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Condensation of Steam

Steam is frequently used as heating medium. The mean temperature difference is given here:  A pure, saturated, vapour will condense at a fixed temperature, at constant pressure. For an isothermal process such as this, the simple LMTD can be used in equation 1.3; no correction factor for multiple passes is needed. The LMTD will be given by:

ΔT m =            (t2 – t1)               (eq. 1.34)

ln [(Tsat-t1)/(Tsat-t2)]

Where T sat  =     saturation temperature of the vapour,

            t1     =     inlet coolant temperature,

            t2     =     outlet coolant temperature.

When the condensation process is not exactly isothermal but the temperature change is small; such as where here is a significant change in pressure; the LMTD can still be used but the temperature correction factor will be needed for multipass condensers. The appropriate terminal temperatures should be used in the calculation.

Effect of scale formation

In most heat exchangers some scale formation will take place on both sides of the heat transfer surface after the heat exchanger has been in use for some time (unless scale inhibition mechanism is in place). This introduces two additional resistances in the heat flow path. Thus the total thermal resistance becomes:

ΣR        = Ri + Rsi +Rw + Rso + Ro                         (eq. 1.27)

Where Rsi = thermal resistance due to scale formation on inside surface of inner pipe, m2 C/W

        Rso = thermal resistance due to scale formation on outside surface of inner pipe, m2 C/W

(We can here consider no scaling on outside surface of inner pipe)

Since it is difficult to ascertain accurately the thickness and thermal conductivity of the scale formed, the effect of scale deposit on heat flow is generally taken into account by specifying an equivalent scale heat transfer coefficient, hs.
 
The reciprocal of the scale heat transfer coefficient is called the fouling factor. If hsi and hso denote the heat transfer coefficient for the scale formed on the inside and outside surface of he inner pipe, then:

Rsi        =   1/Aihsi                                                           (eq. 1.29)

Rso        =   1/Aohso                                                         (eq. 1.30)

And q = .                          (ti - to)                                     (eq. 1.31)

               1/Aihi + 1/Aihsi + ln (ro/ri) +1/Aohso + 1/Aoho

                                           2πLKw

Ui  = .                                          1                                                (eq. 1.32)

                 1/hi + 1/hsi + Ai ln (ro/ri) + [Ai/Ao]. 1/hso + [Ai/Ao]. 1/ho

                                                  2πLKw

or

Ui  =                                             1                                            (eq. 1.33)  

            1/hi + 1/hsi + [ri/kw] ln (ro/ri) + [ri/ro]. 1/hso + [ri/ro]. 1/ho

The fouling factor (1/hs) for some representative applications are listed in following table:

Fluid

Fouling factor (1/hs) (m2/K/W)

Distilled water

0.000086

Sea water

0.000172

Well water

0.000344

Treated boiler feed water

0.000172

Fuel oil and crude oil

0.00086

Steam, non-oil bearing

0.00009

Overall heat transfer coefficient

An important parameter in the design and monitoring of heat exchangers is the overall heat transfer coefficient, U, between the two fluids. A value for U can be easily obtained by knowing the followings:

  1. Mass flow of the fluid,
  2. Specific heat of the fluid,
  3. Difference in temperature of the fluid across the heat exchanger,
  4. Inlet and outlet temperature of both the fluids involved in heat exchanger and
  5. Area of the heat transfer surface.

From the equation         Q = m.cp. Δt

Calculating Q in watts:

Where m = mass flow in kg/hr of kg/sec

            Cp= specific heat or heat content in KJ/Kg K

            Δt= Temperature difference across the heat exchanger.

Calculating LMTD from the following equation with the inlet and outlet temperatures of the hot and cold fluid across the heat exchanger and with the knowledge of type of heat exchanger using:

(For multipass heat exchangers)

ΔT m =    (T2-t1) – (T1 – t2)                                                                           

               ln [(T2-t1)/(T1-t2)]

Where ΔT m = log mean temperature difference,

            T1  = inlet shell side fluid temperature,

            T2  = outlet shell side fluid temperature,

            t1   = inlet tube side temperature,

            t2   = outlet tube-side temperature,

 OR

(for single pass heat exchanger)

Δtm = (T1-t2) +(T2-t1)                                    

                        2

We can calculate U from the equation                Q = UA Δtm

Where Q = heat transferred per unit time (W)

            U = the overall heat transfer coefficient (W/m2  oC)

            A –heat-transfer area, m2,

            Δtm = the mean temperature difference, the temperature driving force, oC.

U =      .   Q   .                                                                                                 (eq. 1.14)

            A Δtm

Elaborated method for calculating U values

A general expression for U can be easily obtained as follows. Consider a double pipe heat exchanger in which one fluid through the inner pipe and the other fluid through the annular space between space between the two pipes.

Let       L          = length of heat exchanger, m

            ri          = inside radius of inner pipe, m

            ro          = outside radius of inner pipe, m

            Ai         = inside surface area of inner pipe (2πriL), m2.

            Ao        = outside surface area of inner pipe (2πroL), m2.

            hi          = film coefficient of heat transfer at inside surface of inner pipe, W/m2 C

            ho         = film coefficient of heat transfer at outside surface of inner pipe, W/m2 C

            kw        = thermal conductivity of inner pipe wall, W/m C.

            ti          = temperature of fluid flowing through the inner pipe, C

to          = temperature of fluid flowing through the annular space between the two  pipes, C

Ri         = thermal resistance of fluid film at the inside surface of inner pipe, m2 C/W

Ro        = thermal resistance of fluid film at the outside surface of inner pipe, m2 C/W     

Rw        = thermal resistance of inner pipe, m2 C/W

(i)         The rate of heat transfer between the two fluids is given by:

                        q          = ti - to                        (eq 1.15)

                                         ΣR

Where              ΣR       = Ri + Ro +Rw               (eq. 1.16)

Since                Ri         =   1/Aihi                     (eq. 1.17)

                        Rw        = ln (ro/ri)                     (eq. 1.19)

                                        2πLKw

                        Ro        = 1/Aoho                      (eq. 1.20)

Hence

                        q          = .          (ti - to)                           (eq. 1.21)

                                       1/Aihi + ln (ro/ri) + 1/Aoho

                                                     2πLKw

(ii) If Ui and Uo denote respectively the overall heat transfer coefficient based on unit area of the inside and outside surfaces of the inner pipe, then

                        q          = AiUi (ti - to)  = AoUo (ti - to)        (eq. 1.22)

from eq. 1.21 and 1.22  Ui       = .                          1                         (eq. 1.23)

                                                1/hi + Ai ln (ro/ri) + [Ai/Ao]. 1/ho

                                                                2πLKw

                        Uo        = .                          1                                  (eq. 1.24)

                                       [Ao/Ai].1/hi + [Ao/2πL].ln (ro/ri) + 1/ho

                                                                                  Kw

(iii) Since Ai = 2πriL and Ao = 2πroL, eq. c and d can also be written as:

            Ui         = .                          1                             (eq. 1.25)

                           1/hi + [ri/kw] ln (ro/ri) + [ri/ro]. 1/ho

Uo       = .                          1                              (eq. 1.26)

                 [ro/ri].1/hi + [ro/ Kw].ln (ro/ri) +  1/ho

L.M.T.D. Correction factors

Estimation of the “true temperature difference” from the logarithmic mean temperature difference by applying a correction factor to allow for the departure from true counter-current flow:

ΔT m = Ft ΔT tm                          (eq – 1.10)

Where ΔT m = true temperature difference, the mean temperature difference for use in the design eq. 1.3.

            Ft      = the temperature correction factor.

The correction factor is a function of the shell and tube fluid temperatures, the number of tube and shell passes. It is normally correlated as a function of two dimensionless temperature ratios:

            R   = (T1 – T2)                (eq – 1.11)

                       (t2 – t1)

and

            S   =  (t2 – t1)                (eq – 1.12)

                     (T1 – t1)

R is equal to the shell-side fluid flow-rate times the fluid mean specific heat; divided by the tube-side fluid flow-rate times the tube-side fluid specific heat.

S is a measure of the temperature efficiency of the exchanger.

For a 1 shell: 2 tube pass exchanges, the correction factor is given by:

Ft  =  .                       √(R2 + 1)  ln [(1 – S)/(1 – RS)]                      .      (eq  - 1.13)

         (R – 1) ln [(2 – S (R + 1 - √(R2 + 1))/ (2 – S (R + 1 - √(R2 + 1))]

The derivation of equation 1.9 is for a 1 shell : 2 tube pass exchanger can be used for any exchanger with an even number of tube passes. Plots of the same can be seen in the books.

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