Tutorial

Introduction

ht is the heat transfer component of the Chemical Engineering Design Library (ChEDL). Functions are provided to calculate heat transfer in a variety of situations, generally using dimensionless factors such as Reynolds and Prandtl number, and giving results in terms of dimensionless heat transfer coefficient, the Nusselt number. The ‘dimensional’ heat transfer coefficient may then be determined

$$h = \frac{k\cdot \text{Nu}}{L}$$

Design philosophy

Like all libraries, this was developed to scratch my own itches. Since its public release it has been found useful by many others, from students across the world to practicing engineers at some of the world’s largest companies.

The bulk of this library’s API is considered stable; enhancements to functions and classes will still happen, and default methods when using a generic correlation interface may change to newer and more accurate correlations as they are published and reviewed.

To the extent possible, correlations are implemented depending on the highest level parameters. The Nu_conv_internal correlation does not accept pipe diameter, velocity, viscosity, density, heat capacity, and thermal conductivity - it accepts Reynolds number and Prandtl number. This makes the API cleaner and encourages modular design.

All functions are desiged to accept inputs in base SI units. However, any set of consistent units given to a function will return a consistent result; for instance, a function calculating volume doesn’t care if given an input in inches or meters; the output units will be the cube of those given to it. The user is directed to unit conversion libraries such as pint to perform unit conversions if they prefer not to work in SI units.

The standard math library is used in all functions except where special functions from numpy or scipy are necessary. SciPy is used for root finding, interpolation, scientific constants, ode integration, and its many special mathematical functions not present in the standard math library. The only other required library is the fluids library, a sister library for fluid dynamics. No other libraries will become required dependencies; anything else is optional.

There are two ways to use numpy arrays with ht. Easiest to use is a vectorized module, which wraps all of the ht functions with np.vectorize. Instead of importing from ht, the user can import from ht.vectorized:

>>> from ht.vectorized import *
>>> LMTD([100, 101], 60., 30., 40.2)
array([43.20040929, 43.60182765])

It is possible to switch back and forth between the namespaces with a subsequent import:

>>> from ht import *

The second way is Numba. This optional dependency provides the speed you expect from NumPy arrays - or better. In some cases, much better. The tutorial for using it is at ht.numba, but in general use it the same way but with a different import.

>>> from ht.numba_vectorized import *

Insulation

Insulating and refractory materials from the VDI Heat Transfer Handbook and the ASHRAE Handbook: Fundamentals have been digitized and are programatically available in ht. Density, heat capacity, and thermal conductivity are available although not all materials have all three.

The actual data is stored in a series of dictionaries, building_materials, ASHRAE_board_siding, ASHRAE_flooring, ASHRAE_insulation, ASHRAE_roofing, ASHRAE_plastering, ASHRAE_masonry, ASHRAE_woods, and refractories. A total of 390 different materials are available. Functions have been written to make accessing this data much more convenient.

To determine the correct string to look up a material by, one can use the function nearest_material:

>>> from ht import *
>>> nearest_material('stainless steel')
'Metals, stainless steel'
>>> nearest_material('mineral fibre')
'Mineral fiber'

Knowing a material’s ID, the functions k_material, rho_material, and Cp_material can be used to obtain its properties.

>>> wood = nearest_material('spruce')
>>> k_material(wood)
0.09
>>> rho_material(wood)
400.0
>>> Cp_material(wood)
1630.0

Materials which are refractories, stored in the dictionary refractories, have temperature dependent heat capacity and thermal conductivity between 400 °C and 1200 °C.

>>> C = nearest_material('graphite')
>>> k_material(C)
67.0
>>> k_material(C, T=800)
62.9851975

The limiting values are returned outside of this range:

>>> Cp_material(C, T=8000), Cp_material(C, T=1)
(1588.0, 1108.0)

Radiation

The Stefan-Boltzman law is implemented as q_rad. Optionally, a surrounding temperature may be specified as well. If the surrounding temperature is higher than the object, the calculated heat flux in W/m^2 will be negative, indicating the object is picking up heat not losing it.

>>> q_rad(emissivity=1, T=400)
1451.613952
>>> q_rad(.85, T=400, T2=305.)
816.7821722650002
>>> q_rad(.85, T=400, T2=5000) # ouch
-30122590.815640796

A blackbody’s spectral radiance can also be calculated, in units of W/steradian/square metre/metre. This calculation requires the temperature of the object and the wavelength to be considered.

>>> blackbody_spectral_radiance(T=800., wavelength=4E-6)
1311694129.7430933

Heat exchanger sizing

There are three popular methods of sizing heat exchangers. The log-mean temperature difference correction factor method, the ε-NTU method, and the P-NTU method. Each of those are cannot size a heat exchanger on their own - they do not care about heat transfer coefficients or area - but they must be used first to determine the thermal conditions of the heat exchanger. Sizing a heat exchanger is a very iterative process, and many designs should be attempted to determine the optimal one based on required performance and cost. The P-NTU method supports the most types of heat exchangers; its form always requires the UA term to be guessed however.

LMTD correction factor method

The simplest method, the log-mean temperature difference correction factor method, is as follows:

$$Q = UA\Delta T_{lm} F_t$$

Knowing the outlet and inlet temperatures of a heat exchanger and Q, one could determine UA as follows:

>>> dTlm = LMTD(Tci=15, Tco=85, Thi=130, Tho=110)
>>> Ft = F_LMTD_Fakheri(Tci=15, Tco=85, Thi=130, Tho=110, shells=1)
>>> Q = 1E6 # 1 MW
>>> UA = Q/(dTlm*Ft)
>>> UA
15833.566307803789

This method requires you to know all four temperatures before UA can be calculated. Fakheri developed a general expression for calculating Ft; it is valid for counterflow shell-and-tube exchangers with an even number of tube passes; the number of shell-side passes can be varied. Ft is always less than 1, approaching 1 with very high numbers of shells:

>>> F_LMTD_Fakheri(Tci=15, Tco=85, Thi=130, Tho=110, shells=10)
0.9994785295070708

No other expressions are available to calculate Ft for different heat exchanger geometries; only the TEMA F and E exchanger types are really covered by this expression. However, with results from the other methods, Ft can always be back-calculated.

Log mean temperature are available for both counterflow (by default) and co-current flow. This calculation does not depend on the units of temperature provided.

>>> LMTD(Thi=100, Tho=60, Tci=30, Tco=40.2)
43.200409294131525
>>> LMTD(100, 60, 30, 40.2, counterflow=False)
39.75251118049003

Effectiveness-NTU method

This method uses the formula $Q=\epsilon C_{min}(T_{h,i}-T_{c,i})$. The main complication of this method is calculating effectiveness epsilon, which is a function of the mass flows, heat capacities, and UA $\epsilon = f(NTU, C_r)$. The effectiveness-NTU method is implemented in in effectiveness_from_NTU and NTU_from_effectiveness. The supported heat exchanger types are somewhat limited; they are:

• 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

To illustrate the method, first the individual methods will be used to determine the outlet temperatures of a heat exchanger. After, the more convenient and flexible wrapper effectiveness_NTU_method is shown. 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:

>>> 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, 0.14242142266335814)
>>> NTU, eff, Q
(2.160072595281307, 0.8312180361425988, 131675.32715043944)
>>> Tco, Tho
(110.59007415639887, 116.38592564614977)

That was not very convenient. The more helpful wrapper effectiveness_NTU_method needs only the heat capacities and mass flows of each stream, the type of the heat exchanger, and one combination of the following inputs is required:

• Three of the four inlet and outlet stream temperatures

• 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

• Temperatures for the cold outlet and hot inlet and UA

The function returns all calculated parameters for convenience as a dictionary.

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': 0.2848428453267163,
 'NTU': 1.1040839095588,
 'Q': 192850.0,
 'Tci': 15,
 'Tco': 85,
 'Thi': 130,
 'Tho': 110.06100082712986,
 'UA': 3041.751170834494,
 'effectiveness': 0.6086956521739131}

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': 0.2848428453267163,
 'NTU': 1.1040834845735028,
 'Q': 192849.96310220254,
 'Tci': 15,
 'Tco': 84.99998660697007,
 'Thi': 130,
 'Tho': 110.06100464203861,
 'UA': 3041.75,
 'effectiveness': 0.6086955357127832}