# Heat exchanger sizing and rating (ht.hx)¶

ht.hx.DBundle_for_Ntubes_HEDH(N, Do, pitch, angle=30)[source]

A rough equation presented in the HEDH for estimating the tube bundle diameter necessary to fit a given number of tubes. No accuracy estimation given. Only 1 pass is supported.

$D_{bundle} = (D_o + (\text{pitch})\sqrt{\frac{1}{0.78}}\cdot \sqrt{C_1\cdot N})$

C1 = 0.866 for 30° and 60° layouts, and 1 for 45 and 90° layouts.

Parameters
Nfloat

Number of tubes, [-]

Dofloat

Tube outer diameter, [m]

pitchfloat

Pitch; distance between two orthogonal tube centers, [m]

anglefloat

The angle the tubes are positioned; 30, 45, 60 or 90, [degrees]

Returns
DBundlefloat

Outer diameter of tube bundle, [m]

Notes

Easily reversed from the main formulation.

References

1

Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1983.

Examples

>>> DBundle_for_Ntubes_HEDH(N=928, Do=.028, pitch=.036, angle=30)
1.183993079564


Determine the bundle diameter required to fit a specified number of tubes in a heat exchanger. Uses the highly accurate method of [1], which takes into account pitch, number of tube passes, angle, and tube diameter. The method is analytically correct when used in the other direction (calculating number of tubes from bundle diameter); in reverse, it is solved by bisection.

Parameters
Ntubesint

Total number of tubes that fit in the heat exchanger, [-]

Dofloat

Tube outer diameter, [m]

pitchfloat

Pitch; distance between two orthogonal tube centers, [m]

Ntpint

Number of tube passes, [-]

anglefloat, optional

The angle the tubes are positioned; 30, 45, 60 or 90, [degrees]

Returns
DBundlefloat

Outer diameter of tube bundle, [m]

Notes

This function will fail when there are more than 100,000 tubes. There are a range of correct diameters for which there can be the given number of tubes; a number within that range is returned as found by bisection.

References

1

Phadke, P. S., Determining tube counts for shell and tube exchangers, Chem. Eng., September, 91, 65-68 (1984).

Examples

>>> DBundle_for_Ntubes_Phadkeb(Ntubes=782, Do=.028, pitch=.036, Ntp=2, angle=45.)
1.1879392959379533

ht.hx.DBundle_min(Do)[source]

Very roughly, determines a good choice of shell diameter for a given tube outer diameter, according to figure 1, section 3.3.5 in [1].

Parameters
Dofloat

Tube outer diameter, [m]

Returns
DShellfloat

Shell inner diameter, optional, [m]

Notes

This function should be used if a tube diameter is specified but not a shell size. DShell will have to be adjusted later, once the area requirement is known. This function is essentially a lookup table.

References

1

Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1983.

Examples

>>> DBundle_min(0.0254)
1.0

ht.hx.D_baffle_holes(do, L_unsupported)[source]

Determines the diameter of holes in baffles for tubes according to TEMA [1]. Applies for all geometries.

Parameters
dofloat

Tube outer diameter, [m]

L_unsupportedfloat

Distance between tube supports, [m]

Returns
dBfloat

Baffle hole diameter, [m]

References

1

Standards of the Tubular Exchanger Manufacturers Association, Ninth edition, 2007, TEMA, New York.

Examples

>>> D_baffle_holes(do=.0508, L_unsupported=0.75)
0.0516
>>> D_baffle_holes(do=0.01905, L_unsupported=0.3)
0.01985
>>> D_baffle_holes(do=0.01905, L_unsupported=1.5)
0.019450000000000002

ht.hx.D_for_Ntubes_VDI(N, Ntp, Do, pitch, angle=30)[source]

A rough equation presented in the VDI Heat Atlas for estimating the size of a tube bundle from a given number of tubes, number of tube passes, outer tube diameter, pitch, and arrangement. No accuracy estimation given.

$OTL = \sqrt{f_1 z t^2 + f_2 t \sqrt{z} - d_o}$
Parameters
Nfloat

Number of tubes, [-]

Ntpfloat

Number of tube passes, [-]

Dofloat

Tube outer diameter, [m]

pitchfloat

Pitch; distance between two orthogonal tube centers, [m]

anglefloat

The angle the tubes are positioned; 30, 45, 60 or 90

Returns
DBundlefloat

Outer diameter of tube bundle, [m]

Notes

f1 = 1.1 for triangular, 1.3 for square patterns f2 is as follows: 1 pass, 0; 2 passes, 22; 4 passes, 70; 8 passes, 105. 6 tube passes is not officially supported, only 1, 2, 4 and 8. However, an estimated constant has been added to support it. f2 = 90.

References

1

Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010.

Examples

>>> D_for_Ntubes_VDI(N=970, Ntp=2., Do=0.00735, pitch=0.015, angle=30.)
0.5003600119829544

ht.hx.F_LMTD_Fakheri(Thi, Tho, Tci, Tco, shells=1)[source]

Calculates the log-mean temperature difference correction factor Ft for a shell-and-tube heat exchanger with one or an even number of tube passes, and a given number of shell passes, with the expression given in [1] and also shown in [2].

$F_t=\frac{S\ln W}{\ln \frac{1+W-S+SW}{1+W+S-SW}}$
$S = \frac{\sqrt{R^2+1}}{R-1}$
$W = \left(\frac{1-PR}{1-P}\right)^{1/N}$
$R = \frac{T_{in}-T_{out}}{t_{out}-t_{in}}$
$P = \frac{t_{out}-t_{in}}{T_{in}-t_{in}}$

If R = 1 and logarithms cannot be evaluated:

$W' = \frac{N-NP}{N-NP+P}$
$F_t = \frac{\sqrt{2}\frac{1-W'}{W'}}{\ln\frac{\frac{W'}{1-W'}+\frac{1} {\sqrt{2}}}{\frac{W'}{1-W'}-\frac{1}{\sqrt{2}}}}$
Parameters
Thifloat

Inlet temperature of hot fluid, [K]

Thofloat

Outlet temperature of hot fluid, [K]

Tcifloat

Inlet temperature of cold fluid, [K]

Tcofloat

Outlet temperature of cold fluid, [K]

shellsint, optional

Number of shell-side passes, [-]

Returns
Ftfloat

Log-mean temperature difference correction factor, [-]

Notes

This expression is symmetric - the same result is calculated if the cold side values are swapped with the hot side values. It also does not depend on the units of the temperature given.

References

1

Fakheri, Ahmad. “A General Expression for the Determination of the Log Mean Temperature Correction Factor for Shell and Tube Heat Exchangers.” Journal of Heat Transfer 125, no. 3 (May 20, 2003): 527-30. doi:10.1115/1.1571078.

2

Hall, Stephen. Rules of Thumb for Chemical Engineers, Fifth Edition. Oxford; Waltham, MA: Butterworth-Heinemann, 2012.

Examples

>>> F_LMTD_Fakheri(Tci=15, Tco=85, Thi=130, Tho=110, shells=1)
0.9438358829645933

ht.hx.L_unsupported_max(Do, material='CS')[source]

Determines the maximum length of a heat exchanger tube can go without a support, according to TEMA [1]. The limits provided apply for the worst-case temperature allowed for the material to be used at.

Parameters
Dofloat

Outer tube diameter, [m]

materialstr

Material type, either ‘CS’ or ‘aluminium’, [-]

Returns
L_unsupportedfloat

Maximum length of unsupported tube, [m]

Notes

The ‘CS’ results is also listed as representing high alloy steel, low alloy steel, nickel-copper, nickel, and nickel-chromium-iron alloys. The ‘aluminium’ results are those of copper and copper alloys and also titanium alloys.

The maximum and minimum tube outer diameter tabulated are 3 inch and 1/4 inch respectively. The result is returned for the nearest tube diameter equal or smaller than the provided diameter, which helps ensures the returned tube length will not be optimistic. However, if the diameter is under 0.25 inches, the result will be optimistic!

References

1

Standards of the Tubular Exchanger Manufacturers Association, Ninth edition, 2007, TEMA, New York, p 5.4-5.

Examples

>>> L_unsupported_max(Do=.0254, material='CS')
1.88

ht.hx.NTU_from_P_E(P1, R1, Ntp, optimal=True)[source]

Returns the number of transfer units of a TEMA E type heat exchanger with a specified (for side 1) thermal effectiveness P1, heat capacity ratio R1, the number of tube passes Ntp, and for the two-pass case whether or not the inlets are arranged optimally. The supported cases are as follows:

• 1-1 TEMA E, shell fluid mixed

• 1-2 TEMA E, shell fluid mixed (this configuration is symmetric)

• 1-2 TEMA E, shell fluid split into two steams individually mixed

• 1-3 TEMA E, shell and tube fluids mixed, one parallel pass and two counterflow passes (efficient)

• 1-3 TEMA E, shell and tube fluids mixed, two parallel passes and one counteflow pass (inefficient)

• 1-N TEMA E, shall and tube fluids mixed, efficient counterflow orientation, N an even number

Two of these cases have analytical solutions; the rest use numerical solvers of varying quality.

The analytical solution to 1-1 TEMA E, shell fluid mixed (the same as pure counterflow):

$NTU_1 = - \frac{1}{R_{1} - 1} \ln{\left (\frac{P_{1} R_{1} - 1}{P_{1} - 1} \right )}$

1-2 TEMA E, shell fluid mixed:

$NTU_1 = \frac{2}{\sqrt{R_{1}^{2} + 1}} \ln{\left (\sqrt{\frac{P_{1} R_{1} - P_{1} \sqrt{R_{1}^{2} + 1} + P_{1} - 2}{P_{1} R_{1} + P_{1} \sqrt{R_{1}^{2} + 1} + P_{1} - 2}} \right )}$
Parameters
P1float

Thermal effectiveness of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

R1float

Heat capacity ratio of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

Ntpint

Number of tube passes, 1, 2, 3, 4, or an even number [-]

optimalbool, optional

Whether or not the arrangement is configured to give more of a countercurrent and efficient (True) case or an inefficient parallel case, [-]

Returns
NTU1float

Thermal number of transfer units of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

Notes

For odd numbers of tube passes greater than 3, an exception is raised.

For the 2 pass, unoptimal case, a bounded solver is used with NTU1 between 1E-11 and 100; the solution to any feasible P1 was found to lie in there. For the 4 or a higher even number of pass case, the upper limit on NTU1 is 1000; this solver works pretty well, but as NTU1 reaches its limit the change in P1 is so small a smaller but also correct solution is often returned.

For both the optimal and unoptimal 3 tube pass case, a solution is only returned if NTU1 is between 1E-11 and 10. These functions are extremely mathematically frustrating, and as NTU1 rises above 10 catastrophic cancellation quickly results in this expression finding a ZeroDivisionError. The use of arbitrary prevision helps little - quickly 1000 digits are needed, and then 1000000 digits, and so one. Using SymPy’s rational number support works better but is extremely slow for these complicated solutions. Nevertheless, so long as a solution is between 1E-11 and 10, the solver is quite robust.

Examples

>>> NTU_from_P_E(P1=.58, R1=1/3., Ntp=2)
1.0381979240816719

ht.hx.NTU_from_P_G(P1, R1, Ntp, optimal=True)[source]

Returns the number of transfer units of a TEMA G type heat exchanger with a specified (for side 1) thermal effectiveness P1, heat capacity ratio R1, the number of tube passes Ntp, and for the two-pass case whether or not the inlets are arranged optimally. The supported cases are as follows:

• One tube pass (tube fluid split into two streams individually mixed, shell fluid mixed)

• Two tube passes (shell and tube exchanger with shell and tube fluids mixed in each pass at the cross section), counterflow arrangement

• Two tube passes (shell and tube exchanger with shell and tube fluids mixed in each pass at the cross section), parallelflow arrangement

Parameters
P1float

Thermal effectiveness of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

R1float

Heat capacity ratio of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

Ntpint

Number of tube passes, 1 or 2 [-]

optimalbool, optional

Whether or not the arrangement is configured to give more of a countercurrent and efficient (True) case or an inefficient parallel case (only applies for two passes), [-]

Returns
NTU1float

Thermal number of transfer units of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

Notes

For numbers of tube passes greater than 1 or 2, an exception is raised.

Although this function allows the thermal effectiveness desired to be specified, it does not mean such a high value can be obtained. An exception is raised which shows the maximum possible effectiveness obtainable at the specified R1 and configuration.

>>> NTU_from_P_G(P1=1, R1=1/3., Ntp=2)
Traceback (most recent call last):
ValueError: No solution possible gives such a high P1; maximum P1=0.954545 at NTU1=10000.000000


Of the three configurations, 1 pass and the optimal 2 pass have monotonic functions which allow for a bounded solver to work smoothly. In both cases a solution is searched for between NTU1 values of 1E-11 and 1E-4.

For the 2 pass unoptimal solution, a bounded solver is first use, but the upper bound on P1 and the upper NTU1 limit is calculated from a pade approximation performed with mpmath.

Examples

>>> NTU_from_P_G(P1=.573, R1=1/3., Ntp=1)
0.9999513707759524

ht.hx.NTU_from_P_H(P1, R1, Ntp, optimal=True)[source]

Returns the number of transfer units of a TEMA H type heat exchanger with a specified (for side 1) thermal effectiveness P1, heat capacity ratio R1, the number of tube passes Ntp, and for the two-pass case whether or not the inlets are arranged optimally. The supported cases are as follows:

• One tube pass (tube fluid split into two streams individually mixed, shell fluid mixed)

• Two tube passes (shell fluid mixed, tube pass mixed between passes)

• Two tube passes (shell fluid mixed, tube pass mixed between passes, inlet tube side next to inlet shell-side)

Parameters
P1float

Thermal effectiveness of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

R1float

Heat capacity ratio of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

Ntpint

Number of tube passes, 1, or 2, [-]

optimalbool, optional

Whether or not the arrangement is configured to give more of a countercurrent and efficient (True) case or an inefficient parallel case, [-]

Returns
NTU1float

Thermal number of transfer units of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

Notes

For numbers of tube passes greater than 1 or 2, an exception is raised.

Only numerical solutions are available for this function. For the case of 1 tube pass or the optimal 2 tube pass, the function is monotonic and a bounded solver is used with NTU1 between 1E-11 and 100; it will find the solution anywhere in that range.

For the non-optimal 2 pass case, the function is not monotonic and a pade approximation was used to obtain a curve of NTU1s which give the maximum P1s which is used as the upper bound in the bounded solver. The lower bound is still 1E-11. These solvers are all robust.

Examples

>>> NTU_from_P_H(P1=0.573, R1=1/3., Ntp=1)
0.9997628696891168

ht.hx.NTU_from_P_J(P1, R1, Ntp)[source]

Returns the number of transfer units of a TEMA J type heat exchanger with a specified (for side 1) thermal effectiveness P1, heat capacity ratio R1, and the number of tube passes Ntp. The supported cases are as follows:

• One tube pass (shell fluid mixed)

• Two tube passes (shell fluid mixed, tube pass mixed between passes)

• Four tube passes (shell fluid mixed, tube pass mixed between passes)

Parameters
P1float

Thermal effectiveness of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

R1float

Heat capacity ratio of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

Ntpint

Number of tube passes, 1, 2, or 4, [-]

Returns
NTU1float

Thermal number of transfer units of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

Notes

For numbers of tube passes that are not 1, 2, or 4, an exception is raised.

For the 1 tube pass case, a bounded solver is used to solve the equation numerically, with NTU1 ranging from 1E-11 to 1E3. NTU1 grows extremely quickly near its upper limit (NTU1 diverges to infinity at this maximum, but because the solver is bounded it will only increase up to 1000 before an exception is raised).

>>> NTU_from_P_J(P1=.995024, R1=.01, Ntp=1)
13.940758768266656
>>> NTU_from_P_J(P1=.99502487562189, R1=.01, Ntp=1)
Traceback (most recent call last):
ValueError: No solution possible gives such a high P1; maximum P1=0.995025 at NTU1=1000.000000


For the 2 pass and 4 pass solution, a bounded solver is first use, but the upper bound on P1 and the upper NTU1 limit is calculated from a pade approximation performed with mpmath. These normally do not allow NTU1 to rise above 100.

Examples

>>> NTU_from_P_J(P1=.57, R1=1/3., Ntp=1)
1.0003070138879664

ht.hx.NTU_from_P_basic(P1, R1, subtype='crossflow')[source]

Returns the number of transfer units of a basic heat exchanger type with a specified (for side 1) thermal effectiveness P1, and heat capacity ratio R1. The supported cases are as follows:

• Counterflow (ex. double-pipe) [analytical]

• Parallel (ex. double pipe inefficient configuration) [analytical]

• Crossflow, single pass, fluids unmixed [numerical]

• Crossflow, single pass, fluid 1 mixed, fluid 2 unmixed [analytical]

• Crossflow, single pass, fluid 2 mixed, fluid 1 unmixed [analytical]

• Crossflow, single pass, both fluids mixed [numerical]

The analytical solutions, for those cases they are available, are as follows:

Counterflow:

$NTU_1 = - \frac{1}{R_{1} - 1} \ln{\left (\frac{P_{1} R_{1} - 1}{P_{1} - 1} \right )}$

Parallel:

$NTU_1 = \frac{1}{R_{1} + 1} \ln{\left (- \frac{1}{P_{1} \left(R_{1} + 1\right) - 1} \right )}$

Crossflow, single pass, fluid 1 mixed, fluid 2 unmixed:

$NTU_1 = - \frac{1}{R_{1}} \ln{\left (R_{1} \ln{\left (- \left(P_{1} - 1\right) e^{\frac{1}{R_{1}}} \right )} \right )}$

Crossflow, single pass, fluid 2 mixed, fluid 1 unmixed

$NTU_1 = - \ln{\left (\frac{1}{R_{1}} \ln{\left (- \left(P_{1} R_{1} - 1\right) e^{R_{1}} \right )} \right )}$
Parameters
P1float

Thermal effectiveness of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

R1float

Heat capacity ratio of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

subtypefloat

The type of heat exchanger; one of ‘counterflow’, ‘parallel’, ‘crossflow’, ‘crossflow approximate’, ‘crossflow, mixed 1’, ‘crossflow, mixed 2’, ‘crossflow, mixed 1&2’.

Returns
NTU1float

Thermal number of transfer units of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

Notes

Although this function allows the thermal effectiveness desired to be specified, it does not mean such a high value can be obtained. An exception is raised when this occurs, although not always a helpful one.

>>> NTU_from_P_basic(P1=.99, R1=.1, subtype='parallel')
Traceback (most recent call last):
ValueError: math domain error


For the ‘crossflow approximate’ solution the function is monotonic, and a bounded solver is used within the range of NTU1 from 1E-11 to 1E5.

For the full correct ‘crossflow’ solution, the initial guess for newton’s method is obtained by the ‘crossflow approximate’ solution; the function may not converge because of inaccuracy performing the numerical integral involved.

For the ‘crossflow, mixed 1&2’ solution, a bounded solver is first use, but the upper bound on P1 and the upper NTU1 limit is calculated from a pade approximation performed with mpmath.

Examples

>>> NTU_from_P_basic(P1=.975, R1=.1, subtype='counterflow')
3.984769850376482

ht.hx.NTU_from_P_plate(P1, R1, Np1, Np2, counterflow=True, passes_counterflow=True, reverse=False)[source]

Returns the number of transfer units of a plate heat exchanger with a specified side 1 heat capacity ratio R1, side 1 number of transfer units NTU1, number of passes on sides 1 and 2 (respectively Np1 and Np2).

For all cases, the function also takes as arguments whether the exchanger is setup in an overall counter or parallel orientation counterflow, and whether or not individual stream passes are themselves counterflow or parallel.

The 20 supported cases are as follows. (the first number of sides listed refers to side 1, and the second number refers to side 2):

• 1 pass/1 pass parallelflow

• 1 pass/1 pass counterflow

• 1 pass/2 pass

• 1 pass/3 pass or 3 pass/1 pass (with the two end passes in parallel)

• 1 pass/3 pass or 3 pass/1 pass (with the two end passes in counterflow)

• 1 pass/4 pass

• 2 pass/2 pass, overall parallelflow, individual passes in parallel

• 2 pass/2 pass, overall parallelflow, individual passes counterflow

• 2 pass/2 pass, overall counterflow, individual passes parallelflow

• 2 pass/2 pass, overall counterflow, individual passes counterflow

• 2 pass/3 pass or 3 pass/2 pass, overall parallelflow

• 2 pass/3 pass or 3 pass/2 pass, overall counterflow

• 2 pass/4 pass or 4 pass/2 pass, overall parallel flow

• 2 pass/4 pass or 4 pass/2 pass, overall counterflow flow

For all except the simplest cases numerical solutions are used.

1 pass/1 pass counterflow (also 2/2 fully counterflow):

$NTU_1 = - \frac{1}{R_{1} - 1} \ln{\left (\frac{P_{1} R_{1} - 1}{P_{1} - 1} \right )}$

1 pass/1 pass parallel flow (also 2/2 fully parallelflow):

$NTU_1 = \frac{1}{R_{1} + 1} \ln{\left (- \frac{1}{P_{1} \left(R_{1} + 1\right) - 1} \right )}$
Parameters
P1float

Thermal effectiveness of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

R1float

Heat capacity ratio of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

Np1int

Number of passes on side 1 [-]

Np2int

Number of passes on side 2 [-]

counterflowbool

Whether or not the overall flow through the heat exchanger is in counterflow or parallel flow, [-]

passes_counterflowbool

In addition to the overall flow direction, in some cases individual passes may be in counter or parallel flow; this controls that [-]

reversebool

Used internally only to allow cases like the 1-4 formula to work for the 4-1 flow case, without having to duplicate the code [-]

Returns
NTU1float

Thermal number of transfer units of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

Notes

The defaults of counterflow=True and passes_counterflow=True will always result in the most efficient heat exchanger option, normally what is desired.

If a number of passes which is not supported is provided, an exception is raised.

For more details, see temperature_effectiveness_plate.

Examples

Three passes on side 1; one pass on side 2; two end passes in counterflow orientation.

>>> NTU_from_P_plate(P1=0.5743, R1=1/3., Np1=3, Np2=1)
0.9998336056090733

ht.hx.NTU_from_UA(UA, Cmin)[source]

Returns the Number of Transfer Units for a heat exchanger having UA, and with Cmin heat capacity rate.

$NTU = \frac{UA}{C_{min}}$
Parameters
UAfloat

Combined Area-heat transfer coefficient term, [W/K]

Cminfloat

The heat capacity rate of the smaller fluid, [W/K]

Returns
NTUfloat

Thermal Number of Transfer Units [-]

Notes

Used with the effectiveness method for heat exchanger design.

References

1

Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> NTU_from_UA(4400., 22.)
200.0

ht.hx.NTU_from_effectiveness(effectiveness, Cr, subtype='counterflow', n_shell_tube=None)[source]

Returns the Number of Transfer Units of a heat exchanger at a specified heat capacity rate, effectiveness, and configuration. The following configurations are supported:

• Counterflow (ex. double-pipe)

• Parallel (ex. double pipe inefficient configuration)

• Shell and tube exchangers with even numbers of tube passes, one or more shells in series (TEMA E (one pass shell) only)

• Crossflow, single pass, fluids unmixed

• Crossflow, single pass, Cmax mixed, Cmin unmixed

• Crossflow, single pass, Cmin mixed, Cmax unmixed

• Boiler or condenser

These situations are normally not those which occur in real heat exchangers, but are useful for academic purposes. More complicated expressions are available for other methods. These equations are confirmed in [1], [2], and [3].

For parallel flow heat exchangers:

$NTU = -\frac{\ln[1 - \epsilon(1+C_r)]}{1+C_r}$

For counterflow heat exchangers:

$NTU = \frac{1}{C_r-1}\ln\left(\frac{\epsilon-1}{\epsilon C_r-1}\right)$
$NTU = \frac{\epsilon}{1-\epsilon} \text{ if } C_r = 1$

For TEMA E shell-and-tube heat exchangers with one shell pass, 2n tube passes:

$(NTU)_1 = -(1 + C_r^2)^{-0.5}\ln\left(\frac{E-1}{E+1}\right)$
$E = \frac{2/\epsilon_1 - (1 + C_r)}{(1 + C_r^2)^{0.5}}$

For TEMA E shell-and-tube heat exchangers with more than one shell pass, 2n tube passes (this model assumes each exchanger has an equal share of the overall NTU or said more plainly, the same UA):

$\epsilon_1 = \frac{F-1}{F-C_r}$
$F = \left(\frac{\epsilon C_r-1}{\epsilon-1}\right)^{1/n}$
$NTU = n(NTU)_1$

For cross-flow (single-pass) heat exchangers with both fluids unmixed, there is no analytical solution. However, the function is monotonically increasing, and a closed-form solver is implemented as ‘crossflow approximate’, guaranteed to solve for $$10^{-7} < NTU < 10^5$$. The exact solution for ‘crossflow’ uses the approximate solution’s initial guess as a starting point for Newton’s method. Some issues are noted at effectivenesses higher than 0.9 and very high NTUs, because the numerical integral term approaches 1 too quickly.

For cross-flow (single-pass) heat exchangers with Cmax mixed, Cmin unmixed:

$NTU = -\ln\left[1 + \frac{1}{C_r}\ln(1 - \epsilon C_r)\right]$

For cross-flow (single-pass) heat exchangers with Cmin mixed, Cmax unmixed:

$NTU = -\frac{1}{C_r}\ln[C_r\ln(1-\epsilon)+1]$

For cases where Cr = 0, as in an exchanger with latent heat exchange, flow arrangement does not matter:

$NTU = -\ln(1-\epsilon)$
Parameters
effectivenessfloat

The thermal effectiveness of the heat exchanger, [-]

Crfloat

The heat capacity rate ratio, of the smaller fluid to the larger fluid, [-]

subtypestr, optional

The subtype of exchanger; one of ‘counterflow’, ‘parallel’, ‘crossflow’ ‘crossflow approximate’, ‘crossflow, mixed Cmin’, ‘crossflow, mixed Cmax’, ‘boiler’, ‘condenser’, ‘S&T’.

n_shell_tubeNone or int, optional

The number of shell and tube exchangers in a row, [-]

Returns
NTUfloat

Thermal Number of Transfer Units [-]

Notes

Unlike ht.hx.effectiveness_from_NTU, not all inputs can calculate the NTU - many exchanger types have effectiveness limits below 1 which depend on Cr and the number of shells in the case of heat exchangers. If an impossible input is given, an error will be raised and the maximum possible effectiveness will be printed.

>>> NTU_from_effectiveness(.99, Cr=.7, subtype='5S&T')
Traceback (most recent call last):
Exception: The specified effectiveness is not physically possible for this configuration; the maximum effectiveness possible is 0.974122977755.


References

1

Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

2

Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002.

3(1,2)

Holman, Jack. Heat Transfer. 10th edition. Boston: McGraw-Hill Education, 2009.

Examples

Worst case, parallel flow:

>>> NTU_from_effectiveness(effectiveness=0.5881156068417585, Cr=0.7, subtype='parallel')
5.000000000000


Crossflow, somewhat higher effectiveness:

>>> NTU_from_effectiveness(effectiveness=0.8444821799748551, Cr=0.7, subtype='crossflow')
5.000000000000


Counterflow, better than either crossflow or parallel flow:

>>> NTU_from_effectiveness(effectiveness=0.9206703686051108, Cr=0.7, subtype='counterflow')
5.0


Shell and tube exchangers:

>>> NTU_from_effectiveness(effectiveness=0.6834977044311439, Cr=0.7, subtype='S&T')
5.000000000000
>>> NTU_from_effectiveness(effectiveness=0.9205058702789254, Cr=0.7, n_shell_tube=50, subtype='S&T')
4.999999999999


Overall case of rating an existing heat exchanger where a known flowrate of steam and oil are contacted in crossflow, with the steam side mixed, known inlet and outlet temperatures, and unknown UA (based on example 10-8 in [3]):

>>> Cp_oil = 1900 # J/kg/K
>>> Cp_steam = 1860 # J/kg/K
>>> m_steam = 5.2 # kg/s
>>> m_oil = 1.45 # kg/s
>>> Thi = 130 # °C
>>> Tci = 15 # °C
>>> Tco = 85 # °C # Design specification
>>> Q = Cp_oil*m_oil*(Tci-Tco)
>>> dTh = Q/(m_steam*Cp_steam)
>>> Tho = Thi + dTh
>>> Cmin = calc_Cmin(mh=m_steam, mc=m_oil, Cph=Cp_steam, Cpc=Cp_oil)
>>> Cmax = calc_Cmax(mh=m_steam, mc=m_oil, Cph=Cp_steam, Cpc=Cp_oil)
>>> Cr = calc_Cr(mh=m_steam, mc=m_oil, Cph=Cp_steam, Cpc=Cp_oil)
>>> effectiveness = -Q/Cmin/(Thi-Tci)
>>> NTU = NTU_from_effectiveness(effectiveness, Cr, subtype='crossflow, mixed Cmax')
>>> UA = UA_from_NTU(NTU, Cmin)
>>> U = 200 # Assume this was calculated; would actually need to be obtained iteratively as U depends on the exchanger geometry
>>> A = UA/U
>>> Tho, Cmin, Cmax, Cr
(110.06100082712986, 2755.0, 9672.0, 0.2848428453267163)
>>> effectiveness, NTU, UA, A
(0.608695652173, 1.1040839095, 3041.75117083, 15.2087558541)

ht.hx.Ntubes(DBundle, Do, pitch, Ntp=1, angle=30, Method=None)[source]

Calculates the number of tubes which can fit in a heat exchanger. The tube count is effected by the pitch, number of tube passes, and angle.

The result is an exact number of tubes and is calculated by a very accurate method using number theory by default. This method is available only up to 100,000 tubes.

Parameters
DBundlefloat

Outer diameter of tube bundle, [m]

Dofloat

Tube outer diameter, [m]

pitchfloat

Pitch; distance between two orthogonal tube centers, [m]

Ntpint, optional

Number of tube passes, [-]

anglefloat, optional

The angle the tubes are positioned; 30, 45, 60 or 90, [degrees]

Returns
Nint

Total number of tubes that fit in the heat exchanger, [-]

Other Parameters
Methodstring, optional

One of ‘Phadkeb’, ‘HEDH’, ‘VDI’ or ‘Perry’

Examples

>>> Ntubes(DBundle=1.2, Do=0.025, pitch=0.03125)
1285


The approximations are pretty good too:

>>> Ntubes(DBundle=1.2, Do=0.025, pitch=0.03125, Method='Perry')
1297
>>> Ntubes(DBundle=1.2, Do=0.025, pitch=0.03125, Method='VDI')
1340
>>> Ntubes(DBundle=1.2, Do=0.025, pitch=0.03125, Method='HEDH')
1272

ht.hx.Ntubes_HEDH(DBundle=None, Do=None, pitch=None, angle=30)[source]

A rough equation presented in the HEDH for estimating the number of tubes in a tube bundle of differing geometries and tube sizes. No accuracy estimation given. Only 1 pass is supported.

$N = \frac{0.78(D_{bundle} - D_o)^2}{C_1(\text{pitch})^2}$

C1 = 0.866 for 30° and 60° layouts, and 1 for 45 and 90° layouts.

Parameters
DBundlefloat

Outer diameter of tube bundle, [m]

Dofloat

Tube outer diameter, [m]

pitchfloat

Pitch; distance between two orthogonal tube centers, [m]

anglefloat

The angle the tubes are positioned; 30, 45, 60 or 90, [degrees]

Returns
Nfloat

Number of tubes, [-]

Notes

Seems reasonably accurate.

References

1

Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1983.

Examples

>>> Ntubes_HEDH(DBundle=1.184, Do=.028, pitch=.036, angle=30)
928

ht.hx.Ntubes_Perrys(DBundle, Do, Ntp, angle=30)[source]

A rough equation presented in Perry’s Handbook [1] for estimating the number of tubes in a tube bundle of differing geometries and tube sizes. Claimed accuracy of 24 tubes.

Parameters
DBundlefloat

Outer diameter of tube bundle, [m]

Doint

Tube outer diameter, [m]

Ntpfloat

Number of tube passes, [-]

anglefloat

The angle the tubes are positioned; 30, 45, 60 or 90, [degrees]

Returns
Ntint

Number of tubes, [-]

Notes

Perry’s equation 11-74. Pitch equal to 1.25 times the tube outside diameter No other source for this equation is given. Experience suggests this is accurate to 40 tubes, but is often around 20 tubes off.

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. New York: McGraw-Hill Education, 2007.

Examples

>>> Ntubes_Perrys(DBundle=1.184, Do=.028, Ntp=2, angle=45)
803


Using tabulated values and correction factors for number of passes, the highly accurate method of [1] is used to obtain the tube count of a given tube bundle outer diameter for a given tube size and pitch.

Parameters
DBundlefloat

Outer diameter of tube bundle, [m]

Dofloat

Tube outer diameter, [m]

pitchfloat

Pitch; distance between two orthogonal tube centers, [m]

Ntpint

Number of tube passes, [-]

anglefloat, optional

The angle the tubes are positioned; 30, 45, 60 or 90, [degrees]

Returns
Ntint

Total number of tubes that fit in the heat exchanger, [-]

Notes

For single-pass cases, the result is exact, and no tubes need to be removed for any reason. For 4, 6, 8 pass arrangements, a number of tubes must be removed to accommodate pass partition plates. The following assumptions are involved with that:

• The pass partition plate is where a row of tubes would have been. Only one or two rows are assumed affected.

• The thickness of partition plate is < 70% of the tube outer diameter.

• The distance between the centerline of the partition plate and the centerline of the nearest row of tubes is equal to the pitch.

This function will fail when there are more than 100,000 tubes. [1] tabulated values up to approximately 3,000 tubes derived with number theory. The sequesnces of integers were identified in the On-Line Encyclopedia of Integer Sequences (OEIS), and formulas listed in it were used to generate more coefficient to allow up to 100,000 tubes. The integer sequences are A003136, A038590, A001481, and A057961. The generation of coefficients for A038590 is very slow, but the rest are reasonably fast.

The number of tubes that fit generally does not increase one-by-one, but by several.

>>> Ntubes_Phadkeb(DBundle=1.007, Do=.028, pitch=.036, Ntp=2, angle=45.)
558
>>> Ntubes_Phadkeb(DBundle=1.008, Do=.028, pitch=.036, Ntp=2, angle=45.)
574


Because a pass partition needs to be installed in multiple tube pass shells, more tubes fit in an exchanger the fewer passes are used.

>>> Ntubes_Phadkeb(DBundle=1.008, Do=.028, pitch=.036, Ntp=1, angle=45.)
593


References

1(1,2)

Phadke, P. S., Determining tube counts for shell and tube exchangers, Chem. Eng., September, 91, 65-68 (1984).

Examples

>>> Ntubes_Phadkeb(DBundle=1.200-.008*2, Do=.028, pitch=.036, Ntp=2, angle=45.)
782

ht.hx.Ntubes_VDI(DBundle=None, Ntp=None, Do=None, pitch=None, angle=30.0)[source]

A rough equation presented in the VDI Heat Atlas for estimating the number of tubes in a tube bundle of differing geometries and tube sizes. No accuracy estimation given.

Parameters
DBundlefloat

Outer diameter of tube bundle, [m]

Ntpfloat

Number of tube passes, [-]

Dofloat

Tube outer diameter, [m]

pitchfloat

Pitch; distance between two orthogonal tube centers, [m]

anglefloat

The angle the tubes are positioned; 30, 45, 60 or 90, [degrees]

Returns
Nfloat

Number of tubes, [-]

Notes

No coefficients for this method with Ntp=6 are available in [1]. For consistency, estimated values were added to support 6 tube passes, f2 = 90.. This equation is a rearranged form of that presented in [1]. The calculated tube count is rounded down to an integer.

References

1(1,2)

Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010.

Examples

>>> Ntubes_VDI(DBundle=1.184, Ntp=2, Do=.028, pitch=.036, angle=30)
966

ht.hx.P_NTU_method(m1, m2, Cp1, Cp2, UA=None, T1i=None, T1o=None, T2i=None, T2o=None, subtype='crossflow', Ntp=1, optimal=True)[source]

Wrapper for the various P-NTU method function calls, which can solve a heat exchanger. The heat capacities and mass flows of each stream and the type of the heat exchanger are always required. As additional inputs, one combination of the following inputs is required:

• Three of the four inlet and outlet stream temperatures.

• Temperatures for the side 1 outlet and side 2 inlet and UA

• Temperatures for the side 1 outlet side 2 outlet and UA

• Temperatures for the side 1 inlet and side 2 inlet and UA

• Temperatures for the side 1 inlet and side 2 outlet and UA

Computes the total heat exchanged as well as both temperatures of both streams.

Parameters
m1float

Mass flow rate of stream 1 (shell side = 1, tube side = 2), [kg/s]

m2float

Mass flow rate of stream 2 (shell side = 1, tube side = 2), [kg/s]

Cp1float

Averaged heat capacity of stream 1 (shell side), [J/kg/K]

Cp2float

Averaged heat capacity of stream 2 (tube side), [J/kg/K]

UAfloat, optional

Combined Area-heat transfer coefficient term, [W/K]

T1ifloat, optional

Inlet temperature of stream 1 (shell side), [K]

T1ofloat, optional

Outlet temperature of stream 1 (shell side), [K]

T2ifloat, optional

Inlet temperature of stream 2 (tube side), [K]

T2ofloat, optional

Outlet temperature of stream 2 (tube-side), [K]

subtypestr, optional

The subtype of exchanger; one of ‘E’, ‘G’, ‘H’, ‘J’, ‘counterflow’, ‘parallel’, ‘crossflow’, ‘crossflow, mixed 1’, ‘crossflow, mixed 2’, or ‘crossflow, mixed 1&2’. For plate exchangers ‘Np1/Np2’ where Np1 is the number of side 1 passes and Np2 is the number of side 2 passes

Ntpint, optional

For real heat exchangers (types ‘E’, ‘G’, ‘H’, and ‘J’), the number of tube passes needss to be specified as well. Not all types support any number of tube passes.

optimalbool, optional

For real heat exchangers (types ‘E’, ‘G’, ‘H’, and ‘J’), there is often a more countercurrent (optimal) way to arrange the tube passes and a more parallel (optimal=False) way to arrange them. This controls that.

Returns
resultsdict
• Q : Heat exchanged in the heat exchanger, [W]

• UA : Combined area-heat transfer coefficient term, [W/K]

• T1i : Inlet temperature of stream 1, [K]

• T1o : Outlet temperature of stream 1, [K]

• T2i : Inlet temperature of stream 2, [K]

• T2o : Outlet temperature of stream 2, [K]

• P1 : Thermal effectiveness with respect to stream 1, [-]

• P2 : Thermal effectiveness with respect to stream 2, [-]

• R1 : Heat capacity ratio with respect to stream 1, [-]

• R2 : Heat capacity ratio with respect to stream 2, [-]

• C1 : The heat capacity rate of fluid 1, [W/K]

• C2 : The heat capacity rate of fluid 2, [W/K]

• NTU1 : Thermal Number of Transfer Units with respect to stream 1 [-]

• NTU2 : Thermal Number of Transfer Units with respect to stream 2 [-]

Notes

The main equations used in this method are as follows. For the individual expressions used to calculate P1, see the See Also section.

$Q = P_1 C_1 \Delta T_{max} = P_2 C_2 \Delta T_{max}$
$\Delta T_{max} = T_{h,i} - T_{c,i} = |T_{2,i} - T_{1,i}|$
$R_1 = \frac{C_1}{C_2} = \frac{T_{2,i} - T_{2,o}}{T_{1,o} - T_{1, i}}$
$R_2 = \frac{C_2}{C_1} = \frac{T_{1,o} - T_{1, i}}{T_{2,i} - T_{2,o}}$
$R_1 = \frac{1}{R_2}$
$NTU_1 = \frac{UA}{C_1}$
$NTU_2 = \frac{UA}{C_2}$
$NTU_1 = NTU_2 R_2$
$NTU_2 = NTU_1 R_1$
$P_1 = \frac{T_{1,o} - T_{1,i}}{T_{2,i} - T_{1,i}}$
$P_2 = \frac{T_{2,i} - T_{2,o}}{T_{2,i} - T_{1,i}}$
$P_1 = P_2 R_2$
$P_2 = P_1 R_1$
$C_1 = m_1 Cp_1$
$C_2 = m_2 Cp_2$

Once P1 has been calculated, there are six different cases for calculating the other stream temperatures depending on the two temperatures provided. They were derived with SymPy.

Two known inlet temperatures:

$T_{1,o} = - P_{1} T_{1,i} + P_{1} T_{2,i} + T_{1,i}$
$T_{2,o} = P_{1} R_{1} T_{1,i} - P_{1} R_{1} T_{2,i} + T_{2,i}$

Two known outlet temperatures:

$T_{1,i} = \frac{P_{1} R_{1} T_{1,o} + P_{1} T_{2,o} - T_{1,o}}{P_{1} R_{1} + P_{1} - 1}$
$T_{2,i} = \frac{P_{1} R_{1} T_{1,o} + P_{1} T_{2,o} - T_{2,o}}{P_{1} R_{1} + P_{1} - 1}$

Inlet 1 known, outlet 2 known:

$T_{1,o} = \frac{1}{P_{1} R_{1} - 1} \left(P_{1} R_{1} T_{1,i} + P_{1} T_{1,i} - P_{1} T_{2,o} - T_{1,i}\right)$
$T_{2,i} = \frac{P_{1} R_{1} T_{1,i} - T_{2,o}}{P_{1} R_{1} - 1}$

Outlet 1 known, inlet 2 known:

$T_{1,i} = \frac{P_{1} T_{2,i} - T_{1,o}}{P_{1} - 1}$
$T_{2,o} = \frac{1}{P_{1} - 1} \left(R_{1} \left(P_{1} T_{2,i} - T_{1,o}\right) - \left(P_{1} - 1\right) \left(R_{1} T_{1,o} - T_{2,i}\right)\right)$

Input and output of 2 known:

$T_{1,i} = \frac{1}{P_{1} R_{1}} \left(P_{1} R_{1} T_{2,i} - T_{2,i} + T_{2,o}\right)$
$T_{1,o} = \frac{1}{P_{1} R_{1}} \left(P_{1} R_{1} T_{2,i} + \left(P_{1} - 1\right) \left(T_{2,i} - T_{2,o}\right)\right)$

Input and output of 1 known:

$T_{2,i} = \frac{1}{P_{1}} \left(P_{1} T_{1,i} - T_{1,i} + T_{1,o}\right)$
$T_{2,o} = \frac{1}{P_{1}} \left(P_{1} R_{1} \left(T_{1,i} - T_{1,o}\right) + P_{1} T_{1,i} - T_{1,i} + T_{1,o}\right)$

References

1

Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002.

2

Thulukkanam, Kuppan. Heat Exchanger Design Handbook, Second Edition. CRC Press, 2013.

3

Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

Examples

Solve a heat exchanger with the UA specified, and known inlet temperatures:

>>> from pprint import pprint
>>> pprint(P_NTU_method(m1=5.2, m2=1.45, Cp1=1860., Cp2=1900,
... subtype='E', Ntp=4, T2i=15, T1i=130, UA=3041.75))
{'C1': 9672.0,
'C2': 2755.0,
'NTU1': 0.314490281224,
'NTU2': 1.104083484573,
'P1': 0.173081161436,
'P2': 0.60763738417,
'Q': 192514.714242,
'R1': 3.5107078039,
'R2': 0.28484284532,
'T1i': 130,
'T1o': 110.095666434,
'T2i': 15,
'T2o': 84.878299180,
'UA': 3041.75}


Solve the same heat exchanger as if T1i, T2i, and T2o were known but UA was not:

>>> pprint(P_NTU_method(m1=5.2, m2=1.45, Cp1=1860., Cp2=1900, subtype='E',
... Ntp=4, T1i=130, T2i=15, T2o=84.87829918042112))
{'C1': 9672.0,
'C2': 2755.0,
'NTU1': 0.31449028122,
'NTU2': 1.10408348457,
'P1': 0.173081161436,
'P2': 0.60763738417,
'Q': 192514.714242,
'R1': 3.5107078039,
'R2': 0.2848428453,
'T1i': 130,
'T1o': 110.095666434,
'T2i': 15,
'T2o': 84.878299180,
'UA': 3041.7499999}


Solve a 2 pass/2 pass plate heat exchanger with overall parallel flow and its individual passes operating in parallel and known outlet temperatures. Note the overall parallel part is trigered with optimal=False, and the individual pass parallel is triggered by appending ‘p’ to the subtype. The subpass counterflow can be specified by appending ‘c’ instead to the subtype, but this is never necessary as it is the default.

>>> pprint(P_NTU_method(m1=5.2, m2=1.45, Cp1=1860., Cp2=1900., UA=300,
... T1o=126.7, T2o=26.7, subtype='2/2p', optimal=False))
{'C1': 9672.0,
'C2': 2755.0,
'NTU1': 0.0310173697270,
'NTU2': 0.10889292196,
'P1': 0.0289452959747,
'P2': 0.101618476467,
'Q': 32200.0503078,
'R1': 3.5107078039,
'R2': 0.28484284532,
'T1i': 130.029202885,
'T1o': 126.7,
'T2i': 15.0121414490,
'T2o': 26.7,
'UA': 300}

ht.hx.Pc(x, y)[source]

Basic helper calculator which accepts a transformed R1 and NTU1 as inputs for a common term used in the calculation of the P-NTU method for plate exchangers.

Returns a value which is normally used in other calculations before the actual P1 is calculated. Nominally used in counterflow calculations

$P_c(x, y) = \frac{1 - \exp[-x(1 - y)]}{1 - y\exp[-x(1 - y)]}$
Parameters
xfloat

A modification of NTU1, the Thermal Number of Transfer Units [-]

yfloat

A modification of R1, the thermal effectiveness [-]

Returns
zfloat

Just another term in the calculation, [-]

Notes

Used with the P-NTU plate method for heat exchanger design. At y =-1, this function has a ZeroDivisionError but can be evaluated at the limit to be $$z = \frac{x}{1+x}$$.

References

1

Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002.

2

Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

Examples

>>> Pc(5, .7)
0.920670368605

ht.hx.Pp(x, y)[source]

Basic helper calculator which accepts a transformed R1 and NTU1 as inputs for a common term used in the calculation of the P-NTU method for plate exchangers.

Returns a value which is normally used in other calculations before the actual P1 is calculated.

$P_p(x, y) = \frac{1 - \exp[-x(1 + y)]}{1 + y}$
Parameters
xfloat

A modification of NTU1, the Thermal Number of Transfer Units [-]

yfloat

A modification of R1, the thermal effectiveness [-]

Returns
zfloat

Just another term in the calculation, [-]

Notes

Used with the P-NTU plate method for heat exchanger design. At y = -1, this function has a ZeroDivisionError but can be evaluated at the limit to be z = x

References

1

Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002.

2

Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

Examples

>>> Pp(5, .4)
0.713634370024

ht.hx.UA_from_NTU(NTU, Cmin)[source]

Returns the combined area-heat transfer term for a heat exchanger having a specified NTU, and with Cmin heat capacity rate.

$UA = NTU C_{min}$
Parameters
NTUfloat

Thermal Number of Transfer Units [-]

Cminfloat

The heat capacity rate of the smaller fluid, [W/K]

Returns
UAfloat

Combined area-heat transfer coefficient term, [W/K]

Notes

Used with the effectiveness method for heat exchanger design.

References

1

Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> UA_from_NTU(200., 22.)
4400.0

ht.hx.baffle_thickness(Dshell, L_unsupported, service='C')[source]

Determines the thickness of baffles and support plates in TEMA HX [1]. Applies to latitudinal baffles along the diameter of the HX, but not longitudinal baffles parallel to the tubes.

Parameters
Dshellfloat

Shell inner diameter, [m]

L_unsupportedfloat

Distance between tube supports, [m]

servicestr

Service type, C, R or B, [-]

Returns
tfloat

Baffle or support plate thickness, [m]

Notes

No checks are implemented to ensure the given shell size is TEMA compatible. The baffles do not need to be strongas the pressure is almost the same on both of their sides. L_unsupported is a design choice; the more baffles in a given length, the higher the pressure drop.

References

1

Standards of the Tubular Exchanger Manufacturers Association, Ninth edition, 2007, TEMA, New York.

Examples

>>> baffle_thickness(Dshell=.3, L_unsupported=50, service='R')
0.0095

ht.hx.calc_Cmax(mh, mc, Cph, Cpc)[source]

Returns the heat capacity rate for the maximum stream having flows mh and mc, with averaged heat capacities Cph and Cpc.

$C_c = m_cC_{p,c}$
$C_h = m_h C_{p,h}$
$C_{max} = \max(C_c, C_h)$
Parameters
mhfloat

Mass flow rate of hot stream, [kg/s]

mcfloat

Mass flow rate of cold stream, [kg/s]

Cphfloat

Averaged heat capacity of hot stream, [J/kg/K]

Cpcfloat

Averaged heat capacity of cold stream, [J/kg/K]

Returns
Cmaxfloat

The heat capacity rate of the larger fluid, [W/K]

Notes

Used with the effectiveness method for heat exchanger design. Technically, it doesn’t matter if the hot and cold streams are in the right order for the input, but it is easiest to use this function when the order is specified.

References

1

Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> calc_Cmax(mh=22., mc=5.5, Cph=2200, Cpc=4400.)
48400.0

ht.hx.calc_Cmin(mh, mc, Cph, Cpc)[source]

Returns the heat capacity rate for the minimum stream having flows mh and mc, with averaged heat capacities Cph and Cpc.

$C_c = m_cC_{p,c} C_h = m_h C_{p,h} C_{min} = \min(C_c, C_h)$
Parameters
mhfloat

Mass flow rate of hot stream, [kg/s]

mcfloat

Mass flow rate of cold stream, [kg/s]

Cphfloat

Averaged heat capacity of hot stream, [J/kg/K]

Cpcfloat

Averaged heat capacity of cold stream, [J/kg/K]

Returns
Cminfloat

The heat capacity rate of the smaller fluid, [W/K]

Notes

Used with the effectiveness method for heat exchanger design. Technically, it doesn’t matter if the hot and cold streams are in the right order for the input, but it is easiest to use this function when the order is specified.

References

1

Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> calc_Cmin(mh=22., mc=5.5, Cph=2200, Cpc=4400.)
24200.0

ht.hx.calc_Cr(mh, mc, Cph, Cpc)[source]

Returns the heat capacity rate ratio for a heat exchanger having flows mh and mc, with averaged heat capacities Cph and Cpc.

$C_r=C^*=\frac{C_{min}}{C_{max}}$
Parameters
mhfloat

Mass flow rate of hot stream, [kg/s]

mcfloat

Mass flow rate of cold stream, [kg/s]

Cphfloat

Averaged heat capacity of hot stream, [J/kg/K]

Cpcfloat

Averaged heat capacity of cold stream, [J/kg/K]

Returns
Crfloat

The heat capacity rate ratio, of the smaller fluid to the larger fluid, [W/K]

Notes

Used with the effectiveness method for heat exchanger design. Technically, it doesn’t matter if the hot and cold streams are in the right order for the input, but it is easiest to use this function when the order is specified.

References

1

Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> calc_Cr(mh=22., mc=5.5, Cph=2200, Cpc=4400.)
0.5

ht.hx.check_tubing_TEMA(NPS=None, BWG=None)[source]
>>> check_tubing_TEMA(2, 22)
False
>>> check_tubing_TEMA(0.375, 22)
True

ht.hx.effectiveness_NTU_method(mh, mc, Cph, Cpc, subtype='counterflow', Thi=None, Tho=None, Tci=None, Tco=None, UA=None, n_shell_tube=None)[source]

Wrapper for the various effectiveness-NTU method function calls, which can solve a heat exchanger. The heat capacities and mass flows of each stream and the type of the heat exchanger are always required. As additional inputs, one combination of the following inputs is required:

• Three of the four inlet and outlet stream temperatures.

• Temperatures for the cold outlet and hot inlet and UA

• Temperatures for the cold outlet and hot outlet and UA

• Temperatures for the cold inlet and hot inlet and UA

• Temperatures for the cold inlet and hot outlet and UA

Parameters
mhfloat

Mass flow rate of hot stream, [kg/s]

mcfloat

Mass flow rate of cold stream, [kg/s]

Cphfloat

Averaged heat capacity of hot stream, [J/kg/K]

Cpcfloat

Averaged heat capacity of cold stream, [J/kg/K]

subtypestr, optional

The subtype of exchanger; one of ‘counterflow’, ‘parallel’, ‘crossflow’ ‘crossflow, mixed Cmin’, ‘crossflow, mixed Cmax’, ‘boiler’, ‘condenser’, ‘S&T’, or ‘nS&T’ where n is the number of shell and tube exchangers in a row

Thifloat, optional

Inlet temperature of hot fluid, [K]

Thofloat, optional

Outlet temperature of hot fluid, [K]

Tcifloat, optional

Inlet temperature of cold fluid, [K]

Tcofloat, optional

Outlet temperature of cold fluid, [K]

UAfloat, optional

Combined Area-heat transfer coefficient term, [W/K]

n_shell_tubeNone or int, optional

The number of shell and tube exchangers in a row, [-]

Returns
resultsdict
• Q : Heat exchanged in the heat exchanger, [W]

• UA : Combined area-heat transfer coefficient term, [W/K]

• Cr : The heat capacity rate ratio, of the smaller fluid to the larger fluid, [W/K]

• Cmin : The heat capacity rate of the smaller fluid, [W/K]

• Cmax : The heat capacity rate of the larger fluid, [W/K]

• effectiveness : The thermal effectiveness of the heat exchanger, [-]

• NTU : Thermal Number of Transfer Units [-]

• Thi : Inlet temperature of hot fluid, [K]

• Tho : Outlet temperature of hot fluid, [K]

• Tci : Inlet temperature of cold fluid, [K]

• Tco : Outlet temperature of cold fluid, [K]

Examples

Solve a heat exchanger to determine UA and effectiveness given the configuration, flows, subtype, the cold inlet/outlet temperatures, and the hot stream inlet temperature.

>>> from pprint import pprint
>>> pprint(effectiveness_NTU_method(mh=5.2, mc=1.45, Cph=1860., Cpc=1900,
... subtype='crossflow, mixed Cmax', Tci=15, Tco=85, Thi=130))
{'Cmax': 9672.0,
'Cmin': 2755.0,
'Cr': 0.284842845326,
'NTU': 1.104083909,
'Q': 192850.0,
'Tci': 15,
'Tco': 85,
'Thi': 130,
'Tho': 110.0610008271,
'UA': 3041.75117083,
'effectiveness': 0.608695652173}


Solve the same heat exchanger with the UA specified, and known inlet temperatures:

>>> pprint(effectiveness_NTU_method(mh=5.2, mc=1.45, Cph=1860., Cpc=1900,
... subtype='crossflow, mixed Cmax', Tci=15, Thi=130, UA=3041.75))
{'Cmax': 9672.0,
'Cmin': 2755.0,
'Cr': 0.284842845326,
'NTU': 1.104083484573,
'Q': 192849.9631022,
'Tci': 15,
'Tco': 84.9999866069,
'Thi': 130,
'Tho': 110.0610046420,
'UA': 3041.75,
'effectiveness': 0.608695535712}

ht.hx.effectiveness_from_NTU(NTU, Cr, subtype='counterflow', n_shell_tube=None)[source]

Returns the effectiveness of a heat exchanger at a specified heat capacity rate, number of transfer units, and configuration. The following configurations are supported:

• Counterflow (ex. double-pipe)

• Parallel (ex. double pipe inefficient configuration)

• Shell and tube exchangers with even numbers of tube passes, one or more shells in series

• Crossflow, single pass, fluids unmixed

• Crossflow, single pass, Cmax mixed, Cmin unmixed

• Crossflow, single pass, Cmin mixed, Cmax unmixed

• Boiler or condenser

These situations are normally not those which occur in real heat exchangers, but are useful for academic purposes. More complicated expressions are available for other methods. These equations are confirmed in [1], [2], and [3].

For parallel flow heat exchangers:

$\epsilon = \frac{1 - \exp[-NTU(1+C_r)]}{1+C_r}$

For counterflow heat exchangers:

$\epsilon = \frac{1 - \exp[-NTU(1-C_r)]}{1-C_r\exp[-NTU(1-C_r)]},\; C_r < 1$
$\epsilon = \frac{NTU}{1+NTU},\; C_r = 1$

For TEMA E shell-and-tube heat exchangers with one shell pass, 2n tube passes:

$\epsilon_1 = 2\left\{1 + C_r + \sqrt{1+C_r^2}\times\frac{1+\exp [-(NTU)_1\sqrt{1+C_r^2}]}{1-\exp[-(NTU)_1\sqrt{1+C_r^2}]}\right\}^{-1}$

For TEMA E shell-and-tube heat exchangers with more than one shell pass, 2n tube passes (this model assumes each exchanger has an equal share of the overall NTU or said more plainly, the same UA):

$\epsilon = \left[\left(\frac{1-\epsilon_1 C_r}{1-\epsilon_1}\right)^2 -1\right]\left[\left(\frac{1-\epsilon_1 C_r}{1-\epsilon_1}\right)^n - C_r\right]^{-1}$

For cross-flow (single-pass) heat exchangers with both fluids unmixed, there is an approximate and an exact formula. The approximate one is:

$\epsilon = 1 - \exp\left[\left(\frac{1}{C_r}\right) (NTU)^{0.22}\left\{\exp\left[C_r(NTU)^{0.78}\right]-1\right\}\right]$

The exact solution for crossflow (fluids unmixed) uses SciPy’s quad to perform an integral (there is no analytical integral solution available). $$I_0(v)$$ is the modified Bessel function of the first kind. This formula was developed in [4].

$\epsilon = \frac{1}{C_r} - \frac{\exp(-C_r \cdot NTU)}{2(C_r NTU)^2} \int_0^{2 NTU\sqrt{C_r}} \left(1 + NTU - \frac{v^2}{4C_r NTU}\right) \exp\left(-\frac{v^2}{4C_r NTU}\right)v I_0(v) dv$

For cross-flow (single-pass) heat exchangers with Cmax mixed, Cmin unmixed:

$\epsilon = \left(\frac{1}{C_r}\right)(1 - \exp\left\{-C_r[1-\exp(-NTU)]\right\})$

For cross-flow (single-pass) heat exchangers with Cmin mixed, Cmax unmixed:

$\epsilon = 1 - \exp(-C_r^{-1}\{1 - \exp[-C_r(NTU)]\})$

For cases where Cr = 0, as in an exchanger with latent heat exchange, flow arrangement does not matter:

$\epsilon = 1 - \exp(-NTU)$
Parameters
NTUfloat

Thermal Number of Transfer Units [-]

Crfloat

The heat capacity rate ratio, of the smaller fluid to the larger fluid, [-]

subtypestr, optional

The subtype of exchanger; one of ‘counterflow’, ‘parallel’, ‘crossflow’ ‘crossflow approximate’, ‘crossflow, mixed Cmin’, ‘crossflow, mixed Cmax’, ‘boiler’, ‘condenser’, ‘S&T’

n_shell_tubeNone or int, optional

The number of shell and tube exchangers in a row, [-]

Returns
effectivenessfloat

The thermal effectiveness of the heat exchanger, [-]

Notes

Once the effectiveness of the exchanger has been calculated, the total heat transferred can be calculated according to the following formulas, depending on which stream temperatures are known:

If the inlet temperatures for both sides are known:

$Q=\epsilon C_{min}(T_{h,i}-T_{c,i})$

If the outlet temperatures for both sides are known:

$Q = \frac{\epsilon C_{min}C_{hot}C_{cold}(T_{c,o}-T_{h,o})} {\epsilon C_{min}(C_{hot} +C_{cold}) - (C_{hot}C_{cold}) }$

If the hot inlet and cold outlet are known:

$Q = \frac{\epsilon C_{min}C_c(T_{co}-T_{hi})}{\epsilon C_{min}-C_c}$

If the hot outlet stream and cold inlet stream are known:

$Q = \frac{\epsilon C_{min}C_h(T_{ci}-T_{ho})}{\epsilon C_{min}-C_h}$

If the inlet and outlet conditions for a single side are known, the effectiveness wasn’t needed for it to be calculated. For completeness, the formulas are as follows:

$Q = C_c(T_{c,o} - T_{c,i}) = C_h(T_{h,i} - T_{h,o})$

There is also a term called $$Q_{max}$$, which is the heat which would have been transferred if the effectiveness was 1. It is calculated as follows:

$Q_{max} = \frac{Q}{\text{effectiveness}}$

References

1

Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

2

Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002.

3(1,2)

Holman, Jack. Heat Transfer. 10th edition. Boston: McGraw-Hill Education, 2009.

4

Triboix, Alain. “Exact and Approximate Formulas for Cross Flow Heat Exchangers with Unmixed Fluids.” International Communications in Heat and Mass Transfer 36, no. 2 (February 1, 2009): 121-24. doi:10.1016/j.icheatmasstransfer.2008.10.012.

Examples

Worst case, parallel flow:

>>> effectiveness_from_NTU(NTU=5, Cr=0.7, subtype='parallel')
0.5881156068417585


Crossflow, somewhat higher effectiveness:

>>> effectiveness_from_NTU(NTU=5, Cr=0.7, subtype='crossflow')
0.84448217997


Counterflow, better than either crossflow or parallel flow:

>>> effectiveness_from_NTU(NTU=5, Cr=0.7, subtype='counterflow')
0.920670368605


One shell and tube heat exchanger gives worse performance than counterflow, but they are designed to be economical and compact which a counterflow exchanger would not be. As the number of shells approaches infinity, the counterflow result is obtained exactly.

>>> effectiveness_from_NTU(NTU=5, Cr=0.7, subtype='S&T')
0.683497704431
>>> effectiveness_from_NTU(NTU=5, Cr=0.7, subtype='S&T', n_shell_tube=50)
0.920505870278


Overall case of rating an existing heat exchanger where a known flowrate of steam and oil are contacted in crossflow, with the steam side mixed (example 10-9 in [3]):

>>> U = 275 # W/m^2/K
>>> A = 10.82 # m^2
>>> Cp_oil = 1900 # J/kg/K
>>> Cp_steam = 1860 # J/kg/K
>>> m_steam = 5.2 # kg/s
>>> m_oil = 0.725 # kg/s
>>> Thi = 130 # °C
>>> Tci = 15 # °C
>>> Cmin = calc_Cmin(mh=m_steam, mc=m_oil, Cph=Cp_steam, Cpc=Cp_oil)
>>> Cmax = calc_Cmax(mh=m_steam, mc=m_oil, Cph=Cp_steam, Cpc=Cp_oil)
>>> Cr = calc_Cr(mh=m_steam, mc=m_oil, Cph=Cp_steam, Cpc=Cp_oil)
>>> NTU = NTU_from_UA(UA=U*A, Cmin=Cmin)
>>> eff = effectiveness_from_NTU(NTU=NTU, Cr=Cr, subtype='crossflow, mixed Cmax')
>>> Q = eff*Cmin*(Thi - Tci)
>>> Tco = Tci + Q/(m_oil*Cp_oil)
>>> Tho = Thi - Q/(m_steam*Cp_steam)
>>> Cmin, Cmax, Cr
(1377.5, 9672.0, 0.1424214226633)
>>> NTU, eff, Q
(2.16007259528, 0.831218036142, 131675.3271504)
>>> Tco, Tho
(110.5900741563, 116.3859256461)


Alternatively, if only the outlet temperatures had been known:

>>> Tco = 110.59007415639887
>>> Tho = 116.38592564614977
>>> Cc, Ch = Cmin, Cmax # In this case but not always
>>> Q = eff*Cmin*Cc*Ch*(Tco - Tho)/(eff*Cmin*(Cc+Ch) - Ch*Cc)
>>> Thi = Tho + Q/Ch
>>> Tci = Tco - Q/Cc
>>> Q, Tci, Thi
(131675.3271504, 14.99999999999, 130.0000000000)

ht.hx.get_tube_TEMA(NPS=None, BWG=None, Do=None, Di=None, tmin=None)[source]
ht.hx.shell_clearance(DBundle=None, DShell=None)[source]

Looks up the recommended clearance between a shell and tube bundle in a TEMA HX [1]. Either the bundle diameter or the shell diameter are needed provided.

Parameters
DBundlefloat, optional

Outer diameter of tube bundle, [m]

DShellfloat, optional

Shell inner diameter, [m]

Returns
cfloat

Shell-tube bundle clearance, [m]

Notes

Lower limits are extended up to the next limit where intermediate limits are not provided.

References

1

Standards of the Tubular Exchanger Manufacturers Association, Ninth edition, 2007, TEMA, New York.

Examples

>>> shell_clearance(DBundle=1.245)
0.0064

ht.hx.size_bundle_from_tubecount(N, Do, pitch, Ntp=1, angle=30, Method=None)[source]

Calculates the outer diameter of a tube bundle containing a specified number of tubes. The tube count is effected by the pitch, number of tube passes, and angle.

The result is an exact number of tubes and is calculated by a very accurate method using number theory by default. This method is available only up to 100,000 tubes.

Parameters
Nint

Total number of tubes that fit in the heat exchanger, [-]

Dofloat

Tube outer diameter, [m]

pitchfloat

Pitch; distance between two orthogonal tube centers, [m]

Ntpint, optional

Number of tube passes, [-]

anglefloat, optional

The angle the tubes are positioned; 30, 45, 60 or 90, [degrees]

Returns
DBundlefloat

Outer diameter of tube bundle, [m]

Other Parameters
Methodstring, optional

One of ‘Phadkeb’, ‘HEDH’, ‘VDI’ or ‘Perry’

Notes

The ‘Perry’ method is solved with a numerical solver and is very unreliable.

Examples

>>> size_bundle_from_tubecount(N=1285, Do=0.025, pitch=0.03125)
1.1985676402390355

ht.hx.temperature_effectiveness_TEMA_E(R1, NTU1, Ntp=1, optimal=True)[source]

Returns temperature effectiveness P1 of a TEMA E type heat exchanger with a specified heat capacity ratio, number of transfer units NTU1, number of tube passes Ntp, and whether or not it is arranged in a more countercurrent (optimal configuration) way or a more parallel (optimal=False) case. The supported cases are as follows:

• 1-1 TEMA E, shell fluid mixed

• 1-2 TEMA E, shell fluid mixed (this configuration is symmetric)

• 1-2 TEMA E, shell fluid split into two steams individually mixed

• 1-3 TEMA E, shell and tube fluids mixed, one parallel pass and two counterflow passes (efficient)

• 1-3 TEMA E, shell and tube fluids mixed, two parallel passes and one counteflow pass (inefficient)

• 1-N TEMA E, shall and tube fluids mixed, efficient counterflow orientation, N an even number

1-1 TEMA E, shell fluid mixed:

$P_1 = \frac{1 - \exp[-NTU_1(1-R_1)]}{1 - R_1 \exp[-NTU_1(1-R_1)]}$

1-2 TEMA E, shell fluid mixed (this configuration is symmetric):

$P_1 = \frac{2}{1 + R_1 + E\coth(E\cdot NTU_1/2)}$
$E = [1 + R_1^2]^{1/2}$

1-2 TEMA E, shell fluid split into two steams individually mixed:

$P_1 = \frac{1}{R_1}\left[1 - \frac{(2-R_1)(2E+R_1B)}{(2+R_1)(2E-R_1/B)} \right]$
$E = \exp(NTU_1)$
$B = \exp(-NTU_1 R_1/2)$

1-3 TEMA E, shell and tube fluids mixed, one parallel pass and two counterflow passes (efficient):

$P_1 = \frac{1}{R_1} \left[1 - \frac{C}{AC + B^2}\right]$
$A = X_1(R_1 + \lambda_1)(R_1 - \lambda_2)/(2\lambda_1) - X_3 \delta - X_2(R_1 + \lambda_2)(R_1-\lambda_1)/(2\lambda_2) + 1/(1-R_1)$
$B = X_1(R_1-\lambda_2) - X_2(R_1-\lambda_1) + X_3\delta$
$C = X_2(3R_1 + \lambda_1) - X_1(3R_1 + \lambda_2) + X_3 \delta$
$X_i = \exp(\lambda_i NTU_1/3)/(2\delta),\;\; i = 1,2,3$
$\delta = \lambda_1 - \lambda_2$
$\lambda_1 = -\frac{3}{2} + \left[\frac{9}{4} + R_1(R_1-1)\right]^{1/2}$
$\lambda_2 = -\frac{3}{2} - \left[\frac{9}{4} + R_1(R_1-1)\right]^{1/2}$
$\lambda_3 = R_1$

1-3 TEMA E, shell and tube fluids mixed, two parallel passes and one counteflow pass (inefficient):

$P_2 = \left[1 - \frac{C}{(AC + B^2)}\right]$
$A = \chi_1(1 + R_2 \lambda_1)(1 - R_2\lambda_2)/(2R_2^2\lambda_1) - E -\chi_2(1 + R_2\lambda_2)(1 - R_2\lambda_1)/(2R^2\lambda_2) + R/(R-1)$
$B = \chi_1(1 - R_2\lambda_2)/R_2 - \chi_2(1 - R_2 \lambda_1)/R_2 + E$
$C = -\chi_1(3 + R_2\lambda_2)/R_2 + \chi_2(3 + R_2\lambda_1)/R_2 + E$
$E = 0.5\exp(NTU_2/3)$
$\lambda_1 = (-3 + \delta)/2$
$\lambda_2 = (-3 - \delta)/2$
$\delta = \frac{[9R_2^2 + 4(1-R_2))]^{0.5}}{R_2}$
$\chi_1 = \frac{\exp(\lambda_1 R_2 NTU_2/3)}{2\delta}$
$\chi_2 = \frac{\exp(\lambda_2 R_2 NTU_2/3)}{2\delta}$

1-N TEMA E, shall and tube fluids mixed, efficient counterflow orientation, N an even number:

$P_2 = \frac{2}{A + B + C}$
$A = 1 + R_2 + \coth(NTU_2/2)$
$B = \frac{-1}{N_1}\coth\left(\frac{NTU_2}{2N_1}\right)$
$C = \frac{1}{N_1}\sqrt{1 + N_1^2 R_2^2} \coth\left(\frac{NTU_2}{2N_1}\sqrt{1 + N_1^2 R_2^2}\right)$
$N_1 = \frac{N_{tp}}{2}$
Parameters
R1float

Heat capacity ratio of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

NTU1float

Thermal number of transfer units of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

Ntpint

Number of tube passes, 1, 2, 3, 4, or an even number[-]

optimalbool, optional

Whether or not the arrangement is configured to give more of a countercurrent and efficient (True) case or an inefficient parallel case, [-]

Returns
P1float

Thermal effectiveness of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

Notes

For odd numbers of tube passes greater than 3, an exception is raised. [2] actually has a formula for 5 tube passes, but it is an entire page long. The convention for the formulas in [1] and [3] are with the shell side as side 1, and the tube side as side 2. [2] has formulas with the opposite convention.

References

1

Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002.

2(1,2)

Thulukkanam, Kuppan. Heat Exchanger Design Handbook, Second Edition. CRC Press, 2013.

3

Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

Examples

>>> temperature_effectiveness_TEMA_E(R1=1/3., NTU1=1., Ntp=1)
0.5870500654031314

ht.hx.temperature_effectiveness_TEMA_G(R1, NTU1, Ntp, optimal=True)[source]

Returns temperature effectiveness P1 of a TEMA G type heat exchanger with a specified heat capacity ratio, number of transfer units NTU1, and of number of tube passes Ntp. For the two tube pass case, there are two possible orientations, one inefficient and one efficient controlled by the optimal option. The supported cases are as follows:

• One tube pass (tube fluid split into two streams individually mixed, shell fluid mixed)

• Two tube passes (shell and tube exchanger with shell and tube fluids mixed in each pass at the cross section), counterflow arrangement

• Two tube passes (shell and tube exchanger with shell and tube fluids mixed in each pass at the cross section), parallelflow arrangement

1-1 TEMA G, tube fluid split into two streams individually mixed, shell fluid mixed (this configuration is symmetric):

$P_1 = A + B - AB(1 + R_1) + R_1 AB^2$
$A = \frac{1}{1 + R_1}\{1 - \exp(-NTU_1(1+R_1)/2)\}$
$B = \frac{1 - D}{1 - R_1 D}$
$D = \exp[-NTU_1(1-R_1)/2]$

1-2 TEMA G, shell and tube exchanger with shell and tube fluids mixed in each pass at the cross section:

$P_1 = (B - \alpha^2)/(A + 2 + R_1 B)$
$A = -2 R_1(1-\alpha)^2/(2 + R_1)$
$B = [4 - \beta(2+R_1)]/(2 - R_1)$
$\alpha = \exp[-NTU_1(2+R_1)/4]$
$\beta = \exp[-NTU_1(2 - R_1)/2]$

1-2 TEMA G, shell and tube exchanger in overall parallelflow arrangement with shell and tube fluids mixed in each pass at the cross section (this is only shown in [2], and the stream convention in it is different but converted here; P1 is still returned):

$P_2 = \frac{(B-\alpha^2)}{R_2(A - \alpha^2/R_2 + 2)}$
$A = \frac{(1-\alpha)^2}{(R_2-0.5)}$
$B = \frac{4R_2 - \beta(2R_2 - 1)}{2R_2 + 1}$
$\alpha = \exp\left(\frac{-NTU_2(2R_2-1)}{4}\right)$
$\beta = \exp\left(\frac{-NTU_2(2R_2+1)}{2}\right)$
Parameters
R1float

Heat capacity ratio of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

NTU1float

Thermal number of transfer units of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

Ntpint

Number of tube passes, 1 or 2, [-]

optimalbool, optional

Whether or not the arrangement is configured to give more of a countercurrent and efficient (True) case or an inefficient parallel case (only applies for two passes), [-]

Returns
P1float

Thermal effectiveness of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

Notes

For numbers of tube passes greater than 1 or 2, an exception is raised. The convention for the formulas in [1] and [3] are with the shell side as side 1, and the tube side as side 2. [2] has formulas with the opposite convention.

References

1

Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002.

2(1,2)

Thulukkanam, Kuppan. Heat Exchanger Design Handbook, Second Edition. CRC Press, 2013.

3

Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

Examples

>>> temperature_effectiveness_TEMA_G(R1=1/3., NTU1=1., Ntp=1)
0.5730149350867675

ht.hx.temperature_effectiveness_TEMA_H(R1, NTU1, Ntp, optimal=True)[source]

Returns temperature effectiveness P1 of a TEMA H type heat exchanger with a specified heat capacity ratio, number of transfer units NTU1, and of number of tube passes Ntp. For the two tube pass case, there are two possible orientations, one inefficient and one efficient controlled by the optimal option. The supported cases are as follows:

• One tube pass (tube fluid split into two streams individually mixed, shell fluid mixed)

• Two tube passes (shell fluid mixed, tube pass mixed between passes)

• Two tube passes (shell fluid mixed, tube pass mixed between passes, inlet tube side next to inlet shell-side)

1-1 TEMA H, tube fluid split into two streams individually mixed, shell fluid mixed:

$P_1 = E[1 + (1 - BR_1/2)(1 - A R_1/2 + ABR_1)] - AB(1 - BR_1/2)$
$A = \frac{1}{1 + R_1/2}\{1 - \exp[-NTU_1(1 + R_1/2)/2]\}$
$B = \frac{1-D}{1-R_1 D/2}$
$D = \exp[-NTU_1(1-R_1/2)/2]$
$E = (A + B - ABR_1/2)/2$

1-2 TEMA H, shell and tube fluids mixed in each pass at the cross section:

$P_1 = \frac{1}{R_1}\left[1 - \frac{(1-D)^4}{B - 4G/R_1}\right]$
$B = (1+H)(1+E)^2$
$G = (1-D)^2(D^2 + E^2) + D^2(1 + E)^2$
$H = [1 - \exp(-2\beta)]/(4/R_1 -1)$
$E = [1 - \exp(-\beta)]/(4/R_1 - 1)$
$D = [1 - \exp(-\alpha)]/(4/R_1 + 1)$
$\alpha = NTU_1(4 + R_1)/8$
$\beta = NTU_1(4-R_1)/8$

1-2 TEMA H, shell and tube fluids mixed in each pass at the cross section but with the inlet tube stream coming in next to the shell fluid inlet in an inefficient way (this is only shown in [2], and the stream 1/2 convention in it is different but converted here; P1 is still returned):

$P_2 = \left[1 - \frac{B + 4GR_2}{(1-D)^4}\right]$
$B = (1 + H)(1 + E)^2$
$G = (1-D)^2(D^2 + E^2) + D^2(1 + E)^2$
$D = \frac{1 - \exp(-\alpha)}{1 - 4R_2}$
$E = \frac{\exp(-\beta) - 1}{4R_2 +1}$
$H = \frac{\exp(-2\beta) - 1}{4R_2 +1}$
$\alpha = \frac{NTU_2}{8}(4R_2 -1)$
$\beta = \frac{NTU_2}{8}(4R_2 +1)$
Parameters
R1float

Heat capacity ratio of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

NTU1float

Thermal number of transfer units of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

Ntpint

Number of tube passes, 1, or 2, [-]

optimalbool, optional

Whether or not the arrangement is configured to give more of a countercurrent and efficient (True) case or an inefficient parallel case, [-]

Returns
P1float

Thermal effectiveness of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

Notes

For numbers of tube passes greater than 1 or 2, an exception is raised. The convention for the formulas in [1] and [3] are with the shell side as side 1, and the tube side as side 2. [2] has formulas with the opposite convention.

References

1

Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002.

2(1,2)

Thulukkanam, Kuppan. Heat Exchanger Design Handbook, Second Edition. CRC Press, 2013.

3

Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

Examples

>>> temperature_effectiveness_TEMA_H(R1=1/3., NTU1=1., Ntp=1)
0.5730728284905833

ht.hx.temperature_effectiveness_TEMA_J(R1, NTU1, Ntp)[source]

Returns temperature effectiveness P1 of a TEMA J type heat exchanger with a specified heat capacity ratio, number of transfer units NTU1, and of number of tube passes Ntp. The supported cases are as follows:

• One tube pass (shell fluid mixed)

• Two tube passes (shell fluid mixed, tube pass mixed between passes)

• Four tube passes (shell fluid mixed, tube pass mixed between passes)

For 1-1 TEMA J shell and tube exchangers, shell and tube fluids mixed:

$P_1 = \frac{1}{R_1}\left[1- \frac{(2-R_1)(2E + R_1 B)}{(2+R_1) (2E - R_1/B)}\right]$

For 1-2 TEMA J, shell and tube fluids mixed. There are two possible arrangements for the flow and the number of tube passes, but the equation is the same in both:

$P_1 = \left[1 + \frac{R_1}{2} + \lambda B - 2\lambda C D\right]^{-1}$
$B = \frac{(A^\lambda +1)}{A^\lambda -1}$
$C = \frac{A^{(1 + \lambda)/2}}{\lambda - 1 + (1 + \lambda)A^\lambda}$
$D = 1 + \frac{\lambda A^{(\lambda-1)/2}}{A^\lambda -1}$
$A = \exp(NTU_1)$
$\lambda = (1 + R_1^2/4)^{0.5}$

For 1-4 TEMA J, shell and tube exchanger with both sides mixed:

$P_1 = \left[1 + \frac{R_1}{4}\left(\frac{1+3E}{1+E}\right) + \lambda B - 2 \lambda C D\right]^{-1}$
$B = \frac{A^\lambda +1}{A^\lambda -1}$
$C = \frac{A^{(1+\lambda)/2}}{\lambda - 1 + (1 + \lambda)A^\lambda}$
$D = 1 + \frac{\lambda A^{(\lambda-1)/2}}{A^\lambda -1}$
$A = \exp(NTU_1)$
$E = \exp(R_1 NTU_1/2)$
$\lambda = (1 + R_1^2/16)^{0.5}$
Parameters
R1float

Heat capacity ratio of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

NTU1float

Thermal number of transfer units of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (shell side = 1, tube side = 2) [-]

Ntpint

Number of tube passes, 1, 2, or 4, [-]

Returns
P1float

Thermal effectiveness of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

Notes

For numbers of tube passes that are not 1, 2, or 4, an exception is raised. The convention for the formulas in [1] and [3] are with the shell side as side 1, and the tube side as side 2. [2] has formulas with the opposite convention.

References

1

Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002.

2

Thulukkanam, Kuppan. Heat Exchanger Design Handbook, Second Edition. CRC Press, 2013.

3

Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

Examples

>>> temperature_effectiveness_TEMA_J(R1=1/3., NTU1=1., Ntp=1)
0.5699085193651295

ht.hx.temperature_effectiveness_air_cooler(R1, NTU1, rows, passes, coerce=True)[source]

Returns temperature effectiveness P1 of an air cooler with a specified heat capacity ratio, number of transfer units NTU1, number of rows rows, and number of passes passes. The supported cases are as follows:

• N rows 1 pass

• N row N pass (up to N = 5)

• 4 rows 2 passes

For N rows 1 passes ([2], shown in [1] and [3]):

$P = \frac{1}{R} \left\{1 - \left[\frac{N\exp(NKR)} {1 + \sum_{i=1}^{N-1}\sum_{j=0}^i {{i}\choose{j}}K^j \exp(-(i-j)NTU/N) \sum_{k=0}^j \frac{(NKR)^k}{k!}}\right]^{-1}\right\}$

For 2 rows 2 passes (cited as from [4] in [1]):

$P_1 = \frac{1}{R}\left(1 -\frac{1}{\xi}\right)$
$\xi = \frac{K}{2} + \left(1 - \frac{K}{2}\right)\exp(2KR)$
$K = 1 - \exp\left(\frac{-NTU}{2}\right)$

For 3 rows / 3 passes (cited as from [4] in [1]):

$\xi = K\left[1 - \frac{K}{4} - RK\left(1 - \frac{K}{2}\right)\right] \exp(KR) + \exp(3KR)\left(1 - \frac{K}{2}\right)^2$
$K = 1 - \exp\left(\frac{-NTU}{3}\right)$

For 4 rows / 4 passes (cited as from [4] in [1]):

$\xi = \frac{K}{2}\left(1 - \frac{K}{2} + \frac{K^2}{4}\right) + K\left(1 - \frac{K}{2}\right) \left[1 - \frac{R}{8}K\left(1 - \frac{K}{2}\right)\exp(2KR)\right] + \exp(4KR)\left(1 - \frac{K}{2}\right)^3$
$K = 1 - \exp\left(\frac{-NTU}{4}\right)$

For 5 rows / 5 passes (cited as from [4] in [1]):

$\xi = \left\{K \left(1 - \frac{3}{4}K + \frac{K^2}{2}- \frac{K^3}{8} \right) - RK^2\left[1 -K + \frac{3}{4}K^2 - \frac{1}{4}K^3 - \frac{R}{2}K^2\left(1 - \frac{K}{2}\right)^2\right]\right\}\exp(KR) + \left[K\left(1 - \frac{3}{4}K + \frac{1}{16}K^3\right) - 3RK^2\left(1 - \frac{K}{2}\right)^3\right]\exp(3KR)+ \left(1 - \frac{K}{2}\right)^4 \exp(5KR)$

For 4 rows / 2 passes (cited as from [4] in [1]):

$P_1 = \frac{1}{R}\left(1 -\frac{1}{\xi}\right)$
$\xi = \left\{\frac{R}{2}K^3[4 - K + 2RK^2] + \exp(4KR) + K\left[1 - \frac{K}{2} + \frac{K^2}{8}\right] \left[1 - \exp(4KR)\right] \right\}\frac{1}{(1+RK^2)^2}$
$K = 1 - \exp\left(\frac{-NTU}{4}\right)$
Parameters
R1float

Heat capacity ratio of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (process fluid side) [-]

NTU1float

Thermal number of transfer units of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (process fluid side) [-]

rowsint

Number of rows of tubes in the air cooler [-]

passesint

Number of passes the process fluid undergoes [-]

coercebool

If True, the number of passes or rows, if otherwise unsupported, will be replaced with a similar number to allow the calculation to proceed, [-]

Returns
P1float

Thermal effectiveness of the heat exchanger in the P-NTU method, calculated with respect to stream 1 (process fluid side) [-]

Notes

For the 1-pass case, the exact formula used can take a while to compute for large numbers of tube rows; 100 us for 20 rows, 1 ms for 50 rows. Floating point rounding behavior can also be an issue for large numbers of tube passes, leading to thermal effectivenesses larger than one being returned:

>>> temperature_effectiveness_air_cooler(1e-10, 100, rows=150, passes=1.0)
1.000026728092962


Furthermore, as a factorial of the number of tube counts is used, there comes a point where standard floats are not able to hold the intermediate calculations values and an error will occur:

>>> temperature_effectiveness_air_cooler(.5, 1.1, rows=200, passes=1.0)
Traceback (most recent call last):
...
OverflowError: int too large to convert to float


References

1(1,2,3,4,5,6)

Thulukkanam, Kuppan. Heat Exchanger Design Handbook, Second Edition. CRC Press, 2013.

2

Schedwill, H., “Thermische Auslegung von Kreuzstromwarmeaustauschern, Fortschr-Ber.” VDI Reihe 6 19, VDI, Germany, 1968.

3

Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1983.

4(1,2,3,4,5)

Nicole, F. J. L.. “Mean temperature difference for heat exchanger design.” Council for Scientific and Industrial Research, Special Report Chem. 223, Pretoria, South Africa (1972).

Examples

>>> temperature_effectiveness_air_cooler(.5, 2, rows=2, passes=2)
0.7523072855817072

ht.hx.temperature_effectiveness_basic(R1, NTU1, subtype='crossflow')[source]

Returns temperature effectiveness P1 of a heat exchanger with a specified heat capacity ratio, number of transfer units NTU1, and of type subtype. This function performs the calculations for the basic cases, not actual shell-and-tube exchangers. The supported cases are as follows:

• Counterflow (ex. double-pipe)

• Parallel (ex. double pipe inefficient configuration)

• Crossflow, single pass, fluids unmixed

• Crossflow, single pass, fluid 1 mixed, fluid 2 unmixed

• Crossflow, single pass, fluid 2 mixed, fluid 1 unmixed

• Crossflow, single pass, both fluids mixed

For parallel flow heat exchangers (this configuration is symmetric):

$P_1 = \frac{1 - \exp[-NTU_1(1+R_1)]}{1 + R_1}$

For counterflow heat exchangers (this configuration is symmetric):

$P_1 = \frac{1 - \exp[-NTU_1(1-R_1)]}{1 - R_1 \exp[-NTU_1(1-R_1)]}$

For cross-flow (single-pass) heat exchangers with both fluids unmixed (this configuration is symmetric), there are two solutions available; a frequently cited approximation and an exact solution which uses a numerical integration developed in [4]. The approximate solution is:

$P_1 \approx 1 - \exp\left[\frac{NTU_1^{0.22}}{R_1} (\exp(-R_1 NTU_1^{0.78})-1)\right]$

The exact solution for crossflow (single pass, fluids unmixed) is:

$\epsilon = \frac{1}{R_1} - \frac{\exp(-R_1 \cdot NTU_1)}{2(R_1 NTU_1)^2} \int_0^{2 NTU_1\sqrt{R_1}} \left(1 + NTU_1 - \frac{v^2}{4R_1 NTU_1} \right)\exp\left(-\frac{v^2}{4R_1 NTU_1}\right)v I_0(v) dv$

For cross-flow (single-pass) heat exchangers with fluid 1 mixed, fluid 2 unmixed:

$P_1 = 1 - \exp\left(-\frac{K}{R_1}\right)$
$K = 1 - \exp(-R_1 NTU_1)$

For cross-flow (single-pass) heat exchangers with fluid 2 mixed, fluid 1 unmixed:

$P_1 = \frac{1 - \exp(-K R_1)}{R_1}$
$K = 1 - \exp(-NTU_1)$

For cross-flow (single-pass) heat exchangers with both fluids mixed (this configuration is symmetric):

$P_1 = \left(\frac{1}{K_1} + \frac{R_1}{K_2} - \frac{1}{NTU_1}\right)^{-1}$
$K_1 = 1 - \exp(-NTU_1)$
$K_2 = 1 - \exp(-R_1 NTU_1)$
Parameters
R1float

Heat capacity ratio of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

NTU1float

Thermal number of transfer units of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

subtypefloat

The type of heat exchanger; one of ‘counterflow’, ‘parallel’, ‘crossflow’, ‘crossflow approximate’, ‘crossflow, mixed 1’, ‘crossflow, mixed 2’, ‘crossflow, mixed 1&2’.

Returns
P1float

Thermal effectiveness of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

Notes

The crossflow case is an approximation only. There is an actual solution involving an infinite sum. This was implemented, but found to differ substantially so the approximation is used instead.

References

1

Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002.

2

Thulukkanam, Kuppan. Heat Exchanger Design Handbook, Second Edition. CRC Press, 2013.

3

Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

4

Triboix, Alain. “Exact and Approximate Formulas for Cross Flow Heat Exchangers with Unmixed Fluids.” International Communications in Heat and Mass Transfer 36, no. 2 (February 1, 2009): 121-24. doi:10.1016/j.icheatmasstransfer.2008.10.012.

Examples

>>> temperature_effectiveness_basic(R1=.1, NTU1=4, subtype='counterflow')
0.9753412729761263

ht.hx.temperature_effectiveness_plate(R1, NTU1, Np1, Np2, counterflow=True, passes_counterflow=True, reverse=False)[source]

Returns the temperature effectiveness P1 of side 1 of a plate heat exchanger with a specified side 1 heat capacity ratio R1, side 1 number of transfer units NTU1, number of passes on sides 1 and 2 (respectively Np1 and Np2).

For all cases, the function also takes as arguments whether the exchanger is setup in an overall counter or parallel orientation counterflow, and whether or not individual stream passes are themselves counterflow or parallel.

The 20 supported cases are as follows. (the first number of sides listed refers to side 1, and the second number refers to side 2):

• 1 pass/1 pass parallelflow

• 1 pass/1 pass counterflow

• 1 pass/2 pass

• 1 pass/3 pass or 3 pass/1 pass (with the two end passes in parallel)

• 1 pass/3 pass or 3 pass/1 pass (with the two end passes in counterflow)

• 1 pass/4 pass

• 2 pass/2 pass, overall parallelflow, individual passes in parallel

• 2 pass/2 pass, overall parallelflow, individual passes counterflow

• 2 pass/2 pass, overall counterflow, individual passes parallelflow

• 2 pass/2 pass, overall counterflow, individual passes counterflow

• 2 pass/3 pass or 3 pass/2 pass, overall parallelflow

• 2 pass/3 pass or 3 pass/2 pass, overall counterflow

• 2 pass/4 pass or 4 pass/2 pass, overall parallel flow

• 2 pass/4 pass or 4 pass/2 pass, overall counterflow flow

For all plate heat exchangers, there are two common formulas used by most of the expressions.

$P_p(x, y) = \frac{1 - \exp[-x(1 + y)]}{1 + y} P_c(x, y) = \frac{1 - \exp[-x(1 - y)]}{1 - y\exp[-x(1 - y)]}$

The main formulas used are as follows. Note that for some cases such as 4 pass/2 pass, the formula is not shown because it is that of 2 pass/4 pass, but with R1, NTU1, and P1 conversions.

For 1 pass/1 pass paralleflow (streams symmetric):

$P_1 = P_p(NTU_1, R_1)$

For 1 pass/1 pass counterflow (streams symmetric):

$P_1 = P_c(NTU_1, R_1)$

For 1 pass/2 pass (any of the four possible configurations):

$P_1 = 0.5(A + B - 0.5ABR_1)$
$A = P_p(NTU_1, 0.5R_1)$
$B = P_c(NTU_1, 0.5R_1)$

For 1 pass/3 pass (with the two end passes in parallel):

$P_1 = \frac{1}{3}\left[B + A\left(1 - \frac{R_1 B}{3}\right)\left(2 - \frac{R_1 A}{3}\right)\right]$
$A = P_p\left(NTU_1, \frac{R_1}{3}\right)$
$B = P_c\left(NTU_1, \frac{R_1}{3}\right)$

For 1 pass/3 pass (with the two end passes in counterflow):

$P_1 = \frac{1}{3}\left[A + B\left(1 - \frac{R_1 A}{3}\right)\left(2 - \frac{R_1 B}{3}\right)\right]$
$A = P_p\left(NTU_1, \frac{R_1}{3}\right)$
$B = P_c\left(NTU_1, \frac{R_1}{3}\right)$

For 1 pass/4 pass (any of the four possible configurations):

$P_1 = \frac{1-Q}{R_1}$
$Q = \left(1 - \frac{AR_1}{4}\right)^2\left(1 - \frac{BR_1}{4}\right)^2$
$A = P_p\left(NTU_1, \frac{R_1}{4}\right)$
$B = P_c\left(NTU_1, \frac{R_1}{4}\right)$

For 2 pass/2 pass, overall parallelflow, individual passes in parallel (stream symmetric):

$P_1 = P_p(NTU_1, R_1)$

For 2 pass/2 pass, overall parallelflow, individual passes counterflow (stream symmetric):

$P_1 = B[2 - B(1 + R_1)]$
$B = P_c\left(\frac{NTU_1}{2}, R_1\right)$

For 2 pass/2 pass, overall counterflow, individual passes parallelflow (stream symmetric):

$P_1 = \frac{2A - A^2(1 + R_1)}{1 - R_1 A^2}$
$A = P_p\left(\frac{NTU_1}{2}, R_1\right)$

For 2 pass/2 pass, overall counterflow and individual passes counterflow (stream symmetric):

$P_1 = P_c(NTU_1, R_1)$

For 2 pass/3 pass, overall parallelflow:

$P_1 = A + B - \left(\frac{2}{9} + \frac{D}{3}\right) (A^2 + B^2) - \left(\frac{5}{9} + \frac{4D}{3}\right)AB + \frac{D(1+D)AB(A+B)}{3} - \frac{D^2A^2B^2}{9}$
$A = P_p\left(\frac{NTU_1}{2}, D\right)$
$B = P_c\left(\frac{NTU_1}{2}, D\right)$
$D = \frac{2R_1}{3}$

For 2 pass/3 pass, overall counterflow:

$P_1 = \frac{A + 0.5B + 0.5C + D}{R_1}$
$A = \frac{2R_1 EF^2 - 2EF + F - F^2} {2R_1 E^2 F^2 - E^2 - F^2 - 2EF + E + F}$
$B = \frac{A(E-1)}{F}$
$C = \frac{1 - A}{E}$
$D = R_1 E^2 C - R_1 E + R_1 - \frac{C}{2}$
$E = \frac{3}{2R_1 G}$
$F = \frac{3}{2R_1 H}$
$G = P_c\left(\frac{NTU_1}{2}, \frac{2R_1}{3}\right)$
$H = P_p\left(\frac{NTU_1}{2}, \frac{2R_1}{3}\right)$

For 2 pass/4 pass, overall parallel flow:

$P_1 = 2D - (1 + R_1)D^2$
$D = \frac{A + B - 0.5ABR_1}{2}$
$A = P_p\left(\frac{NTU_1}{2}, \frac{R_1}{2}\right)$
$B = P_c\left(\frac{NTU_1}{2}, \frac{R_1}{2}\right)$

For 2 pass/4 pass, overall counterflow flow:

$P_1 = \frac{2D - (1+R_1)D^2}{1 - D^2 R_1}$
$D = \frac{A + B - 0.5ABR_1}{2}$
$A = P_p\left(\frac{NTU_1}{2}, \frac{R_1}{2}\right)$
$B = P_c\left(\frac{NTU_1}{2}, \frac{R_1}{2}\right)$
Parameters
R1float

Heat capacity ratio of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

NTU1float

Thermal number of transfer units of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

Np1int

Number of passes on side 1 [-]

Np2int

Number of passes on side 2 [-]

counterflowbool

Whether or not the overall flow through the heat exchanger is in counterflow or parallel flow, [-]

passes_counterflowbool

In addition to the overall flow direction, in some cases individual passes may be in counter or parallel flow; this controls that [-]

reversebool

Used internally only to allow cases like the 1-4 formula to work for the 4-1 flow case, without having to duplicate the code [-]

Returns
P1float

Thermal effectiveness of the heat exchanger in the P-NTU method, calculated with respect to stream 1 [-]

Notes

For diagrams of these heat exchangers, see [3]. In all cases, each pass is assumed to be made up of an infinite number of plates. The fluid velocities must be uniform across the plate channels, and the flow must be uniformly distributed between the channels. The heat transfer coefficient is also assumed constant.

The defaults of counterflow=True and passes_counterflow=True will always result in the most efficient heat exchanger option, normally what is desired.

If a number of passes which is not supported is provided, an exception is raised.

References

1

Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002.

2

Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

3

Kandlikar, S. G., and R. K. Shah. “Asymptotic Effectiveness-NTU Formulas for Multipass Plate Heat Exchangers.” Journal of Heat Transfer 111, no. 2 (May 1, 1989): 314-21. doi:10.1115/1.3250679.

4

Kandlikar, S. G., and R. K. Shah. “Multipass Plate Heat Exchangers Effectiveness-NTU Results and Guidelines for Selecting Pass Arrangements.” Journal of Heat Transfer 111, no. 2 (May 1, 1989): 300-313. doi:10.1115/1.3250678.

Examples

Three passes on side 1; one pass on side 2; two end passes in counterflow orientation.

>>> temperature_effectiveness_plate(R1=1/3., NTU1=1., Np1=3, Np2=1)
0.5743514352720835


If the same heat exchanger (in terms of NTU1 and R1) were operating with sides 1 and 2 switched, a slightly less efficient design results.

>>> temperature_effectiveness_plate(R1=1/3., NTU1=1., Np1=1, Np2=3)
0.5718726757657066