# 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.

### 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.

### 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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