Efficiency calculations
Using water as the liquid to be heated, assume an inlet water temperature of Tin (°F), an outlet water temperature of Tout (°F), a liquid or slurry heat capacity of cp
(MMBtu/lb•°F) and a liquid or slurry flowrate of Q (lb/h).

The heat transferred to the solution (Hnet, in MMBtu/h) is calculated (neglecting the volume of water lost out the stack) by: Hnet = Qcp(Tout – Tin) (1)
The thermal efficiency, E, of the system is the useful heat divided by Hgross (also in MMBtu/h), the heat added (assuming the amount of fuel consumed is known):
E = Hnet/Hgross (2)

For natural gas, the heat from the fuel is taken to be the volume of gas multiplied by the higher heating value of the gas.
Multiple units are often used to provide additional capacity.

Furthermore, intermediate streams of liquid may be withdrawn when the liquid has reached a certain concentration
or temperature. For example, one installation has three burners, each rated at 13.5 MMBtu/h, in series. The temperatures of the tanks are 105°F, 140°F and 170°F. The plant uses approximately 600–800 gal/min at 105°F as hot water for washing in the plant and an additional 400–600 gal/min at 170°F to drive a chemical reaction.

Efficiency calculations for multiple units involve adding up the water streams where heat is removed. For the above example: Hnet = cp[Q1(Tout,1 – Tin) + Q2(Tout,2 – Tin)] (3) where Q1 and Tout,1 are the flowrate and temperature of
water taken from the first unit and Q2 and Tout,2 are the flowrate and temperature of water taken from the last unit. Some of the water in the saturated vapor comes from the combustion process. For aqueous solutions or slurries, the stack gas will require additional water from the liquid to become saturated. As the gas temperature increases, the amount of water vapor required to saturate the gas stream increases exponentially, as demonstrated by psychrometric data (see table). Water vapor out the stack represents lost heat.

Therefore, the overall efficiency of submerged combustion depends on the temperature of the gas leaving the liquid medium. For example, the table shows that for an increase in stack gas temperature from 95°F to 104°F, the enthalpy increases from 40.5 to 54.4 Btu/lb dry air, a difference of 13.9 Btu/lb.

However, an increase of another 9°F to 113°F translates to an 18.3 But/lb increase, from 54.4 to 72.7 Btu/lb. The enthalpy increase of a stack temperature change from 104°F to 113°F is 32% more than that from 95°F to 104°F, and the difference increases faster as temperatures increase (Figure 1).

As the heated liquid temperature increases, the stack gas temperature also goes up. Heat lost out the stack is mostly
water vapor. For example, 1 lb of saturated dry stack gas at 86°F contains 0.027 lb water, while air at 149°F contains 0.206 lb water. The theoretical efficiency (assuming negligible heat losses) of a single burner that heats water or an aqueous solution is 97.6% at 86°F and 77.4% at 149°F. This efficiency is essentially independent of inlet water temperature.

Based on similar calculations, an efficiency curve can be plotted as a function of the outlet stack temperature
(Figure 2). At lower stack temperatures, the system efficiency approaches 100%, because the enthalpy of ambient air at a higher temperature and humidity contributes to the overall energy balance.

If the overall system efficiency is to
be kept above 90%, the stack temperature must stay below 124°F. Including other variables, such as ambient air temperature and humidity, amount of excess air, air pressure (elevation), and different types of fuel, should result in
more-precise efficiency calculations.
Note also that the efficiency of a submerged combustion unit is 0% at approximately 187°F. At this temperature, all of the heat from combustion is used for saturating the stack gas. Therefore, 187°F is theoretically the highest temperature achievable by submerged combustion technology for

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