Design of a Tubular Heat Exchanger

 

 

 

 

 

 

 

 

 

 

 

 

The primary objective in using a heat exchanger is to transfer thermal energy from one fluid to another.  We will use the following assumptions:

1.    Heat transfer is under steady-state conditions.

2.    The overall heat-transfer coefficient is constant throughout the length of pipe.

3.    There is no axial conduction of heat in the metal pipe.

4.    The heat exchanger is well insulated.  The heat exchange is between the two liquid streams flowing in the heat exchanger. There is negligible heat loss to the surroundings.

Recall that change in heat energy in a fluid stream, if its temperature changes from T1 to T2, is:

 

                                              

 

where = mass flow rate of a fluid (kg/s), cp = specific heat of a fluid (kJ/kg°C), and the temperature change of a fluid is from some inlet temperature T1 to an exit temperature T2.

 

Consider a tubular heat exchanger, as shown in figure. A hot fluid, H, enters the heat exchanger at location (1) and it flows through the inner pipe, exiting at location (2). Its temperature decreases from TH, inlet to TH,exit . The second fluid, C, is a cold fluid that enters the annular space between the outer and inner pipes of the tubular heat exchanger at location (1) and exits at location (2). Its temperature increases from TC,inlet to TC,exit. The outer pipe of the heat exchanger is covered with an insulation to prevent any heat exchange with the surroundings. Because the heat transfer occurs only between fluids H and C, the decrease in the heat energy of fluid H must equal the increase in the energy of fluid C.  Therefore, conducting an energy balance, the rate of heat transfer between the fluids is:

 

                  

where cpH is the specific heat of the hot fluid (kJ/kgºC), cpC is the specific heat of the cold fluid (kJ/kgºC),  is the mass flow rate of the hot fluid (kg/s), and is the mass flow rate of the cold fluid (kg/s).

Equation is useful to determine the inlet and exit temperatures of the two fluid streams, and to determine the mass flow rate of either fluid stream, provided all other conditions are known. But, this equation does not provide us with any information about the size of the heat exchanger required for accomplishing a desired rate of heat transfer, and we cannot use it to determine how much thermal resistance to heat transfer exists between the two fluid streams. For those questions, we need to determine heat transfer perpendicular to the flow of the fluid streams, as discussed in the following.

Consider a thin slice of the heat exchanger, as shown in figure. We want to determine the rate of heat transfer from fluid H to C, perpendicular to the direction of the fluid streams. For this thin slice of the heat exchanger, the rate of heat transfer, dq, from fluid H to fluid C may be expressed as:

 

                                                       

 

where DT is the temperature difference between fluid H and fluid C. Note that this temperature difference, DT, varies from location (1) to (2) of the heat exchanger. At the inlet of the fluid streams, location (1), the temperature difference, DT, is TH,inlet – TC,inlet and on the exit side, location (2), it is TH,exit – TC,exit . To solve Eq. we can substitute only one value of DT, or its average value that represents the temperature gradient perpendicular to the direction of the flow. We will develop the following mathematical analysis to determine a value of DT that will correctly identify the "average" temperature difference between the fluids H and C as they flow through the heat exchanger.

The temperature difference, DT, between the two fluids H and C is

                                                  

Where TH is the temperature of the hot stream and TC is that of the cold stream. For a small differential ring element as shown in Figure 4.28, using energy balance for the hot stream H we get

                               

 

and, for cold stream C in the differential element,

                                       

In Eq. , dTH is a negative quantity, therefore we added a negative sign to obtain positive value for dq. Solving for dTH and dTC, we obtain

                                        

and

                                         

Then subtracting Eq.   from Eq.

            

Using Eqs and   and substituting in Eq.

                              

 

Integrating Eq. from locations (1) to (2) shown in Figure,

              

Noting that

                                

we get

                            

 

Substituting Eq. in Eq.

 

         

Rearranging terms in Eq. ,

 

                           

Substituting Eq.  in Eq 

 

We obtain,

                               

Rearranging terms,

                                              

 

                                                

where

 

                                            

          DTlm is called the log mean temperature difference. Equation is used to design a heat exchanger and determine its area and the overall resistance to heat transfer, as illustrated in the following examples.