Chapter 3.1: Heat Exchanger Analysis Using LMTD method

Steady Flow Steady Flow Energy Equation (SSSF-EE) Neglecting potential and kinetic energy changes and in the absence of the external work, the SSSF-EE is reduced to Multiplying the two sides by For the hot fluid side, is negative (heat rejected) For the cold side, is positive (heat added) h h h h

Note that eqns.(1) and (2) are independent of the flow arrangement and heat exchanger type. Another useful expression can be established using the overall heat transfer coefficient U that is composed of the thermal resistances of inside flow, separating wall material and outside flow. This rate equation is of the form: Where is an appropriate mean temperature difference.

Logarithmic Mean Temperature Difference Parallel-Flow Heat Exchanger: The hot and cold fluid temperature distributions associated with a parallel-flow heat exchanger are shown in this figure. The temperature difference T is initially large but decays rapidly with increasing x. The outlet temperature of the cold fluid never exceeds that of the hot fluid. The form of Tm may be determined by applying an energy balance to differential elements in the hot and cold fluids. Each element is of length dx and heat transfer area dA.

Assumptions: 1. The heat exchanger is insulated from its surroundings, in which case the only heat exchange is between the hot and cold fluids. 2. Axial conduction along the separating wall is negligible. 3. Potential and kinetic energy changes are negligible. 4. The fluid properties are constant. 5. The overall heat transfer coefficient is constant. Let, Applying an energy balance to each of the differential elements, it follows that and

The heat transfer across the surface area dA may be given by where, T = Th – Tc is the local temperature difference between the hot and the cold fluids. , by differentiation Substituting eqns. (4) and (5) into eqn.(7) results in Substituting for from eqn.(6) into eqn.(8) and integrating across the heat exchanger, we obtain

Substituting for Ch and Cc from eqns. (1) and (2) into eqn Substituting for Ch and Cc from eqns.(1) and (2) into eqn.(9) , it follows that For the parallel-flow heat exchanger: Comparing the above expression with eqn.(3), we conclude that the appropriate average temperature difference is a

logarithmic mean temperature difference (Tlm) or (LMTD) where For the parallel-flow heat exchanger, the endpoint temperature differences are defined as

Counter-flow Heat Exchanger: The hot and cold fluid temperature distributions associated with a counter-flow heat exchanger are shown in the figure. Note that the temperature of the cold fluid Tc,o may now exceed the outlet temperature of the hot fluid Th,o. The following equations are used for the counter-flow arrangement

where, For the counter-flow heat exchanger the endpoint temperature differences must be now defined as For the special case of T1 = T2 , But by the application of L’Hospital rule

Example 1: In a concentric double pipe heat exchanger, the inlet and outlet temperatures of the hot fluid are, respectively, Th,i = 260oC and Th,o = 140oC, while for the cold fluid they are Tc,i = 70oC and Tc,o = 125oC. Calculate the logarithmic mean temperature difference for (a) parallel-flow arrangement, and (b) counter-flow arrangement. Data: Find: (a) , (b) Solution: The temperature profiles for the counter-flow and parallel-flow arrangements are illustrated in the sketch. (a) For the parallel-flow arrangement heat exchanger:

(b) For the counter-flow arrangement heat exchanger:

Comment: For the same inlet and outlet temperatures, the log mean temperature for counter-flow is larger than that for parallel-flow: Hence the surface area required to achieve a prescribed heat transfer rate is smaller for the counter-flow than for the parallel-flow arrangement, assuming the same value of U . Arithmetic Mean Temperature Difference It is of interest to compare the LMTD of T1 and T2 with their arithmetic mean:

Special Operating Conditions: Equal Heat Capacity Rates: A comparison of the logarithmic and arithmetic mean temperature differences as a function of the ratio (T1/T2) is presented in the following table: Special Operating Conditions: Equal Heat Capacity Rates: The figure shows a counter-flow heat exchanger for which the heat capacity rates are equal (Cc =Ch), or . The temperature difference T must then be a constant throughout the 3 2 1.7 1.5 1.2 1 T1/T2 1.10 1.04 1.023 1.0137 1.0028 Tam/Tlm

heat exchanger, thereby . Infinite Heat Capacity Rates: a) Condensation: (in a condenser) In case of condensing a vapor, the hot fluid undergoes a change of phase from gas to liquid phase. Its temperature remains constant throughout the heat exchanger (isothermal process), while the temperature of the cold fluid increases. During condensation, the hot fluid loses Latent heat ,while the cold fluid gains sensible heat.

a) Evaporation: (in an evaporator or boiler) In an evaporator or a boiler, the cold fluid undergoes a change in phase from liquid to gas phase. The cold fluid gains A latent heat for evaporation, while the hot fluid loses sensible heat.

Correction Factor (F): The logarithmic mean temperature difference developed above is not applicable for the heat transfer analysis of cross-flow and multi-pass heat exchangers. Therefore, a correction factor (F) must be introduced so that the simple LMTD for the counter-flow regime can be adjusted to represent the corrected temperature difference Tcorr for the cross-flow and multi-pass arrangements as The rate of heat transfer is given then by The correction factor is less than unity for cross-flow and multi-pass arrangements. It is unity for true counter-flow heat exchanger. It represents the degree of departure of true mean temperature difference from the LMTD for the counter-flow.

Algebraic expressions for the correction factor F have been developed for various shell-and-tube and cross-flow heat exchanger configurations, and the results are represented graphically. Selected results are shown in the following figures for common heat exchanger configurations. T = the fluid temperature in the shell side, t = the fluid temperature in the tube-side. With this convention it does not matter whether the hot fluid or the cold fluid flows through the shell or the tubes. If the temperature change of one fluid is negligible, . Such would be the case of phase change (evaporation or condensation)

shell-and-tube heat exchanger with one shell and any multiple of two tube passes

shell-and-tube heat exchanger with two shell passes and any multiple of four tube passes

Single pass, cross-flow heat exchanger with both fluids unmixed

Single pass, cross-flow heat exchanger with one fluid mixed and the other unmixed