Principles of Food and Bioprocess Engineering (FS 231) Exam 2 Part A -- Closed Book (50 points)

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1 Principles of Food and Bioprocess Engineering (FS 231) Exam 2 Part A -- Closed Book (50 points) 1. Are the following statements true or false? (20 points) a. Thermal conductivity of a substance is a measure of how much energy is required to raise the temperature of unit mass (1 kg) of the substance by 1 C b. Insulators have low values of thermal resistance c. The magnitude of h for free convection between a sphere and the surrounding air depends on the diameter of the sphere d. A material with a high value of specific heat conducts heat very quickly from its surface to its center e. Increasing the flow rate of the product or hot water in a double tube heat exchanger will increase the overall heat transfer coefficient for transfer of energy between hot water and the product f. When Biot number is greater than 40, the surface temperature of the solid object is very close to the surrounding fluid temperature g. When Biot number is less than 0.1, the external resistance to heat transfer is negligible h. In using the Heisler chart for a sphere undergoing forced convection, the characteristic dimension (dc) to be used is the diameter of the sphere i. When rapid initial cooling of a product in a double tube heat exchanger is desired, a countercurrent mode of operation is better suited than a co-current mode j. The time taken for the center of a finite cylinder (of radius r and height h ) to attain a certain temperature is equal to the sum of the times taken by the centers of an infinite cylinder (of radius r ) and an infinite slab (of thickness h ) to attain that same temperature

2 2. There may be more than one correct. Answer. Circle ALL correct answers. (5 points) For forced convection, a. The magnitude of h depends predominantly on the temperature difference between the surface of the solid object and the surrounding fluid b. The magnitude of h increases with an increase in the velocity of the fluid c. The magnitude of h increases with an increase in the viscosity of the fluid d. The amount of energy transferred between the hot object and surrounding fluid depends on the temperature difference between the surface of the solid object and the surrounding fluid 3. There may be more than one correct. Answer. Circle ALL correct answers. (5 points) During unsteady state heat transfer, the temperature ratio, (TR) = (T a - T )/(T a - T i), can be greater than 1.0 for: a. Heating b. Cooling c. Either heating or cooling for any value of N Bi d. Either heating or cooling, but only when N Bi < 0.1 e. Either heating or cooling, but only when N Bi > 40 f. Neither heating nor cooling 4. There may be more than one correct. Answer. Circle ALL correct answers. (4 points) During heating of a canned food product, a. The internal (conductive) resistance to heat transfer is generally negligible b. The external (convective) resistance to heat transfer is generally negligible c. Neither the internal (conductive) nor the external (convective) resistance to heat transfer are negligible d. Both the internal (conductive) and the external (convective) resistance to heat transfer are negligible e. The rate of increase in temperature of the product increases with an increase in the specific heat of the product 5. There may be more than one correct. Answer. Circle ALL correct answers. (4 points) When the length of a double tube heat exchanger operating in a co-current mode is doubled, which of the following parameters can potentially increase significantly? a. h between the product and the inside wall of the inner pipe b. Reynolds number of the product c. Overall heat transfer coefficient d. Total energy transferred e. Logarithmic mean temperature difference f. Logarithmic mean area

3 6. Give approximate values for the following (with SI units): (12 points) a. Convective heat transfer coefficient for free convection between a hot cylinder and still air b. Convective heat transfer coefficient for forced convection in a pipe under turbulent flow conditions c. Thermal conductivity of a metal d. Thermal conductivity of an insulator e. Specific heat of water at room temperature f. Viscosity of water at room temperature

4 Principles of Food and Bioprocess Engineering (FS 231) Exam-2 Part B -- Open Book (50 points) 1. A well-insulated tubular heat exchanger (I.D. and O.D. of inner pipe are 3.4 cm and 3.5 cm resp., k = 16 W/m K; I.D. and O.D. of outer pipe are 4.6 cm and 4.7 cm resp., k = 16 W/m K; Length = 60 m) is operating in a co-current mode. The heating medium (c p = 4100 J/kg K) enters the outer tube at a flow rate of 5.5 kg/s and a temperature of 95 C. The product (c p = 3800 J/kg K, = Pa s, k = 0.5 W/m K) enters the inner tube at the rate of 1.5 kg/s at 15 C and exits at 65 C. a. What is the convective heat transfer coefficient between the product and the inner wall of the inner pipe? b. What is the overall heat transfer coefficient for transfer of energy from the hot water to the product? c. What is the convective heat transfer coefficient between the heating medium and the outer wall of the inner pipe? 2. a. A tuna processor hires an FS 231 student to determine the appropriate dimensions of a cylindrical can to heat process tuna in a retort. The options presented to the student by the processor are a cylindrical can (Can #1) of radius 3 cm and height 16 cm OR a cylindrical can (Can #2) of radius 6 cm and height 4 cm. In order to determine the convective heat transfer coefficient between the hot steam in the retort and the surface of the product, the student creates two solid aluminum cylinders having the same dimensions as those provided by the tuna processor and subject them to heating from ALL SIDES by steam (at 125 C) and measures the center temperature of the cylinders as a function of time till the centers of both cylinders reached 110 C. After the experiments were conducted and the data was analyzed, the student concluded that the convective heat transfer coefficients in both situations were approximately the same. Which cylinder took a longer time for its center to reach 110 C? Explain how you arrived at the answer. b. A second FS 231 student used a free convection analysis to theoretically calculate the 2 convective heat transfer coefficient between steam and Can #1 and found it to be 75 W/m K. The student also determines that the density, specific heat, and thermal conductivity of the 3 product are 950 kg/m, 4000 J/kg K, and 0.5 W/m K respectively. The product was then placed in Can #1 when it was at 25 C and the can was placed in a retort (where the steam temperature was 125 C) at 3 pm right before the student went to take the FS 231 exam. What will be the center temperature of the product at 5:45 pm (when the student completes the FS 231 exam)?

5 Exam #2 (Solutions) Part A 1. a. False b. False c. True d. False e. True f. True g. False h. False i. False j. False 2. B, D 3. F 4. C 5. D, F 6. a W/m 2 -K b ,000 W/m 2 -K c W/m-K d W/m-K e J/kg-K -6-6 f. 200 x x 10 Pa-s

6 Exam #2 (Solutions) Part B 1. a. This is a forced convection problem. So, we begin be determining the Reynolds number., L = 60 m Thus, the flow is turbulent N Nu = 0.023(N Re) (N Pr) ( b w) Since the wall temp. is unknown, we assume b w = 1 Here, N Pr = (c p)( )/k = (3800) (0.004)/0.5 = 30.4 Thus, N Nu = Also, N Nu = hid c/kf Substituting N = 147.6, d = m, k = 0.5 W/m-K yields: h = 2171 W/m 2 -K Nu c f i b. Since the tubular heat exchanger is well-insulated, all the heat lost by hot water is gained by the product. Thus, Solving, we get: T ho = 82.4 C The energy transferred from hot water to the product is given by: Q = 1.5 (4200) (65-15) = W Also, Q = U A lm T lm (1) T lm = ( T 1 - T 2)/[ln( T 1/ T 2)] (2) Here, T 1 = = 80 C, T 2 = = 17.4 C Substituting these in equation 2, we get: T lm = 41 C Also, A lm = (A o - A i)/[ln(a o/a i)] (3) Here, A i = 2 ril, A o = 2 rol with r i = m, r o = m, L = 60 m 2 2 Thus, A i = 6.4 m, A o = 6.6 m Substituting these in equation 3, we get: A lm = 6.5 m 2 Substituting this in equation (1), we get: U = 1182 W/m 2 -K c. The expression for total thermal resistance for transfer of energy from hot water to the product is given by: Here, r = ( )/2 = m k = 16 W/m K h i = 2171 W/m 2 K A i = 6.4 m, A o = 6.6 m, A lm = 6.5 m U = 1182 W/m 2 K Solving, we get: h = 2839 W/m 2 K o

7 2. a. This is an unsteady state heat transfer problem. Since heat transfer in an aluminum cylinder is being considered, the lumped parameter analysis can be used. Thus, For Can #1 and Can #2, the TR, h, r, and c p are the same. Thus, the product of (A/V) and t is the same for both cans. Thus, the can having a smaller A/V ratio will have a higher t. 2 2 For a cylinder, A/V = (2 r + rl)/( r L) = 2 (r + L)/rL For Can #1, r = 0.03 m, L = 0.16 m; thus, A/V = 79.2 m -1 For Can #2, r = 0.06 m, L = 0.04 m; thus, A/V = 83.3 m -1 Since the A/V value is smaller for Can #1, it will have a higher t value. b. This is an unsteady state heat transfer problem with a finite cylinder. A finite cylinder is obtained by the intersection of an infinite cylinder and an infinite slab. For infinite cylinder: d c = 0.03 m, k = 0.5 W/m-K N Bi = hd c/k = 4.5 Thus, k/hd c = 0.22 For infinite slab: d c = 0.08 m, k = 0.5 W/m-K N Bi = hd c/k = 12 Thus, k/hd c = Thus, we use the Heisler chart. = k/( c p) = x 10 m /s t = 2 hrs and 45 mins = 165 min = 9900 s 2 2 N Fo = t/d c = 1.45 for the infinite cylinder and N Fo = t/d c = 0.2 for the infinite slab Thus, TR = for the infinite cylinder and TR = 0.8 for infinite slab Thus, TR for finite cylinder = (0.005) (0.8) = = (T - T ) / (T - T ) T = C i

1. Nusselt number and Biot number are computed in a similar manner (=hd/k). What are the differences between them? When and why are each of them used?

1. Nusselt number and Biot number are computed in a similar manner (=hd/k). What are the differences between them? When and why are each of them used?. During unsteady state heat transfer, can the temperature