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### Designing a U tube Heat Exchanger for Waster Heat Recovery

Heat absorb by cold fluid of U tube heat Exchanger
In U tube heat exchanger the cold fluid usually moves inside the tubes of the U tube heat exchanger whereas the hot fluid surrounds the cold fluid inside the U tube heat exchanger (Asawari, 2016). This gives greater surface area to the cold fluid to absorb heat from hot fluid as each tube of U tube heat exchanger containing the cold fluid is surrounded by the hot fluid. The heat transfer to the cold fluid can be calculated in different methods, one is to due heat exchanger design parameters to calculated the heat transfer and temperature output of the cold fluid coming out of U tube heat exchanger and second is to use fluid working parameters in U tube heat exchanger to calculate the heat transfer to the fluid (Durges 2014). A second method is used here as U tube heat exchanger dimensions are not known at this stage. The heat transfer in Watt to the cold fluid of U tube heat exchanger can be calculated using the cold fluid mass flow rate inside U tube heat exchanger, specific heat of cold fluid and the inlet and outlet temperature of the cold fluid. Complete equation to calculate heat transfer to cold fluid in U tube heat exchanger is shown below (T.D Eastop, 1993)
Q= m_c*  C_pc*( T_co-  T_ci )
Q is the heat transfer in Watt
m_c is cold fluid mass flow rate in Kg/sec
C_pc is cold fluid specific heat in J/Kg.K
T_co is cold fluid outlet temperature in degree centigrade
T_ci   is the cold fluid inlet temperature in degree centigrade Heat Rejected by hot fluid U tube heat Exchanger
In U tube heat exchanger the hot fluid moves in the shell of U tube heat exchanger and it surrounds the tubes of U tube heat exchanger (Asawari, 2016). This layout helps better transfer of heat from hot fluid to cold fluid in U tube heat exchanger. The hot fluid will enter the U tube heat exchanger at high temperature and leave the heat exchanger at lower temperature and heat will be transfer from the hot fluid. The methods of calculating the heat lost by the hot fluid is very much same as that of the heat absorb by the cold fluid (Durges 2014). This is case the final temperature is lower as compared to initial temperature due to which during calculation the final temperature will be subtracted from initial temperature to get the positive temperature difference. Other parameter which includes mass flow rate of hot fluid and specific heat of hot fluid have their own specific values for hot fluid moving in shell of U tube heat exchanger (T.D Eastop, 1993).
Q= m_h*  C_ph*( T_hi-  T_ho )
Q is the heat transfer in Watt
m_h is hot fluid mass flow rate in kg/sec
C_ph is hot fluid specific heat in J/kg.K
T_ho is hot fluid outlet temperature in degree centigrade
T_hi   is the hot fluid inlet temperature in degree centigrade

Heat Lost in U Tube Heat Exchanger
According to the laws of thermodynamics it is not possible for a system to transfer all its heat to work and energy (Asawari, 2016). Some of the heat present in the system will be lost to environment, friction and any other losses present in system. Similar to that the heat exchange happening in U tube heat exchanger will not be an ideal case where all heat rejected by hot fluid will be absorb by the cold fluid. Some of the energy will lose to the environment (Durges 2014). Loss of heat can be from the shell of the U tube heat exchanger where heat of hot fluid is absorb by shell material and then transfer to the surrounding environment.  Similar to this some of the heat will be lost with the hot fluid going out of the U tube heat exchanger due to irregular flow of hot or cold fluid inside U tube heat exchanger (T.D Eastop, 1993).
For an ideal case
Q_(tansfer )= Q_(rejected )= Q_absorb
For practical case
Q_(tansfer )= Q_absorb<Q_(rejected )
It also can be said
Q_(rejected )=Q_absorb+Q_lost

Heat Transfer from hot to cold fluid of U tube heat Exchanger
In U tube heat exchanger the ability of the heat exchanger to transfer heat between hot fluid and cold fluid depends on the design parameters of the U tube heat exchanger (Asawari, 2016). The heat transfer by the U tube heat exchanger depends on the heat transfer coefficient of the U tube heat exchanger, Area of the U tube heat exchanger available for the exchanger of the heat and temperature difference between the input and output of the U tube heat exchanger called the log mean temperature difference of the U tube heat exchanger (Durges 2014).  The below mention equation of the heat transfer in U tube heat exchanger design parameter dependent and can be utilised to determine the design parameters like required area of the U tube heat exchanger.
Q= U*n*A* ∆T_lmtd
Q is heat transfer in Watt
U is over all heat transfer coefficient in W/m^2 K
A is tube area of heat exchanger in m^2
∆T_lmtd is long mean temperature difference in degree centigrade
n Number of tubes

Area of Tube of U tube heat Exchanger
Based on the value of area obtained from the above equation of the heat transfer by the U tube heat exchanger, the value of the area required by U tube heat exchanger to transfer the heat can be calculated (Durges 2014). Utilizing the value of area of U tube heat exchanger required for heat transfer and the standard dimensions of U tube heat exchanger the number of tubes required to have the necessary area of U tube heat exchanger can be calculated (Asawari, 2016).
A = π*n* d* l
A is area of tube in square meter m^2
d is diameter of tube in meter m
l is the length of the tube from inlet to outlet in meter m
n is the number of tubes

Log mean temperature Difference of U tube heat Exchanger
A single U tube heat exchanger is dealing with two different types of fluids which are entering and leaving the U tube heat exchanger at very different temperatures (Singh 2013). In real life application the heat rejected and absorb by the hot and cold fluid respectively will be different so temperature difference of none of the fluid can be used in the calculation of U tube heat exchanger design parameters calculation. So the temperature difference which considers both fluid temperatures called the log mean temperature difference can be utilised. Log mean temperature difference can be calculated using following formula.
∆T_lmtd=( ∆T_(2 )- ∆T_1)/ln⁡((∆T_2)/(∆T_1 ))
∆T_lmtd is log mean temperature difference
∆T_(2 ) is temperature difference at outlet
∆T_1 is temperature difference at inlet

Overall heat transfer coefficient of U tube heat Exchanger
In a U tube heat exchanger the heat transfer from the hot fluid to the cold fluid happens through a medium of metal pipe where heat first transfer from hot fluid to the outer surface of the tube using the mechanism of heat transfer called convection heat transfer (Rajesh Ghosh 2013). After this the heat transfer from outer tube surface to inner tube surface using the mechanism of heat transfer called conductive heat transfer (Singh 2013). In last heat first transfer from tube inner surface to cold fluid using the mechanism of heat transfer called convection heat transfer. The transfer of heat using convection, conduction and convection depends on the heat transfer coefficient of convection, conduction and convection. The first convective heat transfer coefficient depends on the hot fluid and tube material relation and thermal properties. The conductive heat transfer coefficient depends on tube thickness and tube material thermal conductivity. The second convective heat transfer coefficient depends on the cold fluid and tube material relation and thermal properties. In design of U tube heat exchanger all three different heat transfer coefficient needed to be considered. An overall heat transfer coefficient of U tube heat exchanger can be calculated using the below mention formula.
1/U=1/(h (tube fluid))+x/(k (tube))+1/(h (shell fluid))
Where
U is overall heat transfer coefficient W/m^2 K
H is convective heat transfer coefficient W/m^2 K
K is conductive heat transfer coefficient W/mK

Pressure Drop in U tube heat Exchanger
The tube side pressure drop is the sum of the pressure drop through the tubes, the pressure drop through the channels and pressure drop through the 180 degree bend,
∆P_T=(f*〖m_h〗^2*l*n)/(2*ρ*g*di)+(4*n〖*v〗^2)/(2*ρ*g)+(k*ρ*u^2)/2
f is constant
mh is tube mass flow rate
l is length of tube
n is number of tubes
ρ is density of tube fluid
g is gravity
di is tube internal diameter

Tube Pitch of U tube heat Exchanger
As more than one tube is required in U tube heat exchanger so the distance at which they can be installed in the U tube heat exchanger should be perfect and calculated so that proper heat transfer can be take place (Rajesh Ghosh 2013). There are two different patterns in which tubes can be placed inside the shell of U tube heat exchanger (Singh 2013). One is the square arrangement where tubes are place in such a manner that they made square cross section at their inlet and outlet. Second is the triangular arrangement where tubes are place in such a manner that they made triangular cross section at their inlet and outlet (Singh 2013). The pitch of the tube basically depends on the outer diameter of the tubes and can be calculated as follow.
Pt=1.25*Tube outer diameter