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Log Mean Temperature Difference

Before equation (eq – 1.7) can be used to determine the heat transfer area required for a given duty, an estimate of the mean temperature difference Δtm must be made. This will normally be calculated from the terminal temperature differences: the difference in the fluid temperatures at the inlet and outlet of the exchanger. The well-known “logarithmic mean” temperature difference (LMTD) is only applicable to sensible heat transfer in true co-current or counter current flow. For counter flow the LMTD is given by:

In a heat exchanger the temperatures of the hot and cold fluids keep on changing from point to point along the length of the exchanger. The question therefore arises what value of the temperature difference should be used to compute the rate of heat flow. The need of mean temperature difference which, when multiplied by the overall coefficient of heat transfer and the appropriate area, will give the correct heat flow, originated. An expression for the mean temperature difference

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,

The equation is the same for co-current flow, but the terminal temperature differences will be (t1 – t1) and (T2 – t2). Strictly, equation (eq – 1.9) will only apply when there is no change in the specific heats, the overall heat-transfer coefficient is constant, and there are no heat losses.

In most shell and tube exchangers the flow will be a mixture of co current, counter current and cross flow. Fig 1.3 show typical temperature profiles for an exchanger with one shell pass and two tube passes (a 1: 2 exchanger).

Designing a test heat exchanger

As the physical layout of the exchanger cannot be determined until the area is known the design of an exchanger is of necessity a trial and error procedure. The steps in a typical design procedure are given below:

  1. Define the duty: heat transfer rate, fluid flow-rates, temperatures.
  2. Collect together the fluid physical properties required: density, viscosity, thermal conductivity.
  3. Decide on the type of exchanger to be used.
  4. Select a value for the overall coefficient, U.
  5. Calculate the mean temperature difference, Δtm .
  6. Calculate the area required from equation Q = UA Δtm
  7. Decide the exchanger layout.
  8. Calculate the individual coefficients.
  9. Calculate the U and compare it with selected U value. If the calculated value differs significantly then return to step 6.
  10. Optimize the design: repeat steps 4 to 10, as necessary, to determine the cheapest exchanger that will satisfy the duty. Usually that will be one with the smallest area.

Typical values of the overall heat-transfer coefficient for various types of heat exchanger are given below:

Shell and tube exchangers

Hot fluid

Cold fluid

U (W/m2  oC)

Heat exchangers

Water

Water

800-1500

Steam

Water

1500-4000

Coolers

Organic solvents

Water

250-750

Light oils

Water

350-900

Heavy oils

Water

60-300

Gases

Water

20-300

Condensers

Aqueous vapors

Water

1000-1500

Organic vapors

Water

700-1500

Organics (some non-condensables)

Water

500-700

Vacuum condensers

Water

200-500

Jacketed vessels

Water

Dilute aqueous sol.

200-500

Water

Light organics

200-300

Types of Heat Exchangers

The principal types of heat exchangers used in chemical and allied industries which will be discussed in this chapter, are listed below:

  1. Double-pipe exchanger: the simplest type, used for cooling and heating.
  2. Shell and tube exchangers: used for all applications
  3. Plate and frame exchangers (plate heat exchangers): used for heating and cooling.
  4. Plate-fin exchangers.
  5. Spiral heat exchangers.
  6. Air-cooled: coolers and condensers.
  7. Direct contact: cooling and quenching.
  8. Agitated vessels.
  9. Fired heaters.

Heat exchanger types are also classified according to the direction of flow of the hot and cold fluids with respect to each other, or according to the temperature distribution of the two fluids along the exchanger length. Thus, we may have the following types of heat exchanger:

a) Parallel-flow exchanger
b) Counter-flow exchanger
c) Cross-flow exchanger
d) Condenser or evaporators

Parallel-flow exchangers: The hot fluid and cold fluids flow in the same direction, hence the name parallel-flow. Many devices, such as water heaters, oil heaters and oil coolers, etc., belong to this class.
The temperature difference between hot and cold fluid keeps on decreasing from inlet to exit as shown in following fig.1.2

Counter-flow Exchangers: In this case the fluids flow through exchanger in opposite directions, hence the name counter flow. The temperature distribution in counter-flow exchanger is shown in fig.1.3 Below:

It can be seen that the temperature difference between the two fluids remains more nearly constant as compared to the parallel-flow type. This arrangement gives maximum heat transfer rate for a given surface area. If the fluid flows through the exchanger only once, it is called a single pass heat exchanger. In many designs, one or both fluids may traverse the exchanger more than once. Such exchangers are called multi-pass exchangers.

Cross-flow exchangers: Here the two fluids flow at right angles to each other. Two different arrangements of this exchanger are commonly used. In one case, each of the fluids is unmixed as it flows through the exchanger. As a result, the temperatures of the fluids leaving the exchanger are not uniform. An automobile radiator is an example of this type of exchanger. In other case, one fluid is perfectly mixed while the other is unmixed as it flows through the exchanger.

Condenser: In a condenser the condensing fluid (hot fluid) remains at constant temperature throughout the exchanger while the temperature of the colder fluid gradually increases from inlet to outlet. Similarly in an evaporator the boiling fluid (cold fluid) remains at constant temperature while the hot fluid temperature gradually decreases. The temperature distribution in condenser is shown below. Since the temperature of one of these fluids remains constant, it is immaterial whether the two fluids flow in the same direction of opposite direction.

Basic design procedure and theory

The general equation for heat transfer across a surface is:

Q = UA Δtm                                                                                           (eq – 1.7)

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.

The overall coefficient is the reciprocal of the overall resistance to heat transfer, which is the sum of several individual resistances, is given by:

1 = 1 + 1   + (doIn(do/di)) + do >< 1 + do >< 1                                 (eq – 1.8)

Uo  ho   hod               2kw           di       hid   di       hi

Where   Uo = the overall coefficient based on the outside area of the tube, W/m2  oC,

            ho = outside fluid film coefficient, W/m2  oC,

            hi = inside fluid film coefficent, W/m2  oC,

            hod = outside dirt coefficient (fouling factor), W/m2  oC,

            hid = inside dirt coefficient, W/m2  oC,

            kw = thermal conductivity of the tube wall material, W/m2  oC,

            di = tube inside diameter, m,

            do= tube outside diameter, m.

The magnitude of the individual coefficients will depend on the nature of the transfer process (conduction, convection, condensation, boiling or radiation), on the physical properties of fluids, on the fluid flow-rates, and on the physical arrangement of the heat transfer surface.

Heat transfer within cooling system (heat exchanger)

The process represented in the above figure (1.1) is one in which hot fluid is cooled by water, which itself heated without any loss of exchanged heat. Industrial heat exchangers consist of a number of tubes enclose in a shell. Exchangers with cooling water in the tubes, and hot product in the shell are the most satisfactory. Fouling often occurs if water is circulated through the shell, because the velocity of the water stream is lower in this design. Tubes are much easier to clean than the shell. Matters are also so arranged that the pressure of the product being cooled is higher than that of cooling water, so that the water cannot leak into the hot product, and damage the equipment.

In the figure (t1) temperature of CW inlet; (t2) temperature of CW outlet; (T1) temperature of process inlet; (T2) Temperature of process outlet. Here we will consider CW as cooling water and Process as P.

Now  U = ΔHp/ΔtmA                                                                                                (eq-1.0)

Where:             U = net effective overall heat transfer coefficient (Kcal/0C-h-m2)

                        ΔH = difference in enthalpy of P at T1 and T2 (Kcal/kg)

                        P = flow rate of the product (m3/h)

                        A = area of heat transfer surface (m2)

Δtm = the average of the temperature differences at both ends of the exchanger (0C)

Δtm = (T1-t2) +(T2-t1)                                     (eq – 1.5)

                                                2

Here it is assumed that the temperature of the fluid is in the shell falls continuously and uniformly from T1 to T2, while that of the water inside the tubes rises similarly from t1 to t2. With this assumption it is permissible to use the average of the terminal differences for the mean temperature difference. In more complicated heat exchangers, however, it is necessary to use the log mean temperature difference, and when calculation the value for multipass exchangers correction factors also must be applied to (Δtm).

(Δt)log e =     (Δt)max - (Δt)min                        = (T2-t1) – (T1 – t2)                (eq – 1.6)

                  log e [(Δt)max/(Δt)min]                    log e [(T2-t1)/(T1-t2)]

(see topic LMTD for detailed description)

In a water-tube exchanger U is likely to decrease gradually because of accumulating deposits, or because of scale formation on the tubes. Referring to the figure, the effect of these events on the heat transfer coefficient can be predicted qualitatively. If a thin layer of insulating scale forms on either side of the tube T2 rises and t2 falls as less heat passes from P to CW through the insulating layer. Thus ΔH decreases, and as both t1 and T1 are unaffected by conditions within the heat exchanger, Δtm increases. The net result is that the heat transfer coefficient becomes smaller.

If deposits slow the flow of water, t2 and T2 both rise. In this event, however, Δtm    may increase and ΔH may decrease by such small amounts that the effect on U may not be significant.

The reciprocal of the heat transfer coefficient is called the “fouling resistance,” this number multiplied by one thousand is the “fouling factor.” Except in unusual circumstances the effect of fouling resistances can never be exactly known, as fouling within an actual heat exchanger is seldom uniform, and also the net effect is a combination of conditions on both sides of the heat transfer surface.

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