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Fabrizio Sarghini » 24.Heat Exchangers - I Part


Heat Exchangers

A heat exchanger is a piece of equipment aimed to transfer energy in the form of internal heat (enthalpy) between two or more fluids, or between a surface and a fluid, between a particle and a fluid through a thermal contact without external interactions and without external work .

The fluid can be composed of a single component or mixtures.

Typical applications are heating and cooling of a fluid of interest, the evaporation or condensation of a single or multi-component flow, the recovery or dissipation of heat from a system.

In the case of the agro-food industry, the goal can be a process of sterilization, pasteurization, fractionation, distillation, concentration, crystallization, or to simply control a process.

Heat Exchangers

The simplest type of heat exchanger is in direct mixing.

The most common type of heat exchanger is instead the so-called recovery heat exchanger.

The simplest heat exchanger is the tube in a tube heat exchanger.

Depending on the warm and cold flow direction, we can define two major configurations: concurrent flow heat exchanger and countercurrent flow heat exchanger.

Heat Exchanger

The simplest type of heat exchanger is in direct mixing.

The most common type of heat exchanger is instead the so-called recovery heat exchanger.

The simplest heat exchanger is the tube in a tube heat exchanger.

Depending on the warm and cold flow direction, we can define two major configurations: concurrent flow heat exchanger and countercurrent flow heat exchanger.

Heat Exchangers

Concurrent Flow – In this exchange system, the two fluids flow in the same direction.

This diagram presents a generic representation of this exchange system, with two parallel tubes containing fluid separated by a  thermoconductive wall.

This diagram presents a generic representation of this exchange system, with two parallel tubes containing fluid separated by a thermoconductive wall.


Heat Exchangers

Countercurrent Flow - In this exchange system, the two fluids flow in the opposite direction.

This diagram presents a generic representation of this exchange system, with two parallel tubes containing fluid separated by a  thermoconductive wall.

This diagram presents a generic representation of this exchange system, with two parallel tubes containing fluid separated by a thermoconductive wall.


Heat Exchangers

Concurrent flow heat exchanger 
temperature diagram.

Concurrent flow heat exchanger temperature diagram.


Heat Exchangers

Countercurrent flow heat exchanger temperature diagram.

Countercurrent flow heat exchanger temperature diagram.


Heat Exchangers

Condenser.

Condenser.


Heat Exchangers

Evaporator.

Evaporator.


Heat Exchangers

dq=U\cdot dA\cdot \Delta t

  • For the hot fluid (sub index H), we have

dq=-m_Hdh_H=m_Hc_{PH}dT_H

where mH is the mass flow (kg/h), hH the specific enthalpy, cPH is the specific heat at constant pressure (kcal / kg ºC) and T the average temperature (ºC) of the fluid;

  • for the cold fluid (sub index C), we have

dq=\pm m_c dh_C=\pm m_Cc_{PC}dT_C

where flow positive sign applies to the concurrent ( positive slope of the temperature gradient), while negative applies to the countercurrent (negative slope of the gradient of temperature).

Heat Exchangers

Equating the two expressions, we can therefore write that

dq=-m_Hc_{PH}dT_H=\pm m_C c_{PC}dT_C

We can then introduce the thermal capacity per hour (kcal/hºC) for hot fluid CH=mHCPH and CC=mCCPC for the cold one,

-C_HdT_H=\pm C_cdT_C

Assuming that these two thermal capacity allocations are constant along the whole heat exchanger, the integration of this equality is quite simple: select for the integration, the input section (characterized by temperatures TCI and THI) and a generic section (characterized by TC and TH), we obtain

-C_H(T_H-T_{HI})=C_C(T_C-T_{CI})

taking into account the ± in the integration process.

Heat Exchangers

In the generic section the difference in temperature between hot and cold fluid, adding and subtracting the term CHTC at first member and rearranging, we obtain

T_H-T_C=-\Biggl(1+\frac{C_C}{C_H}\Biggr)T_C+\frac {C_C}{C_H}T_{CI}+T_{HI}

and
dq=U\cdot dA\cdot \Delta T=U\vdot dA\cdot (T_H-T_C)

dq=U\cdot dA\Delta T=U\cdot dA\cdot \biggl[ -\biggl(1+\frac{C_C}{C_H}\Biggr)T_C+\frac {C_C}{C_H}T_{CI}+T_{HI}\Biggr]

C_CdT_C=U\cdot dA\cdot\Biggl[-\Biggl(1+\frac{C_C}{C_H}\Biggr)T_C+\frac {C_C}{C_H}T_{CI}+T_{HI}\Biggr]

Heat Exchangers

\frac{dT_C}{\Biggl[-\Biggl(1+\frac{ {C_C}}{C_H}\Biggr)T_C+\frac{C_C}{C_H}T_{CI}+T_{HI}\Biggr]}=\frac{U\cdot dA}{C_C}

\int_{T_{CI}}^{T_{CO}}\frac{dT_C}{\Biggl[-\Biggr(1+\frac{C_C}{C_H}\Biggr)T_C+\frac{C_C}{C_H}T_{CI}+T_{HI}\Biggr]}=\int_0^{A_{tot}}\frac {U\cdot dA}{C_C}

Where at the inflow A=0 and T=TCI and at the outflow A=Atot and T=TCO, obtaining
\ln \frac{\Biggl[-\Biggr(1+\frac{C_C}{C_H}\Biggr)T_{CO}+\frac{C_C}{C_H}T_{CI}+T{HI}\Biggr]}{\Biggl[-\Biggr(1+\frac{C_C}{C_H}\Biggr)T_{CI}+\frac{C_C}{C_H}T_{CI}+T{HI}\Biggr]}=-\Biggl(\frac 1{C_C}-\frac 1 {C_H}\Biggl)UA_{tot}

Heat Exchangers

This relation holds for any section of the HE including the outflow, where we can write

\frac{C_F}{C_U}=\frac{T_{FU}-T_{FI}}{T_{CU}-T_{CI}}

Considering that q=-C_C(T_C-T_{CI})=C_F(T_F-T_{FI})
we can write:

\ln \frac{T_{CU}-T_{FU}}{T_{CI}-T_{FI}}=\biggl[(T_{CU}-T_{FU})-(T_{CI}-T_{FI})\biggr]\frac{U\cdot A_{tot}}q

and rearranging

q=\biggl[(T_{CU}-T_{FU})-(T_{CI}-T_{FI})\biggr]\frac{U\cdot A_{tot}}{\ln \frac{T_{CU}-T_{FU}}{T_{CI}-T_{FI}}}

Heat Exchangers

Observing that

\Delta T_{min}=\Delta T_b=T_{CU}-T_{FI}

and

\Delta T_{max}=\Delta T_{\alpha}=T_{CI}-T_{FI}

it results:

q=U\cdot A_{tot}\frac{\Delta T_b-\Delta T_{\alpha}}{\ln \frac {\Delta T_b}{\Delta T_\alpha}}

Defining the log mean temperature difference (also known as LMTD)

\Delta T_{LMTD}=\frac{\Delta T_b-\Delta T_\alpha}{\ln\frac{\Delta T_b}{\Delta T_{\alpha}}}

we obtain for both concurrent and countercurrent heat exchanger

q=U\cdot A\cdot \Delta T_{LMTD}

Heat Exchangers

Limitations:

  • the assumption that the rate of change for the temperature of both fluids is proportional to the temperature difference is valid for fluids with a constant specific heat and  is a good description of fluids changing temperature over a relatively small range. If the specific heat changes, the LMTD approach is no longer accurate;
  • the LMTD is not applicable to condensers and reboilers where the latent heat associated to phase change makes the hypothesis invalid;
  • it has also been assumed that the heat transfer coefficient (U) is constant, and not a function of temperature.
  • the LMTD is a steady-state concept, and cannot be used in dynamic analysis.
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