Heat Transfer

1. Heat is flowing steadily in a metal plate whose shape is an infinite rectangle occupying the-a <z <a, y> 0 of the (r, y) plane. The temperature at the point (z,y) is denoted by u(x, y). The sides z = ta are insulated, the temperature approaches zero as yoo, while the side y = 0 is maintained at a fixed temperature -T for a <r <0 and T for 0 < z <a. It is known that u(r,y) satisfies the Laplace equation 8²u F²₂ + = 0 Jy2 i. Sketch by hand the configuration of the metal plate and specify all boundary conditions corresponding to each side of the metal plate ii. Use the method of separation to obtain the solution in the form: یران u(x, y) 1 ΑΣ Σ B n-0 Analyse all three cases of a separation constants (A <,,> 0). Coefficients A, B, C and D (D is a trigonometric expression) have to be calculated and highlighted in your assignment. Full marks are awarded for a complete step by step proof. iii. Take 7 the temperature from part a) to be equal to the last two figures of your student Monash ID number (if ID XXXXXX31, take T-31; ID XXXXXX09, take T-9; ID XXXXX1100, take T=10). And take a = 1. In MATLAB, on the same graph plot the partial sum up to the 50th harmonic of u(x,y) for 10 relevant values y = 0,0.01, 0.02, and continuing with any y of your own choice. Label and ADD a legend to the graph and publish the graph of your solution, and attach it to the assignment. iii. For what value y does the temperature drop to 10% of the initial temperature for 0 < <a? TOTAL=4+18+10+1+2[ncat] =35 marks

1. (60 pts) A plane wall is a composite of two materials, A and B. Material A is 0.5 cm thick and has a thermal material A in contact with the insulated boundary needs to below 90 °C, so that the material A can be stable is applied to material A such that 15 W/cm is generated as heat. The surface temperature T, of the conductivity of k, = 20 W/(mK). The back side of material A is thermally insulated. Electrical current and do not melt. Material B is 0,3 cm thick and has a thermal conductivity of k, = 30 W/(mK). The surface of solid B is exposed to air. The surface temperature, 7,, of material B is 70 °C. The bulk air a) Use shell balance to derive the following temperature profile of material A. The process is at steady temperature, T, is constant at 25 °C. state. T=- -x² + ₂x + G₂ b) Determine the constants G₁ and C₂. Clearly show the boundary conditions used. c) What is the heat flux? Unit must be in SI unit. e) d) Determine the temperature T, (unit in °C) at the interface of materials A and B. of the flowing a Determine the temperature T, (unit in °C) at x = 0 of the insulated surface. Q™ 2KA

2. (13 pts) A long cylinder of certain material with a diameter of 3 inch is initially at a uniform temperature of 80 °F. The cylinder is then placed in a medium at 1000 °F with an associated convective heat-transfer coefficient of 5 Btu/(hr ft²-F). Material properties of the cylinder: the thermal conductivity 213 Btu/(hr ft "F), density 555 lbm/ft, and the specific heat capacity 0.092 Btu/(lbm-F). Can we assume that the temperature gradient within the cylinder is quite small leading to an approximately uniform temperature? Provide your reasoning.

3. (15 pts) You are going to boil an egg in water for breakfast. The egg can be approximated as a sphere with a diameter of 4 cm. Initially, the egg was at room temperature of 20 °C. What is the corresponding Fourier number F. (dimensionless time), if the egg temperature everywhere needs to be at least 65 °C? Indicate how you obtain the F. in the figure.

4. (12 pts) An infinite long fin with rectangular cross section is exposed to the ambient air with T-40 "C and the heat transfer coefficient of 10 W/(m2K). The fin is fabricated from stainless steel with a thermal conductivity of 17 W/(m K). The fin length, width, and thickness are 90 cm, 5 cm and 0.3 cm. respectively. The fin base temperature 7,280 °C. Determine the heat lost by the fin (unit in W). abou

Air enters a pipe (K = 36 BTU/hr ft °F) at a speed of 20 ft/s. The pipe has a length of 80 feet, an internal diameter of 9" and a thickness of 1/8"; Air enters at 400°F and 40 psia and leaves at 200°F. The air exchanges heat with water at ambient pressure, which circulates outside the pipe. The inlet water temperature is 52°F and the outlet temperature is 180ºF. The recommended speed of the water is 350 ft/minute. Water flows countercurrent relative to air. Include in your analysis the effect of fouling on the heat exchanger. The properties of air and water are shown in the following table, as well as the contamination caused by circulating fluids. Re critical (gases) = 1x10^5 Re critical (liquids) = 2300 Tm=Taverage 1) The thermodynamic heat of the air in BTU/hour; 2) The mass flow of water necessary to transfer the thermodynamic heat of the air, in lbm/hour; 3) The dynamic input length in feet 4) The convective coefficient of air 5) The convective coefficient of water 6) The global or total coefficient of Heat Transfer U; 7) The LMTD of the heat exchanger;

C. Which type of heat exchanger provides a higher heat-transfer rate and by what %? 5. (6 pts) Numerical Solution for a Counter-flow Heat Exchanger (by Daisy Fuchs) A countercurrent, shell-and-tube heat exchanger with an effectiveness of 54.3% uses a coolant fluid to cool hot oil. You know that there is one shell pass and four tube passes. Unfortunately, some of the temperature probes and flow meters have broken. The outlet temperature of the oil, flowing at 15 kg/s, is 60 °C, although the inlet temperature is unknown. The coolant enters the shell side of the heat exchanger at 15 °C and exits the heat exchanger at 45 °C. Although the exact flow rate of coolant is unknown, you do know that it is 1 greater than 9 kg/s. The specific heat of the oil is 2.2 kJ/m²-K and the specific heat of the coolant is 4.2 kJ/m²K. The overall heat-transfer coefficient (based on the outside surface area of the tubes) is 1200 W/m²-K and the outside heat-transfer area of the tubes with the correction factored applied is 26.3 m². A. What are two different methods that you can use to solve for parameters in a shell-and-tube heat exchanger? When would you use one method over the other? B. What is the temperature of the hot oil at the inlet of the heat exchanger? Solve the non-linear equation for this temperature using both the secant method in Excel and a built-in solver in Excel. Include a PDF of your spreadsheet. Your spreadsheet should be easy to follow, have all variables labeled (including units), and your name should be on the spreadsheet. Hint: You may need to combine the two methods from Part A to generate an equation for the unknown entrance temperature of the oil. C. What is the exact mass flow rate of the coolant? D. What is the correction factor for this system? What is the outside surface area of the tubes? E. What is the number of transfer units of this heat exchanger? F. If the heat exchange was co-current and had the same number of transfer units, what would the effectiveness be?

2. (1 pt) Gas-phase and Liquid-phase Diffusivities Using Welty et al., look up the diffusivity of carbon dioxide in air at T = 273 K and P = 1 atm and 2 atm and in water at 7 = 283 K and 293 K. Comment on why the following are true: (1) the gas-phase diffusivities are much larger than the liquid-phase diffusivities, (ii) the gas-phase diffusivity decreases with increasing pressure, and (iii) the liquid-phase diffusivity increases with temperature (also true of the gas- phase diffusivity).

3. (1 pt) Effectiveness Factor and NTU Chart for Heat Exchangers Figure 22.23 (c) in Welty et al. is a graph of effectiveness factor (a) vs. the number of transfer units (NTU) for different ratios Cmin/Cmax- Note that & 73% for Cmin/Cmax = 0.25 and NTU = 2. Suppose that Cmin is then doubled (e.g., by doubling the mass flow rate of the fluid with lower capacity coefficient C. What is the new & if Cmax A and U are unchanged?

4. (1 pt) Heating of a Product in Parallel Flow or Counterflow Heat Exchanger: NTU Method Consider a concentric-tube heat exchanger that is 3 m in length. The inside diameter of the inner tube is 0.01 m and the inside diameter of the outer tube is 0.02 m. The wall thickness of the inner tube can be neglected. A fluid enters the inner tube at 10 °C and flows at an average velocity of 0.3 m/s. Water enters the annular region between the tubes at 50 °C and flows at an average velocity of 0.1 m/s. The overall heat- transfer coefficient is 350 W/m²-K. Properties of the fluid: p = 1300 kg/m³, c = 2500 J/kgK. Properties of water: p = 1000 kg/m³, c = 4200 J/kgK. Using the NTU method, determine the temperature of the fluid exiting the tube if: A. It is a counterflow heat exchanger. B. It is a parallel-flow heat exchanger. C. Which type of heat exchanger provides a higher heat-transfer rate and by what %?