated with the internal flow. (a) Determine the tube length L that meets a design requirement that the tube be twice as long as the thermal entrance length associated with the highest Reynolds number of interest. Evaluate water properties at 305 K. (b) Determine the water outlet temperature, Tm,o, that is expected to be associated with ReD 2000. Evaluate the heating water (water in cross flow over the tube) properties at 330 K. (c) Calculate the pressure drop from the entrance to the exit of the tube for ReD 2000. (d) Based on the calculated flow rate and pressure drop in the tube, estimate the height of a column of water (at 300 K) needed to supply the necessary pressure at the tube entrance and the time needed to collect 0.1 liter of water. Discuss how the outlet temperature of the water flowing from the tube, Tm,o, might be measured. 8.11

Determine the tube diameter that corresponds to a 10% reduction in the convection heat transfer coeffi- cient for thermal and momentum accommodation coefficients of t 0.92 and p 0.89, respectively. Determine the channel spacing, a, that is associated with a 10% reduction in h using the same accommodation coefficients. The gas is air at T 350 K and atmospheric pressure for both the tube and the parallel plate configurations. The flow is laminar and fully developed with constant surface heat flux.

An experiment is devised to measure liquid flow and convective heat transfer rates in microscale channels. The mass flow rate through a channel is determined by measuring the amount of liquid that has flowed through the channel and dividing by the duration of the experiment. The mean temperature of the outlet fluid is also measured. To minimize the time needed to perform the experiment (that is, to collect a significant amount of liquid so that its mass and temperature can be accurately measured), arrays of microchannels are typically used. Consider an array of microchannels of circular cross section, each with a nominal diameter of 50 m, fabricated into a copper block. The channels are 20 mm long, and the block is held at 310 K. Water at an inlet temperature of 300 K is forced into the channels from a pressurized plenum, so that a pressure difference of 2.5  106 Pa exists from the entrance to the exit of each channel. In many microscale systems, the characteristic dimensions are similar to the tolerances that can be controlled during the manufacture of the experimental apparatus. Hence, careful consideration of the effect of machining tolerances must be made when interpreting the experimental results. (a) Consider the case in which three microchannels are machined in the copper block. The channel diameters exhibit some deviation due to manufacturing constraints and are of actual diameter 45 m, 50 m, and 55 m, respectively. Calculate the mass flow rate through each of the three channels, along with the mean outlet temperature of each channel. (b) If the water exiting each of the three channels is collected and mixed in a single container, calculate the average flow rate through each of the three channels and the average mixed temperature of the water that is collected from all three channels. (c) The enthusiastic experimentalist uses the average flow rate and the average mixed outlet temperature to analyze the performance of the average (50 m) diameter channel and concludes that flow rates and heat transfer coefficients are increased and decreased, respectively, by about 5% when forced convection occurs in microchannels. Comment on the validity of the experimentalists conclusion. 560