Shell-and-Tube Heat Exchangers Unit Operations Laboratory - Sarkeys E111 February 11 th & 18 th, 2015 ChE Section 3

Transcription

1 Shell-and-Tube Heat Exchangers Unit Operations Laboratory - Sarkeys E111 February 11 th & 18 th, 2015 ChE Section 3 Eric Henderson Eddie Rich Xiaorong Zhang Mikey Zhou

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3 ABSTRACT Shell-and-tube heat exchangers were the focal point of this experiment, and were examined to evaluate the properties of pressure drop and heat transfer. Tube-side pressure drop data for a single 1-4 heat exchanger was collected for flow rates between and lb/hr to determine the portion of the overall tube-side pressure drop that was due to tube boxes was between 26% and 45.6% respectively, while the overall tube-side pressure drop due to skin friction was between 74% and 55.4% respectively. Shell-side pressure drop data was collected for five heat exchangers with different number of baffles for flow rates ranging from 2610 to lb/hr to determine that there were approximately 9 baffles in a heat exchanger that had an unknown number of baffles. A 1-4 heat exchanger with no baffles was used as a steam condenser to find a linear relationship between the convective film heat transfer coefficient of water flowing tube-side, and the convective film heat transfer coefficient inside the long straight tubes of the condenser. This linear relationship was used to find that the heat transfer coefficient ranged from 1.86 to BTU/hft 2 F for steam at 50% of its maximum flow rate, while the tube-side flow rate ranged from 4.8 to lb/hr. Data was also collected for steam at 100% of its maximum flow rate to get a heat transfer coefficient range of to BTU/hft 2 F, while the tube-side flow rate ranged from 4.77 to lb/hr. 2

4 INTRODUCTION AND THEORY In this experiment, shell-and-tube heat exchangers were examined to better understand the causes of tube-side and shell-side pressure drop, and heat transfer of flowing water. This experiment was performed in two parts on separate occasions. The purpose of the first part of the experiment was to determine the amount of pressure drop due to skin friction and tube boxes for the tube-side flow, and to develop an equation correlating shell-side pressure drop to the number of baffles in heat exchangers. Tube boxes enclose each end of the shell of a heat exchanger and redirects flow so the tube-side fluid can make another pass. Baffles are metal plates inside the shell that alter shell-side flow to increase heat transfer. The purpose of the second part of the experiment was to evaluate the convective film heat transfer coefficient for water flowing through long straight tubes in a steam condenser. Tubular heat exchangers are used in a vast majority of industrial and chemical process, which is the reason their designs are so highly developed. 1 The basic function of heat exchangers is to transfer heat from a flowing fluid of high temperature, to a flowing fluid with a lower temperature. A typical shell-and-tube heat exchanger consists of a shell that is enclosed at each end by stationary heads as shown in Figure 1. Stationary heads are also known as tube boxes. Inside the shell are tubesheets at each end of the exchanger in which a guide rod and baffle spacers are attached to support the baffles and tubes. Two major aspects when designing shell-and-tube heat exchangers are determining if the tube-side and shell-side flows should be single-pass or multipass, and the number of baffles to place inside the shell along with their configuration. This experiment used 1-4 heat exchangers which indicate the number of passes the shell-side and tubeside fluid make through the system. For example, the shell-side made one pass through the heat exchanger, while the tube-side made four passes. There were twenty 3

5 FIGURE 1. Single pass 1-1 shell-and-tube heat exchanger. 2 FIGURE Shell-and tube heat exchanger. 11 copper tubes inside each heat exchanger, with each pass going through five of the tubes. Twenty solid dummy tubes were also placed inside the heat exchanger to prevent a path of lower resistance for the shell-side fluid to flow through from forming. The dummy tubes did not participate in heat transfer. The first part of the experiment was split between finding the tube-side pressure drop of a single heat exchanger, and finding the number of baffles in an unknown heat exchanger using pressure drop data from five heat exchangers of known number of baffles. Since all of the four pass heat exchangers only differed in the number of baffles, tube-side pressure drop data only needed to be recorded for one heat exchanger. Tube-side pressure drop is caused by skin friction inside the pipes, and change in flow direction inside the tube boxes. The skin friction of the tubes causes a pressure drop known as head loss h L, which can be calculated using the Fanning friction factor f f, length of tubes L, diameter of tubes D, velocity of fluid through tubes ʋ, and the gravitational constant g. 3 L ʋ h L = 2f 2 f = ΔP D g ρ (1.1) The Fanning friction factor can be calculated using equation (1.2), assuming that the flow is turbulent and the pipe is smooth. 4 1 f f = 4.0 log 10 (Re f f ) 0.40 (1.2) 4

6 The Reynolds number Re represents a dimensionless ratio of the inertial forces to the viscous forces in fluid flow. 5 L is the length of pipe, ʋ is the velocity of the fluid, ρ is the density of the fluid, and µ is the viscosity of the fluid. Re = Lʋρ µ (1.3) The tube-side mass flow rate was calculated using the tube-side Venturi meter calibration curve, and then was converted into the velocity of the fluid by knowing the density of water and the diameter of the tube. ln (y) = ln (x) (1.4) y = water flowrate ( lb hr ) x = Δ P (cmhg) At each end of the heat exchanger, there are tube boxes that redirect the flow in the opposite direction so the fluid can make another pass through the exchanger. These tube sheets cause a significant pressure drop as well and can be calculated by subtracting the pressure drop due to head loss from the overall measured tube-side pressure drop. P tube boxes = P total P friction (1.5) The single-segmental baffles used in the heat exchangers for this experiment are semicircular metal plates attached to a guide rod and copper tubes in a perpendicular manner inside the shell of the exchanger. 6 Baffles are used to increase heat transfer by increasing the average velocity and causing crossflow of the shell-side fluid over the tubes. 1 Although baffles increase heat transfer, they cause a significant pressure drop. This pressure drop was used in the first part of the experiment to determine the number of baffles in an unknown heat exchanger using pressure drop data from heat exchangers with a known number of baffles. The Venturi meter calibration equation for the shell-side flow was used to calculate the shell-side mass flow rate. ln(y) = 0.52 ln(x) (1.6) 5

7 y = water flowrate ( lb hr ) x = Δ P (cmhg) An equation had to be developed to find the number of baffles in the unknown heat exchanger. Equations (1.7) and (1.8) were given in the lab manual 12, and utilized to develop the equation to find the number of baffles N, where a, b, y and z are constants. P = a (flow rate) z (1.7) a = b N y (1.8) Applying equations (1.7) and (1.8) to the calibration curves for shell-side flow, the number of unknown baffles can be evaluated once the constants are calculated. log( P) = log(a) + z log(flow rate) (1.9) log(a) = log(b) + y log (N) (1.10) For the second part of the experiment, water was passed tube-side through a heat exchanger while steam was passed shell-side to heat up the water. The convective film heat transfer coefficient for the water flowing tube-side was of interest. To calculate the heat transfer coefficient, a different form of the Dittus-Boelter equation was used. 7 Nu = h id k = αre0.8 Pr 1/3 (1.11) The Reynolds number is used in the calculations, and is found the same way as in part one of the experiment. The Prandtl number Pr, is a dimensionless ratio of the molecular diffusivity of momentum to the molecular diffusivity of heat, where µ is the viscosity of the fluid, c p is the heat capacity of the fluid, and k is the thermal conductivity of the fluid. 8 Pr = µc p k (1.12) The form of the Dittus-Boelter equation used was given in the lab manual 12, where U i is the overall heat transfer coefficient. 9 6

8 1 U i = 1 k (1.13) D Re0.8 Pr 1/3 The overall heat transfer coefficient is found by calculating the heat gained in the system using equation (1.14), and then solving for the overall heat transfer coefficient in equation (1.15). 9 Δq = m c p ΔT L (1.14) U = q AΔT (1.15) L The mass flow rate m of the water is found the same way as in the first part of the experiment. The heat capacity c p of water fluctuates slightly with temperature which must be accounted for. The logarithmic-mean temperature difference ΔT L is a very important fluid heat transfer equation to help determine how well a heat exchanger performs. 10 LMTD = ΔT L = ΔT 1 ΔT 2 ln ( ΔT 1 ΔT2 ) (1.16) From equation (1.13), a Wilson plot of 1 U i vs. 1 k D Re0.8 Pr 1/3 was created to find the slope of the trendline α, generated from the experimental data. The convective film heat transfer coefficient was then calculated using equation (1.17). Apparatus and Procedures 1 h = (1.17) α k D Re0.8 Pr 1/3 Equipment Description The heat exchanger unit is comprised of six, 1-4 pass shell-and-tube heat exchangers, a single-pass heat exchanger, and a steam condenser. In a 1-4 pass heat exchanger, water flows through the length of the shell four times before exiting. For a single-pass heat exchanger, water enters tube-side and just flows through the length of the shell once before exiting. In each of the heat exchangers there are a different number of baffles which, on average, are inches thick. 7

9 FIGURE 3. Apparatus The distance between each baffle varies depending on the number of baffles in the exchanger. The tubes within the exchangers are made of copper, and have an outer diameter of 0.3 inches and an inner diameter of 0.25 inches. The walls of the tubes are inches thick, and the center-to-center distance between tubes is inches. Each tube is 30 inches long, measuring from tube-sheet to tube-sheet. Not all of the tubes in the exchangers are open-ended. Some solid, close-ended tubes are included to be used as dummy tubes, which serve to prevent short-circuiting of the shell-side flow. Since there is no flow through these dummy tubes, they provide no heat transfer surface. Shell-side, the outer and inner diameters are and 3.06 inches, respectively. As for the steam condenser, there are no shell baffles and the tube-sheet to tube-sheet length is 23.3 inches. The condenser s role in the second week of the experiment is to use steam to heat up the process water; it is not used in the first week. There are three feeds into the heat exchanger unit: process water, cooling water, and steam. The manual valves (indicated in green in FIGURE 4) for each feed must first be opened before the user can manipulate the flow rate at which water will run through the valves via process control valves on the HeatX computer program, which is the software used in this experiment for data collection. 8

10 FIGURE 4. Schematic illustrating fluid flow excluding heat exchangers These process control valves generate a current signal that is sent to the current-topneumatic transducers, which then converts the signal to an air pressure between 3 and 15 psig. This pressure closes the valve the desired amount, which can range from 0% open to 100% open. Temperatures at the tube-in, tube-out, and shell-out locations of each heat exchanger can be read from the Thermo Electric thermocouple on the front of the apparatus. The Omega Engineering rotary thermocouple switch is used to control which exchanger s temperature reading is displayed. The figure above is a depiction of the front of the apparatus. The process water flows tube-side through the condenser where it is heated by steam that flows in shell-side. The process water then flows tube-side through the heat exchangers where it is cooled by the cooling water that flows in through the heat exchangers shell-side. There are two Venturi meters in the apparatus. One Venturi meter is placed at a location before the process water enters the condenser, and the other is placed at a location before the cooling water enters the heat exchangers. Two pressure transducers send signals to the computer that allow the pressure drop across these Venturi meters to be read continuously; this data is then used to calculate tube-side and shell-side flow rate. A third pressure transducer allows for the reading of pressure drop across any heat exchanger. To obtain these readings from the HeatX program, the heat exchanger s corresponding tubes must be attached to the appropriate pressure 9 FIGURE 5. Process control valves on HeatX program

11 sockets on the front of the apparatus (FIGURE 6). As can be seen, there are nine pairs of highand low-pressure tubes. Although not shown in the figure, there is a sheet of paper in the laboratory pinned to the front of the apparatus that specifies which pair of tubes correspond to each particular FIGURE 6. Schematic of apparatus front panel exchanger. Operating Procedures: For week one of the experiment, the subject of interest was the pressure drop on the tubeside and shell-side of the exchanger. Because all the heat exchangers were similar in the tube-side direction, to analyze tube-side pressure drop it was only necessary to collect data for the 11-baffle exchanger. Shell-side pressure drop data was recorded for the 25, 21, 17, 13, 11, and X (unknown number) baffle heat exchangers. In order to measure the pressure drops, a pair of tubes corresponding to each exchanger has to be attached to the high pressure and low pressure sockets on the front of the apparatus. The high pressure socket is where the tube connected to the entrance of the heat exchanger is attached, and the low pressure socket is where the tube connected to the exit of the heat exchanger is attached. To turn on the water flow, the process water manual valve must first be opened. With this valve open, the process control valve on the HeatX program can be manipulated to vary the flow rate of water entering. In this lab, the process control valve was initially set to 100% open, and was decreased in increments of 10% until the valve was 0% open. 10

12 Pressure drop data across the heat exchanger and the Venturi was recorded at each process control valve setting. The pressure drop data fluctuated too quickly to get an accurate reading, so the data was exported to Excel to be averaged, and this value was then recorded. Collection of the pressure drop data is nearly identical for both the tube-side and the shell-side procedure. The only difference is that when examining shell-side pressure drop, the cold water feed is used instead of the process water. Consequently, on the HeatX program, the cold water valve is varied instead of the process water valve. Because the flow rates were being changed from trial to trial, it was important to make sure the system reached steady-state before collecting data. Thus, data was not recorded until the temperature reading on the HeatX program had stabilized. In week two, the objective was to determine the convective film heat transfer coefficient inside the tubes. In this part, the condenser and the steam feed would actually be used. Before the steam was allowed to flow through the unit, process water was run through the condenser and each heat exchanger. Cooling water was also run through each exchanger to lower the temperature of the exiting process water. After water had been flowing for a few minutes, steam was allowed to flow through the condenser to heat up the process water. On top of the condenser is a pressure gauge that was used to monitor the pressure within the condenser. This pressure had to be kept within 1-3 psig; if the limit was exceeded, the steam vent valve must be opened to release some pressure from the condenser. With each fluid running, the experiment began with the process water at 100% open. The steam temperature and the temperatures of the entering and exiting process water were read off of the thermocouple on the front of the apparatus. The pressure drop across the Venturi was acquired from the HeatX computer program, just as it was done in the first week. The process water valve was then closed in increments of 10% until the valve was completely shut, and the temperature and pressure drop data were recorded for each trial. After all runs were 11

13 completed, the steam supply valve was closed, allowing the condenser to cool down. Finally, the process water and cooling water supply valves were shut off. The primary safety concern with this experiment arises from working with steam. Because the temperature of the steam was so high, it was necessary to wear gloves while working with the condenser. RESULTS & DISCUSSION In order to obtain data for correlating pressure drop and water flow rate, varying pressure differentials and drops were recorded via computer, while the amount of water fed into the tube side of the 11 baffle heat exchanger was decreased from 100% to 0% in 10% increments. The results from data analysis showed that an increase in the water flow rate caused an increase in the pressure drop within the tubes, and consequently, a decrease in frictional head loss (Figure 7). This trend is consistent with the relationship in Equation 1.4. Pertaining to the pressure drop with the tube side feed, a greater total pressure drop than pressure drop due to friction was expected. As water flow rate varied from to lb/hr, the total pressure drop varied from 0.04 to 1.16 inches H2O, whereas the frictional pressure drop varied from 0.62 to 3.87 inches H2O. These ranges generate two curves, and by taking the difference between these two curves, the pressure due to boxes (Boxes P) could then be found. The pressure drop due to the boxes ranged between 0.25 to 0.57 inches H2O, increasing with increasing flow rate. Figure 7 shows an example of this. Shell side feed for a heat exchanger also involves a relationship between pressure drop and water flow rate, which is dependent on the number of baffles present in the heat exchanger. 12

14 Log P [in H2O] y = x R² = y = x R² = Log of Flow Rate [lb/hr] Boxes P = 0.55 Total Frictional FIGURE 7. A log-log plot for an 11 baffle heat exchanger with tube side feed showing the total and frictional pressure drops vs. the flow rate of water. A heat exchanger with an unknown number of baffles (X) was presented and by plotting the pressure drop vs. flow rate for heat exchangers with a known number of baffles (25, 21, 17, 13 & 11), an estimation for X could be deduced. Figure 4 indicates that increasing the number of baffles will increase pressure drop at a fixed flow rate. In order to calculate the actual number of baffles in X, a relationship between pressure drop and flow rate from Equations 1.7 and 1.8 must be used, along with the knowledge from Bernoulli s equation that the theoretical slope of the lines for the heat exchangers is two. Table 1 can then be constructed by taking slope (Z) and intercept (log(a)) values from Figure 4. By utilizing Equations 1.7, 1.8, and the linest values from Table 1, the number of baffles in the unknown heat exchanger, X, is calculated to be 8.53, or about 9 baffles. It may also be noted that the linest values of log(a) and log(baffles) in Table 1 correspond to the slope and the intercept values from Figure 9. The rounding up of X from 8.53 to 9 is caused from 13

15 log (a) Log P [psig] estimation of trends seen in Figure 8, and by the curvature in Figure 9, which can be attributed to 25 Baffles 21 baffles 17 Baffles 13 baffles 11 Baffles X Baffles y = 1.978x y = x y = x y = x y = x y = x R² = R² = R² = R² = R² = R² = Log of Flow Rate [lb/hr] -0.4 the pressure losses in both the entrance and exit nozzles for the heat exchangers. Other error considerations during this part of the experiment arose from uncertainty in measurement of the differential pressure measurements. The small sample size associated with Figure 7 led to error prorogation analysis in the range that data points should lie in, as illustrated by the vertical error bars shown in Figure Baffles Z a log baffles log a y = x R² = E E E E TABLE Values used to 5.05E-09 ultimately determine the FIGURE 8. A log-log plot of the pressure drop vs. flow number rate FIGURE for 25, 17, 9. A 11, log-log and an plot unknown log of (# a of vs. Baffles) (X) the number of of baffles in a heat of baffles exchanger. X in X E-08 #VALUE! for each heat exchanger during shell side flow.

16 h (Btu/(h*ft2* F) h (Btu/(h*ft2* F) 1/Ui 1/Ui For the subsequent part of the experiment, the amount of water heated by steam within the condenser was varied. By doing this, the convective film heat transfer coefficient for the tube side could be evaluated, as quantified by Equation For comparison, the amount of steam within the condenser was also varied. A Wilson plot (Figures 10) was then constructed using the inverse of the overall heat transfer coefficient in order to find a value for the alpha constant in Equation 1.11, which the slope of the plotted data. An average alpha value was used to calculate the convective film heat transfer coefficient, since the exchanger was run at several steam flow rates. Observed trends consisted of the amount of heat transferring from the steam to the water increased (228.1 to 263,010.8 BTU/hr for 50% steam) as the water flow rate increased (4.8 to lb/hr). Another trend also showed the overall heat transfer coefficient increasing (1.86 to BTU/hft 2 F for 50% steam) in relation to increasing water flow rate, which is consistent with the theory since Wilson Plot 50% Steam Wilson Plot 100% Steam y = x R² = y = x R² = (k/di*re0.8*pr1/3) (k/di*re0.8*pr1/3) Tube-Side Convective Heat Transfer Coefficient for 50% Steam y = x R² = Water Flow Rate (lb/hr) Tube-Side Convective Heat Transfer Coefficient for 100% Steam y = x R² = Flow Rate (lb/hr) FIGURE % vs. 100% steam flow rate Wilson and tube-side convective heat transfer coefficient plots. 15

17 the coefficient is directly proportional to the amount of heat transferred. By comparison, the heat transferring from the steam to the water increased from 15,016.1 to 439,442.9 BTU/hr, and the coefficient increased from to BTU/hft 2 F for 100% steam flow rate. The experimental value for the overall coefficient was found to have an error of 11.3% for 50% steam and 47.1% for 100% steam, when compared to literature values 6. This error is in large part due to the roughness of the pipe and the uncertainty of the measurements. Other errors could arise from uncertainty in the measurement of the differential pressure across the heat exchanger, which would also result in error associated with calculated flow rates of water. CONCLUSION 1. Tube-side pressure drop attributed to the tube boxes was between 26% and 45.6% of the total pressure drop, while the pressure drop due to skin friction was between 74% and 55.4% of the total pressure drop, between flow rates of and lb/hr. 2. The unknown heat exchanger was found to have approximately 9 baffles by forming an equation correlating the shell-side pressure drop to the number of baffles. 3. The linear relationship between the convective film heat transfer coefficient and the tubeside flow rate was used to find that the heat transfer coefficient ranged between 1.86 and BTU/hft 2 F, while the tube-side flow rate ranged from 4.8 to lb/hr for steam at 50% of its maximum flow rate. For steam at 100% of its maximum flow rate, the heat transfer coefficient ranged from to BTU/hft 2 F, while the tube-side flow rate ranged from 4.77 to lb/hr 16

18 REFERENCES 1. McCabe, W. L., Smith, J. C., and Harriott, P. Unit Operations of Chemical Engineering, 7th ed. New York: McGraw Hill, 2005, p Mukherjee, R. Effectively Design Shell-and-Tube Heat Exchangers. CEP. Feb p. 3. Print 3. Welty, J. R., Wicks, C. W., Wilson, R. E., and Rorrer, G. L. Fundamentals of Momentum, Heat, and Mass Transfer, 5th ed. New Jersey: Wiley, 2008, p Welty, J. R., Wicks, C. W., Wilson, R. E., and Rorrer, G. L. Fundamentals of Momentum, Heat, and Mass Transfer, 5th ed. New Jersey: Wiley, 2008, p Welty, J. R., Wicks, C. W., Wilson, R. E., and Rorrer, G. L. Fundamentals of Momentum, Heat, and Mass Transfer, 5th ed. New Jersey: Wiley, 2008, p Bouhairie, Salem. Selecting Baffles for Shell-and-Tube Heat Exchangers. CEP. Feb p Print 7. McCabe, W. L., Smith, J. C., and Harriott, P. Unit Operations of Chemical Engineering, 7 th ed. New York: McGraw Hill, 2005, p Welty, J. R., Wicks, C. W., Wilson, R. E., and Rorrer, G. L. Fundamentals of Momentum, Heat, and Mass Transfer, 5 th ed. New Jersey: Wiley, 2008, p Welty, J. R., Wicks, C. W., Wilson, R. E., and Rorrer, G. L. Fundamentals of Momentum, Heat, and Mass Transfer, 5 th ed. New Jersey: Wiley, 2008, p McCabe, W. L., Smith, J. C., and Harriott, P. Unit Operations of Chemical Engineering, 7 th ed. New York: McGraw Hill, 2005, p "How Hot Is Too Hot?" Hydraulics & Pneumatics. N.p., 13 June Web. 22 Feb University of Oklahoma, Ch E 3432: Shell and Tube Heat Exchanger. Norman, Spring

19 Appendices Part 1 Sample Calculations To try to minimize the variations in the pressure readings from the HeatX software, after allowing a few seconds for the flow to become steady, the averages were recorded for each pressure reading. The average of the values was used to find the differential pressure in the pressure drop across each heat exchanger an meter. At 0% flow, there was still a pressure drop and differential pressure change because the flow was not completely obstructed by the actuator. The measurements at 0% flow on the HeatX software, the correction values, were subtracted from the appropriate pressure changes in the heat exchangers an meter. The units given from the readings were in psig, and were converted to cmhg using the conversion factor of: 76 cmhg 14.7 psig Tube-side The tube-side with 11 baffles sample data from our measurements will be used for 100% and 0% flow rates: The pressure corrections for Heat Exchanger (HX) using the 0% values: 100%: d = ( 0.17) = psig The pressure corrections for Venturi using the 0% values: 100%: d = 1.25 ( 0.14) = psig The correcte dp is converted to cmhg: 100%: d = psig 76cmHg = cmhg 14.7 psig The tube-side flow rate was calculated using the tube-side flow rate equation for the Venturi meter, which was given in the lab instructions 12 : 18

20 ln(y) = ln(x) Equation 1 y = e ln(x) where y = water flow rate ( lb ) and x = (cmhg) hr Total flow rate: y = e ln(10.676) = lb/hr The log base 10 was taken of the pressure drop for heat exchanger and total flow rate: log(corrected P) : log(13.802) = log(total flow rate): log( ) = There are 5 separate tubes in each pass, so we divided the total flow rate by 5: Equation 2 m = Individual flow rate: m = lb hr = lb 5 hr Total Flow Rate Density and viscosity values were obtained and interpolated for 60 C Welty 3. This data was used to find the volumetric flow rate (Q), velocity (v), and Reynolds number (Re). These equations were obtained from the same text: 5 Temp μ ρ Inner Tube CS Tube Tube ( F) (lb/s*ft) (lbm/ft 3 ) Diameter (ft) Area(ft 2 ) Length(in) E E Table 2. Water properties for 60 Q ( ft3 ) = m s (lb m ) 1 hr 1 hr ρ ( lb m ft 3 ) 3600 s Equation 3 19

21 Q ( ft3 s ) = (lb m hr ) 1 Re = ( lb m ft 3) ft 3 s ) 1 hr ft3 = s s v ( ft ) = Q ( π Equation 4 s [D (ft)]2 4 v ( ft ft ( 3 s ) = s ) ft π = [ (in)]2 s Re = ρ (lbm ft3 ) D (ft) v (ft s ) Equation 5 μ ( lb m ft s ) ( lb m ft3) (ft) (ft s ) ( lb = m ft s ) The equation relating the Reynolds number and the Fanning friction factor for turbulent flow in a smooth pipe was obtained for Welty 2 : 1 f f = 4 log(re f f ) 0.4 Equation 6 All of the terms were moved to one side and the known Reynolds number and a guessed friction factor were plugged in. The Solver function in Excel was used to get as close to 0 as possible for this equation: 0 = 4 log (Re f f ) f f 0 = 4 log ( f f ) f f = The length of the tube was given in the lab instructions as 23.3 in. for one pass. The following head loss equation was also obtained from Welty 3 : f f 20

22 h L (ft H 2 O) = 2 f f h L (ft H 2 O) = Length of pipe (in) D(in) (in) 0.25 (in) [v ( ft s )] 2 [ ( ft s )] ( ft s 2) 1 g ( ft s 2) This head loss was converted to a pressure drop due to skin friction in psig with: P f (psig) = h L (ft H 2 O) ρ H2 O ( lb m ft 3) 1 ft2 144 in 2 = ft H 2 O P f (psig) = (ft H 2 O) ( lb m ft 3) 1 ft2 = psig 144 in2 The graph for the log of the pressure drop due to skin friction was shown below and the difference between the total pressure drop and the pressure drop due to skin friction was calculated to equal the pressure drop due to the boxes. The pressure drop due to the boxes was divided by the total pressure drop and multiplied by 100% was the percentage of the total pressure drop: P boxes = log ( P total ) log ( P friction ) Equation 7 P boxes = log ( psig) log (9.455 psig) = psig Shell-side P boxes P total 100% psig 100% = 31.5% psig For the shell-side, the exact same calculations were used and the flow rate was found with a slightly different calibration equation as it is a separate Venturi meter: ln(y) = 0.52 ln(x) Equation 8 21

23 where y = flow rate ( lb ) and x = (cmhg) hr The flow rates and pressure drops were plotted log-log. The slope of the lines should be the same, so the intercepts for the trend lines were briefly adjusted so the slope was equalized to 2 across all 6 heat exchangers as this is the exponent shown in Bernoulli s equation: Figure 8: The log-log plot of the pressure drop versus flow rate through 6 heat exchangers with varying number of baffles, including the trend lines. The following equations relating the pressure drop, flow rate, and the number of baffles were given in the lab instructions: P = a (flow rate) z Equation 9 a = b N y Equation 10 where z, b, and y are constants and N is the number of baffles Applied to the equations 1 and 8: log( P) = log(a) + z log(flow rate) log(a) = log(b) + y log (N) Equation 11 Equation 11 was used as a template for the trend lines from figure 2. z = 2 The intercepts in the trend lines represent the log(a). These values were plotted against the log(baffles). Figure 9: The log-log plot of a-values versus the number of baffles. The plot shows curvature due to pressure drop at the entrance and exit but these were not incorporated to produce a linear trend. The a-values for calculated for all heat exchangers (using data from x baffles in this calculation): log(a) =

24 log (a) a = 10 a = = Applying equation 4 to the trend line in Figure 9, we can find b and y. The intercept is log(b) and the slope is y: y = log(b) = log (b) b = 10 b = = Based on the b and y values and equation 2, the number of baffles in the x heat exchanger can be estimated: a = b N y (eq. 10) N = ( a b ) 1 y N = ( ) = baffles Shell-side error propagation Shell Side(25 Baffles) (psig) (cmhg) 100% Trial % Trial average Table 3. Shell-side trials for average P and calculations. 23

25 The standard deviation formula can be applied to data: σ = (x i x ) 2 N 1 Equation 12 where x i is the measured data point, x is the mean, and N is the number of data points Applying the standard deviation to the sample data above: σ = ( )2 + ( ) = psig The standard deviation of must be converted to cmhg from psig using the previous conversion: σ = psig 76 cmhg = cmhg 14.7 psig For a function, F, dependent on two variables, the error can be propagated using: σ F = F 2 x σx 2 + F 2 y σy 2 Equation 13 Error propagation was needed for the flow rate because of the fluctuations in the pressure drop. y = FR and x = ln y = 0.52 ln x y = e x 0.52 y x = e x 0.48 σ y = y 2 σ2 x x σ FR = [e (29.1) 0.48 ] = lb h 24

26 Since flow rate and pressure had discrepancies, error propagation for a was done: a = where z = 2 from Figure 8 FRz a P = 1 FR 2 and a FR = 2 FR 3 σ a = a 2 σ 2 P P + a 2 2 σ FR FR 1 σ a = ( ) = N error can be found by using the formula below: 1 N = a y and b and y are be constant b N a = y b1 a y 1 y σ N = N 2 σ2 a a 1 1 σ N = ( ( ) ) ( ) 2 =

27 Part 2 Sample Calculations Trial 1 Trial 2 Tube Side (psig) Temp. In ( F) Temp. Out ( F) Temp. Steam( F) (psig) Temp. In ( F)) Temp. Out ( F) Temp. Steam( F) 100% Table 4. Tube-side condenser trial data for average and T bulk calculations. The averaged pressures: 100%: Average = = 1.42 psig The pressures were converted to cmhg: 100%: average in cmhg = 1.42 psig 76cmHg 14.7 psig = 7.367cmHg The average temperatures: 100%: Average T steam = 100%: Average T in = = = %: Average T out = = 73 The bulk temperatures for the process water in and out: 100% T bulk1 = T bulk = T in + T out = 67

28 100% T bulk2 = The average of the two T bulk values was used for our T: 100% average T bulk = = 67.5 Plugging average into Equation 2, the flow rate can be found as: = Total flow rate: y = e ln(7.354) = lb hr single flow rate: m = lb hr = lb 5 hr Water Properties ρ (lb/ft 3 ) cp (Btu/lbm* F) µ (lbm/ft*s) k (Btu/h*ft* F) Pr T=60( F) T=80( F) Table 5. Water properties of water at different temperatures Using interpolation method to find the water properties at 67.25( F) which is the bulk temperature: y = y 1 y 0 x 1 x 0 (x x 0 ) + y 0 Equation 14 ρ = 62.2 lb m ft lb m ft 3 ( ) lb m ft 3 = lb m ft BTU lb cρ = m 1 BTU lb m ( ) + 1 BTU lb m = BTU lb m 27

29 μ = lb m ft s lb m ft s ( ) lb m ft s = lb m ft s BTU hr ft 0.34 BTU hr ft BTU k = ( ) hr ft Pr = BTU = hr ft ( ) = Tube Side Temp. In ( F) Temp. Out ( F) Steam Temp. ( F) 100% Table 6. tube-side temperature LMTD expression: LMTD trial 1 = LMTD = ((T steam T out ) (T steam T in )) ln((t steam T out )/(T steam T in )) (( ) ( )) ln(( ) ( )) = Using Equation 3, 4, and 5, volumetric flow rate, velocity and Re become: Q ( ft3 s 100% = (lb m hr ) ( lb m ft 3) v ( ft ft ( 3 s 100% = s ) ft π = [ (in)]2 s 28 1 hr ft3 = s s

30 100% = ( lb m ft3) (ft) (ft s ) ( lb = m ft s ) Calculating for the Wilson Plot, k D i, Re 0.8, Pr 1 3 and heat gained by the water are necessary: k D 100% = BTU hr ft (ft) = BTU hr Re 100% = = Pr 1 100% = = Heat Gained = Flowrate cp avg LMTD Heat Gained = ( lb hr ) ( BTU lb m ) = BTU/hr The Wilson Plot is necessary for the calculation of h and to check data accuracy: Wilson plot: 1 k Re D Pr3 i vs 1 U i where 1/U i = avg LMTD contact area/heat gained contact area = D i π L 4passes 5 tubes = (ft) π 1.942(ft) 4 5 = ft 2 for 100%: 1 U i = ft BTU/hr = hr ft 2 /BTU 1 for 100%: k D Re 0.8 Pr 1 = 3 i ( BTU hr ) = Figure 10. Wilson Plot was used for h calculations. 29

31 Linest Table 7. Computation of linest values. for 100%: h = = Heat-transfer error propagation k D i Re 0.8 Pr BTU (ft ( ) ) hr = BTU/(hr ft 2 ) Using Equations 12 and 13 to find the errors in the flow rate with the following sample data: Tube Side (psig) Temp. In ( F) Temp. Out ( F) Steam Temp. ( F) Bulk Temp. ( F) % Trial % Trial average Table 8. The measured pressures and temperatures of the condenser water in and out and steam. from Equation 12 σ = ( )2 +( ) 2 = psig and from Equation1 2 1 ln y 1 = ln ( x ) was obtained y 2 x+σ Total flow rate:ln = ln ( y 2 Then y 2 = lb hr ) σ FR = y 2 y 1 = = lb hr For single flow: σ SFR = σ FR lb = hr 30

32 Standard deviation for bulk temperature was obtained using equation 12: σ Bulk Temp = ( )2 + ( ) = Equation 14 was used to interpolate the physical properties of water. To find the error in these properties due to the estimated error in the bulk temperature, the partial derivative of this general equation is found to be: y x = y 1 y 0 x 1 x 0 where y is the physical property and x is the temperature σ y = y 2 σ x 2 x = ( y 1 y 2 0 ) σ2 x 1 x x 0 σ ρ = lb m ft 3 σ cp = Btu lb m σ μ = Using the Equation 13 on Equations 3, 4, and 5: lb m ft s σ k = Btu hr ft σ Pr = σ v = ft s σ h = σ Re = Btu hr ft 2 31

33 Flow Rate % d HX Venturi Tube Side - 11 Baffles d Venturi d Venturi (cm Hg) d Flow Rate (lb/hr) log of flow rate HX T ( F) Was ln (y) = ln (x) where y = water flowrate (lb/hr), x = D P (cm Hg) Tube Water Diameter Density (ft) (lb m /ft 3 ) Water Viscosity (lb m /ft*s) CS Tube Area (ft 2 ) HX Hot Correction Correction D/e Flow Rate % Flow Per Tube (lb/hr) Volumetric Flow Rate (ft 3 /s) Velocity (ft/s) Re Fanning Head Loss H L ΔP f (inches log of ΔP f ΔP Friction Factor boxes (ft) H 2 O) #DIV/0! #DIV/0! % Total ΔP f Solver for ff 32

34 Flow Rate % Hg) (lb/hr) Flow Rate % Hg) (lb/hr) Flow Rate HX HX HX d HX d HX d HX Venturi Venturi Venturi 25 Baffles d Venturi 17 Baffles d Venturi 11 Baffles d Venturi d Venturi (cm d Venturi (cm d Venturi (cm d Flow Rate d Flow Rate d Flow Rate log of flow rate log of flow rate log of flow rate % Hg) (lb/hr) T ( F) T ( F) T ( F) 33

35 Flow Rate % Hg) (lb/hr) Flow Rate d Venturi % Hg) (lb/hr) Flow Rate HX HX HX d HX d HX d HX Venturi Venturi Venturi X (unknown) Baffles 13 Baffles d Venturi 21 Baffles d Venturi d Venturi (cm d Venturi (cm d Venturi (cm d Flow Rate d Flow Rate d Flow Rate log of flow rate log of flow rate log of flow rate % Hg) (lb/hr) T ( F) T ( F) T ( F) 34

36 Flow Rate % (cm Rate Flow Rate % Venturi 25 Baffles - Trial 2 (cm Rate Flow Rate % HX HX HX d HX d HX d HX Venturi Venturi 17 Baffles - Trial 2 11 Baffles - Trial 2 (cm Rate d Flow d Flow d Flow log of flow rate log of flow rate log of flow rate T ( F) T ( F) T ( F)

37 Flow Rate % (cm Rate Flow Rate % (cm Rate Flow Rate % HX HX HX d HX d HX d HX X (unknown) Baffles - Trial 2 Venturi Venturi Venturi 13 Baffles - Trial 2 21 Baffles - Trial 2 (cm Rate d Flow d Flow d Flow log of flow rate log of flow rate log of flow rate T ( F) T ( F) T ( F)