The geometry of the water-air heat exchanger in this system is shown in Figure 1. The warmed water from the condenser (hot fluid) is introduced through water pipes, while the air (cold fluid) flows across the outer surface of the pipes. These two working fluids move perpendicular to each other. To enhance the heat transfer, thin plates are attached to the heat exchanger to regulate the airflow. As a result, this unmixed cross flow compact heat exchanger works similar to a car radiator, and the heat is dissipated from the warm water to the air.

In this heat exchanger, the water is dispersed accordingly to the small, horizontal pipes from a huge central pipe. The copper is chosen as the pipe material due to its high conductivity.

If the specific heat of water is assumed to be constant, the energy balance of the cooling water leads to: (1) Q ˙ = m ˙ water c p , water ( T water , in − T water , exit ) = C max ( T water , in − T water , exit )where C_max is the maximum heat capacity rate of the heat exchanger and Q̇ is the heat transfer rate of the system.

Since warm water is only energy source of the system, mass flow rate and inlet temperature of the water are usually specified, while the water temperature at the exit and heat transfer rate of the heat exchanger usually should be determined. In this paper, water is assumed to be the fluid with high heat capacity rate.

The classic NTU-effectiveness method has been adopted to design this heat exchanger. To calculate the Number of Transfer Unit (NTU), it is necessary to determine the overall heat transfer coefficient (based on the water side surface area), U, which can be calculated with the following equation, if the conduction resistance of the pipes is neglected: (2) U = h o h i A air h o A air + h i A s , waterwhere h_i is the heat transfer coefficient between the water and inner surface of pipes, which can be calculated with the following equation [7]: (3) Nu = h i D i k water = 0.023 Re 0.8 Pr 0.4

And h_o is the heat transfer coefficient between the outer surfaces of the pipes and air, which can be calculated with the following equation [8]: (4) Nu = h o D o k air = 0.3 + 0.62 Re 1 / 2 Pr 1 / 3 [ 1 + ( 0.4 / Pr ) 2 / 3 ] 1 / 4 [ 1 + ( Re 282,000 ) 5 / 8 ] 4 / 5

Based on Equations (3) and (4), as well as the range of the air flow rate and water flow rate of this system, it can be concluded that h_i ≫ h_o, thus Equation (2) can be simplified as (5) U ≈ h o A air A s , water = h o A nofin + η fin A fin A s , waterwhere η_fin is the fin efficiency, A_fin, A_nofin is air side surface area of the fins, air side non-finned surface area, respectively.

The Number of the Transfer Unit of this heat exchanger thus can be calculated as: (6) NTU = U A s , water m ˙ air c air = Un π DL m ˙ air c airwhere n is the number of the water pipes, and D, L is the diameter and length of the water pipes, respectively.

For this unmixed cross flow heat exchanger, the effectiveness is given by [9]: (7) ε = 1 − exp { NTU 0.22 c [ exp ( − cNTU 0.78 ) − 1 ] }where c is the capacity ratio

The actual rate of heat transfer is then given by: (8) Q ˙ = ε Q ˙ max = ε m ˙ air c p , air ( T water , in − T ∞ )

Therefore the air temperature at the bottom of the chimney T_b can be determined with: (9) T b = T ∞ + Q ˙ m ˙ air c p , air

And the exit temperature of the water can be calculated with Equations (1) and (8): (10) T water , exit = T water , in + ε m ˙ air c p , air ( T water , in − T ∞ ) m water ˙ c p , water

The heat exchanger thus can be installed inside the chimney (shown in Figure 2). This arrangement can better facilitate the air flow towards the turbine, which sits on the top of the chimney. As the air passes through the layers of the heat exchanger and absorbs the heat, it reaches the highest temperature at the bottom entrance of the chimney structure.

The chimney physically constrains the air just like a nozzle. A converging chimney as the one shown in Figure 3 has a variable cross sectional diameter for which the fluid velocity changes because the flow area is different from one section of the chimney to the other. As the cross sectional area decreases with height, the velocity increases. The fluid flow is governed by the conservation of mass as well as the conservation of the energy.

To harvest meaningful kinetic energy from the airflow, the March number of the airflow will exceed 0.3 and thus the airflow can no longer be treated as incompressible fluid.

Based on the continuity equation, the air flow satisfies: (11) ρ b V b A b = ρ t V t A t

To further simplify the calculations, the air is taken as an ideal gas and the chimney is assumed to be isentropic (no heat losses and friction losses are considered), therefore: (12) T t T b = ( ρ t ρ b ) k − 1

Finally based on the energy balance, if potential energy is negligible (13) h b + V b 2 2 = h t + V t 2 2

Since V b 2 2 ≪ V t 2 2

With the assumption that the specific heat is constant, Equation (13) becomes (14) c p , air T b = c p , air T t + V t 2 2

Thus, if the properties of the air at the bottom of the chimney and the bottom to top cross-sectional area ratio are known, with Equations (11), (12) and (14), the velocity of the air exiting the chimney V_t can be calculated with the following: (15) V t V b ( A b / A t ) = ( 2 c p , air T b 2 c p , air T b − V t 2 ) 1 / ( k − 1 )

The exit velocity calculated using the above equation are ideal and does not take into account friction losses, gravity, and heat losses through the chimney wall.

The selected turbine in this system is a horizontal turbine, which is mounted in an inverted configuration.

In order to harvest the most amount of power from the chimney, multiple wind turbines can be used in series or in parallel. Figure 4 illustrates a turbine configuration at the top of the chimney,

After the air is warmed by the heat exchanger, it rises and gains speed through the chimney. The highest velocity is reached at the top of the chimney where it is focused through the first axial turbine. This turbine produces power by converting kinetic energy to electrical energy. The air exiting each of the turbine stages is roughly one third of the velocity entering, due to the inefficiency of the turbine and its inability to harvest all the power from the air stream. For this reason, multiple stages are used to gain the maximum benefit. Up to three stages have been investigated and it has been determined that the third stage could be unnecessary if the minimal beneficial gain can be found from the additional stage.

The total power output by a wind turbine is given by: (16) p output = 1 2 m air ˙ ( V air , t 2 − 1 9 V air , t 2 ) η turbine = 4 9 m ˙ air V air , t 2 η turbinewhere η_turbine is turnine efficiency.

Water pumps are needed to pump the cooling water to the heat exchanger. The total pump work required by the system can be easily calculated with the water flow rate as well as geometry of the pipes.

Fans are needed to force the ambient air flow over the warm water pipes. Air flow actually can be initiated purely through natural convection and based on some studies [10], an air velocity of 0.4 m/s can be achieved purely through natural convection. As discussed later in this paper, this velocity is too low for the air to obtain enough velocity at the exit of the chimney for the turbines harvesting meaningful kinetic energy. Therefore, fans are necessary to produce forced convection between the water and airflow.

Heat transfer rate of the heat exchanger represents the rate of actual amount of the heat picked by the airflow from the condenser water. Parameters such as air velocity, surface area of the heat exchanger can significantly affect the heat transfer rate. Figure 5 shows the relationship between actual heat transfer rate and air velocity. If the water pipes used has a length of 24 m and a diameter of 28 cm, for a fixed water flow rate of 2000 kg/s and inlet water temperature of 57 °C, the heat transfer rate increases steadily when air velocity increases from 0.1 m/s to 5 m/s. Figure 5 also shows how the number of the water pipes will affect the heat transfer. When air velocity is 5 m/s, if 60 water pipes are used, the heat transfer rate will be 6.09 MW, while if 240 water pipes are used, the heat transfer rate can reach to 21.62 MW. This can be well explained by the increased surface areas with the additional pipes, but more pipes used will lead to higher construction costs as well as more pump work consumed.

Another parameter which affects the heat transfer rate is the inlet air temperature, which is usually the same as the ambient temperature. Figure 6 shows that with a lower inlet air temperature, the higher heat transfer rate can be achieved. For the heat exchanger, the “worst” situation occurs in the hottest day in the summer time with an inlet ambient temperature of over 30 °C. The temperature difference between the two fluids is the smallest comparing with other times of the year. As a result, the heat transfer rate of the heat exchanger is the smallest in this situation. For other days, especially during winter time, the heat transfer rate would be higher due to the bigger temperature difference between the water and the air. Therefore, the heat transfer rate of the heat exchanger would be higher on winter than the hottest days.

The effectiveness of this heat exchanger is related with NTU, thus related to the total surface areas of the heat exchanger. As shown in Figure 7, the more water pipes are used, the higher effectiveness will be. Another factor which affects the effectiveness is the air velocity. The higher the air velocity is, the lower its effectiveness will be. This is due to the fact that a higher air exit temperature can be achieved with a lower air velocity. But a lower air velocity will lead to a lower heat transfer rate as well. For example, for a natural convection that has induced an air velocity of 0.2 m/s, its total heat transfer rate of is only 2.35 MW even 240 water pipes are used (shown in Figure 5), although it has a relatively high effectiveness of 0.7273 (shown in Figure 7).

Lower effectiveness leads to higher exit temperature difference between the water and the air. Figure 8 shows how the air velocity affects the temperature difference between these two fluids at the exit of the heat exchanger. It can be seen that with an air velocity less than 0.2 m/s (natural convection induction), the temperature difference could reach as low as 5 K. At the same time it can be seen that the water exit temperature is still high and that shows a low heat transfer rate.

Air flow accelerates through the converging chimney due to the variations of the cross section area and the air density. Figure 9 shows the relationship between the air velocities at the top and bottom of the chimney. It can be seen that, with a higher cross section area ratio, air accelerates faster. Since the chimney is proposed to be subsonic, the air velocity at the top of the chimney is expected to be lower than 343 m/s. It can be seen that from Figure 9, with a cross section ratio of 100, the air velocity can reach 300 m/s at the top with a bottom velocity of 1.92 m/s.

Figure 10 shows the air density ratio between inlet and exit of the chimney. This ratio is approximately unity when the air velocity is lower than 50 m/s and can reach as high as 1.55 with an air velocity of 300 m/s.

The chimney is assumed to be isentropic. When the air flow accelerates through the chimney, air temperature decreases while the thermal energy is converted to the kinetic energy. This temperature variation will be negligible if the air velocity is low and air can thus be treated as incompressible. Figure 11 shows a case for a chimney with a cross section area ratio of 100 and a very low air inlet temperature at the heat exchanger (−23 °C), the air temperatures at the bottom of the chimney, T_b, can be lifted as shown in the figure, and it can be found that the temperature difference between T_b and T_f can be as high as 45 °C if a bottom air velocity of 1.92 m/s is initiated, which will lead to a 300 m/s air velocity at the top of the chimney.

Table 1 shows the power outputs of a two-stage wind turbine with two different heat exchanger air velocities. The system operates at a steady flow condition, with a water flow rate of 700 kg/s, and a water inlet temperature of 57 °C. The air inlet temperature at the heat exchanger is 300 K. The heat exchanger has 240 water pipes (each has a length of 24 m and a diameter of 28 cm). The cross section area ratio of the chimney is 50.

It can be found that with a heat exchanger air velocity of 2.75 m/s, the total power output of the system is about 5.7 MW, while with a heat exchanger air velocity of 3.85 m/s, the total power output will reach 19.7 MW.

Pump work is related to the water flow rate as well as size and number of the water pipes used in the heat exchanger. For a water flow rate of 2000 kg/s, if 120 water pipes are used in the heat exchanger, the total pump work required is about 73.7 kW, which is very small compared with the total power output of the turbine.

For the fans, if its isentropic efficiency is 80%, for the cases in section 3.5, the total fan work consumption is 19.02 kW for an introduced air velocity of 3.85 m/s, which is also negligible, compared the power output of 19.7 MW of the system.

The benefits of the proposed system directly come from the electricity generated. Since the only energy source is the waste heat, which is obtained at no costs, the major costs come from the financial costs and operating costs such as maintenance and pump work consumptions.

Based on the 19.7 MW power output in section 3.3, with a price of 0.04 $/kwh, the system can generate an additional revenue of 6.9 million dollars for the power plant. If the construction cost is 40 million dollars, for a 25-year loan with an interest rate of 5.0%, the financial cost is about 2.77 million dollars per year, if the operating cost is 2 million dollars per year, the system still may generate a net profit of over 2 million dollars per year.

The system actually can produce a power output more than 19.7 MW, considering the air velocity at the exit of the chimney can be well above 252.4 m/s if higher heat exchanger air velocity is introduced.