Chegg Guided Solutions for Process Systems Analysis and Control 3rd Edition: Chapter 8

Chapter: Problem:
2.5 Stars(3 users)
  • Step 1 of 1

    The two-tank heating process shown in Fig. P8.1 consists of two identical, well- stirred tanks in series. A flow of heat can enter tank 2. At time t = 0, the flow rate of heat to tank 2 suddenly increases according to a step function to 1000 Btu/min, and the temperature of the inlet water T drops from 60° F to 52° F according to a step function. These changes in heat flow and inlet water temperature occur simultaneously.

    (a)   Develop a block diagram that relates the outlet temperature of tank 2 to the inlet temperature to tank 1 and the flow of heat to tank 2.

    (b)   Obtain an expression for T2' (s) where T2' is the deviation in the temperature of tank 2. This expression should contain numerical values of the parameters.

    (c)  Determine T2(2) and T2(∞).

    (d)   Sketch the response T2' (t) versus t.

    Initially, Ti=T1= T2= 60° F and q = 0. The following data apply: w = 250 lb/min

    holdup volume of each tank = 5 ft3

    density of fluid = 50 lb/ft3

    heat capacity of fluid = 1 Btu/(lb) (° F)

    Solution 8.1)

    We can write energy balances for each tank:

    where T1is the temperature of tank 1, T2is the temperature of tank 2, p, is the fluid density, Cpis the fluid heat capacity, w is the mass flow rate through the tanks, Tiis the temperature of the fluid entering tank 1, and q is the heat added to the inner tank. Note that the volume, V, of each tank is the same.

    At steady state the above energy balances reduce to:

    where the subscript s denotes a steady state value. Subtracting the steady state equations from the corresponding transient equation, defining the following deviation variables:


    wCp(T’i(s)T’1(s))=pVCpT’1(s) Tank 1

    wCp (T’i(s)T’2(s))+Q’ (s)=pVCpsT’2(s) Tank 2

    Now substitute the following numerical values for the material properties and process conditions and rearrange:

    p VCp=(50 lb/ft3)(5 ft3)(1 Btu/lb/F)=250 Btu/F

    wCp={250 lb/min){1 Btu/lb/F)=250 Btu/F/min

    where time has units of minutes. A simulation of this model using Simulink, is:

    Note at t=2minutes, T’2=-1.29, therefore T2=60-1.29=58.7F, and T2()=60-4=56F.

    Inverting this with Matlab:


    T2s =




    T2t =


    t=2; T2=60+eval(T2t)

    T2 =


    t=1000; T2=60+eval(T2t)

    T2 =


    We can also find the final value by using the final value theorem and multiplying T2'(s) by s and letting s approach 0. This gives a value of T2'=-4 so T2= T2'+60=56.

Select Your Problem

Step-by-step solutions for this book and 2,500 others
Already used by over 1 million students to study better

Get Study Help from Chegg
