# 1. (Problem 8.23 in the Book)

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1 1. (Problem 8.23 in the Book) SOLUTION Schematic 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 x q s ( x) q o sin L where x is distance measured from the tube inlet. Fluid at an inlet temperature, T m,i flows through the tube at a rate of m. 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, T m,o ; c) the axial distribution of the wall temperature, T s (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 =10,000 W/m 2. Fluid passing through the tube has a flow rate of kg/s, a specific heat of 4180 kj/kgk, an entrance temperature of 25 C, and a convection coefficient of 1000 W/m 2 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. Assumptions 1) Steady state conditions, 2) Applicability of Eq and 3) Turbulent, fully developd flow. Analysis (a) The total rate of heat transfer from the tube to the fluid is

2 (b) the fluid outlet temperature follows from the overall energy balance with knowledge of the total heat rate (c) The axial distribution of the wall temperature can be determined from the rate equation Where, by combining expressions of part (a) and (b), T m,x (x) is Hence, substituting Eq (4) into (3), find (d) To determine the location of the maximum wall temperrature x, where T x (x ) = T s,max, set At this location, the wall temperature is (e) Consider the prescribed conditions for which to compute and plot T m (x) and T s (x), Using Eqs. (4) and (5), the results are plotted as follows

3 The effect of a lower convection coefficient is to increase the wall temperaure. The position of the maximum temperature, T s,max, moves away from the tube exit with decreasing convection coefficient. Comments 1. Because the fow is fully developed and turbulent, assuming h is constant along the entire length of the tube is reasonable. 2. To determine whether the T x (x) distribution has a maximum (rather than a minimum), one 2 2 should evaluate d Ts ( x) dx to show the value is indeed negative.

4 2. (Problem 8.38 in the book) 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 surface. Consider conditions for which air enters a 50-mm-diameter tube at a pressure of 5 atm, a temperature of T m,i = 17 C, and a flow rate of m = 0.03kg/s, while saturated steam at bars condenses on the outer surface of the tube. If the length of the annulus is L = 5 m, what are the outlet temperature T m,o and pressure p o of the air? What is the mass rate at which condensate leaves the annulus. SOLUTION Assumptions 1) Steady state, 2) Outer surface of annulus is adiabatic, 3) Ideal gas with negligible viscous dissipation and pressure variation, 4) Fully-developed flow through the tube, 5) Smooth tube surface and 6) Constant properties. Properties Analysis With a uniform surface temperature, the air outlet temperrature is

5 The rate of heat transfer to the air is And the rate of condenstaion is then Comments 1. With Tm ( Tm, i Tm, o ) / 2 331K, the initial estimate of 325 K is reasonable and iteration is not necessary. 2. For a steam flow rate of 0.01 Kg/Sec, approximately 10 % of the outflow would be in the form of saturated liquid. 3. With L/D i = 100, it is reasonable to assume fully developed flow throughout the tube.

6 3. (Problem 8.53 in the book) 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 air flow rate of 0.01 kg/s is maintained in each tube, determine the air outlet temperature T m,o and the total hear rate q for the tube. b) The air outlet temperature may be controlled by adjusting the tube mass flow rate. Compute and plot T m,o as a function of m for m 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? Schematic Assumptions 1) Steady state, 2) Ideal gas with negligible viscous dissipation and pressure variation, and 3) Negligible tube wall thermal resistance Properties Analysis (a) For m 0. 01Kg/s the Reynold number is 12,810. Hence the flow is turbulent. If fully developed flow is assumed throughout the tube, the Dittus-Boelter correlation may be used to obtain the average Nusselt number.

7 (b) The effect of flowrate on the outlet temperature is plotted below Although h and hence the heat rate increase with increasing m, the increase in q is not linearly proportional to the increase in m and T m,o decreases with increasing m. A flowrate of m = 0.05 Kg/s is not large enough to provide the desired outlet temperautre of 75 C and to achieve this value, a flowrate of Kg/s would be needed. At such a flowrate, N = 1 Kg/s/0.0678Kg/s = tubes would be needed to satisfy the process air requirement. Alternatively, a lower flowrate could be supplied to a larger number of tubes and the discharge mixed with ambient air to satisfy the desired conditions. Requirements of this option are that Where m is the flowrate per tube. Using a larger number of tubes with a smaller flowrate per tube would reduce flow pressure losses and hence provide for reduced operaing costs. Comments With L/D = 5 m/0.05 m = 100, the assumption of fully developed conditions throughout the tube is reasonable.

8 4. (Problem 8.61 in the book) 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 T m,i = 47 C. a) If the tube surface is maintained at a uniform temperature of 27 C, what is the outlet temperature of water, T m,o? To obtain the properties of water, assume an average mean temperature of T m = 300K. 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 T f = 350 K. c) In the foregoing calculations, were the assumed values of T m and T f appropriate? If not, use properly evaluated properties and recomputed T m,o for the conditions of part (b). Schematic Assumptions 1) Steady state, 2) Incompressible liquid with negligible viscous dissipation and negligible axial conduction, 3) Fully developed flow and thermal conditions for internal flow, and 4) Negligible tube wall thermal resistance. Properties Analysis (a) For the constant wall temperature cooling process, Ts = 27 C, the water outlet temperature can be determined from Eq 8.41b, with P = D To estimate the convection coefficient, characterize the flow evaluating properties at T m = 300 K

9 Hence, the flow is turbulent and assuming fully developed (L/D = 200) and using Dittus-Boelter correlation, Eq. 8.60, find h i Substitute h i into Eq. 1, find (b) For the air heating process, T =100 C, the water outlet temperature follows from Eq. 8.45a, To estimate h o, use the Churchill-Bernstein correlation Eq. 7.54, for cross flow over a cylinder using properties at T f = 350 K Assuming h i = 9080 W/m 2 -K as calculated from part (a), fing U and then T m,o (c) The analyses of part (b) are performed considering the appropriate temperatures to evaluate thermophysical properties. For internal and ecternal flow, respectively, Where the average tube wall temperature is evaluated from the thermal circuit,

10 The results of the analyses are summarized in the table along with the results from parts (a) and (b) Note that since ho hi, U is controlled by the value of o h which is evaluated near 350 K for both parts (b) and (c). Hence, it follows that T m,o is not very sensitive to h i which, as seen above, is sensitive to the value of T m.

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