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# Condenser Evaporator (TL-2P)

Models heat exchange between a thermal liquid network and a network that can undergo phase change

• Library:
• Simscape / Fluids / Fluid Network Interfaces / Heat Exchangers

## Description

The Condenser Evaporator (TL-MA) block models a heat exchanger with one thermal liquid network, which flows between ports A1 and B1, and one two-phase fluid network, which flows between ports A2 and B2. The heat exchanger can act as a condenser or as an evaporator. The fluid streams can be aligned in parallel, counter, or cross-flow configurations.

Example Heat Exchanger for Refrigeration Applications

You can model the thermal liquid side as flow within tubes, flow around the two-phase fluid tubing, or by an empirical, generic parameterization.

The block uses the Effectiveness-NTU (E-NTU) method to model heat transfer through the shared wall. Fouling on the exchanger walls, which increases thermal resistance and reduces the heat exchange between the two fluids, is also modeled. You can also optionally model fins on both the thermal liquid and two-phase fluid sides. Pressure loss due to viscous friction on both sides of the exchanger can be modeled analytically or by generic parameterization, which you can use to tune to your own data.

You can model the two-phase fluid side as flow within a tube or a set of tubes. The two-phase fluid tubes use a boundary-following model to track the sub-cooled liquid (L), vapor-liquid mixture (M), and super-heated vapor (V) in three zones. The relative amount of space a zone occupies in the system is called a zone length fraction within the system.

Zone Length Fractions in the Two-Phase Fluid Piping

The sum of the zone length fractions in the two-phase fluid tubing equals `1`. Port Z returns the zone length fractions as a vector of physical signals for each of the three phases: [L, M, V].

### Heat Exchanger Configuration

The heat exchanger effectiveness is based on the selected heat exchanger configuration, the fluid properties in each phase, the tube geometry and flow configuration on each side of the exchanger, and the usage and size of fins.

Flow Arrangement

The Flow arrangement parameter assigns the relative flow paths between the two sides:

• `Parallel flow` indicates the fluids are moving in the same direction.

• `Counter flow` indicates the fluids are moving in parallel, but opposite directions.

• `Cross flow` indicates the fluids are moving perpendicular to each other.

Thermal Mixing

When Flow arrangement is set to ```Cross flow```, use the Cross flow arrangement parameter to indicate whether the two-phase fluid or thermal liquid flows are separated into multiple paths by baffles or walls. Without these separations, the flow can mix freely and is considered mixed. Both fluids, one fluid, or neither fluid can be mixed in the cross-flow arrangement. Mixing homogenizes the fluid temperature along the direction of flow of the second fluid, and varies perpendicular to the second fluid flow.

Unmixed flows vary in temperature both along and perpendicular to the flow path of the second fluid.

Sample Cross-Flow Configurations

Note that the flow direction during simulation does not impact the selected flow arrangement setting. The ports on the block do not reflect the physical positions of the ports in the physical heat exchange system.

All flow arrangements are single-pass, which means that the fluids do not make multiple turns in the exchanger for additional points of heat transfer. To model a multi-pass heat exchanger, you can arrange multiple Condenser Evaporator (TL-2P) blocks in series or in parallel.

For example, to achieve a two-pass configuration on the two-phase fluid side and a single-pass configuration on the thermal liquid side, you can connect the two-phase fluid sides in series and the thermal liquid sides to the same input in parallel (such as two Mass Flow Rate Source blocks with half of the total mass flow rate), as shown below.

Flow Geometry

The Flow geometry parameter sets the thermal liquid flow arrangement as either inside a tube or set of tubes, or perpendicular to a tube bank. You can also specify an empirical, generic configuration. The two-phase fluid always flows inside a tube or set of tubes.

When Flow geometry is set to ```Flow perpendicular to bank of circular tubes```, use the Tube bank grid arrangement parameter to define the two-phase fluid tube bank alignment as either `Inline` or `Staggered`. The red, downward-pointing arrow indicates the direction of thermal liquid flow. The Inline figure also shows the Number of tube rows along flow direction and the Number of tube segments in each tube row parameters. Here, flow direction refers to the thermal liquid flow, and tube refers to the two-phase fluid tubing. The Length of each tube segment in a tube row parameter is indicated in the Staggered figure.

Fins

The heat exchanger configuration is without fins when the Total fin surface area parameter is set to `0 m^2`. Fins introduce additional surface area for additional heat transfer. Each fluid side has a separate fin area.

### Effectiveness-NTU Heat Transfer

The heat transfer rate is calculated for each fluid phase. In accordance with the three fluid zones that occur on the two-phase fluid side of the heat exchanger, the heat transfer rate is calculated in three sections.

The heat transfer in a zone is calculated as:

`${Q}_{zone}=ϵ{C}_{\text{Min}}\left({T}_{\text{In,2P}}-{T}_{\text{In,TL}}\right),$`

where:

• CMin is the lesser of the heat capacity rates of the two fluids in that zone. The heat capacity rate is the product of the fluid specific heat, cp, and the fluid mass flow rate. CMin is always positive.

• TIn,2P is the zone inlet temperature of the two-phase fluid.

• TIn,TL is the zone inlet temperature of the thermal liquid.

• ε is the heat exchanger effectiveness.

Effectiveness is a function of the heat capacity rate and the number of transfer units, NTU, and also varies based on the heat exchanger flow arrangement, which is discussed in more detail in Effectiveness by Flow Arrangement. The NTU is calculated as:

`$NTU=\frac{z}{{C}_{\text{Min}}R},$`

where:

• z is the individual zone length fraction.

• R is the total thermal resistance between the two flows, due to convection, conduction, and any fouling on the tube walls:

`$R=\frac{1}{{U}_{\text{2P}}{A}_{\text{Th,2P}}}+\frac{{F}_{\text{2P}}}{{A}_{\text{Th,2P}}}+{R}_{\text{W}}+\frac{{F}_{\text{TL}}}{{A}_{\text{Th,TL}}}+\frac{1}{{U}_{\text{TL}}{A}_{\text{Th,TL}}},$`

where:

• U is the convective heat transfer coefficient of the respective fluid. This coefficient is discussed in more detail in Two-Phase Fluid Correlations and Thermal Liquid Correlations.

• F is the Fouling factor on the two-phase fluid or thermal liquid side, respectively.

• RW is the Thermal resistance through heat transfer surface.

• ATh is the heat transfer surface area of the respective side of the exchanger. ATh is the sum of the wall surface area, AW, and the Total fin surface area, AF:

`${A}_{\text{Th}}={A}_{\text{W}}+{\eta }_{\text{F}}{A}_{\text{F}},$`

where ηF is the Fin efficiency.

The total heat transfer rate between the fluids is the sum of the heat transferred in the three zones by the subcooled liquid (QL), liquid-vapor mixture (QM), and superheated vapor (QV):

`$Q=\sum {Q}_{\text{Z}}={Q}_{\text{L}}+{Q}_{\text{M}}+{Q}_{\text{V}}.$`

Effectiveness by Flow Arrangement

The heat exchanger effectiveness varies according to its flow configuration and the mixing in each fluid. Below are the formulations for effectiveness calculated in the liquid and vapor zones for each configuration. The effectiveness is $\epsilon =1-\mathrm{exp}\left(-NTU\right)$ for all configurations in the mixture zone.

• When Flow arrangement is set to `Parallel flow`:

`$ϵ=\frac{1-\text{exp}\left[-NTU\left(1+{C}_{\text{R}}\right)\right]}{1+{C}_{\text{R}}}$`

• When Flow arrangement is set to `Counter flow`:

`$ϵ=\frac{1-\text{exp}\left[-NTU\left(1-{C}_{\text{R}}\right)\right]}{1-{C}_{\text{R}}\text{exp}\left[-NTU\left(1-{C}_{\text{R}}\right)\right]}$`

• When Flow arrangement is set to `Cross flow` and Cross flow arrangement is set to ```Both fluids unmixed```:

`$ϵ=1-\text{exp}\left\{\frac{NT{U}^{\text{0}\text{.22}}}{{C}_{\text{R}}}\left[\text{exp}\left(-{C}_{\text{R}}NT{U}^{\text{0}\text{.78}}\right)-1\right]\right\}$`

• When Flow arrangement is set to `Cross flow` and Cross flow arrangement is set to ```Both fluids mixed```:

`$ϵ={\left[\frac{1}{1-\text{exp}\left(-NTU\right)}+\frac{{C}_{\text{R}}}{1-\text{exp}\left(-{C}_{\text{R}}NTU\right)}-\frac{1}{NTU}\right]}^{-1}$`

When one fluid is mixed and the other unmixed, the equation for effectiveness depends on the relative heat capacity rates of the fluids. When Flow arrangement is set to ```Cross flow``` and Cross flow arrangement is set to either ```Thermal Liquid 1 mixed & Two-Phase Fluid 2 unmixed``` or ```Thermal Liquid 1 unmixed & Two-Phase Fluid 2 mixed```:

• When the fluid with Cmax is mixed and the fluid with Cmin is unmixed:

`$ϵ=\frac{1}{{C}_{\text{R}}}\left(1-\text{exp}\left\{-{C}_{R}\left\{1-\mathrm{exp}\left(-NTU\right)\right\}\right\}\right)$`

• When the fluid with Cmin is mixed and the fluid with Cmax is unmixed:

`$ϵ=1-\text{exp}\left\{-\frac{1}{{C}_{\text{R}}}\left[1-\text{exp}\left(-{C}_{\text{R}}NTU\right)\right]\right\}$`

CR denotes the ratio between the heat capacity rates of the two fluids:

`${C}_{\text{R}}=\frac{{C}_{\text{Min}}}{{C}_{\text{Max}}}.$`

### Two-Phase Fluid Correlations

Heat Transfer Coefficient

The convective heat transfer coefficient varies according to the fluid Nusselt number:

`$U=\frac{\text{Nu}k}{{D}_{\text{H}}},$`

where:

• Nu is the zone mean Nusselt number, which depends on the flow regime.

• k is the fluid phase thermal conductivity.

• DH is tube hydraulic diameter.

For turbulent flows in the subcooled liquid or superheated vapor zones, the Nusselt number is calculated with the Gnielinski correlation:

`$\text{Nu}=\frac{\frac{{f}_{D}}{8}\left(\text{Re}-1000\right)\text{Pr}}{1+12.7\sqrt{\frac{f}{8}}\left({\text{Pr}}^{2/3}-1\right)},$`

where:

• Re is the fluid Reynolds number.

• Pr is the fluid Prandtl number.

For turbulent flows in the liquid-vapor mixture zone, the Nusselt number is calculated with the Cavallini-Zecchin correlation:

`$\text{Nu}=\frac{{\text{aRe}}_{\text{SL}}^{b}{\text{Pr}}_{\text{SL}}^{c}\left\{{\left[\left(\sqrt{\frac{{\rho }_{\text{SL}}}{{\rho }_{\text{SV}}}}-1\right){x}_{\text{Out}}+1\right]}^{1+b}-{\left[\left(\sqrt{\frac{{\rho }_{\text{SL}}}{{\rho }_{\text{SV}}}}-1\right){x}_{\text{In}}+1\right]}^{1+b}\right\}}{\left(1+b\right)\left(\sqrt{\frac{{\rho }_{\text{SL}}}{{\rho }_{\text{SV}}}}-1\right)\left({x}_{\text{Out}}-{x}_{\text{In}}\right)}.$`

where:

• ReSL is the Reynolds number of the saturated liquid.

• PrSL is the Prandtl number of the saturated liquid.

• ρSL is the density of the saturated liquid.

• ρSV is the density of the saturated vapor.

• a= 0.05, b = 0.8, and c= 0.33.

For laminar flows, the Nusselt number is set by the Laminar flow Nusselt number parameter.

For transitional flows, the Nusselt number is a blend between the laminar and turbulent Nusselt numbers.

Empirical Nusselt Number Formulation

When the Heat transfer coefficient model parameter is set to `Colburn equation`, the Nusselt number for the subcooled liquid and superheated vapor zones is calculated by the empirical the Colburn equation:

`$\text{Nu}=a{\text{Re}}^{b}{\text{Pr}}^{c},$`

where a, b, and c are defined in the Coefficients [a, b, c] for a*Re^b*Pr^c in liquid zone and Coefficients [a, b, c] for a*Re^b*Pr^c in vapor zone parameters.

The Nusselt number for liquid-vapor mixture zones is calculated with the Cavallini-Zecchin equation, with the coefficients specified in the Coefficients [a, b, c] for a*Re^b*Pr^c in mixture zone parameter.

Pressure Loss

The pressure loss due to viscous friction varies depending on flow regime and configuration. The calculation uses the overall density, which is the total two-phase fluid mass divided by the total two-phase fluid volume.

For turbulent flows, when the Reynolds number is above the Turbulent flow lower Reynolds number limit, the pressure loss due to friction is calculated in terms of the Darcy friction factor. The pressure differential between port A2 and the internal node I2 is:

`${p}_{\text{A2}}-{p}_{\text{I2}}=\frac{{f}_{\text{D,A}}{\stackrel{˙}{m}}_{\text{A2}}|{\stackrel{˙}{m}}_{\text{A2}}|}{2\rho {D}_{\text{H}}{A}_{\text{CS}}^{2}}\left(\frac{L+{L}_{\text{Add}}}{2}\right),$`

where:

• $\stackrel{˙}{m}$A2 is the total flow rate through port A2.

• fD,A is the Darcy friction factor, according to the Haaland correlation:

`${f}_{\text{D,A2}}={\left\{-1.8{\text{log}}_{\text{10}}\left[\frac{6.9}{{\text{Re}}_{\text{A2}}}+{\left(\frac{{ϵ}_{\text{R}}}{3.7{D}_{\text{H}}}\right)}^{1.11}\right]\right\}}^{\text{-2}},$`

where εR is the two-phase fluid pipe Internal surface absolute roughness. Note that the friction factor is dependent on the Reynolds number, and is calculated at both ports for each liquid.

• L is the Total length of each tube on the two-phase fluid side.

• LAdd is the two-phase fluid Aggregate equivalent length of local resistances, which is the equivalent length of a tube that introduces the same amount of loss as the sum of the losses due to other local resistances in the tube.

• ACS is the total tube cross-sectional area.

The pressure differential between port B2 and internal node I2 is:

`${p}_{\text{B2}}-{p}_{\text{I2}}=\frac{{f}_{\text{D,B}}{\stackrel{˙}{m}}_{\text{B2}}|{\stackrel{˙}{m}}_{\text{B2}}|}{2\rho {D}_{\text{H}}{A}_{\text{CS}}^{2}}\left(\frac{L+{L}_{\text{Add}}}{2}\right),$`

where $\stackrel{˙}{m}$B2 is the total flow rate through port B2.

The Darcy friction factor at port B2 is:

`${f}_{\text{D,B2}}={\left\{-1.8{\text{log}}_{\text{10}}\left[\frac{6.9}{{\text{Re}}_{\text{B2}}}+{\left(\frac{{ϵ}_{\text{R}}}{3.7{D}_{\text{H}}}\right)}^{1.11}\right]\right\}}^{\text{-2}}.$`

For laminar flows, when the Reynolds number is below the Laminar flow upper Reynolds number limit, the pressure loss due to friction is calculated in terms of the Laminar friction constant for Darcy friction factor, λ. λ is a user-defined parameter when Tube cross-section is set to `Generic`, otherwise, the value is calculated internally. The pressure differential between port A2 and internal node I2 is:

`${p}_{\text{A2}}-{p}_{\text{I2}}=\frac{\lambda \mu {\stackrel{˙}{m}}_{\text{A2}}}{2\rho {D}_{\text{H}}^{2}{A}_{CS}}\left(\frac{L+{L}_{\text{Add}}}{2}\right),$`

where μ is the two-phase fluid dynamic viscosity. The pressure differential between port B2 and internal node I2 is:

`${p}_{\text{B2}}-{p}_{\text{I2}}=\frac{\lambda \mu {\stackrel{˙}{m}}_{\text{B2}}}{2\rho {D}_{\text{H}}^{2}{A}_{CS}}\left(\frac{L+{L}_{\text{Add}}}{2}\right).$`

For transitional flows, the pressure differential due to viscous friction is a smoothed blend between the values for laminar and turbulent pressure losses.

Empirical Pressure Loss Formulation

When Pressure loss model is set to ```Pressure loss coefficient```, the pressure losses due to viscous friction are calculated with an empirical pressure loss coefficient, ξ.

The pressure differential between port A2 and internal node I2 is:

`${p}_{\text{A2}}-{p}_{\text{I2}}=\frac{1}{2}\xi \frac{{\stackrel{˙}{m}}_{\text{A2}}|{\stackrel{˙}{m}}_{\text{A2}}|}{2\rho {A}_{\text{CS}}^{2}}.$`

The pressure differential between port B2 and internal node I2 is:

`${p}_{\text{B2}}-{p}_{\text{I2}}=\frac{1}{2}\xi \frac{{\stackrel{˙}{m}}_{\text{B2}}|{\stackrel{˙}{m}}_{\text{B2}}|}{2\rho {A}_{\text{CS}}^{2}}.$`

### Thermal Liquid Correlations

Heat Transfer Coefficient for Flows Inside One or More Tubes

When the thermal liquid Flow geometry is set to `Flow inside one or more tubes`, the Nusselt number is calculated according to the Gnielinski correlation in the same manner as two-phase subcooled liquid or superheated vapor. See Heat Transfer Coefficient for more information.

Heat Transfer Coefficient for Flows Across a Tube Bank

When the thermal liquid Flow geometry is set to `Flow perpendicular to bank of circular tubes`, the Nusselt number is calculated based on the Hagen number, Hg, and depends on the Tube bank grid arrangement setting:

`$\text{Nu}=\left\{\begin{array}{cc}0.404L{q}^{\text{1/3}}{\left(\frac{\text{Re}+1}{\text{Re}+1000}\right)}^{0.1},& Inline\\ 0.404L{q}^{1/3},& Staggered\end{array}$`

where:

`$Lq=\left\{\begin{array}{cc}1.18\text{Pr}\left(\frac{4{l}_{\text{T}}/\pi -D}{{l}_{\text{L}}}\right)\text{Hg}\left(\text{Re}\right),& Inline\\ 0.92\text{Pr}\left(\frac{4{l}_{\text{T}}/\pi -D}{{l}_{\text{D}}}\right)\text{Hg}\left(\text{Re}\right),& Staggeredwith{l}_{L}\ge D\\ 0.92\text{Pr}\left(\frac{4{l}_{\text{T}}{l}_{\text{L}}/\pi -{D}^{2}}{{l}_{\text{L}}{l}_{\text{D}}}\right)\text{Hg}\left(\text{Re}\right),& Staggeredwith{l}_{L}`

• D is the Tube outer diameter.

• lL is the Longitudinal tube pitch (along flow direction), the distance between the tube centers along the flow direction. Flow direction refers to the thermal liquid flow.

• lT is the Transverse tube pitch (perpendicular to flow direction), shown in the figure below. The transverse pitch is the distance between the centers of the two-phase fluid tubing in one row.

• lD is the diagonal tube spacing, calculated as ${l}_{\text{D}}=\sqrt{{\left(\frac{{l}_{\text{T}}}{2}\right)}^{2}+{l}_{\text{L}}^{2}}.$

The longitudinal and transverse pitch distances are the same for both grid bank arrangement types.

Cross-Section of Two-Phase Fluid Tubing with Pitch Measurements

Empirical Nusselt Number Forumulation

When the Heat transfer coefficient model is set to `Colburn equation` or when Flow geometry is set to `Generic`, the Nusselt number is calculated by the empirical the Colburn equation:

`$\text{Nu}=a{\text{Re}}^{b}{\text{Pr}}^{c},$`

where a, b, and c are the values defined in the Coefficients [a, b, c] for a*Re^b*Pr^c parameter.

Pressure Loss for Flow Inside Tubes

When the thermal liquid Flow geometry is set to `Flow inside one or more tubes`, the pressure loss is calculated in the same manner as for two-phase flows, with the respective Darcy friction factor, density, mass flow rates, and pipe lengths of the thermal liquid side. See Pressure Loss for more information.

Pressure Loss for Flow Across Tube Banks

When the thermal liquid Flow geometry is set to `Flow perpendicular to bank of circular tubes`, the Hagen number is used to calculate the pressure loss due to viscous friction. The pressure differential between port A1 and internal node I1 is:

`${p}_{\text{A1}}-{p}_{\text{I1}}=\frac{1}{2}\frac{{\mu }^{2}{N}_{\text{R}}}{\rho {D}^{2}}\text{Hg}\left(\text{Re}\right),$`

where:

• μTL is the fluid dynamic viscosity.

• NR is the Number of tube rows along flow direction. This is the number of two-phase fluid tube rows along the thermal liquid flow direction.

The pressure differential between port B1 and internal node I1 is:

`${p}_{\text{B1}}-{p}_{\text{I1}}=\frac{1}{2}\frac{{\mu }^{2}{N}_{\text{R}}}{\rho {D}^{2}}\text{Hg}\left(\text{Re}\right).$`

Empirical Pressure Loss Formulation

When the Pressure loss model is set to ```Euler number per tube row``` or when Flow geometry is set to `Generic`, the pressure loss due to viscous friction is calculated with a pressure loss coefficient, in terms of the Euler number, Eu:

`$\text{Eu}=\frac{\xi }{{N}_{R}},$`

where ξ is the empirical pressure loss coefficient.

The pressure differential between port A1 and internal node I1 is:

`${p}_{\text{A1}}-{p}_{\text{I1}}=\frac{1}{2}{N}_{R}Eu\frac{{\stackrel{˙}{m}}_{\text{A1}}|{\stackrel{˙}{m}}_{\text{A1}}|}{2\rho {A}_{\text{CS}}^{2}}.$`

The pressure differential between port B1 and internal node I1 is:

`${p}_{\text{B1}}-{p}_{\text{I1}}=\frac{1}{2}{N}_{R}Eu\frac{{\stackrel{˙}{m}}_{\text{B1}}|{\stackrel{˙}{m}}_{\text{B1}}|}{2\rho {A}_{\text{CS}}^{2}}.$`

### Conservation Equations

Two-Phase Fluid

The total mass accumulation rate in the two-phase fluid is defined as:

`$\frac{d{M}_{\text{2P}}}{dt}={\stackrel{˙}{m}}_{\text{A2}}+{\stackrel{˙}{m}}_{\text{B2}},$`

where:

• M2P is the total mass of the two-phase fluid.

• $\stackrel{˙}{m}$A2 is the mass flow rate of the fluid at port A2.

• $\stackrel{˙}{m}$B2 is the mass flow rate of the fluid at port B2.

The flow is positive when flowing into the block through the port.

The energy conservation equation relates the change in specific internal energy to the heat transfer by the fluid:

`${M}_{2P}\frac{d{u}_{2P}}{dt}+{u}_{2P}\left({\stackrel{˙}{m}}_{A2}+{\stackrel{˙}{m}}_{B2}\right)={\varphi }_{\text{A2}}+{\varphi }_{\text{B2}}-Q,$`

where:

• u2P is the two-phase fluid specific internal energy.

• φA2 is the energy flow rate at port A2.

• φB2 is the energy flow rate at port B2.

• Q is heat transfer rate, which is positive when leaving the two-phase fluid volume.

Thermal Liquid

The total mass accumulation rate in the thermal liquid is defined as:

`$\frac{d{M}_{\text{TL}}}{dt}={\stackrel{˙}{m}}_{\text{A1}}+{\stackrel{˙}{m}}_{\text{B1}}.$`

The energy conservation equation is:

`${M}_{TL}\frac{d{u}_{TL}}{dt}+{u}_{TL}\left({\stackrel{˙}{m}}_{A1}+{\stackrel{˙}{m}}_{B1}\right)={\varphi }_{\text{A1}}+{\varphi }_{\text{B1}}+Q,$`

where:

• ϕA1 is the energy flow rate at port A1.

• ϕB1 is the energy flow rate at port B1.

The heat transferred to or from the thermal liquid, Q, is equal to the heat transferred from or to the two-phase fluid.

## Ports

### Conserving

expand all

Inlet or outlet port associated with the thermal liquid.

Inlet or outlet port associated with the thermal liquid.

Inlet or outlet port associated with the two-phase fluid.

Inlet or outlet port associated with the two-phase fluid.

### Output

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Three-element vector of the zone length fractions in the two-phase fluid channel, returned as a physical signal. The vector takes the form [L, M, V], where L is the sub-cooled liquid, M is the liquid-vapor mixture, and V is the superheated vapor.

## Parameters

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### Configuration

Flow path alignment between heat exchanger sides. The available flow arrangements are:

• `Parallel flow`. The flows run in the same direction.

• `Counter flow`. The flows run parallel to each other, in the opposite directions.

• `Cross flow`. The flows run perpendicular to each other.

Select whether each of the fluids can mix in its channel. Mixed flow means that the fluid is free to move in the transverse direction as it travels along the flow path. Unmixed flow means that the fluid is restricted to travel only along the flow path. For example, a side with fins is considered an unmixed flow.

#### Dependencies

To enable this parameter, set Flow arrangement to ```Cross flow```.

Thermal resistance of the wall separating the two sides of the heat exchanger. The wall thermal resistance, wall fouling, and the fluid convective heat transfer coefficient influence the amount of heat transferred between the flows.

Flow area at the thermal liquid port A1.

Flow area at the thermal liquid port B1.

Flow area at the two-phase fluid port A2.

Flow area at the two-phase fluid port B2.

### Thermal Liquid 1

Thermal liquid flow path. The flow can run externally over a set of tubes or internal to a tube or set of tubes. You can also specify a generic parameterization based on empirical values.

Number of thermal liquid tubes. More tubes result in higher pressure losses due to viscous friction, but a larger amount of surface area for heat transfer.

#### Dependencies

To enable this parameter, set Flow geometry to `Flow inside one or more tubes`.

Total length of each thermal liquid tube.

#### Dependencies

To enable this parameter, set Flow geometry to `Flow inside one or more tubes`.

Cross-sectional shape of one tube. Set to `Generic` to specify an arbitrary cross-sectional geometry.

#### Dependencies

To enable this parameter, set Flow geometry to `Flow inside one or more tubes`.

Internal diameter of the cross-section of one tube. The cross-section and diameter are uniform along the tube. The size of the diameter influences the pressure loss and heat transfer calculations.

#### Dependencies

To enable this parameter, set Flow geometry to `Flow inside one or more tubes` and Tube cross-section to `Circular`.

Internal width of the cross-section of one tube. The cross-section and width are uniform along the tube. The width and height influence the pressure loss and heat transfer calculations.

#### Dependencies

To enable this parameter, set Flow geometry to `Flow inside one or more tubes` and Tube cross-section to `Rectangular`.

Internal height of one tube. The cross-section and height are uniform along the tube. The width and height influence the pressure loss and heat transfer calculations.

#### Dependencies

To enable this parameter, set Flow geometry parameterization of ```Flow inside one or more tubes``` and Tube cross-section to `Rectangular`.

Smaller diameter of the annular cross-section of one tube. The cross-section and inner diameter are uniform along the tube. The inner diameter influences the pressure loss and heat transfer calculations. Heat transfer occurs through the inner surface of the annulus.

#### Dependencies

To enable this parameter, set Flow geometry parameterization of ```Flow inside one or more tubes``` and Tube cross-section to `Annular`.

Larger diameter of the annular cross-section of one tube. The cross-section and outer diameter are uniform along the tube. The outer diameter influences the pressure loss and heat transfer calculations.

#### Dependencies

To enable this parameter, set Flow geometry to `Flow inside one or more tubes` and Tube cross-section to `Annular`.

Internal flow area of each tube.

#### Dependencies

To enable this parameter, set Flow geometry to `Flow inside one or more tubes` and Tube cross-section to `Generic`.

Perimeter of the tube cross-section that the fluid touches. The cross-section and perimeter are uniform along the tube. This value is applied in pressure loss calculations.

#### Dependencies

To enable this parameter, set Flow geometry to `Flow inside one or more tubes` and Tube cross-section to `Generic`.

Tube perimeter for heat transfer calculations. This is often the same as the tube perimeter, but in cases such as the annular cross-section, this may be only the inner or outer diameter, depending on the heat-transferring surface. The cross-section and tube perimeter are uniform along the tube.

#### Dependencies

To enable this parameter, set Flow geometry to `Flow inside one or more tubes` and Tube cross-section to `Generic`.

Method of pressure loss calculation due to viscous friction. Different models are available for different flow configurations. The settings are:

• ```Correlation for flow inside tubes```. Use this setting to calculate the pressure loss with the Haaland correlation.

• `Pressure loss coefficient`. Use this setting to calculate the pressure loss based on an empirical loss coefficient.

• `Euler number per tube row`. Use this setting to calculate the pressure loss based on an empirical Euler number.

• ```Correlation for flow over tube bank```. Use this setting to calculate the pressure loss based on the Hagen number.

The pressure loss models available depend on the Flow geometry setting.

#### Dependencies

When Flow geometry is set to `Flow inside one or more tubes`, Pressure loss model can be set to either:

• ```Pressure loss coefficient```.

• ```Correlation for flow inside tubes```.

When Flow geometry is set to ```Flow perpendicular to bank of circular tubes```, Pressure loss model can be set to either:

• ```Correlation for flow over tube bank```.

• ```Euler number per tube row```.

When Flow geometry is set to `Generic`, the Pressure loss model parameter is disabled. Pressure loss is calculated empirically with the Pressure loss coefficient, delta_p/(0.5*rho*v^2) parameter.

Empirical loss coefficient for all pressure losses in the channel. This value accounts for wall friction and minor losses due to bends, elbows, and other geometry changes in the channel.

The loss coefficient can be calculated from a nominal operating condition or be tuned to fit experimental data. It is defined as:

`$\xi =\frac{\Delta p}{\frac{1}{2}\rho {v}^{2}},$`

where Δp is the pressure drop, ρ is the thermal liquid density, and v is the flow velocity.

#### Dependencies

To enable this parameter, set either:

• Flow geometry to ```Flow inside one or more tubes``` and Pressure loss model to ```Pressure loss coefficient```.

• Flow geometry to `Generic`.

Combined length of all local resistances in the tubes. This is the length of tubing that results in the same pressure losses as the sum of all minor losses in the tube due to such things as bends, tees, or unions. A longer equivalent length results in larger pressure losses.

#### Dependencies

To enable this parameter, set Flow geometry to `Flow inside one or more tubes` and Pressure loss model to ```Correlations for flow inside tubes```.

Mean height of tube surface defects. A rougher wall results in larger pressure losses in the turbulent regime for pressure loss calculated with the Haaland correlation.

#### Dependencies

To enable this parameter, set Flow geometry to `Flow inside one or more tubes` and either:

• Pressure loss model

• Heat transfer coefficient model

to ```Correlation for flow inside tubes```.

Largest Reynolds number that indicates laminar flow. Between this value and the Turbulent flow lower Reynolds number limit, the flow regime is transitional.

#### Dependencies

To enable this parameter, set Flow geometry to `Flow inside one or more tubes` and Pressure loss model to ```Correlation for flow inside tubes```.

Smallest Reynolds number that indicates turbulent flow. Between this value and the Laminar flow upper Reynolds number limit, the flow regime is transitional between the laminar and turbulent regimes.

#### Dependencies

To enable this parameter, set Flow geometry to `Flow inside one or more tubes` and Pressure loss model to ```Correlation for flow inside tubes```.

Coefficient in pressure loss equations for viscous friction in laminar flows. This parameter is also known as the shape factor. The default value corresponds to a circular tube cross-section.

#### Dependencies

To enable this parameter, set Flow geometry to `Correlation for flow inside tubes`, Tube cross section to `Generic`, and Pressure loss model to ```Correlation for flow inside tubes```.

Method of calculating the heat transfer coefficient between the fluid and the wall. The available settings are:

• `Colburn equation`. Use this setting to calculate the heat transfer coefficient with user-defined variables a, b, and c of the Colburn equation.

• ```Correlation for flow over tube bank```. Use this setting to calculate the heat transfer coefficient based on the tube bank correlation using the Hagen number.

• ```Correlation for flow inside tubes```. Use this setting to calculate the heat transfer coefficient for pipe flows with the Gnielinski correlation.

#### Dependencies

To enable this parameter, set Flow geometry to either:

• ```Flow perpendicular to bank of circular tubes```.

• ```Flow inside one or more tubes```.

Three-element vector containing the empirical coefficients of the Colburn equation. The Colburn equation is a formulation for calculating the Nusselt Number. The general form of the Colburn equation is:

`$\text{Nu}=a{\text{Re}}^{b}{\text{Pr}}^{c}.$`

When the Heat transfer coefficient model is set to `Colburn equation` and Flow geometry is set to ```Flow inside one or more tubes```, or Flow geometry is set to `Generic`, the default Colburn equation is:

`$\text{Nu}=0.023{\text{Re}}^{0.8}{\text{Pr}}^{1/3}.$`

When the Heat transfer coefficient model is set to ```Colburn equation``` and Flow geometry is set to ```Flow perpendicular to bank of circular tubes```, the default Colburn equation is:

`$\text{Nu}=0.27{\text{Re}}^{0.63}{\text{Pr}}^{0.36}.$`

#### Dependencies

To enable this parameter, set:

1. Flow geometry to either:

• ```Flow inside one or more tubes```

• ```Flow perpendicular to bank of circular tubes```

and Heat transfer coefficient model to ```Colburn equation```.

2. Flow geometry to `Generic`.

Ratio of convective to conductive heat transfer in the laminar flow regime. The fluid Nusselt number influences the heat transfer rate and depends on the tube cross-section.

#### Dependencies

To enable this parameter, set Flow geometry to `Flow inside one or more tubes`, Tube cross-section to `Generic`, and Heat transfer parameterization to ```Correlation for flow inside tubes```.

Alignment of tubes in a tube bank. Rows are either in line with their neighbors, or staggered.

• `Inline`: All tube rows are located directly behind each other.

• `Staggered`: Tubes of the one tube row are located at the gap between tubes of the previous tube row.

Tube alignment influences the Nusselt number and the heat transfer rate.

#### Dependencies

To enable this parameter, set Flow geometry to ```Flow perpendicular to bank of circular tubes```.

Number of two-phase fluid tube rows in a tube bank. The rows are aligned with the direction of thermal liquid flow.

#### Dependencies

To enable this parameter, set Flow geometryto ```Flow perpendicular to bank of circular tubes```.

Number of two-phase fluid tubes in each row of a tube bank. This measurement is perpendicular to the thermal liquid flow.

#### Dependencies

To enable this parameter, set Flow geometry to ```Flow perpendicular to bank of circular tubes```.

Length of each tube that spans a tube row. All tubes in a tube bank are the same length.

#### Dependencies

To enable this parameter, set Flow geometry to ```Flow perpendicular to bank of circular tubes```.

Outer diameter of a two-phase fluid tube. The cross-section is uniform along a tube and so the diameter is constant throughout. This value influences the losses in the flow across a tube bank due to viscous friction.

#### Dependencies

To enable this parameter, set Flow geometry to ```Flow perpendicular to bank of circular tubes```.

Distance between tube centers of the two-phase fluid tubes, aligned with the direction of flow of the thermal liquid.

#### Dependencies

To enable this parameter, set Flow geometry to ```Flow perpendicular to bank of circular tubes```.

Distance between the tube centers in a row of two-phase fluid tubes. This measurement is perpendicular to the thermal liquid flow direction. See Heat Transfer Coefficient for Flows Across a Tube Bank for more information.

#### Dependencies

To enable this parameter, set Flow geometry to ```Flow perpendicular to bank of circular tubes```.

Empirical coefficient for pressure drop across one tube row. The Euler number is the ratio between pressure drop and fluid momentum:

`$\text{Eu}=\frac{\Delta p}{N\frac{1}{2}\rho {v}^{2}},$`

where N is the Number of tube rows along flow direction, Δp is the pressure drop, ρ is the thermal liquid mixture density, and v is the flow velocity.

Each tube row is located in a plane perpendicular to the thermal liquid flow.

#### Dependencies

To enable this parameter, set Flow geometry to ```Flow perpendicular to bank of circular tubes``` and Pressure loss model to `Euler number per tube row`.

Smallest total flow area between inlet and outlet. If the channel is a collection of ducts, tubes, slots, or grooves, the minimum free-flow area is the sum of the smallest areas.

#### Dependencies

To enable this parameter, set Flow geometry to `Generic`.

Total area of the heat transfer surface, excluding fins.

#### Dependencies

To enable this parameter, set Flow geometry to `Generic`.

Total volume of thermal liquid in the heat exchanger.

#### Dependencies

To enable this parameter, set Flow geometry to `Generic`.

Additional thermal resistance due to fouling layers on the surfaces of the wall. In real systems, fouling deposits grow over time. However, the growth is slow enough to be assumed constant during the simulation.

Total heat transfer surface area of both sides of all fins. For example, if the fin is rectangular, the surface area is double the area of the rectangle.

The total heat transfer surface area is the sum of the channel surface area and the effective fin surface area, which is the product of the Fin efficiency and the Total fin surface area.

Ratio of actual heat transfer to ideal heat transfer through the fins.

Thermal liquid pressure at the start of the simulation.

Temperature in the thermal liquid channel at the start of the simulation. This parameter can be a scalar or a two-element vector. A scalar value represents the mean initial temperature in the channel. A vector value represents the initial temperature at the inlet and outlet in the form [`inlet`, `outlet`]. The block calculates a linear gradient between the two ports. The inlet and the outlet ports are identified according to the initial flow direction.

### Two-Phase Fluid 2

Number of two-phase fluid tubes.

Total length of each two-phase fluid tube.

Cross-sectional shape of a tube. Use `Generic` to specify an arbitrary cross-sectional geometry.

Internal diameter of the cross-section of one tube. The cross-section and diameter are uniform along the tube. The size of the diameter influences the pressure loss and heat transfer calculations.

#### Dependencies

To enable this parameter, set Tube cross-section to `Circular`.

Internal width of the cross-section of one tube. The cross-section and width are uniform along the tube. The width and height influence the pressure loss and heat transfer calculations.

#### Dependencies

To enable this parameter, set Tube cross-section to `Rectangular`.

Internal height of one tube. The cross-section and height are uniform along the tube. The width and height influence the pressure loss and heat transfer calculations.

#### Dependencies

To enable this parameter, set Tube cross-section to `Rectangular`.

Smaller diameter of the annular cross-section of one tube. The cross-section and inner diameter are uniform along the tube. The inner diameter influences the pressure loss and heat transfer calculations. Heat transfer occurs through the inner surface of the annulus.

#### Dependencies

To enable this parameter, set Tube cross-section to `Annular`.

Larger diameter of the annular cross-section of one tube. The cross-section and outer diameter are uniform along the tube. The outer diameter influences the pressure loss and heat transfer calculations.

#### Dependencies

To enable this parameter, set Tube cross-section to `Annular`.

Internal flow area of each tube.

#### Dependencies

To enable this parameter, set Tube cross-section to `Generic`.

Perimeter of the tube cross-section that the fluid touches. The cross-section and perimeter are uniform along the tube. This value is applied in pressure loss calculations.

#### Dependencies

To enable this parameter, set Tube cross-section to `Generic`.

Tube perimeter for heat transfer calculations. This is often the same as the tube perimeter, but in cases such as the annular cross-section, this may be only the inner or outer diameter, depending on the heat-transferring surface. The cross-section and tube perimeter are uniform along the tube.

#### Dependencies

To enable this parameter, set Tube cross-section to `Generic`.

Method of pressure loss calculation due to viscous friction. The settings are:

• `Pressure loss coefficient`. Use this setting to calculate the pressure loss based on an empirical loss coefficient.

• ```Correlation for flow inside tubes```. Use this setting to calculate the pressure loss based on the pipe flow correlation.

Empirical loss coefficient for all pressure losses in the channel. This value accounts for wall friction and minor losses due to bends, elbows, and other geometry changes in the channel.

The loss coefficient can be calculated from a nominal operating condition or be tuned to fit experimental data. It is defined as:

`$\xi =\frac{\Delta p}{\frac{1}{2}\rho {v}^{2}},$`

where Δp is the pressure drop, ρ is the two-phase fluid density, and v is the flow velocity.

#### Dependencies

To enable this parameter, set Pressure loss model to ```Pressure loss coefficient```.

Combined length of all local resistances in the tubes. This is the length of tubing that results in the same pressure losses as the sum of all minor losses in the tube due to such things as bends, tees, or unions. A longer equivalent length results in larger pressure losses.

#### Dependencies

To enable this parameter, set Pressure loss model to ```Correlation for flow inside tubes```.

Mean height of tube surface defects. A rougher wall results in larger pressure losses in the turbulent regime for pressure loss calculated with the Haaland correlation.

#### Dependencies

To enable this parameter, set either:

• Pressure loss model

• Heat transfer coefficient model

to ```Correlation for flow inside tubes```.

Largest Reynolds number that indicates laminar flow. Between this value and the Turbulent flow lower Reynolds number, the flow regime is transitional.

#### Dependencies

To enable this parameter, set Pressure loss model to ```Correlations for tubes```.

Smallest Reynolds number that indicates turbulent flow. Between this value and the Laminar flow upper Reynolds number limit, the flow regime is transitional.

#### Dependencies

To enable this parameter, set Pressure loss model to ```Correlations for tubes```.

Coefficient in pressure loss equations for viscous friction in laminar flows. This parameter may also be known as the shape factor. The default value corresponds to a circular tube cross-section.

#### Dependencies

To enable this parameter, set Pressure loss model to ```Correlations for tubes```.

Method of calculating the heat transfer coefficient between the fluid and the wall. The available settings are:

• `Colburn equation`. Use this setting to calculate the heat transfer coefficient with user-defined variables a, b, and c. In the liquid and vapor zones, the heat transfer coefficient is based on the Colburn equation. In the liquid-vapor mixture zone, the heat transfer coefficient is based on the Cavallini-Zecchin equation.

• ```Correlation for flow inside tubes```. Use this setting to calculate the heat transfer coefficient for pipe flows. In the liquid and vapor zones, the heat transfer coefficient is calculated with the Gnielinski correlation. In the liquid-vapor mixture zone, the heat transfer coefficient is calculated with the Cavallini-Zecchin equation.

Three-element vector containing the empirical coefficients of the Colburn equation. Each fluid zone has a distinct Nusselt number, which is calculated by the Colburn equation per zone. The general form of the Colburn equation is:

`$\text{Nu}=a{\text{Re}}^{b}{\text{Pr}}^{c}.$`

#### Dependencies

To enable this parameter, set Heat transfer coefficient model to ```Colburn equation```.

Three-element vector containing the empirical coefficients of the Cavallini-Zecchin equation. Each fluid zone has a distinct Nusselt number, which is calculated in the mixture zone by the Cavallini-Zecchin equation:

`$\text{Nu}=\frac{{\text{aRe}}_{\text{SL}}^{b}{\text{Pr}}_{\text{SL}}^{c}\left\{{\left[\left(\sqrt{\frac{{\rho }_{\text{SL}}}{{\rho }_{\text{SV}}}}-1\right){x}_{\text{Out}}+1\right]}^{1+b}-{\left[\left(\sqrt{\frac{{\rho }_{\text{SL}}}{{\rho }_{\text{SV}}}}-1\right){x}_{\text{In}}+1\right]}^{1+b}\right\}}{\left(1+b\right)\left(\sqrt{\frac{{\rho }_{\text{SL}}}{{\rho }_{\text{SV}}}}-1\right)\left({x}_{\text{Out}}-{x}_{\text{In}}\right)}.$`

#### Dependencies

To enable this parameter, set Heat transfer coefficient model to ```Colburn equation```.

Three-element vector containing the empirical coefficients of the Colburn equation. Each fluid zone has a distinct Nusselt number, which is calculated by the Colburn equation per zone. The general form of the Colburn equation is:

`$\text{Nu}=a{\text{Re}}^{b}{\text{Pr}}^{c}.$`

#### Dependencies

To enable this parameter, set Heat transfer coefficient model to ```Colburn equation```.

Ratio of convective to conductive heat transfer in the laminar flow regime. The fluid Nusselt number influences the heat transfer rate and depends on the tube cross-section.

#### Dependencies

To enable this parameter, set Heat transfer coefficient model to ```Correlation for flow inside tubes```.

Additional thermal resistance due to fouling layers on the surfaces of the wall. In real systems, fouling deposits grow over time, however, the growth is slow enough to be assumed constant during the simulation.

Total heat transfer surface area of both sides of all fins. For example, if the fin is rectangular, the surface area is double the area of the rectangle.

The total heat transfer surface area is the sum of the channel surface area and the effective fin surface area, which is the product of the Fin efficiency and the Total fin surface area.

Ratio of actual heat transfer to ideal heat transfer through the fins.

Quantity used to describe the initial state of the fluid: temperature, vapor quality, vapor void fraction, specific enthalpy, or specific internal energy.

Fluid pressure at the start of the simulation.

Temperature in the two-phase fluid channel at the start of simulation. This parameter can be a scalar or a two-element vector. A scalar value represents the mean initial temperature in the channel. A vector value represents the initial temperature at the inlet and outlet in the form [`inlet`, `outlet`]. The block calculates a linear gradient between the two ports. The inlet and the outlet ports are identified according to the initial flow direction.

#### Dependencies

To enable this parameter, set Initial fluid energy specification to `Temperature`.

Vapor mass fraction in the two-phase fluid channel at the start of simulation. This parameter can be a scalar or a two-element vector. A scalar value represents the mean initial vapor quality in the channel. A vector value represents the initial vapor quality at the inlet and outlet in the form [`inlet`, `outlet`]. The block calculates a linear gradient between the two ports. The inlet and the outlet ports are identified according to the initial flow direction.

#### Dependencies

To enable this parameter, set Initial fluid energy specification to ```Vapor quality```.

Vapor volume fraction in the two-phase fluid channel at the start of simulation. This parameter can be a scalar or a two-element vector. A scalar value represents the mean initial void fraction in the channel. A vector value represents the initial void fraction at the inlet and outlet in the form [`inlet`, `outlet`]. The block calculates a linear gradient between the two ports. The inlet and the outlet ports are identified according to the initial flow direction.

#### Dependencies

To enable this parameter, set Initial fluid energy specification to ```Vapor void fraction```.

Enthalpy per unit mass in the two-phase fluid channel at the start of simulation. This parameter can be a scalar or a two-element vector. A scalar value represents the mean initial specific enthalpy in the channel. A vector value represents the initial specific enthalpy at the inlet and outlet in the form [`inlet`, `outlet`]. The block calculates a linear gradient between the two ports. The inlet and the outlet ports are identified according to the initial flow direction.

#### Dependencies

To enable this parameter, set Initial fluid energy specification to ```Specific enthalpy```.

Internal energy per unit mass in the two-phase fluid channel at the start of simulation. This parameter can be a scalar or a two-element vector. A scalar value represents the mean initial specific internal energy in the channel. A vector value represents the initial specific internal energy at the inlet and outlet in the form [`inlet`, `outlet`]. The block calculates a linear gradient between the two ports. The inlet and the outlet ports are identified according to the initial flow direction.

#### Dependencies

To enable this parameter, set Initial fluid energy specification to ```Specific internal energy```.

## References

[1] 2013 ASHRAE Handbook - Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., 2013.

[2] Çengel, Yunus A. Heat and Mass Transfer: A Practical Approach. 3rd ed, McGraw-Hill, 2007.

[3] Shah, R. K., and Dušan P. Sekulić. Fundamentals of Heat Exchanger Design. John Wiley & Sons, 2003.

[4] White, Frank M. Fluid Mechanics. 6th ed, McGraw-Hill, 2009.

## Version History

Introduced in R2020b


text:
manpagez: man pages & more
info emacs
Home | html | info | man
 [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ]

### 30.9.2 LaTeX Editing Commands

LaTeX mode, and its variant, SliTeX mode, provide a few extra features not applicable to plain TeX.

C-c C-o

Insert ‘\begin’ and ‘\end’ for LaTeX block and position point on a line between them (tex-latex-block).

C-c C-e

Close the innermost LaTeX block not yet closed (tex-close-latex-block).

In LaTeX input, ‘\begin’ and ‘\end’ commands are used to group blocks of text. To insert a ‘\begin’ and a matching ‘\end’ (on a new line following the ‘\begin’), use C-c C-o (tex-latex-block). A blank line is inserted between the two, and point is left there. You can use completion when you enter the block type; to specify additional block type names beyond the standard list, set the variable latex-block-names. For example, here's how to add ‘theorem’, ‘corollary’, and ‘proof’:

 (setq latex-block-names '("theorem" "corollary" "proof"))

In LaTeX input, ‘\begin’ and ‘\end’ commands must balance. You can use C-c C-e (tex-close-latex-block) to insert automatically a matching ‘\end’ to match the last unmatched ‘\begin’. It indents the ‘\end’ to match the corresponding ‘\begin’. It inserts a newline after ‘\end’ if point is at the beginning of a line.

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text:
# State faraday law of electro magnetic induction?

There are two laws in Faraday’s Laws of Electromagnetic Induction. The first law defines emf induction in a conductor, while the second law quantifies the emf produced.

## Faraday’s First Law of Electromagnetic Induction

Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced. If the conductor circuit is closed, a current is induced which is called induced current.

### Changing the Magnetic Field Intensity in a Closed Loop

Mentioned here are a few ways to change the magnetic field intensity in a closed loop:

• By rotating the coil relative to the magnet.
• By moving the coil into or out of the magnetic field.
• By changing the area of a coil placed in the magnetic field.
• By moving a magnet towards or away from the coil.

## Faraday’s Second Law of Electromagnetic Induction

Faraday’s second law of electromagnetic induction states that the induced emf in a coil is equal to the rate of change of flux linkage.

The flux is the product of the number of turns in the coil and the flux associated with the coil. The formula of Faraday’s law is given below:

$$\varepsilon =-N\frac{\Delta \phi }{\Delta t}$$


text:
×

No. of nodes in a segment tree.

 0 No. of nodes in a recursive segment tree of an array of size $n$ is $2n-1$. But still choosing $2n- 1$ as the size of segment tree gives a segmentation fault, and I have seen it in many places that it's recommended to use size of segment tree as $4n$. But why? Can someone clarify on this? asked 22 Jun '18, 15:31 825●1●13 accept rate: 13%

 1 @pshishod2645 , you can have a look to this link for a formal proof https://stackoverflow.com/questions/28470692/how-is-the-memory-of-the-array-of-segment-tree-2-2-ceillogn-1 answered 22 Jun '18, 15:41 121●7 accept rate: 6% Thanks @aman_robotics (22 Jun '18, 15:51)
 0 @pshishod2645 , If you like to build memory efficient program then you can build segment tree in iterative format. Then it will never use more than 2N memory and if you also want to have lazy propagation then it will require a total of 2N + N = 3*N memory ( extra N for lazy array ). :) answered 22 Jun '18, 16:16 4★meet2mky 201●3 accept rate: 26% Iterative version of segment tree is clear to me, I was just having a doubt about it's recursive version. (22 Jun '18, 16:30)
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# Debian – Apache2 | http.conf How to set AllowOverride All on a directory

apache-http-serverdebiandebian-wheezyhttpd

I have to set AllowOveride to All on my web server, but in my httpd.conf file, there is no <Directory> tag for me to place it in.

I never seen this before. What can I do? Where is the file I have to change?

I am on a LAMP server, and Debian7, with Apache2.

Thanks!

The file that you are looking for is probably in /etc/apache2/sites-enabled/000-default.conf or similar.

You can create the Directory tags yourself in /etc/httpd.conf, /etc/apache2/sites-enabled/000-default.conf, or in a new file placed in /etc/apache2/sites-enabled/example.conf

Below is what you need to include in one of the files mentioned above:

<Directory /path/to/directory-name>
AllowOverride All
</Directory>

Apache will load this directory block from any of the configuration files. I imagine that you are probably used to putting this in the default block that had previously been located in the /etc/apache2/httpd.conf file. The default DocumentRoot and Directory tags still exist but are probably located in /etc/apache2/sites-available/000-default.conf.

You may also consider creating a new configuration file for your specific website in /etc/apache2/sites-available/example.conf. Then placing within that file, something like this:

<VirtualHost *:80>
ServerName site-name.com
ServerAlias other-name.com www.site-name.com
DocumentRoot /path/to/application
<Directory />
AllowOverride All
</Directory>
</VirtualHost>

By default Apache will load configuration from any file that is located in /etc/apache2/sites-enabled/* that ends in .conf


text:
# Define starting line number when using mcode.sty to display Matlab code and referring to the code

I want to include parts of my Matlab code in my LaTeX document using mcode.sty. Here's a MWE:

\documentclass[b5paper, 10pt, twoside, openright, dvipsnames,cmyk]{report}
\usepackage{listings}   % Is this package necessary?
\usepackage[framed,numbered,autolinebreaks]{mcode}
\begin{document}
Figure~\ref{fig:binarysearch} doesnt work!
\begin{figure}
\label{fig:binarysearch}
\lstinputlisting{pics/met/binarysearch.m}
\caption[Matlab code for the half-interval search method]
{Matlab code for the half-interval search method}
\end{figure}
\end{document}

Is there a way to define which line number I want to start on for each included code section? Let's say I want to start on line 6. How is this accomplished? The default is (naturally) 1.

I've placed the code in a figure environment, and I want to refer to it as a figure. However, referring in the text to the figure doesn't work. The compiled file writes ?? instead of the figure reference.

• You should have a look at the matlab-prettifier package; it improves upon mcode in many ways. See this answer, for instance. – jub0bs May 12 '15 at 12:29
• I read your answer, @Jubobs, but I didn't really get the hang of it. What specifically is the difference between mcode and your package? Is it possible to explain it with a few sentences? – ROLF May 12 '15 at 14:34
• There wouldn't be enough space in a Stack-Overflow comment :p I refer you to section 2 of the matlab-prettifier doc, in which you will find a point-by-point comparison of the alternatives (mcode, minted, etc.). Basically, matlab-prettifier aims to be easier to use and more consistent than mcode. – jub0bs May 12 '15 at 16:07
• @Jubobs: When using mcode, there seems to be less space per line in LaTeX compared to Matlab. I've fittet my code into one line in Matlab, but the text must be on two lines in my document. Is this the case for matlab-prettifier too? – ROLF Jun 7 '15 at 19:07
• That has little to do with mcode or matlab-prettifier; rather, it depends on the font (family, size, etc.) and the text width you use; you can use the listings' option called linewidth to tweak the latter to your liking. – jub0bs Jun 7 '15 at 21:04

The following code solves the first question:

\lstinputlisting[firstnumber=6]{{pics/met/binarysearch.m}}

firstnumber defines the starting line number.

The second question: Wrong order of the code. The following code is correct:

\begin{figure}[htb]
\centering
\lstinputlisting[firstnumber=6]{pics/met/binarysearch.m}
\caption[Matlab code for the half-interval search method]
{Matlab code for the half-interval search method}
\label{fig:binarysearch}
\end{figure}


text:
## Tuesday, May 11, 2004

### Simulation programs

"\Data\bm\Bead_sim_tracking" is the original one. The beam is droven by the mouse.
"\Data\bm\Bead_sim_tracking2" is the one I showed to U. Automatic single beam with different speed.


text:
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# If m and n are both positive integers is (m + n^4)^(1/6) an integer?

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If m and n are both positive integers is (m + n^4)^(1/6) an integer?  [#permalink]

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If m and n are both positive integers is $$\sqrt[6]{m + n^4}$$ an integer?

(1) $$m = n^4 + n^5$$

(2) $$m = n^4(n^2 - 1)$$

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Re: If m and n are both positive integers is (m + n^4)^(1/6) an integer?  [#permalink]

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21 Mar 2018, 06:34
Bunuel wrote:
If m and n are both positive integers is $$\sqrt[6]{m + n^4}$$ an integer?

(1) $$m = n^4 + n^5$$

(2) $$m = n^4(n^2 - 1)$$

1)
n^4 + n^5 = m

$$\sqrt[6]{m + n^4}$$ = $$\sqrt[6]{n^4 + n^5 + n^4}$$ = $$\sqrt[6]{n^5 + 2n^4}$$ = insufficient

2) $$m = n^4(n^2 - 1)$$ = $$m = n^6 - n^4$$
$$\sqrt[6]{m + n^4}$$ = $$\sqrt[6]{n^6 - n^4 + n^4}$$ = n

(B) imo
Re: If m and n are both positive integers is (m + n^4)^(1/6) an integer? &nbs [#permalink] 21 Mar 2018, 06:34
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text:
# Dense path in square, constructed from grid

Working on this stack overflow question led me to ask another, related question (here) .

This first attempt was shown unsuccessful in the answer to it, so I try a rather different approach here. Let $D=\bigg\lbrace \frac{i}{2^j}\bigg | j \geq 1, 0 < i < 2^j \bigg\rbrace$ be the set of all dyadic numbers in $]0,1[$.

Let us say that a (finite or infinite) sequence $u=(u_n)$ of elements in $D^2$ is rectangular if all the vectors $\overrightarrow{u_{k}u_{k+1}}$ are horizontal or vertical.
When $u$ is finite of length $n$, we have a polygonal path $p_u: [0,1] \to ]0,1[^2$ such that $p_u$ induces a bijection between $[\frac{k-1}{n},\frac{k}{n}]$ and the segment $[u_ku_{k+1}]$, for each $k$. When $u$ is infinite, $u=(u_k)_{k \geq 1}$, we have an “infinite” polygonal path $p_u$ such that $p_u$ induces a bijection between $[k,k+1]$ and the segment $[u_ku_{k+1}]$, for each $k \geq 1$.

It is easily seen for any $\varepsilon >0$, any finite rectangular sequence $u$ such that $p_u$ is injective (let us call these good finite sequences for short) can be extended to form a larger good finite sequence that’s $\varepsilon$-dense (formally, this means that if $(u_1,u_2,u_3, \ldots ,u_p)$ is a good finite sequence, then there is a $q>p$ and values $u_{p+1},u_{p+2}, \ldots ,u_q$ in $D^2$ such that the sequence $(u_1,u_2,u_3, \ldots ,u_q)$ is again a good finite sequence, and for any $x\in [0,1]^2$ there is an index $i$ between $1$ and $q$ such that $||x-u_i|| \lt \varepsilon$ ). We deduce that there exist countably infinite, rectangular sequences $u$ that are dense in $[0,1]^2$ and such that $p_u$ is injective (let us call these good infinite sequences).

Let $(u_k)_{k\geq 1}$ be a good infinite sequence. Let $P$ be the path followed by the sequence, i.e. the union of all the segments $[u_ku_{k+1}]$ for $k\geq 1$. Is $Z=]0,1[^2 \setminus P$ path connected ? A related question : is it true that for any $z\in Z$, there is an open segment containing $z$ and contained in $Z$ ?

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I don't really understand the part where you say that any finite rectangular sequence whose path is injective can be extended to form a larger sequence. What exactly do you mean by "extended"? Also, in the next sentence, what is the "associated polygonal path" for an infinite sequence? – Dan Shved Dec 4 '12 at 12:27
@DanShved : I edited the post, hope it is clearer now. – Ewan Delanoy Dec 4 '12 at 15:20

$Z$ may not be path-connected, and the number of its path-connected components may even be uncountable.

Instead of working with dyadics, I find it easier to use triadic numbers to work out an explicit example (and it makes no real difference : just use any order preserving bijection between the two to recover your original context) Let $T_n = \{(a/3^n, b/3^n), 0 < a,b < 3^n \}$.

First, pick $u_0, \ldots u_3$ in $T_1$ so that they make a U shape. Then, pick $u_4, \ldots u_{16}$ in $T_2$ by extending the line $u_2 u_3$ by $1/9$, then go back inside the U and "follow the walls" until you've passed on every point in $T_2$ and are back at a corner of $T_2$. Then pick $u_{17} \ldots u_{53}$ in $T_3$ using the same procedure, etc.

You will notice that there are big areas of the square where you are always moving along the same axis : the intersection of $P$ with those areas consist of a bunch of horizontal (or vertical) lines. Thus the path-connected components of $Z$ in those areas must also be made of horizontal (or vertical lines).

There is a half $H$ of the square where every path-connected component of $Z \cap H$ or $P \cap H$ have the same U shape. You can parametrize those components by a real number in $(0;1/2]$. As you look progressively into the other (more complicated) half of the square, what happens is that you're extending those components, twisting them, and connecting some of them. Those connections are of the form "if $x$ is in some interval $I_n$, connect component $x$ with component $r_n \pm x$" where $r_n$ is a rational (triadic) number and $I_n$ an interval with rational (halves of triadic numbers, I think) endpoints.

Then the path-connected components of $Z$, together with $P$, correspond to the quotient of $(0;1/2]$ modulo those connections. $P$ corresponds to the class of $1/3$ and has an endpoint ($u_0$). The path-connected component corresponding to the class of $1/2$ will have countably many endpoints and nodes. And all the other path-connected components will have no endpoint and no node (they can't form a loop since that would disconnect the square, so there is a continuous bijection from $\Bbb R$ into them). Since each equivalence class is countably infinite and $(0;1/2]$ is uncountable, the number of equivalence classes (and thus of path-connected components) has the same cardinality as $(0;1/2]$. Also, every one of them is dense in the square, and so choosing to remove any one of them instead of $P$ will give you the same partition of the square into path-connected subsets.

Assuming you start at $u_0 = (1/3,2/3)$ and go down to form the first U shape, we get $u_3 = (2/3,2/3), u_4 = (2/3,7/9), u_5 = (5/9,7/9), \ldots u_{16} = (8/9,8/9), u_{17} = (8/9,25/27), \ldots u_{53} = (26/27,26/27) \ldots$

Note $\overline{x}$ the component of the point $(x,1/2)$, looking at the bottom half of the square you have the equivalence $\overline{x} \sim_0 \overline{1-x}$. This makes an endpoint at $(1/2,1/2)$.

The other reflections take place on the upper half and they have pairwise disjoint support, so from almost any point at $(x,1/2)$ you can travel through the top half to get to a unique $(y,1/2)$ or travel through the bottom half to get to $(1-x,1/2)$.

Next you have $\overline x \sim_1 \overline{2/3 - x}$ for $x \in [1/6 ; 1/2]$, making the endpoint at $u_0$; and $\overline x \sim_2 \overline{11/9 - x}$ for $x \in [1/2 ; 13/18]$, making an endpoint at $(11/18,13/18)$ and a T node at $(1/2,5/6)$.

Since $11/18 \sim_0 7/18 \sim_1 5/18 \sim_0 13/18 \sim_2 1/2$, these are all on the same component (1/2's) (and this chain of equivalences describe an actual path from $(11/18,13/18)$ to the T node which is directly tied to the other endpoint $(1/2,1/2)$).

On the next step, you have $\overline x \sim_3 \overline{8/9 - x}$ for $x \in [1/18 ; 1/6] \cup [13/18 ; 5/6]$ and $\overline x \sim_4 \overline{47/27 - x}$ for $x \in [5/6 ; 49/54]$, making an endpoint at $x=47/54$ and a T node at $x=5/6$.

And so on : $\overline x \sim_{2n+1} \overline{1-3^{-n-1} - x}$ for $x \in [\frac 1 2 3^{-n-1} ; \frac 1 2 3^{-n}] \cup [1- \frac 5 2 3^{-n-1} ; 1- \frac 1 2 3^{-n}]$ and $\overline x \sim_{2n+2} \overline{2-7.3^{-n-2} - x}$ for $x \in [1- \frac 1 2 3^{-n}; 1- \frac 5 2 3^{-n-2}]$, making an endpoint at $x = 1- \frac 7 2 3^{-n-2}$ and a T node at $x = 1- \frac 1 2 3^{-n}$.

So hopefully, this will help to draw the picture.

I would like to say that $\overline{x} = \overline{y}$ if and only if $x-y$ or $x+y$ is a triadic number, but that would require more analysis on what those reflections do in terms of the base 3 expansion of $x$.

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So my answer looks like a version of this answer : math.stackexchange.com/a/234999/17445 – mercio Dec 4 '12 at 16:41
That’s a good synthesis of a lot of things that would be awkward to write out formally. I certainly need to check and work out the details before accepting this as an answer. – Ewan Delanoy Dec 4 '12 at 20:33
Why does the equivalence class of $\frac{1}{2}$ have countably many endpoints and nodes ? All I see is that it has an endpoint at $(\frac{1}{2},\frac{4}{3})$, the middle of $[u_8u_9]$. – Ewan Delanoy Dec 5 '12 at 6:08


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# q-product semiring

q-product is defined as

$x \otimes _q y = (x^{1-q}+y^{1-q}-1)^{1/(1-q)}$

Observation:

• $(+,\otimes_\infty)$ is min-plus tropical semiring on the segment $[0,1]$
• $(+,\otimes_1)$ is R
• $(+,\otimes_{-\infty})$ is max-plus tropical semiring on $[0,\infty]$

Consider the following "generalized distributivity" law:

$x \otimes _q (y+z) = (x \otimes _p y)+(x \otimes _p z)$

I'm looking for closed form expression for $p$ in terms of $q$, or this is not feasible?

-

It is very easy to see that this is not feasible. It suffices to try $q=0$, so we'd look for a function $f(\cdot,\cdot)$ such that $x+y+z-1=f(x,y)+f(x,z)$. As we can switch variables, this would yield $f\equiv const$, which is impossible.


text:
Direct variation calculator mathpapa

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text:
# how much crushed stone calculator

## 2020 Crushed Stone Calculator: 3/4 inch rock HomeAdvisor

A good rule of thumb for the depth of crushed stone projects is to calculate a minimum of 2-4 inches. Some projects need more than this, but this is a good starting point. Note that you’ll need to convert your depth into feet by dividing it by 12.

## Gravel Calculator Calculate how much gravel you need

Note that a gravel's density depends on this, so make sure to consult the manufacturer's technical specifications for the estimated density which you should then input in our gravel calculator. Typical gravel density (dry) is between 1520 and 1680 kg/m3 (95 to 105 lbs/ft3). If it is mixed with sand the density is around 1920 kg/m3 or 120 lbs/ft3 Crushed stone is often used as a material for producing

## Landscape Materials Calculator Peel Landscape Depot

This calculator is only intended to be used as a means of estimating how much product you may need for a particular job. Whilst every care is taken to ensure the accuracy of the information as a guide for costing, no responsibility is accepted by Peel Landscape Depot for its accuracy.

## Coverage Calculator Southwest Boulder & Stone

Select the material, desired size, and the area you need to cover to find out how much of that material you’ll need.

## Sub base stone, mot and scalpings calculator

2011-12-5  This form will calculate the approximate metric tonnes of unconsolidated DTp1 required for the entered area. Please enter the dimensions in the white fields below. The calculations will be completed when you leave the last input field.

## Hardcore/Type1 Calculator Sand & Gravel Direct

Hardcore/Type1 Calculator Sand & Gravel Direct. Home / Calculator / Hardcore/Type1 Calculator.

## Stone Calculator Crushed Stone Cubic Yard Calculator

Our Stone calculator will help you estimate how many Cubic Yards of Stone you need for your desired coverage area. The crushed stone calculator offers 4 "Box" area fields and 2 "Circular" area fields for you to calculate multiple areas simultaneously (back yard,

## How much crushed rock do I need calculator?

In the construction world, most materials are measured in cubic yards. Multiply the length (L), in feet, by the width (W), in feet, by the height (H), in feet, and divide by 27. This will tell you how many cubic yards of crushed stone you need. When using this equation, make sure all of your measurements are in feet.

## Calculate Crushed Stone Amount

It is composed of pea-sized stones that have been rounded over time through natural weathering. The individual pebbles range in size from 1/8 inch to 3/8 inch and come in many colors. Calculate Crushed Stone Amount. Gravel Calculator calculate how much gravel you need Crushed stone is often used as a material for producing concrete.

## How Much Crushed Stone Do I Need for My Project? THE

2018-11-9  Crushed stone is derived from larger rocks that have been crushed or broken down by a crushing machine. Crushed stone is used for a number of different construction and landscaping purposes. The Stone Construction Co., Inc delivers a wide variety of different materials for your home improvement and construction needs.

## How much crush Run do I need? FindAnyAnswer

2020-2-6  How much crush Run do I need? A rule of thumb for many crushed stone projects is to have a minimum depth of 2 to 4 inches, although some projects require more. For example, a patio that's 12-foot square, with a 2-inch base of crushed stone, will need 0.89 cubic yards. Simply so, how much crushed stone do I need calculator?

## How many bags of crushed rock do I need? AnswersToAll

2021-3-20  How many bags of crushed rock do I need? In the construction world, most materials are measured in cubic yards. Multiply the length (L), in feet, by the width (W), in feet, by the height (H), in feet, and divide by 27. This will tell you how many cubic yards of crushed stone you need.

## Landscape Materials Calculator Peel Landscape Depot

This calculator is only intended to be used as a means of estimating how much product you may need for a particular job. Whilst every care is taken to ensure the accuracy of the information as a guide for costing, no responsibility is accepted by Peel Landscape Depot for its accuracy.

## Sub base stone, mot and scalpings calculator

2011-12-5  Sub Base Estimator. This form will calculate the approximate metric tonnes of unconsolidated DTp1 required for the entered area. Please enter the dimensions in the white fields below.

## Bulk Material Calculator Contractors Stone Supply

2020-4-24  Most of our bulk materials, with the exception of mulch, are sold by the weight. The following are approximate weights for most of our bulk materials. Sand 1.10 1.25 tons (2,200 2,500 lb.) per cubic yard. Planting Mix 1 ton (2,000 lb.) per cubic

## Calculate #57 Granite Stone cubic yards / Tons

Calculate #57 Granite Stone. Type in inches and feet of your project and calculate the estimated amount of Granite Stone in cubic yards, cubic feet and Tons, that your need for your project. The Density of #57 Granite Stone: 2,410 lb/yd³ or 1.21 t/yd³ or 0.8 yd³/t.

## Convert Crushed Stone From Tons to m3: Online Calculator

2019-11-27  Crushed stone 5-10, 5-20 mm: 0.74 cubic meters in one ton. 20-40 millimeters: 0.76 m3. 25-60, 40-70 mm: 0.72 m3. Before making a decision about buying or using crushed stone in construction, you should double-check the variety by checking all the details with your supplier. However, this calculator

## Stone Calculator Crushed Stone Cubic Yard Calculator

Our Stone calculator will help you estimate how many Cubic Yards of Stone you need for your desired coverage area. The crushed stone calculator offers 4 "Box" area fields and 2 "Circular" area fields for you to calculate multiple areas simultaneously (back yard,

## Gravel Calculator Stone Warehouse

Gravel Calculator Our gravel calculator provides a rough guide to how much gravel you will need to cover a specific area (such as a driveway). It should help you estimate the amount of stone you will need. The depth will depend on the size of the material as well as its intended use.

## Sub base stone, mot and scalpings calculator

2011-12-5  This form will calculate the approximate metric tonnes of unconsolidated DTp1 required for the entered area. Please enter the dimensions in the white fields below. The calculations will be completed when you leave the last input field.

## Stone Tonnage Calculator Rohrer's Incorporated

Stone Tonnage Calculator rohrers-admin 2018-06-06T15:51:08-04:00 Tonnage calculations are based on averages and should be used as estimates. Actual amounts needed may vary.

## Calculator, Landscape Materials, Mulch, Stone, Grass Seed

Use this handy calculator to determine how much bulk material you need. Works great for calculating the required amount of soil, mulch, stone, grass seed, sand and more. Seasonal Hours: M-F: 8:00 5:00 Sat: 8:00 Noon Quantity Calculator. E-mail Us. Call 269.375.8000

## How many tonnes of crushed rock are in a cubic meter

How much crushed stone do I need calculator? Multiply the length (L), in feet, by the width (W), in feet, by the height (H), in feet, and divide by 27. This will tell you how many cubic yards of crushed stone you need. When using this equation, make sure all of your measurements are in feet.

## Concrete & Crushed Stone Calculator Piercon Group

2021-8-18  Calculate the amount of concrete or crushed stone required for your project. Saison 2021 : Carrières ouvertes Carrières PCM inc. Princeville,St-Cyrille,St-Nicéphore,St-Wenceslas,Warwick,St-Rosaire

## HELPFUL CALCULATOR Shelby Crushed Stone

SHELBY CRUSHED STONE. 10830 Blair Road. Medina, NY 14103 EAGLE HARBOR SAND & GRAVEL. 4780 Eagle Harbor Road. Albion, NY 14411 We gladly accept: MasterCard, Visa, Discover, American Express Webmail Login

## how much crushed stone calculator

Convert Crushed Stone From Tons to m3: Online . Crushed stone is one of the basic building materials used in the construction of houses, roads, fences and so on Your local seller may indicate prices in tons or cubic meters Suppose you are waiting for the delivery of two tons of crushed stone, but do not know how much it is in m3 Then you will need our online calculator!


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# Parallel loop [python-like zip] in Tikz

An easy way to write axis labels in Tikz is:

\foreach \x/\xtext in {-1,-0.5,0.5,1}
\draw[very thin,gray] (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\xtext$};

Is it possible to have something similar but where \x and \xtext do not get the values from the same list as follows ?

\foreach \x in {-1,-0.5,0.5,1}/\xtext in {-10,-5,5,10}
\draw[very thin,gray] (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\xtext$};

• Are you talking about this: \x/\xtext in {a/A,b/B,c/C}? Jan 26, 2013 at 18:03
• Does this come close with individual arrays ? tex.stackexchange.com/questions/89135/loop-for-includegraphics/… Jan 26, 2013 at 18:09
• have you thought about using pgfplots? Jan 26, 2013 at 18:37
• yes but here I am using tikz to have more control. Jan 26, 2013 at 20:44

Here a simple code. Is this what you want?

\begin{tikzpicture}
\draw[very thin,gray] (-1,0)--(1,0);
\foreach \x/\xtext in {-1/a_0,-0.5/a_1,0.5/a_2,1/a_3}{
\draw[very thin,gray] (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\xtext$};
}
\end{tikzpicture}

• yes, it is nice. I have to experiment. Jan 26, 2013 at 20:44
• @pluton I know this is not the core of this question, but in this case, one could also use the count key: \foreach \x[count=\xIndex from 0] and then use $a_{\xIndex}$. Jul 12, 2013 at 16:29

If your lists are from different sources, then you would need to merge them as follows:

\documentclass{article}
\usepackage{tikz}
\makeatletter
% \specialmergetwolists{<coupler>}{<list1>}{<list2>}{<return macro>}
% \specialmergetwolists*{<coupler>}{<listcmd1>}{<listcmd2>}{<return macro>}
\protected\def\specialmergetwolists{%
\begingroup
\@ifstar{\def\cnta{1}\@specialmergetwolists}
{\def\cnta{0}\@specialmergetwolists}%
}
\def\@specialmergetwolists#1#2#3#4{%
\def\tempa##1##2{%
\edef##2{%
\ifnum\cnta=\@ne\else\expandafter\@firstoftwo\fi
\unexpanded\expandafter{##1}%
}%
}%
\tempa{#2}\tempb\tempa{#3}\tempa
\def\cnta{0}\def#4{}%
\foreach \x in \tempb{%
\xdef\cnta{\the\numexpr\cnta+1}%
\gdef\cntb{0}%
\foreach \y in \tempa{%
\xdef\cntb{\the\numexpr\cntb+1}%
\ifnum\cntb=\cnta\relax
\xdef#4{#4\ifx#4\empty\else,\fi\x#1\y}%
\breakforeach
\fi
}%
}%
\endgroup
}
\makeatother

\begin{document}
\specialmergetwolists{/}{-1,-0.5,0.5,1}{a_0,a_1,a_2,a_3}\clist
% If the lists are in macros, use the star (*) form:
% \def\alist{-1,-0.5,0.5,1}
% \def\blist{a_0,a_1,a_2,a_3}
% \specialmergetwolists*{/}\alist\blist\clist

\begin{tikzpicture}
\draw[very thin,gray] (-1,0)--(1,0);
\foreach \x/\y in \clist{
\draw[very thin,gray] (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\y$};
}
\end{tikzpicture}
\end{document}

• Just what I needed! The semantics are like Python's zip: the merged list is truncated to match the shorter of the two input lists: \specialmergetwolists{/}{-1,-0.5,0.5}{a_0,a_1,a_2,a_3}\clist puts -1/a_0,-0.5/a_1,0.5/a_2 in \clist. Jul 11, 2013 at 1:26
• This should become part of tikz-pgf. Aug 30, 2020 at 15:25
• what if one list is a macro and one is a sequence of integers? Apr 4, 2021 at 15:08


text:
# NCERT Solutions for Class 10 Maths

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## Select Chapter

Euclid’s division lemma, Fundamental Theorem of Arithmetic – statements after reviewing work done earlier and after illustrating and motivating through examples. Proofs of results – irrationality of √2, √3, √5 , decimal expansions of rational numbers in terms of terminating/non-terminating recurring decimals.

Zeros of a polynomial. Relationship between zeros and coefficients of a polynomial with particular reference to quadratic polynomials. Statement and simple problems on division algorithm for polynomials with real coefficients.

Pair of linear equations in two variables. Geometric representation of different possibilities of solutions/inconsistency. Algebraic conditions for number of solutions. Solution of pair of linear equations in two variables algebraically – by substitution, by elimination and by cross multiplication. Simple situational problems must be included. Simple problems on equations reducible to linear equations may be included.

Standard form of a quadratic equation $ax^2 + bx + c = 0, (a ≠ 0)$ . Solution of quadratic equations
(only real roots) by factorization and by completing the square, i.e., by using quadratic formula. Relationship between discriminant and nature of roots. Problems related to day-to-day activities to be incorporated.

Motivation for studying AP. Derivation of standard results of finding the nth term and sum of first n terms.

Definitions, examples, counterexamples of similar triangles.
1. (Prove) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.
3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.
4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.

5. (Motivate) If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.
6. (Motivate) If a perpendicular is drawn from the vertex of the right angle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.
7. (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides.
8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
9. (Prove) In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right triangle.

Review the concepts of coordinate geometry done earlier including graphs of linear equations. Awareness of geometrical representation of quadratic polynomials. Distance between two points and section formula (internal). Area of a triangle.

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0° and 90°. Values (with proofs) of the trigonometric ratios of 30°, 45° and 60°. Relationships between the ratios.
Trigonometric Identities: Proof and applications of the identity $sin^{2}A+cos^{2}A=1$ . Only simple identities to be given. Trigonometric ratios of complementary angles.