The turbo-generators and thermoelectric generator system developed by the designing and installing of turbo-generators after exhaust manifold and installing the thermoelectric generator in the next position. Therefore, this section outlines the process of design for turbo-generators, followed by turbo-generator development.
3.3. Specific Design
In the first step, the dimension design of the parts of the rotor used the details following the instructions as shown in
Table 3 and
Figure 3.
The equation below describes the relationships of the theoretical work for turbines:
where
is the theoretical work by energy per mass of exhaust gas
,
is the value of adiabatic exhaust gases,
is the constant value of the exhaust gases relative to the gas constant to the molecular weight of the gas,
is the pressure of exhaust gases behind the turbine,
is the pressure of exhaust gases in front of the turbine,
is the temperature of exhaust gases flowing into the turbine, and
is the temperature of exhaust gases flowing out from the turbine.
The following equation explains the size of the work occurring in the turbine, which is related to the decrease of the temperature after the exhaust gas flows through the turbine.
where
is the theoretical power
,
is the intensity of exhaust gas flow
, and
is the efficiency of the turbine in isentropic.
The Euler equation shows the energy transfer in the rotor, which can be represented as a product of the torque by the angular velocity:
where
is the energy transfer,
is the heat transfer per unit mass flow,
is the angular velocity,
is the mass rate of flow,
is the rate of change of angular momentum,
is the specific heat at a constant pressure, and
10-T20 is the temperature difference of the inlet and outlet.
3.4. Formula for One Dimensional Calculation
Once the energy transfer has been determined, the mass flow of exhaust gas required to achieve the specified power can be calculated from:
The blade speed at the inlet can be calculated from the velocity triangles, in which the relative work output is:
The absolute velocity at the inlet:
where
is absolute gas angle at radius.
The radius at the inlet of the rotor has the value of:
The relative velocity at the inlet:
The rotor shroud radius and hub radius are:
The rotor hub radius:
where
is the hub-tip ratio at the inlet impeller.
The mean of the exit radius is equal to its square value using the Balje diagram:
Figure 4 shows the Balje diagram used to calculate the value of the exit radius. The value of
efficiency is in range 0.2–0.8 and the recommended value of
is in range 0.53–0.66.
Consequently, the blade speed at the mean exit radius is:
The blade width at the outlet:
Using mass balance, the inlet blade height can be determined as:
Using the velocity diagram for calculating the flow angle of relative velocity at the exit:
The flow angle of relative velocity at the hub and shroud at the outlet are:
Figure 5 shows an overview of the equations used for calculating the dimensions of the turbine, respectively, where the process for calculating values is continuously correlated using the MATLAB program.
3.7. Drawing and Construction of a Thermoelectric Generator
The output voltage is directly proportional to the temperature change, which is the principle of a thermoelectric generator using the phenomenon characteristics known as the Seebeck effect [
16], and displayed as the following equation:
where
α is the Seebeck coefficient
and
is the temperature difference of two sides of the surface in
.
The Reynolds number derived from the value of the heat source from the engine exhaust gas is:
where
is the density,
is viscosity,
is the equivalent diameter, and
is the viscosity of the fluid flowing through the tube.
The heat transfer coefficient of the hot side is determined by:
The Nusselt number is defined as the ratio of convection heat transfer to fluid conduction heat.
where
is Nusselt number,
is thermo conductivity of exhaust, and
is the hydraulic diameter.
The following equation shows convection on the plate:
where Re is the Reynolds number and Pr is the Prandtl number.
The heat conversion efficiency of waste heat recovery is calculated using the energy from the thermoelectric generator divided by the heat inserted into the thermoelectric generator.
where is the conversion efficiency, is the thermoelectric generator power output, is the exhaust gas mass flow rate, is the exhaust gas specific heat, is the exhaust gas system inlet temperature, and is the exhaust gas system outlet temperature.
The equation to calculating the power generated by the thermoelectric generator is:
where
N is the number of thermoelectric couples employed,
is the Seebeck coefficient,
is, respectively, the electric current,
is the thermoelectric leg temperature difference, and
is the value of the thermoelectric resistivity couple.
The equation for the system efficiency,
η, can be calculated by:
where
is the thermoelectric generator maximum output power and
is the power of the engine.
The efficiency of a TE module used as a generator can be approximated by the following relationship, where
is a material property,
is cold temperature,
is hot temperature, and
is
.
where
is the Seebeck coefficient corresponding to
,
is the Seebeck coefficient corresponding to
,
is thermal conductivity corresponding to
,
is thermal conductivity corresponding to
,
is electrical resistivity the corresponding to
, and
is electrical resistivity corresponding to
.
The power output is:
where
is the electrical current in the generator circuit,
is the electric resistance of semiconductor couple,
is the heat absorbed from heat source, and
is the heat absorbed from the cold source.
Power outputs and conversion efficiencies were calculated applying the numerical results, the electric current and absorbed heat shown in the equations below:
where
is the output power,
is the number of thermoelectric elements in the module, and
is the Seebeck coefficient.
is the length of the legs and
is the cross-sectional area.
The equation for the thermocouple conversion efficiency is:
where
is the absorbed and
is the output power.
Another important part of the process for designing a thermo-electric generator is finding the heat transfer correlations, which is calculated from the heat of the engine exhaust flowing through the exhaust pipe. The design will focus on the suitability and ease of installation with real engines in a limited space to make the best heat transfer efficiency and to not extremely affect the pressure of the exhaust.
Figure 8 illustrates the heat transfer correlations of the thermo-electric generator.
Energy balance of cooling heat exchanger calculated by:
where
is the heat transfer coefficient for the coolant heat exchanger,
is the heat transfer area for the coolant side,
is the mass flow rate,
is the coolant water temperature, and
is the temperature of the cold side of the thermoelectric module.
Energy balance of exhaust heat exchanger calculated by:
where
is the heat transfer coefficient for the exhaust heat exchanger,
is the heat transfer area for the exhaust side,
is the mass flow rate,
is the specific heat of the exhaust,
is the temperature of the exhaust, and
is the temperature of the hot side of the thermoelectric module.
Heat and cooling transfer of the system calculated by:
where
is the Seebeck coefficient,
is the internal resistance,
is the thermal conductance of the module, and
is the total current of the generator.
Calculations for efficiency of a thermo-electric generator:
The exhaust gas mass flow rate was 0.1024 kg/s, the exhaust gas specific heat was , and the supplied heat was: ,
The thermoelectric module in
Figure 9 composed of two units with dimensions of 240 mm × 100 mm × 300 mm. It is a combination of three main parts: The water cooling box, heat exchange box, and thermoelectric plate
The water cooling box has eight parts with dimensions of 70 mm × 40 mm × 300 mm. It was used as a thermo-electric plate to reduce the heat by removing the heat from the water using the radiator. The heat exchange box has dimensions of 100 mm × 100 mm × 300 mm to transfer the heat from hot water to the 80 thermoelectric plates. The design will focus on the suitability and ease of installation with real engines in limited space to make the best heat transfer efficiency and to not extremely affect the pressure of exhaust.