Definition of Terms and Units:
E = Mean Heat Transfer Rate or Heat Load (Btu/hr)
ED= Design Heat Load (Btu/hr)
U= Overall Heat Transfer Coefficient (Btu/(hr-ft2-°F)) A= Heat Transfer Surface Area of Heat Exchanger (ft2)
ΔTM = Mean Temperature Difference between
Steam and Water (°F)
QW= Volumetric Flow Rate of Water (GPM)
QS= Steam Load or Steam Capacity (lbs/hr)
Cp= Specific Heat Capacity of Water (Btu/(lb-°F))
TS= Saturated Steam Temperature (°F)
TB= Back pressure Equivalent Saturated
Steam Temperature (°F)
Formula 1: Mean Heat Transfer Rate (E) of Heat Exchanger
E = U A ΔTM
The Heat Transfer Rate E(in Btu/hr) that takes place in a Heat Exchanger (HX) is a function of the Surface Area A(ft2), the average temperature difference ΔTM(°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 Uvalues 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 = ED U ΔTM
This formula is used to calculate the surface area (size) of the heat exchangerʼs internal tube or plates based on the design (maximum) heat load (ED) and average temperature difference (ΔTM) between the steam and water. Since ΔTM
is directly proportional to the steam pressure inside the HX shell, the specific steam pressure used to heat the water at EDwill determine the HX size. From the above formula, it can be seen that ΔTMis 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 ( ΔTM) between Steam and Water
ΔTM= ( TS - To) + (TS - Ti ) 2
This formula gives the average of the temperature differences between the steam and water at the outlet of the HX (Ts– To) and at the inlet of the HX (Ts– Ti).
Formula 4: Saturated Steam Temperature (TS) as function of Mean Temperature Difference
Ts= ΔTM+ T WM Where,T WM= (To+ Ti)/2
This formula is derived by solving Formula 3for TS. It is useful for determining the steam temperature when the mean
temperature difference (ΔTM) is known. For example, the steam temperature at minimum load can be determined by
solving Formula 1for ΔTMwhen E = Emin, and then substituting ΔTMinto the above formula. Once TSis known, the
pressure inside the HX shell can be determined from the Saturated Steam Table. Formula 5: Heat Load (E)
E = Qw x 500 x C px ΔTw = Qw x 500 x (T o– Ti) [ C p = 1.0 Btu/(lb-°F)]
To= Outlet Water Temperature (°F)
Ti= Inlet Water Temperature (°F)
ΔTW= Temperature Rise of Water (°F) = To– Ti
TWM= Mean Water Temperature (°F) = (To+ Ti)/2
LH= Latent Heat of Saturated Steam (Btu/lb)
P1= Control Valve Inlet Pressure (PSIA)
P2= Control Valve Outlet Pressure (PSIA)
ΔP= Control Valve Differential Pressure (PSI) = P1– P2
Cv= Control Valve Flow Coefficient
Formulas for Heat Exchanger System using a Modulating Control Valve
Formula 6: Steam Load (QS) as function of Heat Load
QS = E LH
The steam load or capacity (QSin lbs/hr) is dependent on the heat load (Ein Btu/hr) and the latent heat (LHin Btu/lb) the
steam contains. The Latent Heat of saturated steam is dependent on the steam pressure. Consult the Saturated Steam Table in Engineering Section. LH is typically approximated to 1,000 Btu/lb.
Formula 7: Steam Load (QS) as function of Water Flow Rate
QS = QW x 500 x ( To – Ti ) QS = QW x ΔTW
LH 2
This formula is derived by substituting the right side of Formula 5for Ein Formula 6. It can be used for calculating the steam load directly from the flow rate of water to be heated.
Formula 8: Water Flow Rate (Qw) as function of Heat Load
QW = E 500 x ( T o – Ti)
This formula is derived by solving Formula 5for Qw. It is useful for determining the water flow rate thru the HX at the
stall point (Qw-stall). This is explained in the following HX example (see part M).
Formula 9: Percent Stall Load
% S tall Loa d = TB – TWM x 100 Where TWM= To + Ti
TS– TWM 2
This formula is used to calculate the percentage of Full Heat Load (ED) at which heat exchanger stall will occur.
Since water flow rate is proportional to heat load (seeFormula 8), the % Stall Load can be used to calculate the water flow rate at stall (see Formula 10).
Formula 10: Water Flow Rate at Stall (Qw-stall)
Qw-stall= Qw-full load x (% Stall Load)/100
Where, Qw-full load= Water flow rate at design (maximum) heat load (ED) = Maximum water flow rate
This formula is used in conjunction with Formula 9to calculate the water flow rate at which heat exchanger stall will occur without having to know the size of the HX.
Formula 11: Control Valve Steam Capacity (QS)at Sub-Critical Flow
For ΔP < 0.42 P1: 11a: QS= 2.1 Cv 11b: Cv = QS 2.1
These formulas are applied when the pressure drop across the control valve (ΔP) is less than the critical pressure drop (0.42P1).
Formula 12: Control Valve Steam Capacity (QS) at Critical Flow
For ΔP > 0.42 P1: 12a: QS = 1.71 Cv P1 12b: Cv= QS 1.71 P1
When the pressure drop across the valve (ΔP) is greater than or equal to the critical pressure drop (0.42 P1), the steam
capacity (QS) depends only on the valve inlet pressure (P1). The flow rate at this condition is called the critical flow.
For a constant inlet pressure, the critical flow is the maximum capacity of the valve. The above formulas are derived from Formula 11aby using the critical pressure drop (ΔP = 0.42 P1) and differential pressure (ΔP = P1– P2) formulas to
ΔP(P1 + P2)
ΔP(P1 + P2)
(approximation for LH = 1,000 Btu/lb)