_______________________ Last Name, First CHE426: Problem set #511. A thermometer having a time constant of 0.2 min is placed in a temperature bath, and after the thermometer comes to equilibrium with the bath, the temperature of the bath is increased linearly with time at a rate of 1°/min. What is the difference between the indicated temperature and the bath temperature (a) 0.1 min, (b) 1.0 min after the change in temperaturebegins?(c) What is the maximum deviation between indicated temperature and bath temperature, andwhen does it occur?(d) Plot the forcing function and response on the same graph. After a long enough time, by how many minutes does the response lag the input?2. A mercury thermometer bulb is ½ in. long by 1/8 in. diameter. The glass envelope is very thin. Calculate the time constant in water flowing at 10 ft/sec at a temperature of 100oF. You can assume all resistance to heat flow is in the outside water and all heat capacity is in mercury. The heat transfer coefficient for water can be estimated from the following equation:hDk = 0.19140.618 1/ 3Re Pr(Note: Pr = pCkm)Data: Water: Cp = 1.0 Btu/lboF, k = 0.36 Btu/fthroF,  = 0.72 cp = 0.72×2.42 lb/ fthrMercury: Cp = 0.33 Btu/lboF, specific gravity = 13.63.1 A thermometer having first-order dynamics with a time constant of 1 min is at 100oF. The thermometer is suddenly placed in a bath at 110oF at t = 0 and left there for 1 min, after which it is immediately returned to a bath at 100oF.(a) Draw a sketch showing the variation of the thermometer reading with time.(b) Calculate the thermometer reading at t = 0.5 min and at t = 2.0 min.4. A mercury thermometer, which has been on a table for some time, is registering the room temperature, 75oF. Suddenly, it is placed in a 400oF oil bath. The following data are obtained for the response of the thermometer.Time, sec 0 1 2.5 5 8 10 15 30Thermometer T, oF 75 107 140 205 244 282 328 385Estimates the thermometer time constant.5.1 A thermometer having a time constant of 1 min is initially at 50oC. It is immersed in a bath maintained at 100oC at t = 0. Determine the temperature reading at t =1.2 min.6. The level in a tank responds as a first order system with changes in the inlet flow. Given the following level vs. time data that was gathered after the inlet flow was increased quickly from 1.5 gal/min to 4.8 gal/min, determine the transfer function that relates the height in the tank to the inlet flow. Be sure to use deviation variables and include units on the steady state gain and the time constant.Time (min) Level (ft) Time (min) Level (ft) Time (min) Level (ft)0 4.8 1.2423 8.9354 2.4847 11.45130.138 5.3673 1.3804 9.2805 2.6227 11.66120.2761 5.9041 1.5184 9.6071 2.7607 11.85990.4141 6.412 1.6564 9.9161 ………….. …………..0.5521 6.8927 1.7945 10.2085 14.3558 15.32610.6902 7.3475 1.9325 10.4853 14.4938 15.3280.8282 7.7779 2.0705 10.7471 14.6319 15.32970.9663 8.1852 2.2086 10.9949 14.7699 15.33131.1043 8.5706 2.3466 11.22947.2 The bypass cooling system shown is designed so that the total flow of 200 gpm of a liquidwith heat capacity of 0.8 Btu/lboF is split under the normal conditions, 20 % going aroundthe bypass and 80 % going through the cooler. Process inlet and outlet temperature underthese conditions are 250 and 150oF. Inlet and outlet water temperatures are 80 and 120oF.Process side pressure drop the exchanger is 10 psi. The control valves have linear trim andare designed to be half open at design rates with a 15 psi drop over the bypass valve and 10psi drop over the cooler valve. Liquid density is constant at 62.3 lb/ft3.What will the valve positions be if the total process flow is reduced to 25 percent of designand the process outlet temperature is held at 150oF?W a t e rP r o c e s ss t r e a mC o o l e d s t r e a mT ( t )iT ( t )T CS e t p o i n t( S P )T TP r o c e s s b y p a s s8. Express the function given the graph in the t-domain9. Given f(s) = (1  2e-s + e-2s)/s2 find f(t)10. Find the inverse ofF(s) = 228 20ss s+- + 11. Determine the transfer functions for this system at s = 0.5. 12. Consider the following control system with all instrumentation in electronic (4 to 20 mA) TTTCFTHeatexchangerCooler90 FoHot oilCooled oil 70 FoRefrigerant50 Fo a) If the temperature transmitter has a range of 50-100oF, determine the value from the temperature transmitter. b) If the range of the orifice-differential pressure flow transmitter on the water line is 0-2000 gpm, determine the value from the flow transmitter for a water flow rate of 900 gpm.References1. D.R. Coughanowr and S. LeBlanc, Process Systems Analysis and Control, McGraw-Hill, 3nd edition, 2008.2. Process Modeling, Simulation, and Control for Chemical Engineers by