HEAT EXCHANGER FORMULAS & EXAMPLE Formulas for Heat Exchanger System using a Modulating Control Valve
Definition of Terms and Units:
E = Mean Heat Transfer Rate or Heat Load (Btu/hr) E D = Design Heat Load (Btu/hr) U = Overall Heat Transfer Coefficient (Btu/(hr-ft 2 -°F)) A = Heat Transfer Surface Area of Heat Exchanger (ft 2 ) Δ T M = Mean Temperature Difference between Steam and Water (°F) Q W = Volumetric Flow Rate of Water (GPM) Q S = Steam Load or Steam Capacity (lbs/hr) C p = Specific Heat Capacity of Water (Btu/(lb-°F))
T o = Outlet Water Temperature (°F) T i = Inlet Water Temperature (°F)
Δ T W = Temperature Rise of Water (°F) = T o – T i T WM = Mean Water Temperature (°F) = (T o + T i )/2 LH = Latent Heat of Saturated Steam (Btu/lb) P 1 = Control Valve Inlet Pressure (PSIA) P 2 = Control Valve Outlet Pressure (PSIA) Δ P = Control Valve Differential Pressure (PSI) = P 1 – P 2 C v = Control Valve Flow Coefficient
T S = Saturated Steam Temperature (°F) T B = Back pressure Equivalent Saturated Steam Temperature (°F)
Formula 1: Mean Heat Transfer Rate (E) of Heat Exchanger E = U A Δ T M
The Heat Transfer Rate E (in Btu/hr) that takes place in a Heat Exchanger (HX) is a function of the Surface Area A (ft 2 ), the average temperature difference Δ T M (°F) between the steam and water, and the overall heat transfer coefficient U . The above formula can be used to calculate the heat loads for a HX based on the steam temperature inside the HX shell. This formula, when solved for A , can be used to size the HX (see Formula 2 ). Typical U values used for a steam to water HX range from 120 for stainless steel to over 200 for copper.
Formula 2: Heat Transfer Surface Area (A) of Heat Exchanger A = E D U Δ T M
This formula is used to calculate the surface area (size) of the heat exchanger’s internal tube or plates based on the design (maximum) heat load ( E D ) and average temperature difference ( Δ T M ) between the steam and water. Since Δ T M is directly proportional to the steam pressure inside the HX shell, the specific steam pressure used to heat the water at E D will determine the HX size. From the above formula, it can be seen that Δ T M is inversely proportional to A (the surface area). Therefore, the higher the steam pressure, the smaller the HX size, and vice versa. Formula 3: Mean Temperature Difference ( Δ T M ) between Steam and Water Δ T M = ( T S - T o ) + (T S - T i ) 2 This formula gives the average of the temperature differences between the steam and water at the outlet of the HX ( T s – T o ) and at the inlet of the HX ( T s – T i ). Formula 4: Saturated Steam Temperature (T S ) as function of Mean Temperature Difference T s = Δ T M + T WM Where, T WM = (T o + T i )/2 This formula is derived by solving Formula 3 for T S . It is useful for determining the steam temperature when the mean temperature difference ( Δ T M ) is known. For example, the steam temperature at minimum load can be determined by solving Formula 1 for Δ T M when E = E min , and then substituting Δ T M into the above formula. Once T S is known, the pressure inside the HX shell can be determined from the Saturated Steam Table.
Formula 5: Heat Load (E) E = Q w x 500 x C p x Δ T w = Q w x 500 x (T o – T i )
[ C p = 1.0 Btu/(lb-°F)] The above formula shows that the heat load for the HX depends on the water flow rate ( Q w ) and the water temperature rise ( Δ T w = T o – T i ).
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